Abstract

The equality of the reflected and the transmitted irradiances by a beam splitter that consists of a thin absorbing coating (typically a metallic film) on a transparent plate is considered. The absorption and the phase difference between the reflected and transmitted fields are also studied. The lack of reversibility of this beam splitter introduces an asymmetry that is discussed for a Michelson interferometer.

© 1995 Optical Society of America

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References

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  1. W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 8.6.
  2. A. L. Fymat, “Jones’s matrix representation of optical instruments. I: Beam splitters,” Appl. Opt. 10, 2499–2505 (1971).
    [CrossRef] [PubMed]
  3. J. J. Monzón, L. L. Sánchez-Soto, “Some properties of absorbing beam splitters,” Optik 97, 71–74 (1994).
  4. H. A. McLeod, Thin-Film Optical Filters (Hilger, Bristol, 1986), p. 148.
  5. F. Pi, G. Orriols, “Energy balance in the superposition of light waves with lossless beam splitters,” Am. J. Phys. 53, 667–670 (1985).
    [CrossRef]
  6. Ref. 4, pp. 398–407.
  7. K. W. Raine, M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” Opt. Acta 25, 549–558 (1978).
    [CrossRef]
  8. F. Parmigiani, “Phase dependence of the Michelson interferometer outputs on the absorbing beam splitter thickness,” Opt. Commun. 38, 319–324 (1981).
    [CrossRef]
  9. U. Oppenheim, “Semireflecting silver films for infrared interferometry,” J. Opt. Soc. Am. 46, 628–633 (1956).
    [CrossRef]
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.
  11. J. J. Monzón, L. L. Sánchez-Soto, E. Bernabeu, “Influence of coating thickness on the performance of a Fabry–Perot interferometer,” Appl. Opt. 30, 4126–4132 (1991).
    [CrossRef] [PubMed]
  12. Ref. 4, pp. 46–48.
  13. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  14. J. J. Monzón, L. L. Sánchez-Soto, “On the definition of absorption for a Fabry–Perot interferometer,” Pure Appl. Opt. 1, 219–226 (1992).
    [CrossRef]
  15. J. J. Monzón, L. L. Sánchez-Soto, “Optical performance of absorber structures for thermal detectors,” Appl. Opt. 33, 5137–5141 (1994).
    [CrossRef] [PubMed]
  16. J. J. Monzón, L. L. Sánchez-Soto, “Reflected fringes in a Fabry–Perot interferometer with absorbing coatings,” J. Opt. Soc. Am. A 12, 132–136 (1995).
    [CrossRef]
  17. G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
    [CrossRef]
  18. F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
    [CrossRef]
  19. G. A. Vanasse, H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 309.
  20. E. D. Palik, N. Ginsburg, H. B. Rosenstock, R. T. Holm, “Transmittance and reflectance of a thin absorbing film on a thick substrate,” Appl. Opt. 17, 3345–3347 (1978).
    [CrossRef] [PubMed]

1995 (1)

1994 (2)

J. J. Monzón, L. L. Sánchez-Soto, “Optical performance of absorber structures for thermal detectors,” Appl. Opt. 33, 5137–5141 (1994).
[CrossRef] [PubMed]

J. J. Monzón, L. L. Sánchez-Soto, “Some properties of absorbing beam splitters,” Optik 97, 71–74 (1994).

1992 (1)

J. J. Monzón, L. L. Sánchez-Soto, “On the definition of absorption for a Fabry–Perot interferometer,” Pure Appl. Opt. 1, 219–226 (1992).
[CrossRef]

1991 (1)

1986 (2)

G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
[CrossRef]

F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
[CrossRef]

1985 (1)

F. Pi, G. Orriols, “Energy balance in the superposition of light waves with lossless beam splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

1981 (1)

F. Parmigiani, “Phase dependence of the Michelson interferometer outputs on the absorbing beam splitter thickness,” Opt. Commun. 38, 319–324 (1981).
[CrossRef]

1978 (2)

E. D. Palik, N. Ginsburg, H. B. Rosenstock, R. T. Holm, “Transmittance and reflectance of a thin absorbing film on a thick substrate,” Appl. Opt. 17, 3345–3347 (1978).
[CrossRef] [PubMed]

K. W. Raine, M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” Opt. Acta 25, 549–558 (1978).
[CrossRef]

1971 (1)

1956 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Bernabeu, E.

