Intuitive understanding of the connection between Pancharatnam–Berry optical beam deflectors and polarization volume holograms

Fenglin Xi; Philip Bos

doi:10.1364/AO.478269

1. INTRODUCTION

Polarization volume hologram (PVH) gratings and Pancharatnam–Berry phase devices (PPD) have wide application in optical systems. For example, a pancake optics design using a PVH has been proposed as one of the potential solutions for a lightweight headset in a virtual reality (VR) system [1]. PPD devices, proposed in [2,3], have been used in optical beam steering devices [4] and metamaterial [5–8].

These two types of devices, when considered separately, have much different optical effects and intuitive explanations as shown in Fig. 1. In the figure, as in the remaining discussion, it is assumed that the thickness of each device is close to $\lambda /({2}\Delta {n})$. The optical response of PPD devices is to have a polarization dependent steering of light between two angles symmetric about the input angle of light, which has been made clear using the Jones calculus. PVH devices are similar in their optical properties to Bragg reflectors except that light of orthogonal polarization states is either deflected to the Bragg angle, or not deflected. PPD devices have a relatively high efficiency for a wide range of incident light angles, while PVH devices have a narrower range of incident angles that yield high efficiency optical deflection.

Fig. 1. Comparison of response of (a) PPD and (b) PVH that have positive Px. Px is defined as positive if along the ${+}x$ direction, the director is rotated such that using the right-hand rule the thumb is pointing to the ${+}z$ direction. Px is defined as negative if along the ${+}x$ direction, the director is rotated such that using right-hand rule the thumb is pointing to the ${-}z$ direction. The intensity of light in the direction of the dashed arrows is close to zero when the device thickness ${D} = \lambda /({2}\Delta {n})$.

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For the given structure, Px is positive, and the grating vector is along the ${+}x$ direction, as shown in Fig. 1 with the black arrow. The grating vector angle $\phi$ makes ${+}{90}^\circ$ with the ${+}z$ axis. The angle $\tau$ describes the director rotation about the $z$ axis due to Px. ${\theta _0}$ is the incident angle in the grating. When the incident angle has projections on the ${+}x$ and ${+}z$ axes, the incident angle is positive (in Fig. 1, ${\theta _0}$ is positive). The polarization state of light represented by the red arrow is right-hand circular polarization (RCP), while the green arrow is left-hand circular polarization (LCP). For PPD, with RCP incidence the zeroth and ${-}{1}$st diffraction orders are observed. With LCP incidence, the zeroth and ${+}{1}$st diffraction orders are observed. For PVH, with RCP incidence only the zeroth diffraction order is observed. With LCP incidence, the zeroth and ${+}{1}$st diffraction orders are observed. The intensity of the two diffraction orders depends on the device thickness, while the highest diffract efficiency (${-}{1}$st or ${+}{1}$st order) is reached for the case of the device thickness of $\lambda /({2}\Delta {n})$.

Both of these effects can be quantitatively modeled using the rigorous coupled wave approach (RCWA) or by using the finite difference time domain (FDTD) method. The two methods are rigorous without approximations and give the same accurate result [9], but these computational approaches do not provide an intuitive analytical connection between the PPD and the PVH type devices, or a predictive analytical expression to explain their properties.

For isotropic materials, Gaylord and Moharam [10] considered a similar question in exploring the connection between Raman–Nath and Bragg type devices. They used an approximate coupled wave analysis to provide an intuitive derivation of the “$\rho$” parameter, which is used for distinguishing the Raman–Nath grating and Bragg grating effects in [11].

In this paper, the approach of Gaylord is extended to birefringent materials using a beam propagation method to clearly show the analytical connection between the optical properties of PPD and PVH devices.

2. APPROXIMATE COUPLED WAVE APPROACH FOR BIREFRINGENCE MATERIAL

In this work, the approach of Gaylord and Moharam for isotropic material [10] and the “$\rho$” parameter in [11] is extended to birefringence materials using a modified version of the dielectric tensor derivation of Sasaki et al. [12].

