## Abstract

We propose specially designed double anisotropic polarization diffraction gratings capable of producing a selective number of diffraction orders and with selective different states of polarization. Different polarization diffraction gratings are demonstrated, including linear polarization with horizontal, vertical and ± 45° orientations, and circular R and L polarization outputs. When illuminated with an arbitrary state of polarization, the system acts as a complete polarimeter where the intensities of the diffraction orders allow measurement of the Stokes parameters with a single shot. Experimental proof-of-concept is presented using a parallel-aligned liquid crystal display operating in a double pass architecture.

© 2016 Optical Society of America

## 1. Introduction

The controlled generation and detection of polarized light plays a crucial role in many optical systems. Some cases require the controlled generation of different states of polarization by means of polarization state generators (PSG). Polarimetry techniques involve the determination of the state of polarization through the measurement of the intensity of a light beam projected onto different polarization state analyzers (PSA) [1]. Complete Stokes polarimetry plays an important role in many types of optical sensing methods.

There are some interesting emerging areas requiring optical polarizing sensing. These include orbital angular momentum (OAM) and vector beam multiplexing [2–4] as well as Stokes vector scrambling in optical fiber systems [5]. A surprising new area involves use of polarization in analysis of human polarization sensitivity [6].

Typically, Stokes analysis requires measurements with linear polarizers having orientations 0°, ± 45°, and 90°, and right circular (RCP) and left circular (LCP) analyzers. In general, these measurements are made sequentially. Thus, if the state of polarization of the light beam changes within the measurement interval, the results are not reliable. Single shot polarization measurements are well suited for this case.

One of the newest methods to generate polarization states is based on polarization diffraction gratings (PDG). These are diffraction gratings based on a one-dimensional local periodic variation of the polarization transmission [7]. They are designed usually to be either polarizer or waveplate periodic structures, where the orientation of the transmission axis of the polarizer [8,9], or the principal axis of the wave-plate is periodically rotated [10]. They have been fabricated with liquid crystals with special periodic distributions of the director axis [11], with subwavelength patterns written with electron-beam lithography [12], or by femto-laser direct writing [13]. Another approach is the use of liquid crystal displays (LCDs), devices where the birefringence can be controlled in each pixel of the matrix. LCDs have been used to display different types of PDGs, including binary birefringence gratings in ferroelectric LCDs [14,15], or blazed birefringence gratings in parallel-aligned nematic LCDs, to generate duplicators [16] and triplicators [17].

The use of such PDG as elements for Stokes polarimetry was already proposed in the initial works [8,9]. A Fourier transform polarimeter based on a PDG fabricated using subwavelength gratings has been demonstrated in [18] in the near IR range. However, none of the proposed PDG designs provide a single-shot characterization of the above mentioned six typical polarization components used in polarimetry.

In this work, we present different designs for PDGs generating different diffraction orders where the state of polarization can be controlled at will. In particular, we present a grating that produces six diffraction orders with the six above-mentioned typical states employed in polarimetry. Therefore, the same system, in an inverted configuration, is capable to measure the Stokes parameters of the input light beam in a single shot.

This powerful technique is obtained by combining two previously reported major advances in order to create a combined anisotropic diffraction grating acting on two orthogonal linear polarizations. First, we use an optical system where the display is operated in a reflection configuration [19]. The light beam traverses twice through the display, and the screen is divided in two halves to implement two different gratings. The introduction of a quarter-wave plate between the display and the mirror allows switching the polarization component that is modulated on each pass, thus providing a full polarization control. Here, we apply this optical setup to display a double grating, one grating acting on the horizontal linear polarization, and another acting on the vertical linear polarization.

Then, we apply a powerful technique useful to design phase-only gratings with selected target diffraction orders. Romero and Dickey introduced the mathematical formulation for these optimized phase-only gratings in [20,21]. We have recently used this approach to generate phase-only gratings with LCDs showing control of the intensity of the target diffraction orders [22], as well as control of the phase of the orders [23].

