Abstract

Two or more unstable optical resonators can be coupled together by sharing optical elements. The result is then a single compound resonator with multiple outputs. For identical coupled cavities, the transverse structure of the outputs would be identical. In general, there will be misalignments and other aberrations that will vary from cavity to cavity. The cumulative effects of such aberrations are treated using both analytical and numerical approaches. It is shown that the average output of a multioutput resonator is the same as the output of a single uncoupled resonator with aberrations equal to the average of those contained in the multioutput resonator.

© 1983 Optical Society of America

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References

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  1. M. B. Spencer, W. E. Lamb, Phys. Rev. A5, 893 (1972).
  2. A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [CrossRef] [PubMed]
  3. Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).
  4. A. Gerrand, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

1974 (1)

1972 (2)

Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).

M. B. Spencer, W. E. Lamb, Phys. Rev. A5, 893 (1972).

Burch, J. M.

A. Gerrand, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Gerrand, A.

A. Gerrand, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Lamb, W. E.

M. B. Spencer, W. E. Lamb, Phys. Rev. A5, 893 (1972).

Siegman, A. E.

Spencer, M. B.

M. B. Spencer, W. E. Lamb, Phys. Rev. A5, 893 (1972).

Appl. Opt. (1)

Phys. Rev. (1)

M. B. Spencer, W. E. Lamb, Phys. Rev. A5, 893 (1972).

Sov. J. Quantum Electron. (1)

Yu. A. Ananev, Sov. J. Quantum Electron. 1, 565 (1972).

Other (1)

A. Gerrand, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

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Figures (5)

Fig. 1
Fig. 1

Examples of two-coupled half-symmetric unstable resonators. Resonators share either (a) a common flat or (b) a common feedback mirror.

Fig. 2
Fig. 2

N coupled half-symmetric unstable resonators

Fig. 3
Fig. 3

Equivalent lens train for the compound resonator.

Fig. 4
Fig. 4

Single half-symmetric unstable resonator and its phase-perturbation plane.

Fig. 5
Fig. 5

Full round-trip segments of the unfolded lens trains representing (a) the HSUR and (b) the CHSUR. Tilt operations are applied at the plane coincident with a given mirror.

Tables (2)

Tables Icon

Table I Individual Tilts and the Corresponding Eigenvalues for Two Coupled Resonators Together with the Respective Averages

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Table II Tilts and the Corresponding Eigenvalues for a Single Resonator

Equations (24)

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x 1 = [ 1 + L 2 L ( M 1 ) ] x , x 2 = [ 1 + ( 1 L 2 L ) ( M 1 ) ] x .
Δ ϕ ( x ) = k = 1 δ k x k .
Δ Φ ( x ) = Δ ϕ ( x 1 M ) + Δ ϕ ( x 2 M ) .
Δ Φ T ( x ) = l = 1 Δ Φ ( x / M l ) = l = 1 k = 1 δ k [ ( x 1 / M l ) k + ( x 2 / M l ) k ] .
Δ Φ T ( x ) = k = 1 δ k 1 M k 1 ( x 1 k + x 2 k ) .
Δ Φ ( x ) = n = 1 N [ Δ ϕ 1 , n ( x 1 M n ) + Δ ϕ 2 , n ( x 2 M n ) ] .
Δ Φ T ( x ) = l = 1 n = 1 N [ Δ ϕ 1 , n ( x 1 M ( l 1 ) N + n ) + Δ ϕ 2 , n ( x 2 M ( l 1 ) N + n ) ] = l = 1 n = 1 N k = 1 [ δ 1 , n ( k ) ( x 1 M ( l 1 ) N + n ) k + δ 2 , n ( k ) ( x 2 M ( l 1 ) N + n ) k ] .
Δ Φ T ( x ) = k = 1 M k N M k N 1 n = 1 N [ δ 1 , n ( k ) ( x 1 M n ) k + δ 2 , n ( k ) ( x 2 M n ) k ] .
Δ Φ T ( x ) = k = 1 δ ( k ) M k 1 [ x 1 k + x 2 k ] ,
[ Δ Φ T ( x ) ] a υ = k = 1 [ δ ( k ) ] a υ M k 1 ( x 1 k + x 2 k ) .
R = ( x α 1 ) ,
M = ( A B x 0 C D α 0 0 0 1 ) .
( x α 1 ) = ( A B x 0 C D α 0 0 0 1 ) ( x α 1 ) .
( A B x 0 C D α 0 0 0 1 ) = ( 1 0 0 0 1 δ 1 0 0 1 ) ( 1 2 L 0 0 1 0 0 0 1 ) · ( 1 0 0 0 1 δ 1 0 0 1 ) ( 1 0 0 1 f 1 0 0 0 1 ) ,
( A B x 0 C D α 0 0 0 1 ) = ( 1 2 L f 2 L 2 δ 1 L 1 f 1 2 δ 1 0 0 1 ) .
x 0 = 2 δ 1 f ,
α 0 = δ 1 .
( A B x 0 C D α 0 0 0 1 ) = [ ( 1 2 L f ) 2 2 L f 4 L ( 1 L f ) 4 L [ δ 1 ( 1 L f ) + δ 2 ] 2 f ( 1 L f ) 1 2 L f 2 [ δ 1 ( 1 L f ) + δ 2 ] 0 0 1 ] .
α 0 = δ 1 ( 1 L f ) + δ 2 2 ( 1 L 2 f ) ,
x 0 = f [ δ 1 ( 1 L f ) + δ 2 ] 1 L 2 f .
α ¯ 0 = 1 2 ( α 0 + α 0 ) ,
x ¯ 0 = 1 2 ( x 0 + x 0 ) ,
α ¯ 0 = 1 2 ( δ 1 + δ 2 ) ,
x ¯ 0 = ƒ ( δ 1 + δ 2 ) .

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