Generalized elliptical retarder design and construction using nematic and cholesteric phase liquid crystal polymers

Sawyer Miller; Linan Jiang; Stanley Pau

doi:10.1364/OE.456874

1. Introduction

Retarders are essential polarization components used in polarimetry [1–4], display technology [5–7], intereferometry [8,9], and quantum optics [10,11]. Common commercial types of retarders are classified as linear retarders, where the eigenpolarizations of these optics are linear polarization states [12]. Linear retarders can be made of birefringent materials, such as transparent uniaxial crystals [12], nematic phase liquid crystal polymer (LCP) [13], film with form birefringence [14], and metamaterial [15]. Other common retarders are circular retarders which exhibit optical activity. The eigenpolarizations of these optics are limited to the circular polarization states. Circular retarders can be made of materials with chiral symmetry, such as glucose, organic materials, and chiral crystals [12]. Both linear and circular retarders are special cases of the generalized elliptical retarder (${ER}$), which has orthogonal elliptical eigenpolarzations. Elliptical retarders are not as common as linear or circular retarders, but have been explored for uses in polarimetry and phase compensators to eliminate polarization aberrations [16].

In this work, the concept of the ${ER}$ is described using the Pauli spin matrices. An arbitrary elliptical retarder, with eigenpolarizations located at any position on the Poincaré sphere, can be realized using multiple layers of linear and circular retarders. The linear and circular retarders used in this work are LCP and cholesteric phase liquid crystal polymer (ChLCP), respectively. We find through simulation and experimental verification that there exists a large achromatic region of circular retardance in ChLCP outside of the Bragg reflection region. Additionally, using an eigen-decomposition of the elliptical retarder Jones matrix, eigenpolarizations and retardance can be specified and then fabricated using linear and circular retarders, each of which can be made from either LCP or ChLCP.

2. Theory

All retarders can be decomposed into their linear and circular retardance components, and as such the eigenpolarizations can be determined as well [12]. Of importance to this work is the inverse problem, where two eigenpolarizations and a retardance are specified, resulting in an ${ER}$. Eigenpolarizations are important for retarders as they define the axis of rotation on the Poincaré sphere for polarization state change. All pure retarders have orthogonal eigenpolarizations due to the unitary nature of the Jones matrices representing these retarders. The axis of rotation is constructed by connecting the two orthogonal eigenpolarizations on the Poincaré sphere. Figure 1 shows three different retarders and their operation on the Poincaré sphere. Figure 1(a) shows the axis of rotation for linear retarders defined by their linear eigenpolarizations: a 0/90-degree orientated retarder is depicted in dark blue, while a 45/135-degree linear retarder is depicted in light blue. Notice that the axis of rotation for these retarders is limited to the ${S_1/S_2}$ plane of the Poincaré sphere. The use of a linear retarder allows for polarization states changing not only in longitude (i.e., polarization major axis rotation) but also in latitude (i.e., polarization ellipticity change). Figure 1(b) depicts a circular retarder, producing circular retardance or optical activity. The two eigenpolarizations are circular; therefore, the axis of rotation on the Poincaré sphere is limited to the ${S_3}$ axis. Only longitudinal changes on the Poincaré sphere can occur (i.e., polarization major axis orientations). Figure 1(c) shows an arbitrary elliptical retarder. The axis of rotation on the Poincaré sphere is not limited to the ${S_1}$, ${S_2}$, or ${S_3}$ axes. Changes in latitude and longitude on the Poincaré sphere are possible with this type of retarder.

Fig. 1. (a) Linear retarders can be represented on the Poincaré sphere with their axes of rotation being limited to the ${S_1}$/${S_2}$-plane. Fast-axis orientations of 0/90 degrees are shown in dark blue and 45/135 are shown in light blue. Changes in latitude and longitude on the Poincaré sphere can be achieved with linear retarders. (b) Circular retarders are represented on the Poincaré sphere with their axis of rotation being limited to the ${S_3}$ axis. Circular retarders only change the longitude on the Poincaré sphere. (c) A generalized elliptical retarder can be represented with its axis of rotation being defined by a line connecting the two orthogonal eigenpolarizations. Elliptical retarders can change the latitude and longitude on the Poincaré sphere.

Download Full Size | PDF

The Jones matrix of ${ER}$ can be constructed using eigen-decomposition, where the eigenpolarizations of the retarder are as follows [12,17]:

(1)$$\vec{E_1} = \begin{pmatrix} \mathrm{cos}\,\theta \\ \mathrm{sin}\,\theta\, e^{i\delta} \end{pmatrix}\,\,,\,\, \vec{E_2} = \begin{pmatrix} -\mathrm{sin}\,\theta\, e^{{-}i\delta} \\ \mathrm{cos}\,\theta \end{pmatrix}$$

Here, ${\theta }$ is an auxiliary angle representing the ellipticity and major-axis orientation of the fast eigenpolarization; ${\delta }$ represents the phase delay between horizontal and vertical components of the eigenpolarization. Using the two eigenpolarizations, the Jones matrix of ${ER}$ can be constructed, where the eigenvalues correspond to the retardance between the two eigenpolarizations.

(2)$$\begin{aligned} ER &= \begin{bmatrix} \mathrm{cos}^2\,\theta e^{i\frac{\phi}{2}} + \mathrm{sin}^2\,\theta e^{{-}i\frac{\phi}{2}} & -i\mathrm{sin}\,2\theta \,\mathrm{sin}\frac{\phi}{2} e^{{-}i\delta} \\ -i\mathrm{sin}\,2\theta \,\mathrm{sin}\frac{\phi}{2} e^{i\delta} & \mathrm{sin}^2\,\theta\, e^{i\frac{\phi}{2}} + \mathrm{cos}^2\,\theta e^{{-}i\frac{\phi}{2}}\end{bmatrix}\\ & = \begin{bmatrix} \mathrm{cos}\,\theta & -\mathrm{sin}\,\theta e^{{-}i\delta}\\ \mathrm{sin}\,\theta e^{i\delta} & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\phi/2} & 0\\ 0 & e^{{-}i\phi/2} \end{bmatrix} \begin{bmatrix} \mathrm{cos}\,\theta & \mathrm{sin}\,\theta e^{{-}i\delta}\\ -\mathrm{sin}\,\theta e^{i\delta} & \mathrm{cos}\,\theta \end{bmatrix} \end{aligned}$$

${\phi }$ represents the retardance introduced by ${ER}$ for the eigenpolarizations. It should be noted that the equation above can be reduced to the Jones matrix of a linear retarder if ${\delta = 0}$, and for a circular retarder if ${\theta = \pi /4}$ and ${\delta = \pi /2}$.

