Abstract

A recently introduced optimal control theory method for optical waveguide design is applied to Y-branch waveguides and Mach–Zehnder modulators. The method simultaneously optimizes many parameters in a chosen design scheme; computational effort scales mildly with the number of parameters considered. Significant improvement in guiding efficiency relative to intuitively reasonable initial parameter choices is obtained in all cases.

© 1999 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  2. D. Marcuse, Theory of Dielectric Waveguides, 2nd ed. (Academic, New York, 1991).
  3. See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).
  4. S. Shi, H. Rabitz, “Quantum mechanical optimal control of physical observables in microsystems,” J. Chem. Phys. 92, 364–376 (1990);P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, M. Shayegan, “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B 49, 11,100–11,110 (1994).
    [CrossRef]
  5. D. K. Pant, R. D. Coalson, M. I. Hernandez, J. Campos-Martinez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16(2), 292–300 (1998).
    [CrossRef]
  6. D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
    [CrossRef]
  7. D.-S. Min, “Channeling devices for high speed signals in integrated optics and circuits,” Ph.D. dissertation (University of Pittsburgh, Pittsburgh, Pa., 1998).
  8. O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
    [CrossRef]
  9. M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
    [CrossRef]
  10. H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
    [CrossRef]
  11. H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
    [CrossRef]
  12. L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
    [CrossRef]
  13. D. A. McQuarrie, Quantum Chemistry (University Science, Mill Valley, Calif., 1983); E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970).
  14. G. B. Hocker, W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [CrossRef] [PubMed]
  15. K. S. Chiang, “Analysis of optical fibers by the effective index method,” Appl. Opt. 25, 348–354 (1986).
    [CrossRef] [PubMed]
  16. Note that V(x, z) ≅ -kΔn(x, z), where Δn(x, z) ≡ neff(x, z) - n0 is the deviation of the index from the reference value n0, which in the waveguides of interest here is much less than the reference value itself, i.e., Δn/n0 ≪ 1.
  17. M. D. Feit, J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]
  18. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  19. Criteria other than output quality can in principle be included in the cost function. Some possibilities were considered in Ref. 5. Here for simplicity we assume that output quality is the only important issue.
  20. Other criteria for optimal guiding are possible, for example, maximization of the integrated beam intensity within the boundaries of the guide at its output. There exists an appropriate projection operator analogous to the one given in Eq. (5) for each optimal guiding criterion.
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).
  22. To check that the paraxial equation yields an accurate approximation to the full Helmholtz equation, we utilized the numerical method presented in Ref. 23 to solve the latter equation for several parameter sets considered in this paper. The results were found to be essentially identical to those obtained from the paraxial equation, giving us confidence that the paraxial equation suffices for the systems of interest here. Development of an OCT procedure for direct optimization of wave propagation according to the Helmholtz equation presents an interesting problem for further research.
  23. S. Banerjee, A. Sharma, “Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields,” J. Opt. Soc. Am. A 6, 1884–1894 (1989).
    [CrossRef]
  24. The calculation presented in Fig. 8, with five adjustable parameters, required approximately 50 iterations and took approximately 90 min of CPU time on a low-end work station.
  25. J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
    [CrossRef]
  26. J. S. Cites, P. R. Ashley, “High performance Mach-Zehnder modulators in multiple quantum well GaAs/AlGaAs,” J. Lightwave Technol. 12, 1167–1173 (1994).
    [CrossRef]

1998 (1)

1997 (2)

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

1996 (1)

H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
[CrossRef]

1995 (1)

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

1994 (2)

H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
[CrossRef]

J. S. Cites, P. R. Ashley, “High performance Mach-Zehnder modulators in multiple quantum well GaAs/AlGaAs,” J. Lightwave Technol. 12, 1167–1173 (1994).
[CrossRef]

1990 (2)

S. Shi, H. Rabitz, “Quantum mechanical optimal control of physical observables in microsystems,” J. Chem. Phys. 92, 364–376 (1990);P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, M. Shayegan, “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B 49, 11,100–11,110 (1994).
[CrossRef]

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

1989 (1)

1986 (1)

1985 (1)

O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
[CrossRef]

1981 (1)

1980 (1)

1977 (1)

Ashley, P. R.

J. S. Cites, P. R. Ashley, “High performance Mach-Zehnder modulators in multiple quantum well GaAs/AlGaAs,” J. Lightwave Technol. 12, 1167–1173 (1994).
[CrossRef]

Banerjee, S.

Burns, W. K.

Campos-Martinez, J.

Chan, H. P.

H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
[CrossRef]

Cheng, S. Y.

H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
[CrossRef]

Cherin, A. H.

See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).

Chiang, K. S.

Chu, P. L.

H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
[CrossRef]

Chung, P. S.

H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
[CrossRef]

Cites, J. S.

J. S. Cites, P. R. Ashley, “High performance Mach-Zehnder modulators in multiple quantum well GaAs/AlGaAs,” J. Lightwave Technol. 12, 1167–1173 (1994).
[CrossRef]

Coalson, R. D.

