Abstract

Experimental results are presented for a computed tomography imaging spectrometer (CTIS) with imposed spatial–spectral modulation on the image scene. This modulation structure on the CTIS tomographic dispersion created substantial gains in spectral reconstruction resolution after standard iterative, nonlinear, inversion techniques were used. Modulation limits system ambiguities, so high-frequency spectral and low-frequency spatial scene data could be recovered. The results demonstrate how spatial modulation acts as a high-frequency spectral deconvolver for the snapshot hyperspectral imager technology.

© 2006 Optical Society of America

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2005

2004

2002

R. Z. Stodilka, E. J. Soares, and S. J. Glick, "Characterization of tomographic sampling in hybrid PET using the Fourier cross-talk matrix," IEEE Trans. Med. Imaging 21, 1468-1478 (2002).
[CrossRef]

1999

1997

1995

1994

H. H. Barrett and H. Gifford, "Cone-beam tomography with discrete data sets," Phys. Med. Biol. 39, 451-476 (1994).
[CrossRef] [PubMed]

H. H. Barrett, "Noise properties of the EM algorithm. I. Theory," Phys. Med. Biol. 39, 833-846 (1994).
[CrossRef] [PubMed]

H. R. Morris, C. C. Hoyt, and P. J. Treado, "Imaging spectrometers for fluorescence and Raman microscopy: acousto-optic and liquid-crystal-tunable filters," Appl. Spectrosc. 48, 857-866 (1994).
[CrossRef]

1993

1991

T. Okamoto and I. Yamaguchi, "Simultaneous acquisition of spectral image information," Opt. Lett. 16, 1277-1279 (1991).
[CrossRef] [PubMed]

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

1982

L. A. Sheep and Y. Vardi "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

1949

C. E. Shannon, "Communication in the presence of noise," Proc. IEEE 37, 10-21 (1949).

Backlund, J.

W. R. Johnson, D. W. Wilson, G. H. Bearman, and J. Backlund, "An all-reflective computed tomography imaging spectrometer," in Instruments, Science, and Methods for Geospace and Planetary Remote Sensing, C.A.Nardell, P.G.Lucey, J.-H.Yee, and J.B.Garvin, eds., Proc. SPIE 5660, 88-97 (2004).

Barbastathis, G.

Barrett, H. H.

H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, "Objective assessment of image quality. Fisher information, Fourier cross talk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995).
[CrossRef]

H. H. Barrett, "Noise properties of the EM algorithm. I. Theory," Phys. Med. Biol. 39, 833-846 (1994).
[CrossRef] [PubMed]

H. H. Barrett and H. Gifford, "Cone-beam tomography with discrete data sets," Phys. Med. Biol. 39, 451-476 (1994).
[CrossRef] [PubMed]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Basty, S.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. L. Dereniak, "Hyperspectral imaging for astronomy and space surveillance," in Imaging Spectrometry IX, S.S.Chen and P.E.Lewis, eds., Proc. SPIE 5159, 380-391 (2003).

Bearman, G.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Bearman, G. H.

W. R. Johnson, D. W. Wilson, G. H. Bearman, and J. Backlund, "An all-reflective computed tomography imaging spectrometer," in Instruments, Science, and Methods for Geospace and Planetary Remote Sensing, C.A.Nardell, P.G.Lucey, J.-H.Yee, and J.B.Garvin, eds., Proc. SPIE 5660, 88-97 (2004).

Bulygin, F. V.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

Chao, T. H.

T. H. Chao, H. Zhou, X. Xia, and S. Serati, "Hyperspectral imaging using electro-optic Fourier transform spectrometer," in Optical Pattern Recognition XV, D.P.Casasent and T.-H.Chao, eds., Proc. SPIE 5437, 163-170 (2004).

Denny, J. L.

Dereniak, E.

J. Garcia and E. Dereniak, "Mixed expectation reconstruction technique," Appl. Opt. 38, 3745-3748 (1999).
[CrossRef]

C. Volin and E. Dereniak, "Signal to noise analysis of the Computed Tomography Imaging Spectrometer," in Imaging Spectrometry IV, M.R.Descour and S.S.Shen, eds., Proc. SPIE 3438, 107-113 (1998).

