Abstract

Imaging characteristics are analyzed for a wavefront coding system suffering from off-axis aberrations such as primary astigmatism and primary coma. Some analytical expressions for the optical transfer function are obtained by using the stationary-phase method. These expressions give the relationship between the optical transfer function and the off-axis aberrations. Some cases are computed and illustrated graphically. It is shown that the wavefront coding system has a high tolerance to primary astigmatism and a low sensitivity to primary coma.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am 50, 749-753 (1960).
    [CrossRef]
  2. M. Mino and Y. Okano, "Improvement in the OTF of a defocused optical system through the use of shaded apertures," Appl. Opt. 10, 2219-2225 (1971).
    [CrossRef] [PubMed]
  3. J. Ojeda-Castaneda, P. Andres, and A. Diaz, "Annular apodizers for low sensitivity to defocus and to spherical aberration," Opt. Lett. 11, 487-489 (1986).
    [CrossRef] [PubMed]
  4. J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, "High focal depth by apodization and digital restoration," Appl. Opt. 27, 2583-2586 (1988).
    [CrossRef] [PubMed]
  5. J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, "Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer," Appl. Opt. 28, 2666-2670 (1989).
    [CrossRef] [PubMed]
  6. J. Ojeda-Castaneda and L. R. Berriel-Valdos, "Zone plate for arbitrarily high focal depth," Appl. Opt. 29, 994-997 (1990).
    [CrossRef] [PubMed]
  7. D. Zalvidea and E. E. Sicre, "Phase pupil functions for focal-depth enhancement derived from a Wigner distribution function," Appl. Opt. 37, 3623-3627 (1998).
    [CrossRef]
  8. S. Mezouari and A. R. Harvey, "Phase pupil functions for control defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
    [CrossRef] [PubMed]
  9. H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-5662 (2001).
    [CrossRef]
  10. E. R. Dowski, Jr. and W. Thomas Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  11. H. B. Wach, E. R. Dowski, and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
    [CrossRef]
  12. S. Mezouari and A. R. Harvey, "Wavefront coding for aberration compensation in thermal imaging systems," in Novel Optical Systems Design and Optimization IV, J. M. Sasian and P. K. Manhart, eds., Proc. SPIE 4442, 34-42 (2001).
    [CrossRef]
  13. R. Narayanswamy and A. E. Baron, "Applications of wavefront coded imaging," in Computational Imaging II, C. A. Bouman and E. L. Miller, eds., Proc. SPIE , 5299, 163-173 (2004).
    [CrossRef]
  14. S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," in Novel Optical Systems Design and Optimization V, J. M. Sasian and R. J. Koshel, eds., Proc. SPIE 4768, 21-31 (2002).
    [CrossRef]
  15. S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
    [CrossRef]
  16. Z. L. Xu and W. X. Chen, Methods and Applications of Asymptotic Analysis (National Defence Industry Press, 1991).
  17. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1985).

2004

R. Narayanswamy and A. E. Baron, "Applications of wavefront coded imaging," in Computational Imaging II, C. A. Bouman and E. L. Miller, eds., Proc. SPIE , 5299, 163-173 (2004).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
[CrossRef]

2003

2002

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," in Novel Optical Systems Design and Optimization V, J. M. Sasian and R. J. Koshel, eds., Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

2001

S. Mezouari and A. R. Harvey, "Wavefront coding for aberration compensation in thermal imaging systems," in Novel Optical Systems Design and Optimization IV, J. M. Sasian and P. K. Manhart, eds., Proc. SPIE 4442, 34-42 (2001).
[CrossRef]

H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-5662 (2001).
[CrossRef]

1998

1995

1990

1989

1988

1986

1971

1960

W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am 50, 749-753 (1960).
[CrossRef]

Andres, P.

Baron, A. E.

R. Narayanswamy and A. E. Baron, "Applications of wavefront coded imaging," in Computational Imaging II, C. A. Bouman and E. L. Miller, eds., Proc. SPIE , 5299, 163-173 (2004).
[CrossRef]

Berriel-Valdos, L. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1985).

Cathey, W. T.

Cathey, W. Thomas

Chen, W. X.

Z. L. Xu and W. X. Chen, Methods and Applications of Asymptotic Analysis (National Defence Industry Press, 1991).

Diaz, A.

Dowski, E. R.

Gan, F.

Harvey, A. R.

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
[CrossRef]

S. Mezouari and A. R. Harvey, "Phase pupil functions for control defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
[CrossRef] [PubMed]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," in Novel Optical Systems Design and Optimization V, J. M. Sasian and R. J. Koshel, eds., Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

S. Mezouari and A. R. Harvey, "Wavefront coding for aberration compensation in thermal imaging systems," in Novel Optical Systems Design and Optimization IV, J. M. Sasian and P. K. Manhart, eds., Proc. SPIE 4442, 34-42 (2001).
[CrossRef]

Mezouari, S.

