Abstract

To be able to design optical systems (e.g., variable focus or zoom lenses) made from liquid-crystal devices, it is necessary to be able to ray trace in a birefringent medium where the angle of the optical axis is a function of position in the device. To our knowledge, the theory required to achieve this has not previously been published, and we derive a suitable algorithm and give some examples of its use.

© 2007 Optical Society of America

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References

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2006

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

2004

S. Kuiper and H. H. W. Hendriks, "Variable-focus liquid lens for minature cameras," Appl. Phys. Lett. 85, 1128-1130 (2004).
[CrossRef]

D. Y. Zhang, N. Justis, and Y. H. Lo, "Integrated fluidic adaptive zoom lens," Opt. Lett. 29, 2855-2857 (2004).
[CrossRef]

D. Wick and T. Martinez, "Adaptive optical zoom," Opt. Eng. (Bellingham) 43, 8-9 (2004).
[CrossRef]

2001

2000

L. G. Commander, S. E. Day, and D. R. Selviah, "Variable focal length microlenses," Opt. Commun. 177, 157-170 (2000).
[CrossRef]

J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, "Aberrations introduced by a lens made from a birefringent material," Appl. Opt. 39, 592-596 (2000).
[CrossRef]

1999

A. F. Naumov, G. D. Love, M. Yu. Loktev, and F. L. Vladimirov, "Control optimization of spherical modal liquid crystal lenses," Opt. Express 4, 344-352 (1999).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. A. Sethian, "Fast marching methods," SIAM Rev. 41, 199235 (1999).
[CrossRef]

1998

1997

R. R. Shannon, The Art and Science of Optical Design (Cambridge U. Press, 1997).

1994

1993

1990

1982

Agarwal, A. K.

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

Äyräs, P.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Beebe, D. J.

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Chipman, R. A.

Commander, L. G.

L. G. Commander, S. E. Day, and D. R. Selviah, "Variable focal length microlenses," Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Davidson, N.

Day, S. E.

L. G. Commander, S. E. Day, and D. R. Selviah, "Variable focal length microlenses," Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Dong, L.

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

Duncan, A. J.

Friedman, N.

Ghatak, A. K.

Giridhar, M. S.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Guralnik, I. R.

Haddock, J. N.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Hendriks, H. H. W.

S. Kuiper and H. H. W. Hendriks, "Variable-focus liquid lens for minature cameras," Appl. Phys. Lett. 85, 1128-1130 (2004).
[CrossRef]

Hillman, L. W.

Honkanen, S.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Jiang, H.

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

Justis, N.

Kaplan, A.

Kippelen, B.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Kuiper, S.

S. Kuiper and H. H. W. Hendriks, "Variable-focus liquid lens for minature cameras," Appl. Phys. Lett. 85, 1128-1130 (2004).
[CrossRef]

Kumar, D. V.

Lesso, J. P.

Li, G.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Liang, Q.-T.

Lo, Y. H.

Loktev, M. Yu.

Love, G. D.

Major, J. N.

Martinez, T.

D. Wick and T. Martinez, "Adaptive optical zoom," Opt. Eng. (Bellingham) 43, 8-9 (2004).
[CrossRef]

Mathine, D. L.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

McClain, S. C.

Meredith, G. R.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Naumov, A. F.

Padgett, M. J.

Peyghambarian, N.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Purvis, A.

Schwiegerling, J.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Selviah, D. R.

L. G. Commander, S. E. Day, and D. R. Selviah, "Variable focal length microlenses," Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Sethian, J. A.

J. A. Sethian, "Fast marching methods," SIAM Rev. 41, 199235 (1999).
[CrossRef]

Shannon, R. R.

R. R. Shannon, The Art and Science of Optical Design (Cambridge U. Press, 1997).

Sharma, A.

Sibbett, W.

Valley, P.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Vdovin, G.

Vladimirov, F. L.

Wick, D.

D. Wick and T. Martinez, "Adaptive optical zoom," Opt. Eng. (Bellingham) 43, 8-9 (2004).
[CrossRef]

Willib, G.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Zhang, D. Y.

