1. Introduction

Polarization changes of light are all around us due to many natural properties of light reflection, transmission or propagation dependence on the polarizations, phase changes depending on the polarization even in the simple case of Fresnel reflection as well as in more complicated cases such as in anisotropic media or scattering effects. However, detectors, similar to our eyes, are not sensitive to polarization and some polarization manipulation elements are needed to observe the effects of polarization changes. Ellipsometry is the technique of quantifying the polarization ellipse of polarized light, while polarimetry is more general term in which partial polarization properties are also extracted [1]. Ellipso-polarimetric (EP) imaging refers to a technique that can measure the polarization ellipse and the Stokes or Mueller matrix elements in one system. Usually, it is done by rotating polarizers [2,3] or for faster operation using photoelastic modulators [4,5]. However, photoelastic modulators (PAs) are wavelength dependent, bulky and expensive, while mechanical rotation of polarizers is slow and prone to vibrations. The use of liquid crystal variable retarders (LCVRs) for polarimetric and ellipsometric imaging has increased recently as they are more compact, cost-effective, operate with low voltage and low power consumption following their long term development for displays [6–8]. In addition to extracting ellipsometric and partial polarimetric parameters, a full Mueller matrix polarimetry systems have been proposed based on integrating LCVRs in the polarization state generator (PSG) and polarization state analyzer (PSA) [9–11]. By integrating polarizer and single LCVR in the PSA and the PSG, the Mueller matrix components can be reconstructed partially, while the rest of the parameters can be extracted by adding single quarter waveplate in each of the PSG and PSA [9]. An optimized Mueller matrix polarimeter can be achieved by including two LCVRs in the PSG and the PSA [10,11]. In addition to the extensive use of LC devices in the optical range, various devices based LC have been developed in the terahertz region such as a broadband tunable waveplate [12] and beam shaping Q-plates [13,14].

One of the limitations of standard LCVRs as well as PAs is their wavelength dependence, therefore not suitable for white light ellipso-polarimetry which demands using narrow spectral band [15]. The use of narrow spectral band is a noise source since the low intensity of the incident light can lead to low contrast images [16]. Therefore, broadening the light spectrum can improve the images contrast while retarders achromatization is an essential requirement to keep the accuracy.

In many applications, the EP images taken using ambient light can reveal characteristics, not seen with standard images [3]. EP imaging examples using ambient light include polarimetric mapping of the sky [17], outdoor imaging capability utilizing sunlight based polarization lattices [18], polarized imaging underwater [19–21], through fog and in biomedical applications such as skin moles imaging [22–25] as well as eye retina imaging [26].

In a previous letter [27] we have demonstrated a tunable achromatic waveplate made of two LC retarders oriented at 90 degrees to each other and operated at different voltages (Fig. 1(a)). The basic concept relies on the fact that the LC birefringence dispersion is varied with the applied voltage. The total retardation of the AWP device equals to the difference between the two retardations of the LC cells that comprise it. By applying the suitable voltages on the LC retarders, the AWP operates as achromatic half waveplate (AHWP), achromatic quarter waveplate (AQWP) and achromatic full waveplate (AFWP). The flexibility, simplicity, tunability and possibility of choosing the achromaticity range within the same device makes it very attractive for many photonic applications with the potential to replace bulky crystal based AWPs.

 

Fig. 1 (a) Schematic diagram of the AWP device consisting of two nematic LC cells with optical axes ${\text{OA}}_1$and ${\text{OA}}_2$ at 90 degrees to each other. (b) The LC cells structure of the old AWP: The two retarders are built with anti-parallel configuration. (c) The LC cells structure of the improved AWP: the first retarder is replaced with a parallel pi-cell configuration while the 2nd part is composed of a cascade of two anti-parallel alignment cells oriented as mirror images one to the other. The pi-cell is the thicker one held at fixed voltage to avoid its effect on the switching time and any instabilities during voltage variations

Download Full Size | PDF

In this paper first we report on extending the viewing angle, wider spectral range and improved speed of the LC AWP and then on integrating it with an imaging camera. Then we prove that tuning the AWP between its three operation modes is enough to extract the ellipsometric parameters: sin(2ψ), cos(Δ) and the polarimetric Stokes parameters ${\text{S}}_1$ and ${\text{S}}_3$ by assuming polarized light while error estimation shows that standard devices give good results for many applications. Finally, integration with a mobile phone camera is demonstrated and potential applications for biometric, remote sensing and archaeological imaging are presented.

