Abstract

The creation of a field-controlled variation of the index of refraction in a liquid crystal cell has been analyzed and experimentally verified. To obtain a spherical lens utilizing a simple electrode structure and capable of focusing arbitrary incoming polarizations requires four flat nematic liquid crystal cells. With electrodes fabricated well within the current capability of photolithography, near diffraction-limited performance in terms of the optical transfer function is predicted. The focusing capability of a liquid crystal lens was demonstrated using a single cell with linear transparent electrodes. A plano-convex cylindrical lens for a single incoming polarization was thus created. While the cell had a crude electrode structure, it affirmed all the major qualitative predictions. The fringing along the edge of the electrodes required for eventually obtaining near diffraction-limited performance was observed.

© 1984 Optical Society of America

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References

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  1. J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1972).
  2. V. Freedericksz, V. Zolina, Trans. Faraday Soc. 29, 919 (1933).
    [CrossRef]
  3. H. Gruller, G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
    [CrossRef]
  4. H. Deuling, Mol. Cryst. Liq. Cryst. 19, 123 (1972).
    [CrossRef]
  5. F. J. Kahn, Appl. Phys. Lett. 20, No. 5, 199 (Mar.1972).
    [CrossRef]
  6. M. F. Schiekel, K. Fahrenschon, Appl. Phys. Lett. 19, No. 10, 391 (Nov.1971).
    [CrossRef]
  7. F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
    [CrossRef]
  8. E. B. Priestley et al., Introduction to Liquid Crystals (Plenum, New York, 1974).
  9. L. T. Creagh, Proc. IEEE 61, 814 (1973).
    [CrossRef]
  10. P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).
  11. S. T. Kowel, D. S. Cleverly, “A Liquid Crystal Adaptive Lens,” in Proceedings, NASA Conference on Optical Information Processing for Aerospace Applications, Hampton, Va. (1981).
  12. D. S. Cleverly, “Creation of a Lens by Field-Controlled Variation of the Index of Refraction in a Liquid Crystal,” Ph.D. Dissertation, Syracuse U., Syracuse, N.Y. (1982).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, No. 4 (Feb.1971).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

1973 (2)

F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
[CrossRef]

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[CrossRef]

1972 (3)

H. Gruller, G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[CrossRef]

H. Deuling, Mol. Cryst. Liq. Cryst. 19, 123 (1972).
[CrossRef]

F. J. Kahn, Appl. Phys. Lett. 20, No. 5, 199 (Mar.1972).
[CrossRef]

1971 (2)

M. F. Schiekel, K. Fahrenschon, Appl. Phys. Lett. 19, No. 10, 391 (Nov.1971).
[CrossRef]

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, No. 4 (Feb.1971).
[CrossRef]

1933 (1)

V. Freedericksz, V. Zolina, Trans. Faraday Soc. 29, 919 (1933).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Cleverly, D. S.

S. T. Kowel, D. S. Cleverly, “A Liquid Crystal Adaptive Lens,” in Proceedings, NASA Conference on Optical Information Processing for Aerospace Applications, Hampton, Va. (1981).

D. S. Cleverly, “Creation of a Lens by Field-Controlled Variation of the Index of Refraction in a Liquid Crystal,” Ph.D. Dissertation, Syracuse U., Syracuse, N.Y. (1982).

Creagh, L. T.

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[CrossRef]

deGennes, P. G.

P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

Deuling, H.

H. Deuling, Mol. Cryst. Liq. Cryst. 19, 123 (1972).
[CrossRef]

Fahrenschon, K.

M. F. Schiekel, K. Fahrenschon, Appl. Phys. Lett. 19, No. 10, 391 (Nov.1971).
[CrossRef]

Freedericksz, V.

V. Freedericksz, V. Zolina, Trans. Faraday Soc. 29, 919 (1933).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gruller, H.

H. Gruller, G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[CrossRef]

Helfrich, W.

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, No. 4 (Feb.1971).
[CrossRef]

Kahn, F. J.

F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
[CrossRef]

F. J. Kahn, Appl. Phys. Lett. 20, No. 5, 199 (Mar.1972).
[CrossRef]

Kowel, S. T.

