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

Journal of the Korean Physical Society, Vol. 67, No. 5, September 2015, pp. 792∼799

Vectorial Polarization Modes Platform Realized with Jones Vectors in Mathematica Yong-Dae Choi Department of Microbial & Nanomaterials, Mokwon University, Taejeon 302-729, Korea

Hee-Joong Yun∗ Korean Institute of Science and Technology Information, Taejeon 305-806, Korea (Received 21 April 2015, in final form 31 July 2015) The fundamental concept in physics of polarization propagation of electromagnetic waves is newly understood to be a cardinal keyword in quantum cryptography transport technology and the Standard Model. Interactive visualization of the propagation mechanism of polarized electromagnetism in a medium with its helicity has received attention recently from scientists in the age of information and communication. This study presents a new dynamic polarization platform that presents the polarization modes of a transverse electromagnetic wave by using Jones vectors calculations in the symbolic program Mathematica to convert the state of polarization through the arrangement of the optical elements. The platform simulates a propagation process that satisfies Maxwell’s two vector equations precisely with the vectorial nature of the electromagnetic wave. PACS numbers: 07.05.Tp, 41.20.Jb, 03.50.De, 02.70.-C Keywords: Polarization mode platform, Mathematica Simulation, Jones vector, Polarizer, Helicity DOI: 10.3938/jkps.67.792

I. INTRODUCTION

cularly [10], right-hand circularly [11], and right-handed [12,13] circular polarization. Right circular polarization and right-handed circular polarization are different polarizations; i.e., they are different polarizations of opposite helicity although the names are similar. For this reason, it may be necessary to modify the physics concept and to reorganize the scheme of polarization if necessary. In fact, the helicity of polarization has not been defined in modern optics; it clarifying the spin direction either from the point of view of the source or from the point of view of the receiver is sufficient, so the helicity will depend on the point of view. We know by duality that light may be considered not only as an EM wave but also as a stream of particles called photons. As a boson, photons with helicities of ±1 mediate the electromagnetic force between electrically charged particles in quantum electrodynamics (QED). Adjusting the physics concepts of electromagnetic waves to satisfy Maxwell’s equations to the level of a gauge boson is necessary in the Standard Model. We propose that a polarized EM wave should be described in oscillating vector fields in a constant direction or should be varied in a regular method if it has a phase difference between components of the vector fields [14– 16]. That we establish a propagation principle of the EM wave in a well-defined visualization scheme is described. In a previous paper [17], we demonstrated a new dynamic polarization platform visualizing of polarization in oscil-

At this time, electromagnetic (EM) radiation is one of the most important intellectual resources for mankind. In particular, the polarization characteristic of the EM wave has become a cardinal key word in the knowledgeinformation-oriented society. The coherent characteristics of the polarized EM wave are utilized in photonics, and on radio, optical, and space communication. Recently, the quantum key distribution (QKD) has been using the phase of polarization for quantum communication [1–3]. Astronomers of BICEP2 announced their detection of signature patterns of polarized light in the cosmic microwave background (CMB), which is the strongest confirmation yet of cosmic inflation theory [4–7]. The scope of application of polarizations has expanded explosively this decade with the development of communication technology. Because the adoption of polarization has extended to various categories in physics, physics concepts and characteristics have been modified and shifted accordingly. The nomenclature in polarization differs from field to field or from text to text so that even students in physics are puzzled. For example, there are similar customary nomenclatures for right circular polarization: right circularly [8,9], clockwise cir∗ E-mail:

[email protected]; Fax: +82-42-869-0699

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Vectorial Polarization Modes Platform Realized with Jones Vectors · · · – Yong-Dae Choi and Hee-Joong Yun

lating vector fields satisfying Maxwell’s vector equations by using Mathematica. Mathematica is a computational software program used in many scientific, engineering, mathematical and computing fields, and is based on symbolic mathematics [14,18]. The remarkable symbolic, numerical, and graphical capabilities of Mathematica are merged into a single rich and powerful highlevel programming language, thus providing a wide range of useful built-in special functions such as Graphics3D, Table[Arrow], Evaluate[Jm.E1], and Manipulate for a dynamic platform simulating the polarized vector fields with Jones matrices calculations [8,19]. In this paper, we describe a polarization mode manipulating platform with considerably improved polarization characteristics with an instructive scheme of a platform for students and researchers to examine the physical properties of polarization modes. The platform cleverly presents the simulation of dynamic polarization modes by using Jones vector calculations in Mathematica to convert polarizing mode with an arrangement of the optical devices. Furthermore, the platform ensures the orthogonality of the advancing Poynting vector in the graphic mode and promptly cross-checks the graphics with the helicity of the numerical simulations in Mathematica. The platform jdpmp performs the simulations interactively in Mathematica. Even if Mathematica has not been installed in your computer, you can perform a jdpmp. cdf file instead in a CDF player (free download) [20].

