spatial heterodyne spectrometer: modeling and ...fixed mirror re- imaging lenses figure 1. the shs...

Spatial heterodyne spectrometer: modeling andinterferogram processing for calibrated spectral

radiance measurements

Cara P. Perkins, John P. Kerekes, Michael G. Gartley

Rochester Institute of Technology, 54 Lomb Memorial Drive, Rochester, NY, USA

ABSTRACT

This work presents a radiometric model of a spatial heterodyne spectrometer (SHS) and acorresponding interferogram-processing algorithm for the calculation of calibrated spectral radiancemeasurements. The SHS relies on Fourier Transform Spectroscopy (FTS) principles, and sharesdesign similarities with the Michelson Interferometer. The advantages of the SHS design, includingthe lack of moving parts, high throughput, and instantaneous spectral measurements, make itsuitable as a field-deployable instrument. Operating in the long-wave infrared (LWIR), the imagingSHS design example included provides the capability of performing chemical detection based onreflectance and emissivity properties of surfaces of organic compounds. This LWIR SHS modeloutputs realistic, interferometric data and serves as a tool to find optimal SHS design parametersfor desired performance requirements and system application. It also assists in the data analysisand system characterization. The interferogram-processing algorithm performs flat-fielding andphase corrections as well as apodization before recovering the measured spectral radiance fromthe recorded interferogram via the Inverse Fourier Transform (IFT). The model and processingalgorithm demonstrate results comparable to those in the literature with a noise-equivalent change intemperature of 0.35K. Additional experiments show the algorithm’s real-time processing capability,indicating the LWIR SHS system presented is feasible.

Keywords: hyperspectral, LWIR, FTIR, SHS, infrared, FTS, interferometry

1. INTRODUCTION

The Fourier Transform Spectrometer (FTS) has long been used in spectroscopic studies.1–5

The main advantage of the FTS over typical dispersive spectrometers is that it does not requirethe use of an entrance slit, but a wide, circular aperture instead. The term, optical throughput,describes the amount of space through which light can enter an optical system. The large apertureof the FTS provides a much higher optical throughput than the thin entrance slit of the dispersivespectrometer, allowing much more radiant energy to enter and, thus, increasing the signal-to-noiseratio.

The Spatial Heterodyne Spectrometer (SHS) is a specific type of FTS, known for having a veryhigh throughput. For this reason, it is often used to study emission lines.6,7 An SHS is very similarto another, more well-known FTS, the Michelson Interferometer (MI).5 A MI uses a beam-splitterto divide the incoming light into two interferometer arms, each containing a mirror. The mirrors

Further author information: (Send correspondence to C.P.P.)C.P.P.: E-mail: [email protected], Telephone: 1 978 400 1409J.P.K.: E-mail: [email protected], Telephone: 1 585 475 6996M.G.G.: E-mail: [email protected], Telephone: 1 585 475 5612

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reflect the two beams back through the beam-splitter to be recombined at the detector. Afterthe wavefronts recombine, we have interference as a function of the difference in the optical pathlength of the two beams. One mirrored-arm moves in increments to measure the interference ateach optical path-length difference (OPD) within the system range, such that the measurementis taken temporally. An SHS also uses a beam-splitter, but its interferometer arms are fixed andcontain diffraction gratings in place of mirrors, creating the interference pattern for all OPDs. TheSHS design collects entire interference measurements simultaneously and without any moving parts,making it more compact and robust.8

Most SHS designs operate in the visible and ultraviolet regions of the electromagnetic spectrumand have astronomical and atmospheric applications,9–11 but an SHS developed for the long-waveinfrared (LWIR) region is of particular interest. Organic compounds have distinct spectral signaturesin the thermal infrared (IR) region of the electromagnetic spectrum. Spectra collected by a LWIRSHS can be used to perform material detection processes. Although efforts have been madeto develop a long-wave infrared (LWIR) SHS,12,13 the particular area of research is, generally,understudied. The Naval Research Laboratory has designed the only successful LWIR SHS,13 theSpatial Heterodyne Imager for Chemical and Atmospheric Detection (SHIMCAD).