Appl. Opt.

Appl. Phys. Lett.

S. Kuiper and H. H. W. Hendriks, "Variable-focus liquid lens for minature cameras," Appl. Phys. Lett. 85, 1128-1130 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Nature

L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, "Adaptive liquid microlenses activated by stimuli responsive hydrogels," Nature 442, 551-554 (2006).
[CrossRef] [PubMed]

Opt. Commun.

L. G. Commander, S. E. Day, and D. R. Selviah, "Variable focal length microlenses," Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Opt. Eng. (Bellingham)

D. Wick and T. Martinez, "Adaptive optical zoom," Opt. Eng. (Bellingham) 43, 8-9 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Willib, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, "Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications," Proc. Natl. Acad. Sci. U.S.A. 103, 6100-6104 (2006).
[CrossRef] [PubMed]

SIAM Rev.

J. A. Sethian, "Fast marching methods," SIAM Rev. 41, 199235 (1999).
[CrossRef]

Other

R. R. Shannon, The Art and Science of Optical Design (Cambridge U. Press, 1997).

ZEMAX Development Corporation, www.ZEMAX.com.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Varioptic, http://www.varioptic.com.

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Figures (3)

Fig. 1
Fig. 1

Various vectors are shown for the extraordinary ray. The wave vector k, Poynting unit vector (ray) u, crystal axis c, and the electric and displacement fields E and D are all in the same plane (shown as the x z plane here, without loss of generality for a uniaxial crystal). The ordinary ray, with the same direction of propagation k, would have its electric field along the y axis.

Fig. 2
Fig. 2

Schematic of the geometry of our propagator for the ray. The ray is indicated by a dashed curve; the dotted curve is the curve to which the wave vector is tangent.

Fig. 3
Fig. 3

Ray-trace results, showing OPDs and ray heights, for the sagittal (left) and tangential rays; solid curves are on the axis and dashed curves are at a 1 ° field angle.

Tables (2)

Tables Icon

Table 1 Comparison of Refraction Algorithms a

Tables Icon

Table 2 Comparison of Wedge Models a

Equations (36)

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

ϵ = [ ϵ e 0 0 0 ϵ o 0 0 0 ϵ o ] ,
E ( r ) = e ( r ) e ι k 0 S ( r ) ,
H ( r ) = h ( r ) e ι k 0 S ( r ) ,
S × h + ϵ e = 0 ,
S × e μ h = 0 ,
ϵ e S = 0 ,
( e S ) S S 2 e + μ ϵ e = 0 .
S = S ( cos θ , sin θ , 0 ) ,
S 2 = n o 2 ,
S 2 = n e 2 n o 2 n o 2 sin 2 θ + n e 2 cos 2 θ ,
n e f f = n e n o ( n o 2 sin 2 θ + n e 2 cos 2 θ ) 1 2 ,
tan α = ( n e 2 n o 2 ) tan θ n e 2 + n o 2 tan 2 θ ,
d r d s = R ( α ) S S .
u = d r d s ,
d d s n e f f u = d d s R S .
d d s n e f f u = ( u ) R S ,
= 1 2 n e f f [ ( R S 2 ) ] 1 n e f f ( R S × × R S ) ,
= n e f f u × × n e f f u ,
d u d s = u × × u .
d r d s = k ,
n e f f = k = S ,
d d s n e f f k = n e f f ,
n e f f = n e f f θ ( 1 1 ( c k ) 2 ) ( c k ) .
n e f f θ = n e f f 3 ( n e 2 n o 2 ) cos θ sin θ n e 2 n o 2 .
( c 1 n e f f S ) .
k t n e f f = constant ,
d r d s = k ,
n e f f = d k d s = n e f f k ( k ) n e f f .
d ϕ = κ d r ,
ϕ = ray ( κ cos α ) d s ,
OPD = ray ( n e f f cos α ) d s
S ( r ) = n e f f r cos α .
ϕ = i = 1 8 a i r i ,
c x = x r sin ϕ ,
c y = x r sin ϕ ,
c z = cos ϕ .

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