Abstract

Pupil-phase engineering is a design process in which pupil-phase masks are optimized for performance characteristics by minimizing a cost function. To reduce computational complexity for optimizing focal depth, a cost function based on the second derivative of the optical transfer function with respect to misfocus at the origin is proposed. Efficient formulas for computing this metric are derived, and a design is presented to demonstrate that this metric can be used to predict insensitivity of the system to large values of misfocus.

© 2007 Optical Society of America

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References

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  1. J. O. Castañeda, L. R. B. Valdos, and E. Montes, "Ambiguity function as a design tool for high focal depth," Appl. Opt. 27, 790-795 (1988).
    [CrossRef]
  2. J. O. Castañeda, R. Ramos, and A. N. Isgleas, "High focal depth by apodization and digital restoration," Appl. Opt. 27, 2583-2586 (1988).
    [CrossRef]
  3. E. Dowski and W. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  4. S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," Proc. SPIE 5108, 1-12 (2003).
    [CrossRef]
  5. S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, "Pupil phase optimization for extended-focus, aberration-corrected imaging systems," Proc. SPIE 5559, 335-345 (2004).
    [CrossRef]
  6. V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," Proc. SPIE 5205, 348-357 (2003).
    [CrossRef]
  7. K. Brenner, A. Lohmann, and J. O. Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  9. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).
  10. D. Bertsekas, Nonlinear Programming (Athena Scientific, 1999).

2004

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, "Pupil phase optimization for extended-focus, aberration-corrected imaging systems," Proc. SPIE 5559, 335-345 (2004).
[CrossRef]

2003

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," Proc. SPIE 5205, 348-357 (2003).
[CrossRef]

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

1995

1988

1983

K. Brenner, A. Lohmann, and J. O. Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Appl. Opt.

Opt. Commun.

K. Brenner, A. Lohmann, and J. O. Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Proc. SPIE

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, "Engineering the pupil phase to improve image quality," Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, and J. van der Gracht, "Pupil phase optimization for extended-focus, aberration-corrected imaging systems," Proc. SPIE 5559, 335-345 (2004).
[CrossRef]

V. Pauca, R. Plemmons, S. Prasad, and T. Torgersen, "Integrated optical-digital approaches for enhancing image restoration and focus invariance," Proc. SPIE 5205, 348-357 (2003).
[CrossRef]

Other

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

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

D. Bertsekas, Nonlinear Programming (Athena Scientific, 1999).

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Figures (3)

Fig. 1
Fig. 1

H a ( u , v , τ ) H a ( u , v , 0 ) E versus τ for five phase masks. The legend gives the corresponding value calculated for M(a). Mask coefficients a in descending M are [ 0.9 , 62 , 87 , 65 , 36 ] , [ 0.5 , 33 , 76 , 50 , 22 ] , [ 66 , 87 , 75 , 7 , 80 ] , [ 39 , 10 , 48 , 35 , 28 ] , and [ 49 , 25 , 134 , 280 , 412 ] .

Fig. 2
Fig. 2

Same as Fig. 1 over a larger range of τ.

Fig. 3
Fig. 3

(a) Test pattern, (b) test pattern image with a misfocus of 10, (c) PSF of the designed phase mask, (d) image in (b) recovered after wavefront coding with (c).

Equations (58)

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N 2 log ( N )
N × N
P ( x ) = P * ( x )
θ ( x , y )
x y
θ ( x , y ) = θ ( x , y )
P ( x )
P ( x ) = { 1 2 exp [ j ( θ ( x ) + τ x 2 ) ] for   | x | 1 0 , otherwise ,
θ ( x )
f ( x )
A f ( α , γ ) = f ( x + α 2 ) f * ( x α 2 ) exp ( j 2 π γ x ) d x ,
f *
H ( u , τ ) = A P ( u , u τ π ) ,
A f ( α , γ ) = f ^ ( r + γ 2 ) f ^ * ( r γ 2 ) exp ( j 2 π α r ) d r ,
f ^
P ^ ( r + γ / 2 ) P ^ * ( r γ / 2 )
γ = 0
τ = 0
H ( u , τ ) = h ( r + u τ 2 π ) h * ( r u τ 2 π ) exp ( j 2 π u r ) d r .
2 τ 2 H ( u , 0 ) = ( u 2 π ) 2 ( h * 2 h r 2 2 h r h * r + h 2 h * r 2 ) × exp ( j 2 π u r ) d r = ( u 2 π ) 2 F { h * 2 h r 2 2 h r h * r + h 2 h * r 2 } ,
F { }
H ( u , v , τ )
h ( r , s )
h r s = h * 2 h r s h r h * s h s h * r + h 2 h * r s ,
h d d = h * 2 h d 2 2 h d h * d + h 2 h * d 2 ,
d = r
d = s
2 τ 2 H ( u , v , 0 ) = ( u 2 π ) 2 F { h r r } + ( v 2 π ) 2 F { h s s } + 2 u v ( 2 π ) 2 F { h r s } .
h ( r , s )
h h *
θ ( x , y )
θ ( x , y )
a 1 ( x 3 + y 3 ) + a 2 ( x 2 y + x y 2 ) + a 3 ( x 5 + y 5 ) + a 4 ( x 4 y + x y 4 ) + a 5 ( x 3 y 2 + x 2 y 3 ) .
H a
a = [ a 1 , a 2 , a 3 , a 4 , a 5 ]
H a ( u , v , τ ) H a ( u , v , 0 ) + 2 H a ( u , v , 0 ) τ 2 τ 2 2 .
M ( a ) = | | 2 τ 2 H a ( u , v , 0 ) | | E ( n m | 2 τ 2 H a ( u n , v m , 0 ) | 2 ) 1 / 2 ,
E
H a ( u , v , τ ) H a ( u , v , 0 ) E M ( a ) τ 2 / 2
H a ( u , v , τ ) H a ( u , v , 0 ) E
M 1 = 3.3318
M 5 = .5357
M 1 / M 5
τ 4
S ( a ) = K ( 1 Ψ ( a ) ) 1 exp ( η ( Ψ ( a ) Ψ ˜ ) ) + 1 ,
Ψ ( a )
Ψ ˜
C ( a ) = M ( a ) + S ( a )
500 a i 500
a = [ 74.98 , 112.038 , 247.559 , 491.998 , 440.664 ]
( a = [ 21.65 , 41.91 , 338.2 , 318.6 , 586.66 ] )
( a 1 = 90 )
H a ( u , v , τ ) H a ( u , v , 0 ) E
[ 0.9 , 62 , 87 , 65 , 36 ]
[ 0.5 , 33 , 76 , 50 , 22 ]
[ 66 , 87 , 75 , 7 , 80 ]
[ 39 , 10 , 48 , 35 , 28 ]
[ 49 , 25 , 134 , 280 , 412 ]

Metrics