Abstract

The addition of one or more partially reflecting mirrors to a traditional two-mirror Fabry–Perot interferometer results in a multimirror Fabry–Perot interferometer. A superposition of all possible multiple reflected beams is described with a general theory for multimirror interferometers, featuring matrices analogous to the theory of multilayer thin films. However, the parameters in the matrix elements are mirror reflection coefficients and spacings instead of the usual refractive indices and layer thicknesses of thin films. The transmission characteristics of two-, three-, and four-mirror Fabry–Perot optical filters are described. It is shown that a suitable choice of reflection coefficients results in transmission properties that can be described approximately with Butterworth profiles, which are known from network analysis of electrical circuits.

© 1985 Optical Society of America

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References

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  1. R. E. Loughhead, R. J. Bray, N. Brown, “Instrumental profile of a triple Fabry−Perot interferometer for use in solar spectroscopy,” Appl. Opt. 17, 415−419 (1978).
    [Crossref] [PubMed]
  2. F. L. Roesler, in Methods of Experimental Physics, L. Marton, ed. (Academic, New York, 1974), Vol. 12A, p. 540.
  3. C. Roychoudhuri, M. Hercher, “Stable multipass Fabry−Perot interferometer: design and analysis,” Appl. Opt. 16, 2514−2520 (1977).
    [Crossref] [PubMed]
  4. W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
    [Crossref]
  5. S. Chandra, “Very high frequency intensity oscillations in coupled optical cavities,” Appl. Opt. 21, 3069−3070 (1982).
    [Crossref] [PubMed]
  6. H. A. MacLeod, Thin Film Optical Filters (Hilger, London, 1969).
  7. V. K. Aatre, Network Theory and Filter Design (Wiley, New Delhi, 1980).
  8. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).
  9. E. Hecht, A. Zajac, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1977).
  10. W. Gunning, “Double cavity electrooptic Fabry–Perot tunable filter,” Appl. Opt. 21, 3129–3131 (1982).
    [Crossref] [PubMed]
  11. H. J. Pain, The Physics of Vibrations and Waves, 3rd ed. (Wiley, New York, 1983).

1983 (1)

W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
[Crossref]

1982 (2)

W. Gunning, “Double cavity electrooptic Fabry–Perot tunable filter,” Appl. Opt. 21, 3129–3131 (1982).
[Crossref] [PubMed]

S. Chandra, “Very high frequency intensity oscillations in coupled optical cavities,” Appl. Opt. 21, 3069−3070 (1982).
[Crossref] [PubMed]

1978 (1)

1977 (1)

Aatre, V. K.

V. K. Aatre, Network Theory and Filter Design (Wiley, New Delhi, 1980).

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

Bray, R. J.

Brown, N.

Chandra, S.

Gunning, W.

W. Gunning, “Double cavity electrooptic Fabry–Perot tunable filter,” Appl. Opt. 21, 3129–3131 (1982).
[Crossref] [PubMed]

Hecht, E.

E. Hecht, A. Zajac, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1977).

Hercher, M.

Logan, R. A.

W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
[Crossref]

Loughhead, R. E.

MacLeod, H. A.

H. A. MacLeod, Thin Film Optical Filters (Hilger, London, 1969).

Olsson, N. A.

W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
[Crossref]

Pain, H. J.

H. J. Pain, The Physics of Vibrations and Waves, 3rd ed. (Wiley, New York, 1983).

Roesler, F. L.

F. L. Roesler, in Methods of Experimental Physics, L. Marton, ed. (Academic, New York, 1974), Vol. 12A, p. 540.

Roychoudhuri, C.

Tang, W. T.

W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

Zajac, A.

E. Hecht, A. Zajac, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1977).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

W. T. Tang, N. A. Olsson, R. A. Logan, “Highspeed direct single frequency modulation with large tuning rate and frequency excursion in cleaved-coupled-cavity semiconductor lasers,” Appl. Phys. Lett. 42, 650−655 (1983).
[Crossref]

Other (6)

F. L. Roesler, in Methods of Experimental Physics, L. Marton, ed. (Academic, New York, 1974), Vol. 12A, p. 540.

H. A. MacLeod, Thin Film Optical Filters (Hilger, London, 1969).

V. K. Aatre, Network Theory and Filter Design (Wiley, New Delhi, 1980).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

E. Hecht, A. Zajac, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1977).

H. J. Pain, The Physics of Vibrations and Waves, 3rd ed. (Wiley, New York, 1983).

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Figures (13)

Fig. 1
Fig. 1

The amplitudes at mirrors i and i + 1 out of a stack of N mirrors.