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
[CrossRef]

S. Mezouari and A. R. Harvey, "Phase pupil functions for control defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
[CrossRef] [PubMed]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," in Novel Optical Systems Design and Optimization V, J. M. Sasian and R. J. Koshel, eds., Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

S. Mezouari and A. R. Harvey, "Wavefront coding for aberration compensation in thermal imaging systems," in Novel Optical Systems Design and Optimization IV, J. M. Sasian and P. K. Manhart, eds., Proc. SPIE 4442, 34-42 (2001).
[CrossRef]

Mino, M.

Muyo, G.

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
[CrossRef]

Narayanswamy, R.

R. Narayanswamy and A. E. Baron, "Applications of wavefront coded imaging," in Computational Imaging II, C. A. Bouman and E. L. Miller, eds., Proc. SPIE , 5299, 163-173 (2004).
[CrossRef]

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Okano, Y.

Ramos, R.

Sicre, E. E.

Tepichin, E.

Wach, H. B.

Wang, H.

Welford, W. T.

W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am 50, 749-753 (1960).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1985).

Xu, Z. L.

Z. L. Xu and W. X. Chen, Methods and Applications of Asymptotic Analysis (National Defence Industry Press, 1991).

Zalvidea, D.

Appl. Opt.

J. Opt. Soc. Am

W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am 50, 749-753 (1960).
[CrossRef]

Opt. Lett.

Proc SPIE

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," in Optical Design and Engineering, L. Mazuray, P. J. Rogers, and R. Wartmann, eds., Proc SPIE 5249, 238-248 (2004).
[CrossRef]

Proc. SPIE

S. Mezouari and A. R. Harvey, "Wavefront coding for aberration compensation in thermal imaging systems," in Novel Optical Systems Design and Optimization IV, J. M. Sasian and P. K. Manhart, eds., Proc. SPIE 4442, 34-42 (2001).
[CrossRef]

R. Narayanswamy and A. E. Baron, "Applications of wavefront coded imaging," in Computational Imaging II, C. A. Bouman and E. L. Miller, eds., Proc. SPIE , 5299, 163-173 (2004).
[CrossRef]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," in Novel Optical Systems Design and Optimization V, J. M. Sasian and R. J. Koshel, eds., Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Other

Z. L. Xu and W. X. Chen, Methods and Applications of Asymptotic Analysis (National Defence Industry Press, 1991).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Comparison of the results of approximation and numerical calculation. (a), (b) u = 0 , W 20 = W 22 = 2 λ , θ = 90 ° . (c), (d) u = 0 , W 20 = W 22 = 2 λ , θ = 30 ° . (e), (f) v = 0 , W 20 = W 31 = 2 λ . (g), (h) u = v , W 20 = W 31 = 2 λ . For all cases, α = 60 λ / 2 π . Dotted curves represent the results of approximation. Solid curves represent the results of numerical calculation. The curves near zero are the errors.

Fig. 2
Fig. 2

(Color online) Curves of the MTF and the phase of the OTF with defocus and astigmatism along the azimuths of (a), (b) 0°, and (c), (d) 45°. Solid curve, W 20 = 0 λ , W 22 = 0 λ ; dashed–dotted curve, W 20 = 1 λ , W 22 = 1 λ ; dotted curve, W 20 = 2 λ , W 22 = 2 λ ; dashed curve, W 20 = 3 λ , W 22 = 1 λ . For all cases, α = 60 λ / 2 π , and θ = 30 ° .

Fig. 3
Fig. 3

(Color online) Curves of the MTF and the phase of the OTF with defocus and coma along the azimuths of (a), (b), 0°, (c), (d) 90°, and (e), (f) 45°. Solid curve, W 20 = 0 λ , W 31 = 0 λ ; dashed-dotted curve, W 20 = 1 λ , W 31 = 1 λ ; dotted curve, W 20 = 2 λ , W 31 = 2 λ ; dashed curve, W 20 = 3 λ , W 31 = 1 λ . For all cases, α = 60 λ / 2 π .

Equations (44)

