Terahertz dispersion using multi-depth phase modulation grating

Qiu-jie Yang; Jing-guo Huang; Zhen-yang Xiao; Zhi-ming Huang; Rong Shu; Zhi-ping He

doi:10.1364/OE.27.012732

1. Introduction

Terahertz (THz) spectrum detection of free-space targets suffers from low emitter powers [1–3] and a signal transmission susceptible to environmental disturbance [4,5]. Therefore, a real-time detection scheme for weak signals should be adopted in THz spectrum detection for free-space targets. The Fellgett and Jacquinot advantages [6–8] can be used in Fourier-transform spectroscopy (FTS) to obtain higher signal powers [9,10]; they are ideal methods for weak signal detection. However, Fourier-transform technology depends on the continuous scanning of the dynamic mirror, and spectra cannot be generated in real time. In addition, a 50% energy loss of the beam splitter in FTS is unacceptable for the THz detection of free-space targets. Therefore, the development of novel FTS systems is crucial.

Much research in visible and infrared (IR) wavelengths has concluded that multichannel Fourier-transform spectroscopy, which combines the multichannel detection technique and Fourier-transform interferometric spectroscopy, is a potential solution. This idea was originally proposed by Okamoto et al. [11]. They designed an FTS device without mechanical moving parts by using a triangular common path and a self-scanning photodiode array to generate and detect interferograms. The self-scanning photodiode array as a multichannel detector provided the spectrometer with the ability to perform time-resolved spectroscopy. Then, various types of interferometers were used for this type of spectrometry. Hashimoto and Kawata [12] developed a compact Fourier-transform IR spectrometer with a Savart plate to obtain multi-channel interferograms in a stationary manner. The spectral resolution of the prototype system was 27.6 cm^–1 between 2000 and 5000 cm^–1. Ebizuka et al. [13] devised a multichannel Fourier-transform spectrometer that incorporates a Wollaston prism and a polarizing interferometer combined with two Savart plates and a phase-retarding plate. Komisarek et al. [14] improved this considerably using a Wollaston prism array and a two-dimensional (2D) photodetector array. Li et al. [15] proposed a Fourier-transform imaging spectropolarimeter covering the 450–1000-nm spectral range. The system used two birefringent retarders and a Wollaston polarizing interferometer to obtain interferogram patterns. All spectra could be obtained by array detectors. Zhang et al. [16] provided a high-throughput static channeled interference imaging spectropolarimeter over the spectral range of 480–960 nm based on a Savart polariscope. These studies prove that multichannel Fourier-transform spectroscopy is a feasible scheme for real-time detection in the THz region.

In these instruments, the Wollaston prism or birefringent retarder works as a phase modulator, they are the key devices to generate interference signals in synchronization. However, these transmitting optical elements such as the Wollaston prism and birefringent retarder are currently unavailable in terahertz range. Some other devices aiming at controlling the phase of THz wave are developed these days. Luo et al. [17] and Mirzaei et al [18] designed Fourier phase gratings for generating 2 × 2 and 2 × 4 beams at 1.25THz and 1.4THz separately. Wang et al [19] presented a reflective liquid crystal terahertz plates with sub-wavelength metal grating and metal ground plane electrodes to achieve beam steering and polarization conversion. Yang et al [20] designed a reflective spatial phase shifter operating at THz regime above 325GHz to realize a tunable THz reflective phase shifter. However, these phase modulators do not fit for multichannel Fourier-transform spectroscopy. The Fourier phase grating, designed based on the Fourier series expansion theory, is used for passive quasi-optical multiplexing devices. The THz phase shifter based on liquid crystal is suitable for the reconfigurable reflect array. The THz phase grating based on liquid crystal is used for reducing the beam splitting ratio of the zeroth-order diffraction to the first-order diffraction.

Fortunately, the development of diffractive optical elements in time-modulated Fourier spectroscopy has given us a path forward. Hall et al. [21] first described a high-resolution, far-IR lamellar grating interferometer that operates in either a single-beam or double-beam differencing mode of operation. Manzardo et al. [22] presented a lamellar grating interferometer realized with microelectromechanical system technology. Motion is performed by an electrostatic comb drive actuator fabricated by silicon micromachining, particularly by silicon-on-insulator technology. Yu et al. [23] presented a lamellar grating-based Fourier-transform micro-spectrometer in which an electromagnetic actuator is used to drive the mobile facets of the lamellar grating to move bi-directionally. Ayerden et al. [24] developed a Fourier-transform IR spectroscopy system using a microelectromechanical system in the target range of 2.5–16.0 μm with a spectral resolution of 15–20 cm⁻¹. The core was a lamellar grating interferometer and was fast, compact, and achromatic. Although the lamellar grating in the above studies cannot directly be used in multichannel Fourier-transform spectroscopy, the method of obtaining interferograms can be used for reference. This was the main point of this study.

