Xin Cheng,1,2 Jing Wang,1,* Qingsheng Xue,1 Yongfeng Hong,1 and Shi Li3
1Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences,Changchun 130033, China
2Graduate University of Chinese Academy of Sciences, Beijing 100039, China3Institute of Graphics and Image, Hangzhou Dianzi University, Hangzhou 310018, China
The imaging spectrometer is a well-established tech-nique for achieving the spatial and spectral informa-tion of the target simultaneously. One of theimportant features of anext-generation imaging spec-trometer is wide field of view (FOV), which has re-ceived less attention until recently. A wide fieldimaging spectrometer that explored the Earth atmo-sphere from lowEarth orbithadanFOVaswideas70°[1]; another one designed for coastal ocean detectionhad a large FOVof 36° [2]. However, themain reasonswide FOV can be obtained for them are low operationheight and short focal length, as we know that, with agiven format of focal plane array (FPA), short focallength usually allows an imaging spectrometer toachieve wide FOV. In general, the FOVof an imagingspectrometer is primary restricted by the format ofthe FPA. With a larger-sized one, a wider FOV could
be obtained, just as the Maritime HyperspectralImager designed at the Naval Research Laboratory,which had a 4° field from 600km altitude [3]. Conse-quently, increasing the size of the FPA by geometri-cally splicing multiple FPA components in thespatial direction will eventually increase systemFOV. According to Cook’s description, an ultrawideFOV imaging spectrometer based on reflective tripletform has been successfully built at Raytheon [4]. Theslit format is 65mm, and dozens of slits and FPA com-ponents were utilized to realize the wide FOV. How-ever, with the enlargement of the FOV, the dimensionandweight of the spectrometer are increased, and thespectral distortion, typically termed smile and key-stone, are also increased. Wide FOV can be realizedin a number of other ways includingmultibarrel [5,6],multispectrometer [7], and even multiple imagingspectrometers using field splicing [8]. Among them,the middle one shows an enormous potential in wideFOV application, which proposes behind relays onsplitting the huge FOV in smaller portions, each ofthem reimaging onto a reasonable size detector
through an individual spectrometer. In this arrange-ment, although the distortion would be controlledreadily, the volume and weight does not decreaseessentially.
In general, the optomechanical configuration de-termines the dimension and weight of the remotesensing instrument; thus, increasing the FOV physi-cally will induce a bulky structure, especially for theimaging spectrometer, which also generates a greatspectral distortion. Here we show an FOV-folding ap-proach that can significantly enlarge the FOV of theimaging spectrometer but still make them compactand meanwhile keep the spectral distortion in micro-meter magnitude. This method uses several lineararrays of imaging fiber bundles to substitute forthe slit and connect fore-optics and spectrometerflexibly. Fiber bundles working as a luminous energytransmission component reformatting for spectro-meters to compose a nonimaging fiber-optic spectro-meter is very well known. It also was traditionallyused in medical imaging systems and remote inspec-tion devices such as flexible endoscopes involving aconfocal microscope as an image transmission com-ponent. In terms of the application in integral fieldspectroscopy (IFS), fiber bundles were used as a fieldsplitting component to transform a two-dimensionalgeometry at the focal plane of the telescope into theone-dimensional geometry at the entrance of thespectrograph [9,10]. The goal of using fiber bundlesin IFS is to record simultaneously three-dimensionalinformation of an extended object, while the goal ofour proposed technology is to extend the FOV on thecross-track direction when the imaging spectrometerworks in a pushbroom scan. When used in a nonima-ging system, the effect of the lattice structure of fiberbundle distribution on the spatial resolution neednot be considered; however, it cannot be neglectedin an imaging system. The fiber bundle is a well-known discrete sampling component, combining withthe detector to make the imaging fiber-optic spectro-meter (IFOS) become a double-sampling imagingsystem. For a complex sampling imaging systemwith a coupling component, the influence of fiber cou-pling on imaging performance must be taken intoconsideration. Therefore, it is necessary to calculatethemodulation transfer function (MTF) for the IFOS.In Section 2, we describe the FOV-folding techniqueusing fiber bundles; in Section 3 we analyze the effectof fiber coupling on the MTF of the imaging spectro-meter, and then build a cascade MTF model to eval-uate the imaging performance for the IFOS. Anillustrative example is given in Section 4.
