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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App. Opt. (3)

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]

E. R. Dowski and W. T. Cathey, �??Extended depth of field through wave-front coding,�?? App. Opt. 34, 1859-1866 (1995).
[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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