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.

© 1975 Optical Society of America

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References

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  1. A. Yariv, IEEE J. QE-9, 919 (1973).
    [Crossref]
  2. F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys. Lett. 22, 190 (1973).
    [Crossref]
  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).
    [Crossref]
  5. D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
    [Crossref]
  6. K. O. Hill, Appl. Opt. 13, 1853 (1974).
    [Crossref] [PubMed]
  7. M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).
    [Crossref] [PubMed]
  8. J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
    [Crossref]
  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)2whereas the widths of the regions of stationary phase increase only as |C|l(i.e., as l1/2); thus degrees of freedom is proportional to |C|l or to l1/2.
  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 μ m. 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 μ m and period 0.163 μ m, 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 μ m at one end of the filter to 0.172 μ m 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).
    [Crossref]

1974 (4)

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

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

K. O. Hill, Appl. Opt. 13, 1853 (1974).
[Crossref] [PubMed]

M. Matsuhara and K. O. Hill, Appl. Opt. 13, 2886 (1974).
[Crossref] [PubMed]

1973 (2)

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

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

1958 (1)

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

1954 (1)

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

Dabby, F. W.

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

Flanders, D. C.

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

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.

Kestenbaum, A.

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

Kogelnik, H.

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

Matsuhara, M.

Pierce, J. R.

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

Saifi, M. A.

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

Schmidt, R. V.

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

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, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

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).

Shubert, R.

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

Unger, H. -G.

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

Yariv, A.

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

Appl. Opt. (2)

Appl. Phys. Lett. (2)

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

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

Bell Syst. Tech. J. (1)

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

IEEE J. (1)

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

J. Appl. Phys. (2)

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

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

Other (4)

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)2whereas the widths of the regions of stationary phase increase only as |C|l(i.e., as l1/2); thus degrees of freedom is proportional to |C|l or to l1/2.

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 μ m. 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 μ m and period 0.163 μ m, 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 μ m at one end of the filter to 0.172 μ m at the other, a broad-band filter of 50 nm bandwidth results having reflectance R= 0.97 for λ = 475–525 nm.

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).

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Figures (6)

FIG. 1
FIG. 1

Variation of normalized chirp as a function of the normalized distance employed in the synthesis of band-rejection filters. The solid line [labelled Eq. (13)] is for a constant coupling coefficient and the dashed line [labelled Eq. (15)] is for a coupling coefficient that follows the raised-cosine distribution (see text).

FIG. 2
FIG. 2

Frequency-response curves of band-rejection filters synthesized with a constant coupling coefficient and with R0 = 0.999. The solid curve is for the case |C|l = 100, and the dashed curve is for |C|l = 10. R is the power-reflection coefficient.

FIG. 3
FIG. 3

Frequency-response curves of band-rejection filters synthesized with a coupling coefficient that follows the raised-cosine distribution and with R0 = 0.999. The solid and dashed curves are for |C0|l = 100 and 10, respectively.

FIG. 4
FIG. 4

Variation of the normalized chirp as a function of the normalized distance employed in the synthesis of a power-discriminator filter (solid curve) and an amplitude-discriminator filter (dashed curve) with a constant coupling coefficient.

FIG. 5
FIG. 5

Frequency-response characteristics of the synthesized power discriminator for |C|l = 100 (solid curve) and |C|l = 10 (dashed curve).

FIG. 6
FIG. 6

Frequency-response characteristics of the synthesized amplitude discriminator for |C|l = 100 (solid curve) and |C|l = 10 (dashed curve).

Equations (31)

