Non-paraxial idealized polarizer model

An idealized polarizer model that works without the structural andmaterial 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). Vol. 26, No. 8 | 16 Apr 2018 | OPTICS EXPRESS 9840 #323448 https://doi.org/10.1364/OE.26.009840 Journal © 2018 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).


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][3][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 where E in x/y and E out x/y are the input field in the isotropic medium in front of the polarizer and the 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][12][13] can be used to realize the transformation between the two domains with high efficiency.

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 axis, it is possible to handle polarizers with any other orientation, as shown in Fig. 1 (c).Then, 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  the dielectric permittivity tensor can be written as 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 is the Fourier transform of E x (ρ, z) (similar rules apply for the other field components).In addition, we define η 0 = µ 0 / 0 as a scaling factor for the magnetic field [15], k 0 = 2π/λ as the wavenumber in vacuum, and n x = k x /k 0 and n y = k y /k 0 as normalized values of k x and k y for convenience.Eq. ( 3) represents a set of ordinary differential equations, whose general solution can be found in analytical form Ẽx (κ, z) Ẽy (κ, z) where we use the notation for some of the matrix elements, we define the dielectric permittivity tensor can be written as 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 d dz Ẽx (κ, z) Ẽy (κ, z) is the Fourier transform of E x (ρ, z) (similar rules apply for the other field components).In addition, we define η 0 = µ 0 / 0 as a scaling factor for the magnetic field [15], k 0 = 2π/λ as the wavenumber in vacuum, and n x = k x /k 0 and n y = k y /k 0 as normalized values of k x and k y for convenience.Equation (3) represents a set of ordinary differential equations, whose general solution can be found in analytical form Ẽx (κ, z) Ẽy (κ, z) where we use the notation for some of the matrix elements, we define 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 Ṽ ⊥ (κ, z) = Ẽx , Ẽy , η 0 Hx , η 0 Hy T , and introducing each column of the 4 × 4 matrix in Eq. ( 4) as a polarization mode vector we can rewrite Eq. ( 4) as 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 for an E-type polarizer that transmits TM mode only.That means only one mode is maintained with either pure TE polarization W TE + (O-type) or pure TM polarization W TM + (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 where the TM and TE polarization mode vectors in this case are defined as with and with the identical propagation constant In Eqs.(11)(12)(13)(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 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 for an O-type polarizer, and for an E-type polarizer.In contrast to the expression given in Eq. ( 1) which is defined for the E x and E y 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 By substituting Eq. ( 18) into Eq.( 16), we obtain 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 WB (κ).This gives rise to polarization crosstalk between the input and output field components.Moreover, the anti-diagonal element WB (κ) is dependent on the spatial frequency κ and, as defined in Eq. ( 13), WB = n x n y /(− + n 2 x ) is non-zero only when k x 0 and k y 0, since n x and n y are just normalized values of them.
The result in (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 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 WB (κ) in both Eqs.(19) and (20) vanishes.That leaves a constant 2 × 2 matrix in the k domain, and 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.

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.

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. Eq. ( 19) in the k domain.The algorithmic process as well as the simulation results in both domains are visualized in Fig. 2.
For any input field given in the spatial domain, it is always possible to define the input field in the k domain with the help of the Fourier transform.In our case, with the Gaussian input field, the Fourier transform can even be evaluated analytically, and the results are still Gaussian distributions, as shown in Fig. 2 (a).This is the input of the numerical algorithm.Then, we follow Eq. ( 19) and multiply a 2 × 2 matrix on each spatial frequency κ.Since the 2 × 2 matrix has a non-zero anti-diagonal element WB (κ), it converts the Ẽx component from the input to the Ẽy component of the output, as in Fig. 2 (c).The resulting Ẽy of the output is modulated by WB (κ) in the k domain.Thus, we see that non-zero values only appear in the off-axis regions, because only there k x k y 0. In addition, an asymmetry in the amplitude of Ẽy can be seen as well, and that is caused by the denominator in the expression of WB (κ) which takes the form (− + n 2 x ).An inverse Fourier transform can be employed to visualize the field also in the spatial domain, as is shown in Fig. 2 (b) and (d), and further analysis can be done afterwards.

