Abstract

Analytical methods are presented for transverse mode analysis of a laser resonator having spherical mirrors with a Gaussian reflectivity profile. The modes of this type of resonator have a form similar to that of the conventional Gaussian modes, but it is necessary to define an additional beam parameter to meet the self-consistency requirement for resonator modes. Stability of both the conventional complex beam parameter and the additional parameter is discussed. It is predicted mathematically that small perturbations in the new beam parameter will cause the intensity profile of the higher-order modes to evolve into that of the fundamental mode. Mode losses and discrimination are also discussed. The results may be useful in the design of regenerative laser amplifiers.

© 1986 Optical Society of America

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References

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  1. L. W. Casperson, S. D. Lunman, “Gaussian Modes in High Loss Laser Resonators,” Appl. Opt. 14, 1193 (1975).
    [CrossRef] [PubMed]
  2. A. Yariv, P. Yeh, “Confinement and Stability in Optical Resonators Employing Mirrors with Gaussian Reflectivity Tapers,” Opt. Commun. 13, 370 (1975).
    [CrossRef]
  3. N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. PHys. 10, 1439 (1965).
  4. H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2344 (1970).
  5. J. A. Arnaud, “Optical Resonators in the Approximation of Gauss,” Proc. IEEE 62, 1561 (1974).
    [CrossRef]
  6. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 115–121.
  7. U. Ganiel, A. Hardy, “Eigenmodes of Optical Resonators with Mirrors Having Gaussian Reflectivity Profiles,” Appl. Opt. 15, 2145 (1976).
    [CrossRef] [PubMed]
  8. A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 113–117.
  9. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 235–239.
  10. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 294–304.
  11. A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 135–139.

1976 (1)

1975 (2)

L. W. Casperson, S. D. Lunman, “Gaussian Modes in High Loss Laser Resonators,” Appl. Opt. 14, 1193 (1975).
[CrossRef] [PubMed]

A. Yariv, P. Yeh, “Confinement and Stability in Optical Resonators Employing Mirrors with Gaussian Reflectivity Tapers,” Opt. Commun. 13, 370 (1975).
[CrossRef]

1974 (1)

J. A. Arnaud, “Optical Resonators in the Approximation of Gauss,” Proc. IEEE 62, 1561 (1974).
[CrossRef]

1970 (1)

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2344 (1970).

1965 (1)

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. PHys. 10, 1439 (1965).

Arnaud, J. A.

J. A. Arnaud, “Optical Resonators in the Approximation of Gauss,” Proc. IEEE 62, 1561 (1974).
[CrossRef]

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 115–121.

Casperson, L. W.

Ganiel, U.

Hardy, A.

Lunman, S. D.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 235–239.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 294–304.

Vakhimov, N. G.

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. PHys. 10, 1439 (1965).

Yariv, A.

A. Yariv, P. Yeh, “Confinement and Stability in Optical Resonators Employing Mirrors with Gaussian Reflectivity Tapers,” Opt. Commun. 13, 370 (1975).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 135–139.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 113–117.

Yeh, P.

A. Yariv, P. Yeh, “Confinement and Stability in Optical Resonators Employing Mirrors with Gaussian Reflectivity Tapers,” Opt. Commun. 13, 370 (1975).
[CrossRef]

Zucker, H.

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2344 (1970).

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Zucker, “Optical Resonators with Variable Reflectivity Mirrors,” Bell Syst. Tech. J. 49, 2344 (1970).

Opt. Commun. (1)

A. Yariv, P. Yeh, “Confinement and Stability in Optical Resonators Employing Mirrors with Gaussian Reflectivity Tapers,” Opt. Commun. 13, 370 (1975).
[CrossRef]

Proc. IEEE (1)

J. A. Arnaud, “Optical Resonators in the Approximation of Gauss,” Proc. IEEE 62, 1561 (1974).
[CrossRef]

Radio Eng. Electron. PHys. (1)

N. G. Vakhimov, “Open Resonators with Mirrors Having Variable Reflection Coefficients,” Radio Eng. Electron. PHys. 10, 1439 (1965).

Other (5)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 115–121.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 113–117.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), pp. 235–239.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), pp. 294–304.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 135–139.

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Figures (8)

Fig. 1
Fig. 1

ABCD matrices for a few simple optical elements. Gaussian beam is assumed to be incident from the left.

Fig. 2
Fig. 2

Plots of the normalized beam spot size k w s 2 / 2 L vs Fresnel number N for various values of ρ = L/Rm.

Fig. 3
Fig. 3

Plots of the normalized wave-front curvature Rs/L vs N.

Fig. 4
Fig. 4

Plots of the real part of the normalized secondary beam parameter k w s 2 / 2 L vs N. For resonators that have confined beam modes at large N, these curves approach each other as N increases.