Following [10], to find the diffraction efficiency, the starting point is the Helmholtz equation:

(1)$${\nabla ^2}E + {k^2}\varepsilon E = 0.$$

${E},\;{ k}$, and $\varepsilon$ are the electric field, wave vector, and dielectric tensor. To use Eq. (1) to show the evolution of the amplitude and polarization states of light as it propagates in a periodic birefringent structure, light is considered to be decomposed into plane waves with defined directions corresponding to the allowed diffraction orders under the Floquet condition. Further, the periodic birefringent layer is considered to be discretized into layers that are parallel to the plane of the film. We can think the film consists of ${ N}$ layers, each with a very small thickness $d$ ($d = {D}/{N}$).

Two assumptions are used to simplify the calculation:

1. The $k$ vector directions of the $s$ and $p$ modes for each layer are assumed to be at the same angle.
2. Boundary effects are ignored.

Equations are here derived to find the amplitude and polarization state of each Floquet order in a particular layer, given the amplitude and polarization state of each of the orders in the layer before it.

The calculation steps shown in Fig. 2 that are detailed in Supplement 1, give the equation that relates the values of the complex wave amplitude for the $l$th diffraction order, ${ S}_l^{\rm{layer}}$, in a layer to those in the next layer (${\rm layer} + {1}$). Specifically, the complex wave amplitude for the $l$th diffraction order ${S}_l^{\rm layer + 1}$, in the light propagation direction, is

(2)$$\begin{split}{S}_l^{\rm layer + 1} &= {S}_l^{\rm{layer}} - \frac{{i\pi d}}{{2{A_l}}}\left\{{\frac{{2l({m - l} )}}{{{\Lambda ^2}}}{S}_l^{\rm{layer}} + \left({\frac{2}{{{\lambda ^2}}}} \right)\left({{\varepsilon _a} - n_{\rm{ave}}^2} \right)}\right.\\&\quad\times\left.{{S}_l^{\rm{layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}_{l + 1}^{\rm{layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_{l - 1}^{\rm{layer}}} \right\}.\end{split}$$

Fig. 2. Calculation steps in this work to derive Eq. (2).

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The details of this derivation are in Supplement 1, and the definitions of the variables in Eq. (2) are defined below.

In Eq. (2), the complex amplitude of ${{S}_l}$ in a layer is given by

{S_l}\; = R({{\theta _l}} ){\left[{\begin{array}{*{20}{c}}{{J_{\textit{sl}}}}\\{{J_{\textit{pl}}}}\\0\end{array}} \right]_k}.

${J}$ is the Jones vector describing the polarization state of light in a frame with the $z$ axis along the $k$ vector direction. ${\theta _l}$ is the diffraction angle for the $l$th diffraction order. $R({{\theta _l}})$ is the rotation matrix from the considered ray’s $k$ vector frame to the lab frame:

R({{\theta _l}} ) = \left({\begin{array}{*{20}{c}}{\cos({{\theta _l}} )}&\,\,\,0&\,\,\,{- \sin({{\theta _l}} )}\\0&\,\,\,1&\,\,\,0\\{\sin({{\theta _l}} )}&\,\,\,0&\,\,\,{\cos({{\theta _l}} )}\end{array}} \right).

$m$ is defined as

m = \frac{{2{n_{\rm{ave}}}\Lambda}}{\lambda}\cdot \cos({{\theta _0} - \phi} ).

${{n}_{\rm{ave}}}$ is the average index in the grating. In this work, in terms of the extraordinary and ordinary refractive index ${{n}_e}$ and ${{n}_o}$,

{n_{\rm{ave}}} = \sqrt {\frac{{({{n}_e^2 + {n}_o^2} )}}{2}} .

$\Lambda$ is the grating pitch. $\lambda$ is the vacuum wavelength of the incident beam. The value of $m$ is equal to an integer when the Bragg condition is met.

${{A}_l}$ is

{A_l} = \frac{{{n_{\rm{ave}}}\; \cos({{\theta _0}} )}}{\lambda} - \frac{{l\; \cos(\phi )}}{\Lambda}.

Also, the components of the dielectric tensor that appear in Eq. (2) are ${\varepsilon_a,\;{{\rm M}_ +}}$, and ${\rm M} $-, where ${\varepsilon _a}$ is the part of the dielectric tensor that is not a function of the periodicity of the material (thus not a function of $\tau$).