II. MATHEMATICA SIMULATION FOR THE CONVERTING POLARIZATION MODES 1. Electromagnetic Waves in Solids

The propagation process of electromagnetic waves in  and B  solids is different from that in a vacuum, for E of the electromagnetic waves interact with electrons in a solid [8,10]. In particular, as the electromagnetic waves are harmonic plane waves, the fields may pull or push the electrons in an orbital of the solid, which is responsible for inducing a dipole moment P and a magnetization  in the solid. If we assume that the medium is not a M magnetic material and that J = 0, the Helemholtz wave equation in a solid [9,10] is  + ∇ × (∇ × E)

 1 ∂2E ∂ 2 P = −μ0 2 , 2 2 c ∂t ∂t

(1)

 If we suppose the EM wave has a where P = χ0 E.  solution of the form as Eˆ0 ei(k·r−ωt) , we can rewrite Eq. (1) using a wave vector k: 2 2 k × (k × E)  = − ω χE,   +ω E c2 c2

(2)

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 which indicates that This is a vector equation for E, the propagation process depends on the component of the electric susceptibility χ tensor of a crystal. Thus,  solution of Eq. (2) in a solid as an we will write the E inhomogeneous plane wave solution as follows [8–11]: ˆ r −ωt) ˆ r, t) = (ε1 E1 + ε2 E2 ) ei(K· E( 

= E0 (ε1 + ε2 eiε ) ei(k·r−ωt)   A  ei(k·r−ωt) . = B + iC

(3)

ˆ field vector may be presented as a As shown above, the E Jones vector, {A, B ± iC}, with a complex vector amplitude oscillating in an inhomogeneous plane [8,11]. Here, the wave vector k = kn forms a real mutually orthogonal ˆ = k + i set of unit vectors, such as (ε1 , ε2 , n), where K α is a complex wave vector and N = n + iκ a complex refraction index. Because Eq. (3) is a solution of Eq. (2) as an homogeneous plane harmonic wave, it should be K = ωc N . Then we get the relations α = ωc κ and k = ωc n, which result in a propagation speed v = nc which is different along different directions in the medium. Therefore, there will be a cumulative phase difference ε between two waves as they emerge in uniaxial crystals (Quartz, Jircon, Calcite, etc.). After the wave travels a distance d, the phase difference is ε = ωc d(n2 − n1 ) between the Ex wave and the Ey wave when the radiation is propagating along the k direction. If the ε is zero while  is real, a vector, such as the Jones the amplitude of E vector {A, B} is responsible for the linear polarization; otherwise, the complex amplitude vector is responsible for the elliptically polarized Jones vector as {A, B ± iC}. Specifically, if B = 0 and A = C, then the wave is a circularly polarized Jones vector such as {1,i}. A quarterwave plate is a thin birefringent crystal, the thickness of which has been adjusted to produce a ±π/4 phase difference between the ordinary and the extraordinary rays at the operating wavelength. We desire a matrix that will transform the elements E0x eiϕx into E0x ei(ϕx +εx ) and E0y eiϕy into E0y ei(ϕy +εy ) ;  iε     e x 0 Eox eiϕx Eox ei(ϕx +εx ) = , (4) 0 eiεy Eoy eiϕy Eoy ei(ϕy +εy ) where εx and εy represent the advances in phases of the Ex - and the Ey -component of the incident light. For example, a quarter-wave plate (QWP) matrix transforms the {1, 1} Jones vector to the {1, i} Jones vector:      1 0 1 1 e−iπ/4 = e−iπ/4 , (5) 0 i 1 i causes the phase of Ey to lead the Ex by π/2(i = eiπ/2 ) resulting in a right-handed circularly polarized wave. This is the case of fast axis vertical (FA vertical). In the fast axis horizontal (FA horizontal) case, we get a {1, −i} Jones vector. Similarly, we can determine the