2. Device structure and performance

The AWP used in this work has a significant improvement over the previously reported one in that its two LCVR parts have wider field of view, shorter response time and their combination allowed wider spectral range. Figure 1(a) shows the main principle of the AWP design schematically. The angle between the optical axis of the 1st part (${\text{OA}}_1$) and that of the 2nd part (${\text{OA}}_2$) is 90 degrees. The old AWP consisted of two LCVRs with anti-parallel alignment configuration as shown in Fig. 1(b). The first part used Merck BL036 LC material with a thickness of $27\text{μm}$ while the second used Merck E7 LC material with a thickness of $50\mu m$. Figure 1(c) shows schematically the LCVRs structure of the two parts in the improved AWP. The improved 1st part uses Merck BL036 LC material with a thickness of $28\mu m$ and parallel pi-cell configuration [28] (Fig. 1(c)) while the improved 2nd part is composed of a cascade of two cells using Merck E7 LC with a thickness of $25\mu m$, and anti-parallel alignment with the optical axes at the same direction. To obtain the wider field of view, the two devices of the 2nd part are oriented as mirror images one to the other [29] (Fig. 1(c)). Splitting the thicker retarder into two symmetric cells also decreases the response time in addition to improving the viewing angle due to the mirror image orientation geometry. Appendix A expounds the viewing angle comparison between device based parallel and anti-parallel cells with an emphasis on the advantage of AWP based parallel cells.

The performance of the AWP was tested by analyzing its transmission spectrum as described elsewhere [27], when it is sandwiched between polarizer and analyzer while the optical axis of the 1st part of the AWP is fixed at 45 degrees relative to the polarizer as Figs. 2(a) and 2(b) show. The transmission spectrum has been collected in two cases: crossed and parallel polarizer-analyzer orientations where the transmission can be expressed by ${\text{sin}}^2\left({\text{Γ}}_{\text{AWP}}/2\right)$ and ${\text{cos}}^2\left({\text{Γ}}_{\text{AWP}}/2\right)$ respectively. ${\text{Γ}}_{\text{AWP}}={\text{Γ}}_1-{\text{Γ}}_2$ presents the total retardation of the AWP and ${\text{Γ}}_1,{\text{Γ}}_2$ present the retardation of the first and the second part respectively. When the AWP operates as AHWP, the transmission under crossed polarizers case is 100% and under parallel polarizers it is 0%. On the other hand, if the AWP operates as AFWP, the transmission in both cases switches in opposite manner. Finally, when the AWP operates as AQWP, the transmission in both cases is equal to 50%. To see the phase retardation of the AWP, the measured transmission is normalized to transmission of the substrates (including the ITO and the alignment layers). To extract the normalization reference line, we apply a series of voltages on both parts of the AWP and collect the transmission spectrum. When the applied voltages are not suitable for the achromatic performance, we get many fringes in the spectrum with the maximum contrast. Then, we can extract the reference line that connects the maximum points of the different fringes. The more different voltages are applied, the more accurate the reference.

 

Fig. 2 (a) and (b) show schematically, the AWP sandwiched between two polarizers while the optical axis of the first part (violet arrow) is rotated 45 degrees with respect to the polarizer (P) represented by the red arrow. The yellow arrow represents the analyzer oriented in (a) parallel or (b) crossed orientation with respect to the polarizer respectively. (c) Normalized measured transmission of the device performance: the suffixes “-p” and “-c” symbolize measurements in case of parallel and crossed polarizers respectively. AFWP, AHWP and AQWP present measurements when the device operates as: achromatic full, half and quarter wave plate respectively. The applied voltage on the 1st part for the three operation modes is fixed at 2 Volts while on the 2nd part the voltages are: 2.325, 2.365 and 2.4 Volts for AHWP, AQWP and AFWP operation modes respectively.

Download Full Size | PDF

As seen from Fig. 2, the three operation modes of the AWP are achievable by applying suitable voltages on the two parts in the range of 480-750 nm. Generally, if we apply a very high voltage on the LC device it operates as AFWP (especially for thick devices), however in our case this is achieved at low voltage. Low operation voltage is an advantage and essential requirement for compactness such as when integrating the device with a smartphone camera. Practically, the AWP tunes between the three operation modes by applying different voltages on the second part while the applied voltage on the first part is constant. By fixing the applied voltage on the first part before the tuning between the three modes starts, the response time of the AWP is dependent only on the second part. Thus, using one LCVR with pi-cell configuration in the first part improves the viewing angle and saves separating it into two LCVRs to improve the response time. Keeping it at fixed voltage also helps in avoiding instabilities and lack of repeatability that may arise while switching the voltage between different values.

3. Ellipsometry analysis

For the ellipsometric imaging the incident light first passes through a fixed polarizer, then it interacts with the target (transmitted or reflected) and finally passes through the tunable retarder followed by a fixed analyzer. Using Jones matrices, the output can be represented as:

Here ${\text{J}}_{\text{in}}$ represents the incident light after the polarizer, ${\text{J}}_{\text{Ret}}$ represents the retarder at 0 degrees with respect to the optical axis, ${\text{J}}_{\text{target}}$ represents the target and $\text{P}\left(\text{β}\right)$ represents analyzer at $\text{β}$ degrees relative to the optical axis. This equation shows that this is a tantamount to the case of a target between two polarizers with the phase difference shift by the retardation. By choosing linear polarizer at 45 degrees and $\text{β}=-45$, the intensity is:

By performing measurements at the three states of the tunable AWP, from the last equation we can extract the ellipsometric parameters:

