Abstract

Like all short-wavelength optics, whispering-gallery mirrors that are intended for use with soft x rays require a high-quality surface finish. A coupled-mode description of scatter in whispering-gallery mirrors provides a convenient way of understanding the effects of surface imperfections and of establishing fabrication tolerances. Different kinds of imperfection have different effects on mirror performance and can be tolerated to different degrees. Figure errors, for example, are not critical, but periodic toolmarks are potentially disastrous. The acceptable amounts of random waviness and microscopic roughness depend on the correlation lengths involved. Overall, the tolerances are strict but less severe than those for normal-incidence optics.

© 1992 Optical Society of America

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References

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  1. A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).
  2. A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).
  3. Lord Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).
    [CrossRef]
  4. Lord Rayleigh, “Further applications of Bessel’s functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100–109 (1914).
    [CrossRef]
  5. J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part I: Diffraction of whispering-gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
    [CrossRef]
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), pp. 446–452.
  7. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
    [CrossRef]
  8. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  9. J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part III: Mirror shapes,” submitted to IEEE J. Quantum Electron.
  10. M. B. Priestly, Spectral Analysis and Time Series (Academic, New York, 1981).
  11. H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 4 101, 209–214 (1954).
  12. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987), Section 5.3.
  13. J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part II: Mirrors with non-uniform curvature,” IEEE J. Quantum Electron. 28, 254–264 (January1992).
    [CrossRef]
  14. R. Dashen, D. Wurmser, “A new theory for scattering from a surface” J. Math. Phys. 32, 971–985 (1991).
    [CrossRef]
  15. R. Dashen, D. Wurmser, “Approximate versions of the scattering amplitude,” J. Math. Phys. 32, 986–996 (1991).
    [CrossRef]
  16. R. Dashen, D. Wurmser, “Applications of the new scattering formalism: The Dirichlet boundary condition,” J. Math. Phys. 32, 997–1003 (1991).
    [CrossRef]
  17. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  18. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

1992 (1)

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part II: Mirrors with non-uniform curvature,” IEEE J. Quantum Electron. 28, 254–264 (January1992).
[CrossRef]

1991 (4)

R. Dashen, D. Wurmser, “A new theory for scattering from a surface” J. Math. Phys. 32, 971–985 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Approximate versions of the scattering amplitude,” J. Math. Phys. 32, 986–996 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Applications of the new scattering formalism: The Dirichlet boundary condition,” J. Math. Phys. 32, 997–1003 (1991).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part I: Diffraction of whispering-gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[CrossRef]

1988 (1)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

1985 (1)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

1979 (1)

1954 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 4 101, 209–214 (1954).

1914 (1)

Lord Rayleigh, “Further applications of Bessel’s functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100–109 (1914).
[CrossRef]

1910 (1)

Lord Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), pp. 446–452.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987), Section 5.3.

Bennett, J. M.

J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. 69, 31–47 (1979).
[CrossRef]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Braud, J. P.

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part II: Mirrors with non-uniform curvature,” IEEE J. Quantum Electron. 28, 254–264 (January1992).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part I: Diffraction of whispering-gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part III: Mirror shapes,” submitted to IEEE J. Quantum Electron.

Dashen, R.

R. Dashen, D. Wurmser, “Approximate versions of the scattering amplitude,” J. Math. Phys. 32, 986–996 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Applications of the new scattering formalism: The Dirichlet boundary condition,” J. Math. Phys. 32, 997–1003 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “A new theory for scattering from a surface” J. Math. Phys. 32, 971–985 (1991).
[CrossRef]

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 4 101, 209–214 (1954).

Elson, J. M.

Hagelstein, P. L.

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part II: Mirrors with non-uniform curvature,” IEEE J. Quantum Electron. 28, 254–264 (January1992).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part I: Diffraction of whispering-gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part III: Mirror shapes,” submitted to IEEE J. Quantum Electron.

Kozhevnikov, I. V.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Priestly, M. B.

