Abstract

The coupled-mode theory (CMT) for optical waveguides is reviewed, with emphasis on the analysis of coupled optical waveguides. A brief account of the recent development of the CMT for coupled optical waveguides is given. Issues raised in the debates of the 1980’s on the merits and shortcomings of the conventional as well as the improved coupled-mode formulations are discussed. The conventional coupled-mode formulations are set up in a simple, intuitive way. The rigorous CMT is established on the basis of a linear superposition of the modes for individual waveguides. The cross-power terms appear logically as a result of modal nonorthogonality. The cross power is necessary for the self-consistency of the CMT for dissimilar waveguides. The nonorthogonal CMT, though more complicated, yields more-accurate results than the conventional orthogonal CMT for most practical applications. It also leads to the prediction of cross talk in directional couplers. The conventional orthogonal CMT is, however, reliably accurate for describing the power coupling between two weakly coupled, nearly identical waveguides. For dissimilar waveguides, a self-consistent orthogonal CMT can be derived by a redefinition of the coupling coefficients, and it predicts the coupling length and therefore the power exchange between the waveguides accurately if the two waveguides are far apart. Three typical coupler configurations—the uniform, the grating-assisted, and the tapered—are examined in detail. The accuracy, scope of validity, limitations, and extensions of the coupled-mode formulations are discussed in conjunction with each configuration. To verify the arguments in the discussions, comparisons with the exact analytical solutions and the rigorous numerical simulations are made.

© 1994 Optical Society of America

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  92. W. P. Huang, J. Hong, Z. M. Mao, “An improved coupled-mode formulation for grating-assisted co-directional couplers,” IEEE J. Quantum Electron. (to be published).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1993 (1)

S. Lessard, W. P. Huang, “Assessment of coupled-mode theory for tapered optical coupler,” J. Lightwave Technol. 11, 405–407 (1993).
[CrossRef]

1992 (9)

W. P. Huang, B. E. Little, C. L. Xu, “On phase-matching and power coupling in grating-assisted couplers,” IEEE Photon. Technol. Lett. 4, 151–153 (1992).
[CrossRef]

J. Hong, W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. 10, 873–881 (1992).
[CrossRef]

W. P. Huang, B. E. Little, “Power exchange in tapered optical couplers,” IEEE J. Quantum Electron. 27, 1932–1938 (1992).
[CrossRef]

W. P. Huang, S. Lessard, “Wavefront-tilt in nonparallel optical waveguides,” J. Lightwave Technol. 10, 316–322 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, J. Hong, “A transfer matrix approach based on local modes for coupled waveguides with periodic perturbations,” J. Lightwave Technol. 10, 1367–1374 (1992).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A scalar coupled-mode theory with vector correction,” J. Quantum Electron. 28, 184–193 (1992).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 10, 1254–1263 (1992).

G. Griffle, M. Itzkovich, A. A. Hardy, “Coupled-mode formulations for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 28, 985–994 (1992).

1991 (6)

G. Griffle, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

W. P. Huang, B. E. Little, S. K. Chaudhuri, “A new approach to grating-assisted couplers,” J. Lightwave Technol. 9, 721–727 (1991).
[CrossRef]

W. P. Huang, W. Y. Lit, “Nonorthogonal coupled-mode theory of grating-assisted codirectional couplers,” J. Lightwave Technol. 9, 845–852 (1991).
[CrossRef]

H. A. Haus, W P. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

Y. Chen, A. W. Snyder, “Grating-assisted couplers,” Opt. Lett. 16, 217–219 (1991).
[CrossRef] [PubMed]

R. R. A. Syms, “Improved coupled-mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8, 1062–1069 (1991).
[CrossRef]

1990 (7)

D. G. Hall, “Coupled-mode theory for corrugated optical waveguides,” Opt. Lett. 15, 619–621 (1990).
[CrossRef] [PubMed]

D. Marcuse, “Radiation loss of grating-assisted directional coupler,” J. Lightwave Technol. 8, 675–684 (1990).

W. P. Huang, S. K. Chaudhuri, “Variational coupled-mode theory of optical couplers,” J. Lightwave Technol. 8, 1565–1570 (1990).
[CrossRef]

Y. Cai, T. Mizumoto, Y. Naito, “Analysis of the coupling characteristics of a tapered coupled waveguide system,” J. Lightwave Technol. 8, 90–98 (1990).
[CrossRef]

W. P. Huang, H. A. Haus, “Self-consistent vector coupled-mode theory for tapered optical waveguides,” J. Lightwave Technol. 8, 922–926 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part I: the identical-core,” J. Lightwave Technol. 8, 823–831 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part II: the nonidentical-core,” J. Lightwave Technol. 8, 832–837 (1990).
[CrossRef]

1989 (13)

J. P. Donnelly, L. A. Molter, H. A. Haus, “The extinction ratio in optical two-guide coupler Δβ switches,” IEEE J. Quantum Electron. 25, 924–932 (1989).
[CrossRef]

H. S. Huang, H. C. Chang, “Analytical expressions for the coupling between two optical fiber cores with a-power refractive-index distribution,” J. Lightwave Technol. 7, 694–702 (1989).
[CrossRef]

Y. Chen, “Solutions to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

W. P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” J. Lightwave Technol. 7, 920–924 (1989).
[CrossRef]

R. R. A. Syms, R. G. Peall, “The digital optical switch: analogous directional coupler devices,” Opt. Commun. 68, 235–238 (1989).
[CrossRef]

H. A. Haus, W. P. Huang, “Mode coupling in tapered structures,” J. Lightwave Technol. 7, 729–730 (1989).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

A. W. Snyder, Y. Chen, A. Ankiewicz, “Coupled waves on optical fibers by power conservation,” J. Lightwave Technol. 7, 1400–1406 (1989).
[CrossRef]

Y. Shama, A. Hardy, E. Marom, “Multimode coupling of unidentical waveguides,” J. Lightwave Technol. 7, 420–425 (1989).
[CrossRef]

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

H. S. Huang, H. C. Chang, “Vector coupled-mode analysis of coupling between two identical optical fiber cores,” Opt. Lett. 14, 90–92 (1989).
[CrossRef] [PubMed]

H. A. Haus, W P. Huang, A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14, 1222–1224 (1989).
[CrossRef] [PubMed]

Y. Shama, E. Marom, A. Hardy, “Analysis of power transfer in nonsymmetric directional couplers,” Appl. Opt. 28, 990–994 (1989).
[CrossRef] [PubMed]

1988 (16)

A. Ankiewicz, A. Altintas, A. W. Snyder, “Polarization properties of evanescent couplers,” Opt. Lett. 13, 524–525 (1988).
[CrossRef] [PubMed]

Z. H. Wang, S. R. Seshadri, “Asymptotic theory of guided modes in two parallel, identical dielectric waveguides,” J. Opt. Soc. Am. A 5, 782–792 (1988).
[CrossRef]

F. Tian, Y. Z. Wu, P. A. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” J. Lightwave Tech-nol. 6, 304–311 (1988).
[CrossRef]

C. Vassello, “About coupled-mode theories for dielectric waveguides,” J. Lightwave Technol. 6, 294–303 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1988).
[CrossRef]

W. Streifer, “Comment on ‘Fundamental error of recent coupled mode formulations,’” Electron. Lett. 22, 718–719 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Coupled mode theory neglects polarization phenomena” (reply to Ref. 26), Electron. Lett. 22, 720–721 (1988).
[CrossRef]

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 162–163; erratum, 428 (1988).

