Abstract

We describe a simple method to extend the depth of field of a conventional camera by inserting a transparent annular ring in front of the pupil of the lens. The insertion of the ring creates an unbalanced optical path difference across the lens aperture, which partitions the pupil and leads to an extended depth of field. This system is analyzed by diffraction and random process theory. Experiments are reported that are in good agreement with the theory.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Chi, K. Chu, and N. George, “Polarization coded aperture,” Opt. Express 14, 6634-6642 (2006).
    [CrossRef]
  2. W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 355.
  3. A. E. Conrady, Applied Optics and Optical Design, Part I(Dover, 1957), pp. 136.
  4. A. E. Conrady, Applied Optics and Optical Design, Part II(Dover, 1960), pp. 585.
  5. Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.
  6. E. N. Leith and G. J. Swanson, “Optical processing techniques in incoherent light,” Proc. SPIE 388, 38 (1983).
  7. E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145-149 (1957).
  8. W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  11. W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 100.
  12. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459-8467 (1997).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1995).
  14. P. J. Davis and I. Polonsky, “Numerical interpolation, differentiation, and integration,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), pp. 887-888.
  15. R. Barakat, “Total illumination in a diffraction image containing spherical aberration,” J. Opt. Soc. Am. 51, 152-157 (1961).
  16. R. Barakat and M. V. Morello, “Computation of the totalilluminance (encircled energy) of an optical system from the design data for rotationally symmetric aberrations,” J. Opt. Soc. Am. 54, 235-240 (1964).
  17. V. N. Mahajan, “Aberrated point spread functions for rotationally symmetric aberrations,” Appl. Opt. 22, 3035-3041(1983).
  18. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297-303 (1967).
  19. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511-518 (1972).
  20. W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273 (2003).
    [CrossRef]

2006 (1)

2005 (1)

W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).

2003 (1)

1997 (1)

1983 (2)

E. N. Leith and G. J. Swanson, “Optical processing techniques in incoherent light,” Proc. SPIE 388, 38 (1983).

V. N. Mahajan, “Aberrated point spread functions for rotationally symmetric aberrations,” Appl. Opt. 22, 3035-3041(1983).

1972 (1)

1967 (1)

1964 (1)

1961 (1)

1957 (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145-149 (1957).

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1995).

Braat, J.

Chi, W.

Chu, K.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part I(Dover, 1957), pp. 136.

A. E. Conrady, Applied Optics and Optical Design, Part II(Dover, 1960), pp. 585.

Davis, P. J.

P. J. Davis and I. Polonsky, “Numerical interpolation, differentiation, and integration,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), pp. 887-888.

Frieden, B. R.

George, N.

Goodman, J. W.

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

Helstrom, C. W.

Leith, E. N.

E. N. Leith and G. J. Swanson, “Optical processing techniques in incoherent light,” Proc. SPIE 388, 38 (1983).

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145-149 (1957).

Mahajan, V. N.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Morello, M. V.

Polonsky, I.

P. J. Davis and I. Polonsky, “Numerical interpolation, differentiation, and integration,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), pp. 887-888.

Rayleigh, Lord

Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 355.

W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 100.

Swanson, G. J.

E. N. Leith and G. J. Swanson, “Optical processing techniques in incoherent light,” Proc. SPIE 388, 38 (1983).

Wolf, E.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145-149 (1957).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1995).

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

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

Opt. Commun. (1)

W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).

Opt. Express (1)

Proc. Phys. Soc. B (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145-149 (1957).

Proc. SPIE (1)

E. N. Leith and G. J. Swanson, “Optical processing techniques in incoherent light,” Proc. SPIE 388, 38 (1983).

Other (9)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1995).

P. J. Davis and I. Polonsky, “Numerical interpolation, differentiation, and integration,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., M.Abramowitz and I.A.Stegun, eds. (Dover, 1972), pp. 887-888.

W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 355.

A. E. Conrady, Applied Optics and Optical Design, Part I(Dover, 1957), pp. 136.

A. E. Conrady, Applied Optics and Optical Design, Part II(Dover, 1960), pp. 585.

Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

W. J. Smith, Modern Optical Engineering, 3rd ed. (SPIE Press), p. 100.

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 (12)

Fig. 1
Fig. 1

Proposed imaging system to extend depth of field: (a) the inserted plate; (b) the modified imaging system.

Fig. 2
Fig. 2

Conventional imaging system.

Fig. 3
Fig. 3

Phase plate model for derivation of the transmission function of the glass plate.

Fig. 4
Fig. 4

PSF examples of a perfect lens for three object distances: (a) unpartitioned pupil, (b) partitioned pupil with unbalanced OPD, Eq. (23).

Fig. 5
Fig. 5

(a) Peak and (b) FWHM of the central lobe of PSF of a lens before the partition (dashed curve) and after the partition (solid curve).

