Non-paraxial idealized polarizer model - OSA Publishing

Vol. 26, No. 8 | 16 Apr 2018 | OPTICS EXPRESS 9840

Non-paraxial idealized polarizer model S ITE Z HANG , 1,2,* H ENRI PARTANEN , 3 C HRISTIAN H ELLMANN , 2,4 AND F RANK W YROWSKI 1 1 Applied

Computational Optics Group, Institute of Applied Physics, Friedrich Schiller University Jena, Albert-Einstein-Straße 15, 07743 Jena, Germany 2 LightTrans International UG, Kahlaische Straße 4, 07745 Jena, Germany 3 University of Eastern Finland, Yliopistokatu 2, FI-80100 Joensuu, Finland 4 Wyrowski Photonics UG, Kahlaische Straße 4, 07745 Jena, Germany

* [email protected]

Abstract: An idealized polarizer model that works without the structural and material information is derived in the spatial frequency domain. The non-paraxial property is fully included and the result takes a simple analytical form, which provides a straight-forward explanation for the crosstalk between field components in non-paraxial cases. The polarizer model, in a 2 × 2-matrix form, can be conveniently used in cooperation with other computational optics methods. Two examples in correspondence with related works are presented to verify our polarizer model. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (260.0260) Physical optics; (260.1440) Birefringence; (260.5430) Polarization.

References and links 1. R. C. Jones, “A new calculus for the treatment of optical systems I. description and discussion of the calculus,” J. Opt. Soc. Am. 31(7), 488–493 (1941). 2. Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt. 23(18), 3188–3195 (1984). 3. A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett. 34(20), 3160–3162 (2009). 4. J. Korger, T. Kolb, P. Banzer, A. Aiello, C. Wittmann, C. Marquardt, and G. Leuchs, “The polarization properties of a tilted polarizer,” Opt. Express 21(22), 27032–27042 (2013). 5. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5-6), 449–466 (2011). 6. P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE 0307, 13–21 (1982). 7. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982). 8. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10(5), 966–973 (1993). 9. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017). 10. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 7 42122 (2017). 11. Z. Wang, S. Zhang, and F. Wyrowski, “The semi-analytical fast Fourier transform,” Proc. DGaO, P2 (2017). 12. L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86-92 (1969). 13. F. Wyrowski and C. Hellmann, “The geometric Fourier transform,” Proc. DGaO, A37 (2017). 14. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). 15. G. D. Landry and T. A. Maldonado, “Gaussian beam transmission and reflection from a general anisotropic multilayer structure,” Appl. Opt. 35(30), 5870–5879 (1996). 16. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13(5), 1024–1035 (1996). 17. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20(4), 655–660 (2003). 18. S. Zhang, C. Hellmann, and F. Wyrowski, “Algorithm for the propagation of electromagnetic fields through etalons and crystals,” Appl. Opt. 56(15), 4566–4576 (2017). 19. Fast physical optics software “Wyrowski VirtualLab Fusion,” LightTrans GmbH, Jena, Germany. 20. The simulation example on polarizer in focal region, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=439 21. F. Wyrowski, “Unification of the geometric and diffractive theories of electromagnetic fields,” Proc. DGaO, A36 (2017). 22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A253, 358–379 (1959).

#323448 Journal © 2018

https://doi.org/10.1364/OE.26.009840 Received 16 Feb 2018; revised 23 Mar 2018; accepted 23 Mar 2018; published 5 Apr 2018


23. The simulation example on Stokes parameters measurement behind a tilted polarizer, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=441 24. S. Zhang, D. Asoubar, C. Hellmann, and F. Wyrowski, “Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation,” Appl. Opt. 55(3), 529–538 (2016).

1.

