Platform for manipulating polarization modes

orm for manipulating polarization modes realized with Jones

ors in MATHEMATICA

Technical -Dae Choi1, Bogyeong Kim2,Paper Hee-Joong Yun3†

J. Astron. Space Sci. 32(2), 63-71 (2015) http://dx.doi.org/10.5140/JASS.2015.32.2.63

tment of Microbial and Nanomaterials, Mokwon University, on 302-729, Korea

Platform for manipulating polarization modes realized ean an Institute of Science and Technology with JonesInformation, vectors in MATHEMATICA

tment of Astronomy and Space Science, Chungnam National University, on 305-764, Korea

eon on 305-806, Korea ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] 1 2

ved

Yong-Dae Choi , Bogyeong Kim , Hee-Joong Yun3† Revised Accepted 1

Department of Microbial and Nanomaterials, Mokwon University, Daejeon 302-729, Korea Department of Astronomy and Space Science, Chungnam National University, Daejeon 305-764, Korea 3 in physics of the propagation of the electromagnetic wave polarization in matter is ndamental damental conception Korean Institute of Science and Technology Information,Daejeon 305-806, Korea 2

understood as the cardinal keyword in free-space quantum communication technology and cosmology in

ysics. Interactive visualization of the propagation mechanism polarized of electromagnetism in a medium The fundamental conception inofphysics the propagation of the

electromagnetic wave polarization in matter is newly understood as the cardinal keyword in free-space quantum communication technology and cosmology in astrophysics. s helicity has accordingly received attention from scientists exploiting the protocol of quantum key Interactive visualization of the propagation mechanism of polarized electromagnetism in a medium with its helicity has tion (QKD) to guarantee unconditional security in cryptography communication. We have provided a accordingly received attention from scientists exploiting the protocol of quantum key distribution (QKD) to guarantee unconditional in cryptography communication. We have c polarization platform for presentingsecurity the polarization modes of a transverse electromagnetic wave, provided a dynamic polarization platform for presenting the polarization modes of a transverse electromagnetic wave, converting the state of polarization through the arrangement ing the state of polarization through the arrangement of optical elements, using Jones vectors calculations of optical elements, using Jones vectors calculations in Methematica. The platform graphically simulates the mechanism of hematica. The platform graphically the mechanism of production and propagation of the production andsimulates propagation of the polarized waves in a medium while satisfying Maxwell's equations.

ed waves in a medium while satisfying Maxwell's equations.

Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity

rds: polarization modes, cosmology, Jones vector, wave plate, helicity

1. INTRODUCTION

to receive the binary signals, which are modulated by two circular polarizations. FSO communication employing Polarization is a coherent characteristic of the electromagnetic a binary polarization shift keying coherent modulation rization is a coherent characteristic of the electromagnetic (EM) wave in a medium. In a plane (EM) wave in a medium. In a plane wave, both the electric scheme are utilized in atmospheric turbulence channel ⃗⃗ and ⃗⃗ of field vector vector E field vector vector B (Tang et al. 2010) and free-space quantum key distribution both the electric field and magnetic magnetic field ofthe the electromagnetic electromagnetic radiation radiation always oscillates parallel to a fixed direction in (QKD) by rotation-invariant twisted photons are used oscillates parallel to a fixed direction in space. Light of such character is said to be linearly space. Light of such character is said to be linearly polarized, to guarantee unconditional security in cryptographic and maintains a constant direction of oscillation, and communication (Vallone 2014). On 17 March 2014, 1 does vary spatially in a regular manner, producing either astronomers from the Harvard Smithsonian Center for elliptically polarized or circularly polarized light (Fowles Astrophysics announced their detection of signature 1975; Jackson 1975; Pedrotti & Pedrotti 1987). Polarization patterns of polarized light in the Cosmic Microwave characteristic have been used in radio transmission to Background (CMB) (Ade 2014; Calvin 2014; CfA 2014). reduce interference between channels, particularly at The team hunted for a special type of polarization called VHF frequencies and beyond (Masayoshi 2007; Yao et al. ‘B-mode’ which represents a twisting or ‘curl’ pattern in 2007). Free-space communication has forced the use of the polarized orientations of the ancient light. This is the circular polarization, which has the fundamental advantage strongest confirmation yet of the cosmic inflation theory of precluding disturbance from the reflections of signals (Boyle 2006). Gravitational waves squeeze space as they (Leitch et al. 2002; Elser et al. 2009). Free-space optical travel, and the squeezing produces a distinct pattern in the (FSO) communication utilizes a spatial diversity receiver CMB. Gravitational waves have a “handedness” much like

TRODUCTION

Received Mar 1, 2015 Revised Apr 7, 2015 Accepted Apr 8, 2015 †Corresponding Author E-mail: [email protected], ORCID: 0000-0002-1273-5974 Tel: +82-42-821-7492, Fax: +82-42-821-8891

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http:// creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Copyright © The Korean Space Science Society

63

http://janss.kr plSSN: 2093-5587 elSSN: 2093-1409

distribution (QKD) to guarantee unconditional security inplane cryptography communic Fowlesfields 1975).may In 2.1 particular, although electromagnetic waves an harmonic wave,isthe fields may pull orthe push the electrons inofthe orbital of a solid, which respons Electromagnetic waves solids pull or push the electrons ininthe orbital aare solid, which isreceived responsible for indus helicity has accordingly attention from Fowles 1975). In particular, although the electromagnetic waves are an harmonic plane with wave,itsthe dynamic polarization platform for presenting the polarization modes of a trans fields maypolarization pull or push the electrons inJones the orbital of a solid, which is responsible for inducing dipole Keywords: modes, cosmology, vector, plate, propagating of waves in solids is that in the va distribution (QKD) toassume guarantee unconditional security in cr ⃗ process ⃗M ⃗⃗ helicity ⃗The ⃗M ⃗⃗ electromagnetic moment P and magnetization in solid. If different we assume that the me moment P and inwave the solid. Ifthewe that the from medium is not fields may pull or push the electrons in the orbital of a solid, whichmagnetization is responsible for inducing dipole Platform for realiz converting the statemanipulating of polarization throughpolarization the arrangement of modes optical elements, usi J. Astron. Space Sci. 32(2), 63-71 (2015) ⃗ and magnetization ⃗M ⃗⃗ in the solid. If moment P we assume that theplatform medium not a magnetic dynamic polarization forisin presenting the polarizati ⃗ and B ⃗ of the electromagnetic E waves interact with the electrons a solid (Pedrotti &P ⃗ and magnetization ⃗M ⃗⃗ in the solid. moment P If and we assume that the is notequation a magnetic material and Jmedium =0, thewave Helmholtz wave equation in a solid (Fowles 1975, in Methematica. The platform simulates the mechanism of Jackso produc material J=0, the Helmholtz in agraphically solid (Fowles 1975, Jackson 1975) is MATHEMATICA vectors in converting the 1975, state ofJackson polarization through the arrangement o material and J=0, the Helmholtz equation in although a solid (Fowles 1975) is 1. INTRODUCTION Fowles 1975).wave In particular, the electromagnetic waves are an harmonic plan waves in while satisfying Maxwell's equations. material and J=0, the Helmholtz wave equation in a solid (Fowles 1975,polarized Jackson 1975) is a 1medium ∂2 ⃗P 1 ∂∂22⃗P⃗E ∂2 ⃗E

⃗ × (in⃗∇2Methematica. ⃗E )൅ 02 2The ⃗ × ( ⃗∇ −u ሺͳሻ the ሺm ∇ simulates light wave and can have left- and right-handed polarization. ×2E⃗⃗E )൅ (1) = platform 0 ∂t 2graphically ∇ ⃗× 2 = −uc 1 ∂ ∂t2ofwave ∂t c2 ∂∂tP Polarization is2a coherent ofthetheelectrons electromagnetic (EM) in awhich medium. In a plane for ind fields in the orbital a solid, is responsible ⃗ may ⃗∇pull × (characteristic × ⃗Eor )൅push ∇ 2 ∂t2 = −u0 ∂t2 ሺͳሻ ⃗ ⃗ ∂ P 1Z) ∂2mediate E c In the Standard Model, the weak∇ bosons (W±, 1 2 3† ⃗ × ( ⃗∇ × ⃗E )൅ 2 2 = −u Yong-Dae manipulating ሺͳሻChoi , Bogyeong polarized waves in a medium whilerealized satisfying 0 ∂t2 for Platform polarization with⃗⃗equa Jo Kim , modes Hee-Joong Yun Maxwell's c ∂t −⍵ 2 2 aaform ⃗P ⃗E ⃗ .magnetization ⃗wave ⃗we ⃗M ⃗⃗modes, Keywords: polarization cosmology, Jones vector, wave plate, helicity ̂0 ei(𝒦𝒦 ⃗Pelectric ⃗Efield where = 𝜒𝜒𝜀𝜀 If we suppose the EM will be form of solution such a the weak interactions between different flavors (allboth quarks where = If we suppose the EM wave will be of wave, the vector E and magnetic field vector B of the electromagnetic radiation moment P and in the solid. If we assume that the medium is .r⃗no ⍵ ⍵ where = 𝜒𝜒𝜀𝜀 . If suppose the EM wave will be a form of solution such as E ⃗⃗ 0 0 ⃗ i(𝒦𝒦 .r −⍵t) ̂ ⃗ . If we suppose the EM wave will ⃗⃗⃗ ×be( ⃗k ⃗E of ⃗ ⃗ where ⃗P= 𝜒𝜒𝜀𝜀0E a form solution such as E e , we can k × )൅ E = − χ E 0 2 2 ⃗ i(𝒦𝒦 .r −⍵t) c c ̂0 e ⃗ = 𝜒𝜒𝜀𝜀0E ⃗ . Ifresults and leptons). where Experimental have that , we wecan can rewrite Eq. (1) using a P we suppose theshown EM wave willall be a formsolution of solutionsuch such as as E always oscillates parallel toMATHEMATICA a1Department fixed in⃗ space. Light such is said to be linearly vectors in and J=0, the Helmholtz equation in acharacter solidMokwon (Fowles 1975, Jackson 1975) is ⃗ ofpolarization ofaMicrobial and Nanomaterials, University, rewrite Eq. (1)k⃗direction using wave vector k rewrite Eq.material (1) using a wave vector k wave Keywords: modes, cosmology, Jones vector, wave produced and observed neutrinos have left-handed vector rewritehelicity, Eq. (1) using awave wave vector

302-729, Korea It thenDaejeon becomes a vector equation for ⃗E, which indicates that the propa