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Journal of the Korean Physical Society, Vol. 67, No. 5, September 2015

corresponding Jones matrix for a half-wave plate (HWP) or an eighth-wave plate (EWP) or an arbitrary phase of a retarded field. As a result, vector fields are linearly polarized if  ε = the phase shift of the two components of E, mπ (m = 0, ±1, ±2, ...), circularly polarized if ε = ±π/2 + mπ (m = 0, ±1, ±2, ...), and elliptically polarized if |ε| > 0, |ε| =  π/2, and |ε| + m π (m = 0, ±1, ±2, ...). The Jones vectors and the Jones matrices for several optical elements are summarized in Table 1. The general form of a matrix representing a phase retarder transforms the elements by the matrix operation as follows:         an bn a2 b2 a1 b1 A A = . (6) ... cn dn c2 d2 c1 d1 B B Now, we want to present a new simulation for presenting the producing and polarizing modes from a Jones calculations corresponding to the physical arrangement of optical elements in Mathematica. In a plane wave,  always oscillates parallel to the the electric field vector E fixed direction in space. Light with such characteristics is said to be linearly polarized. If the linearly-polarized light passes through a quarter-wave plate, ellipticalpolarized light emerges. The same can be said of the  which maintains an orientation magnetic field vector B, perpendicular to the electric field vector such that the  ×B  is everywhere the direction of wave direction of E

propagation. Thus, the possibility of polarizing light is essentially due to its transverse character. Therefore, the Mathematica simulation should show a transverse  × B,  with its vectorial behaviors dynamcharacter of E ically satisfying Maxwell’s wave equations.

2. Numerical Simulations of Polarization Modes in Mathematica

We used Mathematica to implement the interactive Jones matrix calculations and animations for the generation and the propagation of the polarization modes in a solid state. First, we desire to confirm that the complex ˆ of Eq. (3) with Jones vectors in matter vector field E satisfies Maxwell’s vector equations, Eqs. (7) and (8) in both numeric and graphic simulations: ˆ ·E ˆ = 0, K ˆ ·B ˆ = 0, K ˆ ×E ˆ = ωB ˆ K =E  × H,  S

(7) (8)

ˆ E, ˆ and B ˆ are all complex vectors and E  and where K,     B are real vectors with B = μH, H being the magnetic intensity vector in matter. We examine the process of polarization-mode generation with the normalized Jones vectors calculations in the Mathematica simulation as in Mathematica code 1 below:

Mathematica code 1 In[11]:= In[12]:= In[13]:= In[14]:= In[13]:= In[16]:= In[17]:= In[18]:= In[19]:= In[21]:= In[22]:= In[23]:= In[24]:= In[25]:= In[26]:= In[27]:=

√ √ E0 = 1/ 2 {1, 0}+ 1/ 2 {0, 1}; LP = {Cos[α],Sin[α]}; √ LCP = 1/√2 {1, -I }; RCP = 1/ 2 {1, I }; LW = {{ Cos[β]2 ,Sin[β] Cos[β]}, {Sin[β] Cos[β],Sin[β]2 }}; QWV = {{1, 0}, {0, I}}; QWH = {{1, 0}, {0, -I}} HWV = {{1, 0}, {0, -1}}; EWV = {{1, 0}, {0, Exp[I 1/4 π ] }}; EWH = {{1, 0}, {0, Exp[-I 1/4π]}}; E0*LP·LW /.{α ->1/4π,β->1/3π} E0*LP·LW /.{α ->1/4π,β->3/4π} E0*LP·HWV /.α->1/4π E0*LP·QWV /.α->1/4π E0*LP·EWH /.α ->1/4π E0*LP·QWH /.α->1/4π E0*RCP·EWH /.α->1/4π

Mathematica returns the Jones vector calculations: Out[21]:= Out[22]:= Out[23]:= Out[24]:= Out[25]:= Out[26]:= Out[27]:=

{0.0341506,0.591506 } {0.,0.} {1/2,-1/2 } {1/2, i/2 } {0.5, 0.353553- 0.353553 i } {1.35914, -1.35914 i } {0.5, 0.353553+ 0.353553 i }

The Mathematica simulations show the emerging polarization mode through the optical devices arrangement,

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

2(a)) 2(b)) 3(a)) 3(b)) 3(c)) 1(a)) 1(b))

which is promptly checked in the animating platform. For

Vectorial Polarization Modes Platform Realized with Jones Vectors · · · – Yong-Dae Choi and Hee-Joong Yun

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Table 1. Summary of Jones vectors and Jones matrices for some linear optical elements of polarizers and wave plates [8,10]. Polarized light linearly

Jones vectors Cos[α]

right(left)-handed circularly right(left)-handed elliptically Polarizers and Wave plates general phase retarder

