Abstract

Thermal self-emission contributes to interferograms measured with Fourier-transform infrared spectrometers. If the beam-splitter is almost transparent, the complex spectral amplitude that is due to the detector port emission is opposite that of the input port, whereas the amplitude that is due to the beam-splitter emission is in quadrature. The situation of an absorbing beam splitter is examined here. The volume beam-splitter emission is modeled by a superposition of dipole sources spread in an absorbing film. Angular polarization correlations are taken into account. It is found that the phase relations between the complex spectral amplitudes are affected. Numerical data are given for experimental conditions adapted to those of the airborne limb sounder MIPAS-FT.

© 2001 Optical Society of America

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References

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  1. 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]
  2. Ch. Weddigen, C. E. Blom, M. Höpfner, “Phase corrections for the emission sounder MIPAS-FT,” Appl. Opt. 32, 4586–4589 (1993).
    [CrossRef] [PubMed]
  3. C. E. Blom, M. Höpfner, Ch. Weddigen, “Correction of phase anomalies of atmospheric emission spectra by the double-differencing method,” Appl. Opt. 35, 2649–2652 (1996).
    [CrossRef] [PubMed]
  4. O. Trieschmann, Ch. Weddigen, “Thermal emission from dielectric beam splitters in Michelson interferometers: a schematic analysis,” Appl. Opt. 39, 5834–5842 (2000).
    [CrossRef]
  5. J.-M. Thériault, “Beam-splitter layer emission in Fourier-transform infrared interferometers,” Appl. Opt. 37, 8348–8351 (1998).
    [CrossRef]
  6. B. Carli, L. Palchetti, P. Raspollini, “Effect of beam-splitter emission in Fourier transform spectroscopy,” Appl. Opt. 38, 7475–7480 (1999).
    [CrossRef]
  7. Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 13, Sec. 5.2, pp. 641–656.
  9. Ref. 8, Chap. 1.1.1, pp. 1–2.

2000 (1)

1999 (1)

1998 (1)

1996 (1)

1993 (1)

1988 (1)

Blom, C. E.

C. E. Blom, M. Höpfner, Ch. Weddigen, “Correction of phase anomalies of atmospheric emission spectra by the double-differencing method,” Appl. Opt. 35, 2649–2652 (1996).
[CrossRef] [PubMed]

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

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 13, Sec. 5.2, pp. 641–656.

Buijs, H.

Carli, B.

Fergg, F.

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Fischer, H.

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Gulde, Th.

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Höpfner, M.

Howell, H. B.

LaPorte, D. D.

Palchetti, L.

Piesch, Ch.

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Raspollini, P.

Revercomb, H. E.

Smith, W. L.

Sromovsky, L. A.

Thériault, J.-M.

Trieschmann, O.

Weddigen, Ch.

Wildgruber, G.

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 13, Sec. 5.2, pp. 641–656.

Appl. Opt. (6)

Other (3)

Th. Gulde, Ch. Piesch, C. E. Blom, H. Fischer, F. Fergg, G. Wildgruber, “The airborne MIPAS infrared emission experiment,” in Proceedings of the First International Airborne Remote Sensing Conference and Exhibition (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1994), Vol. II, pp. 301–311.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 13, Sec. 5.2, pp. 641–656.

Ref. 8, Chap. 1.1.1, pp. 1–2.

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Figures (7)

Fig. 1
Fig. 1

Ray tracings for the classical beams, (a) balanced and (b) unbalanced, and (c) for beam-splitter emission.

Fig. 2
Fig. 2

Inhomogeneous waves involved in the description of multiple reflection of (a) the classical beams, superscript c; of (b) radiation emitted in the absorbing BS, superscript e; and of (c), (d) retroreflected radiation, superscript r. Filled triangles point into the direction of phase propagation; open triangles, into the direction of decreasing intensity. The quantities Rklj and Tklj are coefficients of reflection and transmission, respectively.

Fig. 3
Fig. 3

Magnitudes of the coefficients of reflection Rklj and transmission Tklj according to Eqs. (34)–(37). Solid curves and dotted curves for |Roer| and |Toer| refer to an absorbing BS with a number of absorption a = 0.05; dashed curves refer to a transparent BS with a = 0. The results for a = 0 are identical to those given in Ref. 4. The characteristics of the BS are adapted to the MIPAS-FT (see Subsection 2.D).

Fig. 4
Fig. 4

Phase changes that are due to absorption in the BS as defined in Eq. (44). The characteristics of the BS are adapted to the MIPAS-FT. Solid curves refer to a number of absorption a = 0.05. For a transparent BS, a = 0, all phase changes are zero by definition.

Fig. 5
Fig. 5

Amplitudes of the unmodulated (subscript 0) and modulated components for the balanced (superscript b) and unbalanced (superscript u) beams, as given by Eqs. (45) and (51). The characteristics of the BS are adapted to the MIPAS-FT. Solid curves refer to a number of absorption a = 0.05; dashed curves, to a = 0. The auxiliary quantities Zeu [Eq. (49)] and Zmu [Eq. (50)] are shown for a = 0.05 as dotted and dashed–dotted curves, respectively.

