Abstract

Regarding the design problem of corrugated planar optical waveguide filters, a new numerical method is presented consisting of a direct numerical solution of the coupled Gel’fand–Levitan–Marchenko integral equations. This method, which uses leapfrogging in space and time, is exact in principle and avoids some difficulties encountered in previously derived analytical methods of solution. Straightforward numerical calculations permit the design of several classes of filters such as Butterworth, Chebyshev, Cauer (elliptic) and others, as presented in the paper. The accuracy of our proposed method of design is checked in several ways, mainly through the numerical solution of the corresponding direct-scattering problem (Riccati differential equation).

© 2002 Optical Society of America

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  1. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical waveguide filters: synthesis,” J. Opt. Soc. Am. 64, 804–809 (1975).
    [CrossRef]
  2. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
    [CrossRef]
  3. P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett. 1, 43–45 (1977).
    [CrossRef]
  4. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979), pp. 66–79.
  5. G. H. Song, S. Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
    [CrossRef]
  6. G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  7. G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).
  8. P. Frangos, D. L. Jaggard, “A numerical solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
    [CrossRef]
  9. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).
  10. V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSSR 104, 635–698 (1955).
  11. M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).
  12. I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
    [CrossRef]
  13. H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
    [CrossRef]
  14. D. L. Jaggard, K. E. Olson, “Numerical reconstruction for dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).
    [CrossRef]
  15. D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,” IEEE Trans. Antennas Propag. 35, 934–936 (1987).
    [CrossRef]
  16. P. V. Frangos, “One-dimensional inverse scattering: exact methods and applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1986).
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).
  18. M. E. Van Valkenburg, Analog Filter Design (Saunders College Publishing, Harcourt Brace Jovanovich College Publishers, New York, 1982).

1991 (1)

P. Frangos, D. L. Jaggard, “A numerical solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

1987 (1)

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,” IEEE Trans. Antennas Propag. 35, 934–936 (1987).
[CrossRef]

1985 (2)

1983 (1)

G. H. Song, S. Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

1982 (1)

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

1977 (1)

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

1975 (1)

1974 (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

1960 (1)

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

1955 (2)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSSR 104, 635–698 (1955).

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Cross, P. C.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).

Frangos, P.

P. Frangos, D. L. Jaggard, “A numerical solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

Frangos, P. V.

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,” IEEE Trans. Antennas Propag. 35, 934–936 (1987).
[CrossRef]

P. V. Frangos, “One-dimensional inverse scattering: exact methods and applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1986).

Ge, D. B.

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Hill, K. O.

Jaggard, D. L.

P. Frangos, D. L. Jaggard, “A numerical solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,” IEEE Trans. Antennas Propag. 35, 934–936 (1987).
[CrossRef]

D. L. Jaggard, K. E. Olson, “Numerical reconstruction for dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).
[CrossRef]

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Kay, I.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

Kogelnik, H.

P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett. 1, 43–45 (1977).
[CrossRef]

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979), pp. 66–79.

Kritikos, H. N.

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Marchenko, V. A.

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSSR 104, 635–698 (1955).

Matsuhara, M.

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Olson, K. E.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Shin, S. Y.

G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
[CrossRef]

G. H. Song, S. Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Song, G. H.

G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
[CrossRef]

G. H. Song, S. Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).

Van Valkenburg, M. E.

M. E. Van Valkenburg, Analog Filter Design (Saunders College Publishing, Harcourt Brace Jovanovich College Publishers, New York, 1982).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).

Watanabe, A.

Am. Math. Soc. Trans. (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

Commun. Pure Appl. Math. (1)

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSSR 104, 635–698 (1955).

IEEE Trans. Antennas Propag. (2)

P. Frangos, D. L. Jaggard, “A numerical solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

D. L. Jaggard, P. V. Frangos, “The electromagnetic inverse scattering problem for layered dispersionless dielectrics,” IEEE Trans. Antennas Propag. 35, 934–936 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Proc. IEEE (2)

G. H. Song, S. Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numeric reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Stud. Appl. Math. (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Other (5)

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980).

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979), pp. 66–79.

P. V. Frangos, “One-dimensional inverse scattering: exact methods and applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1986).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1995).

M. E. Van Valkenburg, Analog Filter Design (Saunders College Publishing, Harcourt Brace Jovanovich College Publishers, New York, 1982).

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Figures (5)

Fig. 1
Fig. 1

(a) Geometry of the corrugated optical planar waveguide. (b) Side view of the corrugated slab waveguide. Here nf (nc) is the film (cover) index of refraction, v2(z, k) the incident (forward) wave, and v1(z, k) the reflected (backward) wave.

Fig. 2
Fig. 2

(a) Discretization of the ηξ plane. The kernels are found at the discrete points (circles), d is the grid spacing, and p is the order of the diagonals shown in the figure. (b) Points at which the kernel values are related through Eqs. (16). (c) Points at which the kernel values are related through Eqs. (20).

Fig. 3
Fig. 3

(a) Coupling potential obtained through the proposed leapfrogging method (Qleap) compared with that obtained by the method proposed by Song and Shin5,6 (Qss) (graphs are indistinguishable). Here the one-pole reflection coefficient of Eq. (21) is considered. (b) Reconstructed r(k) through the numerical integration of the Riccati differential equation (D.E.) [Eq. (10)] compared with the original r(k) of Eq. (21).