Y. Wu, “Discussion of HS formulation using equivalent current theory,” Electron. Lett. 24, 376–377 (1988).
[CrossRef]

A. W. Snyder, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
[CrossRef]

R. R. A. Syms, R. G. Peall, “Explanation of asymmetric switch response of three-arm directional couplers in Ti:LiNbO3using strong coupling theory,” Opt. Commun. 66, 260–264 (1988).
[CrossRef]

J. P. Donnelly, H. A. Haus, L. A. Molter, “Cross power and crosstalk in waveguide couplers,” J. Lightwave Technol. 6, 257–268 (1988).
[CrossRef]

Y. Tomabechi, K. Matsumura, “Improved analysis for the coupling characteristics of two rectangular dielectric waveguides laid in different layers,” IEEE J. Quantum Electron. 24, 2359–2361 (1988).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Scalar strong coupled mode theory for slowly-varying waveguide arrays,” Opt. Commun. 67, 421–424 (1988).
[CrossRef]

1987 (9)

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three-guide optical coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

D. Marcuse, “Directional couplers made of nonidentical asymmetrical slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. LT-5, 268–273 (1987).
[CrossRef]

H. A. Haus, W P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” J. Lightwave Technol. LT-5, 16–23 (1987).
[CrossRef]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “Reformulation of coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Fundamental error of recent coupled mode formulations,” Electron. Lett. 23, 1097–1098 (1987).
[CrossRef]

W. Streifer, “Coupled mode theory,” Electron. Lett. 23, 216–217 (1987).
[CrossRef]

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. LT-5, 174–183 (1987).
[CrossRef]

1986 (11)

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

C. Vassallo, “Condensed formula for coupling coefficients between parallel dielectric waveguides,” Electron. Lett. 23, 304–306 (1986).

E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

H. A. Haus, “Coupled-mode theory revisited,” in Fiber Optics, Optoelectronics and Laser Applications in Science and Engineering, Proc. Soc. Photo-Opt. Instrum. Eng. (1986).

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 14, 324–336 (1986).
[CrossRef]

J. R. Qian, “Generalized coupled-mode equations and applications to fiber couplers,” Electron. Lett. 22, 304–306 (1986).
[CrossRef]

A. Ankiewicz, A. W. Snyder, X. Zheng, “Coupling between parallel optical fiber cores—critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[CrossRef]

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[CrossRef]

A. Hardy, M. Osiński, W. Streifer, “Application of coupled-mode theory to nearly parallel waveguide systems,” Electron. Lett. 22, 1249–1250 (1986).
[CrossRef]

A. Hardy, W. Streifer, M. Osinski, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef] [PubMed]

1985 (3)

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[CrossRef] [PubMed]

H. A. Haus, N. A. Whitaker, “Elimination of cross talk in optical directional couplers,” Appl. Phys. Lett. 46, 1–3 (1985).
[CrossRef]

A. Hardy, W. Streifer, “Coupled-mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

1984 (1)

K. Chen, S. Wang, “Cross-talk problems in optical directional couplers,” Appl. Phys. Lett. 44, 166–168 (1984).
[CrossRef]

1979 (1)

A. Milton, W. K. Burns, “Mode coupling in tapered optical waveguide structures and electro-optic switches,” IEEE Trans. Circ. Syst. CS-26, 1020–1028 (1979).
[CrossRef]

1978 (1)

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. QE-14, 843–847 (1978).
[CrossRef]

1976 (1)

H. Kogelnik, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

1973 (2)

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–842 (1973).

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

1972 (2)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

1970 (1)

A. W. Snyder, “Coupling of modes on a tapered dielectric cylinder,” IEEE Trans. Microwave Theor. Technol. MTT-18, 383–392 (1970).
[CrossRef]

1965 (1)

A. W. Snyder, “Surface mode coupling along a tapered dielectric rod,” IEEE Trans. Antennas Propag. AP-13, 821–822 (1965).
[CrossRef]

1955 (1)

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

1954 (2)

J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719, (1954).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

Alferness, R. C.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[CrossRef]

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. QE-14, 843–847 (1978).
[CrossRef]

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Altintas, A.

A. W. Snyder, A. Ankiewicz, A. Altintas, “Coupled mode theory neglects polarization phenomena” (reply to Ref. 26), Electron. Lett. 22, 720–721 (1988).
[CrossRef]

A. Ankiewicz, A. Altintas, A. W. Snyder, “Polarization properties of evanescent couplers,” Opt. Lett. 13, 524–525 (1988).
[CrossRef] [PubMed]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Fundamental error of recent coupled mode formulations,” Electron. Lett. 23, 1097–1098 (1987).
[CrossRef]

Ankiewicz, A.

A. W. Snyder, Y. Chen, A. Ankiewicz, “Coupled waves on optical fibers by power conservation,” J. Lightwave Technol. 7, 1400–1406 (1989).
[CrossRef]

A. Ankiewicz, A. Altintas, A. W. Snyder, “Polarization properties of evanescent couplers,” Opt. Lett. 13, 524–525 (1988).
[CrossRef] [PubMed]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Coupled mode theory neglects polarization phenomena” (reply to Ref. 26), Electron. Lett. 22, 720–721 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Fundamental error of recent coupled mode formulations,” Electron. Lett. 23, 1097–1098 (1987).
[CrossRef]

A. Ankiewicz, A. W. Snyder, X. Zheng, “Coupling between parallel optical fiber cores—critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[CrossRef]

Baets, R.

J. Willems, J. Haes, R. Baets, G. Sztefka, H. P. Nolting, “Eigenmode propagation analysis of radiation losses in waveguides with discontinuities and grating-assisted couplers,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 229.

Buhl, L. L.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[CrossRef]

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Burns, W. K.

A. Milton, W. K. Burns, “Mode coupling in tapered optical waveguide structures and electro-optic switches,” IEEE Trans. Circ. Syst. CS-26, 1020–1028 (1979).
[CrossRef]

Burrus, C. A.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Cai, Y.

Y. Cai, T. Mizumoto, Y. Naito, “Analysis of the coupling characteristics of a tapered coupled waveguide system,” J. Lightwave Technol. 8, 90–98 (1990).
[CrossRef]

Chang, H. C.

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part II: the nonidentical-core,” J. Lightwave Technol. 8, 832–837 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part I: the identical-core,” J. Lightwave Technol. 8, 823–831 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Vector coupled-mode analysis of coupling between two identical optical fiber cores,” Opt. Lett. 14, 90–92 (1989).
[CrossRef] [PubMed]

H. S. Huang, H. C. Chang, “Analytical expressions for the coupling between two optical fiber cores with a-power refractive-index distribution,” J. Lightwave Technol. 7, 694–702 (1989).
[CrossRef]

Chaudhuri, S. K.