Sin[α] √ 1 1/ 2 ±i √  A 1/ A2 + B 2 + C 2 B±iC

Jones matrices  eiεx 0 M= eiεy

0 M=

quarter-wave plate (V/H)

M = e∓iπ/4

eighth-wave plate (V/H)

±1 ±1

Cos2 [β] Sin[β]Cos[β] 2 [β] Sin[β]Cos[β] Sin



linearly polarizer (β, TA)

half -wave plate (V/H)

Helicty 0



1 0

0 ±i  1 0 M = e∓iπ/2 0 −1 

M=

1 0 0 e±iπ/4

Table 2. Helicities summarized from the combination of the optical devices. The list shows helicities simulated from the helicity [JvC,JmC] module, where JvC indicates the polarizer and JmC the wave plate. The first value in the list is the helicity in the E1 block and the 2nd is the helicity of the E2 block. polarizer (JvC) wave plate (JmC) 1. 2. 3. 4. 5. 6. 7.

1. linear polarizer (LP)

2. left-circular polarizer (LCP)

3. right-circular polarizer (RCP)

0, 0 0, 1 0, −1 0, 0 0, 0 0, 1 0, −1

−1, 0 −1, 0 −1, 0 −1, 1 −1, 1 −1, −1 −1, −1

1, 0 1, 0 1, 0 1, −1 1, −1 1, 1 1, 1

linear polarizer (LW) quarter-wave plate vertical (QWV) quarter-wave plate horizontal (QWH) half-wave plate vertical (HWV) half-wave plate horizontal (HWH) eighth-wave plate vertical (EWV) eighth-wave plate vertical (EWH)

instance, an arrangement of In[27] E0*RCP·EWH/.α->1/4π produces Out[27], that is, right-handed ellipticallypolarized light (RHEP) as shown in Fig. 1(b). From the combination of polarizers, we can switch the linearly polarized light based on the relative angles between the two polarizers; a linearly polarized wave emerges in Out[21], but no wave emerges as Out[22], as shown in Fig. 2(a) and (b).

3. Helicity of the Elliptically-polarized Wave

The handedness of an elementary particle depends on the correlation between its spin and momentum [15,21]. ˆ = σ · p/p, so the The helicity operator is defined as h helicity of a photon is ±, and is expressed as briefly ±1 if we let  = 1. If the spin and the momentum of

a photon are parallel, then they can be said to righthanded or have a helicity of +1. If they are antiparallel, then they can be said to be left-handed or have a helicity of −1. We may also adopt for modern optics, as the circularly-polarized electromagnetic wave is just a helical motion with helicity. The helicity of the polarized electromagnetic wave is a critical factor in modern communication technology and photonics [3,4,21]. However, determining the helicity of the polarization is a complex issue because we need to be aware of the vector state accurately unless we may observe the helicity in practice [21]. To evaluate the helicity of the polarized waves in progress, we have provided a Mathematica module, helicity [JvC ,JmC ]:=Module [{E1, E2,dp1,dp2}], for a train of two optical elements. If you type in two numbers assigned to the polarizer JvC and the wave plate JmC in the module, then helicity [JvC,JmC]

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Journal of the Korean Physical Society, Vol. 67, No. 5, September 2015

Fig. 1. (Color online) Emerging (a) the left-handed circular polarization (LHCP) in E0*LP·QWH optical device array. (b) Right-handed elliptical polarization (RHEP) in E0*RCP·EWH array.

returns the helicities with a list of phase shifts. The helicities from a combination of optical devices are summarized in Table 2. The outputs of the helicity [JvC,JmC] of Mathematica code 2 show the helicity in the blocks respectively, with its phase shifts as below: Mathematica code 2 In[81]:=

helicity[1,2] Out[81]:= {0., 0., 0.} helicity1 = 0 {1.5708, 1.5708, 1.5708} helicity2 = +1

(Fig. 3(b))

In[82]:=

helicity[1,4] Out[82]:={0., 0., 0.} helicity1 = 0 {3.14159, 3.14159, 3.14159} helicity2 = 0

(Fig. 3(a))

In[83]:=

helicity[1,7] Out[83]:={0., 0., 0.} helicity1 = 0 {−0.785398, −0.785398, −0.785398} helicity2 = -1

(Fig. 3(c))

