Abstract

Approximate expressions of the ray aberrations for off-axis collimated beams and free form phase plates with a small derivative magnitude are derived with the defocus aberration taken into account. The cubic phase plate, which is one of the most commonly used phase plates in wavefront coding imaging systems, is illustrated as an example. The approximate expressions for the upper and lower boundaries of ray map, and the spot size in the vicinity of the focal plane are derived. The sensitivity to the defocus aberration and the variation of the induced aberrations with respect to the field positions are analyzed with derived approximate expressions as well. Some characteristics unmentioned before are derived, showing a good agreement with the exact aberrations. Finally some useful guidelines are given for the design of imaging systems with phase plates.

© 2007 Optical Society of America

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  1. E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866(1995).
    [CrossRef] [PubMed]
  2. G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
    [CrossRef]
  3. S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
    [CrossRef]
  4. S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
    [CrossRef]
  5. F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
    [CrossRef]
  6. W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41:6080-6092 (2002).
    [CrossRef] [PubMed]
  7. T. A. Mitchell and J. M. Sasian, "Variable aberration correction using axially translating phase plates," Proc. SPIE 3705, 209-220 (1999).
    [CrossRef]
  8. S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
    [CrossRef]
  9. S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).
  10. S. Mezouari and A. R. Harvey, "Phase pupil functions for control defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003).
    [CrossRef] [PubMed]
  11. N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A, Pure Appl. Opt. 5, S157-S163 (2003).
  12. S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
    [CrossRef]
  13. G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1985).
  15. W. Z. Zhang, Z. Ye, T. Y. Zhao, Y. P. Chen, and F. H. Yu, "Point spread function characteristics analysis of the wavefront coding system," Opt. Express 15, 1543-1552 (2007).
    [CrossRef] [PubMed]

2007 (1)

2006 (2)

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

2005 (2)

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

2004 (1)

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

2003 (3)

S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

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

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A, Pure Appl. Opt. 5, S157-S163 (2003).

2002 (2)

W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41:6080-6092 (2002).
[CrossRef] [PubMed]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

2001 (1)

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

1999 (1)

T. A. Mitchell and J. M. Sasian, "Variable aberration correction using axially translating phase plates," Proc. SPIE 3705, 209-220 (1999).
[CrossRef]

1995 (1)

Andersson, M.

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

Bosch, S.

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

Cathey, W. T.

Chen, Y. P.

Chi, W.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A, Pure Appl. Opt. 5, S157-S163 (2003).

de la Fuente, M.

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

Dowski, E. R.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41:6080-6092 (2002).
[CrossRef] [PubMed]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866(1995).
[CrossRef] [PubMed]

Ferré-Borrull, J.

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

George, N.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A, Pure Appl. Opt. 5, S157-S163 (2003).

Gómez-Morales, F.

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

Harvey, A. R.

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," 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, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Huckridge, D.

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

Johnson, G. E.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Mezouari, S.

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," 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, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Mitchell, T. A.

T. A. Mitchell and J. M. Sasian, "Variable aberration correction using axially translating phase plates," Proc. SPIE 3705, 209-220 (1999).
[CrossRef]

Muyo, G.

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

Sasian, J. M.

T. A. Mitchell and J. M. Sasian, "Variable aberration correction using axially translating phase plates," Proc. SPIE 3705, 209-220 (1999).
[CrossRef]

Sherif, S. S.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

Silveira, P. E. X.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Singh, A.

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

Tudela, R.

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

Ye, Z.

Yu, F. H.

Zhang, W. Z.

Zhao, T. Y.

Appl. Opt. (2)

J. Mod. Opt. (1)

S. Mezouari and A. R. Harvey, "Combined amplitude and phase filters for increased tolerance to spherical aberration," J. Mod. Opt. 50, 2213-2220 (2003).

J. Opt. A, Pure Appl. Opt. (1)

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A, Pure Appl. Opt. 5, S157-S163 (2003).

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (7)

T. A. Mitchell and J. M. Sasian, "Variable aberration correction using axially translating phase plates," Proc. SPIE 3705, 209-220 (1999).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

G. Muyo, A. Singh, M. Andersson, D. Huckridge, and A. R. Harvey, "Optimized thermal imaging with a singlet and pupil plane encoding: experimental realization," Proc. SPIE 6395, 63950M (2006).
[CrossRef]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

F. Gómez-Morales, R. Tudela, J. Ferré-Borrull, S. Bosch, and M. de la Fuente, "Pupil filters for wavefront coding: off axis performances," Proc. SPIE 5962, 596237 (2005).
[CrossRef]

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Other (1)

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

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Figures (14)

Fig. 1.
Fig. 1.

