Abstract

Theoretical and experimental results are reported on the thermal lensing effect caused by the radial thermal gradient present in optically pumped Nd: YAG laser rods. The presented theory is in agreement with the experimental observations. The results reveal that a Nd:YAG rod under pumped light becomes a positive lens with two focal lengths. The temperature dependent variation of the refractive index constitutes the major contribution of the thermal lensing. The stress dependent variation of the refractive index modifies the focal length about 20%. The effect of end-face curvature caused by an elongation of the rod is less than 6%.

© 1970 Optical Society of America

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References

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  1. L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
    [CrossRef]
  2. H. S. Carslow, Conduction of Heat in Solids (Clarendon Press, Oxford, England, 1952), p. 204.
  3. M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965).
  4. J. F. Nye, Physical Properties of Crystals (Oxford University Press, London, 1964).
  5. H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965).
  6. E. Snitzer, C. G. Young, in Lasers, A. K. Levine, Ed. (M. Dekker, New York, 1968), Vol. 2.
  7. D. A. La Marre, “High Performance Laser Research,” Report AD840913American Optical Corp., June1968.
  8. R. W. Dixon, J. Appl. Phys. 38, 5149, 1967.
    [CrossRef]
  9. “YAG:Nd Data,” Union Carbide Corporation, Electronics Div., Crystal Products Dept.
  10. J. D. Foster, L. M. Osterink, Appl. Opt. 7, 2428 (1968).
    [CrossRef] [PubMed]
  11. W. Koechner, Appl. Opt. 9, 1427 (1970).
  12. H. Kogelnik, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  13. W. Koechner, D. K. Rice, IEEE J. Quantum Electron. QE-6, 557 (1970).
    [CrossRef]
  14. S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).
  15. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

1970 (2)

W. Koechner, Appl. Opt. 9, 1427 (1970).

W. Koechner, D. K. Rice, IEEE J. Quantum Electron. QE-6, 557 (1970).
[CrossRef]

1968 (2)

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

J. D. Foster, L. M. Osterink, Appl. Opt. 7, 2428 (1968).
[CrossRef] [PubMed]

1967 (1)

R. W. Dixon, J. Appl. Phys. 38, 5149, 1967.
[CrossRef]

1966 (1)

H. Kogelnik, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1965 (1)

H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965).

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965).

Carslow, H. S.

H. S. Carslow, Conduction of Heat in Solids (Clarendon Press, Oxford, England, 1952), p. 204.

Dixon, R. W.

R. W. Dixon, J. Appl. Phys. 38, 5149, 1967.
[CrossRef]

Foster, J. D.

J. D. Foster, L. M. Osterink, Appl. Opt. 7, 2428 (1968).
[CrossRef] [PubMed]

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Koechner, W.

W. Koechner, D. K. Rice, IEEE J. Quantum Electron. QE-6, 557 (1970).
[CrossRef]

W. Koechner, Appl. Opt. 9, 1427 (1970).

Kogelnik, H.

H. Kogelnik, Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965).

La Marre, D. A.

D. A. La Marre, “High Performance Laser Research,” Report AD840913American Optical Corp., June1968.

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford University Press, London, 1964).

Osterink, L. M.

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

J. D. Foster, L. M. Osterink, Appl. Opt. 7, 2428 (1968).
[CrossRef] [PubMed]

Rice, D. K.

W. Koechner, D. K. Rice, IEEE J. Quantum Electron. QE-6, 557 (1970).
[CrossRef]

Snitzer, E.

E. Snitzer, C. G. Young, in Lasers, A. K. Levine, Ed. (M. Dekker, New York, 1968), Vol. 2.

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965).

Young, C. G.

E. Snitzer, C. G. Young, in Lasers, A. K. Levine, Ed. (M. Dekker, New York, 1968), Vol. 2.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Bell. Syst. Tech. J. (1)

H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965).

IEEE J. Quantum Electron. (1)

W. Koechner, D. K. Rice, IEEE J. Quantum Electron. QE-6, 557 (1970).
[CrossRef]

J. Appl. Phys. (1)

R. W. Dixon, J. Appl. Phys. 38, 5149, 1967.
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other (7)

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951).

“YAG:Nd Data,” Union Carbide Corporation, Electronics Div., Crystal Products Dept.

E. Snitzer, C. G. Young, in Lasers, A. K. Levine, Ed. (M. Dekker, New York, 1968), Vol. 2.

D. A. La Marre, “High Performance Laser Research,” Report AD840913American Optical Corp., June1968.

H. S. Carslow, Conduction of Heat in Solids (Clarendon Press, Oxford, England, 1952), p. 204.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1965).

J. F. Nye, Physical Properties of Crystals (Oxford University Press, London, 1964).

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Figures (6)

Fig. 1
Fig. 1

Crystal orientation for a YAG rod (top) and orientation of indicatrix of a thermally stressed YAG rod in a plane perpendicular to the rod axis (bottom).

Fig. 2
Fig. 2

Experimental setup for the measurement of the effective focal length with a gas laser.

Fig. 3
Fig. 3

Effective focal length of YAG crystal as a function of lamp input power. The two theoretical focal lengths according to Eq. (31) are plotted (solid lines) together with experimental values obtained with a gas laser (Δ) and values obtained from resonator theory (○).

Fig. 4
Fig. 4

Stability diagram of an optical resonator. Shaded areas indicate regions of unstable operation. Points A, B and C correspond to plane parallel, confocal, and concentric resonators, respectively.

Fig. 5
Fig. 5

Experimental setup for the measurement of the end face curvature of a Nd:YAG crystal.

Fig. 6
Fig. 6

Theoretical (solid lines) and experimental (○) values of the end face curvature of a YAG rod as a function of lamp input power.

