Abstract

Phase-only liquid-crystal spatial light modulators provide a powerful means of wavefront control. With high resolution and diffractive (modulo 2π) operation, they can accurately represent large-dynamic-range phase maps. As a result, they provide an excellent means of producing electrically controllable, dynamic, and repeatable aberrations. However, proper calibration is critical to achieving accurate phase maps. Several calibration methods from previous literature were considered. With simplicity and accuracy in mind, we selected one method for each type of necessary calibration. We augmented one of the selected methods with a new step that improves its accuracy. After calibrating our spatial light modulator with our preferred methods, we evaluated its ability to produce aberrations in the laboratory. We studied Zernike polynomial aberrations using interferometry and Fourier-transform-plane images, and atmospheric aberrations using a Shack–Hartmann wavefront sensor. These measurements show the closest agreement with theoretical expectations that we have seen to date.

© 2007 Optical Society of America

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References

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  1. M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
    [CrossRef]
  2. J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
    [CrossRef]
  3. Boulder Nonlinear Systems, User Manual: 512x512 SLM System, 2nd ed. (2003).
  4. M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
    [CrossRef]
  5. Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
    [CrossRef]
  6. S. Harris, "Characterization and application of a liquid crystal beam steering device," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. L. Sutherland, D. W. Prather, and I. Cindrich, eds., Proc. SPIE 4291 , 109-119 (2001).
    [CrossRef]
  7. R. J. Brecha and J. M. O'Hare, Optical Radiation and Matter, unpublished book used as a text at University of Dayton (2004).
  8. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  9. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).
  10. MATLAB R2006a The MathWorks (2006), http://www.mathworks.com.
  11. D. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).
  12. J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).
  13. N. W. Hart, "Characterizing static aberrations of liquid crystal spatial light modulators," M.S. thesis (Michigan Technological University, 2005).
  14. S. Bishop and W. Bishop, Elementary Linear Algebra, 4th ed. (Brooks-Cole, 1996).
  15. R. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
  16. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  18. D. C. Dayton, S. L. Browne, S. P. Sandven, J. D. Gonglewski, and A. V. Kudryashov, "Theory and laboratory demonstrations on the use of a nematic liquid-crystal phase modulator for controlled turbulence generation and adaptive optics," Appl. Opt. 37, 5579-5589 (1998).
  19. J. D. Widiker and E. P. Magee, "Open-loop simulations of atmospheric turbulence using the AdAPS interface," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589404 (2005).
  20. S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978).
  21. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

2005 (2)

J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).

J. D. Widiker and E. P. Magee, "Open-loop simulations of atmospheric turbulence using the AdAPS interface," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589404 (2005).

2004 (2)

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

2001 (1)

S. Harris, "Characterization and application of a liquid crystal beam steering device," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. L. Sutherland, D. W. Prather, and I. Cindrich, eds., Proc. SPIE 4291 , 109-119 (2001).
[CrossRef]

2000 (1)

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

1998 (1)

1994 (1)

Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

1976 (1)

Bishop, S.

S. Bishop and W. Bishop, Elementary Linear Algebra, 4th ed. (Brooks-Cole, 1996).

Bishop, W.

S. Bishop and W. Bishop, Elementary Linear Algebra, 4th ed. (Brooks-Cole, 1996).

Brecha, R. J.

R. J. Brecha and J. M. O'Hare, Optical Radiation and Matter, unpublished book used as a text at University of Dayton (2004).

Browne, S. L.

Clifford, S. F.

S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978).

Dayton, D. C.

DeSandre, L. F.

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Dymale, R. C.

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Giles, M. K.

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gruneisen, M. T.

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Harriman, J.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

Harris, S.

S. Harris, "Characterization and application of a liquid crystal beam steering device," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. L. Sutherland, D. W. Prather, and I. Cindrich, eds., Proc. SPIE 4291 , 109-119 (2001).
[CrossRef]

Hart, N. W.

N. W. Hart, "Characterizing static aberrations of liquid crystal spatial light modulators," M.S. thesis (Michigan Technological University, 2005).

Jacobs, A. A.

J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).

Jimenez, R.

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

Kudryashov, A. V.

Linnenberger, A.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

Lu, G.

Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Lubin, D. L.

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Magee, E. P.

J. D. Widiker and E. P. Magee, "Open-loop simulations of atmospheric turbulence using the AdAPS interface," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589404 (2005).