Downs, M. J.

K. W. Raine, M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” Opt. Acta 25, 549–558 (1978).
[CrossRef]

Fymat, A. L.

Ginsburg, N.

Holm, R. T.

McLeod, H. A.

H. A. McLeod, Thin-Film Optical Filters (Hilger, Bristol, 1986), p. 148.

Monzón, J. J.

Oppenheim, U.

Orriols, G.

F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
[CrossRef]

G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
[CrossRef]

F. Pi, G. Orriols, “Energy balance in the superposition of light waves with lossless beam splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Palik, E. D.

Parmigiani, F.

F. Parmigiani, “Phase dependence of the Michelson interferometer outputs on the absorbing beam splitter thickness,” Opt. Commun. 38, 319–324 (1981).
[CrossRef]

Pi, F.

F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
[CrossRef]

G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
[CrossRef]

F. Pi, G. Orriols, “Energy balance in the superposition of light waves with lossless beam splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Raine, K. W.

K. W. Raine, M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” Opt. Acta 25, 549–558 (1978).
[CrossRef]

Rosenstock, H. B.

Sakai, H.

G. A. Vanasse, H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 309.

Sánchez-Soto, L. L.

Schmidt-Iglesias, C.

G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
[CrossRef]

F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 8.6.

Vanasse, G. A.

G. A. Vanasse, H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 309.

Am. J. Phys. (1)

F. Pi, G. Orriols, “Energy balance in the superposition of light waves with lossless beam splitters,” Am. J. Phys. 53, 667–670 (1985).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

G. Orriols, C. Schmidt-Iglesias, F. Pi, “Optical bistability in two-beam interferometric devices with in-phase outputs,” Opt. Acta 33, 7–11 (1986).
[CrossRef]

K. W. Raine, M. J. Downs, “Beam-splitter coatings for producing phase quadrature interferometer outputs,” Opt. Acta 25, 549–558 (1978).
[CrossRef]

Opt. Commun. (2)

F. Parmigiani, “Phase dependence of the Michelson interferometer outputs on the absorbing beam splitter thickness,” Opt. Commun. 38, 319–324 (1981).
[CrossRef]

F. Pi, C. Schmidt-Iglesias, G. Orriols, “Thermally induced optical multistability with a beam splitter in the degenerate two-beam interferometer configuration,” Opt. Commun. 58, 133–138 (1986).
[CrossRef]

Optik (1)

J. J. Monzón, L. L. Sánchez-Soto, “Some properties of absorbing beam splitters,” Optik 97, 71–74 (1994).

Pure Appl. Opt. (1)

J. J. Monzón, L. L. Sánchez-Soto, “On the definition of absorption for a Fabry–Perot interferometer,” Pure Appl. Opt. 1, 219–226 (1992).
[CrossRef]

Other (7)

G. A. Vanasse, H. Sakai, “Fourier spectroscopy” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, p. 309.

H. A. McLeod, Thin-Film Optical Filters (Hilger, Bristol, 1986), p. 148.

Ref. 4, pp. 398–407.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 8.6.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Ref. 4, pp. 46–48.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

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Figures (9)

Fig. 1
Fig. 1

Reflection and transmission of a plane wave in a BS that comprises an absorbing film on a transparent plate with plane-parallel boundaries.

Fig. 2
Fig. 2

Curves of equal reflectance and transmittance, for p and s polarizations, for a silver film on a glass plate. Coating thickness d 1 is in angstroms, and plate phase thickness β2 is in radians.

Fig. 3
Fig. 3

Curves of equal reflectance and transmittance fs = T as a function of the wavelength (in micrometers) and β2 for the same BS as in Fig. 2 (p polarization).

Fig. 4
Fig. 4

Phase difference τ − ρ (in radians) as a function of the plate phase thickness β2. For each value of β2 we have chosen d 1 that gives the 50/50 BS condition (s polarization). The other data are as in Fig. 2. In the marked point (β2 = π) the BS behaves exactly reversibly.

Fig. 5
Fig. 5

Sum of the BS absorptances versus d 1 (in angstroms) and β2 (in radians) for a gold film on a glass plate. θ0 = 45°, λ = 6888 Å, N 1 = 0.16 − 3.80i, and N 2 = 1.51, s polarization.

Fig. 6
Fig. 6

Same BS as in Fig. 1 in a Michelson interferometer. M’s, mirrors.

