Double-channel angular-multiplexing polarization holography with common-path and off-axis configuration

Lu Han; Zhen-Jia Cheng; Yang Yang; Ben-Yi Wang; Qing-Yang Yue; Cheng-Shan Guo

doi:10.1364/OE.25.021877

1. Introduction

Digital holography (DH), due to its superior ability in measuring and imaging the complex amplitude distribution (including both the amplitude and phase information) of an object wavefront, has drawn much attention in recent years and has been widely applied in many significant fields such as quantitative phase microscopy, contour measurement and nondestructive tests [1–8]. According to the relative orientation between the object and the reference beams in recording geometry, DHs are often divided into two categories: on-axis (or in-line) and off-axis (or off-line). It is well known that the on-axis configuration can reduce the spatial bandwidth of the recorded holograms and thus make full use of the resolving power of the image sensor; but the reconstructed images often suffer from the disturbances of the zero-order and twin images, which need to be removed by using phase shifting operation or other time-consuming elimination algorithms [9–18]. In off-axis DHs, the zero-order and twin images are well separated from the required image in spatial frequency domain and so they can be removed from a single hologram simply by a digital spatial filtering, but at the cost of a high bandwidth demand on the imaging sensor. As the spatial resolution and the pixel number of image sensors have been significantly increased in recent years (for example, the pixel size of a current commercial image sensor can be close to 1 μm and the pixel number can reach more than 20 megapixels), off-axis digital holography is becoming a powerful tool in dynamic phase measuring and imaging applications [19–25].

DHs with common-path and off-axis configuration denote a class of holographic recording architectures, in which both the imaging and the reference beams for off-axis interferometric recording propagate nearly through the same optical path and the same imaging lenses [26–33]. Compared with the non-common path DHs such as those based on Mach-Zehnder or Michelson interferometers, common-path configurations are more stable, relatively insensitive to mechanical vibrations and air fluctuations, and require fewer optical elements. One-arm lateral shearing interferometers (LSIs) [33] are the most widely used common-path configurations. In conventional LSI-based common-path DHs, the holograms are mainly formed by interference between two beams sheared from the same object beam passing through the same optical path and imaging system. Because both the two beams involved in the interference are disturbed by the object, the reconstructed image will be disrupted by some additional undesired terms (such as the duplicated image), along with the zero-order and twin images. To overcome this drawback, some modified LSI systems used for common-path off-axis DHs have been developed in recent years [34–41]. For example, Mico et al. [34] proposed a common-path off-axis DH system based on spatially-multiplexed LSI, in which the input plane of the system is spatially divided into two parts in side-by-side configuration: the object region under test and a black region for the reference; at the same time, a one-dimensional grating is inserted between the input and output planes of the system to realize the overlap and off-axis interference of the object and reference beams at the output plane. Seo et al. [35] also proposed a similar modified LSI for the same purpose nearly at the same time, in which an optical window glass was used as the shearing element. Because the common-path and off-axis configurations, based on the modified LSI, have the advantages of simple structure and high anti-disturbing ability and provide a low-cost way to convert a regular microscope into a holographic one, they have drawn much attention most recently and have been applied in quantitative phase imaging and surface inspection [36–41].

In this paper, we propose a common-path off-axis polarization holographic (COPH) imaging system based on a modified double-channel LSI configuration. In the proposed system, the input plane of the imaging system is spatially divided into three windows: an object window (or object window) and two reference windows; two orthogonal linear polarizers are placed, respectively, on the two reference windows; in the path space between the input and output planes of the imaging system is inserted a specially designed two-dimensional (2-D) cross grating. Thus, the object beam passing through the object window and the two orthogonal polarized reference beams passing through the two reference windows can overlap each other at the output plane of the system and form a double-channel angular-multiplexing polarization hologram (DC-AM-PH). Using this system, the complex amplitude distributions (including both the amplitude and phase information) of the two orthogonal polarized components of the object image can be recorded and retrieved by one single-shot DC-AM-PH at the same time.

