Abstract

We show that a deformable mirror with four point actuators is suitable for the simultaneous correction of astigmatism and defocus in a given optical system by adequately choosing the relative position of the actuators. An analytical model is developed that describes adequately the mirror deformation as a function of actuator position, showing that it is possible to continuously tune the weight of each aberration. Experimental measurements with a single adjuster deformable mirror assembly confirm the validity of the model.

© 2007 Optical Society of America

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  1. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, "Generation and characterization of the highest laser intensities (1022 W/cm2)," Opt. Lett. 29,2837-2839 (2004).
    [CrossRef]
  2. B. Wattellier, J. Fuchs, J. P. Zou, K. Abdeli, H . Pepin, and C. Haefner, "Repetition rate increase and diffractionlimited focal spots for a nonthermal-equilibrium 100-TW Nd:glass laser chain by use of adaptive optics," Opt. Lett. 29,2494-2496 (2004).
    [CrossRef] [PubMed]
  3. C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)
  4. H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)
  5. J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)
  6. Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
    [CrossRef]
  7. J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
    [CrossRef] [PubMed]
  8. J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
    [CrossRef]
  9. J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
    [CrossRef]
  10. C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).
  11. S. Timoshenko and S. Woinowski-Krieger, Theory of plates and shells, McGraw-Hill, New York, 1959.
  12. J. C. Wyant and K. Creath, Basic wavefront aberration theory for optical metrology, (Academic Press, 1992).
  13. IDEA - Interferometric Data Evaluation Algorithms, http://www.optics.tugraz.at/idea/idea.html

2006 (4)

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
[CrossRef] [PubMed]

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

2004 (3)

2002 (1)

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

1998 (2)

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).

Abdeli, K.

Akahane, Y.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Aoyoma, M.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Bahk, S.-W.

Borneis, S.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Chvykov, V.

Collier, J.

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).

Danson, C.

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

Edwards, C.

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

Fuchs, J.

Fukuda, Y.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Gaul, E.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Geissel, M.

Haefner, C.

Häfner, C.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Hawkes, S.

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).

Headley, D.

J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
[CrossRef] [PubMed]

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

Hernandez-Gomez, C.

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).

Heuck, H.-M.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Hooker, C.

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

Kalintchenko, G.

Kiriyama, H.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Kudriaschow, A

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Kudryashov, A. V.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Kühl, T.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Ma, J.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Maksimchuk, A.

Mourou, G. A.

Pepin, H

Pepler, D.

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

Planchon, T. A.

Porter, J.

J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
[CrossRef] [PubMed]

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

Rambo, P.

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
[CrossRef] [PubMed]

Ramsey, M.

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

Ross, I.

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

Rousseau, P.

Schwarz, J.

J. Schwarz, M. Geissel, P. Rambo, J. Porter, D. Headley, and M. Ramsey, "Development of a variable focal length concave mirror for on-shot thermal lens correction in rod amplifiers," Opt. Express 14, 10957-10969 (2006)
[CrossRef] [PubMed]

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

Sheldakova, J. V.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Smith, I.

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

Wattellier, B.

Wiewior, P.

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Winstone, T.

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

Wittrock, U

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Yamakawa, K.

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Yanovsky, V.

Zou, J. P.

Appl. Phys. B (1)

J. Schwarz, M. Ramsey, D. Headley, P. Rambo, I. Smith, and J. Porter, "Thermal lens compensation by convex deformation of a flat mirror with variable annular force," Appl. Phys. B 82,275-281 (2006).
[CrossRef]

CLF Report (3)

J. Collier, C. Hooker, S. Hawkes, and C. Edwards, "Adaptive optics for the petawatt upgrade," CLF Report 2001/2002, 181-182 (2002)

C. Hernandez-Gomez, J. Collier, S. Hawkes, C. Danson, I. Ross, T. Winstone, and D. Pepler, "Wavefront analysis of the vulcan laser system," CLF Report 97/98, 150 (1998)

C. Hernandez-Gomez, J. Collier, and S. Hawkes,"Vulcan intensity increase by wavefront quality improvement," CLF Report 97/98, 153 (1998).

GSI Scientific Report (1)

H.-M. Heuck, S. Borneis, E. Gaul, C. Häfner, A . Kudriaschow, T. Kühl, P. Wiewior, and U . Wittrock, "Beam diagnostics and adaptive optics for PHELIX," GSI Scientific Report 2003, 124 (2004)

Opt. Commun. (1)

J. Schwarz, M. Ramsey, I. Smith, D. Headley, and J. Porter, "Low order adaptive optics on Z-Beamlet using a single actuator deformable mirror," Opt. Commun. 264,203-212 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

Y. Akahane, J. Ma, Y. Fukuda,M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, "Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2," Rev. Sci. Instrum. 77,023102 (2006).
[CrossRef]

Other (3)

S. Timoshenko and S. Woinowski-Krieger, Theory of plates and shells, McGraw-Hill, New York, 1959.

J. C. Wyant and K. Creath, Basic wavefront aberration theory for optical metrology, (Academic Press, 1992).

IDEA - Interferometric Data Evaluation Algorithms, http://www.optics.tugraz.at/idea/idea.html

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Figures (10)

Fig. 1.
Fig. 1.

Deformable mirror geometry, with a the mirror radius, b the radius where the point forces P/2 are applied, at θ = 0,π, and the mirror is simply supported at the two edge points θ = ±π/2.

Fig. 2.
Fig. 2.

