Abstract

A mode index lens and a lens formed by a spherical depression in the substrate of the waveguide have opposite aberrations and can be combined in an element with considerably reduced aberrations. Only small differences in the mode index are required for the optimum combination.

© 1974 Optical Society of America

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References

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  1. R. Shubert, Y. H. Harris, J. Opt. Soc. Am. 61, 154 (1971).
    [CrossRef]
  2. T. van Duzer, Proc. IEEE 58, 1230 (1970).
    [CrossRef]
  3. G. C. Righini, V. Russo, S. Sattini, G. Toraldo di Francia, Appl. Opt. 11, 1442 (1972).
    [CrossRef] [PubMed]
  4. J. E. Goell, R. D. Standley, Bell Syst. Tech. J. 48, 3445 (1969).
  5. M. J. Rand, R. D. Standley, Appl. Opt. 11, 2482 (1972).
    [CrossRef] [PubMed]
  6. T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, Appl. Opt. 12, 1240 (1973).
    [CrossRef] [PubMed]
  7. H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
    [CrossRef]
  8. E. G. Lean, “Acousto-optic Interaction in Guided Wave Structures,” in Integrated Optics, M. Barnoski, Ed. (Plenum Press, New York—to be published).
  9. L. P. Boivin, Appl. Opt. 13, 391 (1974).
    [CrossRef] [PubMed]

1974

1973

1972

1971

1970

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

1969

J. E. Goell, R. D. Standley, Bell Syst. Tech. J. 48, 3445 (1969).

Andrews, R. A.

Boivin, L. P.

Chandross, E. A.

H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
[CrossRef]

Giallorenzi, T. G.

Ginther, R.

Goell, J. E.

J. E. Goell, R. D. Standley, Bell Syst. Tech. J. 48, 3445 (1969).

Harris, Y. H.

Kirk, R.

Lean, E. G.

E. G. Lean, “Acousto-optic Interaction in Guided Wave Structures,” in Integrated Optics, M. Barnoski, Ed. (Plenum Press, New York—to be published).

Rand, M. J.

Righini, G. C.

Russo, V.

Sattini, S.

Shubert, R.

Standley, R. D.

M. J. Rand, R. D. Standley, Appl. Opt. 11, 2482 (1972).
[CrossRef] [PubMed]

J. E. Goell, R. D. Standley, Bell Syst. Tech. J. 48, 3445 (1969).

Tomlinson, W. J.

H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
[CrossRef]

Toraldo di Francia, G.

Ulrich, R.

H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
[CrossRef]

van Duzer, T.

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

Weber, H. W.

H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
[CrossRef]

West, E. J.

Appl. Opt.

Appl. Phys. Lett.

H. W. Weber, R. Ulrich, E. A. Chandross, W. J. Tomlinson, Appl. Phys. Lett. 20, 143 (1972).
[CrossRef]

Bell Syst. Tech. J.

J. E. Goell, R. D. Standley, Bell Syst. Tech. J. 48, 3445 (1969).

J. Opt. Soc. Am.

Proc. IEEE

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

Other

E. G. Lean, “Acousto-optic Interaction in Guided Wave Structures,” in Integrated Optics, M. Barnoski, Ed. (Plenum Press, New York—to be published).

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Figures (6)

Fig. 1
Fig. 1

Geometry of a spherical depression (or protrusion lens). (a) Top view in the plane of the waveguide. (b) Side view defining the angle θ, which characterizes the depth of the depression. (c) Spherical triangle formed by the incident ray and the pole of the depression.

Fig. 2
Fig. 2

Normalized focal length f/Rc vs normalized distance x/Rc of the incident ray from the axis. The parameter ng/ng′ is the ratio of the mode indices inside and outside of the depression.

Fig. 3
Fig. 3

Ratio of the mode indices ng/ng′ inside and outside of the depression for best compensation of the aberration and normalized power Rc/f for an uncompensated (ng/ng′ = 1) depression lens and a depression lens with optimum compensation plotted vs the angle θ.

Fig. 4
Fig. 4

Dispersion curves for the first three TE modes for a waveguide of Corning 7059 glass on a quartz substrate.

Fig. 5
Fig. 5

Rays propagating through a spherical depression. Top:θ = 19° and no compensation (ng/ng′ = 1). Bottom:θ = 19° and ng/ng′ = 1.04.

Fig. 6
Fig. 6

Normalized focal length f/Rc vs normalized distance x/Rc of the incident beam from the axis for a spherical depression before and after compensation of the aberrations. Points are experimental; curves are theoretical result. θ = 19°, Rc = 2.5 mm, λ = 0.633 μ.

Equations (5)

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f sin ψ = R c sin ( ϕ f ψ ) ,
ψ = ϕ i + ϕ f π ,
ϕ f = π + ϕ i 2 tan 1 ( cos θ  tan α ) ,
sin α = ( n g / n g ) sin ϕ i ,
sin ϕ i = x / R c .

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