Abstract

A procedure is presented to design an aspheric corrective mirror to remove the effects of thermally induced optical aberrations in end-pumped solid-state lasers. The design is based on solving the inverse problem of bending a thin plate of variable thickness; i.e., given the plate deflection profile a thickness profile must be calculated by solving the differential equation for bending. The advantage of this type of aberration correction is the fact that it can be scaled to different pump powers during operation while still matching the aspheric profile in question. Guidelines for fabrication of the mirror are also presented.

© 1992 Optical Society of America

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References

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  1. W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988).
  2. A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
    [CrossRef]
  3. S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).
  4. C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
    [CrossRef]
  5. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed. (Dover, New York, 1927).
  6. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).
  7. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).
  8. R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.
  9. V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
    [CrossRef]

1992

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

1991

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

1990

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

Apollonov, V. V.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Bowers, M. S.

S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

Chetkin, S. A.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Cousins, A. K.

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

Fields, R. A.

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

Fincher, C. L.

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).

Innocenzi, M. E.

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

Klein, C. A.

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988).

Love, A. E. H.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed. (Dover, New York, 1927).

Lowenthal, D. D.

S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

Ostrovskaya, L. M.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).

Rodin, V. N.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Rose, T. S.

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).

Tidwell, S. C.

S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

Vdovin, G. V.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

Yura, H. Y.

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

IEEE J. Quantum Electron.

A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

Opt. Eng.

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 29, 343–350 (1990).
[CrossRef]

Sov. J. Quantum Electron.

V. V. Apollonov, G. V. Vdovin, L. M. Ostrovskaya, V. N. Rodin, S. A. Chetkin, “Active correction of a thermal lens in a solid-state laser. I. Metal mirror with a controlled curvature of the central region of the reflecting surface,” Sov. J. Quantum Electron. 21, 116–118 (1991).
[CrossRef]

Other

W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988).

S. C. Tidwell, M. S. Bowers, A. K. Cousins, D. D. Lowenthal, “Scaling output power of end-pumped solid-state lasers,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991).

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed. (Dover, New York, 1927).

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1990).

R. A. Fields, T. S. Rose, M. E. Innocenzi, H. Y. Yura, C. L. Fincher, “Diode laser end-pumped neodymium lasers: the road to higher powers,” in Proceedings of the Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, D.C., 1990), pp. 301–308.

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Figures (8)

Fig. 1
Fig. 1

Top-hat pump distribution in an end-pumped laser rod.

Fig. 2
Fig. 2

Rod temperature profiles for different pump spot radii.

Fig. 3
Fig. 3

Top-hat aberration profile and best-fit Gaussian mirror deformation profile (a/b = 0.5).

Fig. 4
Fig. 4

Deformable mirror assembly.

Fig. 5
Fig. 5

Example deformation profile.

Fig. 6
Fig. 6

Example mirror thickness profile.

Fig. 7
Fig. 7

Control volume used to derive the thickness matching condition at pump discontinuity.

Fig. 8
Fig. 8

Aberration profile for Gaussian pump and best-fit Gaussian profile (a/b = 0.5).

Tables (2)

Tables Icon

Table 1 Coefficients of Best-Fit Gaussians (Top-Hat Pump OPD’s)

Tables Icon

Table 2 Coefficients of Best-Fit Gaussians (Gaussian Pump OPD’s)

Equations (25)

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OPD = χ L T ,
χ = ( 1 + ν ) ( n 0 - 1 ) α T + n T .
T = Q 4 π k L { - 2 ln ( a b ) + 1 - ( r a ) 2 ,             r a - 2 ln ( r b ) ,             r > a
OPD = Δ w 1 - 2 ln ( a / b ) { - 2 ln ( a b ) + 1 - ( r a ) 2 ,             r a - 2 ln ( r b ) ,             r > a
D ( r ) = E h 3 ( r ) 12 ( 1 - ν 2 ) .
φ ( r ) = - d w d r .
D d d r ( d φ d r + φ r ) + d D d r ( d φ d r + ν φ r ) = - 1 2 p 0 r .
h = h 0 exp [ - 1 6 β ( r b ) 2 ] ,
φ ( r ) = - 6 ( 1 - ν 2 ) b 3 p 0 E h 0 3 ( 3 - ν ) β ( r b ) exp [ 1 2 β ( r b ) 2 ] ,
w ( r ) - w ( b ) = 6 ( 1 - ν 2 ) b 4 p 0 E h 0 3 ( 3 - ν ) β 2 × { exp [ 1 2 β ] - exp [ 1 2 β ( r b ) 2 ] } .
k 0 + k 1 exp [ 1 2 β ( r b ) 2 ] = 1 1 - 2 ln ( a / b ) { - 2 ln ( a b ) + 1 - ( r a ) 2 , r α - 2 ln ( r b ) , r > a ,
p 0 = E h 0 3 ( 3 - ν ) β 3 6 ( 1 - ν 2 ) b 4 w ( 0 ) - w ( b ) 1 - exp [ ½ β ] .
1 r r ( r T r ) = { - Q ( π a 2 L k ) , r a 0 , r > a .
d d r ( d φ d r + φ r ) d d r [ 1 r d d r ( r d T d r ) ] = 0
d φ d r + ν φ r = Δ w 1 - 2 ln ( a / b ) { 2 ( 1 + ν ) / a 2 , r < a - 2 ( 1 - ν ) / r 2 , r > a } .
h ( r ) = h 0 { ( 1 - ν ) 1 / 3 [ C - ( r a ) 2 ] 1 / 3 , r < a ( 1 + ν ) 1 / 3 [ C + 1 2 ( r a ) 4 ] 1 / 3 , r > a ,
h 0 [ 3 2 p 0 a 4 E 1 - 2 ln ( a / b ) Δ w ] 1 / 3 .
M r + = M r - .
D + ( d φ d r + ν φ r ) + = D - ( d φ d r + ν φ r ) - ,
D + D - = - 1 + ν 1 - ν < 0.
I = I 0 exp [ - 2 ( r a ) 2 ] ,
OPD = Δ w γ + E 1 [ 2 ( b a ) 2 ] + ln [ 2 ( b a ) 2 ] × { γ + E 1 [ 2 ( b a ) 2 ] + ln [ 2 ( b a ) 2 ] , r a E 1 [ 2 ( b a ) 2 ] - E 1 [ 2 ( r a ) 2 ] - ln [ ( r b ) 2 ] , r > a .
k 0 + k 1 exp [ 1 2 β ( r b ) 2 ] = 1 γ + E 1 [ 2 ( b a ) 2 ] + ln [ 2 ( b a ) 2 ] × { γ + E 1 [ 2 ( b a ) 2 ] + ln [ 2 ( b a ) 2 ] , r a E 1 [ 2 ( b a ) 2 ] - E 1 [ 2 ( r a ) 2 ] - ln [ ( r b ) 2 ] , r > a .
a ( r ) = d φ d r + ν φ r , b ( r ) = d d r ( d φ d r + φ r ) , c ( r ) = ½ p 0 r , D h ( r ) = exp [ - r b ( r ) a ( r ) d r ] .
h ( r ) = { [ 12 ( 1 - ν ) 2 E ] D h ( r ) [ C - r c ( r ) a ( r ) d r D h ( r ) ] } 1 / 3 ,

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