Fig. 2
Fig. 2

Transmission of a two-mirror FPI with r12 = r22 = 0.5 as a function of ϕ.

Fig. 3
Fig. 3

The transmission of a three-mirror FPI with r12r22 = r32 = 0.5 and ϕ1 = ϕ2 = ϕ.

Fig. 4
Fig. 4

Transmission of a three-mirror FPI with r1 = r3 = 0.837 and ϕ1 = ϕ2 = ϕ around ϕ = π/2 and for various values of r22.

Fig. 5
Fig. 5

Transmission of a three-mirror FPI with r12 = r32 = 0.7, r22 = 0.97, and ϕ1/ϕ2 = 11.

Fig. 6
Fig. 6

Transmission of a three-mirror FPI with r12 = r32 = 0.7, r22 = 0.97, and ϕ1/ϕ2 = 11/9.

Fig. 7
Fig. 7

Transmission of a three-mirror FPI with r12 = r32 = 0.7, r22 = 0, and ϕ1/ϕ2 = 11/9; this is actually a two-mirror FPI with the same width as the three-mirror FPI of Fig. 6.

Fig. 8
Fig. 8

Transmission of a three-mirror FPI with r12 = 0.36, r32 = 0.64, and r22 = 0.36, 0.63, 0.895, 0.98.

Fig. 9
Fig. 9

Transmission of a three-mirror FPI with r12 = 0.6, r22 = 0.972, r32 = 0.8, and ϕ1/ϕ2 = 11. Note the decreased sidelobes and broadened peaks as compared with Fig. 5.

Fig. 10
Fig. 10

Transmission of a three-mirror FPI with r12 = 0.8, r22 = 0.972, r32 = 0.6, and ϕ1/ϕ2 = 11. Note the increased sidelobes and narrower peaks as compared to Fig.5.

Fig. 11
Fig. 11

Transmission of a four-mirror FPI with r12 = r22 = r32 = r42 = r2 = 0.5 and ϕ1 = ϕ2 = ϕ3 = ϕ.

Fig. 12
Fig. 12

Transmission of a four-mirror FPI with r12 = r42 = 0.7 and ϕ1 = ϕ2 = ϕ3 = ϕ around ϕ= π/2 and for various values of r22 = r32.

Fig. 13
Fig. 13

Flat-top or approximate Butterworth transmission for two-, three-, and four-mirror FPI’s with the same bandwidth.

Equations (49)