For slanted transmissive polarization volume hologram (T-PVH) [13], it is

{\varepsilon _a} = \left[ \begin{array}{ccc} {{n}}_{o}^{2}+\frac{\Delta {{N}^{2}}{{{\sin}}^{2}}( \phi )}{2} &\,\,\, 0 & \,\,\,\frac{\Delta {{N}^{2}}( {\sin}( \phi ){\cos}( \phi ) )}{2} \\ 0 & \,\,\,{{n}}_{o}^{2}+\frac{\Delta {{N}^{2}}}{2} & \,\,\,0 \\ \frac{\Delta {{N}^{2}}({\sin}( \phi ){\cos}( \phi ) )}{2} & \,\,\,0 & \,\,\,{{n}}_{o}^{2}+\frac{\Delta {{N}^{2}}{{{\cos}}^{2}}( \phi )}{2} \end{array} \right].

$\Delta {{N}^2}$ is defined as

\Delta N^2 = n_e^2 - n_o^2.

${{\rm M}_ +}$ and ${\rm M}$- are matrices:

\begin{split}{M_ +}& = \left[{\begin{array}{*{20}{c}}{{{{\sin}}^2}(\phi )}&{- i{\sin}(\phi )}&{\left({{\sin}(\phi ){\cos}(\phi )} \right)}\\{- i{\sin}(\phi )}&{- 1}&{- i{\cos}(\phi )}\\{\left({{\sin}(\phi ){\cos}(\phi )} \right)}&{- i{\cos}(\phi )}&{{{{\cos}}^2}(\phi )}\end{array}} \right], \\ {M_ -}& = \left[{\begin{array}{*{20}{c}}{{{{\sin}}^2}(\phi )}&{i{\sin}(\phi )}&{\left({{\sin}(\phi ){\cos}(\phi )} \right)}\\{i{\sin}(\phi )}&{- 1}&{i{\cos}(\phi )}\\{({{\sin}(\phi ){\cos}(\phi )} )}&{i{\cos}(\phi )}&{{{{\cos}}^2}(\phi )}\end{array}} \right].\end{split}

At the Bragg condition, the incident, deflected, and grating vector forms a vector triangle [14]. Thus, the Bragg diffraction order is ${+}{1}$st order when the incident angle is positive ($m = {1}$). The Bragg diffraction order is ${-}{1}$st order when the incident angle is negative ($m = - {1}$).

For the case of positive incident angle, and grating vector of 90°, writing out Eq. (2) for the ${-}{1}$st, zeroth, and ${+}{1}$st orders,

(3)$$\begin{split}&{S}_{- 1}^{\rm layer + 1} = {S}_{- 1}^{{\rm layer}} - \frac{{i\pi d}}{{2{A_{- 1}}}}\\&\left\{{\left({\frac{2}{{{\lambda ^2}}}} \right)\left({{\varepsilon _a} - {n}_{{\rm ave}}^2} \right){S}_{- 1}^{{\rm layer}} + \frac{{- 4}}{{{\Lambda ^2}}}{S}_{- 1}^{{\rm layer}} }+ {\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}^{{\rm layer}}_0} \right\}, \\ &{S}_0^{\rm layer + 1} = {S}_0^{{\rm layer}} - \frac{{i\pi d}}{{2{A_0}}}\\&\left\{{\left({\frac{2}{{{\lambda ^2}}}} \right)\left({{\varepsilon _a} - {n}_{{\rm ave}}^2} \right){S}_0^{{\rm layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}^{{\rm layer}}_1 }+ {\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}^{{\rm layer}}_{- 1}} \right\}, \\ &{S_1}^{\rm layer + 1} = {S}_1^{{\rm layer}} - \frac{{i\pi d}}{{2{A_1}}}\left\{{\left({\frac{2}{{{\lambda ^2}}}} \right)\left({{\varepsilon _a} - {n}_{{\rm ave}}^2} \right){S}_l^{{\rm layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}^{{\rm layer}}_0} \right\}.\end{split}$$

These equations can be used to find the output amplitude and polarization state of light exiting a film of many layers. It turns out that this simple method provides quite accurate results compared with RCWA, at least for the case in which the grating $k$ vector is in the plane of the layers ($\phi = {90}^\circ$). A comparison between the proposed beam propagation method and RCWA can be found in Supplement 1. Given the verification of this method to predict the amplitude and phase of light exiting a film of a periodic birefringent media, we can use the simple form of these equations to analyze the evolution of light going through the film as a function the parameters they contain.