Malacara, D.

D. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

Mansell, J. D.

J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).

Maynard, M.

J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).

Noll, R.

O'Hare, J. M.

R. J. Brecha and J. M. O'Hare, Optical Radiation and Matter, unpublished book used as a text at University of Dayton (2004).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Raha, J.

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Rotge, J. R.

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Sandven, S. P.

Serati, S.

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

Seward, A.

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

Vorontsov, M. A.

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Widiker, J. D.

J. D. Widiker and E. P. Magee, "Open-loop simulations of atmospheric turbulence using the AdAPS interface," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589404 (2005).

Yu, F. T.

Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Zhang, Z.

Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

Z. Zhang, G. Lu, and F. T. Yu, "Simple method for measuring phase modulation in liquid crystal televisions," Opt. Eng. 33, 3018-3022 (1994).
[CrossRef]

M. T. Gruneisen, L. F. DeSandre, J. R. Rotge, R. C. Dymale, and D. L. Lubin, "Programmable diffractive optics for wide-dynamic-range wavefront control using liquid-crystal spatial light modulators," Opt. Eng. 43, 1387-1393 (2004).
[CrossRef]

Proc. SPIE (4)

J. Harriman, A. Linnenberger, and S. Serati, "Improving spatial light modulator performance through phase compensation," in Advanced Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5553, 58-67 (2004).
[CrossRef]

J. D. Mansell, A. A. Jacobs, and M. Maynard, "Development of an adaptive optics test-bed for relay mirror applications," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589401 (2005).

M. K. Giles, A. Seward, M. A. Vorontsov, J. Raha, and R. Jimenez, "Setting up a liquid-crystal phase screen to simulate atmospheric turbulence," in High-Resolution Wavefront Control: Methods, Devices, and Applications II, M. T. Gruneisen, J. D. Gonglewski, and M. A. Vorontsov, eds., Proc. SPIE 4124, 89-97 (2000).
[CrossRef]

J. D. Widiker and E. P. Magee, "Open-loop simulations of atmospheric turbulence using the AdAPS interface," in Advanced Wavefront Control: Methods, Devices, and Applications III, M. T. Gruneisen, J. D. Gonglewski, and M. K. Giles, eds., Proc. SPIE 5894, 589404 (2005).

Proc. SPIE 4291 (1)

S. Harris, "Characterization and application of a liquid crystal beam steering device," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. L. Sutherland, D. W. Prather, and I. Cindrich, eds., Proc. SPIE 4291 , 109-119 (2001).
[CrossRef]

Other (12)

R. J. Brecha and J. M. O'Hare, Optical Radiation and Matter, unpublished book used as a text at University of Dayton (2004).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 2000).

MATLAB R2006a The MathWorks (2006), http://www.mathworks.com.

D. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

N. W. Hart, "Characterizing static aberrations of liquid crystal spatial light modulators," M.S. thesis (Michigan Technological University, 2005).

S. Bishop and W. Bishop, Elementary Linear Algebra, 4th ed. (Brooks-Cole, 1996).

Boulder Nonlinear Systems, User Manual: 512x512 SLM System, 2nd ed. (2003).

S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1998).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (11)

Fig. 1
Fig. 1

Setup for performing the phase-to-command-value calibration.

Fig. 2
Fig. 2

Irradiance contrast factor, C f , as a function of the two polarizer angles. The + indicates the maximum contrast factor of 0.67, and the ○ marks the value C f = 0.5 that we actually used in our experiment.

Fig. 3
Fig. 3

(a) Normalized data for our phase-to-command-value calibration. (b) Corresponding unwrapped phase.

Fig. 4
Fig. 4

Twyman–Green interferometer arrangement used in measuring our SLM's static aberration.

Fig. 5
Fig. 5

(a) Processed interferogram (after filtering and scaling) with assigned phase values in units of waves. (b) Mock interferogram created by taking the cosine of the solved aberration.

Fig. 6
Fig. 6

Flat-phase interferograms captured (a) without first compensating for the SLM's static aberration, (b) after static aberration compensation.

Fig. 7
Fig. 7

Variety of interferograms demonstrating the benefits of static aberration compensation. From left to right the columns correspond to results without compensation, with compensation, and theory.

Fig. 8
Fig. 8

Experimental setup for PSF measurements.

Fig. 9
Fig. 9

Measured and theoretical Strehl-ratio values for Zernike modes 2–34.

