Abstract

The recently developed coherence theory of stable laser resonators is applied to an ideal plane-parallel Fabry–Perot resonator. We make use of an explicit, integral-type, biorthogonal expansion of the exact, non-Hermitian, scalar propagation kernel. The resonator admits various spatially partially coherent modes because of the degeneracy of the conventional transverse modes. The spatial second-order modes of the Fabry–Perot resonator containing only homogeneous plane waves correspond to an eigenvalue of unity, and several types of such partially coherent modes are identified. These include the diagonal superpositions of the coherent Fox–Li modes and the general expressions for the cross-spectral densities associated with the propagation-invariant modes and the self-imaging modes. Some special cases are analyzed as illustrations.

© 1994 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  2. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  3. J. Turunen, A. Vasara, A. T. Friberg, “Propagation-invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [CrossRef]
  4. G. C. Sherman, “Introduction to the angular spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
    [CrossRef]
  5. E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  6. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,”J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  7. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).
  8. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  9. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,”J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]
  10. Since we are dealing with kernels and wave fields that are not square integrable, we will generally replace the sum of the coherent modes in the Mercer expansion by an integral over a distribution of such orthogonal modes. In this context, see also Eqs. (35) and (46) and the discussions immediately following them.
  11. See, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. Press, Cambridge, 1965), Chap. 5.
  12. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  13. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989).
    [CrossRef]
  14. F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  15. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  16. W. D. Montgomery, “Self-imaging objects of infinite aperture,”J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  17. G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
    [CrossRef]
  18. P. Swaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
    [CrossRef]
  19. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  20. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), formula (9.1.21).

1991 (1)

1989 (2)

1988 (1)

1987 (2)

F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1984 (1)

1983 (1)

G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
[CrossRef]

1982 (1)

1981 (1)

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1976 (1)

1975 (1)

1972 (1)

1967 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Agarwal, G. S.

Carter, W. H.

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Friberg, A. T.

Goldberg, R. R.

See, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. Press, Cambridge, 1965), Chap. 5.

Gori, F.

F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Indebetouw, G.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Mandel, L.

Marchand, E. W.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Montgomery, W. D.

Pavadoni, C.

F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Sherman, G. C.

G. C. Sherman, “Introduction to the angular spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[CrossRef]

Swaykowski, P.

Turunen, J.

Vasara, A.

Wolf, E.

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

G. Indebetouw, “Self-imaging through a Fabry–Perot resonator,” Opt. Acta 30, 1463–1471 (1983).
[CrossRef]

Opt. Commun. (2)

F. Gori, G. Guattari, C. Pavadoni, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (5)

G. C. Sherman, “Introduction to the angular spectrum representation of optical fields,” in Applications of Mathematics in Modern Optics, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.358, 31–38 (1982).
[CrossRef]

Since we are dealing with kernels and wave fields that are not square integrable, we will generally replace the sum of the coherent modes in the Mercer expansion by an integral over a distribution of such orthogonal modes. In this context, see also Eqs. (35) and (46) and the discussions immediately following them.

See, for example, R. R. Goldberg, Fourier Transforms (Cambridge U. Press, Cambridge, 1965), Chap. 5.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), formula (9.1.21).

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Figures (7)

Fig. 1
Fig. 1

Ideal Fabry–Perot resonator formed by mirrors A and B.

Fig. 2
Fig. 2

Equation (48) with λ/2d = 0.01 presented for some small values of n: solid line, n = 0; dashed curves, n = ±1; dashed–dotted curves, n = ±2; dotted curves, n = ±5. The curves corresponding to positive values of n are below the straight solid line.

Fig. 3
Fig. 3

Illustration of the circles of radii α and β in the ξ and η spaces, on which the amplitudes associated with the self-imaging two-beam interference example assume nonzero (singular) values. When the angular dependence on each circle is taken to be a single Fourier-series term, the interfering wave fields are Bessel beams.

Fig. 4
Fig. 4

Axial cross sections of the optical intensities [in units of (k/2π)2] corresponding to Eq. (51) as a function of (a) kx and (b) ky: solid curves, r = 0; dashed curves, r = 1; dashed–dotted curves, r = 2; dotted curves, r = 5. The parameters are α = 0.44, 〈|aα|2〉 = 〈|az|2〉 = 1.0, and a α * a z = 0.5.

Fig. 5
Fig. 5

Three-dimensional contour plot of the optical intensity corresponding to the case r = 1 of Fig. 4.