3. EFFECT OF THE INCIDENT LIGHT POLARIZATION STATE ON THE RESPONSE OF PPD AND PVH DEVICES

In Eq. (2), on the right-hand side in the bracket, there are two terms containing the complex amplitude for the considered diffraction order in the previous layer (${S}_{l}^{{\rm layer}}$), and terms containing the complex amplitude for the adjacent diffraction orders (${S}_{l + 1}^{{\rm layer}}$ and ${S}_{l - 1}^{{\rm layer}}$).

There is an extra term [the term with $({{\varepsilon _a} - {n}_{{\rm ave}}^2})$ in the brackets of Eqs. (2) and (3)] as compared the equation for an isotropic material in [10]. When the grating vector is close to 90°, and light is normally incident (electric field has a small component along the $z$ direction), this term is very close to 0.

With the above assumptions, Eq. (2) becomes Eq. (4), and Eq. (3) becomes Eq. (5):

(4)$$\begin{split}{S}_l^{\rm layer + 1} &= {S}_l^{{\rm layer}} - \frac{{i\pi d}}{{2{A_l}}}\left\{{\frac{{2l({m - l} )}}{{{\Lambda ^2}}}{S}_l^{{\rm layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}_{l + 1}^{{\rm layer}} }\right.\\&\qquad\qquad\quad\qquad+\left.{ \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_{l - 1}^{{\rm layer}}} \right\}, \\ {S_0}^{{\rm layer} + 1} &= {S}_0^{{\rm layer}} - \frac{{i\pi d}}{{2{A_0}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}_{+ 1}^{{\rm layer}} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_{- 1}^{{\rm layer}}} \right\},\end{split}$$

(5)$$\begin{split}{S}_{+ 1}^{\rm layer + 1} &= {S}_{+ 1}^{{\rm layer}} - \frac{{i\pi d}}{{2{A_1}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_0^{{\rm layer}}} \right\},\\{S}_{- 1}^{\rm layer + 1} &= {S}_{- 1}^{{\rm layer}} - \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{- 4}}{{{\Lambda ^2}}}{S}_{- 1}^{{\rm layer}} + \frac{{\Delta {N^2}{M_ -}S_0^{{\rm layer}}}}{{2{\lambda ^2}}}} \right\}.\end{split}$$

For slanted T-PVH, when the grating vector angle $\phi = {90}^\circ$, ${\rm M} +$ and ${\rm M} -$ are simplified to

{M_ +} = \left[{\begin{array}{*{20}{c}}1&\;\;{- i}&\;\;0\\{- i}&\;\;{- 1}&\;\;0\\0&\;\;0&\;\;0\end{array}} \right], \quad {M_ -} = \left[{\begin{array}{*{20}{c}}1&\;\;i&\;\;0\\i&\;\;{- 1}&\;\;0\\0&\;\;0&\;\;0\end{array}} \right].

When the incident polarization state is LCP, and the incident angle is close to 0° (normal incidence), Eq. (5) yields

\begin{split}{S}_1^{\rm layer + 1} &= {S}_1^{{\rm layer}} - \frac{{i\pi d}}{{2{A_1}}}\left\{{\frac{{\Delta {N^2}{M_ +}S_0^{{\rm layer}}}}{{2{\lambda ^2}}}} \right\} \\ &= {S}_1^{{\rm layer}} - \frac{{i\pi d{A_0}\Delta {N^2}}}{{4{\lambda ^2}{A_1}}}\left[{\begin{array}{*{20}{c}}1&\,\,\,{- i}&\,\,\,0\\{- i}&\,\,\,{- 1}&\,\,\,0\\0&\,\,\,0&\,\,\,0\end{array}} \right]\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right] \\ &= {S}_1^{{\rm layer}} - \frac{{i\pi d{A_0}\Delta {N^2}}}{{2{\lambda ^2}{A_1}}}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right].\end{split}

We can see from here that for the ${+}{1}$st order beam, excited from the zeroth order LCP incident beam, is RCP.