Introduction

As one of the most widely used optical components, polarizers can be found in almost every modern optical system. They are designed for paraxial fields under normal incidence, and under such conditions a polarizer can be described by its Jones matrix [1]. Several works have been carried out with the aim to go beyond this paraxial limitation; one subset of these works, [2–4], relies on a geometric model. The original theory was from Fainman and Sharmir [2], and it was adapted by Korger et al. [4] to connect it better with the physical nature of polarizers. A common aspect in these works is the three-dimensional representation for all vectorial quantities, which is correct but somehow complicated and even inconvenient to use alongside many other computational optics methods. As a matter of fact, to fully represent any electromagnetic field in an isotropic homogeneous medium, it is enough to use two field components only [5]. As a consequence, the propagation through a polarizer can be fully described by out in Ex (ρ, z out ) Ex (ρ, zin ) Cxx Cxy = , (1) Eyout (ρ, z out ) Eyin (ρ, zin ) Cyx Cyy in and E out are the input field in the isotropic medium in front of the polarizer and the where Ex/y x/y output field in the isotropic medium behind it, the matrix operator C includes possible crosstalk between field components, and ρ = (x, y) is the transverse spatial variable. The formulation in Eq. (1) is valid in general, without any restriction. As we will show later, the Jones matrix method [1] is a special case of Eq. (1), in which the operators are replaced by constant numbers, and its application is restricted to paraxial cases. The works from Yeh [6, 7] and Gu [8] extended the Jones matrix method to general cases while maintaining the 2 × 2-matrix form. They represent the polarizer as a uniaxial crystal slab, and with such a model the physical nature of a polarizer can be well understood. The extended Jones matrix method has been used in many applications; for example, it was recently applied by Martínez-Herrero et al. for their study of the polarizer effect with highly focused fields [9, 10]. Although an idealized polarizer model was discussed in [7, 10], the application of such a model requires the knowledge, or at least an estimation, of the refractive indices of the polarizer. This fact makes it, in our opinion, not a complete idealization. Since the material information of the polarizer is usually unknown or used as a free parameter in practice, it may cause difficulties when applying their method. In this paper, we would like to find a completely idealized polarizer model that does not require any knowledge of the structural and material properties of the polarizer. By following the idea of S-matrix [16, 17], we first study the field inside the polarizer, which can be treated as a plate of uniaxially anisotropic material. By doing that, two independent polarization modes are found, and then the idealized polarized model can be directly defined with respect to the modes. The resulting model maintains the 2 × 2-matrix form. It takes the non-paraxial effects into consideration, because the question is investigated in the spatial frequency domain (k domain), where the angular dependency should be automatically included via the relation between spatial frequency and angle of incidence. To deal with general fields with different kinds of wavefronts, several advanced Fourier transform techniques [11–13] can be used to realize the transformation between the two domains with high efficiency.


2.

Idealized polarizer model

Following [6], a polarizer can be modeled as a thin plate of uniaxially anisotropic material, with its optic axis lying parallel to the plate. Without loss of generality, we set up a Cartesian coordinate system so that the z axis is perpendicular to the plate and the x axis is parallel to the optic axis (o.a.), as is shown in Fig. 1. By applying a coordinate system rotation around the z optic axis (o.a.), as is shown in Fig. 1. By applying a coordinate system rotation around the z axis, it is possible to handle polarizers with any other orientation, as shown in Fig. 1 (c). Then, axis, it is possible to handle polarizers with any other orientation, as shown in Fig. 1 (c). Then, y

d

x o.a.

o.a.

y

y x z (a) three-dimensional view

o.a.

y x

z (b) y-z section

(c) x-y section

Fig.Fig. 1. Uniaxial crystal model for a for polarizer. A Cartesian coordinate system system x-y-z isx-y-z set is set 1. Uniaxial crystal model a polarizer. A Cartesian coordinate up as shown in (a), with the x axis along the optic axis (o.a). The polarizer plate has a up as shown in (a), with the x axis along the optic axis (o.a). The polarizer plate has a permittivity tensor and a thickness of d. The embedding medium has a permittivity . A permittivity tensor and a thickness of d. The embedding medium has a permittivity . A rotated polarizer can be treated with the help of coordinate transformations, as shown in (c).

rotated polarizer can be treated with the help of coordinate transformations, as shown in (c).

the dielectric permittivity tensor can be written as

the dielectric permittivity tensor can be written as

0 0 e 0 0 = 0 © oe 0 , ª = 0 o 0 ® , 0 00 0o

(2)

(2) o ¬ « with the subscripts “o” and “e” for ordinary and extraordinary respectively. Following Berreman’s with the subscripts “o” and “e” for ordinary and extraordinary respectively. Following Berreman’s 4 × 4-matrix formulation [14], Maxwell’s equations can be formulated in the k domain as follows 4 × 4-matrix formulation [14], Maxwell’s equations can be formulated in the k domain as follows

0 0 nx ny /o 1 − n2x /o 2 E˜x (κ, z)˜ E˜ (κ, z) 0 0 n2x ny /o 1 − n / ˜x E˜x (κ, z) Ex (κ, z) d E©y (κ, z) 0 −1 + ny /o 2−n x ny /o x Eo˜y (κ, ª 0© ª ª © z) ˜ , ˜ = ik d E 0 0 −1 + n / −n n / Ey (κ, z) (κ, z) n ® x y ηo H y0 o z) ®® =0 ik−n −o + n2x 0 2 0 ®®˜ x(κ, z) dz η0 H˜ xy(κ, x y ®, 0 η0z)H˜ x (κ, z) ® H˜ xz) (κ, z) ® nx 0 0 − −n ® 2 x nn dz η0 Hη˜ y0(κ, y n −o + 0 ˜ (κ, H η n 0 0 y e x y y 2 0 0 « e − ny nx ny « η0 H˜ y (κ, z) ¬ ¬ « η0 H˜ y (κ, (3)z) ¬ (3) where κ = (k x, k y ) is the transverse variable in the k domain, E˜x (κ, z) = F[Ex (ρ, z)] is the ˜ where κ = (k , k ) is the transverse variable in the k domain, E (κ, z) = F[E (ρ, z)] is the x of yE x (ρ, z) (similar rules apply for the other field components). x x Fourier transform In addition, (similar rulesfor apply the other field components). we weFourier define transform η0 =p µ0of /0Exas(ρ,a z) scaling factor the for magnetic field [15], k0 = 2π/λInasaddition, the define η = µ / as a scaling factor for the magnetic field [15], k = 2π/λ as the wavenumber wavenumber of k x and k y for 0 in vacuum, 0 0 and n x = k x /k 0 and ny = k y /k 0 as normalized values 0 convenience. a setnof ordinary differential equations, general solution in vacuum,Eq. and(3)nxrepresents = k x /k0 and k y /k0 as normalized values whose of k x and k y for convenience. y = canEquation be found(3) in analytical representsform a set of ordinary differential equations, whose general solution can be

