Compact ultrahigh resolution interferometric spectrometer

Qinghua Yang

doi:10.1364/OE.27.030606

1. Introduction

Whether for scientific research or industrial applications, military or civil applications, earth remote sensing or space exploration, spectroscopy is of great significance and value. One of the most important performance criteria of spectrometers is spectral resolution (resolving power). The gratings used in either reflection or transmission commonly provide resolving powers from 1000 to 100,000 [1–5], but high spectral resolution gratings are generally large in physical size. For grating spectrometers, the higher the resolution, the larger the physical size and the more expensive the cost. More importantly, resolving power higher than 500,000 can only be provided by interferometric-type spectrometers.

For ultrahigh resolution spectral measurements in the infrared region, Michelson-type interferometers [6–11] still suffer three drawbacks: (1) a very large maximum optical path difference is needed, which means a very large maximum displacement of the moving mirror, which makes the instruments very large in physical size and limits their application where space and mass are at a premium; (2) a large travel of the moving part will reduce the stability against various disturbances; (3) an acceptable solid angle decreases with the increase of optical path difference, which causes the optical throughput to decrease as the spectral resolution increases.

Fabry-Perot interferometers (FPI) [12–21] provide a compact and cost-effective method to get ultrahigh spectral resolution. The FPI faces the problem of overlapping orders, which can be separated either by a fixed narrowband filter [22], a grating [23], or by another FPI [24–30], however, none of these methods can obtain resolving power higher than 1,000,000. A method using a standard Michelson interferometer to separate the FPI overlapping orders [31,32] can provide resolving power higher than 1,000,000 in near-infrared, short-wave infrared or mid-wave infrared region, however, this method still suffers three drawbacks: (1) a large travel of the moving mirror of Michelson interferometer is still needed, which greatly increases spectral measurement time and reduces the stability; (2) the biggest problem associated with the use of a standard Michelson interferometer as a Fourier transform spectrometer (tilt of moving mirror during scanning) is not solved, which greatly reduces the stability; (3) a drive system for moving mirror of Michelson interferometer is needed, which makes the instrument more complex and reduces the stability.

This paper proposes a spectrometer that can achieve ultrahigh spectral resolution, short measurement time and small physical size. After a description of the principle, preliminary numerical simulations are given by two examples with resolving power higher than 1,000,000 in short-wave infrared region. Finally, the conclusion is given.

2. Principle

Figure 1 shows the optics of the compact ultra-high resolution interferometric spectrometer (UHRIS), which consists of a FPI, a static stepped-mirror interferometer (SMI), a collimating lens, a collecting lens, and an area-array detector. The static SMI is suitable for high resolution spectral measurement in narrow spectral bands in the infrared region [33–36]. The SMI comprises one beam splitter and two fixed stepped mirrors with a shape of stairs: the stepped mirror M1 has small steps, the stepped mirror M2 has tall steps, the tall step height corresponds to the sum of the small step heights, the directions of the steps of two stepped mirrors are orthogonal, and so the SMI acquires a spatial array of optical path difference [35]. Namely, the SMI acquires a spatial sampling of the interferogram. In the UHRIS, the spectral resolution of the SMI only needs to be enough to separate the overlapping orders of the FPI. The FPI is first assigned a given spacing and a first spectrum is obtained from the SMI, then the FPI is assigned the next spacing position and the next spectrum is obtained from the SMI, and so on until the full spectral range is covered, finally the multiple spectra obtained from the SMI constitute an ultra-high resolution spectrum.

Fig. 1 Optics of the UHRIS combining a FPI and a static SMI: (a) Basic optical layout and (b) Equivalent perspective view.

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For the normal incidence and vacuum medium between the FPI plates, the maximum transmission equation of the FPI is given by

2 d = m λ .

where

d

is the plate spacing of the FPI,

λ

is the wavelength of light, and m is the order that is an integer.

The free spectral range (FSR) in wavenumber of the FPI is given by

F S R_{σ} = \frac{1}{2 d} .

The FSR in wavelength of the FPI is given by

F S R_{λ} = \frac{λ^{2}}{2 d} .

For the normal incidence and vacuum medium between the FPI plates, the transmittance function of the FPI can be written as [12,37]

T_{F P I} (σ, d) = \frac{{(1 - R)}^{2}}{1 + R^{2} - 2 R \cos (4 π σ d)} .

where

σ

is the wavenumber (

σ = 1 / λ

),

R

is the reflectance of the FPI plates.