2.1 Jones matrix decomposition with Pauli matrices

Jones matrices have the ability to be decomposed in two different useful ways with Pauli spin matrices proposed by Whitney [18]. The first method uses a weighted sum of Pauli matrices to decompose a Jones matrix into the linear and circular components for both diattenuation and retardance. The decomposition is shown below:

(3)$$J = \begin{bmatrix} j_{xx} & j_{xy} \\ j_{yx} & j_{yy} \end{bmatrix} =c_0\,\sigma_0+c_1\,\sigma_1+c_2\,\sigma_2+c_3\,\sigma_3$$

where the Pauli spin matrices are:

(4)$$\sigma_0 = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\,,\, \sigma_1 = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}\,,\, \sigma_2 = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\,,\, \sigma_3 = \begin{bmatrix} 0 & -i\\ i & 0 \end{bmatrix}\,\,\,$$

${\sigma _0}$ represents a non-polarizing and non-absorbing identify matrix, ${\sigma _1}$ represents a half-wave of linear retardance between 0- and 90-degree light, ${\sigma _2}$ represents a half-wave of linear retardance between 45- and 135-degree light, and ${\sigma _3}$ represents a half-wave of circular retardance between left and right circular light. The coefficients, ${c_i}$ are complex in general, with the real part relating to diattenuation and the imaginary part relating to phase.

Using the Pauli matrix decomposition, it is much easier to see how cascading retarders can lead to other optical retardance properties. For example:

(5)$$\sigma_1\cdot\sigma_2 =i\sigma_3$$

From this, it is easily shown that cascading linear retarders with unaligned fast-axes produces a circular retardance component, thus resulting in an elliptical retarder. Additionally, a linear retarder followed by a circular retarder produces an elliptical retarder by the same analysis. Using the same Pauli matrices, an even more useful decomposition is found by taking the logarithm of the Jones matrix to produce a weighted summation of the Pauli matrices. However, the coefficients in this decomposition relate directly to the amount of retardance and diattenuation found in the polarization component. The decomposition of an elliptical retarder using the matrix logarithm produces the form [12,19]:

(6)$$2i\,\mathrm{ln}(J) = \alpha\sigma_0+\phi_H\sigma_1+\phi_{45}\sigma_2+\phi_L\sigma_3$$

where, ${\alpha }$ is a polarization independent absolute phase, and ${\phi _H, \phi _{45}, \phi _L}$ are the amounts of horizontal, 45-degree, and left circular retardance present in the elliptical retarder respectively. Given the amount of retardance, ${\phi }$ desired in the elliptical retarder from Eq. (2), the corresponding retardance components obey the following:

(7)$$\phi = \sqrt{\phi_H^2+\phi_{45}^2+\phi_L^2}$$

The eigenpolarizations of ${J}$ from Eq. (2) can be expressed as Stokes parameters through the retardance definitions from above as:

(8)$$\vec{E_{V1}}= \frac{1}{\phi} \begin{pmatrix} \phi \\ \phi_H \\ \phi_{45} \\ \phi_L \end{pmatrix}\;\;,\; \vec{E_{V2}}= \frac{1}{\phi} \begin{pmatrix} \phi \\ -\phi_H \\ -\phi_{45} \\ -\phi_L \end{pmatrix}$$

Additionally, Eq. (1) can be expressed as Stokes parameters:

(9)$$\vec{E_{S1}} = \begin{pmatrix} 1 \\ \mathrm{cos}\,2\theta \\ \mathrm{sin}\,2\theta \,\mathrm{cos}\,\delta\\ \mathrm{sin}\,2\theta \,\mathrm{sin}\,\delta \end{pmatrix}\;\;,\; \vec{E_{S2}} = \begin{pmatrix} 1 \\ -\mathrm{cos}\,2\theta \\ -\mathrm{sin}\,2\theta \,\mathrm{cos}\,\delta\\ -\mathrm{sin}\,2\theta \,\mathrm{sin}\,\delta \end{pmatrix}$$

With Eqs. (8) and (9), a relation between the desired eigenpolarizations and needed retardance components can be expressed. Azimuth (${\alpha }$) and ellipticity (${\epsilon }$) value of the Poincaré sphere can be related to the Stokes parameters in Eq. (9) as the following:

(10)$$\mathrm{tan}\, \delta = \frac{\mathrm{tan}\,2\epsilon}{\mathrm{sin}\,2\alpha}\,,\,\,\,\mathrm{cos}\,2\theta= \mathrm{cos}\,2\alpha\,\mathrm{cos}\,2\epsilon$$

It should be noted that the retardance components specified in Eq. (8) defining the eigenpolarizations are a result of the cascaded retarders. Multiplication of the Pauli matrices must be carried through to understand the full effect of any produced linear or circular retardance. As demonstrated by Eq. (5), cascaded retarders with differing orientations and eigenpolarizations will produce a combination of linear and circular retardance which may not be present in any of the individual retarders. It is also observed that there is no single unique combination of linear and circular retarders, which may produce the desired eigenpolarizations. Given the nature of the Pauli matrix multiplication shown in Eq. (5), combinations of both linear and circular retarders are viable in producing an elliptical retarder with the same properties. Eigenpolarization of an elliptical retarder can be calculated from a Mueller matrix utilizing the Lu-Chipman Decomposition [20]. Utilizing the pure retarder Mueller matrix from the decomposition (${R_m}$), the following operation can be performed to calculate the retardance (${\phi }$) produced by the elliptical retarder.