Fig. 4
Fig. 4

(a) Magnitude of the reflection coefficient of Chebyshev filters of orders m=2, 3, and 4. One can observe the ripple in the passband (Ref. 18, Chap. 8). (b) Coupling potential q(z) calculated through the leapfrogging algorithm for the case of Chebyshev filters of order m=2, 3, and 4. (c) Recovered reflection coefficients r(k) (magnitude) by use of Riccati equation (10) for the coupling potential corresponding to the Chebyshev filter of order 4 (m=4), showing very good design accuracy.

Fig. 5
Fig. 5

(a) Magnitude of the reflection coefficient for the Cauer (elliptic) filters of orders 3, 4, and 5. (b) Coupling potentials obtained for the case of Cauer (elliptic) filters of orders 3, 4, and 5, with the proposed leapfrogging algorithm. The truncation point5,6 is taken here at zt=15. (c) Recovered reflection coefficients (magnitude) |r(k)| by use of Riccati equation (10) for the case of the Cauer (elliptic) filter of order 3.

Equations (56)

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h(z)=ho+he(z)sin2πzΛ+θ(z),
q(z)=Ahe(z)exp[-iθ(z)],
dν1(z, k)dz+ikv1(z, k)=q(z)v2(z, k),
dv2(z, k)dz-ikv2(z, k)=±q*(z)v1(z, k),
r(k)=limz-ν1(z, k)ν2(z, k)exp(2ikz),
K1*(z, y)-yzK2(z, y)R(y+y)dy=0,
-K2*(z, y)+R(z+y)
+-yzK1(z, y)R(y+y)dy=0,
R(z)=12π-r(k)exp(-ikz)dk.
q(z)=-2K2*(z, z),z0.
2K1(z, y)z2-2K1(z, y)y2
=2K2*(z, z)K2(z, y)y-K2(z, y)z
-2K2(z, y)dK2*(z, z)dz,
2K2(z, y)z2-2K2(z, y)y2
=2K2(z, z)K1(z, y)z-K1(z, y)y
2K1(z, y)dK2(z, z)dz.
K1(z,-z)=0,
K2(z,-z)=R*(0),
dρ(z, k)dz=-2ikρ(z, k)+q(z)q*(z)ρ2(z, k),
ξ=(z+y)/2,
η=(z-y)/2.
K˜1*(ξ, η)20ξK˜2(ξ+η, ξ-ξ)R˜(2ξ)dξ=0,
-K2*(ξ, η)+R˜(2ξ)+20ξK˜1(ξ+η, ξ-ξ)R˜(2ξ)dξ
=0.
K˜1(0, η)=0,
K˜2(0, η)=R˜*(0),
2K˜1ξη=-2K˜2*(ξ, 0)ξ=ξ+ηK˜2η-2K˜2(ξ, η)dK˜2(ξ, 0)dξξ=ξ+η,
2K˜2ξη=2K˜2*(ξ, 0)ξ=ξ+ηK˜1ξ2K˜1(ξ, η)dK˜2(ξ, 0)dξξ=ξ+η,
q(ξ)=-2K˜2*(ξ, 0).
Km,n(1)=±2d v=2m-1Kv+n-1,m-v+1(2)R˜2ν-1±dKm+n-1,1(2)R˜2m-1*,
Km,n(2)=R˜2m-1+2dv=2m-1Kv+n-1,m-v+1(1)R˜2v-1+dAm+n-1,1(1)R˜2m-1*,
K1,n(1)=0
K1,n(2)=R˜1*
Km+1,n+1(1)=Km,n+1(1)+Km+1,n(1)-Km,n(1)-2[Km+n,1(2)]*(Km,n+1(2)-Km,n(2))-2Km,n(2)(Km+n+1,1(2)-Km+n,1(2))*,
Km+1,n+1(2)=Km,n+1(2)+Km+1,n(2)-Km,n(2)2Km+n,1(2)×(Km+1,n(1)-Km,n(1))2Km,n(1)(Km+n+1,1(2)-Km+n,1(2)),
qm=-2[Km,1(2)]*.
Km+1,n+1(1)Km,n+1(1)+Km+1,n(1)-Km,n(1),
Km+1,n+1(2)=Km,n+1(2)+Km+1,n(2)-Km,n(2).
r(k)=0.5ik+i,
r(k)=0.5(1+k)j=1101-kexp[-iπ(2j-1)/20].
r(s)=r0n=0Ncn(s/ξc)n,
(N=2)
r(k)=1k2+i2k-1,
(N=3)
r(k)=1-ik3+2k2+2ik-1,
(N=4)
r(k)=1-k4-2.613ik3+3.414k2+2.613ik-1.
|r(k)|2=11+δ2Tm2(k),
(m=2)r(k)=1-k2-1.112ik+1.117,
(m=3)r(k)=0.5ik3-k2-1.25ik+0.5,
(m=4)
r(k)=1.2558k4+2.16ik3-3.31k2-2.86ik+1.26,
|r(k)|2=11+δ2Rm2(k, L),
(m=3)r(k)=0.1203(-k2+8.559)(-ik+0.7928)(-k2-0.6725ik+1.2984),
(m=4)r(k)=(-k2+7.5834)(-k2+41.6025)602.3(-k2-1.0234ik+0.4765)(-k2-0.3752ik+1.1377),
(m=5)r(k)=0.0046205(-k2+4.365)(-k2+10.5677)(-ik+0.392612)(-k2-0.58054ik+0.525)(-k2-0.19255ik+1.03402)

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