Our final goal is to establish a multichannel Fourier-transform spectroscopy working in THz band used for mineral exploration on planetary, and the schematic is shown in Fig. 1. The THz radiation from the planetary surface is collected and converged at the focus of Cassegrain telescope. After optical filtering, it is collimated by an off-axis parabolic mirror. Then the parallel beam is diffracted by a novel multi-depth phase modulation grating (MPMG) which consists of many grating cells with different groove depths. 0-order diffraction light with different optical path difference from each grating cell is picked up in synchronization by a detector array. The THz spectrum of the mineral could be obtained by FFT. The key device in the FTS is the MPMG.

Fig. 1 The diagram of mineral exploration on planetary by a multichannel Fourier-transform spectroscopy. The THz radiation form the planetary surface is collected and converged at the focus of Cassegrain telescope. After optical filtering, it is collimated by an off-axis parabolic mirror. Then the parallel beam is diffracted by an MPMG. 0-order diffraction light for each grating cell is converged at the focus of the lens 1. After filtered by the stop 2, the 0-order diffraction light is collimated by the lens 2. Finally, 0-order diffraction light with different optical path difference from each grating cell converged by lens array is picked up in synchronization by a detector array.

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In this paper, we propose a novel MPMG in the THz region, which is inspired by lamellar gratings. The MPMG consists of a series of grating cells with different groove depths equal to the different locations of the motive mirror in an FTS system. As a reflective wavefront-splitting interferometric device, it can realize the real-time detection of weak THz signals in free space by replacing the transmissive amplitude-splitting interferometric device in a multi-channel Fourier-transform interferometer. We calculate the Fraunhofer diffraction field distribution and grating equation of an MPMG based on Fresnel–Kirchhoff diffraction theory in Section 2. The influence of grating parameters on the Fraunhofer diffraction field distribution is discussed, and the diffraction efficiency of the MPMG is calculated in Section 3. Section 4 presents experimental data from the MPMG and discusses the results, while our conclusion is presented in Section 5.

2. Diffraction field distribution of MPMG

2.1. MPMG

An MPMG for THz waves consists of a series of grating cells with different groove depths equal to the different locations of the motive mirror in an FTS system. The grating cells consist of pairs of crest and trough planes parallel to each other. Schematics of the one-dimensional (1D) and 2D MPMG, both composed of four grating cells, are illustrated in Figs. 2(a) and 2(b). A grating cell with two pairs of crest and trough planes is illustrated in Fig. 2(c). h₁, h₂, h₃, and h₄ represent the different cell depths in Figs. 2(a) and 2(b), and they change in a constant gradient.

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

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The maximum optical path difference of the MPMG can be determined by the spectral resolution, as shown in Eq. (1):

δ_{\max} \geq \frac{1}{2 δ υ},

where

δ_{max}

is the maximum optical path difference and

δυ

is the spectral resolution.

The number of grating cells in an MPMG can be determined by the Nyquist sampling theorem:

N \geq 2 δ_{\max} (σ_{\max} - σ_{\min}),

where N is the number of grating cells and

σ_{max}

and

σ_{min}

indicate the maximum and minimum wavenumbers of the measured signal, respectively.

2.2. Diffraction field distribution of MPMG

A grating cell is equivalent to a transmission grating placed in the x–y plane. The transmission grating consists of a series of alternating planes with zero and $φ$ phase differences. When oblique incidence occurs, as shown in Fig. 3(a), assuming that $\vec{k}$ represents the wave vector, the angle between the projection of $\vec{k}$ on the x–z plane and $\vec{k}$ is Y, and the angle between the projection of $\vec{k}$ on the x–z plane and the x-axis is $θ$ ; therefore the phase path difference introduced can be expressed as Eq. (3):

φ {ρ, τ} = \frac{4 π h {ρ, τ}}{λ \cos Y \sin θ}, ρ = 1, 2, 3, \dots, τ = 1, 2, 3, \dots,

where

λ

denotes wavelength,

h {ρ, τ}

denotes the trough depth of at different locations on the x–y plane,

{ρ, τ}

denotes the sequence of the grating cell in an MPMG,

ρ

denotes the sequence of row and

τ

denotes sequence of column.

Fig. 3 (a) Reflection grating simulated as a plane transmission grating. (b) Simulation diagram.

