Real-time measurement of the liquid-crystal optic-axis angle and effective refractive index distribution based on a common-path interferometer

Qinnan Zhang; Mingyu Gong; Jiaosheng Li; Wenjie Li; Xiaoxu Lu; Liyun Zhong; Jindong Tian

doi:10.1364/OE.27.019474

1. Introduction

Liquid crystal (LC) is one of the most fascinating optical materials, owing to its special polarization-modulation and phase-modulation characteristics [1–5]. LC materials have been widely used in various optical devices, e.g., liquid-crystal displays (LCDs) [6–9], variable-delay devices [10,11], gratings [12,13], microlenses [14–16], wedges [17–19], Fresnel zone plates [20], and spatial light modulators [21–23]. In addition, with the development of two-dimensional (2D) metasurface devices, LC materials have application potential for electrically controlled metasurface materials [24].

However, all of these depend on the 2D control of the LC molecules’ extraordinary refractive index and optic-axis angle by an external electric field. In addition, the optic-axis angle of these LC optical devices may be modulated with comparatively fast switching frequencies greater than 1 kHz. Moreover, many LC devices must be driven with a bipolar electric field to prevent ionic screening effects, as they cannot be held in a static state [25]. Therefore, the real-time measurement of the optical properties distribution in LC materials is crucial for developing and testing LC optical devices.

Usually, the intensity values of light fields in different detection states are used to calculate the optic-axis angle of an LC molecule through rotating the polarized optical elements. However, actual applications cannot achieve real-time measurements, because the polarized elements must be rotated multiple times [26,27]. In addition, the results will be affected by power fluctuations of the light source. In recent years, the Mueller matrix method [28–31] has been commonly used by most conventional polarization-based techniques, because it can rapidly detect the average LC director. However, it does not detect an LC molecule’s optic-axis angle, and the optical system is very complex.

Additionally, some polarization imaging techniques based on polarized digital holography have been proposed to extract a spatially resolved Jones matrix of anisotropic samples [32–36], and the Jones matrix has been used for describing the optical properties of polarization-sensitive materials. However, these methods cannot achieve real-time measurements either, because they need to separately record the intensity information of ± 45° polarized incident light through the sample. Recently, the double-source Mach-Zehnder interferometer has been used to achieve real-time 2D Jones matrix maps and phase maps [37]; however, multiple light sources are required and the measurement system is very complex. In summary, finding more dedicated measurement systems to simultaneously measure both the optic-axis angle and effective refractive index in real-time has become a crucial problem.

In this study, a common-path interferometer for real-time measurement of the liquid-crystal optic-axis angle and the effective refractive index distribution is proposed. It can be realized by adding a polarizer and polarization camera to a general optical microscope. The proposed method has the following advantages. By implementing only single-exposure imaging without changing any optical elements, real-time measuring can be realized, and the measurement process and system can be simplified greatly. In addition, the measurement results will not be affected by light-source power fluctuations or a non-uniform spatial distribution. Therefore, this method is suitable for rapidly testing the optical properties of electrically controlled 2D thin liquid-crystal devices. In the following section, we will introduce the proposed method in detail.

2. Principle analysis

As shown in Fig. 1, the common-path interferometer system is modified with a fluorescence microscope (ECLIPSE Ti, Nikon, Japan). The light first goes through a narrow bandpass filter (central wavelength is 589 nm, $Δ λ$ is 10 nm), a polarizer, an LC sample (E7, King Optronics Co., Ltd., China), and a microscope objective (40 × , N.A. is 0.65), and is collected by a polarization camera (CM-HS, 4D Technology Company, USA).

Fig. 1 Schematic of the common-path interferometer system, including a narrow bandpass filter (NBF), polarizer (P), liquid crystal sample (LC sample), microscope objective (MO) and polarization camera; the direction of the light is parallel to the z axis, and the direction of the polarizer is parallel to the y axis.