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 10, 1254–1263 (1992).

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A scalar coupled-mode theory with vector correction,” J. Quantum Electron. 28, 184–193 (1992).
[CrossRef]

W. P. Huang, B. E. Little, S. K. Chaudhuri, “A new approach to grating-assisted couplers,” J. Lightwave Technol. 9, 721–727 (1991).
[CrossRef]

W. P. Huang, S. K. Chaudhuri, “Variational coupled-mode theory of optical couplers,” J. Lightwave Technol. 8, 1565–1570 (1990).
[CrossRef]

Chen, K.

K. Chen, S. Wang, “Cross-talk problems in optical directional couplers,” Appl. Phys. Lett. 44, 166–168 (1984).
[CrossRef]

Chen, Y.

Y. Chen, A. W. Snyder, “Grating-assisted couplers,” Opt. Lett. 16, 217–219 (1991).
[CrossRef] [PubMed]

A. W. Snyder, Y. Chen, A. Ankiewicz, “Coupled waves on optical fibers by power conservation,” J. Lightwave Technol. 7, 1400–1406 (1989).
[CrossRef]

Y. Chen, “Solutions to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A scalar coupled-mode theory with vector correction,” J. Quantum Electron. 28, 184–193 (1992).
[CrossRef]

Chuang, S. L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” J. Lightwave Tech-nol. 6, 304–311 (1988).
[CrossRef]

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. LT-5, 174–183 (1987).
[CrossRef]

Cross, P. S.

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. QE-14, 843–847 (1978).
[CrossRef]

Donnelly, J. P.

J. P. Donnelly, L. A. Molter, H. A. Haus, “The extinction ratio in optical two-guide coupler Δβ switches,” IEEE J. Quantum Electron. 25, 924–932 (1989).
[CrossRef]

J. P. Donnelly, H. A. Haus, L. A. Molter, “Cross power and crosstalk in waveguide couplers,” J. Lightwave Technol. 6, 257–268 (1988).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three-guide optical coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

Griffle, G.

G. Griffle, M. Itzkovich, A. A. Hardy, “Coupled-mode formulations for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 28, 985–994 (1992).

G. Griffle, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

Haes, J.

J. Willems, J. Haes, R. Baets, G. Sztefka, H. P. Nolting, “Eigenmode propagation analysis of radiation losses in waveguides with discontinuities and grating-assisted couplers,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 229.

Hall, D. G.

D. G. Hall, “Coupled-mode theory for corrugated optical waveguides,” Opt. Lett. 15, 619–621 (1990).
[CrossRef] [PubMed]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Hardy, A.

Y. Shama, A. Hardy, E. Marom, “Multimode coupling of unidentical waveguides,” J. Lightwave Technol. 7, 420–425 (1989).
[CrossRef]

Y. Shama, E. Marom, A. Hardy, “Analysis of power transfer in nonsymmetric directional couplers,” Appl. Opt. 28, 990–994 (1989).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 162–163; erratum, 428 (1988).

W. Streifer, M. Osinski, A. Hardy, “Reformulation of coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

A. Hardy, W. Streifer, M. Osinski, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef] [PubMed]

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 14, 324–336 (1986).
[CrossRef]

A. Hardy, M. Osiński, W. Streifer, “Application of coupled-mode theory to nearly parallel waveguide systems,” Electron. Lett. 22, 1249–1250 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, “Coupled-mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “A critical review of coupled mode theory,” in Integrated Optical Circuit Engineering V, M. A. Mentzer, ed., Proc. Soc. Photo-Opt. In-strum. Eng.835, 178 (1987).
[CrossRef]

Hardy, A. A.

G. Griffle, M. Itzkovich, A. A. Hardy, “Coupled-mode formulations for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 28, 985–994 (1992).

Haus, H. A.

H. A. Haus, W P. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

W. P. Huang, H. A. Haus, “Self-consistent vector coupled-mode theory for tapered optical waveguides,” J. Lightwave Technol. 8, 922–926 (1990).
[CrossRef]

J. P. Donnelly, L. A. Molter, H. A. Haus, “The extinction ratio in optical two-guide coupler Δβ switches,” IEEE J. Quantum Electron. 25, 924–932 (1989).
[CrossRef]

H. A. Haus, W. P. Huang, “Mode coupling in tapered structures,” J. Lightwave Technol. 7, 729–730 (1989).
[CrossRef]

W. P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” J. Lightwave Technol. 7, 920–924 (1989).
[CrossRef]

H. A. Haus, W P. Huang, A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14, 1222–1224 (1989).
[CrossRef] [PubMed]

J. P. Donnelly, H. A. Haus, L. A. Molter, “Cross power and crosstalk in waveguide couplers,” J. Lightwave Technol. 6, 257–268 (1988).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three-guide optical coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

H. A. Haus, W P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” J. Lightwave Technol. LT-5, 16–23 (1987).
[CrossRef]

H. A. Haus, “Coupled-mode theory revisited,” in Fiber Optics, Optoelectronics and Laser Applications in Science and Engineering, Proc. Soc. Photo-Opt. Instrum. Eng. (1986).

H. A. Haus, N. A. Whitaker, “Elimination of cross talk in optical directional couplers,” Appl. Phys. Lett. 46, 1–3 (1985).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J.1984).

H. A. Haus, “Electron beam waves in microwave tubes,” in Proceedings of the Symposium on Eectronic Waveguides (Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1958).

Heismann, F.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

Hong, J.

W. P. Huang, J. Hong, “A transfer matrix approach based on local modes for coupled waveguides with periodic perturbations,” J. Lightwave Technol. 10, 1367–1374 (1992).
[CrossRef]

J. Hong, W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. 10, 873–881 (1992).
[CrossRef]

W. P. Huang, J. Hong, Z. M. Mao, “An improved coupled-mode formulation for grating-assisted co-directional couplers,” IEEE J. Quantum Electron. (to be published).

Huang, H. S.

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part I: the identical-core,” J. Lightwave Technol. 8, 823–831 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Analysis of optical fiber directional coupling based on the HE11 modes. Part II: the nonidentical-core,” J. Lightwave Technol. 8, 832–837 (1990).
[CrossRef]

H. S. Huang, H. C. Chang, “Vector coupled-mode analysis of coupling between two identical optical fiber cores,” Opt. Lett. 14, 90–92 (1989).
[CrossRef] [PubMed]

H. S. Huang, H. C. Chang, “Analytical expressions for the coupling between two optical fiber cores with a-power refractive-index distribution,” J. Lightwave Technol. 7, 694–702 (1989).
[CrossRef]

Huang, W P.

H. A. Haus, W P. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

H. A. Haus, W P. Huang, A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14, 1222–1224 (1989).
[CrossRef] [PubMed]

H. A. Haus, W P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” J. Lightwave Technol. LT-5, 16–23 (1987).
[CrossRef]

Huang, W. P.