(11)$$\phi = \mathrm{cos}^{{-}1}\Big(\frac{\mathrm{Tr}(R_m)}{2}-1\Big)$$

where Tr is the trace operation. The individual retardance parameters shown in Eq. (8) can be calculated as:

(12)$$(\phi_H,\phi_{45},\phi_L) = \frac{\phi}{2\,\mathrm{sin}\,\phi}(m_{23}-m_{32},m_{31}-m_{13},m_{21}-m_{12})$$

where ${m}$ are the matrix elements from ${R_m}$ and the matrix element notation is consistent with Chipman [12].

2.2 Elliptical retarder using linear retarder of a different orientation

As shown in Eq. (5), cascading linear retarders with different fast-axis orientations causes a circular retardance component to be produced. In return, the overall retardance will be elliptical. Linear retarders are often made from uniaxial materials, such as nematic phase LCP. For example, we consider two quarter-wave linear retarders [12,21], the first with the fast-axis oriented at 0 degrees, the second with the fast-axis oriented at 45 degrees. These cascaded retarders produce eigenpolarizations of ${\vec {E_{N1}}=(1,1/\sqrt {3},1/\sqrt {3},1/\sqrt {3})^T}$ and ${\vec {E_{N2}}=(1,-1/\sqrt {3},-1/\sqrt {3},-1/\sqrt {3})^T}$. To illustrate the effect of the cascaded linear retarders, it is useful to visualize the rotation on the Poincaré sphere. Figure 2(a) shows a schematic of the two layers of nematic LCP acting as linear retarders with different fast-axis orientations. The two-layer retarder operates as a unidirectional elliptical retarder with light travelling from top to bottom. 22.5-degree linear polarized light is incident as an example to show the change in polarization state. Figure 2(b) depicts the polarization state rotation along the Poincaré sphere. 22.5-degree linearly polarized light is input at Point 1 and is rotated along an arc on the Poincaré sphere to Point 2. Notice this arc is centered around the axis of rotation defined by the eigenpolarizations of the first linear retarder, horizontal and vertical polarizations. The polarization state is then rotated again from Point 2 to Point 3 by the second linear retarder. Again, notice the light blue arc and how it is centered on the axis defined by the eigenpolarizations of the second linear retarder, 45-degree and 135-degree polarized light. Finally, the effective polarization transformation can be illustrated using the produced elliptical retarder and the eigenpolarizations of ${\vec {E_{N1}}}$ and ${\vec {E_{N2}}}$. The effective polarization change is shown in the yellow arc and connects Points 1 and Points 3; the yellow arc is centered on the axis defined by ${\vec {E_{N1}}}$ and ${\vec {E_{N2}}}$.

Fig. 2. (a) Two linear retarders with different fast-axis orientations and thicknesses can be used to produce an elliptical retarder. Incident light travels from top to bottom. (b) Polarization evolution of a Stokes parameter through two linear retarders with different fast-axis orientations on the Poincaré sphere. The initial Stokes state is shown with Point 1, intermediate Stokes state is shown with Point 2, and final Stokes state is shown with Point 3. The axes of rotation on the Poincaré sphere are also plotted for the two linear retarders and resulting elliptical retarder.

Download Full Size | PDF

2.3 Elliptical retarder using a linear retarder and circular retarder

Previously, we showed in Eq. (5) that elliptical retarders can be constructed with a cascade of linear and circular retarders [12,22]. Multiplication of the Pauli matrices results in a combination of both linear and circular retardance components. Many optical components produce circular retardance or optical activity, notably chiral materials or organic materials such as glucose. Other materials possessing circular retardance are twisted nematic (TN) LCP [23]. ChLCP is primarily used for its selective reflection of circularly polarized light meeting the Bragg condition. ChLCP and TN are nearly identical [13,23]; however, the twisting rate of ChLCP is much higher, on the order of a wavelength, making Bragg diffraction possible. Outside of the reflection region, at shorter wavelengths than in the Bragg regime, we observe a wavelength band of nearly achromatic circular retardance, while linear retardance and diattenuation are at close to zero levels. To the best of our knowledge, this band of circular retardance and its utility as a circular retarder have not been reported before in the literature. Outside of the Bragg regime, at longer wavelengths, an achromatic region of circular retardance is not found since the linear birefringence of the material trends toward zero. Figure 3(a) shows the optical properties of a ChLCP film modeled with the Berreman 4x4 method [13,24,25]. The plot shows the magnitude of circular polarizance to be close to unity from 900nm to 1000nm, where the film satisfies the Bragg condition. In addition, both linear and circular retardances are shown. The circular polarizance and the linear retardance are close to zero in the wavelength band from 550nm-700nm, whereas the circular retardance has a nonzero magnitude and is slowly varying according to our calculation. Figure 3(b) plots the circular retardance as a function of wavelength and film thickness in the achromatic region of interest. The change in circular retardance at a single wavelength is approximately linear as a function of thickness. Near achromatic behavior is still observed as the magnitude in circular retardance increases.

Fig. 3. (a) Optical properties of ChLCP are simulated using the Berreman 4x4 method. Plot is the circular diattenuation of the stack in blue as well as linear and circular retardance of the stack in orange. At shorter wavelengths outside the Bragg regime, a spectral band of near achromatic circular retardance is observed from 550nm-700nm. Diattenuation and linear retardance both have little to no magnitude. (b) Circular retardance of ChLCP is plotted as functions of wavelength and film thickness in the achromatic region of interest. The circular retardance increases linearly as a function of thickness from 0 microns to 10 microns.

Download Full Size | PDF

The magnitude of the circular retardance is a function of film thickness as well as the linear birefringence of the material. A simple analytical solution for the circular retardance as a function of thickness and birefringence does not exist; however, simulations show that increasing thickness and linear birefringence increases the circular retardance in the region of interest.

