Abstract

The effects of rounding of the edge on the focal properties of spherical-depression geodesic lenses are calculated. The focal length for paraxial rays is increased very slightly for slight rounding. For greater rounding, a decrease in focal length results from the increased optical path. Edge-rounding has very little effect on lens aberrations.

© 1976 Optical Society of America

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References

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  1. E. Spiller, J. S. Harper, Appl. Opt. 13, 2105 (1974).
  2. T. van Duzer, Proc. IEEE 58, 1230 (1970).
  3. D. W. Vahey, C. M. Verber, V. E. Wood, in Digest of Technical Papers, 1976 OSA/IEEE Integrated Optics Conference, p. MD2-1.
  4. C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).
  5. D. W. Vahey, J. R. Link, Battelle Columbus Labs., unpublished work.
  6. K. S. Kunz, J. Appl. Phys. 25, 642 (1954).
  7. I. A. Viktorov, Sov. Phys. Acoust. 7, 13 (1961).
  8. G. C. Righini, V. Russo, S. Sottini, G. Toraldo di Francia, Appl. Opt. 12, 1477 (1973); A. Scheggi, G. Toraldo di Francia, Alta Freq. 29, 438 (1960).
  9. P. Franklin, Methods of Advanced Calculus (McGraw-Hill, New York, 1944), p. 454, for example.

1976 (1)

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).

1974 (1)

1973 (1)

1970 (1)

T. van Duzer, Proc. IEEE 58, 1230 (1970).

1961 (1)

I. A. Viktorov, Sov. Phys. Acoust. 7, 13 (1961).

1954 (1)

K. S. Kunz, J. Appl. Phys. 25, 642 (1954).

Franklin, P.

P. Franklin, Methods of Advanced Calculus (McGraw-Hill, New York, 1944), p. 454, for example.

Harper, J. S.

Kunz, K. S.

K. S. Kunz, J. Appl. Phys. 25, 642 (1954).

Link, J. R.

D. W. Vahey, J. R. Link, Battelle Columbus Labs., unpublished work.

Righini, G. C.

Russo, V.

Sottini, S.

Spiller, E.

Toraldo di Francia, G.

Vahey, D. W.

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).

D. W. Vahey, C. M. Verber, V. E. Wood, in Digest of Technical Papers, 1976 OSA/IEEE Integrated Optics Conference, p. MD2-1.

D. W. Vahey, J. R. Link, Battelle Columbus Labs., unpublished work.

van Duzer, T.

T. van Duzer, Proc. IEEE 58, 1230 (1970).

Verber, C. M.

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).

D. W. Vahey, C. M. Verber, V. E. Wood, in Digest of Technical Papers, 1976 OSA/IEEE Integrated Optics Conference, p. MD2-1.

Viktorov, I. A.

I. A. Viktorov, Sov. Phys. Acoust. 7, 13 (1961).

Wood, V. E.

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).

D. W. Vahey, C. M. Verber, V. E. Wood, in Digest of Technical Papers, 1976 OSA/IEEE Integrated Optics Conference, p. MD2-1.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28, 514 (1976).

J. Appl. Phys. (1)

K. S. Kunz, J. Appl. Phys. 25, 642 (1954).

Proc. IEEE (1)

T. van Duzer, Proc. IEEE 58, 1230 (1970).

Sov. Phys. Acoust. (1)

I. A. Viktorov, Sov. Phys. Acoust. 7, 13 (1961).

Other (3)

P. Franklin, Methods of Advanced Calculus (McGraw-Hill, New York, 1944), p. 454, for example.

D. W. Vahey, C. M. Verber, V. E. Wood, in Digest of Technical Papers, 1976 OSA/IEEE Integrated Optics Conference, p. MD2-1.

D. W. Vahey, J. R. Link, Battelle Columbus Labs., unpublished work.

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Figures (5)

Fig. 1
Fig. 1

Cross section (heavy line) of spherical-depression lens with rounded rim.

Fig. 2
Fig. 2

Coordinates on toroidal rim region.

Fig. 3
Fig. 3

Top view of spherical-depression lens with rounded lip, showing typical light path and nomenclature of limiting angles in several regions traversed. Displacement of input beam from center line = y; focal length for beam with this displacement = l(y). The angle Λs is measured on the spherical surface, while Λi and Λo are measured on the plane.

Fig. 4
Fig. 4

Normalized focal length L of geodesic lens of apex angle 39.4° as function of normalized beam displacement Y for various degrees of edge-rounding A. (a) Change in focal length, ΔL = L(A = 0.01) − L(A = 0), for lens rounded with rim radius 1% of original lens radius in waveguide plane. (b) Focal length for lens with abrupt edge and for 10% rounding.

Fig. 5
Fig. 5

Focal length for paraxial rays as a function of degree of rim rounding for a geodesic lens with apex angle 39.4°. The focal length is increased slightly for values of A less than 0.03.

Equations (21)

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B = [ 1 + 2 A tan ( θ o / 2 ) ] 1 / 2 ,
cos α = ( A + cot θ o ) / ( A + csc θ o ) .
ϕ = C 2 - a C 1 d θ ( b - a sin θ ) [ ( b - a sin θ ) 2 - C 1 2 ] 1 / 2 ,
ϕ = C 2 - a C 1 b ( b 2 - C 1 2 ) 1 / 2 [ θ - a b 2 b 2 - C 1 2 b 2 - C 1 2 cos θ ] +
tan Λ = ( b a - sin θ ) d ϕ d θ .
ϕ = ϕ 1 + ( a / b ) tan ϕ 1 [ θ + ( a / b ) ( 2 + tan 2 ϕ 1 ) ( 1 - cos θ ) ] .
ϕ 2 = ϕ 1 + ( a / b ) tan ϕ 1 [ α + ( a / b ) ( 2 + tan 2 ϕ 1 ) ( 1 - cos α ) ] ,
tan Λ s = ( b / a - sin α ) × [ ( a / b ) tan ϕ 1 ] [ 1 + ( a / b ) ( 2 + tan 2 ϕ 1 ) sin α ] .
cot ( Δ / 2 ) = tan Λ s cos α
Δ + ϕ 2 + ϕ 3 = π .
ϕ = ϕ 4 + a / b tan Λ o [ θ + ( a / b ) ( 2 + tan 2 Λ o ) ( 1 - cos θ ) ] ;
Λ o = ϕ 1 ;
ϕ 4 = π - Δ - 2 ϕ 2 + ϕ 1 .
l = b ( sin ϕ 4 cot Ψ + cos ϕ 4 ) ,
l = - b sin ϕ 1 csc ( Δ + 2 ϕ 2 ) ,
l = - y csc ( Δ + 2 ϕ 2 ) ,
l ( y 0 ) = ½ b [ 1 - cos α + ( a / b ) ( α - sin α cos α ) + 2 ( a 2 / b 2 ) ( 1 - cos 3 α ) ] - 1 .
r 1 = x 1 i ^ + y 1 j ^ + z 1 k ^ , x 1 = - ( b - a sin θ ) cos Φ ( θ ) , y 1 = ( b - a sin θ ) sin Φ ( θ ) , z 1 = a cos θ .
r 2 = x 2 i ^ + y 2 j ^ + z 1 k ^ , x 2 = - ( b - a sin θ ) cos Φ o , y 2 = ( b - a sin θ ) sin Φ o .
Λ = cos - 1 [ t 1 · t 2 / ( t 1 t 2 ) ] .
cos Λ = [ 1 + ( b / a - sin θ ) 2 ( d Φ / d θ ) 2 ] - 1 / 2 ,

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