2. FOV-Folding Method Using Fiber Bundles
For imaging spectrometer work in a pushbroom scan,the slit is projected on the ground and scanned for-ward with the platformmotion. The width of the scanin the cross-track direction is the cross-track field ofview (CFOV). As for the IFOS, the slit is replaced by alinear array of imaging fiber bundles; using thecharacteristics of flexibility and separability, we can
segment the wide CFOVof the telescope or multitele-scope into several smaller sub-CFOVs at the inputend on the imaging plane of the telescope(s), as showninFig. 1(a), and then fold andarrange themat the out-put end on the objective plane of the spectrometer inthe orthogonal direction to theCFOV [see Fig. 1(b)]. Itshould be noted that the intervals between each fiberbundle must be greater than the dispersive width ofthe spectrometer so that the spectral images of thesub-CFOVs do not overlap with the adjacent one.All sub-CFOVs are then imaged by the common spec-trometer and depend on wavelengths on a commondetector [see Fig. 1(c)]. After that, all of the spectralimages of the sub-CFOVs are spliced into an integerin sequence by using the technique of image stitching[see Fig. 1(d)]. To achieve the same CFOV, the novelarrangement allows a more compact structure thanthat of a slit imaging spectrometer because all sub-CFOVs of the telescope share a common spectrometerand a single focal plane array. Moreover, the spectraldistortion would be controlled easily since all the sub-CFOVs are imaged independently.
3. MTF Evaluation for IFOS
In general the shift invariant (or isoplanatic) as-sumption for the sampling imaging system is nolonger valid. The traditional approach defining theMTF of a sampled imaging system is to analyze theimpulse response by sampling and discrete Fouriertransform [11,12]. The main results involving sam-pling and imaging effects were not suited for the cas-cade system; thus, the traditional method of MTFanalysis cannot clearly account for the degenerationof the MTF caused by the effect factors. In a later sec-tion we will develop an MTF model for the IFOS,from which one can clearly find the effects on theMTF from the sampling process and fiber coupling.
For the imaging spectrometer designed for Earthobservation, in most cases, the spectrometers havea magnification of 1× so that the slit width ismatched with the size of the detector pixel. As for
Fig. 1. (Color online) Diagrammatic sketch of distribution of fiberor spectral images (a) on the imaging plane of the telescope (at theinput end of fiber bundles), (b) on the objective plane of the spectro-meter (at the output end of fiber bundles), (c) on the imaging planeof the spectrometer, and (d) after image stitching.
the fiber-based imaging spectrometer, the image offiber bundles (pseudo slit) is generally formed to cov-er two rows of pixels on the FPA. Therefore, in theperfect world, each fiber image for a certain wave-length would be fully matched with the correspond-ing 2 × 2 detector pixels and be sampled by them asshown in Fig. 2(a). However, the departure of the fi-ber image from the ideal location will be generated inthe presence of the spectrometer distortion, whichgives rise to a disorder to the sampling process ofthe detector, as shown in Fig. 2(b), and hence a degra-dation of imaging performance. It therefore is neces-sary to estimate the MTF for the IFOS. In Fig. 2(b),the vertical component of the departure is the well-known smile, and the horizontal component istermed “offset k,” with the maximum value markedas “offset kM ,” which are the products of the fact thatthe spectrometer is not distortion-free and its reduc-tion ratio is not strictly equal to 1.
