Useof theSagnac interferometer for spectroscopynowhas nearly a three-decade history (e.g., [1–6]), evenleading to its use in space as a hyperspectral imager[7]. This application of the Sagnac interferometer ispart of a larger class of Fourier transform spectro-meters (FTSs) that sample an interference pattern si-multaneously by using a detector array, rather thantemporally as in the Michelson interferometer.Interferometer spectrometers in general share
characteristics that differ substantially from disper-sive or filter-based spectrometers. Among the mostimportant from a performance standpoint are themultiplex, or Fellgett, advantage and the through-put, or Jacquinot, advantage [8–10]. The Fellgett ad-vantage arises from the simultaneous measurementof all spectral bands (today shared by array disper-sive spectrometers), while the Jacquinot advantagearises from the fact that an FTS does not require
a slit for its measurement, and so admits substan-tially more light into the system. “Advantage” inthese cases is something of a misnomer and only ap-plies to systems that are dominated by read noise;that is, photon noise is negligible, as will be discussedfurther below. However, in the read-noise-dominatedcase the advantage can be substantial.
An important and maturing class of read-noise-dominated detector is the microbolometer array [11].At present these arrays are large in pixel count(arrays up to 640 × 480 are widely available) andin physical size (more than 2 cm2). Their advertisedsensitivities hover near 85mK at 30Hz frame ratesand f -number near 1, though in our experience per-formance is significantly better. The sensitivity ofthese arrays has already given rise to IR multispec-tral sensors. In terms of data return, the most prolificmicrobolometer multispectral imaging system is theRaytheon THEMIS sensor currently in orbit aroundthe planet Mars [12]. THEMIS is a step-filter time-delay and integration (TDI) system employing aRaytheon 320 × 244 microbolometer array that col-lects six 1 μmwide bands used to map the mineralogy
of nearly the entire surface of Mars at 100m spatialresolution.Recognizing both the increasing sensitivity of mi-
crobolometers and the advantages of FTSs over dis-persive or filter approaches for read-noise-dominatedsystems, we initiated a research program to investi-gate the suitability of spatial FTSs and microbol-ometers for long-wavelength IR (LWIR)multispectralandhyperspectral imaging [6]. In this paperwe reportresults obtained with our most recent prototype. Weshow that the signal-to-noise ratios (SNRs) obtainedare consistent with a very simple model of sensorperformance and demonstrate the advantage of thistechnique for spectroscopy.We present fieldmeasure-mentsmadewith the systemdirectly comparedwith adispersive hyperspectral imaging (HSI) system. Fi-nally, we present the implications of this technologyfor HSI using photon-noise-dominated detectors inthe LWIR (8–14 μm) spectral region.
2. Sensor
Our Sagnac interferometer realization is arranged ina box format with three fold mirrors arranged at 90°angles and a ZnSe beam splitter (Figs. 1 and 2). Thebox arrangement, as opposed to triangle path typi-cally used in Sagnac spectral interferometers, coun-terintuitively has a slightly shorter total path length,reducing vignetting. The box format using circular
optics has a roughly 5∶1 aspect ratio of total lengthto aperture [13].
The interferogram is produced from a smoothly,nearly linearly, varying optical path difference acrossthe detector array. The path difference is imposed by atilt of the beam following the reflected path throughthe interferometer relative to the transmitted path(Fig. 1). The beam tilt is imposed by a rotation ofthe beam splitter as opposed to an offset of a mirror[1,2] because we find that this technique does not sig-nificantly shift the zeropathdifferencewhen the slopeof the optical path difference is varied, somewhat sim-plifying instrument adjustment and alignment.
Spatial interferometers have been used both withand without slits. We use a Sagnac interferometerin a manner similar to that used by Horton [14] fora Mach–Zehnder interferometer to preserve theJacquinot advantage [15], a mode that does not usea slit or anamorphic optics. In this method, the sceneis viewed through the interferometer (similar to animaging Michelson interferometer), and the interfer-ence fringes are superimposed on the scene. The spec-tral data are reconstructed from relatively extensiveresampling of the data discussed below. The slit,
Fig. 1. (Color online) Schematic of the Sagnac interferometer lay-out. Top, in the interferometer, the input beam is divided intotransmitted (bold) and reflected (fine) paths. A slight rotationshears the beams, as it has a larger effect on the reflected path.Bottom, the scene is viewed through the interferometer with acamera.
