Abstract

A complex spectrum arises from the Fourier transform of an asymmetric interferogram. A rigorous derivation shows that the rms noise in the real part of that spectrum is indeed given by the commonly used relation σR = 2X ×NEP/(ηAΩ τN), where NEP is the delay-independent and uncorrelated detector noise-equivalent power per unit bandwidth, ±X is the delay range measured with N samples averaging for a time τ per sample, η is the system optical efficiency, and AΩ is the system throughput. A real spectrum produced by complex calibration with two complex reference spectra [Appl. Opt. 27, 3210 (1988)] has a variance σL 2 = σR 2 + σc 2(L h - L s)2/(L h - L c)2 + σh 2(L s - L c)2/(L h - L c)2, valid for σR, σc, and σh small compared with L h - L c, where L s, L h, and L c are scene, hot reference, and cold reference spectra, respectively, and σc and σh are the respective combined uncertainties in knowledge and measurement of the hot and cold reference spectra.

© 2003 Optical Society of America

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  1. J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, A. Benz, A. Huber, M. Mayor, eds. (Geneva Observatory, Sauverny, Switzerland, 1985), pp. 1–61.
  2. H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
    [CrossRef]
  3. H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.
  4. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
    [CrossRef] [PubMed]
  5. J.-M. Thériault, “Beam splitter layer emission in Fourier-transform infrared interferometers,” Appl. Opt. 37, 8348–8351 (1998).
    [CrossRef]
  6. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).
  7. J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).
  8. Research Systems, Inc., IDL Reference Guide (Research Systems, Inc., Boulder, Colo., 1995), Vol. 1.
  9. W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981), p. 764.
  10. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 15.
  11. J. J. Tuma, Engineering Mathematics Handbook, 2nd ed. (McGraw-Hill, New York, 1979), p. 95.
  12. R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).
  13. E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).
  14. S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, San Diego, Calif., 2001).
  15. D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

1998

1988

Abrams, M. C.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, San Diego, Calif., 2001).

Best, F. A.

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Boreman, G. D.

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

Brault, J. W.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, A. Benz, A. Huber, M. Mayor, eds. (Geneva Observatory, Sauverny, Switzerland, 1985), pp. 1–61.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, San Diego, Calif., 2001).

Buchholtz, G. M.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Buijs, H.

Conrath, B. J.

R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).

Crowe, D. G.

D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

Davis, S. P.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, San Diego, Calif., 2001).

Dedecker, R. G.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Dereniak, E. L.

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

Dirkx, T. P.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Fry, P. M.

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

Hanel, R. A.

R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).

Herbsleb, R. A.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Howell, H. B.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
[CrossRef] [PubMed]

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Jennings, D. E.

R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).

Kaplan, W.

W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981), p. 764.

Knudteson, R. O.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

LaPorte, D. D.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
[CrossRef] [PubMed]

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

Limperis, T.

D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

Mathews, J.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 15.

Mudar, J.

D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

Norton, P. R.

D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Revercomb, H. E.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
[CrossRef] [PubMed]

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

Samuelson, R. E.

R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).

Short, J. F.

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Smith, W. L.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
[CrossRef] [PubMed]

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

Sromovsky, L. A.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210–3218 (1988).
[CrossRef] [PubMed]

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Thériault, J.-M.

Tuma, J. J.

J. J. Tuma, Engineering Mathematics Handbook, 2nd ed. (McGraw-Hill, New York, 1979), p. 95.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Walker, R. L.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

Appl. Opt.

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

Research Systems, Inc., IDL Reference Guide (Research Systems, Inc., Boulder, Colo., 1995), Vol. 1.

W. Kaplan, Advanced Mathematics for Engineers (Addison-Wesley, Reading, Mass.1981), p. 764.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 15.

J. J. Tuma, Engineering Mathematics Handbook, 2nd ed. (McGraw-Hill, New York, 1979), p. 95.

R. A. Hanel, B. J. Conrath, D. E. Jennings, R. E. Samuelson, Exploration of the Solar System by Infrared Remote Sensing (Cambridge U. Press, Cambridge, UK, 1992).

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, San Diego, Calif., 2001).

D. G. Crowe, P. R. Norton, T. Limperis, J. Mudar, “Detectors,” in Electro Optical Components, W. D. Rogatto, ed., (SPIE, Bellingham, Wash., 1993), pp. 175–283.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, A. Benz, A. Huber, M. Mayor, eds. (Geneva Observatory, Sauverny, Switzerland, 1985), pp. 1–61.

H. E. Revercomb, L. A. Sromovsky, P. M. Fry, F. A. Best, D. D. LaPorte, “Demonstration of imaging Fourier transform spectrometer (FTS) performance for planetary and geostationary Earth observing,” in Hyperspectral Remote Sensing of the Land and Atmosphere, W. L. Smith, Y. Yasuoka, eds., Proc. SPIE4151, 1–10 (2001).
[CrossRef]

H. E. Revercomb, W. L. Smith, R. O. Knudteson, F. A. Best, R. G. Dedecker, T. P. Dirkx, R. A. Herbsleb, G. M. Buchholtz, J. F. Short, H. B. Howell, “AERI—Atmospheric Emitted Radiance Interferometer,” in Eighth Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 180–182.