Dereniak, E. L.

M. R. Descour, C. E. Volin, E. L. Dereniak, T. M. Gleeson, M. F. Hopkins, D. W. Wilson, and P. D. Maker, "Demonstration of a computed-tomography imaging spectrometer using a computer-generated hologram disperser," Appl. Opt. 36, 3694-3698 (1997).
[CrossRef] [PubMed]

M. R. Descour and E. L. Dereniak, "Computed-tomography imaging spectrometer: experimental calibration and reconstruction results," Appl. Opt. 34, 4817-4826 (1995).
[CrossRef] [PubMed]

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. L. Dereniak, "Hyperspectral imaging for astronomy and space surveillance," in Imaging Spectrometry IX, S.S.Chen and P.E.Lewis, eds., Proc. SPIE 5159, 380-391 (2003).

M. R. Descour, E. L. Dereniak, and A. C. Dubey, "Mine detection using instantaneous spectral imaging ," in Detection Technologies for Mines and Minelike Targets, A. C. Dubey, I. Cindrich, J. Ralston, and K. Rigano, eds., Proc. SPIE 2496, 286-304 (1995).

Descour, M. R.

Dillner, U.

R. Riesenberg and U. Dillner, "Hadamard imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R.Descour and S.S.Shen, eds., Proc. SPIE 3753, 203-213 (1999).

Dubey, A. C.

M. R. Descour, E. L. Dereniak, and A. C. Dubey, "Mine detection using instantaneous spectral imaging ," in Detection Technologies for Mines and Minelike Targets, A. C. Dubey, I. Cindrich, J. Ralston, and K. Rigano, eds., Proc. SPIE 2496, 286-304 (1995).

Ewing, W. S.

J. E. Murguia, T. D. Reeves, J. M. Mooney, W. S. Ewing, and F. D. Shepherd, "A compact visible/near infrared hyperspectral imager," in Infrared Detectors and Focal Plane Arrays VI, E.L.Dereniak and R.E.Sampson, eds., Proc. SPIE 4028, 457-468 (2000).

Fletcher-Holmes, D. W.

A. R. Harvey and D. W. Fletcher-Holmes, "Multispectral imaging in a snapshot," in Spectral Imaging: Instrumentation, Applications, and Analysis, G.H.Bearman, A.Mahadevan-Jansen, and R.M.Levenson, eds., Proc. SPIE 5694, 110-119 (2004).

Garcia, J.

Gaskill, J.

J. Gaskill, Linear Systems, Fourier transforms, and Optics (Wiley, 1978).

Georges, J.

J. Georges, Designing a Non-scanning Snapshot Hyperspectral Imager (University of Arizona Press, 2001).

Gifford, H.

H. H. Barrett and H. Gifford, "Cone-beam tomography with discrete data sets," Phys. Med. Biol. 39, 451-476 (1994).
[CrossRef] [PubMed]

Gleeson, T. M.

Glick, S. J.

R. Z. Stodilka, E. J. Soares, and S. J. Glick, "Characterization of tomographic sampling in hybrid PET using the Fourier cross-talk matrix," IEEE Trans. Med. Imaging 21, 1468-1478 (2002).
[CrossRef]

Harvey, A. R.

A. R. Harvey and D. W. Fletcher-Holmes, "Multispectral imaging in a snapshot," in Spectral Imaging: Instrumentation, Applications, and Analysis, G.H.Bearman, A.Mahadevan-Jansen, and R.M.Levenson, eds., Proc. SPIE 5694, 110-119 (2004).

Hege, K.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. L. Dereniak, "Hyperspectral imaging for astronomy and space surveillance," in Imaging Spectrometry IX, S.S.Chen and P.E.Lewis, eds., Proc. SPIE 5159, 380-391 (2003).

Hopkins, M. F.

Hoyt, C. C.

Hsieh, J.

