Abstract

Simple expressions are given for the halfwidth and finesse of a Fabry-Perot interferometer fringe broadened by defects. The expressions are unrestricted as to defect magnitude and have good accuracy for most practical forms of defect. The results are applicable to mirror surface defects, parallelism errors, angular dispersion of illumination, finite spectral linewidth, and fluctuations in interferometer spacing as well as most combinations of two or more types of broadening defect. An experimental investigation of broadening due to parallelism errors confirms the theory.

© 1984 Optical Society of America

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References

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  1. R. Chabbal, “Recherche des Meilleures Conditions d’Utilisation d’un Spectromètre Fabry-Perot,” J. Rech. CNRS. 5, 138 (1953).
  2. G. Hernandez, “Analytical Description of a Fabry-Perot Photoelectric Spectrometer. 2: Numerical Results,” Appl. Opt. 9, 1591 (1970).
    [CrossRef] [PubMed]
  3. R. M. Hill, “Some Fringe-Broadening Defects in a Fabry-Perot Etalon,” Opt. Acta 10, 141 (1963).
    [CrossRef]
  4. J. V. Ramsay, “Aberrations of Fabry-Perot Interferometers When Used as Filters,” Appl. Opt. 8, 569 (1969).
    [CrossRef] [PubMed]
  5. C. Dufour, R. Picca, “Sur l’Interféromètrie Fabry-Perot. Importance des Imperfections des Surfaces,” Rev. Opt. 24, 19 (1945).
  6. V. N. Del Piano, A. F. Quesada, “Transmission Characteristics of Fabry-Perot Interferometers and a Related Electrooptic Modulator,” Appl. Opt. 4, 1386 (1965).
    [CrossRef]
  7. J. Stoner, “PEPSIOS Purely Interferometric High-Resolution Scanning Spectrometer. III: Calculation of Interferometer Characteristics by a Method of Optical Transients,” J. Opt. Soc. Am. 56, 370 (1966).
    [CrossRef]
  8. M. A. Khashan, “Analytical Determination of Linewidths Using the Fabry-Perot Spectrometer,” Physica 98C, 93 (1979).
  9. J. Meaburn, Detection and Spectrometry of Faint Light (Reidel, Dordrecht, Holland, 1976), p. 117.
  10. H. C. Burger, P. H. van Cittert, “Wahre und Scheinbare Breite von Spektrallinien,” Z. Phys. 49, 58 (1927).
  11. W. H. Steel, Interferometry (Cambridge U.P., New York, 1983), p. 151.
  12. H. A. McLeod, Thin-Film Optical Filters (Hilger, London, 1969), pp. 88–110.
  13. P. Jacquinot, “New Developments in Interference Spectroscopy,” Rep. Progr. Phys. 23, 267 (1960).
    [CrossRef]
  14. Ref. 11, p. 145.
  15. J. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. Opt. 40, 231 (1961).
  16. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.
  17. W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 318.

1980 (1)

W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
[CrossRef]

1979 (1)

M. A. Khashan, “Analytical Determination of Linewidths Using the Fabry-Perot Spectrometer,” Physica 98C, 93 (1979).

1970 (1)

1969 (1)

1966 (1)

1965 (1)

1963 (1)

R. M. Hill, “Some Fringe-Broadening Defects in a Fabry-Perot Etalon,” Opt. Acta 10, 141 (1963).
[CrossRef]

1961 (1)

J. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. Opt. 40, 231 (1961).

1960 (1)

P. Jacquinot, “New Developments in Interference Spectroscopy,” Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

1953 (1)

R. Chabbal, “Recherche des Meilleures Conditions d’Utilisation d’un Spectromètre Fabry-Perot,” J. Rech. CNRS. 5, 138 (1953).

1945 (1)

C. Dufour, R. Picca, “Sur l’Interféromètrie Fabry-Perot. Importance des Imperfections des Surfaces,” Rev. Opt. 24, 19 (1945).

1927 (1)

H. C. Burger, P. H. van Cittert, “Wahre und Scheinbare Breite von Spektrallinien,” Z. Phys. 49, 58 (1927).

Bairnsfather, H.

W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 318.

Burger, H. C.

H. C. Burger, P. H. van Cittert, “Wahre und Scheinbare Breite von Spektrallinien,” Z. Phys. 49, 58 (1927).

Chabbal, R.

R. Chabbal, “Recherche des Meilleures Conditions d’Utilisation d’un Spectromètre Fabry-Perot,” J. Rech. CNRS. 5, 138 (1953).

Clothier, W. K.

W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
[CrossRef]

Connes, J.

J. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. Opt. 40, 231 (1961).

Del Piano, V. N.

Dufour, C.

C. Dufour, R. Picca, “Sur l’Interféromètrie Fabry-Perot. Importance des Imperfections des Surfaces,” Rev. Opt. 24, 19 (1945).

Hernandez, G.

Hill, R. M.

R. M. Hill, “Some Fringe-Broadening Defects in a Fabry-Perot Etalon,” Opt. Acta 10, 141 (1963).
[CrossRef]

Jacquinot, P.

P. Jacquinot, “New Developments in Interference Spectroscopy,” Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

Khashan, M. A.

M. A. Khashan, “Analytical Determination of Linewidths Using the Fabry-Perot Spectrometer,” Physica 98C, 93 (1979).

McLeod, H. A.

H. A. McLeod, Thin-Film Optical Filters (Hilger, London, 1969), pp. 88–110.

Meaburn, J.

J. Meaburn, Detection and Spectrometry of Faint Light (Reidel, Dordrecht, Holland, 1976), p. 117.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.

Picca, R.