For ${{S}_{- 1}}$,

\begin{split}{S}_{- 1}^{\rm layer + 1} &= {S}_{- 1}^{{\rm layer}} - \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{- 4}}{{{\Lambda ^2}}}{S}_{- 1}^{{\rm layer}} + \frac{{\Delta {N^2}{M_ -}{S}_0^{{\rm layer}}}}{{2{\lambda ^2}}}} \right\} \\ &= {S}_{- 1}^{{\rm layer}} + \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{- 4}}{{{\Lambda ^2}}}{S}_{- 1}^{{\rm layer}}} \right\} \\&\quad- \frac{{i\pi d{A_0}\Delta {N^2}}}{{4{\lambda ^2}{A_1}}}\left[{\begin{array}{*{20}{c}}1&\,\,\,i&\,\,\,0\\i&\,\,\,{- 1}&\,\,\,0\\0&\,\,\,0&\,\,\,0\end{array}} \right]\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right] \\ &= {S}_{- 1}^{{\rm layer}} + \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{- 4}}{{{\Lambda ^2}}}{S}_{- 1}^{{\rm layer}}} \right\} - \frac{{i\pi d{A_0}\Delta {N^2}}}{{4{\lambda ^2}{A_1}}}\left[{\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right].\end{split}

We can see here that there is no energy flowing into the ${-}{1}$st order.

The same calculation steps can be applied for RCP incidence, in which case the ${-}{1}$st order is activated instead of the ${+}{1}$st order.

Note that when the incident angle is negative($m = \; - {1}$), the same calculation can be applied to find whether the same incident polarization state will activate the same diffraction order as when the incident angle is positive.

To summarize the polarization response for a PPD/PVH structure with positive Px,

1. LCP incidence can only activate the ${+}{1}$st order.
2. RCP incidence can only activate the ${-}{1}$st order.

4. DISTINCTION BETWEEN PPD AND PVH DEVICES

In [11], the “$\rho$” parameter is derived for an isotropic material that is the ratio of the two terms for the complex amplitude ${{S}_{l}}$. In this section, the $\rho$ parameter is extracted for a birefringent material from Eq. (2) applying a similar method as for the isotropic material. In this section a positive angle of incidence is assumed.

In Eq. (5), we can see that the change in the zeroth and ${+}{1}$st orders is related to their coupling with adjacent orders. But the change in the ${-}{1}$st order has a different dependence. It is related to its coupling with the adjacent zeroth order, and to itself.

It will be shown below that, for the ${-}{1}$st order, the ratio of the self-coupling term and the coupling to zeroth order terms is significant to differentiating the PPD and PVH optical effects. The ratio is

\begin{split}{\rm ratio} &= \frac{{\left({\frac{4}{{{\Lambda ^2}}}} \right)}}{{\left({\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}} \right)}}= \frac{{4{\lambda ^2}}}{{\left({\frac{{{\Lambda ^2}({{n}_e^2 - {n}_o^2} )}}{2}} \right)}} \\&= \frac{{4{\lambda ^2}}}{{\left({\frac{{{\Lambda ^2}({{n}_e + {n_o}} )({{n_e} - {n_o}} )}}{2}} \right)}} = \frac{{4{\lambda ^2}}}{{\left({{\Lambda ^2}\cdot {n}_{{\rm ave}}^\prime \cdot \Delta n} \right)}},\end{split}

where

n_{{\rm ave}}^\prime = \frac{{\left({{n_{ e}} + {n_o}} \right)}}{2}.

This ratio will be designated as the “$\rho$” parameter for birefringence material,

\rho = \frac{4*\lambda^2}{\Lambda^2 * n_{\rm ave}^\prime * \Delta n}.

The effect of the value of the “$\rho$” parameter can be clearly seen by looking at the propagation of light in the birefringent periodic material though its first two layers [the first two iterations of Eq. (5)].

Light incident on the first layer only has energy in the zeroth order. With RCP incident light, using Eq. (5), the complex amplitudes of the zeroth, ${+}{1}$st, and ${-}{1}$st orders exiting the first layer are

\begin{split}{S}_0^{\rm layer = 1} &= {S}_0^{\rm layer = 0} = \frac{1}{{\sqrt 2}}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right], \\ {S}_{+ 1}^{\rm layer = 1} &= - \frac{{i\pi d}}{{2{A_1}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_0^{\rm layer = 0}} \right\} \\ &= - \frac{{i\pi d}}{{2{A_1}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}\frac{1}{{\sqrt 2}}{M_ +}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right]} \right\} = 0, \\ {S}_{- 1}^{\rm layer = 1} &= - \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}_0^{\rm layer = 0}} \right\} \\ &= - \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{{\Delta {N^2}}}{{2\sqrt 2 {\lambda ^2}}}{M_ -}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right]} \right\} \\ &= - \frac{{i\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right] = \frac{{\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right]\exp\!\left({- i\frac{\pi}{2}} \right).\end{split}

As shown in Section 3, with the assumption that the incident angle is close to 0, energy does not flow to the ${+}{1}$st order with RCP incidence. We therefore only need to calculate the zeroth and ${-}{1}$st orders when exiting the second layer.

