Abstract

We present an analysis of the perturbed Gaussian transverse modes of an antiresonant-ring laser cavity useful for cavity dumping, mode locking, and other applications. A plane-wave analysis of the antiresonant ring with a 50-50 beam splitter indicates that the coupling out the output port of the ring should be identically zero. A perturbed Gaussian mode analysis of a stable antiresonant-ring cavity shows, however, that if the ring is not optically symmetric (in the sense that the paraxial ray matrix is not symmetric in the two directions around the ring) then the modes of the cavity consist of a perturbed mixture of stable Gaussian modes, with a significant interference output or leakage from the output port of the ring. The existence of this leakage output has also been confirmed experimentally and by numerical Fox and Li calculation of the lowest mode of the unbalanced ring. This represents an unusual situation in which there is transverse mode discrimination without there being any diffractive output coupling.

© 1976 Optical Society of America

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References

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  1. A. E. Siegman, “An Antiresonant Ring Interferometer for Coupled Laser Cavities, Laser Output Coupling, Mode Locking, and Cavity Damping,” IEEE J. Quantum Electron. QE-9, 247 (1973).
    [CrossRef]
  2. U. S. Patent No. 3, 869, 210, “Laser System with an Antiresonant Ring,” 4March1975, A. E. Siegman.
  3. W. R. Trutna, Shinan-Chur Sheng, and A. E. Siegman, “Laser Cavity Dumping Using the Antiresonant Ring” (unpublished).
  4. Stuart A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. 60, 1168 (1970).
    [CrossRef]
  5. G. A. Deschamps, “Ray Techniques in Electromagnetics, Proc. IEEE 60, 1022 (1972).
    [CrossRef]
  6. P. Baues, “Huygens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” J. Optoelectron. 1, 37 (1966).
  7. P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” J. Optoelectron. 1, 103–108 (1969).
  8. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

1973 (1)

A. E. Siegman, “An Antiresonant Ring Interferometer for Coupled Laser Cavities, Laser Output Coupling, Mode Locking, and Cavity Damping,” IEEE J. Quantum Electron. QE-9, 247 (1973).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray Techniques in Electromagnetics, Proc. IEEE 60, 1022 (1972).
[CrossRef]

1970 (1)

1969 (1)

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” J. Optoelectron. 1, 103–108 (1969).

1966 (1)

P. Baues, “Huygens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” J. Optoelectron. 1, 37 (1966).

Baues, P.

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” J. Optoelectron. 1, 103–108 (1969).

P. Baues, “Huygens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” J. Optoelectron. 1, 37 (1966).

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Collins, Stuart A.

Deschamps, G. A.

G. A. Deschamps, “Ray Techniques in Electromagnetics, Proc. IEEE 60, 1022 (1972).
[CrossRef]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Sheng, Shinan-Chur

W. R. Trutna, Shinan-Chur Sheng, and A. E. Siegman, “Laser Cavity Dumping Using the Antiresonant Ring” (unpublished).

Siegman, A. E.

A. E. Siegman, “An Antiresonant Ring Interferometer for Coupled Laser Cavities, Laser Output Coupling, Mode Locking, and Cavity Damping,” IEEE J. Quantum Electron. QE-9, 247 (1973).
[CrossRef]

U. S. Patent No. 3, 869, 210, “Laser System with an Antiresonant Ring,” 4March1975, A. E. Siegman.

W. R. Trutna, Shinan-Chur Sheng, and A. E. Siegman, “Laser Cavity Dumping Using the Antiresonant Ring” (unpublished).

Trutna, W. R.

W. R. Trutna, Shinan-Chur Sheng, and A. E. Siegman, “Laser Cavity Dumping Using the Antiresonant Ring” (unpublished).

IEEE J. Quantum Electron. (1)

A. E. Siegman, “An Antiresonant Ring Interferometer for Coupled Laser Cavities, Laser Output Coupling, Mode Locking, and Cavity Damping,” IEEE J. Quantum Electron. QE-9, 247 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Optoelectron. (2)

P. Baues, “Huygens’ Principle in Inhomogeneous, Isotropic Media and a General Integral Equation Applicable to Optical Resonators,” J. Optoelectron. 1, 37 (1966).

P. Baues, “The Connection of Geometrical Optics with the Propagation of Gaussian Beams and the Theory of Optical Resonators,” J. Optoelectron. 1, 103–108 (1969).

Proc. IEEE (1)

G. A. Deschamps, “Ray Techniques in Electromagnetics, Proc. IEEE 60, 1022 (1972).
[CrossRef]

Other (3)

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

U. S. Patent No. 3, 869, 210, “Laser System with an Antiresonant Ring,” 4March1975, A. E. Siegman.

W. R. Trutna, Shinan-Chur Sheng, and A. E. Siegman, “Laser Cavity Dumping Using the Antiresonant Ring” (unpublished).

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Figures (5)

FIG. 1
FIG. 1

Basic structure of the antiresonant-ring laser cavity. Laser tubes, modulators, and other elements may be located anywhere within the cavity.