⃗ rewrite Eq. (1) using a wave vector k

1. INTRODUCTION ⃗ ⃗ and all antineutrinos have right-handed helicity (Aad et ∂2 P 1 ∂2 E ⃗ × ( ⃗∇ 1× 2⃗E )൅ 2Jones 2 = −u0 ሺͳሻ ∇ Platform for manipulating polarization modes realized ⍵with 2 ∂t⍵2 ∂t2 Chungnam c Space 2 University, 2 ⃗ ⃗ ⃗ ⃗ ⃗ Department of Astronomy and Science, 2 ⍵E 2 2⍵E 2 3† ⍵ 2 ⍵2 National − ⍵ሺʹሻ al. 2012; CfA 2014). The helicity of the elementary particle (2) 1k × ( k × E )൅ ⍵ 2 ⃗2⍵22⍵χ 2×2( = ⃗k𝜒𝜒. ⃗kTherefore, ⃗kac2susceptibility ⃗kHee-Joong ⃗E2the ⃗ (EM) ⃗E Yun ⃗ electromagnetic ⍵ ⍵⃗× ( × )൅ the ሺʹሻ c × E )൅ = − χ E c2⃗Ewe = −will cሺʹሻ ሺʹሻ E the component of electric write ⃗ ⃗ ⃗ ⃗ Yong-Dae Choi , Bogyeong Kim , ⍵ ⍵ 2 2 ⃗ ⃗ ⃗ ⃗ ⃗ 2 2 Polarization is coherent characteristic of wav 2 χ ⃗ ⃗ 2 3 ⃗ ⃗ ⃗ 2 2 k × k ( × k ( × k E × )൅ E )൅ E = E = − − χ E χ E ሺʹሻ 2 2⍵ 2 22 ⍵ 2⍵ 3 ⃗⍵× (305-764, ⃗k k Daejeon Korea ⃗E )൅ ⃗ ⃗ ( k × E )൅ E = − χ E c c 3 ⍵× ⍵ 2 2 2 2 ⍵ ⍵ k × E = − χ E ሺʹሻ ⍵ ⍵ ⍵ ⍵ 2 2 ⃗ ⃗ 2 2 ⃗ ⃗ ⃗ ⃗ ⃗ 1. INTRODUCTION ⃗ ⃗ ⃗ ⃗⃗ .r⃗− c2 c χ cሺʹሻ 2c ⃗ cሺʹሻ c ⍵ ⃗ ⃗ ⃗ k × ( k × E )൅ E = − χ E ሺʹሻ ⃗ ⃗ k × ( k × E )൅ E = − χ E ⃗ ⃗ ⃗ could be a keyword to determine 2the Standard Model. ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⍵ 2 2 3 c c k × ( k × E )൅ E = − E ሺʹሻ ̂0 ei(𝒦𝒦 ⃗ ⃗ k × ( k × E )൅ E = − χ E k × ( k × E )൅ E = − χ E ሺʹሻ k × ( k × E )൅ E = − χ E ሺʹሻ 2 2 2 2 2 22 where P = 𝜒𝜒𝜀𝜀 E . If we suppose the wave will be a form of solution such as E ⍵ ⍵ 2⃗ 2 EM 2 cc 2 0c c c c c ⃗ 2 2 2 2 ⃗ ⃗ ⃗ ⍵ ⍵ ⍵ ⍵ c c c c ⃗ ⃗ c k × ( k × E )൅ E = − χ E ሺ ⃗ ⃗ ⃗ vectors 2MATHEMATICA 2 E = 3− 2 χ E ⃗ ሺʹሻ ⃗k ⃗ ××((⃗k⃗k××⃗Ein ⃗E)൅ ⃗⃗ ==⍵−− ⃗E ⃗ k × 2 ⍵ k )൅ E (then k × E )൅ ሺʹሻ ሺʹሻ c2indicates cthat ⃗ ⃗ c2It as cinhomogeneous 2= 2EE 22χχ , which the propagation process It becomes a vector equation for E ⃗ ⃗ ⃗ ⃗ The scope ofk⃗ application polarization has expanded then becomes a vector equation for , which indicates wave, both the electric field vector E and magnetic field vector B of the Korean Institute of Science and Technology Information, c c c c , which indicates that It then becomes a vector equation for E solid an plane wave solution below (Pedrotti & Pred × ( ⃗k × ⃗E )൅ of E − χ E ሺʹሻ , which indicates that the propagation It then becomes a vector equation for E 2 ⃗Eindicates ⃗Ewhich ⃗E⃗E, ,Polarization ⍵2 It1 then ⍵becomes , which that the propagation process Itbecomes then⃗⃗becomes aequation vector equation for c2 c2 ⃗ that the propagation process varies wit then becomes aa vector equation for , Mokwon which indicates indicates that that the the propagation propagation process varv ItIt then aDaejeon vector vector equation for for is aindicates coherent characteristic ofprocess the electro ⃗k ×a ⃗Evector ⃗E equation ⃗⃗for ⃗which ⃗Eሺʹሻ k becomes × ( communication = − for χfor E equation using indicates indicates indicates 305-806, afor Nanomaterials, that Korea Department of Microbial and University, ,which that the propagation process varies with then becomes avector vector equation , which which that the propagation process varies with then becomes avector equation Ewhich ⃗E⃗EEq. rewrite (1) wave k ,vector which indicates that the propagation process varies with Itvector becomes afor , vector which indicates that the propagation process varies with It then becomes ,the indicates that the propagation process varies with then becomes equation ,E the propagation process varies with ItItItItthen aa)൅ 2equation explosively this decade with the development of that propagation process varies with the component cthen c2 for ⃗ , becomes which indicates that the propagation process varies with then becomes athat vector equation for Eprocess ⃗E⃗EIt ⃗ always oscillates parallel to a fixed direction in space. Light of such chara Daejeon 302-729, Korea , , which which indicates indicates that the the propagation propagation process varies varies with with ItIt then then becomes becomes aa vector vector equation equation for for ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] , which indicates that the propaga It then a vector equation for E ⃗ ⃗ component of electric susceptibility 𝜒𝜒.the Therefore, the component ofthesusceptibility electric 𝜒𝜒. will Therefore, we will the E will solution ofsolu E the component of electric susceptibility 𝜒𝜒. Therefore, we will write E⃗write Jackson 1975) ⃗E , which indicates that the propagation process varies with It then becomes a Accordingly, vector equation it foris⃗Efundamentally ⃗E ⃗the solution ofmagnetic Eq.we (2) infiE thecomponent component of electric susceptibility 𝜒𝜒. Therefore, write the solution of the component of electric 𝜒𝜒.⃗⃗ Therefore, we will write technology. important of electric susceptibility . susceptibility Therefore, will write the wave, both thewewe electric field vector and 1 2the 3† solution of of Eq. the component of𝜒𝜒.𝜒𝜒. of electric electric susceptibility susceptibility 𝜒𝜒. 𝜒𝜒. Therefore, Therefore, we we will will write write the the⃗EE Esolution Received Revised Accepted ⃗E(2) ⃗E ⃗E ⃗Evaries Eq. the of electric susceptibility Therefore, we will write the Esolution solution of Eq. the,component component ofelectric electric Therefore, we will write the E solution Yong-Dae Choi Bogyeong Kim , ⃗Esusceptibility Hee-Joong Yun solution the component susceptibility 𝜒𝜒.write Therefore, we will the solution ofwrite Eq. (2) in a(2) thethe component electric susceptibility 𝜒𝜒.electric will thethe solution of of Eq. (2) in ain ain aa of Eq. (2) in a the component of electric susceptibility 𝜒𝜒.Therefore, Therefore, we will write the of Eq. (2) in of 𝜒𝜒. Therefore, we will write ,2susceptibility which of indicates that thewe propagation process with It then becomes acomponent vector of equation for ⃗ Department of Astronomy and Space Science, National University, ofplane Eq. below (2) in1plane a(Pedrotti component susceptibility 𝜒𝜒.solid Therefore, we will write the Eansolution ⃗E⃗E solution for application of a manipulation to of understand solution Eq. (2) a as solid asChungnam inhomogeneous solution of ofof Eq. Eq. (2) ininsolid ain a wave the the component component of of electric electric susceptibility susceptibilitythe 𝜒𝜒. 𝜒𝜒. scheme Therefore, Therefore, we weelectric will will write write the the an inhomogeneous wave solution below (Pedro as an(2) inhomogeneous plane wave solution &Fowles Predrotti ⃗ ⃗Ean ⃗⃗a .r solid as an inhomogeneous plane solution below (Pedrotti &to Predrotti 1987, 1975 solid as inhomogeneous plane wave solution below (Pedrotti & Predrotti Fo the component of susceptibility 𝜒𝜒. Therefore, will the ⃗we solution of (2) inelectric a plane the component of electric susceptibility 𝜒𝜒. Therefore, we will write the −⍵t) solid as an Eq. inhomogeneous wave solution below (Pedrotti & Predrotti always oscillates parallel fixed ̂plane Daejeon 305-764, Korea (r (ε )Fowles solid solid as as an an inhomogeneous inhomogeneous plane wave wave solution below & & Predrotti Predrotti 1987, Fowle E ,& t) Ebelow +1987, ε31987; E(Pedrotti direction Fowles write 1987, 1987, in 1987, space. Fow EFo ei(𝒦𝒦 1 1987, 11987, 2(Pedrotti 2Fowles solid as an inhomogeneous plane wave solution below &= Predrotti Fowles 1975, solid asan an inhomogeneous plane wave solution below (Pedrotti & Predrotti 1987, 1975, as an inhomogeneous plane wave solution below (Pedrotti & Predrotti 1987, 1975, solid as an inhomogeneous plane wave solution below (Pedrotti & Predrotti Fowles 1975, solid as an inhomogeneous plane wave solution below (Pedrotti &solution Predrotti 1987, Fowles 1975, solid as inhomogeneous plane wave solution below (Pedrotti Predrotti Fowles 1975, ⃗E(Pedrotti the physics of polarization’s conception and thesolid process wave solution below (Pedrotti & Pedrotti 1975; solution of Eq. (2) in a the component of electric susceptibility 𝜒𝜒. Therefore, we will write the solid as an inhomogeneous plane wave solution below (Pedrotti & Predrotti 1987, Fowles 1975, 1 solid solid as as an an inhomogeneous inhomogeneous plane plane wave wave solution solution below below (Pedrotti (Pedrotti & & Predrotti Predrotti 1987, 1987, Fowles Fowles 1975, 1975, Jackson 1975) Jackson 1975) Jackson 1975) 3 Department of Microbial and Nanomaterials, Mokwon University, The fundamental in physics the propagation of the electromagnetic wave solid of as producing an inhomogeneous plane wave solution below (Pedrotti & Predrotti 1987, Fowles 1975,andconception Jackson Jackson 1975) polarization modes. Presently the definition Jackson 1975) Korean Institute of Science Technology Information, solid as 1975) an inhomogeneous plane wave ofsolution below (Pedrotti & Predro 1 ⃗ Jackson Jackson 1975) 1975) 1975) Jackson 1975) 1975) Jackson 1975) Jackson 1975) Jackson 1975) Daejeon Korea solid as302-729, an Jackson inhomogeneous plane Jackson wave solution below (Pedrotti & Predrotti 1987, Fowles 1975, = E ei∆ε (ε ) ei(k.r⃗−⍵t) Daejeon 305-806, Korea 0 1 + ε2 Jackson 1975) ⃗⃗ .r⃗−⍵t) techn ⃗⃗ .r ⃗⃗ .r⃗−⍵t) i(𝒦𝒦 Jackson Jackson 1975) 1975) ⃗ −⍵t) i(𝒦𝒦 of polarization has been modified, and its nomenclature i(𝒦𝒦 newly understood as the cardinal keyword in free-space quantum communication ̂ ̂ ̂ (r (ε ) (r (ε ) ⃗⃗ E , t) = E + ε E e E , t) = E + ε E ሺ͵ሻ e (r (ε ) E , t) = .r⃗2−⍵t) 1 1 ሺ͵ሻ eε1 E1⃗⃗) ⃗ ei(𝒦𝒦 Jackson 1975) 2 2 ሺ͵ሻ 2 ̂(r,1t)E1=+(εε12 EE1⃗⃗2 + ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] E i(𝒦𝒦 −⍵t) 2 ⃗⃗2i(𝒦𝒦 ⃗⃗−⍵t) 2e ⃗⃗(r ̂E ⃗⃗i(𝒦𝒦 ⃗⃗,i(𝒦𝒦 ⃗⃗⃗−⍵t) ⃗⃗−⍵t) i(𝒦𝒦 .r⃗2−⍵t) i(𝒦𝒦 .r⃗.r .r⃗−⍵t) i(𝒦𝒦 .r⃗i(𝒦𝒦 −⍵t) Jackson 1975) i(𝒦𝒦 .r⃗.r (ε )2eሺ͵ሻ .r⃗E .r⃗−⍵t) −⍵t) E t) = E + ε E ሺ͵ሻ ̂ ̂ ̂ ̂Science, ̂ ̂E ̂ Department of Astronomy and Spaceof Chungnam National University, ̂(r (r (ε ) 1 1 2 E , t) = + ε E ሺ͵ሻ e (rphysics (ε ) (r (r (ε (ε Jackson 1975) ) ) E , t) = E + ε E ሺ͵ሻ e upgraded, which can be confusing to students (r (ε ) E , t) , t) = = E E + + ε ε E E ሺ͵ሻ ሺ͵ሻ e (r (ε ) (ε ) (r (ε ) , t) = E + ε E ሺ͵ሻ e E , t) = E + ε E e E , t) = E + ε E ሺ͵ሻ e E , t) = E + ε E ሺ͵ሻ e 1 1 2 2 1 1 2 2 1 1 1 1 2 2 2 ⃗⃗ 1 1 2 2 Received Accepted 111 11 22 22 i(𝒦𝒦.r⃗Revised −⍵t) ⃗⃗⃗⃗.r⃗.r⃗−⍵t) ̂ i(𝒦𝒦 −⍵t) Interactive of thei(k⃗propagation mechanism of polarizedi(k⃗electro A + ε22astrophysics. E2 ) e visualization i∆ε ሺ͵ሻ ⃗r−⍵t) ̂E ̂ i(k ⃗⃗.r⃗∙−⍵t) 1ሺ͵ሻ (r (r, ,t)t)==(ε (ε EE ++ εKorea ))⃗⃗ee.ri(𝒦𝒦 −⍵t) has E ε EE22i(𝒦𝒦 producing E (r ,t) = (ε 1 Eሺ͵ሻ Daejeon i(k ⃗ −⍵t) ))been modes. Presently = theE0definition polarization (ε1 + ε2 eof )E0e((ε1.r⃗i∆ε ̂(rFor = E0ee(ε + ε2 ei∆εand ) e its.r⃗−⍵t == + εe2i(k e⃗ i∆ε (ε1 305-764, .r⃗−⍵t) E , t) =instance, E111+11there ε2 E222) are customary polarization ሺ͵ሻ e similar and researchers. 1modified, = E + ε e (ε ) B + iC ⃗ ⃗ ⃗⃗ ⃗⃗ ⃗ 0 1 2 ⃗ ⃗ ⃗ ⃗.r⃗.r⃗−⍵t) i(𝒦𝒦.r⃗−⍵t) i∆ε i(ki∆ε .r⃗i∆ε −⍵t) ⃗ .r ⃗⃗ .r ⃗ −⍵t) i(k .r −⍵t) ⃗i(k i(𝒦𝒦 .r i∆ε i(k .r −⍵t) i∆ε i(k .r⃗−⍵t) −⍵t) i∆ε .r⃗i(k i∆ε i(k i∆ε −⍵t) ̂ ⃗ ⃗ i∆ε i(k i(k −⍵t) −⍵t) ̂ (r (ε ) E , t) = E + ε E ሺ͵ሻ e = E e + ε e (ε ) = E e + ε e (ε ) (r (ε ) = E e + ε e = E e + ε e = E e + ε e (ε ) (ε ) (ε ) )polarization t) E E2e and (3) exploiting the p e from with accordingly ee0⃗E.rhas = =the +22εits ε2ei∆ε ehelicity 2has ) i(k producing polarization of 1⃗modes. 1 2 Presently 2 1,= 0(ε its scientists E = E + ε2received eε)2 e)attention 0= 1 2+ (ε 0E 1definition 0E 1(ε 1+ 12εe+ 11 ⃗ −⍵t) 1(ε 11been 2modified, ⃗.r⃗.r⃗−⍵t) nomenclature for 3the right circularly right Korean Institute of and Information, i∆ε i∆ε Technology i(k −⍵t) =modified, E00(ε + ε22can eitsbe 00 to0A ) econfusing nomenclature upgraded, students ofi(k⃗physics researchers. For instance, 1which = = EEdefinition ++ εpolarization: ε22ofeeof has (ε11Science )i(k )⃗ e.r⃗ei(k producing producingpolarization polarizationmodes. modes. Presently Presently the definition been been modified, and and its ⃗r−⍵t) ⃗ ∙ ⃗r−⍵t) ⃗ ∙ ⃗rA −⍵t) A and 00(ε i(k i(k −⍵t) = (the propagation )A = e of(∙the = Ethe epolarization ε2 ei∆ε has ) polarization ⃗ ∙ ⃗r) = ( ) e e Daejeon 305-806, Korea 0 (ε1 + The fundamental conception in physics of electromagnetic wave polarization in ma i(k −⍵t) B + iC⃗ ∙ ⃗r−⍵t) For circularly (Fowles 1975; Jackson 1975),upgraded, right-circularly distribution to security B⃗ + iC communicatio = (unconditional e iC in cryptography i(k .r⃗−⍵t) B⃗ )⃗ ⃗+ ⃗ ∙i(k A guarantee ⃗i(k ⃗−⍵t) nomenclature which confusing to of and instance, AA⃗ A i(k ⃗r(QKD) = E0 (εcan ε2= ei∆ε r⃗−⍵t) ∙∙⃗⃗r⃗physics )(A A students i∆ε i(k.r⃗−⍵t) ∙−⍵t) i(k ∙⃗r⃗r−⍵t) −⍵t) 1 +be AA Acircularly BE )+)(ε)eei(k ⃗rε iC = )researchers. e= i(k ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] ⃗r−⍵t) (=== ) e))i(k i(k ∙ ⃗r∙−⍵t) ∙ −⍵t) the (e(For = = e)ei(k ) e ( e ⃗ ∙ ⃗r(−⍵t) e + e ) A there are similar customary nomenclature for right polarization: right circularly (Fowles i(k = ( = ( ( e ⃗ ⃗ nomenclature nomenclature upgraded, upgraded, which which can can be be confusing confusing to to students students of of physics physics and and researchers. researchers. For instance, instance, 0 1 2 A A i(k(Reitz ∙ ∙⃗r⃗r−⍵t) −⍵t) et newly B + in B iC + BBB+ +iC iCcardinal =B (B ) e keyword (Pedrotti & PedrottiReceived 1987), right-hand circularly al. understood ++ iC iC theiC free-space communication technology and cosmol ⍵ 2 Accepted == (( ))⃗ e∙ei(k Revised B++for + iC iC B iCquantum ⃗r−⍵t) dynamic platform presenting the polarization a transvers ⍵2 B as iC polarization nomenclature = there ( Aare ei(k ̂ + above, BB + + )k iC ⃗iC As field may be presented Jonesofvector (Jo ⃗ ⃗ shown similar customary for right circularly (Fowlesas the modes × ( ⃗kGeorgi × ⃗E )൅ the circularly E polarization: ሺʹሻ vectorright ⃗ ∙ ⃗r−⍵t) B& +Wolf iC 1999; 2 EA= − c2 χ Ei(k 1993), and right-handed (Born 1982) c visualization = )circularly e Interactive 1975, 1975), right-circularly (Pedrotti &propagation Predrotti 1987), right-hand circularly (Reitz et al. there thereare aresimilar similarcustomary customarynomenclature nomenclatureforforthe the circularlypolarization: polarization: right right circularly (Fowles (Fowles (Jackson right right ⃗ ∙electromagnetism astrophysics. of the mechanism of polarized in a mJ A circularly ⃗ i(k r −⍵t) B + iC ̂ converting the state of polarization through the arrangement of optical elements, using As shown above, E field vector may be presented=̂ as (the Jones vector (Jones 1941) with a comple and )al. ̂ shown AsE above, E field vector beaspresented as the Jones Right circularly AsAs above, field vector may be presented the(Jones shown fieldmay vector may be presented asethe Jones vector (Jones 19 1975, Jackson polarization 1975), right-circularly (Pedrotti &shown Predrotti 1987), right-hand circularly (Reitz etmay ̂above, circularly polarization. Bwith + iC As shown above, E field vector be presented as the Jones vector 1941) with ⃗E, which indicates that the propagation process varies It (Pedrotti then becomes a vector equation for vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. T 1993), and right-handed (Born & Wolf 1999, Georgi 1982) circularly polarization. ̂ ̂ 1975, 1975,Jackson Jackson1975), 1975),right-circularly right-circularly (Pedrotti & & Predrotti Predrotti 1987), 1987), right-hand right-hand circularly circularly (Reitz (Reitz et et al. al. ̂ ̂ with its helicity has accordingly received attention from scientists exploiting theRight protocol of quantum As shown above, E field vector may be presented as the Jones vector (Jones 1941) with acircularly complex As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex The fundamental conception in physics of the propagation of the electromagnetic wave polarization in matter is As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex ̂ ̂ As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex in Methematica. The platform graphically simulates the mechanism of production right-handed polarization are different types of polarization, Jones vector (Jones 1941) with a complex vector amplitude ⃗ ̂ ̂ ̂ vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. The wave vector k = k n As shown above, E field vector may be presented as the Jones vector (Jones 1941) with a complex As As shown shown above, above, E E field field vector vector may may be be presented presented as as the the Jones Jones vector vector (Jones (Jones 1941) 1941) with with a ̂ ̂E ̂ field As As shown shown above, above, E field vector vector may may be be presented presented as as the the Jones Jones vector vector (Jones (Jones 1941) 1941) with with a a complex complex As shown above, E field vector may be presented as the Jones vector (Jones 1941) with 1993), and right-handed (Born Wolf 1941) 1999, with Georgi 1982) circularly polarization. circularly vectorBamplitude {A, Right B േiniC} in an plane. inhomogeneous vector amplitude േ iC} oscillating an oscillating inhomogeneous The wa ̂ field vector may be As shown above,with E presented as the Jones vector& (Jones a guarantee complex right-handed vector amplitude {A, the B േ{A, oscillating inin ancryptography inhomogeneous plane. TheWe⃗wave vecto ⃗EiC} although similar names. Recently, the visualization of {A, B ± iC} oscillating in an inhomogeneous plane. The distribution (QKD) to unconditional security communication. have prov polarization and polarization are different types of polarization, although with similar solution of Eq. (2) in a the component of electric susceptibility 𝜒𝜒. Therefore, we will write newly understood asGeorgi the cardinal keyword in free-space quantum communication technology and cosmology in 1993), 1993),and and right-handed right-handed (Born (Born &As & Wolf Wolf 1999, 1999, Georgi 1982) 1982) circularly circularly polarization. polarization. Right Right circularly circularly ̂ ⃗k =⃗k⃗kThe ̂amplitude polarized waves in plane. ain1941) medium while satisfying Maxwell's vector amplitude Ban േ iC} oscillating anplane. inhomogeneous plane. wave = k n̂ K and ainhomogeneous real mutually orthogonal unit vectors (n̂n̂= ε1⃗⃗⃗⃗ ̂i 𝛼𝛼). isk Here vector {A,{A, B േ iC} oscillating inform inhomogeneous The wave vector k= vector amplitude {A, േേpresented iC} oscillating in anan inhomogeneous plane. The wave vector vector amplitude BBB േ iC} oscillating in an plane. The wave vector = kk shown above, E field vector may be as{A, the Jones vector (Jones with aThe complex ⃗kequations. ⃗kn̂= 2+, n vector amplitude {A, iC} oscillating inhomogeneous vector ̂,kn̂εvector vector amplitude {A, േ iC} oscillating in an inhomogeneous plane. The = ̂ ⃗kplane. and form awave real mutually unit vectors ( wave εwave ,n ̂vector ).similar Here K a comple vector {A, BB േ iC} oscillating inin anB inhomogeneous plane. wave = k n̂kkn 1 , εwith 2vector ⃗kേ ⃗korthogonal and right-handed polarization are different polarization, although vector vector amplitude amplitude {A, {A, B േ types iC} in in an an inhomogeneous inhomogeneous plane. The The wave wave vector vector vector vector amplitude amplitude propagating {A, {A, BB േേ iC} iC} oscillating oscillating ininhas an anamplitude inhomogeneous inhomogeneous plane. plane. The Thewave wave vector vector == koscillating n̂mutually n̂oscillating polarization in polarization matter drawn physicist's vector =iC} kkof and form aThe real mutually orthogonal ⃗⃗⃗⃗ ̂ and form a real mutually orthogonal unit vectors ( ε , ε , n ̂ ). vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. The wave vector and form a real orthogonal unit vectors ( ε , ε , n ̂ ). Here K = k ⃗ 1 2 1 2 dynamic polarization platform for presenting the polarization modes of a transverse electromagnetic vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. The wave vector k = k n ̂ ⃗⃗⃗⃗k + (Jone ̂vector astrophysics. visualization ofpolarization, the propagation mechanism of polarized in a medium names. Recently, the visualization of polarization propagating in( εmatter has Here drawnK physicist's ̂electromagnetism and form aabove, real mutually orthogonal unit vectors , n̂ the ). = i 𝛼𝛼 is solid as Interactive an inhomogeneous plane wave solution below (Pedrotti &E Predrotti 1987, may Fowles 1975, polarization polarizationand andright-handed right-handedpolarization polarization are are different different types typesof of polarization, although although with with similar similar As shown field vector be presented Jones 1 , ε̂2as ⃗⃗⃗⃗k wave ̂complex ⃗kε⃗⃗⃗⃗ ⃗⃗⃗⃗ ̂⃗⃗⃗⃗ ̂(K and orthogonal form a real mutually orthogonal unit vectors ε= ,= ,⃗⃗⃗⃗ n̂index. Here K = + i 𝛼𝛼a is aFor complex andand form aേ real mutually unit vectors (and εi1(κ ). Here K =K kK + i+ 𝛼𝛼k). is a𝛼𝛼 complex vector amplitude {A, B oscillating in an inhomogeneous plane. wave vector = n ̂ and form real mutually orthogonal unit vectors (,(ε(ε1N ,εεn̂εThe ).n Here = k + i 𝛼𝛼 is a complex attention for potential applications in modern physics and unit vectors ). Here = is a complex form aaaiC} real mutually orthogonal unit vectors ,complex ,,εn̂n̂,+ ). Here k= i 𝛼𝛼 is a complex 1K 2+i wave vector =n i κ refraction index. Eq. (3) ⃗⃗⃗⃗ 21ε ̂ ̂ wave vector and N =n + refraction For Eq. (3) to be solution of Eq. (2 2 2 and form real mutually orthogonal unit vectors , ε n ̂ ). Here k + i is a complex and form a real mutually orthogonal unit vectors ( , , ̂ ). Here k + i 𝛼𝛼 is a complex ⃗⃗⃗⃗kdrawn ̂has names. Recently, polarization invector matter and form a (real orthogonal vectors ( and εorthogonal εwave ,22n̂ ). K = ioptical is,εcomplex a).n complex ⃗⃗⃗⃗ ̂K ̂of= ⃗⃗⃗⃗k ⃗⃗⃗⃗ ⃗⃗⃗⃗ ̂ refraction ̂Jones 1 ,11 2through and and form form aa real real mutually mutually orthogonal orthogonal unit unit vectors vectors (εεthe , ,εmutually ε22visualization , ,n̂n̂attention ).). Here Here K k⃗⃗⃗⃗kunit ++ ii𝛼𝛼 𝛼𝛼 isis apropagating a complex complex converting state of polarization the arrangement using vectors calcul and Nand =nkey +of κ index. wave vector +unit iHere κ quantum complex index. For (3) to be and and form form a=drawn real athe mutually mutually orthogonal unit vectors vectors (+ ,𝛼𝛼εε1physicist's n̂index. ̂). ). Here Here K K = kEq. + i 1997, 𝛼𝛼aii solutio 𝛼𝛼is is a F with its helicity has accordingly from scientists exploiting the protocol of ⃗⃗⃗⃗kdrawn ̂has 2,refraction 2 ̂= Keywords: modes, cosmology, wave plate, helicity for potential applications in modern physics information technology (Tamm and Recently, form a real mutually orthogonal unit vectors ( ε1 ,received ε211, n̂attention ). Here K = + ireal 𝛼𝛼mutually is aand complex Jackson 1975) vector =n + iN κ=n complex refraction For Eq. (3) to be information technology (Tamm 1997; Mooleskamp Stokes vector and N orthogonal =polarization n+i is the complex refraction index. For form awave real unit vectors (ε(i1εJones , n̂,elements, Here K = k+ + 𝛼𝛼 is names. names. Recently, the the visualization visualization of ofpolarization polarization propagating propagating in& inand matter matter has physicist's physicist's 1⍵, ε2vector, ⍵ wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. 𝛼𝛼ൌ (2) wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. (2) ⃗⃗⃗⃗ wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. (2) wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. (2) ̂ vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. The ⍵ and form aattention realwave mutually orthogonal unit vectors ( ε , ε , n ̂ ). Here K = k + i 𝛼𝛼 is a complex . Then we get the relations: 𝜅𝜅 as a homogeneous plane harmonic wave, it should be ǡ 𝒦𝒦ൌ 1 2 vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. (2) wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution of Eq. (2) for potential applications in modern physics information technology (Tamm 1997, ⍵ cw ⍵ cshould vector andindex. N =n +For isecurity κin complex refraction index. For Eq. (3) to be solution ofwave, Eq.production to . Then as aand homogeneous plane harmonic wave, it should be ǡ(2) it(3) 𝒦𝒦ൌ Methematica. The platform graphically simulates the of and 2015; Yunand & Choi Eq. (3) be aiand solution ofharmonic Eq. (2) as aahomogeneous plane distribution (QKD) wave to guarantee unconditional in to cryptography We have provided amechanism wave wave vector vector and NN=n =n2013). ++iiκκ complex complex refraction refraction index. For Eq. Eq. (3) (3) to be aa2015, solution solution of of Eq. Eq. (2) ⍵For as2013). a homogeneous plane should ǡsolution 𝒦𝒦ൌ to .c we Then werelati get as ato homogeneous plane wave, itharmonic be ǡ Eq. 𝒦𝒦ൌ wave wave vector vector and N =n N =n +communication. i+ κ κ (2) complex (2) complex refraction refraction index. index. For Eq. (3) bebe be apropagation athe solution ot ⃗⃗abe & Stokes Yun & Choi i(𝒦𝒦 .r⃗solution −⍵t) wavefor vector and N =n + i κ complex refraction For Eq. (3) to be of Eq. attention attention forpotential potential applications applications ininmodern modernphysics physics technology technology (Tamm (Tamm 1997, 1997, ̂and c . Then get as a homogeneous plane harmonic wave, it should be ǡ 𝒦𝒦ൌ c (rand (εMooleskamp ) ⍵ ⍵ E ,index. t)information =information E + ε E ሺ͵ሻ e ⍵ ⍵ wave vector and N =n + i κ complex refraction index. For Eq. (3) to be a solution ⍵ ⍵ ⍵ ⍵ 1 1 2 2 c𝜅𝜅, ⍵ harmonic ⍵ ⍵ ⍵ the . Then we get relations: 𝛼𝛼ൌmedium 𝜅𝜅, as a homogeneous harmonic itshould should bebe, ǡwe that 𝒦𝒦ൌ We have providedwave a dynamic polarization modes platform wave, . Then we get it. Then we get theEq. relations: 𝛼𝛼ൌ 𝜅𝜅, as aas homogeneous plane harmonic wave, itplane should be ǡbe Eq. . Then get the relations: 𝛼𝛼ൌ 𝜅𝜅,⍵the asas homogeneous plane harmonic wave, itmodes should ǡwave, 𝒦𝒦ൌ . we get the relations: 𝛼𝛼ൌ homogeneous plane wave, it should be ǡ𝒦𝒦ൌ while and Naaaa=n +& i κStokes refraction index. For (3) to be aThen solution of (2) ⍵ ⍵ ̂ = ൌ 𝑛𝑛, which result in propagation speeds are different along the ⍵ ⍵ harmonic ⍵ polarized waves aa medium Maxwell's equations. Mooleskamp 2015, Yun & Choi 2013). cthe c K c⍵ dynamic vector polarization for complex the polarization of abe transverse wave, . Then we get the relations: 𝜅𝜅,𝜅𝜅,, n homogeneous plane harmonic wave, should ǡǡ𝒦𝒦ൌ 𝒦𝒦ൌ 𝒦𝒦ൌ . Then we get the relations: ̂ ). inHere homogeneous plane harmonic it should be 𝒦𝒦ൌ c electromagnetic csatisfying c𝛼𝛼ൌ and form real mutually orthogonal unit vectors (𝛼𝛼ൌ εcthe , εdirection we get relations: as ashould homogeneous plane harmonic wave, ititin should be ǡ𝛼𝛼ൌ 1𝜅𝜅, ⍵ ⍵ ⍵ c𝛼𝛼ൌ cwave, ⍵Then ⍵ ⍵ ⍵ . . Then Then we we get get the the relations: relations: 𝛼𝛼ൌ 𝜅𝜅, 𝜅𝜅, as asaahomogeneous homogeneous plane plane harmonic harmonic wave, wave,as ititplatform should be be ǡǡpresenting 𝒦𝒦ൌ 𝒦𝒦ൌ cc. cc 2 c c We have provided a dynamic polarization modes platform for simulating polarization modes with 1. INTRODUCTION Mooleskamp Mooleskamp & & Stokes Stokes 2015, 2015, Yun Yun & & Choi Choi 2013). 2013). ⍵ ⃗ ൌ 𝑛𝑛, which result propagation speeds are different along the cc as ccand . that .are Then Then we we get get theare the relations: relation as a homogeneous awehomogeneous plane harmonic wave, wave, should itpropagation should be be ǡ that ǡspeeds 𝒦𝒦ൌ 𝒦𝒦ൌ ൌinitin 𝑛𝑛, which result in propagation speeds that differen ൌ𝛼𝛼ൌ 𝑛𝑛,harmonic which result that different along the direc get the 𝜅𝜅, as a homogeneous planepolarization harmonic wave,modes it shouldwith be ǡ Jones 𝒦𝒦ൌ c . ⍵ for simulating matrices relations: which result in propagation ⃗plane i∆ε i(k ⍵c.r−⍵t) =asEThen c it should e )relations: 𝑛𝑛, which propagation speeds are along the in t c cresult in c cdifferent ⍵ ⍵ ⍵ . direction Then we getdirection the relatio harmonic wave, be ⍵ǡin 𝒦𝒦ൌ 0a(εhomogeneous 1 + ε 2 e ൌ cplane . Then weJones get the relations: 𝛼𝛼ൌ 𝜅𝜅, the in as a homogeneous plane harmonic wave, itin⍵ should beofǡspeeds 𝒦𝒦ൌ converting the state ofൌ polarization through the arrangement optical elements, using vectors calculations 𝑛𝑛, which result in propagation speeds that are different along the in the medium. ⍵⍵ We have a dynamic polarization modes platform for simulating polarization with 𝑛𝑛,provided which result inൌ propagation are different along thethe direction medium. c the ൌ 𝑛𝑛, which which result propagation speeds that are different along the direction the medium. ൌ 𝑛𝑛, result in propagation speeds that are different along direction in the medium. ⃗ ⍵ Therefore, there will be aN cumulative phase difference between the two components of(3) thetoE cthat c ∆𝜀𝜀modes calculations, corresponding to the physical arrangement of speeds that are different along the direction in medium. ⍵⍵ c cൌ wave vector and =n + i κ complex refraction index. For Eq. cc 𝑛𝑛, which result in propagation speeds that are different along the direction in the medium. ൌ 𝑛𝑛, which result in propagation speeds that are different along the direction in the medium. which result matrices in propagation speeds that are different along the direction inofthe Jones calculations, corresponding the physical arrangement of optical elements, in a in We Wehave have provided provided awhich adynamic dynamic polarization polarization modes modes platform platform for for simulating simulating polarization polarization modes modes with with ⍵ ൌ ൌ 𝑛𝑛,𝑛𝑛, which result result in in propagation propagation speeds speeds that are are different different along along the direction direction ininPolarization the the medium. medium. ⍵the ⍵ is a to coherent characteristic themedium. electromagnetic (EM) wave cthat c ccwhich result in propagation speeds thatc are different along Therefore, there will be a cumulative phase difference ∆𝜀𝜀 bet ൌ 𝑛𝑛, the direction in the medium. Therefore, there will be a cumulative phase difference ∆𝜀𝜀 between the two c ⃗ A 𝑛𝑛, 𝑛𝑛, which which result in in propagation propagation speeds speeds that are are different different along along the the direction direction in in thethm ൌ⍵ൌ i(k ∙ ⃗rresult −⍵t) Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity in a Methematica. Thematrices platformcalculations, graphically simulates the of production and ofthat the optical elements, in Mathematica computing environment Therefore, will be aapropagation cumulative phase difference ∆ there will cumulative phase difference ∆𝜀𝜀 between the ⍵ Therefore, there will a be cumulative phase difference ∆𝜀𝜀 the two componen c there = ( )mechanism eas cthat cTherefore, corresponding to the physical arrangement of optical elements, in abetween field vector they emerge inbe uniaxial crystals (Quartz, Calcite, etc.). After wave has traveled ൌ 𝑛𝑛, which result in propagation speeds that are different the direction in ⃗ th ൌ 𝑛𝑛,Jones which result in propagation are different along the direction in the medium. ⃗along ⃗ the ⃗E Therefore, there will be adifference cumulative phase difference ∆𝜀𝜀components between components of the E Therefore, there will be abe cumulative difference between the two components ofthe the E Therefore, there will cumulative phase difference ∆𝜀𝜀 between the two components oftwo the there will be aaspeeds cumulative phase ∆𝜀𝜀 between the two of the E B + iCphase c Mathematica computing environment (Mathematica 2015). coptical Jones Jonesmatrices matrices calculations, calculations, corresponding corresponding to toTherefore, the thephysical physical arrangement arrangement of of optical elements, elements, ininboth a two a∆𝜀𝜀 ⃗E ⃗⍵vector ⃗ ⃗ ⃗ Therefore, there will be a cumulative phase difference ∆𝜀𝜀 between the two components of the E Therefore, there will be a cumulative phase difference ∆𝜀𝜀 between the two components of the . Then we as a homogeneous plane harmonic wave, it should be ǡ 𝒦𝒦ൌ there will be a cumulative phase difference ∆𝜀𝜀 between the two components of the E (Mathematica 2015). between the components of the field vector as they wave, the electric field vector E and magnetic field B of the eleg ⃗ ⃗ Therefore, Therefore, there there will will be be aa cumulative cumulative phase phase difference difference ∆𝜀𝜀 ∆𝜀𝜀 between between the theequations. two two components components of ofas the the E E field vector as they emerge in uniaxial crystals (Quartz, Calcite, field vector they emerge in uniaxial crystals (Quartz, Calcite, etc.). After the polarized waves in a medium while satisfying Maxwell's c ⃗ ⍵ crystals (Quartz, Calcite, etc.). After the wave ha Therefore, there will be a cumulative Mathematica phase difference ∆𝜀𝜀 between the two components of the E field vector as they emerge in uniaxial computing environment (Mathematica 2015). Therefore, Therefore, there there will will be be a cumulative a cumulative phase phase difference difference ∆𝜀𝜀 ∆𝜀𝜀 between between the the two two components component field vector as they emerge in uniaxial crystals (Quartz, Calcite, etc.). Afte distance d, the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) between E wave and E wave when tho 2 1 x y field vector as they emerge in uniaxial crystals (Quartz, Calcite, etc.). After the wave has traveled a field vector emerge in uniaxial crystals Calcite, etc.). After wave has traveled a aaAfter field vector as they they emerge in uniaxial uniaxial crystals (Quartz, Calcite, etc.). After the wave has traveled field vector as emerge in crystals (Quartz, Calcite, etc.). After wave traveled emerge in uniaxial crystals (Quartz, Calcite, etc.). ⃗ has 𝑐𝑐 thethe Therefore, there will beasa they cumulative phase difference ∆𝜀𝜀(Quartz, between the two components of the E Mathematica Mathematicacomputing computingenvironment environment (Mathematica (Mathematica 2015). 2015). Therefore, there will be atraveled cumulative phase difference ∆𝜀𝜀 between the component ⍵has field as they emerge inin uniaxial crystals (Quartz, Calcite, etc.). After the wave traveled aa ⍵two field vector asthey they emerge uniaxial crystals (Quartz, Calcite, etc.). the wave has traveled vector as emerge in uniaxial (Quartz, Calcite, etc.). After the wave has traveled aLight always oscillates tothe aAfter fixed in space. of such characte ⍵ field field vector vector as as they they emerge emerge inin uniaxial uniaxial crystals crystals (Quartz, (Quartz, Calcite, Calcite, etc.). etc.). After After the thecrystals wave wave has has traveled aa parallel ⍵direction distance d, phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) betwee distance d, the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) between E wave and 2 1 2 1 x 1. INTRODUCTION field vector as they emerge in uniaxial crystals (Quartz, Calcite, etc.). After the wave has traveled a difference 𝑛𝑛,⍵emerge which result ina⍵propagation speeds are different along ൌ the wave has traveled distance d, the phase difference 𝑐𝑐 thethe distance d, phase is ∆𝜀𝜀 = 𝑑𝑑(nCalcite, )that between ExAfter wave and Ehas wav 𝑐𝑐1⍵ 2-n ythe ⍵the field field vector as as they emerge in in uniaxial uniaxial crystals crystals (Quartz, (Quartz, Calcite, etc.). etc.). After has trd ̂emerge c⍵they ⃗kFOR ⃗ wave 𝑐𝑐 E As shown E vector may be presented as the vector (Jones 1941) with aand complex radiation is ∆𝜀𝜀 propagating the∆𝜀𝜀 the ∆𝜀𝜀=0 the amplitude of wave E th distance d,vector phase difference = 𝑑𝑑(n -nand between E wave wave when the distance d,field the phase difference isthe ∆𝜀𝜀 = 𝑑𝑑(n )-n Edirection. wave E wave the d, the difference is ∆𝜀𝜀 =while 𝑑𝑑(n -n Eisx real, wave distance d, the the phase difference ∆𝜀𝜀 =Jones 𝑑𝑑(n -nbetween )After between E wave and wave when the d, phase difference isisdistance = between E wave wave when the 2.MATHEMATICA SIMULATION CONVERTING POLARIZATION 2has 1)If xwhen field vector above, as distance they in uniaxial crystals (Quartz, Calcite, etc.). the traveled 2-n 122along xwave y E ⍵𝑑𝑑(n ⍵ ⍵ 1)phase 1is xx THE yy a 2and 1)Eybetween ⍵⍵ 𝑐𝑐 crystals 𝑐𝑐 = 𝑐𝑐𝑐𝑐 𝑑𝑑(n 𝑐𝑐 Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity field vector as they emerge in uniaxial (Quartz, Calcite, etc.). After the wave has 2. MATHEMATICA SIMULATION FOR THE d, the phase difference is ∆𝜀𝜀 -n ) between E wave and E wave when the distance the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) between E wave and E wave when the is between wave and wave when the distance d, the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) between E wave and E wave when the 2 1 x y 2 1 x y 2 1 x y 1 distance distance d,d, the the phase phase difference difference isis ∆𝜀𝜀 ∆𝜀𝜀⍵== 𝑑𝑑(n 𝑑𝑑(n-n -n11)) between between EExx wave wave and and EEyisy wave whencharacteristic the the 𝑐𝑐THE 𝑐𝑐when 𝑐𝑐 when ⃗kthe ⃗k direction. Polarization awave coherent the wave while in aIfmedium. In w a⃗ 𝑐𝑐𝑐𝑐 2-n12)2 between distance d, the phase difference is 2.MATHEMATICA ∆𝜀𝜀 = 𝑑𝑑(n Ex wave and EFOR the radiationalong is⃗ofpropagating along theIf(EM) direction. the amplitud ∆𝜀𝜀=0 SIMULATION POLARIZATION radiation isCONVERTING propagating the ∆𝜀𝜀=0 ⍵ ⍵electromagnetic y wave ⍵an d, 𝑐𝑐{A, radiation isphase propagating along the kvector direction. ∆𝜀𝜀=0 while the amplitude of Ei vector is d, responsible for the linearly such as𝑑𝑑(n vector {A, B}, theEamplitude ⃗kphase distance distance the thephase difference difference is is ∆𝜀𝜀 ∆𝜀𝜀 =the =the 𝑑𝑑(n -n -n )kIf⃗1n̂between )the between Ethe wave and wavew CONVERTING POLARIZATION MODES vector amplitude iC} oscillating in inhomogeneous plane. The wave = radiation is propagating direction. Ifthe ∆ =0between Therefore, there will be apolarized cumulative difference ∆𝜀𝜀 2Jones 21the x Ewave xotherwise y Ewave ythe 2.MATHEMATICA 2.MATHEMATICA SIMULATION SIMULATION FOR THE THE CONVERTING POLARIZATION distance d,FOR the phase difference isradiation ∆𝜀𝜀 = 𝑑𝑑(n E wave and Ealong wave when ⃗ and ⃗ ⃗ ⃗ ⃗ ⃗ 2-n 1) between x ⃗k ythe isPOLARIZATION propagating along the direction. If ∆𝜀𝜀=0 while the amplitude of E is real, thetw 𝑐𝑐 𝑐𝑐 MODES ⍵ radiation isB propagating along the k direction. If the ∆𝜀𝜀=0 while the amplitude of E is real, radiation isേCONVERTING propagating along the k direction. If the ∆𝜀𝜀=0 while the amplitude of E is real, the radiation is propagating along the k direction. If the ∆𝜀𝜀=0 while amplitude of E is real, the ⃗ 𝑐𝑐 radiation is propagating along the k direction. If the ∆𝜀𝜀=0 while the am ⃗ ⃗ ⃗k⃗kdirection. ⃗ ⃗ distance d, the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n -n ) between E wave and E wave is propagating along the k If the ∆𝜀𝜀=0 while the amplitude of E is real, the ⃗k⃗k direction. 2 1 x y ⃗ ⃗ radiation propagating along the direction. If the ∆𝜀𝜀=0 while the amplitude of E is real, the radiation is propagating along the direction. If the ∆𝜀𝜀=0 while the amplitude of E is real, the ⃗ ⃗ radiation radiation isis propagating propagating along along the the direction. If If the the ∆𝜀𝜀=0 ∆𝜀𝜀=0 while while the the amplitude amplitude of of E E is is real, real, the the vector isis responsible for theJones linearly polarized such{A, as Jones vecto is responsible for the linearly polarized such as Jones vector B}, otherw while amplitude real, vector responsible for both the electric field Eelliptically and magnetic field vector B ofasthe electromagnetic rad 𝑐𝑐the ⃗theis MODESIf the ∆𝜀𝜀=0 while thewave, the complex vector responsible for the polarized vector {A, BേiC}. Specifically radiation is propagating along the ⃗k direction. amplitude of E real, the vector isvector responsible forvector theof linearly polarized such asis Jones vector {A, B}, otherwise the ̂ and formis avector real mutually orthogonal unit vectors (vector εfor ,such εthe ,such n̂while ). along Here K =⃗k ⃗⃗⃗⃗ k⃗k + iuniaxial 𝛼𝛼 aIfreal, complex ⃗ isE ⃗is field as they emerge in crystals (Quartz, Calcite, etc.). After vector is responsible linearly polarized such as Jones vector {A, B}, otherwise the amplitude 1waves 2such is responsible for the linearly polarized as Jones vector {A, B}, otherwise amplitude isthe vector responsible for the linearly polarized as Jones vector {A, B}, otherwise the amplitude vector isis responsible the linearly as Jones vector {A, B}, otherwise the amplitude isis ⃗kfor ⃗is MODES MODES radiation radiation ispolarized is propagating propagating along the the direction. direction. If thethe the ∆𝜀𝜀=0 ∆𝜀𝜀=0 while while the amplitude amplitude of of E 1.radiation INTRODUCTION along the direction. If the ∆𝜀𝜀=0 the amplitude of E is the 2.1 Electromagnetic in solids the linearly polarized such as Jones vector {A, B}, otherwise 2.1 Electromagnetic waves inpropagating solids isisresponsible for linearly polarized such asas Jones vector {A, B}, otherwise thethe amplitude is isis vector responsible for the linearly polarized such Jones vector {A, B}, otherwise amplitude vector responsible forthe the linearly polarized such as Jones vector {A, B}, otherwise the amplitude vector is responsible for the linearly polarized such asaselliptically Jones vector B}, vector vectorisisresponsible responsiblefor forthe thelinearly linearlypolarized polarized such such as asJones Jonesvector vector {A, {A, B}, B}, otherwise otherwise the the amplitude amplitude is is the complex vector responsible the polarized Jone the complex vector responsible for the elliptically polarized Jones as be {A, always oscillates parallel to athe fixed direction in space. Light offor such character isvector said{A, to ⃗ ⃗linB if B=0 and A=C then the wave is a circularly polarized Jones vector such {1, i}. radiation is propagating along k direction. If the ∆𝜀𝜀=0 while the amplitude of vector is responsible for the linearly polarized such as Jones vector {A, B}, otherwise the amplitude is the complex vector responsible for the elliptically polarized Jones vector as {A, BേiC}. 2.1 Electromagnetic waves infor solids the amplitude isJones the complex vector responsible forBേiC}. the Specifically,E wave vector and Ncomplex =n +the i κvector complex refraction index. For Eq. (3) topolarized bepolarized a solution of Eq. (2) ⍵ Specifically, the complex vector responsible for the elliptically polarized Jones vector as {A, the complex vector responsible the elliptically polarized vector as {A, BേiC}. Specifically, the complex vector responsible for the elliptically polarized Jones vector as {A, BേiC}. Specifically, the responsible for the elliptically polarized Jones vector as {A, BേiC}. vector is responsible for linearly polarized such as Jones vector {A, B}, otherwise the amplitude is vector vector is responsible is responsible for for the the linearly linearly such such as as Jones Jones vector vector {A, {A, B}, B}, otherwise otherwise the the amp am Polarization is athe coherent characteristic of the electromagnetic (EM) wave in awaves medium. In a{A, plane The propagating process of electromagnetic in solids is∆𝜀𝜀 different from that the vacuum, since 2.1 2.1Electromagnetic Electromagneticwaves wavesinin solids solids distance d, the phase difference is = 𝑑𝑑(n between ExJones wavei}. a 2-n 1) in vector responsible for the elliptically polarized vector BേiC}. Specifically, vector responsible for the elliptically polarized Jones vector as {A, BേiC}. thecomplex complex vectorJones responsible for the elliptically polarized Jones vector {A, BേiC}. the thecomplex complex vector vectorresponsible responsible for forthe the elliptically polarized polarized Jones vector vector as asthe {A, {A, BേiC}. BേiC}. Specifically, Specifically, if Jones B=0 and A=C then the wave isSpecifically, aSpecifically, circularly polarized if B=0 and A=C then the wave isas aasas circularly Jones vector such as {1,vect 𝑐𝑐polarized Thevector propagating process of elliptically electromagnetic waves in elliptically polarized Jones vector {A, B±iC}. Specifically, A {A, quarter wave plate is athe thin birefringent crystal the thickness of which hasJones been adjusted t complex vector responsible for the elliptically polarized vector as 1 the complex responsible for the The elliptically polarized Jones vector as BേiC}. Specifically, if B=0 and A=C then wave is a circularly polarized Jones vector such as {1, i}. ⍵ ⍵ vector is responsible for the linearly polarized such as Jones vector {A, B}, otherwise the a propagating process of electromagnetic waves in solids is different from that in the vacuum, since B=0 and A=C then the wave isJones aThen circularly polarized Jones vector as {1, i}. responsible . we get the relations: 𝛼𝛼ൌ 𝜅𝜅, such as acomplex homogeneous plane harmonic wave, should be ǡ vector 𝒦𝒦ൌ ifwaves B=0 and A=C then the wave iscomplex athat circularly polarized vector such as {1, i}. polarized B=0 and A=C then the wave circularly polarized Jones vector such as {1, i}. ififvacuum, B=0 and A=C then the wave isis aa in circularly polarized Jones vector such as {1, i}. the vector responsible for elliptically polarized Jones vector as {A, BേiC}. Specifically, ⃗E⃗ifthe ⃗it ⃗since the the complex responsible for for the the elliptically elliptically polarized Jones Jones vector vector as {A, {A, BേiC}. BേiC}. SpeS c since c in a polarized The Thepropagating propagating process ofofelectromagnetic electromagnetic waves in solids solids is different different from from that in the the vacuum, vacuum, solids isprocess different from that the since and of ifvector B=0 and A=C then the wave is aas circularly Jones and B of the electromagnetic waves interact with the electrons solid (Pedrotti &as Predrotti 1987, wave, both the electric field vector magnetic field vector B of the electromagnetic radiation ififin B=0 and A=C then the wave isis aaa{1, circularly polarized Jones vector such as {1, i}.i}. if B=0 and A=C then the wave circularly Jones vector as {1, B=0 andis A=C then the wave is circularly polarized Jones vector such {1, i}. is ififB=0 B=0and and A=C A=C then thenthe the wave wave isisaain circularly circularly polarized polarized Jones Jones vector vector such such as as {1, i}. A quarter plate a thin birefringent crystal the thickh Apolarized quarter wave plate issuch a wave thin birefringent crystal the thickness ofthe which produce a i}. േ 𝜋𝜋/4 phase difference between the ordinary and extraordinary rays at the operatin ⃗k radiation is propagating along the direction. If the ∆𝜀𝜀=0 while ampl if B=0 and A=C then the wave is a circularly polarized Jones vector such as {1, i}. A quarter wave plate is a thin birefringent crystal the thickness of which has been if B=0 and A=C then the wave is a circularly polarized Jones vector such asS ⃗ ⃗ E and Bwith of the electromagnetic waves interact with thethin electrons inthe a solid (Pedrotti &been Predrotti 1987, the complex vector responsible for elliptically polarized Jones vector as {A, BേiC}. the electromagneticif waves the electrons in a vector such as {1, i}. ⍵interact A quarter wave plate is a birefringent crystal the thickness of which has been adjusted to A quarter wave plate is a thin birefringent crystal the thickness of which has adjusted to A quarter wave plate is a thin birefringent crystal the thickness of which has been adjusted to A quarter wave plate is a thin birefringent crystal the thickness of which has been adjusted to B=0oscillates and A=C then theelectrons wave is in a in circularly polarized Jones vector such asthe {1, i}. ൌ 𝑛𝑛, which result infixed propagation speeds that are different along direction in the medium. ⃗E ⃗Eand ⃗ B ⃗ ofofthe Fowles 1975). In particular, although the electromagnetic waves are an harmonic plane wave, the always parallel to a direction in space. Light of such character is said to be linearly if B=0 if B=0 and and A=C A=C then then the the wave wave is a is circularly a circularly polarized polarized Jones Jones vector vector such such as as {1, {1, i}. i}. andB theelectromagnetic electromagnetic waves waves interact interact with with the the electrons a solid a solid (Pedrotti (Pedrotti & & Predrotti Predrotti 1987, 1987, AA quarter wave plate is a thin birefringent crystal the thickness of which has been adjusted to c birefringent A A quarter quarter wave wave plate plate is is a a thin thin birefringent crystal crystal the the thickness thickness of of which which has has been been adjusted adjusted to to A quarter wave plate is a thin birefringent crystal the thickness of which has been adjusted to quarter wave plate of is which a thin has birefringent crystal the thickness of𝜋𝜋/4 which hasdifference been to produce phase between ordinary andra produce a wave േto𝜋𝜋/4 phase between the adjusted ordinary andtheextraordinary solid (Pedrotti & Pedrotti 1987; Fowles 1975). particular, A quarter plate isadifference aേbetween thin birefringent crystal the A quarter wave plate is a thin birefringent crystal the In thickness been produce aresponsible േadjusted 𝜋𝜋/4wave phase difference the ordinary and extraordinary rays th vector isquarter for the linearly polarized as Jones {A, B}, Fowles 1975). particular, although the electromagnetic waves are ana ordinary harmonic plane wave, the A plate is thin crystal thevector ofatoth wh 4birefringent produce a and േcrystal 𝜋𝜋/4 phase difference between the and extraordinary rays atthickness the i}. operating produce a േaaisേ 𝜋𝜋/4 phase difference between the ordinary and extraordinary rays at Jones the operating if B=0 A=C then the wave is aand circularly polarized vector such as {1, produce േIn 𝜋𝜋/4 phase difference between ordinary and extraordinary rays at such the operating produce 𝜋𝜋/4 phase difference between the ordinary extraordinary rays at the operating A quarter wave plate a thin birefringent the thickness of which has been adjusted to fields may pull or push the electrons in the orbital of a solid, which is responsible for inducing dipole Fowles Fowles1975). 1975). In particular, particular, although although the theelectromagnetic electromagnetic waves waves are arean an harmonic harmonic plane wave, wave, the the although waves are harmonic plane thickness of which has been adjusted toat produce aof±of π/4 ⃗the produce aaaordinary േ 𝜋𝜋/4 phase difference between ordinary and extraordinary thethe operating A A quarter quarter wave wave plate plate is is a thin a thin birefringent birefringent crystal crystal the thickness thickness which which hashas been been adja Therefore, there will bean aordinary cumulative phase difference ∆𝜀𝜀 between the two components ofrays the E 1 plane produce produce aaInേ േthe 𝜋𝜋/4 𝜋𝜋/4electromagnetic phase phase difference difference between between the the and and extraordinary extraordinary rays rays at atthe the the operating operating produce േ 𝜋𝜋/4 phase difference between the ordinary and extraordinary rays at operating produce േ 𝜋𝜋/4 phase difference between the ordinary and extraordinary rays at the operating produce a േ 𝜋𝜋/4 phase difference between the pull ordinary andtheextraordinary raysorbital at theof operating fieldsthe may or push electrons in the aേsolid, which is responsible for inducing dipole wave, the fields mayproduce pull or apush electrons in the orbital phaseplate difference between the ordinary andthickness extraordinary complex vector responsible the elliptically polarized as { 4 vector 4 ordinary produce adipole phase difference between the and extraordin േorbital 𝜋𝜋/4ofphase difference between thethe ordinary and extraordinary rays atfor operating A quarter wave is𝜋𝜋/4 thin crystal ofisJones which has been ⃗ ⃗M ⃗⃗adifference 4the fields fieldsmay maypull pullororpush pushthe theelectrons electrons in in the the orbital of a a solid, solid, which which is is responsible responsible for for inducing inducing dipole wavelength. Weextraordinary wavelength. desire anot matrix that desire will a om t moment P and magnetization in thebirefringent solid. Ifthe we assume that the medium aWe magnetic field vector as they emerge in uniaxialproduce crystals (Quartz, Calcite, etc.). After the wave has traveled a produce a േ a േ 𝜋𝜋/4𝜋𝜋/4 phase phase difference between between the the ordinary ordinary and andextraordinary rays rays at atthe the 4 4 4operating 4 of a solid, which is responsible for inducing dipole moment rays at the wavelength. We desire a matrix that will iφ iφxx i(ε i(εxx+φ +φxx)) ⃗ 4and ⃗ ⃗⃗ 4 moment P magnetization M in the solid. If we assume that the medium is not a magnetic wavelength. wavelength. We Weand desire desire aa matrix that thatwave will will transform transform the the element element EE0x into into EE ee 4 4matrix 4 0x)) eeJones 0x B=0 A=C then the axxinto circularly vector asi(ε{1 iφ i(ε +φ iφ iφ i(ε iφ i(ε +φ +φ +φ )0xy )rays ⍵that 4 we produce aif േ 𝜋𝜋/4 phase difference between the and extraordinary at the yy x y+ x xx xxy+φ x ) xand x and ⃗ P ⃗and ⃗⃗ ⃗M ⃗⃗ inin magnetization the solid. we assume the the element into into E0y intosuch .and EThe Eand wavelength. wavelength. wavelength. We We We desire wavelength. desire desire matrix aathe matrix matrix We desire will will will anot transform matrix transform thatthe will the the element element transform Eand EE0x the eEiφ eis eelement into into EEordinary EEE0x eethe epolarized eei(ε into E and ei(εeexiφ moment moment P and and magnetization magnetization⃗M inthe the solid. If Ifwe assume assume that that the medium medium is istransform not abetween amagnetic magnetic material and Jthat =0, the Helmholtz wave equation in (Fowles 1975, Jackson 1975) is 0y 0y 0y e gen 0x 0x 0x 0x 0x 0x distance d,solid. the phase difference isathat ∆𝜀𝜀 =that 𝑑𝑑(n -n )transform Eelement wave when x wave ya solid 𝑐𝑐 4 2 1 iφ iφyy i(ε i(εyy+φ +φyy)) 4 medium is not a magnetic material =0, the theHelmholtz Helmholtz into general form of4 aisa matrix representing ee equation into into EEin a esolid e .. The The The general general form form matrix matrix representing representing aa phase phase retarder retarder w EE wave material and and J=0, (Fowles 1975, Jackson 1975) 4of