Exiting the second layer, for the zeroth order,

\begin{split}{S}_0^{\rm layer = 2} &= {S}_0^{\rm layer = 1} - \frac{{i\pi d}}{{2{A_0}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ -}{S}_{+ 1}^{\rm layer = 1} + \frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_{- 1}^{\rm layer = 1}} \right\} \\ &= {S}_0^{\rm layer = 1} - \frac{{i\pi d}}{{2{A_0}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}{S}_{- 1}^{\rm layer = 1}} \right\} \\ &= {S}_0^{\rm layer = 0} - \frac{{i\pi d}}{{2{A_0}}}\left\{{\frac{{\Delta {N^2}}}{{2{\lambda ^2}}}{M_ +}\!\left({- \frac{{i\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right]} \right)} \right\} \\ &= \frac{1}{{\sqrt 2}}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right] - \frac{1}{{4\sqrt 2 {A_0}{A_{- 1}}}}{\left({\frac{{\pi d\Delta {N^2}}}{{{\lambda ^2}}}} \right)^2}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right].\end{split}

The phase of the zeroth order remains unchanged after propagating through the first two layers.

Exiting the second layer, for the ${-}{1}$st order,

\begin{split}{{S}}_{-1}^{\rm layer=2}&={{S}}_{-1}^{\rm layer=1}-\frac{i\pi d}{2{{A}_{-1}}}\left\{ \fbox{${{\frac{-4}{\Lambda}S_{-1}^{\rm layer = 1}}}$} +\frac{\Delta {{N}^{2}}}{2{{\lambda }^{2}}}{{M}_{-}}{{S}}_{0}^{\rm layer=1} \right\}\\ &= - \frac{{i\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right] - \frac{{i\pi d}}{{2{A_{- 1}}}}\left\{{\frac{4}{{{\Lambda ^2}}}\frac{{i\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right] }\right.\\&\quad+ \left.{\frac{{\Delta {N^2}}}{{2\sqrt 2 {\lambda ^2}}}{M_ -}\left[{\begin{array}{*{20}{c}}1\\{- i}\\0\end{array}} \right]} \right\}\\ &=\frac{\pi d\Delta {{N}^{2}}}{2\sqrt{2}{{\text{A}}_{-1}}{{\lambda }^{2}}}\left[ \begin{array}{*{20}{l}} 1 \\ i \\ 0 \\ \end{array} \right]\exp\!\left( -i\frac{\pi }{2} \right)+\fbox{$\frac{{{\pi }^{2}}{{d}^{2}}\Delta {{N}^{2}}}{\sqrt{2}{{\left( {{A}_{-1}} \right)}^{2}}{{\lambda }^{2}}{{\Lambda }^{2}}}\left[ \begin{array}{*{20}{l}} 1 \\ i \\ 0 \\ \end{array} \right]$} \\&\quad +\frac{{\pi d\Delta {N^2}}}{{2\sqrt 2 {A_{- 1}}{\lambda ^2}}}\left[{\begin{array}{*{20}{c}}1\\i\\0\end{array}} \right]\exp\!\left({- i\frac{\pi}{2}} \right).\end{split}

The term coupling from the zeroth order in the second layer has the same phase as the ${-}{1}$st order in the first layer. However, the self-coupling term (boxed) has phase of zero that is ${-}\pi /{2}$ different from the ${-}{1}$st order in the first layer.

Fig. 3. Phase of the ${+}{1}$st (blue curve), zeroth (orange curve), and ${-}{1}$st (yellow curve) order waves for (a) $\rho = {0.012}$, (b) 0.094, (c) 1.2, and (d) 9.3. $x$ axis is thickness with units of ($\Delta {n}*{ D}/\lambda$); $y$ axis is the phase of the three diffraction orders.

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If the “$\rho$” parameter is small, the self-coupling term is negligible, which results in the phase of the ${-}{1}$st order being constant as it propagates from layer to layer through the material. However, if the “$\rho$” parameter is large, the self-coupling term is not negligible. In this case the phase of the ${-}{1}$st order changes when it propagates from layer to layer, and its phase is changing with respect to the other orders.