Here ${\text{I}}_{\text{F}}$, ${\text{I}}_{\text{Q}}$ and ${\text{I}}_{\text{H}}$ represent the measured intensity when the retarder operates as AFWP, AQWP and AHWP respectively. Furthermore, part of the Stokes parameters can be extracted by these three measured intensities. Since ${\text{I}}_{\text{F}}$ and${\text{I}}_{\text{Q}}$ are equivalent to measuring the intensity using linear polarizer at 0 (${\text{I}}_0$) and 90 (${\text{I}}_{90}$) degrees respectively, ${\text{S}}_1$ can be calculated by:

${\text{S}}_0$ represents a regular nonpolarized image and it is equal to the average of ${\text{I}}_{\text{H}}$ and ${\text{I}}_{\text{F}}$ (${\text{S}}_0=\left[{\text{I}}_{\text{H}}+{\text{ I}}_{\text{F}}\right]/2$). ${\text{I}}_{\text{Q}}$ measurement is equivalent to measuring the intensity using right circular polarizer (${\text{I}}_{\text{R}}$). To calculate S3 we need the left circular polarizer measurement (${\text{I}}_{\text{L}}$), which can be extracted by the relation: ${\text{I}}_{\text{H}}+{\text{I}}_{\text{F}}={\text{I}}_{\text{R}}+{\text{I}}_{\text{L}}\Rightarrow {\text{I}}_{\text{L}}={\text{I}}_{\text{H}}+{\text{I}}_{\text{F}}-{\text{I}}_{\text{R}}$. Then ${\text{S}}_3$ can be calculated by:

Deeper look in the equations exposes more relations between the extracted parameters: ${\text{S}}_1$, ${\text{S}}_3$, $\text{cos}\left(\text{Δ}\right)$ and $\text{sin}\left(2\text{Ψ}\right)$. First observation shows that ${\text{S}}_1$and $\text{sin}\left(2\text{Ψ}\right)$ have a common denominator. An additional step in calculating $\text{sin}\left(2\text{Ψ}\right)$ shows:$\text{sin}\left(2\text{Ψ}\right)=\sqrt{{\text{S}}_1{}^2+{\text{S}}_3{}^2}$, in other words, $\text{sin}\left(2\text{Ψ}\right)$ includes partially the degree of polarization. Second observation shows that $\text{cos}\left(\text{Δ}\right)$ has some similarity to ${\text{S}}_1$ (common numerator) but normalized by a factor as:

This relation enables $\text{cos}\left(\text{Δ}\right)$ images to provide different details of the target images such as the EP camera results shown in the following sections. Considering the deviation around the perfect operation modes versus wavelength (see Fig. 2), the EP parameters contain an error within 5% in the visible band especially $\text{cos}\left(\text{Δ}\right)$ as described in appendix B.

4. Imaging systems

This section presents the proposed imaging systems based AWP while the following section shows the results. Transmission mode EP imaging system is presented in Fig. 3(a). In this setup, the emerging light from a light source (L) passes through a diffuser (D) with a small aperture, then it illuminates the transparent target (T) using lens pair (${\text{LP}}_1$) including a linear polarizer (P). Another lens pair (${\text{LP}}_2$) images the target through the AWP to get more uniform effect of the device. Finally, a zoom camera (ZC) with an analyzer (A) captures the intermediate image on the AWP. The polarizer, AWP and analyzer are fixed relative to the optical axis at 0, 45 and 90 degrees respectively. In the imaging system, we used IDS camera with 1.3-megapixel CMOS sensor (model: UI-3240ML-NIR-GL) and zoom lens of magnification: 0.7X-4X. Wire grid polarizing films were used as polarizers and analyzers. ${\text{LP}}_1$ and ${\text{LP}}_2$ were chosen with a magnification of 1:5 and 4:1 respectively. The reflection imaging setup is presented in Fig. 3(b). The illumination path consists of an annular light source (AL) for uniform illumination on the target (T). AL is followed by an annular linear polarizer (AP) with suitable internal radius, to polarize the incident light without interfering with the imaging path. For that, we used a fiber optic ring light guide connected to a halogen lamp source (Dolan-Jenner MI-157) with IR filter (750nm). Figure 3(c) shows integrating the AWP with a smartphone camera. The smartphone camera is followed by attached X8 zoom lens for remote sensing imaging then by an analyzer and the AWP.

 

Fig. 3 Ellipso-polarimetric imaging systems built both with different and similar illumination paths: (a) transmission, and (b) reflection modes. The illumination path in (a) is composed of: light source (L), diffuser (D), lens pair (LP) and polarizer (P). The illumination path in (b) is composed of: annular light source (AL) followed by an annular linear polarizer (AP). T represents the imaged target (fingerprint as an example). The common imaging path is composed of: lens pair, AWP, analyzer (A) and zoom camera (ZC). (c) presents integration of the AWP with a smartphone using an analyzer (A) and attached zoom lens X8 (Z).

