Abstract

A quasi-paraxial ray tracing calculation is used to investigate the illumination requirements of confocal cavities. It is shown that the additional reflections (four per round trip as compared with two for the traditional Fabry-Perot interferometer) do not degrade the resolving power of the etalon. The need to clarify the concept of free spectral range is established and a tentative revision is suggested and justified. Construction and performance details of a 15-cm confocal interferometer are given. With a free spectral range C/2R0 of 1 GHz, a finesse of 333 has been achieved yielding a peak width of 3 MHz and resolving power of 1.58 × 108 (at 6328 Å).

© 1968 Optical Society of America

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References

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  1. R. Chabbal, J. Rech, Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24, 138 (1953).
  2. P. Connes, Rev. Opt. 35, 39 (1956).
  3. P. Connes, J. Phys. Radium 19, 262 (1958).
    [CrossRef]
  4. D. A. Jackson, Proc. Roy. Soc. London Series A 263, 289 (1961).
    [CrossRef]
  5. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  6. J. E. Mack, D. P. McNutt, F. L. Roesler, R. Chabbal, Appl. Opt. 2, 873 (1963).
  7. D. R. Herriott, Appl. Opt. 2, 861 (1963).
    [CrossRef]
  8. J. Haisma, G. Bouwhuis, Phys. Rev. Lett. 12, 287 (1964).
    [CrossRef]
  9. R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  10. J. R. Johnson, J. Opt. Sco. Amer. 56, 1447A (1966).
  11. W. K. Kahn, Polytechnic Institute of Brooklyn Research Rept. PIBMRI–1285–65.
  12. J. R. Pierce, Theory and Design of Electron Beams (D. Van Nostrand Company, Inc., New York, 1954), pp. 194–197.
  13. M. Hercher, Appl. Opt. 7, 951 (1968).
    [CrossRef] [PubMed]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.8.4.
  15. G. Toraldo di Francia, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.
  16. J. R. Johnson, Appl. Opt. 6, 1930 (1967).
    [CrossRef] [PubMed]
  17. J. R. Johnson, IEEE J. Quantum Electron. QE-4, 37 (1968).
    [CrossRef]

1968

M. Hercher, Appl. Opt. 7, 951 (1968).
[CrossRef] [PubMed]

J. R. Johnson, IEEE J. Quantum Electron. QE-4, 37 (1968).
[CrossRef]

1967

1966

J. R. Johnson, J. Opt. Sco. Amer. 56, 1447A (1966).

1964

1963

1961

D. A. Jackson, Proc. Roy. Soc. London Series A 263, 289 (1961).
[CrossRef]

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

1958

P. Connes, J. Phys. Radium 19, 262 (1958).
[CrossRef]

1956

P. Connes, Rev. Opt. 35, 39 (1956).

1953

R. Chabbal, J. Rech, Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24, 138 (1953).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.8.4.

Bouwhuis, G.

J. Haisma, G. Bouwhuis, Phys. Rev. Lett. 12, 287 (1964).
[CrossRef]

Boyd, G. D.

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

Chabbal, R.

J. E. Mack, D. P. McNutt, F. L. Roesler, R. Chabbal, Appl. Opt. 2, 873 (1963).

R. Chabbal, J. Rech, Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24, 138 (1953).

Connes, P.

P. Connes, J. Phys. Radium 19, 262 (1958).
[CrossRef]

P. Connes, Rev. Opt. 35, 39 (1956).

Fork, R. L.

Gordon, J. P.

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

Haisma, J.

J. Haisma, G. Bouwhuis, Phys. Rev. Lett. 12, 287 (1964).
[CrossRef]

Hercher, M.

Herriott, D. R.

Jackson, D. A.

D. A. Jackson, Proc. Roy. Soc. London Series A 263, 289 (1961).
[CrossRef]

Johnson, J. R.

J. R. Johnson, IEEE J. Quantum Electron. QE-4, 37 (1968).
[CrossRef]

J. R. Johnson, Appl. Opt. 6, 1930 (1967).
[CrossRef] [PubMed]

J. R. Johnson, J. Opt. Sco. Amer. 56, 1447A (1966).

Kahn, W. K.

W. K. Kahn, Polytechnic Institute of Brooklyn Research Rept. PIBMRI–1285–65.

Kogelnik, H.

Mack, J. E.

McNutt, D. P.

Pierce, J. R.

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

Rech, J.

R. Chabbal, J. Rech, Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24, 138 (1953).

Roesler, F. L.

Toraldo di Francia, G.

G. Toraldo di Francia, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.8.4.

Appl. Opt.

Bell Syst. Tech. J.

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

Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24

R. Chabbal, J. Rech, Centre Nat. Rech. Sci. Lab. Bellevue Parts No. 24, 138 (1953).

IEEE J. Quantum Electron.