In [9], the second term (red boxed) is called a “dephasing” term. This effect can be illustrated by plotting the phase of each of the considered three orders as a function of film thickness as shown in Fig. 3.

When the “$\rho$” parameter is very small, the ${-}{1}$st order has a constant phase [Fig. 3(a)]. As the “$\rho$” parameter keeps increasing, the variation of the phase of the ${-}{1}$st order across the thickness of the device starts to be obvious [Fig. 3(c)]. When the “$\rho$” parameter is large, we can see that the phase of the ${-}{1}$st order is rapidly changing relative to the other two orders [Fig. 3(d)].

From Eq. (5), we can see that the energy is only transferred between adjacent orders. Further, if the ${-}{1}$st order does not have a constant phase with respect to the zeroth order, then as this diffraction order travels through the periodic media the energy transferred to the ${-}{1}$st order from the zeroth order is averaged out to zero. It can also be seen that relative phase between the ${+}{1}$st order and the zeroth order is approximately constant, so energy is transferred between these two orders independent of the value of the “$\rho$” parameter.

The physical explanation of this rapidly varying phase of the ${-}{1}$st order with respect to the zeroth order for the case of a large “$\rho$” value can be roughly seen by considering the simplified ray trace diagrams in Fig. 4.

Fig. 4. Figure shows an incident light beam to a periodic media, and the resulting diffracted orders for PPD (left) and PVH (right).

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For PPD, because the grating period is large, there is not expected to be a large phase difference for the ${+}{1}$st (long dashes) and ${-}{1}$st (dotted line) order beams. For PVH, with a much smaller grating period, the zeroth order deflected beam (solid line) and the ${+}{1}$st order beam traverse regions of similar index of refraction and are expected to have a similar phase variation as they propagate through the media. However, the ${-}{1}$st order sees a greater variation in the index of refraction, and therefore is expected to have a changing phase relationship to the zeroth order beam.

Note that when the incident angle is negative, $m = - {1}$, the ${+}{1}$st order has a dephasing term (instead of the ${-}{1}$st order for the positive incident angle). Following the same calculation in this section, the ${+}{1}$st order will not have a constant phase with a large “$\rho$” parameter because of the self-coupling effect illustrated.

Fig. 5. Transmitted light intensity ($y$ axis) for the structure shown in Fig. 1 versus thickness [$x$ axis, with units of ($\Delta { n}*{\rm D}/\lambda$)]. The input light is RCP, and the incident angle of light is ${+}{10}^\circ$. The transmitted light into the ${+}{1}$st order is shown by the blue curve; into the 0th order by the red curve; and into the ${-}{1}$st order by the yellow curve. (a) $\rho = {1}$; (b) $\rho = {2}$; (c) $\rho = {4}$; and (d) $\rho = {8}$.

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5. EFFECT THE “$\rho$” PARAMETER AND THE TRANSITION FROM PPD TO PVH BEHAVIOR

In this section, we evaluate the effect of the “$\rho$” parameter on the PVH to PPD transition when RCP light incident on the ${+}{\rm Px}$ structure (as in Fig. 1).

In Fig. 5(a), where the value of “$\rho$” is low, it is seen, where the value of $\Delta {n}*{ D}/\;\lambda = {1/2}$, that nearly all intensity is transferred from the incident zeroth order to the deflected ${-}{1}$st order as expected for a PPD device. However, as the value of “$\rho$” increases [Figs. 5(b)–5(d)], less power is transferred from the incident zeroth order to the ${-}{1}$st order, showing the transition from PPD type optical response to PVH type optical response.

Also, for the PPD device, the oscillation of power between the allowed diffracted order and the zeroth order as a function of thickness is made clear.

6. SUMMARY AND DISCUSSION

In this work, the approximate coupled wave approach is extended from isotropic to birefringent materials. Compared with the RCWA, the approximate approach has its limitations (close to Bragg incidence and small vertical grating vector), but is more intuitively understandable.

