1 Introduction

Due to the lack of active radiation sources and beam-splitting devices with wide spectrum in terahertz band, there are few kinds of spectral imaging instruments in terahertz band. In particular, as the key component of spectral imaging instrument, the optical splitter directly affects the performance, structural complexity, weight and volume of the instrument^[1-3]. The existing optical splitters of visible and infrared spectroscopic instruments include prism, amplitude grating, acousto-optic tuned filter, etc. However, these beam-splitting devices, which are suitable for visible and infrared bands, cannot meet the requirements of spectral detection and imaging in terahertz band. Firstly, as a classical optical splitter, the prism has the advantages of wide free spectrum, simple structure, large amount of light, high energy utilization and easy stray-light suppression. However, with the increase of the wavelength in terahertz band, the penetration of electromagnetic waves will increase and the dispersion of materials will become extremely weak. This means that it is no longer feasible to obtain fine spectra by material dispersion in terahertz spectrum detection^[4-5]. Secondly, the beam splitting of amplitude grating depends on the diffraction effect. However, the spectral range of a single grating is limited by the overlap of adjacent secondary diffraction spectra. In order to improve the utilization of light energy, the flare grating is adopted and the limited bandwidth of its wavelength further reduces its free spectral range. The terahertz bandwidth (30 μm~3 mm) is nearly 6 000 times the visible bandwidth (0.38 μm~0.75 μm) and 100 times the infrared bandwidth (0.75 μm~30 μm)^[6-8]. This determines that the beam-splitting devices in terahertz band must have a wide free spectral range, so the flared grating and amplitude modulated grating are not applicable to terahertz band. Finally, the acousto-optic tuned filter relies on acousto-optic effect to realize diffraction, and has the advantages of small volume, high crystal diffraction rate and large field of view. However, no acousto-optic crystal suitable for terahertz band has been found in the published literature^[9-11].

The weak-signal feature of terahertz detection requires terahertz beam splitter with high energy efficiency. To meet the requirements of terahertz spectral imaging for wide spectral range, high efficiency and real-time detection of spectrometers, a Multi-depth Phase Modulation Grating (MPMG) in terahertz band is proposed in this paper^[12]. The phase modulation of incident light is realized by introducing optical path difference resulted from the change of groove depth, so that different regions of reflecting terahertz wave front have different phase information. The typical characteristic of MPMG is that its 0th-order diffraction light carries the phase information due to groove depth modulation and therefore has the ability of splitting light.

2 Diffraction field distribution of MPMG

The MPMG is composed of a series of grating cells whose groove depths vary in an equal gradient. The one-dimensional MPMG and two-dimensional MPMG, whose structures are shown in Fig. 1(a) and Fig. 1(b) respectively, are both composed of four grating cells. The groove depths of grating cells, namely h₁, h₂, h₃ and h₄, vary in an equal gradient. Each grating cell is composed of two pairs of crest reflection planes and groove reflection planes parallel to each other, as shown in Fig. 1(c). There is no difference in performance between one-dimensional MPMG and two-dimensional MPMG, but the latter's structure is compact and conducive to the miniaturization of spectral instruments.

图 1. MPMG示意图(a)一维MPMG，(b)二维MPMG(c)光栅单元

Fig. 1. Schematic diagram of MPMG. (a) 1D MPMG, (b) 2D MPMG, (c) grating cell

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The grating cells are equivalent to a series of planes staggered by different phase differences, as shown in Fig. 2. When the beam is obliquely incident to the MPMG, the angle between the projection of the wave vector $\vec k$ on the plant (x, z) and the vector $\vec k$ is ψ, and the angle between this projection and the axis x is θ. The phase difference thereby introduced to the crest reflection planes and groove reflection planes is:

图 2. (a) MPMG等效为平面透射光栅示意图 (b) MPMG衍射示意图

Fig. 2. (a) Reflection grating simulated as a plane transmission grating; (b) MPMG diffraction diagram

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When the terahertz wave is incident in the direction parallel to the paper, that is ψ=0, the normalized light intensity distribution after the MPMG diffraction is shown in Eq. (2). The crest reflection planes and groove reflection planes are considered to be with the same width during calculation.

α and β represent the diffraction angle in the x direction and the diffraction angle in the y direction respectively; $\vec k$ is the wave vector; w is the width of crest reflection planes and groove reflection planes; l is the grating length; and n is the number of crest reflection plane and groove reflection plane pairs.