This research provides an SHS model in support of LWIR SHS research and general SHSinstrument design processes. The model can be used to choose design parameters for an instrument,given performance criteria or characterize a particular SHS design. The model generates realisticinterferometric data - interferograms, which are the Fourier Transform (FT) of the radiancespectrum of the observed scene. This work also provides an interferogram processing algorithm forthe conversion of the SHS interferograms into calibrated radiance spectra with real-time processingcapability. The model and processing algorithm may be used as generic tools for a variety of SHSdesigns and applications, but the results presented assume a LWIR SHS design.

2. SPATIAL HETERODYNE SPECTROSCOPY

Figure 1 compares the SHS to the MI - the difference between the two designs being the diffractiongratings in the SHS. Each diffraction grating creates a wave-front of dispersed wavenumbers. Thetwo wave-fronts recombine at the detector and create interference fringes, known as Fizeau fringes,which represent the real part of the FT of the source’s spectrum about the heterodyne wavenumber.The heterodyne wavenumber is that which produces parallel wave-fronts (i.e. the beam entersand exits the diffraction grating interface at the same angle) with zero spatial frequency, and isknown as the Littrow wavenumber. A row of detector elements collects the entire interferencepattern simultaneously.8 A detector array can also be used along with a platform scanner to collecthyperspectral imagery.

The SHS disperses the different wavenumber components of the input via the diffraction gratingequation:

k[sin (θL) + sin (θL − γ)] = m/d, (1)

where k is the wavenumber of the incident light, θL is the Littrow angle (a property of the diffractiongratings), γ is the angle that the output wave-front makes with the normal to the detector, m isthe order of diffraction, and 1/d is the density of the grating grooves.14 The Littrow wavenumber,kL, is found by solving for k when γ is set to zero.15 We can then calculate the Littrow angle ofthe diffraction gratings, θL, the necessary diffracting grating width, W , for a desired number of



Fixed Mirror

Re- ImagingLenses

Figure 1. The SHS (B.) design is almost identical to that of the MI (A.). The main difference is that themirrors in the MI are replaced by diffraction gratings tilted by the Littrow angle in the SHS.12

interferogram samples, N , the maximum position, xmax, along the detector, the spectral resolution,dk, and the spatial sample spacing, dx.12

The wavenumber-dependent Fizeau fringes of the interferogram have spatial frequencies:

νF = 4(k − kL) tan (θL). (2)

This gives a number of fringes per centimeter on the detector. By the Nyquist theorem, we mustsample at at least two times the number of fringes per centimeter to recover a non-aliased signal.16

The interferogram as a function of position, x, along the detector is given by:

I(x) =

∫ ∞−∞

1

2L(k)[1 + cos (2π · 4(k − kL) tan θLx)]dk, (3)

where L(k) is the spectral radiance measured at the input and L(−k) is the same function as L(k)but flipped.15 As in all FTS instruments, we can use the inverse Fourier transform (IFT) of theunbiased interferogram to recover L(k).

3. SHS MODEL AND INTERFEROGRAM SIMULATION

3.1 SHS Model Outline

Figure 2 shows our general, simulated SHS system. The sample spectrum provides the systeminput, the SHS measures its interferogram, and the algorithm processes the data to best-recoverthe input spectral radiance. Depending on the SHS application, post-processing may be needed.

The SHS Model is described in blocks:

1. System Input

2. Entrance Optics

3. Interferometer



GeneratedSpectra

Recorded fSpectra

Entrance Inter -Optics r--- ferometer --> Exit Optics

Display Data Processing

Results Analysis (ifany)

F Algorithm,Quantization

Detection

F

i

Inter-ferogram

Figure 2. The entire SHS system consists of the input, or the scene spectral radiance, the SHS itself, andprocessing software to determine the measured spectral radiance.17

4. Exit Optics

5. Detection

6. System Output

3.2 Input

The user-defines the SHS operational parameters:

• minimum accepted wavenumber, kmin

• maximum accepted (Littrow) wavenumber, kL

• number of interferogram samples, N

• diffraction grating order, m

• diffraction grating groove density, d

Using the above parameters, the model calculates all additional necessary SHS parameters definedin Section 2. The model generates a high resolution interferogram with NH number of samples,which is re-sampled at N samples by the detection scheme.