For the single fiber the assumption of a linearspace invariant has been proved to be established[13] based on which MTF model is built up. Forthe sake of simplicity, the following analysis is lim-ited to the one-dimensional case, and the expressioncan be expanded to the two-dimensional case also. Ingeneral, two MTFs are involved in the sampling pro-cess [14]: the integral MTF as a result of pixel size,MTFint, and the fictitious sampling MTF caused byspatial sampling rate, MTFsam. Reference [13] indi-cates that the aperture function of the integral sur-face could be regarded as the fictitious point-spreadfunction (PSF) for a sampling component. In the caseof a perfect match [see Fig. 3(a)], the entire fiber coreis sampled by the corresponding 2 × 2 pixels; thus,the aperture function of core is the fiber PSF, andthe integral MTF of sampling process is given by
MTFðf Þa−int ¼ jFðf ÞjjGðf Þj; ð1Þ
where Fðf Þ and Gðf Þ are the Fourier transforms ofthe aperture function of fiber core and rectangulardetector pixel, respectively. Here they are well-known functions expressed as
Fðf Þ ¼ J1ð2πrf Þπrf ; ð2Þ
Gðf Þ ¼ sincðRf Þ; ð3Þwhere r and R are the radius of the cross sections ofthe fiber core and fiber, respectively, with pixel pitchequal to R, and J1 is the Bessel function of the firstkind and of the first order. Combining Eqs. (1)–(3), weobtain the integral MTF of a perfect match,
MTFðf Þa−int ¼J1ð2πrf Þ
πrf sincðRf Þ: ð4Þ
In the case of a mismatch shown in Figs. 3(b) and3(c), according to the theory about misalignment infiber bundles, the average MTF model of the sam-pling process of a detector is given by [15]
MTFðf Þm−int ¼ jFðf ÞjjG�ðf ÞjjPðf Þj; ð5Þ
where Pðf Þ is the Fourier transform of distributionfunction of stochastic departures from the match.Since the offset k and the smile described in Fig. 2(b)are far less than fiber bundle width, the offset ki ofthe ith couple of fiberandcorrespondingdetectorpixelcan be regarded as uniform distribution from zero tothe maximum value kM and described by
pðkiÞ ¼�1=kM ; 0 ≤ ki ≤ kM;0; elsewhere: ð6Þ
Equation(6)canbeexpressedinanotherformlikethis:
pðkiÞ ¼1kM
rect�ki − kM=2
kM
�: ð7Þ
Thus, the Fourier transform of pðkiÞ is
jPðf Þj ¼ sincðkMf Þ; ð8Þwith neglecting the parameter for the right term.Combining Eqs. (2), (3), (5), and (8), we determinethe average MTF of the sampling process to be
MTFðf Þm−int ¼J1ð2πrf Þ
πrf sincðRf ÞsincðkMf Þ: ð9Þ
It is to be noted that Eq. (9) accounts for roughevaluation when the setoff kM is greater than fiber
Fig. 2. Match state between the image of fiber bundle and FPAfor a certain wavelength (only half the fiber bundle is drawn; theother half gives symmetrical results): (a) perfectmatch and (b)mis-match (for clear illustration, each square includes 2 × 2 samplingpixels).
Fig. 3. Match state for a single fiber and corresponding pixelswhen (a) kM ¼ 0, (b) 0 < kM ≤ R − r, and (c) kM > R − r.
cladding thickness. There are some details that mustbe considered. In the case of mismatch but with theoffset kM less than fiber cladding thickness, as shownin Fig. 3(b), the offset just introduces a phase factorafter the Fourier transform, which does not contri-bute to MTF, and the entire core is still sampled bythe corresponding pixels; thereby the MTF is thesame as that of perfect match and expressed byEq. (4). However, for the case of the offset kM greaterthan fiber cladding thickness, as shown in Fig. 3(c),the accurate MTF expression,
MTFðf Þm−int ¼J1ð2πrf Þ
πrf sincðRf Þsinc½ðkM − Rþ rÞf �;ð10Þ
should be used to replace Eq. (9). The third term ofthe right polynomial in Eq. (10) can be regarded asthe mismatch MTF, while the product of other twoterms represents the integral MTF of perfect match.So the average integral MTF is expressed as Eq. (4)when the entire fiber core is sampled by the corre-sponding pixels and as Eq. (10) when the fiber coreis partially sampled. For fiber bundles and detector,both are sampling components and have their ownsampling MTFs, which are well known and respec-tively given by the sinc formulas:
MTFf−sam ¼ sincð2Rf Þ; ð11Þ
MTFd−sam ¼ sincðRf Þ: ð12Þ
Therefore, combining the Eqs. (4), (11), and (12),when the maximum offset kM is less than the fibercladding thickness, the system MTF of the IFOScan be expressed:
MTFðf Þa−sys ¼J1ð2πrf Þ
πrf sincð2Rf Þsinc2ðRf Þ; ð13Þ
and when the maximum offset kM is greater than thefiber cladding thickness, it is expressed by
MTFðf Þm−sys ¼J1ð2πrf Þ
πrf sincð2Rf Þsinc2ðRf Þsinc½ðkM− Rþ rÞf �: ð14Þ
Compared with the MTF of the slit imaging spec-trometer, this model has three additional terms: theintegral, the samplingMTFof fiber core, and themis-match MTF; the product of them can be regarded ascoupling MTF owing to the introduction of fiberbundles.