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

P ( x , y ) = { exp [ j k α ( x 3 + y 3 ) ] x , y 1 0 otherwise ,
OTF ( u , v ) 1 4 ( π 3 k α u π 3 k α v ) 1 / 2  exp [ j k ( α u 3 4 + α v 3 4 W 20 2 u 3 α W 20 2 v 3 α ) ] exp [ j   sgn ( u ) π / 4 + j   sgn ( v ) π / 4 ] u 0 , v 0 1 4 ( π 3 k α u π 3 k α v ) 1 / 2  exp [ j k α u 3 4 + j k α v 3 4 + j sgn ( u ) π + sgn ( v ) π 4 ] u 0 , v 0 ,
sgn ( x ) = { 1 1 x > 0 x < 0 .
P ( x , y ) = { exp [ j φ ( x , y ) ] 0 x , y 1 otherwise ,
OTF ( u , v ) = ( 1 u / 2 ) 1 u / 2 ( 1 v / 2 ) 1 v / 2 P ( x + u / 2 , y + v / 2 ) P * ( x u / 2 , y v / 2 ) d x d y 1 1 1 1 P ( x , y ) 2 d x d y ,
OTF ( u , v ) { π 12 k α ( 1 u v ) 1 / 2  exp { j [ ϕ u ( u ) + ϕ v ( v ) ] } u 0 , v 0 , 1 2 ( π 3 k α u ) 1 / 2 sin [ k W 22 u  sin ( 2 θ ) ] k W 22 u   sin ( 2 θ )  exp [ j ϕ u ( u ) ] u 0 , v = 0 , 1 2 ( π 3 k α v ) 1 / 2 sin [ k W 22 v   sin ( 2 θ ) ] k W 22 v   sin ( 2 θ )  exp [ j ϕ v ( v ) ] u = 0 , v 0 ,
P ( x , y ) = { exp [ j φ ( x , y ) ] 0 x , y 1 otherwise ,
OTF ( u , v ) = ( 1 u / 2 ) 1 u / 2 ( 1 v / 2 ) 1 v / 2 P ( x + u 2 , y + v 2 ) P * ( x u 2 , y v 2 ) d y d x 1 1 1 1 P ( x , y ) 2 d x d y .
OTF ( u , v )
{ A 1 ( u , v ) exp [ j φ 1 ( u , v ) ] u 0 , v 0 , A 2 ( u , v ) exp [ j k α v 3 4 + j   sgn ( v ) π 4 ] u = 0 , v 0 , A 3 ( u , v ) exp [ j φ 3 ( u , v ) ] u 0 , v = 0 ,
A 1 ( u , v ) = 1 4 ( π 3 k ( α + W 31 ) u ) 1 / 2 ( π 3 k α v + k W 31 u k W 31         2 v 2 / 3 u ( α + W 31 ) ) 1 / 2 , ϕ 1 ( u , v ) = k ( α + W 31 ) u 3 4 + k α v 3 4 + k W 31 v 2 u 4 k W 20         2 u 3 ( α + W 31 ) k [ W 20 v W 20 W 31 v / 3 ( α + W 31 ) ] 2 3 α v + W 31 u W 31         2 v 2 / 3 u ( α + W 31 ) + sgn ( u ) π 4 + sgn [ 3 k α v + k W 31 u k W 31         2 v 2 / 3 u ( α + W 31 ) ] π 4 ,
A 2 ( u , v ) = 1 4 ( π 3 k α v ) 1 / 2 1 1 exp [ j k ( W 20 + W 31 x ) 2 v 3 α ] d x , A 3 ( u , v ) = 1 4 ( π 3 k ( α + W 31 ) u ) 1 / 2 1 1   exp ( j k W 31 u y 2 ) d y , ϕ 3 ( u , v ) = k α + W 31 4 u 3 k W 20         2 u 3 ( α + W 31 ) + sgn ( u ) π 4 .
y s = W 20 + W 22 3 α ,
OTF ( u , v ) = 1 4 ( 1 u / 2 ) 1 u / 2 exp [ j ϕ 1 ( x ) ] d x × ( 1 v / 2 ) 1 v / 2 exp [ j ϕ 2 ( y ) ] d y ,
OTF u ( u ) = ( 1 u / 2 ) 1 u / 2   exp [ j ϕ 1 ( x ) ] d x , u 2.
OTF u ( u ) [ 2 π ϕ 1 ( x s ) ] 1 / 2   exp [ j ϕ 1 ( x s ) ± j π 4 ] ,
( / x s ) ϕ 1 ( x s ) = 0 ,
x s = u W 20 + ( u cos 2 θ + 0.5 v   sin   2 θ ) W 22 3 α u ,   u 0.
OTF u ( u ) ( π 3 k α u ) 1 / 2 exp ( j k α u 3 4 + sgn ( u ) j π 4 ) exp { j k [ u W 20 + ( u cos 2 θ + 0.5 v   sin   2 θ ) W 22 ] 2 3 α u } ,   u 0.