Our results show that a ChLCP film can be utilized as an achromatic circular retarder, which can be integrated into a multi-layer LCP stack. An elliptical retarder can be realized using a combination of LCP and ChLCP layers. As an example, Fig. 4 shows a quarter-wave LCP linear retarder followed by a circular retarder with ${\pi /3}$ radians of circular retardance made with ChLCP. The two-layer retarder operates as a unidirectional elliptical retarder with incident light travelling from top to bottom. To illustrate the polarization state change, the rotations of the incident 22.5-degree linearly polarized light is plotted on the Poincaré sphere in Fig. 4. Figure 4(a) shows a schematic of the film construction, with a nematic LCP layer with thickness and fast-axis orientation. This is followed by the ChLCP film. Figure 4(b) shows the evolution of the polarization state change with 22.5-degree linearly polarized light incident. From Point 1, the quarter-wave plate, oriented at 45 degrees, rotates the linear polarized state to an elliptical state at Point 2. Notice that the axis rotation for the dark blue arc is centered on the axis defined by the eigenpolarizations of the quarter waveplate. From Point 2 to Point 3, the polarization state is rotated by the ChLCP by circular retardance. Again, notice that the light blue arc depicting the polarization state change is centered around the ${S_3}$-axis, the eigenpolarizations of the circular retarder. Finally, the yellow arc shows what the effective polarization rotation is by use of the elliptical retarder. The yellow arc is centered around the axis defined by the eigenpolarizations of the cascaded retarders, ${\vec {E_{C1}}=(1,0.4472,0.7746,0.4472)^T}$ and ${\vec {E_{C2}}=(1,-0.4472,-0.7746,-0.4472)^T}$.

Fig. 4. A linear retarder and a circular retarder (ChLCP) can be combined to produce an elliptical retarder. Incident light travels from top to bottom. (b) Polarization evolution of a Stokes parameter through two linear retarders with different fast-axis orientations on the Poincaré sphere. The initial Stokes state is shown with Point 1, intermediate Stokes state is shown with Point 2, and final Stokes state is shown with Point 3. The axes of rotation on the Poincaré sphere are also plotted for the linear retarder and circular retarder as well as the resulting elliptical retarder.

Download Full Size | PDF

2.4 Inverse design

Solutions for an ${ER}$ have been demonstrated using two layers of liquid crystal polymers, using both nematic and cholesteric phases. However, the analysis of the retarder uses Pauli matrices to determine the net effect of the retarder LCP stack. Of equal importance is the inverse problem: how an arbitrary ${ER}$ with desired eigenpolarizations and retardance can be constructed using a combination of linear and circular retarders. By further decomposing Eq. (2), a solution exists where every parameter of the ${ER}$ can be controlled by a single layer. These parameters include the retardance magnitude as well as the eigenpolarizations.

First, it can be determined that the second matrix of Eq. (2) is a linear retarder using the symmetric phase convention, with the fast-axis oriented at 90 degrees. Changing the retardance of this layer modifies the magnitude of retardance of the produced ${ER}$. Second, the first and third matrices in Eq. (2) can be further decomposed into simple unitary matrices, which represent familiar Jones matrices. By taking the first matrix of the product in Eq. (2), we have:

(13)$$\begin{aligned} ER_1 & = \begin{bmatrix} \mathrm{cos}\,\theta & -\mathrm{sin}\,\theta e^{{-}i\delta}\\ \mathrm{sin}\,\theta e^{i\delta} & \mathrm{cos}\,\theta \end{bmatrix}\\ & = \begin{bmatrix} e^{{-}i\delta/2} & 0\\ 0 & e^{i\delta/2} \end{bmatrix} \begin{bmatrix} \mathrm{cos}\,\theta & -\mathrm{sin}\,\theta\\ \mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\delta/2} & 0\\ 0 & e^{{-}i\delta/2} \end{bmatrix} \end{aligned}$$

Eq. (13) shows that the decomposition of the first matrix in Eq. (2) can be represented by a linear retarder with retardance ${\delta }$ having a fast-axis alignment of 0 degrees, a left circular retarder with retardance ${2\theta }$, and a linear retarder with retardance ${\delta }$ having a fast-axis alignment of 90 degrees. Similarly, the last matrix in Eq. (2) can be decomposed in a similar fashion to:

(14)$$\begin{aligned} ER_3 & = \begin{bmatrix} \mathrm{cos}\,\theta & \mathrm{sin}\,\theta e^{{-}i\delta}\\ -\mathrm{sin}\,\theta e^{i\delta} & \mathrm{cos}\,\theta \end{bmatrix}\\ & = \begin{bmatrix} e^{{-}i\delta/2} & 0\\ 0 & e^{i\delta/2} \end{bmatrix} \begin{bmatrix} \mathrm{cos}\,\theta & \mathrm{sin}\,\theta\\ -\mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\delta/2} & 0\\ 0 & e^{{-}i\delta/2} \end{bmatrix} \end{aligned}$$

where the only difference between Eqs. (13) and (14) is the sign of the circular retarders. We have a left circular retarder in Eq. (13) and a right circular retarder in Eq. (14). Combining Eqs. (2), (13), and (14) results in a seven-layer LCP system, with five linear retarders and two circular retarders.

By slightly changing the form of the eigenpolarizations, the seven-layer system can be simplified even further to a five-layer system. The eigenpolarizations in Eq. (1) can be rewritten as:

(15)$$\vec{E_1} = \begin{pmatrix} \mathrm{cos}\,\theta \\ \mathrm{sin}\,\theta\, e^{i\delta} \end{pmatrix} = e^{i/\frac{\delta}{2}} \begin{pmatrix} \mathrm{cos}\,\theta\, e^{{-}i/\frac{\delta}{2}} \\ \mathrm{sin}\,\theta\, e^{i/\frac{\delta}{2}} \end{pmatrix}\,,\,\,\, \vec{E_2} = \begin{pmatrix} -\mathrm{sin}\,\theta\, e^{{-}i\delta} \\ \mathrm{cos}\,\theta \end{pmatrix} = e^{{-}i/\frac{\delta}{2}} \begin{pmatrix} -\mathrm{sin}\,\theta\, e^{{-}i/\frac{\delta}{2}} \\ \mathrm{cos}\,\theta\, e^{i/\frac{\delta}{2}} \end{pmatrix}$$

Using the rewritten eigenpolarizations from Eq. (15), the first matrix in Eq. (2) becomes:

(16)$$\begin{aligned} ER_1 & = \begin{bmatrix} \mathrm{cos}\,\theta e^{{-}i/\frac{\delta}{2}} & -\mathrm{sin}\,\theta e^{{-}i/\frac{\delta}{2}}\\ \mathrm{sin}\,\theta e^{i/\frac{\delta}{2}} & \mathrm{cos}\,\theta e^{i/\frac{\delta}{2}} \end{bmatrix}\\ & = \begin{bmatrix} e^{{-}i\delta/2} & 0\\ 0 & e^{i\delta/2} \end{bmatrix} \begin{bmatrix} \mathrm{cos}\,\theta & -\mathrm{sin}\,\theta\\ \mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \end{aligned}$$

A similar analysis can be done for the third matrix in Eq. (2):

(17)$$\begin{aligned} ER_3 & = \begin{bmatrix} \mathrm{cos}\,\theta e^{i/\frac{\delta}{2}} & \mathrm{sin}\,\theta e^{{-}i/\frac{\delta}{2}}\\ -\mathrm{sin}\,\theta e^{i/\frac{\delta}{2}} & \mathrm{cos}\,\theta e^{{-}i/\frac{\delta}{2}} \end{bmatrix}\\ & = \begin{bmatrix} \mathrm{cos}\,\theta & \mathrm{sin}\,\theta\\ -\mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\delta/2} & 0\\ 0 & e^{{-}i\delta/2} \end{bmatrix} \end{aligned}$$

Finally, by combining Eqs. (2), (16), and (17), the same ${ER}$ equation from Eq. (2) is calculated.

(18)$$\begin{aligned}ER = \begin{bmatrix} e^{{-}i\delta/2} & 0\\ 0 & e^{i\delta/2} \end{bmatrix} \begin{bmatrix} \mathrm{cos}\,\theta & -\mathrm{sin}\,\theta\\ \mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\phi/2} & 0\\ 0 & e^{{-}i\phi/2} \end{bmatrix} & \begin{bmatrix} \mathrm{cos}\,\theta & \mathrm{sin}\,\theta\\ -\mathrm{sin}\,\theta & \mathrm{cos}\,\theta \end{bmatrix} \begin{bmatrix} e^{i\delta/2} & 0\\ 0 & e^{{-}i\delta/2} \end{bmatrix} \end{aligned}$$

In this representation, each layer has a commonly used Jones matrix and can be realized using different phases of LCP. The eigenpolarizations as well as the retardance magnitude of the ${ER}$ can be changed by variation of the thicknesses of the individual layers. Table 1 below summarizes what each layer in the five-layer film stack controls. This design has more layers compared with that in the previous sections and is more complex to fabricate, but each layer in this design has a simpler interpretation, and there is a direct relation between the parameters of the individual layers and the characteristics of the ${ER}$.

Table 1. A five-layer solution to the inverse design problem is presented. The type of retarder, the required retardance, alignment, and functionality for each layer are included.

View Table | View all tables in this article

We note that the matrices representing the circular retarders in Eq. (19) are very close to the coordinate rotation matrices. Thus, the fabrication of the ${ER}$ can be simplified further. By rotating the central linear retarder by an angle ${\theta }$, the five-layer system can be reduced to a three-layer system as shown below in Eq. (19).

(19)$$ER = \begin{bmatrix} e^{{-}i\delta/2} & 0\\ 0 & e^{i\delta/2} \end{bmatrix} \begin{bmatrix} e^{i\frac{\phi}{2}}\mathrm{cos}^2\theta+e^{{-}i\frac{\phi}{2}}\mathrm{sin}^2\theta & i\mathrm{sin}\,2\theta\,\mathrm{sin}\,\frac{\phi}{2}\\ i\mathrm{sin}\,2\theta\,\mathrm{sin}\,\frac{\phi}{2} & e^{{-}i\frac{\phi}{2}}\mathrm{cos}^2\theta+e^{i\frac{\phi}{2}}\mathrm{sin}^2\theta \end{bmatrix} \begin{bmatrix} e^{i\delta/2} & 0\\ 0 & e^{{-}i\delta/2} \end{bmatrix}$$

This is a three-layer solution to the inverse problem, where the eigenpolarizations and the retardance of the ${ER}$ are related to a single layer’s thickness and orientation. Table 2 summarizes the three-layer design.

Table 2. A three-layer solution for the inverse design elliptical retarder problem is presented. Included are the type of retarder, the required retardance, alignment, and functionality for each layer.

View Table | View all tables in this article

2.5 Bidirectional designs

So far, the designs described assume unidirectional behavior: the eigenpolarizations are different whether the propagation of light is forward or backwards. Designs where the eigenpolarizations are the same regardless of propagation direction, or bidirectional designs, can be realized using either multiple linear retarders or a combination of linear and circular retarder configurations.

For the multiple linear retarder design, three requirements must be met for the eigenpolarizations to remain constant in bi-directional operation. They include: (1) Each of the individual fast-axes for each of the individual linear retarders must have a constant angle difference between them. (2) The individual retarders must have the same magnitude in linear retardance, or have linear retardances that are symmetrical through the stack. For example, in a three-layer stack, retarders 1 and 3 must have the same linear retardance or all three retarders must have the same linear retardance. (3) The fast-axes of each of the individual linear retarders must have both x- and y-symmetry in the local coordinates. These three requirements ensure that the same retarder is encountered whether propagating from the first retarder to the last retarder in the stack, or from the last retarder to the first retarder in the stack.

Similar requirements must be met for the linear/circular retarder configuration. These requirements include: (1) The type of retarder at a position within the retarder stack must be the same whether the propagation is forward or backward. For example, if the retarder stack consists of a linear-circular-linear design, the type of retarder is the same whether forward or backward propagation occurs. (2) The fast-axis of the linear retarders must retain x- and y-symmetry in the local coordinates with a symmetric offset in fast-axis orientations. (3) The magnitude of the retardance for each layer must be symmetric throughout the stack, allowing for the same elliptical retarder to be encountered in both forward and backward propagation. U*sing the ChLCP as a circular retarder works well for the bidirectional design, as the same magnitude and sign of circular retardance is encountered regardless of propagation direction.

