Abstract

The theoretical characteristics of a 1-D model of a steady-state laser with uniform saturable gain and distributed losses are calculated by iterative solution, using integration of a series representation of the two-way intensity distribution over the length of the laser. Results for this model and for the model with point losses at the end mirrors show that the output can be the same for both models over a range of the point loss factor. This factor becomes unique if the losses are also set equal, but the internal intensity distributions are similar only when the loss is low.

© 1987 Optical Society of America

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References

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  1. W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
    [CrossRef]
  2. Spontaneous emission is not included in the model. As the losses increase and the laser approaches threshold, the model predictions are increasingly in error since more gain is available for spontaneous emission and amplification. Multimode oscillation also affects the saturation characteristics. See, for example, L. W. Casperson, “Threshold Characteristics of Multimode Laser Oscillations,” J. Appl. Phys. 46, 5194 (1975).
    [CrossRef]
  3. If the requirement of equal potential losses is dropped and other values of t and a are used in the point loss model, the internal distribution must still vary much more over the length of the laser to provide both high output and high loss at the boundaries.

1975 (1)

Spontaneous emission is not included in the model. As the losses increase and the laser approaches threshold, the model predictions are increasingly in error since more gain is available for spontaneous emission and amplification. Multimode oscillation also affects the saturation characteristics. See, for example, L. W. Casperson, “Threshold Characteristics of Multimode Laser Oscillations,” J. Appl. Phys. 46, 5194 (1975).
[CrossRef]

1965 (1)

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Casperson, L. W.

Spontaneous emission is not included in the model. As the losses increase and the laser approaches threshold, the model predictions are increasingly in error since more gain is available for spontaneous emission and amplification. Multimode oscillation also affects the saturation characteristics. See, for example, L. W. Casperson, “Threshold Characteristics of Multimode Laser Oscillations,” J. Appl. Phys. 46, 5194 (1975).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

J. Appl. Phys. (2)

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

Spontaneous emission is not included in the model. As the losses increase and the laser approaches threshold, the model predictions are increasingly in error since more gain is available for spontaneous emission and amplification. Multimode oscillation also affects the saturation characteristics. See, for example, L. W. Casperson, “Threshold Characteristics of Multimode Laser Oscillations,” J. Appl. Phys. 46, 5194 (1975).
[CrossRef]

Other (1)

If the requirement of equal potential losses is dropped and other values of t and a are used in the point loss model, the internal distribution must still vary much more over the length of the laser to provide both high output and high loss at the boundaries.

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Figures (6)

Fig. 1
Fig. 1

Wave intensities in an asymmetrical laser with values at the crossing and the end mirrors labeled.

Fig. 2
Fig. 2

Output intensity from one end of a symmetrical laser with g0L = 10.

Fig. 3
Fig. 3

Output intensity from a single-ended laser with g0L = 10.

Fig. 4
Fig. 4

Output intensity from one end of a symmetrical laser with g0L = 40.

Fig. 5
Fig. 5

Right-traveling wave intensity distribution in a symmetrical laser with g0L = 40. (Iterated solutions for two values of αL and t, —; exponential approximation, - - - - -; point loss solutions, – – –).

Fig. 6
Fig. 6

Output intensities of symmetrical and single-ended lasers vs g0L with α/g0 fixed at 0.1 and r = 0.3.

Equations (23)

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g ( z ) = g 0 / ( 1 + β + + β - ) .
1 β + d β + d z = - 1 β - d β - d z = g ( z ) - α .
β + β - = C = β 0 2 = β 1 β 4 = β 2 β 3 .
β 3 = r 2 β 2 ,             β 1 = r 1 β 4 .
r 1 = 1 - a 1 - t 1 ,             r 2 = 1 - a 2 - t 2 .
( 1 + 1 β + + β 0 2 β + 2 ) d β + d z = g 0 - α ( 1 + β + + β - ) .
β 0 ( 1 - r 2 r 2 + 1 - r 1 r 1 ) - ½ ln ( r 1 r 2 ) = g 0 L - α L - α 0 L ( β + + β - ) d z .
1 β + d β + d ζ = g 0 L 1 + β + + β 0 2 / β + - α L ,
- 1 β - d β - d ζ = g 0 L 1 + β - + β 0 2 / β - - α L .
β - ( - ζ ) = β + ( ζ )
( 1 + β + + β - ) d β + / d ζ = [ g 0 L - α L ( 1 + β + + β - ) ] β + .
β + ( ζ ) = n = 0 I n ζ n / n ! ,
I 1 = ( s - α L ) β 0 , I 2 = ( s - α L ) 2 β 0 , I 3 = ( s - α L ) 2 [ s 1 + 2 β 0 - α L ] β 0 , I 4 = ( s - α L ) 3 [ s ( 1 - 6 β 0 1 + 2 β 0 ) - α L ] β 0 ,
s = g 0 L / ( 1 + 2 β 0 ) .
σ = 1 + ( s - α L ) 2 2 2 · 3 ! + ( s - α L ) 3 2 4 · 5 ! [ s ( 1 - 6 β 0 1 + 2 β 0 ) - α L ] + .
β 0 = r 1 / 2 ( g 0 L + ln r - α L ) 2 ( 1 - r - α L σ r 1 / 2 ) .
β 2 t = ( t / 2 ) [ g 0 L + ln ( 1 - a - t ) ] / ( t + a ) ,
β 2 t = ( t / 2 ) [ g 0 L - α L + ln ( 1 - t ) ] / [ t + α L ( 1 - t ) 1 / 2 σ ] .
β + 2 + β 0 2 β + + ln β + β 0 = ζ [ g 0 L - α L - 2 α L β 0 σ ( ζ ) ] ,
σ ( ζ ) = 1 + ( s - α L ) 2 ζ 2 3 ! + ( s - α L ) 3 [ s ( 1 - 6 β 0 1 + 2 β 0 ) - α L ] ζ 4 5 ! + .
I ( ζ ) = β 0 exp [ ( s - α L ) ζ ]
s - α L ln ( 1 / r ) .
β 0 1 2 [ g 0 L α L + ln ( 1 / r ) - 1 ] .

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