Spatial heterodyne spectrometer based on the Mach–Zehnder interferometer

Optics Communications 355 (2015) 239–245

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Spatial heterodyne spectrometer based on the Mach–Zehnder interferometer Qisheng Cai a,n, Bin Xiangli a,b, Shusong Du b a b

Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, Anhui 230027, China Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing 100094, China

art ic l e i nf o

a b s t r a c t

Article history: Received 16 April 2015 Received in revised form 21 June 2015 Accepted 28 June 2015

Spatial heterodyne spectroscopy (SHS) is a new kind of Fourier-transform spectroscopic technique capable of very high spectral resolution. In this paper, a spatial heterodyne spectrometer based on the Mach–Zehnder interferometer (MZ–SHS) is proposed. It is modified by replacing one mirror in the Mach–Zehnder interferometer with a diffraction grating. This technique retains many of the advantages of traditional SHS. Moreover, the spatial frequency of the interferogram is strictly linear with wavenumber. We describe the concept of the new MZ–SHS and elaborate the exact expression of the interferogram. Also, a design example and two kinds of imitated interferograms are presented in this paper. One is simulated in MATLAB and the other is generated in ZEMAX using ray tracing method. The retrieved spectra from these two interferograms show a good agreement with the theoretical results. & 2015 Elsevier B.V. All rights reserved.

Keywords: Spatial heterodyne spectroscopy Fourier transform Interferometer Simulation

1. Introduction Spatial heterodyne spectroscopy (SHS) is a new kind of Fouriertransform spectroscopic (FTS) technique. It retains the advantages of FTS [1], such as simultaneous measurement of all wavelengths (multiplex or Felgett advantage), intrinsic wavelength calibration and the best instrumental line-shape of any spectrometer (Connes advantages). Besides that, SHS is capable of very high spectral resolution near a selected wavenumber [2], and there are no moving parts in it. The first SHS instrument was developed by Roesler and Harlander in 1990s [3,4]. It is based on a Michelson interferometer modified by replacing the mirrors in each arm with diffraction gratings [5,6]. This technique has been used in many areas of applications. For instance, the Spatial Heterodyne Imager for Mesospheric Radicals (SHIMMER) was designed for remote sensing of the global distribution of the hydroxyl radical OH in the Earth’s middle atmosphere [7]. The first generation SHIMMER instrument flew as a Middeck experiment on Space Shuttle STS-112 in October 2002 [8], and the second generation which used a monolithic interferometer was launched on board the Space Test Program Satellite-1 (STPSat-1) on March 9, 2007 [9]. Recently, Harlander and Englert et al. have developed a Doppler Asymmetric Spatial Heterodyne (DASH) spectroscopy technique for upper atmospheric wind and temperature observations [10,11]. This will be n Correspondence to: Room 802, Main Building of Academy of Opto-Electronics, No.9, DengZhuangNan Road, Haidian District, Beijing city, Beijing 100094, China. Fax: þ86 1082178674. E-mail address: [email protected] (Q. Cai).

http://dx.doi.org/10.1016/j.optcom.2015.06.068 0030-4018/& 2015 Elsevier B.V. All rights reserved.

used in the Michelson Interferometer for Global High-resolution Thermospheric Imaging (MIGHTI) which is one of the four instruments on the NASA Ionospheric Connection (ICON) Explorer mission scheduled for launch in 2017 [12]. SHS is very suitable for high spectral resolution, narrow spectral band applications. As the bandpass is limited by the sampling number of the detector [2], many technologies have been proposed to extend the spectral band [13,14]. In fact, SHS technique can work at broad spectral band conditions without changing any elements except for a lower spectral resolution. In the basic theory of SHS, there is a linear approximation for the dependence of spatial frequency of the interferogram with wavenumber. This approximation is justified for high spectral resolution, narrow spectral band applications. However, at low spectral resolution, the non-linear relationship becomes evident and this should be handled in the data analysis by making the wavenumber axis nonlinear. In this paper, we propose a new spatial heterodyne spectrometer based on the Mach–Zehnder interferometer (MZ–SHS). It retains the high resolving power and robustness advantages of SHS. Moreover, the relationship between spatial frequency and wavenumber is strictly linear. Thus, it is suitable in broad spectral band applications without any non-linear data processing. In Section 2, we review the basic theory of SHS. The principle of MZ– SHS is elaborated in Section 3. Section 4 presents a design example and the simulation of MZ-SHS. The conclusions are drawn in Section 5.