Here we combine two such optimized phase-only diffraction gratings to generate a PDG using the optical architecture in [19]. As a result we demonstrate the capability to generate diffraction gratings where we can control the number of diffraction orders, and where the polarization content can be designed at will. We show different cases. One of them demonstrates the implementation of a PDG that generates the six polarization components usually applied in polarimetry. We show afterwards a proof-of-concept that this approach can allow the measurement of the Stokes parameters in a single shot.

## 2. Experimental system and gratings design

The LCD device used in the experiments is a parallel-aligned (PAL) liquid crystal spatial light modulator (LC-SLM) (Seiko Epson, 640 × 480 pixels, 42 μm pixel pitch). Each pixel acts as an electrically controlled wave-plate with a phase shift that exceeds 2π radians when illuminated with an Ar laser at the wavelength of 514.5 nm [24]. As mentioned earlier the LCD is used in a reflection configuration as shown in Fig. 1, in order to achieve full polarization control [19]. Input light linearly polarized at 45° illuminates the system, so there is equal weight of horizontal and vertical polarization components.

The LCD director axis is vertically oriented. Thus, in the first passage, light is modulated only in the vertical linear polarization component, while the horizontal linear component remains unaffected. A quarter-wave plate located behind the SLM produces a 90° rotation of polarization upon reflection. Then, in the second passage through the display, the initial linear horizontal polarization is now modulated by the SLM. This way we can implement a PDG by splitting the LCD screens in two halves, each one displaying a phase-only grating, one acting on the horizontal polarization component (grating H) and another one acting on the vertical polarization component (grating V). Note that, because the SLM and the mirror are located at the focal length distance from the lens L1, the first grating is imaged on the second grating with unit magnification. The final lens (L2) forms the Fourier transform on the CCD plane. Alternatively, the different beams could remain collimated, but propagating in different directions.

These two gratings have a diameter of 240 pixels and are designed following the procedure in [20,21]. This procedure allows the design of a phase-only grating that produces a selected number of diffraction orders with selective intensity and phase control. Following this approach, we start by defining the target diffraction orders as:

*k*. The parameters ${\mu}_{k}$ and ${\alpha}_{k}$ are respectively magnitude and phase numerical parameters associated to each diffraction order

*k*, and $\gamma $ defines the grating’s period.

The function *s*(*x*) is, in general, a complex valued grating. Then, the grating is made phase-only as:

The total efficiency of the optimized grating is defined as:

Therefore, the design of the optimized grating involves the following steps: 1) defining the desired target orders *k*; 2) defining the desired magnitudes $\left|{c}_{k}\right|$ and phases ${\beta}_{k}$ of the target diffraction orders; and finally 3) finding the numerical parameters ${\mu}_{k}$ and ${\alpha}_{k}$ that provide these desired target orders with the maximum possible efficiency $\eta $. The optimization of Eqs. (1-4) with the selected contraints can be performed with advanced numerical tools. We used the SOLVER routine in Microsoft Excel, which employs a generalized reduced gradient algorithm [25]. As a result, a phase-only grating exp[*iφ*(*x*)] is obtained in accordance to Eqs. (2)-(3). This phase-only function can be viewed as a phase look-up table (LUT) that transforms the linear phase grating, with phase values *ϕ* in the range [0,2π], into the corresponding phases *φ*(*x*) of the optimized phase-only grating. As a result a phase mapping LUT *φ*(*ϕ*) is obtained, that can be applied to other phase-only functions [23].

In most of the cases, this technique has been applied to laser splitting problems, and only the magnitude values |*c _{m}*| are of interest. However, the technique has also been used to also achieve phase control between the different diffraction orders, as we examined in [23]. Here we apply this full control to design polarization gratings. The ability to control the magnitude and the phase of the diffraction orders in the two orthogonal components of the electric field permits this control of the state of polarization.

## 3. Experimental results

We present in this section some examples that demonstrate the flexibility of the proposed system and the capability to generate diffraction orders with different states of polarization to act as a polarization state generator (PSG).