D. K. Pant, R. D. Coalson, M. I. Hernandez, J. Campos-Martinez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16(2), 292–300 (1998).
[CrossRef]

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

Feit, M. D.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Fleck, J. A.

Hanaizumi, O.

O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
[CrossRef]

Hatami-Hanza, H.

H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
[CrossRef]

Hernandez, M. I.

Hocker, G. B.

Hu, M. H.

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

Huang, J. Z.

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Jones, K. L.

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

Kawakami, K.

O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
[CrossRef]

Koren, U.

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

Lagasse, P. E.

Langer, D. W.

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

Lederer, M. J.

H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
[CrossRef]

Levy, M.

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Waveguides, 2nd ed. (Academic, New York, 1991).

McQuarrie, D. A.

D. A. McQuarrie, Quantum Chemistry (University Science, Mill Valley, Calif., 1983); E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970).

Miller, B. I.

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

Min, D.-S.

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

D.-S. Min, “Channeling devices for high speed signals in integrated optics and circuits,” Ph.D. dissertation (University of Pittsburgh, Pittsburgh, Pa., 1998).

Miyagi, M.

O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
[CrossRef]

Osgood, R. M.

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

Pant, D. K.

D. K. Pant, R. D. Coalson, M. I. Hernandez, J. Campos-Martinez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16(2), 292–300 (1998).
[CrossRef]

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Rabitz, H.

S. Shi, H. Rabitz, “Quantum mechanical optimal control of physical observables in microsystems,” J. Chem. Phys. 92, 364–376 (1990);P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, M. Shayegan, “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B 49, 11,100–11,110 (1994).
[CrossRef]

Scarmozzino, R.

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

Sharma, A.

Shi, S.

S. Shi, H. Rabitz, “Quantum mechanical optimal control of physical observables in microsystems,” J. Chem. Phys. 92, 364–376 (1990);P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, M. Shayegan, “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B 49, 11,100–11,110 (1994).
[CrossRef]

Soldano, L. B.

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

van der Donk, J.

Van Roey, J.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

Zucker, J. E.

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (1)

H. P. Chan, S. Y. Cheng, P. S. Chung, “Low-loss wide-angle symmetric Y-branch waveguide,” Electron. Lett. 32(7), 652–654 (1996);“Novel design of low-loss wide-angle asymmetric Y-branch waveguides,” Microwave Opt. Technol. Lett. 111(2), 87–89 (1996).
[CrossRef]

Fiber Integr. Opt. (1)

D.-S. Min, D. W. Langer, D. K. Pant, R. D. Coalson, “Wide angle low-loss waveguide branching for integrated optics,” Fiber Integr. Opt. 16, 331–342 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. Hanaizumi, M. Miyagi, K. Kawakami, “Wide Y-junctions with low losses in three dimensional dielectric optical waveguides,” IEEE J. Quantum Electron. QE-21(2), 168–173 (1985).
[CrossRef]

IEEE Photon. Lett. (1)

H. Hatami-Hanza, P. L. Chu, M. J. Lederer, “A new low-loss wide angle Y-branch configuration for optical dielectric slab waveguides,” IEEE Photon. Lett. 6(4), 528–530 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. E. Zucker, K. L. Jones, B. I. Miller, U. Koren, “Miniature Mach-Zehnder InGaAsP quantum well waveguide interferometers for 1.3 µm,” IEEE Photon. Technol. Lett. 2(1), 32–34 (1990).
[CrossRef]

M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, R. M. Osgood, “A low-loss and compact waveguide Y-branch using refractive index tapering,” IEEE Photon. Technol. Lett. 9(2), 203–205 (1997).
[CrossRef]

J. Chem. Phys. (1)

S. Shi, H. Rabitz, “Quantum mechanical optimal control of physical observables in microsystems,” J. Chem. Phys. 92, 364–376 (1990);P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, M. Shayegan, “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B 49, 11,100–11,110 (1994).
[CrossRef]

J. Lightwave Technol. (3)

D. K. Pant, R. D. Coalson, M. I. Hernandez, J. Campos-Martinez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16(2), 292–300 (1998).
[CrossRef]

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

J. S. Cites, P. R. Ashley, “High performance Mach-Zehnder modulators in multiple quantum well GaAs/AlGaAs,” J. Lightwave Technol. 12, 1167–1173 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (11)

The calculation presented in Fig. 8, with five adjustable parameters, required approximately 50 iterations and took approximately 90 min of CPU time on a low-end work station.

Criteria other than output quality can in principle be included in the cost function. Some possibilities were considered in Ref. 5. Here for simplicity we assume that output quality is the only important issue.

Other criteria for optimal guiding are possible, for example, maximization of the integrated beam intensity within the boundaries of the guide at its output. There exists an appropriate projection operator analogous to the one given in Eq. (5) for each optimal guiding criterion.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992).

To check that the paraxial equation yields an accurate approximation to the full Helmholtz equation, we utilized the numerical method presented in Ref. 23 to solve the latter equation for several parameter sets considered in this paper. The results were found to be essentially identical to those obtained from the paraxial equation, giving us confidence that the paraxial equation suffices for the systems of interest here. Development of an OCT procedure for direct optimization of wave propagation according to the Helmholtz equation presents an interesting problem for further research.