Both of the design presented in Section 2.2 and the three-layer design presented in Section 2.4 can meet the bidirectional requirements given careful consideration of the retardance for each layer as well as the alignment. Performing a Pauli matrix decomposition is a quick way to check whether or not the design is an elliptical retarder in either propagation direction. The design presented in Section 2.3 does not meet the requirements for a bidirectional retarder, but with the addition of a properly selected linear retarder following the circular retarder, a bi-directional design can be produced. Finally, the five-layer design presented in Section 2.4 does not meet the bi-directional requirements, as the ChLCP used as circular retarders has the same handedness regardless of propagation direction, thereby breaking symmetry constraints.

3. Experimental methods and materials

3.1 Materials and characterization

Wafers of 1.5-inch diameter and 500${\mu m}$ thickness used in this work are made from polished soda lime glass, selected for its excellent optical transmission in the visible spectrum and non-polarizing behavior. Before fabrication, each wafer is cleaned and prepared using oxygen plasma treatment. Two different phases of liquid crystal polymer are used in the work: the nematic A-plate material, model RMM141C, and the cholesteric phase material, model RMM1695, both commercially available (EMD Performance Materials, a unit of Merck KGaA, Darmstadt, Germany). Both materials are dissolved in toluene to allow for spin-coating application. The photoalignment layer consists of SD1 dissolved in N,N dimethylformamide (DMF) [26]. Polarization measurements are carried out using an Axometrics AxoScan Mueller matrix polarimeter (Axometrics, Hunstville, AL, USA), and thickness measurements are carried out using a surface profilmeter (Dektak 150, Veeco, Plainville, NY, USA). Spectral transmission measurements are carried out using a Cary 5000 spectrometer (Agilent Technologies, Santa Clara, CA, USA).

3.2 Fabrication

Each of the designs presented requires multiple layers of LCP. For simplicity, fabrication of a single layer is presented below, and added layers can be added on top by addition of multiple wafers, barrier layers using silicon dioxide deposition [3,4], or thin intermediate twisting layers [27].

1. The substrate is cleaned using oxygen plasma for 1 minute.
2. SD1 dissolved in DMF at a 0.38% weight concentration is dispensed through a 0.2${\mu }$m PTFE filter and spin-coated onto the substrate at 3000 RPM for 30 seconds. The wafer is then baked on a hot plate at 150${^\circ }$C for 5 minutes.
3. The SD1 is then aligned using a UV dichroic polarizer (Boulder Vision Optik, Boulder, CO, USA) and a 365nm UV light source with 150mJ/cm${^2}$ energy for 5 minutes.
4. Either the nematic or cholesteric phase material is dispensed through a 0.2${\mu }$m PTFE filter and spin-coated onto the aligned SD1 layer at 3000 RPM for 1 minute. The nematic material rests on the substrate at room temperature for 1 minute post spin-coating, whereas the cholesteric material is baked on a hot plate at 75${^\circ }$C for 1 minute post spin-coating.
5. The LCP material is then cured under nitrogen atmosphere for 5 minutes using the 365nm light source with 150mJ/cm${^2}$ energy.
6. Additional layers can now be added, either using additional substrates or by depositing a layer of SiO${_2}$ and repeating Steps 2-5 again.

4. Results

As a demonstration of the utility of the ChLCP, an ${ER}$ using the theory from Section 2.3 is fabricated. Both a nematic and cholesteric layer are fabricated, and then subsequently stacked to produce an elliptical retarder. The ChLCP used has a reflection band centered at 850nm. This allows for the use of the achromatic circular retardance to be centered in the visible spectrum. Shown in Fig. 5(a) are simulated transmission and polarization properties of the single-layer film using the Berreman 4x4 method. The ChLCP is simulated in the wavelength range of 400nm-1200nm. From the Berreman 4x4 method, both Jones and Mueller matrices can be derived detailing relevant polarization properties, namely circular and linear retardance, as well as polarization dependent transmission arising from Bragg diffraction. Our calculation can be compared with measurement. Transmission data is taken in the wavelength region of 400nm-1200nm and polarization data is taken in the region of 400nm-800nm. The transmission band and linear retardance calculated in simulation fit quite well with measurement. Film thickness of the fabricated ChLCP sample was measured to be 5.5${\mu }$m, while the best fit in simulation used a thickness of 5.3${\mu }$m. Additionally, good agreement of the circular retardance curves is found between simulation and measurement. Slight deviation between the two occurs starting at 500nm and is attributed to material dispersion, which is not considered in the Berreman 4x4 calculation. The indices of refraction for the ChLCP material are found to be 1.57 and 1.43 for the extraordinary and ordinary indices respectively. Additionally, the pitch of the ChLCP is calculated to be 560nm.

Fig. 5. (a) Simulated and measured properties of the ChLCP film used. Transmission data (simulation and measurement) is plotted in blue, while retardance data is plotted in orange (simulation and measurement). Simulation results are in good agreement with measurement results, without consideration of material dispersion. (b) Circular retardance is plotted for a single-layer and double-layer of ChLCP material. The double coated layer has close to double the circular retardance of the single coated layer. Measurements confirm simulations showing approximately linearly increasing circular retardance with increasing layer thickness.

Download Full Size | PDF

Unfortunately, a closed form solution in finding the circular retardance as a function of thickness, refractive indices, and wavelength was not found. Our calculations show that the circular retardance increases approximately linearly with increasing thickness, as shown in Fig. 3(b). This behavior is confirmed from measurements of fabricated samples made of single and double layers of ChLCP. The double-layer sample has ChLCP on both sides of the wafer. Figure 5(b) shows the circular retardance as a function of wavelength for single- and double-layer ChLCP. The circular retardance of the film is approximately doubled, as expected and the achromatic behavior is preserved. Due to slight differences in the fabrication among the two layers (i.e. spin-speed, spin-acceleration, bake time), the circular retardance is not precisely doubled.