OTF u ( 0 ) = 2   sin [ k W 22 v   sin ( 2 θ ) ] / [ k W 22 v   sin ( 2 θ ) ] .
OTF v ( v ) = ( 1 v / 2 ) 1 v / 2 exp [ j ϕ 2 ( y ) ] d y ,
OTF v ( v ) ( π 3 k α v ) 1 / 2   exp [ j k α v 3 4 + sgn ( v ) j π 4 ] exp { j k [ v W 20 + ( v sin 2 θ + 0.5 u   sin   2 θ ) W 22 ] 2 3 α v } , v 0
OTF ( u , v ) = 1 4 ( 1 u / 2 ) 1 u / 2 ( 1 v / 2 ) 1 v / 2 exp [ j ϕ ( x , y ) ] × d y d x ,
ϕ ( x , y ) = k [ 3 ( α + W 31 ) x 2 u + 2 W 20 x u + 3 α y 2 v + W 31 y 2 u + 2 W 20 y v + 2 W 31 x y v + ( α + W 31 ) u 3 / 4 + α v 3 / 4 + u v 2 W 31 / 4 ] .
OTF ( u , v ) = C 4 ( 1 v / 2 ) 1 v / 2   exp [ j ϕ 1 ( y ) ] d y
× ( 1 u / 2 ) 1 u / 2 exp [ j ϕ 2 ( x , y ) ] d x ,
( 1 u / 2 ) 1 u / 2   exp [ j ϕ 2 ( x , y ) ] d x [ 2 π ϕ 2 ( x s ) ] 1 / 2 × exp { j ϕ 2 ( x s , y ) + j sgn [ ϕ 2 ( x s ) ] } ,
( / x s ) ϕ 2 ( x s , y ) = 0 ,
x s = W 20 u + W 31 y v 3 ( α + W 31 ) u u 0.
( 1 u / 2 ) 1 u / 2 exp [ j ϕ 2 ( x , y ) ] d x [ π 3 k ( α + W 31 ) u ] 1 / 2 exp [ j k ( W 20 u + W 31 y v ) 2 3 ( α + W 31 ) u + sgn ( u ) j π 4 ] u 0 .
OTF ( u ,   v ) C 4 [ π 3 k ( α + W 31 ) u ] 1 / 2  exp [ sgn ( u ) j π 4 ] ( 1 v / 2 ) 1 v / 2 exp [ j ϕ 1 ( y ) j k ( W 20 u + W 31 y v ) 2 3 ( α + W 31 ) u ] d y u 0.
ϕ 3 ( y ) = ϕ 2 ( y ) - k ( W 20 u + W 31 y v ) 2 3 ( α + W 31 ) u .
( / y s ) ϕ 3 ( y s ) = 0 ,
y s = W 20 v W 20 W 31 v / 3 ( α + W 31 ) 3 α v + W 31 u W 31 2 v 2 / 3 u ( α + W 31 ) .
OTF ( u ,   v ) C 4 [ π 3 k ( α + W 31 ) u ] 1 / 2  exp [ sgn ( u ) j π 4 ] [ 2 π ϕ 3 ( y s ) ] 1 / 2  exp { j ϕ 3 ( y s ) + sgn [ ϕ 3 ( y s ) ] } u 0 , v 0 C 4 [ π 3 k ( α + W 31 ) u ] 1 / 2 [ π 3 k α v + k W 31 u k W 31 2 v 2 / 3 u ( α + W 31 ) ] 1 / 2 × exp { j k [ W 20 v W 20 W 31 v / 3 ( α + W 31 ) ] 2 3 α v + W 31 u W 31 2 v 2 / 3 u ( α + W 31 ) j k W 20 2 u 3 ( α + W 31 ) + j θ } , u 0,   v 0,
ϕ ( x , y ) = k [ 3 ( α + W 31 ) x 2 u + 2 W 20 x u + W 31 y 2 u + ( α + W 31 ) u 3 / 4 ] ,
OTF ( u ,   v ) = 1 4 exp [ j k ( α + W 31 ) u 3 / 4 ] 1 1   exp ( j k W 31 u y 2 ) d y ( 1 u / 2 ) 1 u / 2   exp { j k [ 3 ( α + W 31 ) x 2 u + 2 W 20 x u ] } d x .
OTF ( u ,   v ) 1 4 exp [ j k ( α + W 31 ) u 3 / 4 ]
× 1 1 exp ( j k W 31 u y 2 ) d y [ π 3 k ( α + W 31 ) u ] 1 / 2 × exp [ j k W 20 2 u 3 ( α + W 31 ) + sgn ( u ) j π 4 ] ,
u 0.
ϕ ( x , y ) = k [ 3 α y 2 v + 2 W 20 y v + 2 W 31 x y v + α v 3 / 4 ] ,
OTF ( u ,   v ) = 1 4 exp [ j k α v 3 / 4 ] 1 1 ( 1 v / 2 ) 1 v / 2   exp { j k [ 3 α y 2 v + 2 W 20 y v + 2 W 31 x y v ] } d y d x .
OTF ( u ,   v ) 1 4 ( π 3 k α v ) 1 / 2 exp [ j k α v 3 / 4 + sgn ( v ) j π / 4 ] × 1 1   exp [ j k ( W 20 + W 31 x ) 2 v 3 α ] d x ,
v 0.

Metrics