The reflective finesse of the FPI is given by

F_{r} = \frac{π \sqrt{R}}{1 - R} .

For the normal incidence and vacuum medium between the FPI plates, the interferogram of the UHRIS can be expressed as

\begin{array}{l} I_{U H R I S} (σ, d, x) = \int_{0}^{\infty} B (σ) \cdot T_{F P I} (σ, d) \cdot [1 + \cos (2 π σ x)] d σ \\ = \int_{0}^{\infty} \frac{{(1 - R)}^{2} B (σ) [1 + \cos (2 π σ x)]}{1 + R^{2} - 2 R \cos (4 π σ d)} d σ . \end{array}

where

x

is the optical path difference of the SMI, and

B (σ)

is the input spectral intensity at a wavenumber

σ

.

First of all, according to the desired spectral resolution $δ σ_{U H R I S}$ and spectral range $σ_{\min} \leq σ \leq σ_{\max}$ , select a particular FPI spacing $d_{0}$ , let free spectral range $F S R_{σ (0)} = 1 / (2 d_{0})$ be equal to $P$ times the spectral resolution (in wavenumber) $δ σ_{S M I}$ of the SMI at a center wavenumber $σ_{0}$ , i.e., $F S R_{σ (0)} = P \cdot δ σ_{S M I}$ , so the spectral resolution of the SMI is given by

δ σ_{S M I} = \frac{1}{2 P d_{0}} .

where the value of

P

is determined by making the spectral broadening in the SMI negligible as compared with

F S R_{σ (0)} = 1 / (2 d_{0})

. In general

P \geq 3

and

P

does not need to be an integer.

Secondly, the FPI needs to scan $N (P < N \leq F_{r})$ steps in order to cover the full spectral range of $σ_{\min} \leq σ \leq σ_{\max}$ , and this scanning is generally carried out by using a piezoelectric device. The free spectral range $F S R_{σ (0)} = 1 / (2 d_{0})$ needs to be scanned $N (P < N \leq F_{r})$ times with a scanning interval $δ σ_{U H R I S}$ in wavenumber, so the wavenumber position of center spectral peak moves from $σ_{0} - F S R_{σ (0)} / 2$ to $σ_{0} + F S R_{σ (0)} / 2$ , namely, the wavenumber position of center spectral peak moves from $σ_{0} + (- N / 2) \cdot δ σ_{U H R I S}$ to $σ_{0} + (N / 2 - 1) \cdot δ σ_{U H R I S}$ . Thus, the spectral resolution of the UHRIS can be expressed as

δ σ_{U H R I S} = \frac{F S R_{σ (0)}}{N} = \frac{1}{2 N d_{0}} .

where

P < N \leq π \sqrt{R} / (1 - R)

and, for convenience, let

N

be an even number.

According to Eqs. (7) and (8), it can be obtained that

δ σ_{U H R I S} = \frac{P}{N} δ σ_{S M I} .

The resolving power of the UHRIS can be written as

R_{U H R I S} = \frac{λ}{△ λ} \approx \frac{λ}{F S R_{λ} / N} = m N = \frac{2 N d}{λ} .

At the first FPI spacing position $d_{1}$ (i.e., at the first scanning position of the FPI), the maximum transmission wavenumbers of the FPI are $σ_{\min} \leq σ = m / (2 d_{1}) \leq σ_{\max}$ (m is an integer), a first spectrum is obtained from Fourier transform of the first interferogram obtained from the SMI. At the $k -th$ FPI spacing position $d_{k}$ (i.e., at the $k -th$ scanning position of the FPI), where $- N / 2 \leq k \leq N / 2 - 1$ , the maximum transmission wavenumbers of the FPI are $σ_{\min} \leq σ = m / (2 d_{k}) \leq σ_{\max}$ (m is an integer), the $k -th$ spectrum is obtained from Fourier transform of the $k -th$ interferogram obtained from the SMI. At the $N -th$ FPI spacing position $d_{N}$ (i.e., at the $N -th$ scanning position of the FPI), the maximum transmission wavenumbers of the FPI are $σ_{\min} \leq σ = m / (2 d_{N}) \leq σ_{\max}$ (m is an integer), the $N -th$ spectrum is obtained from Fourier transform of the $N -th$ interferogram obtained from the SMI. A total of $N$ spectra are obtained from the SMI, and these $N$ spectra constitute an ultra-high resolution spectrum.