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According to the Fresnel–Kirchhoff diffraction integral, as shown in Fig. 3(b), the complex amplitude of a point P_i(x_i, y_i, z_i) on the x–y plane parallel to the diffraction screen can be described as Eq. (4).

u (P_{i}) = \frac{E}{i λ} \iint \frac{\exp [i k (r + s)] \exp [- i φ (ρ, τ)]}{r s} \frac{\cos (\vec{n}, \vec{r}) - \cos (\vec{n}, \vec{s})}{2} d x d y,

where r denotes the distance from the light source at P_s(x_s, y_s, z_s) to a point Q(x₀, y₀, 0) on the diffraction screen, and s denotes the distance from the point Q(x₀, y₀, 0) to a point P_i(x_i, y_i, z_i) on the observation plane, as shown in Fig. 3(b). The approximate process is

u (P_{i}) = \frac{E \cos χ}{i λ} \iint \frac{\exp [i k (r + s)] \exp [- i φ (ρ, τ)]}{r^{'} s^{'}} d x d y .

With the help of Taylor expansion and taking the first two into account, r and s can be simplified to Eq. (8):

r \approx r^{'} - \frac{x_{s} x_{0} + y_{s} y_{0}}{{r^{'}}^{2}}, s \approx s^{'} - \frac{x_{i} x_{0} + y_{i} y_{0}}{{s^{'}}^{2}} .

Substituting Eq. (6) into Eq. (5), it can be simplified to Eq. (7):

u (P_{i}) = K \iint \exp [- ik (p x + q y)] \exp [- i φ (ρ, τ)] d x d y,

where

K = \frac{E \cos χ}{i λ} \frac{\exp (i k r^{'})}{r^{'}} \frac{\exp (i k s^{'})}{s^{'}}

,

p = \frac{x_{s}}{r^{'}} + \frac{x_{i}}{s^{'}}

, and

q = \frac{y_{s}}{r^{'}} + \frac{y_{i}}{s^{'}}

.

When parallel light is incident, the contribution of all small planes to the image plane can be calculated according to Born and Wolf's Principles of Optics [25].

First, the integral of the whole diffraction screen is divided into the sum of several integrals according to the phase distribution:

\begin{array}{l} u (P_{i}) = K \sum_{n = 0}^{n - 1} \int_{- \frac{l}{2}}^{\frac{l}{2}} \exp (- ik q y) d y {\int_{- \frac{w}{2}}^{\frac{w}{2}} \exp [- ik p (x + n d)] d x + \int_{- \frac{c}{2}}^{\frac{c}{2}} \exp [- ik p (x + w + n d)] \exp (- i φ) d x} \\ = \frac{2 K \sin (\frac{k q l}{2})}{k q} {\frac{\sin (\frac{n k d p}{2})}{\sin (\frac{k d p}{2})} [\frac{2 \sin (k p \frac{w}{2})}{k p} + \frac{2 \sin (k p \frac{c}{2})}{k p} \exp [- i(φ + k p w)]] \exp [\frac{- i (n - 1) k d p}{2}]} . \end{array}

Where $w$ denotes the width of crest plane, $c$ denotes the width of the trough plane, $l$ denotes the length of grating cell, $n$ denotes the number of pairs of crest and trough planes in a grating cell. These parameters are shown in the Fig. 4.

Fig. 4 Schematic of structural parameters for a grating cell.

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The intensity on the detection plane is the product of the field amplitude and its complex conjugation. Thus, the intensity at P_i(x_i, y_i, z_i) can be expressed as Eq. (9):

\begin{array}{l} I (P_{i}) = K^{2} l^{2} \sin c^{2} (\frac{k q l}{2}) \frac{\sin^{2} (\frac{n k d p}{2})}{\sin^{2} (\frac{k d p}{2})} \\ \cdot [w^{2} {sinc}^{2} (k p \frac{w}{2}) + c^{2} {sinc}^{2} (k p \frac{c}{2}) + 2 w c sinc (k p \frac{w}{2}) sinc (k p \frac{c}{2}) \cos (φ + k p w)] . \end{array}

When the widths of two adjacent small planes are equal, Eq. (9) can be simplified to

I (P_{i}) = 4 K^{2} l^{2} w^{2} \sin c^{2} (\frac{k q l}{2}) {sinc}^{2} (k p \frac{w}{2}) \frac{\sin^{2} (\frac{n k d p}{2})}{\sin^{2} (\frac{k d p}{2})} \cos^{2} (\frac{k p w + φ}{2}) .

For the convenience of discussion, the relationship between the diffraction angle and diffraction intensity is established. In Fraunhofer diffraction,

\sin α = \frac{x_{i}}{s^{'}} \approx \frac{x_{i}}{f}, \sin β = \frac{y_{i}}{s^{'}} \approx \frac{y_{i}}{f}, \sin α^{'} = - \frac{x_{s}}{r^{'}}, \sin β^{'} = - \frac{y_{s}}{r^{'}} .

where

α

and

β

, the directional cosine of the angles between QP_i with the x-axes and QP_i with the y-axes, respectively, are the 2D diffraction angles.