2. Principle of the system

Figure 1 shows the thematic of our proposed COPH system. It is essentially an imaging system mainly composed of an objective lens L1 and a tube lens L2. The main improvements are: a special limiting screen P1 with three windows is placed at the input plane and a 2D cross grating CG is inserted between L1 and L2. The object S to be tested is put in the central window (named object window). Two linear polarizer with orthogonal polarization directions, PR1 and PR2, are respectively attached behind the other two windows (named reference windows). Thus, after passing through this limiting screen P1, the input beam (its polarization state can be adjusted by a half wave plate WP, as required) is divided into three parts: the object beam passing through the object window and the two orthogonal polarized reference beams passing through the two reference windows. These three parallel transmission beams co-pass through the objective lens L1, the cross grating CG and the tube lens L2 and are finally imaged on the output plane P2. Due to the diffraction effect of the cross grating CG inserted between L1 and L2, the complex amplitude distribution of the output field at the output plane P2 will be composed of some replica images separated from each other, each corresponding to one of the diffraction orders from the grating CG. If the separation distance of the windows, the grating parameters and other system parameters are set to satisfy some conditions, one image of the object beam will be superimposed with the images of the two reference beams at the center part of the output plane as shown in Fig. 1, and form a DC-AM-PH recording the complex amplitude information of two orthogonal polarized components of the object image, which can be reconstructed by conventional spatial filtering algorithm used in angular-multiplexing off-axis holography [42]. In the following paragraphs, we will give first its theoretical analysis and then give its verification by some experiments in next section.

Fig. 1 Schematic of the proposed COPH system

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We know there are two approaches to describe polarized beams and polarization-sensitive materials: using Stokes vector and Mueller matrix or using Jones vector and Jones matrix [43]. The Stokes formalism is usually used for intensity superposition problems, while the Jones calculus is often applicable for complex amplitude superposition problems. Like most of holographic polarization imaging methods [44–50], we also use Jones formalism to describe the measured parameters in the following analysis.

Supposing the input beam is first linearly polarized along x axis (horizontal) and the transmission directions of the linear polarizers PR1 and PR2 are set, respectively, along + 45° and −45°, the Jones vector of the field passing through the limiting screen P1 can be expressed as

u_{1} (x, y) = r e c t (\frac{x}{a}, \frac{y}{a}) [\begin{matrix} E_{x} (x, y) \\ E_{y} (x, y) \end{matrix}] + R_{1} r e c t (\frac{x - b}{a}, \frac{y}{a}) [\begin{matrix} 1 \\ 1 \end{matrix}] + R_{2} r e c t (\frac{x}{a}, \frac{y - b}{a}) [\begin{matrix} 1 \\ - 1 \end{matrix}],

where,

r e c t (x / a, y / a)

is a square window function with the width of a, R₁ and R₂ are the amplitudes of the beams respectively passing through the two reference windows, and b is the central distance of the reference windows away from the object window along x or y axis, respectively; while

E_{x} (x, y)

and

E_{y} (x, y)

are the complex amplitudes of two Jones vector components of the object field passing through the object window, which also can be respectively expressed by Jones matrix parameters of the object as

E_{x} = A_{x} J_{x x} (x, y)

and

E_{y} = A_{x} J_{y x} (x, y)

(where A_x corresponds to the amplitude of the input horizontally polarized beam). As indicated above, the complex amplitude distribution of the output field at the image plane P2 will be composed of some replica images separated from each other, due to the diffraction effect of the cross grating CG. Considering only the replicas corresponding to the zero order and the two + 1 orders respectively along x and y directions, as shown at the image plane P2 of Fig. 1, the output complex amplitude distribution can be written as

\begin{matrix} u_{2} (x, y) = D_{0} u_{1} (\frac{x}{M}, \frac{y}{M}) + D_{1 x} u_{1} (\frac{x + d_{g}}{M}, \frac{y}{M}) \exp [- j 2 π ξ_{g} x] \\ + D_{1 y} u_{1} (\frac{x}{M}, \frac{y + d_{g}}{M}) \exp [- j 2 π ξ_{g} y_{0}], \end{matrix}

in which,

M = f_{2} / f_{1}

is the amplification of the image system, f₁ and f₂ are, respectively, the focal lengths of L1 and L2; D₀, D_1x and D_1y are three constants related to the amplitude of the input beam and the diffraction efficiencies of the cross grating; the shift distance d_g and the tilt frequency parameter ξ_g (corresponding to the carrier frequency of the reference beams) of the replicas depend on the configuration parameters of the grating and the imaging system, which can be determined by the following formulas [51]:

d_{g} = \frac{λ f_{2}}{T}, and ξ_{g} = \frac{Δ}{f_{2} T},

where λ is the wavelength of the input beam, T is the period of the cross grating, and Δ is the distance of the cross grating away from the front focal plane of the tube lens as shown in Fig. 1. It can be seen from Eq. (3) that the carrier frequency or the off-axis angle (

\sin θ = λ ξ_{g}

) of the reference beams can be simply aligned by changing the distance parameter Δ.