Fig. 10
Fig. 10

Experimental setup used to measure atmospheric phase written onto our calibrated SLM.

Fig. 11
Fig. 11

Measured and theoretical phase structure functions for atmospheric phase screens with D / r 0 = 5 (bottom pair), 10 (middle pair), and 20 (top pair).

Tables (1)

Tables Icon

Table 1 Zernike Coefficients for SLM Static Aberration

Equations (30)

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

E in = ( cos   θ 1 sin   θ 1 ) ,
W SLM = [ 1 0 0 exp ( i ϕ SLM ) ] ,
M = 1 ,
W WP = [ 1 0 0 exp ( i ϕ WP ) ] ,
P = ( cos 2 θ 2 sin   θ 2   cos   θ 2 sin   θ 2   cos   θ 2 sin 2 θ 2 ) ,
E out = P W WP M W SLM E in
= ( cos 2 θ 2 sin   θ 2   cos   θ 2 sin   θ 2   cos   θ 2 sin 2 θ 2 ) ( 1 0 0 exp [ i ( ϕ SLM + ϕ WP ) ] ) ( cos   θ 1 sin   θ 1 ) = ( cos   θ 1 cos 2 θ 2 + sin   θ 1   sin   θ 2   cos   θ 2   exp [ i ( ϕ SLM + ϕ WP ) ] sin   θ 1   sin   θ 2   cos   θ 2 + sin   θ 1 sin 2 θ 2   exp [ i ( ϕ SLM + ϕ WP ) ] ) ( A + B   exp [ i ( ϕ SLM + ϕ WP ) ] B + C   exp [ i ( ϕ SLM + ϕ WP ) ] ) ,
I = E out * · E out
= { A + B   exp [ i ( ϕ SLM + ϕ WP ) ] } × { A + B   exp [ i ( ϕ SLM + ϕ WP ) ] } + { B + C   exp [ i ( ϕ SLM + ϕ WP ) ] } × { B + C   exp [ i ( ϕ SLM + ϕ WP ) ] }
= C 0 + C f   cos ( ϕ SLM + ϕ WP ) ,
C 0 = A 2 + 2 B 2 + C 2
= cos 2 θ 1 cos 4 θ 2 + 2 sin 2 θ 1 sin 2 θ 2 cos 2 θ 2 + sin 2 θ 1 sin 4 θ 2 ,
C f = 2 ( A B + B C )
= cos   θ 1   sin   θ 1   sin   θ 2 cos 3 θ 2 + sin 2 θ 1   cos   θ 2 sin 3 θ 2 .
I 1 = C 0 + C f   cos ( ϕ SLM ) .
I 2 = C 0 C f   sin ( ϕ SLM ) .
ϕ SLM = tan 1 C 0 I 2 I 1 C 0 ,
P ( x ) = m = 0 M a m x m 1 + n = 1 N b n x n ,
c ( ϕ SLM )
= 274.6 ϕ SLM 3 71.13 ϕ SLM 2 + 358.2 ϕ SLM + 31.31 8.257 ϕ SLM 2 + 7.793 ϕ SLM + 1 ,
I ( x , y ) = 2 | E ( x , y ) | 2 { 1 + cos [ Δ ϕ ( x , y ) ] } ,
ϕ j = i = 1 N a i Z i ( x j , y j ) ,
ϕ = Z a ,
ϕ a b e r r ( x , y ) = i = 4 N a i Z i ( x , y ) .
ϕ ( r , θ ) = ( 28 r 7 24 r 5 ) sin ( 5 θ ) ,
D ϕ ( R 1 , R 2 ) = [ ϕ ( R 1 ) ϕ ( R 2 ) ] 2 ,
D ϕ ( R ) = 6.88 ( R r 0 ) 5 / 3 ,
a i a j = 0.0072 ( D r 0 ) 5 / 3 ( 1 ) ( n i + n j 2 m i ) / 2 [ ( n i + 1 ) ( n j + 1 ) ] 1 / 2 π 8 / 3 δ m i m j × Γ ( 14 / 3 ) Γ [ ( n i + n j 5 / 3 ) / 2 ] Γ [ ( n i n j + 17 / 3 ) / 2 ] Γ [ ( n j n i + 17 / 3 ) / 2 ] Γ [ ( n i + n j + 23 / 3 ) / 2 ] ,
a i a j = 0 ,
a = U b ,

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