The displacement of the $k -th$ FPI plate spacing from the spacing $d_{0}$ is given by

Δ d_{k} = \frac{- k d_{0}}{m N + k} = \frac{- k d_{0}}{2 d_{0} σ_{0} N + k} .

where

- N / 2 \leq k \leq N / 2 - 1

.

The spectral resolution of the SMI is also determined by

δ σ_{S M I} = \frac{1}{2 \cdot x_{\max}} .

where

x_{\max}

is the maximum optical path difference of the SMI.

Based on Eqs. (7) and (12), it can be obtained that

x_{\max} = P d_{0} .

According to the Nyquist-Shannon sampling criterion [38,39], for a spectral bandwidth $Δ σ = σ_{\max} - σ_{\min}$ , the sampling interval is given by

χ < \frac{1}{2 Δ σ} .

The height of the small steps is $h_{1}$ , and the small step mirror M1 has $s_{1}$ steps. The tall step mirror M2 has $s_{2}$ steps, and the height of the tall steps is $h_{2} = h_{1} \times s_{1}$ . The sampling interval for each of the $N$ interferograms produced by the SMI is

χ = 2 \times h_{1} .

The measurement spectral bandwidth of the SMI is calculated by

Δ σ \leq \frac{1}{4 h_{1}} .

The number of sampling points for each of the N interferograms produced by the SMI is

K = s_{1} \times s_{2} .

The maximum optical path difference of the SMI is $x_{\max} = χ \times K + 2 l_{0} = 2 \times h_{1} \times K + 2 l_{0}$ , where $l_{0}$ is the distance between the image of stepped mirror M2 to beam splitter and the stepped mirror M1 as shown in Fig. 1. Let $l_{0} = 0$ , it can be obtained that

K = \frac{P d_{0}}{2 h_{1}} .

Table 1 shows the main advantages and disadvantages of the UHRIS compared with the other two types of spectrometers with resolving power higher than 1,000,000 in short-wave infrared region.

Table 1. Comparisons of the UHRIS with two other types of spectrometers with resolving power higher than 1,000,000 in near-infrared, short-wave infrared or mid-wave infrared region

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It is well known that the throughput (Jacquinot) [40,41] and multiplex (Fellgett) [42,43] advantages are the two basic benefits of interferometric over dispersive spectrometers in the infrared region. The FPI has the throughput advantage as well as the Michelson-type interferometers used for Fourier transform spectrometer in the infrared region [7]. The combination of a FPI and a static SMI makes the UHRIS capable of achieving high optical throughput and therefore achieving high signal-to-noise ratio in the infrared spectral region.

Compared with the combination of a FPI and a fixed narrowband filter, the UHRIS can obtain much higher spectral resolution. Compared with the combination of a FPI and a reflection grating, the UHRIS can obtain higher spectral resolution and higher signal-to-noise ratio in the infrared spectral region [7,44,45].

3. Preliminary numerical simulation with two examples

When the reflectance of the FPI plates is $R = 0.95$ , the reflective finesse of the FPI is $F_{r} = π \sqrt{R} / (1 - R) \approx 61.2$ , so the number of scanning steps of the FPI is $N \leq 61$ .

3.1. The first example

The first example is that a source spectra covers a central wavenumber $σ_{0} = 6250 {cm}^{- 1}$ (wavelength 1.6 µm) and a spectral bandwidth $Δ σ = 30 {cm}^{- 1}$ (a wavenumber range from 6235 cm⁻¹ to 6265 cm⁻¹), and the desired spectral resolution is $δ σ_{U H R I S} = 0.005 {cm}^{- 1}$ . Based on Eqs. (7) and (8), some key parameters of the UHRIS are shown in Table 2. In order to cover the full wavenumber range of 6235 cm⁻¹ to 6265 cm⁻¹, from Eq. (11), the FPI plate spacing is approximately assigned from 1.9999616 cm to 2.00004 cm, namely, the FPI should scan N = 50 steps of approximate 16 nm in step interval. Due to assembling constraints and limitations, the sampling interval cannot be lower than 100 µm and the small step height cannot be lower than 50 µm [35], so the measurement spectral bandwidth of the SMI cannot be greater than 50 cm⁻¹. The main parameters of the SMI in the UHRIS are shown in Table 3. According to Eq. (10), the resolving power of the UHRIS for the wavelength range from 0.8 µm to 2 µm is shown in Fig. 2.