α^{'}

and

β^{'}

are the angles of incidence.

The normalized intensity expression at any point P_i in the radiation field of the MPMG can be calculated.

\begin{array}{l} I (P_{i}) = \sin c^{2} [\frac{k l (\sin β + \sin β^{'})}{2}] {sinc}^{2} [\frac{k w (\sin α + \sin α^{'})}{2}] \\ \cdot \frac{\sin^{2} [\frac{n k d (\sin α + \sin α^{'})}{2}]}{\sin^{2} (\frac{k d (\sin α + \sin α^{'})}{2})} \cos^{2} [\frac{k w (\sin α + \sin α^{'}) + φ}{2}] . \end{array}

When the incident light is parallel to the optical axis, the above formula can be changed to Eq. (13).

I (P_{i}) = \sin c^{2} (\frac{k l \sin β}{2}) {sinc}^{2} (\frac{k w \sin α}{2}) \frac{\sin^{2} [\frac{n k d \sin α}{2}]}{\sin^{2} (\frac{k d \sin α}{2})} \cos^{2} (\frac{k w \sin α + φ}{2}) .

2.3. Grating equation of MPMG

The grating equation of an MPMG can be extracted from the multi-beam interference factor, $\frac{\sin^{2} [\frac{n k d \sin α}{2}]}{\sin^{2} (\frac{k d \sin α}{2})}$ as shown in Eq. (14).

d \sin α =m λ, d = w + c, m = 0, \pm 1, \pm 2, \dots .

Equation (14) demonstrates that the diffraction order distribution of a grating cell only depends on grating constant d, similar to that [21] in the amplitude-modulated grating.

The multi-beam interference factor is zero for those $α$ satisfying Eq. (15). There are $t - 1$ zeros between two adjacent principal maxima.

k d \sin α = 2 π (m+ \frac{m'}{t}), m = 0, \pm 1, \pm 2, \dots; m' = 1, 2, \dots, t - 1; t = 0, 1, 2, \dots, m .

The angular distance between two adjacent zeros is expressed in Eq. (16). The angular distance between the main maximum and an adjacent zero is also the same.

Δ α = \frac{λ}{t d \cos α} .

The results indicate that the larger the number of grooves, the smaller the half-angle width of the main maximum, that is, the sharper the diffraction fringes.

2.4. Conclusion

The Fraunhofer diffraction field distribution, see Eq. (13), and grating equation, see Eq. (14), of an MPMG are calculated based on Fresnel–Kirchhoff diffraction theory. The grating equation of an MPMG shown in Eq. (14) is same as the traditional grating. But the Fraunhofer diffraction field distribution shown in Eq. (13) is rather different from that of the traditional grating. Compared with the diffraction intensity distribution equation of the traditional gratings, there is an additional phase modulation factor $\cos^{2} (\frac{k w \sin α + φ}{2})$ for MPMG. It means that the diffraction intensity distribution after MPMG is modulated by cell depth.

3. Analysis of diffraction characteristics for MPMG

3.1. Effect of grating parameters on Fraunhofer diffraction field distribution

According to Eq. (13), the intensity distribution of MPMG was simulated by MATLAB. The influence of different parameters on the distribution of the diffraction field is shown in Fig. 5. Figures 5(a)–5(c) present the intensity distribution curves of the cells of n = 4, 8, and 12, respectively, with $w / λ = 2$ for phase steps $φ = π / 2, π, 3π / 2, 2 π$ . Figure 5(d) shows the curve of the intensity distribution relation with the width of the crest or trough surface at $n = 8; φ = 2 π$ .

Fig. 5 Influence of different parameters on the intensity distribution of grating cell diffraction field. (a) Phase difference, (b) phase difference, (c) phase difference, and (d) $w / λ$ .The transverse axis represents the diffraction along x direction and longitudinal axis represents normalized light intensity.

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Figure 5(a) shows the influence of the phase difference $φ$ on the diffraction pattern. With the increase in $φ$ , the energy transfers between the 0th and ± 1st order. This implies that varying the depth of the grating cell only affects the energy transfer between the 0th and ± 1st order but has no effect on the other physical parameters; for example, the peak half-width of the zeroth order. The same behavior can be found in Figs. 5(b) and 5 (c).