J. Hsieh, Computed Tomography, Principles, Design, Artifacts and Recent Advances (SPIE Optical Engineering Press, 2003).

Humayun, M.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Javaheri, M

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Johnson, B.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Johnson, W.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. L. Dereniak, "Hyperspectral imaging for astronomy and space surveillance," in Imaging Spectrometry IX, S.S.Chen and P.E.Lewis, eds., Proc. SPIE 5159, 380-391 (2003).

Johnson, W. R.

W. R. Johnson, D. W. Wilson, G. H. Bearman, and J. Backlund, "An all-reflective computed tomography imaging spectrometer," in Instruments, Science, and Methods for Geospace and Planetary Remote Sensing, C.A.Nardell, P.G.Lucey, J.-H.Yee, and J.B.Garvin, eds., Proc. SPIE 5660, 88-97 (2004).

Karpukhin, D. V.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

Kearney, K. J.

K. J. Kearney and Z. Ninkov, "Characterization of a digital micromirror device for use as an optical mask in imaging and spectroscopy," in Spatial Light Modulators, R.L.Sutherland, ed., Proc. SPIE 3292, 81-92 (1998).

Levin, G. G.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

Liu, W.

Maker, P. D.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2001).

Mooney, J. M.

J. E. Murguia, T. D. Reeves, J. M. Mooney, W. S. Ewing, and F. D. Shepherd, "A compact visible/near infrared hyperspectral imager," in Infrared Detectors and Focal Plane Arrays VI, E.L.Dereniak and R.E.Sampson, eds., Proc. SPIE 4028, 457-468 (2000).

Morris, H. R.

Murguia, J. E.

J. E. Murguia, T. D. Reeves, J. M. Mooney, W. S. Ewing, and F. D. Shepherd, "A compact visible/near infrared hyperspectral imager," in Infrared Detectors and Focal Plane Arrays VI, E.L.Dereniak and R.E.Sampson, eds., Proc. SPIE 4028, 457-468 (2000).

Myers, K. J.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Ninkov, Z.

K. J. Kearney and Z. Ninkov, "Characterization of a digital micromirror device for use as an optical mask in imaging and spectroscopy," in Spatial Light Modulators, R.L.Sutherland, ed., Proc. SPIE 3292, 81-92 (1998).

O'Connell, D.

K. Hege, D. O'Connell, W. Johnson, S. Basty, and E. L. Dereniak, "Hyperspectral imaging for astronomy and space surveillance," in Imaging Spectrometry IX, S.S.Chen and P.E.Lewis, eds., Proc. SPIE 5159, 380-391 (2003).

Okamoto, T.

Psaltis, D.

Reeves, T. D.

J. E. Murguia, T. D. Reeves, J. M. Mooney, W. S. Ewing, and F. D. Shepherd, "A compact visible/near infrared hyperspectral imager," in Infrared Detectors and Focal Plane Arrays VI, E.L.Dereniak and R.E.Sampson, eds., Proc. SPIE 4028, 457-468 (2000).

Riesenberg, R.

R. Riesenberg and U. Dillner, "Hadamard imaging spectrometer with micro slit matrix," in Imaging Spectrometry V, M.R.Descour and S.S.Shen, eds., Proc. SPIE 3753, 203-213 (1999).

Sadda, S.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Serati, S.

T. H. Chao, H. Zhou, X. Xia, and S. Serati, "Hyperspectral imaging using electro-optic Fourier transform spectrometer," in Optical Pattern Recognition XV, D.P.Casasent and T.-H.Chao, eds., Proc. SPIE 5437, 163-170 (2004).

Shannon, C. E.

C. E. Shannon, "Communication in the presence of noise," Proc. IEEE 37, 10-21 (1949).

Sheep, L. A.

L. A. Sheep and Y. Vardi "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

Shepherd, F. D.

J. E. Murguia, T. D. Reeves, J. M. Mooney, W. S. Ewing, and F. D. Shepherd, "A compact visible/near infrared hyperspectral imager," in Infrared Detectors and Focal Plane Arrays VI, E.L.Dereniak and R.E.Sampson, eds., Proc. SPIE 4028, 457-468 (2000).

Soares, E. J.

