Abstract

Analytical formulas for the synthesis of optical-waveguide filters having arbitrary spectral-response characteristics are derived from coupled-mode formalism. Use of these general formulas is illustrated by design of several filters, one of which is a linear power discriminator. The synthesis yields the functional dependence of spatial-perturbation period on the distance along the direction of wave propagation in the waveguide filter. The coupled-mode equations for the functional perturbation forms as determined by the synthesis process were solved numerically to find the actual response characteristics of the filter designs. Excellent agreement was found between the desired characteristics and those of the synthesized filters.

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  1. A. Yariv, IEEE J. QE-9, 919 (1973).
  2. F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).
  3. D. C. Flanders, R. V. Schmidt, and C. V. Shank, in Digest of Technical Papers, Integrated Optics Conference, New Orleans (Optical Society of America, Washington, D. C., 1974).
  4. R. Shubert, J. Appl. Phys. 45, 209 (1974).
  5. D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
  6. K. O. Hill, Appl. Opt. 13, 1853 (1974).
  7. M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).
  8. J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
  9. For simplicity, we assume that B(z) [i.e., δd(z)], is a monotonically increasing function of z.
  10. Degrees of freedom is defined to be the number of regions of stationary phase (see Ref. 6) contained within the filter length l. For a given desired filter response (| C | 2l = const), l increases as (| C | l)2 whereas the widths of the regions of stationary phase increase only as | C | l (i.e., as l½); thus degrees of freedom is proportional to | C | l or to l½.
  11. A value | C | l = 100 is achievable in a planar waveguide with symmetric index profile (n = 1.5, 1.55, 1.5) of thickness 0. 6 µm. Such a guide supports a single TE mode at λ = 500 nm. For a sinusoidal waveguide-thickness modulation at one of the boundaries having amplitude 0. 1 µm and period 0. 163 µm, we calculate a contradirectional mode-coupling coefficient | C | = 220 cm-1. The required waveguide-filter length in this case is thus 0.455 cm. If the periodicity is assumed to vary linearly from 0. 155 µm at one end of the filter to 0.172 µm at the other, a broad-band filter of 50 nm bandwidth results having reflectance R = 0.97 for λ = 475–525 nm.
  12. H. -G. Unger, Bell Syst. Tech. J. 37, 899 (1958).

Dabby, F. W.

F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).

Flanders, D. C.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).

D. C. Flanders, R. V. Schmidt, and C. V. Shank, in Digest of Technical Papers, Integrated Optics Conference, New Orleans (Optical Society of America, Washington, D. C., 1974).

Hill, K. O.

M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).

K. O. Hill, Appl. Opt. 13, 1853 (1974).

Kestenbaum, A.

F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).

Kogelnik, H.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).

Matsuhara, M.

M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).

Pierce, J. R.

J. R. Pierce, J. Appl. Phys. 25, 179 (1954).

Saifi, M. A.

F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).

Schmidt, R. V.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).

D. C. Flanders, R. V. Schmidt, and C. V. Shank, in Digest of Technical Papers, Integrated Optics Conference, New Orleans (Optical Society of America, Washington, D. C., 1974).

Shank, C. V.

D. C. Flanders, R. V. Schmidt, and C. V. Shank, in Digest of Technical Papers, Integrated Optics Conference, New Orleans (Optical Society of America, Washington, D. C., 1974).

D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).

Shubert, R.

R. Shubert, J. Appl. Phys. 45, 209 (1974).

Unger, H. -G.

H. -G. Unger, Bell Syst. Tech. J. 37, 899 (1958).

Yariv, A.

A. Yariv, IEEE J. QE-9, 919 (1973).

Other (12)

A. Yariv, IEEE J. QE-9, 919 (1973).

F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).

D. C. Flanders, R. V. Schmidt, and C. V. Shank, in Digest of Technical Papers, Integrated Optics Conference, New Orleans (Optical Society of America, Washington, D. C., 1974).

R. Shubert, J. Appl. Phys. 45, 209 (1974).

D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).

K. O. Hill, Appl. Opt. 13, 1853 (1974).

M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).

J. R. Pierce, J. Appl. Phys. 25, 179 (1954).

For simplicity, we assume that B(z) [i.e., δd(z)], is a monotonically increasing function of z.

Degrees of freedom is defined to be the number of regions of stationary phase (see Ref. 6) contained within the filter length l. For a given desired filter response (| C | 2l = const), l increases as (| C | l)2 whereas the widths of the regions of stationary phase increase only as | C | l (i.e., as l½); thus degrees of freedom is proportional to | C | l or to l½.

A value | C | l = 100 is achievable in a planar waveguide with symmetric index profile (n = 1.5, 1.55, 1.5) of thickness 0. 6 µm. Such a guide supports a single TE mode at λ = 500 nm. For a sinusoidal waveguide-thickness modulation at one of the boundaries having amplitude 0. 1 µm and period 0. 163 µm, we calculate a contradirectional mode-coupling coefficient | C | = 220 cm-1. The required waveguide-filter length in this case is thus 0.455 cm. If the periodicity is assumed to vary linearly from 0. 155 µm at one end of the filter to 0.172 µm at the other, a broad-band filter of 50 nm bandwidth results having reflectance R = 0.97 for λ = 475–525 nm.

H. -G. Unger, Bell Syst. Tech. J. 37, 899 (1958).

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