Figures 5(a)–5(c) demonstrate the influence of the number of pairs of crest and trough planes on the diffraction pattern. Increasing the number n, the half-angle width of the main maximum decreases significantly. Further, the diffraction pattern becomes sharper and the diffraction phenomenon becomes clearer. Therefore, appropriately increasing the number of crest and trough planes in the grating cells will be beneficial for signal detection.

Figure 5(d) shows the influence of $w / λ$ on the diffraction pattern. Increasing the ratio $w / λ$ results in a smaller half-angle width of the main maximum and diffraction aperture angles.

3.2. Diffraction efficiency of MPMG

For mth-order diffraction light of a grating cell, the normalized expression of light intensity can be calculated using Eqs. (14) and (13), which is shown in Eq. (17):

I_{m} = \sin c^{2} (\frac{m π}{2}) \frac{\sin^{2} (m n π)}{\sin^{2} (m π)} \cos^{2} (\frac{m π + φ}{2}) = \frac{2 n^{2}}{π^{2}} \sin^{2} (\frac{m π}{2}) [1 + {(- 1)}^{m} \cos (φ)] .

For m = 0 and ± 1, the diffraction efficiency is

η_{0} = \frac{1 + \cos (φ)}{2}, η_{\pm 1} = \frac{2}{π^{2}} [1 - \cos (φ)] .

In order to verify the theoretical calculation results above, the diffraction efficiency is recalculated by Fourier optics theory.

The screen function of a grating cell including many pairs of crest and trough planes can be expressed by Eq. (19).

t (x) = \frac{1}{2 w} rect (\frac{x}{w}) * comb (\frac{x}{2 w}) rect (\frac{y}{l}) + \frac{1}{2 w} rect (\frac{x}{w}) * comb (\frac{x + w}{2 w}) rect (\frac{y}{l}) \exp (i \frac{4 π}{λ} h) .

Where $rect (x, y)$ is rectangle function and $comb (x, y)$ is comb function. The definition of rectangle function and comb function could be found in [26]. Assuming that the incident light $u_{0} (x, y)$ is a plane wave, the field distribution $u (x, y)$ after a grating cell is the product of incident light $u_{0} (x, y)$ and screen function $t (x, y)$ , according to linear system theory.

u (x, y) = t (x, y) u_{0} (x, y) .

According to Fourier optics theory and convolution theory [27,28], the spectral distribution function after the grating can be expressed as follows:

U (f_{x}, f_{y}) = ℑ {u (x, y)} = ℑ {t (x, y)} * ℑ {u_{0} (x, y)} .

Where $ℑ {}$ is the sign of Fourier transform, f_x and f_y are spatial frequency for incident light u₀(x, y). For the plane wave u₀(x, y), its frequency spectrum can be considered as unity. Then, the spectral distribution function after a grating cell can be expressed as

U (f_{x}, f_{y}) = ℑ {u (x, y)} = ℑ {t (x, y)} .

That is, the spectral distribution function after a grating cell is actually the Fourier transform of its screen function. The Fourier transform of the grating function is shown in the following formula:

\begin{array}{l} ℑ {t (x, y)} = \frac{1}{2 w} ℑ {rect (\frac{x}{w}) * comb (\frac{x}{2 w}) rect (\frac{y}{l})} \\ + \frac{1}{2 w} ℑ {rect (\frac{x}{w}) * comb (\frac{x + w}{2 w}) rect (\frac{y}{l}) \exp (- i φ)} . \end{array}

Using the convolution theorem, the upper formula can be written as follows:

\begin{array}{l} ℑ {t (x, y)} = \frac{1}{2 w} ℑ {rect (\frac{x}{w}) * comb (\frac{x}{2 w}) rect (\frac{y}{l})} \\ + \frac{\exp (- i φ)}{2 w} ℑ {rect (\frac{x}{w}) * comb (\frac{x + w}{2 w}) rect (\frac{y}{l})} \\ = l \sin c (l f_{y}) \sum_{n = - \infty}^{\infty} \frac{1}{2} sinc (\frac{n}{2}) + l \exp (- i φ) sinc (l f_{y}) \sum_{n = - \infty}^{\infty} \frac{1}{2} sinc (\frac{n}{2}) \exp (inπ) . \end{array}

Therefore, the amplitude of the mth-order diffraction is

U_{m} (f_{x}, f_{y}) = \frac{l \sin c (l f_{y})}{2} \sin c (\frac{m}{2}) [1 + {(- 1)}^{m} \exp (- i φ)] .

Then, the diffraction efficiency of the mth-order diffraction is

η_{m} = \frac{U_{m} {(f_{x}, f_{y})}^{2}}{U {(f_{x}, f_{y})}^{2}} = \frac{1}{2} \sin c^{2} (\frac{m}{2}) [1 + {(- 1)}^{m} \cos (φ)] .