Fig. 6
Fig. 6

Axial cross sections of the magnitudes of the complex degree of spatial coherence associated with the wave fields characterized by Eq. (51) as a function of (a) kx1 and (b) ky1, when ρ2 = 0. The parameter values are the same, and the different line types correspond to the same integers r as in Fig. 4.

Fig. 7
Fig. 7

Three-dimensional contour plot of the degree of spatial coherence corresponding to the case r = 1 of Fig. 6.

Equations (52)

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U 1 ( ρ ) = A ( q ) exp ( i k q · ρ ) d 2 q ,
U 2 ( ρ ) = A ( q ) exp { i k [ q · ρ + m ( q ) 2 d ] } d 2 q ,
m ( q ) = { 1 - q 2 if q 1 i q 2 - 1 if q > 1 .
U 2 ( ρ ) = L ( ρ , ρ ) U 1 ( ρ ) d 2 ρ ,
L ( ρ , ρ ) = k 2 ( 2 π ) 2 exp { i k [ q · ( ρ - ρ ) + m ( q ) 2 d ] } d 2 q
L ( ρ , ρ ) φ ( q , ρ ) d 2 ρ = α ( q ) φ ( q , ρ ) .
φ ( q , ρ ) = k 2 π exp ( i k q · ρ ) ,
α ( q ) = exp [ i k m ( q ) 2 d ] ,
L * ( ρ , ρ ) χ ( q , ρ ) d 2 ρ = β ( q ) χ ( q , ρ ) ,
χ ( q , ρ ) = k 2 π exp ( i k q · ρ ) ,
β ( q ) = exp [ - i k m * ( q ) 2 d ] .
L ( ρ , ρ ) = α ( q ) φ ( q , ρ ) χ * ( q , ρ ) d 2 q ,
φ * ( q , ρ ) χ ( q , ρ ) d 2 q = δ ( 2 ) ( ρ - ρ ) ,
φ * ( q 1 , ρ ) χ ( q 2 , ρ ) d 2 ρ = δ ( 2 ) ( q 1 - q 2 ) .
W ( r 1 , r 2 ) = U * ( r 1 ) U ( r 2 ) ,
W ( ρ 1 , ρ 2 ) L * ( ρ 1 , ρ 1 ) L ( ρ 2 , ρ 2 ) d 2 ρ 1 d 2 ρ 2 = σ W ( ρ 1 , ρ 2 ) .
σ > 0.
μ ( ρ 1 , ρ 2 ) = W ( ρ 1 , ρ 2 ) [ I ( ρ 1 ) I ( ρ 2 ) ] - 1 / 2 ,
I ( ρ ) = W ( ρ , ρ )
α * ( q 1 ) α ( q 2 ) w ( q 1 , q 2 ) φ * ( q 1 , ρ 1 ) φ ( q 2 , ρ 2 ) d 2 q 1 d 2 q 2 = σ W ( ρ 1 , ρ 2 ) ,
w ( q 1 , q 2 ) = W ( ρ 1 , ρ 2 ) χ ( q 1 , ρ 1 ) χ * ( q 2 , ρ 2 ) d 2 ρ 1 d 2 ρ 2 .
[ σ - α * ( ξ ) α ( η ) ] w ( ξ , η ) = 0.
σ = α * ( ξ ) α ( η ) ,
σ ξ , η = exp [ i 2 k d ( 1 - η 2 - 1 - ξ 2 ) ] .
2 k d ( 1 - η 2 - 1 - ξ 2 ) = 2 π n ,
W σ ( ρ 1 , ρ 2 ) = D ( σ ) w ( ξ , η ) φ * ( ξ , ρ 1 ) φ ( η , ρ 2 ) d 2 ξ d 2 η .
σ ξ , η = exp ( - 2 k d η 2 - 1 ) exp ( - i 2 k d 1 - ξ 2 ) ,
σ ξ , η = exp ( - 2 k d ξ 2 - 1 ) exp ( i 2 k d 1 - η 2 ) .
2 k d 1 - ξ 2 = 2 π n ,
2 k d 1 - η 2 = 2 π m ,