3.3 Entrance Optics

The first block corresponding to the SHS is the Entrance Optics block, where we incorporate theinstrument self-emission, field of view (FOV), and optical transmission. The instrument self-emissionis modeled as a blackbody (BB) radiator with unit emissivity and temperature, T1. The output ofthe Entrance Optics is:

L′(kn) = Ω · τ1(k)Lsamp(kn) + Lself,1(kn, T1), (4)

where Ω is the system field-of-view, τ1(k) is the transmission function of the Entrance Optics, kn isthe discrete variable for wavenumber, Lsamp(kn) is the input sample spectrum, and Lself,1(kn, 290)is the self-emission term of the entrance optics.



3.4 SHS Interferometer

The next block is the SHS Interferometer block, where we account for the diffraction gratingefficiencies and any vibrational and phase errors. The measured radiance at the detector is:

L′′(k) = (1

2ηA +

1

2ηB)L′(k), (5)

where ηA and ηB are the efficiencies of diffraction gratings A and B, respectively. The vibrationscause the Fizeau frequency to become:

νF = 2(k − kL) tan θL + 2(k − (kL + ke)) tan (θL + θe), (6)

where θe is the angle of error between the Littrow angles of the two gratings and ke is the Littrowwavenumber associated with the diffraction grating Littrow angle, θL + θe. The high-resolutioninterferogram calculated in the Interferometer block is:

I(xm) =

NH/2−1∑n=−NH/2

1

2L′′(kn)[1 + cos (2πνF (xm + xe) + φ(k, x))]dkH , (7)

where xm is the discrete variable for position on the detector, ranging from −xmax to +xmax, kn isthe discrete wavenumber variable within the user-specified wavenumber range, and xe and φ(k, x)are the errors associated with vibrations along the detector and phase shifts, respectively.

3.5 Exit Optics

The high resolution interferogram is then passed through the Exit Optics, which includes asecond transmission function and another self-emission term, also modeled as a BB but with atemperature, T2. The output of the Exit Optics is:

I ′(xm) = τ2,avg(k)I(xm) + Ωτ2(k)avg∑n

Lself,2(kn, T2)dk, (8)

where τ2,avg(k) is the mean of the Exit Optics transmission.

3.6 Detection

The last block in the SHS model is the Detection block, where we re-sample the high resolutioninterferogram and add noise. The re-sampling occurs at the user-specified number of samples,N , spectral resolution, dk, and sample spacing, dx. The Detection block also accounts for thesensitivity variation across the detector and the system noise, such that the output becomes:

I ′′(x) = SFPA(x)I ′resampled(x) + Inoise(x), (9)

where SFPA(x) describes the pixel-to-pixel sensitivity variation and Inoise(x) is the system noise inthe interferogram domain.

We calculate a standard deviation for each noise source in the system and add it to our measuredsignal in a Gaussian fashion. Our measured interferogram, I(x), has irradiance units of W/m2,



which we convert to a number of electrons incident on the detector.18 We calculate the noisestandard deviations for shot noise, calibration noise, bit error rate noise, quanitization noise,18

detector and readout noise, and electronics noise.5 Once the noise has been added to the signalusing the Gaussian distribution, we convert the total signal back to irradiance units.