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Equations (57)

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Imx=0 ηνηmν-1Lνdν+0 ηνLνcos2πνxdν,
ην=12 ηbνηoν,
Bν=ηνLν, Ix=Imx-0 ηmν-1Bνdν.
Ix=0 Bνcos2πνxdν.
Ix=0 Bνcos2πνx-ϕνdν
Ix=0 Bνcosϕνcos2πνxdν+0 Bνsinϕνsin2πνxdν,
Cν=Bνcosϕνfor ν>00for ν<0, Sν=Bνsinϕνfor ν>00for ν<0.
Ix=- Cνcos2πνxdν+- Sνsin2πνxdν.
Cν=Ce+Co, Ceν=12Cν+C-ν, Coν=12Cν-C-ν,
Sν=Se+So, Seν=12Sν+S-ν, Soν=12Sν-S-ν.
Ix=-Ceν+iSoνcos2πνxdν-i -Ceν+iSoνsin2πνxdν,
Gν=Ceν+iSoν,
Ix=- Gνexp-2πiνxdν.
Gν=- Ixexp+2πiνxdx,
Gν=12 Bνexpiϕνfor ν>012 B-νexp-iϕ-νfor ν<0,
Lν=ην-1Bν=2ηνexp-iϕν- Ixexp+2πiνxdx.
- Ixexp2πiνxdx=Gν+Oν,
Vh=RdGh+O=Rdη2Lh expiϕ+RdO hot,
Vc=RdGc+O=Rdη2Lc expiϕ+RdO cold,
Vs=RdGs+O=Rdη2Ls expiϕ+RdO scene.
R expiϕ=Vh-VcLh-Lc, R=|Vh-Vc|Lh-Lc,
RdO=Vc-RLc exp iϕ,
Ls=Lc+ReVs-VcVh-VcLh-Lc.
Hnk=0N-1 hk exp2πikn/N,
hk=1Nn=0N-1 Hn exp-2πikn/N.
k=0N-1exp2πikn-m/N=Nδnm,
Hν=-X+X Ixexp2πiνxdx.
Hν=Gν * 2X sinc2Xν,
Hνk=-N/2N/2-1 Ixkexp2πiνxkΔ,
νn=n/2X, n=-N/2, -N/2+1,, N/2-1.
Hνn=2XNk=-N/2N/2-1 Ixkexp2πink/N.
k=-N/2-1 Ixkexp2πink/N=k=-N/2+NN-1×Ixk-Nexp2πink/N,
Hνn=2XNk=0N-1 Isxkexp2πink/N,
Isxk=Ixk=I2XN kfor k=0, 1,, N/2-1,Ixk-N=I2XN k-Xfor k=N/2,, N-1,
Hn=Hνn=Hn2Xfor n=0, 1,, N/2-1,Hνn-N=Hn2X-N2Xfor n=N/2,, N-1.
PSx=AΩIx,
Ix=PxAΩ.
Hνn=2XNk=0N-1 Ixkexp2πink/N,
Dνncos ϕν=2ηReHνn, Dνnsin ϕν=2ηImHνn.
σR2EDn cos ϕnu2 =4η22XN2Ek=0N-1 Ixkcos2πnk/N×k=0N-1 Ixkcos2πnk/N,
σR2=4η22XN2k=0N-1 EIxk2cos22πnk/N
EIxkIxk=EIxk2δk,k,
σR=4XσI/η2Nfor n04XσI/ηNfor n=0,
σI2=EIxk2=1AΩ2 EPxk2=1AΩ2NEP22τ,
NESRσR=2X 2ηNEPAΩ2τN=NEPηAΩΔντN,
D*=Ad1/2NEP,
NESR=2Ad1/2ηAΩΔνD*2τN=1ηAΩΔνD*AdτN1/2.
D,n=Dνnexpiϕνn+2η Hνn.
ED,nD,n*=ED2+4 HνnHνn*η2+cross terms =D2+2σR2,
L=ReLs-Lc+s-cexp-iϕLh-Lc+h-cexp-iϕLh-Lc-Ls-Lc,
L=Res exp-iϕ=Rescos ϕ+Imssin ϕ.
σL2=EL2=ERes2 cos2 ϕ+EIms2sin2 ϕ=4η2 EReH2=σR2,
LRes-cexp-iϕ-Ls-LcLh-Lc×Reh-cexp-iϕ.
ERes-ccos ϕ+Ims-csin ϕ2=σR2+σc2,
EReh-ccos ϕ+Imh-csin ϕ2=σh2+σc2,
ERes-cReh-ccos2 ϕ+Ims-cImh-csin2 ϕ=σc2.
σL2=EL2=σR2+σc2Lh-LsLh-Lc2+σh2Ls-LcLh-Lc2,

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