R. Z. Stodilka, E. J. Soares, and S. J. Glick, "Characterization of tomographic sampling in hybrid PET using the Fourier cross-talk matrix," IEEE Trans. Med. Imaging 21, 1468-1478 (2002).
[CrossRef]

Stodilka, R. Z.

R. Z. Stodilka, E. J. Soares, and S. J. Glick, "Characterization of tomographic sampling in hybrid PET using the Fourier cross-talk matrix," IEEE Trans. Med. Imaging 21, 1468-1478 (2002).
[CrossRef]

Takahashi, A.

Treado, P. J.

Updike, P.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Vardi, Y.

L. A. Sheep and Y. Vardi "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

Vishnyakov, G. N.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

Volin, C.

C. Volin, "MWIR Spectrometer Operating Theory," Ph.D. dissertation (University of Arizona Press, 2000).

C. Volin and E. Dereniak, "Signal to noise analysis of the Computed Tomography Imaging Spectrometer," in Imaging Spectrometry IV, M.R.Descour and S.S.Shen, eds., Proc. SPIE 3438, 107-113 (1998).

Volin, C. E.

Wagner, R. F.

Walsh, A.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Wilson, D.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Wilson, D. W.

M. R. Descour, C. E. Volin, E. L. Dereniak, T. M. Gleeson, M. F. Hopkins, D. W. Wilson, and P. D. Maker, "Demonstration of a computed-tomography imaging spectrometer using a computer-generated hologram disperser," Appl. Opt. 36, 3694-3698 (1997).
[CrossRef] [PubMed]

W. R. Johnson, D. W. Wilson, G. H. Bearman, and J. Backlund, "An all-reflective computed tomography imaging spectrometer," in Instruments, Science, and Methods for Geospace and Planetary Remote Sensing, C.A.Nardell, P.G.Lucey, J.-H.Yee, and J.B.Garvin, eds., Proc. SPIE 5660, 88-97 (2004).

Wolfe, W.

W. Wolfe, Introduction to Imaging Spectrometers (SPIE Optical Engineering Press, 1997).
[CrossRef]

Xia, X.

T. H. Chao, H. Zhou, X. Xia, and S. Serati, "Hyperspectral imaging using electro-optic Fourier transform spectrometer," in Optical Pattern Recognition XV, D.P.Casasent and T.-H.Chao, eds., Proc. SPIE 5437, 163-170 (2004).

Xie, J.

J. Xie, G. Bearman, D. Wilson, B. Johnson, A. Walsh, S. Sadda, P. Updike, M Javaheri, and M. Humayun, "Snap-shot retinal imaging spectroscopy," in Anatomy & Pathology/Visual Psychophysics/Physiological Optics, Publ. 4283/B651 (Association for Research in vision and Ophthalmology, 2005).

Yamaguchi, I.

Zhou, H.

T. H. Chao, H. Zhou, X. Xia, and S. Serati, "Hyperspectral imaging using electro-optic Fourier transform spectrometer," in Optical Pattern Recognition XV, D.P.Casasent and T.-H.Chao, eds., Proc. SPIE 5437, 163-170 (2004).

Appl. Opt.

Appl. Spectrosc.

IEEE Trans. Med. Imaging

L. A. Sheep and Y. Vardi "Maximum likelihood reconstruction for emission tomography," IEEE Trans. Med. Imaging MI-1, 113-122 (1982).
[CrossRef]

R. Z. Stodilka, E. J. Soares, and S. J. Glick, "Characterization of tomographic sampling in hybrid PET using the Fourier cross-talk matrix," IEEE Trans. Med. Imaging 21, 1468-1478 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Opt. Spectrosc.

F. V. Bulygin, G. N. Vishnyakov, G. G. Levin, and D. V. Karpukhin, "Spectrotomography--a new method of obtaining spectrograms of 2-D objects," Opt. Spectrosc. (USSR) 71, 561-563 (1991).

Phys. Med. Biol.