${{\rm{sinc}} ^2}\left( {\dfrac{{{{k}}l\sin \beta }}{2}} \right) $ and ${{\rm{sinc}} ^2}\left( {\dfrac{{{{k}}w\sin \alpha }}{2}} \right) $ are used to describe the rectangular aperture diffraction factor, which has a maximum value in 0th-order diffraction; $\dfrac{{{{\sin }^2}\left[ {\dfrac{{n{\rm{k}}d\sin \alpha }}{2}} \right]}}{{{{\sin }^2}\left(\dfrac{{{\rm{k}}d\sin \alpha }}{2}\right)}}$ is used to describe the multiple-beam interference factor. So the grating equation of MPMG is

That is, the diffraction order distribution of the grating only depends on the grating constant d. When the diffraction light with the same order, the higher d corresponds to a larger diffraction angle.

${\cos ^2}\left( {\dfrac{{{{k}}w\sin \alpha + \varphi }}{2}} \right) $ represents the modulation of the additional phase introduced by groove depth on diffraction field intensity.

3 Analysis of MPMG diffraction characteristics

As known from Eq. (3), the intensity distribution of diffraction field along the X-axis is related to the grating constant d (d=2w), the number of crest reflection plane and trough reflection plane pairs (n), and the groove depth (h). The Fig. 3 (color online) depicts the distribution of Fraunhofer diffraction field along the X-axis under different design parameter conditions. As can be seen from Figs. 3(a), 3(b) and 3(c), when only groove depth is variable, the phase modulation introduced by groove depth will enable the energy of diffraction field to shift between level 0th- and ±1st-order. When the phase difference φ is 2π, the energy of Fraunhofer diffraction field is all concentrated at the 0th-order; when the phase difference φ is π, the energy of Fraunhofer diffraction field is all concentrated at the ±1st-order; when the phase difference φ is π/2 or 3π/2, the energy of Fraunhofer diffraction field is evenly distributed at the 0th- and ±1st-order. The comparison of Figs. 3(a), 3(b) and 3(c) show that, with the increase of crest reflection plane and groove reflection plane pairs, the flare angle of each diffraction order will decrease. The number of crest reflection plane and trough reflection plane pairs is independent of the diffraction angle of each diffraction order. When discussing the effect of the grating constant d (d=2w) on diffraction field distribution, w/λ is chosen to avoid the influence of the introduced wavelength. As can be seen from Fig. 3(d), with the increase of w/λ, the primary maximum angular width and diffraction angle of each diffraction order will decrease and the diffraction phenomenon will become less obvious.

图 3. 不同相位调制下沿x轴方向的光强分布

Fig. 3. Light intensity distributions along the x axis under different phase modulation conditions

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4 Testing of MPMG diffraction characteristics

We established a set of experimental facilities for evaluating the diffraction characteristics of MPMG in the laboratory, including a high-power THz radiation source system, Fig. 4(a), a THz laser collimation and transmission system Fig. 4(b) and a grating testing system, as shown in Fig. 4(c). In the Fig. 4, the Lens 1 and Lens 2 constitute a set of beam expanders that represent the collimation and transmission path of the system; and the Lens 3, Lens 4, MPMG and Lens 5 constitute the testing light path of MPMG diffraction characteristics. In the experiment, a tunable Optical Parametric Oscillator (OPO) with the wavelength of 1 066-1 078 nm and a 1 064 nm Nd:YAG laser were used to produce high-power THz radiation under difference frequency effect^[13], as shown in Fig. 4(a). The spot size of the pumped laser beam is about 4 mm. This THz laser beam produced by nonlinear difference frequency effect is different from a visible or infrared laser beam. Laser Gaussian beam can be treated as parallel light under the following two conditions: first, the radius of beam waist is much larger than the laser wavelength; second, the wavefront radius is much larger than the laser wavelength. Under the former condition, the spot size will remain approximately unchanged during the laser transmission; under the latter condition, the laser wavefront can be approximated to a plane. The wavelength of THz laser is equivalent to the size of beam waist, and the spot size of THz laser is approximately proportional to the transmission distance^[14]. The divergence angle of self-made difference-frequency THz source obtained through experimental measurement is 12°. Therefore, in the experiment, a lens group was used to collimate the THz beams in the optical path shown in Fig. 4(b). The lens group is equivalent to a set of beam expanders in the Fig. 4(c)^[15-16]. After being collimated by the lens group, the divergence angle of THz laser beam is 0.1°. The grating test system consists of three high density polye thylene lenses, a rectangular stop, a MPMG and a THz detector, as shown in Fig. 4(e). The 1D MPMG used in the experiment is composed of 8 grating cells, each of which includes five pairs of crest planes and groove planes. The MPMG parameters are given in Table 1 and the MPMG picture is shown in Fig. 4(d). The THz detector used in the experiment is the second generation of quasi-optical detector (2dl 12c LS 2500 A1) purchased from Advanced Compound Semiconductor Technologies (ACST, Hanau, Germany). The photosensitive surface of the detector is encapsulated with a convergence mirror with a diameter of 2 mm to achieve higher light-energy collection efficiency, as shown in Fig. 4(f).