S. Lessard, W. P. Huang, “Assessment of coupled-mode theory for tapered optical coupler,” J. Lightwave Technol. 11, 405–407 (1993).
[CrossRef]

W. P. Huang, S. Lessard, “Wavefront-tilt in nonparallel optical waveguides,” J. Lightwave Technol. 10, 316–322 (1992).
[CrossRef]

W. P. Huang, J. Hong, “A transfer matrix approach based on local modes for coupled waveguides with periodic perturbations,” J. Lightwave Technol. 10, 1367–1374 (1992).
[CrossRef]

W. P. Huang, B. E. Little, C. L. Xu, “On phase-matching and power coupling in grating-assisted couplers,” IEEE Photon. Technol. Lett. 4, 151–153 (1992).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A scalar coupled-mode theory with vector correction,” J. Quantum Electron. 28, 184–193 (1992).
[CrossRef]

J. Hong, W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. 10, 873–881 (1992).
[CrossRef]

W. P. Huang, B. E. Little, “Power exchange in tapered optical couplers,” IEEE J. Quantum Electron. 27, 1932–1938 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 10, 1254–1263 (1992).

W. P. Huang, W. Y. Lit, “Nonorthogonal coupled-mode theory of grating-assisted codirectional couplers,” J. Lightwave Technol. 9, 845–852 (1991).
[CrossRef]

W. P. Huang, B. E. Little, S. K. Chaudhuri, “A new approach to grating-assisted couplers,” J. Lightwave Technol. 9, 721–727 (1991).
[CrossRef]

W. P. Huang, H. A. Haus, “Self-consistent vector coupled-mode theory for tapered optical waveguides,” J. Lightwave Technol. 8, 922–926 (1990).
[CrossRef]

W. P. Huang, S. K. Chaudhuri, “Variational coupled-mode theory of optical couplers,” J. Lightwave Technol. 8, 1565–1570 (1990).
[CrossRef]

H. A. Haus, W. P. Huang, “Mode coupling in tapered structures,” J. Lightwave Technol. 7, 729–730 (1989).
[CrossRef]

W. P. Huang, H. A. Haus, “Power exchange in grating-assisted couplers,” J. Lightwave Technol. 7, 920–924 (1989).
[CrossRef]

W. P. Huang, J. Hong, Z. M. Mao, “An improved coupled-mode formulation for grating-assisted co-directional couplers,” IEEE J. Quantum Electron. (to be published).

Itzkovich, M.

G. Griffle, M. Itzkovich, A. A. Hardy, “Coupled-mode formulations for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 28, 985–994 (1992).

Kawakami, S.

H. A. Haus, W P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” J. Lightwave Technol. LT-5, 16–23 (1987).
[CrossRef]

Koch, T. L.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Kock, T. L.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
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H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.
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Koren, U.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Lee, D. L.

D. L. Lee, Electromagnetic Principle of Integrated Optics (Wiley, New York, 1986).

Lessard, S.

S. Lessard, W. P. Huang, “Assessment of coupled-mode theory for tapered optical coupler,” J. Lightwave Technol. 11, 405–407 (1993).
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W. P. Huang, S. Lessard, “Wavefront-tilt in nonparallel optical waveguides,” J. Lightwave Technol. 10, 316–322 (1992).
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Lit, W. Y.

W. P. Huang, W. Y. Lit, “Nonorthogonal coupled-mode theory of grating-assisted codirectional couplers,” J. Lightwave Technol. 9, 845–852 (1991).
[CrossRef]

Little, B. E.

W. P. Huang, B. E. Little, C. L. Xu, “On phase-matching and power coupling in grating-assisted couplers,” IEEE Photon. Technol. Lett. 4, 151–153 (1992).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 10, 1254–1263 (1992).

W. P. Huang, B. E. Little, “Power exchange in tapered optical couplers,” IEEE J. Quantum Electron. 27, 1932–1938 (1992).
[CrossRef]

W. P. Huang, B. E. Little, S. K. Chaudhuri, “A new approach to grating-assisted couplers,” J. Lightwave Technol. 9, 721–727 (1991).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Mao, Z. M.

W. P. Huang, J. Hong, Z. M. Mao, “An improved coupled-mode formulation for grating-assisted co-directional couplers,” IEEE J. Quantum Electron. (to be published).

Marcatili, E.

E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[CrossRef]

Marcuse, D.

D. Marcuse, “Radiation loss of grating-assisted directional coupler,” J. Lightwave Technol. 8, 675–684 (1990).

D. Marcuse, “Directional couplers made of nonidentical asymmetrical slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. LT-5, 268–273 (1987).
[CrossRef]

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–842 (1973).

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

Marom, E.

Y. Shama, E. Marom, A. Hardy, “Analysis of power transfer in nonsymmetric directional couplers,” Appl. Opt. 28, 990–994 (1989).
[CrossRef] [PubMed]

Y. Shama, A. Hardy, E. Marom, “Multimode coupling of unidentical waveguides,” J. Lightwave Technol. 7, 420–425 (1989).
[CrossRef]

Martyak, M. J. R.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

Matsumura, K.

Y. Tomabechi, K. Matsumura, “Improved analysis for the coupling characteristics of two rectangular dielectric waveguides laid in different layers,” IEEE J. Quantum Electron. 24, 2359–2361 (1988).
[CrossRef]

Miller, B. I.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Miller, S. E.

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719, (1954).

Milton, A.

A. Milton, W. K. Burns, “Mode coupling in tapered optical waveguide structures and electro-optic switches,” IEEE Trans. Circ. Syst. CS-26, 1020–1028 (1979).
[CrossRef]

Mizumoto, T.

Y. Cai, T. Mizumoto, Y. Naito, “Analysis of the coupling characteristics of a tapered coupled waveguide system,” J. Lightwave Technol. 8, 90–98 (1990).
[CrossRef]

Molter, L. A.

J. P. Donnelly, L. A. Molter, H. A. Haus, “The extinction ratio in optical two-guide coupler Δβ switches,” IEEE J. Quantum Electron. 25, 924–932 (1989).
[CrossRef]

J. P. Donnelly, H. A. Haus, L. A. Molter, “Cross power and crosstalk in waveguide couplers,” J. Lightwave Technol. 6, 257–268 (1988).
[CrossRef]

Naito, Y.

Y. Cai, T. Mizumoto, Y. Naito, “Analysis of the coupling characteristics of a tapered coupled waveguide system,” J. Lightwave Technol. 8, 90–98 (1990).
[CrossRef]

Nolting, H. P.

J. Willems, J. Haes, R. Baets, G. Sztefka, H. P. Nolting, “Eigenmode propagation analysis of radiation losses in waveguides with discontinuities and grating-assisted couplers,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 229.

Osinski, M.

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 162–163; erratum, 428 (1988).

