Abstract

The theory, design, and use of the confocal spherical mirror Fabry-Perot interferometer (FPS) is described in detail. Topics covered include performance of an FPS for small departures from the confocal mirror separation, optimization of the (resolution) × (light gathering power) product, factors limiting realizable finesse, mode matching considerations, alignment procedures, and general design considerations. Two specific instruments are described. One is a versatile spectrum analyzer with piezoelectric scanning; the other is a highly stable etalon with fixed spacing. Examples of the performance of these instruments are given.

© 1968 Optical Society of America

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Errata

Michael Hercher, "Errata: The Spherical Mirror Fabry-Perot Interferometer," Appl. Opt. 7, 1336_1-1336 (1968)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-7-7-1336_1

References

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  1. P. Connes, Rev. Opt. 35, 37 (1956).
  2. P. Connes, J. Phys. Radium 19, 262 (1958).
    [CrossRef]
  3. P. Connes, in Quantum Electronics and Coherent LightP. A. Miles, Ed. (Academic Press, Inc., New York, 1964), p. 198ff.
  4. W. H. Steel, Interferometry (Cambridge University Press, Cambridge, 1967), p. 123.
  5. K. M. Baird, G. R. Hanes, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press, Inc., New York, 1967), Vols. 4 and 5, p. 350.
  6. R. L. Fork, D. R. Herriott, H. Kogelnik, Appl. Opt. 3, 1471 (1964).
    [CrossRef]
  7. G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 453 (1961).
  8. P. Laures, Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]

1967

1964

1961

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 453 (1961).

1958

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

1956

P. Connes, Rev. Opt. 35, 37 (1956).

Baird, K. M.

K. M. Baird, G. R. Hanes, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press, Inc., New York, 1967), Vols. 4 and 5, p. 350.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 453 (1961).

Connes, P.

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

P. Connes, Rev. Opt. 35, 37 (1956).

P. Connes, in Quantum Electronics and Coherent LightP. A. Miles, Ed. (Academic Press, Inc., New York, 1964), p. 198ff.

Fork, R. L.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 453 (1961).

Hanes, G. R.

K. M. Baird, G. R. Hanes, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press, Inc., New York, 1967), Vols. 4 and 5, p. 350.

Herriott, D. R.

Kogelnik, H.

Laures, P.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge University Press, Cambridge, 1967), p. 123.

Appl. Opt.

Bell Sys. Tech. J.

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 453 (1961).

J. Phys. Radium

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

Rev. Opt.

P. Connes, Rev. Opt. 35, 37 (1956).

Other

P. Connes, in Quantum Electronics and Coherent LightP. A. Miles, Ed. (Academic Press, Inc., New York, 1964), p. 198ff.

W. H. Steel, Interferometry (Cambridge University Press, Cambridge, 1967), p. 123.

K. M. Baird, G. R. Hanes, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic Press, Inc., New York, 1967), Vols. 4 and 5, p. 350.

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Figures (17)

Fig. 1
Fig. 1

General ray path in a spherical mirror Fabry-Perot interferometer.

Fig. 2
Fig. 2

(a) Ray path in an FPS in the paraxial approximation (reentrant rays), (b) aberrated ray path, showing intersection of rays at point P.

Fig. 3
Fig. 3

Near confocal FPS fringe patterns. At each value of , the solid curves give the radii of the circular interference fringes for the case of a monochromatic source and a bright fringe on axis. The dashed line shows the spot size radius ρs for a finesse of 100, and the dotted line defines the zone of best focus as a function of . (Appendix I shows how to change the scales for different wavelengths and mirror separations.)

Fig. 4
Fig. 4

Calculated distribution of light in an FPS fringe pattern for a monochromatic source and various values for F, the finesse. Note the broad central fringe ( = 0, r = 10 cm).

Fig. 5
Fig. 5

Calculated FPS instrumental profiles for two different detector aperture radii. These correspond to the spectra which would be recorded using a monochromatic source in the scanning mode of operation ( = 0, r = 10 cm). (a) ρ = 0.05, (b) ρ = 0.2.

Fig. 6
Fig. 6

Generalized picture of a spectrometer or monochrometer.

Fig. 7
Fig. 7

FPS instrumental transmission as a function of the (absorption:transmission) ratio of the mirror coatings.