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The LC sample is dripped on a slide, and the alignment layer is used on the slide. 1-micron Silica microspheres are tiled on the edge of the slide, and a coverslip is used to cover the slide. With a 20°C temperature, the LC sample is in the nematic phase state, the refractive indices of ordinary light $n_{o}$ is 1.517, the refractive indices of extraordinary light $n_{e}$ is 1.741 and $Δ n$ is 0.224. A polarization-mask layer is added in front of the camera. The mask corresponds to the pixel units of the camera area, one by one. Each four-pixel point is used as a period, and the polarization mask of the four-pixel points is in a different direction, 0°, 45°, 90°, and 135°, as shown in Fig. 1. The four different polarization directions can be acquired in one shot. In our method, the intensity distributions recorded with the 0°, 45°, 90° and 135° polarizations are used to calculate the distributions of the optic-axis angle and the refractive index of the LC sample.

Nematic liquid crystal, as an anisotropic material, has similar optical properties to uniaxial crystals. LC’s molecular structure is rod-like. In particular, the optical axis of the LC is accordant with the molecule long axis. The optical properties can be described with the Jones matrix. The propagating direction of the light is parallel to the z axis, and the direction of the polarizer is parallel to the y axis. The Jones matrix of LC sample can be expressed as follows:

G = [\begin{matrix} A \cos^{2} θ + A \sin^{2} θ e^{i δ} & A \cos θ \cdot \sin θ - A \cos θ \cdot \sin θ e^{i δ} \\ A \cos θ \cdot \sin θ - A \cos θ \cdot \sin θ e^{i δ} & A \sin^{2} θ + A \cos^{2} θ e^{i δ} \end{matrix}],

where

A

is the amplitude,

θ

is the angle between the LC molecule optic-axis angle and the x axis,

δ

is phase difference between ordinary light and extraordinary light,

δ = (n_{o} - n') \cdot d \cdot 2 π / λ

,

d

is the LC thickness,

n_{o}

and

n_{e}

are the refractive indices of ordinary light and extraordinary light, respectively,

n'

is the effective refractive index,

λ

is the central wavelength of the light source. When the vertical polarized light passes through the LC sample, the corresponding the complex amplitude of the optical field in the polarization camera can be expressed as follows:

E = [\begin{matrix} \cos β & \sin β \end{matrix}] G [\begin{matrix} 0 \\ 1 \end{matrix}],

where

β

is the angle between the polarization direction of the polarization mask in the polarization camera and the x axis. (

β

is 0°, 45°, 90°, and 135°). Then, the interference patterns of ordinary and extraordinary light are recorded by the camera. The corresponding the complex amplitude of the optical field can be expressed as follows:

_{$E_{1}$} is the complex amplitude of the optical field in the polarization camera with 0° polarization direction,

E_{1} = A \cos θ \cdot \sin θ e - A \cos θ \cdot \sin θ e^{i δ} .

_{$E_{2}$} is the complex amplitude of the optical field in the polarization camera with 90° polarization direction,

E_{2} = A \sin^{2} θ + A \cos^{2} θ e^{i δ} .

_{$E_{3}$} is the complex amplitude of the optical field in the polarization camera with 45° polarization direction,

E_{3} = \frac{\sqrt{2}}{2} (A \cos θ \cdot \sin θ - A \cos θ \cdot \sin θ e^{i δ} + A \sin^{2} θ + A \cos^{2} θ e^{i δ})

_{$E_{4}$} is the complex amplitude of the optical field in the polarization camera with 135° polarization direction,

E_{4} = \frac{\sqrt{2}}{2} (- A \cos θ \cdot \sin θ + A \cos θ \cdot \sin θ e^{i δ} + A \sin^{2} θ + A \cos^{2} θ e^{i δ})

And the corresponding intensity distributions are $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ . They can be expressed as follows:

_{$I_{1}$} is the intensity distribution in the polarization camera with 0° polarization direction,

I_{1} = E_{1} \cdot E_{1}^{*} = 2 A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot [1 - \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ})] .

