Abstract

A modeling study aimed at characterizing the radiometric properties of a double-beam Fourier-transform infrared interferometer is presented. Measurements showed that the two responsivities associated with each interferometer channel are different in certain spectral regions. This anomaly was attributed to a dissymmetry between the optical transmissions of the two plates that form the beam splitter. This dissymmetry is primarily responsible for the instrument residual emission. A secondary cause of residual emission is attributed to the relative alignment of the two input optics. Both effects were taken into account in a model that gives the instrument residual emission in terms of the beam splitter temperature. Actual results indicate that in the 7–14-µm window the instrument residual emission can be modeled with an absolute radiometric error smaller than 0.5 K (blackbody at 290 K). The model was used to develop an automatic calibration procedure that yields radiance errors smaller than 0.05 µW/cm2 sr cm-1 in the 7–14-µm band. The radiometric stability of the interferometer was analyzed.

© 1999 Optical Society of America

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References

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  1. W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for the retrieval of temperature and absorbing constituents from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
    [CrossRef] [PubMed]
  2. D. Lubin, A. S. Simpson, “The Longwave emission signature of urban pollution: radiometric FTIR measurement,” Geophys. Res. Lett. 21, 37–40 (1994).
    [CrossRef]
  3. W. F. J. Evans, E. Puckrin, “An observation of the atmospheric emission spectrum of CFC-11,” Geophys. Res. Lett. 21, 2381–2384 (1994).
    [CrossRef]
  4. M. L. Polak, J. L. Hall, K. C. Herr, “Passive Fourier-transform infrared spectroscopy of chemical plumes: an algorithm for quantitative interpretation and real-time background removal,” Appl. Opt. 34, 5406–5412 (1995).
    [CrossRef] [PubMed]
  5. R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
    [CrossRef]
  6. D. S. Flanigan, “Prediction of the limits of detection of hazardous vapors by passive infrared with the use of MODTRAN,” Appl. Opt. 35, 6090–6098 (1996).
    [CrossRef] [PubMed]
  7. R. V. Allen, F. J. Mucray, X. Liu, “Mid-infrared measurements of the atmospheric emission over the South Pole using a radiometrically calibrated Fourier transform Spectrometer,” Appl. Opt. 35, 1523–1530 (1996).
    [CrossRef] [PubMed]
  8. J.-M. Thériault, C. Bradette, J. Gilbert, “Atmospheric remote sensing with a ground-based spectrometer system,” in Infrared Technology and Applications XXII, B. F. Andresen, M. Strojnik, M. S. Scholl, eds., Proc. SPIE2744, 664–672 (1996).
    [CrossRef]
  9. Also see recent papers in Proceedings of the SPIE Conference on Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763 (1996), Proc. SPIE3082 (1997), Proc. SPIE3383 (1998).
  10. J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
    [CrossRef]
  11. A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
    [CrossRef]
  12. R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
    [CrossRef]
  13. P. Fellgett, “Les progrès Récents en spectroscopie interférentielle, Bellevue,” Colloq. Int. C. N. R. S. p. 53 (1957).
  14. T. F. Zehnpfennig, O. Shepherd, S. Rappaport, W. P. Reidy, G. Vanasse, “Background suppression in double-beam interferometry,” Appl. Opt. 18, 1996–2002 (1979).
    [CrossRef] [PubMed]
  15. G. A. Vanasse, R. E. Murphy, F. H. Cook, “Double-beaming technique in Fourier spectroscopy,” Appl. Opt. 15, 290–291 (1976).
    [CrossRef] [PubMed]
  16. O. Shepherd, A. G. Hurd, R. B. Wattson, H. J. P. Smith, G. A. Vanasse, “Spectral measurements of stack effluents using a double-beam interferometer with background suppression,” Appl. Opt. 20, 3972–3980 (1981).
    [CrossRef] [PubMed]
  17. H. Krenn, I. Roschger, G. Bauer, “Dual-beam interferometer for optical difference measurements,” Appl. Opt. 23, 3065–3074 (1984).
    [CrossRef] [PubMed]
  18. 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]
  19. C. Weddigen, C. E. Blom, M. Höpfner, “Phase corrections for the emission sounder MIPAS-FT,” Appl. Opt. 32, 4586–4589 (1993).
    [CrossRef] [PubMed]
  20. J. Schreiber, T. Blumenstock, H. Ficher, “Effects of the self-emission of an IR Fourier-transform spectrometer on measured absorption spectra,” Appl. Opt. 35, 6203–6209 (1996).
    [CrossRef] [PubMed]
  21. J.-M. Thériault, “Beam splitter layer emission in Fourier-transform infrared Interferometer,” Appl. Opt. 37, 8348–8351 (1998).
    [CrossRef]
  22. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

