Abstract

We analyze a class of custom-mode laser cavities with aspheric mirrors that become either asymptotically flat or hemispheric away from the optical axis. Previous work classifies the modes of such cavities as either bound or unbound. We develop an analytic approximation for the losses and frequencies of such modes. The bound modes have losses that diminish exponentially as the size of the cavity mirror increases and have frequencies that become independent of the mirror size. On the other hand, unbound modes have losses that diminish asymptotically as the inverse third power of the mirror width and frequencies that converge toward those of the unperturbed (flat or hemispheric) cavity with increasing mirror width. Finally, we show good agreement between our model and numeric cavity eigenvalue calculations.

© 2007 Optical Society of America

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References

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  1. P. A. Bélanger and C. Paré, "Optical resonators using graded-phase mirrors," Opt. Lett. 16(14), 1057-1059 (1991).
    [CrossRef] [PubMed]
  2. J. R. Leger, D. Chen, and G. Mowry, "Design and performance of diffractive optics for custom laser resonators," Appl. Opt. 34, 2498-2509 (1995).
    [CrossRef] [PubMed]
  3. C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
    [CrossRef] [PubMed]
  4. M. Kuznetsov, M. Stern, and J. Coppeta, "Single transverse mode optical resonators," Opt. Express 13, 171-181 (2005).
    [CrossRef] [PubMed]
  5. L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, 1969).
  6. L. A. Weinstein, The Theory of Diffraction and the Factorization Method (Golem, Boulder, 1969).
  7. V. V. Lyubimov and I. B. Orlova, "Approximate Calculation of Oscillations in Resonators with Concave Mirrors," Opt. Spectrosc. 29, 310-313 (1970).
  8. L.-W. Chen and L. B. Felsen, "Coupled-Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. 9, 1102-1113 (1973).
    [CrossRef]
  9. G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).
  10. J. Goodman, Fourier Optics, 2nd ed. (McGraw Hill, 1995).
  11. V. V. Lyubimov and I. B. Orlova, "Oscillations in a Tilted-Mirror Resonator," Opt. Spectrosc. 30, 409-411 (1971).
  12. C. E. Santana and L. B. Felsen, "Ray-Optical Calculation of Edge Diffraction in Unstable Resonators," IEEE Trans. Microwave Theory Tech. MTT-26, 101-108 (1978).
    [CrossRef]

2005 (1)

1995 (1)

1992 (1)

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

1991 (1)

1978 (1)

C. E. Santana and L. B. Felsen, "Ray-Optical Calculation of Edge Diffraction in Unstable Resonators," IEEE Trans. Microwave Theory Tech. MTT-26, 101-108 (1978).
[CrossRef]

1973 (2)

L.-W. Chen and L. B. Felsen, "Coupled-Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. 9, 1102-1113 (1973).
[CrossRef]

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).

1971 (1)

V. V. Lyubimov and I. B. Orlova, "Oscillations in a Tilted-Mirror Resonator," Opt. Spectrosc. 30, 409-411 (1971).

1970 (1)

V. V. Lyubimov and I. B. Orlova, "Approximate Calculation of Oscillations in Resonators with Concave Mirrors," Opt. Spectrosc. 29, 310-313 (1970).

Bélanger, P. A.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

P. A. Bélanger and C. Paré, "Optical resonators using graded-phase mirrors," Opt. Lett. 16(14), 1057-1059 (1991).
[CrossRef] [PubMed]

Chen, D.

Chen, L.-W.

L.-W. Chen and L. B. Felsen, "Coupled-Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. 9, 1102-1113 (1973).
[CrossRef]

Coppeta, J.

Felsen, L. B.

C. E. Santana and L. B. Felsen, "Ray-Optical Calculation of Edge Diffraction in Unstable Resonators," IEEE Trans. Microwave Theory Tech. MTT-26, 101-108 (1978).
[CrossRef]

L.-W. Chen and L. B. Felsen, "Coupled-Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. 9, 1102-1113 (1973).
[CrossRef]

Gagnon, L.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

Kuznetsov, M.

Leger, J. R.

Lyubimov, V. V.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).

V. V. Lyubimov and I. B. Orlova, "Oscillations in a Tilted-Mirror Resonator," Opt. Spectrosc. 30, 409-411 (1971).

V. V. Lyubimov and I. B. Orlova, "Approximate Calculation of Oscillations in Resonators with Concave Mirrors," Opt. Spectrosc. 29, 310-313 (1970).

Mowry, G.

Orlova, I. B.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).

V. V. Lyubimov and I. B. Orlova, "Oscillations in a Tilted-Mirror Resonator," Opt. Spectrosc. 30, 409-411 (1971).

V. V. Lyubimov and I. B. Orlova, "Approximate Calculation of Oscillations in Resonators with Concave Mirrors," Opt. Spectrosc. 29, 310-313 (1970).