J. R. Johnson, IEEE J. Quantum Electron. QE-4, 37 (1968).
[CrossRef]

J. Opt. Sco. Amer.

J. R. Johnson, J. Opt. Sco. Amer. 56, 1447A (1966).

J. Phys. Radium

P. Connes, J. Phys. Radium 19, 262 (1958).
[CrossRef]

Phys. Rev. Lett.

J. Haisma, G. Bouwhuis, Phys. Rev. Lett. 12, 287 (1964).
[CrossRef]

Proc. Roy. Soc. London Series

D. A. Jackson, Proc. Roy. Soc. London Series A 263, 289 (1961).
[CrossRef]

Rev. Opt.

P. Connes, Rev. Opt. 35, 39 (1956).

Other

W. K. Kahn, Polytechnic Institute of Brooklyn Research Rept. PIBMRI–1285–65.

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

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), Sec. 8.8.4.

G. Toraldo di Francia, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963), p. 157.

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Figures (14)

Fig. 1
Fig. 1

Geometry of symmetric nonconfocal cavity.

Fig. 2
Fig. 2

Confocal cavity showing the trajectory of a ray when fourth order terms are considered.

Fig. 3
Fig. 3

Confocal cavity showing AB type and CD type beams.

Fig. 4
Fig. 4

Photograph showing the interferometer package without covers or electronics.

Fig. 5
Fig. 5

Schematic of the interferometer package identifying the several major components.

Fig. 6
Fig. 6

Display due to a laser oscillating in three adjacent axial modes (ΔνL = 300 MHz); two successive orders are shown with near ideal illumination.

Fig. 7
Fig. 7

Same presentation as Fig. 6, but with improper illumination.

Fig. 8
Fig. 8

Display due to a laser oscillating in both (0, 0) and (0, 1) spatial modes; scan, one free spectral range.

Fig. 9
Fig. 9

Display showing complex mode structure from a laser with a nearly confocal cavity; scan, one free spectral range.

Fig. 10
Fig. 10

Time exposure showing gain profile with Lamb dip: at most two and at least one mode oscillating.

Fig. 11
Fig. 11

Display similar to Fig. 10, but with at most four and at least three modes oscillating.

Fig. 12
Fig. 12

Display showing gain profile exhibiting little or no mode interaction.

Fig. 13
Fig. 13

Display showing the effect of 3.39–0.6328-μ competition on the gain profile at 0.6328 μ.

Fig. 14
Fig. 14

Display due to two independent single frequency lasers both oscillating at very near the same frequency.

Equations (62)