Fig. 6
Fig. 6

PSF examples of an aberrated lens (a) without or (b) with partition, W 040 = 0.55 , W 060 = 0.15 .

Fig. 7
Fig. 7

Original object.

Fig. 8
Fig. 8

Blurred images when the object shown in Fig. 7 is moved along the optical axis and the lens is focused at the “0” position. A segment of the image is digitally magnified and placed next to the image.

Fig. 9
Fig. 9

Recovery from images shown in Fig. 8.

Fig. 10
Fig. 10

PSF measured over different object positions.

Fig. 11
Fig. 11

Blurry images before digital processing.

Fig. 12
Fig. 12

Recovery from images shown in Fig. 11.

Equations (28)

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

h 12 ( x , y , ξ , η ; ν ) = exp [ i k ( z 1 + z 2 + ξ 2 + η 2 2 z 1 + x 2 + y 2 2 z 2 ) ] ν 2 c 2 z 1 z 2 × Σ P ( u , v ) exp [ i k u 2 + v 2 2 ( 1 z 1 + 1 z 2 1 f ) ] exp { i k [ ( ξ z 1 + x z 2 ) u + ( η z 1 + y z 2 ) v ] } d u d v ,
I ( x , y ) = S ( ξ , η ; ν ) | o ( ξ , η ) | 2 | h 12 ( x , y , ξ , η ; ν ) | 2 d ξ d η d ν ,
| h 12 ( x , y , ξ , η ) | 2 = S ( ν ) | h 12 ( x , y , ξ , η ; ν ) | 2 d ν .
P ( u , v ) = { P 1 ( u , v ) , ( u , v ) Σ 1 ; P 2 ( u , v ) , ( u , v ) Σ 2 .
| h 12 ( x , y , ξ , η ) | 2 = S ( ν ) | h 12 ( 1 ) ( x , y , ξ , η ; ν ) + h 12 ( 2 ) ( x , y , ξ , η ; ν ) | 2 d ν ,
h 12 ( 1 ) ( x , y , ξ , η ; v ) = exp [ i k ( z 1 + z 2 + ξ 2 + η 2 2 z 1 + x 2 + y 2 2 z 2 ) ] ν 2 c 2 z 1 z 2 × Σ 1 P 1 ( u , v ) exp [ i k u 2 + v 2 2 ( 1 z 1 + 1 z 2 1 f ) ] exp { i k [ ( ξ z 1 + x z 2 ) u + ( η z 1 + y z 2 ) v ] } d u d v , h 12 ( 2 ) ( x , y , ξ , η ; v ) = exp [ i k ( z 1 + z 2 + ξ 2 + η 2 2 z 1 + x 2 + y 2 2 z 2 ) ] ν 2 c 2 z 1 z 2 × Σ 2 P 2 ( u , v ) exp [ i k u 2 + v 2 2 ( 1 z 1 + 1 z 2 1 f ) ] exp { i k [ ( ξ z 1 + x z 2 ) u + ( η z 1 + y z 2 ) v ] } d u d v ,
d = { d 1 , ( u , v ) Σ 1 d 2 , ( u , v ) Σ 2
t plate ( u , v ) = { exp ( i k n d 1 ) , ( u , v ) Σ 1 exp [ i k ( d 1 d 2 + n d 2 ) ] , ( u , v ) Σ 2 .
P ( p ) ( u , v ) = { P 1 ( u , v ) exp ( i k n d 1 ) , ( u , v ) Σ 1 ; P 2 ( u , v ) exp [ i k ( d 1 d 2 + n d 2 ) ] , ( u , v ) Σ 2 ,
h 12 ( p ) ( x , y , ξ , η ; ν ) = h 12 ( 1 ) ( x , y , ξ , η ; ν ) exp ( i k n d 1 ) + h 12 ( 2 ) ( x , y , ξ , η ; ν ) exp [ i k ( d 1 d 2 + n d 2 ) ] .
| h 12 ( p ) ( x , y , ξ , η ) | 2 = S ( ν ) | h 12 ( 1 ) ( x , y , ξ , η ; ν ) exp ( i k n d 1 ) + h 12 ( 2 ) ( x , y , ξ , η ; ν ) exp [ i k ( d 1 d 2 + n d 2 ) ] | 2 d ν = | h 12 ( 1 ) ( x , y , ξ , η ) | 2 + | h 12 ( 2 ) ( x , y , ξ , η ) | 2 + cross term ,