Table 2. Some key parameters of the UHRIS for the first example

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Table 3. Main parameters of the SMI in the UHRIS for the first example

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Fig. 2 Resolving power of the UHRIS for the wavelength range from 0.8 µm to 2 µm when $d_{0} = 2 cm$ and $N = 50$ .

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Suppose that a source spectrum contains only wavenumber 6249.745 cm⁻¹, 6249.75 cm⁻¹, 6249.755 cm⁻¹, 6249.995 cm⁻¹, 6250 cm⁻¹ (1.6 µm), 6250.005 cm⁻¹, 6250.245 cm⁻¹, 6250.25 cm⁻¹ and 6250.255 cm⁻¹. Thus, the FPI needs to scan three steps with approximate 16 nm step interval, and three interferograms produced by the SMI of the UHRIS are shown in Fig. 3. The first interferogram ( $d = 1 .9999984 cm$ ) contains only wavenumber 6249.755 cm⁻¹, 6250.005 cm⁻¹ and 6250.255 cm⁻¹. The second interferogram ( $d = 2 cm$ ) contains only wavenumber 6249.75 cm⁻¹, 6250 cm⁻¹ and 6250.25 cm⁻¹. The third interferogram ( $d = 2 .0000016 cm$ ) contains only wavenumber 6249.745 cm⁻¹, 6249.995 cm⁻¹ and 6250.245 cm⁻¹. Figure 4 shows the spectrum obtained from Fourier transform of the three interferograms in Fig. 3, and the spectral resolution is 0.005 cm⁻¹.

Fig. 3 Three interferograms produced by the SMI of the UHRIS with a spectral resolution 0.005 cm⁻¹.

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Fig. 4 Spectrum obtained from Fourier transform of the three interferograms in Fig. 3.

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Table 4 shows some parameters of both the UHRIS and a standard Michelson interferometer for the same spectral resolution 0.005 cm⁻¹.

Table 4. Some parameters of both the UHRIS and a standard Michelson interferometer for the same spectral resolution 0.005 cm⁻¹

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3.2. The second example

The second example is that a source spectra covers a central wavenumber $σ_{0} = 4000 {cm}^{- 1}$ (wavelength 2.5 µm) and a spectral bandwidth $Δ σ = 30 {cm}^{- 1}$ (a wavenumber range from 3985 cm⁻¹ to 4015 cm⁻¹), and the desired spectral resolution is $δ σ_{U H R I S} = 0 .004 {cm}^{- 1}$ . Some key parameters of the UHRIS for the second example are shown in Table 5. In order to cover the full wavenumber range from 3985 cm⁻¹ to 4015 cm⁻¹, the FPI plate spacing is approximately assigned from 2.4999400 cm to 2.5000625 cm, namely, the FPI should scan N = 50 steps of approximate 25 nm in step interval. Let $P = 4$ , the spectral resolution of the SMI of the UHRIS for the second example is still $δ σ_{S M I} = 0 .05 {cm}^{- 1}$ , and therefore the main parameters of the SMI in the UHRIS for the second example are still shown in Table 3.

Table 5. Some key parameters of the UHRIS for the second example

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Suppose that a source spectrum contains only wavenumber 3999.796 cm⁻¹, 3999.8 cm⁻¹, 3999.804 cm⁻¹, 3999.996 cm⁻¹, 4000 cm⁻¹ (wavelength 2.5 µm), 4000.004 cm⁻¹, 4000.196 cm⁻¹, 4000.2 cm⁻¹ and 4000.204 cm⁻¹. The FPI needs to scan three steps with approximate 25 nm step interval, and three interferograms produced by the SMI of the UHRIS for the second example are shown in Fig. 5. The first interferogram ( $d = 2 .4999975 cm$ ) contains only wavenumber 3999.804 cm⁻¹, 4000.004 cm⁻¹ and 4000.204 cm⁻¹. The second interferogram ( $d = 2 .5 cm$ ) contains only wavenumber 3999.8 cm⁻¹, 4000 cm⁻¹ (2.5 µm) and 4000.2 cm⁻¹. The third interferogram ( $d = 2 .5000025 cm$ ) contains only wavenumber 3999.796 cm⁻¹, 3999.996 cm⁻¹ and 4000.196 cm⁻¹. Figure 6 shows the spectrum obtained from Fourier transform of the three interferograms in Fig. 5, and the spectral resolution is 0.004 cm⁻¹.