Fig. 8
Fig. 8

Computed FPS characteristics as a function of detector aperture radius ρ for different values of F, the finesse. Arrows indicate the value of ρs in each case ( = 0, r = 10 cm). The maxima in the curves shown in (d) define the aperture radius giving the best compromise between resolving power and peak transmitted power.

Fig. 9
Fig. 9

(a) Scan and fringe displays of an FPS in normal operation. Note the secondary fringe pattern. (b) Scan and fringe displays of a very nearly mode-matched FPS. The alignment of the FPS relative to the source has been adjusted to eliminate the secondary fringe pattern, resulting in a doubling of both the free spectral range and the instrumental transmission. ( ~ 20 μ, r = 5 cm; three-mode laser source.)

Fig. 10
Fig. 10

A versatile FPS instrument. (a) Optical layout showing the FPS etalon, lens L1, and detector aperture; (b) arrangement for scanning; (c) arrangement for observing and recording fringe pattern (detector removed).

Fig. 11
Fig. 11

Schematic of a highly stable fixed mirror FPS etalon: (1) outer case (Al alloy); (2) end plates with windows; (3) Cer-Vit etalon spacer; (4) fused quartz mirrors; (5) fixed mirror cell (Invar); (6), (7) adjustable mirror cell (Invar); (8) ports in etalon spacer; (9) phosphor bronze springs holding etalon; (10) fixture for evacuating chamber and pressure scanning.

Fig. 12
Fig. 12

An FPS spectrum analyzer for scanning or static mode of operation: (1) removable detector (photodiode); (2) soft O-ring for mounting FPS etalon; (3) quartz mirrors (r = 5 cm); (4) piezo-electric transducer/etalon spacer; (5) scanning voltage terminal; (6) auxiliary lens (focal point is between mirrors); (7) mounting flange; (8), (9) adjustable mirror cell; (10) fixed mirror cell; (11) outer case. (The mirror cells are designed to compensate for the thermal expansion of the etalon spacer.)

Fig. 13
Fig. 13

Typical fringe patterns in the vicinity of confocal separation for a 10-cm FPS. In each case, the source is a single mode He–Ne laser. Variations in the fringe patterns in each horizontal row were obtained by making small changes (<λ/4) in the mirror separation. (a) = −70 μ, (b) = 0, (c) = 70 μ.

Fig. 14
Fig. 14

Observed instrumental profiles for different detector aperture diameters D. Light source was a 1-cm wide collimated single mode laser beam. ( ~ 0; r = 10 cm.) The 0.3-cm aperture is clearly the best compromise between signal amplitude and resolution.

Fig. 15
Fig. 15

Spectra of a single mode laser obtained with 5-cm and 10-cm scanning FPS instruments. Top: 10-cm FPS, measured finesse F ~ 154; middle: 5-cm (broadband mirrors), F ~ 148; bottom: single mode He–Ne gain profile, showing Lamb’s dip (see text).

Fig. 16
Fig. 16

Spectra of an adjustable multimode He–Ne gas laser. Various numbers of modes were excited by adjusting the laser mirrors. The 5-cm FPS shown in Fig. 12 was used. 330 MHz/cm.

Fig. 17
Fig. 17

Top left: spectrum of a He–Ne gas laser operating in three axial modes and the TEM00 transverse mode. ( ~ 20 μ, r = 5 cm, free spectral range 1500 MHz). Top right: same as above, but with an additional TEM01 mode in oscillation. Bottom: spectrum of a 10-nsec single mode pulse from a Q-switched ruby laser (see text). ( ~ 0, r = 10 cm, free spectral range 750 MHz).

Tables (2)

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Table I List of Symbols

Tables Icon

Table II Characteristics of Some Multilayer Reflective Coatings

Equations (52)