Equations (58)

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T ( r ) = T 0 - A 0 r 2 / 4 K ,
A 0 = P a / π r 0 2 L ,
P a = η P in .
T ( r ) = T 0 - η P in r 2 / 4 K π L r 0 2 .
n ( r ) = n 0 + Δ n ( r ) T + Δ n ( r ) ,
n ( r ) = n 0 + ( δ n / δ T ) [ T ( r ) - T 0 ] .
n ( r ) = n 0 - ( P in r 2 η / 4 K π L r 0 2 ) ( δ n / δ T ) ,
Δ n ( r ) T = c 1 ( r 2 / r 0 2 ) ,
c 1 = - ( η P in / 4 K π L ) ( δ n / δ T ) .
Δ B i j = p i j k l k l ( i , j , k , l = 1 , 2 , 3 ) ,
( B ° + Δ B x * x * ) x * 2 + ( B ° + Δ B y * y * ) y * 2 = 1.
B ° = ( n 0 2 ) - 1
Δ n x * = - 1 2 ( n 0 3 ) Δ B x * x * ,             Δ n y * = - 1 2 ( n 0 3 ) Δ B y * y * ,
Δ n ( r ) x * = c 2 + c 3 ( r 2 / r 0 2 ) ,
Δ n ( r ) y * = c 4 + c 5 ( r 2 / r 0 2 ) ,
n ( r ) x * = n 0 + c 2 + ( c 1 + c 3 ) ( r 2 / r 0 2 ) ,
n ( r ) y * = n 0 + c 3 + ( c 1 + c 5 ) ( r 2 / r 0 2 ) .
f = b ( 2 n 0 sin 2 L / b ) - 1 .
n = n 0 ( 1 - 2 r 2 / b 2 ) ,
h = b / 2 n 0 tan ( L / b ) .
n 0 = n 0 + c 2 ,
b 2 = - 2 ( n 0 + c 2 ) r 0 2 / ( c 1 + c 3 ) .
sin ( 2 L / b ) 2 L / b and n 0 n 0 .
f = b 2 / 4 n 0 L .
h = L / 2 n 0 .
l ( r ) = l 0 + ( δ l / δ T ) ( T ( r ) - T 0 ) ,
δ l / δ T = α l 0 ,
R = - ( d 2 l / d r 2 ) - 1 .
R = 2 K π L r 0 2 / α η P in l 0 .
f = R / 2 ( n 0 - 1 ) .
f = ( 1 f + 1 f ) - 1 = [ 4 n 0 L b 2 + 2 ( n 0 - 1 ) R ] - 1 .
f 1 = 2 K π r 0 2 P in η [ δ n δ T - n 0 3 α c x 48 ( 1 - ν ) + 2 ( n 0 - 1 ) α l 0 L ] - 1 .
f 1 = 1.41 / P in ,
f 2 = 2.0 / P in ,
δ n δ T > n 0 3 α c x 48 ( 1 - ν ) > 2 ( n 0 - 1 ) α l 0 L
G 1 = a 1 a 2 [ 1 - d 2 f - 1 R 1 ( d 1 + d 2 - d 1 d 2 f ) ] ,
G 2 = a 2 a 1 [ 1 - d 1 f - 1 R 2 ( d 1 + d 2 - d 1 d 2 f ) ] ,
a 1 = a 2 , R 1 = R 2 , and d 1 = d 2 = d .
G 1 = G 2 = 1 - d / f .
f = d / 2.
R = 2 t d 1 / d 0 - 1 ,
Δ B x * x * = 1 6 [ p 11 ( 3 r + Φ + 2 x ) + p 12 ( 3 r + 5 Φ + 4 z ) + 2 p 44 ( 3 r - Φ - 2 z ) ] ,
Δ B y * y * = 1 6 [ p 11 ( r + 3 Φ + 2 z ) + p 12 ( 5 r + 3 Φ + 4 z ) - 2 p 44 ( r - 3 Φ + 2 z ) ] ,
r = S [ ( 3 ν - 1 ) - ( 7 ν - 1 ) r 2 / r 0 2 ] ,
Φ = S [ ( 3 ν - 1 ) - ( 5 ν - 3 ) r 2 / r 0 2 ] ,
z = S [ 2 ( ν - 1 ) - 4 ( ν - 1 ) r 2 / r 0 2 ] .
S = α A 0 r 0 2 / ( 1 - ν ) 16 K ,
Δ B x * x * = S ( c x = c x r 2 / r 0 2 ) / 6 ,
Δ B y * y * = S ( c y - c y r 2 / r 0 2 ) / 6 ,
S = α η P in / ( 1 - ν ) 16 K π L ,
c x = c y = 4 ( 3 ν - 1 ) ( p 11 + 2 p 12 + p 44 ) + 4 ( ν - 1 ) ( p 11 + 2 p 12 - 2 p 44 ) ,
c x = 3 ( 7 ν - 1 ) ( p 11 + p 12 + 2 p 44 ) + ( 5 ν - 3 ) ( p 11 + 5 p 12 - 2 p 44 ) + 8 ( ν - 1 ) ( p 11 + 2 p 12 - 2 p 44 ) ,
c y = ( 7 ν - 1 ) ( p 11 + 5 p 12 - 2 p 44 ) + 3 ( 5 ν - 3 ) ( p 11 + p 12 + 2 p 44 ) + 8 ( ν - 1 ) ( p 11 + 2 p 12 - 2 p 44 ) .
Δ n x * = c 2 + c 3 r 2 / r 0 2 ,
Δ n y * = c 4 + c 5 r 2 / r 0 2 ,
c 2 = c 4 = - n 0 3 S c x / 12 ,
c 3 = n 0 3 S c x / 12 ,
c 5 = n 0 3 S c y / 12.

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