Fig. 5 Three interferograms produced by the SMI of the UHRIS with a spectral resolution 0.004 cm⁻¹ (the second example).

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Fig. 6 Spectrum obtained from Fourier transform of the three interferograms in Fig. 5.

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The resolving power of the UHRIS for the wavelength range from 0.8 µm to 2.5 µm for the second example is shown in Fig. 7.

Fig. 7 Resolving power of the UHRIS for the wavelength range from 0.8 µm to 2.5 µm when $d_{0} = 2 .5 cm$ and $N = 50$ (the second example).

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Table 6 shows some parameters of both the UHRIS and a standard Michelson interferometer for the same spectral resolution 0.004 cm⁻¹.

Table 6. Some parameters of both the UHRIS and a standard Michelson interferometer for the same spectral resolution 0.004 cm⁻¹

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According to Eq. (10), by adjusting the values of $d$ (the plate spacing of the FPI) and $N$ (the number of scanning steps of the FPI), the UHRIS can be scaled to achieve resolving power higher than 1,000,000 in near-infrared (NIR) or mid-wave infrared (MWIR) region.

4. Conclusion

The principle of a compact UHRIS is described in detail, the results of preliminary numerical simulation with two examples are shown, and the comparisons of the UHRIS with two other types of ultra-high resolution spectrometers are given (see Tables 1, 4 and 6). Compared with Michelson-type interferometers used for ultrahigh resolution spectral measurements, the UHRIS has much smaller physical size, higher stability and shorter measurement time. Since the SMI of the UHRIS is static, does not require a drive system and acquires a spatial sampling of the interferogram, compared with the combination of a FPI and a Michelson-type interferometer, the UHRIS has much shorter measurement time, higher stability and compactness. Due to the use of the SMI, there is also a tradeoff for the UHRIS, i.e., the spectral range is narrow. The UHRIS is a unique concept to provide resolving power higher than 1,000,000 in near-infrared, short-wave infrared or mid-wave infrared region while achieving short measurement time and small physical size.

Funding

National Natural Science Foundation of China (NSFC) (61605151).

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	main advantages of the UHRIS	disadvantages of the UHRIS
UHRIS versus Michelson-type interferometers	much smaller physical size, higher stability, shorter measurement time	spectral range is narrow
UHRIS versus the combination of a FPI and a Michelson-type interferometer	much shorter measurement time, higher stability, more compact	spectral range is narrow

	main advantages of the UHRIS	disadvantages of the UHRIS
UHRIS versus Michelson-type interferometers	much smaller physical size, higher stability, shorter measurement time	spectral range is narrow
UHRIS versus the combination of a FPI and a Michelson-type interferometer	much shorter measurement time, higher stability, more compact	spectral range is narrow

Compact ultrahigh resolution interferometric spectrometer

Abstract

1. Introduction

2. Principle

3. Preliminary numerical simulation with two examples

3.1. The first example

3.2. The second example

4. Conclusion

Funding

References

Cited By

Figures (7)

Tables (6)

Equations (18)

Optics Express

	The UHRIS	Michelson interferometer
spectral resolution	0.005 cm⁻¹	0.005 cm⁻¹
central value for plate spacing of the FPI in the UHRIS	$d_{0} = 2 cm$
maximum optical path difference of the SMI in the UHRIS	$x_{\max} = 10 cm$
maximum optical path difference of the standard Michelson interferometer		100 cm

	The UHRIS	Michelson interferometer
spectral resolution	0.004 cm⁻¹	0.004 cm⁻¹
central value for plate spacing of the FPI in the UHRIS	$d_{0} = 2 .5 cm$
maximum optical path difference of the SMI in the UHRIS	$x_{\max} = 10 cm$
maximum optical path difference of the standard Michelson interferometer		125 cm