4. INTERFEROGRAM PROCESSING CONCEPTS

After the SHS model outputs the realistic, interferometric data, the interferogram processingalgorithm performs the following tasks:

• Flat-fielding correction

• Phase correction

• Bias removal

• Apodization

• Inverse Fourier Transform

• Noise reduction

• Spectral Radiance Calibration

4.1 Flat-fielding

Flat-fielding is the process of correcting the interferometric data for variations in the pixel-to-pixel sensitivity across the detector. These variations are usually constant of the detectorand need only to be characterized after the instrument is built. There are a number of differentflat-fielding techniques.19 Our method uses two measurements of the same source - one made withone interferometer arm blocked and the second made with the opposite arm blocked. We divide ourcollected interferograms by the unit-normalized mean of the two single-arm measurements. Theflat-field corrected interferogram is then:

Iff (x) =I(x)(

IA(x)+IB(x)2

)normalized

, (10)

where I(x) is any measured interferogram and IA(x) and IB(x) are the measurements of the knownsource taken while blocking arms B and A, respectively.

4.2 Phase Correction

There are two different phase errors to be corrected in the processing algorithm - a wavenumber-dependent and a wavenumber-independent error. The former can be characterized once theinstrument is built.19,20 The latter causes the shift of the interferogram center-burst from the zeropath difference (ZPD) location. We can correct this phase error by implementing the FT shifttheorem.

If the center-burst is located a distance, +xc from the x = 0cm, then we can shift theinterferogram back to ZPD by:



Ip(x) = IFTFTI(x) · FTδ(x+ xc), 16 (11)

where the FT of δ(x+ xc) is exp (2π · xc · x). Ben-David et al.21 derive an approximation of thelocation of xc by:

xc = dx

[∑i i|I(xi)|∑i |I(xi)|

− i0], (12)

where i0 is the current sample index of the apparent ZPD in the recorded interferogram. Anestimate of i0 can be found by locating the index of the maximum of the recorded interferogram.

4.3 Bias Removal

From Equation 3, we see that the measured interferogram is the sum of modulated andunmodulated terms. The modulated term (the cosine term) is the interferogram, by definition,whereas the unmodulated term is called the bias and may be neglected. We determine theinterferogram bias by averaging the samples on either side of the interferogram center-burst. Wethen subtract it from the measured interferogram so that we are left with the modulated term only.

4.4 Apodization

The measured interferogram is cut off abruptly at the −xmax and +xmax values due to thedetector’s finite size. This is often referred to as signal truncation. Truncation causes non-zeroamplitudes belonging to the lower frequencies located at the higher frequencies after we performthe IFT.16 To reduce the effects of signal truncation we can multiply the measured interferogramby an apodization function that slowly decays to zero at the endpoints. We perform apodizationwith the Hamming window, as in the SHIMCAD system.13 The Hamming window reduces ourspectral resolution by a factor of about 1.82 due to its wide full-width half-maximum (FWHM) inthe spectral domain.22

4.5 Inverse Fourier Transform

As we have said, taking the IFT of the measured interferogram gives the measured spectrum.We expect the recovered spectral radiance to be a real and even function because the interferogramis actually the FT of the mirrored spectrum. In many cases, the recorded interferogram is notperfectly symmetric and the IFT is not a real and even function. This causes an induced phase errorwe need to correct by taking the magnitude of the spectrum after the IFT to maintain the spectrumpower.16 We also need to scale the IFT to account for the instrument optics and resolution suchthat the raw recovered spectral radiance is:

Lrecovered(kn) =2

Ωdkn|IFT (I(xm))|, (13)

where the factor of 2 accounts for the fact that as the diffracted beams travel back through thebeam-splitter, only half the total radiance reaches the detector, while the other half travels backtoward the system aperture. Finally, as we said above, the measured interferogram is actually theFT of the mirrored spectrum. We can discard the first N/2 samples because we only need the halfof the recovered spectral radiance that corresponds to +k.



4.6 Noise Reduction

Some filters, such as the mean and median filter,16 can help reduce noise in the data. Normally,such filtering would take place at this point in the processing algorithm. The noise reduction methodused in this work, however, is sample averaging, which is performed as the raw interferograms arerecorded.