W. Streifer, M. Osinski, A. Hardy, “Reformulation of coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

A. Hardy, W. Streifer, M. Osinski, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef] [PubMed]

A. Hardy, M. Osiński, W. Streifer, “Application of coupled-mode theory to nearly parallel waveguide systems,” Electron. Lett. 22, 1249–1250 (1986).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “A critical review of coupled mode theory,” in Integrated Optical Circuit Engineering V, M. A. Mentzer, ed., Proc. Soc. Photo-Opt. In-strum. Eng.835, 178 (1987).
[CrossRef]

Peall, R. G.

R. R. A. Syms, R. G. Peall, “The digital optical switch: analogous directional coupler devices,” Opt. Commun. 68, 235–238 (1989).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

R. R. A. Syms, R. G. Peall, “Explanation of asymmetric switch response of three-arm directional couplers in Ti:LiNbO3using strong coupling theory,” Opt. Commun. 66, 260–264 (1988).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Scalar strong coupled mode theory for slowly-varying waveguide arrays,” Opt. Commun. 67, 421–424 (1988).
[CrossRef]

Pierce, J. R.

J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

Qian, J. R.

J. R. Qian, “Generalized coupled-mode equations and applications to fiber couplers,” Electron. Lett. 22, 304–306 (1986).
[CrossRef]

Raybon, G.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Schelkunoff, S. A.

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

Seshadri, S. R.

Shakir, S.

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 14, 324–336 (1986).
[CrossRef]

Shama, Y.

Y. Shama, A. Hardy, E. Marom, “Multimode coupling of unidentical waveguides,” J. Lightwave Technol. 7, 420–425 (1989).
[CrossRef]

Y. Shama, E. Marom, A. Hardy, “Analysis of power transfer in nonsymmetric directional couplers,” Appl. Opt. 28, 990–994 (1989).
[CrossRef] [PubMed]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Shen, Y. R.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

Snyder, A. W.

Y. Chen, A. W. Snyder, “Grating-assisted couplers,” Opt. Lett. 16, 217–219 (1991).
[CrossRef] [PubMed]

A. W. Snyder, Y. Chen, A. Ankiewicz, “Coupled waves on optical fibers by power conservation,” J. Lightwave Technol. 7, 1400–1406 (1989).
[CrossRef]

H. A. Haus, W P. Huang, A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14, 1222–1224 (1989).
[CrossRef] [PubMed]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Coupled mode theory neglects polarization phenomena” (reply to Ref. 26), Electron. Lett. 22, 720–721 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1988).
[CrossRef]

A. Ankiewicz, A. Altintas, A. W. Snyder, “Polarization properties of evanescent couplers,” Opt. Lett. 13, 524–525 (1988).
[CrossRef] [PubMed]

A. W. Snyder, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Fundamental error of recent coupled mode formulations,” Electron. Lett. 23, 1097–1098 (1987).
[CrossRef]

A. Ankiewicz, A. W. Snyder, X. Zheng, “Coupling between parallel optical fiber cores—critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[CrossRef]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

A. W. Snyder, “Coupling of modes on a tapered dielectric cylinder,” IEEE Trans. Microwave Theor. Technol. MTT-18, 383–392 (1970).
[CrossRef]

A. W. Snyder, “Surface mode coupling along a tapered dielectric rod,” IEEE Trans. Antennas Propag. AP-13, 821–822 (1965).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Storz, F.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 162–163; erratum, 428 (1988).

W. Streifer, “Comment on ‘Fundamental error of recent coupled mode formulations,’” Electron. Lett. 22, 718–719 (1988).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “Reformulation of coupled-mode theory of multiwaveguide systems,” J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

W. Streifer, “Coupled mode theory,” Electron. Lett. 23, 216–217 (1987).
[CrossRef]

A. Hardy, W. Streifer, M. Osinski, “Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, S. Shakir, W. Streifer, “Coupled-mode equations for two weakly guiding single-mode fibers,” Opt. Lett. 14, 324–336 (1986).
[CrossRef]

A. Hardy, M. Osiński, W. Streifer, “Application of coupled-mode theory to nearly parallel waveguide systems,” Electron. Lett. 22, 1249–1250 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Analysis of phased-array diode lasers,” Opt. Lett. 10, 335–337 (1985).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, “Coupled-mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “A critical review of coupled mode theory,” in Integrated Optical Circuit Engineering V, M. A. Mentzer, ed., Proc. Soc. Photo-Opt. In-strum. Eng.835, 178 (1987).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, “Improved coupled-mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8, 1062–1069 (1991).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Comparison between strong coupling theory and experiment for three-arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

R. R. A. Syms, R. G. Peall, “The digital optical switch: analogous directional coupler devices,” Opt. Commun. 68, 235–238 (1989).
[CrossRef]

R. R. A. Syms, R. G. Peall, “Explanation of asymmetric switch response of three-arm directional couplers in Ti:LiNbO3using strong coupling theory,” Opt. Commun. 66, 260–264 (1988).
[CrossRef]

R. G. Peall, R. R. A. Syms, “Scalar strong coupled mode theory for slowly-varying waveguide arrays,” Opt. Commun. 67, 421–424 (1988).
[CrossRef]

Sztefka, G.

J. Willems, J. Haes, R. Baets, G. Sztefka, H. P. Nolting, “Eigenmode propagation analysis of radiation losses in waveguides with discontinuities and grating-assisted couplers,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 229.

Tian, F.

F. Tian, Y. Z. Wu, P. A. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Tomabechi, Y.

Y. Tomabechi, K. Matsumura, “Improved analysis for the coupling characteristics of two rectangular dielectric waveguides laid in different layers,” IEEE J. Quantum Electron. 24, 2359–2361 (1988).
[CrossRef]

Tsang, L.

L. Tsang, S. L. Chuang, “Improved coupled-mode theory for reciprocal anisotropic waveguides,” J. Lightwave Tech-nol. 6, 304–311 (1988).
[CrossRef]

Vassallo, C.

C. Vassallo, “Condensed formula for coupling coefficients between parallel dielectric waveguides,” Electron. Lett. 23, 304–306 (1986).

Vassello, C.

C. Vassello, “About coupled-mode theories for dielectric waveguides,” J. Lightwave Technol. 6, 294–303 (1988).
[CrossRef]

Wang, S.

K. Chen, S. Wang, “Cross-talk problems in optical directional couplers,” Appl. Phys. Lett. 44, 166–168 (1984).
[CrossRef]

Wang, Z. H.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Whitaker, N.

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three-guide optical coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

Whitaker, N. A.

H. A. Haus, W P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” J. Lightwave Technol. LT-5, 16–23 (1987).
[CrossRef]

H. A. Haus, N. A. Whitaker, “Elimination of cross talk in optical directional couplers,” Appl. Phys. Lett. 46, 1–3 (1985).
[CrossRef]

Willems, J.

J. Willems, J. Haes, R. Baets, G. Sztefka, H. P. Nolting, “Eigenmode propagation analysis of radiation losses in waveguides with discontinuities and grating-assisted couplers,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 229.

Wu, Y.

Y. Wu, “Discussion of HS formulation using equivalent current theory,” Electron. Lett. 24, 376–377 (1988).
[CrossRef]

Wu, Y. Z.

