Abstract

This paper analyzes thermally induced mirror deformations and their resulting wavefront distortions which occur under the conditions of radially nonuniform mirror heating. The analysis is adaptable to heating produced by any radially nonuniform incident radiation. Specific examples of radiation distributions which are considered are the cosine squared and the gaussian and TEM(0, 1) laser distributions. Deformation effects are examined from two aspects, the first of which is the reflected wavefront radial phase distortion profile caused by the thermally induced surface irregularities at the mirror face. These phase distortion effects appear as aberrations in noncoherent optical applications and as the loss of spatial coherence in coherent applications. The second aspect is the gross wavefront bending due to mirror curvature effects. The analysis considers substrate material, geometry, and cooling in order to determine potential deformation controlling factors. Substrate materials are compared, and performance indicators are suggested to aid in selecting an optimum material for a given heating condition. Deformation examples are given for materials of interest and specific absorbed power levels.

© 1970 Optical Society of America

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References

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  1. A. J. Chapman, Heat Transfer (Macmillan, New York, 1967), 2nd ed.
  2. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959), 2nd ed.
  3. W. H. Giedt, Principles of Engineering Heat Transfer (Van Nostrand, Princeton, N. J., 1957).
  4. S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951), 2nd ed.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959), 2nd ed.

Chapman, A. J.

A. J. Chapman, Heat Transfer (Macmillan, New York, 1967), 2nd ed.

Giedt, W. H.

W. H. Giedt, Principles of Engineering Heat Transfer (Van Nostrand, Princeton, N. J., 1957).

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951), 2nd ed.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959), 2nd ed.

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951), 2nd ed.

Other (4)

A. J. Chapman, Heat Transfer (Macmillan, New York, 1967), 2nd ed.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959), 2nd ed.

W. H. Giedt, Principles of Engineering Heat Transfer (Van Nostrand, Princeton, N. J., 1957).

S. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1951), 2nd ed.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Radial power density distribution.

Fig. 3
Fig. 3

Enclosed power distribution.

Fig. 4
Fig. 4

Radial phase distortion for b/w = 1.

Fig. 5
Fig. 5

Radial phase distortion for b/w = 5.

Fig. 6
Fig. 6

Radial phase distortion for b/w = 1.

Fig. 7
Fig. 7

Radial phase distortion for b/w = 5.

Tables (3)

Tables Icon

Table I Radial Power Density and Enclosed Power Distributions

Tables Icon

Table II Radius of Curvature for General Case

Tables Icon

Table III Radius of Curvature for Three Materials of Interest in Tens of Kilometers

Equations (21)

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δ ( R ) = W ( 0 ) W ( R ) [ κ 1 ( κ 2 b 2 R 2 ) 1 4 ]
W ( R ) = ( b 2 α k ) i = 0 ψ i Ω i ( J 0 ( β i R ) J 0 ( β i ) G ( R ) G ( 1 ) ) ,
ψ i = P 0 Δ i / { J 0 ( β i ) ( C s 2 + β i 2 ) [ β i sinh β i D + ( 1 + M ) C s × cosh β i D ] } , Ω i = β i sinh β i D + M C s ( cosh β i D 1 ) , G ( R ) = { [ 4 ( 1 ν ) C ] I 0 ( π R / D ) + ( π R / D ) I 1 ( π R / D ) } ,
C = [ ( 1 2 ν ) ( π / D ) I 0 ( π / D ) + ( π / D ) 2 I 1 ( π / D ) ( π / D ) I 0 ( π / D ) I 1 ( π / D ) ] , Λ i = 2 0 1 p ( R ) R J 0 ( β i R ) d R .
κ = ( 6 α K D 3 ) i = 0 ψ i ω i ,
ω i = [ 2 + M C s D + ( D β i 2 M C s / β i ) sinh β i D + ( M C s D 2 ) cosh β i D ] .
T ( R , Z ) = b k i = 0 β i ψ i [ J 0 ( β i R ) / J 0 ( β i ) ] ( β i cosh β i D Z + M C s sinh β i D Z ) ,
C s = k f N nu / k .
C s = 8.63 × 10 4 ( P 0 b 2 / M ) 1 5 k if N gr N pr < 10 8
= 6.13 × 10 4 ( P 0 b 2 / M ) 1 4 k if N gr N pr > 10 8
C b = k f N nu / k ,
N nu = 0.664 N pr 1 3 N re 1 2 if N re < 4 × 10 5 ,
0.036 N pr 1 3 N re 4 5 if N re > 4 × 10 5 .
υ air = 16.7 ( P 0 b 2 M 4 ) 0.4 / b
υ water = 2.54 × 10 4 ( P 0 b 2 M 4 ) 0.4 / b
υ water = 1.28 × 10 4 ( P 0 M 3 ) 0.5
z ( R , Z ) = W ( R , Z ) / Z = E 1 [ σ z ν ( σ r + σ θ ) ] + α [ T ( R , Z ) T ( 1,0 ) ] ,
S ( r , z ) = [ a 0 I 0 ( a 2 r ) a 1 I 1 ( a 2 r ) ] cos a 2 z ,
σ r = ν 2 S / z 2 S / z r 2 , σ θ = ν 2 S / z ( 1 / r ) 2 S / z r , σ z = ( 2 ν ) 2 S / z 3 S / z 3 , τ r z = ( 1 ν ) 2 S / r 3 S / r z 2 .
σ r ( r , z ) = a 1 a 2 3 { [ 1 2 ν C ] I 0 ( a 2 r ) + [ ( a 2 r ) 2 + C ] I 1 ( a 2 r ) / ( a 2 r ) } sin a 2 z , σ θ ( r , z ) = a 1 a 2 3 [ ( 1 2 ν ) I 0 ( a 2 r ) C I 1 ( a 2 r ) / ( a 2 r ) ] sin a 2 z , σ z ( r , z ) = a 1 a 2 3 { [ ( 4 2 ν ) C ] I 0 ( a 2 r ) + ( a 2 r ) I 1 ( a 2 r ) } sin a 2 z , τ r z ( r , z ) = a 1 a 2 3 { ( a 2 r ) I 0 ( a 2 r ) + [ 2 2 ν C ] I 1 ( a 2 r ) } cos a 2 z , a 0 = a 1 C , a 1 = ( E 1 + ν ) ( D b 2 π ) 0 z T ( b , z ) d z / G ( 1 ) [ cos ( a 2 z ) 1 ] ,
r ( r , z ) = κ z + c = E 1 [ σ r ν ( σ z + σ θ ) ] + α [ T ( r , z ) T ( 1,0 ) ] ,

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