Abstract

The effects of the coating thickness on the physical performance of a Fabry–Perot interferometer (FP) are investigated. The FP is modeled as three media separated by two thin films and not merely by two interfaces. We show that the transmitted intensity obeys an Airy function, but not the reflected intensity because of the appearance of a complex factor accounting for the coupling between the reflected waves in the coatings. The Stokes relations are generalized for this model. We study the dependence of the phase lag of reflection in the coatings on the angle of incidence, which causes a shift in the position of the intensity maxima. We discuss as well the properties of the FP with an absorbing medium in the cavity, defining a merit function that optimizes the compromise between peak transmission and finesse.

© 1991 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 7.
  2. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Sec. 16.7.
  3. C. R. Pidgeon, S. D. Smith, “Resolving power of multilayer filters in nonparallel light,” J. Opt. Soc. Am. 54, 1459–1466 (1964).
    [CrossRef]
  4. P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method for the study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. GPO, Washington, D.C., 1964), pp. 157–200.For a more recent review of the model see R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.
  5. G. Hernandez, “Fabry-Perot with an absorbing etalon cavity,” Appl. Opt. 24, 3062–3067 (1985).
    [CrossRef] [PubMed]
  6. J. B. Kumer, T. C. James, “Effect of substrate absorption on the performance of solid Fabry-Perot etalons,” Appl. Opt. 27, 4800–4801 (1988).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Sec. 7.6.2.
  8. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 337.
  9. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  10. W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 9.3.1.

1988 (1)

1985 (1)

1964 (1)

Azzam, R. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 337.

Bashara, N. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 337.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 7.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Sec. 7.6.2.

Hayfield, P. C. S.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method for the study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. GPO, Washington, D.C., 1964), pp. 157–200.For a more recent review of the model see R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Hernandez, G.

James, T. C.

Kumer, J. B.

Pidgeon, C. R.

Smith, S. D.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 9.3.1.

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Sec. 16.7.

White, G. W. T.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method for the study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. GPO, Washington, D.C., 1964), pp. 157–200.For a more recent review of the model see R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 7.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Sec. 7.6.2.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (7)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 7.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Sec. 16.7.

P. C. S. Hayfield, G. W. T. White, “An assessment of the stability of the Drude–Tronstad polarized light method for the study of film growth on polycrystalline metals,” in Ellipsometry in the Measurements of Surfaces and Thin Films, E. Passaglia, R. R. Stromberg, J. Kruger, eds., Natl. Bur. Stand. Misc. Publ. 256 (U.S. GPO, Washington, D.C., 1964), pp. 157–200.For a more recent review of the model see R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Sec. 7.6.2.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), p. 337.

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

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Sec. 9.3.1.

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Figures (9)

Fig. 1
Fig. 1

Reflection and transmission of a plane wave in a FP when the coatings are considered as two thin films with parallel-plane boundaries.

Fig. 2
Fig. 2

Reflection phase shifts (in radians) for p polarization as a function of the angle of incidence for the spacer–coating–ambient system (λ = 6888 Å, N2 = 1.5, N0 = 1) for (a) a silver coating with N1 = 0.14–4.44i and (b) an aluminum coating with N1 = 1.74–8.21i, both with a thickness of 500 Å.

Fig. 3
Fig. 3

Variation of the reflective finesse ℱ versus coating thickness (in angstroms) and angle of incidence for an FP with silver coatings and a transparent spacer (N2 = 1.5, λ = 6888 Å) for p polarization.

Fig. 4
Fig. 4

Dependence of ℱ on the wavelength and the coating thickness for (a) silver and (b) aluminum coatings and angle of incidence ϑ = 3°. (The other data are the same as in Fig. 3.)

Fig. 5
Fig. 5

Variation of the peak transmission versus coating thickness and angle of incidence of the same data as in Fig. 3.

Fig. 6
Fig. 6

Dependence of T max on the wavelength and the coating thickness for the same data as in Fig. 4.