iφ iφyy iφi(ε +φ +φ i(εyquarter 2E 2P y yy+φ y ) .yy))..E y )form ⃗is ⃗birefringent wave arepresenting thin crystal the thickness 1 ∂form transform the will elements transform by the elements matrix opera by into into into EE0y EE1975, e1975, ei(ε e⃗eyyi(ε into The The The eA general general general . (form form The of⃗E of general of a)൅ aplate matrix a matrix matrix representing of a ∂matrix arepresenting phase retarder phase will will retarder willof whic E0y EEin material materialand andJ=0, J=0,the theHelmholtz Helmholtzwave waveequation equation in aesolid aiφeesolid (Fowles (Fowles Jackson Jackson 1975) 1975) is is+φ∇ 0y 0y 0y 0y ⃗ × ⃗∇ ⃗2the × =representing −u is a real, aphase phase the retarder retarder by athe ሺͳሻ radiation is propagating along0y the k isdirection.0y If a thephase ∆𝜀𝜀=0 while the will amplitude of0 ∂tE wave equation in a solid (Fowles 1975; Jackson 1975) retarder elements matrix c2 ∂t2transform 2E 2P ⃗ ⃗ 4 ∂ 1 ∂ transform theelements elements by bythe the asfollows follows ⃗ × ( ⃗∇transform ⃗E )൅ 2the ×matrix = −u matrix matrix as operation operation as ሺͳሻ ∇ iφ 2∂ ⃗ 2E ⃗ transform ⃗ 2the ⃗ the 0 ∂t 2operation iεxand extraordinary ∂2∂P E P transform transform the elements elements elements transform the matrix operation operation by2Jones the asvector as as follows follows follows follows ∂t c operation produce amatrix േ 𝜋𝜋/4 phase difference betweenis the ordinary 0 ) ( (Ee0xiεxe x ⃗ ∇ ⃗××( ⃗∇ ⃗E)൅)൅12is1∂2 responsible ⃗⃗ .r⃗−⍵t) for the linearly such as {A,EM B},wave otherwise thea form amplitude ( ⃗∇××⃗Evector −u −u by polarized by by the the the =elements ሺͳሻ matrix ሺͳሻ ∇ (̂e i(𝒦𝒦 00 ⃗P ⃗ 2 2== 2 2 where 𝜒𝜒𝜀𝜀 E . If we suppose the will be of can iφy ∂t∂t c c∂t∂t iεy , we 0 iφ iφxx i(ε i(εsolution +φ +φxx)) such as E00e x x E e iεiεxx e 0 e ee ee ⃗⃗ EE0x EE0x 0y 0i(ε 0iφi(ε iφxiφ iφxx ((ee i(ε +φ +φ ) x0x i(ε== )0x i(𝒦𝒦 .r⃗−⍵t) )) x x+φ x( x( xsolution x)) E x +φ xE ̂ ⃗ ⃗ (4) (4) ) ) ) ) ( ( iε iε iε iε e e e e e e e e E E E E E E E where P = 𝜒𝜒𝜀𝜀 E . If we suppose the EM wave will be a form of such as e , we can x x x x iφ iφ i(ε i(εyy+φ +φyy)) 0 0x 0x 0xpolarized 0x0x 0x 0xiε 0x iε 0 http://dx.doi.org/10.5140/JASS.2015.32.2.63 0 0 0 0 e e e e y y the complex vector responsible for the elliptically Jones vector as {A, BേiC}. Specifically, y y 64 ⃗⃗ ⃗⃗ E E e e E E e e i(𝒦𝒦 .r⃗) −⍵t) .r⃗−⍵t) 0, (we eei(εiφi(ε =),== =) (4) (4) (solution (solution ( )Eq. )as ) as 0y 0y )y) (0we (can ) +φ ()) ) (4)(4) ( ((̂0E ( ̂e0i(𝒦𝒦 ⃗ ⃗ ⃗y+φ where where⃗P=⃗P=𝜒𝜒𝜀𝜀𝜒𝜒𝜀𝜀 wewesuppose supposethe theEM EMwave wavewill willbebea form a formofof such such can iφ i(εy +φy0y )0y k 0E0.E. If If yywave yy+φ y ) yy)) E yy (1) EE EE0using eeiφeeyiφ E Evector EE eeeeyi(ε where εx 4and εwhere εx andthe represent εy advance repres 0 rewrite 00 eiεeeyiεiε eaiε 0y 0y 0y 0y 0y 0y 0y 0y e y ⃗ Eq. the (1) wave using is a wave vector k if B=0 andrewrite A=C then a circularly polarized Jones vector such as {1, i}. where where εεxx and and εεyy represent represent the the advance advancein in phase phase of of EExx--and and EEyy--component component of ofthe theincident incidentlight. light ⃗ k ⃗ rewrite rewriteEq. Eq.(1)(1)using usinga wave a wavevector vectork an example, consider a quarter-wave consider plate a qu where where where εxεεxand εx andthethe and εywhere εεyy represent represent represent εthe advance advance represent in in in phase the phase phase advance ofof of ExE -Exand and and phase EyE -Eycomponent of component Ex- andofEof of the component the incident incident incident light. of light. light. the Asan As incident Asexample, light. As x--in y-- component y- the x and yadvance 0y 0y