Polarizer in focal region
Martínez-Herrero et al. studied how ideal polarizers affect highly focused fields in [10].The focusing effect by a high-NA microscope objective was modeled with the Richards-Wolf integral in their work.In our example, we set up a similar optical system as in Fig. 3, and the simulation files including the source codes can be found in [20].
The input is a linearly polarized (along the x axis) plane wave, with a wavelength of 633 nm.The plane wave is truncated in a circular shape with the diameter of 24 mm.An aspheric lens (No. 49113) from Edmund optics, with an NA of 0.66, focuses the plane wave to the focal plane 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, For any input field given in the spatial domain, it is always possible to define the input field in the k domain with the help of the Fourier transform.In our case, with the Gaussian input field, the Fourier transform can even be evaluated analytically, and the results are still Gaussian distributions, as shown in Fig. 2(a).This is the input of the numerical algorithm.Then, we follow Eq. ( 19) and multiply a 2 × 2 matrix on each spatial frequency κ.Since the 2 × 2 matrix has a non-zero anti-diagonal element WB (κ), it converts the Ẽx component from the input to the Ẽy component of the output, as in Fig. 2(c).The resulting Ẽy of the output is modulated by WB (κ) in the k domain.Thus, we see that non-zero values only appear in the off-axis regions, because only there k x k y 0. In addition, an asymmetry in the amplitude of Ẽy can be seen as well, and that is caused by the denominator in the expression of WB (κ) which takes the form (− + n 2 x ).An inverse Fourier transform can be employed to visualize the field also in the spatial domain, as is shown in Fig. 2(b) and Fig. 2(d), and further analysis can be done afterwards.

Polarizer in focal region
Martínez-Herrero et al. studied how ideal polarizers affect highly focused fields in [10].The focusing effect by a high-NA microscope objective was modeled with the Richards-Wolf integral in their work.In our example, we set up a similar optical system as in Fig. 3, and the simulation files including the source codes can be found in [20].
The input is a linearly polarized (along the x axis) plane wave, with a wavelength of 633 nm.The plane wave is truncated in a circular shape with the diameter of 24 mm.An aspheric lens (No. 49113) from Edmund optics, with an NA of 0.66, focuses the plane wave to the focal plane 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 x y z plane wave aspheric lens polarizer α 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 x axis.and also show the E z component which is calculated from E x and E y [5].It should be noted that the actual calculation only involves Ẽx and Ẽy as in Eq. ( 19) in the k domain.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, and also show the E z component which is calculated from E x and E y [5].It should be noted that the actual calculation only involves Ẽx and Ẽy as in Eq. ( 19) in the k domain.
A high-NA focusing lens may introduce polarization crosstalk [22] and, as a result, a non-zero E y component is seen in Fig. 4(a), at the focal plane but in front of the polarizer.In the high-NA focusing situation, a relatively strong E z component appears and therefore an elliptical shape is seen in the intensity distribution.When the polarizer is parallel to the x axis [Fig.4(b)], the E x component is transmitted while the E y component is not.When the polarizer is orthogonally placed with respect to the input field polarization [Fig.4(c)], the E x component does not pass while the E y component may transmit due to crosstalk.In this case, the amplitude of E y 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 E x component.Due to the same reason as explained in the first example, the resulting E y component is non-zero only in the off-axis regions and shows a slight asymmetry.The resulting intensity distributions in Fig. 4 are in good agreement with those in [9].

Tilted polarizer
Korger et al. studied the interaction of a polarizer with an obliquely incident wave both experimentally and theoretically.They used a geometric model to predict the effect of the polarizer.We set up a similar optical system as in Fig. 3, and the simulation files including the source codes can be found in [23].A linearly polarized (along the y axis) plane wave, with a wavelength of 633 nm 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 [24].We program, in addition to the polarizer, a customized detector to calculate the normalized Stokes parameters.On the detector plane, the calculated 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 x axis.and also show the E z component which is calculated from E x and E y [5].It should be noted that the actual calculation only involves Ẽx and Ẽy as in Eq. ( 19) in the k domain.A high-NA focusing lens may introduce polarization crosstalk and, as a result, a non-zero E y component is seen in row (a) of Fig. 4, at the focal plane but in front of the polarizer.In the high-NA focusing situation, a relatively strong E z component appears and therefore an elliptical shape is seen in the intensity distribution.When the polarizer is parallel to the x axis [row (b) in Fig. 4], the E x component is transmitted while the E y component is not.When the polarizer is orthogonally placed with respect to the input field polarization [row (c) in Fig. 4], the E x component does not pass while the E y component may transmit due to crosstalk.In this case, the amplitude of E y 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 is caused by the crosstalk effect from the input E x component.Due to the same reason as explained in the first example, the resulting E y component is non-zero only in the off-axis regions and shows a slight asymmetry.The resulting intensity distributions in Fig. 4 are in good agreement with those in [9].