图 4. (a) 高功率辐射源系统；(b) 激光准直与发射系统；(c) 实验方案；(d) 一维MPMG；(e) 光栅测试系统；(f) 探测器照片

Fig. 4. (a) High-power THz radiation source system. (b) Laser collimation and transmission system. (c) Schematic of experiment. (d) 1D MPMG. (e) Grating-testing system. (f) Photograph of the THz detector.

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The test method is described below. After being collimated by the lens group, the 0.5 THz radiation is parallelly incident to and diffracted by the MPMG (θ = 60°, ψ = 0°). After being converged by the THz lens, the diffracted waves at the 0th- and ±1st-order are detected by the THz detector in the focal plane of the lens. A thin rectangular stop is placed on the front surface of the MPMG to ensure that only one cell is effectively illuminated by each incident THz wave. By moving the detector in proper order, the diffraction intensities at the 0th- and ±1st-order can be recorded. By repeating this operation for each grating cell, the intensities of all the grating cell at the 0th- and ±1st-order can be obtained. In order to eliminate the influence of laser jitter on the measurement results, the measurement results of multiple tests (20 measurements) are averaged. By normalizing the measured data, the 0th- and ±1st-order diffraction efficiencies of 8 grating cells can be obtained.

By changing the OPO wavelength, the diffraction intensity of the 0.34 THz radiation wave in 8 grating cells is measured. The theoretical simulation curves and measurement results of diffraction efficiencies of 0.5 THz and 0.34 THz radiation in the grating cells are shown in Fig. 5. It can be seen that the experimental results are in agreement with the theoretical simulation results, that demonstrates the diffraction characteristics of MPMG, that is, the 0th-order diffracted light of MPMG carries the phase information and its diffraction intensity is modulated by the phase introduced by groove depth.

图 5. 光栅单元0级和1级衍射效率的模拟和测试结果(a) 0.34 THz (b) 0.5 THz

Fig. 5. Simulation and test results of 0th- and 1st-order diffraction efficiency for each grating cell at (a) 0.34 THz and (b) 0.5 THz

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5 Conclusion

In this paper, a new MPMG in THz band, which is composed of a series of grating cells, is presented. The groove-depth gradients of these grating cells correspond to different positions of the moving mirrors in the Fourier transform spectrum system. The calculation results of Fraunhofer diffraction field distribution and diffraction efficiency of MPMG show that, the 0th-order diffracted light of MPMG carries the phase information and its diffraction intensity is modulated by the phase introduced by groove depth. We develop the MPMG composed of eight grating cells and test its 0th- and 1st-order diffraction efficiencies at 0.5 THz and 0.34 THz. The test results agree well with the simulation results. Therefore, we believe that the 0th-order diffracted light of MPMG carries the phase information and its diffraction intensity is modulated by the phase introduced by groove depth.

1 引　言

太赫兹波段的光谱成像仪器种类稀少，这是由于太赫兹宽谱主动辐射光源与太赫兹宽谱分光器件缺乏所共同导致的。其中分光器件作为光谱成像仪器的关键部件，直接影响仪器的性能、结构的复杂程度、重量和体积等^[1-3]。目前，可见、红外光谱仪器的分光器件有：棱镜、振幅光栅、声光调谐滤波器等。然而，这些适用于可见、红外波段的分光器件并不能满足太赫兹波段物质的谱探测与成像对分光器件的需求。棱镜作为经典的分光器件具有自由光谱范围宽、结构简单、通光量大，能量利用率高，杂散光易于抑制的优点，然而在太赫兹波段，随着波长的增加，电磁波的穿透性能增强，材料的色散现象变的极其微弱，这意味着通过材料色散获取精细光谱的方式在太赫兹谱探测中不在可行^[4-5]。振幅光栅是依靠衍射效应分光的，然而受相邻衍射次级光谱重叠的限制，单块光栅的光谱范围有限，为了提高光能利用率而采用的闪耀光栅，其闪耀波长的带宽有限，进一步缩小了光栅的自由光谱范围。太赫兹（30 μm~3 mm）的频段带宽是可见（0.38 μm~0.75 μm）频段带宽的近6 000倍，是红外（0.75 μm~30 μm）频段带宽的100倍^[6-8]。这决定了太赫兹波段的分光器件必须具备宽自由光谱范围，因此闪耀光栅、振幅调制光栅不适合用作太赫兹波段的分光器件。声光调谐滤波器依靠声光效应实现衍射分光，其具有体积小，晶体衍射率高、大视场的优点，但目前尚未在公开报道的文献中查阅到适用于太赫兹波段的声光晶体^[9-11]。