0y 0y

Mathematica computing environment (Mathem polarized waves1 in a medium while satisfying Maxwell's equations. eneous plane. The wave vector ⃗k = k n̂ Department of Microbial and Nanomaterials, Mokwon University, ̂ = ⃗⃗⃗⃗k realized and form Platform a real mutually unit vectorspolarization ( ε1 , ε2 , n̂ ). Here K + i 𝛼𝛼 is a complex fororthogonal manipulating modes with Jones 1 Department of Microbial and Nanomaterials, Mokwon University, Daejeon 302-729, Korea ⃗⃗⃗⃗ ̂ ε2 , n̂ ). Here K = k + i 𝛼𝛼 is a complex Daejeon 302-729, Koreaindex. 2.MATHEMATICA SIMULATION FO x Eq. x +φx ) ofand 2 (2) vectoraand N =nthat + i κwill complex For (3)Eto0xbeeai(ε solution Eq. wavelength. wave We desire matrix transform therefraction element E0x eiφ into polarization modes, Jones wave plate,Chungnam helicity Nationa Department ofcosmology, Astronomy and vector, Space Science, Yong-Dae Choi etKeywords: al. Manipulating Polarization Modes Realized with Jones Vectors MATHEMATICA vectors in 2 Daejeon 305-764, Korea iφ iφxx i(εxx+φ +φxx)) dex. For Eq. (3) to be a solution We of Eq. (2) aa matrix producing polarization modes. P Department oftransform Astronomy and Space Science, National wavelength. wavelength. We desire desire matrix that that will will transform the the element element EE0x into into +φ EE0x eei(ε and and ⍵ ⍵University, 0x ee Chungnam 0x +φy )plane x xrelations: x) MODES y .eiφiφ Then we 𝛼𝛼ൌ c 𝜅𝜅, as a E homogeneous harmonic wave, it305-764, should ǡ 𝒦𝒦ൌ We desire that will transform the element Eciφ into Eget exi(ε and eiφy wavelength. into ei(ε .a matrix The general form of a be matrix representing a phase retarder E0ywavelength. 0x 0xi(εthe +φ i(ε )will Daejeon Korea 0ydesire x einto x Einto x ) x +φ xproducing 3and polarization modes. Presently the definition of polariza avelength. We desire We a matrix a matrix that will that transform will transform the element the element E e E e E e and 0x 0x 0x 0x ⍵ ⍵ Korean Institute of Science andnomenclature Technology Information, iφ iφyy i(ε +φyy)) upgraded, which can theeerelations: 𝛼𝛼ൌ 𝜅𝜅,i(ε yy+φ 𝒦𝒦ൌ c . Then we getEE0y into into EE0y .. The The general general form form of of aa matrix matrix representing representing aa phase phase retarder retarder will will305-806, Korea 0y cy ) ee iφy 0y i(εy +φ 3 Daejeon ⍵ 1.retarder INTRODUCTION 2.1 Thus, Electromagnetic waves solids eelements into E e matrix .operation The general form of matrix representing phase Ethe ⍵ operation as follows direction of wave propagation. possibility Institute of2a) Science and Technology Information, 1 x Korean 3† 0y 0y ൌi(ε 𝑛𝑛,the which result in speeds that are different along thea the direction in the will medium. iφ ⃗ in transform by asEfollows nomenclature upgraded, which canthe be confusing to students x +φ iφthat iφ +φ i(ε )y +φ k × ( ⃗k ×of⃗Ephysi )൅ y y transform ythe y ) .EChoi , general Bogyeong Yun matrix will element epropagation into ei(ε and c yYong-Dae ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣheejy@r 0x 0x general The form form of aKim matrix of xKorea a, Hee-Joong matrix representing representing a phase a phase retarder retarder will will c there are similar customary nomen 0y eE0y einto Einto 0y eE0y e . The Daejeon 305-806, transform transform the the elements elements by by the the matrix matrix operation operation as as follows follows of polarizing light is essentially due to its transverse different along the direction in the medium. Received Revised Accepted Polarization is a coherent characteristic of the electromagnetic (EM) wave in The propagating process of electromagnetic wa transform the elements by the matrix operation as follows ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] there similar customary nomenclature forshould the right circularly pola +φy ) x +φx ) will ⃗ the iεx there Therefore, will be a cumulative difference ∆𝜀𝜀 between the two components of theareE E0x E eiφx Received ei(ε . The general form of 0 matrix representing a phase phase retarder character. Therefore, Mathematica simulation 0x follows It then becomes a vector for 1975, Jackson 1975), equation right-circula Revised Accepted ansform transform the elements the elements theaby matrix operation operation follows as iφ iφxx i(εxx+φ +φxx)) (4) (4) (e by1Department )the( matrix =and ) as (EE0x iε iεxx iφ eei(ε E)E) 0x ee i(εy +φ 0x 0x 0 0 e e iε of Microbial Nanomaterials, Mokwon University, y y iφ i(ε +φ ) ⃗ y x x x ⃗ and ⃗ with ⃗ ×and ⃗waves ∆𝜀𝜀 between the two components e0x 0 eeiεx of 0E = = (4) (4) (the (0y Ee E0xiεiε e)) (( E0y iφ ) ) ( ( ) ) show the transverse character of the its vectorial E B of the electromagnetic interac wave, both the electric field vector E magnetic field vector B of the elec e E 1975, Jackson 1975), right-circularly (Pedrotti & Predrotti 1987), ri iφ i(ε i(ε +φ +φ ) ) y yy field vector as( they 302-729, emerge (Quartz, Calcite, After(4) the wave has traveled a EEcrystals e yy+φ EE0y ee yy yetc.). 00 (inKorea eeuniaxial Daejeon = e(i(ε ) the matrix operation as follows 0y 0y 1993), and right-handed & i(ε +φ the component of electric iεy ) iφ iφexiφy ) 0y i(ε ) xy +φ y fundamental conception in has physics the propagation of the(Born electrom producing polarization modes. Presently theThe definition of polarization beenof modified, and susceptibility its xE0y x x x)) E e iεx iεx 0 Ee E E e e e e E behaviors dynamically satisfying Maxwell's wave equations. 0y 0x 0x 0x 0x e 0 has Calcite, After Fowles 1975). In1999, particular, the el oscillates parallel to a fixed direction in isspace. Light of although such = = (of (4)always (εeythe (wave ) ) (0traveled ( iφay ) The 1993), and right-handed (Born & Wolf Georgi 1982)character circular ⍵ xconception 2 iεi(ε) +φ whereetc.). εx eand represent the)ε of advance iny(phase -𝑑𝑑(n and -between component of(4) the incident light. Aslight. iφ i(εyE )y)+φ )in fundamental of--the propagation the electromagnetic in matter iφdistance yChungnam y yof x Department and Space Science, National University, where where εεyxx xand and and represent the advance advance in in phase phase of EEphysics and EE component component of ofy of the the incident incident light. As Aswave polarization eAstronomy Erepresent E eE ei(ε d,e0 phase ∆𝜀𝜀 =0y +φ -nE and E wave the represent the advance of In addition towhen the transverse the ethe E0x polarization and right-handed pol 0where eiεxEy)ε0y xx-- and yxy wave newly asphysics the cardinal keyword in free-space quantum 2in 1)phase yy difference 0y e isthe solid as anpolarizing inhomogeneous planecommu wave 0y 0 ) ( E0x where upgraded, which can belight. confusing to understood students ofcharacter andofresearchers. For instance, 𝑐𝑐 of Enomenclature ε and ε represent the advance in phase and E component of the incident As = (4) ) ( ) x y x y Daejeon 305-764, Korea iφy i(εy +φy ) iεy producing polarization modes. Presently definition polarization hascosmology been modified, and its E0y fields may pullinor push the electrons in the orb ebetween - and component of the incident light. Ascardinal an example, wave, thethe helicity of aofpolarized wave is a critical factor in Ex ewave andEE wave when the polarization and right-handed polarization are different types of pol 0yy -e newly understood as the keyword in free-space quantum communication technology and 1 astrophysics. Interactive visualization of the propagation mechanism of po an where example, consider a3represent quarter-wave (QWP) which makes ∆𝜀𝜀=𝜋𝜋/2. We may write the Jones names. Recently, the visualizat an an example, example, consider consider aplate a quarter-wave quarter-wave plate plate (QWP) (QWP) which which makes makes ∆𝜀𝜀=𝜋𝜋/2. ∆𝜀𝜀=𝜋𝜋/2. We We may may write write the the Jones Jonescircularly Jackson 1975) ⃗ ⃗ there are similar customary nomenclature for the right polarization: right circularly (Fowles here εx and ε ε and represent ε the advance the advance in phase in phase of E of and E E and component E component of the of incident the incident light. light. As As radiation is propagating along the k direction. If the ∆𝜀𝜀=0 while the amplitude of E is real, the x x y y yconsider y consider a quarter-wave (QWP) which quantum cryptography communication technology or Korean ofplate Science and Technology Information, anxexample, a Institute quarter-wave plate (QWP) which makes makes ∆𝜀𝜀=𝜋𝜋/2. Wecan may the Jones nomenclature upgraded, bewrite confusing ofelectromagnetism physicsthe and researchers. Formagnetization instance, ⃗ and ⃗M ⃗⃗ in the so astrophysics. Interactive visualization of which the propagation mechanismto ofstudents polarized in a medium sent the advance in phase ofDaejeon Ex⃗- and 305-806, Ey- component of the incident light. As names. Recently, visualization of polarization propagating moment P with its helicity received attention from scientists exin Korea 2 has accordingly 2 𝜀𝜀=0 whileMthe amplitude ofwrite Etransforming isthe real, the the attention for potential ⍵ ⍵ We may Jones matrix, M transforming the Jones cosmology in modern physics. matrix, matrix, M M transforming the Jones Jones vector vector ∆𝜀𝜀=𝜋𝜋/2 ∆𝜀𝜀=𝜋𝜋/2 to to produce produce right-handed right-handed circular circular polarizing polarizing light light as as matrix, transforming the Jones vector ∆𝜀𝜀=𝜋𝜋/2 to produce right-handed circular polarizing light as ̂ሺʹሻ 1975, Jackson 1975), right-circularly (Pedrotti &×isPredrotti 1987), right-hand circularly (Reitz et, t) al.=applicatio ⃗ ⃗ (r (ε1 E1 + ⃗ ⃗ ⃗ E vector is responsible for the linearly polarized such as Jones vector {A, B}, otherwise the amplitude k × ( k E )൅ E = − χ E n example, an example, consider consider a quarter-wave a quarter-wave platevector (QWP) plate (QWP) which makes makes ∆𝜀𝜀=𝜋𝜋/2. ∆𝜀𝜀=𝜋𝜋/2. We may Wepolarizing write may the Jones the Jones ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] matrix, M transforming the Jones ∆𝜀𝜀=𝜋𝜋/2 towhich produce right-handed circular light asexploiting 2 cprotocol c2 quantum there areaccordingly similar customary nomenclature for the right circularly polarization: right circularly (Fowles with its helicity has received attention fromwrite scientists of key attention forthepotential applications in modern physics andequation inform makes to We produce right-handed circular may write the Jones uarter-wave (QWP)which which makes ∆𝜀𝜀=𝜋𝜋/2. distribution (QKD) tomaterial guarantee unconditional security inwave cryptography and J =0, the Helmholtz Received Revised Accepted 11 es vector {A,plate B}, vector otherwise the amplitude is Mooleskamp Stokes 2015, Yun 1 −i −i ππ 1993), andJones right-handed (Born & WolfSpecifically, 1999, Georgi 1982) circularly polarization. Right & circularly 1 responsible 44 −i πvector 1 1 0 0 0 0 e e thepolarizing complex for the elliptically polarized vector as {A, BേiC}. −iπ/4 −iπ/4 =var E0 −i π ⃗Eright-hand 4 Jones light 2.2becomes Polarization modes Mathematica simulation distribution guarantee unconditional security in cryptography communication. We have providedthat a(Reitz 1 ))to 0==1produce 04as e the M= M= ee(QKD) ((to )) right-handed QWP, QWP, FA FAIt vertical vertical (5) , which indicates the propagation process then a(5) vector equation for matrix, matrix, M transforming M transforming the Jones vector ∆𝜀𝜀=𝜋𝜋/2 right-handed circular circular polarizing polarizing light as light as 1987), 1975, Jackson right-circularly (Pedrotti & Predrotti circularly et 0to produce dynamic polarization platform theal. polarization modes Mooleskamp & Stokes 2015, Yunfor& presenting Choi 2013). M= ( to e−iπ/4 (ii1414∆𝜀𝜀=𝜋𝜋/2 QWP, FA1975), vertical (5) ) (=(01vector ⃗ o 00 iQWP, i as ππ () 1 ∂2 E e Jones ∆𝜀𝜀=𝜋𝜋/2 produce polarizing light 1right-handed M= eee−iπ/4 ) FA vertical (5) (eSpecifically, ) = circular ⃗ ⃗ ⃗ 0 0 ed Jonesvector vector as {A, BേiC}. 0 i We have provided a dynamic i π × ( similar ∇ × E )൅ 2 2 p= 0 i i π polarization and right-handed areofdifferent typeselectromagnetic of polarization, although ∇with c ∂t 0 1 then e04 theewave 4 if B=0 and isdynamic a circularly polarized Jones vector such asthe {1,Wolf i}.polarization polarization platform for presenting polarization modes a transverse wave, 1 A=C ⃗ 1993), and right-handed (Born & 1999, Georgi 1982) circularly polarization. Right circularly converting the state of polarization through the arrangement of optical ele solution of Eq. the component of electric susceptibility 𝜒𝜒. Therefore, we will write the E = The fundamental conception in physics of the propagation of the electromagnetic wave polarization in matter is −i π −i π have a dynamic polarization modes platform for sim Wethe have usedWe Mathematica implement interactive 4 is 40 case This This is the the case0 of of fast fast axis axis vertical(FA vertical). Similarly, Similarly, we we can can determine determine the corresponding corresponding Jones Jonesprovided to 1vertical(FA 0 1 0 vertical). e e −iπ/4 −iπ/4 1 0 Jones matrices calculations, corr 0 ) =such es vector as {1, i}. −iπ/4 M=is(A(of M= = e = e ( ( ) ) QWP, QWP, FA vertical FA vertical (5) (5) ( ) ) (5) names. Recently, the visualization of polarization propagating in matter has drawn physicist's eThis ) QWP, FA vertical (5) ⃗ ⃗ the case of fast axis vertical(FA vertical). Similarly, we can determine the corresponding Jones 1 1 This case fast axis vertical(FA vertical). Similarly, we can determine the corresponding Jones converting the state of polarization through the arrangement of optical elements, using Jones vectors calculations where P = 𝜒𝜒𝜀𝜀 E . If we suppose the EM wave 1 is the wave is a thin birefringent crystal the of which hasmatrix been adjusted to and 0 i 0 keyword ipolarization π plate 0 quarter in cosmology Methematica. The platform simulates the1987, mechanism calculations animations forgraphically the generation andthickness right-handed polarization are different types of in polarization, although with the 0cardinal in free-space quantum solid communication technology and asJones an inhomogeneous wave solution below (Pedrotti &similar Fowle 0i newly 0 ei4understood ei4π asplate(HWP) Jones matrices calculations, corresponding toPredrotti the physical arrange ei4π equations Eq. (6,7)plane in both numeric and graphic simulations. Mathematica matrix matrix for for aaadjusted half-wave half-wave plate(HWP) or or eighth-wave eighth-wave plate plate (EWP) (EWP) or orEarbitrary arbitrary phase phaseand of of retarded. retarded. Jones Jones iφ i(ε +φ iφ i(ε computing environm xxx into xxx+φ xx)x ) and e thickness matrix of wavelength. which has been to wavelength. We desire a matrix that will transform the element E e into e We desire a matrix that will transform the element E e E e 0x 0x attention for potential applications in modern physics and information technology (Tamm 1997, in Methematica. The platform simulates the mechanism of production and propagation of the 0x orgraphically 0x and propagation of the polarization modes in the solid state. for a half-wave plate(HWP) or eighth-wave plate (EWP) arbitrary phase of retarded. Jones equations Eq. (6,7) in both numeric and graphic simulations. ⃗ produce a astrophysics. േ 𝜋𝜋/4 phase difference between ordinary extraordinary at the Jones operating matrix for avertical). half-wave plate(HWP) or eighth-wave plate or and arbitrary phase ofrays retarded. rewritehas Eq. (1) using aMaxwell's wave vector k polarized waves in medium while satisfying equations. Interactive visualization of the the(EWP) propagation mechanism of polarized in(6,7) a medium names. Recently, the visualization ofelectromagnetism polarization propagating in a matter drawn physicist's equations Eq. in both numeric and graphic simulations. Jackson 1975) is Similarly, wevertical(FA can determine the corresponding Jones Mathematica computing environment (Mathematica 2015). hisvertical(FA isThis the is case theofcase fast of axis fast vertical(FA axis vertical). vertical). Similarly, Similarly, we can wedetermine can the corresponding the corresponding Jones ̂ ̂of iφ +φ ) axiswave matrices derived derived for for various various wave plates plates are are summarized summarized in indetermine Table Table 1. 1. isinto the case fast vertical (FA Similarly, First, wewill desire toJones confirm that complex vector As shown above, E field vector may be equations Eq. (6,7) in both +φ y ̂ ∙numeric ̂and ̂ simulations. ̂ = the ̂graphic ̂field y y eiφmatrices E0y0y ei(εi(εyyof The general form ofvertical). amedium matrix representing representing a phase phase retarder will E0y0y eThis yy into y)y. . The 𝒦𝒦 E 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B E e general form of a matrix a retarder E y and extraordinary rays at the operating polarized waves in a while satisfying Maxwell's equations. Mooleskamp & Stokes 2015, Yun & Choi 2013). matrices derived wave plates are summarized in Table 1. scientists exploiting equations (6,7) in both and simulations. equations Eq. (6,7) in both numeric Eq. graphic simulations. inE numeric and graphic ̂ ∙E ̂numeric ̂ ×graphic ̂information ̂technology ̂both ̂ withforitsvarious helicity has accordingly received from protocol ofEq. quantum key 𝒦𝒦 =equations 𝒦𝒦 ∙and E = 0,(6,7) 𝒦𝒦 = B matricesorderived for wave plates are summarized inattention Table 1.for for potential applications in the modern and ⃗ −⍵t) .r canvarious determine the corresponding Jones matrix (3) with Jones innumeric the ate(HWP) eighth-wave plateplate(HWP) (EWP) or eighth-wave arbitrary phaseplate ofattention retarded. Jones ̂ε0,2∙matter ̂ ̂ ⍵ equations Eq. (6,7) in(ε both and graphic simulations. ̂2 )= ̂ = ̂ = ̂ ̂physics (r 𝒦𝒦 0,⃗⃗satisfy 𝒦𝒦 ∙E 0, Maxwell's ×1997, B E ,Jones t)vectors = EE the (Tamm simulations. 𝒦𝒦 simulations. E ⍵ ሺ͵ሻ ei(𝒦𝒦 matrix matrix for a half-wave for awe half-wave plate(HWP) or operation eighth-wave (EWP) plate or arbitrary ora halfarbitrary phase phase ofEq. retarded. ofequations retarded. Jones 1 Ein 1+ Eq. (6,7) both numeric and graphic equations Eq. (6,7) in both numeric and graphic transform theelements elementsby byor thematrix matrix operation follows 4 (EWP) 2.MATHEMATICA SIMUL vector amplitude {A, B േ iC} oscillatin ̂ ̂ ̂ transform the the asasfollows ̂ ̂ ̂ ̂ 𝒦𝒦 ∙in Epolarization = 0, 𝒦𝒦 ∙modes, Ein =both 0, 𝒦𝒦 × E = ⍵ B equations Eq. (6,7) in both Weor have a dynamic polarization modes platform for simulating polarization modes with ⃗S(6) ⃗a𝒦𝒦 ⃗(7) Keywords: cosmology, Jones vector, wave plate, helic wave plate (HWP) or eighth-wave plate (EWP) arbitrary vector equations Eqs. and both numeric and = E × distribution (QKD) to guarantee unconditional security in provided cryptography communication. We have provided ̂ ̂ ̂graphic equations equations Eq. equations equations equations (6,7) Eq. Eq. equations in (6,7) Eq. (6,7) Eq. both equations equations Eq. (6,7) in (6,7) equations in numeric (6,7) both Eq. both equations (6,7) both in Eq. numeric both Eq. numeric both Eq. and (6,7) in (6,7) numeric numeric Eq. graphic both numeric and and (6,7) in0, numeric both graphic and graphic and both simulations. in numeric numeric graphic both graphic numeric and simulations. simulations. and simulations. simulations. and simulations. graphic graphic and graphic simulations. graphic simulat ̂(6,7) ̂ ̂ ̂∙ simulation ̂ ̂E ̂E ̂ ̂ ̂ ̂in ̂in ̂numeric ̂ ̂= ̂simulatio ∙⃗H 𝒦𝒦 ∙and = 0, 𝒦𝒦 × E = ⍵ B 𝒦𝒦 ∙in E = 0, 𝒦𝒦 ∙= E =in 0, 𝒦𝒦 ×graphic E 𝒦𝒦 = ∙and ⍵E B 0, 𝒦𝒦 E = 0,simu 𝒦𝒦 Mooleskamp & Stokes 2015,equations Yun & Choi 2013). ⃗ ⃗ ⃗ ⃗ us wave plates are summarized in Table 1. equations Eq. (6,7) both numeric and graphic simulations. ⃗ S = E × H 2.MATHEMATICA SIMULATION FOR THE CONVER ̂ ̂ ̂ i∆ε i(k .r⃗−⍵t) ̂ ̂ ̂ ̂ iφ i(ε +φ ) iφ i(ε +φ ) equations Eq. equations (6,7) in Eq. both (6,7) numeric in both and numeric graphic and simulations. graphic simula x x x 𝒦𝒦 ∙ E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B equations Eq. (6,7) in both numeric and graphic simulations. iε x x x ⃗ ⃗ ⃗ ⃗ = E e + ε e (ε ) Table Table 1. 1. Summary Summary of of Jones Jones vectors vectors in in the the most most common common and and Jones Jones matrices matrices of of the the wave wave plates plates e e E E Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity iεxxx plates x summarized xfor x various matrices matrices derived derived for various for wave wave plates are summarized ine Table in Table 1. 1. eare E0x0x E0x phase of various retarded. Jones matrices derived wave graphic simulations. S E H 0x 0𝒦𝒦 1= 2× equations Eq. (6,7) in both numeric and graphic simulations. 0 e ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ equations Eq. (6,7) in both numeric and graphic simulations. 0 e ̂ and form a real mutually orthogonal ∙ E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B 𝒦𝒦 ∙ E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B Jones matrices calculations, corresponding the(6,7) physical arrangement of MODES optical elements, in a 𝒦𝒦 ∙u (4)of ( Jones equations equations Eq. Eq.(6,7) (6,7)into in both both numeric and and graphicsimulations. simulations. dynamic polarization for the modes of a transverse electromagnetic wave, Table 1. Summary vectors iniφiφ the common and Jones matrices the waveequations == ((presenting (4) (of )) (( platform ))most )) )polarization ⃗̂̂̂̂ ⃗⃗̂̂̂graphic i(εy+φ +φ Eq. both numeric and graphic simulations. iε = E × y y ̂ ̂𝒦𝒦 ̂for ̂ ̂̂0, ̂⃗𝒦𝒦 ̂ ̂ ̂ ̂ ̂ ̂ ̂̂ ̂⃗0, ̂𝒦𝒦 ̂= ̂ ̂ ̂ ̂and ̂0, ̂⃗Snumeric ̂ ̂ ̂𝒦𝒦 ̂ ̂ ̂∙E ̂= ̂0, ̂𝒦𝒦 ̂ ̂ ̂⍵ ̂ ̂ ̂B ̂, 0, ̂ ̂ ̂𝒦𝒦 ̂H ̂E ̂× ̂(6 ̂⍵ ̂plates ̂in, both ̂E ⃗𝒦𝒦 ⃗B ⃗̂⃗ = y)y provided a dynamic polarization EE0y EE0y0yWe eei(εyyhave 𝒦𝒦 ∙numeric 𝒦𝒦 = 0, ∙ ∙in E 𝒦𝒦 E𝒦𝒦 𝒦𝒦 = = ∙𝒦𝒦 ∙̂ ∙H 0, E⃗E E 0,graphic ∙complex = 𝒦𝒦 = E = 𝒦𝒦 𝒦𝒦 = 0, ∙0, ∙E ∙polarization 𝒦𝒦 E E 0, 𝒦𝒦 = 𝒦𝒦 = = 𝒦𝒦 ∙ ∙ × 0, E ∙ 𝒦𝒦 E ∙ 0, 0, E 𝒦𝒦 E = ∙ 𝒦𝒦 𝒦𝒦 = E = 0, = 0, × 0, ∙ = × 0, ⍵ 𝒦𝒦 E 𝒦𝒦 E E 𝒦𝒦 B = = × 𝒦𝒦 × ∙ ∙ 0, E E × E ⍵ ∙ = = B E E = ∙ = ⍵ 0, = × ⍵ E 𝒦𝒦 0, B 𝒦𝒦 E ⍵ = B 𝒦𝒦 = × B × ⍵ E × E B = ⍵ = ⍵E B 0y ee yy 1. 00 eeiεyyy in modes platform simulating modes with are where 𝒦𝒦 , E and B are both vectors and E ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ plates are summarized Table S = E × H equations Eq. (6,7) and simulations. ̂ ̂ ̂ ̂ ̂ ̂ ̂ S S = E × H = E × H ̂𝒦𝒦 ̂̂0,and ̂̂⃗B ̂B ̂̂⃗∙ = ̂E ̂= ̂⍵ ̂B= ̂are ̂= Table 1. Summary of Jones in the most common and Jones matrices of where the wave plates 𝒦𝒦 ∙⃗× Eand 0, E⃗r−⍵t) = =∙ × ̂ ,𝒦𝒦 ̂∙ ,E ̂𝒦𝒦 ⃗0, ⃗⃗⍵ MODES 𝒦𝒦 ∙vectors E 0, 𝒦𝒦 𝒦𝒦 ∙∙𝒦𝒦 E =× 0,E 𝒦𝒦 E = E 0,H 𝒦𝒦 ×E (Pedrotti (Pedrotti 𝒦𝒦 E and B are both E ,𝒦𝒦 and re and andvectors phase phase retarders retarders & & Predrotti Predrotti 1987). 1987). ̂ ̂̂̂ ̂̂Âcomplex ̂̂ ̂̂= ̂= ⃗ ⃗ ⃗ equations Eq. (6,7) in both numeric graphic simulations. i(k ∙̂ = 0, ∙ E = 0, 𝒦𝒦 E = ⍵ B (6) S E × H ̂ ̂ ̂ ̂ ̂ ̂ converting the state of polarization through the arrangement of optical elements, using Jones vectors calculations ̂ ̂ ̂ ̂ wave vector and N =n + i κ com ̂ ̂ ⃗ ⃗ ⃗ ⃗ Mathematica computing environment (Mathematica 2015). ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ = ( ) e ⃗ 𝒦𝒦 ∙ E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B (6) 𝒦𝒦 ∙ Ein 0,E𝒦𝒦INTRODUCTION = 0, 𝒦𝒦 × E = ⍵ B(6) andS = H where 𝒦𝒦=,both ,𝒦𝒦 and B0,𝒦𝒦 are both vectors and E (6) 𝒦𝒦∙numeric E∙∙ E⃗SE == 0, ∙̂= E∙ E = =0, 0,𝒦𝒦 ×̂ × EBE= =(6) ⍵⍵BElectromagnetic B ̂ (6) light. 1. and Predrotti 1987). ⃗graphic ⃗0, ⃗𝒦𝒦complex ⃗⃗S𝒦𝒦 ⃗H Wexphase desire now to(Pedrotti create a& new simulation E × = E × Eq. and simulations. ̂ ̂2.1 ̂ ̂, waves where and εεyretarders represent theadvance advance phase -and andEEyy-presenting component the incident light. As B + iC where and represent ininphase ofofEExxplates -xmatrices ofofequations the incident As y-component 𝒦𝒦 ∙⃗⃗⃗⃗⃗E = 𝒦𝒦 ∙E =⃗ 0, 𝒦𝒦 × E =⍵ B y and ̂⃗⃗(6,7) ̂ ,the ̂ both ⃗⃗in ⃗⃗ areinr Jones calculations, corresponding toE physical arrangement of⃗⃗⃗H optical elements, a⃗B where 𝒦𝒦 ,Eq. and B are both complex vectors and E , and H es vectors in the mostεεxcommon Jonesthe matrices of the wave ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ equations (6,7) in numeric and graphic simulations. S S S S S S S S S S S = E × = = H E E = × = × H = E H E × × E H = × H H E = × = H = E E × × E = H × H E H × H ̂ ̂ ̂ ̂ ̂ ̂ ̂ is a magnetic intensity vector in the matter. We examine where H ̂ ̂B ⃗ , ⃗B ⃗ ̂ , and ̂ ⃗Ŝ = ⃗solids ⃗ ×both ̂in ̂E ̂complex ⃗̂⃗H⃗ , and ⃗B complex ⃗ and 𝒦𝒦, Electromagnetic ∙E 𝒦𝒦 ∙⃗Sare E= = 0, × E =⃗H⃗⍵ Band (6)rH (Pedrotti where 𝒦𝒦 , = B E and phase retarders &polarizing Predrotti 1987). 1.the INTRODUCTION = E E where E and are∙0, both where complex 𝒦𝒦 ,=⃗S⃗H ,in vectors and are and H are vect ⃗∙vector ⃗E Polarized Polarized light light Jones Jones vectors vectors helicity waves in Methematica. The platform graphically simulates mechanism of production propagation of the a means of producing modes from atheJones E equations Eq.helicity (6,7) both numeric and graphic simulations. ⃗H ⃗𝒦𝒦 ̂and ̂ both ̂× ̂ ̂×𝒦𝒦 ̂vectors a, ⃗E intensity in the matter. We th( where ⃗⃗2.1 ⃗H ⃗⃗⃗⃗SPolarization 𝒦𝒦 E =⃗H 0, 𝒦𝒦 E 0, 𝒦𝒦 ×B E = ⍵ofEB able Table 1. Summary 1. Summary of Jones of Jones vectors vectors in the in most common most common and Jones and∆𝜀𝜀=𝜋𝜋/2. Jones matrices matrices the of wave plates avectors homogeneous wave, ̂⃗S⃗Sis ̂plates ⃗plane ⃗B ⃗⃗(7) ⃗SB (7) ⃗E ⃗× ⃗H ⃗both the (7) ̂ ̂and ̂ Polarized light Jones vectors helicity ,matter. H are where 𝒦𝒦 , ⃗S= E and are and = =magnetic E × (7) (7) ⃗× is coherent characteristic the electromagnetic where 𝒦𝒦 , harmonic E ,examine and B are propagating process of electro ⃗magnetic anexample, example, consider quarter-wave plate (QWP) which makes ∆𝜀𝜀=𝜋𝜋/2. Weof write thewave Jones may consider aaquarter-wave plate (QWP) which makes We may write the Jones =,𝒦𝒦 E × = E ×𝒦𝒦 H,∙⃗H (7) ̂ ̂⃗are ̂⃗as ̂acomplex ̂The ̂ is a⃗̂B intensity vector in theE We exam where H ̂ ̂ ̂ ⃗ ⃗ ⃗ ⃗ ̂ ̂ ̂ ⃗ ⃗ ⃗ ⃗ Mathematica computing environment (Mathematica 2015). E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × E = ⍵ B (6 where , E and B both complex vectors and E , B and H where 𝒦𝒦 , E , and are both complex vectors and E , B and H are ers (Pedrotti & an Predrotti 1987). ⃗ ⃗ ⃗ calculation corresponding towhile the satisfying physical arrangement Ŝ̂= Evector Ĥ0, Eq.