#### 3.1 Polarization duplicator

As a first simple example we start by demonstrating the action of the system as a polarization beam splitter with 100% efficiency. For that purpose, we address a linear phase grating on each side of the SLM. The periods of both gratings are the same, but each grating diffracts in an opposite direction. Figure 2(a) shows the image addressed to the SLM as well as the linear phase profile encoded onto each grating. Note that, due to the mirror inversion on the reflected beam, the same linear grating as shown in Fig. 2(a) diffracts in opposite angle for the reflected beam. Each blazed grating acts on a different polarization component, and the result is the generation of the ± 1 diffraction orders, with orthogonal linear states of polarization.

Figure 2(b) shows the experimental results captured by the CCD camera. The capture indicated as “NoA” corresponds to the absence of a polarization analyzer before the CCD camera. This capture shows the generation of two equally intense diffraction orders. The other captures correspond to the case where an analyzer is placed in front of the CCD detector, including a linear polarizer at orientations 0°, 45°, 90° and 135°, and right circular (RCP) and left circular (LCP) polarizers. These analyzers are indicated on top of the figure. These results verify that the + 1 diffraction order is linearly polarized with vertical orientation, since it is completely cancelled when the linear analyzer is oriented horizontally. On the contrary, the −1 diffraction order is linearly polarized with horizontal orientation, as it can be verified by noting that it disappears when the analyzer is vertically oriented.

Note, therefore, that this simple PDG acts as a polarization beam splitter with ± 1 orders (as opposed to the PDG in [15] where we used the + 1 and 0 orders).

#### 3.2 Polarization quadruplicators

A second more interesting case is presented in this subsection. Now we design quadruplicator gratings, i.e., gratings that generate four diffraction orders with the same intensity, but where we can control the states of polarization for each order.

A first such case is presented in Fig. 3. Here we designed a PDG to produce four diffraction orders: ± 2 orders and ± 1 orders. The + 2 order is designed to generate the vertical linear polarization while the −2 order is designed to generate the horizontal linear polarization. The ± 1 orders are designed to generate the RCP and LCP polarizations.

To produce these polarization states we need to properly design the V and H gratings. The optimized phase-only grating V acting on the vertical linear polarization is designed with the following restrictions: 1) The intensity of the + 2 order is twice that of the ± 1 orders, i.e., |*c*_{+2}|^{2} = 2|*c*_{+1}|^{2} = 2|*c*_{-1}|^{2}; 2) the phase relations at the positive and negative first diffraction orders are fixed to be ${\beta}_{+1}^{V}=\pi /2$ and ${\beta}_{-1}^{V}=0$ radians. Figure 3(a) shows the phase LUT profile that must be applied to a linear phase grating to obtain this desired response. The efficiency of this grating is *η* = 0.86. Table 1 gives the numerical values for the *μ _{k}* and

*α*numerical parameters that provide these values.

_{k}The grating H is similar, but designed with slightly different restrictions: 1) The intensity of the −2 order is twice that of the ± 1 orders, i.e., |*c*_{-2}|^{2} = 2|*c*_{+1}|^{2} = 2|*c*_{-1}|^{2} and phase values ${\beta}_{+1}^{H}=0$ and ${\beta}_{-1}^{H}=\pi /2$ (they are interchanged compared to the V grating due to the mirror inversion in the optical system) The efficiency is again $\eta =0.86$.

The combination of these V and H gratings results in a PDG that generates four diffraction orders with equal intensity but four different states of polarization. Since the + 2 order is only selected as a target at the V grating, this order only has vertical polarization component. Similarly, since the −2 order is only selected at the H grating, this order only has horizontal linear polarization component.

On the contrary, the ± 1 orders are selected target orders for both V and H gratings. Therefore, they will receive the contributions from both gratings. But the + 1 diffraction order gains an additional phase of *π*/2 radians for the vertical component, while the order −1 gains a phase of *π*/2 for the horizontal component. Therefore, the superposition of the two diffraction patterns generates LCP and RCP polarizations at these two diffraction orders. Figure 3(a) shows the two gratings as well as the grating phase profiles over a single period.