Ray aberrations introduced by the phase plate.

Fig. 2.
Fig. 2.

Approximate ray aberrations and absolute errors for (a) X component and (b) Y component.

Fig. 3.
Fig. 3.

(a) Tangential rays map. (b) Comparisons between the exact and approximate boundaries and spot size.

Fig. 4.
Fig. 4.

Tangential (a) and sagittal (b) ray aberrations in the presence of defocus aberrations with the field angles of 30° and 0° for X and Y dimensions.

Fig. 5.
Fig. 5.

Tangential (a) and sagittal (b) ray aberrations in the presence of defocus aberrations with the field angles of 50° and 0° for X and Y dimensions.

Fig. 6.
Fig. 6.

Tangential (in solid lines) and sagittal (in dotted lines) ray aberrations enlarged by off-axis collimated beams of different field angles (a), and ratios between the exact and approximate enlarged scales (b).

Fig. 7.
Fig. 7.

Scales enlarged by off-axis collimated beams calculated from Eq. (17).

Fig. 8.
Fig. 8.

Spot diagram and its three approximate boundaries.

Fig. 9.
Fig. 9.

Spot diagrams (in black dots) and approximate boundaries (in red lines) for 25 field positions. The X and Y field angles are listed on the left column and top row, respectively.

Fig. 10.
Fig. 10.

Spot diagrams from Code V (in black dots) and approximate boundaries (in red lines) for 25 field positions. The X and Y field angles are listed on the left column and top row, respectively.

Fig. 11.
Fig. 11.

Spot diagrams from Code V in the presence of field curvature for 25 field positions. The X and Y field angles are listed on the left column and top row, respectively.

Fig. 12.
Fig. 12.

Spot diagrams from Code V in the presence of a small field distortion for 25 field positions. The X and Y field angles are listed on the left column and top row, respectively.

Fig. 13.
Fig. 13.

Spot diagrams from Code V in the presence of a large field distortion for 25 field positions. The X and Y field angles are listed on the left column and top row, respectively.

Fig. 14.
Fig. 14.

Distortion grid from Code V for the lens module.

Tables (1)

Tables Icon

Table 1. Included angles between the first and second approximate boundaries for 25 field positions. The X and Yfield angles are listed in the left column and top row, respectively.

Equations (23)