̂⃗H in both and graphic simulations. ⃗ (6,7) ̂, and ̂vectors ̂the ̂,̂× ̂complex ̂complex aE intensity in the matter. We examine th where 𝒦𝒦 ∙E E = 𝒦𝒦 ∙̂both = 𝒦𝒦 × E ⍵⃗Bvectors B ̂electromagnetic ̂𝒦𝒦 ̂is ̂ ̂numeric ̂ ̂ ̂ ̂𝒦𝒦 ̂ ̂ ̂where ̂ ̂𝒦𝒦 ̂THE ̂𝒦𝒦 ̂process ̂ ̂̂,both ̂ ̂ ̂ ̂vector ⃗= ⃗complex ⃗ ,in ⃗Bvectors ⃗B ⃗E ⃗H ⃗vectors ⃗ ⃗B ⃗Bvectors ⃗B ⃗E ⃗and ⃗differen ⃗real ⃗BH ⃗an ⃗E ⃗H ⃗aE Cos Cos polarized waves in aJones medium Maxwell'scharacteristic equations. equations polarization modes with Jones vectors calculations the ̂̂ ̂normalized ̂ ̂and ⃗in ⃗Bexa where where where 𝒦𝒦 , where where E⃗H ,where and ,, E where 𝒦𝒦 ,̂The ,⃗H B ,⃗̂magnetic 𝒦𝒦 ,FOR and E E are ,where ,E where and B and ,E both where are are B 𝒦𝒦 ,⃗B ,𝒦𝒦 ,both and B are both are E E ,0, are ,E both and B and ,complex complex E and are B vectors complex and complex B are are complex are vectors B both both complex are both vectors vectors E complex and and complex and E and E and E E E and and ,waves and , vectors ,and ,and are , is and H H are are ⃗B ⃗H Polarization isααa))2.MATHEMATICA coherent ofSIMULATION the (EM) wave in aE medium. In aboth plane where 𝒦𝒦 ,complex where and 𝒦𝒦 B𝒦𝒦 E are ,both and B complex are both vectors and vectors , ,and propagating of electromagnetic solids Polarized light helicity ⃗H ⃗E = × (⃗aH CONVERTING POLARIZATION llinearly inearly polarized polarized ((toproduce 0⃗where 0𝒦𝒦 ̂̂⃗Sand ̂B ⍵ ̂E ̂,𝒦𝒦 ̂ ̂both magnetic intensity in the matter. ̂ ⃗and ⃗BWe ⃗v aand magnetic intensity vector is asimulations. in magnetic the matter. intensity examine vector in th where where 𝒦𝒦 ∙,E = 0, 𝒦𝒦 ∙Epresented = 0, × E = ⍵ B (6) Cos α1987). matrix, transforming theJones Jones vector ∆𝜀𝜀=𝜋𝜋/2 produce right-handed circular polarizing light as 𝒦𝒦 ,wave, ,anumeric and Bcomplex are both complex E ,real and polarization with the normalized Jones calculations in the M (Pedrotti (Pedrotti matrix, MMtransforming the vector ∆𝜀𝜀=𝜋𝜋/2 right-handed circular polarizing light as ⃗ ̂B ̂where and phase and phase retarders retarders & in Predrotti &vectors 1987). ̂ ̂,, ,modes ̂is ⃗ ⃗ ⃗ ⃗and ⃗⃗,vectors ⃗B ⃗BH ⃗⃗and ⃗⃗We ̂is ⃗ vectors ⃗where ⃗ ̂equations ̂ ̂ ⃗ ⃗ ⃗ ⃗ Sin αα ൌ 𝑛𝑛, which result in propagation where where 𝒦𝒦 𝒦𝒦 , , E E and B B are are both both complex complex vectors vectors and and E E , and and H H are are real vectors vector of optical elements the Mathematica computing where , and are both vectors and , and As shown above, E field vector may be as the Jones vector (Jones 1941) with a c E and B of the electromagnetic w both the electric field vector and magnetic field vector S = E × H is a magnetic H Eq. (6,7) in both and graphic where 𝒦𝒦 , E , and are both complex vectors and E , B are real vectors w linearly polarized (Predrotti ) toSin 0 ⃗ ⃗ is ⃗aboth magnetic intensity vector in the matter. We examine t ̂𝒦𝒦 ̂, , E ̂E ̂H ⃗ ⃗ ⃗ ⃗ ̂,where ̂⃗⃗ are ⃗ ⃗ ⃗ ⃗ polarization modes with the normalized Jones vectors calculations i c , B and H are real vectors w where and B complex vectors and E where 𝒦𝒦 ,where andwhere B are both complex vectors and E , B and H are real vectors Jones vectors helicity ⃗ ⃗ ⃗ ⃗ Sin α ⃗a𝒦𝒦 = Eand × H ̂ ̂Jones ̂ ̂in= ̂matter. is magnetic intensity vector in the matter. exam H ismodes magnetic the WeE examine t HThe ̂with ̂ ̂ ⃗electrons ⃗B the 𝒦𝒦 ∙a⃗⃗E = 𝒦𝒦 ∙are E = 0,vector 𝒦𝒦 ×magnetic E ⍵ B (6) ,We and where ,Ŝboth ,̂of B both vectors and polarization with the normalized vectors calculations in the M ⃗intensity ⃗aintensity ⃗ in ⃗and ⃗ and ⃗E ⃗H ⃗aelectromagnetic ⃗H ⃗graphic below. input code is shown below: 111 a plane environment. −i−i In wave, the the electric field are vectors =µ where is aintensity magnetic intensity and B of the electromagnetic waves interact with the in a electric fieldvector vector E field B the radiation = E × is magnetic isvector acomplex intensity intensity in vector matter. in We m where equations equations Eq.magnetic (6,7) in Eq. both numeric numeric and simulations. Coswave, α both ⃗,real ⃗H ⃗aand ⃗H ⃗(6,7) ⃗H ⃗⃗H ⃗Mathematica ⃗H ⃗is ⃗H ⃗0, ⃗H ⃗graphic ⃗H ⃗is ⃗H 11MODES is magnetic isvector is aE aTHE is magnetic magnetic is awhere amodes intensity magnetic a⃗Smagnetic is magnetic intensity intensity is magnetic is aa⃗⃗H vector intensity magnetic asimulations. is magnetic intensity magnetic avector in magnetic vector the vector intensity vector in intensity inbelow: matter. intensity the the vector in invector in matter. the matter. the vector vector We the vector in matter. matter. examine vector matter. the We We in in matter. the examine We the We examine in the We the matter matte exam exam the ma We ex pr where where where where where where where where where H where where ̂⃗H ̂ ̂ ⃗intensity ⃗B ⃗⃗in polarization with the normalized Jones vectors calculations polarization modes with the polarization normalized Jones modes vectors with the calculations normalized in Jones the M , and H are real ve where 𝒦𝒦 E , B are both complex vectors and E 02.MATHEMATICA 0 () Jones e 44π4π circularly linearly polarized ) 0 ⃗ ⃗ ⃗ ⃗ 11101/√2 −iπ/4 0 e ⃗ ⃗ ⃗ ⃗ ⃗ Fowles 1975). In particular, alth always oscillates parallel to a fixed direction in space. Light of s right(left)-handed right(left)-handed circularly 1/√2 ( ( ) ) ±1 ±1 below. The Mathematica input code is shown polarization modes with th ⃗ ⃗ −iπ/4 S = E × H (7 SIMULATION FOR CONVERTING POLARIZATION is is a a magnetic magnetic intensity intensity vector vector in in the the matter. matter. We We examine examine the the process process where where H H −iπ/4 Polarized Polarized light light Jones vectors vectors helicity helicity vector amplitude {A, B േ iC} oscillating in an inhomogeneous plane. The wave vector k ̂ ̂ ̂ ̂ ̂ ̂ ̂ is a magnetic intensity vector in the matter. We exa where H M= = e ( ) QWP, FA vertical (5) ( ̂both ̂ ,intensity ̂∙Mathematica ⃗calculations ⃗⃗ inare M= ( = modes, eα (cosmology, ) QWP, FA vertical (5)where ⃗⃗⃗⃗helicity ) Sin Therefore, there will a process cumulative p 𝒦𝒦 E =examine 0, 𝒦𝒦normalized ∙ and E = 0,matter. 𝒦𝒦 ×simulations. E = ⍵ B below: (6re Cos always α ±i ±i , ⃗Bbe and H 𝒦𝒦 ,the E and B are both complex vectors and E 11i1π is apolarization magnetic vector in the examine the where modes with the Jones vectors theof M Keywords: polarization Jones vector, plate,⃗H below. The input code isWe shown circularly 1/√2 ±1 oscillates parallel to the fixed direction inwave space. vector in matter. We the process of generating equations Eq. (6,7) Eq. (6,7) numeric in both and numeric graphic simulations. graphic ⃗equations i)i ( right(left)-handed ) 0(00±i isis apolarization intensity vector in the matter. We the process of where H a𝒦𝒦 magnetic intensity vector in the matter. We examine process where H ̂magnetic ̂ ̂of ⃗examine ⃗BJones ⃗the ⃗ Jones polarization modes with the normalized Jones vectors calculations in modes with the normalized Jones vectors calculations inreal the 00 eei44π4 where , in E , The and B are both complex vectors and E ,̂electromagnetic and H are ⃗∙E ⃗H ⃗0, Sin α Fowles 1975). In particular, although the waves av((M always oscillates parallel to a fixed direction in space. Light such character is said to be linearly below. Mathematica input code is0, shown below: polarization polarization modes with modes the normalized with the normalized vectors calculatio vector ⃗H ⃗ ⃗S0, = E × ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ is a magnetic intensity vector in the matter. We exa where ̂ ̂ ⃗ ⃗ ⃗ ⃗ 𝒦𝒦 ∙ E = 𝒦𝒦 𝒦𝒦 ∙ E = = 0, 𝒦𝒦 𝒦𝒦 ∙ E × = E = 𝒦𝒦 ⍵ B × E = ⍵ B (6) where 𝒦𝒦 , E , and B are both complex vectors and E , B and H are re polarization polarization polarization polarization polarization modes polarization polarization modes modes with polarization modes polarization modes the polarization with modes with polarization normalized with modes the with the with normalized modes the normalized modes the with modes the normalized normalized Jones modes normalized with the with with Jones normalized vectors Jones the the with the Jones normalized Jones normalized vectors the vectors Jones normalized calculations normalized vectors vectors Jones vectors calculations calculations Jones Jones vectors calculations Jones calculations in calculations Jones vectors vectors the vectors calculat in Mathe in vecto the the cal in ca in fields may pull or the electro 2.1 waves in solids below. The Mathematica The Mathematica input code is shown below: Lightright(left)-handed of such character said linearly polarized. polarization modes with the normalized Jones below. The Mathematica input below. code The is shown Mathematica below: input isiuniaxial shown AA and ⃗H ⃗ a,Mathematica polarization polarization modes modes with with the thenormalized normalized Jones Jones calculations in in the the Mathematica Mathematic ⃗⃗⃗⃗ ̂emerge 22 Electromagnetic code #1 Cos αisCos 1α) to be is abelow. magnetic intensity vector in the We examine the where vector they form real mutually orthogonal unit vectors (vectors εvectors ,n ̂calculations ).E Here K = kcalculations +in 𝛼𝛼 is pro ave ci ̂with ̂Mathematica ⃗ as ⃗B ⃗code ⃗1push MODES polarization modes normalized Jones vectors right(left)-handed elliptically 1/√A 1/√A222+ + BB + CC22determine ((polarization ))the ±1 1 , εand 2matter. where 𝒦𝒦 E ,⃗H and B are both complex vectors , vectors and H are real ⃗S𝒦𝒦 ⃗E ⃗H ⃗with This1isisthe the caseofoffast fastaxis axiselliptically vertical(FA vertical). we+ determine Jones linearly linearly polarized polarized ( 1/√2()( vertical). 0corresponding 0̂Mathematica modes normalized Jones vectors calculations in the Mathematica sim ⃗±1 Acan This case vertical(FA we can corresponding Jones below. The input code is field shown below: right(left)-handed circularly )2 + BSimilarly, ±1 = E × 2Similarly, ̂The ̂ ̂#1 ̂𝒦𝒦 ̂∙the ̂ ̂ ̂ ̂pull ̂vectors ̂The ̂ ̂shown ̂ isthe athe magnetic intensity vector in the matter. We examine th where code B Bpolarization + + ii CCthe 𝒦𝒦 ∙the E = 0, 𝒦𝒦 ∙ ∙E == 0, 0, 𝒦𝒦 E ×vectors = E 0, = 𝒦𝒦 ⍵ B ×isas E = ⍵ B (6) which polarization modes with normalized Jones calculations inshown the Mathematica siis( right(left)-handed elliptically 1/√A + through C ( )quarter±1 modes with normalized Jones calculations in the Mathematica Sin αlight Sin fields may or push the electrons in the orbital of a solid, ±iα±1 below. The below. Mathematica Mathematica input code input is code below: is shown below below. The Mathematica input code below: below. Mathematica input code is shown below: ly 1/√2 (If the ) linearly polarized passes a calculations in the Mathematica simulation below. The ⃗ ⃗ Mathematica code #1 1isE a magnetic intensity vector in the matter. We examine the pr where H B + i C ̂ ̂ ̂ ⃗ ⃗ ⃗ ⃗ polarization modes with the normalized Jones vectors calculations ±i , B and H are real v where 𝒦𝒦 , , and B are both complex vectors and E ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ The propagating process of electromagnetic waves in solids isMathematica different from that in the vacuum, since moment Pis and magnetization ⃗H ⃗Jones below. below. The The Mathematica input input code code is is shown shown below: below: below. below. below. The below. below. below. Mathematica The Mathematica The The The below. below. Mathematica Mathematica below. The input below. Mathematica The The code input input The Mathematica input The is input code Mathematica code shown input Mathematica code is code is input code shown shown below: isis input input code shown is shown input below: shown below: code input is code below: code shown below: below: is code shown is shown below: shown is below: below: belo SThe Scomplex =Mathematica E × =Mathematica E × H (7) is=n abelow. magnetic intensity vector in the matter. We examine where Mathematica code #1 wave vector and N iMathematica κ H refraction index. For Eq. (3) to beshown abelow: solution of matrix forplate, ahalf-wave half-wave plate(HWP)or oreighth-wave eighth-wave plate (EWP) arbitrary phase of retarded. distance d, the phase difference is matrix for a1. plate(HWP) plate (EWP) ororarbitrary phase ofin retarded. Jones polarization modes with the normalized Jones vectors calculations in the Mathem Mathematica code #1 th INTRODUCTION 2.1 Electromagnetic waves solids below. The Mathematica input code isvectors shown below: E0 =+ 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= wave elliptically light The Mathematica input code isshown shown below: Polarizer Polarizer & & Wave Wave Plate Plate polarized Jones Jones matrices matrices ⃗H ⃗ 𝒦𝒦 below. The Mathematica input code is below: Mathematica code #1 magnetic intensity vector in the examine where Mathematica code #1 Mathematica code #1 ̂is, E 12Jones21matrices ̂aE0 ̂ ⃗ ,We ⃗B ⃗⃗we are Aemerges. polarization modes with the normalized Jones calculations inthereal thepro Mv where , and B are both complex vectors and E and H 2 ⃗E ⃗M ⃗⃗matter. below. The Mathematica input code is shown below: ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ below. The Mathematica input code is shown below: moment P and magnetization in the solid. If assume tha A = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= Polarizer Wave Plate S S = E × H = × H (7) right(left)-handed right(left)-handed 1/√2 ( 1/√2 ) ( ) ±1 ±1 2 circularly 2 &circularly 2 right(left)-handed elliptically 1/√A + B + C ( ) ±1 code #1the lly 1/√A + B derived +derived C be (for ) wave polarization modes with the normalized Jones vectors calculations in Helmholtz the the Mathe Mathematica Mathematica code #1 code #1 ⃗E ⃗ ,ofwhich below. Mathematica input code is⃗⍵matter. shown below: E0 = with 1/√2{1, 0}+ ±i are ±i same can said theplates magnetic fieldBin vector +Table iand C 1.1. B the electromagnetic interact electrons in abe solid (Pedrotti &⃗Then Predrotti 1987, matrices for various wave plates are±1 summarized in Table ⃗H ⃗Mathematica material Jwe =0, the w matrices various summarized code #1 Mathematica code #1 magnetic intensity vector in examine pr B+ i C of ̂where ̂is,waves ̂(EM) ̂In[11]:= ̂aMathematica ̂The ⃗𝒦𝒦ൌ ⃗⃗,We ⃗Bare ⃗the ⃗ vectors vacuum, . relations: as a,where homogeneous plane harmonic it1/√2{0,1}; should ǡradiation the polarization with the normalized Jones calculations in the M where 𝒦𝒦 E ,In[11]:= and 𝒦𝒦 B E are ,waves and both B are complex both complex vectors and vectors E ,#1 Bcvectors and Eand H and real H are real w ⃗k 1/√ In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; Mathematica Mathematica code code #1 #1 Polarization is a coherent characteristic of the electromagnetic wave in amodes medium. In awave, plane The propagating process of electromagnetic in solids is different that in the since E0get =along 1/√2{1, 0}+ In[11]:= below. The Mathematica input code isfrom shown below: E0 =Mathematica 1/√2{1, 0}+ 1/√2{0,1}; isand propagating the dir Mathematica Mathematica Mathematica Mathematica Mathematica code Mathematica #1 code code Mathematica Mathematica code #1 code Mathematica #1 code Mathematica #1 #1 code #1 code #1 code code #1 #1 code #1 Mathematica code #1 polarization modes with the normalized Jones vectors calculations in the Mathem below. The Mathematica input code is shown below: In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; E0 = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= maintains an orientation perpendicular to the electric E0 = 1/√2{1, 0}+ 1/√2{0,1}; E0 = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[11]:= ⃗ ⃗ material and J =0, the Helmholtz wave equation in a solid (Fowles 197 equations Eq. (6,7) in both numeric and graphic simulations. Mathematica code #1 is a magnetic intensity vector in the matter. We examine the pr where H ̂ ̂ ̂ ̂ ̂ ̂ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Mathematica code #1 A A , B and H , B are and real H vectors are real w where 𝒦𝒦 , where E , and 𝒦𝒦 , B E are , and both B are complex both complex vectors and vectors E and E In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; 2 2 2 2 2 2 E0 = 1/√2{1, E0 0}+ = 1/√2{1, 1/√2{0,1}; 0}+ 1/√2{0,1}; In[11]:= In[11]:= Fowles 1975). In particular, although the electromagnetic waves are an harmonic plane wave, the below. The Mathematica input code is shown below: Polarizer & Wave Plate Jones Jones matrices Mathematica #1 E0 = 1/√2{1, 1/√2{0,1}; Jones vectors 2calculations In[11]:= ⍵ right(left)-handed right(left)-handed elliptically elliptically 1/√A + matrices B ++CB ⃗+( C⃗ ) ±1 below. ±1code Mathematica code #1 20}+normalized In[12]:= LP =in{Cos[α], Sin[𝛽𝛽 ⃗ ×Mathem ⃗∇the ⃗E modes with the the ×om ∇ ⃗(CB)is ⃗In[12]:= In[15]:= =intensity {{Cos β,both Cosβ Sinβ}, {and Cosβ Sinβ, Sin βexamine }};1987, The Mathematica input ismatter. shown below: LP =1/√2{1, {Cos[α], Sin[𝛽𝛽]}; E0 =0}+ 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= E0 =magnetic 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= vector responsible for the linearly pola ൌ 𝑛𝑛, which result in propagation speeds that are different along the direction in(the field vector the 1/√A direction of ×and In[11]:= E0 ;the E B and of+everywhere the waves interact with the electrons in a code solid (Pedrotti &is Predrotti wave,such both that the electric field vector E+ magnetic field vector B of electromagnetic radiation E0 E0 =LW =1/√2{1, 0}+ 1/√2{0,1}; 1/√2{0,1}; In[11]:= In[11]:= ⃗H ⃗polarization ⃗H ⃗the iB i C electromagnetic equations Eq. (6,7) numeric graphic simulations. Mathematica code #1 is a magnetic is a intensity vector in vector in the We matter. We the examine process p where where 2=in 2 c 2 2 In[12]:= LP {Cos[α], Sin[𝛽𝛽]}; ⃗ ⃗ In[12]:= LP ==={Cos[α], Sin[𝛽𝛽]}; In[12]:= LP = {Cos[α], In[15]:= LW {{Cos β,input Cosβ Sinβ}, {shown Cosβ Sinβ, Sin β }}; The Mathematica code is below: ∂ P 1Sin[𝛽𝛽]}; ∂ E 2 ̂ E0 = 1/√2{1, E0 E0 1/√2{1, E0 1/√2{1, E0 0}+ = E0 = 1/√2{1, 1/√2{0,1}; 1/√2{1, = 0}+ 1/√2{1, E0 0}+ = 1/√2{0,1}; 1/√2{0,1}; 1/√2{1, 0}+ E0 0}+ E0 0}+ = E0 1/√2{0,1}; = 1/√2{0,1}; 1/√2{1, 1/√2{1, = E0 1/√2{0,1}; 0}+ 1/√2{1, = 1/√2{1, 1/√2{0,1}; 0}+ 0}+ 0}+ 1/√2{0,1}; 1/√2{0,1}; 0}+ 1/√2{0,1}; 1/√2{0,1}; In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= In[11]:= Table1.1.Summary SummaryofofJones Jonesvectors vectorsininthe themost mostcommon commonand andJones Jonesmatrices matricesbelow. of the wave plates ̂ ̂ ̂ ̂ ̂ ̂ Mathematica code #1 2 Table of the wave plates In[12]:= LP In[12]:= = {Cos[α], LP Sin[𝛽𝛽]}; = {Cos[α], Sin[𝛽𝛽]}; ⃗Jones ⃗E𝒦𝒦)൅Sinβ, Eβ, = 0, 𝒦𝒦 ∙× =⃗∇{ × 0, ×for E= ⍵βB modes the normalized vectors calculations in (responsible −u the 2Mathe β,the Cosβ op ∇ fields maywhere pull or push electrons in orbital of𝒦𝒦 a ∙solid, which isEmatter. inducing dipole In[15]:= LW {{Cos Cosβ Sinβ}, Cosβ Sin }}; E0 == 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[12]:= LP ={Cos[α], {Cos[α], Sin[𝛽𝛽]}; 0the In[12]:= LP =the { with }; ⃗H ⃗polarization ⃗H ⃗ LP 2 examine 2 =We In[15]:= LW =2 {{Cos E0 ==the 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= Mathematica code #1 is a magnetic is a magnetic intensity intensity vector in vector the in the We matter. examine process where ∂t ∂t c 2 2 In[12]:= In[12]:= LP = = {Cos[α], Sin[𝛽𝛽]}; Sin[𝛽𝛽]}; E0 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[12]:= LP =to{Cos[α], Sin[𝛽𝛽]}; In[12]:= LP = {{Cos {Cos[α], Sin[𝛽𝛽]}; the complex vector responsible for the th e In[16]:= QWV =a#1 {{1, 0}, {0,̂ I}}; =Sinβ, {{1, -I}}}; Fowles 1975). In In[11]:= particular, although electromagnetic waves areisQWH an harmonic plane the parallel tomatrices a fixed direction in space. Light of such character is said LW = β,be Cosβ Sinβ}, {difference Cosβ Sin β̂wave, }}; below. Thethe Mathematica input code below: E0 =In[15]:= 1/√2{1, 0}+ 1/√2{0,1}; ⃗P ⃗ Mathematica Polarizer Polarizer & Wave & Wave Platealways Plate oscillates Jones Jones matrices ̂linearly ̂shown ̂0},{0, ̂ Sin[𝛽𝛽]}; ̂ Mathematica code where = 𝜒𝜒𝜀𝜀 If we suppose 2the Therefore, there will phase between the 2Sinβ, 2E 0βE.two 𝒦𝒦 ∙LP E2Sin[𝛽𝛽]}; = 0, 𝒦𝒦 ∙LW E = 0, 𝒦𝒦 = ⍵ B polarization polarization modes with modes the normalized with the Jones vectors Jones calculations vectors calculations in Math 2 2Sin 2{components 2 (Pedrotti&&Predrotti In[15]:= LW =β, {{Cos β, {Sin[𝛽𝛽]}; Cosβ Sin andphase phaseretarders retarders(Pedrotti Predrotti1987). 1987). E0 =2{Cos[α], 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[16]:= QWV ={{Cos {{1, 0}, {0, I}}; QWH = {{1, 0},{0, In[12]:= In[12]:= In[12]:= LP In[12]:= = In[12]:= {Cos[α], In[12]:= LP LP = In[12]:= LP {Cos[α], LP {Cos[α], Sin[𝛽𝛽]}; LP ==cumulative In[12]:= In[12]:= {Cos[α], In[12]:= LP {Cos[α], Sin[𝛽𝛽]}; In[12]:= =normalized {Cos[α], LP Sin[𝛽𝛽]}; LP ==Sin[𝛽𝛽]}; {Cos[α], {Cos[α], = LP =Sinβ}, {Cos[α], Sin[𝛽𝛽]}; Sin[𝛽𝛽]}; In[15]:= LW ===be Cosβ Sinβ}, In[15]:= Cosβ = Sinβ, {{Cos β, β-I}}}; }}; Sinβ}, Cosβ Sinβ, Sso In[15]:= LW {{ , Sin[𝛽𝛽]}; }, {Cos[α], {{Sin[𝛽𝛽]}; ,∆𝜀𝜀 }};Cosβ and In[15]:= LW In[15]:= =Cosβ {{Cos LW β, =Cosβ {{Cos Sinβ}, β,× Cosβ Sinβ}, Sinβ, { }}; Sin Cosβ βin Sinβ, }};the Sin β }} Mathematica code #1 2{Cos[α], 2{ Cosβ ⃗In[12]:= ⃗ ⃗⃗ Jones In[16]:= QWV = {{1, 0}, {0, In[12]:= LP = Sin[𝛽𝛽]}; 2solid. 21/√2{0,1}; 2 2 Table 1. Summary of Jones vectors in the most common In[16]:= QWV = {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; ⃗ LP = {Cos[α], Sin[𝛽𝛽]}; ⃗ ⃗ ⃗ below. The Mathematica input code is shown below: moment and P and magnetization M in the If we assume that the medium is not a magnetic E0 = 1/√2{1, 0}+ In[11]:= In[15]:= LW = {{Cos β, Cosβ Sinβ}, { Cosβ Sinβ, Sin β }}; ⃗Jones SSinβ}, = E H In[15]:= In[15]:= LW LW = ={{Cos {{Cos β,β,Cosβ Cosβ Sinβ}, {× {Cosβ Sinβ, Sinβ, Sin Sin βvectors β}}; }}; Mathematica code #1 =1/√2{0,1}; 𝜒𝜒𝜀𝜀 .Sinβ}, we suppose the EM wave beMathematica awave form soluti 20E 2will 2 ⃗P 2A=C In[12]:= LP =modes {Cos[α], Sin[𝛽𝛽]}; polarization polarization modes the normalized with the normalized vectors Jones calculations in the in the Math s if B=0 and then the isofahas circul fields may pull or push thefield electrons inasthe orbital of a= solid, which isIfCosβ responsible for inducing dipole In[15]:= {{Cos β, Cosβ Sinβ}, {Sinβ, Cosβ Sinβ, Sin βAfter }}; In[15]:= LW ==LW {{Cos β, Cosβ {(Quartz, Sin βcalculations }}; In[12]:= LP =In[11]:= {Cos[α], Sin[𝛽𝛽]}; In[17]: =with HW {{1, 0}, {0, -1}}; E0 =emerge 1/√2{1, 0}+ In[16]:= QWV I}}; QWH ={{1, {{1, 0},{0, -I}}}; In[16]:= QWV =where {{1, 0}, I}}; QWH =Cosβ 0},{0, -I}}}; matrices of light the wave plates and phase retarders (Pedrotti & Pedrotti 1 1987). vector they in uniaxial Calcite, the wave 2 QWV 2{Cos[α], 2{0, 22QWV 2⃗crystals 2 2 2 2 2below: 2 rewrite 22etc.). 2{{1, 2 -I}}}; 2= {{1, 220},{0, 2 vecto 2-I}} Eq. using aSin wave Polarized Jones vectors helicity Polarized light Jones vectors helicity In[16]:= In[16]:= =LW {{1, QWV 0}, {0, = {{1, 0}, QWH {0, =Cosβ QWH 0},{0, -I}}}; In[12]:= LP =code Sin[𝛽𝛽]}; In[16]:= =β, {{1, 0}, {0, I}}; QWH = {{1, 0},{0, ⃗Sinput ⃗H ⃗Sinβ}, below. The below. Mathematica Mathematica input is shown code below: is shown In[17]: =The HW ==In[15]:= {{1, 0}, -1}}; E0 = 0}+ 1/√2{0,1}; In[11]:= In[16]:= QWV {{1, 0}, {0, I}}; In[16]:= QWH QWV =β,{{{1, 0},{0, {{1, 0}, -I}}}; {0, QWH =ββ{{1, 0}, In[15]:= In[15]:= In[15]:= LW In[15]:= In[15]:= =1/√2{1, In[15]:= {{Cos LW LW = LW LW {{Cos β, {{Cos LW = Cosβ In[15]:= = In[15]:= {{Cos {{Cos =In[15]:= LW β, {{Cos Sinβ}, Cosβ In[15]:= Cosβ = LW β, LW β, {{Cos Cosβ Sinβ}, LW β, Cosβ {Sinβ}, =× =Cosβ Cosβ {{Cos {{Cos = β, Sinβ}, {{Cos {Cosβ = Sinβ, {Sinβ}, Cosβ Cosβ {{Cos β, Cosβ {Sinβ}, β, Sin Cosβ Cosβ Sinβ, Cosβ Sinβ, {=I}}; Cosβ Cosβ β, βSinβ}, Sinβ}, Cosβ Sinβ, Sin Sinβ, }}; {Sin Sinβ}, Cosβ Sinβ, βSinβ}, Sin β{I}}; Sin {}}; Sinβ, }}; Cosβ Sin {2I}}; β(1) βCosβ }}; {β}}; Sinβ, Sin Sinβ, Cosβ }}; Sinβ, β=Sin Sinβ, }}; Sin Sin }}; β}}; }}; βtra = E Mathematica code #1 2 2 In[17]: = HW {{1, 0}, {0, -1 2 2 Polarized light Jones vectors helicity In[15]:= LW = {{Cos β,QWH Cosβ Sinβ}, { Cosβ Sinβ, Sin }}; In[16]:= QWV QWV == ={{1, {{1, 0}, 0}, {0, I}}; I}}; QWH = ={{1, 0},{0, -I}}}; -I}}}; In[12]:= LP =wave {Cos[α], Sin[𝛽𝛽]}; In[17]: HW {{1, 0}, {0, -1}}; E0 =equation 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[15]:= LW ==In[16]:= {{Cos Cosβ {̂ Cosβ Sinβ, Sin ββ2 }}; In[17]:= HW =Sinβ}, {{1, 0}, {0, -1}}; 2β, 2{{1, In[16]:= QWV = {{1, 0}, {0, I}}; QWH =0},{0, {{1, 0},{0, -I}}}; material and J=0, the Helmholtz in aEq. solid (Fowles 1975, Jackson 1975) is ⃗ βplate ⃗{{1, 2 In[15]:= LW {{Cos β, Cosβ Sinβ}, {{0, Cosβ Sinβ, Sin ̂ ̂={Cos[α], ⃗⃗1/4𝜋𝜋 rewrite using athe vector k A wave isand a- thin birefr ⃗ and ⃗M ⃗⃗E0 ⍵ In[12]:= LP =code , ⃗Band are r 𝒦𝒦 ,Cosβ E ,0}+ and B are both complex vectors and E In[16]:= QWV =Sin[𝛽𝛽]}; 0}, {0, I}}; QWH =quarter -I}}}; In[16]:= = {{1, 0}, {0, I}}; = {{1, 0},{0, -I}}}; In[18]: = EWV I=QWH 1/4𝜋𝜋 }}; EWH = {{1, 0}, IH }} In[15]:=In[11]:= LW =where {{Cos β,QWV Sinβ}, {{{1, Cosβ Sinβ, Sin β}}; }}; below. The below. Mathematica Mathematica input code input is(1) shown code below: iswave shown below: moment P magnetization in the solid. If we assume that medium is not a0},{0, magnetic In[17]: HW {{1, 0}, {0, -1}}; ==The 1/√2{1, 1/√2{0,1}; Cosαα Cos Mathematica #1 2Exp[ 2 {0, Exp[ In[17]: = HW In[17]: == {{1, HW 0}, {0, = {{1, -1}}; 0}, {0, -1}}; distance d, the phase difference is ∆𝜀𝜀 𝑑𝑑(n -n ) between E wave EI{{1, wave w linearlypolarized polarized 0 2 1 x y1/4𝜋𝜋 llinearly inearly (( Sin α)) 0In[12]:= In[15]:= LW = {{Cos β, Cosβ Sinβ}, { Cosβ Sinβ, Sin β }}; LP = {Cos[α], Sin[𝛽𝛽]}; In[17]: HW = {{1, 0}, {0, -1}}; In[18]:= EWV = {{1, 0}, Exp[ 1/4 }}; In[18]: = EWV = {{1, 0}, {0, I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, Exp[ }}; In[17]: = HW = {{1, 0}, {0, -1}}; In[17]: = HW = {{1, 0}, {0, -1}}; polarized 0 In[18]: EWV =0},{0, 0}, {0, In[16]:= In[16]:= In[16]:= QWV In[16]:= In[16]:= In[16]:= QWV =QWV {{1, In[16]:= QWV QWV = 0}, = QWV {{1, {{1, In[16]:= In[16]:= {0, In[16]:= 0}, QWV {{1, I}}; 0}, = In[16]:= {{1, {0, {0, 0}, QWV QWV = QWH I}}; I}}; 0}, QWV {{1, {0, = QWV {0, = I}}; I}}; 0}, QWH = {{1, {{1, = {{1, I}}; {{1, {0, = 0}, QWH 0}, QWH = {{1, 0},{0, I}}; = 0}, QWH {{1, {0, {{1, {0, 0}, = {0, I}}; = I}}; QWH 0},{0, -I}}}; {{1, 0},{0, {{1, {0, I}}; QWH 0},{0, I}}; QWH 0},{0, = -I}}}; -I}}}; QWH 0},{0, {{1, = QWH -I}}}; = -I}}}; 0},{0, {{1, {{1, = -I}}}; {{1, = 0},{0, 0},{0, -I}}}; {{1, 0},{0, -I}}}; -I}}}; -I}}} -I 𝑐𝑐QWH Sin α 2 2 In[17]: In[17]: = = HW HW = = {{1, {{1, 0}, 0}, {0, {0, -1}}; -1}}; ̂ ̂ ̂ ⃗ ⃗ ⃗ ⃗ 2EWV In[12]:= LP == {Cos[α], , B H are where 𝒦𝒦 ,{0, E1code ,∂β, and BSin[𝛽𝛽]}; are both vectors and E -I}}}; ⃗ Cosβ ⃗{{1, In[16]:= QWV {{1, 0}, {0,complex I}}; QWH = 0},{0, In[16]:= QWV ==LW {{1, {0, QWH 0},{0, -I}}}; == {{1, 0}, {0, Exp[ I Sin 1/4𝜋𝜋 EWH = {{1, 0},and {0, Exp[ - I 1r In[15]:= {{Cos Sinβ}, { Cosβ Sinβ, β }}; }};{{1, ∂=2 -1}}; E P E0 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[17]: ==0}, HW =I}}; {{1, 0}, {0, Mathematica #1 #1 2a In[16]:= QWV {{1, 0}, QWH =0}, {{1, ⃗ Mathematica ⃗∇{{1, ⃗EIn[18]: produce േ}}; 𝜋𝜋/4EWH phase difference bet × (LP × )൅ =2= −u Sinβ}, {{1, 0},{0, I=-1}}; 0},{0, {Jackson Cosβ EWV -I}}}; }}; {0, EWH {{1, Sin ሺͳሻ ∇ In[15]:= LW ={0, {{Cos β, Cosβ Sinβ, β{0, In[12]:= = Sin[𝛽𝛽]}; In[17]: HW {{1, {0, In[17]: ={Cos[α], HW =I}}; 0}, -1}}; In[19]: EWH ={{1, {{1, 0}, {0, -1}}; In[16]:= QWV =code QWH -I}}}; 0{0, where EWH =I}}; {{1, 0}, {0, Exp[ 1/4 }}; In[18]: EWV 1/4𝜋𝜋 = {{1, 0}, Exp[ -examine I0}, 1/4𝜋𝜋 }}; In[18]: =∂t EWV In[18]: ={{1, 0}, = Exp[ 0}, Ithe 1/4𝜋𝜋 Exp[ }}; I{0, 1/4𝜋𝜋 = }}; {{1, EWH {0, = {{1, Exp[ 2= and55 J=0, the Helmholtz wave equation solid (Fowles 1975, 1975) is ⃗H ⃗0}, ∂t c2in is a magnetic intensity vector matter. We 2a2QWV 2in= 11 ) material 5 In[19]: = EWH = {{1, 0}, {0, ⃗ ⃗ In[16]:= I}}; QWH {{1, 0},{0, -I}}}; In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, Exp[ -}}; In[15]:= LW = {{Cos β, Cosβ Sinβ}, { Cosβ Sinβ, Sin β }}; right(left)-handed circularly 1/√2 ( ±1 E0 = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[19]: = EWH = {{1, 0}, {0, -1}}; radiation is propagating along the k direction. If the ∆𝜀𝜀=0 while the amplitude of E isIt1r In[18]: = EWV = {{1, 0}, {0, Exp[ In[18]: I 1/4𝜋𝜋 = EWV }}; EWH = {{1, = 0}, {{1, {0, 0}, Exp[ {0, Exp[ I 1/4𝜋𝜋 I }}; 1/4𝜋𝜋 EWH In[17]: = In[17]: In[17]: HW In[17]: = In[17]: = = In[17]: HW {{1, HW = = In[17]: = 0}, HW = HW {{1, {{1, HW {0, In[17]: = In[17]: = 0}, {{1, -1}}; 0}, In[17]: {{1, = HW = {{1, {0, = In[17]: {0, 0}, 0}, = HW -1}}; HW 0}, -1}}; {{1, {0, {0, = HW = {0, = -1}}; -1}}; 0}, HW {{1, {{1, = -1}}; {{1, {0, = 0}, 0}, {{1, -1}}; 0}, {0, {0, 0}, {0, -1}}; -1}}; {0, -1}}; -1}}; right(left)-handed circularly 1/√2 ( ±i) ±1 Mathematica linearly polarized ±1Mathematica In[18]: In[18]: = = EWV EWV =code ={{1, {{1,0}, 0}, {0,Exp[ Exp[I 1/4𝜋𝜋 I 1/4𝜋𝜋}}; }};EWH EWH= ={{1, {{1,0}, 0}, {0,Exp[ Exp[- I- 1/4𝜋𝜋 I31/4𝜋𝜋}}; }}; code #1 #1 2 {0, 2{0, ±i In[15]:= LW = {{Cos β, Cosβ Sinβ}, { Cosβ Sinβ, Sin β }}; In[17]:= In[16]:= HW = {{1, 0}, {0, -1}}; In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; In[17]: = HW = {{1, 0}, {0, -1}}; In[19]: = EWH = {{1, 0}, {0, -1}}; QWV = {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; In[19]:= EWH {{1, 0}, {0, -1}}; In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}t ⃗ ⃗ is magnetic intensity vector in matter. We where H In[17]:=In[15]:= HW = {{1, {0, 2=a {{1, 21/√2{1, 2 ⃗P ⃗⃗ = In[19]: ={0,I}}; EWH =I=Exp[ {{1, EWH 0}, {0, = 2EWH {{1, -1}}; 0},the -1}}; ⃗E ⃗ −⍵t) In[16]:= 0}, {0, QWH =I }}; {{1, -I}}}; i(𝒦𝒦 .r In[18]: =0}, {{1, 0}, {0, 1/4𝜋𝜋 }}; EWH {{1, 0}, {0,examine - I }} 1/ LW {{Cos Cosβ Sinβ}, {Exp[ Cosβ Sinβ, Sin β0},{0, }}; =the EWV ={{1, {{1, 1/4𝜋𝜋 = {{1, 0}, {0, Exp[ -Exp[ I 1/4𝜋𝜋 In[20]:= l1 ==-1}}; E0*LP; l2=LW ;In[19]: =IfE0 HW =In[18]: {{1, 0}, {0, -1}}; ∂will 1/√2{1, E0 0}+ =∂QWV 1/√2{0,1}; 0}+ 1/√2{0,1}; In[11]:= In[19]: ==10}, EWH =β,EWV 0}, {0, -1}}; ̂{0, ⃗P= In[11]:= ⃗ ⃗ × In[20]:= = E0*LP; l2=LW where 𝜒𝜒𝜀𝜀In[17]: we suppose EM wave be a the form of solution such as E ,l1otherwise we can ⃗∇=× ⃗E 0E.