Download Full Size | PDF

5. Results

To demonstrate the performance of the transmission imaging setup, we first used a transparent retardation film as an imaging target (Fig. 4(a)). This target consists of two different areas: half waveplate (HWP) film area and full waveplate (FWP) film area. The (FWP) film area produced by a double layer of HWP. Figures 4(b)-4(d) show the three captured images at different AWP operation modes: AHWP, AQWP and AFWP respectively.

 

Fig. 4 The result of transmission imaging setup by imaging a target (left) composed of two different films: HWP and FWP at 45 degrees to the optical axis:(a) non-polarized image, (b-d) the captured images at different operation mode of the AWP: AHWP, AQWP and AFWP respectively. (e-i) show the extracted images: (e) ${\text{S}}_1$, (f) ${\text{S}}_3$, (g) $\sin \left(2\text{Ψ}\right)$, (h)$\text{cos}\left(\text{Δ}\right)$ and (i) $\left|\text{Δ}\right|$.

Download Full Size | PDF

Since the polarizer and analyzer are crossed to each other, When the AWP operates as AHWP (Fig. 4(b)), the system is acting like imaging target between parallel polarizers. In this case, the FWP area in the target is seen bright while the HWP area is seen dark. On the contrary, when the AWP operates as AFWP (Fig. 4(d)), the system is acting like imaging target between crossed polarizers. Then, the FWP area in the target is seen dark while the HWP area is seen bright. Finally, when the AWP operates as AQWP (Fig. 4(c)), the linear analyzer works as right circular analyzer. Because of that, the intensities in the two areas (FWP and HWP) are the same and equal to the middle gray level. Figures 4(e)-4(h) show the images of ${\text{S}}_1$ ${\text{S}}_3$, $\text{cos}\left(\text{Δ}\right)$ and $\text{sin}\left(2\text{Ψ}\right)$ respectively. In the images of: ${\text{S}}_1$, ${\text{S}}_3$ and $\text{cos}\left(\text{Δ}\right)$ the values are between −1 and 1, while they are represented by gray levels between black and white respectively. The values of $\text{sin}\left(2\text{Ψ}\right)$ are between 0 (black) and 1 (white) indicating nonpolarized and full polarized light respectively.

Since the FWP area in the target keeps the polarization in the same state (the light passes through a linear polarizer), The FWP area is expected to appear white (1) in ${\text{S}}_1$ image. On the other hand, the HWP area is expected to appear dark (−1) in ${\text{S}}_1$, meaning that the linear polarization rotated to the perpendicular state. The small variation from the expected values originates from the HWP film capability to rotate the polarization without affecting its linearity. This also appears in $\text{sin}\left(2\text{Ψ}\right)$ image that has values very close to 1, which means polarized light regardless of its sign. Since the two areas have linear polarizations, ${\text{S}}_3$ image appears mostly in the middle gray level indicating 0 value. The $\text{cos}\left(\text{Δ}\right)$ image also fits the expectations since: $\text{cos}\left({\text{Δ}}_{\text{FWP}}\right)=1$ (white) and $\text{cos}\left({\text{Δ}}_{\text{FWP}}\right)=1$(dark). Comparison between ${\text{S}}_1$ and $\text{cos}\left(\text{Δ}\right)$ images shows that $\text{cos}\left(\text{Δ}\right)$ image has better contrast than ${\text{S}}_1$. We can understand the contrast difference by looking back to Eq. (8). Since $\text{sin}\left(2\text{Ψ}\right)$ includes positive fraction values and $\text{cos}\left(\text{Δ}\right)$ is achieved by dividing ${\text{S}}_1$ on $\text{sin}\left(2\text{Ψ}\right)$, $\text{cos}\left(\text{Δ}\right)$ compensates the deviation from expected values and magnifies the contrast between the positive and negative values of ${\text{S}}_1$ Physically the $\text{cos}\left(\text{Δ}\right)$ images represent phase imaging between the two orthogonal polarizations and therefore expected to give higher contrast when phase variations exist. The sub-image $\left|\text{Δ}\right|$ is calculated using the arccosine numeric function and its values between $\pi$ (white) and 0 (dark) then normalized by $\pi$. It is seen from Fig. 4(i) that the image gray levels in $\left|\text{Δ}\right|$ are the opposite from the $\text{cos}\left(\text{Δ}\right)$ image. This operation converts the image to different mapping which can improve the extracted image as we will see in the next examination. This example demonstrates in fact the ability of our camera to provide a quantitative measure of the phase retardation.

To demonstrate the performance of the reflection imaging setup, a finger leaned on a transparent glass substrate (as a target), then fingerprint images have been captured at the different AWP operation modes. The captured images and the extracted EP images are presented in Fig. 5. It is noticed that the extracted EP images (Figs. 5(e)-5(i)) divulge more details with different contrast levels. By looking closer to ${\text{S}}_1$ we can distinguish between the lines better, but contrast improvements are needed while ${\text{S}}_3$ expose less details. This contrast problem originates from the fact that ${\text{S}}_1$ here has values with small variation around 0 (middle gray level). Since$\text{sin}\left(2\text{Ψ}\right)$ image includes the variation in ${\text{S}}_1$ and ${\text{S}}_3$, and the 0 value is mapped to black, the contrast improved. The $\text{cos}\left(\text{Δ}\right)$ image shows better contrast and details while the nonlinear conversion to $\left|\text{Δ}\right|$ image (Fig. 5(i)) improves the result much more.