Figure 3(b) shows the experimental results captured by the camera. The left panel shows the CCD capture in the absence of a polarization analyzer. We can verify the production of four diffraction orders with the same intensity. Note that there are some other residual light (in this case less than 15%) being diffracted onto other diffraction orders (mainly the zero and the ± 4 orders).

In order to verify the different states of polarization generated at the four target diffraction orders, we introduced the linear polarizer analyzer at angles 0°, 45°, 90° and 135°, and the RCP and LCP analyzers. When the linear polarizer analyzer is oriented in the vertical direction (second column), we can verify the production of the + 2, + 1 and −1 orders. The order −2 is not produced, thus showing that this order is horizontally polarized. Similarly, the + 2 order is not produced when the linear analyzer is oriented horizontal, thus showing that this order is vertically polarized. The circular polarization on the ± 1 orders is verified when placing the circular polarizers. Order + 1 vanishes when the LCP analyzer is selected, while order −1 vanished when the RCP analyzer is used. Therefore, these results show the production of four different polarizations on the four selected target orders.

A second quadruplicator grating design is presented in Fig. 4. In this case, again, the PDG is designed to produce four diffraction orders but now we operate with the ± 3 and ± 1 diffraction orders. In this example we want to generate linear polarization states at 45° and 135°, and the RCP and LCP circular polarization states.

To accomplish these states, the most efficient solution that we found consists in the two following different gratings: 1) The V grating is constrained to have ± 3 and ± 1 diffraction orders with the same intensities, and relative phases of ${\beta}_{+3}^{V}=\pi $ and ${\beta}_{+1}^{V}={\beta}_{-1}^{V}={\beta}_{-3}^{V}=0$ radians; 2) the H grating has the same constraint for intensity values, but the relative phases are forced to be ${\beta}_{+3}^{H}={\beta}_{-1}^{H}=0$, and ${\beta}_{+1}^{H}=\pi /2$ and ${\beta}_{-3}^{H}=-\pi /2$. Therefore, ± 3 and ± 1 diffraction orders will be generated by both H and V gratings, with the same intensities, and with relative phase differences $\Delta {\beta}_{+3}^{}={\beta}_{+3}^{H}-{\beta}_{+3}^{V}=-\pi $, $\Delta {\beta}_{+1}^{}={\beta}_{+1}^{H}-{\beta}_{+1}^{V}=\pi /2$, $\Delta {\beta}_{-1}^{}={\beta}_{-1}^{H}-{\beta}_{-1}^{V}=0$, and $\Delta {\beta}_{-3}^{}={\beta}_{-3}^{H}-{\beta}_{-3}^{V}=-\pi /2$.

Figure 4(a) shows the two phase LUT profiles that must be applied to a linear phase grating to obtain these two gratings, as well as the images addressed to the SLM to produce the combined PDG. Table 2 provides the numerical parameters that generate these phase profiles. Note that both V and H gratings produce the same four target diffraction orders. The efficiency of these two gratings is the same *η* = 0.86, so it is ensured that both horizontal and vertical polarization components have equal weight on these four diffraction orders.

As before, the selected phase relations will fix the state of polarization of these superpositions. Linear polarization states oriented at 45° and 135° are produced at the −1 and + 3 diffraction orders, where the phase difference in the superposition is zero and π radians respectively. Circular polarization states will be produced at the + 1 and −3 diffraction orders, where the phase difference is ± π/2.

The experimental results in Fig. 4(b) confirm these expected responses. Again, the first column shows the results without a polarization analyzer, and the four target diffraction orders appear with the same intensity. The expected states of polarization are shown when placing the polarization analyzers in front of the CCD camera. Now, when the linear polarizer analyzer is oriented either vertical or horizontal, these two polarization components show the same intensity on the four diffraction orders. On the contrary, when the linear analyzers are oriented at 45° and 135°, the + 3 and −1 orders are cancelled respectively, showing that these two diffraction orders are linearly polarized at the orthogonal angles respectively. Finally, the + 1 and −3 orders are cancelled when using the circular polarizers as analyzers, showing that these two orders are circularly polarized, one with RCP and the other with LCP states.