H. H. Barrett and H. Gifford, "Cone-beam tomography with discrete data sets," Phys. Med. Biol. 39, 451-476 (1994).
[CrossRef] [PubMed]

H. H. Barrett, "Noise properties of the EM algorithm. I. Theory," Phys. Med. Biol. 39, 833-846 (1994).
[CrossRef] [PubMed]

Proc. IEEE

C. E. Shannon, "Communication in the presence of noise," Proc. IEEE 37, 10-21 (1949).

Other

D. Malacara, Optical Shop Testing (Wiley, 2001).

A. R. Harvey and D. W. Fletcher-Holmes, "Multispectral imaging in a snapshot," in Spectral Imaging: Instrumentation, Applications, and Analysis, G.H.Bearman, A.Mahadevan-Jansen, and R.M.Levenson, eds., Proc. SPIE 5694, 110-119 (2004).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

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Figures (13)

Fig. 1
Fig. 1

Generalized optical layout of the CTIS.

Fig. 2
Fig. 2

CTIS tomographic dispersion pattern, showing a complex landscape in panchromatic 0th order in the center while angular chromatic dispersion creates the surrounding higher orders.

Fig. 3
Fig. 3

0th order and one higher-order projection, showing regions of ambiguity (i.e., detector pixels where the angles from multiple wavelengths are incident.)

Fig. 4
Fig. 4

Since though a frequency spatial–spectral cube at one spectral frequency (assuming a 3 × 3 grating). Each line represents a plane in 3D space. The lines shown are the intercepts of the spatial frequency planes with the particular spectral frequency. Each line signifies a CTIS projection given by the bracketed numbers. Each line has a thickness, so there are regions of ambiguity in spatial frequency space and hence regions of ambiguity in CTIS measurements.

Fig. 5
Fig. 5

Slice though a frequency spatial–spectral cube at one spectral frequency (assuming a simple 5 × 5 grating). See Fig. 4.

Fig. 6
Fig. 6

Field stop patterns used for evaluating spatial–spectral transfer functions: (a) simulating a standard fully illuminated field stop, (b) sinusoidal spatial modulation in the x direction, (c) cross or girded sinusoidal modulation in both x and y. A characteristic spectral signature (HgAr calibration target) was placed only in regions shown in white; black regions had a low uniform background from the simulated detector.

Fig. 7
Fig. 7

Spectral frequency plot, showing strengths of the spectral frequencies transferred by the full-field, lowest spatial frequency CTIS. The lack of frequency strength is due to the missing cone. NORM SQRT, normal square root.

Fig. 8
Fig. 8

Spectral frequency plot showing successful probing of spectral frequencies in kx . As no spatial frequencies were used in the y axis, no spectral frequency information is transferred in ky.

Fig. 9
Fig. 9

Spectral frequency plot showing successful probing of spectral frequencies in both kx and ky with the crossed modulation pattern used as a field stop versus the standard full field (Fig. 7). We believe that higher frequencies are dampened owing to the overlapping nature of the crossed sinusoids.

Fig. 10
Fig. 10

Matching grid patterns for practical application of crossed sinusoidal patterns. A duty cycle of 50% is used, so only two images are required.

Fig. 11
Fig. 11

Images of a HgAr calibration source that nearly fills the field stop. (a) coadded and coregistered data cube raster, showing some artifacts that are due to grid summation. (b) Standard CTIS coregistered data cube raster.

Fig. 12
Fig. 12

Plots obtained from same spatial location near the center of the image. The large broad spectral region centered at 650 nm is regained. Some missing cone artifacts are still present, indicating future work to optimize the modulation frequency will be needed.

Fig. 13
Fig. 13

Plots obtained from same spatial location near the center of the image as for Fig. 12. Most of the prominent HgAr lines have been regained. Calculations place the resolution near the center of the field of view near 6.5 nm for the modulation case, while the flat field has a resolution close to 30 nm.