Fig. 2. (Color online) Photograph of the Sagnac LWIRHSI sensor.The cover has been removed to reveal the interferometer cube(black object toward the bottom) and the camera (dark cylinder to-ward the top). For scale, the sensor is 13 in. (33 cm) high.
reimaging, and cylindrical lenses developed by [2]using a Sagnac interferometer remove the FTS ad-vantages, though data reduction is simpler with themethod of [2].The camera used to collect the data for this paper
is a FLIR (forward-looking IR) Photon 320 × 244 ele-ment microbolometer array running at 30Hz; wehave also used Indigo Merlin and BAE Systems cam-eras. Using the camera as delivered and a tempera-ture adjustable flat-plate blackbody, we measuredthe sensitivity at 21mK with its stock f =1:4 optics;the camera is substantially more sensitive than ad-vertised and produces very useful multispectral andhyperspectral performance.
3. Data Recovery and Reduction
Each pixel in a row samples a different optical pathdifference; so an individual frame does not containthe information to recover the spectrum of any sceneelement. The sensor is scanned across the scene toenable the system to sample all optical path differ-ences for each scene element [14]. If the sensor is wellaligned with the scan system and the scanning isvery regular with respect to the frame rate, data re-covery is very straightforward. However, our applica-tion is airborne imaging without sensor stabilization,so there is complex image motion in pitch, roll, andyaw. Minnett and Sellar [16] reported shipbornemeasurements of visible range spectral propertiesof arctic sea ice performed by using a Sagnac inter-ferometer operated in this mode taken under condi-tions of complex platformmotion, and they presentedpreliminary results showing color data recovery.In our experiments, because on average there is
only about 1 pixel offset between subsequent images(scan rates or airspeed is adjusted to this averagecondition), we exploit this redundancy to measurethe image–image correlation to develop a model ofthe sensor motion with respect to the scene. The off-sets in pitch, roll, and yaw are then used to resamplethe data set to align the interferograms. These offsetscould also be derived from an inertial measurementunit, but we have not used this method to deriveimage motion.Figure 3 shows data before and after the roll cor-
rection using the scene-derived image motion. Theroll correction simply slides each interferogram byusing a subpixel resampling algorithm according tothe roll error measurement (which is in units ofpixels). The pitch error is somewhat less obvious in-tuitively. Figure 4 shows a slice through the roll-corrected cube with the x axis being time (frame)and the y axis being the cross-track spatial axis.The wavy lines through the data are the path of in-dividual scene elements through the interferogram.The waves are due to aircraft pitch, and to a lesserdegree, variations in airspeed. The observed errorsare equivalent to variations in frame rate, so thepitch correction uses the pitch error measurementto resample the data to a constant apparent data col-lection rate, with the result shown in Fig. 4.
Once the data are aligned so that each scene ele-ment has a complete interferogram, processing fol-lows conventional methods. Low-spatial-frequencyartifacts are removed by fitting each interferogramwith a high-order polynomial, and subtracting thisfunction from the interferogram. A triangle apodiza-tion function is applied to each interferogram. Theinterferograms are converted to the spectral domainby using the method of Mertz [17]. Spectra arecalibrated to radiance by using observations of aflat-plate blackbody observed at two temperaturesbracketing the scene. Using the same apodizationand transformation to the spectral domain as forthe scene data, the blackbody observations are usedto construct a two-point correction to radiance.Wave-length calibration is accomplished by observing theblackbody at an elevated temperature (70°C) witha narrow-bandpass filter interposed. After transfor-mation to the spectral domain, the frequency of thefilter, and zero frequency, are used to compute thewavelengths.
4. Data Examples
We have collected data from both stationary plat-forms and from the air. One test compared resultsfrom the Sagnac Microbolometer LWIR HSI withdata simultaneously collected by using the Univer-sity of Hawaii airborne hyperspectral imager
Fig. 3. Top, image of raw airborne data over suburban Honoluluobtained with the interferometer. Bottom, image corrected byusing frame-to-frame correlation.
Fig. 4. Top, slice through the interference cube. The x axis is theframe and the y axis is the cross-track position. The wavy lines arethe track of scene elements while the aircraft undergoes pitch andvelocity variations. Bottom, the same data after resampling to aconstant apparent time sampling.
cryogenic HSI system from a fixed location on a bal-cony of a building on the University of Hawaii cam-pus. Figure 5 shows spectral signatures derived froma variety of target panels at a range of ∼30m fromthe two sensors illustrating the similarity. Figure 6shows a multispectral image derived from that40-band data set as well as an airborne multispectralimage.
