Abstract

The influence of the geometric relationship between laser pump and mode distributions on the performance of low-gain lasers has been analyzed. By considering cases in which the pump size is varied for a constant mode size and the mode size is varied for a constant pump size, a figure of merit has been defined through which laser performance can be optimized. The analysis shows that for any value of the pump radius R0, the figure of merit is a maximum when w, the 1/e radius of the Gaussian resonator mode, is approximately equal to R0.

© 1981 Optical Society of America

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References

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  1. W. B. Bridges, IEEE J. Quantum Electron. QE-4, 820 (1965).
  2. D. G. Hall, R. J. Smith, R. R. Rice, Appl. Opt. 19, 3041 (1980).
    [CrossRef] [PubMed]
  3. L. W. Casperson, Appl. Opt. 19, 422 (1980).
    [CrossRef] [PubMed]

1980

1965

W. B. Bridges, IEEE J. Quantum Electron. QE-4, 820 (1965).

Appl. Opt.

IEEE J. Quantum Electron.

W. B. Bridges, IEEE J. Quantum Electron. QE-4, 820 (1965).

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Figures (6)

Fig. 1
Fig. 1

Plots of sI1 vs F for α = 1 and β = 0, 0.3, and 0.5. β = 0 corresponds to uniform pumping over an area π R 0 2. The other two curves show the effects of a donut-shaped pump distribution.

Fig. 2
Fig. 2

Threshold value of the pump parameter F as a function of α = R0/w for β = 0, 0.3, and 0.5.

Fig. 3
Fig. 3

Plots of sI1 (proportional to circulating power) as a function of F (proportional to pump power). The symbol δ indicates on-axis pumping [see Eq. (8)], MMG indicates pumping with a mode-matched Gaussian [see Eq. (9)], and the other three curves are for the cylindrical pumping case for values R0/w = 1, 2, and 2.

Fig. 4
Fig. 4

Plots of s P / ( π R 0 2 ) as a function of G for cylindrical pumping with α = 0.5, 1.0, and 2.0. Note that both variables are normalized by π R 0 2 in this figure, while in Figs. 13 the variables contain a normalization of πw2.

Fig. 5
Fig. 5

Threshold value of G and normalized slope efficiency as functions of the laser mode size w. The labels on the left-hand vertical axis refer to the threshold curve labeled Gth. The right-hand vertical axis refers to the other curve.

Fig. 6
Fig. 6

Plot of the figure of merit A as a function of α = R0/w. For any given R0, the maximum value of A occurs for wR0.

Equations (25)

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d I d z = g h 0 I 1 + s I η I ,
I ( r , ϕ , z ) = P ( z ) f ( r , ϕ , z ) ,
f ( r , ϕ , z ) = 2 π w 2 ( z ) exp [ 2 r 2 / w 2 ( z ) ] ,
d P d Z = Q ( z ) η P ( z ) ,
Q ( z ) = 0 2 π 0 g h 0 P ( z ) f ( r , ϕ , z ) 1 + s P ( z ) f ( r , ϕ , z ) r d r d ϕ ,
0 2 π 0 g h 0 ( r ) r d r d ϕ = G 0 ,
g h 0 = { G 0 / ( π R 0 2 ) 0 r R 0 , 0 r > R 0 ,
g h 0 = G 0 δ ( x ) δ ( y ) ,
g h 0 = 2 G 0 π w 2 exp ( 2 r 2 / w 2 ) .
g h 0 = { 0 r < r 0 , G 0 π ( R 0 2 r 0 2 ) r 0 r R 0 , 0 r > R 0 .
Q ( z ) = G 0 w 2 2 s ( R 0 2 r 0 2 ) ln [ 1 + 2 s I 1 exp ( 2 r 0 2 / w 2 ) 1 + 2 s I 1 exp ( 2 R 0 2 / w 2 ) ] ,
Q ( z ) l / P ( z ) = ( 1 R r ) + ( 1 R l ) + η l ,
F = 2 G 0 l / ( π w 2 ) ( 1 R r ) + ( 1 R l ) + η l
F 8 s I 1 ( α 2 β 2 ) ln [ 1 + 4 s I 1 exp ( 2 β 2 ) 1 + 4 s I 1 exp ( 2 α 2 ) ] = 1 ,
F t h = 2 ( α 2 β 2 ) exp ( 2 β 2 ) exp ( 2 α 2 ) .
G = ( G 0 / H ) / ( π R 0 2 ) ,
H = [ ( 1 R r ) + ( 1 R l ) + η l ] / l .
G = 4 s P π w 2 { ln 1 + 4 s P / ( π w 2 ) 1 + [ 4 s P / ( π w 2 ) ] exp ( 2 α 2 ) } 1 .
G t h = [ 1 exp ( 2 α 2 ) ] 1 .
d ( s P ) d G ( π w 2 / 2 ) tanh ( α 2 ) .
G 0 / H = 4 s P R 0 2 w 2 / ln { 1 + 4 s P / ( π w 2 ) 1 + [ 4 s P / ( π w 2 ) ] exp ( 2 R 0 2 / w 2 ) } ,
( G 0 / H ) t h = π R 0 2 1 exp ( 2 R 0 2 / w 2 ) ,
d ( s P ) d ( G 0 / H ) 1 2 w 2 R 0 2 tanh ( R 0 2 / w 2 ) .
A = [ d ( s P ) / d ( G 0 / H ) ] / ( G 0 / H ) t h ,
A = ( w 2 / R 0 2 ) tanh ( R 0 2 / w 2 ) [ 1 exp ( 2 R 0 2 / w 2 ) ] / ( 2 π R 0 2 ) .

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