Two-step method for fast phase-shifting digital holography using ferroelectric liquid crystal retarder

Shuhei Yoshida; Keizo Nakayama

doi:10.1364/OSAC.2.001908

1. Introduction

In digital holography (DH) [1], the phase-shifting method [2,3] is widely used to extract object light by removing any undiffracted and conjugate light from the interference fringes, and when compared with the Fourier transform method [4], the phase-shifting method obtains wavefronts with higher spatial resolution. Current phase-shifting methods involve changing the optical path length using elements such as a piezoelectric transducer (PZT) [2,3], a rotating wave plate [5], or an electro-optic (EO) modulator [6]. For phase shifting using a PZT or wave plate, closed-loop control and mechanical operations are required, and high-speed measurement of the interference fringes is difficult, and while the other phase-shifting method using the EO modulator does enable high-speed phase shifting, there is an associated cost disadvantage. In addition, optical systems using multiple image sensors [7–9], phase-shifting arrays [10] or polarization imaging cameras [11–18] have been proposed to measure the phase-shifted fringes using a single exposure. While these methods can measure the high-speed interference fringes, there is another problem in that optical and imaging systems required are complex and thus prevent the image sensor’s spatial resolution from being used fully.

In this paper, we propose a simple and fast phase-shifting DH system using a ferroelectric liquid crystal (FLC) retarder. Nematic liquid crystals (NLCs) are conventionally used as phase retarders [19,20], but the rise time of a typical NLC is several tens of ms, thus causing the drawback of slow response speeds. Because the FLC response speed is 1 ms or less for both rising and falling signals, higher phase shift speeds are possible when using the FLC retarder as compared with the NLC [21].

The NLC without spontaneous polarization is monostable, and continuous phase modulation of the NLC can be performed by varying the applied voltage. In contrast, the FLC with spontaneous polarization is bistable; basically, this means that only binary phase modulation of the FLC is possible. Therefore, we propose a novel two-step phase-shifting method that is applicable to FLC retarders. The wavefronts of several samples are measured using the proposed method and we verify the method’s effectiveness experimentally.

2. Principle

In the planar aligned NLC cell, when a voltage is applied in the film thickness direction, dielectric polarization occurs and the liquid crystal molecules then tilt in the film thickness direction. At this time, because the inclination of these molecules varies with the applied voltage, the molecule inclination can be adjusted continuously using the applied voltage. In general, the FLC assumes a helical structure, but when the FLC is injected into a sufficiently thin cell, this helical structure disappears as a result of the surface stabilizing effect and the FLC exhibits bistability. Because this surface-stabilized FLC cell has no helical structure, the spontaneous polarization can be used fully and very-high-speed responses are then possible. These states are illustrated in Fig. 1. The two stable states of the surface-stabilized FLC can be switched by changing the direction in which the voltage is applied. Comparison of the two stable states shows that the FLC molecules appear to be rotating within the plane.

Fig. 1. Motion of liquid crystal molecules. (a) In the planar aligned NLC, the molecules are oriented perpendicular to the film thickness direction when no voltage is applied, but the molecules tilt in the film thickness direction depending on the magnitude of the applied voltage. NLCs are usually driven with AC voltages to prevent charge accumulation. (b) In the surface-stabilized FLC, the injected molecules rotate in the in-plane direction, depending on the direction in which the electric field is applied.

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The response time of the FLC with spontaneous polarization is faster than that of the NLC, but it is difficult to control the molecular orientation continuously using the applied voltage. Therefore, when the FLC is used as a phase retarder, the conventional phase-shifting method cannot be applied in its current form. In this research, we propose a novel phase-shifting method that is applicable to FLC retarders with bistability.

2.1 Two-step phase-shifting method for FLC retarders

In the proposed method, the wavefront is reconstructed from a pre-measured reference light intensity and two interference fringes that have been phase-shifted using the FLC retarder. The proposed method is classified as a two-step phase-shifting method [22,23].