F. Tian, Y. Z. Wu, P. A. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, B. E. Little, C. L. Xu, “On phase-matching and power coupling in grating-assisted couplers,” IEEE Photon. Technol. Lett. 4, 151–153 (1992).
[CrossRef]

Yariv, A.

G. Griffle, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Ye, P. A.

F. Tian, Y. Z. Wu, P. A. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Young, M. G.

R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” in Integrated Photonics Research, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 308.

Zheng, X.

A. Ankiewicz, A. W. Snyder, X. Zheng, “Coupling between parallel optical fiber cores—critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

E. A. J. Marcatili, L. L. Buhl, R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” Appl. Phys. Lett. 49, 1692–1693 (1986).
[CrossRef]

K. Chen, S. Wang, “Cross-talk problems in optical directional couplers,” Appl. Phys. Lett. 44, 166–168 (1984).
[CrossRef]

H. A. Haus, N. A. Whitaker, “Elimination of cross talk in optical directional couplers,” Appl. Phys. Lett. 46, 1–3 (1985).
[CrossRef]

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, M. J. R. Martyak, “Grating assisted InGaAsP/InP vertical co-directional coupler filter,” Appl. Phys. Lett. 55, 2011–2013 (1989).
[CrossRef]

Bell Syst. Tech. J. (3)

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–842 (1973).

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J. 33, 661–719, (1954).

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

Electron. Lett. (9)

C. Vassallo, “Condensed formula for coupling coefficients between parallel dielectric waveguides,” Electron. Lett. 23, 304–306 (1986).

A. W. Snyder, A. Ankiewicz, “Fibre couplers composed of unequal cores,” Electron. Lett. 22, 1237–1238 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Fundamental error of recent coupled mode formulations,” Electron. Lett. 23, 1097–1098 (1987).
[CrossRef]

W. Streifer, “Coupled mode theory,” Electron. Lett. 23, 216–217 (1987).
[CrossRef]

W. Streifer, “Comment on ‘Fundamental error of recent coupled mode formulations,’” Electron. Lett. 22, 718–719 (1988).
[CrossRef]

A. W. Snyder, A. Ankiewicz, A. Altintas, “Coupled mode theory neglects polarization phenomena” (reply to Ref. 26), Electron. Lett. 22, 720–721 (1988).
[CrossRef]

Y. Wu, “Discussion of HS formulation using equivalent current theory,” Electron. Lett. 24, 376–377 (1988).
[CrossRef]

J. R. Qian, “Generalized coupled-mode equations and applications to fiber couplers,” Electron. Lett. 22, 304–306 (1986).
[CrossRef]

A. Hardy, M. Osiński, W. Streifer, “Application of coupled-mode theory to nearly parallel waveguide systems,” Electron. Lett. 22, 1249–1250 (1986).
[CrossRef]

Fiber Optics, Optoelectronics and Laser Applications in Science and Engineering (1)

H. A. Haus, “Coupled-mode theory revisited,” in Fiber Optics, Optoelectronics and Laser Applications in Science and Engineering, Proc. Soc. Photo-Opt. Instrum. Eng. (1986).

IEEE J. Quantum Electron. (14)

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

Y. Tomabechi, K. Matsumura, “Improved analysis for the coupling characteristics of two rectangular dielectric waveguides laid in different layers,” IEEE J. Quantum Electron. 24, 2359–2361 (1988).
[CrossRef]

G. Griffle, M. Itzkovich, A. A. Hardy, “Coupled-mode formulations for directional couplers with longitudinal perturbation,” IEEE J. Quantum Electron. 28, 985–994 (1992).

G. Griffle, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

W. P. Huang, B. E. Little, “Power exchange in tapered optical couplers,” IEEE J. Quantum Electron. 27, 1932–1938 (1992).
[CrossRef]

Y. Chen, “Solutions to full coupled wave equations of nonlinear coupled systems,” IEEE J. Quantum Electron. 25, 2149–2153 (1989).
[CrossRef]

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three-guide optical coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

J. P. Donnelly, L. A. Molter, H. A. Haus, “The extinction ratio in optical two-guide coupler Δβ switches,” IEEE J. Quantum Electron. 25, 924–932 (1989).
[CrossRef]

H. Kogelnik, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396–401 (1976).
[CrossRef]

E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

F. Tian, Y. Z. Wu, P. A. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. QE-14, 843–847 (1978).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. P. Huang, B. E. Little, C. L. Xu, “On phase-matching and power coupling in grating-assisted couplers,” IEEE Photon. Technol. Lett. 4, 151–153 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. W. Snyder, “Surface mode coupling along a tapered dielectric rod,” IEEE Trans. Antennas Propag. AP-13, 821–822 (1965).
[CrossRef]

IEEE Trans. Circ. Syst. (1)

A. Milton, W. K. Burns, “Mode coupling in tapered optical waveguide structures and electro-optic switches,” IEEE Trans. Circ. Syst. CS-26, 1020–1028 (1979).
[CrossRef]

IEEE Trans. Microwave Theor. Technol. (1)

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Figures (16)

Fig. 1
Fig. 1

Schematic diagram of a uniform directional coupler.

Fig. 2
Fig. 2

Percentage difference between the coupling coefficients ( K ¯ 12 K ¯ 21 * ) / 2 ( K ¯ 12 + K ¯ 21 * ) as a function of the separation for TE modes of an asymmetric slab coupler. n1 = n3 = n5 = 3.200 and n2 = 3.250. The index n4 varies as δn = n2n4 = 0.01,0.1,0.2. The widths of the two slabs are d2 = d4 × 1.0 μm. The wavelength is λ = 1.5 μm.

Fig. 3
Fig. 3

Effective indices of the symmetriclike and the antisymmetriclike composite modes of the uniform directional couplers: (a) identical waveguides, (b) dissimilar waveguides. Solid curves, exact solutions; dashed curves, nonorthogonal CMT; dashed–dotted curves, orthogonal CMT. The parameters are the same as those in Fig. 2 except that (a) n2 = 3.25 and (b) n2 = 3.23.

Fig. 4
Fig. 4

Electric-field patterns of the symmetriclike and the antisymmetriclike composite modes of the uniform directional couplers made of identical waveguides shown in Fig. 3 (a). Solid curves, exact solutions; dashed curves, nonorthogonal CMT; dotted curves, orthogonal CMT; dashed–dotted curves, waveguide modes. (a) 2S = 1.0 μm, (b) 2S = 0.2 μm.

Fig. 5
Fig. 5

Electric-field patterns of the symmetriclike and the antisymmetriclike composite modes of the uniform directional couplers made of dissimilar waveguides shown in Fig. 3(a). Solid curves, exact solutions; dashed curves, nonorthogonal CMT; dotted curves, orthogonal CMT; dashed–dotted curves, waveguide modes. (a) 2S = 1.0 μm; (b) 2S = 0.2 μm.

Fig. 6
Fig. 6

Input and output structures assumed for the guided power in an individual waveguide: (a) Guided power in waveguide 1, (b) guided power in waveguide 2.