∇ 0 e {0, A ( )൅ = −u ሺͳሻ polarization modes with normalized Jones vectors calculations in the In[19]: In[19]: = = EWH EWH = = {{1, {{1, 0}, 0}, {0, {0, -1}}; -1}}; A 2 2 2 In[12]:= LP {Cos[α], Sin[𝛽𝛽]}; In[17]: HW -1}}; vector is responsible for the linearly polarized such as Jones vector {A, B}, the amp In[19]: = EWH = {{1, 0}, {0, -1}}; In[16]:= QWV = {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; 0 In[20]:= l1 = E0*LP; l2=LW ; 2 2 2 In[20]:= l1 = E0*LP; l2=LW ; In[19]: = EWH = {{1, 0}, {0, -1}}; In[19]: = EWH = {{1, 0}, -1}}; 2 2 2 In[18]: = In[18]: In[18]: EWV In[18]: = In[18]: = In[18]: = EWV EWV = {{1, = In[18]: = EWV EWV = 0}, = {{1, EWV {{1, In[18]: = In[18]: {0, = = In[18]: 0}, EWV {{1, Exp[ 0}, {{1, = = = In[18]: {{1, {0, {0, = 0}, EWV 0}, EWV I = Exp[ 1/4𝜋𝜋 Exp[ 0}, {{1, = EWV {0, {0, = {0, EWV = Exp[ I Exp[ 0}, }}; I {{1, 1/4𝜋𝜋 {{1, = 1/4𝜋𝜋 Exp[ {{1, {0, EWH I = 0}, I 0}, 1/4𝜋𝜋 }}; 1/4𝜋𝜋 {{1, Exp[ }}; I 0}, {0, 1/4𝜋𝜋 {0, = EWH EWH 0}, }}; {0, {{1, Exp[ }}; I Exp[ 1/4𝜋𝜋 }}; {0, EWH Exp[ EWH = 0}, = I {{1, EWH I Exp[ 1/4𝜋𝜋 {{1, }}; 1/4𝜋𝜋 {0, I = 1/4𝜋𝜋 = 0}, EWH {{1, Exp[ 0}, {{1, I = }}; }}; 1/4𝜋𝜋 {{1, {0, {0, }}; 0}, EWH 0}, EWH = Exp[ I 0}, Exp[ }}; {{1, EWH {0, 1/4𝜋𝜋 {0, = {0, EWH = Exp[ Exp[ 0}, {{1, I {{1, = }}; I 1/4𝜋𝜋 Exp[ 1/4𝜋𝜋 {{1, {0, = 0}, 0}, I I {{1 Ex 1/4 -}} 1/ 0} {}M I{ 2 2 2 right(left)-handed elliptically 1/√A ++BB ++CC (( ) ±1 right(left)-handed elliptically ±1 ∂t c ∂t right(left)-handed elliptically 1/√A ) ±1 2 2 5 In[18]:= In[17]: EWV ==HW {{1, 0}, {0, Exp[ IIE0*LP; 1/4𝜋𝜋 }}; EWH ==Exp[ {0, Exp[ -- II;1/4𝜋𝜋 }}; In[16]:= =Exp[ {{1, 0}, {0, I}}; QWH = {{1, -I}}}; In[18]: = EWV =Sinβ}, {{1, 0}, {0, I0}, 1/4𝜋𝜋 EWH = {{1, 0}, {0, Exp[ - I 1 In[15]:= LW {{Cos β, Cosβ Cosβ Sinβ, Sin β }}; }}; In[20]:= = l2=LW ;{{1, {{1, 0}, {0, -1}}; E0 == 1/√2{1, E0 0}+ =QWV 1/√2{1, 1/√2{0,1}; 0}+ 1/√2{0,1}; In[11]:= In[11]:= BB++i iCC In[19]: == EWH =l1 {{1, 0}, {0, In[20]:= l1-1}}; ={ In[20]:= E0*LP; l1 = l2=LW E0*LP; ;0},{0, l2=LW In[18]:= LP EWV {{1, 0}, {0, 1/4𝜋𝜋 }}; EWH {{1, 0}, {0, Exp[ 1/4𝜋𝜋 }}; polarization modes with the normalized Jones vectors calculations in the M In[12]:= In[12]:= = {Cos[α], LP = Sin[𝛽𝛽]}; {Cos[α], Sin[𝛽𝛽]}; In[17]: = HW = {{1, 0}, {0, -1}}; In[19]: = EWH = {{1, 0}, {0, -1}}; In[19]: = EWH = {{1, 0}, {0, -1}}; In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[16]:= QWV {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[22]:= E0*LP/. ->1/4 In[18]:= EWV =vector {{1, 0}, {0, I l2=LW 1/4𝜋𝜋 }}; In[20]:= l1 =E0*LP; E0*LP; ; EWH = {{1, 0},2 {0, Exp[ - I 1/4𝜋𝜋 }}; ⃗=E0*LP; 2Exp[ In[20]:= In[20]:= l1=l1 = l2=LW l2=LW ;the rewrite Eq. (1) using acomplex wave k Polarized light Joneseiεmatrices ⃗⃗Jones x5 the vector responsible for elliptically polarized vector as {A, ⃗ −⍵t) .r In[18]: =In[19]: =-1}}; {{1, 0}, {0, Exp[ 1/4𝜋𝜋 }}; EWH {{1, 0}, BേiC}. {0, Exp[Spec -I In[15]:= LW {{Cos Cosβ Sinβ}, {{0, Cosβ Sin βi(𝒦𝒦 }}; l1 == E0*LP; l2=LW ;Sinβ, iε In[17]: =In[20]:= HW =In[20]:= {{1, 0}, {0, -1}}; below. The code is shown below: In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋=;= 00 l1 = E0*LP; l2=LW ;-1}}; In[20]:= l1 =I{{1, E0*LP; l2=LW In[19]: = In[19]: In[19]: EWH In[19]: =In[19]: In[19]: = EWH EWH ={{1, =will =Mathematica EWH EWH =β, 0}, =EWV {{1, EWH {{1, In[19]: {0, = =In[19]: 0}, EWH {{1, 0}, {{1, = In[19]: {{1, {0, {0, =0}, EWH 0}, EWH =input 0}, -1}}; = {{1, EWH {0, ={0, EWH = -1}}; -1}}; 0}, {{1, {{1, = -1}}; {{1, {0, = 0}, 0}, -1}}; 0}, {0, {0, -1}}; -1}}; {0, -1}}; -1}}; ̂{0,e0}, iεxx where Polarizer WavePlate Plate Jonesmatrices ematrices If we =suppose the EM be aIn[19]: form of solution as E , we=;can &&Wave Jones 0y ))) ⃗P= 𝜒𝜒𝜀𝜀0⃗E. In[12]:= ggPolarizer M= In[19]: EWH ==EWV {{1, 0}, {0, eneral phase retarder M= LP In[12]:= = ={Cos[α], LP =In[19]: Sin[𝛽𝛽]}; {Cos[α], Sin[𝛽𝛽]}; In[17]: = 2wave HW {{1, 0}, {0, -1}}; =-1}}; EWH =22]:= {{1, 0}, {0, -1}}; In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 In[16]:= QWV =E0*LP∙QWV/. {{1, 0}, {0, I}}; = such {{1, 0},{0, -I}}}; In[ E0*LP/.𝛼𝛼 ->1/4 E0*LP/.𝛼𝛼 𝜋𝜋 0 ->1/4 𝜋𝜋 In[23]:= ->1/4 In[18]: =l1 {{1, 0}, {0, Exp[ I QWH 1/4𝜋𝜋 }}; Exp[ - I 1/4𝜋𝜋 }}; general eneralphase phaseretarder retarder M= (((e0e0iεx eeiεiε In[20]:= = =E0*LP; l2=LW ;In[𝜋𝜋22]:= yy 0 In[19]: = EWH {{1, 0}, {0, -1}}; iε 2 2EWH = {{1, 2 0}, {0, In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1 general phase retarder M= ( 0 e iεy ) In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; In[15]:= LW In[15]:= = {{Cos LW β, = Cosβ {{Cos Sinβ}, β, Cosβ { Cosβ Sinβ}, Sinβ, { Cosβ Sin β Sinβ, }}; Sin β }}; In[20]:= l1 = E0*LP; l2=LW ; general phase retarder In[20]:= l1 = E0*LP; l2=LW ; In[17]: HW = {{1, 0}, {0, -1}}; In[19]: = EWH = {{1, 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 below. The Mathematica input code is shown below: In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[ In[ 22]:= 22]:= E0*LP/.𝛼𝛼 E0*LP/.𝛼𝛼->1/4 ->1/4𝜋𝜋𝜋𝜋 0 e ifIn[18]: B=0 In[23]:= and A=C then wave is al2=LW circularly polarized vector as {1, i}. }}; In[19]: = EWH = 0}, In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 In[16]:= QWV I}}; QWH =;l2=LW {{1, 0},{0, -I}}}; LP∙QWH/. ->1/4 =In[24]:= EWV =E0* {{1, 0}, {0, Exp[ Il2=LW 1/4𝜋𝜋 }}; = {{1, 0}, {0, Exp[ - I 1/4𝜋𝜋 In[ In[ E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 22]:= E0*LP/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 𝜋𝜋 In[20]:= In[20]:= In[20]:= l1 =In[20]:= E0*LP; In[20]:= l1l1 ==the E0*LP; In[20]:= l1 E0*LP; l1 l2=LW l1 =E0*LP; In[20]:= E0*LP; In[20]:= =In[23]:= E0*LP; In[20]:= l1 l2=LW ;->1/4 ={{1, In[20]:= E0*LP; l1l1 l2=LW = l1 ;= l2=LW ;E0*LP; E0*LP; =𝜋𝜋{0, l1E0*LP; = ;-1}}; E0*LP; ;EWH l2=LW l2=LW ;Jones l2=LW l2=LW ; ;->1/4 ; ->1/4 ;𝜋𝜋such 22 ⃗22]:= 2In[20]:= 2= 2E0*LP∙QWV/.𝛼𝛼 2 rewrite Eq. (1) using a wave vector k Cos β Cosβ Sinβ E0*LP∙QWV/.𝛼𝛼 In[23]:= ->1/4 𝜋𝜋 𝜋𝜋 2 In[20]:= l1 = E0*LP; l2=LW ; Cos β Cosβ Sinβ In[18]: = β, EWV =Sinβ}, {{1, 0}, {0, Exp[ ICosβ 1/4𝜋𝜋 }}; {{1, In[24]:= 0}, {0, E0* Exp[LP∙QWH/.𝛼𝛼 - I 1/4𝜋𝜋 }}; In[15]:= LW LW Cosβ {{Cos β, Cosβ {-1}}; Cosβ Sinβ}, Sinβ, {->1/4 Sin βSinβ, }};EWH Sin β= }}; l1code =->1/4 E0*LP; l2=LW ; 𝜋𝜋 In[17]: ={{Cos HW =In[20]:= 0}, {0, -1}}; =={{1, {{1, 0}, {0, In[23]:= In[EWH 22]:= E0*LP/.𝛼𝛼 𝜋𝜋 linearly polarizer(𝛽𝛽,TA) M= ( 5 Cos 2β5 Cosβ2 Sinβ)) In[20]:= In[19]: l1In[15]:= === E0*LP; l2=LW ;E0*LP∙QWV/.𝛼𝛼 ->1 Mathematica linearly polarizer(𝛽𝛽,TA) M= 3#1 linearly polarizer(𝛽𝛽,TA) M= ((Cosβ In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋I->1/4 𝜋𝜋-1}}; Cos β Cosβ 2βSinβ) Sinβ Sin In[19]: = E0*LP∙QWV/.𝛼𝛼 EWH =0}, {{1, 0}, {0, In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 In[16]:= QWV In[16]:= = {{1, QWV 0}, {0, = {{1, I}}; QWH {0, I}}; = {{1, QWH 0},{0, {{1, -I}}}; 0},{0, In[ 22]:= E0*LP/.𝛼𝛼 𝜋𝜋 In[25]:= E0* LP∙HW /.->1/4 ->1/4 EWV = {{1, {0,->1/4 Exp[ 1/4𝜋𝜋 }};=EWH = {{1, -I}}}; 0}, {0, Exp[ - I 1/4𝜋𝜋 }}; In[20]:=In[18]: l1 ==In[23]:= E0*LP; l2=LW ;is0}, In[24]:= E0* LP∙QWH/.𝛼𝛼 linearly polarizer(β,TA) In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋𝜋𝜋𝜋𝜋 Sin Cosβ Sinβ Sinβ Sin22ββ ) linearly polarizer(𝛽𝛽,TA) M= (Cosβ A 22]:= quarter wave plate a𝜋𝜋 thin birefringent crystal the𝜋𝜋 thickness of which has been adju In[20]:= l1 = E0*LP; l2=LW ; In[17]: HW = {{1, 0}, {0, -1}}; In[19]: =In[23]:= EWH = {{1, 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 In[ In[ In[ In[ In[ In[ In[ In[ In[ In[ In[ 22]:= E0*LP/.𝛼𝛼 22]:= 22]:= 22]:= E0*LP/.𝛼𝛼 E0*LP/.𝛼𝛼 22]:= ->1/4 E0*LP/.𝛼𝛼 E0*LP/.𝛼𝛼 22]:= E0*LP/.𝛼𝛼 ->1/4 ->1/4 22]:= 22]:= E0*LP/.𝛼𝛼 22]:= ->1/4 ->1/4 𝜋𝜋 𝜋𝜋 ->1/4 E0*LP/.𝛼𝛼 22]:= E0*LP/.𝛼𝛼 𝜋𝜋 E0*LP/.𝛼𝛼 𝜋𝜋 ->1/4 𝜋𝜋 E0*LP/.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 𝜋𝜋 E0*LP∙QWV/.𝛼𝛼 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 In[24]:= E0* LP∙QWH/.𝛼𝛼 Cosβ Sinβ Sin β In[24]:= E0* In[24]:= LP∙QWH/.𝛼𝛼 E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 Mathematica code In[ E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[19]: = EWH =0}, {{1, 0}, {0, In[ 22]:= E0*LP/.𝛼𝛼 In[16]:= QWV In[16]:= = {{1, 0}, {0, = {{1, I}}; 0}, QWH {0, I}}; =𝜋𝜋->1/4 {{1, QWH 0},{0, = EWH {{1, 0},{0, E0* LP∙EWV /. ->1/4 //N In[18]: =In[26]:= EWV = {{1, {0, Exp[ I-1}}; 1/4𝜋𝜋 }}; = {{1,-I}}}; 0}, {0, Exp[ E0* - I 1/4𝜋𝜋 }};/.𝛼𝛼 ->1 In[25]:= LP∙HW l1 =QWV E0*LP; l2=LW ;#1 In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋-I}}}; 1 00 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋𝜋𝜋 In[22]:= 22]:= In[20]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[24]:= E0* E0* LP∙QWH/.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋 ∓iπ/4 ( 1 quarter-wave plate(V/H) M= e∓iπ/4 In[20]:= l1 =LP∙QWH/.𝛼𝛼 E0*LP; ; 𝜋𝜋𝜋𝜋 In[17]: = produce HW In[17]: {{1, =aEWH HW 0}, {0, = {{1, -1}}; 0}, {0,l2=LW -1}}; In[19]: ==In[24]:= =E0* {{1, 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4the 𝜋𝜋 ordinary and extraordinary rays at the o quarter-wave plate(V/H) M= (011 ± 0i0))) In[22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 E0 = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= In[25]:= E0* LP∙HW /.𝛼𝛼 In[24]:= LP∙QWH/.𝛼𝛼 ->1/4 quarter-wave plate(V/H) quarter-wave plate(V/H) M= ee∓iπ/4 േ 𝜋𝜋/4 phase difference between 3 ∓iπ/4 ( 0 ± i In[ 22]:= E0*LP/.𝛼𝛼 In[27]:= E0* LP∙EWH /. ->1/4 //N In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, quarter-wave plate(V/H) M= e (0 ± i ) In[25]:= E0* In[25]:= LP∙HW E0* /.𝛼𝛼 LP∙HW ->1/4 𝜋𝜋 /.𝛼𝛼 ->1/4 𝜋𝜋Exp[𝜋𝜋 - I 1/4𝜋𝜋 }}; In[20]:= l1 = E0*LP; l2=LW ; In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[23]:= In[23]:= In[23]:= E0*LP∙QWV/.𝛼𝛼 In[23]:= In[23]:= In[23]:= E0*LP∙QWV/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 In[23]:= E0*LP∙QWV/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 In[23]:= In[23]:= ->1/4 In[23]:= E0*LP∙QWV/.𝛼𝛼 In[23]:= 𝜋𝜋 ->1/4 ->1/4 E0*LP∙QWV/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 E0*LP∙QWV/.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋 𝜋𝜋 E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 𝜋𝜋 ->1/4 𝜋𝜋 ->1/4 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 𝜋𝜋 𝜋𝜋 In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 In[24]:= 𝜋𝜋 E0* LP∙QWH/.𝛼𝛼 ->1/4 In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 0 ±i In[26]:= E0* LP∙EWV /.𝛼𝛼 -> In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋𝜋𝜋/.𝛼𝛼 In[20]:= l1 l2=LW ; ->1/4 In[17]: = In[ HW In[17]: =In[25]:= {{1, =E0*LP/.𝛼𝛼 HW 0}, {0, {{1, -1}}; 0}, {0, -1}}; In[19]: =In[11]:= EWH ===LP∙HW {{1, 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 22]:= ->1/4 𝜋𝜋/.𝛼𝛼 E0 =E0*LP; 1/√2{1, 0}+ 1/√2{0,1}; In[25]:= E0* LP∙HW In[25]:= E0* E0* LP∙HW ->1/4 ->1/4 𝜋𝜋->1/4 𝜋𝜋/.𝛼𝛼 In[24]:= E0* LP∙QWH/.𝛼𝛼 𝜋𝜋->1/4𝜋𝜋𝜋𝜋 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 1 00 ∓iπ/4 ( 1 In[ E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[12]:= LP {Cos[α], In[28]:= l1.l2/.{ ->1/4 ,}}; ->1/3 //N In[18]: = In[20]:= EWV In[18]: =In[24]:= = {{1, 0}, {0, ==->1/4 {{1, Exp[ 0}, ILP∙QWH/.𝛼𝛼 1/4𝜋𝜋 {0, Exp[ IEWH 1/4𝜋𝜋 =}}}; {{1, 0}, {0, = {{1, Exp[ 0},- I{0, 1/4𝜋𝜋 Exp[ }};- I 1/4𝜋𝜋 }}; l122]:= =EWV E0*LP; l2=LW ;Sin[𝛽𝛽]}; half-wave plate(V/H) M= e∓iπ/4 In[24]:= E0* ->1/4 𝜋𝜋 EWH E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[23]:= E0*LP∙QWV/.𝛼𝛼 𝜋𝜋 In[26]:= LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋In[26]:= half-wave plate(V/H) M= ((011 −100))) half-wave plate(V/H) half-wave plate(V/H) M= ee∓iπ/4 In[26]:= E0* LP∙EWV E0* /.𝛼𝛼 LP∙EWV ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 ->1/4 In[19]: = EWH =E0* {{1, 0}, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 𝜋𝜋->1/4 In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋{0, In[25]:= E0* LP∙HW /.𝛼𝛼 −1 ) In[24]:= In[24]:= In[24]:= E0* In[24]:= In[24]:= LP∙QWH/.𝛼𝛼 In[24]:= E0* In[24]:= LP∙QWH/.𝛼𝛼 E0* LP∙QWH/.𝛼𝛼 E0* E0* In[24]:= LP∙QWH/.𝛼𝛼 In[24]:= LP∙QWH/.𝛼𝛼 ->1/4 In[24]:= LP∙QWH/.𝛼𝛼 E0* In[24]:= 𝜋𝜋LP∙QWH/.𝛼𝛼 ->1/4 E0* ->1/4 E0* E0* LP∙QWH/.𝛼𝛼 ->1/4 LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋->1/4 𝜋𝜋E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋LP∙QWH/.𝛼𝛼 𝜋𝜋E0* 𝜋𝜋 LP∙HW ->1/4 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋->1/4 𝜋𝜋 𝜋𝜋 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[25]:= /.𝛼𝛼𝜋𝜋 ->1/4 𝜋𝜋 E0* LP∙EWH /.𝛼𝛼 -> In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ half-wave plate(V/H) M= e∓iπ/4 (00 −1 In[27]:= In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[26]:= In[26]:= E0* E0* LP∙EWV LP∙EWV /.𝛼𝛼 /.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ 0 −1 In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; In[29]:= l1.l2/.{ ->1/4 , ->3/4 } //N In[18]: = EWV In[18]: = = {{1, EWV 0}, {0, = {{1, Exp[ 0}, I 1/4𝜋𝜋 {0, Exp[ }}; I EWH 1/4𝜋𝜋 = }}; {{1, EWH 0}, {0, = {{1, Exp[ 0}, I {0, 1/4𝜋𝜋 Exp[ }};- I 1/4𝜋𝜋 }}; In[20]:= l1 = E0*LP; l2=LW ; In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 E0*LP∙QWV/.𝛼𝛼 4 In[26]:= E0* LP∙EWV In[24]:= In[23]:= E0* LP∙QWH/.𝛼𝛼 𝜋𝜋->1/4 In[25]:= E0*->1/4 LP∙HW ->1/4/.𝛼𝛼 𝜋𝜋 ->1/4 𝜋𝜋ȀȀ 2/.𝛼𝛼𝜋𝜋 2 000 ) In[15]:= LW =->1/4 {{Cos β,/.𝛼𝛼 Cosβ Sinβ}, {𝜋𝜋E0* Cosβ Sinβ, Sin }};𝜋𝜋ȀȀ In[19]: = In[ EWH In[19]: =In[25]:= =E0*LP/.𝛼𝛼 {{1, EWH 0}, {0, =->1/4 {{1, -1}}; 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 22]:= 𝜋𝜋𝜋𝜋LP∙HW In[25]:= E0* /.𝛼𝛼 ->1/4 E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[24]:= E0* LP∙QWH/.𝛼𝛼 In[27]:= LP∙EWH ->1/4 𝜋𝜋ȀȀ In[26]:= E0* LP∙EWV 𝜋𝜋ȀȀ eighth-wave plate(V/H) M= (111 In[27]:= E0* In[27]:= LP∙EWH /.𝛼𝛼 LP∙EWH ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 β->1/4 eighth-wave plate(V/H) eighth-wave plate(V/H) M= ) ±iπ/4 eighth-wave plate(V/H) M= ((001 ee±iπ/4 ) 0 In[20]:= l1 =In[25]:= E0*LP; l2=LW ; 𝜋𝜋 l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 In[24]:= LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[23]:= E0*LP∙QWV/.𝛼𝛼 In[26]:= E0* /.𝛼𝛼 𝜋𝜋ȀȀ In[25]:= In[25]:= In[25]:= E0* In[25]:= LP∙HW In[25]:= E0* E0* In[25]:= LP∙HW E0* LP∙HW E0* /.𝛼𝛼 E0* In[25]:= LP∙HW ->1/4 In[25]:= LP∙HW /.𝛼𝛼 In[25]:= LP∙HW E0* /.𝛼𝛼 In[25]:= ->1/4 ->1/4 LP∙HW /.𝛼𝛼 E0* /.𝛼𝛼 E0* /.𝛼𝛼 E0* ->1/4 ->1/4 𝜋𝜋𝜋𝜋ȀȀ LP∙HW 𝜋𝜋 LP∙HW ->1/4 E0* /.𝛼𝛼 LP∙HW 𝜋𝜋𝜋𝜋LP∙HW ->1/4 𝜋𝜋/.𝛼𝛼 /.𝛼𝛼/.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋 /.𝛼𝛼 ->1/4 𝜋𝜋->1/4 𝜋𝜋 𝜋𝜋/.𝛼𝛼In[28]:= 𝜋𝜋->1/4 𝜋𝜋ȀȀ In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 In[26]:= E0* LP∙EWV In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[27]:= In[27]:= E0* E0* LP∙EWH /.𝛼𝛼 /.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ 2LP∙EWV eighth-wave plate(V/H) M= (0 e±iπ/4 In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋𝜋𝜋 In[15]:= LW =->1/4 {{Cos β, Cosβ Sinβ}, { Cosβ Sinβ, Sin2 β }}; In[19]: = In[24]:= EWH In[19]: = =E0* {{1, EWH 0}, {0, =LP∙EWH {{1, -1}}; 0}, {0, -1}}; In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 𝜋𝜋ȀȀ 5 ) In[ 22]:= E0*LP/.𝛼𝛼 𝜋𝜋->1/4 In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋𝜋𝜋ȀȀ LP∙QWH/.𝛼𝛼 𝜋𝜋->1/4 0 e5±iπ/4 In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 In[26]:= E0* LP∙EWV /.𝛼𝛼 In[20]:= l1 In[20]:= = E0*LP; l1 = l2=LW E0*LP; ; l2=LW ; †Wave Wave plate (V/H) stand for the fast transmission axis of vertical/horizontal. In[16]:= QWV = {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; In[28]:= l1.l2/.{𝛼𝛼 In[28]:= ->1/4 l1.l2/.{𝛼𝛼 𝜋𝜋, 𝛽𝛽 ->1/3 ->1/4 𝜋𝜋} 𝜋𝜋, //N 𝛽𝛽 ->1/3 𝜋𝜋} //N In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ †† plate (V/H) stand for the fast transmission axis of vertical/horizontal. In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 † Wave Wave plate plate (V/H) (V/H) stand stand for for the the fast fast transmission transmission axis axis of of vertical/horizontal. vertical/horizontal. In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋In[26]:= In[25]:= LP∙HW 𝜋𝜋E0* In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[28]:= In[28]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 ->1/4 ->1/4 𝜋𝜋, 𝜋𝜋, 𝛽𝛽/.𝛼𝛼 𝛽𝛽 ->1/3 ->1/3 𝜋𝜋} 𝜋𝜋} //N //N In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[26]:= In[26]:= In[26]:= E0* In[26]:= In[26]:= LP∙EWV In[26]:= E0* E0* In[26]:= LP∙EWV E0* LP∙EWV E0* /.𝛼𝛼 E0* In[26]:= LP∙EWV LP∙EWV ->1/4 In[26]:= LP∙EWV E0* /.𝛼𝛼 /.𝛼𝛼 In[26]:= 𝜋𝜋ȀȀ LP∙EWV ->1/4 E0* ->1/4 E0* /.𝛼𝛼 /.𝛼𝛼 E0* /.𝛼𝛼 LP∙EWV ->1/4 LP∙EWV ->1/4 𝜋𝜋ȀȀ E0* 𝜋𝜋ȀȀ LP∙EWV ->1/4 /.𝛼𝛼 𝜋𝜋ȀȀ LP∙EWV 𝜋𝜋ȀȀ ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 /.𝛼𝛼 /.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ ->1/4 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ𝜋𝜋ȀȀ In[27]:= E0* LP∙EWH ->1/4 In[27]:= 𝜋𝜋ȀȀ LP∙EWH /.𝛼𝛼 ->1/4 In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, ->1/3 𝜋𝜋} //N † Wave plate (V/H) stand for the fast transmission axis of vertical/horizontal. In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[20]:= l1 In[20]:= = E0*LP; l1 LP∙HW =In[28]:= l2=LW E0*LP; ;=l1.l2/.{𝛼𝛼 l2=LW ;𝛽𝛽->1/4 In[16]:= QWV {{1, 0}, {0, I}}; QWH =𝜋𝜋}{{1, 0},{0, -I}}}; In[24]:= E0* LP∙QWH/.𝛼𝛼 𝜋𝜋𝜋𝜋ȀȀ In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 In[26]:= E0* LP∙EWV /.𝛼𝛼 𝜋𝜋ȀȀ E0* /.𝛼𝛼 ->1/4 𝜋𝜋->1/4 In[26]:= In[25]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ 𝜋𝜋, 𝛽𝛽->1/4 ->1/3 //N In[27]:= E0* LP∙EWH /.𝛼𝛼 In[29]:= l1.l2/.{𝛼𝛼 In[29]:= ->1/4 l1.l2/.{𝛼𝛼 𝜋𝜋, 𝛽𝛽 ->3/4 ->1/4 𝜋𝜋} 𝜋𝜋, //N 𝛽𝛽 ->3/4 𝜋𝜋} //N In[In[26]:= In[22]:= 22]:= In[24]:= E0*LP/.𝛼𝛼 E0*LP/.𝛼𝛼 𝜋𝜋 LP∙HW ->1/4 𝜋𝜋 𝜋𝜋, In[17]: =LP∙QWH/.𝛼𝛼 HW =LP∙EWH {{1, 0}, {0, -1}}; In[25]:= E0* /.𝛼𝛼 ->1/4 𝜋𝜋 𝜋𝜋ȀȀ E0*->1/4 𝜋𝜋𝛽𝛽->1/4 In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[27]:= E0* /.𝛼𝛼 E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //N In[28]:= l1.l2/.{𝛼𝛼 ->1/4 ->1/3 𝜋𝜋} //N Mathematica returns calcu In[29]:= In[29]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 ->1/4 ->1/4 𝜋𝜋,->1/4 𝜋𝜋, 𝛽𝛽E0* 𝛽𝛽 ->3/4 ->3/4 𝜋𝜋} 𝜋𝜋} //N //N In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[27]:= In[27]:= In[27]:= E0* In[27]:= In[27]:= LP∙EWH In[27]:= E0* E0* In[27]:= LP∙EWH E0* LP∙EWH E0* /.𝛼𝛼 E0* In[27]:= LP∙EWH In[27]:= LP∙EWH In[27]:= LP∙EWH /.𝛼𝛼 /.𝛼𝛼 In[27]:= 𝜋𝜋ȀȀ LP∙EWH ->1/4 E0* ->1/4 E0* /.𝛼𝛼 /.𝛼𝛼 E0* /.𝛼𝛼 LP∙EWH ->1/4 LP∙EWH ->1/4 𝜋𝜋ȀȀ E0* 𝜋𝜋ȀȀ LP∙EWH ->1/4 /.𝛼𝛼 𝜋𝜋ȀȀ LP∙EWH 𝜋𝜋ȀȀ ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 /.𝛼𝛼 /.𝛼𝛼 ->1/4 ->1/4 𝜋𝜋ȀȀ /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ ->1/4 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ 𝜋𝜋ȀȀ In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 In[28]:= 𝜋𝜋} //N l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //N In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[ In[22]:= 22]:= E0*LP/.𝛼𝛼 ->1/4 E0*LP/.𝛼𝛼 𝜋𝜋 𝜋𝜋 In[17]: = HW =->1/4 {{1, 0}, {0, In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋/.𝛼𝛼 In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋𝛽𝛽-1}}; In[27]:= E0* LP∙EWH In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[27]:= E0* LP∙EWH /.𝛼𝛼 𝜋𝜋ȀȀ In[29]:= l1.l2/.{𝛼𝛼 𝜋𝜋, returns 𝛽𝛽->1/4 ->3/4 𝜋𝜋}returns //N calculations; In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋,->1/4 ->1/3 𝜋𝜋} //N𝜋𝜋ȀȀ Mathematica Mathematica calculations; We polarizing modes from aaa ->1/4 http://janss.kr We desire now to create new simulation presenting means of producing polarizing modes from 65 Wedesire desirenow nowto tocreate createaaanew newsimulation simulationpresenting presentingaaameans meansof ofproducing producing polarizing modes from In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; In[23]:= E0*LP∙QWV/.𝛼𝛼 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 Out[22]:= {1/2, -1/2} In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //N Mathematica returns calculations; Mathematica Mathematica returnscalculations; calculations; We desire now to create a new simulation presenting a means of producing polarizing modes from areturns In[24]:= LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[26]:= E0* LP∙EWV /.𝛼𝛼 𝜋𝜋ȀȀ In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //N In[28]:= In[28]:= In[28]:= l1.l2/.{𝛼𝛼 In[28]:= In[28]:= In[28]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 ->1/4 In[28]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 𝜋𝜋, ->1/4 In[28]:= In[28]:= ->1/4 𝛽𝛽 In[28]:= l1.l2/.{𝛼𝛼 ->1/3 ->1/4 ->1/4 𝜋𝜋, In[28]:= 𝜋𝜋, ->1/4 𝛽𝛽l1.l2/.{𝛼𝛼 𝛽𝛽 l1.l2/.{𝛼𝛼 𝜋𝜋} ->1/3 𝜋𝜋, ->1/3 𝜋𝜋, l1.l2/.{𝛼𝛼 ->1/4 //N 𝛽𝛽𝜋𝜋, 𝛽𝛽l1.l2/.{𝛼𝛼 ->1/3 ->1/3 𝜋𝜋} 𝛽𝛽 𝜋𝜋} ->1/4 ->1/3 ->1/4 𝜋𝜋, //N //N ->1/4 𝜋𝜋} 𝛽𝛽 𝜋𝜋} ->1/3 𝜋𝜋, 𝜋𝜋} ->1/4 //N 𝜋𝜋, //N 𝛽𝛽 𝜋𝜋, 𝛽𝛽//N ->1/3 ->1/3 𝜋𝜋} 𝛽𝛽->1/4 𝜋𝜋,->1/3 //N 𝛽𝛽{{1, 𝜋𝜋} ->1/3 𝜋𝜋} //N //N //N 𝜋𝜋} //N In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 In[29]:= 𝜋𝜋} //N l1.l2/.{𝛼𝛼 𝜋𝜋,𝜋𝜋} 𝛽𝛽 ->3/4 𝜋𝜋} //N Mathematica returns calculations; In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽𝛽𝛽 ->1/3 𝜋𝜋} //N Jones calculation corresponding to the physical arrangement of optical elements in the Mathematica In[18]: = EWV = {{1, 0}, {0, Exp[ I 1/4𝜋𝜋 }}; EWH = 0}, {0, Exp[ I 1/4𝜋𝜋 }}; In[23]:= E0*LP∙QWV/.𝛼𝛼 In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 ->1/4 𝜋𝜋 Out[22]:= {1/2, Out[22]:= 1/2} {1/2, 1/2} In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 Jones calculation corresponding to the physical arrangement of optical elements in the Mathematica In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, ->1/3 𝜋𝜋} //N In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ Jones calculation corresponding to the physical arrangement of optical elements in theIn[29]:= Mathematica Mathematica returns calculations; l1.l2/.{𝛼𝛼 𝜋𝜋,{0, 𝛽𝛽𝜋𝜋->3/4 𝜋𝜋} //N Out[23]:= {1/2, I/2 In[19]: =LP∙EWV EWH =/.𝛼𝛼 {{1, 0}, In[24]:= E0* In[24]:= LP∙QWH/.𝛼𝛼 E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋->1/4 ->1/4 Jones calculation corresponding to the physical arrangement of optical elements in the Mathematica In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 In[26]:= E0* ->1/4 𝜋𝜋ȀȀ Out[22]:= Out[22]:= {1/2, {1/2, 1/2} 1/2} In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋,𝜋𝜋ȀȀ 𝛽𝛽 𝜋𝜋} ->3/4 In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 -1}}; ->3/4 //N𝜋𝜋} //N In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽returns ->1/3 𝜋𝜋} //N Out[22]:= {1/2, 1/2} Mathematica calculations; In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N In[27]:= LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ Mathematica returns calculations; In[29]:= In[29]:= In[29]:= l1.l2/.{𝛼𝛼 In[29]:= In[29]:= In[29]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 In[29]:= l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 l1.l2/.{𝛼𝛼 𝜋𝜋, ->1/4 In[29]:= In[29]:= ->1/4 𝛽𝛽In[29]:= l1.l2/.{𝛼𝛼 ->3/4 ->1/4 ->1/4 𝜋𝜋, In[29]:= 𝜋𝜋, ->1/4 𝛽𝛽l1.l2/.{𝛼𝛼 𝛽𝛽 l1.l2/.{𝛼𝛼 𝜋𝜋} ->3/4 𝜋𝜋, ->3/4 𝜋𝜋, l1.l2/.{𝛼𝛼 ->1/4 //N 𝛽𝛽𝜋𝜋, 𝛽𝛽l1.l2/.{𝛼𝛼 ->3/4 ->3/4 𝜋𝜋} 𝛽𝛽𝜋𝜋} ->1/4 ->3/4 ->1/4 𝜋𝜋, //N //N ->1/4 𝜋𝜋} 𝛽𝛽𝜋𝜋} ->3/4 𝜋𝜋, 𝜋𝜋} ->1/4 //N 𝜋𝜋, //N 𝛽𝛽𝜋𝜋, 𝛽𝛽//N ->3/4 ->3/4 𝜋𝜋} 𝛽𝛽𝜋𝜋,->3/4 //N 𝛽𝛽𝜋𝜋} ->3/4 𝜋𝜋}𝜋𝜋} //N //N//N 𝜋𝜋} //N (Fig. 2, Fig. (Fig. Out[23]:= {1/2, Out[23]:= I/2 {1/2, I/2 3(a)) 2 Mathematica returns calculations; Mathematica returns calculations; Out[22]:= {1/2, 1/2} In[29]:=oscillates l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽->1/4 ->3/4 𝜋𝜋} //N computing environment. In a plane wave, the electric field vector ⃗E⃗ always parallel to the }