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2. Basic theory of SHS A schematic diagram of the basic SHS configuration is show in Fig. 1. It is based on a Michelson interferometer modified by replacing the mirrors in each arm with diffraction gratings [5]. Light transmitted through the collimating lens (L1) becomes a parallel beam, then it is divided into two beams by the beam splitter (BS). The gratings (G1 and G2) in each arm return the beams back to the beam splitter. The two diffracted beams generate an interferogram, and the localization plane of the interferogram is at the gratings. Lens L2 images the pattern onto the detector. In the basic SHS configuration, the two gratings are set at a fixed angle θ with respect to the optical axis, and they have the same distance from the beam splitter along the optical axis. The on-axis rays are incident on the gratings with an angle θ . The diffracted rays for a certain wavelength return back parallel to the optical axis. This wavelength is called the Littrow wavelength. For the Littrow wavelength, the wavefronts of the two beams exiting from the interferometer are perpendicular to the optical axis, and they are completely in phase with each other, producing an interferogram with zero frequency. For a wavelength slightly different from the Littrow wavelength, the diffracted beams return with a small angle ±γ with respect to the optical axis. The recombined wavefronts are crossed with an angle 2γ , and the optical path difference (OPD) of the two wavefronts remains zero at the center of the aperture and reaches a maximum at the edges. The off-Littrow angle γ , related to different wavelength, is determined by the grating equation

σ[sin θ + sin(θ − γ )] = m/d,

I (x) =

∞

{

}

B(σ ) 1 + cos[8π (σ − σ0)x tan θ ] dσ .

ΔU = 2W sin θ ,

(3)

(4)

where W is the width of the gratings. The maximum spectral resolution is

1 1 = 2ΔU 4W sin θ

δσ =

(5)

and the resolving power is

R=

σ = 4Wσ sin θ . δσ

(6)

When the spectral resolution is fixed, the non-aliased spectral range of the spatial heterodyne spectrometer is limited by the number of samples across the interferogram. As the interferogram is imaged on the detector, this sampling number is equal to the pixel number of the detector. If the detector has N pixels, the nonaliased spectral range is

Δσ =

(2)

∫0

Inverse Fourier transform of the interferogram is used to recover the input spectrum. In fact, the recovered spectrum by Eq. (3) is B (σ − σ0 ), and the spectral range is σ0 ± Δσ . The spectral resolution of SHS is determined by the maximum OPD, which is

(1)

where σ is the wavenumber of the incident light, m is the order of diffraction, θ is the Littrow angle, and 1/d is the grating groove density. The spatial frequency of the interferogram is related to the wavenumber of light by

fx = 2σ sin γ ≈ 4 (σ − σ0 ) tan θ ,

where σ0 is the Littrow wavenumber and σ0 = m /2d sin θ . This approximation assumes small γ . For input spectrum B (σ ), the intensity of the interferogram as a function of position x is given by

N ∙δσ . 2

(7)

Under high spectral resolution conditions, the passband of SHS is small and hence the linear approximation given in Eq. (2) is justified. The analyses above is based on the on-axis rays. In fact, to obtain a measurable amount of radiation through the spectrometer, a finite entrance aperture is needed. By including off-axis input angles in the grating equation (Eq. 1), it can be shown that the change in spatial frequency with off axis angle limits the useable solid angle Field-of-View (FOV) Ω of SHS instruments to [4]

Ω=

2π . R

(8)

This is the same value achieved by conventional FTS and Fabry– Perot interferometers.

3. Principle of MZ–SHS

Fig. 1. Schematic diagram of basic SHS configuration. It is based on a Michelson interferometer modified by replacing the mirrors in each arm with diffraction gratings.

Fig. 2 presents a sketch of the optical layout of the proposed MZ–SHS. MZ–SHS is modified by replacing one mirror in the Mach–Zehnder (MZ) interferometer with a diffraction grating. In the modified MZ interferometer, the grating and the mirror are placed at symmetric positions with respect to the beam splitter. The grating is tilted by an angle α . The light transmitted through the collimating lens (L1) becomes a parallel beam, and then it is divided into two beams at the beam splitter. For the on-axis rays, one beam is reflected back to the beam splitter by the mirror, and then exits the interferometer with its wavefront perpendicular to the optical axis, indicated by the exiting wavefront E2. The other beam is incident on the grating with an angle 2α . For a particular wavelength, the diffraction angle is zero, and this wavelength is diffracted parallel to the optical axis. We call it the reference wavelength. The wavefront of the reference wavelength exiting from the interferometer is also perpendicular to the optical axis. The