Substituting Eq. (1) into Eq. (2), we can find that, if the central distance b between the object window and the reference windows is set to satisfy the condition of

b = \frac{d_{g}}{M} = \frac{λ f_{1}}{T},

the images of the beams from the two reference windows with orthogonal linear polarization states will exactly overlap with the image of the object window in the center region highlighted by the red dotted square shown in the image plane P2 of Fig. 1. The total complex amplitude in this region can be expressed as

u_{c} (x, y) = r e c t (\frac{x}{M a}, \frac{y}{M a}) [\begin{matrix} D_{0} J_{x x} (\frac{x}{M}, \frac{y}{M}) + D_{1 x} R_{1} \exp [- j 2 π ξ_{g} x] + D_{1 y} R_{2} \exp [- j 2 π ξ_{g} y] \\ D_{0} J_{y x} (\frac{x}{M}, \frac{y}{M}) + D_{1 x} R_{1} \exp [- j 2 π ξ_{g} x] - D_{1 y} R_{2} \exp [- j 2 π ξ_{g} y] \end{matrix}] .

Its intensity distribution just forms a DC-AM-PH simultaneously recording the two orthogonal linear polarization components of the object beam from the object window, which can be written as

\begin{matrix} H (x, y) = {| D_{0} J_{x x} (\frac{x}{M}, \frac{y}{M}) + D_{1 x} R_{1} \exp [- j 2 π ξ_{g} x] + D_{1 y} R_{2} \exp [- j 2 π ξ_{g} y] |}^{2} \\ + {| D_{0} J_{y x} (\frac{x}{M}, \frac{y}{M}) + D_{1 x} R_{1} \exp [- j 2 π ξ_{g} x] - D_{1 y} R_{2} \exp [- j 2 π ξ_{g} y] |}^{2}, \\ = Y_{0} (x, y) + Y_{1 x} (x, y) + Y_{1 y} (x, y) + Y_{1 x}^{*} (x, y) + Y_{1 y}^{*} (x, y) \end{matrix}

in which,

Y_{0} (x, y) = {| D_{0} J_{x x} (\frac{x}{M}, \frac{y}{M}) |}^{2} + {| D_{0} J_{y x} (\frac{x}{M}, \frac{y}{M}) |}^{2} + 2 {| D_{1 x} R_{1} |}^{2} + 2 {| D_{1 y} R_{2} |}^{2},

and

\begin{array}{l} Y_{1 x} (x, y) = D_{0} D_{1 x} R_{1} [J_{x x} (\frac{x}{M}, \frac{y}{M}) + J_{y x} (\frac{x}{M}, \frac{y}{M})] \exp [j 2 π ξ_{g} x] \\ Y_{1 y} (x, y) = D_{0} D_{1 y} R_{2} [J_{x x} (\frac{x}{M}, \frac{y}{M}) - J_{y x} (\frac{x}{M}, \frac{y}{M})] \exp [j 2 π ξ_{g} y] . \end{array}

Figure 2 gives a typical example of the spatial Fourier transform of the DC-AM-PH described by Eq. (6). It can be seen from Fig. 2 that simultaneously recording of the two polarization components is performed while sharing the dynamic range of an off-axis hologram by multiplexing two off-axis interferograms into a single hologram. This kind of angular multiplexing technique for digital holograms has already been shown to be feasible and has been used for many applications such as holographically recording ultrafast events and increasing super-resolution capabilities of an imaging system [52–55]. So the terms Y_1x and Y_1y in Eq. (8) can be directly extracted from one DC-AM-PH using a spatial filtering algorithm for reconstructing angular multiplexing holograms. Then the Jones parameters J_xx and J_yx under test can be determined by the following formulas

\begin{array}{l} J_{x x} (\frac{x}{M}, \frac{y}{M}) = \frac{1}{2} (\frac{Y_{1 x} (x, y)}{Y_{1 x 0} (x, y)} + \frac{Y_{1 y} (x, y)}{Y_{1 y 0} (x, y)}) \\ J_{y x} (\frac{x}{M}, \frac{y}{M}) = \frac{1}{2} (\frac{Y_{1 x} (x, y)}{Y_{1 x 0} (x, y)} - \frac{Y_{1 y} (x, y)}{Y_{1 y 0} (x, y)}) . \end{array}

Here Y_1x0 and Y_1y0 correspond to the reconstructed values from a background DC-AM-PH, which can be obtained by performing a measurement before the object is placed in the object window.