若要使太赫兹探测到弱信号，则要求太赫兹分光器件必须具备高能量利用效率。针对太赫兹谱成像对宽光谱、高光能利用率、实时探测分光器件的需求，本文提出一种太赫兹立体相光栅(MPMG)分光器件^[12]，通过刻槽深度的变化引入光程差，实现对入射光的相位调制，使反射太赫兹波前的不同区域具有不同的相位信息。该MPMG的典型特征在于，槽深调制使MPMG的零级衍射光携带相位信息，因此MPMG零级衍射光具备分光能力。

2 MPMG的衍射场分布

MPMG由槽深呈等梯度变化的一系列光栅单元组成。一维MPMG、二维MPMG的结构示意图分别如图1(a)、1(b)所示，二者均由4个光栅单元组成，光栅单元的槽深 ${h_1}$、 ${h_2}$、 ${h_3}$、 ${h_4}$呈等梯度变化，光栅单元由两对互相平行的顶、槽反射平面组成，如图1(c)所示。一维立体相位光栅和二维相位光栅在性能上没有差异，但二维相位光栅结构紧凑，利于光谱仪器的小型化。

将光栅单元等效为一系列不同相位差交错排列的平面，如图2所示，当光束斜入射MPMG时，波矢 $\vec k$在 $(x,{\textit{z}})$平面的投影与 $\vec {{k}}$方向的夹角为 $\psi $，波矢 $\vec k$在 $(x,{\textit{z}})$平面的投影与x轴的夹角为 $\theta $，则顶、槽反射面由此引入的相位差为：

当太赫兹波平行于纸面方向入射，即（ $\psi=0$）时，经MPMG衍射后归一化的光强分布表达式如式(2)所示，计算时认为顶反射面和槽反射面的宽度相等。

$\alpha $、 $\beta $分别表示沿x方向的衍射角和沿y方向的衍射角， $\vec k$表示波矢， $w$表示顶反射面和槽反射面的宽度， $l$表示光栅的长度， $n$表示顶反射面和槽反射面的对数。

${{\rm{sinc}} ^2}\left( {\dfrac{{{\rm{k}}l\sin \beta }}{2}} \right)$和 ${{\rm{sinc}} ^2}\left( {\dfrac{{{\rm{k}}w\sin \alpha }}{2}} \right)$用于描述矩孔衍射因子，零级衍射有最大值； $\dfrac{{{{\sin }^2}\left[ {\dfrac{{n{\rm{k}}d\sin \alpha }}{2}} \right]}}{{{{\sin }^2}\left(\dfrac{{{\rm{k}}d\sin \alpha }}{2}\right)}}$用于描述多光束干涉因子，故MPMG的光栅方程为

即光栅的衍射级次分布仅取决于光栅常数 $d$， $d$越大相同级次衍射光的衍射角越大。

${\cos ^2}\left( {\dfrac{{{\rm{k}}w\sin \alpha + \varphi }}{2}} \right)$表示凹槽深度引入的附加相位对衍射场强度的调制。

3 MPMG的衍射特性分析

由式（3）可知，沿x轴的衍射场强度分布与光栅常数 $d$（ $d = 2w$）。顶反射面和槽反射面的对数 $n$以及凹槽深度 $h$有关。图3描述了不同设计参数条件下，夫琅禾费衍射场沿x轴的分布图。从图3(a)、3(b)、3(c)可以看出：当仅有槽深是变量时，槽深引入的相位调制使衍射场的能量在0级和 $ \pm 1$级之间转移；当相位差 $\varphi = 2{\text{π}}$时，夫琅禾费衍射场的能量全部集中在0级，当相位差 $\varphi= {\text{π}}$时，夫琅禾费衍射场的能量全部集中在 $ \pm 1$级，当 $\varphi = {{\text{π}}/ 2},{{3{\text{π}}} / 2}$时，夫琅禾费衍射场的能量平均分布在0级和 $ \pm 1$级。对比图3(a)、3(b)、3(c)可以看出，顶反射面和槽反射面的对数 $n$越多，各衍射级次的张角越小，顶反射面和槽反射面的对数 $n$与各衍射级次的衍射角大小无关。在讨论光栅常数 $d$（ $d = 2w$）对衍射场分布的影响时，为了避免引入波长的影响，选用了 $w/\lambda $。由图3(d)可以看出， $w/\lambda $越大，各衍射级次主极大的角宽度越小，衍射现象越不明显；而且 $w/\lambda $越大，各级衍射级次的衍射角越小。