Relative weight of the Z 3 and Z 11 Zernike polynomial coefficients for a reflected wavefront as a function of the deformable mirror parameter β.

Fig. 3.
Fig. 3.

Calculated wavefront profile and interferogram for a = 37.5 mm, h = 6 mm, β = 0.57. The aperture diameter is normalized to unity, and the load per actuator is P/2 = 75 N

Fig. 4.
Fig. 4.

Radial (left) and tangential (right) stresses for the same parameters used in Fig. 3.

Fig. 5.
Fig. 5.

Failure limited extreme deflections (θ = β, positive values for θ = (0,π), negative values for θ = (π/2,3π/2)) as a function of β, for three mirror thicknesses.

Fig. 6.
Fig. 6.

Deformable mirror geometry, with a the mirror radius, β the radius where the point forces P/2 are applied, at θ = 0,π, and the mirror is simply supported at the two edge points θ = ±π/2.

Fig. 7.
Fig. 7.

Setup of Michelson interferometer for the experimental measurements.

Fig. 8.
Fig. 8.

Experimental results for the deformed wavefront over a range of back actuator positions β. Top: interferograms; Bottom: reconstructed wavefronts (arbitrary scale). The optical aperture is 40 mm in diameter. The circles represent the usable area delimited by the back actuator (red dots) radius.

Fig. 9.
Fig. 9.

Left: 3D plot of the reconstructed wavefront for β = 0.57. Right: residual wavefront after subtracting the corresponding Zernike polynomials given in Eq. (13). The horizontal scale is normalized to the used aperture b = 20 mm. Note that the vertical scale on the right is ten times smaller.

Fig. 10.
Fig. 10.

Curvature ratio as a function of β. Red line - analytical model; squares - interferometric measurements; circles - FEA analysis.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

w 1 , i ( r , θ ) w 1 , i ( ρ ) = Pa 2 8 πD A 0 ρ 2 ,
A 0 = 1 v 2 ( 1 + v ) ( 1 β 2 ) ln β
w 0 , e ( ρ , θ ) = R 0 + m = 1 R m cos ,
R 0 = A 0 + B 0 ρ 2 + C 0 log ρ + D 0 ρ 2 log ρ ,
R 1 = A 1 ρ + B 1 ρ 3 + C 1 ρ 1 + D 1 ρ log ρ ,
and R m = A m ρ m + B m ρ m + C m ρ m + 2 + D m ρ m + 2 .
w 0 , i ( ρ , θ ) = R ' 0 + m = 1 R ' m cos ,
F b ( θ ) = p 2 πb { 1 + m = 1 [ cos m θ + cos m ( θ + π ) ] } ,
F a ( θ ) = p 2 πa { 1 + m = 1 [ cos m ( θ + π 2 ) + cos m ( θ π 2 ) ] } .
D [ 2 w 0 , e ρ 2 + v ( 1 ρ w 0 , e ρ + 1 ρ 2 2 w 0 , e θ 2 ) ] ρ = 1 = 0 and
{ D ρ ( w 0 , e ) 1 ρ θ [ ( 1 v ) D ( 1 ρ 2 w 0 , e ρ θ 1 ρ 2 w 0 , e θ ) ] } ρ = 1 = F a ( θ ) .
w 0 , e = w 0 , i , w 0 , e ρ = w 0 , i ρ and 2 w 0 , e ρ 2 = 2 w 0 , i ρ 2 .
D ρ ( w 0 , e ) ρ = β D ρ ( w 0 , i ) ρ = β = F b ( θ ) .
w 0 , i ( ρ , θ ) = Pa 2 8 πD ( v + 3 ) m = 2,4,6 ( A m ρ m + C m ρ m + 2 ) cos ,
A m = 1 m ( m 1 ) { [ ( m 1 ) β 2 m 8 ( v + 1 ) m ( v 1 ) 2 ] ( v 1 ) β m + ( v + 3 ) β m + 2 ( 1 ) m 2 4 [ 1 2 ( v + 1 ) m ( v 1 ) ] } ,
C m = 1 m ( m + 1 ) { [ m ( 1 β 2 ) + 1 ] ( v 1 ) β m + ( v + 3 ) β m ( 1 ) m 2 4 } .
w i ( ρ , θ ) Pa 2 8 πD [ A 0 + 1 v + 3 ( A 2 + C 2 ρ 2 ) cos 2 θ ] ρ 2 .
W r ( ρ ' , θ ) = Pa 2 4 πD [ a 3 Z 3 ( ρ ' , θ ) + a 4 Z 4 ( ρ ' , θ ) + a 11 Z 11 ( ρ ' , θ ) ] ,
Z 3 ( ρ ' ) = 2 ρ ' 2 1 ,
Z 4 ( ρ ' , θ ) = ρ ' 2 cos 2 θ
Z 11 ( ρ ' , θ ) = ( 4 ρ ' 2 3 ) ρ ' 2 cos 2 θ .
a 3 = β 2 A 0 4 , a 4 = β 2 ( 4 A 2 + 3 β 2 C 2 ) 8 ( v + 3 ) , and a 11 = β 4 C 2 8 ( v + 3 ) .
σ ρρ ( ρ , θ ) = 6 D a 2 h 2 [ 2 w i ρ 2 + v ( 1 ρ w i ρ + 1 ρ 2 2 w i θ 2 ) ] ,
σ θθ ( ρ , θ ) = 6 D a 2 h 2 ( 1 ρ w i ρ + 1 ρ 2 2 w i θ 2 + v 2 w i ρ 2 ) .

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