σ ξ , η = exp [ - 2 k d ( ξ 2 - 1 + η 2 - 1 ) ] .
w ( ξ , η ) = w 0 δ ( 2 ) ( ξ - q ) δ ( 2 ) ( η - q ) ,
W ( ρ 1 , ρ 2 ) = w 0 φ * ( q , ρ 1 ) φ ( q , ρ 2 ) = w 0 k 2 ( 2 π ) 2 exp [ i k q · ( ρ 2 - ρ 1 ) ] .
w ( ξ , η ) = w 0 ( ξ ) δ ( 2 ) ( ξ - η ) ,
W ( ρ 1 , ρ 2 ) = w 0 ( ξ ) φ * ( ξ , ρ 1 ) φ ( ξ , ρ 2 ) d 2 ξ .
μ ( ρ 1 , ρ 2 ) = w ¯ 0 ( ξ ) exp [ i k ξ · ( ρ 2 - ρ 1 ) ] d 2 ξ ,
w ¯ 0 ( ξ ) = w 0 ( ξ ) [ w 0 ( ξ ) d 2 ξ ] - 1 = w 0 ( ξ ) [ ( 2 π ) 2 k 2 I ( ρ ) ] - 1 ..
μ ( ρ 1 , ρ 2 ) = sin k ρ 2 - ρ 1 k ρ 2 - ρ 1 ,
w ( ξ , η ) = w 1 ( ξ , θ ξ , θ η ) δ ( ξ - η ) ,
W ( ρ 1 , ρ 2 ) = 0 1 0 2 π ξ 2 w 1 ( ξ , θ ξ , θ η ) φ * ( ξ , θ ξ , ρ 1 ) × φ ( ξ , θ η , ρ 2 ) d ξ d θ ξ d θ η .
w 1 ( ξ , θ ξ , θ η ) = w 1 ( ξ , θ ξ ) δ ( θ ξ - θ η ) ,
W ( ρ 1 , ρ 2 ) = 0 1 0 2 π ξ 2 w 1 ( ξ , θ ξ ) φ * ( ξ , θ ξ , ρ 1 ) φ ( ξ , θ ξ , ρ 2 ) d ξ d θ ξ .
w 1 ( ξ , θ ξ ) = w 1 ( θ ξ ) δ ( ξ - α ) ,
W ( ρ 1 , ρ 2 ) = A J 0 ( k α ρ 2 - ρ 1 ) ,
w 1 ( ξ , θ ξ , θ η ) = w 1 * ( θ ξ ) w 1 ( θ η ) δ ( ξ - α ) ,
W ( ρ 1 , ρ 2 ) = U * ( ρ 1 ) U ( ρ 2 ) ,
U ( ρ ) = 0 2 π α w 1 ( θ ξ ) φ ( α , θ ξ , ρ ) d 2 θ ξ .
η = [ 1 - ( 1 - ξ 2 + n λ 2 d ) 2 ] 1 / 2 ,
W ( ρ 1 , ρ 2 ) = n l ξ 1 0 2 π w n ( ξ , θ ξ , θ η ) × ξ [ 1 - ( 1 - ξ 2 + n λ / 2 d ) 2 ] 1 / 2 φ * ( ξ , θ ξ , ρ 1 ) × φ { [ 1 - ( 1 - ξ 2 + n λ / 2 d ) 2 ] 1 / 2 , θ η , ρ 2 } × d ξ d θ ξ d θ η ,
W ( ρ 1 , ρ 2 ) = a α 2 k 2 α 2 exp [ i r ( ϕ 2 - ϕ 1 ) ] J r ( k α ρ 1 ) J r ( k α ρ 2 ) + a β 2 k 2 β 2 exp [ i s ( ϕ 2 - ϕ 1 ) ] J s ( k β ρ 1 ) J s ( k β ρ 2 ) + a α * a β k 2 α β ( - i ) r - s exp [ i ( s ϕ 2 - r ϕ 1 ) ] × J r ( k α ρ 1 ) J s ( k β ρ 2 ) + a α a β * k 2 α β ( i ) r - s × exp [ i ( r ϕ 2 - s ϕ 1 ) ] J s ( k β ρ 1 ) J r ( k α ρ 2 ) ,
W ( ρ 1 , ρ 2 ) = a α 2 k 2 α 2 exp [ i r ( ϕ 2 - ϕ 1 ) ] J r ( k α ρ 1 ) J r ( k α ρ 2 ) + a z 2 k 2 ( 2 π ) 2 + a α * a z k 2 2 π α ( - i ) r exp ( - i r ϕ 1 ) J r ( k α ρ 1 ) + a α a z * k 2 2 π α ( i ) r exp ( i r ϕ 2 ) J r ( k α ρ 2 ) .
w ( ξ , η , z ) = w ( ξ , η ) exp [ i k ( 1 - η 2 - 1 - ξ 2 ) z ] .

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