Fig. 6
Fig. 6

Amplitudes of the unmodulated (subscript 0) and the modulated (subscripts cos and sin) components of the signal that are due to volume BSE, as given by Eq. (76). The characteristics of the BS are adapted to the MIPAS-FT. Solid curves refer to a number of absorption a = 0.05; dashed curves, to a = 0. The auxiliary quantities Zev [Eq. (74)] and Zmv [Eq. (75)] are shown for a = 0.05 as dotted and dashed–dotted curves, respectively.

Fig. 7
Fig. 7

Reflection and transmission of the inhomogeneous retroreflected radiation by the absorbing BS. Filled triangles point into the direction of phase propagation; open triangles, into the direction of decreasing intensity.

Equations (96)

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γ=4πnσd cos Θ2,
sin Θ2c=z121+z2+1-z22+a21/2-1/2,
sinh 2α2c=a/cos Θ2c.
g1c=4πσdn cosh α2c/cos Θ2c-tan Θ2c sin Θ1,
g2c=4πσdn sinh α2c.
sin Θ2e=z cos Θ1121+1+a2 cos 2Θ1-1/2.
tanh α1e=1+a2-1tan Θ1/a
sinh 2α2e=a
g1e=4πσdn cosh α2e/cos Θ2e cosh α1e-tan Θ2e sin Θ1,
g2e=4πσdn sinh α2e/cos Θ2e cosh α1e.
z1=sin Θ1 cosh α1r/n,
z2=cos Θ1 sinh α1r/n,
z3=a/2,
cosh α2r=121+z12+z222+4z32+2z1z2z3-z121/2+1+z12+z221/2.
sin Θ2r=z1/cosh α2r,
sinβ2r-Θ2r=z2/sinh α2r.
g1r=4πσdn cosh α2r/cos Θ2r cosh α1r-tan Θ2r sin Θ1,
g2r=4πσdn sinh α2r cos β2r/cos Θ2r cosh α1r.
rklj=bklj-cklj/bklj+cklj,
boec=cos Θ1,
coec=ncosh α2c cos Θ2c+i sinh α2c,
bomc=n cos Θ2cosh α2c+i sinh α2c cos Θ2c,
comc=cos Θ2c.
boer=cosh α1r cos Θ1+i sinh α1r sin Θ1,
coer=ncosh α2r cos Θ2r+i sinh α2r cosβ2r-Θ2r,
bomr=n cos Θ1cosh α2r+i sinh α2r cos β2r,
comr=cos Θ2r cosh α1r.
biee=n cos Θ2ecosh α2e+i sinh α2e,
ciee=cosh α1e cos Θ1-i sinh α1e sin Θ1,
bime=cos Θ2e cosh α1e,
cime=n cos Θ1cosh α2e+i sinh α2e.
tiee=1+riee,
time=1-rimecos Θ2e/cos Θ1.
Rolj=rolj1-expiγj/1-rolj2 expiγj=|Rolj|expiρolj,
Tolj=1-rolj2/1-rolj2 expiγj=|Tolj|expiτolj.
Rile=tilerile/1-rile2 expiγe=|Rile|expiρile,
Tile=tile/1-rile2 expiγe=|Tile|expiτile.
φlj=1/2g1j+τolj-ρolj,  j=c, r,
φle=½g1e+ρile-τile.
φec|a=0=φer|a=0=-π/2,
φmc|a=0=φmr|a=0=+π/2,
φee|a=0=γ/2,
φme|a=0=γ/2+π,
Δφlj=φlj-φlj|a=0.
Ib=I0b+Icosb cos2πσX,
I0b=Icosb=|Roec|2|Toec|2+|Romc|2|Tomc|2exp-g2c,
Iu=I0+Zeu cos2πσX+2φec+Zmu cos2πσX+2φmc,
I0u=½|Roec|4+|Romc|4+|Toec|4+|Tomc|4exp-2g2c,
Zeu=|Roec|2|Toec|2 exp-g2c,
Zmu=|Romc|2|Tomc|2 exp-g2c.
Iu=I0u+Icosu cos2πσX+Isinu sin2πσX,
Icosu=-Zeu cos2Δφec-Zmu cos2Δφmc,
Isinu=Zeu sin2Δφec+Zmu sin2Δφmc.
OP1=Δ-yP sin Θ1+γe1/2-δ/4πσ,
OP2=Δ-yP sin Θ1+γe3/2+δ/4πσ,
OP3=2ΔL+Δ+yP sin Θ1+γe1/2+δ+γr/4πσ,
OP4=2ΔL+Δ+yp sin Θ1+γe3/2-δ+γr/4πσ,
