Non-paraxial idealized polarizer model

Site Zhang; Henri Partanen; Christian Hellmann; Frank Wyrowski

doi:10.1364/OE.26.009840

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

(\begin{array}{l} E_{x}^{out} (ρ, z^{out}) \\ E_{y}^{out} (ρ, z^{out}) \end{array}) = (\begin{matrix} 𝒞_{x x} & 𝒞_{x y} \\ 𝒞_{y x} & 𝒞_{y y} \end{matrix}) (\begin{array}{l} E_{x}^{in} (ρ, z^{in}) \\ E_{y}^{in} (ρ, z^{in}) \end{array}),

where

E_{x / y}^{in}

and

E_{x / y}^{out}

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 𝒞̱ 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 axis, it is possible to handle polarizers with any other orientation, as shown in Fig. 1 (c). Then, the dielectric permittivity tensor can be written as

\underline{∊} = (\begin{matrix} ∊_{e} & 0 & 0 \\ 0 & ∊_{o} & 0 \\ 0 & 0 & ∊_{o} \end{matrix}),

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

\frac{d}{d z} (\begin{array}{l} {\tilde{E}}_{x} (κ, z) \\ {\tilde{E}}_{y} (κ, z) \\ η_{0} {\tilde{H}}_{x} (κ, z) \\ η_{0} {\tilde{H}}_{y} (κ, z) \end{array}) = i k_{0} (\begin{matrix} 0 & 0 & n_{x} n_{y} / ∊_{o} & 1 - n_{x}^{2} / ∊_{o} \\ 0 & 0 & - 1 + n_{y}^{2} / ∊_{o} & - n_{x} n_{y} / ∊_{o} \\ - n_{x} n_{y} & - ∊_{o} + n_{x}^{2} & 0 & 0 \\ ∊_{e} - n_{y}^{2} & n_{x} n_{y} & 0 & 0 \end{matrix}) (\begin{array}{l} {\tilde{E}}_{x} (κ, z) \\ {\tilde{E}}_{y} (κ, z) \\ η_{0} {\tilde{H}}_{x} (κ, z) \\ η_{0} {\tilde{H}}_{y} (κ, z) \end{array}),

where κ = (k_x, k_y) is the transverse variable in the k domain, Ẽ_x(κ, z) = ℱ[E_x(ρ, z)] is the Fourier transform of E_x(ρ, z) (similar rules apply for the other field components). In addition, we define

η_{0} = \sqrt{μ_{0} / ∊_{0}}

as a scaling factor for the magnetic field [15], k₀ = 2π/λ as the wavenumber in vacuum, and n_x = k_x/k₀ and n_y = k_y/k₀ 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

(\begin{array}{l} {\tilde{E}}_{x} (κ, z) \\ {\tilde{E}}_{y} (κ, z) \\ η_{0} {\tilde{H}}_{x} (κ, z) \\ η_{0} {\tilde{H}}_{y} (κ, z) \end{array}) = (\begin{array}{l} 1 & 0 & 1 & 0 \\ W_{B} & W_{D} & W_{B} & - W_{D} \\ 0 & 1 & 0 & 1 \\ W_{C} & W_{B} & - W_{C} & W_{B} \end{array}) (\begin{array}{l} C_{+}^{TM} exp (γ^{TM} z) \\ C_{+}^{TE} exp (γ^{TE} z) \\ C_{-}^{TM} exp (- γ^{TM} z) \\ C_{-}^{TE} exp (- γ^{TE} z) \end{array}),

where we use the notation

W_{B} (κ) = \frac{n_{x} n_{y}}{- ∊_{0} + n_{x}^{2}}, W_{C} (κ) = \frac{1}{1 - n_{x}^{2} / ∊_{o}} \frac{1}{i k_{0}} γ^{TM}, W_{D} (κ) = \frac{1}{- ∊_{o} + n_{x}^{2}} \frac{1}{i k_{0}} γ^{TE};

for some of the matrix elements, we define

γ^{TM} (κ) = i k_{0} \sqrt{∊_{e} - n_{y}^{2} - n_{x}^{2} ∊_{e} / ∊_{o}}, γ^{TE} (κ) = i k_{0} \sqrt{∊_{e} - n_{x}^{2} - n_{y}^{2}} .

as the propagation constants, and

C_{\pm}^{TE / TM}

as the complex coefficients which are to be determined by boundary conditions. By defining the transverse electromagnetic field vector

