Abstract

An analytical approximation is derived which provides a transcendental relationship for the spherically symmetric inhomogeneous index distribution which perfectly focuses infinite objects. Using this expression, the index profiles (called generalized Luneburg lenses) were calculated for lenses having f numbers down to f/1. It is shown through a ray-tracing example that these profiles have sufficient accuracy to provide diffraction-limited performance in the optical wavelength region. It is also shown how these lenses may be utilized in a two-dimensional integrated optics waveguide. Utilizing the focusing property achieved by variations in planar waveguide thickness, circularly symmetric waveguide thickness profiles are derived which have the perfect focusing properties of the generalized Luneburg lenses.

© 1977 Optical Society of America

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References

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  1. S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).
    [CrossRef]
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966), p. 187.
  3. M. C. Hamilton, D. A. Wille, and W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” IEEE 1976 Ultrasonic Symposium Proceedings, Annapolis, Md.
  4. D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
    [CrossRef]
  5. F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
    [CrossRef]
  6. W. H. Southwell, “Inhomogeneous Optical Waveguide Lens Analysis,” J. Opt. Soc. Am. 67, 1004–1009 (1977).
    [CrossRef]
  7. V. J. Krylov, Approximate Calculation of Integrals (Macmillian, New York, 1962), pp. 100–111 and337–340.

1977 (2)

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

W. H. Southwell, “Inhomogeneous Optical Waveguide Lens Analysis,” J. Opt. Soc. Am. 67, 1004–1009 (1977).
[CrossRef]

1974 (1)

F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

1958 (1)

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

Anderson, D. B.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

August, R. R.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

Boyd, J. T.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

Davis, R. L.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

Hamilton, M. C.

M. C. Hamilton, D. A. Wille, and W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” IEEE 1976 Ultrasonic Symposium Proceedings, Annapolis, Md.

Krylov, V. J.

V. J. Krylov, Approximate Calculation of Integrals (Macmillian, New York, 1962), pp. 100–111 and337–340.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966), p. 187.

Miceli, W. J.

M. C. Hamilton, D. A. Wille, and W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” IEEE 1976 Ultrasonic Symposium Proceedings, Annapolis, Md.

Morgan, S. P.

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

Southwell, W. H.

Wille, D. A.

M. C. Hamilton, D. A. Wille, and W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” IEEE 1976 Ultrasonic Symposium Proceedings, Annapolis, Md.

Zernike, F.

F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

J. Appl. Phys. (1)

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quantum Electron. (1)

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical Waveguide Lens Technologies,” J. Quantum Electron., special issue on Integrated Optics,  QE13,275 (1977).
[CrossRef]

Opt. Commun. (1)

F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

Other (3)

V. J. Krylov, Approximate Calculation of Integrals (Macmillian, New York, 1962), pp. 100–111 and337–340.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966), p. 187.

M. C. Hamilton, D. A. Wille, and W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” IEEE 1976 Ultrasonic Symposium Proceedings, Annapolis, Md.

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Figures (6)

FIG. 1
FIG. 1

Cross section of multilayer planar dielectric waveguide. The optical waveguide shown at the left with thickness d has a bulk index of n3. However, the waveguide mode index or effective index differs from n3 and is determined from electromagnetic mode theory. This mode index, which determines the optical propagation in the waveguide, will vary as the ray encounters the overlay region shown. By tailoring the profile of the deposited overlay, a two-dimensional lens may be achieved.

FIG. 2
FIG. 2

Index profiles for generalized Luneburg lenses. From top to bottom, the lenses have full aperture f numbers of f/1, f/1.5, f/2.5, and f/4.5. These correspond to s = 2, 3, 5, and 9.

FIG. 3
FIG. 3

Waveguide overlay thickness profiles for the generalized Luneburg lens profiles shown in Fig. 2 for the waveguide parameters listed in Table V.

FIG. 4
FIG. 4

Ray trace of the s = 2 generalized Luneburg lens using the index profile derived from Eq. (7).

FIG. 5
FIG. 5

Phase error profile in the pupil plane of the s = 2 generalized Luneburg lens using the approximation Eq. (7) to derive the index profile.

FIG. 6
FIG. 6

Intensity diffraction pattern using logarithmic scale in the image surface for the Luneburg lens of Figs. 4 and 5. (Note that this coincides with a diffraction-limited distribution.)

Tables (5)

Tables Icon

TABLE I The function ω(ρ, s) using numerical integration of Eq. (5). s is the focal length (from center of lens).

Tables Icon

TABLE II The function ω(0, s) using Eq. (10) nmax = exp[ω(0, s)].