S. T. Kowel, D. S. Cleverly, “A Liquid Crystal Adaptive Lens,” in Proceedings, NASA Conference on Optical Information Processing for Aerospace Applications, Hampton, Va. (1981).

Meier, G.

H. Gruller, G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1972).

Priestley, E. B.

E. B. Priestley et al., Introduction to Liquid Crystals (Plenum, New York, 1974).

Schadt, M.

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, No. 4 (Feb.1971).
[CrossRef]

Schiekel, M. F.

M. F. Schiekel, K. Fahrenschon, Appl. Phys. Lett. 19, No. 10, 391 (Nov.1971).
[CrossRef]

Schonohorn, H.

F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
[CrossRef]

Taylor, G. N.

F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Zolina, V.

V. Freedericksz, V. Zolina, Trans. Faraday Soc. 29, 919 (1933).
[CrossRef]

Appl. Phys. Lett. (3)

F. J. Kahn, Appl. Phys. Lett. 20, No. 5, 199 (Mar.1972).
[CrossRef]

M. F. Schiekel, K. Fahrenschon, Appl. Phys. Lett. 19, No. 10, 391 (Nov.1971).
[CrossRef]

M. Schadt, W. Helfrich, Appl. Phys. Lett. 18, No. 4 (Feb.1971).
[CrossRef]

Mol. Cryst. Liq. Cryst. (2)

H. Gruller, G. Meier, Mol. Cryst. Liq. Cryst. 16, 299 (1972).
[CrossRef]

H. Deuling, Mol. Cryst. Liq. Cryst. 19, 123 (1972).
[CrossRef]

Proc. IEEE (2)

F. J. Kahn, G. N. Taylor, H. Schonohorn, Proc. IEEE 61, 823 (1973).
[CrossRef]

L. T. Creagh, Proc. IEEE 61, 814 (1973).
[CrossRef]

Trans. Faraday Soc. (1)

V. Freedericksz, V. Zolina, Trans. Faraday Soc. 29, 919 (1933).
[CrossRef]

Other (7)

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1972).

E. B. Priestley et al., Introduction to Liquid Crystals (Plenum, New York, 1974).

P. G. deGennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

S. T. Kowel, D. S. Cleverly, “A Liquid Crystal Adaptive Lens,” in Proceedings, NASA Conference on Optical Information Processing for Aerospace Applications, Hampton, Va. (1981).

D. S. Cleverly, “Creation of a Lens by Field-Controlled Variation of the Index of Refraction in a Liquid Crystal,” Ph.D. Dissertation, Syracuse U., Syracuse, N.Y. (1982).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

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Figures (18)

Fig. 1
Fig. 1

Director orientation for variation of the index of refraction. Voltages increase from top to bottom for positive dielectric anisotropy.

Fig. 2
Fig. 2

Focal length limitations using MBBA.

Fig. 3
Fig. 3

Geometry to calculate change in index of refraction with liquid crystal molecule orientation.

Fig. 4
Fig. 4

Index of refraction variation with changing director orientation in MBBA.

Fig. 5
Fig. 5

Director angle as a function of voltage for MBBA.

Fig. 6
Fig. 6

Single stage of liquid crystal lens.

Fig. 7
Fig. 7

Functions of the four stages in liquid crystal lens. U ¯ x and U ¯ y contain unit polarization vectors in the x and y directions, respectively.

Fig. 8
Fig. 8

Four-stage liquid crystal lens.

Fig. 9
Fig. 9

Liquid crystal electrode configuration for lens performance analysis. The center electrode was chosen to be twice as wide for convenience in calculation.

Fig. 10
Fig. 10

Assumed error in index of refraction.

Fig. 11
Fig. 11

Optical transfer function of liquid crystal lens for low frequency.

Fig. 12
Fig. 12

Optical transfer function of liquid crystal lens for high frequency.

Fig. 13
Fig. 13

Birefringence test results at 200 Hz (λ = 633 nm).

Fig. 14
Fig. 14

Effective change of refraction vs voltage for λ = 457.9 nm.