{\tilde{V}}_{⊥} (κ, z) = {(\begin{matrix} {\tilde{E}}_{x}, & {\tilde{E}}_{y}, & η_{0} {\tilde{H}}_{x}, & η_{0} {\tilde{H}}_{y} \end{matrix})}^{T},

and introducing each column of the 4 × 4 matrix in Eq. (4) as a polarization mode vector

\begin{matrix} W_{+}^{TM} (κ) = {(\begin{matrix} 1, & W_{B}, & 0, & W_{C} \end{matrix})}^{T}, & W_{+}^{TE} (κ) = {(\begin{matrix} 0, & W_{D}, & 1, & W_{B} \end{matrix})}^{T}, \\ W_{-}^{TE} (κ) = {(\begin{matrix} 1, & W_{B}, & 0, & - W_{C} \end{matrix})}^{T}, & W_{-}^{TE} (κ) = {(\begin{matrix} 0, & - W_{D}, & 1, & W_{B} \end{matrix})}^{T}, \end{matrix}

we can rewrite Eq. (4) as

\begin{array}{l} {\tilde{V}}_{⊥} (κ, z) & = W_{+}^{TM} C_{+}^{TM} exp (γ^{TM} z) + W_{+}^{TE} C_{+}^{TE} exp (γ^{TE} z) \\ + W_{-}^{TM} C_{-}^{TM} exp (- γ^{TM} z) + W_{-}^{TE} C_{-}^{TE} exp (- γ^{TE} z) . \end{array}

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.

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).

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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

for an O-type polarizer that transmits TE mode only, or

| exp (γ^{TM} d) | ≃ 1, | exp (γ^{TE} d) | ≃ 0

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

\begin{array}{l} {\tilde{V}}_{⊥}^{in} (κ, z) & = & {\bar{W}}_{+}^{TM} C_{+}^{in, TM} exp (\bar{γ} z) + {\bar{W}}_{+}^{TE} C_{+}^{in, TE} exp (\bar{γ} z), \\ {\tilde{V}}_{⊥}^{out} (κ, z) & = & {\bar{W}}_{+}^{TM} C_{+}^{out, TM} exp (\bar{γ} z) + {\bar{W}}_{+}^{TE} C_{+}^{out, TE} exp (\bar{γ} z), \end{array}

where the TM and TE polarization mode vectors in this case are defined as

\begin{array}{l} {\bar{W}}_{+}^{TM} (κ) = {(\begin{matrix} 1, & {\bar{W}}_{B}, & 0, & {\bar{W}}_{C} \end{matrix})}^{T}, & {\bar{W}}_{+}^{TE} (κ) = {(\begin{matrix} 0, & {\bar{W}}_{D}, & 1, & {\bar{W}}_{B} \end{matrix})}^{T}, \end{array}

with

{\bar{W}}_{B} (κ) = \frac{n_{x} n_{y}}{- ∊ + n_{x}^{2}}, {\bar{W}}_{C} (κ) = \frac{1}{1 - n_{x}^{2} / ∊} \frac{1}{i k_{0}} \bar{γ}, {\bar{W}}_{D} (κ) = \frac{1}{- ∊ + n_{x}^{2}} \frac{1}{i k_{0}} \bar{γ},

and with the identical propagation constant

\bar{γ} (κ) = i k_{0} \sqrt{∊ - n_{x}^{2} - n_{y}^{2}} .

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

(\begin{array}{l} C_{+}^{out, TM} (κ, z^{out}) \\ C_{+}^{out, TE} (κ, z^{out}) \end{array}) = (\begin{matrix} s_{11}^{+ +} (κ) & s_{12}^{+ +} (κ) \\ s_{21}^{+ +} (κ) & s_{22}^{+ +} (κ) \end{matrix}) (\begin{array}{l} C_{+}^{in, TM} (κ, z^{in}) \\ C_{+}^{in, TE} (κ, z^{in}) \end{array}) .

Here the 2 × 2 matrix s̱⁺⁺ 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− zⁱⁿ 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

(\begin{array}{l} C_{+}^{out, TM} (κ, z^{out}) \\ C_{+}^{out, TE} (κ, z^{out}) \end{array}) = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) (\begin{array}{l} C_{+}^{in, TM} (κ, z^{in}) \\ C_{+}^{in, TE} (κ, z^{in}) \end{array})

for an O-type polarizer, and

(\begin{array}{l} C_{+}^{out, TM} (κ, z^{out}) \\ C_{+}^{out, TE} (κ, z^{out}) \end{array}) = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) (\begin{array}{l} C_{+}^{in, TM} (κ, z^{in}) \\ C_{+}^{in, TE} (κ, z^{in}) \end{array})

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

(\begin{array}{l} C_{+}^{in / out, TM} (κ, z^{in / out}) \\ C_{+}^{in / out, TE} (κ, z^{in / out}) \end{array}) = {(\begin{matrix} 1 & 0 \\ {\bar{W}}_{B} (κ) & {\bar{W}}_{D} (κ) \end{matrix})}^{- 1} (\begin{array}{l} {\tilde{E}}_{x}^{in / out} (κ, z^{in / out}) \\ {\tilde{E}}_{y}^{in / out} (κ, z^{in / out}) \end{array}) .