Fig. 2 A typical example of the spatial spectrum of the DC-AM-PH described by Eq. (6).

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In the above analysis, if we rotate the polarization direction of the input linearly polarized beam to y axis (vertical direction), the other two Jones matrix parameters J_xy and J_yy of the object can be measured using a formula similar to Eq. (9). Using all the measured Jones parameters J_xx, J_yx, J_xy and J_yy, we can further image and quantitatively analyze the optical anisotropic properties of the object such as birefringent properties [45–47].

3. Experimental results

For demonstrating the feasibility of our proposed method, we established an experimental setup based on the schematic shown in Fig. 1 and measured the Jones matrix parameters of a liquid crystal display (LCD) panel using the setup. In the experiments, a He-Ne laser, with the wavelength of 632.8 nm, is used as the light source. The 2-D cross grating adopted in the experiments is a cross Ronchi grating of equal widths of opaque and transparent rulings with a period of 8 μm, which is photoetched on a chromium-coated optical glass substrate. The tested object is a twisted-nematic LCD panel, with pixel size of 18 × 18 μm, and pixel number of 1024 × 768, without input and output polarizers, which was placed at the object window. Two film linear polarizers, with orthogonal polarization directions are, respectively, attached behind the two reference windows. Because the object to be imaged is of a relative large size, the pupil diameter of the objective lens and the tube lens used in the experiments are taken as 50 mm, with focal lengths of, respectively, 180 mm and 150mm. To further increase the effective size of the object window under the given field of view (FOV) of the objective lens, the object window designed in the experimental setup was shifted to the lower left of the input plane, as shown in Fig. 3(a), in which the dotted circle indicates the range of the FOV of the objective lens. In this configuration, the size of the object window is about a quarter of the original FOV of the objective lens.

Fig. 3 (a) Photo of the designed input widows attached with the test object. (b) Gray level picture displayed on the measured LCD; scale bar: 1.8 mm. (c) An example of the DC-AM-PHs recorded in experiments; scale bar: 645um.

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Because the polarization modulation properties of individual LCD pixels are a function of an applied bias (or voltage) and the bias is controlled, point by point, by the gray levels (usually ranging from 0 to 255) of a gray-scale picture displayed on the LCD panel, the applied bias is often replaced by the corresponding gray level in characterizing the modulation properties of the LCD. In the experiments, we design a picture with a continuous variation of gray levels from 0 to 255, along one dimension, as shown in Fig. 3(b) (only part of the picture is shown). Thus the information for calculating the Jones matrix parameters of the LCD pixels versus different gray levels (corresponding to different applied bias) can be recorded by a single shot DC-AM-PH through displaying the designed picture on the measured LCD panel. The DC-AM-PHs are recorded by a CCD image sensor, with pixel size of 6.45 × 6.45 μm, and pixel number of 1392 × 1040. Figure 3(c) shows an example of the DC-AM-PHs recorded in our experiments. The region outlined with the red dashed rectangle in Fig. 3(c) corresponds to the parts marked with the dashed rectangles in Figs. 3(a) and 3(b). The procedures for extracting the four Jones matrix parameters from the recorded DC-AM-PHs are as follows: (1) transform the DC-AM-PH, recorded when the polarization direction of the input beam is set to x axis (horizontal direction), into the spatial frequency domain using a fast Fourier transform; (2) crop the two terms Y_1x and Y_1y, respectively, from the spatial spectrum of the DC-AM-PH as shown in Fig. 2, and inversely transform them back to the spatial domain; (3) calculate the Jones parameters J_xx and J_yx, based on Eq. (9); (4) repeat the previous two steps to extract the other two Jones parameters J_xy and J_yy from the DC-AM-PH recorded after we rotate the polarization direction of the input beam to y axis (vertical direction). The terms Y_1x0 and Y_1y0 used in Eq. (9) can be obtained from the corresponding background DC-AM-PHs recorded before the object is inserted into the object window.

Figure 4 gives an example of the measured Jones parameter distributions of the tested LCD panel in the part marked by the dashed rectangle in Fig. 3. Figures 4(a)-4(d) are, respectively, the amplitude distributions of the Jones parameters J_xx, J_xy, J_yx and J_yy, while Figs. 4(e)-4(h) correspond to their phase distributions. In order to quantitatively evaluate the measured values,

Fig. 4 Example of the measurement results. (a)-(d) retrieved amplitude distributions of the four Jones matrix parameters J_xx, J_xy, J_yx and J_yy of the measured LCD; (e)-(h) their respective phase distributions. Scale bar: 1.6 mm.