found in analytical form ˜

Ex (κ, z) C TM exp(γ TM z) 1 0 1 0 +TE ˜ W W W −W z) z) TM E˜y (κ, z) Ex (κ, z) B 1 D 0 B 1 C+TMTEexp(γ D 0 C+ exp(γ = , © © © ª ª TM TM η0 H˜Ex˜y(κ, z)TE z) DC−® exp(−γ (κ,z)z) ® 0 W1B W0D WB1 −W C+TE exp(γ = ® ® TE TE z) TM z) η0 H˜ηy0(κ, C−TM exp(−γ H˜ xz)(κ,z) ® WC 0 WB 1 −WC 0 WB 1 C−® exp(−γ

˜

ª (4) ® ®, ® TE TE ¬ « C− exp(−γ z) ¬

Hy (κ, z) ¬ « WC WB −WC WB where we use«theη0notation where we use the n x nnotation 1 1 1 TM 1 TE y WB (κ) = , WC (κ) = γ , WD (κ) = γ ; (5) 2 2 2 n n 1 1 TE 1 ik ik x y 1 − nx /o 0 −o + nx 10 −o + nx TM WB (κ) =

,

WC (κ) =

γ

,

WD (κ) =

γ

2 −oelements, + n2x −o + n2x ik0 for some of the matrix we define 1 − nx /o ik 0 (6) γ TM (κ) = ik0 e − n2y − n2x e /o , γ TE (κ) = ik0 o − n2x − n2y .

(4)

;

(5)


for some of the matrix elements, we define q γ TM (κ) = ik0 e − n2y − n2x e /o ,

q γ TE (κ) = ik0 o − n2x − n2y .

(6)

as the propagation constants, and C±TE/TM as the complex coefficients which are to be determined by boundary conditions. By defining the transverse electromagnetic field vector T V˜ ⊥ (κ, z) = E˜x, E˜y, η0 H˜ x, η0 H˜ y , and introducing each column of the 4 × 4 matrix in Eq. (4) as a polarization mode vector T W TM + (κ) = (1, W B, 0, WC ) ,

T W TE − (κ) = (1, W B, 0, −WC ) ,

we can rewrite Eq. (4) as

T W TE + (κ) = (0, WD , 1, W B ) ,

T W TE − (κ) = (0, −WD , 1, W B ) ,

TM TM TE TE V˜ ⊥ (κ, z) =W TM z) + W TE + C+ exp(γ + C+ exp(γ z)

TE TM TM TE z) + W TE + W TM − C− exp(−γ − C− exp(−γ z) .

(7)

(8)

Equation (8) is in the form of a sum of different modes, and each mode is characterized by a polarization mode vector W , a propagation constant γ, and a mode coefficient C. It is not hard to identify that the polarization is either in transverse electric (TE) or transverse magnetic (TM) mode. Thus, the superscript “TE/TM” is employed. According to the sign in front of γ, the subscript “+/-” is employed to indicate the propagation direction. As is discussed in [6], one of the modes must suffer a strong attenuation in a good polarizer, which can be expressed as either exp(γ TM d) ' 0 , exp(γ TE d) ' 1 (9) for an O-type polarizer that transmits TE mode only , or exp(γ TM d) ' 1 , exp(γ TE d) ' 0

(10)

for an E-type polarizer that transmits TM mode only. That means only one mode is maintained TM with either pure TE polarization W TE + (O-type) or pure TM polarization W + (E-type) in the polarizer plate. As a result, an O-type polarizer may produce a linearly polarized transverse electric field, while an E-type polarizer produces a transverse magnetic field. To investigate the interaction of the electromagnetic field with the polarizer plate, we also need the knowledge of the field in the embedding isotropic medium, with the permittivity scalar . It can be regarded as a special case of the previous derivation but only with e = o = , so that all the conclusions in Eqs. (4–8) still apply. We can directly write down the input field in front of and the output field behind the polarizer plate, both propagating in the positive direction, in terms of TE and TM modes, as in in,TM in,TE ¯ TM ¯ TE V˜ ⊥ (κ, z) = W exp(γz) ¯ +W exp(γz) ¯ , + C+ + C+

out out,TM out,TE ¯ TM ¯ TE V˜ ⊥ (κ, z) = W exp(γz) ¯ +W exp(γz) ¯ , + C+ + C+

(11)

where the TM and TE polarization mode vectors in this case are defined as ¯ TM ¯ ¯ T W + (κ) = (1, W B, 0, WC ) , with

W¯ B (κ) =

n x ny − +

n2x

,

W¯ C (κ) =

¯ TE ¯ ¯ T W + (κ) = (0, WD , 1, W B ) ,

1 1 γ¯ , 2 1 − nx / ik0

W¯ D (κ) =

1 1 γ¯ , 2 − + nx ik0

(12) (13)


and with the identical propagation constant

q γ(κ) ¯ = ik0 − n2x − n2y .