By substituting Eq. (18) into Eq. (16), we obtain

\begin{array}{l} (\begin{array}{l} {\tilde{E}}_{x}^{out} (κ, z^{out}) \\ {\tilde{E}}_{y}^{out} (κ, z^{out}) \end{array}) & = & (\begin{array}{l} 1 & 0 \\ {\bar{W}}_{B} (κ) & {\bar{W}}_{D} (κ) \end{array}) (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) {(\begin{array}{l} 1 & 0 \\ {\bar{W}}_{B} (κ) & {\bar{W}}_{D} (κ) \end{array})}^{- 1} (\begin{array}{l} {\tilde{E}}_{x}^{in} (κ, z^{in}) \\ {\tilde{E}}_{y}^{in} (κ, z^{in}) \end{array}) \\ = & (\begin{array}{l} 0 & 0 \\ - {\bar{W}}_{B} (κ) & 1 \end{array}) (\begin{array}{l} {\tilde{E}}_{x}^{in} (κ, z^{in}) \\ {\tilde{E}}_{y}^{in} (κ, z^{in}) \end{array}) . \end{array}

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 W̄_B(κ). This gives rise to polarization crosstalk between the input and output field components. Moreover, the anti-diagonal element W̄_B(κ) is dependent on the spatial frequency κ and, as defined in Eq. (13),

{\bar{W}}_{B} = n_{x} n_{y} / (- ∊ + n_{x}^{2})

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 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

\begin{array}{l} (\begin{array}{l} E_{x}^{out} (ρ, z^{out}) \\ E_{y}^{out} (ρ, z^{out}) \end{array}) & = & (\begin{array}{l} ℱ^{- 1} & 0 \\ 0 & ℱ^{- 1} \end{array}) (\begin{matrix} 0 & 0 \\ - {\bar{W}}_{B} (κ) & 1 \end{matrix}) (\begin{array}{l} ℱ & 0 \\ 0 & ℱ \end{array}) (\begin{array}{l} E_{x}^{in} (ρ, z^{in}) \\ E_{y}^{in} (ρ, z^{in}) \end{array}) \\ = & (\begin{array}{l} 0 & 0 \\ - ℱ^{- 1} {\bar{W}}_{B} (κ) ℱ & 1 \end{array}) (\begin{array}{l} E_{x}^{in} (ρ, z^{in}) \\ E_{y}^{in} (ρ, z^{in}) \end{array}), \end{array}

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 W̄_B(κ) in both Eqs. (19) and (20) vanishes. That leaves a constant 2 × 2 matrix

(\begin{array}{l} {\tilde{E}}_{x}^{out} (κ, z^{out}) \\ {\tilde{E}}_{y}^{out} (κ, z^{out}) \end{array}) = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) (\begin{array}{l} {\tilde{E}}_{x}^{in} (κ, z^{in}) \\ {\tilde{E}}_{y}^{in} (κ, z^{in}) \end{array})

in the k domain, and

(\begin{array}{l} {\tilde{E}}_{x}^{out} (ρ, z^{out}) \\ {\tilde{E}}_{y}^{out} (ρ, z^{out}) \end{array}) = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) (\begin{array}{l} {\tilde{E}}_{x}^{in} (ρ, z^{in}) \\ {\tilde{E}}_{y}^{in} (ρ, z^{in}) \end{array})

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.

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.

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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 W̄_B(κ), it converts the Ẽ_x component from the input to the Ẽ_y component of the output, as in Fig. 2 (c). The resulting Ẽ_y of the output is modulated by W̄_B(κ) in the k domain. Thus, we see that non-zero values only appear in the off-axis regions, because only there k_xk_y ≠ 0. In addition, an asymmetry in the amplitude of Ẽ_y can be seen as well, and that is caused by the denominator in the expression of W̄_B(κ) which takes the form ( $- ∊ + n_{x}^{2}$ ). 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.

3.2. 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].

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.

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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, 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 Ẽ_x and Ẽ_y as in Eq. (19) in the k domain.

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.

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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].

3.3. 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 Fig. 5. 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 the tilt angle range from 0 to 90° and in comparison with the reference, the validity of our idealized polarizer model can be verified.

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).

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

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4. 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.

Funding

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).

References and links

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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

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Non-paraxial idealized polarizer model

Abstract

1. Introduction

2. Idealized polarizer model

3. Examples

3.1. Polarizer under divergent beam illumination

3.2. Polarizer in focal region

3.3. Tilted polarizer

4. Summary

Funding

Acknowledgment

Disclosures

References and links

Supplementary Material (1)

Cited By

Figures (6)

Equations (23)

Optics Express