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Δ 0 = ρ 1 2 ρ 2 2 cos 2 θ / r 3 + higher order terms .
Δ = ρ 1 2 ρ 2 2 cos 2 θ / r 3 + 2 ( ρ 1 2 + ρ 2 2 ) / r + higher order terms .
Δ ( ρ ) ρ 4 / r 3 + 4 ρ 2 / r 2 ,
I 1 ( ρ , λ ) = I 0 [ T / ( 1 - R 2 ) ] 2 { 1 + [ 2 R / ( 1 - R 2 ) ] 2 × sin 2 [ δ ( ρ , λ ) / 2 ] } - 1
I 2 ( ρ , λ ) = R 2 I 1 ( ρ , λ )
δ ( ρ , λ ) = ( 2 π / λ ) [ Δ ( ρ ) + 4 ( r + ) ] .
ρ 4 / r 3 + 4 ρ 2 / r 2 = m λ ,
ρ m = [ - 2 r ± ( 4 2 r 2 + m λ r 3 ) 1 2 ] 1 2 .
ρ m = [ ( m - ξ ) λ r 3 ] 1 4 ,
R ν / Δ ν m = λ / Δ λ m ,
Δ ν m = c ( 1 - R 2 ) / 4 π r R .
F Δ ν f / Δ ν m .
Δ ν m = C / 4 r F ,
R = 4 r F / λ .
I ( ρ ) = ( T 1 - R 2 ) I 0 1 + ( 2 F / π ) 2 sin 2 ( δ / 2 ) .
F R = π R / ( 1 - R 2 ) π / 2 ( 1 - R ) .
I ( ν - ν 0 ) ( 1 4 ) [ T / ( 1 - R ) ] 2 I 0 ( ν ) × { 1 + [ 2 ( ν - ν 0 ) F c / 4 r ] 2 } - 1 .
ρ s = ( r 3 λ / F ) 1 4 .
I p ( ν - ν 0 ) = 2 π 0 ρ I ( ξ ) x d x ,
ξ = [ ν - ν 0 ( 1 + x 4 / 4 r 4 ) ]
I ( ξ ) = 1 4 { ( T / ( 1 - R ) 2 I 0 { 1 + ( 2 F / π ) 2 sin - 2 [ π ξ / ( c / 4 r ) ] } - 1 ,
F i = π c τ i / 2 r .
F - 1 = i ( F i ) - 1 ,
F f m / 2.
L D 10 - [ 5 ( ρ 0 2 / r λ ) + 1 ]
F D ( FPP ) D 2 / 2 λ d ,
F L ~ π / 2 L .
P ν = N ν A Ω T 0 .
P = N U T 0 .
T 0 = ( 1 + R 2 ) ( T / [ 1 - R 2 ] ) 2 1 2 [ T / ( 1 - R ) ] 2 , for R 1.
T 0 1 2 [ 1 + ( A / T ) ] - 2
U = [ π ρ s 2 ] [ π ρ s 2 / r 2 ] = π 2 r λ / F ,
( R / U ) F P S 0.7 ( 2 F / π λ ) 2 .
( U R ) F P P 0.7 ( π D 2 / 2 ) .
( U R ) F P S 0.7 ( 4 π 2 r 2 ) .
α = ( λ / F D ) ,
U F P S / U F P P = 4 r d F 2 α 2 / λ 2 ,
r * = ( λ / α ) / 2 F .
ν m n q = ( c / 2 d ) { q + ( 1 / π ) ( 1 + m + n ) × cos - 1 [ ( 1 - d / b 1 ) ( 1 - d / b 2 ) ] 1 2 } .
d = ( c / 2 ν 0 ) { q + ( 1 + m + n ) × cos - 1 [ ( 1 - d / b 1 ) ( 1 - d / b 2 ) ] 1 2 } .
ν m n q [ c / 4 ( r + ) ] [ 2 q + ( 1 + m + n ) ] ,
( r + ) ( c / 4 ν 0 ) [ 2 q + ( 1 + m + n ) ] ,
( r + ) = ( c / 4 ν 0 ) ( 2 l + 1 ) ; l an integer , ( m + n ) even ,
( r + ) = ( c / 4 ν 0 ) ( 2 l ) ; ( m + n ) odd .
( r + ) = c l / 4 ν 0 ; l an integer ,
Δ ν f ( multimode ) c / 4 r
Δ ν f ( single transverse mode ) c / 2 r .
max = π r / 2 ( 1 + m + n ) F
τ = 2 r F / π c = ( 2 π Δ ν m ) - 1 ,
Δ ν ( ν 2 - ν 1 ) = ( ν ¯ / 4 r 4 ) [ ( ρ 2 4 - ρ 1 4 ) M - 4 + 4 r ( ρ 2 2 - ρ 1 2 ) M - 2 ]
( ρ n 4 r n 3 ) + ( 4 n ρ n 2 / r n 2 ) = ( m - ξ ) ,
( ρ s ) n = ( r n 3 / F ) 1 4 .

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