C. Dufour, R. Picca, “Sur l’Interféromètrie Fabry-Perot. Importance des Imperfections des Surfaces,” Rev. Opt. 24, 19 (1945).

Quesada, A. F.

Ramsay, J. V.

Sloggett, G. J.

W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
[CrossRef]

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U.P., New York, 1983), p. 151.

Stoner, J.

van Cittert, P. H.

H. C. Burger, P. H. van Cittert, “Wahre und Scheinbare Breite von Spektrallinien,” Z. Phys. 49, 58 (1927).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 318.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Rech. CNRS. (1)

R. Chabbal, “Recherche des Meilleures Conditions d’Utilisation d’un Spectromètre Fabry-Perot,” J. Rech. CNRS. 5, 138 (1953).

Opt. Acta (1)

R. M. Hill, “Some Fringe-Broadening Defects in a Fabry-Perot Etalon,” Opt. Acta 10, 141 (1963).
[CrossRef]

Opt. Eng. (1)

W. K. Clothier, G. J. Sloggett, H. Bairnsfather, “Precise Reflection Interferometry System for an Absolute Standard of Voltage,” Opt. Eng. 19, 834 (1980).
[CrossRef]

Physica (1)

M. A. Khashan, “Analytical Determination of Linewidths Using the Fabry-Perot Spectrometer,” Physica 98C, 93 (1979).

Rep. Progr. Phys. (1)

P. Jacquinot, “New Developments in Interference Spectroscopy,” Rep. Progr. Phys. 23, 267 (1960).
[CrossRef]

Rev. Opt. (2)

J. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. Opt. 40, 231 (1961).

C. Dufour, R. Picca, “Sur l’Interféromètrie Fabry-Perot. Importance des Imperfections des Surfaces,” Rev. Opt. 24, 19 (1945).

Z. Phys. (1)

H. C. Burger, P. H. van Cittert, “Wahre und Scheinbare Breite von Spektrallinien,” Z. Phys. 49, 58 (1927).

Other (6)

W. H. Steel, Interferometry (Cambridge U.P., New York, 1983), p. 151.

H. A. McLeod, Thin-Film Optical Filters (Hilger, London, 1969), pp. 88–110.

J. Meaburn, Detection and Spectrometry of Faint Light (Reidel, Dordrecht, Holland, 1976), p. 117.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.

Ref. 11, p. 145.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 318.

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Figures (4)

Fig. 1
Fig. 1

Defect distribution functions as a function of normalized phase error.

Fig. 2
Fig. 2

Variation with reflectance R of relative error in (a) peak intensity and (b) halfwidth of the Lorentzian approximation to the ideal Fabry-Perot fringe.

Fig. 3
Fig. 3

Computed dependence of parameter a [Eq. (14)] on normalized defect magnitude. The broken curve shows the variation in α predicted by Eq. (16).

Fig. 4
Fig. 4

Measured relationship between normalized halfwidth and normalized defect magnitude for parallelism errors. Bold circles indicate means of several closely overlapping experimental points, the number being shown by bracketed figures. The curve shows the theoretical relationship [Eq. (19)].

Tables (1)

Tables Icon

Table I Defect Distribution Functions and Their Standard Deviations: All Distribution Functions Except the Gaussian Vanish for |ϕμd| > ϕmax.

Equations (30)

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μ d = - d ( ϕ ) ϕ d ϕ ,
σ d 2 = - d ( ϕ ) ( ϕ 2 - μ d 2 ) d ϕ .
f ( ψ ) = - g ( ψ + ϕ ) d ( ϕ ) d ϕ
= g ( ψ ) * d ( - ψ ) ,
μ f = μ g - μ d .
σ g 2 = - ( ψ - μ g ) 2 g ( ψ ) d ψ / - g ( ψ ) d ψ ,
σ f 2 = σ g 2 + σ d 2 .
W f 2 = W g 2 + W d 2 .
A ( ψ ) = ( 1 - R ) 2 1 - 2 R cos ψ + R 2 .
W 0 = 4 sin - 1 ( 1 - R 2 R )
A ( ψ ) = n = - g ( ψ - 2 n π ) ,
g ( ψ ) = P ln 2 R ψ 2 + ln 2 R .
P = - 2 ( 1 - R ) ( 1 + R ) ln R
W 0 = - 2 ln R .
W 2 = W 0 2 + α 2 σ d 2 .
F = F 0 ( 1 - 1.5 π - 2 F 0 2 σ d 2 ) ,
W = W 0 ( 1 - 6 σ d 2 / W 0 2 ) - 1 ,
f ( ψ ) = 1 2 ϕ max tan - 1 [ 2 ϕ max / W 0 1 + ( 2 ψ W 0 ) 2 - ( ϕ max W 0 ) 2 ] ,
W 2 = W 0 2 + 4 ϕ max 2 .
W 2 = W 0 2 + 12 σ d 2 ,
F = ( F 0 - 2 + F d - 2 ) - 1 / 2 ,
W 2 = W 0 2 + 12 i σ d ( i ) 2 ,
p ( θ ) = { 2 θ θ max 2 0 θ θ max , 0 othewise ,
d ( ϕ ) = { 1 π n θ max 2 - π n θ max 2 ϕ 0 , 0 otherwise .
μ d = - ½ π n θ max 2 ,
σ d = 1 2 2 π n θ max 2 .
ψ 4 π d λ 0 ( 1 - λ - λ 0 λ 0 ) ,
ϕ = - 4 π d ( λ - λ 0 ) λ 0 2 .
σ d = 4 π d λ 0 2 · σ λ ,
σ d = 2 π n · σ λ λ 0 ,

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