Abstract

Analysis of the phase modulation characteristics of electro-optic or acousto-optic waveguide modulators reveals that a significant increase of modulation sensitivity, per unit of applied field, above that of conventional, bulk devices can occur in multimode waveguides with tight confinement. In the limit, an enhancement of as much as a factor of 3 is theoretically attainable. Explicit formulas for relative modulation sensitivity are given for both TE- and TM-mode polarizations, and illustrative curves are presented for a wide variety of structural parameters. Semiquantitative experimental verification of the effect is presented, using results of a GaAs electro-optic thin-film modulator at 10.6 μm.

© 1975 Optical Society of America

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  1. Topical Meeting on Integrated Optics, New Orleans, Louisiana, 21–24 January 1974.
  2. W. B. Oldham and A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
    [CrossRef]
  3. S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
    [CrossRef]
  4. W. S. C. Chang and K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
    [CrossRef]
  5. J. F. Lotspeich, Appl. Opt. 13, 2529 (1974).
    [CrossRef] [PubMed]
  6. P. K. Tien and R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  7. D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
    [CrossRef]
  8. W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
    [CrossRef]
  9. W. E. Martin, J. Appl. Phys. 44, 3703 (1973).
    [CrossRef]
  10. I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
    [CrossRef]
  11. D. B. Anderson and J. T. Boyd, Appl. Phys. Lett. 19, 266 (1971).
    [CrossRef]
  12. P. K. Cheo and M. Gilden, Appl. Phys. Lett. 25, 272 (1974).
    [CrossRef]
  13. P. K. Cheo, Appl. Phys. Lett. 22, 241 (1973).
    [CrossRef]
  14. I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, Ohio, 1971), Sec. 5.
  15. F. S. Chen, Proc. IEEE 58, 1440 (1970).
    [CrossRef]
  16. F. K. Reinhart and B. I. Miller, Appl Phys. Lett. 20, 36 (1972).
    [CrossRef]
  17. S. Namba, J. Opt. Soc. Am. 51, 76 (1961).
    [CrossRef]
  18. S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
    [CrossRef]
  19. A. Reisinger, Appl. Opt. 12, 1015 (1973).
    [CrossRef] [PubMed]
  20. I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
    [CrossRef]
  21. J. T. Boyd, IEEE J. Quantum Electron. QE-8, 788 (1972).
    [CrossRef]
  22. G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
    [CrossRef]
  23. J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
    [CrossRef]

1974 (5)

J. F. Lotspeich, Appl. Opt. 13, 2529 (1974).
[CrossRef] [PubMed]

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

P. K. Cheo and M. Gilden, Appl. Phys. Lett. 25, 272 (1974).
[CrossRef]

S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
[CrossRef]

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

1973 (6)

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

P. K. Cheo, Appl. Phys. Lett. 22, 241 (1973).
[CrossRef]

W. E. Martin, J. Appl. Phys. 44, 3703 (1973).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
[CrossRef]

J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
[CrossRef]

1972 (3)

W. S. C. Chang and K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

F. K. Reinhart and B. I. Miller, Appl Phys. Lett. 20, 36 (1972).
[CrossRef]

J. T. Boyd, IEEE J. Quantum Electron. QE-8, 788 (1972).
[CrossRef]

1971 (1)

D. B. Anderson and J. T. Boyd, Appl. Phys. Lett. 19, 266 (1971).
[CrossRef]

1970 (2)

1967 (2)

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

W. B. Oldham and A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

1965 (1)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

1961 (1)

Anderson, D. B.

D. B. Anderson and J. T. Boyd, Appl. Phys. Lett. 19, 266 (1971).
[CrossRef]

Anderson, W. W.

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

Bahraman, A.

W. B. Oldham and A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

Boyd, J. T.

J. T. Boyd, IEEE J. Quantum Electron. QE-8, 788 (1972).
[CrossRef]

D. B. Anderson and J. T. Boyd, Appl. Phys. Lett. 19, 266 (1971).
[CrossRef]

Brandt, G. B.

G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
[CrossRef]

Carruthers, J. R.

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

Chang, W. S. C.

W. S. C. Chang and K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

Chen, F. S.

F. S. Chen, Proc. IEEE 58, 1440 (1970).
[CrossRef]

Cheo, P. K.