The Jones matrix ${\mathbf {J}}(\phi )$ of the retarder with retardation $\phi$ can be expressed as follows [24]:

(1)$${\mathbf{J}}(\phi)=\begin{bmatrix}\mathrm{e}^{\mathrm{i}\phi/2} & 0 \\ 0 & \mathrm{e}^{-\mathrm{i}\phi/2}\end{bmatrix},$$

where the fast axis and the slow axis of the retarder are the $x$- and $y$-axes, respectively. The Jones matrix ${\mathbf {J}}(\phi ,\theta )$ of the retarder, in which the fast axis is tilted from the $x$-axis by $\theta$, is given by ${\mathbf {J}}(\phi ,\theta )={\mathbf {R}}(-\theta ){\mathbf {J}}(\phi ){\mathbf {R}}(\theta )$ where ${\mathbf {R}}(\theta )$ is taken to be the rotation matrix, which then becomes

(2)$${\mathbf{J}}(\phi,\theta)=\begin{bmatrix}\cos\frac{\phi}{2}+\mathrm{i}\sin\frac{\phi}{2}\cos{2\theta} & \mathrm{i}\sin\frac{\phi}{2}\sin{2\theta} \\ \mathrm{i}\sin\frac{\phi}{2}\sin{2\theta} & \cos\frac{\phi}{2}-\mathrm{i}\sin\frac{\phi}{2}\cos{2\theta}\end{bmatrix}.$$

A retarder in which the slow axis is coincident with the x-axis can be expressed as ${\mathbf{J}}(\phi , {90}^{\circ})$ using Eq. (2). Let ${\mathbf {J}}(\phi ,{90}^{\circ})$ be one stable state of the FLC retarder. When the applied voltage is reversed, the FLC molecules rotate by $\theta$ in the plane and this becomes the other stable state. The FLC retarder in this state can then be expressed as ${\mathbf {J}}(\phi ,{90}^{\circ}+\theta )$. Assuming that ${\mathbf {u}}=[b\ 0]^{\intercal }$ is the incident light on the FLC retarder in the ${\mathbf {J}}(\phi ,{90}^{\circ})$ and ${\mathbf {J}}(\phi ,{90}^{\circ}+\theta )$ states, the transmitted light in the two states, represented by ${\mathbf {u}}_{1}$ and ${\mathbf {u}}_{2}$, respectively, can be expressed as

(3)$$\begin{aligned} {\mathbf{u}}_{1} & ={\mathbf{J}}(\phi,{90}^{\circ})\mathbf{u}=b\begin{bmatrix}\mathrm{e}^{-\mathrm{i}\phi/2}\\0\end{bmatrix}=b\begin{bmatrix}\cos\frac{\phi}{2}-\mathrm{i}\sin\frac{\phi}{2}\\0\end{bmatrix} \\ \mathbf{u}_{2} & =\mathbf{J}(\phi,{90}^{\circ}+\theta)\mathbf{u}=b \begin{bmatrix}\cos\frac{\phi}{2}-\mathrm{i}\sin\frac{\phi}{2}\cos{2\theta} \\ \mathrm{i}\sin\frac{\phi}{2}\sin{2\theta}\end{bmatrix}, \end{aligned}$$

and we let $\mathbf {u}_{1}$ and $\mathbf {u}_{2}$ be the reference light beams. Equation (3) shows the phase of the reference light shifts according to the retardation $\phi$. In addition, the phase and polarization of the reference light are changed by the rotation of the FLC molecules even if the retardation $\phi$ is constant. Assuming that the object light is modulated with respect to the amplitude $a$ and the phase $\psi$ is $\mathbf {u}_{o}=[a\mathrm {e}^{\mathrm {i}\psi }\ 0]^{\intercal }$, the interference fringes $I_{1}$ and $I_{2}$ between the object light beam $\mathbf {u}_{o}$ and the reference light beams $\mathbf {u}_{1}$ and $\mathbf {u}_{2}$, respectively, become

