## 1.

## Introduction

The Fourier transform spectrometer (FTS) has advantages of high throughput, high signal-to-noise ratio, and high spectrum resolution, which are widely used in material identification, chemical analysis, and environmental testing.^{1}2.^{–}^{3} The conventional FTS is based on the Michelson-type interferometer, in which the light is split into two beams and the optical path difference (OPD) is modulated by the shift of a movable mirror.^{4} However, the movable mirror makes the system complex, bulky, and expensive. In the past decades, a lot of works have been focused on the developing of an FTS without moving parts.^{5}6.^{–}^{7}

The spectrometer based on birefringent interference is a typical FTS without moving parts. A liquid crystal (LC) is a good birefringent material because of its large tunable refractive index, low driven voltage, and fast response, all of which have been widely researched in birefringent FTS.^{8}9.10.11.12.13.14.^{–}^{15} The primary methods of LC birefringent FTS are based on a Wollaston prism with spatial interference^{8}9.^{–}^{10} and a binary voltage-controlled LC stages with time interference.^{11}12.13.14.^{–}^{15} For the spatial interference, the fringes are recorded by a detector array, which may increase the error due to the noise difference of the subdetectors. The time interference only needs a single detector and records the fringes by time delay. However, the LC stages configuration contains a lot of interfaces which will increase the reflection loss and result in a decrease of the sensitivity.

In this work, a method to realize a simple birefringent liquid crystal based Fourier transform spectrometer (BLC-FTS) with time interference is proposed. The BLC-FTS contains only one piece of LC, which is driven by a continuous voltage. By continuously changing the refractive index of the extraordinary light, the OPDs of the extraordinary and the ordinary light are generated. The interference is obtained from the exiting beams by polarization selection. In this spectrometer, the reflection loss is low and it is easy to realize a compact spectrometer with few optical elements.

## 2.

## Method and Theory

The configuration of the proposed BLC-FTS is shown in Fig. 1. The collimated light propagates into the device, which is transformed into linearly polarized light by polarizer P1. The vector of P1 has an angle of $\alpha $ to the optical axis of LC, the linear light will be divided into ordinary light and extraordinary light after transiting LC. The refractive index of the extraordinary light is tuned by the voltage when transmitted though the LC with a thickness of $d$, which controls the phase difference of the two lights. The polarization axis of P2 is perpendicular to that of P1 in order to obtain interference between the ordinary light and the extraordinary light. The spectrometer has only a few coaxial optical elements which makes it compact and easily miniaturized.

To understand the theory of the BLC-FTS, the interference in the BLC is studied as below. When the incident light is unpolarized, half of the intensity will transmit through P1. For the ordinary light and the extraordinary light propagating to LC, the amplitudes are

## (1)

$${\mathbf{E}}_{\mathbf{o}}=\mathbf{y}\sqrt{\frac{{I}_{\text{in}}(\lambda )}{2}}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha ,$$## (2)

$${\mathbf{E}}_{\mathbf{e}}=\mathbf{x}\sqrt{\frac{{I}_{\text{in}}(\lambda )}{2}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha ,$$When the lights transmit through the LC with a thickness of $d$, the amplitudes are

## (3)

$${\mathbf{E}}_{\mathbf{o}\mathbf{1}}={\mathbf{E}}_{\mathbf{o}}{e}^{i\frac{2\pi {n}_{o}d}{\lambda}},$$## (4)

$${\mathbf{E}}_{\mathbf{e}\mathbf{1}}={\mathbf{E}}_{\mathbf{e}}{e}^{i\frac{2\pi {n}_{e}d}{\lambda}},$$Polarizer P2 effectively changes the linear polarization basis set for light passing through it. The beam transmitted through P2 will have the combined contributions of the ${\mathbf{E}}_{\mathbf{o}\mathbf{1}}$ and ${\mathbf{E}}_{\mathbf{el}}$ components, each projected from the LC basis set ($x\u2013y$ coordinates) to the P2 polarizer basis set (${x}_{2}\u2013{y}_{2}$ coordinate axes). Since only the $x$ component of the P2 basis set is transmitted through P2, the projected amplitude of the transmitted beam is

## (5)

$${E}_{\text{out}}(\lambda )={\mathbf{x}}_{\mathbf{2}}{\mathbf{E}}_{\mathbf{o}\mathbf{1}}+{\mathbf{x}}_{\mathbf{2}}{\mathbf{E}}_{e\mathbf{1}},$$The result from Eq. (5) yields the transmitted light intensity at the wavelength of $\lambda $ which is

## (6)

$${I}_{\text{out}}(\lambda )={|{E}_{\text{out}}|}^{2}=\frac{1}{2}{|{E}_{\text{in}}(\lambda )|}^{2}\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\text{\hspace{0.17em}}2\alpha [1+\mathrm{cos}(\frac{2\pi \mathrm{\Delta}nd}{\lambda}+\pi \left)\right],$$## 3.