Paré, C.

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

P. A. Bélanger and C. Paré, "Optical resonators using graded-phase mirrors," Opt. Lett. 16(14), 1057-1059 (1991).
[CrossRef] [PubMed]

Santana, C. E.

C. E. Santana and L. B. Felsen, "Ray-Optical Calculation of Edge Diffraction in Unstable Resonators," IEEE Trans. Microwave Theory Tech. MTT-26, 101-108 (1978).
[CrossRef]

Stern, M.

Vinokurov, G. N.

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

L.-W. Chen and L. B. Felsen, "Coupled-Mode Theory of Unstable Resonators," IEEE J. Quantum Electron. 9, 1102-1113 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. E. Santana and L. B. Felsen, "Ray-Optical Calculation of Edge Diffraction in Unstable Resonators," IEEE Trans. Microwave Theory Tech. MTT-26, 101-108 (1978).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Spectrosc. (3)

V. V. Lyubimov and I. B. Orlova, "Approximate Calculation of Oscillations in Resonators with Concave Mirrors," Opt. Spectrosc. 29, 310-313 (1970).

G. N. Vinokurov, V. V. Lyubimov, and I. B. Orlova, "Investigation of the selective properties of open unstable cavities," Opt. Spectrosc. 34, 427-432 (1973).

V. V. Lyubimov and I. B. Orlova, "Oscillations in a Tilted-Mirror Resonator," Opt. Spectrosc. 30, 409-411 (1971).

Phys. Rev. A (1)

C. Paré, L. Gagnon, and P. A. Bélanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992).
[CrossRef] [PubMed]

Other (3)

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, 1969).

L. A. Weinstein, The Theory of Diffraction and the Factorization Method (Golem, Boulder, 1969).

J. Goodman, Fourier Optics, 2nd ed. (McGraw Hill, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Laser cavity unfolded into a lens train.

Fig. 2.
Fig. 2.

Reflected and transmitted modes in a waveguide.

Fig. 3.
Fig. 3.

Eigenvalues of bound (b) and unbound (u) cavity modes when Ψ( ξ ¯ ) = sech ξ ¯ . Solid curve is numeric solution; dashed curve is our method. Phase results are indistinguishable for the bound mode.

Fig. 4.
Fig. 4.

Eigenvalues of bound (b) and unbound (u) cavity modes when Ψ( ξ ¯ ) = sech3 ξ ¯ . Solid curve is numeric solution; dashed curve is our method. Phase results are indistinguishable for the first bound mode.

Fig. 5.
Fig. 5.

Zeros of (s) and paths of integration in the complex s-plane

Equations (52)