4.7 Spectral Radiance Calibration

The spectrum recovered by the IFT is the measured spectral radiance, which is affected bythe instrument self-emission terms and optical functions. We need to calibrate the recoveredspectral radiance to remove the effects of the system offset (the total instrument self-emission) andresponsivity (the total optical efficiency as a function of wavenumber, not to be confused with thedetector responsivity). The total measured spectral radiance is:

L(k) = r(k)[Lscene(k) + g(k)] (14)

and if we know the responsivity, r(k), and system offset, g(k), we can calibrate the measuredspectral radiance to the scene spectral radiance by:

Lscene(k) =L(k)

r(k)− g(k). (15)

The calibration calculations are usually performed within the instrument on a measurement-by-measurement basis using built-in hot and cold BB sources. The hot BB has a temperaturejust above the ambient temperature, while the cold BB has a temperature just below ambienttemperature. If the respective radiance measurements of the hot and cold BB’s are LH(k) andLC(k) at temperatures, TH and TC , then the responsivity is:

r(k) =LH(k)− LC(k)

LBB(TH , k)− LBB(TC , k), (16)

where LBB(TH , k) and LBB(TC , k) are the expected BB radiation measurements of a hot and coldBB at temperatures TH and TC , respectively. The instrument offset is:

g(k) =LC(k)

r(k)− LBB(TC , k). (17)

The calibrated spectral radiance can then be found by substituting the calculated responsivity andoffset terms into Equation 15.1



5. RESULTS AND DISCUSSION

The example SHS design in this work is a LWIR SHS with a spectral range of 950 to 1250cm−1.Additional design parameters are listed in Table 1. We incorporate the example design plans in ourSHS model and simulate realistic interferograms. The model calculates the other SHS parametersnecessary to run the program in MATLAB (Table 2). The next few sections show the results of themodel and interferogram processing algorithm using this SHS design.

Table 1: The user-specified operational parameters for the SHS model.

kmin [cm−1] 950kL [cm−1] 1250

N 128m 1

d [cm−1] 143

Table 2: The additional parameters calculated by the SHS model using the user-specified operationalparameters.

NH 600θL [] 3.2791W [cm] 0.9324xmax [cm] 0.4654dkH [cm−1] 1.00dxH [cm] 0.0016dk [cm−1] 4.6875dx [cm−1] 0.0073

5.1 Spectra Recovery Results

As we have said, one LWIR SHS already exists: SHIMCAD. Enlgert et al.13 measured thetransmittance of methanol gas samples, with pressures ranging from 6.9 to 59.9hPa, placed in frontof a 100C blackbody with the SHIMCAD model. We simulate similar measurements with oursystem using methanol transmittance spectra at 9.6hPa provided by the Pacific Northwest NationalLaboratory (PNNL) Infrared Spectral Library.23 Figure 3 shows the results. For comparison, wehave included the SHIMCAD results for a methanol transmittance sample at 6.9hPa in Figure 4.Our spectral recovery result appears to be comparable to the SHIMCAD result, even though ourspectral resolution is only 4.7cm−1 while the SHIMCAD’s spectral resolution is 2cm−1. This resultdemonstrates that our simulated SHS system achieves results that one would expect with a realLWIR SHS.

5.2 Design Optimization Results

Design optimization refers to the selection of design parameters that best achieve certainperformance requirements. Typically, during the design optimization process, we are interested insystem trade-offs. Some system trade-offs to consider are: spectral range versus spectral resolution,diffraction grating width versus spectral resolution, and throughput versus SNR.



0.9

t 0.8

0.7

g 0.6E.

0.5

0.4

40

E 35

C.4.) 30

25

20

Sample Transmittance

1000 1100 1200

Wavenumber (1 /cm)

Shift- Coisected Inteiferogram

-0.5 0

x (cm)

0.5

40

C.4.) 30

25

20

0.9

t 0.8

0.7

g 0.6E.