 

Fig. 5 The result of reflection imaging setup:(a) non-polarized image of a finger laying on a glass substrate as the imaged target, (b-d) the captured images at different operation modes of the AWP: AHWP, AQWP and AFWL respectively. (e-i) show the extracted images: (e) ${\text{S}}_1$, (f) ${\text{S}}_3$, (g) $\sin \left(2\text{Ψ}\right)$, (h)$\text{cos}\left(\text{Δ}\right)$ and (i)$\left|\text{Δ}\right|$.

Download Full Size | PDF

To demonstrate the performance of a smartphone EP camera, we integrated the AWP with a camera of Samsung S7 as presented in Fig. 3(c). Three images of a morning eastern outdoor scene were captured as before and presented in Fig. 6. The images were analyzed offline using MATLAB. The extracted EP images in this application also fit the expectations. As seen from Fig. 6, the $\text{cos}\left(\text{Δ}\right)$ image shows additional details of the scene and distinguishes between different materials especially if we focus on the building in the left top corner of the image. It seems that $\text{cos}\left(\text{Δ}\right)$ image highlights high-frequency details, such as edges, which is expected as these details usually produce phase retardation. Using a higher zoom lens can improve the contrast since looking for a small area.

 

Fig. 6 The result of integrating the AWP with smartphone camera:(a) non-polarized image of the target that is composed of two different films: HWP and FWP at 45 degrees to the optical axis, (b-d) the captured images at different operation modes of the AWP: AHWP, AQWP and AFWL respectively. (e-i) show the extracted images: (e) ${\text{S}}_1$, (f)${\text{S}}_3$, (g) $\sin \left(2\text{Ψ}\right)$, (h) $\text{cos}\left(\text{Δ}\right)$ and (i) $\left|\text{Δ}\right|$.

Download Full Size | PDF

6. Conclusion

In Conclusion, a novel achromatic EP camera is proposed and demonstrated for several applications using an improved version of tunable AWP. It is based on integrating a camera with a tunable wider field of view tunable LC AWP. The tunable LC AWP operates in three achromatic modes: half, quarter and full waveplate at a visible wide range of 430-780nm. The improvement is achieved using two anti-parallel LC devices oriented as mirror images to each other and the third cell is π-cell. This arrangement makes the field of view symmetric and it widens it to around 15 degrees. Furthermore, it becomes faster because the single devices are thinner. The EP analysis here is based on a simple setup comprising: polarizer, target, tunable waveplate and analyzer. In our setup, as with any other demagnification systems, the angular spread near the camera is the largest, yet locating the AWP near the camera gave good results thanks to its wider field of view. The wide view AWP can be easily used with optical microscopy systems even having high numerical aperture objectives by inserting it between the objective lens and the tube lens where the angular extent is small. In high magnification systems, it is even possible to locate the AWP in front of the camera because then the angular spread is smaller than the field of the view of the AWP (approximately 10-15 degrees). Our setup is equivalent to: polarizer, target and analyzer with phase difference shift by the achromatic retardation. Considering the imperfect performance of the AWP, the error estimation shows that the error is less than 50% in a wide range particularly the $\text{cos}\left(\text{Δ}\right)$ parameter and the AWP can be used into accurate EP system. Three imaging setups were built and demonstrated in transmission and reflection modes and integrated with a smartphone camera. According to the analysis, capturing images at the three modes of the AWP is enough to extract the EP images: $\text{cos}\left(\text{Δ}\right)$ and $\sin \left(2\text{Ψ}\right)$, in addition to ${\text{S}}_1$ and ${\text{S}}_3$. The results are done under the same room temperature conditions; however, a temperature stabilizer should be used when larger temperature fluctuations occur. Another option is to measure the temperature continuously and in a feedback loop to correct the voltages to compensate for the retardation fluctuations due to thermal effects, for example when the EP camera is used for imaging from space.

The performance of the imaging system is demonstrated in the different configurations for biometric, remote sensing and quantitative estimation of anisotropy of transparent objects. In the transmission setup, we imaged a known target composed of HWP and FWP films, while the extracted images fit our expectations with $\text{cos}\left(\text{Δ}\right)$ image distinction. Fingerprint images are taken with the reflection setup, while the extracted images show the potential to expose more details especially $\text{cos}\left(\text{Δ}\right)$ image. A morning eastern outdoor scene was imaged using the smartphone EP camera. The distinction of $\text{cos}\left(\text{Δ}\right)$ image in enhancing the contrast and providing fine details with higher resolution is a significant result with great potential in many applications. The EP camera presented here is characterized by extracting significant surface images using a simple white light source with low voltage controllable device and has potential in many applications such as biometric identification, medical diagnosis, imaging through scattering media, remote sensing, archaeological and industrial inspection of parts and materials. It should be mentioned that the device intellectual property is protected in several patent applications and granted US patents [30,31].