4. Example of the Design
The following example is intended to illustrate theapplication of the CFOV-folding method using fiberbundles. It is a spaceborne IFOS that operates ona spectral range of 420–1000nm with four fiber
bundles replacing the slit and providing a CFOV of11:42° for the focal length of 360mm. The fiber bun-dles segment the 72mm linear field of the Wetherellthree-mirror anastigmat (TMA) telescope into fourequal portions a, b, c, and d in 18mm, as shown inFig. 4. They are folded and arranged at the outputend in the same interval along the orthogonal direc-tion to that of an array of fiber bundles. The spectro-meter reimages the images transmitted fromtelescope through the fiber bundles and dispersesthem on the FPA depending on wavelengths (A, B,C, and D in Fig. 4 are the spectral images of a, b,c, and d, respectively). Both at the input end andthe output end, the linear array of the fiber bundleis fixed in V-shaped grooves (see Fig. 5) on the ima-ging plane of the telescope and the objective plane ofspectrometer, respectively. Utilizing this arrange-ment, we can control the cumulative width error ofthe fiber bundle from the fabrication process.
The telescope in this IFOS has two functions: itforms the images of the ground strip scene on the en-trance surface of the fiber bundles and then couplesthem into fiber bundles. Because of the circle aper-ture of the fiber sampling, ð1 − pi × r2=ð4 × R2ÞÞ ×100% of the available light is lost compared withthe slit spectrometer; however, this is not a fataldrawback for a prism spectrometer. The spectro-meter is an Offner type, with four Féry prisms lo-cated on the two arms symmetrically. It has a smallspectral distortion similar to that of an Offner grat-ing spectrometer and a good linearity of dispersionafter the introduction of two other flint prisms[16]. Both the maximum smile and keystone of thedesigned IFOS are controlled less than 2 μm inCODE V, and the nonlinearity of dispersion is less
Fig. 4. (Color online) Sketch of the layout of the spaceborne IFOS.
than 0.15. To show that the compact structure andlow spectral distortion can be obtained simulta-neously for our designed FOV-folding prism spectro-meter, we compare it under the same conditions withthe classical Offner grating spectrometer, which iswell known for a compact system design and lackof distortion. Because the system size, to a largeextent, influences the image quality and spectral dis-tortion, all of these factors are used for comparison,and the results are listed in Table 1. Here the imagequality is measured in micrometers of RMS wave-front error (WFE) including average WFE and max-imumWFE, and the spectral distortion including thesmile and keystone are measured in micrometers.The slit format in the Offner grating spectrometeris 72mm; some decenters are added to optimizationto achieve better image quality. The designed FOV-folding spectrometer has overwhelming advantagesover the designed volume, which is about one fifthof the slit one. Meanwhile, it has better image, lesssmile, and almost equal keystone relative to the slitone. In addition, the flexible fiber bundles permit thetelescope and spectrometer to be in random arrange-ment and to be more compact.
To show the imaging performance of the designedIFOS and the effect of the fiber coupling on theMTF, we take the example of R ¼ 9 μm and r ¼8 μm and plot the systemMTF without considerationof the MTF of telescope and spectrometer and thencompare them with that of the slit imaging spectro-
meter. The maximum offset kM described in Section 3for fiber bundles a, b, c, and d versus three typicalwavelengths are listed in Table 2, which shows thatthe fiber bundles have almost equal kM at thosewavelengths. Take fiber c as the example and plotMTF curves for the wavelength of 600nm by substi-tuting the kM values listed in Table 2 into Eq. (14);those curves are depicted in Fig. 6. The solid squaresrepresent the MTF of mismatch expressed bysinc½ðkM − Rþ rÞf �, the solid curve represents theMTFof the slit imaging spectrometer, the solid circlesrepresent the coupling MTF of the fiber bundles, andthe solid triangles represent the system MTF. It canbe seen that, compared to the slit one, the systemMTFis degraded by about 0.5 due to the coupling effect ofthe fiber bundles at the Nyquist frequency but is stillacceptable.