To fully realize the generalized elliptical retarder utilizing ChLCP, the design from Section 2.3 is fabricated. The design consists of a quarter-wave plate oriented at 45 degrees and a circular retarder with 60 degrees of retardance. The linear retarder is fabricated using RMM141C and has a retardance of 89.6 degrees at 550nm. The circular retarder is fabricated using two layers of RMM1695 and has a circular retardance of 62.3 degrees at 550nm. By stacking the two fabricated devices as well as obtaining proper alignment of the linear retarder with the polarimeter, the two eigenpolarizations of the fabricated device were measured to be: ${\vec {E_{F1}}=(1,0.446,0.764,0.467)^T}$ and ${\vec {E_{F2}}=(1,-0.446,-0.764,-0.467)^T}$, respectively. Calculation of the measured eigenpolarizations were performed using Eqs. (11) and (12). The Euclidean norm between the design eigenpolarizations, ${\vec {E_{C1}}}$ and ${\vec {E_{C2}}}$, and the measured eigenpolarizations, ${\vec {E_{F1}}}$ and ${\vec {E_{F2}}}$, is equal to 0.0225. The maximum Euclidean norm possible for Stokes parameters is equal to 2 [28]. The slight deviation in eigenpolarizations between fabrication and theory is a result of the ChLCP film being slightly too thick, resulting in a 62.3-degree retardance rather than the desired 60 degrees.

The slight difference in eigenpolarizations is represented on the Poincaré sphere, as shown in Fig. 6(a). Results detailing the axis of rotation for both the design and fabricated samples are shown, and the difference in ellipticity and orientation between the two eigenpolarizations is small.

Fig. 6. (a) Axis of rotation plotted on the Poincaré sphere for both the design and fabricated samples, show excellent agreement. The differences in the ellipticity and major-axis orientation of the respective polarization ellipses is plotted as well. (b) Histogram of the Euclidean norm between the target design and simulated eigenpolarizations is plotted for 10,000 variations calculated in a Monte Carlo simulation. In each variation, the thickness of both the LCP and ChLCP are varied by up to 5%.

Download Full Size | PDF

To better quantify the error found in the fabricated sample, a Monte Carlo simulation was performed with 10,000 variations. For each variation of the design, the thickness of both the LCP and ChLCP layer were varied by up to 5%. The Euclidean norm between each of the simulated eigenpolarizations and ${\vec {E_{C1}}}$ and ${\vec {E_{C2}}}$ was calculated. The distribution is plotted in Fig. 6(b), showing that our fabricated result of 0.0225 is close to the peak of the distribution.

An achromatic quarter-wave plate can replace the linear retarder layer to increase the operating bandwidth of this design, allowing for full use of the achromatic circular retarder layer as shown in Fig. 5. Additionally, our method can be applied to other designs using similar fabrication process and materials.

5. Conclusions

In this work, the Jones matrices of ${ER}$ are decomposed using the Pauli matrices. An inverse design process for any arbitrary ${ER}$ is demonstrated, and a design is fabricated and tested. The proposed inverse design process provides a simplified eigenvalue analysis of the ${ER}$. For example, a three-layer system of linear retarders or a five-layer system consisting of three linear retarders and two circular retarders can be used to construct an ${ER}$, where individual characteristics of each layer, either linear or circular retardance and orientation, directly define the retardance of the ${ER}$ and its eigenpolarizations. The proposed multi-layer inverse design solution is both intuitive and straight-forward. The multi-layer retarders can be realized using nematic LCP as linear retarders and ChLCP as circular retarders. To the best of our knowledge, this is the first theoretical and experimental demonstration of achromatic circular retarder using ChLCP at wavelengths shorter than the center of the Bragg regime. In addition, an ${ER}$ was fabricated using a combination of a quarter-wave linear retarder and a circular retarder with 60 degrees retardance. Our measurement shows the fabricated retarder is within a figure of merit of 0.0225 of the designed eigenpolarizations. Further improvement of this design may include replacing the linear retarder with achromatic linear retarders, thus allowing for the full utilization of the achromatic region of the ChLCP material. ${ER}$ like these have many uses including in interferometry and quantum optics, as well as in imaging and display devices.

Funding

National Science Foundation (1918260, 2053754).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. X. Tu, S. McEldowney, Y. Zou, M. Smith, C. Guido, N. Brock, S. Miller, L. Jiang, and S. Pau, “Division of focal plane red–green–blue full-stokes imaging polarimeter,” Appl. Opt. 59(22), G33–G40 (2020). [CrossRef]

2. D. Gottlieb and O. Arteaga, “Optimal elliptical retarder in rotating compensator imaging polarimetry,” Opt. Lett. 46(13), 3139–3142 (2021). [CrossRef]

3. L. Jiang, S. Miller, X. Tu, M. Smith, Y. Zou, F. Reininger, and S. Pau, “Patterned achromatic elliptical polarizer for short-wave infrared imaging polarimetry,” Opt. Express 30(2), 1249–1260 (2022). [CrossRef]

4. X. Tu, L. Jiang, M. Ibn-Elhaj, and S. Pau, “Design, fabrication and testing of achromatic elliptical polarizer,” Opt. Express 25(9), 10355–10367 (2017). [CrossRef]

5. J. L. Pezzaniti, S. C. McClain, R. A. Chipman, and S.-Y. Lu, “Depolarization in liquid-crystal televisions,” Opt. Lett. 18(23), 2071–2073 (1993). [CrossRef]

6. A. Lizana, N. Martin, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17(10), 8491–8505 (2009). [CrossRef]

7. K. Dev and A. Asundi, “Mueller–stokes polarimetric characterization of transmissive liquid crystal spatial light modulator,” Opt. Lasers Eng. 50(4), 599–607 (2012). Computational Optical Measurement. [CrossRef]

8. R. N. Shagam and J. C. Wyant, “Optical frequency shifter for heterodyne interferometers using multiple rotating polarization retarders,” Appl. Opt. 17(19), 3034–3035 (1978). [CrossRef]

9. J. C. Wyant, “Dynamic interferometry,” Opt. Photonics News 14(4), 36–41 (2003). [CrossRef]

10. A. Lohrmann, C. Perumgatt, and A. Ling, “Manipulation and measurement of quantum states with liquid crystal devices,” Opt. Express 27(10), 13765–13772 (2019). [CrossRef]

11. N. A. Peters, J. T. Barreiro, M. E. Goggin, T.-C. Wei, and P. G. Kwiat, “Remote state preparation: Arbitrary remote control of photon polarization,” Phys. Rev. Lett. 94(15), 150502 (2005). [CrossRef]

12. R. A. Chipman, W. T. Lam, and G. Young, Polarized Light and Optical Systems (CRC, 2019).

13. D. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (John Wiley & Sons, 2010).

14. C. Gu and P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21(7), 504–506 (1996). [CrossRef]

15. M. Nyman, S. Maurya, M. Kaivola, and A. Shevchenko, “Optical wave retarder based on metal-nanostripe metamaterial,” Opt. Lett. 44(12), 3102–3105 (2019). [CrossRef]

16. S. Miller, L. Jiang, and S. Pau, “Birefringent coating to remove polarization dependent phase shift,” U.S. Patent 63/239,184 (1 August 2021).

