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Bifocal optical system for distant object tracking

Xinping Liu, Xianyang Cai, Shoude Chang, and Chander P. Grover

Open Access Open Access

Abstract

A new bifocal optical system used for distant object tracking is proposed. This system combines a birefringent element with a conventional glass lens so that the spot image size and its variation with the axial distance can be controlled according to the requirement of a distant object tracker. The lens design for the tracking application is discussed and an example is given. The new lens system provides a more uniform spot image size and an extended focal depth compared to a conventional lens with one focus.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Schematic of a lens having two foci

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Fig. 2.
Fig. 2. Spot size versus the value of Δs

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Fig. 3.
Fig. 3. Spot size versus the axial distance

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Fig. 4.
Fig. 4. Schematic of the combination of a bifocal lens

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Fig. 5.
Fig. 5. Configuration of an optical system comprising a birefringent lens

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Fig. 6.
Fig. 6. Spot size for e rays oe-13-1-136-i001.jpg

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Fig. 7.
Fig. 7. Spot size of a bifocal lens system oe-13-1-136-i002.jpg

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Equations (19)

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D H ( Ω ) = D 0 ( Ω ) D g ( Ω ) .
I ( r ) = F − 1 [ D 0 ( Ω ) D g ( Ω ) ] = I 0 ( r ) * I g ( r ) ,
I 0 ( r ) = [ 2 · J 1 ( λ · F n · r ) λ · F n · r ] 2 ,
I ( r , Δ s , δ z ) = I 0 ( r ) * I g 1 ( r ) + I 0 ( r ) * I g 2 ( r ) .
I g 1 ( r ) = { 1 π [ 2 F n ( Δ s ⁄ 2 ) + δ z ] 2 r ≤ ( Δ s ⁄ 2 ) + δ z 2 F n 0 others ,
and I g 2 ( r ) = { 1 π [ 2 F n ( Δ s ⁄ 2 ) − δ z ] 2 r ≤ ( Δ s ⁄ 2 ) + δ z 2 F n 0 others .
I ( r , Δ s , δ z ) = ∫ 0 a 1 ( 2 J 1 [ λ . F n . ( r − τ ) ] λ . F n . ( r − τ ) ) 2 d τ + ∫ 0 a 2 ( 2 J 1 [ λ . F n . ( r − τ ) ] λ . F n . ( r − τ ) ) 2 d τ ,
where a 1 = [ ( Δ s ⁄ 2 ) + δ z ] ⁄ 2 F n ,
a 2 = [ ( Δ s ⁄ 2 ) − δ z ] ⁄ 2 F n .
RED ( c , Δ s , δ z ) = ∫ 0 c 2 π · I ( r , Δ s , δ z ) · r · dr ∫ 0 ∞ 2 π · I ( r , Δ s , 0 ) · r · dr .
f 1 o ′ = r 1 ⁄ ( n o − 1 ) ,
f 1 e ′ = r 1 ⁄ ( n e − 1 ) .
Δ d 1 = f 1 o ′ − f 1 e ′ .
Δ d 1 = ( n e − n o ) · r 1 ( n o − 1 ) · ( n e − 1 ) .
Δ d = − ( x 2 ′ ⁄ x 2 ) · Δ d 1 ,
f 2 ′ 2 = − x 2 · x 2 ′ ,
1 f ′ = 1 f 1 e ' + 1 f 2 ′ − ( a + f ′ 2 ) f 1 e ′ f 2 ′ .
a = f 1 e ′ − x 2 .
Δ d = f ′ 2 ( n e − 1 ) ( n e − n o ) ( n o − 1 ) · r 1 .
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