FIG. 2
FIG. 2

Internal beam profile of the lowest-order modes as a function of perturbation strength Δ for A = 0.99817, B = −0.72231. (Area under each curve is normalized to unity.)

FIG. 3
FIG. 3

Leakage beam profiles of the lowest-order mode for the same cases as Fig. 2.

FIG. 4
FIG. 4

Comparison of the perturbed Gaussian mode solution and an iterative Fox and Li computer solution for the lowest-order mode of an asymmetric ARR ring cavity with A = 0.9724, B = −0.6934, Δ/A = 6.4%. Calculated power loss per pass: perturbation analysis = 1.96 × 10−3; iterative computer solution = 1.94 × 10−3.

FIG. 5
FIG. 5

Practical ARR cavity design might be used for mode-locking or cavity-dumping applications at λ = 1.06 μm. The transverse dimensions indicated are the effective beam diameters 3w at various planes, where w is the usual Gaussian spot size.

Tables (1)

Tables Icon

TABLE I Hermite-Gaussian mode expansion for the first four modes of the perturbed antiresonant-ring laser cavity.

Equations (21)

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P out / P in = ( R - T ) 2 = 4 δ 2 .
M R = [ A R ± Δ B R C R A R Δ ] ,
M E 2 = [ A E B E C E D E ] 2 .
M = M E M R M E = [ A ± Δ B C A Δ ] ,
A = D = A R ( A E 2 + B E C E ) + B R A E C E + C R A E B E , B = 2 A R A E B E + B R A E 2 + C R B E 2 , C = 2 A R A E B E + B R C E 2 + C R B E 2 ,
( x ) = - K ( x , x 0 ) 0 ( x 0 ) d x 0 ,
K ( x , x 0 ) = ( j B λ ) 1 / 2 exp ( - j π B λ ( A x 0 2 - 2 x x 0 + D x 2 ) ) .
γ m m ( x ) = 1 2 - [ K CW ( x , x 0 ) + K CCW ( x , x 0 ) ] m ( x 0 ) d x 0 .
m ( x ) = ( 2 / π ) 1 / 4 ( 2 m m ! w ) - 1 / 2 H m ( 2 1 / 2 x / w ) e - j ( k / 2 q ) x 2 , γ m = e - j ( m + 1 / 2 ) ϕ , 1 / q = - j λ / π w 2 = - j ( 1 - A 2 ) 1 / 2 / B , ϕ = tan - 1 [ ( 1 - A 2 ) 1 / 2 / A ] , m = 0 , 1 , 2 , 3 , .
γ m m = H m = ( H CW + H CCW ) m ,
[ H CW ( Δ ) ] m n = [ H CCW ( - Δ ) ] m n = F m n - - H m ( x ) H n ( x 0 ) × exp ( a x 0 2 - b x x 0 + c x 2 ) d x 0 d x ,
a = j ( e j ϕ + Δ ) 2 sin ϕ ,             b = j sin ϕ ,             c = j ( e j ϕ - Δ ) 2 sin ϕ , F m n = ( 2 / π ) 1 / 2 ( 1 4 w ) ( - j 2 m + n m ! n ! B λ ) - 1 / 2 .
H m n ( Δ ) = Z m n l 3 P 2 l 1 + l 3 = n 2 l 2 + l 3 = m 1 l 1 ! l 2 ! l 3 ! ( k 1 l 1 k 2 l 2 + k 2 l 1 k 1 l 2 ) k 3 l 3 ,
k 1 = - ( 1 + 1 a + b 2 4 a 2 α ) ,             k 2 = - ( 1 + 1 α ) , k 3 = - b a α , α C - b 2 4 a ,             Z m n = ( m ! n ! 2 m + n + 2 ) 1 / 2 A ˜ - 1 / 2 , A ˜ e j ϕ + ( 1 2 j ) Δ 2 csc ϕ P = { 0 , m - 4 , m - 2 , m } { 0 , n - 4 , n - 2 , n } .
γ m ( Δ ) = H m m ( Δ ) .
ξ m = 1 - γ m 2 .
ξ 0 1 2 Δ 2 ,             ξ 1 3 ξ 0 ,             ξ 2 7 ξ 0 ,             ξ 3 13 ξ 0 .
m ( x ) = m ( x ) + k m a m k k ( x ) ,
a m k = H k m / ( γ m - γ k ) .
1 2 [ M R ( Δ ) M E - M R ( - Δ ) M E ]
ξ 0 1 2 ( l 1 δ R R 2 cos 1 2 θ ) 2 2.1 × 10 - 4 [ δ R ( cm ) ] 2 , ξ 0 2 ( 1 + sin 2 1 2 θ - l 1 R cos 1 2 θ ) 2 ( δ l 2 R cos 1 2 θ ) 2 10 - 6 [ δ l 2 ( cm ) ] 2 .