Abstract

We describe a modal wavefront sensing technique of using multiplexed holographic optical elements (HOEs). The phase pattern of a set of aberrations is angle multiplexed in a HOE, and the correlated information is obtained with a position sensing detector. The recorded aberration pattern is based on an orthogonal basis set, the Zernike polynomials, and a spherical reference wave. We show that only two recorded holographic patterns for any particular aberration type are sufficient to allow interpolated readout of aberrations to λ/50. In this paper, we demonstrate the capability of detecting errors between ±2λ  PV for each orthogonal set at rates limited only by the speeds of the detection electronics, which could be up to 1   MHz. We show how we take advantage of the unavoidable intermodal and intramodal cross talks in determining the type, amplitude, and orientation of the wavefront aberrations.

© 2008 Optical Society of America

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References

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  1. J. M. Geary, Introduction to Wavefront Sensors, Vol. TT18 of SPIE Tutorial Texts in Optical Engineering (SPIE, 1995).
  2. R. K. Tyson, Introduction to Adaptive Optics, Vol. TT41 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2000).
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    [CrossRef]
  4. K. Chen, Z. Yang, H. Wang, E. Li, F. Yang, and Y. Zhang, "PSD-based Shack-Hartmann wavefront sensor," Proc. SPIE 5639, 87-94 (2004).
    [CrossRef]
  5. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  7. B. Xiong, Z. Wang, Y. Zhang, U. Zhong, W. Zhang, and Q. Yang, "A novel method for evaluation of a thin, phase hologram," Proc. SPIE 2866, 172-176 (1996).
    [CrossRef]
  8. L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, "Standards for reporting the optical aberrations of eyes," in Vision Sciences and Its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  9. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 9.
  10. C. Gu, H. Fu, and J. R. Lien, "Correlation patterns and cross-talk noise in volume holographic optical correlators," J. Opt. Soc. Am. A 12, 861-868 (1995).
    [CrossRef]

2004

K. Chen, Z. Yang, H. Wang, E. Li, F. Yang, and Y. Zhang, "PSD-based Shack-Hartmann wavefront sensor," Proc. SPIE 5639, 87-94 (2004).
[CrossRef]

2000

1996

B. Xiong, Z. Wang, Y. Zhang, U. Zhong, W. Zhang, and Q. Yang, "A novel method for evaluation of a thin, phase hologram," Proc. SPIE 2866, 172-176 (1996).
[CrossRef]

1995

1976

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Proc. SPIE

K. Chen, Z. Yang, H. Wang, E. Li, F. Yang, and Y. Zhang, "PSD-based Shack-Hartmann wavefront sensor," Proc. SPIE 5639, 87-94 (2004).
[CrossRef]

B. Xiong, Z. Wang, Y. Zhang, U. Zhong, W. Zhang, and Q. Yang, "A novel method for evaluation of a thin, phase hologram," Proc. SPIE 2866, 172-176 (1996).
[CrossRef]

Other

L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, "Standards for reporting the optical aberrations of eyes," in Vision Sciences and Its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 9.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. M. Geary, Introduction to Wavefront Sensors, Vol. TT18 of SPIE Tutorial Texts in Optical Engineering (SPIE, 1995).

R. K. Tyson, Introduction to Adaptive Optics, Vol. TT41 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2000).

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Figures (8)

Fig. 1
Fig. 1

(a) Recording and (b) reconstruction of the HOE using a spherical wave.

Fig. 2
Fig. 2

Decomposition of distortions using a prerecorded holographic optical element. The aberration type and amplitude refer to Z j and ϵ j of Eq. (5), respectively.

Fig. 3
Fig. 3

Difference between beam spots P 1 and P 2 as a function of the spherical feedback wave from a lens with relative focal length f P B . The inset shows the extrapolation of the data using Eq. (17).

Fig. 4
Fig. 4

(a), (b) Experimental setup for the recording of the holograms and (c) for the playback. Note that, for clarity, the figure depicts the reading back of the conjugate image.

Fig. 5
Fig. 5

Experimental results of Test 2. Two holograms of spherical waves were recorded, and the differences of the magnitudes of the prerecorded spots are plotted as a function of playback curvature.

Fig. 6
Fig. 6

Placement of the spots on the image plane created by the converging reference waves. All six spots may be located anywhere in all 3D space.