Tilted polarizer
Korger et al. studied the interaction of a polarizer with an obliquely incident wave both experimentally and theoretically.They used a geometric model to predict the effect of the polarizer.We set up a similar optical system as in Fig. 3, and the simulation files including the source codes can be found in [22].A linearly polarized (along the y axis) plane wave, with a wavelength of 633 nm 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 s 1 and s 2 can be seen with large tilt angle θ.This result is in accordance with that from [4].With the full scan through

Summary
We present a model for an idealized polarizer, which is defined and also numerically implemented in the k domain, but which can also provide the results in the spatial domain via Fourier transform.Due to the mathematically concise definition in the k domain, the idealized functionality of a polarizer is clearly stated: it maintains and transmits a certain transverse polarization mode (either TE or TM) from the isotropic medium in front of, to the isotropic medium behind the polarizer.The effect of a polarizer can be described by a 2 × 2 matrix with respect to the field components in an analytical form, and the polarization crosstalk can be clearly seen in the non-zero off-axis element in the 2 × 2 matrix.The traditional Jones matrix method can be recovered as a special case under paraxial illumination.With the effect expressed in a 2×2-matrix form, our model can be conveniently used in combination with other computational methods.In comparison to 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 into the functionality of a polarizer.As a result, it allows us to construct an idealized model regardless of the structural and material information of the actual, physically real polarizer.s 1 and s 2 can be seen with large tilt angle θ.This result is in accordance with that from [4].With the full scan through the tilt angle range from 0 to 90 • and in comparison with the reference, the validity of our idealized polarizer model can be verified.

Summary
We present a model for an idealized polarizer, which is defined and also numerically implemented the k domain, but which can also provide the results in the spatial domain via Fourier transform.Due to the mathematically concise definition in the k domain, the idealized functionality of a polarizer is clearly stated: it maintains and transmits a certain transverse polarization mode (either TE or TM) from the isotropic medium in front of, to the isotropic medium behind the polarizer.The effect of a polarizer can be described by a 2 × 2 matrix with respect to the field components in an analytical form, and the polarization crosstalk can be clearly seen in the non-zero off-axis element in the 2 × 2 matrix.The traditional Jones matrix method can be recovered as a special case under paraxial illumination.With the effect expressed in a 2 × 2-matrix form, our model can be conveniently used in combination with other computational methods.In comparison to 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 into the functionality of a polarizer.As a result, it allows us to construct an idealized model regardless of the structural and material information of the actual, physically real polarizer.

Funding
Thuringian Ministry of Economy, Labor and Technology funded from the European Social Fund (ESF) (2017 SDP 0018).

Fig. 1 .
Fig. 1.Uniaxial crystal model for a polarizer.A Cartesian coordinate system x-y-z is set 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 rotated polarizer can be treated with the help of coordinate transformations, as shown in (c).

Fig. 1 .
Fig. 1.Uniaxial crystal model for a polarizer.A Cartesian coordinate system x-y-z is set 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 rotated polarizer can be treated with the help of coordinate transformations, as shown in (c).

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

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

Fig. 4 .
Fig. 4. Electric field components and intensity 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 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 and y axes.The change in the intensity when α changes from 0 • to 90 • , and to 180 • can be visualized in the animation).The simulation from input field to the results in (a) took 2.5 s; from (a) to the results in (b) or (c) it took 3 s; and the animation generation over 180 different angles took 185 s with multi-core processing enabled.

Fig. 3 .
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 x axis.

2 E m = 100 E m = 100 E m = 0 E m = 30 E m = 30 E m = 1 . 7 EFig. 4 .
Fig. 4. Electric field components and intensity 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 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 and y axes.The change in the intensity when α changes from 0 • to 90 • , and to 180 • can be visualized in the animation).The simulation from input field to the results in (a) took 2.5 s; from (a) to the results in (b) or (c) it took 3 s; and the animation generation over 180 different angles took 185 s with multi-core processing enabled.

Fig. 4 .
Fig. 4. Electric field components and intensity 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 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 and y axes.The change in the intensity when α changes from 0 • to 90 • , and to 180 • can be visualized in the Visualization 1).The simulation from input field to the results in (a) took 2.5 s; from (a) to the results in (b) or (c) it took 3 s; and the animation generation over 180 different angles took 185 s with multi-core processing enabled.

Fig. 5 .
Fig.5.A linearly polarized (along the y axis) plane wave propagated through a titled polarizer.The polarizer is tilted around the y axis by a variable angle of θ as sketched in (a), and the absorbing axis of the polarizer makes a fixed angle of φ = 94.5 • with respect to the y axis within the polarizer plane as in (b).

Fig. 5 . 0 − 1 s 1 =Fig. 6 .
Fig.5.A linearly polarized (along the y axis) plane wave propagated through a titled polarizer.The polarizer is tilted around the y axis by a variable angle of θ as sketched in (a), and the absorbing axis of the polarizer makes a fixed angle of φ = 94.5 • with respect to the y axis within the polarizer plane as in (b).