## Optimal control theory for optical waveguide design: application to *Y*-branch structures

Applied Optics, Vol. 38, Issue 18, pp. 3917-3923 (1999)

http://dx.doi.org/10.1364/AO.38.003917

Enhanced HTML Acrobat PDF (162 KB)

### 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

**OCIS Codes**

(230.1360) Optical devices : Beam splitters

(230.7370) Optical devices : Waveguides

(350.5500) Other areas of optics : Propagation

**History**

Original Manuscript: December 8, 1998

Revised Manuscript: March 18, 1999

Published: June 20, 1999

**Citation**

Dhruv K. Pant, Rob D. Coalson, Marta I. Hernández, and José Campos-Martínez, "Optimal control theory for optical waveguide design: application to Y-branch structures," Appl. Opt. **38**, 3917-3923 (1999)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-18-3917

Sort: Year | Journal | Reset

### References

- 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).
- 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]
- 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]
- D.-S. Min, “Channeling devices for high speed signals in integrated optics and circuits,” Ph.D. dissertation (University of Pittsburgh, Pittsburgh, Pa., 1998).
- 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]
- 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]
- 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]
- 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]
- 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]
- D. A. McQuarrie, Quantum Chemistry (University Science, Mill Valley, Calif., 1983); E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970).
- 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]
- K. S. Chiang, “Analysis of optical fibers by the effective index method,” Appl. Opt. 25, 348–354 (1986). [CrossRef] [PubMed]
- 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.
- 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]
- 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]
- 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.
- 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]
- 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.
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1 |
Fig. 2 |
Fig. 3 |

Fig. 4 |
Fig. 5 |
Fig. 6 |

Fig. 7 |
Fig. 8 |
Fig. 9 |

Fig. 10 |
Fig. 11 |
Fig. 12 |

Fig. 13 |
||

« Previous Article | Next Article »

OSA is a member of CrossRef.