D. A. McQuarrie, Quantum Chemistry (University Science, Mill Valley, Calif., 1983); E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970).

Note that V(x, z) ≅ -kΔn(x, z), where Δn(x, z) ≡ neff(x, z) - n0 is the deviation of the index from the reference value n0, which in the waveguides of interest here is much less than the reference value itself, i.e., Δn/n0 ≪ 1.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

D. Marcuse, Theory of Dielectric Waveguides, 2nd ed. (Academic, New York, 1991).

See, for example, A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, 1983).

D.-S. Min, “Channeling devices for high speed signals in integrated optics and circuits,” Ph.D. dissertation (University of Pittsburgh, Pittsburgh, Pa., 1998).

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Figures (13)

Fig. 1
Fig. 1

Typical Y-branch waveguide.

Fig. 2
Fig. 2

Plot of -|〈ψtarget|ψ(z = L)〉|2 versus MMI length for α = 2. The fixed values of the other parameters are MMI width, 4 µm, and n (wedge), 3.558.

Fig. 3
Fig. 3

-|〈ψtarget|ψ(z = L)〉|2 versus branch angle α for the unoptimized (upper curve) and optimized case (lower curve).

Fig. 4
Fig. 4

Plot of the intensities of the wave function at z = L (unoptimized case) and the target wave function [MMI width, 4 µm; MMI length, 100 µm; n(wedge), 3.558; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.28.

Fig. 5
Fig. 5

Plot of the intensities of the wave function at z = L (optimized case) and the target wave function [MMI width, 3.81 µm; MMI length, 98.4 µm; n(wedge), 3.5662; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.97.

Fig. 6
Fig. 6

Contour plots of the waveguide and the beam (unoptimized case) [MMI width, 4 µm; MMI length, 100 µm; n(wedge), 3.558; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.28.

Fig. 7
Fig. 7

Contour plots of the waveguide and the beam (optimized case) [MMI width, 3.81 µm; MMI length, 98.4 µm; n(wedge), 3.5662; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.97.

Fig. 8
Fig. 8

Contour plots of the waveguide and the beam (optimized case) with two Fourier coefficients [MMI width, 3.75 µm; MMI length, 91.3 µm; n(wedge), 3.561; b 1 = -0.15, b 3 = 0.16; α = 1].

Fig. 9
Fig. 9

Typical Mach–Zehnder waveguide.

Fig. 10
Fig. 10

Plot of the intensities of the wave function at z = L (unoptimized case) and the target wave function for the Mach–Zehnder device [MMI width, 4 µm; MMI length, 55 µm; n(wedge), 3.558; b 2 = 0.0, b 3 = 0.0; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.26.

Fig. 11
Fig. 11

Plot of the intensities of the wave function at z = L (optimized case) and the target wave function for the Mach–Zehnder device [MMI width, 3.7 µm; MMI length, 66.2 µm; n(wedge), 3.568; b 2 = 0.138, b 3 = -0.137; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.96.

Fig. 12
Fig. 12

Contour plots of the waveguide and the beam (unoptimized case) for the Mach–Zehnder device [MMI width, 4 µm; MMI length, 55 µm; n(wedge), 3.558; b 2 = 0.0, b 3 = 0.0; α = 1], |〈ψtarget|ψ(z = L)〉|2 = 0.26.

Fig. 13
Fig. 13

Contour plots of the waveguide and the beam (optimized case) for the M–Z device [MMI width, 3.7 µm; MMI length,66.2 µm; n(wedge), 3.568; b 2 = 0.138, b 3 = -0.137; α = 1],|〈ψtarget|ψ(z = L)〉|2 = 0.96.

Tables (2)

Tables Icon

Table 1 Fixed Structural and Refractive-Index Parameters

Tables Icon

Table 2 Optimized Parameters

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

i ψx, y, zz=-12β0 T2+k2n0n02-nx, y, z2ψx, y, z.
i ψx, zz=-12β02x2+k2n0n02-neff2x, zψx, z.
i ψx, zz=Hˆzψx, z,
Tˆ-12β02x2; Vx, zk2n0n02-neff2x, z.
Oˆ|ψtargetψtarget|.
Jp=-ψL|Oˆ|ψL=-|ψtarget|ψL|2.
Jpj=2Re 0LdzλzHpjψz,
|λL=-iOˆ|ψL.
i |λzz=Hˆz|λz.
λx, L=-iψtarget|ψLψtargetx,
Lout=xmid-W/2,Rout=xmid+W/2,
Lout=xmid-mW/2=LMMI,Rout=xmid+mW/2=RMMI.
Lout=LMMI-x3z1z-a2+j bj sinjπz-a2/z1,Rout=RMMI+x3z1z-a2-j bj sinjπz-a2/z1,Lin=RMMI-x2z1z-a2+j bj sinjπz-a2/z1,Rin=LMMI+x2z1z-a2-j bj sinjπz-a2/z1,
Vx, z=k2n0n02-nx, z2,
HpjVpj-k2n0n2pj,

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