0.5

0.4

Measured SHS Inteiferogramn Spectral Signal -to -noise Ratio

4-----.,-_

0.5 0 0.5

x (cm)

Calibrated Transmittance

'11\T

1000 1100 1200

Wavenumber (1/cm)

650

600

0 550

500

450 -

1000 1100 1200

Wavenumber (1 /cm)

Noise -equivalent Change in Temperature

0.34

0.33

0.32

0.31

0.3

1000 1100 1200

Wavenumber (1/cm)

a, 1.0U0 0.8' 0.6to

v= 0.4~ 0.2

00900 1000 1100 1200

Wavenumber [cm's]

Figure 3. Results of the methanol (9.6hPa) test at a 4.7cm−1 spectral resolution: in clockwise rotation fromtop left, the input methanol gas transmittance spectrum, measured interferogram, wavenumber-dependentSNR, shift-corrected interferogram, recovered spectrum, and wavenumber-dependent NEdT.

Figure 4. Results of the SHIMCAD measurement of methanol transmittance (6.9hPa) at a 2cm−1 spectralresolution.

The spectral resolution, dk, is inversely proportional to the spectral range, ∆k. Depending onthe application, having a higher spectral range may be more important than having a higher spectralresolution (when collecting hyperspectral data for image classification, for example) or vice versa(like when studying emission or absorption lines). Unlike the spectral range, the number of samples- and, therefore, the diffraction grating width - increases with the spectral resolution. Althoughthis relationship makes it easy to increase the resolution without sacrificing the spectral range, itputs a constraint on the portability of the instrument. Increasing the size of the optical elementsmakes the instrument larger and heavier and negates the SHS advantage of having a compact sizeand mechanical robustness. Another main advantage of the SHS is the optical throughput. Thelarger the throughput is, the higher the SNR will be. We can increase the throughput by increasingthe instrument aperture and/or FOV. Again, we have a trade-off between an improved signal andinstrument size. These trade-offs are depicted for our SHS system in Figure 5.



Spectral Range vs. Resolutions4

3ae

1 2FVÓ11e

4 6 8 10 12 q 00 2 4 6 8

Spectral Resolution (1 /cm) Spectral Resolution (1/cm)Throughput vs. Average SNR

Diffraction Grating Width vs. Resolution

110

105

100

r 95

90

3500.2 0.4 0.6 0.8

Throughput (gin zsr)

10

Figure 5. Results of the system trade-offs between spectral range and resolution (A), diffraction gratingwidth and spectral resolution (B), and throughput and SNR (C).

5.3 System Characterization Results

We can characterize an SHS system by determining its performance metrics as functions ofsystem sources of error. Some sources of error to consider are: Littrow angle error, θe, phase error,φ(k, x), calibration error rate, and detector position error, xe.

The system performance metric we consider here is the separation angle between the input andrecovered spectra determined by the spectral angle mapping algorithm. The separation angle isthe performance metric that best describes how the shapes of the two spectra compare, and iscalculated by:

α = cos−1[

Lrecovered(k) · Linput(k)

||Lrecovered(k)|| · ||Linput(k)||

], (18)

where Linput(k) is the input spectral radiance corresponding to a particular recovered radiance,Lrecovered(k), Lrecovered(k) · Linput(k) is their dot product, and ||Lrecovered(k)|| and ||Linput(k)||are their magnitudes. Ideally, we want a separation angle of 0 between our input and recoveredspectra.

The separation angle as a function of the listed system errors is depicted in Figure 6. Recallthat the Littrow angle in our example design is 3.2791 and the interferogram sample spacing, dxis 0.0073cm−1. In all four cases, the separation angle between the input and recovered spectralradiance increases with the system error.

We can also observe the average spectrum recovery percent accuracy as a function of SNR(Figure 7). Figure 7 is a very useful plot if, for example, we need to achieve a specific overall systemaccuracy rate. This plot helps to indicate our desired SNR, which will help us in choosing theintegration time or number of sample averages, etc.

