Abstract

The modes of resonators, which consist of mirrors with Gaussian reflectivity profile and contain saturable gain medium, are analyzed by applying ray matrix techniques. A self-consistent equation is obtained which takes into account the mutual effect of the beam on the medium and that of the medium on the beam. The following approach also provides a first-order approximation to the calculation of beam parameters and diffraction losses for the more common lasers with finite (but large) end mirrors.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  2. U. Ganiel, Y. Silberberg, Appl. Opt. 14, 306 (1975).
    [CrossRef] [PubMed]
  3. H. Statz, C. L. Tang, J. Appl. Phys. 36, 1816 (1965).
    [CrossRef]
  4. A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
    [CrossRef]
  5. A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
    [CrossRef] [PubMed]
  6. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  7. L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
    [CrossRef]
  8. G. J. Ernst, W. J. Witteman, IEEE J. Quantum Electron. QE-9, 911 (1973); Appl. Phys. 6, 297 (1975).
    [CrossRef]
  9. L. W. Casperson, U. Ganiel, IEEE J. Quantum Electron. QE-13, 58 (1977).
    [CrossRef]
  10. A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
    [CrossRef]
  11. L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975).
    [CrossRef] [PubMed]
  12. U. Ganiel, A. Hardy, Appl. Opt. 15, 2145 (1976).
    [CrossRef] [PubMed]
  13. U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
    [CrossRef]
  14. Recently such mirrors were realized and applied to the design of laser resonators; see G. Giuliani, Y. K. Park, R. L. Byer, postdeadline paper presented at the International Quantum Electronics Conference, Boston (1980).
  15. A. Hardy, S. C. Sheng, A. E. Siegman, to be published.
  16. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  17. In fact |P1 + noise|2 = |P3|2. However, for a laser that operates well above threshold, the amplified spontaneous emission might be neglected.
  18. A. E. Siegman, An Introduction to Masers and Lasers (McGraw-Hill, New York, 1971).
  19. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  20. One obtains exactly the same equation assuming ρeffG = 1 where G is the gain per pass, and ρeff=ρ20α2/(α2+W2) is the effective reflectivity of the mirror. That is, ρ0−1|A+B/q| is replaced by ρeff−1/2 in Eq. (4b).
  21. A. E. Siegman, Appl. Opt. 13, 353 (1974).
    [CrossRef] [PubMed]
  22. S. R. Barone, M. C. Newstein, Appl. Opt. 3, (1964).
    [CrossRef]

1977 (1)

L. W. Casperson, U. Ganiel, IEEE J. Quantum Electron. QE-13, 58 (1977).
[CrossRef]

1976 (1)

1975 (4)

U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
[CrossRef]

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975).
[CrossRef] [PubMed]

U. Ganiel, Y. Silberberg, Appl. Opt. 14, 306 (1975).
[CrossRef] [PubMed]

1974 (3)

1973 (1)

G. J. Ernst, W. J. Witteman, IEEE J. Quantum Electron. QE-9, 911 (1973); Appl. Phys. 6, 297 (1975).
[CrossRef]

1968 (1)

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

1966 (1)

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

1965 (2)

H. Statz, C. L. Tang, J. Appl. Phys. 36, 1816 (1965).
[CrossRef]

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

1964 (1)

S. R. Barone, M. C. Newstein, Appl. Opt. 3, (1964).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Barone, S. R.

S. R. Barone, M. C. Newstein, Appl. Opt. 3, (1964).
[CrossRef]

Byer, R. L.

Recently such mirrors were realized and applied to the design of laser resonators; see G. Giuliani, Y. K. Park, R. L. Byer, postdeadline paper presented at the International Quantum Electronics Conference, Boston (1980).

Casperson, L.

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Casperson, L. W.

L. W. Casperson, U. Ganiel, IEEE J. Quantum Electron. QE-13, 58 (1977).
[CrossRef]

L. W. Casperson, S. D. Lunnam, Appl. Opt. 14, 1193 (1975).
[CrossRef] [PubMed]

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

Ernst, G. J.

G. J. Ernst, W. J. Witteman, IEEE J. Quantum Electron. QE-9, 911 (1973); Appl. Phys. 6, 297 (1975).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

Ganiel, U.

L. W. Casperson, U. Ganiel, IEEE J. Quantum Electron. QE-13, 58 (1977).
[CrossRef]

U. Ganiel, A. Hardy, Appl. Opt. 15, 2145 (1976).
[CrossRef] [PubMed]

U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
[CrossRef]

U. Ganiel, Y. Silberberg, Appl. Opt. 14, 306 (1975).
[CrossRef] [PubMed]

Giuliani, G.