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E i + 1 + exp ( i ϕ i ) = t i E i + + r i E i + 1 exp ( + i ϕ i ) ,
E i = r i E i + + t i E i + 1 exp ( + i ϕ i ) ,
E i = ( r i / t i ) E i + 1 + exp ( i ϕ i ) + ( 1 / t i ) E i + 1 exp ( + i ϕ i ) .
( E i + E i ) = 1 / t i [ exp ( i ϕ i ) r i exp ( + i ϕ i ) r i exp ( i ϕ i ) exp ( + i ϕ i ) ] ( E i + 1 + E i + 1 ) .
( E 1 + E 1 ) = 1 t 1 t 2 t N 1 [ exp ( i ϕ 1 ) r 1 exp ( i ϕ 1 ) r 1 exp ( i ϕ 1 ) exp ( i ϕ 1 ) ] × [ exp ( i ϕ 2 ) r 2 exp ( i ϕ 2 ) r 2 exp ( i ϕ 2 ) exp ( i ϕ 2 ) ] × . × [ exp ( i ϕ N 1 ) r N 1 exp ( i ϕ N 1 ) r N 1 exp ( i ϕ N 1 ) exp ( i ϕ N 1 ) ] ( E N + E N )
( E 1 + E 1 ) = 1 t 1 t 2 t N 1 ( A B C D ) ( E N + E N ) ,
t = E N + 1 + / E 1 + = t N E N + / E 1 + = t 1 t 2 t N E N + / ( A E N + + B E N )
t = t 1 t 2 t N / ( A r N B ) .
t = t 1 t 2 / [ exp ( i ϕ 1 ) + r 1 r 2 exp ( + i ϕ 1 ) ] .
T = t t * = t 1 2 t 2 2 / [ 1 + r 1 2 r 2 2 + 2 r 1 r 2 cos ( 2 ϕ 1 ) ] .
ϕ 1 = ( 2 k + 1 ) π / 2 or L 1 = ( 2 k + 1 ) λ / 4 .
t = t 1 t 2 t 3 / [ exp ( i ϕ 1 i ϕ 2 ) + r 1 r 2 exp ( + i ϕ 1 i ϕ 2 ) + r 2 r 3 exp ( i ϕ 1 + i ϕ 2 ) + r 1 r 3 exp ( + i ϕ 1 + i ϕ 2 ) ]
T = t t * = ( t 1 t 2 t 3 ) 2 / D 3 ,
D 3 = 1 + ( r 1 r 2 ) 2 + ( r 2 r 3 ) 2 + ( r 1 r 3 ) 2 + 2 r 1 r 2 ( 1 + r 3 2 ) cos ( 2 ϕ 1 ) + 2 r 2 r 3 ( 1 + r 1 2 ) cos ( 2 ϕ 2 ) + 2 r 1 r 3 cos ( 2 ϕ 1 + 2 ϕ 2 ) + 2 r 1 r 2 2 r 3 cos ( 2 ϕ 1 2 ϕ 2 ) .
T = ( 1 r 2 ) 3 / [ ( 1 r 2 ) 3 + ( r ( 1 + r 2 ) + 2 r x ) 2 ] ,
T = 1 r 2 if ϕ = ( 2 k + 1 ) π / 2 or x = 1 ,
T = ( 1 r 2 ) 3 / ( 1 + 3 r 2 ) 2 if ϕ = k π or x = + 1 .
T = 1 if x = cos ( 2 ϕ ) = ( 1 + r 2 ) / 2 .
T = ( 1 r 1 2 ) 2 ( 1 r 2 2 ) / { ( 1 r 1 2 ) 2 ( 1 r 2 2 ) + [ r 2 ( 1 + r 1 2 ) + 2 r 1 x ] 2 } .
2 r 1 cos ( 2 ϕ ) + r 2 ( 1 + r 1 2 ) = 0 .
r 2 2 r 1 / ( 1 + r 1 2 ) .
T = ( 1 r 1 2 ) 2 ( 1 r 2 2 ) / { 1 + 2 r 1 2 r 2 2 + r 1 4 + 2 r 1 r 2 ( 1 + r 1 2 ) [ cos ( 2 ϕ 1 ) + cos ( 2 ϕ 2 ) ] + 2 r 1 2 cos ( 2 ϕ 1 + 2 ϕ 2 ) + 2 r 1 2 r 2 2 cos ( 2 ϕ 1 2 ϕ 2 ) } .
r 1 + r 3 r 2 ( 1 + r 1 r 3 ) = 0 ,
r 1 = r 2 r 3 1 r 2 r 3 , r 2 = r 1 + r 3 1 + r 1 r 3 , r 3 = r 2 r 1 1 r 1 r 2 .