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

{ cos α , cos β , cos γ } = { cos α 0 + P 1 n 0 T ( F X ) , cos β 0 + P 1 n 0 T ( F Y ) , cos γ 0 + P 0 n 0 + P 1 n 0 T } ,
P 1 = n 0 2 n 1 2 + n 1 2 cos 2 I 2 n 1 cos I 2 ,
cos I 2 = n 0 cos α 0 n 1 T ( F X ) + n 0 cos β 0 n 1 T ( F Y ) + n 0 cos γ 0 + P 0 n 1 T .
{ X = [ f + Δ L ( D Z ) Δ L f ] cos α cos γ Δ L f X Y = [ f + Δ L ( D Z ) Δ L f ] cos β cos γ Δ L f Y .
{ X 0 = [ f + Δ L ( D Z 0 ) Δ L f ] cos α 0 cos γ 0 Y 0 = [ f + Δ L ( D Z 0 ) Δ L f ] cos β 0 cos γ 0 ,
{ Δ X = X X 0 = ( f + Δ L D Δ L f ) ( cos α cos γ cos α 0 cos γ 0 ) + Δ L f ( Z cos α cos γ Z 0 cos α 0 cos γ 0 ) Δ L f X ΔY = Y Y 0 = ( f + Δ L D Δ L f ) ( cos β cos γ cos β 0 cos γ 0 ) + Δ L f ( Z cos β cos γ Z 0 cos β 0 cos γ 0 ) Δ L f Y .
{ Δ X = X X 0 = ( f + Δ L D Δ L f ) ( cos α cos γ cos α 0 cos γ 0 ) + Δ LZ f cos α cos γ Δ L f X Δ Y = Y Y 0 = ( f + Δ L D Δ L f ) ( cos β cos γ cos β 0 cos γ 0 ) + Δ LZ f cos β cos γ Δ L f Y .
{ Δ α = P 1 n 0 T ( F X ) Δ β = P 1 n 0 T ( F Y ) Δγ = P 0 n 0 + P 1 n 0 T .
{ Δ α P 10 n 0 ( F X ) Δ β P 10 n 0 ( F Y ) Δγ cos α 0 Δ α + cos β 0 Δ β cos γ 0 P 10 = n 0 cos γ 0 n 1 2 n 0 2 + n 0 2 cos 2 γ 0 .
{ Δ X ( f + Δ L D Δ L f ) cos 3 γ 0 [ Δ α ( 1 cos 2 β 0 ) + Δ β cos α 0 cos β 0 ] Δ L f X Δ Y ( f + Δ L D Δ L f ) cos 3 γ 0 [ Δ β ( 1 cos 2 α 0 ) + Δ α cos α 0 cos β 0 ] Δ L f Y .
{ Δ X f cos 3 γ 0 [ Δ α ( 1 cos 2 β 0 ) + Δ β cos α 0 cos β 0 ] Δ L f X Δ Y f cos 3 γ 0 [ Δ β ( 1 cos 2 α 0 ) + Δα cos α 0 cos β 0 ] Δ L f Y .
Δ X 3 Af ( n 1 n 0 1 ) X 2 Δ L f X , r X r ,
{ Δ X lower Δ L 2 12 Af 3 ( n 1 n 0 1 ) = ( kW 20 ) 2 3 aπf 0 Δ X upper 3 Af ( n 1 n 0 1 ) r 2 + Δ L f r = 3 a + 2 kW 20 πf 0 .
d 3 Af ( n 1 n 0 1 ) r 2 + Δ L f r + Δ L 2 12 A f 3 ( n 1 n 0 1 ) = ( 3 a + kW 20 ) 2 3 aπf 0 .
{ Δ X f P 10 n 0 cos 3 γ 0 [ ( 1 cos 2 β 0 ) ( 3 AX 2 ) + cos α 0 cos β ( 3 A Y 2 ) ] Δ L f X Δ Y f P 10 n 0 cos 3 γ 0 [ ( 1 cos 2 α 0 ) ( 3 AY 2 ) + cos α 0 cos β ( 3 A X 2 ) ] Δ L f Y .
{ Δ X 3 A P 10 f ( 1 cos 2 β 0 ) n 0 cos 3 γ 0 [ X + n 0 cos 3 γ 0 Δ L 6 AP 10 f 2 ( 1 cos 2 β 0 ) ] 2 + n 0 cos 3 γ 0 Δ L 2 12 AP 10 f 3 ( 1 cos 2 β 0 ) 3 AP 10 f cos α 0 cos β 0 n 0 cos 3 γ 0 Y 2 Δ Y 3 A P 10 f ( 1 cos 2 α 0 ) n 0 cos 3 γ 0 [ Y + n 0 cos 3 γ 0 Δ L 6 AP 10 f 2 ( 1 cos 2 α 0 ) ] 2 + n 0 cos 3 γ 0 Δ L 2 12 AP 10 f 3 ( 1 cos 2 α 0 ) 3 AP 10 f cos α 0 cos β 0 n 0 cos 3 γ 0 X 2 .
{ Δ X 3 A P 10 f ( 1 cos 2 β 0 ) n 0 cos 3 γ 0 X 2 3 A P 10 f cos α 0 cos β 0 n 0 cos 3 γ 0 Y 2 Δ Y 3 A P 10 f ( 1 cos 2 α 0 ) n 0 cos 3 γ 0 Y 2 3 A P 10 f cos α 0 cos β 0 n 0 cos 3 γ 0 X 2 .
{ S X n 1 2 n 0 2 + n 0 2 cos 2 θ n 0 cos θ ( n 1 n 0 ) cos 3 θ S Y n 1 2 n 0 2 + n 0 2 cos 2 θ n 0 cos θ ( n 1 n 0 ) cos 3 θ , S = S X S Y 1 cos 2 θ
{ φ X = tan 1 ( Δ Y ( Y = 0 ) Δ X ( Y = 0 ) ) tan 1 ( cos α 0 cos β 0 1 cos 2 β 0 ) φ Y = tan 1 ( Δ X ( X = 0 ) Δ Y ( X = 0 ) ) tan 1 ( cos α 0 cos β 0 1 cos 2 α 0 ) .
( 1 cos α 0 cos β 0 cos 2 α 0 ) Δ X + ( 1 cos α 0 cos β 0 cos 2 β 0 ) Δ Y 3 AP 10 f r 2 n 0 cos γ 0
{ Δ X a = ( Δ X + X ) ( Δ X 0 + X 0 ) ΔY a = ( ΔY + Y ) ( ΔY 0 + Y 0 ) ,
{ Δ X a = ( 1 + p ) X ( 1 + p 0 ) X 0 Δ Y a = ( 1 + q ) Y ( 1 + q 0 ) Y 0 .
{ Δ X a ( 1 + p 0 ) ( X X 0 ) = ( 1 + p 0 ) Δ X Δ Y a ( 1 + q 0 ) ( Y Y 0 ) = ( 1 + q 0 ) Δ Y .

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