Fig. 15
Fig. 15

Seven-electrode array demonstrating focusing.

Fig. 16
Fig. 16

Effect of seven-element array on cross-polarized component.

Fig. 17
Fig. 17

Seven-element array at reduced voltage.

Fig. 18
Fig. 18

Fringing effect at sides of electrodes.

Tables (4)

Tables Icon

Table I Required Voltages for 1-meter Focal Length Lens with λ = 514.5 nm

Tables Icon

Table II Required Voltages for 5-meter Focal Length Lens with λ = 514.5 nm

Tables Icon

Table III Required Voltages for 1-meter Focal Length Lens with λ = 514.5 nm

Tables Icon

Table IV Required Voltages for 5-meter Focal Length Lens with λ = 514.5 nm

Equations (55)

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ϕ ( x , y ) = k Δ n ( x , y ) ,
n ( r ) = A + B r 2 ,
r 2 = x 2 + y 2 .
n ( r = 0 ) = n e .
n ( r = r o ) = n i ,
n o n i < n e .
n ( r ) = n e + ( n i - n e r o 2 ) r 2 .
t ( r ) = exp ( j k Δ n ) = exp [ ( j k Δ ) ( n e + n i - n e r o 2 ) r 2 ] .
t ( r ) = exp ( j k Δ n e ) exp [ ( - j k Δ ) ( n e - n i r o 2 ) r 2 ] .
t l ( r ) = exp ( j k Δ n ) exp ( - j k 2 f r 2 ) ,
1 f = 2 Δ ( n e - n i ) r o 2 .
0 < 1 f 2 Δ ( n e - n i ) r o 2 .
x θ 2 n e 2 + y θ 2 n o 2 = 1.
n θ 2 = x θ 2 + y θ 2 .
n θ 2 - y θ 2 n e 2 + y θ 2 n o 2 = 1.
y θ = ( 1 - n θ 2 / n e 2 ) / ( 1 / n o 2 - 1 / n e 2 ) .
θ = sin - 1 ( 1 / n θ 2 - 1 / n e 2 ) / ( 1 / n o 2 - 1 / n e 2 ) .
n θ = n e / 1 + sin 2 θ [ ( n e 2 / n o 2 - 1 ) ] .
t ( x , y ) = rect ( x / L ) rect ( y / L ) exp [ j k Δ n ( x ) ] ,
n ( x ) = n e - x 2 2 f .
U ( x , y ) = rect ( y L ) rect ( x L ) exp [ j k Δ ( n e - x 2 2 Δ f ) ] .
U ( x , y ) = rect ( y L ) rect ( x L ) exp - i k x 2 2 Δ f .
n ( y ) = n e - y 2 2 Δ f ,
U = t 1 ( x , y ) t 2 ( x , y ) ,
U = rect ( y L ) rect ( x L ) exp - j k ( x 2 + y 2 ) 2 f .
t ( x , y ) = P ( x , y ) exp [ j k W ( x , y ) ] ,
P ( ξ + λ d i f x 2 , η + λ d i f y 2 ) and P ( ξ - λ d i f x 2 , η - λ d i f x 2 ) .
H ( f x , f y ) = δ ( f x , f y ) exp { j k [ W ( ξ + λ d i f x 2 , η + λ d i f y 2 ) - W ( ξ - λ d i f x 2 , η - λ d i f x 2 ) ] } d ξ d η α ( 0 , 0 ) d ξ d η .
H ( f x , f y ) = α ( f x ) exp { j k [ W ( ξ + λ d i f x 2 ) - W ( ξ - λ d i f x 2 ) ] } d ξ α ( 0 ) d ξ × α ( f y ) exp { j k [ W ( η + λ d i f y 2 ) - W ( η - λ d i f y 2 ) ] } d η α ( 0 ) d η .
x = ± ( 4 m + 1 ) a 2 ;             m = 0 , 1 , 2 , , N + 1 2 ;
