Abstract

The problem of designing waveguide geodesic lenses for integrated optical processing is considered. Geodesic lenses are, at present, the best solution whenever crystals of high refractive index must be used as substrates. A good geodesic lens has to be constituted by a surface of rotation, coupled without discontinuities with the planar optical circuit. The present paper derives, by an analytical procedure, the general expression for the profile of aspherical geodesic lenses able to form perfect geometrical images of the points of two given concentric circles on each other. In particular, the equations that characterize a family of lenses having two conjugate foci external to the lens depression are given. The case of one external and one internal focus is only outlined because it is less important in practice. The advantage of this analytical method is represented by the clear theoretical approach that allows a large flexibility in the lens design without long expensive computations.

© 1979 Optical Society of America

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References

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  1. R. Shubert and J. H. Harris, “Optical guided-wave focusing and diffraction,” J. Opt. Soc. Am. 61, 154–161 (1971).
    [Crossref]
  2. P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
    [Crossref]
  3. G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Thin film geodesic lens,” Appl. Opt. 11, 1442–1443 (1972).
    [Crossref] [PubMed]
  4. F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
    [Crossref]
  5. G. Toraldo di Francia, “Un problema sulle geodetiche delle superfici di rotazione che si presenta nella tecnica delle microonde,” Atti Fondaz. Ronchi 12, 151–172 (1957).
  6. G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Opt. 12, 1477–1481 (1973).
    [Crossref] [PubMed]
  7. G. C. Righini, V. Russo, and S. Sottini, “A family of perfect aspherical geodesic lenses for integrated optical circuits,” J. Quantum Electron. QE-15, 1–4 (1979).
    [Crossref]
  8. B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
    [Crossref]
  9. W. H. Southwell, “Geodesic Optical Waveguide Lens Analysis,” J. Opt. Soc. Am. 67, 1293–1299 (1977).
    [Crossref]
  10. K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Doubly Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
    [Crossref]
  11. M. Bocher, An Introduction to the Study of Integral Equations (Cambridge University, Cambridge, 1909), p. 9.

1979 (1)

G. C. Righini, V. Russo, and S. Sottini, “A family of perfect aspherical geodesic lenses for integrated optical circuits,” J. Quantum Electron. QE-15, 1–4 (1979).
[Crossref]

1978 (2)

B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
[Crossref]

P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
[Crossref]

1977 (1)

1974 (1)

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[Crossref]

1973 (1)

1972 (1)

1971 (1)

1957 (1)

G. Toraldo di Francia, “Un problema sulle geodetiche delle superfici di rotazione che si presenta nella tecnica delle microonde,” Atti Fondaz. Ronchi 12, 151–172 (1957).

1954 (1)

K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Doubly Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
[Crossref]

Ashley, P. R.

P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
[Crossref]

Bocher, M.

M. Bocher, An Introduction to the Study of Integral Equations (Cambridge University, Cambridge, 1909), p. 9.

Chang, W. S. C.

P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
[Crossref]

Chen, B.

B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
[Crossref]

Harris, J. H.

Kunz, K. S.

K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Doubly Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
[Crossref]

Marom, E.

B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
[Crossref]

Morrison, R. J.

B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
[Crossref]

Righini, G. C.

Russo, V.

Shubert, R.

Sottini, S.

Southwell, W. H.

Toraldo di Francia, G.

Zernike, F.

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[Crossref]

Appl. Opt. (2)

Appl. Phys Lett. (1)

B. Chen, E. Marom, and R. J. Morrison, “Diffraction-limited geodesic lens for integrated optics circuits,” Appl. Phys Lett. 33, 511–513 (1978).
[Crossref]

Appl. Phys. Lett. (1)

P. R. Ashley and W. S. C. Chang, “Fresnel lens in a thin-film waveguide,” Appl. Phys. Lett. 33, 490–492 (1978).
[Crossref]

Atti Fondaz. Ronchi (1)

G. Toraldo di Francia, “Un problema sulle geodetiche delle superfici di rotazione che si presenta nella tecnica delle microonde,” Atti Fondaz. Ronchi 12, 151–172 (1957).

J. Appl. Phys. (1)

K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Doubly Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
[Crossref]

J. Opt. Soc. Am. (2)

J. Quantum Electron. (1)

G. C. Righini, V. Russo, and S. Sottini, “A family of perfect aspherical geodesic lenses for integrated optical circuits,” J. Quantum Electron. QE-15, 1–4 (1979).
[Crossref]

Opt. Commun. (1)

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[Crossref]

Other (1)

M. Bocher, An Introduction to the Study of Integral Equations (Cambridge University, Cambridge, 1909), p. 9.

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Figures (6)

FIG. 1
FIG. 1

Waveguide geodesic lens realized as a depression in a substrate covered by a homogeneous guiding film of constant thickness.

FIG. 2
FIG. 2

Rotation surface S, with meridional curve l(r), limited by the parallel R of radius ρ. The path of a geodesic crossing R under an angle ψ is shown.

FIG. 3
FIG. 3

Top view of a geodesic lens with two external conjugate foci P and P′. A ray is drawn crossing the parallels A, B, C, and D under the angles ψ0, ψ1. ψ2, and ψ3 respectively. The parallel C separates the depression surface S from the plane surface S′. The lens profile is sketched on the bottom of the figure.

FIG. 4
FIG. 4

Profile of a generalized geodesic lens with focal length f = 9 mm and F/number = 1.5. It perfectly images two circles of radius 11.25 and 45 mm respectively on each other.

FIG. 5
FIG. 5

Three profiles of generalized geodesic lenses perfectly focusing a collimated beam at the same focal distance f = 9 mm. The lenses have the same depression diameter too, while the rounding widths c-d, as well as the F/numbers = f/2d, are different.