t = t 1 t 2 t 3 t 4 / { exp [ i ( ϕ 1 ϕ 2 ϕ 3 ) ] + r 1 r 2 exp [ i ( ϕ 1 ϕ 2 ϕ 3 ) ] + r 2 r 3 exp [ i ( ϕ 1 + ϕ 2 ϕ 3 ) ] + r 1 r 3 exp [ i ( ϕ 1 + ϕ 2 ϕ 3 ) ] + r 3 r 4 exp [ i ( ϕ 1 ϕ 2 + ϕ 3 ) ] + r 1 r 2 r 3 r 4 exp [ i ( ϕ 1 ϕ 2 + ϕ 3 ) ] + r 2 r 4 exp [ i ( ϕ 1 + ϕ 2 + ϕ 3 ) ] + r 1 r 4 exp [ i ( ϕ 1 + ϕ 2 + ϕ 3 ) ] } .
T = t t * = ( t 1 t 2 t 3 t 4 ) 2 / D 4 ,
D 4 = 1 + ( r 1 r 2 ) 2 + ( r 2 r 3 ) 2 + ( r 1 r 3 ) 2 + ( r 3 r 4 ) 2 + ( r 1 r 2 r 3 r 4 ) 2 + ( r 2 r 4 ) 2 + ( r 1 r 4 ) 2 + 2 cos ( 2 ϕ 1 ) r 1 r 2 ( 1 + r 3 2 + r 3 2 r 4 2 + r 4 2 ) + 2 cos ( 2 ϕ 2 ) r 2 r 3 ( 1 + r 1 2 + r 1 2 r 4 2 + r 4 2 ) + 2 cos ( 2 ϕ 3 ) r 3 r 4 ( 1 + r 1 2 + r 1 2 r 2 2 + r 2 2 ) + 2 cos ( 2 ϕ 1 + 2 ϕ 2 ) r 1 r 3 ( 1 + r 4 2 ) + 4 cos ( 2 ϕ 1 + 2 ϕ 3 ) r 1 r 2 r 3 r 4 + 2 cos ( 2 ϕ 2 + 2 ϕ 3 ) r 2 r 4 ( 1 + r 1 2 ) + 2 cos ( 2 ϕ 1 2 ϕ 2 ) r 1 r 2 2 r 3 ( 1 + r 4 2 ) + 4 cos ( 2 ϕ 1 2 ϕ 3 ) r 1 r 2 r 3 r 4 + 2 cos ( 2 ϕ 2 2 ϕ 3 ) r 2 r 3 2 r 4 ( 1 + r 1 2 ) + 2 cos ( 2 ϕ 1 2 ϕ 2 + 2 ϕ 3 ) r 1 r 2 2 r 3 2 r 4 + 2 cos ( 2 ϕ 1 2 ϕ 2 + 2 ϕ 3 ) r 1 r 2 2 r 4 + 2 cos ( 2 ϕ 1 + 2 ϕ 2 2 ϕ 3 ) r 1 r 3 2 r 4 + 2 cos ( 2 ϕ 1 + 2 ϕ 2 + 2 ϕ 3 ) r 1 r 4 .
T = ( 1 r 2 ) 4 / [ ( 1 r 2 ) 4 + 8 r 2 ( 1 + x ) ( r 2 + x ) 2 ] .
T = 1 if x = 1 or x = r 2 .
cos ( 2 ϕ ) = x = ( 2 + r 2 ) / 3 , T = ( 1 r 2 ) / ( 1 + 5 r 2 / 27 ) ,
B = ( 1 + r 1 2 + r 1 r 2 ) r 2 / r 1 .
T = 1 / { 1 + 2 r 1 2 ( 1 + x ) ( 2 x + B 1 ) 2 / [ ( 1 r 1 2 ) 2 ( 1 r 2 2 ) 2 ] } .
x = 1
x = ( 1 r 2 / r 1 r 1 r 2 r 2 2 ) / 2 = ( 1 B ) / 2 ,
x = ( 3 + B ) / 6 , T = 1 / { 1 + 4 r 1 2 ( 3 B ) 3 / [ 27 ( 1 r 1 2 ) 2 ( 1 r 2 2 ) 2 ] } .
B = r 2 2 + r 2 ( r 1 + 1 / r 1 ) 3 .
T = 1 / [ 1 + cos 2 ( ϕ ) / cos 2 ( ϕ 22 ) ]
T = 1 / [ 1 + cos 4 ( ϕ ) / cos 4 ( ϕ 33 ) ]
r 1 = r 3 , r 2 = 2 r 1 / [ 1 + r 1 2 ] , ϕ 1 = ϕ 2 = ϕ ,
4 cos 4 ( ϕ 33 ) = ( 1 r 1 2 ) 2 ( 1 r 2 2 ) / 4 r 1 2 = ( 1 + x 33 ) 2 .
T = 1 / [ 1 + cos 6 ( ϕ ) cos 6 ( ϕ 44 ) ]
r 1 = r 4 , r 2 = r 3 , r 2 2 + r 2 ( r 1 + 1 / r 1 ) = 3 , ϕ 1 = ϕ 2 = ϕ 3 = ϕ ,
8 cos 6 ( ϕ 44 ) = ( 1 r 1 2 ) 2 ( 1 r 2 2 ) 2 / 8 r 1 2 = ( 1 + x 44 ) 3 .
T = 1 / { 1 + [ cos ( ϕ ) / cos ( ϕ N N ) ] 2 N 2 } ,
S N = tan ( ϕ N N ) ( N 1 ) / 2 .
FWHM = π 2 ϕ N N .
f N = 1 / ( 1 2 ϕ N N / π ) .
T = 1 / { 1 + [ sin ( ϕ ) / sin ( ϕ N N ) ] 2 N 2 } ,
T = 1 / [ 1 + ( ϕ / ϕ N N ) 2 N 2 ] .

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