Abstract

This paper contains an analysis of the behavior of saturable filters, based on models with simple energy level structures. In this analysis, the effects of possible excited state absorption at the irradiating wavelength are considered and are shown to give rise to a nonsaturable component of the absorption coefficient. Expressions for the steady-state transmission of an optically thick saturable absorber, and for the temporal response of an optically thin absorber, are derived. The question of homogeneous vs inhomogeneous broadening of the absorption line is discussed, and it is suggested that spectral hole burning may occur in a thermally broadened line at a sufficiently high irradiance level. The desirable features of a saturable absorber for use in a Q-switched laser system are briefly described, and it is shown how the usual rate equations for a laser system may be amended to include the effects of an intracavity saturable filter.

© 1967 Optical Society of America

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References

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  1. P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
    [CrossRef]
  2. B. H. Soffer, J. Appl. Phys. 35, 2551 (1964).
    [CrossRef]
  3. A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
    [CrossRef]
  4. R. W. Keyes, IBM J. 7, 334 (1963).
    [CrossRef]
  5. J. G. Calvert, J. N. Pitts, Photochemistry (John Wiley & Sons, Inc., New York, 1966). p.318.
  6. F. Gires, IEEE Trans. QE-2, 624 (1966).
  7. M. L. Spaeth, W. R. Sooy, in Quantum ElectronicsConference (Phoenix, 1966), paper 9C-7 (unpublished).
  8. B. H. Soffer, B. B. McFarland, Appl. Phys. Letters 8, 166 (1966).
    [CrossRef]
  9. W. G. Wagner, B. A. Lengyel, J. Appl. Phys. 34, 2040 (1983).
    [CrossRef]
  10. A. Szabo, R. A. Stein, J. Appl. Phys. 36, 1562 (1965).
    [CrossRef]

1983

W. G. Wagner, B. A. Lengyel, J. Appl. Phys. 34, 2040 (1983).
[CrossRef]

1966

F. Gires, IEEE Trans. QE-2, 624 (1966).

B. H. Soffer, B. B. McFarland, Appl. Phys. Letters 8, 166 (1966).
[CrossRef]

A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
[CrossRef]

1965

A. Szabo, R. A. Stein, J. Appl. Phys. 36, 1562 (1965).
[CrossRef]

1964

P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
[CrossRef]

B. H. Soffer, J. Appl. Phys. 35, 2551 (1964).
[CrossRef]

1963

R. W. Keyes, IBM J. 7, 334 (1963).
[CrossRef]

Calvert, J. G.

J. G. Calvert, J. N. Pitts, Photochemistry (John Wiley & Sons, Inc., New York, 1966). p.318.

DeMaria, A. J.

A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
[CrossRef]

Gires, F.

F. Gires, IEEE Trans. QE-2, 624 (1966).

Heynau, H.

A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
[CrossRef]

Kafalas, P.

P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
[CrossRef]

Keyes, R. W.

R. W. Keyes, IBM J. 7, 334 (1963).
[CrossRef]

Lengyel, B. A.

W. G. Wagner, B. A. Lengyel, J. Appl. Phys. 34, 2040 (1983).
[CrossRef]

Masters, J. I.

P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
[CrossRef]

McFarland, B. B.

B. H. Soffer, B. B. McFarland, Appl. Phys. Letters 8, 166 (1966).
[CrossRef]

Murray, E. M. E.

P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
[CrossRef]

Pitts, J. N.

J. G. Calvert, J. N. Pitts, Photochemistry (John Wiley & Sons, Inc., New York, 1966). p.318.

Soffer, B. H.

B. H. Soffer, B. B. McFarland, Appl. Phys. Letters 8, 166 (1966).
[CrossRef]

B. H. Soffer, J. Appl. Phys. 35, 2551 (1964).
[CrossRef]

Sooy, W. R.

M. L. Spaeth, W. R. Sooy, in Quantum ElectronicsConference (Phoenix, 1966), paper 9C-7 (unpublished).

Spaeth, M. L.

