Abstract

A generalized study has been done of the transmission characteristics of a Fabry–Perot interferometer (FPI) illuminated by a Gaussian light beam impinging on it at normal and non-normal incidence. The theoretical approach is based on a plane-wave, angular-spectrum representation of both the incident Gaussian beam and the transmitted beam. Expressions are obtained for the FPI instrumental function and for the spatial distribution of the transmitted beam. Numerical results are presented for the FPI maximum transmission, effective finesse, and spectral displacement of the interference maximum.

© 1995 Optical Society of America

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References

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  1. R. J. Chabbal, “Recherche des meilleures conditions d’utilisation d’un spectromètre Fabry–Perot,” J. Rech. CNRS 24, 138–149 (1953).
  2. G. J. Slogget, “Fringe broadening in Fabry–Perot interferometers,” Appl. Opt. 23, 2427–2432 (1984).
    [CrossRef]
  3. D. P. Mahapatra, S. K. Mattoo, “Exact evaluation of the transmitted amplitude for a Fabry–Perot interferometer with surface defects,” Appl. Opt. 25, 1646–1649 (1986).
    [CrossRef] [PubMed]
  4. V. N. Del Piano, A. F. Quesada, “Transmission characteristics of Fabry–Perot interferometers and a related electrooptic modulator,” Appl. Opt. 4, 1386–1390 (1965).
    [CrossRef]
  5. J. V. Ramsay, “Aberrations of Fabry–Perot interferometers when used as filters,” Appl. Opt. 8, 569–574 (1969).
    [CrossRef] [PubMed]
  6. I. Prikryl, “Analytical description of an imperfect Fabry–Perot étalon,” Appl. Opt. 23, 621–627 (1984).
    [CrossRef] [PubMed]
  7. A. S. Naidenov, I. Sh. Etsin, “Instrumental function of a Fabry–Perot interferometer illuminated by a Gaussian light beam,” Opt. Spectrosc. (USSR) 46, 409–412 (1979).
  8. F. Moreno, F. Gonzalez, “Transmission of a Gaussian beam of low divergence through a high-finesse Fabry–Perot device,” J. Opt. Soc. Am. A 9, 2173–2175 (1992).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1987), Chap. 7, pp. 323–329.
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 48– 51.
  11. J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
    [CrossRef] [PubMed]
  12. Ref. 10, Chap. 2, p. 10.
  13. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  14. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos–Hänchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  15. R. P. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]
  16. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16, pp. 645–646.
  17. D. G. Peterson, A. Yariv, “Interferometry and laser control with solid Fabry–Perot étalons,” Appl. Opt. 5, 985–991 (1966).
    [CrossRef] [PubMed]

1992

1991

1986

1985

1984

1979

A. S. Naidenov, I. Sh. Etsin, “Instrumental function of a Fabry–Perot interferometer illuminated by a Gaussian light beam,” Opt. Spectrosc. (USSR) 46, 409–412 (1979).

1977

1971

1969

1966

1965

1953

R. J. Chabbal, “Recherche des meilleures conditions d’utilisation d’un spectromètre Fabry–Perot,” J. Rech. CNRS 24, 138–149 (1953).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1987), Chap. 7, pp. 323–329.

Bretenaker, F.

Carniglia, C. K.

Chabbal, R. J.

R. J. Chabbal, “Recherche des meilleures conditions d’utilisation d’un spectromètre Fabry–Perot,” J. Rech. CNRS 24, 138–149 (1953).

Cotteverte, J. C.

Del Piano, V. N.

Etsin, I. Sh.

A. S. Naidenov, I. Sh. Etsin, “Instrumental function of a Fabry–Perot interferometer illuminated by a Gaussian light beam,” Opt. Spectrosc. (USSR) 46, 409–412 (1979).

Gonzalez, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 48– 51.

Horowitz, B. R.

Le Floch, A.

Mahapatra, D. P.

Mattoo, S. K.

McGuirk, M.

Moreno, F.

Naidenov, A. S.

A. S. Naidenov, I. Sh. Etsin, “Instrumental function of a Fabry–Perot interferometer illuminated by a Gaussian light beam,” Opt. Spectrosc. (USSR) 46, 409–412 (1979).

Peterson, D. G.

Prikryl, I.

Quesada, A. F.

Ramsay, J. V.

Riesz, R. P.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16, pp. 645–646.

Simon, R.

Slogget, G. J.

Tamir, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1987), Chap. 7, pp. 323–329.

Yariv, A.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Rech. CNRS

R. J. Chabbal, “Recherche des meilleures conditions d’utilisation d’un spectromètre Fabry–Perot,” J. Rech. CNRS 24, 138–149 (1953).

Opt. Spectrosc. (USSR)

A. S. Naidenov, I. Sh. Etsin, “Instrumental function of a Fabry–Perot interferometer illuminated by a Gaussian light beam,” Opt. Spectrosc. (USSR) 46, 409–412 (1979).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1987), Chap. 7, pp. 323–329.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, pp. 48– 51.