#### 3.3 Polarization sextuplicator

Finally, we combine all these previous cases into a single grating capable to create all the six polarization states typically required in polarimetry, i.e., linear polarized light oriented in the vertical and horizontal directions, at 45° and 135°, and the circular LCP and RCP states.

Table 3 give the numerical parameters of the gratings In this case, the most efficient solution we found consists of 1) The V grating is constrained to have ± 3, + 2 and ± 1 diffraction orders, with the + 2 order having twice the intensity as the ± 3 and ± 1 orders, which are all equally intense. Note that the −2 order is not included here. In addition, the phase at these diffraction orders are forzed to be of ${\beta}_{+3}^{V}=-3\pi /4$, ${\beta}_{+1}^{V}=+\pi /2$, ${\beta}_{-1}^{V}=-\pi /4$, and ${\beta}_{-3}^{V}=0$ radians; 2) on the contrary, the H grating is selected to have ± 3, −2 and ± 1 diffraction orders, with the −2 order having twice the intensity as the ± 3 and ± 1 orders, which again are all equally intense. Again note that the + 2 order is not included here. However now the phase at these diffraction orders are constrained to be ${\beta}_{+3}^{H}=+3\pi /4$, ${\beta}_{+1}^{H}=-\pi /2$, ${\beta}_{-1}^{H}=+\pi /4$, and ${\beta}_{-3}^{H}=0$ radians. The efficiencies of these two gratings are the same at $\eta =0.87$.

Therefore, as in previous cases, the ± 2 orders will become vertically and horizontally polarized, since these orders are generated only with one of the two H or V gratings. On the contrary, ± 3 and ± 1 diffraction orders will be generated by both H and V gratings, with the same intensity and with relative phase differences $\Delta {\beta}_{+3}^{}={\beta}_{+3}^{H}-{\beta}_{+3}^{V}=3\pi /2$, $\Delta {\beta}_{+1}^{}={\beta}_{+1}^{H}-{\beta}_{+1}^{V}=-\pi $, $\Delta {\beta}_{-1}^{}={\beta}_{-1}^{H}-{\beta}_{-1}^{V}=\pi /2$, and $\Delta {\beta}_{-3}^{}={\beta}_{-3}^{H}-{\beta}_{-3}^{V}=0$ and producing the linearly polarized outputs at ± 45°, and the circular R and L polarization states.

Figure 5(a) shows the phase LUT profiles corresponding to these two gratings, and the corresponding image addressed to the display.

As before, the experimental results in Fig. 5(b) confirm these expected responses. Again, the first column shows the results without a polarization analyzer, and the six target diffraction orders appear with the same intensity. The expected states of polarization are shown when placing the polarization analyzers in front of the CCD camera. Now, when the linear polarizer analyzer is oriented either vertical or horizontal, the + 2 and −2 orders vanish respectively. On the other hand when the linear analyzers are oriented at 45° and 135°, the + 3 and −1 orders are cancelled respectively, showing that these two diffraction orders are linearly polarized at the orthogonal angles. Finally, the + 3 and −1 orders are cancelled when using the circular polarizers as analyzers, showing that these two orders are circularly polarized, one with RCP and the other with LCP states.

## 4. Use as a snapshot Stokes complete polarimeter

Once the ability of the PDG as a polarization state generator (PSG), generating the six polarization components at six different orders has been demonstrated, we show next that the same grating can be used as polarization state analyzer (PSA) capable to detect all of these six polarization components in a single-shot measurement. Therefore, it can be a useful new method to build single-shot complete Stokes division of amplitude polarimeters [1].