7 Appendix A: Viewing angle comparison between parallel and anti-parallel aligned LC devices

To elucidate the viewing angle difference between antiparallel and parallel cells, we calculated the change in the retardation of AWP device at different incident angles in both cell configurations. The simulation is based on the AWP device composed of BL036 LC cell with 28μm thickness and E7 LC cell with 50μm thickness based on the 4x4 matrix approach [28,32]. The incident angle changes in this calculation in the xz plane while the device is parallel to the xy plane with optical axis rotated 45 degrees with respect to the x-axis as shown in Fig. 7(a). Figures 7(b) and 7(c) show the difference ($\Delta \text{Γ}={\text{Γ}}_{\text{Φ}}-{\text{Γ}}_0$) between the light retardation at an oblique incidence angle (${\text{Γ}}_{\text{Φ}}$) and the retardation at normal incidence (${\text{Γ}}_0$) in the range of 400-800nm. It is seen from Fig. 7(c) that the retardation change as function of incidence angle is far smaller in the case of parallel alignment cell. Hence the field of view is extended to at least 30 degrees for tolerance of 0.1 radians change when parallel aligned cells are used or each cell is divided into two halves oriented as mirror image one to the other. Because of the symmetry, the effect of the change in the incidents angle in the yz plane is the same of change in xz plane.

 

Fig. 7 The viewing angle difference between AWP based antiparallel and parallel cells. (a) Schematically, the light incident angle $\text{Φ}$ in xz plane. The calculated retardation variation from the case of normal incident light ($\Delta \text{Γ}={\text{Γ}}_{\text{Φ}}-{\text{Γ}}_0$) of AWP device based on BL036 LC cell with 28μm thickness and E7 LC cell with 50μm thickness in both configurations: (b)Anti-parallel and (c) Parallel LC cells.

Download Full Size | PDF

The obvious effect of the oblique incident angle is produced in an incidence plane containing the optical axis of the LC retarder and the normal z-axis as shown in Fig. 8. In this case, the oblique incident angle “ Φ ” interferes directly with the LC molecules tilt angle “ ϑ ” of the cell in this plane, while the maximum effect appears when $\vartheta ={45}^{°}$ as explained in [29,33]. The difference between parallel and anti-parallel cells, in this case, appears clearly in the equation of the retardance (assuming uniform tilt angle along the LC retarder):

 

Fig. 8 Schematically, the viewing angle difference between AWP based antiparallel and parallel cells. (a) The plane of incidence composed of: z-axis and optical axis of one of the retarders (${\text{OA}}_1$) while the oblique incident light angle is Φ. (b) The oblique incident angle Φ interferes directly with the LC molecules tilt angle ϑ of the antiparallel cell with thickness d. (c) The oblique incident angle “Φ” interferes directly with the LC molecules tilt angle “ϑ” of two anti-parallel alignment cells with thickness d/2 oriented as mirror images one to the other that can replace one parallel alignment LC cell with thickness d.

Download Full Size | PDF

(9)$$r=\left[n_e(\vartheta +\Phi )-n_o\right]d$$

(10)$$r=\left[n_e(\vartheta +\Phi )-n_o\right]\frac{d}{2}+\left[n_e(\Phi -\vartheta )-n_o\right]\frac{d}{2}=\left[\frac{n_e(\vartheta +\Phi )+n_e(\Phi -\vartheta )}{2}-n_o\right]d$$

Equations (9) and (10) determine the retardance of an antiparallel cell and parallel cell respectively. Here d, $n_o$ and ${\text{n}}_{\text{e}}\left(\vartheta \right)$ represent the cell thickness, the ordinary refractive index and the extraordinary refractive index at molecules tilt angle $\vartheta$ respectively. As seen from Eq. (10), the second part of the parallel cell compensates the effect of the oblique incident angle of the first part. Since the AWP is composed of two parts one perpendicular to the other, the retarder with the optical axis in the incident plane has the dominant effect in the retardance shift. This influence is obvious more when the incident plane includes the optical axis of the thicker 50μm part as seen in Fig. 9. The retardation shift parameter ($\Delta \text{Γ}={\text{Γ}}_{\text{Φ}}-{\text{Γ}}_0$) in this case has been calculated using Eqs. (9) and (10) and $\text{Γ}=2\pi r/\lambda$ while $\lambda$ represents wavelength. Figures 9(a) and 9(b) show the difference between $\Delta \text{Γ}$ for antiparallel and parallel cell respectively for the first 28μm part while Figs. 9(c) and 9(d) show the same for the second 50μm part. Here the field of view is ~10 degrees for a tolerance of 0.1 radians.