For m = 0 and ± 1, the diffraction efficiencies are

η_{0} = \frac{1}{2} [1 + \cos (φ)], η_{\pm 1} = \frac{1}{2} \sin c^{2} (\frac{1}{2}) [1 - \cos (φ)] = \frac{2}{π^{2}} [1 - \cos (φ)] .

The same calculation results can be obtained by the two methods. The conclusion is that the intensity of the 0th and ± 1st order is strongly modulated by the groove depth, that is, the intensity of the 0th- and ± 1st-order diffraction light is different for grating elements with different groove depths. The theoretical results are in good agreement with the simulation results of Fig. 5. The maximum diffraction efficiency for the zeroth order is unity and the minimum is 0. The maximum diffraction efficiency for the ± 1st order is 40.53% and the minimum is 0. Compared to the amplitude-modulated gratings, the zeroth-order diffraction light of an MPMG carries phase information, and its diffraction intensity is modulated by the phase introduced by the groove depth, that is, the characteristic of an MPMG. Therefore, the spectral information of the measured object can be calculated by measuring the intensity of the zeroth-order diffraction light for each grating cell.

3.3. Conclusion

In this part, effect of grating parameters on Fraunhofer diffraction field distribution was discussed and diffraction efficiency of an MPMG was calculated in two ways. The simulation shows that the intensity of the 0th and ± 1st order is strongly modulated by the groove depth, that is, the intensity of the 0th- and ± 1st-order diffraction light is different for grating elements with different groove depths. This is the biggest difference between an MPMG and the traditional grating.

4. Verification of diffraction characteristics of MPMG

4.1. Design and manufacture of MPMG

An MPMG working at 0.3-0.54THz with the resolution of 0.166cm⁻¹ was designed and manufactured. The maximum optical path difference of the MPMG was determined by Eq. (1).

δ_{\max} \geq \frac{1}{2 δ υ} =3 .02 cm .

The number of grating cells N in an MPMG was determined by Eq. (2).

N \geq 2 δ_{\max} (σ_{\max} - σ_{\min}) = 8.

When oblique incidence occurs with Y = 0゜and $θ$ = 60゜, as shown in Fig. 3(a), the max groove depth was determined by Eq. (3).

h_{\max} = \frac{δ_{\max} \cos Y \sin ϕ}{2} =1 .308 cm .

The groove depths of eight cells are in turn:1.3080cm, 1.1445cm, 0.9810cm, 0.8175cm, 0.6540cm, 0.4905cm, 0.3270cm, 0.1635cm. The optical path difference (OPD) introduced by groove depth for each grating cell is determined by Eq. (31).

O P D {ρ, τ} = \frac{2 h {ρ, τ}}{\cos Y \sin ϕ}

According to the influence of cell parameters on the distribution of diffraction field, which was discussed in section 3, the number of pairs of crest and trough planes in a grating cell is 8, the wide of the crest or trough planes is $1.13 λ$ ( $λ$ at 0.34THz), and the length of the crest or trough planes is $45.2 λ$ ( $λ$ at 0.34THz). The parameters of the MPMG are presented in Table 1.

Table 1. Parameters of MPMG

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In Fourier spectroscopy, the number of sampling points is usually four times or more the minimum number of sampling points determined by Nyquist theory. So, an MPMG with 64 cells and the same resolution as that in Table 1 was also made. The parameters of the MPMG with 64 cells are presented in Table 2.

Table 2. Parameters of MPMG

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The MPMG was formed by grooving on metal aluminum. Effect of roughness of Al on reflectivity was discussed detailly in our previous work [29]. Since submicron roughness ( $λ / 10$ at 0.5THz) is enough for THz reflector, and it’s no problem with today's mechanical processing. The MPMG was manufactured by Fulian, a factory located in Shanghai.

