Abstract

As the finesse of a Fabry–Perot optical cavity increases to about 20,000, the fringe width (~10 kHz for a 50-cm cavity) is sufficiently narrow for transverse-mode splittings to be resolved by using a highly stabilized ring dye laser. A perturbative theory interprets this effect as a slight deviation of the cavity from cylindrical symmetry, the magnitude asymmetry being at the level of a few tenths of a nanometer. The understanding of these splittings will permit of accurate optical frequency measurements by the recently proposed optical–radio-frequency divider.

© 1986 Optical Society of America

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References

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  1. R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
    [CrossRef]
  2. R. G. DeVoe, C. Fabre, R. G. Brewer, in Laser Spectroscopy VII, T. Hänsch, R. Shen, eds. (Springer-Verlag, New York, 1985), p. 358.
  3. R. W. P. Drever, Gravitational Radiation, Les Houches 1982 Summer School (North-Holland, Amsterdam, 1983), p. 321.
  4. M. M. Popov, Opt. Spectrosc. (USSR) 25, 170, 213 (1968).
  5. J. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
    [CrossRef] [PubMed]
  6. J. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).
  7. We consider elsewhere2 the mirror-reflection phase-shift correction.
  8. For a vector wave equation treatment, see L. W. Davis, Phys. Rev. A 30, 3092 (1984).
    [CrossRef]
  9. R. G. DeVoe, R. G. Brewer, Phys. Rev. Lett. 50, 1269 (1983).
    [CrossRef]
  10. A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
    [CrossRef]
  11. H. Kogelnik, W. W. Rigrod, Proc. IRE 50, 220 (1962).
  12. In addition to variations of the mirror surface that produce elliptical symmetry, there are of course random variations in the mirror surface at the angstrom level that have no particular symmetry at all. It may be possible to map such variations in the mirror surface by studying splittings of higher-order transverse modes, which display multiplets of p + q + 1 frequencies. Since the Hermite polynomials form a complete set in two dimensions, it should be possible to map an arbitrary surface in this way.

1984 (2)

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[CrossRef]

For a vector wave equation treatment, see L. W. Davis, Phys. Rev. A 30, 3092 (1984).
[CrossRef]

1983 (1)

R. G. DeVoe, R. G. Brewer, Phys. Rev. Lett. 50, 1269 (1983).
[CrossRef]

1982 (1)

A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
[CrossRef]

1970 (1)

J. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

1969 (1)

1968 (1)

M. M. Popov, Opt. Spectrosc. (USSR) 25, 170, 213 (1968).

1962 (1)

H. Kogelnik, W. W. Rigrod, Proc. IRE 50, 220 (1962).

Arnaud, J.

J. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

J. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
[CrossRef] [PubMed]

Brewer, R. G.

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[CrossRef]

R. G. DeVoe, R. G. Brewer, Phys. Rev. Lett. 50, 1269 (1983).
[CrossRef]

A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
[CrossRef]

R. G. DeVoe, C. Fabre, R. G. Brewer, in Laser Spectroscopy VII, T. Hänsch, R. Shen, eds. (Springer-Verlag, New York, 1985), p. 358.

Davis, L. W.

For a vector wave equation treatment, see L. W. Davis, Phys. Rev. A 30, 3092 (1984).
[CrossRef]

DeVoe, R. G.

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[CrossRef]

R. G. DeVoe, R. G. Brewer, Phys. Rev. Lett. 50, 1269 (1983).
[CrossRef]

A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
[CrossRef]

R. G. DeVoe, C. Fabre, R. G. Brewer, in Laser Spectroscopy VII, T. Hänsch, R. Shen, eds. (Springer-Verlag, New York, 1985), p. 358.

Drever, R. W. P.

R. W. P. Drever, Gravitational Radiation, Les Houches 1982 Summer School (North-Holland, Amsterdam, 1983), p. 321.

Fabre, C.

R. G. DeVoe, C. Fabre, R. G. Brewer, in Laser Spectroscopy VII, T. Hänsch, R. Shen, eds. (Springer-Verlag, New York, 1985), p. 358.

Kogelnik, H.

J. Arnaud, H. Kogelnik, Appl. Opt. 8, 1687 (1969).
[CrossRef] [PubMed]

H. Kogelnik, W. W. Rigrod, Proc. IRE 50, 220 (1962).

Popov, M. M.

M. M. Popov, Opt. Spectrosc. (USSR) 25, 170, 213 (1968).

Rigrod, W. W.

H. Kogelnik, W. W. Rigrod, Proc. IRE 50, 220 (1962).

Schenzle, A.