P. K. Cheo and M. Gilden, Appl. Phys. Lett. 25, 272 (1974).
[CrossRef]

P. K. Cheo, Appl. Phys. Lett. 22, 241 (1973).
[CrossRef]

Conroy, J. J.

G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
[CrossRef]

Fukunishi, S.

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Gilden, M.

P. K. Cheo and M. Gilden, Appl. Phys. Lett. 25, 272 (1974).
[CrossRef]

Gottleib, M.

G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
[CrossRef]

Izawa, T.

S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, Ohio, 1971), Sec. 5.

Loh, K. W.

W. S. C. Chang and K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

Lotspeich, J. F.

Martin, W. E.

W. E. Martin, J. Appl. Phys. 44, 3703 (1973).
[CrossRef]

McFee, J. H.

J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
[CrossRef]

McKenna, J.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

Miller, B. I.

F. K. Reinhart and B. I. Miller, Appl Phys. Lett. 20, 36 (1972).
[CrossRef]

Miyazawa, S.

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Nahory, R. E.

J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
[CrossRef]

Namba, S.

Nelson, D. F.

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

Noda, J.

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Oldham, W. B.

W. B. Oldham and A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

Pollack, M. A.

J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
[CrossRef]

Ramaswamy, V.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

Reinhart, F. K.

F. K. Reinhart and B. I. Miller, Appl Phys. Lett. 20, 36 (1972).
[CrossRef]

Reisinger, A.

Schmidt, R. V.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

Stulz, L. W.

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

Tien, P. K.

Turner, E. H.

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, Ohio, 1971), Sec. 5.

Uchida, N.

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

Uehara, S.

S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
[CrossRef]

Ulrich, R.

Yamauchi, Y.

S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
[CrossRef]

Appl Phys. Lett. (1)

F. K. Reinhart and B. I. Miller, Appl Phys. Lett. 20, 36 (1972).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (9)

I. P. Kaminow, V. Ramaswamy, R. V. Schmidt, and E. H. Turner, Appl. Phys. Lett. 24, 622 (1974).
[CrossRef]

G. B. Brandt, M. Gottleib, and J. J. Conroy, Appl. Phys. Lett. 23, 53 (1973).
[CrossRef]

J. H. McFee, R. E. Nahory, and M. A. Pollack, Appl. Phys. Lett. 23, 571 (1973).
[CrossRef]

S. Uehara, Y. Yamauchi, and T. Izawa, Appl. Phys. Lett. 24, 19 (1974).
[CrossRef]

S. Fukunishi, N. Uchida, S. Miyazawa, and J. Noda, Appl. Phys. Lett. 24, 424 (1974).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, E. H. Turner, and L. W. Stulz, Appl. Phys. Lett. 22, 540 (1973).
[CrossRef]

D. B. Anderson and J. T. Boyd, Appl. Phys. Lett. 19, 266 (1971).
[CrossRef]

P. K. Cheo and M. Gilden, Appl. Phys. Lett. 25, 272 (1974).
[CrossRef]

P. K. Cheo, Appl. Phys. Lett. 22, 241 (1973).
[CrossRef]

IEEE J. Quantum Electron. (4)

W. S. C. Chang and K. W. Loh, IEEE J. Quantum Electron. QE-8, 463 (1972).
[CrossRef]

W. B. Oldham and A. Bahraman, IEEE J. Quantum Electron. QE-3, 278 (1967).
[CrossRef]

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

J. T. Boyd, IEEE J. Quantum Electron. QE-8, 788 (1972).
[CrossRef]

J. Appl. Phys. (2)

W. E. Martin, J. Appl. Phys. 44, 3703 (1973).
[CrossRef]

D. F. Nelson and J. McKenna, J. Appl. Phys. 38, 4057 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

F. S. Chen, Proc. IEEE 58, 1440 (1970).
[CrossRef]

Other (2)

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, Ohio, 1971), Sec. 5.

Topical Meeting on Integrated Optics, New Orleans, Louisiana, 21–24 January 1974.

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Figures (8)

FIG. 1
FIG. 1

Schematic representation of the dielectric-slab waveguide. Longitudinal cross section.

FIG. 2
FIG. 2

Relative modulation sensitivity, dng/dn1, as a function of the normalized modal index, ng/n1, for a symmetric waveguide with moderately tight confinement. n2 = n3 = 0.7 n1. The solid curves are TE Eq. (6); the dashed curves are TM Eq. (7). The mode number increases as ng decreases.