M. L. Spaeth, W. R. Sooy, in Quantum ElectronicsConference (Phoenix, 1966), paper 9C-7 (unpublished).

Stein, R. A.

A. Szabo, R. A. Stein, J. Appl. Phys. 36, 1562 (1965).
[CrossRef]

Stetser, D. A.

A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
[CrossRef]

Szabo, A.

A. Szabo, R. A. Stein, J. Appl. Phys. 36, 1562 (1965).
[CrossRef]

Wagner, W. G.

W. G. Wagner, B. A. Lengyel, J. Appl. Phys. 34, 2040 (1983).
[CrossRef]

Appl. Phys. Letters

A. J. DeMaria, D. A. Stetser, H. Heynau, Appl. Phys. Letters 8, 174 (1966).
[CrossRef]

B. H. Soffer, B. B. McFarland, Appl. Phys. Letters 8, 166 (1966).
[CrossRef]

IBM J.

R. W. Keyes, IBM J. 7, 334 (1963).
[CrossRef]

IEEE Trans.

F. Gires, IEEE Trans. QE-2, 624 (1966).

J. Appl. Phys.

W. G. Wagner, B. A. Lengyel, J. Appl. Phys. 34, 2040 (1983).
[CrossRef]

A. Szabo, R. A. Stein, J. Appl. Phys. 36, 1562 (1965).
[CrossRef]

P. Kafalas, J. I. Masters, E. M. E. Murray, J. Appl. Phys. 35, 2349 (1964).
[CrossRef]

B. H. Soffer, J. Appl. Phys. 35, 2551 (1964).
[CrossRef]

Other

J. G. Calvert, J. N. Pitts, Photochemistry (John Wiley & Sons, Inc., New York, 1966). p.318.

M. L. Spaeth, W. R. Sooy, in Quantum ElectronicsConference (Phoenix, 1966), paper 9C-7 (unpublished).

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Figures (8)

Fig. 1
Fig. 1

General three-level energy scheme for a saturable absorber showing excitation and spontaneous decay rates.

Fig. 2
Fig. 2

Temporal response of a saturable absorption coefficient α to a square-topped light pulse.

Fig. 3
Fig. 3

Transmission of an optically thick three-level saturable filter at various levels of irradiance.

Fig. 4
Fig. 4

Energy level scheme for a fast four-level saturable filter with excited state absorption at the wavelength of interest.

Fig. 5
Fig. 5

Transmission of an optically thick, fast, four-level, saturable filter at various levels of irradiance. The dashed curves show the corresponding transmission for the case where the excited state absorption is not taken into account.

Fig. 6
Fig. 6

Representative absorption line, showing peak absorption coefficient and linewidth.

Fig. 7
Fig. 7

Hole burning and saturation in an absorption line at different levels of irradiance (schematic).

Fig. 8
Fig. 8

Typical computer solution for a passively Q-switched laser, showing laser power, laser population inversion, and saturable absorber population difference as functions of time.

Equations (38)