Q. Cai et al. / Optics Communications 355 (2015) 239–245

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the aperture and reaches the maximum at the edges. Also, the localization plane of the interferogram is at the grating. Lens L2 images the interferogram onto the detector. In addition, a diaphragm is set at the grating to adapt the apertures of the two beams. The cross angle β is determined by the grating equation

σ (sin 2α − sin β ) = m/d,

(9)

where α is the tilt angle of the grating, and β is the diffraction angle. The spatial frequency of the interferogram is related to the wavenumber of the incident light by

fx = σ sin β = (σ − σr ) sin 2α,

(10)

where σr is the reference wavenumber and σr = m /d sin 2α . Comparing Eq. (10) with Eq. (2), there is no any approximation here, which means the relationship between the frequency and the wavenumber is strictly linear. This ensures that non-linear data processing is avoided in the broad spectral band applications. The intensity of the interferogram for incident spectral density B (σ ) is:

I (x) = Fig. 2. Optical layout of MZ–SHS. It is modified by replacing one mirror in the Mach–Zehnder interferometer with a diffraction grating.

∫0

∞

{

}

B(σ ) 1 + cos[2π (σ − σr )x sin2α ] dσ .

Again, inverse Fourier transform of the interferogram is used to recover the input spectrum. Similar to SHS, the resolving power of MZ–SHS is determined by the maximum OPD of the two beams. The OPD of MZ–SHS can be calculated from the geometry of Fig. 3.The mirror is set at its symmetric position of the beam splitter. As the mirror and the grating have the same distance from the beam splitter, the symmetric mirror and the grating are crossed at their centers. The angle between the grating and the mirror is α . The OPD between the rays diffracted by the grating (dashed line) and reflected by the mirror (solid line) is

Δu = AB + BC = OB∙ tan α(1 + cos2α ) = OB∙ sin2α. Fig. 3. OPD of MZ–SHS. The dashed line represents the ray diffracted by the grating and the solid line represents the ray reflected by the mirror.

(11)

(12)

If the width of the grating is W , the maximum achievable OPD is

ΔU =

1 W sin2α. 2

(13)

The maximum spectral resolution is

δσ =

1 1 = 2ΔU W sin 2α

(14)

and the resolving power is

R=

→ Fig. 4. Off-axis rays of MZ–SHS. The wavevector of the input off-axis ray is k , the → → diffracted wavevector is k 1, and the reflected wavevector is k 2. ψ and ψi (i = 1, 2) denote the angles between the on-axis rays and the component of the off-axis rays in the dispersion plane (x–z plane).

exiting wavefront E2 and the reference wavefront produce an interferogram with zero frequency. For a wavelength slightly different from the reference wavelength, it is diffracted by the grating with a small diffraction angle β , as E1 shows. The recombined wavefronts are crossed with β , and the OPD is zero at the center of

σ = Wσ sin2α. δσ

(15)

Comparing the resolving power of MZ–SHS with that of the SHS, if the reference wavenumber is equal to the Littrow wavenumber of SHS, sin 2α is about twice as much as sin θ . The resolving power of MZ–SHS is about half of the SHS. This is because we only use one grating in the MZ–SHS. The spectral range of MZ–SHS is also determined by the pixel number of the detector, as Eq. (7) indicates. To determine the FOV of MZ–SHS, the off-axis calculation should be performed. We refer to the FOV analysis for SHS in Ref. 4. Fig. 4 shows the off-axis rays diffracted by the grating and reflected by the mirror respectively. The mirror is set at its symmetric position of beam splitter which is the same as in Fig. 3. Also, a coordinate system is shown in this figure. The origin coincides with the point where the images of the mirror and the grating cross as viewed looking back into the instrument. The y axis is along the grooves of the grating and the z axis is along the normal

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Fig. 5. FOV of MZ–SHS and SHS. The shadow region is for MZ–SHS and the circular region centered at (0,0) is for SHS. As ψ , ϕ are assumed to be small, the acceptable region for MZ–SHS should lie at a small region near (0,0). (a) is for tan α > 2/R and (b) is for tan α ≤ 2/R .

Table 1 MZ–SHS system parameters. Item

Design

Groove density Diffraction order Grating width Detector Incident angle Reference wavelength Spectral range Spectral resolution

1200lines/mm 1 28 mm 2048 pixels α = 22.536° 590 nm 572.55–590 nm 0.017 nm (0.5044 cm  1)

→ of the grating. The wavevector of the input off-axis ray is k , the → → diffracted wavevector is k 1, and the reflected wavevector is k 2. ψ and ψi (i = 1, 2) denote the angles between the on-axis rays and the component of the off-axis rays in the dispersion plane (x–z → plane). The angle between the wavevector k and the dispersion plane is ϕ which is not shown in Fig. 4. → For the outgoing rays, k i can be expressed in the coordinates as

k xi = 2πσ cos ϕi sin ψi

(16a)

k yi = 2πσ sin ϕi

(16b)

k zi = 2πσ cos ϕicos ψi .