Fig. 5
Fig. 5

Plots of the imaginary part of the normalized beam parameter vs N. For resonators with confined beam modes at large N, these curves approach zero as N increases.

Fig. 6
Fig. 6

Plots of the stability factor Fq of the primary beam parameter vs Fresnel number for the selected values of ρ.

Fig. 7
Fig. 7

Plots of the stability factor Fw of the secondary beam parameter vs Fresnel number for the selected values of ρ.

Fig. 8
Fig. 8

Plots of the loss coefficients Λ0,0 (solid curves) and Λ0,1 (dashed curves) for the selected values of ρ.

Equations (68)

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t 2 u - 2 i k u z = 0             k = 2 π n λ ,
u ( x , y , z ) = f [ x w ( z ) ] g [ y w ( z ) ] × exp { - 1 [ P ( z ) + k ( x 2 + y 2 ) 2 q ( z ) + ϕ ( z ) ] } ,
P ( z ) = - i ln ( 1 + z q 0 ) ,
q ( z ) = q 0 + z .
1 q ( z ) = 1 R ( z ) - 2 i k W 2 ( z ) ,
f f + 2 i k x ( d w d z - w q ) f f + g ¨ g + 2 i k y ( d w d z - w q ) g ˙ g - 2 k w 2 d ϕ d z = 0 ,
d w d z - w q = 2 i k w or d w 2 d z - 2 w 2 q 0 + z = 4 i k
1 f d 2 f d t 2 - 2 t d f d t + 1 g d 2 g d τ 2 - 2 τ d g d τ - 2 k w 2 d ϕ d z = 0.
d 2 f d t 2 - 2 t d f d t + 2 n f = 0 ,
d 2 g d τ 2 - 2 τ d g d τ + 2 m g = 0 ,
d ϕ d z = - 2 ( n + m k w 2 ) .
f ( x w ) = H n ( 2 x w )             and             g ( y w ) = H m ( 2 y w ) ,
( q 0 + z ) - 2 d w 2 d z - 2 w 2 ( q 0 + z ) - 3 = 4 i k ( q 0 + z ) - 2
d d z [ ( q 0 + z ) - 2 w 2 ] = 4 i k ( q 0 + z ) - 2 .
w 2 ( z ) = ( q 0 + z ) [ w 0 2 q 0 + ( w 0 2 q 0 2 + 4 i k q 0 ) z ] ,
ϕ ( z ) = - 2 0 z n + m k w 2 ( z ) d z = - ( n + m ) η ( z ) ,
η ( z ) = 2 q 0 2 0 z d z ( q 0 + z ) [ k w 0 2 q 0 + ( k w 0 2 + 4 i q 0 ) z ] ,
η ( z ) = i 2 ln [ k w 0 2 ( q 0 + z ) q 0 k w 0 2 + ( k w 0 2 + 4 i q 0 ) z ] .
u n , m ( x , y , z ) = H n ( 2 x w ) H m ( 2 y w ) × exp { - i [ P ( z ) + k ( x 2 + y 2 ) 2 ( q 0 + z ) - ( n + m ) η ( z ) ] } ,
u n , m ( r , θ , z ) = ( 2 r w ) m L n m ( 2 r 2 w 2 ) sin ( m θ ) × exp { - 1 [ P ( z ) + k r 2 2 ( q 0 + z ) - ( 2 n + m ) η ( z ) ] } ,
q out = A q in + B C q in + D ,
( A B C D ) = ( 1 0 - 2 i k W m 2 1 ) ,
w out 2 = ( q in + L q in ) 2 w in 2 + 4 i L k ( q in + L q in ) ,
w out 2 = w in 2
w out 2 = α w in 2 + β ,
α = ( q in + L q in ) 2             and             β = 4 i L k ( q in + L q in )
w out 2 = α w in 2 + β γ w in 2 + δ ,
( α β γ δ ) = ( α β 0 1 ) ,
( α β 0 1 ) = ( α N β N 0 0 ) ( α N - 1 β N - 1 0 1 ) ( α 1 β 1 0 0 ) ,
q s = A q s + B C q s + D             or             1 q s = C + D / q s A + B / q s ,
1 q s = D - A 2 B ± 1 B [ ( A + D 2 ) 2 - 1 ] 1 / 2 ,