Fig. 7
Fig. 7

Coupling lengths as functions of separation for the TE modes of a slab coupler. Solid curves, exact solutions; dashed curves, nonorthogonal CMT; dashed–dotted curves, orthogonal CMT. (a) Identical waveguides shown in Fig. 4, (b) dissimilar waveguides shown in Fig. 5.

Fig. 8
Fig. 8

Power extinction ratios as functions of separation for the TE modes of a slab coupler. Solid curve, exact solution; dashed curve, nonorthogonal CMT. The parameters are the same as those in Fig. 4.

Fig. 9
Fig. 9

(a) Percentage errors in the coupling lengths for the TM modes of parallel slabs, (b) effective indices of the TM modes of parallel slabs. Curves are taken from Ref. 35. β is the propagation constant, ρ is the width of the slab, and ncl is the refractive index of the cladding.

Fig. 10
Fig. 10

Schematic diagram of a grating-assisted coupler.

Fig. 11
Fig. 11

Phase-matching periods as functions of separation for the TE modes of a slab coupler. Parameters: n1 = 1.0, n2 = 3.3, n3 = 3.2, n4 = 3.5, and n5 = 3.0; d2 = 1.0 μm, d4 = 0.3 μm, 2S = 0.6 μm. λ = 1.5 μm. Solid curve, exact; dashed curve, nonorthogonal CMT; dashed–dotted curve, conventional orthogonal CMT.

Fig. 12
Fig. 12

Power exchange as a function of z. Solid curves, finite-difference BPM; dashed curves, nonorthogonal CMT. (a) 2h = 0.1 μm, (b)2h = 0.2 μm. The parameters are the same as those in Fig. 11.

Fig. 13
Fig. 13

Schematic diagram of a tapered coupler.

Fig. 14
Fig. 14

Illustration of the local modes used in the coupled-mode formulations.

Fig. 15
Fig. 15

Guided power in guide 1 as a function of propagation distance L. The parameters are n1 = n2 = 3.1, n0 = 3.0, w1 = 0.8 μm, and w2 = 0.6 μm. x01 = 0.55 μm, and x02 = 0.45 μm. The wavelength is λ = 1.5 μm. (a) θ = 0.1°, (b) θ = 0.5°. Solid curves, nonorthogonal CMT with tapering effect; dashed curves, nonorthogonal CMT that neglects Nsa; dotted curves, nonorthogonal CMT that neglects F12.

Fig. 16
Fig. 16

Output power from guide 1 as a function of tilt angle. The parameters are the same as those in Fig. 14 except that w2 = w1 = 0.8 μm, x01 = −x02 = 0.7 μm. The propagation distance is L = 200 μm. Solid curve, nonorthogonal CMT that considers the wavefront-tilt effect; dashed–dotted curve, nonorthogonal CMT based on the local waveguide modes; dashed curve, finite-difference BPM.

Equations (131)