17. S. Jiao, “Polarization-sensitive mueller-matrix optical coherence tomography,” Ph.D. thesis, Texas A&M University (2003).

18. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61(9), 1207–1213 (1971). [CrossRef]

19. A. H. Al-Mohy and N. J. Higham, “Improved inverse scaling and squaring algorithms for the matrix logarithm,” SIAM J. Sci. Comput. 34(4), C153–C169 (2012). [CrossRef]

20. S. Y. Lu and R. A. Chipman, “Interpretation of mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

21. C.-J. Yu, “Fully variable elliptical phase retarder composed of two linear phase retarders,” Rev. Sci. Instrum. 87(3), 035106 (2016). [CrossRef]

22. C.-J. Yu and C. Chou, “Characterization of a generalized elliptical phase retarder by using equivalent theorem of a linear phase retarder and a polarization rotator,” in Optical Components and Materials VIII, vol. 7934M. J. F. Digonnet, S. Jiang, J. W. Glesener, and J. C. Dries, eds., International Society for Optics and Photonics (SPIE, 2011), pp. 291–297.

23. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley & Sons, 2010), 2nd ed.

24. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62(4), 502–510 (1972). [CrossRef]

25. D. W. Berreman and T. J. Scheffer, “Reflection and transmission by single-domain cholesteric liquid crystal films: Theory and verification,” Mol. Cryst. Liq. Cryst. 11(4), 395–405 (1970). [CrossRef]

26. V. Chigrinov, H. S. Kwok, H. Takada, and H. Takatsu, “Photo-aligning by azo-dyes: Physics and applications,” Liq. Cryst. Today 14(4), 1–15 (2005). [CrossRef]

27. L. Li, S. Shi, and M. J. Escuti, “Improved saturation and wide-viewing angle color filters based on multi-twist retarders,” Opt. Express 29(3), 4124–4138 (2021). [CrossRef]

28. S. Miller, X. Tu, L. Jiang, and S. Pau, “Polarizing beam splitter cube for circularly and elliptically polarized light,” Opt. Express 27(11), 16258–16270 (2019). [CrossRef]

Layer	Retarder Type	Fast-axis Alignment	Material	Retardance	Function
1	Linear	0 degrees	Nematic A-plate LCP	$δ$	Defines phase delay of eigenpolarization ( $δ$ ).
2	Left Circular	N/A	ChLCP	$2 θ$	Defines major-axis orientation and ellipticity of eigenpolarization ( $θ$ ).
3	Linear	90 degrees	Nematic A-plate LCP	$ϕ$	Defines the retardance magnitude of elliptical retarder ( $ϕ$ ).
4	Right Circular	N/A	ChLCP	$2 θ$	Defines major-axis orientation and ellipticity of eigenpolarization ( $θ$ ).
5	Linear	90 degrees	Nematic A-plate LCP	$δ$	Defines phase delay of eigenpolarization ( $δ$ ).

Layer	Retarder Type	Fast-axis Alignment	Material	Retardance	Function
1	Linear	0 degrees	Nematic A-plate LCP	$δ / 2$	Defines phase delay of eigenpolarization ( $δ$ )
2	Linear	$θ$ degrees	Nematic A-plate LCP	$ϕ$	Defines major axis and ellipticity of eigenpolarization ( $θ$ ); retardance of elliptical retarder ( $ϕ$ )
3	Linear	90 degrees	Nematic A-plate LCP	$δ / 2$	Defines phase delay of eigenpolarization ( $δ$ )

Layer	Retarder Type	Fast-axis Alignment	Material	Retardance	Function
1	Linear	0 degrees	Nematic A-plate LCP	$δ$	Defines phase delay of eigenpolarization ( $δ$ ).
2	Left Circular	N/A	ChLCP	$2 θ$	Defines major-axis orientation and ellipticity of eigenpolarization ( $θ$ ).
3	Linear	90 degrees	Nematic A-plate LCP	$ϕ$	Defines the retardance magnitude of elliptical retarder ( $ϕ$ ).
4	Right Circular	N/A	ChLCP	$2 θ$	Defines major-axis orientation and ellipticity of eigenpolarization ( $θ$ ).
5	Linear	90 degrees	Nematic A-plate LCP	$δ$	Defines phase delay of eigenpolarization ( $δ$ ).

Layer	Retarder Type	Fast-axis Alignment	Material	Retardance	Function
1	Linear	0 degrees	Nematic A-plate LCP	$δ / 2$	Defines phase delay of eigenpolarization ( $δ$ )
2	Linear	$θ$ degrees	Nematic A-plate LCP	$ϕ$	Defines major axis and ellipticity of eigenpolarization ( $θ$ ); retardance of elliptical retarder ( $ϕ$ )
3	Linear	90 degrees	Nematic A-plate LCP	$δ / 2$	Defines phase delay of eigenpolarization ( $δ$ )

Generalized elliptical retarder design and construction using nematic and cholesteric phase liquid crystal polymers

Abstract

1. Introduction

2. Theory

2.1 Jones matrix decomposition with Pauli matrices

2.2 Elliptical retarder using linear retarder of a different orientation

2.3 Elliptical retarder using a linear retarder and circular retarder

2.4 Inverse design

2.5 Bidirectional designs

3. Experimental methods and materials

3.1 Materials and characterization

3.2 Fabrication

4. Results

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (2)

Equations (19)

Optics Express