Ref. 10, Chap. 2, p. 10.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 16, pp. 645–646.

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Figures (8)

Fig. 1
Fig. 1

Transmission of a Gaussian beam through an FPI for three different geometries characterized by parameters A and B. Case I, normal incidence of a diverging beam—the multireflected beams have different cross sections. Case II, incidence at an angle θ of a beam with negligible divergence—the multireflected beams possess almost equal cross sections but suffer a lateral displacement at each round trip. Case III, incidence at an angle θ of a diverging beam—the multireflected beams have different cross sections and suffer a lateral displacement at the same time. n and n0 are the refractive indices of the media inside and outside, respectively, of the interferometer surfaces; h is the interferometer thickness, and the incidence plane is perpendicular to the y axis, which is normal to the page plane.

Fig. 2
Fig. 2

Instrumental-function peak value of an FPI, illuminated by a Gaussian beam, versus nominal finesse for (a) A = 0 and (b) A = 0.1 (continuous curves). The dashed curves correspond to the same quantity for the case of an interferometer illuminated by a Gaussian beam whose surfaces have a root-mean-square roughness σrms = λ/200.

Fig. 3
Fig. 3

Effective finesse of an FPI, illuminated by a Gaussian beam, versus nominal finesse for (a) A = 0 and (b) A = 0.1 (continuous curves). The dashed curves correspond to the same quantity for the case of an interferometer, illuminated by a plane wave, whose surfaces have a root-mean-square roughness σrms = λ/200.

Fig. 4
Fig. 4

Instrumental-function peak displacement of an FPI, illuminated by a Gaussian beam, versus nominal finesse for A = 0.1 (continuous curves). The dashed line corresponds to the same quantity for the case of an interferometer, illuminated by a plane wave, whose surfaces have a root-mean-square roughness σrms = λ/200.

Fig. 5
Fig. 5

Transmitted intensity profiles (r.u., relative units) of an incident Gaussian beam through an FPI of nominal finesse: (a) = 30 and (b) = 100. The dotted curves are the profiles stated by the classical theory. Case I, normal incidence of a diverging beam; case II, non-normal incidence of a collimated beam; case III, non-normal incidence of a diverging beam. Note that case III is not in (b) because the corresponding curve would be practically undistinguishable from zero.

Fig. 6
Fig. 6

Instrumental function of an FPI illuminated by a plane wave (dotted curves) and a Gaussian beam (continuous curves). The interferometer nominal finesse is (a) = 30 and (b) = 100. Cases I, II, and III are as in Fig. 5. The dashed curves represent the instrumental function of the same interferometer, illuminated by a plane wave, whose surfaces have a root-mean-square roughness σrms = λ/200.

Fig. 7
Fig. 7

Transmitted power of an étalon as a function of the incidence angle θ. We calculated the continuous curve, assuming that a collimated Gaussian beam illuminated the étalon. The dashed curve is the corresponding envelop that we calculated, assuming that a collimated square-cross-section uniform beam illuminated the étalon.

Fig. 8
Fig. 8

Reference frames utilized in the derivation of Eq. (5). Oxz and Oxz″ are the reference frames where the impinging and the transmitted beams are represented, respectively. The significance of the rotated reference frame Oxz′ is explained in the text.

Equations (41)