FIG. 6
FIG. 6

Top view of a geodesic lens with one external (P) and one internal (P′)focus. The lens depression surface S is limited by the parallel B. The lens profile is sketched on the bottom of the figure.

Equations (39)

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l ( r ) = ρ π ρ r r dr d d r 0 arccos ( r / ρ ) F ( ψ ) sin ψ ( ρ 2 cos 2 ψ r 2 ) 1 / 2 d ψ .
cos ψ 0 = b a cos ψ 1 = c a cos ψ 2 = k a ,
ϕ 0 = ψ 0 arccos ( a b cos ψ 0 ) .
F A ( ψ 0 ) = π + ϕ 0 = π + ψ 0 arccos ( a b cos ψ 0 ) .
F C ( ψ 2 ) = F A ( ψ 0 ) 2 ϕ 0 2 ϕ 1
ϕ 1 = ψ 1 arccos ( b c cos ψ 1 ) .
F C ( ψ 2 ) = π + 2 ψ 2 arccos ( c b cos ψ 2 ) arccos ( c a cos ψ 2 ) .
d ϕ = k l ( r ) d r r ( r 2 k 2 ) 1 / 2 ,
F D ( ψ 3 ) = F C ( ψ 2 ) 2 ϕ 2 ,
ψ 3 = arccos ( c d cos ψ 2 ) .
l ( r ) = [ 1 + c 2 r 2 h 2 ] , ( c r d )
F D ( ψ 3 ) = π ( 1 c 2 h 2 ) + 2 ( 1 + c 2 h 2 ) ψ 3 2 d 2 h 2 cos ψ 3 sin ψ 3 + 2 d 2 h 2 cos ψ 3 ( c 2 d 2 cos 2 ψ 3 ) 1 / 2 + 2 c 2 h 2 arcsin ( d c cos ψ 3 ) arccos ( d b cos ψ 3 ) arccos ( d a cos ψ 3 ) .
h = c ( π 2 γ 12 sin 2 γ 12 γ 13 + γ 14 ) 1 / 2 .
l ( r ) = ρ π 0 arccos ( r / ρ ) F ( ψ ) cos ψ ( ρ 2 cos 2 ψ r 2 ) 1 / 2 d ψ .
l ( r ) = 1 2 ( 1 + c 2 r 2 h 2 ) + 2 π h 2 ( d 2 r 2 ) 1 / 2 ( c 2 d 2 ) 1 / 2 1 2 π arcsin ( r 2 + b 2 2 d 2 b 2 r 2 ) 1 2 π arcsin ( r 2 + a 2 2 d 2 a 2 r 2 ) + r 2 c 2 π h 2 arcsin ( r 2 + c 2 2 d 2 c 2 r 2 ) , ( 0 r d ) .
z ( r ) = 0 r [ l ( r ) 2 1 ] 1 / 2 d r
z ( r ) = 0 r { [ 1 2 ( 1 + c 2 r 2 h 2 ) + 2 π h 2 ( d 2 r 2 ) 1 / 2 ( c 2 d 2 ) 1 / 2 1 2 π arcsin ( r 2 + b 2 2 d 2 b 2 r 2 ) 1 2 π arcsin ( r 2 + a 2 2 d 2 a 2 r 2 ) + r 2 c 2 h 2 arcsin ( r 2 + c 2 2 d 2 c 2 r 2 ) ] 2 1 } 1 / 2 d r , 0 r d
z ( r ) = z ( d ) + d r [ ( c 2 r 2 h 2 ) 2 + 2 c 2 r 2 h 2 ] 1 / 2 d r , d < r c
z ( r ) = z ( c ) , c < r a .
f = a b / ( a + b ) .
1 / p + 1 / q = 1 / f .
F A ( ψ 0 ) = π + ϕ 1 + ϕ 0 ,
ϕ 0 = ψ 0 π 2 + arcsin ( a b cos ψ 0 ) .
F B ( ψ 1 ) = π + ϕ 1 ϕ 0
F C ( ψ 2 ) = π ϕ 0 ϕ 1 .
r sin α = k ,
ϕ = 2 k ρ r l ( r ) d r r ( r 2 k 2 ) 1 / 2 .
ϕ = F ( ψ ) .
x = ρ 2 k 2 = ρ 2 sin 2 ψ , 0 x ρ 2 b 2
ξ = ρ 2 r 2 , 0 ξ ρ 2 b 2
Φ ( ξ ) = l ( ρ 2 ξ ) 1 / 2 ρ 2 ξ ,
f ( x ) = F [ arcsin ( x / ρ ) ] ( ρ 2 x ) 1 / 2 ,
f ( x ) = 0 x Φ ( ξ ) d ξ ( x ξ ) 1 / 2
l ( r ) = r 2 Φ ( ρ 2 r 2 )
I ( ξ ) = 0 ξ f ( x ) d x ( ξ x ) 1 / 2
Φ ( ξ ) = 1 π d d ξ 0 ξ f ( x ) d x ( ξ x ) 1 / 2 .
I ( ξ ) = 0 ξ F ( arcsin x / ρ ) ( ρ 2 x ) 1 / 2 ( ξ x ) 1 / 2 d x = 2 ρ 0 arccos ( r / ρ ) F ( ψ ) sin ψ ( ρ 2 cos 2 ψ r 2 ) 1 / 2 d ψ .
l ( r ) = a π r d d r 0 arccos ( r / ρ ) F ( ψ ) sin ψ ( ρ 2 cos 2 ψ r 2 ) 1 / 2 d ψ
l ( r ) = ρ π ρ r r dr d d r 0 arccos ( r / ρ ) F ( ψ ) sin ψ ( ρ 2 cos 2 ψ r 2 ) 1 / 2 d ψ