Fig. 7
Fig. 7

Global absorptance of the same BS as in Fig. 5, but now included in a Michelson interferometer versus d 1 and β2 (δ = 3π/4 rad).

Fig. 8
Fig. 8

Phase difference ϒ (in degrees) as a function of d 1 (in angstroms) for β2 = π/6, π/3, π/2, and π rad: (a) gold film, N 1 = 0.16 − 3.80i; (b) nickel film, N 1 = 2.14 − 4.00i. The other data are the same as in Fig. 5.

Fig. 9
Fig. 9

Phase difference ϒ as a function of β2 for several values of d 1: (a) gold film, (b) nickel film. The other data are the same as in Fig. 8.

Equations (22)

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β j = 2 π λ N j d j cos θ j
R fs = r 01 + R 120 exp ( - i 2 β 1 ) 1 + r 01 R 120 exp ( - i 2 β 1 ) = R fs exp ( i ρ fs ) , T fs = t 01 T 120 exp ( - i β 1 ) 1 + r 01 R 120 exp ( - i 2 β 1 ) ,
R sf = R 021 + z 021 r 10 exp ( - i 2 β 1 ) 1 + r 01 R 120 exp ( - i 2 β 1 ) = R sf exp ( i ρ sf ) , T sf = T 021 t 10 exp ( - i β 1 ) 1 + r 01 R 120 exp ( - i 2 β 1 ) ,
R i j k = r i j + r j k exp ( - i 2 β j ) 1 + r i j r j k exp ( - i 2 β j ) = R i j k exp ( i Δ i j k ) , T i j k = t i j t j k exp ( - i β j ) 1 + r i j r j k exp ( - i 2 β j )
z 021 = r 02 r 21 + exp ( - i 2 β 2 ) 1 + r 02 r 21 exp ( - i 2 β 2 ) = z 021 exp ( i Φ 021 ) .
exp ( 4 Im β 1 ) + B fs exp ( 2 Im β 1 ) + C fs = 0 ,
B fs = 2 | r 01 R 120 | cos ( δ 01 - Δ 120 ) - | t 01 T 120 R 120 | 2 , C fs = | r 01 R 120 | 2 ,
d 1 λ 4 π Im ( N 1 cos θ 1 ) ln [ - B fs - ( B fs 2 - 4 C fs ) 1 / 2 2 ] .
B sf = 2 | R 021 r 10 z 021 | cos ( Δ 021 - δ 10 - Φ 021 ) - | T 021 t 10 r 10 z 021 | 2 , C sf = | R 021 r 10 z 021 | 2 .
A fs = 1 - R fs - T , A sf = 1 - R sf - T .
E d = T [ R fs + R sf exp ( i δ ) ] , E s = R fs 2 + T 2 exp ( i δ ) .
A BS = 1 - E d 2 - E s 2 = A fs + A sf - A fs A sf - Ω ( δ ) ,
Ω ( δ ) = 2 T R fs [ R fs cos ( 2 ρ fs - 2 τ - δ ) + R sf cos ( ρ fs - ρ sf - δ ) ] .
Δ I = E d 2 - E s 2 = B + O ( δ ) ,
B = T ( R sf - T ) - R fs ( R fs - T ) , O ( δ ) = 2 T R fs [ R sf cos ( ρ fs - ρ sf - δ ) - R fs cos ( 2 ρ fs - 2 τ - δ ) ] .
Δ I peak = Δ I max - Δ I min = 4 T R fs ( R fs + R sf - 2 R fs R sf cos ϒ ) 1 / 2 ,
ϒ = ρ fs + ρ sf - 2 τ ,
Δ I = - ( R - T ) 2 + 4 R T sin ( ρ - τ - δ ) sin ( ρ - τ ) .
Δ I = 4 R 2 sin ( ρ - τ - δ ) sin ( ρ - τ ) .
Δ I = 4 R 2 cos δ .
Δ I = R fs ( R sf - R fs ) + 2 R fs R fs × [ R sf cos ( ρ fs - ρ sf - δ ) - R fs cos ( 2 ρ fs - 2 τ - δ ) ] .
Δ I = R fs ( R sf - R fs ) + 2 R sf R fs × [ R sf cos ( ρ fs - ρ sf - δ ) - R fs cos ( 2 ρ fs - 2 τ - δ ) ] .

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