Abstract

When a spherical mirror interferometer is illuminated by an off-axis ray of light, the repeated reflections cause the ray to trace a path which lies on the surface of a hyperboloid, with the points of reflection on the mirrors on ellipses. Under special conditions, these ellipses may become circles, with the points of reflection displaced by an angle 2θ after every round trip. When 2νθ = 2μπ, ν and μ being integers, the rays retrace their paths. These ray paths give rise to additional resonances which were observed. Pictures of the points of reflection are reproduced. The theory is in good agreement with the experimental observations. In laser amplifiers these ray paths enable one to obtain long effective path lengths in the active medium which may be contained in a thin annular cylindrical or hyperboloidal shell.

© 1964 Optical Society of America

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References

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  1. P. Connes, Rev. Opt. 35, 37 (1956); J. Phys. Radium 19, 262 (1958).
  2. G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
  3. G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
  4. J. R. Pierce, Theory and Design of Electron Beams (Van Nostrand, New York, 1954), pp. 194–197.
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), 322–332.
  6. D. R. Herriott, Appl. Opt. 2, 865 (1963).
    [CrossRef]

1963

1962

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

1961

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

1956

P. Connes, Rev. Opt. 35, 37 (1956); J. Phys. Radium 19, 262 (1958).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), 322–332.

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Connes, P.

P. Connes, Rev. Opt. 35, 37 (1956); J. Phys. Radium 19, 262 (1958).

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Herriott, D. R.

Kogelnik, H.

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Pierce, J. R.

J. R. Pierce, Theory and Design of Electron Beams (Van Nostrand, New York, 1954), pp. 194–197.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), 322–332.

Appl. Opt.

Bell System Tech. J.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Rev. Opt.

P. Connes, Rev. Opt. 35, 37 (1956); J. Phys. Radium 19, 262 (1958).

Other

J. R. Pierce, Theory and Design of Electron Beams (Van Nostrand, New York, 1954), pp. 194–197.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), 322–332.

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Figures (5)

Fig. 1
Fig. 1

Series of equally spaced thin lenses.

Fig. 2
Fig. 2

Projections of intersection points (xn, yn) lying on circle.

Fig. 3
Fig. 3

Photographs of light patterns on mirrors.

Fig. 4
Fig. 4

Closed ray paths.

Fig. 5
Fig. 5

Spectra of spherical mirror resonator. (a) Axial illumination. (b) Off-axis illumination.

Equations (21)

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x n = x 0 cos n θ + d 4 f d ( x 0 + 2 f x 0 ) sin n θ ,
cos θ = 1 ( d / 2 f ) .
0 < d f < 4
x n = A sin ( n θ + α ) ,
tan α = 4 f d 1 / ( 1 + 2 f x 0 x 0 )
A 2 = 4 f 4 f d ( x 0 2 + d x 0 x 0 + d f x 0 2 ) .
y n = B sin ( n θ + β ) .
A = B
α = β ± π 2 ( or tan α · tan β = 1 ) .
y 0 2 = x 0 2 ( 4 f d 1 ) ,
A 2 = x 0 3 + y 0 2 = 4 f d x 0 2 ,
x 0 = 2 x 0 d = A f d .
φ n = n θ + α .
2 ν θ = 2 π ,
2 ν θ = 2 μ π
T 0 = T e j φ 1 R e 2 j φ ,
F 0 = π R 1 R ,
Q 0 = φ 1 R .
T ν = T e j φ 1 R ν e 2 j ν φ ,
F = π R ν / 2 1 R ν ,
Q = ν φ 1 R ν .

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