(4)$$\begin{aligned} I_{1} & =\vert{\mathbf{u}_{o}+\mathbf{u}_{1}}\vert^{2}=I_{o}+I_{r}+2ab\left(\cos\psi\cos\frac{\phi}{2}-\sin\psi\sin\frac{\phi}{2}\right) \\ I_{2} & =\vert{\mathbf{u}_{o}+\mathbf{u}_{2}}\vert^{2}=I_{o}+I_{r}+2ab\left(\cos\psi\cos\frac{\phi}{2}-\sin\psi\sin\frac{\phi}{2}\cos{2\theta}\right), \end{aligned}$$

where $I_{o}=a^{2}$ and $I_{r}=b^{2}$. From Eq. (4), $\sin \psi$ and $\cos \psi$ can be expressed as

(5)$$\begin{aligned} \sin\psi & =\frac{I_{2}-I_{1}}{4ab\sin^{2}{\theta}\sin{\frac{\phi}{2}}}\\ \cos\psi & =\frac{I_{2}-I_{1}\cos{2\theta}-2(I_{o}+I_{r})\sin^{2}{\theta}}{4ab\sin^{2}{\theta}\cos{\frac{\phi}{2}}}, \end{aligned}$$

and solving the quadratic equation $\sin ^{2}\psi +\cos ^{2}\psi =1$ for $I_{o}$ yields

(6)$$I_{o}=\frac{v-\left({v^{2}-4uw}\right)^{1/2}}{2u},$$

where

(7)$$\begin{aligned} u & =4\sin^{4}{\theta}\sin^{2}{\frac{\phi}{2}}\\ v & =4\sin^{2}{\theta}\sin^{2}{\frac{\phi}{2}} \left(I_{2}-I_{1}\cos{2\theta}+2I_{r}\cos{\phi}\sin^{2}{\theta}\right)\\ w &=\cos^{2}{\frac{\phi}{2}} \left(I_{1}-I_{2}\right)^{2}+\sin^{2}{\frac{\phi}{2}} \left(I_{2}-I_{1}\cos{2\theta}-2I_{r}\sin^{2}{\theta}\right)^{2}. \end{aligned}$$

Here, $u$, $v$, $w$ and the term within the square root of Eq. (6) are always positive because all the terms are squared terms:

(8)$$\begin{aligned} & v=4\sin^{4}{\theta}\sin^{2}{\frac{\phi}{2}}\left[\left({a\cos{\psi}+2b\cos{\frac{\phi}{2}}}\right)^{2}+a^{2}\left({1+\sin^{2}{\psi}}\right)\right]\\ & v^{2}-4uw=256b^{2}\sin^{8}{\theta}\sin^{4}{\frac{\phi}{2}}\cos^{2}{\frac{\phi}{2}}\left(a\cos{\psi}+b\cos{\frac{\phi}{2}}\right)^{2}. \end{aligned}$$

Therefore, $0<(v^{2}-4uw)^{1/2}<v$, and Eq. (6) will always yield a positive $I_{o}$. Furthermore, the phase $\psi$ of the object light is obtained as:

(9)$$\psi=\tan^{-1}\left[{\frac{(I_{2}-I_{1})\cot\frac{\phi}{2}}{I_{2}-I_{1}\cos{2\theta}-2(I_{o}+I_{r})\sin^{2}{\theta}}}\right].$$

As shown above, if $I_{1}$, $I_{2}$, $I_{r}$, $\phi$ and $\theta$ are known, the object light $\mathbf {u}_{o}$ can be reconstructed.

3. Experimental results

To use the FLC as a phase retarder, both the retardation $\phi$ and the rotation angle $\theta$ of the molecules are required. We measured these properties first, and then measured the response speed of the FLC. The CS-1024 liquid crystal manufactured by Chisso Corp. was used as the FLC and was injected into a homogeneous alignment cell with a gap of 2 µm. The FLC retarder was then constructed by attaching electrodes to the cell. The retarder was driven using a direct current voltage of $\pm {18}\,{\textrm{V}}$.