M. B. Priestly, Spectral Analysis and Time Series (Academic, New York, 1981).

Rayleigh, Lord

Lord Rayleigh, “Further applications of Bessel’s functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100–109 (1914).
[CrossRef]

Lord Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).
[CrossRef]

Sagitov, S. I.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987), Section 5.3.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), pp. 446–452.

Turyanskil, A. G.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Vinogradov, A. V.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Wurmser, D.

R. Dashen, D. Wurmser, “A new theory for scattering from a surface” J. Math. Phys. 32, 971–985 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Approximate versions of the scattering amplitude,” J. Math. Phys. 32, 986–996 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Applications of the new scattering formalism: The Dirichlet boundary condition,” J. Math. Phys. 32, 997–1003 (1991).
[CrossRef]

Yakushkin, I. G.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

Zorev, N. N.

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

IEEE J. Quantum Electron. (2)

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part II: Mirrors with non-uniform curvature,” IEEE J. Quantum Electron. 28, 254–264 (January1992).
[CrossRef]

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part I: Diffraction of whispering-gallery modes,” IEEE J. Quantum Electron. 27, 1069–1077 (1991).
[CrossRef]

J. Math. Phys. (3)

R. Dashen, D. Wurmser, “A new theory for scattering from a surface” J. Math. Phys. 32, 971–985 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Approximate versions of the scattering amplitude,” J. Math. Phys. 32, 986–996 (1991).
[CrossRef]

R. Dashen, D. Wurmser, “Applications of the new scattering formalism: The Dirichlet boundary condition,” J. Math. Phys. 32, 997–1003 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Philos. Mag. (2)

Lord Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).
[CrossRef]

Lord Rayleigh, “Further applications of Bessel’s functions of high order to the whispering gallery and allied problems,” Philos. Mag. 27, 100–109 (1914).
[CrossRef]

Proc. Inst. Electr. Eng. Part 4 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. Inst. Electr. Eng. Part 4 101, 209–214 (1954).

Sov. Phys. JETP (2)

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, I. G. Yakushkin, “Phenomenon of total external reflection of x-rays,” Sov. Phys. JETP 62, 1225–1229 (1985).

A. V. Vinogradov, N. N. Zorev, I. V. Kozhevnikov, S. I. Sagitov, A. G. Turyanskil, “X-ray scattering by highly polished surfaces,” Sov. Phys. JETP 67, 1631–1638 (1988).

Other (7)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), pp. 446–452.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987), Section 5.3.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

J. P. Braud, P. L. Hagelstein, “Whispering-gallery laser resonators—Part III: Mirror shapes,” submitted to IEEE J. Quantum Electron.

M. B. Priestly, Spectral Analysis and Time Series (Academic, New York, 1981).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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Figures (4)

Fig. 1
Fig. 1

Coordinates (n, s) used in the description of whispering-gallery modes. The coordinate n represents the distance from the ideal mirror surface, and the coordinate s represents the distance along the surface from some arbitrary initial point s = 0 and is taken in the direction of beam propagation.

Fig. 2
Fig. 2

Transverse structure in the direction that is normal to the mirror surface of whispering-gallery modes, E1(n, s) and E6(n, s). The mirror surface lies toward the left at n = 0.

Fig. 3
Fig. 3

Profile h(s) of a simulated wavy surface along with its autocovariance R(x) and spectral density S(p). Notice that the underlying 400-μm periodicity of the waviness can still be seen in spite of the fairly short 200-μm correlation length. The vertical scales for h(s) and R(x) are in units of the rms roughness δ2, and that for S(p) is in units of μm δ2. Since the vertical and horizontal scales differ, the slopes of the surface profile appear greatly exaggerated.

Fig. 4
Fig. 4

Profile h(s) of a simulated rough surface along with its autocovariance R(x) and spectral density S(p). The vertical scales for h(s) and R(x) are in units of the rms roughness δ2, and that for S(p) is in units of μm δ2. Since the vertical and horizontal scales differ, the slopes of the surface profile appear to be greatly exaggerated.