## Birefringent LC

Determining the incident light spectrum [Eq. (8)] requires analyzing how the refractive index difference of the BLC material ($\mathrm{\Delta}n$) depends on the driven electric field $E$. In this article, the positive mixed nematic LC (from Xi’an Caijing Opto-Electrical Science and Technology Co., Ltd., Xi’an, China) is used as the BLC material. It is loaded in an LC box with an LC thickness of 20 *μ*m. As illustrated in Fig. 1, the angle of $\alpha =45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$, and the wavelength of the incident light is 633 nm. The detected intensity versus the voltage curve is shown in Fig. 2(a), which shows a nonlinear interference.

The refractive index difference of the ordinary light and the extraordinary light is solved by Eq. (6) and the result is shown in Fig. 2(b). It shows that the refractive index difference decreases exponentially with the increase of the driven electric field. The maximum $\mathrm{\Delta}n$ is 0.234 when there is no driven electric field. $\mathrm{\Delta}n$ changes quickly when the electric field is ranged from ${0.8\times 10}^{5}$ to ${2\times 10}^{5}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{v}/\mathrm{m}$. Based on this relationship, the spectrum can be solved by Eq. (8).

## 4.

## Spectrum Experiment

To demonstrate the BLC-FTS experimentally, an LC of 100-*μ*m thickness is used in the system. The incident light is the He-Ne laser and the driven electric field is set as shown in Fig. 2(b). Figure 3(a) shows the intensity of the interference between the ordinary and extraordinary light as the driven electric field is varied. The interference intensity undergoes high frequency oscillation over the range of $E$ where $\mathrm{\Delta}n$ changes most rapidly due to the fast change of the interference light phase. The inset shows the detail of the intensity curve marked in the red dashed box, which indicates a clear interference. In the process of cosine Fourier transforming by Eq. (8), the sampling of the OPD should be equally spaced. Hence, Fig. 3(a) shows that the sample space of the driven electric field is nonlinear, which satisfies the condition that the diffractive index difference space is ${1.5825\times 10}^{-3}$ ($\pi /2$ for phase difference).

The interference intensity is treated with Eq. (8), and the result is given in Fig. 3(b), which shows that the spectrum of the laser light is correctly determined with the full width half maximum (FWHM) of 10.5 nm. The FWHM represents the resolution of the spectrometer. For an FTS, the theoretical resolution is^{16}

In the BLC-FTS, the ${\mathrm{OPD}}_{\mathrm{max}}=\mathrm{\Delta}{n}_{\mathrm{max}}d$. From Eq. (9), the theoretical resolution of the spectrometer with a 100-*μ*m LC is 8.6 nm, which shows a slight deviation from the experimental result. It indicates that the proposed BLC-FTS has a good performance for spectrum detection. Based on the theory, the resolution of the BLC-FTS can be further improved by increasing the LC thickness.

Because the active optical element is a single thin-layered BLC which is coaxial to the thin film polarizers, the spectrometer can be integrated with the detector to form a compact spectrum device. In addition, when the detector is an array with multi-subpixels like the charge coupled device or complementary metal oxide semiconductor, the BLC-FTS will be a candidate for field-widened operation to further enhance the throughput, such as for direct spectral imaging in the microscope.

## 5.

## Conclusion

In conclusion, a method to realize a simple FTS without moving parts based on a single-BLC interference is proposed. The theory of the spectrometer is discussed by analyzing the polarization interference of the BLC. With a positive mixed nematic LC material, the method is experimentally demonstrated. The spectrometer has the advantages of compact size and high throughout, which makes it easily be integrated into complex optical systems to carry out fast spectrum measurements.

## Acknowledgments

This work was supported by Graduate Student Innovation Foundation of Institute of Optics and Electronics, Chinese Academy of Sciences; National Natural Science Foundation of China (11174281, 61275061); West Light Foundation of Chinese Academy of Sciences; Application Development Project of Chongqing (cstc2013yykfC00007) and Fundamental and Advanced Research Projects of Chongqing (cstc2013jcyjC00001).

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