Tables Icon

TABLE III Results of fitting Eq. (7) to the data in Table I. The values listed for sum are the sum of the squares of the errors for the 29 points. p5 was determined by Eq. (11).

Tables Icon

TABLE IV Normalized refractive index profiles for generalized Luneburg lenses.

Tables Icon

TABLE V Overlay thickness in microns for optical waveguide Luneburg lenses. The waveguide parameters are n1 = 1, n2 = 2.1, n3 = 1.565, n4 = 1.47, d = 1.0665 μ, and λ = 0.9 μ.

Equations (19)

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n = exp [ ω ( ρ , s ) ] ,
ω ( ρ , s ) = 1 π ρ 1 arcsin ( x / s ) ( x 2 ρ 2 ) 1 / 2 d x .
n = ( 2 r 2 ) 1 / 2 .
ω ( ρ , s ) = 1 π 0 1 ρ arcsin [ ( y + ρ ) / s ] ( y + 2 ρ ) 1 / 2 y 1 / 2 d y .
ω ( ρ , s ) = 1 π { 2 arcsin ( 1 s ) ( 1 ρ ) 1 / 2 ( 1 + ρ ) 1 / 2 0 1 ρ [ ( s 2 ( y + ρ ) 2 ) 1 / 2 arcsin [ ( y + ρ ) / s ] ( y + 2 ρ ) ] y 1 / 2 d y ( y + 2 ρ ) 1 / 2 } .
ω ( ρ , s ) ρ 1 = 2 π arcsin ( 1 s ) ( 1 ρ ) 1 / 2 .
ω ( ρ , s ) = p 1 ( 1 ρ ) 1 / 2 + p 2 ( 1 ρ ) 3 / 2 + p 3 ( 1 ρ ) 5 / 2 + p 4 ( 1 ρ ) 7 / 2 + p 5 ( 1 ρ ) 9 / 2 .
p 1 = 2 π arcsin ( 1 s ) .
ω ( 0 , s ) = 1 π 0 1 / s arcsin x x d x ,
ω ( 0 , s ) = 1 π [ s 1 + 1 2 × 3 × 3 s 3 + 1 × 3 s 5 2 × 4 × 5 × 5 + 1 × 3 × 5 s 7 2 × 4 × 6 × 7 × 7 + ] .
p s = ω ( 0 , s ) p 1 p 2 p 3 p 4 .
F ( n ) = exp [ ω ( ρ , s ) ] n .
d n d r = n 2 G 2 ( 1 r n ) 1 / 2 + n r G ,
d 2 n d r 2 = 2 n 3 [ G 2 ( 1 r n ) 1 / 2 + H ] [ 2 ( 1 r n ) 1 / 2 + n r G ] 3 + 2 n ( d n d r ) 2 ,
G = p 1 + 3 p 2 ( 1 r n ) + 5 p 3 ( 1 r n ) 2 + 7 p 4 ( 1 r n ) 3 + 9 p 5 ( 1 r n ) 4 ,
H = p 1 + 3 p 2 ( 1 r n ) + 15 p 3 ( 1 r n ) 2 + 35 p 4 ( 1 r n ) 3 + 63 p 5 ( 1 r n ) 4 .
h p = [ 1 ( l / q ) tan ( l k d ) ] tan ( h k t ) + ( h / l ) [ l / q + tan ( l k d ) ] [ 1 ( l / q ) tan ( l k d ) ] ( h / l ) [ l / q + tan ( l k d ) ] tan ( h k t ) , n < n 3 = [ ( q + l ) e 2 l k d + q l ] tan ( h k t ) + ( h / l ) [ ( q + l ) e 2 l k d q + l ] [ ( q + l ) e 2 l k d + q l ] ( h / l ) tan ( h k t ) [ ( q + l ) e 2 l k d q + l ] , n n 3
p = ( n 2 n 1 2 ) 1 / 2 , h = ( n 2 2 n 2 ) 1 / 2 , q = ( n 2 n 4 2 ) 1 / 2 , l = ( n 3 2 n 2 ) 1 / 2 , n < n 3 = ( n 2 n 3 2 ) 1 / 2 , n n 3 .
t = tan 1 [ h [ 1 ( l / q ) tan ( l k d ) ] + ( p h / l ) [ l / q tan ( l k d ) ( h 2 / l ) [ l / q tan ( l k d ) ] p [ 1 ( l / q ) tan ( l k d ) ] ] , n < n 3 = tan 1 [ h [ ( q + l ) e 2 l k d + q l ] + ( p h / l ) [ ( q + l ) e 2 l k d q + l ] ( h 2 / l ) [ ( q + l ) e 2 l k d q + l ] p [ ( q + l ) e 2 l k d + q l ] ] , n n 3 .