5. Comparison of Theory and Measurement
After [8–10], the SNRs of an interferometer(SNRinterferometer) can be expressed as follows (we in-clude dispersive (SNRdispersive) and time-delay and in-tegration (SNRTDI) cases for comparison):
SNRinterferometer ¼Sλ
ffiffiffiffiffiN
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ R
Here Sλ is the signal accumulated by the spectralinstrument of any design in a single frame that con-tributes to an individual spectrum. In a spectrographor other dispersive device, this signal is the finalsignal, but in the Horton-type interferometer andTDI systems some coadding occurs, so the SNR is in-creased by the square root of the number of samples(N) used to contribute to the final signal. The noise isdue to the read noise R and the photon noise; in thecase of the interferometer the photon noise includesthe sum of all the photons received over all bands(Rλ Sλ). Considering the read-noise-dominated case
(whereRλ Sλ is much smaller than R), the SNRs are
SNRinterferometer;read ¼ffiffiffiffiffiN
p SλR
; ð4Þ
SNRdispersive;read ¼ Sλ=R; ð5Þ
SNRTDI;read ¼ffiffiffiffiffiN
p SλR
; ð6Þ
but NTDI ¼ N=M; therefore
SNRTDI;read ¼ffiffiffiffiffiNM
rSλR
: ð7Þ
The multiplex (Fellgett) advantage of the interfer-ometer is proportional to the number of samples used
Fig. 5. Top, brightness temperature spectra of a silicate panelshowing silicate reststrahlen obtained at a range of 30m usingboth the Sagnac interferometer (line with pluses) and the Univer-sity of Hawaii adaptive holographic interferometry LWIRHSI sys-tem (solid line). Bottom, spectra of a marble panel showingabsorption due to the carbonate ion. In the overlap region(9–11:5 μm) the two sensors return very similar signatures. Thesignatures were obtained at slightly different times; so tempera-tures will differ because of changing illumination.
Fig. 6. Top, multispectral (9, 10, 11 μm) image from the Sagnacdata shown in Fig. 5. The yellow panels are carbonates, the bluepanels are silicates. Bottom, airborne IR multispectral IR data ofsuburban Honolulu. Blue rooftops have clay tiles showing silicatereststrahlen features.
to construct the interferogram. The TDI system ap-pears comparable with the interferometer, but theTDI system divides the focal plane into M bands, re-ducing its multiplex advantage by the square root ofthe number of bands. Because N is large for Horton’sapproach (typically many hundreds), the SNR ad-vantage of the interferometer over the dispersive(e.g., grating) system is considerable.We apply this model to SNR measurements made
on the Sagnac HSI system. We compute the SNR ofthe sensor versus wavelength (frequency) by cali-brating a 100 frame observation of a blackbody at300K to spectral radiance, and computing the 1-sigma noise in units of noise equivalent spectralradiance. We then compute the SNR from the spec-tral radiance of the blackbody, divided by the noiseequivalent spectral radiance.To model the SNR we need both the read noise and
the signal in compatible units. After [11], the NEDT(noise equivalent delta temperature) is
NEDT ¼ VN ½4ðf -numberÞ2 þ 1�xτ0ADRðΔP=ΔTÞλ1−λ2
; ð8Þ
NEP ¼ VN=R; ð9Þ
so
NEP ¼ NEDTτ0ADðΔP=ΔTÞλ1−λ24ðf -numberÞ2 þ 1
; ð10Þ
where Vn is the root mean square noise voltage, f -number is the focal ratio of the optics (1.4), τ0 is theoptics transmission (assumed to be 95%), AD is thearea of the pixel (38 μm2), R is the responsivity, and(ðΔP=ΔTÞλ1−λ2) is the change in power with tempera-ture over the range 8–14 μm (2:62 × 10−4 W=cm2 K)[11]. The signal P in units of power is
Pσ ¼ LσΔσADτi; ð11Þ
where Lσ is the spectral irradiance in units ofW=ðm2 cm−1Þ, Δσ is the spectral resolution (10 cm−1),and τi is the interferometer transmission (assuming50%). Finally, SNR ¼ Pσ=NEP.The comparison is shown in Fig. 7. The model SNR
varies smoothly from about 300 at 8 μm to 700 at14 μm. This increase is consistent with the 300Kblackbody signal, which, in units of constant fre-quency, monotonically increases toward longer wave-lengths in this region. (In the more familiar units ofconstant wavelength, the peak is near 10 μm.) Themeasured SNR curve approaches the model curveat 12 μm, but deviates significantly at shorter andlonger wavelengths. Our model neglects the spectralefficiency of the interferometer (but does include thefact that at best only half of the light is transmittedby a Sagnac interferometer). Figure 8 shows the
transmission and reflectance of the beam splitter,and the modulation efficiency of the beam splitter,which is the lower of the reflected or transmittedterms. The dropoff at longer wavelengths is consis-tent with the deviation of the beam splitter from50∶50modulation. The dropoff in sensitivity at shortwavelengths appears to be due to the wavelength-dependent sensitivity of the Photon FLIR camera.Preliminary measurements using a BAE Systemscamera and the interferometer reported here resultsin high sensitivity to 7:5 μm.
