Abstract

A method of designing a lens with an extended focal depth is studied. The lens is a cemented doublet composed of a birefringent lens and a conventional lens. The crystal optical axis of the birefringent lens is perpendicular to the axis of the optical system. By properly selecting the parameters of the birefringent lens and the conventional lens, we can flexibly configure an imaging system that simultaneously has a large focal depth and high resolution. We also provide a theoretical analysis which shows that the focal depth of the lens is approximately 1.5 times that of a conventional lens.

© 2005 Optical Society of America

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References

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  1. J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
    [Crossref]
  2. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
    [Crossref]
  3. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
    [Crossref]
  4. K. Bhattacharya, A. K. Chakraborty, and A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
    [Crossref]
  5. S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
    [Crossref]
  6. X. Liu, X. Cai, S. Chang, and C. P. Grover, “Bifocal optical system for distant object tracking,” Opt. Express 13, 136–141 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-136.
    [Crossref] [PubMed]
  7. S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
    [Crossref]
  8. H. Kikuta and K. Iwata, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A 9, 814–819, (1992).
    [Crossref]
  9. J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
    [Crossref]
  10. Jae-Hyeung Park, Sungyong Jung, Heejin Choi, and Byoungho Lee, “Integral imaging with multiple image planes using a uniaxial crystal plate,” Opt. Express 11, 1862–1875 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1862.
    [Crossref] [PubMed]

2005 (1)

2003 (1)

2001 (1)

2000 (2)

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[Crossref]

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

1998 (1)

S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[Crossref]

1995 (2)

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
[Crossref]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[Crossref]

1994 (1)

1992 (1)

Bandyopadhyay, P.

S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[Crossref]

Bhattacharya, K.

Cai, X.

Cathey, W. T.

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[Crossref]

Chakraborty, A. K.

Chang, S.

Chi, W.

Choi, Heejin

Dowski, E. R.

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[Crossref]

Duncan, A. J.

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

Escalera, J. C.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
[Crossref]

George, N.

Ghosh, A.

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[Crossref]

S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[Crossref]

K. Bhattacharya, A. K. Chakraborty, and A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
[Crossref]

Grover, C. P.

Iwata, K.

Jung, Sungyong

Kikuta, H.

Lee, Byoungho

Lesso, J. P.

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

Liu, X.

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
[Crossref]

Padgett, M. J.

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

Park, Jae-Hyeung

Sanyal, S.

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[Crossref]

S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[Crossref]

Silson, W.

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

Yzuel, M. J.

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
[Crossref]

App. Opt. (3)

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” App. Opt. 34, 1859–1866 (1995).
[Crossref]

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” App. Opt. 39, 2321–2325 (2000).
[Crossref]

J. P. Lesso, A. J. Duncan, W. Silson, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” App. Opt. 39, 592–598 (2000).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Ojeda-Castañeda, J. C. Escalera, and M. J. Yzuel, “Supergaussian rings: Focusing properties,” Opt. Commun. 114, 189–193 (1995).
[Crossref]

Opt. Eng. (1)

S. Sanyal, P. Bandyopadhyay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

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Figures (8)

Fig. 1.
Fig. 1.

Optical system configuration

Fig. 2.
Fig. 2.

Intensity in the focus vs values of B and α

Fig. 3.
Fig. 3.

Variation of value of B vs the value of α

Fig. 4.
Fig. 4.

Variation of I(0,0) vs the value of B

Fig. 5.
Fig. 5.

PSF plots for B=0.9 and α=0.67

Fig. 6.
Fig. 6.

MTF plots for B=0.9 and α=0.67

Fig. 7.
Fig. 7.

MTF versus spatial frequency

Fig. 8.
Fig. 8.

Axial intensity for different value of defocusing

Equations (20)

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ϕ o = n 1 o D 1 x 1 2 + y 1 2 2 f 1 o + n 2 D 2 x 1 2 + y 1 2 2 f 2 ,
ϕ e = n 1 e D 1 x 1 2 + y 1 2 2 f 1 e + n 2 D 2 x 1 2 + y 1 2 2 f 2 ,
1 f 1 o = ( n 1 o 1 ) · ( 1 R 1 1 R 2 ) = ( n 1 o 1 ) · δ C 1 ,
1 f 1 e = ( n 1 e 1 ) · ( 1 R 1 1 R 2 ) = ( n 1 e 1 ) · δ C 1 ,
U o ( x , y ; z ) = exp ( i k z ) λ · z P 1 ( x 1 , y 1 ) · exp [ i k ϕ o ( x 1 , y 1 ) ] · exp ( ik ( x x 1 ) 2 + ( y y 1 ) 2 2 z ) · dx 1 dy 1 ,
U e ( x , y ; z ) = exp ( i k z ) λ · z P 1 ( x 1 , y 1 ) · exp [ i k ϕ o ( x 1 , y 1 ) ] · exp ( ik ( x x 1 ) 2 + ( y y 1 ) 2 2 z ) · dx 1 dy 1 ,
p 1 ( x 1 , y 1 ) = { 1 , ( x 1 2 + y 1 2 ) ρ 2 0 , otherwise ,
I ( x , y , z ) = 2 2 U o ( x , y , z ) + 2 2 U e ( x , y , z ) 2 .
1 f 1 = 1 2 ( 1 f 1 o + 1 f 1 e ) ,
1 f = 1 f 1 + 1 f 2 .
I ( r , Δ z ) = c 1 · 0 1 cos [ π ( A B ξ 2 ) ] · J 0 ( π 2 ρ · r λ · f ξ ) exp [ i π ρ 2 λ · f 2 Δ z · ξ 2 ] · ξ · d ξ 2 ,
A = δ n 1 · D 1 λ ,
B = ρ 2 · δ C 1 · δ n 1 ( 2 λ ) ,
I ( 0 , Δ z ) = c 1 · 0 1 cos [ π ( α B · ξ 2 ) ] · exp [ i π ρ 2 λ · f 2 Δ z · ξ 2 ] · ξ · d ξ 2 .
I ( r , 0 ) = c 1 · 0 1 cos [ π ( α B ξ 2 ) ] · J 0 ( π 2 ρ · r λ · f ξ ) · ξ · d ξ 2 ;
I ( 0 , 0 ) = c 1 4 { sin c ( B π 2 ) · { cos [ ( B 2 α ) π ] } } 2 ,
sin c ( x ) = sin ( x ) x .
c ( x , y ) = { cos { π · [ α B ( x 2 + y 2 ) ] } · exp [ i · π · ρ 2 λ · f 2 Δ z · ( x 2 + y 2 ) ] x 2 + y 2 1 0 otherwise ,
OTF ( v ) = c ( x , y ) · c * ( x v , y ) · dxdy c ( x , y ) · c * ( x , y ) · dxdy ;
MTF ( v ) = OTF ( v ) ,

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