 

Fig. 9 The calculated retardation variation from the case of normal incident light ($\Delta \text{Γ}={\text{Γ}}_{\text{Φ}}-{\text{Γ}}_0$) of LC cells. (a-b) BL036 LC cell with 28μm thickness in both configurations: (a) Anti-parallel cell and (b) Parallel cell. (c-d) E7 LC cell with 50μm thickness in both configurations: $\text{©}$ Anti-parallel and $\left(\text{d}\right)$ Parallel cell.

Download Full Size | PDF

8 Appendix B: The estimated errors in EP parameters that originate from non-ideal LC performance

The error estimation is based on the imperfect performance of the retarder versus wavelength as we saw from Fig. 2(c). This can also be applied to calculate the errors due to voltage instabilities and temperature variations. The error analysis is based on calculating the error on the extracted EP parameters as a function of the error in the measured intensities versus wavelength (like in [34]) using:

(11)$$\Delta f_{Theoretical}=\sqrt{{\left(\frac{\partial f(I_H,I_F,I_Q)}{\partial I_H}\Delta I_H\right)}^2+{\left(\frac{\partial f(I_H,I_F,I_Q)}{\partial I_F}\Delta I_F\right)}^2+{\left(\frac{\partial f(I_H,I_F,I_Q)}{\partial I_Q}\Delta I_Q\right)}^2},$$

where $f\left({\text{I}}_{\text{H}},{\text{I}}_{\text{F}},{\text{I}}_{\text{Q}}\right)$ represents: ${\text{S}}_1$, ${\text{S}}_3$, $\text{cos}\left(\text{Δ}\right)$ or $\text{sin}\left(2\text{Ψ}\right)$. $\Delta {\text{I}}_{\text{F},\text{H},\text{Q}}$ represents the measured intensities deviation from the expected values.

The error of the extracted EP parameters has been calculated according to the testing performance measurement in the case of the device between crossed polarizers that appears in Fig. 2(c). In this case, the expected normalized transmissions are:100, 0 and 50 when the device operates as: achromatic full, half and quarter wave plate respectively. In this case, the deviations from the expected values versus wavelength are: $\Delta {\text{I}}_{\text{H}}\left(\lambda \right)=\left|100-{\text{I}}_{\text{H}}\left(\lambda \right)\right|$, $\Delta {\text{I}}_{\text{Q}}=\left|50-{\text{I}}_{\text{Q}}\left(\lambda \right)\right|$ and $\Delta {\text{I}}_{\text{F}}=\left|{\text{I}}_{\text{F}}\left(\lambda \right)\right|$. Figure 10 shows that the Error estimation is less than 0.1 (5% since all the parameters range from −1 to 1) in the range of 450-700nm particularly in $\text{cos}\left(\text{Δ}\right)$ parameter. Hence quantitative measurements can be done for many applications as already demonstrated in the Figs. and in the future we plan to integrate the AWP into a spectroscopic ellipsometer.

 

Fig. 10 Error estimation in the extracted EP parameters (${\text{S}}_1$, ${\text{S}}_3$, $\text{sin}(2\text{Ψ})$ and $\text{cos}\left(\text{Δ}\right)$) as a result of non-ideal LC performance. The error calculation is based on Eq. (11) and the measured normalized transmission from the testing performance setup (from the case of crossed polarizers that appears in Fig. 2(c)).

Download Full Size | PDF

Funding

Israel Ministry of Science, Technology and Space (87538911).

Acknowledgments

M. Abuleil is funded by the Ministry of Science, Technology and Space, Israel (87538911) and the Kreitman Graduate School.

References

1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland Pub. Co., 1977).

2. S. Y. Berezhna, I. V. Berezhnyy, and M. Takashi, “Dynamic photometric imaging polarizer-sample-analyzer polarimeter: instrument for mapping birefringence and optical rotation,” J. Opt. Soc. Am. A 18(3), 666 (2001). [CrossRef]  

3. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef]   [PubMed]  

4. C.-Y. Han and Y.-F. Chao, “Photoelastic modulated imaging ellipsometry by stroboscopic illumination technique,” Rev. Sci. Instrum. 77(2), 023107 (2006). [CrossRef]  

5. H.-M. Tsai, C.-W. Chen, T.-H. Tsai, and Y.-F. Chao, “Deassociate the initial temporal phase deviation provided by photoelastic modulator for stroboscopic illumination polarization modulated ellipsometry,” Rev. Sci. Instrum. 82(3), 035117 (2011). [CrossRef]   [PubMed]  

6. A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar Stokes vector polarimetry,” Opt. Eng. 38(8), 1402 (1999). [CrossRef]  

7. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43(14), 2824–2832 (2004). [CrossRef]   [PubMed]  

8. J. Ladstein, F. Stabo-Eeg, E. Garcia-Caurel, and M. Kildemo, “Fast near-infra-red spectroscopic Mueller matrix ellipsometer based on ferroelectric liquid crystal retarders,” Phys. Status Solidi 5(5), 1097–1100 (2008). [CrossRef]  

9. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000). [CrossRef]  

10. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). [CrossRef]   [PubMed]  