A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

J. Arnaud, Bell Syst. Tech. J. 49, 2311 (1970).

Opt. Spectrosc. (USSR) (1)

M. M. Popov, Opt. Spectrosc. (USSR) 25, 170, 213 (1968).

Phys. Rev. A (3)

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[CrossRef]

For a vector wave equation treatment, see L. W. Davis, Phys. Rev. A 30, 3092 (1984).
[CrossRef]

A. Schenzle, R. G. DeVoe, R. G. Brewer, Phys. Rev. A 25, 2606 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

R. G. DeVoe, R. G. Brewer, Phys. Rev. Lett. 50, 1269 (1983).
[CrossRef]

Proc. IRE (1)

H. Kogelnik, W. W. Rigrod, Proc. IRE 50, 220 (1962).

Other (4)

In addition to variations of the mirror surface that produce elliptical symmetry, there are of course random variations in the mirror surface at the angstrom level that have no particular symmetry at all. It may be possible to map such variations in the mirror surface by studying splittings of higher-order transverse modes, which display multiplets of p + q + 1 frequencies. Since the Hermite polynomials form a complete set in two dimensions, it should be possible to map an arbitrary surface in this way.

R. G. DeVoe, C. Fabre, R. G. Brewer, in Laser Spectroscopy VII, T. Hänsch, R. Shen, eds. (Springer-Verlag, New York, 1985), p. 358.

R. W. P. Drever, Gravitational Radiation, Les Houches 1982 Summer School (North-Holland, Amsterdam, 1983), p. 321.

We consider elsewhere2 the mirror-reflection phase-shift correction.

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Figures (2)

Fig. 1
Fig. 1

Schematic of the geometry of an optical cavity for which the two concave mirrors have orthogonal radii of curvature (R1, R1′) and (R2, R2′), respectively, the mirror spacing is L, and one mirror is rotated relative to the other about the z axis by the angle θ.

Fig. 2
Fig. 2

TEM01 and TEM10 transverse modes exhibit a splitting of 160 kHz as shown here in dispersion when the cavity length is swept. On this scale, the fringe width of ~9 kHz is not resolved.

Equations (20)

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ν n p q = c 2 L [ n + ( p + 1 2 ) φ 1 / π + ( q + 1 2 ) φ 2 / π ]
ν n 00 = ( c / 2 L ) ( n + φ 0 / π )
φ 0 ( φ 1 + φ 2 ) / 2.
φ 0 = π ( L / c ) [ ( ν n 10 - ν n 00 ) + ( ν n 01 - ν n 00 ) ]
2 ψ x 2 + 2 ψ y 2 = 2 i k ψ z
ψ 00 = exp { - i k 2 [ a ( z ) x 2 + b ( z ) y 2 + 2 c ( z ) x y ] + i φ 0 ( z ) } ,
a ( z ) = z - z 0 + α ( z - z 0 ) 2 - ( α 2 + β 2 ) , b ( z ) = z - z 0 - α ( z - z 0 ) 2 - ( α 2 + β 2 ) , c ( z ) = β ( z - z 0 ) 2 - ( α 2 + β 2 ) , φ ( z ) = i 2 ln [ ( z - z 0 ) 2 - ( α 2 + β 2 ) ] ,
Re ( a ) x 2 + Re ( b ) y 2 + 2 Re ( c ) x y = ( x cos θ / 2 - y sin θ / 2 ) 2 / R 2 + ( x sin θ / 2 + y cos θ / 2 ) 2 / R 2 ,
Re ( a + b ) = 1 R 2 + 1 R 2 , Re ( a - b ) = ( 1 R 2 - 1 R 2 ) cos θ , Re ( c ) = 1 2 ( 1 R 2 - 1 R 2 ) sin θ ,
α = [ ( δ 1 - δ 1 - δ 2 + δ 2 ) k 1 - i ( - δ 1 + δ 1 - δ 2 + δ 2 ) k 2 ] cos θ , β = [ ( δ 1 - δ 1 + δ 2 - δ 2 ) k 1 - i ( - δ 1 + δ 1 + δ 2 - δ 2 ) k 2 ] sin θ , z 0 = i z R + ( δ 1 + δ 1 - δ 2 - δ 2 ) k 1 + i ( δ 1 + δ 1 + δ 2 + δ 2 ) k 2 .
ψ ( 1 ) = [ d ( z ) x + e ( z ) y ] ψ 00
d ( z ) = a ( z ) - γ c ( z ) ,             e ( z ) = - γ b ( z ) + c ( z ) ,
γ = λ cot θ ± ( 1 + λ 2 cot 2 θ ) 1 / 2 ,
ν i = ( c / 2 L ) [ n + φ i / π ] ,             n an integer ,
k L - φ i = n π ,
φ i φ i ( - L 2 ) - φ i ( L 2 )
φ 0 = 2 tan - 1 L 2 Z R - L 8 R z R ( δ 1 + δ 1 + δ 2 + δ 2 ) .
φ 1 , 2 = φ 0 ± Δ φ ,
Δ φ = L 8 R z R [ ( δ 1 + δ 1 - δ 2 + δ 2 ) 2 cos 2 θ + ( δ 1 - δ 1 - δ 2 + δ 2 ) 2 sin 2 θ ] 1 / 2 ,
ν 10 - ν 01 = c 2 L 2 Δ φ π

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