(14)

In Eqs. (11–14), we introduce an over-bar for those quantities in the embedding isotropic medium, so as to differentiate them from those corresponding to the anisotropic polarizer plate. The results above can be directly incorporated in the S-matrix [16, 17] method for the study of the field interaction with (multi-) layer structures, and examples of applications can be found in e.g. [18]. The S-matrix concept is derived with respect to the modes and it connects the input and output, in our case, as out,TM ++ in,TM ++ (κ) C+ (κ, zout ) s11 (κ) s12 C+ (κ, zin ) = . (15) ++ (κ) s ++ (κ) s21 C+out,TE (κ, zout ) C+in,TE (κ, zin ) 22 Here the 2 × 2 matrix s ++ is one of the blocks out of the full S matrix, and it corresponds to the forward transmission channel. To calculate the exact S matrix, the knowledge of the material permittivity and the polarizer plate thickness d = z out − z in is needed. However, the goal of this paper is to find an idealized polarizer model regardless of its structural and material information. This requires appropriate assumptions on the functionality of the polarizer. Following the conclusion stemming from Eqs. (9) and (10), a linear polarization mode is maintained inside the polarizer plate, and, without discussing the exact boundary-matching problem at the surfaces, it makes sense to assume that the maintained polarization mode should be extended to the output isotropic medium as well. Since the medium on the input side is identical to the output, this assumption also holds for the input. With these assumptions, we can directly write down the idealized polarizer model out,TM in,TM C+ (κ, zout ) 0 0 C+ (κ, z in ) = (16) 0 1 C+out,TE (κ, zout ) C+in,TE (κ, zin ) for an O-type polarizer, and out,TM C+ (κ, zout ) 1 = 0 C+out,TE (κ, zout )

0 0

C+in,TM (κ, z in ) C+in,TE (κ, zin )

(17)

for an E-type polarizer. In contrast to the expression given in Eq. (1) which is defined for the Ex and Ey field components, the models in Eqs. (16) and (17) are given with respect to the modes in the k domain, and they reveal the functionality of a polarizer: it should maintain the desired polarization mode from the input and deliver it to the output. Since the 2 × 2 matrices are constant, the conclusion is independent of the spatial frequency κ. These results may look similar to those in the Jones matrix method [1], while the physical interpretation of Eqs. (16) and (17) is different. The relation between field components and mode coefficients is implicitly included in the definition of the polarization vectors in Eq. (12). In most cases, it is the electric fields that are of concern. Thus, we will focus on the discussion on the O-type polarizers and electric fields in what follows, while the case of E-type polarizers can be understood in a similar manner. For given electric field components in the embedding isotropic medium, the corresponding mode coefficients can be calculated via the relation ! ! −1 C+in/out,TM (κ, zin/out ) 1 0 E˜xin/out (κ, zin/out ) = . (18) W¯ B (κ) W¯ D (κ) C+in/out,TE (κ, z in/out ) E˜yin/out (κ, zin/out )


By substituting Eq. (18) into Eq. (16), we obtain out −1 E˜x (κ, zout ) 1 0 0 0 1 0 = E˜yout (κ, zout ) W¯ B (κ) W¯ D (κ) 0 1 W¯ B (κ) W¯ D (κ) in E˜x (κ, z in ) 0 0 = . ¯ E˜yin (κ, z in ) −WB (κ) 1

E˜xin (κ, zin ) E˜yin (κ, zin )

(19) The expression in Eq. (19) defines the polarizer model with respect to the field components in the k domain. In contrast to Eq. (16), the 2 × 2-matrix here is not diagonal any more and it has a non-zero anti-diagonal element W¯ B (κ). This gives rise to polarization crosstalk between the input and output field components. Moreover, the anti-diagonal element W¯ B (κ) is dependent on the spatial frequency κ and, as defined in Eq. (13), W¯ B = nx ny /(− + n2x ) is non-zero only when k x , 0 and k y , 0, since nx and ny are just normalized values of them. The result in Eq. (19) differs from the form in Eq. (1) only by the domain in which it is defined. With the help of the Fourier transform, it is straightforward to write down the polarizer model in the spatial domain as out −1 in Ex (ρ, zout ) Ex (ρ, zin ) F 0 0 0 F 0 = Eyout (ρ, zout ) Eyin (ρ, zin ) −W¯ B (κ) 1 0 F 0 F −1 (20) in Ex (ρ, zin ) 0 0 = , Eyin (ρ, zin ) −F −1W¯ B (κ)F 1 where F and F −1 denote the two-dimensional Fourier transform and its inverse respectively. We have thus completed the task that was the object of this paper. Finally, we would like to additionally consider the case with normal incidence with k x = k y = 0 or the paraxial situation with k x ' 0 and k y ' 0. It is obvious to see that the crosstalk term W¯ B (κ) in both Eqs. (19) and (20) vanishes. That leaves a constant 2 × 2 matrix out in E˜x (κ, zout ) E˜x (κ, zin ) 0 0 = (21) E˜yout (κ, zout ) E˜yin (κ, zin ) 0 1 in the k domain, and