5. Conclusions
We present a method of FOVenlargement for the dis-persive imaging spectrometer by substituting the slitwith a linear array of imaging fiber bundles. Utiliz-ing the flexibility and separability of fiber bundles,we split the wide FOV of the telescope into severalsmall units and then fold and arrange them on theobjective plane of the spectrometer with some inter-vals separated between them. The images trans-ferred from the telescope through fiber bundles arereimaged on a common detector by the spectrometer.This approach can simultaneously decrease the vo-lume of the imaging spectrometer and the spectraldistortion due to wide FOV design at the cost of MTFdegradation caused by a couple function of the fiberbundles. The IFOS is a multisampling imaging sys-tem, and a cascade MTF model is developed to esti-mate the imaging performance. An example of aspaceborne IFOS is designed to illustrate the use ofthe FOV folding, and MTF curves were plotted for it,which indicates that the wide FOV and compact
Fig. 5. Cross section of the V-shaped grooves.
Fig. 6. (Color online) MTF curves of the slit spectrometer andthe designed IFOS when the values of R and r are 9 μm and 8 μm,respectively.
Table 1. Parameters Comparison between the Designed FOV-FoldingSpectrometer and the Classical Offner Grating Spectrometer
structure can be obtained simultaneously at thesacrifice of partial MTF value.
This work was supported by the National Programon Key Basic Research Project (973 Program) ofChina under 2009CB7240020603B.
References1. R. E. Haring, F. Williams, G. Vanstone, and G. Putnam,
“WFIS: a wide field-of-view imaging spectrometer,” Proc. SPIE3759, 305–314 (1999).
2. P. Mouroulis, R. O. Green, and D. W.Wilson, “Optical design ofa coastal ocean imaging spectrometer,”Opt. Express 16, 9087–9096 (2008).
3. R. Lucke and J. Fisher, “The Schmidt–Dyson: a fast space-borne wide-field hyperspectral imager,” Proc. SPIE 7812,78120M (2010).
4. L. G. Cook and J. F. Silny, “Imaging spectrometer trade stu-dies: a detailed comparison of the Offner–Chrisp and reflec-tive triplet optical design forms,” Proc. SPIE 7813, 78130F(2010).
5. J. S. Pazder, M. Fletcher, and C. Morbey, “The optical design ofthe wide field optical spectrograph for the thirty meter tele-scope,” Proc. SPIE 6269, 626932 (2006).
6. E. E. Sabatke, J. H. Burge, and P. Hinz, “Optical design ofinterferometric telescopes with wide fields of view,” Appl.Opt. 45, 8026–8035 (2006).
7. D. Magrin, R. Ragazzoni, G. Gentile, M. Dima, and J. Farinato,“Optical design of a highly segmentedwide field spectrograph,”Proc. SPIE 7428, 7428U (2009).
8. P. Hu, Q. Lu, R. Shu, and J. Wang, “An airborne pushbroomhyperspectral imager with wide field of view,” Chin. Opt. Lett.3, 689–691 (2005).
9. C. Vanderriest, “Integral field spectroscopy with opticalfibres,” in 3D Optical Spectroscopic Methods in Astronomy,G. Comte and M. Marcelin, eds., Vol. 71 of ASP ConferenceSeries, 209–218, (Astronomical Society of the Pacific, 1995).
10. I. R. Parry, “Optical fibres for integral field spectroscopy,”NewAstron. Rev. 50, 301–304 (2006).
11. W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars,“The definition of the OTF and the measurement of aliasingfor sampled imaging system,” Opt. Acta 29, 41–50 (1982).
12. S. K. Park, R. A. Schowengerdt, and M. A. Kaczynski,“Modulation-transfer-function analysis for sampled imagesystem,” Appl. Opt. 23, 2572–2582 (1984).
13. R. Drougard, “Optical transfer properties of fiber bundles,” J.Opt. Soc. Am. 54, 907–914 (1964).
14. K. M. Hock, “Effect of oversampling in pixel arrays,” Opt. Eng.34, 1281–1288 (1995).
15. M. E. Marhic, S. E. Schacham, andM. Epstein, “Misalignmentin imaging multifibers,” Appl. Opt. 17, 3503–3506 (1978).
16. S. Kaiser, B. Sang, J. Schubert, S. Hofer, and T. Stuffler, “Com-pact prism spectrometer of pushbroom type for hyperspectralimaging,” Proc. SPIE 7100, 710014 (2008).
