Abstract

We have observed leaky-mode propagation in Ti-diffused LiNbO3 and LiTaO3 planar waveguides in nonaxial propagation directions when the optic axis lies in the waveguide plane. For LiNbO3 waveguides, we measure both the leak angle and the loss coefficient as a function of propagation direction and present a theoretical model that quantitatively agrees with our observations. The loss coefficient is expressed in a form that shows the dependence on anisotropy and mode dispersion.

© 1978 Optical Society of America

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References

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  1. D. P. Gia Russo, J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
    [Crossref]
  2. D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
    [Crossref]
  3. H. P. Hsu, A. F. Milton, Electron. Lett. 12, 104 (1976).
    [Crossref]
  4. H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
    [Crossref]
  5. M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).
  6. W. K. Burns, J. Warner, J. Opt. Soc. Am. 64, 441 (1974).
    [Crossref]
  7. V. Lemanov, B. V. Sukharev, Sov. Tech. Phys. Lett. 2, 434 (1976).
  8. W. K. Burns, NRL, 1978.
  9. G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
    [Crossref]
  10. I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [Crossref]
  11. H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

1977 (2)

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[Crossref]

M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).

1976 (2)

V. Lemanov, B. V. Sukharev, Sov. Tech. Phys. Lett. 2, 434 (1976).

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 104 (1976).
[Crossref]

1975 (2)

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[Crossref]

1974 (1)

1973 (2)

D. P. Gia Russo, J. H. Harris, J. Opt. Soc. Am. 63, 138 (1973).
[Crossref]

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Burns, W. K.

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

W. K. Burns, J. Warner, J. Opt. Soc. Am. 64, 441 (1974).
[Crossref]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

W. K. Burns, NRL, 1978.

Carruthers, J. A.

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Gia Russo, D. P.

Hamilton, M. C.

M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).

Harris, J. H.

Hocker, G. B.

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

Hsu, H. P.

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[Crossref]

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 104 (1976).
[Crossref]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

Kaminow, I. P.

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Lemanov, V.

V. Lemanov, B. V. Sukharev, Sov. Tech. Phys. Lett. 2, 434 (1976).

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[Crossref]

Miceli, W. J.

M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).

Milton, A. F.

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[Crossref]

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 104 (1976).
[Crossref]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

Sheem, S. K.

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

Sukharev, B. V.

V. Lemanov, B. V. Sukharev, Sov. Tech. Phys. Lett. 2, 434 (1976).

Warner, J.

Willie, D. A.

M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).

Appl. Phys. Lett. (1)

I. P. Kaminow, J. A. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[Crossref]

Electron. Lett. (1)

H. P. Hsu, A. F. Milton, Electron. Lett. 12, 104 (1976).
[Crossref]

IEEE J. Quantum Electron. (3)

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[Crossref]

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[Crossref]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

M. C. Hamilton, D. A. Willie, W. J. Miceli, Opt. Eng. 16, 467 (1977).

Sov. Tech. Phys. Lett. (1)

V. Lemanov, B. V. Sukharev, Sov. Tech. Phys. Lett. 2, 434 (1976).

Other (2)

W. K. Burns, NRL, 1978.

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D. C., 1978), paper WD6-1.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of an X-cut LiNbO3 planar waveguide. The wave propagation direction (z) is at an angle θ with respect to the optic axis (c). (b) The indices of refraction for bulk waves (solid lines) and the approximate effective indices for planar hybrid modes (dashed lines) are shown as a function of the angle θ. Also shown are the measured values of the effective indices that support the hybrid mode model.

Fig. 2
Fig. 2

Measurements of the tilt angle (α) of the leaky wave (a) and the observtion of the polarization rotation with respect to the input (b) confirm that the radiation leakage in X- (or Y-) cut LiNbO3 waveguides is due to anisotropic coupling.

Fig. 3
Fig. 3

The leakage rate theoretically calculated (solid line) shows good agreement with the measured values for a LiNbO3 waveguide. The dashed line represents an approximation that is good at large θ. Wave propagation at θ = 57.3° is perpendicular to the (10.2) cleavage plane for an X-cut plate.

Equations (11)

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( β / k ) e = n e ( θ ) + ( Δ n eff ) e ,
( β / k ) o = n o + ( Δ n eff ) o ;
1 n e 2 ( θ ) = cos 2 θ n o 2 + sin 2 θ n e 2 .
( β / k ) e = n o cos α .
( ± i ) 2 γ o 2 = β 2 - k 2 n o 2 ,
( ± i ) 2 γ e 2 = ( n z n x ) 2 [ β 2 - k 2 n e 2 ( θ ) ] .
α L = S x ( o ) S z ( e ) d x ,
α L = 2 γ o s γ e 2 β ( W + 1 γ e s ) ( γ o 2 + k 2 n o 2 tan 2 θ ) ( γ o γ e + k 2 n o 2 tan 2 θ ) 2 ,
( β / k ) e = n e ( θ ) + b Δ n .
Δ n ( x ) = Δ n exp [ - ( x - W / D ) 2 ] ,
α L = [ 4 ( n o 2 - n e 2 ) 1 / 2 cos 2 θ n o 2 sin θ ] × [ Δ n ( 1 - b ) D ln ( 1 / b ) + ( k 2 Δ n n e b ) - 1 ] .

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