Equations (39)

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E j ( n , s ) = Ai ( n / N + a j ) N A i ( a j ) exp [ i ( k + K j ) s ] .
0 d n E j ( n , s ) E m * ( n , s ) = { 1 for j = m 0 for j m .
a j - [ 3 π 2 ( j - 1 / 4 ) ] 2 / 3 .
γ j = a j 1 / 2 ( λ / π R ) 1 / 3 ,
N = ( λ 2 R / 8 π 2 ) 1 / 3 .
K j = a j π / λ R 2 = a j / D ,
D = ( λ R 2 / π ) 1 / 3 .
c ( j m ) = i ( - 1 ) j + m + 1 0 L d s D h ( s ) N exp [ i ( a j - a m ) s D ] .
h ( s ) = H sin ( π s / L ) .
h ( s ) = 2 δ sin [ ( a j - a m ) s / D ] ,
δ max = λ / ( 4 π 5 ) λ / 28.
R ( x ) = h ( s ) h ( s + x ) .
c ( j m ) 2 = 0 L 0 L d s d s ˜ D 2 h ( s ) h ( s ˜ ) N 2 exp [ i ( a j - a m ) ( s - s ˜ ) D ] .
c ( j m ) 2 = L D 2 N 2 S ( a j - a m D ) ,
S ( p ) = - d x R ( x ) exp ( - i p x ) .
R ( x ) = δ 2 exp ( - x / l ) cos ( 2 π x / u ) .
c ( j m ) 2 = π a j - a m δ 2 L D N 2 .
δ max 1 29 ( λ 5 R ) 1 / 6 .
R ( x ) = δ 2 exp ( - x / l ) ,             S ( p ) = 2 δ 2 l 1 + ( p l ) 2 .
P = m j c ( j m ) 2 = L D 2 N 2 m j 2 δ 2 l 1 + [ ( a j - a m ) l / D ] 2 .
P 2 δ 2 l L D 2 N 2 0 d m 1 + ( 3 π m / 2 ) 4 / 3 ( l / D ) 2
= 2 δ 2 l L D 2 N 2 D 3 / 2 2 l 3 / 2
= 2 L R ( k δ ) 2 ( k l ) 1 / 2 .
δ max 1 31 ( λ 3 l ) 1 / 4 .
2 ψ + k 2 ψ = 0.
C d A ν ^ · ( ψ 1 ψ 2 - ψ 2 ψ 1 ) = 0.
ψ 1 ( n , s ) = m = 1 c m ( s ) E m ( n , s ) ,
c m ( s ) = h ( s ) d n ψ 1 ( n , s ) E m * ( n , s ) .
c j ( 0 ) = 1 , c m ( 0 ) = 0 for m j .
c m ( L ) = c ( j m ) .
ψ 2 ( n , s ) = E m * ( n , s ) .
I entrance = 0 d n ( ψ 1 ψ 2 s - ψ 2 ψ 1 s ) | s = 0 . = 0.
I exit = 0 d n ( - ψ 1 ψ 2 s + ψ 2 ψ 1 s ) | s = L = 2 i ( k + K m ) c m ( L ) .
I surface = 0 L d s ( ψ 1 ψ 2 n - ψ 2 ψ 1 n ) | n = 0 .
ψ 1 ( 0 , s ) - h ( s ) ψ 1 n .
I surface 0 L d s [ - h ( s ) ψ 1 n ψ 2 n ] | n = 0 .
n E j ( n , s ) | n = 0 = ( - 1 ) j + 1 exp [ i ( k + K j ) s ] N N ,
I surface = - ( - 1 ) j + m N 3 0 L d s h ( s ) exp [ i ( K j - K m ) s ] .
c m ( L ) = - i ( - 1 ) j + m 2 ( k + K m ) N 3 0 L d s h ( s ) exp [ i ( K j - K m ) s ] .

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