3.1 Physical properties

The retardation was measured using the Sènarmont method as follows. Linearly polarized light $[1\ 0]^{\intercal }$ with its polarization plane on the $x$-axis is incident on the FLC retarder with $\mathbf {J}(\phi ,{45}^{\circ})$, which has its high-speed axis tilted by 45° from the $x$-axis. The transmitted light from the retarder then enters the $\lambda /4$ plate with $\mathbf {Q}({0}^{\circ})=\mathbf {J}(\pi /2, {0}^{\circ})$ and the linear polarizer with $\mathbf {P}({90}^{\circ}+\varphi )$, which has its transmission axis inclined by $\varphi$ from the $y$-axis. The intensity of the transmitted light from the polarizer is given by

(10)$$I=\sin^{2}\left(\varphi-\frac{\phi}{2}\right).$$

From the above, $\phi =2\varphi$ can be obtained from the rotation angle $\varphi$ of the polarizer when the intensity becomes a minimum. The retardation measurement results are shown in Fig. 2(a). By fitting the measurement results with a $\sin ^{2}$ function, the polarizer rotation angle at which the intensity reaches the minimum value was obtained. When the polarizer rotational angle is $\varphi = {6.8}^{\circ}$, the intensity is minimized, and thus the retardation is $\phi = {13.6}^{\circ}$.

Fig. 2. Measurement results for the properties of the FLC. (a) Retardation. (b) Rotation angle of molecules. The horizontal axis shows the rotation angles of the polarizer (a) and the retarder (b), while the vertical axis shows the intensity.

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The rotation angle of the molecules was measured using the following procedure. After adjusting the FLC retarder to the state of $\mathbf {J}(\phi ,{90}^{\circ})$, the applied voltage was inverted to set the retarder to $\mathbf {J}(\phi ,{90}^{\circ}+\theta )$. When the linearly polarized light $[1\ 0]^{\intercal }$ is incident on the retarder with $\mathbf {J}(\phi ,{90}^{\circ}+\theta )$ and the transmitted light then passes through the linear polarizer with $\mathbf {P}({90}^{\circ})$, the intensity becomes

(11)$$I=\sin^{2}\frac{\phi}{2}\sin^{2}{2\theta}.$$

The retarder rotation angle at which the intensity reaches a minimum is the rotation angle of the molecules (See Appendix A for detailed derivation of Eqs. (10) and (11)). The measurement results for the rotation angle of the molecules are shown in Fig. 2(b). The rotation angle of the FLC retarder at which the intensity has its minimum value was again obtained by fitting with a $\sin ^{2}$ function. The intensity reaches its minimum when the FLC retarder rotation angle is 48°, and the rotation angle of the molecules is thus also 48°.

3.2 Response time

The FLC’s response time was obtained by measuring the intensity given by Eq. (11) while reversing the applied voltage. A photodiode (PD) was used to perform the intensity measurements and the measurement results are shown in Fig. 3, where the rise time $T_{R}= {260}$ µs and the fall time $T_{F}= {210}$ µs. When compared with the performance of the standard NLC retarder, the rise time of the FLC retarder is approximately 100 times faster.

Fig. 3. Measurement results for the response time of the FLC. The horizontal axis represents the time, and the right and left vertical axes represent the voltage applied to the FLC and the output voltage of the PD, respectively.