computing environment. In a plane wave, the electric field vector E always oscillates parallel to the

}

}

J. Astron. Space Sci. 32(2), 63-71 (2015)

Mathematica returns calculations;

provided a Mathematica module, helicity [JvC_,JmC_]:= Module[{E1, E2, dp11, dp12, dp13, dp21, dp22, dp23}]. If you type in two numbers assigned to the polarizer JvC and wave plate JmC in the module, then helicity [JvC, JmC] will return helicity with a list of phase shifts. The outputs of the helicity [JvC, JmC] are shown as below:

Out[22]:= Out[23]:= Out[24]:= 53 + 0.353553 I } Out[25]:= 53 - 0.353553 I }Out[26]:= Out[27]:= .591506}} Out[28]:= Out[29]:=

{1/2, 1/2} (Fig. 3(b)) {1/2, I/2} (Fig. 2, Fig. 3a) (Fig. 3(c)) {1/2, -I/2} (Fig. 3b) (Fig. 3(e)) {1/2, -1/2} (Fig. 3c) Platform for manipulating polarization modes realized with Jones {0.5, 0.353553 + 0.353553 I } (Fig. 3e) (Fig. 3(f)) Mathematica code #2 {0.5, 0.353553 - 0.353553 I } 3f ) (Fig. 3(g)) Platform for(Fig. manipulating polarization modes realized with Jones vectors in MATHEMATICA {0.341506,0.591506}} (Fig. 3g) In[81]:= helicity[1,2] (Fig.3 (a)) (Fig. 3(h)) {0,0} (Fig. 3h) Out[81]:= {0., 0., 0.}} vectors in MATHEMATICA helicity1 = 0 Yong-Dae Choi1, Bogyeong Kim2, Hee-Joong Yun3† The Mathematica calculations show the emerging polarization {1.5708, 1.5708, 1.5708} alculations showmode the emerging through the arrangement throughpolarization the physicalmode arrangement of physical optic elements, helicity2 = +1 1 2 Yong-Dae Choi1Department , Bogyeong Kim , Hee-Joong Yun3† of Microbial and Nanomaterials, Mokwon University, which promptly confirm those Jones vectors from the which promptly confirm those Jones vectors from the animating platform. Daejeon For 302-729, Korea animating platform. For instance, in In[23]:= E0*LP∙QWV/. In[82]:= helicity[1,3] (Fig.3 (b)) 1 2 Department of Microbial Mokwon Department of Out[81]:= Astronomy and Space Science, Chungnam National University, E0*LP∙QWV/. 𝛼𝛼 ->1/4 𝜋𝜋 produces produces Out[23]:= Out[23]:= {1/2, I/2} , that is is thethe right-handed {1/2, I/2}, that right-and Nanomaterials, {0., 0.,University, 0.}} Daejeon 302-729, Korea Daejeon 305-764, Korea handed circularly polarized light (RHCP) as shown in Fig. 2. helicity1 = 0 light (RHCP) as shown in Fig. 2. From the arrangement of linearly polarizers inSpace Science, Chungnam National University, 2 3 Department of Astronomy and From the arrangement of linearly polarizers in right angle no Institute of Science and {-1.5708, -1.5708, -1.5708} Korean Technology Information, Daejeon 305-764, Korea Daejeon 305-806, Korea wave emerging shown as Out[29]:= {0, 0} while the Out[28]:= helicity2 = -1 emerging shown as Out[29]:= {0, 0} while the Out[28]:= {0.341506, ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] 0.591506} 3 {0.341506, 0.591506} which show the linearInstitute polarization Korean ofReceived Science and Technology Revised Information, Accepted Daejeon 305-806, ear polarization(see (seeFigs. Figs.3g3(g) 3(h)). enables us switch onKorea off the In[83]:= andand 3h). ThisThis enables us to to switch on ororoff helicity[2,7] ( E0*LCP.EWH) ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] the polarization by the combination of arrangements ofRevisedOut[83]:= Accepted {-1.5708, -1.5708, -1.5708} Received combination of arrangements of polarizers and wave plates. The fundamental conception in physics of the propagation of the electromagnetic wave polarization in matter is polarizers and wave plates. helicity1 = -1 {- 4.71239, - 4.71239, - 4.71239,} -2.35619} newly understood as the cardinal{-2.35619, keyword in -2.35619, free-space quantum communication technology and cosmology in helicity2 = +1 The fundamental conception in physics of the=electromagnetic wave polarization in matter is of the propagation helicity2 -1 2.3 Helicity of the elliptically polarized wave

astrophysics. Interactive visualization of the propagation mechanism of polarized electromagnetism in a medium newly understood as the cardinal keyword in free-space quantum communication technology and cosmology in with its helicity In[84]:= has accordinglyhelicity[2, received attention the protocol of quantum key The helicity[1, handedness of an elementary particle depends on the 5] ] from scientists exploiting ( E0*LCP.HWH) 2] returns twocorrelation helicitiesastrophysics. with two lists three phaseitsdifferences in the two divisions Interactive visualization of the propagation mechanism of polarized electromagnetism in a medium of an elementaryThe particle depends on the between itsof spin and correlation between its spin and its momentum (Goldhaver et (QKD) Out[84]:= -1.5708, distribution to guarantee {-1.5708, unconditional security-1.5708} in cryptography communication. We have provided a with its polarizer( helicitythe has accordingly received attention from scientists exploiting the protocol of quantum key for the optical deviceand arrangement of linearly JvC=1 ) and quarter-wave plate(helicity1 JmC=2 ). =The al. 1958). If the spin momentum are parallel, particle -1 ver et al. 1958). If the spin and momentum are parallel, the particledynamic can be polarization said platform for presenting the polarization modes of a transverse electromagnetic wave, canhelicity[2, be said to7]bereturns right-handed or have a helicity 1. If they are {- 4.71239, - 4.71239, - 4.71239,}We have provided a distribution (QKD) to guarantee unconditional security in cryptography communication. the helicities of the optical of arrangement of left-circularly polarizer( JvC=2 ) and converting the state of polarization through the arrangement of optical elements, using Jones vectors calculations or have a helicityanti of 1. If they anti parallel, the particle is left-handed parallel, theare particle is left-handed or have a helicity of or -1.have helicity2 = +1 polarization platform for presenting the polarization modes of a transverse electromagnetic wave, eighth-wave plate (JmC=7). It needs todynamic be noticed that the three phase differences are all 1.5708(𝜋𝜋/2) in Methematica. The platform graphically simulates the mechanism of production and propagation of the We may also adopt this definition to modern optics, since the We may also adopt this definition to modern optics, since the circularly polarized converting the state of polarization through the arrangement of optical elements, using Jones vectors calculations circularly polarized electromagnetic wave is just a helical motion The helicity[1, 2] returns two helicities with two lists of of the helicity[1, 2] which interprets the vertical component E2zwaves leadincontinuously thesatisfying horizontal polarized a medium while Maxwell's equations. Methematica. The wave platform graphically simulates the mechanism of production and propagation of the ave is just a with helical motion with helicity. Thein helicity of the polarized helicity. The helicity of the polarized electromagnetic three phase differences in the two divisions for the optical

e elliptically polarized wave

⃗

a constant phase shift (𝜋𝜋/2)technology on the yz plane vector propagatingofinlinearly x1 E2y with is acomponent critical factor in modern communication andof E2 field device arrangement polarizer (JvC=1) and quarterpolarized waves in a medium while satisfying Maxwell's equations. ve is a critical factor in modern communication technology and photonics (Aad et Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity the helicities photonics (Aad et al. 2012; Goldhaver et al. 1958; Rubenhok et wave plate (JmC=2). The helicity[2, 7] returns ⃗ 2 vector rotating in a counter direction perpendicular to this plane with velocity ⍵/k. This results in E al. 2013). However, determination of the helicity is a perplex of the optical arrangement of left-circularly polarizer (JvC=2) et al. 1958, Rubenhok et al. 2013). However, determination of the helicity is a direction. That is, if we grasp our clockwise direction (right-handed, ↺) around the advancing x 1 Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity issue because we need to aware of the vectorial behaviors and eighth-wave plate (JmC=7). It needs to be noticed that the se we need to aware of theunless vectorial behaviors accurately unless we may observe three phase differences are all 1.5708 (π/2) of the helicity[1, 2] 1. INTRODUCTION accurately we may observe propagating polarized wave finger along the spin direction with right hand, then thumb directs the advancing direction of (Goldhaver et al. 1958). Therefore, it is very helpful to simulate which interprets the vertical component E2z lead continuously is a coherent characteristic of the electromagnetic (EM) wave in a medium. In a plane ed wave (Goldhaver et al. wave, 1958).hence Therefore, is very helpful toPolarization simulate propagation the wave isit right-handed polarized wave regardless of viewer . This is very INTRODUCTION advanced polarization modes in the 1. medium, and further, it the horizontal component E2y with a constant phase shift (π/2) ⃗ and ⃗ inofxthe be in more helpfultheto evaluate the helicity operator a block yz plane field vector propagating direction wave, both the on electric field vector E field vector B electromagnetic radiation on modes in thewould medium, and further, it helicity would bethe more helpful toon evaluate the helpful confirming of polarized wave theinend of the the platform asofshown inmagnetic 2 Polarization is a coherent characteristic of the electromagnetic (EM) wave in a medium.1In a plane physics state. perpendicular to this plane with velocity ω/k. This results always oscillates to block a fixedbydirection Figs. 2 and 3. We can correctly determine the helicity of the polarized waveparallel in the E2 seeing in space. Light of such character is said to be linearly a physics state. ⃗ and ⃗ of the Here we simply estimate the helicity calculation in E vector rotating a counter clockwise direction (rightwave,with both athe electric field vector magnetic fieldinvector B electromagnetic radiation 2 the spinshift direction of process the wave;ineither the spin direction is parallel (helicity=+1, or anti parallel of phase the the Mathematica (Wolfram handed, ↺)) around the advancing x1 direction. That is, if we 1 y estimate the helicity with aincalculation of phase shift in the process always oscillates parallel in to the a fixed direction in space. Light of such character is said to be linearly 2015a). We can determine the helicity based on a grasp our finger along the spin direction with right hand, then (helicity=-1, ↻) to the advancing x1 direction (momentum direction). Helicity of E2 can be changed 8 phase shift in the block respectively; phase calculation of the thumb directs 1the advancing direction of propagation wave, simply by the choice of JmC optical through element like In[83] andand In[84] codeshence . The half-wave shifts of the E1=E0*Jv after passing theaspolarizer the waveplate(HWH) is right-handed polarized wave regardless of of E2=Jm∙E1 after the wave plate respectively. For evaluation viewer . This is very helpful in confirming the helicity of the shifts phase –𝜋𝜋 ሾʹǡ ͷሿ eighth-wave plate(EWH) shift phase –𝜋𝜋/4 in helicity[2,7]. of the helicity of the polarized waves in progression, we have polarized wave on the end block of the platform as shown in Furthermore we may easily calculate the helicities of the wave train on the helicity [JvC, JmC]

module for any combination of optical elements in Mathematica, so that we can predict exactly http://dx.doi.org/10.5140/JASS.2015.32.2.63 66 helicity of the producing polarization modes. For more information for the helicity [JvC, JmC] module refer to citation (Yun 2015).

retnuoc a ni gnitator rotcev 2E⃗ ni stluser sihT .k/⍵ yticolev htiw enalp siht ot ralucidneprep noitcerid clockwise direction (right-handed, ↺) around the advancing x1 direction. That is, if we grasp our

ruo psarg ew fi ,si tahT .noitcerid 1x gnicnavda eht dnuora )↺ ,dednah-thgir( noitcerid esiwkcolc finger along the spin direction with right hand, then thumb directs the advancing direction of fo noitcerid gnicnavda eht stcerid bmuht nehtYong-Dae ,dnah thChoi gir het tiwal. noManipulating itcerid nips Polarization eht gnola Modes regnif Realized with Jones Vectors propagation wave, hence the wave is right-handed polarized wave regardless of viewer . This is very yrev si sihT . reweiv fo sseldrager evaw deziralop dednah-thgir si evaw eht ecneh ,evaw noitagaporp helpful in confirming the helicity of the polarized wave on the end block of the platform as shown in

Figs. 1 and 2. We can for the helicity ni nw ohscorrectly sa mroftaldetermine p eht fo kcolthe b dnhelicity e eht no eof vathe w deziralopolarization p eht fo yticilehmodes. eht gnimFor rifnomore c ni luinformation fpleh Figs. 2 and 3. We can correctly determine the helicity of the polarized wave in the E2 block by seeing

polarized wave in the E2 block by seeing the spin direction of [JvC, JmC] module refer to citation (Yun 2015). gniees yb kcolb 2E eht ni evaw deziralop eht fo yticileh eht enimreted yltcerroc nac eW .3 dna 2 .sgiF the spin direction the wave; either the spin directionis isparallel parallel(helicity=+1, (helicity=+1, ↺) or anti parallel theofwave; either the spin direction lell(helicity=-1, arap itna ro )↺),to 1+=the yticadvancing ileh( lellarapx1sdirection i noitcerid nips eht rehtie ;evaw eht fo noitcerid nips eht ) or anti parallel (helicity=-1, ↻) to the advancing x1 direction (momentum direction). Helicity of E2 can be changed 3. DYNAMIC POLARIZATION PLATFORM WITH (momentum direction). Helicity of E2 can be changed simply degnahc eb nac 2E fo yticileH .)noitcerid mutnemom( noitcerid 1x gnicnavda eht ot )↻ ,1-=yticileh( simply by the choice of JmC optical element like as element In[83] and In[84] . The half-wave plate(HWH) JONES VECTORS IN MATHEMATICA by the choice of JmC optical like codes as In[83] and n I s a e k i l t n e mele lacitpo CmJ fo eciohc eht yb ylpmis ) H W H ( e t a l p e v a w f l a h e h T . s e d o c ] 4 8 [ n I d n a ] 3 8 [ In[84] codes. The half-wave plate (HWH) shifts phase –π in shifts phase –𝜋𝜋 ሾʹǡ ͷሿ eighth-wave plate(EWH) shift phase –𝜋𝜋/4 in helicity[2,7]. helicity[2, 5] while eighth-wave plate (EWH) shift phase –π/4 We have provided a dynamic polarization platform with .]7,2[yticileh ni 4/𝜋𝜋– esahp tfihs )HWE(etalp evaw-hthgie ሿͷ ǡʹሾ 𝜋𝜋– esahp stfihs Furthermore weinmay easily calculate the helicitieswe of may the wave train on the helicity [JvC, JmC] helicity[2,7]. Furthermore easily calculate the Jones vectors (jdpmp) in the Graphics3D in Mathematica, ht no ni[JvC, art evJmC] aw ehmodule t fo seiticileh ewhich ht etalusimulates clac ylisae ythe am polarizing ew eromrehtmodes ruF ]Cwave mJ ,Ctrain vJ[ yon ticilthe eh ehelicity helicities of the dynamically while module for any combination of optical elements in Mathematica, so that we can predict exactly for any combination yltcaxe tcof ideoptical rp nac eelements w taht os ,in aciMathematica, tamehtaM ni stnemelepresenting lacitpo fo nothe itanihelicity bmoc ynaofroaf running eludom polarization mode. helicity of the so producing modes. For more information the helicity [JvC, that wepolarization can predict exactly helicity of the for producing TheJmC] platform manipulates three zones graphically using ]CmJ ,CvJ[ yticileh eht rof noitamrofni erom roF .sedom noitaziralop gnicudorp eht fo yticileh

module refer to citation (Yun 2015).

.)5102 nuY( noitatic ot refer eludom Fig. 1 1 .giF

Fig. 1. The starting platform of the dynamic polarization modes platform with Jones vectors srotcev senoJ htiw mroftalp sedom noitaziralop cimanyd eht fo mroftalp gnitrats ehT .1 .giF ⃗ 0 injected to the polarizer (LP) have not passed the polarizer yet. (jdpmp). Unpolarized EM waves E .tey reziralop eht dessap ton evah )PL( reziralop eht ot detcejni 0E⃗ sevaw ME deziralopnU .)pmpdj( 10

01

Fig. 1. The starting platform of the dynamic polarization modes platform with Jones vectors (jdpmp). Unpolarized EM waves →E 0 injected to the polarizer (LP) have not passed the polarizer yet. The simulation will run when you click the ▶ appearing while you spread the ⊕ of t1 panel of the platform.

Fig. 2.

Dynamic polarization modes platform (jdpmp). The picture shown is a snapshot of the right-handed circular polarized (RHCP) → propagation EM wave train. Unpolarized EM waves E 0 injected into the linear polarizer (LP) with 45° to the horizontal transmission axis come from the vacuum and pass through the quarter-wave plate (QWPV), which result in the right-handed circular polarized EM wave train. This picture shows the E0*LP∙QWPV/.α->1/4π process in the Jones vectors manipulation with a helicity of +1.