δ n ( x ) = ( n e - x 2 2 Δ f ) Ideal - ( n e - x o 2 2 Δ f ) Created = x o 2 - x 2 2 Δ f ,
x = ± ( 2 m + 1 ) a 2             m = 1 , 2 , .
δ n ( x ) = - a 2 [ N 2 - ( N - ½ ) 2 2 Δ f ] cos ( π x a ) .
δ n ( x ) = - [ ( N - ¼ ) a 2 2 Δ f ] cos ( π x a ) .
W ( x ) = Δ δ n ( x ) = - a 2 ( N - ¼ 2 f ) cos ( π x a ) .
W ( f x ) = α ( f x ) exp { j k [ W ( ξ + λ d i f x 2 ) - W ( ξ - λ d i f x 2 ) ] } d ξ = α ( f x ) exp ( - j k a 2 N - ¼ 2 f ) { cos [ π ( ξ + λ d i f x 2 ) ] a - cos [ π ( ξ - λ d i f x 2 ) ] a } d ξ .
H ( f x ) = [ 1 - A 2 2 + A 4 4 sinc ( l - λ d i f x a ) Λ ( f x / f c ) ,
A = k a 2 ( N - ¼ ) f sin π λ d i f x 2 a ,
Λ ( x ) = { 1 - x , x 1 0 , otherwise f c = l / ( λ d i ) .
H ( f x , f y ) = H ( f x ) H ( f y ) .
I = I p sin 2 2 ϕ sin 2 ( γ / 2 ) ,
γ = ( 2 π Δ δ n ) / λ ,
δ n = λ 2 d = 0.633 × 10 - 6 m 2 ( 12 × 10 - 6 m ) = 0.026 ( λ = 633 nm ) .
n e - n ( r ) = r 2 / ( 2 Δ f ) .
N ( f x ) = α ( f x ) exp { ( - j k a 2 N - ¼ 2 f ) × [ ( cos π ξ a cos π λ d i f x 2 a - sin π ξ a sin π λ d i f x 2 a ) - ( cos π ξ a cos π λ d i f x 2 a + sin π ξ a sin π λ d i f x 2 a ) ] } d ξ .
N ( f x ) = α ( f x ) exp j k a 2 ( N - ¼ f ) sin π ξ a sin π λ d i f x 2 a d ξ .
A = k a 2 ( N - ¼ ) f sin π λ d i f x 2 a .
N ( f x ) = α ( f x ) exp ( j A sin π ξ a ) d ξ .
α ( f x ) d ξ = - l + λ d i f x / 2 l - λ d i f x / 2 d ξ = l - λ d i f x .
α ( f x ) j A sin π ξ a d ξ = - l + λ d i f x / 2 l - λ d i f x j A sin π ξ a d ξ = - j A a π [ cos π a ( l - λ d i f x 2 ) - cos π a ( - l + λ d i f x 2 ) ] = 0.
- α ( f x ) A 2 2 sin 2 π ξ a d ξ = - - l + λ d i f x / 2 l - λ d i f x / 2 A 2 2 sin 2 π ξ a d ξ = - a π A 2 2 π / 2 a ( - l + λ d i f x ) π / 2 a ( l - λ d i f x ) sin 2 Z d Z = - A 2 a 2 π ( Z 2 - sin 2 Z 4 ) | π a ( l - λ d i f x 2 ) π a ( - l + λ d i f x 2 ) = - A 2 a 2 π { π 2 a ( l - λ d i f x ) - sin 4 [ π a ( l - λ d i f x ) ] + sin 4 [ π a ( l - λ d i f x ) ] } = - A 2 a 2 π { π 2 a ( l - λ d i f x ) - sin 2 [ π a ( l - λ d i f x ) ] } .
N ( f x ) = ( l - λ d i f x ) - A 2 a 2 π { π 2 a ( l - λ d i f x ) + ½ sin [ π a ( l - λ d i f x ) ] } .
H ( f x ) = N ( f x ) α ( 0 ) d ξ = N ( f x ) l ,
H ( f x ) = ( 1 - A 2 4 ) ( l - λ d i f x l ) + A 2 a 4 π l sin π a ( l - λ d i f x ) .
H ( f x ) = [ 1 - A 2 4 + A 2 4 sinc ( l - λ d i f x a ) ] Λ ( f x f c ) .

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