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φ j + 1 = φ j + θ j ( D / R 0 ) ,
θ j + 1 = θ j - 2 φ j + 1 .
φ j + 2 + 2 u φ j + 1 + φ j = 0 ,
u = - cos ψ ,
φ p = [ φ 0 + ( 1 + u ) θ 0 ] sin ρ ψ sin ψ - φ 0 sin ( p - 1 ) ψ sin ψ ,
θ p = - [ ( 1 + 2 u ) θ 0 + 2 φ 0 ] sin p ψ sin ψ - θ 0 sin ( p - 1 ) ψ sin ψ ,
0 < D < 2 R 0 ,
φ p = ( φ 0 + θ 0 ) sin ( p π / 2 ) - φ 0 sin [ ( p - 1 ) π / 2 ] ,
θ p = - ( θ 0 + 2 φ 0 ) sin ( p π / 2 ) - θ 0 sin [ ( p - 1 ) π / 2 ] .
φ p + 4 = φ p , θ p + 4 = θ p ,
p j s ^ j + R 0 n j + 1 + d z ^ - R 0 n ^ j = 0.
ρ j = R 0 [ 1 + u - φ j 2 - u ( 1 + u ) θ j 2 / 2 - ( 1 + u ) θ j φ j ] .
L = R 0 p = 0 3 [ 1 + u - φ p 2 - u ( 1 + u ) θ p 2 / 2 - ( 1 + u ) θ p φ p ] .
L = 4 R 0 + ( 4 + 2 θ 0 2 + 4 θ 0 φ 0 ) ( D - R 0 ) .
Δ = 2 R 0 φ 0 2 ( θ 0 + φ 0 ) .
L = 4 R 0 + R 0 φ 0 2 ( θ 0 + φ 0 ) 2 ,
a B = a 0 t 2 q = 0 r 4 q exp [ i k q ( 4 R 0 + λ ) + i ϕ A B ] = a 0 t 2 exp ( i ϕ A B ) [ 1 - r 4 exp ( i 4 k R 0 ) ] - 1 ,
I B = I 0 T 2 [ ( 1 - R 2 ) 2 + 4 R 2 sin 2 ( 2 k R 0 ) ] - 1 .
a 0 t 2 r 2 exp [ i ( φ A B + 2 k R 0 + π ) ] ,
a D = a 0 t 2 r 2 exp [ i ( ϕ A B + 2 k R 0 + π ) ] × [ 1 - r 4 exp ( i 4 κ R 0 ) ] - 1 ,
I D = I 0 T 2 R 2 [ ( 1 - R 2 ) 2 + 4 R 2 sin 2 ( 2 k R 0 ) ] - 1 ;
I D , max = R 2 I B , max = I 0 R 2 T 2 / ( 1 - R 2 ) 2 ,
ν = p c / 4 R 0 ;
I D , min = R 2 I B , min = I 0 R 2 T 2 / ( 1 + R 2 ) 2
ν = ( p + 1 2 ) ( c / 4 R 0 ) .
I D , max I B , max I 0 / 4.
Δ ν R = ( c / 4 R 0 ) ( 1 - R 2 ) / π R .
a B = a 0 t 2 exp [ i ( ϕ A B + k 0 · r ) ] [ 1 - r 4 exp ( i 4 k R 0 ) ] - 1 ,
a D = a 0 t 2 r 2 exp [ i ( ϕ A B + 2 k R 0 + π + k 1 · r ) ] × [ 1 - r 4 exp ( i 4 k R 0 ) ] - 1 ,
a = a 0 t 2 exp ( i ϕ A B ) { exp ( i k 0 · r ) - r 2 exp [ i ( 2 k R 0 + k 1 · r ) ] } × [ 1 - r 4 exp ( i 4 k R 0 ) ] - 1 .
I = I 0 [ T / ( 1 - R ) ] 2 [ 1 + F sin 2 ( k R 0 + k x sin ψ ) ] × [ 1 + F sin 2 ( k R 0 ) ] - 1 [ 1 + F cos 2 ( k R 0 ) ] - 1 ,
F = 4 R / ( 1 - R ) 2 ;
k R 0 = s π / 2 ,
k R 0 = 2 q ( π / 2 ) ,
I = I 0 [ T / ( 1 - R ) ] 2 [ 1 + F sin 2 ( k x sin ψ ) ] ( 1 + F ) - 1
k R 0 = ( 2 q + 1 ) π / 2
I = I 0 [ T / ( 1 - R ) ] 2 [ 1 + F cos 2 ( k x sin ψ ) ] ( 1 + F ) - 1 .
I = I 0 [ T / ( 1 + R ) ] 2 I 0 ,             s even ,
I = I 0 [ T / ( 1 - R ) ] 2 I 0             s odd .
I = I 0 [ T / ( 1 - R ) ] 2 [ 1 + F cos 2 ( k R 0 ) ] - 1 .
I max = I 0 T 2 / ( 1 - R ) 2 ,
ν = ( q + 1 2 ) ( c / 2 R 0 ) .
Δ ν = c / 2 R 0 ,
Δ ν R = ( c / 2 R 0 ) ( 1 - R ) / π R 1 / 2 .
ν = q ( c / 2 R 0 ) ,
I min - I 0 T 2 / ( 1 + R 2 ) 2 .
Δ ν R / Δ ν R = 1 - ( 1 - R ) 2 / 8 + .
ψ = q π / N ,
D = 2 R sin 2 ( q π / 2 N ) ,
ν q m n = ( c / 2 R 0 ) [ q + ( m + n + 1 ) / 2 ] .
I = [ T / ( 1 - R ) ] 2 [ 1 + F sin 2 ( k R 0 ) ] - 1 [ 1 + F cos 2 ( k R 0 ) ] - 1 × 0 2 π d β 0 ψ 0 d ψ I ( ψ ) [ 1 + F sin 2 ( k R 0 + k x sin ψ ) ] .
I = [ T / ( 1 - R ) ] 2 I 0 ( 1 + F / 2 ) [ 1 + F sin 2 ( k R 0 ) ] - 1 × [ 1 + F cos 2 ( k R 0 ) ] - 1 .
T 2 / 2 ( 1 - R ) 2 .
Δ L = ( r r 0 ) 2 / R 0 3 .
r r 0 / R ( λ R 0 / 2 N ) 1 2 ,
L = 4 R 0 + ( 4 + 2 θ 0 2 + 4 θ 0 φ 0 ) ( D - R 0 ) .
φ 0 = 0 ,
Δ L = 2 θ 0 2 ( D - R 0 ) ,
Δ ϕ = ( 4 π ν / c ) ( D - R 0 ) θ 0 2 .
( 4 π ν / c ) D - R 0 θ 0 2 π / 2 N ,
( D - R 0 ) / R 0 ( 4 θ 0 2 R ) - 1 ,
D - R 0 2.1 μ

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