In[84]:=

(Fig. 1(a))

helicity[1,3] Out[84]:={0., 0., 0.} helicity1 = 0 {−1.5708, −1.5708, −1.5708} helicity2 = -1 In[85]:=

helicity[3,7] (Fig. 1(b)) Out[85]:={-4.71239, -4.71239, -4.71239 }

Fig. 2. (Color online) Emerging of (a) linear polarized wave in the optical device array combined with a 1st polarizer(α = 1/4π)and a 2nd polarizer(β = 1/3π). (b) No emerging wave while combined with right angle between two polarizers(α = 1/4π, β = 3/4π).

helicity1 = +1 {0.785398, 0.785398, 0.785398} helicity2 = +1

One should notice that the phase shifts are the same in the block. For instance, Out[81]of the In[81] helicity [1,2]shows that the helicity of the E1 block is 0 and that of the E2 block is +1 because the three phase shifts are all 0 in block E1 while they are 1.5708 (π/2) rad in the E2 block, which result in a linearly polarized (LP) wave in block E1 and a right-handed circularly polarized (RHCP) mode in block E2. In the case of In[84],the helicity [1,7] shows that the phase shift of E2 block is {−0.785398, −0.785398, −0.785398}(−1/4π); that is, constant phase shift of the −1/4π rad occur in block E2 and create a left-handed elliptical polarization (LHEP) mode. Thus, the module helicity [JvC,JmC] help to confirm the polarization mode by using the numerical simulation in Mathematica. For more information about the helicity [JvC,JmC]module, refer to Ref. [23].

Vectorial Polarization Modes Platform Realized with Jones Vectors · · · – Yong-Dae Choi and Hee-Joong Yun

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Fig. 3. (Color online) Producing different polarized modes from the different physical arrangements: (a,d) a linearly-polarized wave in the E0*LP·HWV optical devices array, (b, e)a right-handed circular polarization (RHCP) in the E0*LP·QWV array, and (c, f) a left-handed elliptical polarization (LHEP) in the E0*LP·EWH array. The snapshots of the upper level are viewed at ViewPoint (2.647, −1.611, 1.176). The snapshots of the lower level are viewed at ViewPoint (1, 0, 0) to identify the orthogonality of the vector fields of the polarizing modes.

III. DYNAMIC POLARIZATION PLATFORM WITH THE JONES VECTOR IN MATHEMATICA We have provided a new dynamic polarization platform jdpmp with the Jones vector and Graphics3D in Mathematica,which dynamically simulates the polarizing modes in presenting the helicity of a running polarization mode. The platform manipulates three zones graphically by using the Piecewise function of Mathematica in order to select the polarizer and wave plate for the desired polarization mode. To present the transverse characteristic property that satisfies Eq. (3), together with Eqs. (7) and (8), we used the Arrow function in Mathematica to draw the vector array: Table [Arrow [{{x1,0,0},{x1,Ey,Ez}}]]in the orthogonal {x1, Ey, Ez} coordinate system. Further, in the same way as for the  in a solid was calculated from magnetic field, vector B    of Eq. (7). Graphics3D can the relation B = 1/μk × E  (blue) and B  (red) fields draw the vector array of the E of every point in the block just as for real polarization mode propagation in the k direction, in our case advancing in the x1 direction. This simulation scheme differs basically from that of animation representing the envelope of the polarization  field only by using the Animate funcpropagation of the E tion of the graphic tools: they are superficial interpretations that overlook the vectorial nature of the propagation process of the polarized EM wave in a medium [16, 22]. The platform jdpmp starts with clicking the button  on pop up ⊕ of the t1 panel on the platform in Fig. 1. On choosing the polarizer and the wave plate from the panel of the platform, the platform realistically shows

the polarization mode automatically with the vector ar  ray of E(blue) and B(red) fields in vectorial transverse behavior. While the program is running, we change the polarizer or wave plate, then the changed animations run continuously. Even if the animation is stopped, we can change the configuration of the platform and observe the polarizing mode of the changed state. While the platform is running, we confirm the helicity and the phase shifts of the running polarization mode on the panel and observe the spin direction of the field vectors along the  and the orange trace of the helical propagation of the E  B vector fields at the edge of the platform. Figure 1(a) is a snapshot of left-handed circular polarization (LHCP) in the train of optical devices array E0*LP·QWH, and Fig. 1(b) is that of the right-handed elliptical polarization (RHEP) in E0*RCP·EWH. Figure 1(a) is the graphic version of the numerical simulations In[26] and In[84] while Fig. 1(b) is that of In[27] and In[85] . Similarly, the other figures are also matched to the numeric simulations referred in the figure number. In practice, Out[26] and Out[84] give the complete information on the polarization modes; it confirms the LP mode ˆ = 0, and confirms the in E1 block for ε = 0, and h ˆ = −1 LHCP mode in the E2 block for ε = −π/2 and h numerically. Figure 2(a) shows the emerging wave of the linearly polarized wave (LP) in the optical device with two polarizers (α = 1/4π, β = 1/3π) (a)). However, Fig. 2(b) shows no emerging wave while making right angle between the transmission axes of two two polarizers (α = 1/4π, β = 3/4π) (b)). Thus controlling the phase of polarized EM waves is available for the phase encoding in the quantum key distribution (QKD) of the unconditional security in quantum communication tech-