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3.3 Wavefront reconstruction

Wavefront measurements were performed based on the properties of the FLC retarder that were measured in the previous section. Figure 4 shows the optical system that is used for the proposed method. This system has a simple configuration, in which the FLC retarder is added to a Mach-Zehnder interferometer. In this study, we used a Mach-Zehnder interferometer for measurement of transmission samples, but a Michelson interferometer can be used for reflection samples. An external cavity laser diode (ECLD; Nichia NUV603E) with a center wavelength of $\lambda = {405}\,{\textrm{nm}}$ was used as the light source. The ECLD has a sufficiently narrow spectral linewidth of 130 kHz and is equipped with a mode sensor to maintain a single longitudinal mode. An anamorphic prism pair (AP) was used to correct the incident beam shape from elliptical to circular, and an optical isolator was used to prevent the light from being returned to the source. A half-wave plate (HWP) was used to align the polarization plane of the incident beam with the optical axis of the FLC retarder. A spatial filter (SF) was then used to remove any noise and distortion from the wavefront. The beam that passed through the SF was then collimated to a diameter of approximately 20 mm. The image sensor was a Baumer VCXU-02M, which has a pixel size of 4.8 µm and resolution of $640\times 480$.

Fig. 4. Optical setup for phase-shifting digital holography using FLC retarder. ECLD: external cavity laser diode; AP: anamorphic prism pair; SF: spatial filter; BS: beam splitter; M: mirror; OL: object lens with focal length $f= {20}\,{\textrm{mm}}$; IL: imaging lens with $f= {80}\,{\textrm{mm}}$; CMOS: complementary metal-oxide-semiconductor image sensor.

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First, the measurement accuracy of the proposed method was evaluated. A random binary phase mask with phases of $0$ and $\pi$ at an operating wavelength of 405 nm was used for the evaluation. The pixel size of the phase mask is 15.6 µm. In this experiment, the reference light intensity $I_{r}$ was measured when the shutter on the object light path was closed, and the interference fringes were measured while reversing the applied voltage at 500 µs intervals after the shutter was opened. A microcontroller unit was used to perform timing control. The measurement results for the interference fringes are shown in Fig. 5(a) and (b), and the reconstructed phase distribution of the object light and part of its cross-section are shown in Fig. 5(c) and (d). The intensity change caused by the phase shift is small because the retardation provided by the FLC retarder is also small. The results in Fig. 5 show that the reconstructed phase corresponds to the concavo-convex shape of the phase mask. The values for $\psi _{U}$ and $\psi _{L}$ shown in Fig. 5 (d) are $\psi _{U}=0.172\pi$ and $\psi _{L}=-0.848\pi$, respectively. Because the phase difference is $1.02\pi$, the phase mask structure is reconstructed accurately. When 10 points were extracted at random from the center part with a small aberration, the average value and standard error of the phase were $0.996\pm 0.0107\pi$. These results show that the accuracy of the proposed method is comparable to a previous study using NLC [20]. This demonstrates that the proposed method is effective for wavefront measurement applications.

Fig. 5. Measurement results for the interference fringes and the reconstructed wavefront. (a) Interference fringe $I_{1}$. (b) Interference fringe $I_{2}$. (c) Phase distribution $\psi$ of the reconstructed object light. (d) Cross-section of the phase distribution along the solid red line in part (c).

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Next, we demonstrate the effectiveness of the proposed method for use on biological samples. Figure 6 shows the reconstructed phases of object light that passed through samples of the epidermis of an onion (Allium cepa L.) and the stem of a loofah (Luffa cylindrica (L.) Roem.). The reconstructed phases show that the cell structures can be visualized using the proposed method. In addition, the time required to measure two interference fringes is 1 ms, and it is thus possible to measure the fringes at speeds as high as several tens of times of that of the conventional method using an NLC. The results above demonstrate that the two-step phase-shifting method using the FLC can reconstruct wavefronts with high speed and high accuracy. The measurement experimental results using a Paramecium as a moving object are shown in Fig. 7. Figure 7(a) and (b) show the reconstructed phase of the object light at $t = {0}\,{\textrm{ms}}$ and 3 ms, respectively. These results show that the motion of the moving object such as a Paramecium can be captured with high time resolution by the proposed method.

Fig. 6. Reconstructed phases for biological samples. (a) Onion epidermal layer. (b) Stem of a loofah. The phase distributions were unwrapped using Ghiglia and Romero’s method [25].