FIG. 3
FIG. 3

Modal dispersion curves, Eq. (1), for TE modes 5 and 12 of a symmetric waveguide modulator operating at 10.6 μm, with n2 =n3 = 2.3 and a zero-signal core index n1 = 3.3 (solid curves). The dashed curves result from a hypothetical finite index increase of 0.10 in n1, due to an applied signal field. The resulting changes Δng of modal index are indicated at various values of core thickness b, to illustrate the modulation-sensitivity-enhancement effect.

FIG. 4
FIG. 4

Relative modulation sensitivity of a symmetric waveguide with very tight confinement. n2 = n3 = 0.3n1. (———) TE, (– – –) TM.

FIG. 5
FIG. 5

Maximum enhancement factor for TE modes as a function of b1 for various values of n2, with n3 fixed at 0.3n1.

FIG. 6
FIG. 6

Symmetric As2S3-GaAs-As2S3 waveguide modulator used for modulation measurements at 10.6 μm. Vertical dimensions are not to scale.

FIG. 7
FIG. 7

Modulation results due to beam deflection in the experimental modulator of Fig. 6. Electrode length L = 4.39 mm, electrode effective gap d = 51 μm, applied signal voltage V3 = 70 Vpp, λ0 = 10.6 μm. (●) TE, (■) TM. Equation (12) predicted ΔI/Ia = 0.045 for TE mode m = 0.

FIG. 8
FIG. 8

Results of irradiance modulation as a function of mode number, measured with the modulator of Fig. 6. Operating parameters are the same as those of Fig. 7. Degradation of modulation index Γm with increasing mode number is due to increasing disparity of output-coupling angle between interacting TE and TM modes. Equation (14) predicts Γm = 0.169 for mode m = 0.

Equations (17)

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b = 1 h { ϕ 12 + ϕ 13 + m π } ,
ϕ 12 = arctan K 12 p 2 h , ϕ 13 = arctan K 13 p 3 h ,
K 12 = 1 ;             K 13 = 1             for TE modes K 12 = ( n 1 n 2 ) 2 ;             K 13 = ( n 1 n 3 ) 2             for TM modes .
h 2 = k 1 2 - β 2 = ( n 1 2 - n g 2 ) k 0 2 , p 2 2 = β 2 - k 2 2 = ( n g 2 - n 2 2 ) k 0 2 , p 3 2 = β 2 - k 3 2 = ( n g 2 - n 3 2 ) k 0 2 .
d n g d n 1 = - ( b n 1 ) / ( b n g ) .
d n g d n 1 = n 1 n g { ( b + p 2 h 2 + p 2 2 + p 3 h 2 + p 3 2 ) / ( b + 1 p 2 + 1 p 3 ) } .
d n g d n 1 = n 1 n g { [ b + ( k 1 2 - 2 h 2 k 1 2 ) ( K 12 p 2 h 2 + K 12 2 p 2 2 + k 13 p 3 h 2 + K 13 2 P 3 2 ) ] / [ b + ( K 12 p 2 ) h 2 + p 2 2 h 2 + K 12 2 p 2 2 + ( K 13 p 3 ) h 2 + p 3 2 h 2 + K 13 2 p 3 2 ] } .
M 1 - 1 π arctan ( n 2 2 - n 3 2 n 1 2 - n 2 2 ) 1 / 2 + 2 b λ 1 [ 1 - ( n 2 n 1 ) 2 ] 1 / 2 .
Δ I I a = 1.59 π L Δ n g λ 0 ,
r i j = ( 0 - 2 3 r - 1 3 r 0 2 3 r - 1 3 r 0 0 2 1 3 r 0 - 1 3 r 0 - 1 3 r 0 0 - 2 3 r 0 0 ) ,
x = 1 ¯ 10 , y = 1 ¯ 1 ¯ 2 , z = 111 ,
Δ n 1 = 1 2 n 1 3 1 3 r 41 E 3 = 1 2 n 1 3 1 3 r 41 ( V 3 / d ) ,
Δ I I a = 1.59 π L n 1 3 r 41 V 3 2 3 λ 0 d ,
Δ I / I a = 0.045             ( for m = 0 ) ,
Γ m = 2 π L λ 0 ( Δ n g TE - Δ n g TM ) ,
Γ m = π L n 1 3 3 r 41 V 3 λ 0 d
Γ m = 0.169             for m = 0.