(16c)

The interferogram generated by the two outgoing rays is

I (x) =

∫0

∞

⎧ ⎡⎛ → → ⎞ →⎤⎫ B(σ )⎨1 + cos⎢⎜k1 − k2⎟∙r ⎥⎬dσ . ⎠ ⎦⎭ ⎣⎝ ⎩

(17)

The grating equation for the off-axis rays is given by

σ cos ϕ⎡⎣ sin(2α + ψ ) − sin ψ1⎤⎦ = m/d.

(18)

For the diffracted rays, we can get

sin ψ1 = sin (2α + ψ ) −

σr sin 2α σ cos ϕ

ϕ1 = − ϕ.

(19a) (19b)

For the reflected rays, we can get

ψ2 = ψ

(20a)

ϕ2 = − ϕ.

(20b)

Fig. 6. MZ–SHS model set up in ZEMAX. A build-in collimated light source is used, thus no collimating lens is in this model.

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Fig. 7. Interferogram of the monochromatic source (589 nm): (a) simulated in MATLAB and (b) traced in ZEMAX.

Fig. 8. Recovered spectra of the monochromatic source: (a) from the simulated interferogram in MATLAB and (b) from the traced interferogram in ZEMAX.

As the localization plane of the interferogram is at the grating, where z = 0, and ϕ1 = ϕ2 = − ϕ , Eq. (17) can be rewritten as

I (x) =

∫0

∞

B(σ ) 1 + cos ⎡⎣(k x1 − k x2)x⎤⎦ dσ .

{

}

(21)

Expanding Eq. (19a) to second order in angles, we can get

⎡ ⎛ ϕ2 + ψ 2 ⎞⎤ + ψ tan α⎟ ⎥. k x1 − k x2 ≈ 2π sin 2α ⎢σ − σr − σ ⎜ ⎢⎣ 2 ⎝ ⎠ ⎥⎦

(22)

Eq. (22) indicates that the off-axis rays with angle (ψ ,ϕ ) introduce a phase shift comparing to the on-axis rays. The maximum phase shift at the longest path difference should be less than π , which implies that

⎛ ϕ2 + ψ 2 ⎞ + ψ tan α⎟ ≤ π . 2πσx max sin 2α⎜ 2 ⎝ ⎠

(23)

Considering the resolving power of MZ–SHS (Eq. 15), we can rewrite Eq. (23) as

−

ϕ2 + ψ 2 1 1 ≤ + ψ tan α ≤ . R 2 R

(24)

This is a circular equation, and the radius of the inner circle is

r1 =

tan2 α − 2/R ,

r2 =

tan2

the

radius

α + 2/R . If tan α >

of

the

outer

circle

is

2/R , the angles for (ψ , ϕ) should lie

between these two circles. If tan α ≤ 2/R , The inner circle disappears. The dashed circles centered at O1 (− tan α , 0) in Fig. 5 (a) and (b) shows these two conditions. As ψ , ϕ are assumed to be small, the acceptable region should lie between these two circles at a small region near (0,0). The regions faraway from (0,0) are invalid. Comparing to the FOV of SHS, the solid angle Ω ≈ π (ψ 2 + ϕ2), Eq. 8 can be rewritten as

ϕ2 + ψ 2 1 ≤ . 2 R

(25)

This is also a circular equation centered at (0,0), and its radius is

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Fig. 9. Interferogram of the low pressure sodium lamp: (a) simulated in MATLAB and (b) traced in ZEMAX.

Fig. 10. Recovered spectra of the low pressure sodium lamp: (a) from the simulated interferogram in MATLAB and (b) from the traced interferogram in ZEMAX.

SHS. To improve the sensitivity of MZ–SHS, we can set a smaller α to increase the useable FOV. There is another notable difference between SHS and MZ–SHS. As the basic SHS is based on a Michelson interferometer, half of the light is reflected back to the source. While, MZ–SHS is a dual output system. We can record the fringes at both exits of the beam splitter to improve the energy efficiency and the signal-to-noise ratio.

