Abstract

Gradient-index ray-tracing techniques are used to evaluate performance of inhomogeneous optical waveguide lenses. Using the thin-film waveguide parameters including the lens thickness profile, the phase error and diffraction pattern are derived. The procedure is applied to a classical Luneburg index profile for which exact results are known. The results indicate that better than diffraction-limited accuracy can be achieved with reasonable computer running times. A second example demonstrates the procedure on lens profiles approximating generalized Luneburg lenses (image surface is outside the lens).

© 1977 Optical Society of America

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References

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  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 Integrated Optics,  QE13, 275 (1977).
    [Crossref]
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966), p. 187.
  3. P. K. Tien, “Light waves in Thin Films and Integrated Optics,” Appl. Opt. 10, 2395 (1971).
    [Crossref] [PubMed]
  4. L. Montagnino, “Ray Tracing in Inhomogeneous Media,” J. Opt. Soc. Am. 58, 1667 (1968).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 137.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.
  7. S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).The exact overlayer profiles yielding Morgan’s solutions for the generalized Luneburg lens will be published in a forthcoming paper by Southwell.
    [Crossref]

1977 (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 Integrated Optics,  QE13, 275 (1977).
[Crossref]

1971 (1)

1968 (1)

1958 (1)

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).The exact overlayer profiles yielding Morgan’s solutions for the generalized Luneburg lens will be published in a forthcoming paper by Southwell.
[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 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 Integrated Optics,  QE13, 275 (1977).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 137.

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 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 Integrated Optics,  QE13, 275 (1977).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.

Luneburg, R. K.

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

Montagnino, L.

Morgan, S. P.

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).The exact overlayer profiles yielding Morgan’s solutions for the generalized Luneburg lens will be published in a forthcoming paper by Southwell.
[Crossref]

Tien, P. K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 137.

Appl. Opt. (1)

J. Appl. Phys. (1)

S. P. Morgan, “General Solution of the Luneburg Lens Problem,” J. Appl. Phys. 29, 1358 (1958).The exact overlayer profiles yielding Morgan’s solutions for the generalized Luneburg lens will be published in a forthcoming paper by Southwell.
[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 Integrated Optics,  QE13, 275 (1977).
[Crossref]

Other (3)

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

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 137.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.

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Figures (10)

FIG. 1
FIG. 1

Cross section of multilayer planar dielectric waveguide. Optical propagation is from left to right.

FIG. 2
FIG. 2

Ray path geometry showing ray position vector r and the ray vector s. The scalar s is the distance along the path and the vector s is a unit vector tangent to the path.

FIG. 3
FIG. 3

Plot of the ray paths traced through the classical Luneburg lens described by Eq. (31).

FIG. 4
FIG. 4

Plot of the ray-trace error as a function of the step size Δs for a 2 cm radius classical Luneburg lens shown in Fig. 3. It is shown that extrapolation techniques can yield required accuracy without using extremely small step size.

FIG. 5
FIG. 5

Wave-front phase error in radians in the exit pupil of the Luneburg lens of Fig. 3 as derived from the ray trace technique using λ = 0.633 μm. This residual phase error is due to the numerical ray tracing, but may be made as small as desired at the expense of more computer running time. However, the error shown is within the diffraction limit.

FIG. 6
FIG. 6

Diffraction pattern showing the log of the intensity in the image surface for the Luneburg lens of Figs. 3 and 5. This pattern agrees with the sine function squared.

FIG. 7
FIG. 7

Phase error of the Luneburg lens of Fig. 3 defocused. by 0. 5 μm.

FIG. 8
FIG. 8

Diffraction pattern resulting from defocused Luneburg lens shown in Fig. 7. A 64-point fast Fourier transform algorithm was used to generate this pattern.

FIG. 9
FIG. 9

Waveguide layer profile represented by cos0.18 dependence (upper curve) and cos0.25 dependence (lower curve). Referring to Fig. 1, the other waveguide parameters used to trace this lens are n1 = 1, n2 = 2.1, n3 = 1.565, n4 = 1.47, d = 0.75 μm, and λ = 0.633 μm.

FIG. 10
FIG. 10

Ray plots of waveguide lens described in Fig. 9, (a) withcos0.18 dependence profile (b) withcos0.25 dependence profile (c) withcos0.211 dependence profile.

Tables (1)

Tables Icon

TABLE I Exit height y for various step sizes h.

Equations (36)

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tan ( h k t + ϕ ) = h / p ,
tan ϕ = ( h / l ) tan γ ,
tan ( l k d + γ ) = l / q ,
p = ( n e 2 n 1 2 ) 1 / 2 ,
h = ( n 2 2 n e 2 ) 1 / 2 ,
q = ( n e 2 n 4 2 ) 1 / 2 ,
l = { ( n 3 2 n e 2 ) 1 / 2 n e < n 3 , ( n e 2 n 3 2 ) 1 / 2 n e n 3 ,
F ( n e ) = [ ( 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 ) ) 1 + k p , n e < n 3 F ( n e ) = ( [ ( 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 ] ) 1 + k p , n e n 3 .
n e i + 1 = n e i F ( n e i ) d F / d n ,
r = x î + y ĵ , S = lim Δ s 0 Δ r Δ s = α î + β ĵ .
r ( s 0 + Δ s ) = r ( s 0 ) + d r d s | s = s 0 Δ s + 1 2 ! d 2 r d s 2 | s = s 0 ( Δ s ) 2 + 1 3 ! d 2 r d s 3 | s = s 0 ( Δ s ) 3 + .
s ( s 0 + Δ s ) = s ( s 0 ) + K ( Δ s ) + 1 2 d K d s ( Δ s ) 2 + .
d d s ( n d r d s ) = n ,
n = n x î + n y ĵ .
K = ( 1 / n ) [ Δ n ( n s ) s ] .
K = f ( r ) [ r ( r s ) s ] ,
f ( r ) = 1 n r d n d r .
d K d s = 1 r d f d r ( s r ) [ r ( r s ) s ] f [ ( r K ) s + ( r s ) K ] ,
d g d s = s g , g ( r ) = 1 r d g d r r .
d n d r = 1 2 Δ t [ n ( t + Δ t ) n ( t Δ t ) ] ,
d 2 n d r 2 = 1 ( Δ t ) 2 [ n ( t + Δ t ) 2 n ( t ) + n ( t Δ t ) ] ,
F = r 0 2 r 2 ,
( Δ s ) e = ( Δ s ) 0 F / d f d ( Δ s ) | Δ s = Δ s 0 .
F ( Δ s ) = 2 r d r d ( Δ s ) = 2 r s .
y ( h ) = y ( 0 ) + d y d h h + 1 2 d 2 y d h 2 h 2 + .
y 1 = a 1 + a 2 h + a 3 h 2 , y 2 = a 1 + 1 2 a 2 h + 1 4 a 3 h 2 , y 3 = a 1 + 1 4 a 2 h + 1 16 a 3 h 2 .
y = H a ,
a = H 1 y ,
H = ( 1 h h 2 1 1 2 h 1 4 h 2 1 1 4 h 1 16 h 2 ) .
H 1 = ( 1 3 2 8 3 80 400 320 4266.7 12 800 8533.3 ) .
y = 1 3 y 1 2 y 2 + 8 3 y 3 .
n = n 0 [ 2 ( r / r 0 ) 2 ] 1 / 2 ,
y = T β .
T = c 1 β 2 + c 2 β 4 + c 3 β 6 + .
T β = ( 2 c 1 β + 4 c 2 β 3 + 6 c 3 β 5 + ) .
P = exp [ i 2 π T ( y / y 0 ) / λ ] ,