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d ψ A d z = C exp ( - j 0 z B d z ) ψ B , d ψ B d z = C * exp ( j 0 z B d z ) ψ A ,
B = 2 ( β - n π / d ) , β = ( β A + β B ) / 2 ,
β = β 0 ( 1 + δ β ) = 2 π λ 0 ( 1 + δ β ) , n π d = n π d 0 ( 1 - δ d ) ,
β 0 = n π / d 0 .
B = 4 π λ 0 ( δ β + δ d ) .
δ d ( z ) z = z δ β = - δ β .
R ( δ β ) = 1 - exp [ - λ 0 2 C 2 / d ( δ d ) d z ] z = z δ β .
d ( δ d ) d z = - λ 0 2 C 2 / log [ 1 - R ( - δ d ) ]             at z = z δ β .
δ d ( 0 ) δ d ( z ) log [ 1 - R ( - δ d ) ] d ( δ d ) = - 1 2 λ 0 0 z C 2 d z .
R ( δ β ) = { R 0 , δ β 1 δ β δ β 2 0 , elsewhere
δ d ( z ) = δ d ( 0 ) - λ 0 C 2 2 log ( 1 - R 0 ) z .
δ d ( 0 ) = - δ β 2 , δ d ( l ) = - δ β 1 ,
1 2 λ 0 C 2 l = ( δ β 2 - δ β 1 ) log ( 1 - R 0 ) - 1 , δ β 2 + δ d ( z ) δ β 2 - δ β 1 = z l ,             0 z l .
C ( z ) = 2 C 0 sin 2 ( π z l ) ,             0 z l .
1 2 λ 0 C 0 2 l = ( δ β 2 - δ β 1 ) log ( 1 - R 0 ) - 2 / 3 , δ β 2 + δ d ( z ) δ β 2 - δ β 1 = z l - 2 3 π sin ( 2 π z l ) + 1 12 π sin ( 4 π z l ) ,             0 z l .
R ( δ β ) = { ( δ β - δ β 1 ) / ( δ β 2 - δ β 1 ) , δ β 1 δ β δ β 2 0 ,     otherwise .
1 2 λ 0 C 2 l = δ β 2 - δ β 1 , δ β 2 + δ d ( z ) δ β 2 - δ β 1 { 1 - log [ δ β 2 + δ d ( z ) δ β 2 - δ β 1 ] } = z l .
R ( δ β ) = { [ ( δ β - δ β 1 ) / ( δ β 2 - δ β 1 ) ] 2 , δ β 1 δ β δ β 2 0 ,     otherwise .
i 2 λ 0 C 2 l = ( δ β 2 - δ β 1 ) log ( e / 2 ) 2 , log [ ( e 2 ) 2 · ( 2 - x ) 2 - x e 2 ( 1 - x ) x x ] / log ( e 2 ) 2 = z l ,
x = δ β 2 + δ d ( z ) δ β 2 - δ β 1 .
B = α · ( z - z δ β ) ,
α = 4 π λ 0 [ d d z δ d ( z ) ] z = z δ β ,
d ψ A d x = b exp ( - j 0 x α x d x ) ψ B , d ψ B d x = b * exp ( j 0 x α x d x ) ψ A , b = C exp ( - j 0 z δ β B d z ) ,
d 2 ψ A d x 2 + j α x d ψ A d x - b b * ψ A = 0 ,
ψ A ( x ) = x - 1 / 2 exp ( - 1 4 j α x 2 ) [ G · W κ , μ ( 1 2 j α x 2 ) + H · W - κ , μ ( - 1 2 j α x 2 ) ] ,
κ = - 1 4 + j ( b b * / 2 α ) ,             μ = 1 4 .
ψ B ( x ) = 1 b x - 3 / 2 exp ( 1 4 j α x 2 ) × [ G · { ( - 1 2 - 2 κ ) W κ , μ ( 1 2 j α x 2 ) - 2 W κ + 1 , μ ( 1 2 j α x 2 ) } + H · { ( - 1 2 - 2 κ ) W - κ , μ ( - 1 2 j α x 2 ) - 2 ( μ + κ + 1 2 ) ( μ - κ - 1 2 ) W - κ - 1 , μ ( - 1 2 j α x 2 ) } ] .
ψ A ( 0 ) = P ( x ) ψ A ( x ) + Q ( x ) * ψ B ( x ) , ψ B ( 0 ) = Q ( x ) ψ A ( x ) + P ( x ) * ψ B ( x ) ,
P ( x ) = [ exp ( π 2 | b b * 2 α | ) · | α 2 x 2 | j b b * / 2 α ] / [ π - 1 / 2 · Γ ( 1 2 + j b b * 2 α ) ] , Q ( x ) = - [ exp ( π 2 | b b * 2 α | ) · | α 2 x 2 | j b b * / 2 α ] / [ π - 1 / 2 · b x b x · { | b b * 2 α | exp ( j π 2 α α ) } 1 / 2 · Γ ( j b b * 2 α ) ] .
ψ A ( L ) = exp ( - π b b * / α ) , ψ B ( - L ) = [ 1 - exp ( - 2 π b b * / α ) ] 1 / 2 ,
T = exp [ - λ 0 2 C 2 / | d d z δ d | ] z = z δ β , R = 1 - T .