Exout (ρ, zout ) Eyout (ρ, zout )

=

0 0

0 1

Exin (ρ, zin ) Eyin (ρ, zin )

(22)

in the spatial domain. The result in Eq. (22) is identical to that of the Jones matrix method [1]. It is evident that the polarizer model in the Jones matrix method should be regarded as a special case under paraxial conditions. 3.

Examples

In the physical optics simulation and design software VirtualLab Fusion [19], we followed Eq. (19) to implement the idealized O-type polarizer model using a “programmable component”. Such a component may work in combination with other physical-optics simulation techniques. In this section, we present three examples. With the first one we illustrate the numerical algorithm and with the other two we verify our simulation results by comparing them against references in literature. All simulations are performed on a mobile workstation with Intel Core i7-4910MQ processor at 2.90 GHz and 32 GB memory. 3.1.

Polarizer under divergent beam illumination

It has been addressed in [2] that when a polarizer is illuminated by a diverging Gaussian beam that is linearly polarized orthogonally to the polarizer, the transmitted intensity is non-zero and shows


a quadrupolar pattern. Such an effect can be explained well with our theory. Using the example below, we discuss the polarization crosstalk effect and also explain the numerical algorithm. A linearly polarized (along the x axis) Gaussian field at its waist is selected as the input. The wavelength is set to 633 nm and has a full divergence angle of 40°, which leads to a waist radius of 553 nm. The input field propagates through a polarizer that is along the y axis i.e. orthogonal to the input polarization. We describe the polarizer effect by directly following Eq. (19) in the k domain. The algorithmic process as well as the simulation results in both domains are visualized in Fig. 2. | E˜x (k x , k y )|

| E˜y (k x , k y )|

|Ex (x, y)|

100

|Ey (x, y)|

100

(b)

(a) 1 5 × 106 m

0.75 µm

0

0 2.9

2.6

(d)

(c)

y

ky 0

kx

0

x

Fig.2.2.Simulation Simulationof of diverging diverging Gaussian crossed polarizers: input Fig. Gaussianfield fieldpropagating propagatingthrough through crossed polarizers: input Gaussian field in (a) the k domain and (b) the spatial domain; output field behind the Gaussian field in (a) the k domain and (b) the spatial domain; output field behind the polarizer polarizer in (c) the k domain and (d) the spatial domain. The simulation of this example in (c) the k domain and (d) the spatial domain. The simulation of this example took 2 s. took 2 s.

For any input field given in the spatial domain, it is always possible to define the input field in the with k domain. Theofalgorithmic as well simulation in both inEq. the (19) k domain the help the Fourierprocess transform. In as ourthecase, with theresults Gaussian input domains are visualized in Fig. field, the Fourier transform can2.even be evaluated analytically, and the results are still Gaussian For any input field given the spatial it of is always possiblealgorithm. to define the input distributions, as shown in Fig.in2(a). This isdomain, the input the numerical Then, wefield follow in the k domain with the help of the Fourier transform. In our case, with the Gaussian input Eq. (19) and multiply a 2 × 2 matrix on each spatial frequency κ. Since the 2 × 2 matrix has a field, the Fourier transform can even be evaluated analytically, and the results are still Gaussian ˜ non-zero anti-diagonal element W¯ B (κ), it converts the E˜x component from the input to the Ey distributions, as shown in Fig. 2 (a). This is the input of the numerical algorithm. Then, we component of the output, as in Fig. 2(c). The resulting E˜y of the output is modulated by W¯ B (κ) follow Eq. (19) and multiply a 2 × 2 matrix on each spatial frequency κ. Since the 2 × 2 matrix in the k domain. Thus, we see that non-zero only appear in the off-axis regions, because ¯ B (κ),values has a non-zero anti-diagonal element W it converts the E˜x component from the input to only there k k , 0. In addition, an asymmetry in the amplitude E˜youtput can beisseen as well, x y the E˜y component of the output, as in Fig. 2 (c). The resulting E˜y ofofthe modulated byand 2 ). An ¯ that is caused by the denominator in the expression of W (κ) which takes the form (− + n B ¯ x WB (κ) in the k domain. Thus, we see that non-zero values only appear in the off-axis regions, inverse Fourier transform can visualize the fieldamplitude also in the because only there k x ky 0. be In employed addition, antoasymmetry in the of spatial E˜y can domain, be seen asas is shown and Fig. and further in analysis can be done ¯ afterwards. well, in andFig. that2(b) is caused by 2(d), the denominator the expression of W B (κ) which takes the form (− + n2x ). An inverse Fourier transform can be employed to visualize the field also in the spatial