5.4 Model Validation

This research provides two tools for SHS studies: the SHS model and the interferogram processingalgorithm. A useful SHS model outputs realistic data and a useful interferogram processing algorithm



40

Average Separation Angle

10

1.2

1.1

1

0.9

0.8

0.7

0.05 0.1 0.15 0.2

Littrow Angle Error (degrees)Average Separation Angle

33

3

E 23mbd 2

6 13


0.05 0.1 0.15 0.2 0.25

Phase Error (radians)


03

0.005 0.01 0.015 0.02 0.025

Calibration Error Rate

2

1

03 1 13 2 23 3 33Vibrational (x) Error (on) x 10

Average Spectrum Recovery Accuracy vs. SNR100

° 95

t 90d

850200 400

SNR

600 800

Figure 6. Results of the separation angle as a function of the Littrow angle error (A), phase error (B),calibration error rate (C), and position, x, error (D).

Figure 7. Results of the average spectrum recovery percent accuracy as a function of SNR.

has real-time capability. There are multiple performance metrics we can use to describe our system:noise-equivalent change in temperatute (NEdT) and SNR, for example. We expect an SNR in thehundreds and an NEdT within 0.1 to 0.5K for LWIR systems.24 From the plot in Section 5.1, weknow that we are achieving an average SNR of about 500 and an NEdT of about 0.32K. Theseinitial results demonstrate the validity of our realistic SHS model.

5.5 Real-time Processing Capability

The SHS is suitable as a portable, field-deployable instrument. For on-the-move measurements,specifically, it is required to perform real-time interferogram processing of the SHS data. Thealgorithm is implemented using MATLAB Release 2011a on a MacBook Pro with an Intel Core i5processor at 2.3GHz. The processor is found to operate at 117 mega floating-point operations persecond (MFLOPS) using NovaBench software. The Tesla C2075 GPU by Nvidia operates at 515GFLOPS.25 The Tesla GPU provides a processing speed increase factor of 4,444. The algorithmruns for about 0.2 seconds per interferogram on the MacBook Pro. Theoretically, the algorithm



would only take about 45 microseconds per interferogram on the Tesla GPU. Assuming that ourSHS platform is moving at 30mph with a pixel ground sample distance of 7.5cm, an integrationtime of 500 microseconds, and an image size of 128 by 128 pixels, we would need the algorithm toconvert one row of pixel interferograms in less than one second. Using the Tesla GPU, we shouldbe able to convert an entire image of interferograms into calibrated spectral data in about 0.74seconds, achieving our desired processing speed. This particular GPU is not the fastest on themarket and it is also possible to use multiple GPUs at one time. Therefore, it is reasonable to saythat the interferogram processing algorithm has real-time capability.17

6. CONCLUSIONS

The field of spatial heterodyne spectroscopy is still fairly new. Specifically, the area of LWIRspatial heterodyne spectroscopy is rather understudied. To encourage and assist LWIR SHS studies,this research provides a general, radiometric SHS model and interferogram processing algorithmthat can be used to analyze LWIR designs as well as any other SHS designs. The results presentedare for a LWIR design and indicate mostly accurate spectral recovery with realistic quantitiesfor SNR and NEdT. We demonstrate the use of the model and algorithm to perform designoptimization and system characterization as well as validate the model with methanol transmittancespectra, achieving results comparable to SHIMCAD, the only LWIR SHS. Finally, our interferogramprocessing algorithm proves to have real-time capability. Overall, our model and algorithm simulaterealistic, useful results that demonstrate the feasibility of a LWIR SHS design with accurate spectralrecovery results.

ACKNOWLEDGMENTS

Portions of the project or effort depicted herein were/are sponsored by the U.S. Army underSubcontract No. SC60721-1811-001 and W911SR-13-C-0005. The content of the information doesnot necessarily reflect the position or the policy of the Government and no official endorsement isor should be inferred.

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spatial heterodyne spectrometer: modeling and ...fixed mirror re- imaging lenses figure 1. the shs...

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