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g ( ξ ) = e e i Φ ( ξ ) 2 πi M 2 M 2 e i Φ ( ξ ' ) g ( ξ ' ) exp [ i 2 ( ξ ξ ' ) 2 ] ' .
g ( ξ ) = e i χ e i Φ ˜ ( ξ ) 2 πi M 2 M 2 e i Φ ˜ ( ξ ' ) g ( ξ ' ) exp [ i 2 ( ξ ξ ' ) 2 ] ' .
g ( ξ ) = e i χ ˜ e i Φ ˜ ( ξ ) 2 πi M 2 M 2 e i Φ ˜ ( ξ ' ) g ( ξ ' ) exp [ i 2 ( ξ ξ ' ) 2 ] ' .
s j 2 = 4 π ( p + j ) ,
g ( ξ ) = e 2 πi 0 g ( ξ ' ) exp [ i 2 ( ξ ξ ' ) 2 ] '
R jk e i ( 1 + i ) β ( s j + s k ) 2 2 s j s j + s k where β 8.24
s 0 = πm M + ( 1 + i ) β
B C = S A D
A D = R B C where R = ( R u 0 0 R v )
det ( I SR ) = 0 .
R u = u ( M 2 ) u + ( M 2 ) R 0 ( s u ) and R v = v + ( M 2 ) v ( M 2 ) R 0 ( s v ) ,
s u = 1 u + u + ξ ξ = M 2
S = ( r t t r ) and R = e isM R 0 I ,
ψ ( ξ ̅ ) = e ε 2 πi e i Φ ( ξ ̅ ε ) e i Φ ( ξ ' ̅ ε ) ψ ( ξ ' ̅ ) exp [ i 2 ε 2 ( ξ ̅ ξ ' ̅ ) 2 ] d ξ ' ̅
= e e i Φ ( ξ ̅ ε ) n ( 2 2 ) n h ( 2 n ) ( ξ ̅ ) n !
h ( ξ ̅ ) = e ( ξ ̅ ε ) ψ ( ξ ̅ )
Φ ( ξ ̅ ε ) = arg { e ikl ε πi Ψ ( ξ ' ̅ ) exp [ i ε 2 ( ξ ̅ ξ ' ̅ ) 2 ] d ξ ' ̅ }
= kl + arg n i n ( ε 2 ) 2 n Ψ ( 2 n ) ( ξ ̅ ) n ! ,
Φ ( ξ ) = arg e ikl πi e iξ 2 f ( ξ ) e i ξ 2 e i 2 ξξ .
Φ ˜ ( ξ ̅ ε ) = π 4 + ( ξ ̅ ε ) 2 arg { e ikl e i ( ξ ̅ ε ) 2 επ Ψ ˜ ( ξ ' ̅ ) exp [ i ε 2 ( ξ ̅ ξ ' ̅ ) 2 ] d ξ ̅ }
=− kl + arg n i n ( ε 2 ) 2 n Ψ ˜ ( 2 n ) ( ξ ̅ ) n ! .
Φ ( ξ ̅ ε ) = 1 4 V ( ξ ̅ ) ε 2 + O ( ε 2 ) , where V = 1 Ψ 2 Ψ ξ 2 + 2 πP
Φ ˜ ( ξ ̅ ε ) = 1 4 V ( ξ ̅ ) ε 2 + O ( ε 6 ) , where V = 1 Ψ ˜ 2 Ψ ˜ ξ 2 + 2 πP
ψ = ψ 0 + ε 0 ψ 1 + ε 4 ψ 2 + and χ = χ 0 + ε 2 χ 1 + ε 4 χ 2 +
ψ 0 + ( E V ) ψ 0 = 0
s ̅ = 1 ε πm δ M + ( 1 + i ) β ,
1 e 2 2 β ( πm δ 0 ) 2 ( a ε + M + β ) 3 ,
arg ( e ) = χ ' ( πm δ 0 ) 2 2 ( a ε + M + β ) 2 .
t = b σ ̅ σ ̅ p + O ( 1 ) .
r = b σ ̅ σ ̅ p + O ( 1 ) and consequently λ + = 2 b σ ̅ σ ̅ p + O ( 1 ) .
t = b σ ̅ σ ̅ p + O ( 1 ) so that λ = 2 b σ ̅ σ ̅ p + O ( 1 ) .
σ ̅ = σ ̅ p + 2 b R 0 e σ ̅ p ,
1 e 2 4 ε 2 b σ ̅ p e σ ̅ p ( M + β ) ε sin ( εβ σ ̅ p ) ,
χ ε 2 2 [ σ ̅ p 2 4 b σ ̅ p e σ ̅ p ( M + β ) ε cos ( εβ σ ̅ p ) ] .
Ψ ( ξ ̅ ) = sech n ξ ̅ , V ( ξ ̅ ) = n ( n + 1 ) sech 2 ξ ̅ ,
t ( s ̅ ) = Π m = 1 n s ̅ + i σ ̅ m s ̅ i σ ̅ m
g ( ξ ) = 0 for ξ < 0 ,
( K g ) ( ξ ) = 0 for ξ > 0 ,
K ( ξ ) = δ ( ξ ) e 2 πi e i ξ 2 2 .
g ( ξ ) = e is j ξ + h ( ξ ) .
h ˜ ( s ) e isξ ds 2 π = e i s j ξ for ξ < 0
K ˜ ( s ) h ˜ ( s ) e isξ ds 2 π = 0 for ξ > 0 .
K ˜ ( s ) = K ˜ + ( s ) K ˜ ( s )
h ˜ ( s ) = i K ˜ _ ( s j ) K ˜ + ( s ) ( s + s j ) K ˜ ( s ) = i K ˜ ( s j ) ( s + s j ) K ˜ ( s )
h ( ξ ) = h ˜ ( s ) e isξ ds 2 π = k R jk e i s k ξ for ξ > 0 ,
R jk = i K ˜ ( s j ) K ˜ + ( s k ) ( s j + s k ) s k .
K ˜ ( s ) = e U ( s ) and K ˜ + ( s ) = e U + ( s ) .
log K ˜ ( s ) = U ( s ) + U + ( s ) = log [ 1 exp [ s s " χ " + i ( χ ' + 1 2 ( s " 2 s ' 2 ) ) ] ] ,
U + ( s ) = 1 2 πi Γ log K ˜ ( t ) t s dt for Re s > 0 and Im s 0
U ( s ) = 1 2 πi Γ log K ˜ ( t ) t s dt for Re s < 0 and Im s 0 .
U + ( s ) = U ( s ) = U s p = 1 2 πi log ( i 2 πp t 2 2 ) dt t s i ,
R jk = i e U s j p + U s k p ( s j + s k ) s k .

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