3.2. Polarizer in focal region domain, as is shown in Fig. 2 (b) and (d), and further analysis can be done afterwards.

Martínez-Herrero et al. studied how ideal polarizers affect highly focused fields in [10]. The 3.2. Polarizer focal region focusing effect by ainhigh-NA microscope objective was modeled with the Richards-Wolf integral al. studied ideal polarizers affect highlyasfocused The inMartínez-Herrero their work. In ouretexample, wehow set up a similar optical system in Fig. fields 3, andinthe[10]. simulation focusing effect by a high-NA microscope objective was modeled with the Richards-Wolf integral files including the source codes can be found in [20]. in their work. our example, we set up a similar optical system as in Fig.a3,wavelength and the simulation The input is aInlinearly polarized (along the x axis) plane wave, with of 633 nm. files including the source codes can be found in [20]. The plane wave is truncated in a circular shape with the diameter of 24 mm. An aspheric lens inputfrom is a linearly (along axis) plane wave,the with a wavelength of focal 633 nm. (No.The 49113) Edmundpolarized optics, with an the NAxof 0.66, focuses plane wave to the plane The plane wave is truncated in a circular shape with the diameter of 24 mm. An aspheric lens an 12.5 mm behind the lens. A rotatable O-type polarizer is placed at the focal plane, forming (No. 49113) from Edmund optics, with an NA of 0.66, focuses the plane wave to the focal plane angle α with respect to the x axis, and the surrounding medium is air. The simulation of light 12.5 mm behind the lens. A rotatable O-type polarizer is placed at the focal plane, forming an angle α with respect to the x axis, and the surrounding medium is air. The simulation of light propagating and focusing by the lens is done by the 2nd-generation field tracing technique from the software VirtualLab [19, 21], and only the polarizer needs to be programmed. We orientate the polarizer in both parallel (α = 0◦ ) and orthogonal (α = 90◦ ) positions with respect to the input linear polarization. The amplitudes of the field components and intensity distribution behind the polarizer are shown in Fig.4. Here, we present the result directly in the spatial domain,


plane wave aspheric lens

polarizer α y x z Fig. 3. A linearly polarized (along the x axis) plane wave is focused by an aspheric lens,

Fig. 3. A linearly polarized (along the x axis) plane wave is focused by an aspheric lens, and and a linear polarizer is placed at the focal plane, making an angle of α with respect to the a xlinear axis. polarizer is placed at the focal plane, making an angle of α with respect to the x axis.

propagating andthe focusing by the lens isisdone by the from 2nd-generation field tracingbe technique and also show Ez component which calculated Ex and Ey [5]. It should noted thatfrom ˜ thesoftware actual calculation involves and E as in Eq. (19) in the k domain. the VirtualLabonly [19,21], andE˜only the polarizer needs to be programmed. We orientate the x y polarizer in both parallel (α = 0◦ ) and orthogonal (α = 90◦ ) positions with respect2 to the input |Ey (x, y)| |Ez (x, y)| |Ex (x, y)| E(x, y) linear polarization. The amplitudes of the field components and intensity distribution behind Em 100 Em = 30 Em = 100 Em = 1.7 the polarizer are shown in Fig.4. Here, we present the result directly in the spatial domain, and also show the Ez component which is calculated from Ex and Ey [5]. It should be noted that the (a)calculation only involves E˜x and E˜y as in Eq. (19) in the k domain. actual A high-NA focusing lens may introduce polarization crosstalk [22] and, as a result, a non-zero 1 µm is seen in Fig. 4(a), at the focal plane but in front of the polarizer. In the high-NA Ey component focusing situation, a relatively strong Ez component appears and0 therefore an elliptical 0shape is seen in the intensity distribution. When the polarizer is parallel Etom the x axis [Fig. 4(b)],100the Ex Em = 30 Em = 0 Em = 100 component is transmitted while the Ey component is not. When the polarizer is orthogonally placed with respect to the input field polarization [Fig. 4(c)], the Ex component does not pass (b) while the Ey component may transmit due to crosstalk. In this case, the amplitude of Ey appears in a quadrupolar shape which looks similar to that in Fig. 4(a), however, with an even higher amplitude. Thus, it can be concluded that it is caused by the crosstalk effect from the input Ex 0 0 component. Due to the same reason as explained in the first example, the resulting Ey component Em 0.7 Em = 4.6asymmetry. 0 off-axis regions Em = 8.4 is non-zero only Einm =the and shows a slight The resulting intensity distributions in Fig. 4 are in good agreement with those in [9]. (c)

3.3.