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W 13 = I ( ν 13 ) σ 13 ,
n ˙ 1 = - W 13 ( n 1 - n 3 ) + n 3 A 31 + n 2 A 21 , n ˙ 2 = n 3 A 32 - n 2 A 21 , n ˙ 3 = W 13 ( n 1 - n 3 ) - n 3 ( A 32 + A 31 ) .
n ( n 1 - n 3 ) ,
n s = ( 1 + W 13 [ ( 2 + A 32 / A 21 ) / ( A 31 + A 32 ) ] ) - 1 ,
τ = ( 2 τ 31 τ 32 + τ 31 τ 21 ) / ( τ 32 + τ 31 ) ,
n s = ( 1 + I σ 13 τ ) - 1
I s ( σ 13 τ ) - 1 ,
n s = ( 1 + I / I s ) - 1 .
α s ( I ) α 0 n s = α 0 / ( 1 + I / I s ) .
τ = τ 31 τ 32 ( 1 + g 1 / g 3 ) + τ 21 τ 31 τ 31 + τ 32 .
I s = ( 2 σ 13 τ 31 ) - 1 ( for τ 32 )
I s = ( σ 13 τ 21 ) - 1 ( for τ 32 0 ) .
n = n 1 and n 2 = 1 - n 1 ,
n ˙ 1 = n ˙ = - I σ 13 n + ( 1 - n ) / τ 21 .
n ( t ) = n s ( 1 + ( I / I s ) e - ( 1 + I / I s ) t / τ ,
t r = [ τ 21 / ( 1 + I / I s ) ] < τ 21 ,
d I ( x ) / d x = - I { α 0 / [ 1 + I ( x ) / I s ] } ,
α 0 x = ln ( I 0 I x ) + [ I 0 - I ( x ) ] / I s .
ln ( T 0 / T ) + ( I 0 / I s ) ( 1 - T ) = 0 ,
n 1 ( steady - state ) = ( 1 + I / I s ) - 1 , n 2 ( steady - state ) 1 - n 1 = ( I / I s ) / ( 1 + I / I s ) ,
( α + β ) = α 0 n 1 + β 0 ( 1 - n 1 ) ,
[ α ( t ) + β ( t ) ] = β 0 + ( α 0 - β 0 1 + I / I s ) ( 1 + I I s e - ( 1 + I / I s ) t / τ 21 ) ,
d I d x = - I ( α + β ) = - I [ α 0 + β 0 ( I / I s ) 1 + I / I s ] .
ln T - ln T 0 = ( γ 0 - 1 ) ln ( γ 0 + I 0 / I s γ 0 + T I 0 / I s ) ,
γ 0 α 0 / β 0 σ 13 / σ 24 .
N ν = N 0 ( α ν Δ ν h / α 0 Δ ν 0 ) ,
α ν = N 0 σ ν ( homogeneously broadened line ) , α ν = N ν σ ( inhomogeneously broadened line ) .
σ / σ 0 = Δ ν 0 / Δ ν h ,
σ > σ 0 ( Δ ν 0 / δ ν ) = ( α 0 / N 0 ) / Δ ν 0 / δ ν .
α ( t ) = α 0 ( 1 + I 0 / I s ) × ( 1 + I 0 I s e - ( 1 + I 0 / I s ) t / τ ) .
E s = I 0 h ν x 0 ( α ( t ) - α s ) d t = [ I 0 2 h ν ( α 0 x ) ] / [ σ ( I 0 + I s ) 2 ,
I h ~ ( σ τ d ) - 1 .
m ˙ = - I ( t ) σ s m + ( 1 - m ) / τ s ,
m ˙ = - Φ ( t ) c σ s m + ( 1 - m ) / τ s
m s = [ 1 + Φ ( t ) c σ s τ s ] - 1 .
γ s = α s 0 m d .
n ( t ) = [ 1 λ 1 ( λ 1 - λ 2 ) ( A 21 A 31 + A 21 A 32 + λ 1 2 + λ 1 A 31 + λ 1 A 21 ) e λ 1 t + 1 λ 2 ( λ 2 - λ 1 ) ( λ 2 + A 32 + A 31 ) ( λ 2 + A 21 ) e λ 2 t + A 21 ( A 32 + A 31 ) λ 1 λ 2 ] ,
λ 1 = - 1 2 [ ( A 31 + A 32 + A 21 + 2 I σ 13 ) + ( [ A 31 + A 32 + 2 I σ 13 - A 21 ] 2 - 4 W 13 A 32 ) ½ , λ 2 = - 1 2 [ ( A 31 + A 32 + A 21 + 2 I σ 13 ) - ( [ A 31 + A 32 + 2 I σ 13 - A 21 ] 2 - 4 W 13 A 32 ) ½ ] .

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