_{$I_{2}$} is the intensity distribution in the polarization camera with 90° polarization direction,

I_{2} = E_{2} \cdot E_{2}^{*} = A^{2} \sin^{4} θ + A^{2} \cos^{4} θ + 2 A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ}) .

_{$I_{3}$} is the intensity distribution in the polarization camera with 45° polarization direction,

\begin{matrix} I_{3} = E_{3} \cdot E_{3}^{*} = A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot [1 - \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ})] \\ - A^{2} \cos θ \cdot \sin θ \cdot [(\sin^{2} θ - \cos^{2} θ) \cdot (1 - \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ}))] \\ + \frac{1}{2} A^{2} \sin^{4} θ + \frac{1}{2} A^{2} \cos^{4} θ + A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ}) . \end{matrix}

_{$I_{4}$} is the intensity distribution in the polarization camera with 135° polarization direction,

\begin{matrix} I_{4} = E_{4} \cdot E_{4}^{*} = A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot [1 - \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ})] \\ + A^{2} \cos θ \cdot \sin θ \cdot [(\sin^{2} θ - \cos^{2} θ) \cdot (1 - \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ}))] \\ + \frac{1}{2} A^{2} \sin^{4} θ + \frac{1}{2} A^{2} \cos^{4} θ + A^{2} \cos^{2} θ \cdot \sin^{2} θ \cdot \cos (\frac{(n_{o} - n') \cdot d \cdot 2 π}{λ}) . \end{matrix}

Angle

θ

can be calculated through the following Eq. (11):

2 θ = arccot [\frac{I_{3} - I_{4}}{2 I_{1}}],

where the LC molecule’s actual optic-axis angle is

θ + m \cdot 90 ° (m = 0, 1)

, due to the property of the anti-cotangent function. The optic-axis angle’s spatial difference and change can be accurately acquired, and a continuous change of more than 90° can be calculated by using the phase-unpacking method. If the approximate initial optic-axis angle can be determined in advance, an accurate LC molecule’s actual optic-axis angle can be calculated. Even so, the refractive index calculation results are only related to the value of

θ

. Therefore, according to Eq. (7), the effective refractive index and the average LC director can be calculated as follows:

A^{2} = I_{1} + I_{2}

n' = n_{o} - \frac{λ}{2 π \cdot d} \cdot arc \cos (1 - \frac{I_{1}}{2 A^{2} \cdot \cos^{2} θ \cdot \sin^{2} θ})

n' = \frac{n_{o} \cdot n_{e}}{\sqrt{n_{o}^{2} \cdot \sin^{2} ϕ + n_{e}^{2} \cdot \cos^{2} ϕ}},

where

ϕ

is the angle between LC molecule’s optic-axis and the x-y plane, as shown in Fig. 2,

d

can be obtained in advance, and amplitude

A

can be acquired from Eq. (12).

Fig. 2 Schematic diagram of the LC sample.

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The effective refractive index and the average LC director can be calculated using Eqs. (13) and (14), respectively. From the relationship between the effective refractive index and the average LC director, it is obvious that the calculation process has no approximation or hypothesis. The non-uniformity of the light-source spatial distribution is considered in Eq. (12), and angle $θ$ and the effective refractive index can be calculated with only single-exposure imaging. Therefore, the results will not be affected by the light-source power fluctuation or a non-uniform spatial distribution. Thus, a method for the real-time measuring of the LC optic-axis angle and the refractive index distribution can be realized.