Tilted polarizer

Korger et al. studied the interaction of a polarizer with an obliquely incident wave both 0 0 experimentally and theoretically. They used a geometric model to predict the effect of the polarizer. set up afield similar opticaland system as distribution: in Fig. 3, and the simulation files including the Fig.We 4. Electric components intensity (a) in front of the polarizer, (b) behind can the polarizer parallel to theAx linearly axis, and (c) behind the polarizer to the xwave, with source codes be found in [23]. polarized (along theorthogonal y axis) plane axis. The the field components are displayed withpropagates respect to thefirst individual a wavelength of amplitudes 633 nm isofused as the input. The plane wave through a tilted maximum, labeled with m in each sub-figure. All sub-figures share the same scale of the x polarizer and then reaches the detector plane. The polarizer is tilted around the y axis by an angle ◦ ◦ and y axes. The change in the intensity 0when α changes from 0 to 90 , and to 180◦ can of θ, which turns the polarizer onto the x -y plane. And the absorbing axis, i.e. the be visualized in the animation). The simulation from input field to the results in (a) tookoptic axis, of an O-type polarizer makes ofitφtook with to the y axis in theover x 0-y 2.5 s; from (a) to the resultsan in angle (b) or (c) 3 s;respect and the animation generation 180plane. The polarizerdifferent is embedded in air. propagation of the field between the non-parallel planes is done angles took 185The s with multi-core processing enabled. by VirtualLab using the technique in [24]. We program, in addition to the polarizer, a customized detector to calculate the normalized Stokes parameters. On the detector plane, the calculated

z Fig. 3. A linearly polarized (along the x axis) plane wave is focused by an aspheric lens, and a linear polarizer is placed at the focal plane, making an angle of α with respect to the Vol. 26, No. 8 | 16 Apr 2018 | OPTICS EXPRESS 9848 x axis.

and also show the Ez component which is calculated from Ex and Ey [5]. It should be noted that the actual calculation only involves E˜x and E˜y as in Eq. (19) in the k domain. |Ex (x, y)|

|Ey (x, y)|

Em = 100

Em = 1.7

|Ez (x, y)|

Em = 30

Em

E(x, y) 2

100

(a) 1 µm

Em = 100

Em = 0

Em = 30

0

0

Em

100

(b)

A high-NA focusing lens may introduce polarization crosstalk and, as a result, a non-zero Ey component is seen in row (a) of Fig. 4, at the focal plane but in0 front of the polarizer. In the 0 high-NA focusing situation, a relatively strong Ez component appears and therefore an elliptical Em 0.7 shape is seen in Ethe distribution. to the x axis [row (b) Em = 4.6 is parallel 0 Em = 8.4 When the polarizer m = intensity in Fig. 4], the Ex component is transmitted while the Ey component is not. When the polarizer is orthogonally placed with respect to the input field polarization [row (c) in Fig. 4], the Ex (c) component does not pass while the Ey component may transmit due to crosstalk. In this case, the amplitude of Ey appears in a quadrupolar shape which looks similar to that in row (a), however, with an even higher amplitude. Thus, it can be concluded that it is0caused by the crosstalk effect 0 from the input Ex component. Due to the same reason as explained in the first example, the Fig. 4. component Electric fieldis components and intensity distribution: (a) inand frontshows of the apolarizer, (b) resulting non-zero and onlyintensity in the off-axis regions slight asymmetry. Fig. E 4.yElectric field components distribution: (a) in front of the polarizer, (b) behind the polarizer parallel to the x axis, and (c) behind the polarizer orthogonal to the x The resulting intensity distributions in xFig. are in agreement withorthogonal those in [9]. behind the polarizer parallel to the axis,4 and (c)good behind the polarizer to the x axis. The amplitudes of the field components are displayed with respect to the individual axis. The amplitudes of the field components are displayed with respect to the individual maximum, labeled with m in each sub-figure. All sub-figures share the same scale of the x 3.3. maximum, Tilted polarizer with minintheeach sub-figure. sub-figures scale the x ◦ can 90◦◦same , and to 180◦of and y axes.labeled The change intensity when All α changes from share 0◦ to the and axes. Theinchange in the intensity when α changes from 0◦ to ,results and towave 180 can Korger etyvisualized al. studied the of a simulation polarizer with obliquely incident be the interaction animation). The from an input field to 90 the in (a) both tookbeexperivisualized in(a) thetoVisualization input fieldthe to effect the results in (a)180 took We 2.5and s; from the results in 1). (b) aThe orgeometric (c)simulation it took model 3 s;from andto the animation generation over mentally theoretically. They used predict of the polarizer. 2.5 s; fromangles (a) to took the results in (b) or (c) it took 3 s; and the animation generation over 180 different 185 s with multi-core processing enabled. set up a similar optical system as in Fig. 3, and the simulation files including the source codes can different angles took 185 s with multi-core processing enabled.

be found in [22]. A linearly polarized (along the y axis) plane wave, with a wavelength of 633 nm polarizer

plane wave x

detector φ absorb ing axis

θ y

y x z

x (a) three-dimensional view

(b) section view

Fig.5.5. A linearly polarized (along y axis) wave propagated a titled Fig. A linearly polarized (along the ythe axis) plane plane wave propagated throughthrough a titled polarizer. polarizer. The polarizer is tilted around the y axis by a variable angle of θ as sketched in The polarizer is tilted around the y axis by a variable angle of θ as sketched in (a), and the (a), and the absorbing axis of the polarizer makes a fixed angle of φ = 94.5◦ with respect to absorbing axis of the polarizer makes a fixed angle of φ = 94.5◦ with respect to the y axis the y axis within the polarizer plane as in (b). within the polarizer plane as in (b).