11. R. Uberna, “New polarization generator/analyzer for imaging Stokes and Mueller polarimetry,” SPIE Newsroom (2006).

12. L. Wang, X.-W. Lin, W. Hu, G.-H. Shao, P. Chen, L.-J. Liang, B.-B. Jin, P.-H. Wu, H. Qian, Y.-N. Lu, X. Liang, Z.-G. Zheng, and Y.-Q. Lu, “Broadband tunable liquid crystal terahertz waveplates driven with porous graphene electrodes,” Light Sci. Appl. 4(2), e253 (2015). [CrossRef]  

13. W. Ji, C.-H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T.-H. Lin, V. Chigrinov, and Y.-Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6(1), 25528 (2016). [CrossRef]   [PubMed]  

14. S. Ge, P. Chen, Z. Shen, W. Sun, X. Wang, W. Hu, Y. Zhang, and Y. Lu, “Terahertz vortex beam generator based on a photopatterned large birefringence liquid crystal,” Opt. Express 25(11), 12349–12356 (2017). [CrossRef]   [PubMed]  

15. G. Anna, H. Sauer, F. Goudail, and D. Dolfi, “Fully tunable active polarization imager for contrast enhancement and partial polarimetry,” Appl. Opt. 51(21), 5302–5309 (2012). [CrossRef]   [PubMed]  

16. M. Boffety, H. Hu, and F. Goudail, “Contrast optimization in broadband passive polarimetric imaging,” Opt. Lett. 39(23), 6759–6762 (2014). [CrossRef]   [PubMed]  

17. G. Horváth, A. Barta, J. Gál, B. Suhai, and O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002). [CrossRef]   [PubMed]  

18. M. W. Kudenov, M. J. Escuti, E. L. Dereniak, and K. Oka, “White-light channeled imaging polarimeter using broadband polarization gratings,” Appl. Opt. 50(15), 2283–2293 (2011). [CrossRef]   [PubMed]  

19. M. Dubreuil, P. Delrot, I. Leonard, A. Alfalou, C. Brosseau, and A. Dogariu, “Exploring underwater target detection by imaging polarimetry and correlation techniques,” Appl. Opt. 52(5), 997–1005 (2013). [CrossRef]   [PubMed]  

20. J. Han, K. Yang, M. Xia, L. Sun, Z. Cheng, H. Liu, and J. Ye, “Resolution enhancement in active underwater polarization imaging with modulation transfer function analysis,” Appl. Opt. 54(11), 3294–3302 (2015). [CrossRef]   [PubMed]  

21. B. Huang, T. Liu, H. Hu, J. Han, and M. Yu, “Underwater image recovery considering polarization effects of objects,” Opt. Express 24(9), 9826–9838 (2016). [CrossRef]   [PubMed]  

22. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002). [CrossRef]   [PubMed]  

23. A. Safrani, O. Aharon, S. Mor, O. Arnon, L. Rosenberg, and I. Abdulhalim, “Skin biomedical optical imaging system using dual-wavelength polarimetric control with liquid crystals,” J. Biomed. Opt. 15(2), 026024 (2010). [CrossRef]   [PubMed]  

24. O. Aharon, I. Abdulhalim, O. Arnon, L. Rosenberg, V. Dyomin, and E. Silberstein, “Differential optical spectropolarimetric imaging system assisted by liquid crystal devices for skin imaging,” J. Biomed. Opt. 16(8), 086008 (2011). [CrossRef]   [PubMed]  

25. L. Graham, Y. Yitzhaky, and I. Abdulhalim, “Classification of skin moles from optical spectropolarimetric images: a pilot study,” J. Biomed. Opt. 18(11), 111403 (2013). [CrossRef]   [PubMed]  

26. J. M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24(1), 64–66 (1999). [CrossRef]   [PubMed]  

27. M. J. Abuleil and I. Abdulhalim, “Tunable achromatic liquid crystal waveplates,” Opt. Lett. 39(19), 5487–5490 (2014). [CrossRef]   [PubMed]  

28. D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (John Wiley, 2006).

29. Y. Itoh, H. Seki, T. Uchida, and Y. Masuda, “A Double-Layer Electrically Controlled Birefringence Liquid-Crystal Display with a Wide-Viewing-Angle Cone,” Jpn. J. Appl. Phys. 30(Part 2, No. 7B), L1296–L1299 (1991). [CrossRef]  

30. I. Abdulhalim, “Multi-spectral polarimetric variable optical device and imager,” U.S. patent 10,151,34 B2.

31. I. Abdulhalim and M. J. Abuleil, “Tunable achromatic liquid crystal waveplates,” U.S. patent 10,146,095 B2.

32. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1(5), 646–653 (1999). [CrossRef]  

33. A. Hegyi and J. Martini, “Hyperspectral imaging with a liquid crystal polarization interferometer,” Opt. Express 23(22), 28742–28754 (2015). [CrossRef]   [PubMed]  

34. A. Safrani, “Spectropolarimetric method for optic axis, retardation, and birefringence dispersion measurement,” Opt. Eng. 48(5), 053601 (2009). [CrossRef]