OP5=2ΔR+Δ+yp sin Θ1+γe1/2-δ/4πσ,
OP6=2ΔR+Δ+yp sin Θ1+γe3/2+δ/4πσ.
Ax=sin Θ2e sin ϑ cos φ-Time exp2πiσOP1-Rime exp2πiσOP2+Time+1Tomr exp2πiσOP3+Rime+1Tomr exp2πiσOP4+Time+1Romr exp2πiσOP5+Rime+1Romr exp2πiσOP6.
Ay=cos Θ2e sin ϑ sin φTime exp2πiσOP1-Rime exp2πiσOP2-Time+1Tomr exp2πiσOP3+Rime+1Tomr exp2πiσOP4+Time+1Romr exp2πiσOP5-Rime+1Romr exp2πiσOP6.
Az=cos ϑ-Tiee exp2πiσOP1-Riee exp2πiσOP2-Tiee-1Toer exp2πiσOP3-Riee-1Toer exp2πiσOP4-Tiee-1Roer exp2πiσOP5-Riee-1Roer exp2πiσOP6.
Iϑ, φ; xP, yP, zP; ΔL, ΔR, Δ=|Ax+Ay|2+|Az|2.
IxP, yP, zP; ΔL, ΔR, Δ=|Ax|2+|Ay|2+|Az|2
IxP=δd; X=2ΔL-2ΔR=Z1 exp-g2e1/2-δ+Z2 exp-g2e1/2+δ+Z3 cosφee+g1eδ+Z4 cosφme+g1eδ+Z5 cos2πσX+φer+g1eδ+Z6 cos2πσX+φmr+g1eδ.
Z1=|Tiee|21+|Roer|2+|Time|21+|Rome|2+exp-g2e-g2r|Riee|2|Toer|2+|Rime|2|Tomr|2+2 exp-g2e/2-g2r/2|Riee||Tiee||Roer||Toer|cos2πσX+φer+φee+|Rime||Time||Roer||Toer|cos2πσX+φmr+φme,
Z2=exp-g2e|Riee|21+|Roer|2+|Rime|21+|Romr|2+exp-g2r|Tiee|2|Toer|2+|Time|2|Tomr|2+2 exp-g2e/2-g2r/2×|Riee||Tiee||Roer||Toer|cos2πσX+φer-φee+|Rime||Time||Romr||Tomr|cos2πσX+φmr-φme,
Z3=2 exp-g2e|Riee||Tiee|1+exp-g2r|Toer|2+|Roer|2,
Z4=2 exp-g2esin2 Θ2e-cos2 Θ2e|Rime||Time|1+exp-g2r|Tomr|2+|Romr|2,
Z5=2 exp-g2e/2-g2r/2|Roer||Toer||Tiee|2+exp-g2e|Riee|2,
Z6=2 exp-g2e/2-g2r/2sin2 Θ2e-cos2 Θ2e|Romr||Tomr||Time|2+exp-g2e|Rime|2.
IvX=I0v+Zev cos2πσX+φer+Zmv cos2πσX+φmr,
I0v=1-exp-g2e2g2e|Riee|2 exp-g2e+|Tiee|21+|Roer|2+|Toer|2 exp-g2r+|Rime|2exp-g2e+|Time|21+|Romr|2+|Tomr|2 exp-g2e+sing1e/2g1e/2exp-g2e|Riee||Tiee|1+|Roer|2+|Toer|2 exp-g2rcos φee-|Rime||Time|1+|Romr|2+|Tomr|2 exp-g2ecos2Θ2ecos φme,
Zev=exp-g2e/2-g2r/2|Roer||Toer|2 1-exp-g2eg2e×|Riee||Toee|cos φee+sing1e/2g1e/2|Riee|2 exp-g2e+|Tiee|2,
Zmv=exp-g2e/2-g2r/2|Romr||Tomr|2 1-exp-g2eg2e×|Rime||Time|cos φme-sing1e/2g1e/2|Rime|2 exp-g2e+|Time|2cos 2Θ2e.
Iv=I0v+Icosv cos2πσX+Isinv sin2πσX.
Icosv=Zev sin Δφer-Zmv sin Δφmr,
Isinv=Zev cos Δφer-Zmv cos Δφmr.
E=A expikr-ωt.
c2ε0ΔE=εε02E/t2+σE/t
H/t=-curl E/μ0,
c2k2+εω2+iωσ/ε0=0.
k12-k22=n2k02,
2k1k2 cos β=an2k02,
a=e2+σ/ωε0/e1
α=½ asinha/cos β.
EPFi=ik0cosh α1r-x sin Θ1-y cos Θ1-ct+i sinh α1rx cos Θ1-y sin Θ1,
EPFr=ik0cosh α1r-x sin Θ1+y cos Θ1-ct+i sinh α1rx cos Θ1+y sin Θ1,
EPFt=ik0n cosh α2r-x sin Θ2r-y cos Θ2r-ct+i sinh α2rx sinβ2r-Θ2r-y cosβ2r-Θ2r.
cosh α1r sin Θ1=n cosh α2r sin Θ2r,
sinh α1r cos Θ1=n sinh α2r sinβ2r-Θ2r.
Hi,x=-1μ0c boerAi expEPFi,
Hr,x=1μ0c boerAr expEPFr,
Ht,x=-1μ0c boerAt expEPFt,
γ3=nk0dcosh α2r+i sinh α2r cos β2r/cos Θ2r.
γ5=2k0d cosh α1r tan Θ2r sin Θ1.

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