4 MPMG的衍射特性测试

在实验室建立了一套评价MPMG衍射特性的实验装置。验证装置包括高功率THz辐射源系统（图4(a)）、THz激光准直传输系统（图4(b)）和光栅测试系统（图4(c)）。图4(c)中，Lens 1、Lens 2构成一组扩束镜，用于表示系统的准直与传输光路；Lens 3、Lens 4、MPMG与Lens 5构成MPMG衍射特性的测试光路。实验中使用波长为1 066~1 078 nm的可调谐激光器（OPO）和1 064 nm Nd:YAG激光器的差频产生高功率THz辐射^[13]，如图4(a)所示。泵浦激光束的光斑尺寸约为4 mm。这种由非线性差频效应产生的THz激光束与可见、红外激光束不同。对于激光高斯光束，其可以作为平行光进行处理的两个条件是：束腰半径远大于激光波长，波前半径远大于激光波长。前者可确保在激光传输过程中，光斑大小近似不变，后者可确保激光波前近似为平面。THz激光的波长与束腰尺寸相当，其光斑大小与传输距离近似成正比^[14]，通过实验测量，得到自制差频THz源的发散角为12°。因此，实验中使用透镜组来准直THz光束，光路图如图4(b)所示，该部分在原理图4（c）中等效为一组扩束镜^[15-16]。经过透镜组准直后，THz激光束的发散角为0.1°。光栅测试系统由3个高密度聚乙烯透镜、矩形挡板、MPMG和THz探测器组成，如图4(e)所示。实验中使用的1D MPMG由8个光栅单元组成，每个光栅单元包括5对顶面和槽面。MPMG的参数如表1所示，照片如图4(d)所示。实验中使用的THz探测器是从Advanced Compound Semiconductor Technologies (ACST, Hanau, Germany)购买的第二代准光学探测器（2dl 12c LS 2500 A1），探测器光敏面封装有直径为2 mm的会聚镜，以实现更高的光能收集效率，如图4(f)所示。

试验方法具体为：0.5 THz辐射经透镜组的准直，平行入射到MPMG（θ = 60°，ψ = 0°）上，并被MPMG衍射，衍射波经THz透镜汇聚后，在透镜焦平面上用THz探测器对0级和±1级衍射波进行检测。在MPMG的前表面放置一个很薄的矩形挡板，以确保每次只有一个单元被入射的THz波有效地照亮。通过依次移动探测器的位置，记录了0级和±1级的衍射强度。对每个光栅单元重复此操作，得到了所有光栅单元的0级和1级强度。为了消除激光抖动对测量结果的影响，采用多次测试的方法对测量结果进行了平均，所得结果是20次测量的平均值。将所测量数据进行归一化处理，可以得到8个光栅单元的0阶和±1阶衍射效率。

通过改变OPO的波长，测试了8个光栅单元对0.34 THz辐射波的衍射光强。光栅单元对0.5 THz、0.34 THz辐射的衍射效率理论模拟曲线和测量结果如图5所示。由图5可见，实验测量结果和理论模拟结果相符合，证明了MPMG的衍射特性，即MPMG的零级衍射光携带相位信息，其衍射强度由槽深引入的相位调制。

5 结　论

本文介绍了一种THz波段的新型MPMG，它由一系列光栅单元组成，这些光栅单元的槽深梯度与傅立叶变换光谱系统中动镜的不同位置相对应。对MPMG的夫琅和费衍射场分布和衍射效率的计算表明，MPMG的零级衍射光具有相位信息，其衍射强度由槽深引入的相位调制。接着，研制了由8个光栅单元组成的MPMG，对其在0.5 THz、0.34 THz下的零级和一级衍射效率进行了测试，结果与模拟结果吻合较好。由此可知，MPMG的零级衍射光携带相位信息，其衍射强度由槽深引入的相位调制。