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Fig. 7. Reconstructed phases for a moving Paramecium. (a) $t = {0}\,{\textrm{ms}}$. (b) $t = {3}\,{\textrm{ms}}$.

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4. Conclusion

In this study, we have proposed a two-step phase-shifting method when using an FLC as a phase retarder and have verified its effectiveness experimentally. In accuracy evaluation experiments using a phase mask, we clarified that the phase can be reconstructed with an error of approximately 2%. Additionally, it is possible to measure the interference fringes at 1 kHz or more, which is difficult when using the conventional method with the NLC, and the proposed method is suitable for application to measurement of moving objects. Based on the results of measurements of biological samples, the proposed method is expected to be applied in the bioimaging field.

A. Derivation of Eqs. (10) and (11)

Equation (10) is derived as follows. When the linearly polarized light $[1\ 0]^{\intercal }$ enters the retarder with Jones matrix $\mathbf {J}(\phi , {45}^{\circ})$ and $\lambda /4$ plate $\mathbf {Q}({0}^{\circ})=\mathbf {J}(\pi /2, {0}^{\circ})$, we obtain $\mathbf {u}_{Q}$ as the transmitted light as follows:

(12)$$\mathbf{u}_{Q}=\mathbf{Q}({0}^{\circ})\mathbf{J}(\phi,{45}^{\circ})\begin{bmatrix}1\\0\end{bmatrix}=\mathrm{e}^{\pi\mathrm{i}/4}\begin{bmatrix}\cos\frac{\phi}{2} \\ \sin\frac{\phi}{2}\end{bmatrix}.$$

Let $\mathbf {u}_{Q}$ be incident on the linear polarizer with $\mathbf {P}({90}^{\circ}+\varphi )$, and then the transmitted light from the polarizer $\mathbf {u}_{P}(\varphi )$ becomes

(13)$$\begin{aligned} \mathbf{u}_{P}(\varphi)=\mathbf{P}({90}^{\circ}+\varphi)\mathbf{u}_{Q} & =\mathrm{e}^{\pi\mathrm{i}/4} \begin{bmatrix}\sin^{2}\varphi\cos\frac{\phi}{2}-\sin\varphi\cos\varphi\sin\frac{\phi}{2} \\ \cos^{2}\varphi\sin\frac{\phi}{2}-\sin\varphi\cos\varphi\cos\frac{\phi}{2}\end{bmatrix} \\ \mathbf{P}(\varphi) & =\begin{bmatrix}\cos^{2}\varphi & \sin\varphi\cos\varphi \\ \sin\varphi\cos\varphi & \sin^{2}\varphi\end{bmatrix}. \end{aligned}$$

We obtain Eq. (10) as $I=\vert {\mathbf {u}_{P}(\varphi )}\vert^{2}$.

Equation (11) is also derived as the intensity of the transmitted light $\mathbf {u}_{T}$ when the linearly polarized light $[1\ 0]^{\intercal }$ passed through the retarder with Jones matrix $\mathbf {J}(\phi ,{90}^{\circ}+\theta )$ and the polarizer with $\mathbf {P}({90}^{\circ})$:

(14)$$\mathbf{u}_{T}=\mathbf{P}({90}^{\circ})\mathbf{J}(\phi,{90}^{\circ}+\theta)\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}0 \\ \mathrm{i}\sin\frac{\phi}{2}\sin{2\theta}\end{bmatrix}.$$

Eq. (11) is then expressed as $I=\vert{\mathbf {u}_{T}}\vert^{2}$.

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Two-step method for fast phase-shifting digital holography using ferroelectric liquid crystal retarder

Abstract

1. Introduction

2. Principle

2.1 Two-step phase-shifting method for FLC retarders

3. Experimental results

3.1 Physical properties

3.2 Response time

3.3 Wavefront reconstruction

4. Conclusion

A. Derivation of Eqs. (10) and (11)

References

Cited By

Figures (7)

Equations (14)

OSA Continuum