Fig. 11. Interference curve of the on-axis rays (solid line) and a marginal off-axis rays (dashed line). The phase shift at the maximum OPD is π .

r=

2/R . This circle is shown in Fig. 5 as well. The outer circle of

MZ–SHS and the circle of SHS are crossed at (0,

2/R ) and

(0, − 2/R ). When tan α ≤ 2/R , the FOV of MZ–SHS is comparable with SHS. But this means a lower resolution (a small R ) and a small α . Under normal conditions, the resolving power is very high and tan α > 2/R . The acceptable FOV of MZ–SHS is in a narrow band near (0,0). It is small compared to that of SHS. This will reduce the throughput of MZ–SHS and make it less sensitive than

4. Design example and simulation of MZ–SHS In this section, we show a design example and a MZ–SHS model is set up in ZEMAX. Two kinds of imitated interferograms are presented. One is simulated in MATLAB according to Eq. (11) and the other is generated in ZEMAX using ray tracing method. Two light sources are used as the input spectra in the simulation. One is a monochromatic light (589 nm) and the other is a low pressure sodium lamp whose spectrum has two peaks at 589.0 nm and 589.6 nm. Comparing the interferograms obtained in MATLAB and ZEMAX, the correctness of the principle analyzed above can be verified.

Q. Cai et al. / Optics Communications 355 (2015) 239–245

In our design example, the grating has a groove density of 1200lines/mm and the width is 28 mm. The reference wavelength is 590 nm, and the incident angle on the grating is α = 22.536°. Using Eq. 14, we can get the spectral resolution is δσ = 0.5044 cm  1. If the pixel number of the detector is 2048, we can figure out that the spectral range is 572.55–590 nm according to Eq. (7). The system parameters are listed in Table 1. The MZ–SHS model in ZEMAX is show in Fig. 6. There is no collimating lens in this model because a build-in collimated light source in ZEMAX is used. For the monochromatic source, the interferogram simulated in MATLAB is shown in Fig. 7(a), and the interferogram traced in ZEMAX is shown in Fig. 7(b). Comparing these two interferograms, we can find that they have the same spatial frequency. Moreover, by computing the inverse Fourier transform of the interferogram, we can get the recovered spectra, which are shown in Fig. 8(a) and (b). Both interferograms from MATLAB and ZEMAX retrieve the input spectrum correctly. For the low pressure sodium lamp source, the interferogram simulated in MATLAB is shown in Fig. 9(a), and the interferogram traced in ZEMAX is shown in Fig. 9(b). Again, we can find that these two interferograms show the same pattern. The recovered spectra of these two interferograms are shown in Fig. 10 (a) and (b). The results imply that the two peaks of the sodium lamp spectrum are separated correctly. There are about 35 points between these two peaks, as the gap between the two peaks is 0.6 nm, we can calculate that the spectral interval for each point is about 0.017 nm, which agrees with the spectral resolution in theory. To verify the FOV of MZ–SHS analyzed in Section 3, the interferogram of the marginal off-axis rays is traced in ZEMAX. The angles of the marginal off-axis rays are calculated according to Eq. (24). One marginal off-axis angle, for instance, is ψ = − 0.004°, and ϕ = 0.625°. Fig. 11 shows the interference curve of the on-axis rays (solid line) and the marginal off-axis rays (dashed line) at 589 nm. The 0th pixel is the center of the detector where the OPD is zero, and the 1024th pixel is the edge of the detector where the OPD is maximum. It shows that the phase shift at the maximum OPD is π . This agrees with the theoretical results.

5. Conclusions In this paper, a novel spatial heterodyne spectrometer based on the Mach–Zehnder interferometer is presented and the basic principle of this spectrometer is studied. A design example is given and the MZ–SHS model is set up in ZEMAX. The interferogram traced in ZEMAX shows a good agreement with that simulated in MATLAB. The retrieved spectrum of the low pressure sodium lamp recovers the two peaks (589 nm and 589.6 nm) correctly which demonstrates the high resolving power of MZ–SHS. From Eq. (10), we can find that the relationship between the frequency of the interferogram and the wavenumber is strictly linear. Thus this spectrometer is suitable for broad spectral band applications without any non-linear data processing.

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MZ–SHS has several advantages: (1) The non-linear problem is avoided in the broad spectral band conditions. (2) It remains the high spectral resolving power capability as it is still a spatial heterodyne spectrometer. (3) There are no moving parts in it, and the optical elements are simple and easy for assembly. (4) As a dual output system, all the energy entering into the system can be recorded.

Acknowledgements This work is supported by China National Funds for Distinguished Young Scientists of NSFC under Grant 61225024.

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