Abstract

The WKB approximation is used to derive simple equations that predict the shape of the index profile from measured mode indices of a planar optical waveguide. This nondestructive test is a useful tool in the study of diffused guides. The index profile is assumed either to decrease monotonically from the surface or to be symmetrical in the case of a buried guide. The approximation uses straight line segments to connect the measured points. Results are compared with mathematical solutions for exponential, Fermi, and step distributions and with other independent experimental observations of the profile in a nickel-diffused LiNbO3 guide.

© 1976 Optical Society of America

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References

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  1. E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
    [CrossRef]
  2. D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
    [CrossRef]
  3. P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
    [CrossRef]
  4. E. Conwell, Appl. Phys. Lett. 25, 40 (1974).
    [CrossRef]
  5. G. Borg, Acta Math. 78, (1946).
    [CrossRef]
  6. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 285–287.
  7. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 184–193.
  8. Calculated by A. Reisinger. See Appl. Opt. 12, 1015 (1973).
    [CrossRef] [PubMed]
  9. P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
    [CrossRef]
  10. R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
    [CrossRef]
  11. I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [CrossRef]
  12. L. J. Brillson, E. M. Conwell, J. Appl. Phys. 45, 5289 (1974).
    [CrossRef]
  13. W. E. Martin, Appl. Opt. 13, 2112 (1974).
    [CrossRef] [PubMed]
  14. R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), pp. 134–135.

1974 (5)

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

E. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

L. J. Brillson, E. M. Conwell, J. Appl. Phys. 45, 5289 (1974).
[CrossRef]

W. E. Martin, Appl. Opt. 13, 2112 (1974).
[CrossRef] [PubMed]

1973 (4)

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

Calculated by A. Reisinger. See Appl. Opt. 12, 1015 (1973).
[CrossRef] [PubMed]

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

1969 (1)

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

1946 (1)

G. Borg, Acta Math. 78, (1946).
[CrossRef]

Ballman, A. A.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Borg, G.

G. Borg, Acta Math. 78, (1946).
[CrossRef]

Brillson, L. J.

L. J. Brillson, E. M. Conwell, J. Appl. Phys. 45, 5289 (1974).
[CrossRef]

Brown, H.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Carruthers, J. R.

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

Conwell, E.

E. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

Conwell, E. M.

L. J. Brillson, E. M. Conwell, J. Appl. Phys. 45, 5289 (1974).
[CrossRef]

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Kaminow, I. P.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

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

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 285–287.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 285–287.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), pp. 134–135.

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

Martin, R. J.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Martin, W. E.

Reisinger, A.

Riva-Sanseverino, S.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 184–193.

Schmidt, R. V.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

Tien, P. K.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

Acta Math. (1)

G. Borg, Acta Math. 78, (1946).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (6)

P. K. Tien, R. Ulrich, R. J. Martin, Appl. Phys. Lett. 14, 291 (1969).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25, 458 (1974).
[CrossRef]

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

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, A. A. Ballman, H. Brown, Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

E. Conwell, Appl. Phys. Lett. 25, 40 (1974).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

J. Appl. Phys. (1)

L. J. Brillson, E. M. Conwell, J. Appl. Phys. 45, 5289 (1974).
[CrossRef]

Other (3)

R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), pp. 134–135.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 285–287.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 184–193.

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Figures (8)

Fig. 1
Fig. 1

An exponential refractive index profile, its exact mode indices, and four estimated profiles based on the set of mode indices.

Fig. 2
Fig. 2

A slowly changing Fermi function profile (curved line) and the minimum curvature estimate (dots) match closely in depth and slope at the midpoint and in index value at the surface.

Fig. 3
Fig. 3

A steeper Fermi function profile (curved line) is still well matched by the inverse WKB estimates (dots). Two sets of estimates are shown: In one case, the exact value of the surface index (n0) was used as the starting point (crosses); in the other case, the surface index was selected by minimizing the curvature of the estimated profile (dots).

Fig. 4
Fig. 4

An extreme case is provided by the step function. Again, dots indicate the minimum curvature estimate of the actual profile.

Fig. 5
Fig. 5

Upper curves are estimates based on mode index measurements of a Ni-diffused LiNbO3 guide. Repeated measurements on the same sample give an offset between curves due to difficulty in measuring indices absolutely. The more accurate relative index values within a set of modes determines the shape of the profile. The lower curve is the index profile determined from observation of Fizeau fringes in a thin slice of the guide.

Fig. 6
Fig. 6

Nickel distribution as determined by electron beam mi-croprobe analysis of a slice from the sample used in Fig. 5.

Fig. 7
Fig. 7

Setup for observation of interference fringes in slice from waveguide.

Fig. 8
Fig. 8

Fizeau fringes in the Ni-diffused sample. Curvature at the upper right surface is due to rounding of the edge during polishing. The shape of the fringes at the upper left includes both rounding due to polishing and the index changes due to the presence of the Ni diffusion.

Equations (12)

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E y = F ( z ) exp [ i ( k x x ω t ) ] ,
o z m [ n 2 ( z ) n m 2 ] 1 / 2 dz = 4 m 1 8 , m = 1 , 2 , . . . , M
o z m [ n 2 ( z ) n m 2 ] 1 / 2 dz = 2 m 1 8 , m = 1 , 2 , . . . , M .
k = 1 m z k 1 z k [ n 2 ( z ) n m 2 ] 1 / 2 dz = 4 m 1 8 .
n ( z ) n k + ( n k 1 n k ) ( z k z k 1 ) ( z k z ) for z k 1 z z k .
z m = z m 1 + [ ( 3 2 ) ( n m 1 + 3 n m 2 ) 1 / 2 ( n m 1 n m ) 1 / 2 ] { ( 4 m 1 8 ) 2 3 k = 1 m 1 ( n k 1 + n k 2 + n m ) 1 / 2 ( z k z k 1 n k 1 n k ) × [ ( n k 1 n m ) 3 / 2 ( n k n m ) 3 / 2 ] } for m = 2 , 3 , . . . , M ,
z 1 = 9 16 ( n 0 + 3 n 1 2 ) 1 / 2 ( n 0 n 1 ) 1 / 2 .
k = 0 M 2 [ n k + 2 n k + 1 z k + 2 z k + 1 n k + 1 n k z k + 1 z k z k + 2 + z k + 1 2 z k + 1 + z k 2 ] 2
1 2 [ 2 ( n 1 n 0 ) z 1 2 ] 2
δ n n 1 n 2 = 0.05 , δ n n 2 n 3 = 0.07.
δ z 3 z 3 z 2 = 0.12 , δ z 3 z 3 = 0.03.
Δ m λ o 2 = Δ n L + n Δ L for Δ L L , Δ n n 1 ,

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