4.2. Experimental verification

A verification device to evaluate the diffraction characteristics of an MPMG was created. The verification device includes a high-power THz radiation source system, a THz laser collimation and transmission system, and a grating-testing system, as shown in Fig. 6(c). The high-power THz radiation source system emits THz waves by pumping GaSe using a tunable optical parametric oscillator (OPO) laser in the wavelength range of 1066–1078 nm and a 1064-nm ND:YAG laser from difference frequency generation [30], as shown in Fig. 6(a). The spot size of pump laser beams is about 4mm. This THz laser generated by the nonlinear difference-frequency effect is not the same as traditional visible and IR lasers. The wavefront of this laser is a spherical surface. While the wavefront of the THz laser propagates forward, the center of the wavefront propagates forward and its radius increases. This causes the THz laser beam to have a divergence angle that is tens of times larger than that of a visible laser. The divergence angle of the self-made difference-frequency THz source was obtained by experimental measurements and determined to be 12°. As such, lenses were used to collimate the beams, as shown in the experimental photograph [31] in Fig. 6(b). After the collimation and transmission system, its divergence angle was 0.1°. The grating-testing system includes three pieces of THz lenses composed of high-density polyethylene, a rectangular stop, an MPMG, and a THz detector, as shown in Fig. 6(e). The MPMG used in the experiment consists of eight grating cells in line. Each grating cell includes five pairs of crest and trough planes. The parameters of the MPMG are presented in Table 1 and a photograph is shown in Fig. 6(d). The THz detector used in the experiment is a second-generation quasi-optical detector (2DL 12C LS 2500 A1) purchased from Advanced Compound Semiconductor Technologies (ACST, Hanau, Germany), which has an optical collimating lens with a diameter of 12 mm. It also has an integrated amplifier to achieve higher sensitivities. A photograph of the detector is shown in Fig. 6(f).

Fig. 6 (a) High-power THz radiation source system. (b) Laser collimation and transmission system. (c) Schematic of experiment. (d) 1D and 2D MPMG. (e) Grating-testing system. (f) Photograph of the THz detector.

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The test method is described below. The 0.5-THz radiation is collimated by THz lenses and illuminates the MPMG in parallel at 60° (Y = 0°, $θ = 6 0^{°}$ ). Then, the beam is diffracted by the MPMG. After convergence by the THz lens, it is detected by a THz detector on the focal plane of the lens. A special rectangular stop is placed on the front surface of the MPMG to ensure that only one cell is illuminated effectively by the incident THz wave at a time. The stop is so thin that it has little effect on the diffraction beam. The diffraction intensities of the 0th- and ± 1st order were recorded by moving the position of the detector sequentially. This was repeated for each grating cell. Then, the intensities of the 0th and ± 1st order for all grating cells were obtained. To eliminate the influence of laser jitter on measurement, the average values were taken by multiple tests. The results are the average of 20 measurements. After that, the 0th- and ± 1st-order diffraction efficiency for all grating cells can be obtained by normalization.

The diffraction efficiency for all grating cells at 0.34 THz was also tested by changing the wavelength of the OPO. Both the theoretical simulation and experimental results at 0.5 and 0.34 THz are presented in Fig. 7. The red points represent the diffraction efficiency obtained in the test, while the blue points represent the simulated results. By comparing the simulation and experimental results, we determined that the diffraction efficiencies are in good agreement. Therefore, we believe that the established model can accurately simulate diffraction characteristics of an MPMG in the THz band. We are also convinced that the zeroth-order diffraction light of an MPMG carries phase information, and its diffraction intensity is modulated by the phase introduced by the groove depth.

Fig. 7 (a) Zeroth- and first-order diffraction efficiency for each grating cell at 0.34 and (b) 0.5 THz.

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Furthermore, the spectrum of the THz source is retrieved by Fourier transformation. Here, the MPMG used in the experiment consists of 64 grating cells and the parameters of the MPMG are presented in Table 2. Different grating cell with the different groove depth makes the reflective 0-order diffraction light with different OPD. The test conditions are the same as the experiment above. The collimated 0.34 THz radiation illuminates the MPMG in parallel at 60° (Y = 0°, $θ = 6 0^{°}$ ). The OPD for each grating cell was calculated by Eq. (31). Each 0-order diffraction light was measured by changing the position of a rectangular aperture. It takes us at most 15 seconds to complete a measurement. The total time for the measurement of 64 grating cells is 16 minutes. To eliminate the influence of laser jitter on measurement, the average values were taken by multiple tests. The results are the average of 20 measurements. The zeroth-order diffraction intensities for all grating cells at 0.4, and 0.5 THz were also tested by varying the wavelength of the OPO.

The relationship between the optical path difference and light intensity is described in Fig. 8(a). The blue triangle symbol is the intensity obtained at 0.5 THz, the red point is that obtained at 0.4 THz, and the dark block is that obtained at 0.34 THz. Fourier transformation is applied to each curve to obtain the spectral information of the THz source at each time. The Nyquist sampling theorem and spectral calibration are considered to calculating the frequency of terahertz source. The Fourier transform results are presented in Fig. 8(b). The blue curve represents the Fourier transform results for data obtained at 0.5 THz, the red curve represents those for data obtained at 0.4 THz, and the black curve represents those for data obtained at 0.34 THz. By comparing the Fourier transform results and the actual THz frequencies, we determined that they are in good agreement. Therefore, we believe that the MPMG is an excellent phase modulation device. The grating cells with different groove depths in an MPMG are equal to the different locations of the motive mirror in an FTS. As a reflective wavefront-splitting interferometric device, it can realize the real-time detection of weak THz signals in free space by replacing the transmissive amplitude-splitting interferometric device in a multi-channel Fourier-transform interferometer.