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N s = N eff + k 11 X k 12 1 X 2 + k 12 X k 11 1 X 2 ,
N a = N eff + k 11 X k 12 1 X 2 k 12 X k 11 1 X 2 ,
k i j = 1 4 0 μ 0 ( n 2 n j 2 ) Ψ i * Ψ j d a .
N s = N eff + k 12 ,
N a = N eff k 12 ,
L c = λ 2 ( N s N a ) ,
L c = 1 X 2 k 12 X k 11 λ 4 ,
L c = 1 k 12 λ 4 ,
d a 1 d z = j β 1 a 1 ,
d a 2 d z = j β 2 a 2 .
d a 1 d z = j ( β 1 + K 11 ) a 1 j K 12 a 2 ,
d a 2 d z = j ( β 2 + K 22 ) a 2 j K 21 a 1 ,
P ( z ) = | a 1 | 2 + | a 2 | 2 .
d d z P ( z ) = 0.
K 12 = K 21 * = κ ,
a i ( z ) = â i exp ( j β 1 + K 11 + β 2 + K 22 2 z ) .
d â 1 d z = j δ â 1 j κ â 2 ,
d â 2 d z = + j δ â 2 j κ â 1 ,
δ = β 1 + K 11 β 2 K 22 2
d d z A = j H ¯ A ,
A = [ â 1 â 2 ] ,
H ¯ = [ + δ κ κ δ ] .
O t H ¯ O = B ,
B = [ β s 0 0 β a ] ,
O = [ cos ( η / 2 ) sin ( η / 2 ) sin ( η / 2 ) cos ( η / 2 ) ] ,
tan ( η ) = κ δ .
A = OW ,
d d z W = j BW .
A ( z ) = T ( z ) A ( 0 ) ,
T = O [ exp ( j β s z ) 0 0 exp ( j β a z ) ] O 1 ,
t 11 = t 22 * = cos ( S z ) j cos ( η ) sin ( S z ) ,
t 12 = t 21 = j sin ( η ) sin ( S z ) ,
S = δ 2 + κ 2 .
β s = β 0 + S ,
β a = β 0 S ,
β 0 = β 1 + K 11 + β 2 + K 22 2 .
P 1 ( z ) = cos 2 ( S z ) + cos 2 ( η ) sin 2 ( S z ) ,
P 2 ( z ) = sin 2 ( η ) sin 2 ( S z ) .
L c = π / ( 2 S ) ,
P 2 max = sin 2 ( η ) .
E . R . = P 1 min P 2 max ,
E = a 1 ( z ) e 1 + a 2 ( z ) e 2 ,
H = a 1 ( z ) h 1 + a 2 ( z ) h 2 ,
j P i j d a j d z = j j H ¯ i j a j ,
H ¯ i j = P i j β j + K ¯ i j ,
P i j = 1 4 [ e i * × h j + e j × h i * ] d a
K ¯ i j = 1 4 ω 0 ( n ¯ 2 n j 2 ) e i * e j d a
d d z a i * P i j a j = 0.
H ¯ i j = H ¯ j i *
P i j ( β j β i ) = K ¯ i j K ¯ j i * .
P d d z A = j H ¯ A ,
P = [ 1 X X 1 ] .
O t PO = I ,
O t H ¯ O = B ,
O = 1 cos ( α ) [ cos ( η + α 2 ) sin ( η + α 2 ) sin ( η α 2 ) cos ( η α 2 ) ] ,
sin ( α ) = X
κ = K ¯ 12 + K ¯ 21 X ( K ¯ 11 + K ¯ 22 ) 2 ( 1 X 2 ) 1 / 2 .
t 11 = t 22 * = cos ( α ) cos ( S z ) j cos ( η ) sin ( S z ) cos ( α ) ,
t 12 = j sin ( η + α ) sin ( S z ) cos ( α ) ,
t 21 = j sin ( η α ) sin ( S z ) cos ( α ) ,
β 0 = β 1 + β 2 2 + K ¯ 11 + K ¯ 22 X ( K ¯ 12 + K ¯ 21 ) 2 ( 1 X 2 )
e s = cos [ ( η + α ) / 2 ] cos ( α ) e 1 + sin [ ( η α ) / 2 ] cos ( α ) e 2 ,
e a = sin [ ( η + α ) / 2 ] cos ( α ) e 1 + cos [ ( η α ) / 2 ] cos ( α ) e 2 .
κ = K ¯ 12 + K ¯ 21 2 .
β s = β 1 + β 2 + K ¯ 11 + K ¯ 22 2 + [ δ 2 + ( K ¯ 12 + K ¯ 21 2 ) 2 ] 1 / 2 ,
β a = β 1 + β 2 + K ¯ 11 + K ¯ 22 2 [ δ 2 + ( K ¯ 12 + K ¯ 21 2 ) 2 ] 1 / 2 ,
e s = cos ( η / 2 ) e 1 + sin ( η / 2 ) e 2 ,
e a = sin ( η / 2 ) e 1 + cos ( η / 2 ) e 2 .
P ( z ) = 1 4 ( E × H * + E * × H ) d a = i j a i * ( z ) P i j a i ( z ) .
b i ( z ) = a i ( z ) + X a j ( z ) ,
P i ( z ) = | b i ( z ) | 2 = | a i ( z ) + X a j ( z ) | 2 .
P 1 ( z ) = cos 2 ( S z ) + [ cos ( η ) sin ( α ) sin ( α + η ) cos ( α ) ] 2 sin 2 ( S z ) ,
P 2 ( z ) = sin 2 ( α ) cos 2 ( S z ) + sin 2 ( η ) sin 2 ( S z ) .
P 1 ( z ) = R s 2 + R a 2 + 2 R s R a cos [ ( β s β a ) z ] ,
P 2 ( z ) = R s 2 + R a 2 2 R s R a cos [ ( β s β a ) z ] ,
R s = 1 s a 1 1 1 s s ,
R a = 1 a a 1 1 1 a a ,
i j = 1 4 ( e i * × h j + e j × h i * ) d a .
L c = π β s β a ,
P 2 max = ( R s + R a ) 2 ,
P 1 min = ( R s R a ) 2 .
E . R . = X 2 ,
E . R . = ( R s R a R s + R a ) 2 .
K i j = + C i j m exp ( j 2 m π Λ z ) ,
d a 1 d z = j ( β 1 + C 11 0 ) a 1 j C 12 + m exp ( j 2 m π Λ z ) a 2 ,
d a 2 d z = j ( β 2 + C 22 0 ) a 2 j C 21 m exp ( + j 2 m π Λ z ) a 2 .
a 1 = â 1 exp ( j β 1 + C 11 0 + β 2 + C 22 0 2 z ) exp ( j m π Λ z ) ,
a 2 = â 2 exp ( j β 1 + C 11 0 + β 2 + C 22 0 2 z ) exp ( + j m π Λ z ) .
δ = β 1 + C 11 0 β 2 C 22 0 2 m π Λ
κ = C 12 + m = C 21 m .
n 2 ( x , y , z ) = n ¯ 2 ( x , y ) + Δ n 2 ( x , y ) + F m exp ( j 2 m π Λ z ) ,
H = H ¯ + K ,
K i j = 1 4 ω 0 F m exp ( j 2 m π Λ z ) Δ n 2 e i * e j d a .
d d z W = j BW j LW ,
L = O t K O
L i j = 1 4 ω 0 F m exp ( j 2 m π Λ z ) Δ n 2 e i * e j d a ,
L sa = L as * .
W ( z ) = T W ( z ) W ( 0 ) ,
t 11 W = [ cos ( Q z ) j cos ( ϕ ) sin ( Q z ) ] exp [ j ( π / Λ ) z ] ,
t 12 W = sin ( ϕ ) sin ( Q z ) exp [ j ( π / Λ ) z ] ,
t 21 W = sin ( ϕ ) sin ( Q z ) exp [ j ( π / Λ ) z ] ,
t 22 W = [ cos ( Q z ) + j cos ( ϕ ) sin ( Q z ) ] exp [ j ( π / Λ ) z ] ,
Q = ( δ W 2 + κ W 2 ) 1 / 2 ,
tan ( ϕ ) = κ W / δ W ,
δ W = ( β s β a ) / 2 π / Λ
κ W = 1 2 π ω 0 Δ n 2 e s * e a d a
Λ W = π / ( β s β a ) .
Λ A = π / ( β 1 β 2 ) ,
T W = [ cos ( κ W z ) exp [ j ( π / Λ ) z ] sin ( κ W z ) exp [ j ( π / Λ ) z ] sin ( κ W z ) exp [ + j ( π / Λ ) z ] cos ( κ W z ) exp [ + j ( π / Λ ) z ] ] .
L c = N Λ .
T A = O T W O 1 = 1 cos ( α ) [ cos ( κ W L c α ) sin ( κ W L c ) sin ( κ W L c ) cos ( κ W L c + α ) ] .
P 1 ( L c ) = cos 2 ( κ W L c ) ,
P 2 ( L c ) = sin 2 ( κ W L c α ) .
L max = π / 2 + α κ W ,
L min = π / ( 2 κ W ) .
O = [ 1 / 2 1 / 2 1 / 2 1 / 2 ] ,
T = [ cos ( 0 z κ d z ) j sin ( 0 z κ d z ) j sin ( 0 z κ d z ) cos ( 0 z κ d z ) ] .
0 L c κ d z = π 2 .
E ( x , y , z ) = a 1 ( z ) e 1 ( x , y ; z ) + a 2 ( z ) e 2 ( x , y ; z ) , H ( x , y , z ) = a 1 ( z ) h 1 ( x , y ; z ) + a 2 ( z ) h 2 ( x , y ; z ) .
P d d z A = j H ¯ A FA ,
F i j = 1 4 ( e i * × h j z + e j z × h i * ) d a ,
d d z W = j BW NW ,
N = M + P d d z M + M + FM .
N sa = N as * ,
N sa = 1 4 ( e s * × h a z + e a z × h s * ) d a ,
N sa = 1 4 ω 0 β s β a n 2 z e s * e a d a .
a 1 ( z ) = 1 2 { [ 1 X ( 0 ) 1 X ( z ) ] 1 / 2 + [ 1 + X ( 0 ) 1 + X ( z ) ] 1 / 2 } cos ( 1 2 ϕ ) + j 1 2 { [ 1 X ( 0 ) 1 X ( z ) ] 1 / 2 [ 1 + X ( 0 ) 1 + X ( z ) ] 1 / 2 } sin ( 1 2 ϕ ) ,
a 2 ( z ) = 1 2 { [ 1 X ( 0 ) 1 X ( z ) ] 1 / 2 + [ 1 + X ( 0 ) 1 + X ( z ) ] 1 / 2 } cos ( 1 2 ϕ ) j 1 2 { [ 1 X ( 0 ) 1 X ( z ) ] 1 / 2 [ 1 + X ( 0 ) 1 + X ( z ) ] 1 / 2 } sin ( 1 2 ϕ ) ,
ϕ = 0 z ( β s β a 2 ) d z
P 1 ( L ) | a 1 ( L ) | 2 ,
P 2 ( L ) | a 2 ( L ) | 2 .

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