Using the obtained equation from the beam propagation method of the solution to the wave equation, an explanation is provided for the difference of the PPD and PVH gratings. First, the incident polarization state only has the possibility to excite either the ${+}{1}$st or ${-}{1}$st order depending on its polarization state being LCP or RCP (for an assumed ${+}{\rm Px}$ structure as in Fig. 1). Second, for the case in which the incident angle and the grating $k$ vector have the same sign, the distinction between the optical effect of a PPD versus a PVH is related to the changing relative phase between the ${-}{1}$st order and the zeroth order that results in no net energy being transferred from the zeroth order to the ${-}{1}$st order if the “$\rho$” parameter is large. In the case in which the incident angle and the grating $k$ vector are of the opposite sign, the distinction between the optical effect of a PPD versus a PVH is related to the changing relative phase between the ${+}{1}$st order and the zeroth order that results in no net energy being transferred from the zeroth order to the ${+}{1}$st order if the “$\rho$” parameter is large. These results are summarized in Fig. 6 for the case in which the grating $k$ vector and incident angle are both positive.

Fig. 6. Explanation of the different polarization responses of (a) PPD and (b) PVH. The polarization state of light represented by the green arrow is LCP, while the red arrow is RCP. The black arrows represent a diffraction order that is not allowed. In the case of optimized device thickness, no energy is transferred to the direction of the dashed vectors.

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In Fig. 6, as shown in Section 3, energy transfer to the order indicated by the black arrow is prevented because the order is not activated by the incident polarization state. The difference between PPD and PVH devices was shown in Section 4. For the PVH device, the energy transfer to the ${-}{1}$st order is prevented because of the rapid changing phase relationship between this order and the zeroth order incident beam. The polarization explanation as shown in Section 3 and the phase explanation as shown in Section 4 combine to give us a complete answer to why Fig. 1 is observed in reality.

7. LIMITATION OF THE WAVE PROPAGATION SIMULATION

Unlike FDTD and RCWA, the proposed beam propagation method is very transparent in showing the intensity and phase change of each diffraction order when propagating through an anisotropic media. However, the drawback is that the proposed beam propagation method is not as accurate as FDTD and RCWA because of the necessary approximations that we used to arrive at a simple expression of complex amplitude.

The first approximation is that the $k$ vector directions of the $s$ and $p$ modes for each layer are assumed to be at the same angle.

This limits the simulation to be less accurate for large birefringence. In future work, a more accurate beam propagation method can be worked on by studying the $s$ and $p$ modes separately.

The second approximation is the boundary effects are ignored. This means that the reflections at the interfaces are not considered.

The third approximation is that the term with $({{\varepsilon _a} - {n_{{\rm ave}}^2}}){S}_l^{{\rm layer}}$ in the brackets of Eqs. (2) and (3) is assumed to be 0.

When the $\phi = {90}^\circ$, as shown in Supplement 1, the expression simplifies to

\big({{\varepsilon _a} - {n_{{\rm ave}}^2}} \big){S}_l^{{\rm layer}} = \left[{\begin{array}{*{20}{c}}0&\;\;0&\;\;0\\0&\;\;0&\;\;0\\0&\;\;0&\;\;{- \frac{{\Delta {N^2}}}{2}}\end{array}} \right]\left[{\begin{array}{*{20}{c}}{{S_x}}\\{{S_y}}\\{Sz}\end{array}} \right] \\ = - \frac{{\Delta {N^2}}}{2}Sz.

This shows that the third approximation yields a simulation result that is less accurate for large birefringence, large incident angle (nonzero ${Sz}$), and when the grating vector is different from 90°.

Funding

Meta.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [15].

Supplemental document

See Supplement 1 for supporting content.

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Intuitive understanding of the connection between Pancharatnam–Berry optical beam deflectors and polarization volume holograms

Abstract

1. INTRODUCTION

2. APPROXIMATE COUPLED WAVE APPROACH FOR BIREFRINGENCE MATERIAL

3. EFFECT OF THE INCIDENT LIGHT POLARIZATION STATE ON THE RESPONSE OF PPD AND PVH DEVICES

4. DISTINCTION BETWEEN PPD AND PVH DEVICES

5. EFFECT THE “$\rho$” PARAMETER AND THE TRANSITION FROM PPD TO PVH BEHAVIOR

6. SUMMARY AND DISCUSSION

7. LIMITATION OF THE WAVE PROPAGATION SIMULATION

Funding

Disclosures

Data availability

Supplemental document

REFERENCES

Supplementary Material (1)

Data availability

Cited By

Figures (6)

Equations (23)

Applied Optics