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Journal of the Korean Physical Society, Vol. 67, No. 5, September 2015

nology [1–3]. Figure 3 shows snapshots of three polarization modes: a linearly-polarized wave (LP) (Figs. 3(a), 3(d)), a right-handed circularly-polarized wave (RHCP) (Figs. 3(b), 3(e)) and a left-handed elliptically-polarized wave (LHEP) (Figs. 3(c), 3(f)). The upper level of the snapshots (a-c) is viewed at ViewPoint (2.647, −1.611, 1.176), the lower level of the snapshots (d-f) is viewed at ViewPoint (1, 0, 0) to identify a transverse EM wave that satisfies Maxwell’s vector equations: Eqs. (7) and (8). The  B,  and S  are snapshots show that the three vectors E, orthogonal in the platform. Thus, Fig. 3 confirms that the jdpmp platform is simulating the polarization process in the vectorial nature and satisfying Maxwell’s vector equations exactly in a graphic version. Owing to the cross-checking by numerical simulations of Out[23-25] and Out[81-83], we can analyze the polarization modes precisely and predict the emerging polarized waves in the array of optical devices with confidence. The platform will run faster or slower and can be stopped and restart again by a click of the pop-up menu of the t1 panel. During the platform stop, one can change to another mode only by clicking a panel; then, the changed mode will be presented automatically, and at that time, one will save the presenting mode or print out the mode status. As far as we know, other than jdpmp no instructive platform simulates a transverse EM wave that satisfies Maxwell’s equation in the vectorial nature of the polarization modes. A complete version of the jdpmp program, including helicity module, is available in Ref. [23]. If your computer does not have Mathematica installed, you can use jdpmp.cdf instead of Wolfram CDF Player free download [20,23].

IV. CONCLUSION We have provided a new vectorial polarization modes platform jdpmpfor simulating and producing of polarization modes with the Jones vector calculations corresponding to the physical arrangement of the optical elements in Mathematica. The platform simulates the  vector field together with polarization process of the E  by using Arrow vectors so that of the vector field B that the platform animates a transverse EM wave that satisfies Maxwell’s wave equation every point on the advancing axes of the platform. Consequently, the vec and B  (in that order) form a right-hand ortors k, E, thogonal set. The platform can be manipulated interactively to advance the polarized mode while the user clicks the panels of the polarizer and wave plate with the Manipulate function in Mathematica, so that the program simulates the changed mode continuously. All the graphics of the platform are cross checked with numerical simulations in Mathematica. The module helicity [JvC,JmC] returns the helicity of the process on N otebook of Mathematica when the user types in the names of

the polarizer and the wave plate in the module. The module is helpful for students or researchers to inspect the phase difference or the helicity of the polarized mode for various kinds of physics arrangements. We expect the platform jdpmp to be a useful platform for students of physics and researchers to explore the science of polarization.

ACKNOWLEDGMENTS This research was partially supported by the ReSEAT program funded by the Korean Ministry of Education, Science and Technology through grants from the National Research Foundation of Korea and the Korea Lottery Commission

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Vectorial Polarization Modes Platform Realized with Jones Vectors · · · – Yong-Dae Choi and Hee-Joong Yun

[17] H.-J. Yun and Y.-D. Choi, New Physics: Sae Mulli 63, 1118 (2013). [18] Wolfram Mathematica, http://www.wolfram.com /mathematica (2015). [19] R. Clark Jones, J. Opt. Soc. Am. 31, 488 (1941). [20] Wolfram CDF Player for Interactive Computable Document Format, http://www.wolfram.com/cdfplayer/ (2015).

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[21] M. Goldhaver, L. Grodzins and A. W. Sunyar, Phys.Rev. 109, 1015 (1958). [22] Optics Applets, http://www.cabrillo.edu/jmccullough /Applets/Flash/Optics/CircPol.swf (2015). [23] jdpmp programs, http://home.mokwon.ac.kr/∼ heejy /program.htm (2015).