For that purpose, we eliminate the first input polarizer from the system in Fig. 1, and now illuminate the PDG with different states of polarization, generated with a linear polarizer and a quarter-wave plate system. Finally, the output linear polarizer analyzer is selected to be linear and oriented at 45°. Since the PDG converts a linearly polarized input beam, oriented at 45°, into a set of six diffraction orders with their corresponding polarizations, detecting the linear component at 45° at the output is a measurement of the corresponding component in the input beam. If we denote as *I _{k}* (

*k*= 0,45,90,135,RCP,LCP) the intensity of the

*k*-polarized diffraction order, the Stokes parameters that characterize the input beam are directly obtained as

*S*

_{1}=

*I*

_{0}-

*I*

_{90},

*S*

_{2}=

*I*

_{45}-

*I*

_{135}and

*S*

_{3}=

*I*

_{RCP}-

*I*

_{LCP}. These measurements can be made simultaneously for all six polarizations, thus acting as a single-shot polarimeter.

For instance, if the input light is linearly polarized at 45°, then the pattern in Fig. 5(b) is obtained. When the analyzer is set at 45° then the intensity is maximal at the order −3, and zero at the order + 1. But if other input polarization is used on the input beam, the intensity of the diffracted orders is a direct measurement of the weight of the corresponding component in the incoming beam. Experimental evidence is presented in Fig. 6. On the top of each experimental result, we indicate the polarization of the input beam. On the left, the PSA detecting in each diffraction order is indicated.

These experimental results qualitatively verify the capability of the PDG configured in this way to reproduce a Stokes polarimeter. This is clearly visible in the fact that the order measuring the component orthogonal to that of the incoming beam is absent in each case, while the order with the same polarization component has maximum intensity.

## 5. Conclusions

In summary, we have proposed a double anisotropic polarization diffraction grating (PDG) system capable to generate selected diffraction orders with different polarization states that can be defined at will. We apply a technique for designing phase-only gratings with optimal diffraction efficiency that generate diffraction orders with desired relative intensity and phase [20]. We apply them in an optical system using a parallel-aligned nematic LCD operating in a special double passage optical system that allows the modulation of the two orthogonal linear polarization components and achieve full polarization control. In this way we are able to generate arbitrary polarization diffraction gratings. Different PDG examples have been experimentally demonstrated, and the generation of selected diffraction orders with defined different states of polarization has been probed.

We note that because the SLM is a pixelated device, the theoretical designs of the gratings are not exactly reproduced. This can lead to some polarization distortion as the number of target diffraction orders increases, since the total diffraction efficiency is split into a larger number of orders. In spite of the limited spatial resolution of the SLM, we provided a experimental demonstration of the generation of diffraction orders with polarization control. One of the examples included a sextuplicator grating generating six diffraction orders with the six typical polarizations employed in polarimetry. Then, this PDG has been also demonstrated to be useful as a single-shot Stokes polarimeter to perform instantaneous polarization measurements of a light beam. A proof-of-concept of such polarimeter has been provided by using the same sextuplicator PDG with a final analyzer oriented at 45°. The correct identification of the input polarization has been qualitatively demonstrated by the cancellation of the diffraction order with orthogonal polarization.

The use of such system as an accurate quantitative polarimeter will require a deeper study of the polarization distortions introduced by the technique. This should include the analysis of the theoretical designs (note that the use of phase-only gratings imply that not all the energy is distributed on the target orders), the effect of the limited spatial resolution of the SLM, which can degrade the PDG design performance (especially when a large number of target orders is required), as well as the polarization distortion caused by the other optical elements in the experimental system. These polarization distortion sources must be taken into account to achive accurate quantitative polarimetric measurents.

Nevertheless, the presented experimental results demonstrate the ability of the proposed method to generate arbitrary PDG. As mentioned in the introduction, there are numerous applications for this system.

## Acknowledgments

IM and MMSL acknowledge support from Ministerio de Ciencia e Innovación from Spain (ref. FIS2012-39158-C02-02).

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