is used as the input. The plane wave propagates first through a tilted polarizer and then reaches the detector plane. The polarizer is tilted around the y axis by an angle of θ, which turns the polarizer onto the x -y plane. And the absorbing axis, i.e. the optic axis, of an O-type polarizer makes an angle of φ with respect to the y axis in the x -y plane. The polarizer is embedded in air. The propagation of the field between the non-parallel planes is done by VirtualLab using the technique in [23]. We program, in addition to the polarizer, a customized detector to calculate the normalized Stokes parameters. On the detector plane, the calculated Stokes parameters are shown in Fig. 6. A dramatic change in the normalized Stokes parameters s1 and s2 can be seen with large tilt angle θ. This result is in accordance with that from [4]. With the full scan through


Stokes parameters are shown in Fig. 6. A dramatic change in the normalized Stokes parameters

Stokes parameters

s2 = Ex Ey∗ + Ex∗ Ey /S0

s1 = |Ex | 2 − |Ey | 2 /S0

+1

s3 = i(Ex Ey ∗ − Ex∗ Ey )/S0

0

−1

0

20

40

60

tilt angle θ [◦ ]

80

0

20

40

60


80

0

20

40

60

80


Fig. 6.Normalized Stokes parameters in the detector plane behind the tilted polarizer, with Fig. 6.Normalized Stokes parameters in the detector plane behind the tilted polarizer, with S0 = |Ex2| 2 + |Ey2| 2 . The whole simulation over 89 different tilt angles took 46 s with S0 = |Ex | + |Ey | . The whole simulation over 89 different tilt angles took 46 s with multi-core processing enabled. multi-core processing enabled.

s2 can be seen with large tilt angle θ. This result is in accordance with that from [4]. With s14.andSummary the full scan through the tilt angle range from 0 to 90◦ and in comparison with the reference, the We present a model for an idealized polarizer, which is defined and also numerically implemented validity of our idealized polarizer model can be verified. in the k domain, but which can also provide the results in the spatial domain via Fourier transform. to the mathematically concise definition in the k domain, the idealized functionality of a 4.DueSummary polarizer is clearly stated: it maintains and transmits a certain transverse polarization mode We present model an idealized polarizer, which is defined numerically (either TEaor TM)for from the isotropic medium in front of, to and the also isotropic mediumimplemented behind the which also provide thedescribed results in by the aspatial Fourier transform. inpolarizer. the k domain, The but effect of acan polarizer can be 2 × 2domain matrix via with respect to the fieldtocomponents in an analytical and the polarization can befunctionality clearly seenof ina Due the mathematically concise form, definition in the k domain,crosstalk the idealized the non-zero off-axis element in the and 2 × 2transmits matrix. The traditional Jones matrix method be polarizer is clearly stated: it maintains a certain transverse polarization modecan (either recovered a special case under paraxial With the effect expressed in a the 2×2-matrix TE or TM) as from the isotropic medium in illumination. front of, to the isotropic medium behind polarizer. form, ourof model can be can conveniently usedby in acombination computational In The effect a polarizer be described 2 × 2 matrixwith withother respect to the fieldmethods. components to form, previous on non-paraxial polarizer models, thein extended Jones off-axis matrix incomparison an analytical andworks the polarization crosstalk can be clearlye.g. seen the non-zero methodin[6], is derived with respect the modes of polarization, which element theour 2 ×method 2 matrix. The traditional Jonestomatrix method can be recovered asprovides a special a physical insight into the functionality of effect a polarizer. As a in result, allows usform, to construct an case under paraxial illumination. With the expressed a 2 ×it 2-matrix our model idealized model regardless of the structural material information of the actual, physicallyto can be conveniently used in combination withand other computational methods. In comparison real polarizer. previous works on non-paraxial polarizer models, e.g. the extended Jones matrix method [6], our method is derived with respect to the modes of polarization, which provides a physical insight Acknowledgement into the functionality of a polarizer. As a result, it allows us to construct an idealized model regardless theOlga structural and material of the physically We thankof Ms. Baladron-Zorita forinformation her assistance withactual, proofreading andreal her polarizer. general help with the paper. This work is supported by the Thuringian Ministry of Economy, Labor and Funding Technology funded from the European Social Fund. Thuringian Ministry of Economy, Labor and Technology funded from the European Social Fund (ESF) (2017 SDP 0018). Acknowledgment We thank Ms. Olga Baladron-Zorita for her assistance with proofreading and her general help with the paper. Disclosures Site Zhang: LightTrans International UG (E). Christian Hellmann: LightTrans International UG (I,E), Wyrowski Photonics UG (I,E). Frank Wyrowski: LightTrans International UG (I), Wyrowski Photonics UG (I).