Recently such mirrors were realized and applied to the design of laser resonators; see G. Giuliani, Y. K. Park, R. L. Byer, postdeadline paper presented at the International Quantum Electronics Conference, Boston (1980).

Hardy, A.

U. Ganiel, A. Hardy, Appl. Opt. 15, 2145 (1976).
[CrossRef] [PubMed]

U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
[CrossRef]

A. Hardy, S. C. Sheng, A. E. Siegman, to be published.

Kogelnik, H.

Li, T.

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

Lunnam, S. D.

Newstein, M. C.

S. R. Barone, M. C. Newstein, Appl. Opt. 3, (1964).
[CrossRef]

Park, Y. K.

Recently such mirrors were realized and applied to the design of laser resonators; see G. Giuliani, Y. K. Park, R. L. Byer, postdeadline paper presented at the International Quantum Electronics Conference, Boston (1980).

Sheng, S. C.

A. Hardy, S. C. Sheng, A. E. Siegman, to be published.

Siegman, A. E.

A. E. Siegman, Appl. Opt. 13, 353 (1974).
[CrossRef] [PubMed]

A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

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

A. Hardy, S. C. Sheng, A. E. Siegman, to be published.

Silberberg, Y.

U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
[CrossRef]

U. Ganiel, Y. Silberberg, Appl. Opt. 14, 306 (1975).
[CrossRef] [PubMed]

Statz, H.

H. Statz, C. L. Tang, J. Appl. Phys. 36, 1816 (1965).
[CrossRef]

Sziklas, E. A.

Tang, C. L.

H. Statz, C. L. Tang, J. Appl. Phys. 36, 1816 (1965).
[CrossRef]

Witteman, W. J.

G. J. Ernst, W. J. Witteman, IEEE J. Quantum Electron. QE-9, 911 (1973); Appl. Phys. 6, 297 (1975).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Yeh, P.

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

L. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

IEEE J. Quantum Electron. (4)

G. J. Ernst, W. J. Witteman, IEEE J. Quantum Electron. QE-9, 911 (1973); Appl. Phys. 6, 297 (1975).
[CrossRef]

L. W. Casperson, U. Ganiel, IEEE J. Quantum Electron. QE-13, 58 (1977).
[CrossRef]

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-2, 774 (1966).
[CrossRef]

J. Appl. Phys. (1)

H. Statz, C. L. Tang, J. Appl. Phys. 36, 1816 (1965).
[CrossRef]

Opt. Commun. (2)

A. Yariv, P. Yeh, Opt. Commun. 13, 370 (1975).
[CrossRef]

U. Ganiel, A. Hardy, Y. Silberberg, Opt. Commun. 14, 290 (1975).
[CrossRef]

Other (7)

Recently such mirrors were realized and applied to the design of laser resonators; see G. Giuliani, Y. K. Park, R. L. Byer, postdeadline paper presented at the International Quantum Electronics Conference, Boston (1980).

A. Hardy, S. C. Sheng, A. E. Siegman, to be published.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

In fact |P1 + noise|2 = |P3|2. However, for a laser that operates well above threshold, the amplified spontaneous emission might be neglected.

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

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

One obtains exactly the same equation assuming ρeffG = 1 where G is the gain per pass, and ρeff=ρ20α2/(α2+W2) is the effective reflectivity of the mirror. That is, ρ0−1|A+B/q| is replaced by ρeff−1/2 in Eq. (4b).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Resonator’s geometry.

Fig. 2
Fig. 2

α = |γL|2 as a function of log10N for ρ0 = 1 and g = 1: solid line, solution of Eq. (15); dashed line, solution of Eqs. (4a) and (13).

Fig. 3
Fig. 3

Spot size as a function of log10N for ρ0 = 1 and g = 1: curve a, log10(W/α) in an empty (cold) cavity; curve b, log10(W/α) in an active resonator with saturable gain medium; curve c, log 10 [ ( W / α ) N ] in the active resonator; solid line, solution of Eq. (15); dashed line, solution of Eqs. (4a) and (13).

Fig. 4
Fig. 4

L/Rb as a function of log10N for ρ0 = 1 and g = 1: curve a, in an empty (cold) cavity; curve b, in an active resonator with saturable gain medium; solid line, solution of Eq. (15); dashed line, solution of Eqs. (4a) and (13).

Fig. 5
Fig. 5

α = |γL2| vs log10N for ρ0 = 0.8 and g = 1. Other details as in Fig. 2.