w s 2 = α w s 2 + β             or             ω s 2 = β 1 - α .
( A B C D ) = ( 1 - 2 L R m - 2 i L k W m 2 L - 2 R m - 2 i k W m 2 1 ) ,
L q s = L R s - 2 i L k W s 2 = ρ + i 2 π N ± [ ( 1 - ρ - i 2 π N ) 2 - 1 ] 1 / 2 ,
N = k W m 2 2 π L
1 - ρ = | 1 - L R m | < 1.
( α β 0 1 ) = [ ( q 1 + L q 1 ) 2 4 i L q 1 ( q 1 + L q 1 ) ] ( 1 0 0 1 ) ( 1 0 0 1 ) = [ q 1 + L q 1 4 i L k ( q 1 + L q 1 ) 0 1 ] ,
q s = q 1 + L             or             q 1 = q s - L ,
w s 2 = 2 i L k [ ( L q 1 ) - 1 + ( L q 1 + 2 ) - 1 ] or k w s 2 2 L = - i { ( L q s ) - 1 + { [ ( L q s ) - 1 - 1 ] - 1 - 2 } - 1 - 1 } ,
1 q = 1 q s + δ 1 q ,
1 q s + Δ 1 q = C + D ( 1 q s + δ 1 q ) A + B ( 1 q s + δ 1 q ) ,
Δ 1 q = 1 ( A + B q s ) 2 δ 1 q .
| Δ 1 q | = 1 F q 2 | δ 1 q | ,
F q | A + B q s | .
w s 2 + Δ w 2 = α ( w s 2 + δ w 2 ) + β ,
Δ w 2 = α ( δ w 2 ) .
Δ w 2 = α δ w 2 = 1 F w 2 δ w 2 ,
F w = 1 α .
w 2 δ w 2 - w s 2 ,
F q = | 1 - 2 L R m - 2 L i k W m 2 + L q s | = | 1 - 2 ρ - i π N + L q s | ,
F q = | 1 - ρ - i 2 π N ± [ ( 1 - ρ i 2 π N ) 2 - 1 ] 1 / 2 | ,
F w = 1 1 + L q 1 = 1 - L q s = | 1 - ρ - i 2 π N [ ( 1 - ρ - i 2 π N ) 2 - 1 ] 1 / 2 | .
γ n , m = - - | H n ( 2 x w s ) H m ( 2 2 y w s ) | 2 exp [ - 2 ( x 2 - y 2 ) / W s 2 ] d x d y - - | H n ( 2 x w s ) H m ( 2 2 y w s ) | 2 exp [ - 2 ( x 2 - y 2 ) / W s 2 ] d x d y ,
1 W s 2 = 1 W s 2 + 1 W m 2 ,
γ n , m = R 0 2 0 r 2 m | L n m ( 2 r 2 w s 2 ) | 2 exp ( - 2 r 2 / W s 2 r d r 0 r 2 m | L n m ( 2 r 2 w s 2 ) | 2 exp ( - 2 r 2 / W s 2 r d r ,
Λ n , m = 1 - γ n , m ,
γ 0 , 0 = R 0 2 0 exp ( - 2 r 2 / W s 2 ) r d r 0 exp ( - 2 r 2 / W s 2 ) r d r = R 0 2 W s 2 W s 2 ,
γ 0 , 1 = R 0 2 0 r 3 exp ( - 2 r 2 / W s 2 ) d r 0 r 3 exp ( - 2 r 2 / W s 2 ) d r = R 0 2 W s 4 W s 4 .
Λ 0 , 0 = 1 - R 0 2 W s 2 W s 2 ,
Λ 0 , 1 = 1 - R 0 2 W s 4 W s 4 .
D = 1 - γ 0 , 1 γ 0 , 0 .
D = 1 - W s 2 W s 2 .
Λ 0 , 0 = D = 1 - W s 2 W s 2 = 1 - ( 2 L k W s 2 ) ( 2 L k W s 2 ) ,
Λ 0 , 1 = 1 - W s 4 W s 4 = 1 - ( 2 L k W s 2 ) 2 ( 2 L k W s 2 ) 2 ,
2 L k W s 2 = - Im { ρ + i 2 π N ± [ ( 1 - ρ - i 2 π N ) 2 - 1 ] 1 / 2 } = | Im { [ ( 1 - ρ - i 2 π N ) 2 - 1 ] 1 / 2 } - 1 2 π N | ,
2 L k W s 2 - 2 L k W s 2 + 2 L k W m 2 = 2 L k W s 2 + 1 π N = | Im { [ ( 1 - ρ - i 2 π N ) 2 - 1 ] 1 / 2 } - 1 2 π N | .
Λ 0 , 0 = 1 - Im { [ ( 2 π N - 2 π N ρ - i ) 2 - 4 π 2 N 2 ] 1 / 2 } - 1 Im { [ 2 π N - 2 π N ρ - i ) 2 - 4 π 2 N 2 ] 1 / 2 } + 1 ,
Λ 0 , 1 = 1 - ( Im { [ ( 2 π N - 2 π N ρ - i ) 2 - 4 π 2 N 2 ] 1 / 2 } - 1 ) 2 ( Im { [ ( 2 π N - 2 π N ρ - i ) 2 - 4 π 2 N 2 ] 1 / 2 } + 1 ) 2 .

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