| h 12 ( 1 ) ( x , y , ξ , η ) | 2 = S ( ν ) | h 12 ( 1 ) ( x , y , ξ , η ; ν ) | 2 d ν , | h 12 ( 2 ) ( x , y , ξ , η ) | 2 = S ( ν ) | h 12 ( 2 ) ( x , y , ξ , η ; ν ) | 2 d ν ,
cross term = 2 Re { S ( ν ) exp [ i k ( n 1 ) ( d 1 d 2 ) ] h 12 ( 1 ) ( x , y , ξ , η ; ν ) h 12 ( 2 ) * ( x , y , ξ , η ; ν ) d ν } .
cross term = 2 Re { h 12 ( 1 ) ( x , y , ξ , η ; ν 0 ) h 12 ( 2 ) * ( x , y , ξ , η ; ν 0 ) S ( ν ) exp [ i k ( n 1 ) ( d 1 d 2 ) ] d ν } ,
S ( ν ) exp [ i k ( n 1 ) ( d 1 d 2 ) ] d ν = R ( τ ) , τ = | ( n 1 ) ( d 1 d 2 ) c | .
| h 12 ( p ) ( x , y , ξ , η ) | 2 = | h 12 ( 1 ) ( x , y , ξ , η ) | 2 + | h 12 ( 2 ) ( x , y , ξ , η ) | 2 | ( n 1 ) ( d 1 d 2 ) | c τ c = l c ,
Φ defocus ( u , v ) = k ( n 1 ) d n u 2 + v 2 2 z 1 2 ,
P 1 ( d ) ( u , v ) = P 1 ( u , v ) exp [ i k ( n d 1 + ( n 1 ) d 1 n u 2 + v 2 2 z 1 2 ) ] , ( u , v ) Σ 1 ; P 2 ( d ) ( u , v ) = P 2 ( u , v ) exp [ i k ( d 1 d 2 + n d 2 + ( n 1 ) d 2 n u 2 + v 2 2 z 1 2 ) ] , ( u , v ) Σ 1 ,
I ( x , y ) = S ( ν ) | o g ( ξ ˜ , η ˜ ) | 2 p ( x ξ ˜ , y η ˜ , ; ν ) d ξ ˜ d η ˜ d ν = | o g ( ξ ˜ , η ˜ ) | 2 p ( x ξ ˜ , y η ˜ ) d ξ ˜ d η ˜ ,
o g ( ξ ˜ , η ˜ ) = o ( z 2 z 1 ξ , z 2 z 1 η )
p ( x ξ ˜ , y η ˜ ; ν ) = p ( x + z 2 z 1 ξ , y z 2 z 1 η ; ν ) = | h 12 ( x , y , ξ , η ; ν ) | 2
p ( x ξ ˜ , y η ˜ ) = S ( ν ) | h 12 ( x , y , ξ , η ; ν ) | 2 d ν = 4 π 2 S ( ν ) ν 4 c 4 z 1 2 z 2 2 | Σ P ( u , v ) exp [ i k u 2 + v 2 2 ( 1 z 1 + 1 z 2 1 f ) ] exp { i k [ ( ξ z 1 + x z 2 ) u + ( η z 1 + y z 2 ) v ] } d u d v | 2 d ν .
p ( r ) = 4 π 2 S ( ν ) ν 4 c 4 z 1 2 z 2 2 | 0 a exp [ i k ρ 2 2 ( 1 z 1 + 1 z 2 1 f ) ] J 0 ( k r ρ z 2 ) ρ d ρ | 2 d ν ,
d = { d 1 , 0 ρ ε a ; d 2 , ε a < ρ a ,
p ( 1 ) ( r ) = 4 π 2 S ( ν ) ν 4 c 4 z 1 2 z 2 2 | 0 ε a exp [ i k ρ 2 2 ( 1 z 1 + 1 z 2 1 f + n 1 n d 1 z 1 2 ) ] J 0 ( k r ρ z 2 ) ρ d ρ | 2 d ν , p ( 2 ) ( r ) = 4 π 2 S ( ν ) ν 4 c 4 z 1 2 z 2 2 | 0 ε a exp [ i k ρ 2 2 ( 1 z 1 + 1 z 2 1 f + n 1 n d 2 z 1 2 ) ] J 0 ( k r ρ z 2 ) ρ d ρ | 2 d ν .
p ( p ) ( r ) = p ( 1 ) ( r ) + p ( 2 ) ( r ) .
p ( 1 a ) ( r ) = S ( ν ) ν 4 c 4 z 1 2 z 2 2 | 0 2 π 0 ε a exp [ i k ρ 2 2 ( 1 z 1 + 1 z 2 1 f + n 1 n d 1 z 1 2 ) + Φ ( ρ , ϕ ) ] J 0 ( k r ρ z 2 ) ρ d ρ d ϕ | 2 d ν , p ( 2 a ) ( r ) = S ( ν ) ν 4 c 4 z 1 2 z 2 2 | 0 2 π ε a a exp [ i k ρ 2 2 ( 1 z 1 + 1 z 2 1 f + n 1 n d 2 z 1 2 ) + Φ ( ρ , ϕ ) ] J 0 ( k r ρ z 2 ) ρ d ρ d ϕ | 2 d ν ,
p ( p a ) ( r ) = p ( 1 a ) ( r ) + p ( 2 a ) ( r ) .

Metrics