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In order to quantitatively evaluate the measured values, Fig. 5 further gives the curves of the relation between the Jones parameters and the applied gray levels, fitted according to the experimental data shown in Fig. 4. Figure 5(a) gives the curves of the amplitudes of the Jones parameters versus the gray levels, while Fig. 5(b) shows the curves of the corresponding phases versus the gray levels. It can be seen that the amplitudes of the Jones matrix parameters of the tested LCD panel have the relationships of $| J_{x x} | \approx | J_{y y} |$ and $| J_{x y} | \approx | J_{y x} |$ , which is consistent with the theoretical prediction for such a twisted-nematic LCD panel [56,57].

Fig. 5 Fitted curves of the Jones parameters versus the gray levels displayed on the LCD. (a) and (b) are, respectively, the amplitude and the phase values versus gray level.

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For testing the correctness and the usability of the measured Jones parameters, we further calculated the complex amplitude transmittance of the tested LCD panel with both input and output linear polarizers, based on the measured data and the following formula:

t = \cos (θ_{2}) (J_{x x} \cos (θ_{1}) + J_{x y} \sin (θ_{1})) + \sin (θ_{2}) (J_{y x} \cos (θ_{1}) + J_{y y} \sin (θ_{1})),

in which, θ₁ and θ₂ are, respectively, the azimuth angles of the input polarizer axis and the output polarizer axis. At the same time, we also experimentally measured the amplitude transmittance of the tested LCD panel when the same gray-scale picture shown in Fig. 3(b) was displayed on it, under the corresponding polarizer configurations. Figures 6(a) and 6(b) give two examples of the results, in which Fig. 6(a) is the amplitude transmittance

| t |

in the marked part of the LCD panel when the input and output polarizers are set, respectively, to

θ_{1} = 45 °

and

θ_{1} = 90 °

, while Fig. 6(b) is the result when the polarizer axes are taken as

θ_{1} = 45 °

and

θ_{1} = 135 °

. It can be seen that the predicted results, according to the measured Jones parameters, basically agree with the results directly measured in real experiments.

Fig. 6 Comparison between the calculated amplitude transmittance and the measured results. (a) Amplitude transmittance $| t |$ of the LCD panel when the input and output polarizers are set, respectively, to $θ_{1} = 45 °$ and $θ_{1} = 90 °$ ; (b) the result when the polarizer axes are taken as $θ_{1} = 45 °$ and $θ_{1} = 135 °$ .

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4. Conclusion

Our theoretical analysis and experimental results demonstrated that the proposed COPH system, established by implementing only two small modifications to a conventional lens imaging system: a specific three-window screen put on the input plane and a 2D cross grating inserted between the objective lens and tube lens, can realize single-shot holographic measurement of two orthogonally polarized components of the object beam, and so can be used to measure the Jones matrix parameters of polarization-selective or birefringent materials. As an example, we measured the four Jones parameters of a twisted-nematic LCD panel using this system. Based on the measured Jones parameters, we predicted the amplitude transmittance of the tested LCD panel after adding both input and output linear polarizers. The predicted results basically agree with the results directly observed in real experiments. Compared with existing methods for realizing holographic polarization measurement [44–50], the proposed COPH system may has some advantages because of its common-path interferometric architecture and a minimum of optical elements in the experimental setup. For example, the phase measurement accuracy can be improved since both the object and the reference beams for off-axis interferometric recording nearly propagate through the same optical path and the same imaging elements and so the system is insensitive to environmental perturbations due to mechanical or thermal changes. And a low coherent light source can be used as the illumination to eliminate the influence of the coherent noise on the measurements. This method may provide a low-cost way to convert a regular polarization microscope into a holographic polarization one with only minimal modifications and could be applied for quantitative measurements of birefringent and optical anisotropic specimens often required in biological and material science [44–50,58]. But what needs to be indicated is that, the size of the tested object will be limited because of the spatial multiplexing at the input plane.

Funding

National Natural Science Foundation of China (NSFC) (11474186).

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Double-channel angular-multiplexing polarization holography with common-path and off-axis configuration

Abstract

1. Introduction

2. Principle of the system

3. Experimental results

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (10)

Optics Express