Fig. 8 (a) Relationship between optical path difference and zeroth-order diffraction intensity at 0.5, 0.4, and 0.34 THz. (b) Spectrum of the THz source retrieved from zeroth-order diffraction intensity by Fourier transform.

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5. Conclusions

We have introduced a novel MPMG in the THz band, which is composed of a series of grating cells with a constant groove depth gradient corresponding to the different locations of the motive mirror in a Fourier-transform spectroscopy system. The calculation of the Fraunhofer diffraction field distribution and diffraction efficiency of an MPMG shows that the zeroth-order diffraction light of an MPMG carries phase information, and its diffraction intensity is modulated by the phase introduced by the groove depth. An MPMG consisting of eight grating cells was constructed. Its zeroth- and first-order diffraction efficiencies at 0.5 and 0.34 THz were tested, and they were found to agree well with the simulation. We have further derived successfully the spectra of the THz source at 0.5 THz, 0.4 THz and 0.34 THz from the zeroth-order diffraction intensity by Fourier transformation. The good agreement between the Fourier transform results and the actual THz frequencies suggests that the real-time detection of weak THz signals in free space can be realized by replacing the transmissive amplitude-splitting interferometric device with the MPMG in a multi-channel Fourier-transform interferometer.

THz array detector is unavailable resulting zeroth-order diffraction intensity was detected sequentially for each grating cell. The shortage of wide-spectrum terahertz sources limits the evaluation of spectral resolution of the MPMG. These two also limit the multichannel Fourier-transform spectroscopy available in the lab. However, passive spectrometers worked in THz high frequency band (3-10THz) which is full of mineral fingerprints are good instruments for mineral exploration on planetary. Hence the research reported in the paper represents an important step toward multichannel Fourier-transform spectroscopy in THz band and opens the new fields in THz spectrum detection of free-space targets.

Funding

National Science Fund for Distinguished Young Scholars (61625505); Natural Science Foundation of Shanghai (16JC1403400, 18ZR1445500); Innovation Project Fund of Shanghai Institute of Technical Physics (IPFSITP) (CX-158); Key Laboratory Foundation of Chinese Academy of Sciences (CXJJ-17S025).

Acknowledgments

We thank to Dr. Wei Zhou and Dr. Yan-qing Gao for useful discussions regarding to the presented results.

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Parameter names		Parameter names
N	8	n	5
w (mm)	1	l (mm)	40
Y (°)	0	$θ$ (°)	60
h (cm)	h{1} = 0.1635, h{2} = 0.3270, h{3} = 0.4905, h{4} = 0.6540, h{5} = 0.8175, h{6} = 0.9810, h{7} = 1.1445, h{8} = 1.3080

Parameter names		Parameter names
N	64	n	5
w (mm)	1	l (mm)	40
Y (°)	0	$θ$ (°)	60
h (mm)	$h {ρ, τ} = 0.204375 ρ + 0.204375 (τ - 1), ρ = 1, 2, 3, 4, 5, 6, 7, 8; τ = 1, 2, 3, 4, 5, 6, 7, 8$

Parameter names		Parameter names
N	8	n	5
w (mm)	1	l (mm)	40
Y (°)	0	$θ$ (°)	60
h (cm)	h{1} = 0.1635, h{2} = 0.3270, h{3} = 0.4905, h{4} = 0.6540, h{5} = 0.8175, h{6} = 0.9810, h{7} = 1.1445, h{8} = 1.3080

Parameter names		Parameter names
N	64	n	5
w (mm)	1	l (mm)	40
Y (°)	0	$θ$ (°)	60
h (mm)	$h {ρ, τ} = 0.204375 ρ + 0.204375 (τ - 1), ρ = 1, 2, 3, 4, 5, 6, 7, 8; τ = 1, 2, 3, 4, 5, 6, 7, 8$

Terahertz dispersion using multi-depth phase modulation grating

Abstract

1. Introduction

2. Diffraction field distribution of MPMG

2.1. MPMG

2.2. Diffraction field distribution of MPMG

2.3. Grating equation of MPMG

2.4. Conclusion

3. Analysis of diffraction characteristics for MPMG

3.1. Effect of grating parameters on Fraunhofer diffraction field distribution

3.2. Diffraction efficiency of MPMG

3.3. Conclusion

4. Verification of diffraction characteristics of MPMG

4.1. Design and manufacture of MPMG

4.2. Experimental verification

5. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (2)

Equations (31)

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