Fig. 6
Fig. 6

Spot size vs log10N for ρ0 = 0.8 and g = 1. All other details as in Fig. 3.

Fig. 7
Fig. 7

L/Rb vs log10N for ρ0 = 0.8 and g = 1. For the other details see Fig. 4.

Fig. 8
Fig. 8

α = |γL2| vs log10N for confocal resonators with ρ0 = 1 and g = 0. For other details see Fig. 2.

Fig. 9
Fig. 9

Spot size vs log10N for a confocal resonator with ρ0 = 1 and g = 0. All other details as in Fig. 3.

Fig. 10
Fig. 10

L/Rb vs log10N for a confocal resonator with ρ0 = 1 and g = 0. All other details as in Fig. 4.

Fig. 11
Fig. 11

α = |γL2| vs log10N for an unstable resonator with ρ0 = 1 and g = 1.5. Other details as in Fig. 2.

Fig. 12
Fig. 12

Spot size vs log10N for an unstable resonator with ρ0 = 1 and g = 1.5. All other details as in Fig. 3.

Fig. 13
Fig. 13

L/Rb vs log10N for an unstable resonator with ρ0 = 1 and g = 1.5. Other details as in Fig. 4.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

k = 2 π λ n = ( ω / c ) n
k ( r ) = k 0 1 2 k 2 r 2 = k 0 ( 1 1 2 γ 2 r 2 ) ,
U ( r , z ) = P ( z ) exp [ i k 0 r 2 2 q ( z ) ] ,
1 / q ( z 2 ) = C + D / q ( z 1 ) A + B / q ( z 1 ) ,
P ( z 2 ) = ρ 0 [ A + B / q ( z 1 ) ] 1 exp [ i k 0 ( z 2 z 1 ) ] P ( z 1 ) ,
1 / q = D A 2 B ± i B [ 1 ( A + D 2 ) 2 ] 1 / 2 ,
ρ 0 | exp ( i k 0 L ) | = | A + B / q | = | A + D 2 ± i [ 1 ( A + D 2 ) 2 ] 1 / 2 |
A = cos γ L ( 2 / γ ) [ 1 / R + i / ( k 0 a 2 ) ] sin γ L ,
B = 1 / γ sin γ L ,
C = γ sin γ L 2 [ 1 / R + i / ( k 0 a 2 ) ] cos γ L ,
D = cos γ L ,
ε = ε 0 ( 1 + χ ) = ε 0 ( 1 + χ 0 1 + I / I s ) ,
I = I + exp ( 2 r 2 / W + 2 ) + I exp ( 2 r 2 / W 2 ) ,
I I 0 ( 1 2 r 2 / W 2 ) .
k ( r ) k υ [ 1 + 1 2 χ 0 ( I s / I 0 ) + ( 1 / W 2 ) χ 0 ( I s / I 0 ) r 2 ] ,
k 0 = k υ [ 1 + 1 2 χ 0 ( I s / I 0 ) ] .
γ = ( 1 / W ) [ 2 χ 0 ( I s / I 0 ) ] 1 / 2 .
γ = ± ( 1 / W ) [ 2 | χ 0 | ( I s / I 0 ) ] 1 / 2 exp ( i π / 4 ) .
γ = ± ( 1 / W ) ( 2 / k υ L ) 1 / 2 [ ln ( | A + B / q | 2 ρ 0 2 ) ] 1 / 2 exp ( i π / 4 ) .
ρ 0 ( a 2 a 2 + W 2 ) 1 / 2 exp [ 1 2 k υ | χ 0 | ( I s / I 0 ) L ] = 1 ,
1 / q 1 / R b i λ π W 2
( W / a ) 2 = 1 2 = N Re { [ ( γ L ) 2 ( 1 g + i / 2 N ) 2 + 2 γ L ( 1 g + i / 2 N ) cot ( γ L ) ] 1 / 2 } ,
γ L = ± ( a / W ) N 1 / 2 { ln [ 1 + ( W / a ) 2 ρ 0 2 ] } 1 / 2 exp ( i π / 4 ) ,
N = π a 2 λ L , g = 1 L / R .
( L / R b ) = 1 g ± Re { [ ( 1 g + i / 2 N ) 2 ( γ L ) 2 2 γ L ( 1 g + i / 2 N ) cot ( γ L ) ] 1 / 2 } .
L 00 = 1 ρ 0 2 a 2 / ( a 2 + W 2 ) = [ ( 1 ρ 0 2 ) a 2 + W 2 ] / ( a 2 + W 2 ) .

Metrics