1998 (1)

1996 (3)

1995 (1)

1994 (2)

D. Lubin, A. S. Simpson, “The Longwave emission signature of urban pollution: radiometric FTIR measurement,” Geophys. Res. Lett. 21, 37–40 (1994).
[CrossRef]

W. F. J. Evans, E. Puckrin, “An observation of the atmospheric emission spectrum of CFC-11,” Geophys. Res. Lett. 21, 2381–2384 (1994).
[CrossRef]

1993 (1)

1991 (1)

1988 (1)

1984 (1)

1981 (1)

1979 (1)

1976 (1)

1957 (1)

P. Fellgett, “Les progrès Récents en spectroscopie interférentielle, Bellevue,” Colloq. Int. C. N. R. S. p. 53 (1957).

Allen, R. V.

Bauer, G.

Blom, C. E.

Blumenstock, T.

Bradette, C.

J.-M. Thériault, C. Bradette, J. Gilbert, “Atmospheric remote sensing with a ground-based spectrometer system,” in Infrared Technology and Applications XXII, B. F. Andresen, M. Strojnik, M. S. Scholl, eds., Proc. SPIE2744, 664–672 (1996).
[CrossRef]

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

Buijs, H.

Chamberland, M.

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

Combs, R. J.

R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
[CrossRef]

Cook, F. H.

Evans, W. F. J.

W. F. J. Evans, E. Puckrin, “An observation of the atmospheric emission spectrum of CFC-11,” Geophys. Res. Lett. 21, 2381–2384 (1994).
[CrossRef]

Fellgett, P.

P. Fellgett, “Les progrès Récents en spectroscopie interférentielle, Bellevue,” Colloq. Int. C. N. R. S. p. 53 (1957).

Ficher, H.

Flanigan, D. S.

Gilbert, J.

J.-M. Thériault, C. Bradette, J. Gilbert, “Atmospheric remote sensing with a ground-based spectrometer system,” in Infrared Technology and Applications XXII, B. F. Andresen, M. Strojnik, M. S. Scholl, eds., Proc. SPIE2744, 664–672 (1996).
[CrossRef]

Giroux, J.

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

Godfrey, J. P.

R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
[CrossRef]

Hall, J. L.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

Herr, K. C.

Höpfner, M.

Howell, H. B.

Hurd, A. G.

Knapp, R. B.

R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
[CrossRef]

Krenn, H.

Kroutil, R. T.

R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
[CrossRef]

Lachance, R. L.

R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
[CrossRef]

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

Lafond, C.

R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
[CrossRef]

Laporte, D. D.

Liu, X.

Lubin, D.

D. Lubin, A. S. Simpson, “The Longwave emission signature of urban pollution: radiometric FTIR measurement,” Geophys. Res. Lett. 21, 37–40 (1994).
[CrossRef]

Mucray, F. J.

Murphy, R. E.

Polak, M. L.

Puckrin, E.

W. F. J. Evans, E. Puckrin, “An observation of the atmospheric emission spectrum of CFC-11,” Geophys. Res. Lett. 21, 2381–2384 (1994).
[CrossRef]

Rappaport, S.

Reidy, W. P.

Revercomb, H. E.

Roschger, I.

Schreiber, J.

Shepherd, O.

Simpson, A. S.

D. Lubin, A. S. Simpson, “The Longwave emission signature of urban pollution: radiometric FTIR measurement,” Geophys. Res. Lett. 21, 37–40 (1994).
[CrossRef]

Smith, H. J. P.

Smith, W. L.

Sromovsky, L. A.

Thériault, J.-M.

J.-M. Thériault, “Beam splitter layer emission in Fourier-transform infrared Interferometer,” Appl. Opt. 37, 8348–8351 (1998).
[CrossRef]

J.-M. Thériault, C. Bradette, J. Gilbert, “Atmospheric remote sensing with a ground-based spectrometer system,” in Infrared Technology and Applications XXII, B. F. Andresen, M. Strojnik, M. S. Scholl, eds., Proc. SPIE2744, 664–672 (1996).
[CrossRef]

R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
[CrossRef]

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

Vanasse, G.

Vanasse, G. A.

Villemaire, A.

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
[CrossRef]

Wattson, R. B.

Weddigen, C.

Woolf, H. M.

Zehnpfennig, T. F.