Fig. 7
Fig. 7

Peak transmission T max and reflective finesse ℱ for various reflecting coatings: 1, aluminum (λ = 3542 Å); 2, silver (λ = 5166 Å); 3, silver (λ = 6888 Å).

Fig. 8
Fig. 8

Function P versus coating thickness and angle of incidence for the same FP as that in Fig. 3.

Fig. 9
Fig. 9

Variation of function P with the wavelength and the coating thickness for the same FP as that in Fig. 4.

Equations (37)

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E ( Z ) = [ E ( + ) ( z ) E ( ) ( z ) ] .
[ E ( + ) ( z ) E ( ) ( z ) ] = ( S 11 S 12 S 21 S 22 ) [ E ( + ) ( z ) E ( ) ( z ) ] ,
E ( z ) = S E ( z ) ,
S = I 01 L 1 I 12 L 2 I 21 L 1 I 10 ,
I i j = ( 1 / t i j ) ( 1 r i j r i j 1 ) ,
N 0 sin θ 0 = N 1 sin θ 1 = N 2 sin θ 2 .
t i j t i j = 1 r i j 2 ,
r i j = r i j .
L j = [ exp ( i β j ) 0 0 exp ( i β j ) ] ,
β i = 2 π λ N j d j cos θ j .
T overall = 1 S 11 ,
R overall = S 21 S 11 ,
T overall = T 012 T 210 exp ( i β 2 ) 1 R 210 2 exp ( i 2 β 2 ) ,
R overall = R 012 + z 1 R 210 exp ( i 2 β 2 ) 1 R 210 2 exp ( i 2 β 2 ) ,
T ijk = t i j t j k exp ( i β j ) 1 + r i j r j k exp ( i 2 β j ) ,
R ijk = r i j + r j k exp ( i 2 β j ) 1 + r i j r j k exp ( i 2 β j )
z 1 = r 01 r 12 + exp ( i 2 β 1 ) 1 + r 01 r 12 exp ( i 2 β 1 ) .
T 012 T 210 R 012 R 210 = z 1 ,
R 210 = z 2 R 012 ,
z 2 = r 12 + r 01 exp ( i 2 β 1 ) r 01 + r 12 exp ( i 2 β 1 ) ,
T overall = | T 012 | 2 | T 210 | 2 exp ( 2 β 2 I ) 1 + | R 210 | 4 exp ( 4 β 2 I ) 2 | R 210 | 2 exp ( 2 β 2 I ) cos ( 2 β 2 R 2 Δ 210 ) ,
overall = | R 210 | 2 | z 1 exp ( i 2 β 2 ) 1 / z 2 | 2 1 + | R 210 | 4 exp ( 4 β 2 I ) 2 | R 210 | 2 exp ( 2 β 2 I ) cos ( 2 β 2 R 2 Δ 210 ) ,
R ijk = | R ijk | exp ( i Δ ijk ) ,
β 2 = β 2 R + i β 2 I .
T overall = T max 1 1 + F sin 2 ( β 2 R Δ 210 ) ,
T max = | T 012 T 210 exp ( β 2 I ) | 2 [ 1 | R 210 exp ( β 2 I ) | 2 ] 2 ,
F = 4 | R 210 exp ( β 2 I ) | 2 [ 1 | R 210 exp ( β 2 I ) | 2 ] 2 ,
R ̂ 210 = R 210 exp ( β 2 I ) ,
T ̂ 210 = T 210 exp ( β 2 I / 2 ) ,
T ̂ 012 = T 012 exp ( β 2 I / 2 ) ,
T c + c = 1 ,
T c = | T 012 T 210 | ,
c = | R 210 | 2 ,
A c + T c + c = 1 ,
A overall = 1 overall T overall .
= π ( F ) 1 / 2 2 .
T max F = | T 012 T 210 | 2 4 | R 210 | 2 ,

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