Fig. 7
Fig. 7

Output of the PSDs, which is essentially the differences of the spots at the image plane, for playback waves of (a) Z 2 , 0 and (b) Z 2 , 2 and varying ϵ P B .

Fig. 8
Fig. 8

Output of the PSDs for different orientations of the playback waves. (a) Z 2 , 2 ( θ P B = 0 ° ) , (b) Z 2 , + 2 ( θ P B = 0 ° ) , and the playback is at the same orientation as the recorded wave. (c) Z 2 , 2 ( θ P B = 22.5 ° ) and (d) Z 2 , 2 ( θ P B = 10 ° ) .

Equations (21)

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τ ( r S , r R ) m = 1 M U S , m * ( r S , m ) U R , m ( r R , m ) + c.c.
T ( r S , r R ) = T 0 exp [ i ζ τ ( r S , r R ) ] ,
U R , m ( r R , m ) = U R , m , 0 | R m R | exp { i k m [ ( x x R , m ) 2 + ( y y R , m ) 2 + ( z z R , m ) 2 ] 1 / 2 } ,
U S , m ( r S , m ) = U S , m , 0 exp [ i W ( r S , m ) ] .
W ( r S , m ) = j ϵ m , j Z j .
U d ( r d ) τ ( r S , r R , r ) U P ( r P , r ) G ( r d , r ) d r ,
G ( r d , r ) = exp ( i k d | r d r | ) | r d r | exp { i k d [ ( x d x ) 2 + ( y d y ) 2 + ( z d z ) 2 ] 1 / 2 } z d .
[ ( x d x ) 2 + ( y d y ) 2 + ( z d z ) 2 ] 1 / 2
z d [ 1 + ( x d x ) 2 + ( y d y ) 2 2 z d 2 ] + z [ 1 ( x d x ) 2 + ( y d y ) 2 2 z d 2 ] ,
U d ( r d ) d V   exp [ i k S z ( 1 ξ R z R ) ] exp ( i k S ξ R ) × exp [ i k d z ( 1 ξ d z d ) ] exp ( i k d ξ d ) × exp [ i W ( x , y ) ] Π ( x a , y b ) U P ( r p ) ,
ξ α = ( x α x ) 2 + ( y α y ) 2 2 z α .
U d ( r d ) d x d y exp ( i 2 π ξ R λ ) exp ( i 2 π ξ d λ ) × exp [ i W ( x , y ) ] Π ( x a , y b ) U P ( r P ) × sinc [ 2 L λ ( 1 ξ d z d ) 2 L λ ( 1 ξ d z d ) ] .
2 L λ [ ξ R z R ξ d z d ] = 2 L λ [ ( x R x ) 2 + ( y R y ) 2 z R 2 ( x d x ) 2 + ( y d y ) 2 z d 2 ] .
2 L λ [ ξ R z R ξ d z d ] = L λ z R 2 [ x d 2 x R 2 + 2 x ( x d x R ) + y d 2 y R 2 + 2 y ( y d y R ) ] = L λ z R 2 [ 2 δ x ( x d x ) + 2 δ y ( y d y ) ] ,
U d ( x d , y d , z d ) d x d y exp [ i π ( x R x ) 2 + ( y R y ) 2 λ z R ] × [ i π ( x d x ) 2 + ( y d y ) 2 λ z R ] × exp [ i W ( x , y ) ] Π ( x a , y b ) U p ( x p , y p ) × sinc { L λ z R 2 [ 2 δ x ( x d x ) + 2 δ y ( y d y ) ] } .
h ( x d , y d , z d ) f ( x , y ) g ( x d x , y d y ) d x d y ,
f = exp [ i π ( x R x ) 2 + ( y R y ) 2 λ z R ] × exp [ i W ( x , y ) ] Π ( x a , y b ) U p ( x p , y p ) ,
g = exp ( i π x 2 + y 2 λ z R ) sinc [ L λ z R 2 ( 2 x δ x + 2 y δ y ) ] .
V out l / 2 l / 2 x I ( x ) d x .
V out q = L , R x q I q .
g ( x ) = g o f f + q = 1 , 2 B q exp [ ( x x q , o f f ) 2 2 σ q 2 ] .

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