3. Results and discussion

To verify the proposed system, simulations and experiments were conducted. In the simulation, the effective distributions of the refractive index and optic-axis angle in the preset sample are shown in Figs. 3(a) and 3(b). The wavelength is set to 589 nm. The refractive index distribution is a sloping surface with a variation range of 1.517 to 1.741, and the distribution of $θ$ is set as a peaks function with a variation range of 0° to 180°. The LC thickness $d$ is 1 micron; the refractive indices of ordinary light and extraordinary light are set as 1.517 and 1.741, respectively. The intensity distributions of the four polarization directions can be obtained by Eqs. (7)–(10), which are showed in Figs. 3(c)–3(f). And the size of the simulated interferogram is set to 400 × 400 pixels. In addition, a Gaussian white noise with mean zero and standard deviation of 2 is added to each interferogram. Using Eqs. (11)–(14), the distribution of effective refractive index and optic-axis angle $θ$ of the LC sample can be calculated accurately. The results are shown in Figs. 4(a) and 4(b), and the average director is shown in Fig. 4(c). When using the branch-cut method [38,39], the unwrapped optic-axis angle distribution is obtained, which is showed in Fig. 3(d). The calculated results of effective refractive index and optical axis angle distribution showed in Fig. 4, are in good agreement with the preset values, in which the root-mean-square errors (RMSEs) of effective refractive index and optical axis angle are 0.00001 and 0.012°, respectively. Thus, the feasibility of the proposed method is verified by the simulation results.

Fig. 3 (a) Preset effective refractive index (RI) distribution and (b) preset optic-axis angle distribution; the interferogram’s intensity distribution with four polarization directions, (c) 0°, (d) 45°, (e) 90°, and (f) 135°.

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Fig. 4 Simulation results of (a) effective refractive index (RI) distribution, (b) wrapped optic-axis angle distribution, (c) average director distribution, and (d) unwrapped optic-axis angle distribution.

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In the experiment, the incident light goes through the polarizer and LC sample and is collected by the polarization camera, and the intensity distribution of four polarization directions are obtained by the polarization camera. The LC sample is prepared in two glass slides with uniform optic-axis angle films to evaluate the measurement accuracy of the LC optic-axis angle. The thickness is about 1 micron. We rotate the sample and record the intensity distributions using the polarization camera at 10° intervals. According to the direction of the optic-axis angle films, 0° is defined as the x-axis direction. The distributions of the optic-axis angle direction and the effective refractive index can be calculated. The measured angle is between the optical axis directions and the x axis.

The intensity distributions of the LC sample when rotating 40°, 50°, and 60° are shown in Figs. 5(a)–5(c), respectively; the distributions of the optic-axis angle and the effective refractive index are shown in Figs. 5(d)–5(f) and Figs. 5(g)–5(i), respectively. The mean values of the measured optic-axis angle under different rotation angles are shown in Fig. 6(a), and the results are highly consistent with the expected results. The corresponding effective refractive index results are shown in Fig. 6(b). As the sample rotates, the average refractive index maintains high uniformity, and the fluctuation is less than 0.0035.

Fig. 5 Intensity distributions of the LC sample when rotating (a) 40°, (b) 50°, and (c) 60°; the calculated results of the LC optic-axis angle distribution when rotating (d) 40°, (e) 50°, and (f) 60°; and the calculated results of the LC effective refractive index distribution when rotating (g) 40°, (h) 50°, and (i) 60°.

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Fig. 6 (a) Measured optic-axis angles under different rotation angles, and (b) effective refractive index under different rotation angles.

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The RMSEs of the optic-axis angle distribution and effective refractive index distribution are calculated. The mean values are used as the standard values, and the results are shown in Figs. 7(a) and 7(b). It is obvious that the RMSE of the optic-axis angle distribution is less than 5°, and the RMSE of the effective refractive index distribution is less than 0.0025. However, due to the property of the anti-cotangent function, the RMSE increases rapidly when the optic-axis angle is 0° or 90°. If 0° or 90° can be avoided, the RMSE of the optic-axis angle distribution is 0.8°–1.8° and the RMSE of the effective refractive index distribution is 0.0005–0.0001. The above experimental results further prove the feasibility and validity of the proposed method.

Fig. 7 (a) RMSEs of the measured optic-axis angle distribution under different rotation angles, and (b) RMSEs of the effective refractive index distribution under different rotation angles.