67

http://janss.kr

with its helicity has accordingly received attention from for scientists exploiting the protocol offundamental quantum key equations Eq. (6,7) in both numeric and conception in physics of the propagation ofhelicity the elec polarization modes, cosmology, Jones vector, wave plate, attention potential inThe physics and information technology (Tamm 1997, distribution (QKD) to guarantee unconditional security in cryptography communication. Weapplications have Keywords: provided amodern (Rubenhok et al. 2 & Stokes 2015, Yun & 2013). Choi 2013). ⍵2 ⃗ = − ⍵2 χ E ⃗ Mooleskamp k × ⃗E )൅ 2 E ሺʹሻ The fundamental conception in physics of the propagation of the electromagnetic wave polarization in matter is We have provided a dynamic polarization modes platform for simulating polarization modes with to in guarantee security in cryptography communication. We have provided as a the cardinal keyword polarization c we willc write the ⃗E distribution solution of(QKD) Eq. a theunconditional Therefore, newly understood in free-space quantum ̂ and ̂com ̂ ̂ 𝒦𝒦 ∙E =right-han 0, 𝒦𝒦 ∙E dynamic polarization platform for (2) presenting polarization modes a transverse wave, Mooleskamp & Stokes 2015, Yun &modes Choi 2013). 4 presents snapshots ofofthree kinds ofelectromagnetic polarization viewed in the ViewPoint (200, 0, 0) equations Eq. (6,7) in both numeric and graphic simulations. We have provided a Fig. dynamic polarization modes platform for simulating polarization modes with Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity newly understood as the cardinal keyword in free-space quantum communication technology and cosmology in calculations,modes corresponding to the physical arrangement of optical of elements, inRecently, a mechanism dynamic polarization platformvaries forJones presenting the polarization of a transverse electromagnetic wave, astrophysics. Interactive visualization the propagation o names. which indicates that propagation process withmatrices quation for ⃗E,(Pedrotti lution below & Predrotti 1987, Fowles 1975, converting the state ofthe polarization through the arrangement of opticalWe elements, using Jones vectors calculations ⃗S =modes ⃗ × the ⃗H ⃗ withvi have a dynamic modes platform for simulating polarization 1.polarization INTRODUCTION E against thecorresponding propagation direction for provided confirmation of theof transverse behavior and progression ofE the J. Astron. Space Sci. 32(2), 63-71 (2015) ̂ ̂ ̂ ̂ ̂ ̂ ̂ Jones matrices calculations, to the physical arrangement optical elements, in a 𝒦𝒦 ∙ E = 0, 𝒦𝒦 ∙ E = 0, 𝒦𝒦 × = ⍵ B ( Interactive visualization of the propagation mechanism ofusing polarized in a medium convertingastrophysics. the state of polarization through the arrangement ofenvironment optical elements, Joneselectromagnetism vectors Mathematica computing (Mathematica 2015). with its calculations helicity has accordingly received attention scientistsap attention forfrom potential in Methematica. platform thea mechanism of production and propagation of the solution ofsimulates Eq. (2) in sceptibility 𝜒𝜒. Therefore, we willThe write the ⃗E graphically Jones matrices calculations, corresponding to the physical arrangement of optical elements, in a ̂and ̂the ̂ are both comp Poynting vector. (Mathematica It received shows the orthogonal transverseexploiting of the propagating process the waves𝒦𝒦 Polarization coherent characteristic electromagnetic (EM) where ,ofE , and B Mathematica computing environment 2015). with its helicity has accordingly from scientists protocol key ⃗Sis=a(QKD) ⃗of ⃗H ⃗ toofguarantee 1. INTRODUCTION E in Methematica. The platform graphically simulatesattention the mechanism of production andthe propagation of×quantum the distribution unconditional security in cryptograp ⃗⃗ .r⃗−⍵t) polarized waves in a medium while satisfying Maxwell's equations. Mooleskamp & Stokes 201 i(𝒦𝒦 plane E2 ) ewave solution below (Pedrotti ሺ͵ሻ & Predrotti 1987, Fowles 1975, Mathematica computing 2015). ⃗ ==E ⃗and ⃗H ⃗magnetic ⃗ ×E ⃗ ve ⃗H ⃗ ×satisfying the Piecewisepolarized function of Mathematica depending on the the of(Mathematica the graphic version confirms flow ofunconditional Poynting vector. isvisualization the environment visualization of thethe graphic version ofofwhere wave, both electric field vector field ⃗intensity vector B o is a ⃗ magnetic distribution (QKD) while to the guarantee securityThis in cryptography communication. We have provided aS waves in a medium satisfying Maxwell's equations. ̂ ̂ ̂ ⃗ ⃗ dynamic polarization platform for presenting the polarization mode Platform for manipulating polarization mo Polarization is a coherent characteristic the (EM)complex wave in vectors a medium. whereof𝒦𝒦 , Eelectromagnetic , and are both andInEa, plane B and H are real 2.MATHEMATICA SIMULATION FOR THEB CONVERTING POLARIZATION We have provided a dy ⃗ polarizer and wave plate, with the Manipulate function Maxwell’s vector equations Eqs. (6) and (7). The traces of + ε2 ei∆ε ) ei(k.r⃗−⍵t) dynamic polarization platform for vector presenting the polarization modes a POLARIZATION transverse electromagnetic always oscillates parallel to awave, fixed direction in ofoptical such ⃗ state ⃗⃗ polarization polarization modes withLight the normalize 2.MATHEMATICA SIMULATION FOR THE CONVERTING H are dynamically satisfying Maxwell’s equations Eq. (6) and (7). ofThe traces of both E converting the of through thespace. arrangement of ⃗where ⃗ and ⃗H ⃗ are Keywords: polarizationTo modes, cosmology, Jones vector, wave plate, helicity field vector of Mathematica. present the transverse characteristic both and dynamically shown in the wave, both the electric E and magnetic field vector B of the electromagnetic radiation is a magnetic intensity vector in polarization the matter. We examine the p Jones matrices calculation MODES ⃗⃗ .r⃗−⍵t) i(𝒦𝒦 MATHEMATICA vectors in 2.MATHEMATICA SIMULATION FOR THE CONVERTING POLARIZATION ε⃗ 2 E2 ) e Keywords: converting polarization ሺ͵ሻthemodes, state ofcosmology, polarizationJones through the arrangement of optical elements, using Jones vectors calculations A= (ε1 E1 + i(k vector, wave plate, helicity below.graphically The Mathematica input code i inthe Methematica. The platform simulates the mechanis property satisfying Eq. (1) together with Eqs. (6) and (7), we modes at the edge of jdpmp platform. shown in the polarization modes at the edge of the jdpmp platform. ) e ∙ ⃗r−⍵t) 1 MODES always oscillates parallel to a fixed direction in space. of such character is said to Mathematica becalculations linearly computing polarization modesLight with the normalized Jones vectors in the Math + iC en 2.1 for Electromagnetic waves solids ⃗ .r⃗−⍵t) function in Methematica. The platform graphically the simulates thein mechanism of production andfaster propagation of the and stop Arrow in Mathematica drawing The platform will run or slower and MODES = E0 (ε1 +used ε2 ei∆εthe polarized waves in a medium while satisfying Maxwell's equations. ) ei(k The platform will run faster or slower and stop and restart again by clicking the pop-up menu of 1. INTRODUCTION 2.1 Electromagnetic in Ey, solids The input code is shown vector array: Table[ Arrow [ { {x1, 0,waves 0}, {x1, Ez}satisfying } ] ] in the restartbelow. again by clicking the pop-up menu ofMathematica thebelow: t1 panel2code for #1 1Mathematica polarized waves Platform in a medium Maxwell's equations. for manipulating polarization modes realized with Choi1from , Bogyeong Kim , Hee-Joong Yun3 The while propagating process of electromagnetic waves inYong-Dae solids is different that in theJones vacuum, since ⃗ ∙ ⃗r−⍵t) A orthogonal 1. INTRODUCTION i(k 2.1 Electromagnetic waves in solids the t1 panel for more analytical observations while the polarization mode is running. During the {x1, Ey, Ez} coordinate system. In addition, the more analytical observations while the polarization mode = ( ) e esented as the withcharacteristic a complex isThe a1941) coherent the electromagnetic waveis in a medium. a plane propagating process ofofelectromagnetic waves(EM) in solids different fromInthat in the vacuum, since 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= E0 =2.MATHEMATICA B Jones + Polarization iCvector (Jones S #1 ⃗ and B ⃗ calculated Keywords: polarization modes, cosmology, Jonesto vector, wave plate, h same was used for the magnetic field vector is running. During the platform you will&change E of the electromagnetic wavesMathematica interact withcode the electrons in astop, solid (Pedrotti Predrotti 1987, platform stop, you will change to another mode only if you click a panel, then the changed mode will Polarization is a coherent characteristic of the electromagnetic (EM) wave in a medium. In a plane The propagating process of electromagnetic waves in solids is different from that in the vacuum, since 1 MATHEMATICA vectors in In[12]:= LP = {Cos[α], Sin[𝛽𝛽]}; Department of Microbial and Nanomaterials, Mokwon Universi ⃗kelectromagnetic ⃗electric ⃗vector ⃗k n̂of ⃗ the Keywords: modes, cosmology, Jones vector, plate, helicity n an inhomogeneous plane. TheE wave = from the the relation =1/u ×polarization (6) also in solid. another mode only if&you click1987, a panel, then the changed and B of the waves interact with in a solid (Pedrotti Predrotti wave, both field vector E andEq. magnetic field vector B of electrons the wave electromagnetic radiation MODES = 1/√2{1, 0}+ 1/√2{0,1}; In[11]:= Fowles 1975). In particular, although the E0 electromagnetic waves areKorea an harmonic plane wave, the Daejeon 302-729, be presented automatically. AtB time you B the presenting mode orelectrons print out mode 2 & Predrotti 1987, ⃗ and ⃗ that ⃗ (blue) ⃗ can Graphics3D can draw array of the mode will beelectromagnetic presented At that time you E and of the electromagnetic waves interactautomatically. with the inthe a solid (Pedrotti wave, boththe thevector electric field vector E and magnetic field vector of save the radiation In[15]:= LW = {{Cos β, Cosβ Sinβ}, { Co ctor may be the K Jones vector (Jones 1941) with a complex ⃗⃗⃗⃗kparallel 1975). In particular, although electromagnetic waves are an harmonic plane wave, the always oscillates to direction in space. the Light of such character is In[12]:= said to be linearly ̂ =Fowles vectors ( ε1presented , ε2 , n ̂ ). as Here + i 𝛼𝛼 is aa fixed complex 2 LP = {Cos[α], Sin[𝛽𝛽]}; 1. INTRODUCTION fields may pull or push the electrons in the orbital of a solid, which is responsible for inducing dipole Department of Astronomy and Space Science, Chungnam Natio 2.1 Electromagnetic wa (red) fields of every point in the block as real polarization can save the presenting mode or print out the mode status. 1 2 3† status. While the direction platform is you the helicity observing spin direction In[16]:= = {{1, 0}, {0, I}}; QWH Yong-Dae Choi ,Fowles Bogyeong Kim ,confirm Hee-Joong 1975). Incan although theby waves are QWV anofharmonic plane wave, the always oscillates parallel to a fixed in running, space. Light ofparticular, such character isYun said toelectromagnetic be305-764, linearlythe Daejeon Korea 2 2 1. INTRODUCTION ⃗k = may push the electrons of a solid,While whichIn[15]:= is responsible for inducing dipole oscillatingindex. in an inhomogeneous The wave vector n̂ the orbital modes propagating inpull the advancing direction, in our the platform is running, you {can = {{Cos β, Cosβ Sinβ}, Cosβconfirm Sinβ, Sin the β }};helicity x}refraction For Eq. (3) tofields beplane. a solution ofor Eq. (2) 1 k in ⃗ and magnetization ⃗M ⃗⃗ in the LW is⃗ a the coherent characteristic of0},the{0,electromagnet propagating process o moment P solid. If3 Polarization we assume that medium isHW notThe magnetic In[17]: = responsible =a{{1, -1}}; ⃗Institute producing polarization modes. Presently the definition of polarizati the field vectors along thefields orange trace of the helical propagation of the E and B vector fields at the may pull or push the electrons in the orbital of a solid, which is forInformation, inducing dipole case, advancing in the x direction. This simulation scheme by observing the spin direction of the field vectors along Korean of Science and Technology 1 1 ⍵ ⍵ ⃗ ⃗ ⃗⃗ In[16]:= QWV = {{1, 0}, {0, I}}; QWH = {{1, 0},{0, -I}}}; Polarization characteristic ofIfNanomaterials, the wave in In Korea a plane ⃗⃗⃗⃗k𝛼𝛼ൌ moment PK magnetization M of inMicrobial the solid.and weelectromagnetic assume that the(EM) medium is nota amedium. magnetic ̂ rthogonal ( ε1 , εwe ̂get ). the Here =and + i𝜅𝜅,𝛼𝛼1isDepartment a complex . Then relations: isa coherent hould be ǡunit 𝒦𝒦ൌvectors 2, n Daejeon 305-806, Mokwon University, ⃗ ⃗ ⃗ cdiffers basically from that of an c animation representing the orange trace of the helical propagation of the and E and B of the electroma wave,(Fowles both the1975, electric field vector field vect In[18]: = confusing EWV 0}, {0, Exp[ I 1/4𝜋𝜋 andThe J=0,helicities the Helmholtz wave equation a solid Jackson 1975) is ) = {{1, nomenclature upgraded, which can be to magnetic students of physics helicity panel in the case of E1=E0*LP( edge Daejeon of the material platform. ⃗ on ⃗⃗⃗inare 302-729, Korea moment P andthemagnetization incorrect the ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣheejy In[17]:= ⃗HWM= {{1, 0}, solid. {0, -1}};If we assume that the medium is not a magnetic ⃗ in the envelope of the polarization propagation of E field vector fields at the edge of the platform. The helicities on wave, both the electric field vector and magnetic field1975, vector B of the electromagnetic radiation material and J=0, the Helmholtz wave equation a solid (Fowles Jackson 1975) is complex refraction index. For Eq. (3) to be a solution of Eq. (2) Received Revised Accepted 2P In[19]: EWH = {{1, 1975). 0}, {0, -1}}; eeds that are different along the direction in the medium.2 ⃗ ⃗ Fowles In Light particu always oscillates parallel to a =fixed direction incircularly space. ∂similar 1 ∂2 E there are for the right polario ⃗Space ⃗∇Science, ⃗Ehelicity ×J=0, (The × )൅ = −u {0, in customary a solid in of 1/4𝜋𝜋 optical nomenclature case }}; EWH ሺͳሻ ∇ of andand Chungnam University, manipulation theAstronomy jdpmp platform. of other devices are0}, 0arrangements only using the Animate function ofDepartment graphicon⍵tools (Harrison the are of E1=E0*LP(α) In[18]: =National {{1, 0}, Exp[ I the = {{1, {0, Exp[is- I 1/4𝜋𝜋 }}; 2EWV material thehelicities Helmholtz wave equation (Fowles 1975, Jackson 1975) ∂t2correct c2=∂tpanel ⍵ 2 2 ⃗ ⃗ always parallel to𝜅𝜅,a fixed 1 ∂ Korea E direction∂ in P space. Light of such character is said to be linearly 305-764, we getoscillates the relations: 𝛼𝛼ൌ monic wave, it should be ǡ 𝒦𝒦ൌ c . Then = E0*LP; l2=LW ; push ⃗ Daejeon ⃗Ein modes. Presently the definition ofl1polarization has been modifi × Also, ( ⃗∇ × manipulation polarization 1975, on ሺͳሻ fields may pull the c)൅ 0 2producing Mooleskamp & Stokes 2015). this work, we the jdpmp platform. TheIn[20]:= helicities of other ⃗∇ 2 ∂t 2 = −u e difference ∆𝜀𝜀 2015; between the two components of the E 1975), right-circularly (Pedrotti 1987), rig 1 or con 20}, ⃗⃗ .r⃗−⍵t)& Predrotti = EWH =Jackson {{1, calculated promptly theE helicity∂t[JvC, JmC] In[19]: module in will Mathematica. ⃗ {0, -1}};∂2 P ⃗ i(𝒦𝒦 1a∂form E ̂ ⃗ ⃗ where P = 𝜒𝜒𝜀𝜀 . If we suppose the EM wave be of solution such as E e , we can 3 ⃗ ⃗ ⃗ 0 The fundamental conception in physics of the propagation of the elect 0 × ( ∇ × E )൅ = −u ሺͳሻ ∇ In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 𝜋𝜋 0 simulated polarization process withKorean vectorInstitute fields of satisfying of optical on Science andnomenclature Technology Information, 2devices are 1 arrangements ∂t2 calculated c2 ⃗⃗∂tcan upgraded, which be confusing to studentspromptly of physics and⃗ researchers. Fo propagation that are After different direction the medium. ⃗ −⍵t) .r moment andcircularly magnet ̂0and ⃗ = the right-handed (Born &a Wolf 1999, Georgi P 1982) tals (Quartz,speeds Calcite, etc.). the along wave has traveled ain In[20]:= l11993), =asE0*LP; l2=LW ; can where 𝜒𝜒𝜀𝜀0⃗E . If weAs suppose EM wave willthan be ajdpmp, form ofthere solution E ei(𝒦𝒦 , we Daejeon 305-806, Korea far equations as the we know, other is nosuch instructive platform that simulates transverse wave equation andPMaxwell’s vector at every the helicity [JvC, JmC] module in Mathematica. newly understood as the cardinal keyword in free-space quantum com ⃗ rewrite Eq. (1) using⃗a wave⃗ vector k In[23]:= E0*LP∙QWV/.𝛼𝛼 ->1/4 ⃗⃗ .r ⃗ −⍵t) 𝜋𝜋 ǣ൅ͺʹǦͶʹǦͺʹͳǦ͹Ͷͻʹǡ ǣ൅ͺʹǦͶʹǦͺʹͳǦͺͺͻͳǡǦǣ[email protected] ̂0 ei(𝒦𝒦 where P= 𝜒𝜒𝜀𝜀 If we suppose the EMnomenclature wave𝜋𝜋will be for a form of solution suchpolarization: as E , we can there customary thejdpmp, right circularly right circular 0E. are ⍵ In[ 22]:= E0*LP/.𝛼𝛼 ->1/4 pointExduring the Eprocess. Similar numerical has Assimilar far as we know, other than there is noand types ⃗ method polarization and right-handed polarization are different of Helm pola material J=0, the umulative difference ∆𝜀𝜀 between the(1) two components of thek ⃗E = 𝑑𝑑(n2-nphase wave and wave when the Received Revised Accepted 1) between y Eq. EM wave satisfying Maxwell's equation in the vectorial version of polarization modes (Harrison 2015, rewrite using a wave vector astrophysics. Interactive visualization of the propagation mechanism o 𝑐𝑐 In[24]:= E0* LP∙QWH/.𝛼𝛼 ->1/4 𝜋𝜋 been applied once to solve the Lagrange’s equation rewrite in noninstructive platform that simulates a transverse EM wave Jackson right-circularly (Pedrotti & Predrotti 1987), right-hand circularly (R ⃗ Eq. 1975, (1) using a In[23]:= wave1975), vector k E0*LP∙QWV/.𝛼𝛼 ->1/4 𝜋𝜋 visualization names. Recently, the of polarization n uniaxial crystals (Quartz, Calcite, etc.). After the wave has traveled a 2005, Tamm Mooleskamp & Stokes 1997). The complete version ofitsjdpmp.nb program including helicity accordingly received attentionpropagating from scientists ⃗in× ⃗ is2014). 3 with ∇ on. If the ∆𝜀𝜀=0inertial while the amplitude E real, the frame (Kim of & Yun satisfying Maxwell's equation inhas the vectorial version In[25]:= E0* LP∙HWof /.𝛼𝛼 ->1/4 𝜋𝜋 1993), and right-handed (Born & Wolf 1999, Georgi 1982) circularly polarization. Righ ->1/4 𝜋𝜋wave The physics the propagation of LP∙QWH/.𝛼𝛼 the electromagnetic polarization matter is physics ⍵ will attention for potential applications inin modern informa The simulation run when you theclicking ► appearing while t1 of panel ofIn[24]:= the E0* The platform starts by thefundamental button ▶ conception nspread pop the upin3⊕ polarization modes (Harrison 2015; & Stokes distribution (QKD) to Mooleskamp guarantee unconditional security and in cryptograph ence is as ∆𝜀𝜀 = vector 𝑑𝑑(n -n between Eclick and Ey is wave when you the 2{A, 1) B}, x wave In[26]:= E0* LP∙EWV /.𝛼𝛼 ->1/4 d such Jones otherwise the amplitude 𝑐𝑐 ⃗ . 𝜋𝜋ȀȀ where ⃗P= 𝜒𝜒𝜀𝜀0E If we su 13 3 of the t1 panel on the platform innewly Fig. 1. We canaschoose an keyword 2005; Tamm 1997). The complete version of jdpmp.nb In[25]:= E0* LP∙HW /.𝛼𝛼 ->1/4 𝜋𝜋 technology polarization and right-handed polarization are different types of polarization, although w understood the cardinal in free-space quantum communication and cosmology in dynamic for presenting Mooleskamp & polarization Stokes 2015,platform Yun & Choi 2013). the polarization mode platform. In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ ⃗ rotate Jones as {A, BേiC}. Specifically, gtically the ⃗kpolarized direction. If vector the ∆𝜀𝜀=0 while the amplitude of E is real, the incoming angle to the polarizer and the polarizer to program including helicity [JvC, JmC] module is available 2 2 rewrite Eq. (1) has usingdrawn a wav In[26]:= mechanism E0* ->1/4 𝜋𝜋ȀȀ visualization of the propagation of /.𝛼𝛼 polarized electromagnetism inthrough a medium Recently, the LP∙EWV visualization ofstate polarization propagating matter ⃗k × ( ⃗k × ⃗E )൅ ⍵ ⃗E = − ⍵astrophysics. ⃗ Interactive converting the of polarization theinarrangement of χE polarization ሺʹሻ names. We have provided a dynamic polarization modes platform foroptical simu compose an arrangement for the from the site (Yun 2015). If your PC has not installed In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N c2 of devices c2 polarized Jones vector as vector {1, i}. {A, B}, otherwise the amplitude is nearly polarized such assuch Jones Fig.2 In[27]:= E0* LP∙EWH /.𝛼𝛼 ->1/4 𝜋𝜋ȀȀ with its helicity has accordingly received attention from scientists exploiting the protocol of quantum key for potentialyou applications in modern physics and information technology (Ta in Methematica. The platform graphically simulates the mechanis mode indent shown as Fig. 2. While the program is running, attention Mathematica, can run the jdpmp.cdf instead on the to the physical arrangem ⃗E, which indicates that the propagation process varies with Jones matrices calculations, corresponding In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //N It then becomes aofvector equation ent crystal the thickness which has beenfor todistribution le for the elliptically Jones vector asadjusted {A,orBേiC}. Specifically, we polarized will change the polarizer wave plate, with theto changed Wolfram CDF Player (Wolfram 2015). (QKD) guarantee unconditional security in cryptography communication. We have provided a In[28]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->1/3 𝜋𝜋} //N polarized in a medium while satisfying Maxwell's equations. Mooleskamp & Stokes 2015, Yun & waves Choi 2013). Mathematica computing environment (Mathematica Fig. 2. Dynamic polarization modes continuously. platform (jdpmp).Even The picture shown is a⃗ snapshot of the rightMathematica returns2015). calculations; animation the write animation is of Eq. the (2) in a the component of electric susceptibility 𝜒𝜒.operating weifpolarization will the E solution ordinary and extraordinary rays at asthe{1, enisthe a circularly polarized Jonesrunning vector such i}.Therefore, dynamic platform for presenting polarization modes of a transverse In[29]:= l1.l2/.{𝛼𝛼 ->1/4 𝜋𝜋, 𝛽𝛽 ->3/4 𝜋𝜋} //Nelectromagnetic wave, We have provided a dynamic polarization modes platform for simulating polarization m stopped, can change the configuration of the platform Out[22]:= {1/2, 1/2} ⃗ 0 injected handed polarizedwe (RHCP) propagation EM wave train. Unpolarized EM waves E solid circular as an crystal inhomogeneous plane wave solution below (Pedrotti & Predrotti 1987, Fowles 1975, of optical converting the state of polarization through the arrangement elements, using Jones vectors calculations thin birefringent the thickness of which has been adjusted to Mathematica returns calculations; Keywords: polarization modes, cosmology, Jones vector, wave plate, he SUMMARY and observe the polarizing mode of the changed state. Jones4.matrices calculations, corresponding SIMULATION to the Out[23]:= physical arrangement of CONVER optical elem {1/2, I/2 2.MATHEMATICA FOR THE into4Jackson the linear polarizer (LP) withsnapshots 45° to the horizontal transmission axis come from the vacuum and the in Methematica. The platform graphically simulates mechanism of production and propagation of the 1975) Fig. 3 shows of three kinds of polarization Out[22]:= {1/2, 1/2} erence between the ordinary and extraordinary rays at the operating Mathematica computing environment (Mathematica 2015). modes. Figs. plate 3a and 3b shows circularly polarization We have a dynamic polarization mode platform, pass through the quarter-wave (QWPV), which a result in the right-handed polarized EM provided MODES polarized waves in a mediumcircular while satisfying Maxwell's equations. Out[23]:= {1/2, I/2 (Fig. 2, Fig. 3(a)) ⃗⃗ .r⃗−⍵t) i(𝒦𝒦 ̂(r, t) = (ε ) E E + ε E ሺ͵ሻ e 1 1 2 2 1. INTRODUCTION }

}

mode, Figs. 3c and 3d linearly polarization, Figs. 3e and jdpmp, for simulating and producing polarization modes 4 wave train. This picture shows the E0*LP∙QWPV/.𝛼𝛼Ǧ>1/4𝜋𝜋 process in the Jones vectors manipulation Electromagnetic waves in solids ⃗the 3f elliptically polarization according arrangement with the Jones2.1 calculations, corresponding to the physical i∆ε toi(k .r⃗−⍵t) = E0 (ε1 + ε2 e ) e Polarization is FOR a coherent characteristic of the electromagneti 2.MATHEMATICA SIMULATION THE CONVERTING POLARIZ 7 Keywords: polarization modes, cosmology, Jones vector, wave plate, helicity of optical devices. If we utilize linear polarizer JmLinear arrangement of optical elements, in Mathematica. The with a helicity of +1. The propagating process of electromagnetic waves in solids is different ⃗ ⃗ vector [β] instead of a wave plate, passes a linearly MODES platform animates the polarization process the E wave, both the electric field of vector and magnetic field vecto = ( Athen either ) ei(kit∙ ⃗r−⍵t) B + iC ⃗ ⃗ polarized wave or block in the angle β. While making right field togetherEwith the Arrow vectors, so thatwith thethe electrons in a s and B using of the electromagnetic waves interact always oscillates parallel to a fixed direction in space. Light o 1. INTRODUCTION 2.1 Electromagnetic waves in solids angles between two polarizers no wave passes through transverse EM wave advance in the propagation direction, 3. DYNAMIC POLARIZATION PLATFORM WITH JONES VECTORS Fowles 1975). In particular, although the electromagnetic waves are the polarizer plate as shown in Figs. 3g and 3h. Thus, satisfying Maxwell's wave equation at every point on the 1 ̂ As shown above, E field vector may be presented as the Jones vector (Jones characteristic 1941) a of complex Polarization is a coherent the electromagnetic (EM) wavewaves in a medium. plane from that in the vac Thewith propagating process of electromagnetic in solidsInis adifferent IN MATHEMATICA fields pull or pushConsequently, the electrons in the of a solid, which is controlling of phase of polarized light waves is available for advancing axes of may the platform. theorbital vectors ⃗ ⃗ ⃗ ⃗ ⃗ vector amplitude {A, Bencoding േ iC} oscillating in an inhomogeneous The kmagnetic = n̂ field the phase in the quantum keyboth distribution (inkelectromagnetic that order) form right-hand E vector and B , of the waves with theorthogonal electrons in aset. solid (Pedrotti & Pre wave, the plane. electric(QKD) fieldwave vector E , and vector B ofainteract the electromagnetic radiation ⃗ and magnetization ⃗M ⃗⃗ in the solid. If we assume that We have provided a dynamic polarization platform with Jones vectors (jdpmp) in the Graphics3D in moment P of the unconditional security in quantum communication jdpmp can be accomplished graphically in Graphics3D of Fowles In particular, the electromagnetic always parallel to⃗⃗⃗⃗ka fixed direction in space. Light ofalthough such character is said to be waves linearlyare an harmonic plane ̂= and form awhich real mutually orthogonal unit vectors ( ε1 ,oscillates ε2 , n̂ ).while Here K +the i 𝛼𝛼helicity is 1975). a complex Mathematica, simulates the polarizing presenting of a technology (Rubenhok et al. modes 2013). dynamically Mathematica based and on the vector calculations material J=0, numerical the Helmholtz wave equation in a solid (Fowles 1975 fields may pull or push the electrons in the orbital of a solid, which is responsible for indu Fig. 4=npresents snapshots of threethree kinds of polarization in Mathematica. The platform can be manipulated wave polarization vector and Nmode. + i κThe complex refraction index. Eq. (3) to using be a solution of Eq. (2)1 running platform manipulates zonesFor graphically the Piecewise ⃗ ⃗ ∂2 P 1 ∂2 E ⃗ ⃗ ⃗ × ( ∇ × E )൅ = −u0 ∂t2 ∇ modes viewed in the ViewPoint (200, 0, 0) against the dynamically and interactively by advancing the polarized 2 c2 ∂t ⍵ ⍵ ⃗ ⃗ ⃗⃗ moment P and magnetization M in the solid. If we assume that the medium is not . Then the relations: 𝛼𝛼ൌ c 𝜅𝜅, as a homogeneous plane harmonicon wave, polarizer it should and be ǡ wave 𝒦𝒦ൌ plate, function of Mathematica depending with we the get Manipulate function c the transverse propagation directionthe for confirmation of mode while clicking the panels of the polarizer and wave ⃗ . If we suppose the EM wave will be a form of solution where ⃗P= 𝜒𝜒𝜀𝜀0E and progression of the Poynting vector. It shows plate using Manipulate function thatJackson 1975) is ⍵ behavior material and J=0,the the Helmholtz wave equationininMathematica, a solid (Fowles so 1975, of Mathematica. To present transverse characteristic property satisfying Eq.direction (1) together with Eq. ൌ 𝑛𝑛, which result inthepropagation speeds that are different along the in the medium. c the orthogonal transverse of the propagating process of the program will continuously the2 changed mode. ⃗ rewrite Eq. (1) using1simulate a∂wave vector 2E ⃗ ⃗k ∂ P (6) and Eq. (7), we used the Arrow function in Mathematica for drawing the vector array: ⃗ × is ( ⃗∇running × ⃗E )൅ 2the = −u0 ∂tof polarized ሺͳሻ ∇ confirms phase the flow of Poynting vector.the This Whileofthe ⃗ 2 the Therefore, the therewaves will beand a cumulative difference ∆𝜀𝜀 between twoiscomponents the program E c ∂t2 helicity Table[ Arrow [ { {x1, 0, 0}, {x1, Ey, Ez} } ] ] in the orthogonal {x1, Ey, Ez} coordinate system.

In ⃗⃗ .r⃗−⍵ ̂0 ei(𝒦𝒦 ⃗ . aIf we suppose the EM wave will be a form of solution field vector as they emerge in uniaxial crystals (Quartz, Calcite, etc.). After the wave wherehas⃗P=traveled 𝜒𝜒𝜀𝜀0E 3 such as E ⃗ calculated from the68 addition, the same was used for the magnetic field vector B relation ⃗B=1/𝑢𝑢 http://dx.doi.org/10.5140/JASS.2015.32.2.63 ⍵ ⃗ distance d, the phase difference is ∆𝜀𝜀 = 𝑑𝑑(n2-n1) between Ex wave and E when the a wave vector k y waveEq. rewrite (1) using 𝑐𝑐 ⃗k × ⃗E of Eq. (6) also in solid. Graphics3D can draw the vector array of the E ⃗ (blue) and ⃗B (red) ⃗ is real, the radiation is propagating along the ⃗k direction. If the ∆𝜀𝜀=0 while the amplitude of E

Yong-Dae Choi et al. Manipulating Polarization Modes Realized with Jones Vectors

(a) Producing right-handed circularly polarized

(b) Producing left-handed circularly polarized EM

EM wave in the E0 ∗ LP · QW PV arrangement

wave in the E0 ∗ LP · QW PH arrangement

(c) Producing linearly polarized EM wave in the

(d) Producing linearly polarized EM wave in the

E0 ∗ LP · HW PV arrangement

E0 ∗ LP · HW PH arrangement

(e) Producing right-handed elliptically polarized

(f) Producing left-handed elliptically polarized EM

EM wave in the E0 ∗ LP · EW PV arrangement

wave in the E0 ∗ LP · EW PH arrangement

(g) Producing linearly polarized EM wave by the

(h) No producing any polarized EM wave at the

arrangement of two linear polarizers

right angle between two polarizers

Fig. 3. Producing different polarized modes by the different physical arrangements of polarizer and wave plate: (a) RHCP of helicity +1, (b) LHCP of helicity -1, (c) RHEP polarized EM wave, (d) LHEP polarized EM wave, (e) Linear polarized wave, (f) No wave.

1 69

http://janss.kr

J. Astron. Space Sci. 32(2), 63-71 (2015)

(a) LP by the E0*LP.HWV(α − > π /4) array

(b) By the E0*LP.LW(α − > π /4, β − > 3/4π )array

(c) RHCP by the E0*LP.QWV(α − > π /4) array

(d) RHEP by the E0*LP.EWV(α − > π /4) array

→ modes at the ViewPoint (200, 0, 0); (a) LP of Fig. 3c, (b) No wave of Fig.3h, (c) RHCP of Fig. 3a, (d) RHEP of Fig. 3e. It presents E Fig. 4. Snapshots of the polarized → fields by the blue arrows and H fields by the red arrows.

mode is displayed on the panel. The module helicity [JvC, JmC] will return the helicity of the process on Notebook of Mathematica when you're typing in a polarizer and a wave plate. The module is helpful for students or researchers to inspect the phase difference or helicity of a polarized mode for various kinds of physics arrangements. We expect the platform jdpmp will be a useful starting platform for physics students and researchers to explore polarizations in science.

program funded by the Korean Ministry of Education, Science and Technology through, the National Research Foundation of Korea and the Korea Lottery Commission grants.

REFERENCES

REFERENCES Aad G, Abbott B, Abdallah J, Khalek SA, Abdelalim AA et al., forAbdallah the Standard Model Boson Aad G, Search Abbott B, J, Khalek SA,Higgs Abdelalim AAinetthe al., Search for the S -1 Diphoton Decay Channel with 4.9 fd of pp Collision Higgs Boson in the Diphoton Decay Channel with 4.9 fd-1 of pp C Data at √𝑠𝑠 =7 TeV TeV with ATLAS, Rev. Lett. 108, Rev. 111803Lett. 108, 11 =7 with Phys. ATLAS, Phys. http://dx.doi.org/10.1103/PhysRevLett.108.111803 (2012). http://dx.doi.org/10.1103/PhysRevLett.108.111803

ACKNOWLEDGEMENT This research was partially supported by the ReSEAT

http://dx.doi.org/10.5140/JASS.2015.32.2.63

70

Ade PAR, Aikin RW, Bento SJ, Bischoff CA, Bock JJ et al., Detectio Polarization at Degree Angular Scales by BICEP, Phys. Rev. Let (2014). http://dx.doi.org/10.1103/PhysRevLett.112.241101

Born M, Wolf E, Principle of Optics 7th ed. (Cambridge University Press, Cam 795.

Yong-Dae Choi et al. Manipulating Polarization Modes Realized with Jones Vectors

Ade PAR, Aikin RW, Bento SJ, Bischoff CA, Bock JJ et al., Detection of B-Model Polarization at Degree Angular Scales by BICEP, Phys. Rev. Lett. 112, 241101 (2014). http:// dx.doi.org/10.1103/PhysRevLett.112.241101 Born M, Wolf E, Principle of Optics 7th ed. (Cambridge University Press, Cambridge, 1999), 795. Boyle LA, Steinhardt PJ, Turok N, Inflationary predictions for scalar and tensor fluctuations reconsidered, Phys. Rev. Lett. 96, 111301 (2006). http://dx.doi.org/10.1103/ PhysRevLett.96.111301 Harrison DM, Optics Applets [Internet], cited 2015 Feb 15, Available from: http://www.cabrillo.edu/~jmccullough/ Applets/Flash/Optics/CircPol.swf. Calvin W, NASA Technology Views Birth of the Universe [Internet], cited 2014 Mar 17, Available from: http://jpl. nasa.gov/news/news.php?release=2014-082. CfA, First Direct evidence of cosmic inflation [Internet], cited 2014 Mar 17, Available from: http://cfa.harvard.edu. Elser D, Bartley T, Heim B, Wittmann Ch, Sych D et al., Feasibility of free space quantum key distribution with coherent polarization states, New J. Phys. 11, 045014 (2009). http://dx.doi.org/10.1088/1367-2630/11/4/045014 Fowles GR, Introduction to Modern Optics (Holt, Reinhart and Winston, 1975). Georgi H, 1982, Lie Algebra in Particle Physics (The Benjamin INC, 1982), 162. Goldhaver M, Grodzins L, Sunyar AW, Helicity of Neutrinos, Phys. Rev. Lett. 109, 1015 (1958). http://dx.doi.org/10.1103/ PhysRev.109.1015 Jackson JD, Classical Electrodynamics, (John Wiley & Sons, 1975), 274. Jones RC, A New Calculus for the Treatment of Optical Systems, J. Opt. Soc. Am. 31, 488-493 (1941). http://dx.doi. org/ 10.1364/JOSA.31.000488 Kim B, Yun HJ, Pedagogical Mathematica Platform Visualizing the Coriolis Effects in 3-Cell Atmospheric Circulation Model, J. Astron. Space Sci. 31, 91-99 (2014). http://dx.doi. org/ 10.5140/JASS.2014.31.1.91 Leitch EM, Kovac JM, Pryke C, Carlstrom JE, Halverson NW et al., Measurement of Polarization with the Degree Angular Scale Interferometer, Nature 420, 673-771 (2002). http:// dx.doi.org/10.1038/nature01271 Masayoshi T, Cutting-edge terahertz technology, Nat. Photonics 1, 97 (2007). http://dx.doi.org/10.1038/nphoton.2007.3 Mathematica, Wolfram Mathematica [Internet], cited 2015 Feb 15, Available from: http://www.wolfram.com/ mathematica Mooleskamp FE, Stokes KL, Polarization of an Optical Wave through Polarizers and Wave Plates [Internet], cited 2015 Feb 15, Available from: http://demonstrations.wolfram.

com/PolarizationOfAnOptiionsaturlWaveThroughPolariz ersAndWavePlates Pedrotti FL, Pedrotti LS, Introduction to Optics (Prentice-Hall, 1987). Reitz R, Milford FJ, Christy W, Foundation of Electromagnetic Theory, 4th ed. (Addison Wesley, Reading, Massachusetts, 1993). Rubenhok A, Slater JA, Chan P, Lucio-Martinez I, Tittel W, Real-World Two-Photon Interference and Proof-ofPrinciple Quantum Key Distribution Immune to Detector Attacks, Phys. Rev. Lett. 111, 130501 (2013). http://dx.doi. org/10.1103/PhysRevLett.111.130501 Tamm P, A Physicist's Guide to Mathematica (Academic Press, San Diego 1997), 291-293. Tang X, Ghassemlooy Z, Rajbhandari S, Popoola WO, Lee CG et al., Free-space optical communication employing polarization shift keying coherent modulation in atmospheric turbulence channel, in 7th International Symposium, Newcastle, England, 21-23 July 2010. Vallone G, D’Ambrosio V, Sponselli A, Slussarenko S, Lorenzo Marrucci L et al., Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons, Phys. Rev. Lett. 113, 060503 (2014). http://dx.doi.org/10.1103/ PhysRevLett.113.060503 Wolfram, Wolfram CDF Player for Interactive Computable Document Format [Internet], cited 2015 Feb 15, Available from: http://www.wolfram.com/cdf-player Yao XS, Yan LS, Zhang B, Willer AE, Jiang J, All-optic scheme for automatic polarization division demultiplexing, Opt. Express 15, 7407-7414 (2007). http://dx.doi.org/10.1364/ OE.15.007407 Yun HJ, Choi YD, Dynamic Polarization Mathematica Platform Realized with Poynting Vectors, New Phys.: Sae Mulli 63, 1118-1127 (2013). http://dx.doi.org/10.3938/NPSM.63.1118 Yun HJ, jdpmp programs [Internet], cited 2015 Feb 15, Available from: http://home.mokwon.ac.kr/~heejy/program.htm

71

http://janss.kr

Platform for manipulating polarization modes

Platform for manipulating polarization modes

Suggest Documents

Vectorial polarization modes platform realized with Jones vectors in ...

Manipulating Greek musical modes and tempo

Manipulating carriers' spin polarization in the

Independent Manipulating of Orthogonal-Polarization ... - MDPI

Manipulating Transverse Modes of Photons for Quantum Cryptography

Radial Polarization of Periodically Focused Modes in

Manipulating the quantum information of the radial modes of trapped ...

Monitoring and manipulating Higgs and Goldstone modes in a ...

Manipulating Majorana zero modes on atomic rings with an ... - Nature

Manipulating Greek musical modes and tempo ... - Semantic Scholar

Manipulating light polarization with ultrathin ... - APS link manager

Manipulating wave polarization by twisted plasmonic ... - OSA Publishing

microfluidic-based assay platform for studying polarization mechanism ...

USING MACROECONOMIC INDICATORS FOR MANIPULATING

3-C geophone orientation and wave modes polarization - crewes

Dispersion and polarization conversion of whispering gallery modes in ...

Polarization Dependent Coupling of Whispering Gallery Modes in ...

Polarization splitting of optical resonant modes in ... - APS Link Manager

[A164.Ebook] Fee Download Optical Waveguide Modes: Polarization ...

using macroeconomic indicators for manipulating

Manipulating Strings

How Informal Control Modes affect Developers' Trust in a Platform ...

Sensing and Manipulating Built-for-Human Environments

Manipulating Technology for Surplus Extraction: The ...