Appl. Opt. (12)

W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for the retrieval of temperature and absorbing constituents from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
[CrossRef] [PubMed]

M. L. Polak, J. L. Hall, K. C. Herr, “Passive Fourier-transform infrared spectroscopy of chemical plumes: an algorithm for quantitative interpretation and real-time background removal,” Appl. Opt. 34, 5406–5412 (1995).
[CrossRef] [PubMed]

D. S. Flanigan, “Prediction of the limits of detection of hazardous vapors by passive infrared with the use of MODTRAN,” Appl. Opt. 35, 6090–6098 (1996).
[CrossRef] [PubMed]

R. V. Allen, F. J. Mucray, X. Liu, “Mid-infrared measurements of the atmospheric emission over the South Pole using a radiometrically calibrated Fourier transform Spectrometer,” Appl. Opt. 35, 1523–1530 (1996).
[CrossRef] [PubMed]

T. F. Zehnpfennig, O. Shepherd, S. Rappaport, W. P. Reidy, G. Vanasse, “Background suppression in double-beam interferometry,” Appl. Opt. 18, 1996–2002 (1979).
[CrossRef] [PubMed]

G. A. Vanasse, R. E. Murphy, F. H. Cook, “Double-beaming technique in Fourier spectroscopy,” Appl. Opt. 15, 290–291 (1976).
[CrossRef] [PubMed]

O. Shepherd, A. G. Hurd, R. B. Wattson, H. J. P. Smith, G. A. Vanasse, “Spectral measurements of stack effluents using a double-beam interferometer with background suppression,” Appl. Opt. 20, 3972–3980 (1981).
[CrossRef] [PubMed]

H. Krenn, I. Roschger, G. Bauer, “Dual-beam interferometer for optical difference measurements,” Appl. Opt. 23, 3065–3074 (1984).
[CrossRef] [PubMed]

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]

C. Weddigen, C. E. Blom, M. Höpfner, “Phase corrections for the emission sounder MIPAS-FT,” Appl. Opt. 32, 4586–4589 (1993).
[CrossRef] [PubMed]

J. Schreiber, T. Blumenstock, H. Ficher, “Effects of the self-emission of an IR Fourier-transform spectrometer on measured absorption spectra,” Appl. Opt. 35, 6203–6209 (1996).
[CrossRef] [PubMed]

J.-M. Thériault, “Beam splitter layer emission in Fourier-transform infrared Interferometer,” Appl. Opt. 37, 8348–8351 (1998).
[CrossRef]

Colloq. Int. C. N. R. S. (1)

P. Fellgett, “Les progrès Récents en spectroscopie interférentielle, Bellevue,” Colloq. Int. C. N. R. S. p. 53 (1957).

Geophys. Res. Lett. (2)

D. Lubin, A. S. Simpson, “The Longwave emission signature of urban pollution: radiometric FTIR measurement,” Geophys. Res. Lett. 21, 37–40 (1994).
[CrossRef]

W. F. J. Evans, E. Puckrin, “An observation of the atmospheric emission spectrum of CFC-11,” Geophys. Res. Lett. 21, 2381–2384 (1994).
[CrossRef]

Other (7)

R. T. Kroutil, R. J. Combs, R. B. Knapp, J. P. Godfrey, “Infrared interferogram analysis for ammonia detection with passive FT-IR spectrometry,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 86–102 (1996).
[CrossRef]

J.-M. Thériault, C. Bradette, J. Gilbert, “Atmospheric remote sensing with a ground-based spectrometer system,” in Infrared Technology and Applications XXII, B. F. Andresen, M. Strojnik, M. S. Scholl, eds., Proc. SPIE2744, 664–672 (1996).
[CrossRef]

Also see recent papers in Proceedings of the SPIE Conference on Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763 (1996), Proc. SPIE3082 (1997), Proc. SPIE3383 (1998).

J.-M. Thériault, C. Bradette, A. Villemaire, M. Chamberland, J. Giroux, “Differential detection with a double-beam interferometer,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 65–75 (1997).
[CrossRef]

A. Villemaire, M. Chamberland, J. Giroux, R. L. Lachance, J.-M. Thériault, “Radiometric calibration of FT-IR remote sensing instrument,” in Electro-Optical Technology for Remote Chemical Detection and Identification II, M. Fallahi, E. Howden, eds., Proc. SPIE3082, 83–91 (1997).
[CrossRef]

R. L. Lachance, J.-M. Thériault, C. Lafond, A. Villemaire, “Gaseous emanation detection algorithm using a Fourier transform interferometer operating in differential mode,” in Electro-Optical Technology for Remote Chemical Detection and Identification III, M. Fallahi, E. Howden, eds., Proc. SPIE3383, 124–132 (1998).
[CrossRef]

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

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Figures (9)

Fig. 1
Fig. 1

(a) Optical diagram and (b) picture of the CATSI.