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And then, the measuring errors of LC sample at other retardances are discussed. Firstly, we prepared two new LC samples with different retardance, the different LC thickness (0.5 micron and 1.5 micron) are used to obtain the different retardance ( $0.19 λ$ and $0.57 λ$ ). The LC sample is dripped on a slide, and the alignment layer is used on the slide. The 0.5-micron and 1.5-micron Silica microspheres are respectively tiled on the edge of two slides, the coverslips are used to cover the slides, respectively. We rotate the sample and record the intensity distributions using the polarization camera at 10° intervals. And the distributions of the optic-axis angle direction and the effective refractive index are calculated. The root-mean-square errors (RMSEs) of the effective refractive index distribution with the optical-axis angle distribution are shown as in Fig. 8. The results showed that the errors of optical axis angle have hardly changed at different retardance, but the errors of effective refractive index decrease as the retardance increases.

Fig. 8 (a) RMSEs of the measured optic-axis angle distribution under different rotation angles at different retardance, and (b) RMSEs of the effective refractive index distribution under different rotation angles at different retardance.

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In the other hand, as shown in Fig. 9, local characteristics of the LC effective refractive index distribution showed good repeatability when LC sample rotating 40°, 50° and 60°. In the proposed method, the LC thickness is defined as a constant. The angle $ϕ$ of each LC molecule in each pixel has small differences. The nonuniformity of LC thickness and the angle $ϕ$ would cause the departure from ideal in the experimental results. And the experimental results reveal the true characteristics of the LC sample. As shown in Fig. 9, the local characteristics in the effective refractive index distribution rotate at 10° intervals with the rotation of LC sample.

Fig. 9 The effective refractive index distribution of LC sample refractive index when the LC sample rotating (a) 40°, (b) 50°, (c) 60° and the corresponding local distribution (d-f).

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In order to verify the nonuniformity of LC thickness and the angle $ϕ$ of each LC molecule in each pixel would cause the departure from ideal in the experimental results, we measured the LC sample with varying thickness. The schematic diagram of the LC sample is shown as in Fig. 10. And the experimental results of optic-axis angle distribution and effective refractive index distribution are shown as in Fig. 11. The result of the optic-axis angle distribution showed a high error at the area with varying thickness. The result of the effective refractive index distribution showed that the effective refractive index decline with the decrease of thickness. Due to the effect of surface tension, the optic-axis angle $θ$ and angle $ϕ$ would be nonuniform, which is showed in Fig. 10. And the experimental results demonstrated that the nonuniformity of LC thickness and the angle $ϕ$ of each LC molecule in each pixel would cause the departure from ideal in the experimental results.

Fig. 10 Schematic diagram of the LC sample with varying thickness.

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Fig. 11 The experimental results of LC sample with varying thickness, and the corresponding optic-axis angle distribution and effective refractive index distribution.

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4. Conclusion

In this paper, a common-path interferometer for the real-time measurement of the LC optic-axis angle and effective refractive index distribution was proposed. It was realized by adding a polarizer and a polarization camera to a general optical microscope. When measuring the 1-μm LC sample, the RMSEs of the optic-axis angle distribution and effective refractive index distribution showed high measurement accuracy. This method required only single-exposure imaging without changing any optical elements and greatly simplified the measurement process and system. In addition, the measurement results were not affected by light-source power fluctuations or a non-uniform spatial distribution. Therefore, this method is suitable for testing the optical properties of LC optical devices that cannot be held in a static state.

Funding

National Natural Science Foundation of China (NSFC) (61727814, 51775352, 61875059, 61805086); China Postdoctoral Science Foundation (2018M643114).

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Real-time measurement of the liquid-crystal optic-axis angle and effective refractive index distribution based on a common-path interferometer

Abstract

1. Introduction

2. Principle analysis

3. Results and discussion

4. Conclusion

Funding

References

Cited By

Figures (11)

Equations (14)

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