Equations (88)

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( > 50   bands )
5 × 5
2
1
+ 1
+ 2
1
+ 1
3 × 3
g m = s f ( r ) h m ( r ) d r ,
h m ( r )
r ; f
g = H f = k F k H [ ϕ k ( r ) ] m Ψ F ,
F k
H [ ϕ k ( r ) ]
g m = k Ψ m k F k .
ϕ k x , k y , k λ ( r ) = exp [ 2 π i ( ρ k x , k y , k λ r ) ] S ( r ) ,
ρ k x , k y , k λ ( k x δ x k y δ y k λ δ λ ) ,
S ( r ) = rect ( x δ x ) rect ( y δ y ) rect ( λ δ λ ) .
| x | < 1 / 2
1 / 2   for   | x | = 1 / 2
| x | > 1 / 2
δ λ = λ max λ min
δ x   and   δ y
ρ k x , k y , k λ ( f x , f y , f λ ) ,
r ( x f , y f , λ )
ϕ k x , k y , k λ exp [ 2 π i ( x f f x + y f f y + λ f λ ) ] .
exp [ 2 π i ( x f f x + y f f y + λ f λ ) ] exp [ 2 π i ( f x α x + f y α y ) ] exp [ 2 π i ( f λ λ ) ] .
α x = α x + n x Δ x λ ,
α y = α y + n y Δ y λ .
( Δ x , Δ y )
exp [ 2 π i ( f s α x + f s α y ) ] exp [ 2 π i ( f λ λ f x n x Δ x λ f y n y Δ y λ ) ] .
( x s , y s )
exp [ 2 π i ( f s α x + f y α y ) ] exp [ 2 π i ( f λ λ f x n x Δ x λ f y n y Δ y λ ) ] .
f λ λ   f x n x Δ x λ f y n y Δ y λ 0 ;
f λ f x n x Δ x + f y n y Δ y .
f λ f x n x Δ x + f y n y Δ y
β k k
β k k = m = 1 M Ψ mk * Ψ m k = ( H Φ k , H Φ k ) ,
H Φ k
H Φ k
Ψ m k = d q r exp ( 2 π i ρ k r ) h m ( r ) S ( r ) ,
h m
VSF ( v ) = Gauss ( x m Δ w , y m Δ w , λ m Δ w ) ,
h m ( v ) s [ VSF ( v v ) ] d v .
VSF full
β = VSF full ( f x , k , f y , k , f λ , k ) VSF full ( f x , k , f y , k , f λ , k ) × 1 Δ 3 s exp ( 2 π i { [ ( f x , k f x , k ) x ] + [ ( f y , k f y , k ) y ] + [ ( f λ , k f λ , k ) λ ] } ) d x d y d λ ,
β k k V Δ 3 | VSF full ( f x , k , f y , k , f λ , k ) | 2 δ k k ,
δ k k
f λ λ f x n x Δ x λ f y n y Δ y λ 0 ;
f λ        f x n x Δ x + f y n y Δ y .
( n x Δ x ) f x f λ + ( n y Δ y ) f y f λ 1 ,
f λ = 1 / 5
3 × 3
5 × 5
3 × 3
5 × 5
5 × 5
( k x , k y , k λ )
Ψ k x , k y , k λ = H [ ϕ k x , k y , k λ ( r ) ] = H ( ρ k x , k y , k λ r ) S ( r ) ,
k x , k y ,
k z
Ψ cos k x k y k λ H { Re [ ϕ k x k y k λ ( r ) ] } ,
Ψ sin k x k y k λ H { Im [ ϕ k x k y k λ ( r ) ] } .
k
k x
k y
B sin k k = m = 0 M 1 ( Ψ sin k x k y k ) m ( Ψ sin k x k y k ) m | k [ 0 , k λ ) k [ 0 , k λ ) ,
k
k λ
k x
k y
1024 × 768
12   μm × 12   μm
13.68   μm
+ 12 °
12 °
P ( r , λ ) = rect ( x δ x , y δ y ) ,
rect ( a , b ) = rect ( a ) rect ( b )
P mod ( r , λ ) = P ( r , λ ) × { [ rect ( x δ x , y δ y ) ]   comb ( x N , y M ) } ,
rect ( A , B ) = rect ( a ) rect ( b )
comb ( a , b ) = comb ( a ) comb ( b )
comb ( x ) = n = - δ ( x - n ) .
50 %
( 600 1000   nm )
650   nm
30   nm
6.5   nm

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