6. Additional Model Results
The encouraging results using this uncooled detectorraises a question regarding the performance of thissystem using a cooled detector. The photon-noise-limited SNR is strongly dependent on the full-wellcapacity of the array. In the absence of instrumentthermal emission, the SNR relationships amongthe spectral methods are given by
where FW is the full-well capacity, and these equa-tions assume that the signal has been integratedto this level (this formalism is a subset of the presen-tation of [1,2], where the range of behavior betweenphoton noise and read noise dominance wasexamined.]
Fig. 7. Model and measured SNR ratios for the Sagnac LWIRHSI. The smooth curve on top is the model, the irregular curveis the measurement. Deviations of the model and measurementare consistent with the unaccounted beam splitter efficiencyshown in Fig. 8.
In this case the multiplex advantage has been sig-nificantly attenuated by the factor of the number ofbands of interest. For the case of a ten wavenumberresolution sensor (∼50 bands) computed from a 256element interferogram, the multiplex disadvantageis 0.3, and it is unity for a 16 band sensor, so the in-terferometer appears to be somewhat at a disadvan-tage, though amodest one. Two practical engineeringfactors mitigate this disadvantage. First, the inter-ferometer illuminates the detector with high-fluxbroadband radiation, so very deep well direct injec-tion arrays can be used, with full-well capacities ofover 20 × 106 electrons. Deep wells translate into ahigher SNR. In contrast, direct injection arrays donot work well at the low fluxes in cryogenic spectro-graphs, which must employ low background arrayswith much smaller full-well capacities, for example,near 4 × 106 electrons. This difference in full-well ca-pacity gives the interferometer an improvement ofabout a factor of 2 (there is ample signal to providenear-full-well flux for room temperature objects,even at high frame rates, for both methods). Second,because the interferometer views the scene throughlow-emissivity optics, the interferometer need not becooled, unlike a spectrograph. This could be a sub-stantial savings in mass and power over a cryogenicinstrument. The use of a cooled detector and un-cooled interferometer was validated in an imagingMichelson application [18].
To illustrate the potential of this method using acooled detector, Fig. 9 shows the SNR of a photon-noise-dominated (i.e., with a sensitive cryogenicphoton sensitive detector) spectral interferometerwith 10 wavenumber resolution, covering the range8–12 μm, assuming a maximum signal of 10 × 106
electrons on the array. The model shows steeply in-creasing SNR with wavelength, principally owing tothe strong slope of the signal curve that is in units ofphotons and unit frequency. Nevertheless the SNRsare respectable, and there is sufficient flux for exist-ing arrays to deliver this SNR at frame rates above300Hz. The response could be substantially flat-tened, and improved at short wavelengths, if opticalfiltering could smoothly attenuate the flux at longerwavelengths, which contributes to the noise at theshorter wavelengths with less signal.
7. Conclusions
Use of a Sagnac spatial interferometer for infraredHSI is demonstrated by using a microbolometer ar-ray detector. The performance of the sensor is well-represented by a simple parametric model, withSNRs in the range of several hundred and above thatare usable for many applications. The very modestcost of the components of this sensor could greatlyincrease the use of LWIR hyperspectral and multi-spectral imaging. The performance of a version ofthis sensor using a cooled detector but warm opticswould be higher, and capable of obtaining qualitydata at much higher rates, with a potentially morecompact and lower power instrument than disper-sive methods.
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