OptoElectronics Chair - Electrical Engineering

(Tel Aviv University)

 
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Short review of Coupled Mode Theory.

 

Coupled Mode Theory for Parallel Waveguides in its exact (vector) formulation is another representation of Maxwell equations, which is convenient for description of interaction and propagation of guided modes in 1D and 2D arrays of coupled waveguides and lasers. This CMT formulation was developed for the first time for finite arrays by A. A.Hardy and W. Steifer in 1985 ([1] -

[5]) and extended to the case of infinite photonic crystals by A. A. Hardy, V. R.  Shteeman, D. L. Boiko, I. B. Nusinsky, and E. Kapon in 2006 ([6]-[9]).

Note, that CMT can only handle a specific class of 2D photonic crystals, in which guided light propagates in longitudal direction (z-direction).

As a physical model CMT considers a 2D array of coupled parallel waveguides.

CMT assumes first that guided modes of all solitary waveguides are known in advance. An array is considered as an assembly of solitary waveguides, interacting with each other. Therefore, the modes of the solitary waveguides experience perturbation due to the interaction with neighbor waveguides.

Solitary waveguide' guided modes
Solitary waveguides combined into a photonic array
 

Analysis of array modes:

Array modes of a photonic crystal (partial amplitudes and propagation constants), should be found from the CMT eigenequation:

Here, matrix M contains integrals of interaction of guided modes of all solitary waveguides with each other and with the cores of neighbor waveguides, and propagation constants of all solitary waveguides; U0 is a vector of partial amplitudes of the array mode, I is a unity matrix and σ is a propagation constant of the array mode.

 

partial amplitudes, U0, of the array mode

full array mode (U0 times solitary waveguides' modes)

 
Analysis of light propagation along z-axis:

To  analyze a propagation of the optical signal along the z-axe of a photonic crystal one needs to solve a general CMT differential equation with the initial conditions U(0):

 

System of 3 different coupled waveguides. Initial conditions: U(0) = {1 0 1}, i.e. light injected in the 1st and in the 3rd waveguides.

Coupling between the waveguides results in the flow of light from the rightmost and leftmost waveguides to the central waveguide and backward.

 

The unique feature of CMT, which calculates only envelopes of array modes (instead of full modes as in the solution of the Helmholtz equation in finite differences, Plane-wave expansion, FDTD etc), allows for accurate and much faster computations of array modes of finite arrays.

Furthermore,  imposing translational symmetry, one can combine CMT equations with the Bloch theorem. This allows analysis of infinite 2D photonic arrays (lattices or superlattices), as well as point and linear defects in infinite arrays. In addition, time- and CPU-consumption reduces dramatically (down to several minutes as opposed to hours or days for other methods), while computation accuracy remains almost unaffected.

 

 
Articles on Coupled Mode Theory.

 

Exact (vector) CMT:

 
CMT for finite photonic arrays:
[1] A. A. Hardy, W. Streifer. Coupled mode theory of parallel waveguides. IEEE Journal of Lightwave Technology, LT-3, No. 5, pp. 1135-1146 (1985).
[2] A. A. Hardy, W. Streifer, M. Osinski. Coupled-mode equations for multimode waveguide systems in isotropic or anisotropic media. Optics Letters, 11, No. 11, pp. 742-744 (1986)
[3] A. A. Hardy, W. Streifer. Coupled modes of multiwaveguide systems and phased arrays. IEEE Journal of Lightwave Technology, LT-4, No. 1, pp. 90-99 (1986).
[4] A. A. Hardy. A unified approach to coupled mode phenomena. IEEE Journal of Quantum Electronics, 34, No. 7, pp. 1109-1116 (1998).
[5] A. A. Hardy and E. Kapon. Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays. IEEE Journal of Quantum Electronics, 32, pp. 966-971 (1996).

 

CMT extended to infinite photonic arrays:

[6] V. Shteeman, D. Boiko, E. Kapon, A.  A. Hardy. Extension of Coupled Mode analysis to periodic large arrays of identical waveguides for photonic crystals applications. IEEE Journal of Quantum Electronics, 43, No 3, pp. 215-224 (2007).
[7] V. Shteeman, I. Nusinsky, E. Kapon, A. A. Hardy. Extension of Coupled Mode analysis to infinite photonic superlattices. IEEE Journal of Quantum Electronics, 44, No 9 pp. 826-833 (2008).
[8] V. Shteeman, I. Nusinsky, E. Kapon, A. A. Hardy. Analysis of Photonic Crystals With Defects Using Coupled Mode Theory. Submitted to IEEE Journal of Quantum Electronics (2008).
[9] V. Shteeman, I. Nusinsky, E. Kapon, A. A. Hardy. Analysis and design of large-sized 2D photonic crystal devices. Proceedings of IEEEI 25th Convention (2008).
[10] V. Shteeman, A. Hardy, E. Kapon, I. Nusinsky. Analysis of photonic crystals with defects using coupled-mode theory. JOSA B, 26, 6, pp. 1248–1255 (2009).

 

Scalar CMT:

 
[11] A. Snyder. Coupled-Mode Theory for Optical Fibers. Journal of the Optical Society of America, 62, No 11 pp. 1267-1277 (1972).
[12] E. Kapon, J. Katz, A. Yariv. Supermode analysis of phase-locked arrays of semiconductor lasers. Optics Letters, 10, No 4, pp. 125-127 (1984).
[13] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer. Coupled-resonator optical waveguide: a proposal and analysis. Optics Letters,  24, pp. 711-713 (1999).
[14] H. Kogelnik. Theory of dielectric waveguides in Integrated optics (edited by T.Tamir). Springer-Verlag, Berlin, 1975.
 

 

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