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τ ( Φ ) = t 1 t 2 exp ( i Φ / 2 ) 1 - r 1 r 2 exp ( i Φ ) ,
Φ = Φ ( θ ) = 4 π λ n n 0 h [ 1 - ( n 0 n ) 2 sin 2 θ ] 1 / 2 ,
ψ 0 ( x ) = ψ ( x , z 0 ) = - + Ψ 0 ( α ) exp ( i 2 π α x ) d α ,
ψ ( x , z ) = exp [ i 2 π λ ( z - z 0 ) ] - + Ψ 0 ( α ) × exp [ - i π λ ( z - z 0 ) α 2 ] exp ( i 2 π α x ) d α .
ψ t ( x , z ) = exp [ i 2 π λ ( z - z 0 - h cos θ ) ] - + Ψ 0 ( α ) × τ [ Φ ( θ ) ] exp [ - i π λ ( z - z 0 - h cos θ ) α 2 ] × exp ( i 2 π α x ) d α ,
θ = θ + cos - 1 [ 1 - ( λ α ) 2 2 ] = θ + sin - 1 ( λ α ) .
Ψ 0 ( α ) = ( 2 π ) 1 / 4 w 0 exp [ - ( π w 0 α ) 2 ] .
ψ t ( x , z ) = ( 2 π ) 1 / 4 w 0 exp [ i 2 π λ ( z - z 0 - h cos θ ) ] - + exp [ - ( π w 0 α ) 2 ] × t 1 t 2 exp [ i φ θ ( α ) / 2 ] exp [ - i π λ ( z - z 0 - h cos θ ) α 2 ] 1 - r 1 r 2 exp [ i φ θ ( α ) ] exp ( i 2 π α x ) d α ,
φ θ ( α ) = 4 π λ n n 0 h ( 1 - ( n 0 n ) 2 × { λ α cos θ + [ 1 - ( λ α ) 2 2 ] sin θ } 2 ) 1 / 2 .
φ θ ( α ) Φ - 2 A ( π w 0 α ) 2 - 2 B ( π w 0 α ) ,
A = n 0 n h z R cos 2 θ ,
B = n 0 h n w 0 sin 2 θ ,
tan 2 2 θ ( B A ) 2 .
ψ ˜ t ( x , z ) = ( π w 0 ) 1 / 2 ψ t ( π w 0 x , z ) ,
ψ ˜ t ( x , z ) = ( 2 π ) 1 / 4 exp [ i 2 π λ ( z - z 0 - h cos θ ) ] - + exp ( - ξ 2 ) × t 1 t 2 exp [ i ( Φ 2 - A ξ 2 - B ξ ) ] exp ( - i z - z 0 - h cos θ z R ξ 2 ) 1 - r 1 r 2 exp [ i ( Φ - 2 A ξ 2 - 2 B ξ ) ] exp ( i 2 π ξ x ) d ξ ,
T = - + ψ t ( x , z ) 2 d x - + ψ ( x , z 0 ) 2 d x = - ψ t ( x , z ) 2 d x .
T = - + Ψ t ( α ) 2 d α ,
Ψ t ( α ) 2 = Ψ 0 ( α ) 2 × τ ( θ ) 2 ,
T = - + Ψ 0 ( α ) 2 × τ ( θ ) 2 d α .
T = T ( Φ ) = ( 2 π ) 1 / 2 - + exp ( - 2 ξ 2 ) 1 + F sin 2 ( Φ 2 - A ξ 2 - B ξ ) d ξ ,
F = 4 R ( 1 - R ) 2
lim A , B 0 T ( Φ ) = T 0 ( Φ ) = 1 1 + F sin 2 Φ 2 .
F = 2 π FWHM = π 2 sin - 1 ( 1 / F ) π 2 F ( if F 1 ) ,
Δ Φ = π F .
A ξ 2 + B ξ < 4 A + 2 B
2 A + B π 2 F ,
T ( Φ ) - T 0 ( Φ ) 4 ( cos Φ + F π sin Φ ) .
ψ t ( x , z ) = ( 2 π ) 1 / 4 w 0 exp [ i 2 π λ ( z - z 0 - h cos θ ) ] - + exp [ - ( π w 0 α ) 2 ] × T exp [ i φ θ ( α ) / 2 ] exp { i tan - 1 [ R sin φ θ ( α ) 1 - R cos φ θ ( α ) ] } [ 1 + R 2 - 2 R cos φ θ ( α ) ] 1 / 2 exp [ - i π λ ( z - z 0 - h cos θ ) α 2 ] exp ( i 2 π α x ) d α ,
ψ t ( x , z ) ( 2 π ) 1 / 4 w 0 exp [ i 2 π λ ( z - z 0 - h cos θ ) ] × - + exp [ - ( π w 0 α ) 2 ] × exp { - i π λ [ z - z 0 - h cos θ + ( 1 + 2 R 1 / 2 π F ) A z R ] α 2 } × exp { i 2 π α [ x - ( 1 2 + R 1 / 2 π F ) B w 0 ] } d α .
Δ z = ( 1 + 2 R 1 / 2 π F ) A z R ,
Δ x = ( 1 2 + R 1 / 2 π F ) B w 0 .
FSR T ( Φ ) d Φ = FSR T 0 ( Φ ) d Φ .
χ ( x , z ) = exp { i 2 π [ α x + γ ( z - z 0 ) ] } .
( x z ) = ( cos θ - sin θ sin θ cos θ ) ( x z ) , ( α γ ) = ( cos θ - sin θ sin θ cos θ ) ( α γ ) ,
χ ( x , z ) = exp { i 2 π [ α ( x - z 0 sin θ ) + γ ( z - z 0 cos θ ) ] } .
χ t ( x , z 1 + h ) = τ { Φ [ sin - 1 ( λ α ) ] } χ ( x , z 1 ) ,
χ t ( x , z ) = τ { Φ [ sin - 1 ( λ α ) ] } χ ( x , z - h ) .
χ t ( x , z ) = τ [ Φ ( θ ) ] χ ( x + h sin θ , z - h cos θ ) ,
χ t ( x , z ) = τ [ Φ ( θ ) ] χ ( x , z - h cos θ ) ,
γ 1 λ - λ α 2 2 ,
χ t ( x , z ) = τ [ Φ ( θ ) ] exp ( i 2 π α x ) exp [ i 2 π λ ( z - z 0 - h cos θ ) ] × exp [ - i π λ ( z - z 0 - h cos θ ) α 2 ] .

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