Fig. 2
Fig. 2

Calibrated measurement of a blackbody (BBY) at 15 °C (upper curve) and the resulting difference when the two ports are used.

Fig. 3
Fig. 3

(a) Module and (b) phase of the responsivity ratio K1/K2 of the CATSI system (without telescopes and input windows).

Fig. 4
Fig. 4

Comparison of the measured and the modeled instrument residual emission re2 for the CATSI system in the 500–1400-cm-1 spectral region. Here (a) and (b) correspond to the real part, whereas (c) and (d) correspond to the imaginary part of re2. The 1800–2800-cm-1 spectral region shows the same level of agreement.

Fig. 5
Fig. 5

Comparison of the measured rms value of instrument residuals (upper curve) with the rms value obtained after subtraction with the predicted residual (see text).

Fig. 6
Fig. 6

Dependence of the K2 -1 parameter (inverse responsivity) to different beam splitter temperatures. The spectrum corresponding to 304.7 K was recorded 11 days after the others (see text).

Fig. 7
Fig. 7

Comparison of the two calibration methods used to obtain the spectral radiance of a forested mountain (background) situated at 10 km. The actual curve refers to the two-temperature calibration method, and the automatic calibration is based on the beam splitter temperature. The bottom curve (right scale) represents the difference between the two.

Fig. 8
Fig. 8

Ray tracings showing the output amplitudes (A1 and A2) for (a) beams of unit amplitudes incident on input 1 and (b) input 2 for a beam splitter made of an optically thin layer of air squeezed between two substrates properly covered with AR coatings on their external faces.

Fig. 9
Fig. 9

Ray tracings representing the two internal emission components ε1 and ε2 from (a) substrate 1 and (b) substrate 2, respectively.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

S=E1+E2+Elayer,
E1=K1L1+SEin1,
E2=K2L2+SEin2,
S=K2L2+O2,
O2=SEin2+K1K2 L1+K1K2SEin1,
K2=Shot-SambBhot-Bamb,
O2=SambBhot-ShotBambShot-Samb.
S=K2L2-L1.
L2=SK2TBS+L1.
δL=SK2TBS.
L2=SK2TBS+L1-re2TBS
δL=SK2TBS-re2TBS.
raw specε1=K11-Tc1Ts1Tc1Ts1 Bs,
raw specε2=K21-Tc2Ts2Tc2Ts2 Bs
S=E1+E2,
E1=K1L1+1-Tc1Ts1Tc1Ts1 Bs,
E2=K2L2+1-Tc2Ts2Tc2Ts2 Bs.
S=K2L2-L1+re2
S=K2L2-L1+1+K1K2L1+K1K21-Tc1Ts1Tc1Ts1 Bs+1-Tc2Ts2Tc2Ts2 Bs.
K1K2=-Tc1Ts1Tc2Ts2,
re2=1+K1K2L1-Bs.
re2=1+K1K2L1-Bs+αK2-1+γBs,
A1=tc13ts13rttc2ts2+tc1ts1trtc23ts23 expiϕ,
A2=tc22ts22t2tc12ts12+tc24ts24r2 expiϕ,
ts1=ts1 expiσ1, ts2=ts2 expiσ2,  tc1=tc1 expiΓ1, tc2=tc2 expiΓ2.
A1=tc1ts1rtts2tc2ts12tc12 exp2iσ1+2iΓ1+ts22tc22 exp2iσ2+2iΓ2expiϕ
A2=tc22ts22t2ts12tc12 exp2iσ1+2iΓ1+r2ts22tc22 exp2iσ2+2iΓ2expiϕ.
I1=A1A1*=Tc1Ts1R2Ts2Tc2H2Ts12Tc12+H2Ts22Tc22+H2Ts1Tc1Ts2Tc2 cosϕ+ψ,
I2=A2A2*=R2Ts22Tc22H4Ts12Tc12+Ts22Tc22+H2Ts1Tc1Ts2Tc2 cosϕ+ψ-π,
ψ=2σ2+Γ2-σ1-Γ1
H=1-r122r1 sinδ
K1=K1 expiψ=+R2H2Ts12Tc12Ts22Tc22 expiψ,
K2=K2 expiψ-π=-R2H2Ts1Tc1Ts23Tc23 expiψ,
K1=-K2=R2H2Ts14Tc14 expiψ.
ε1=1-Tc1Ts1Bs,
modulationε1=1-Tc1Ts1Bs×R2H2Ts1Tc1Ts22Tc22 cosϕ+ψ,
raw specε1=K11-Tc1Ts1Tc1Ts1 Bs,
raw specε2=K21-Tc2Ts2Tc2Ts2 Bs,
raw specε1=-raw specε2.

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