Abstract

We investigate coupled-mode theory in designing high index contrast silicon-on-insulator waveguide couplers and arrayed waveguides. We develop and demonstrate a method of solution to the inverse problem of reconstructing the coupling matrix from the modal profiles obtained, in this case, from finite-difference frequency-domain field calculations. We show that whereas supermode theory provides a good approximation of the mode profiles, next-to-nearest-neighbor coupling becomes significant at small separation distances between arrayed waveguides. These distances are quantified for three different silicon-on-insulator material platforms. We also point out the phenomenon of field skewing and deformation at small separations.

© 2009 Optical Society of America

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  1. F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
    [CrossRef]
  2. Q. Xu, D. Fattal, and R. G. Beausoleil, "Silicon microring resonators with 1.5-μm radius," Opt. Express 16, 4309-4315 (2008).
    [CrossRef] [PubMed]
  3. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, "Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects," Opt. Express 15, 11934-11941 (2007).
    [CrossRef] [PubMed]
  4. A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
    [CrossRef] [PubMed]
  5. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
    [CrossRef] [PubMed]
  6. W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, "Optical modulation using anti-crossing between paired amplitude and phase resonators," Opt. Express 15, 17264-17272 (2007).
    [CrossRef] [PubMed]
  7. X. Liu, I. Hsieh, X. Chen, M. Takekoshi, J. I. Dadap, N. C. Panoiu, R. M. Osgood,W. M. Green, F. Xia, and Y. A. Vlasov, "Design and fabrication of an ultra-compact silicon on insulator demultiplexer based on arrayed waveguide gratings," in Proceedings of the Conference on Lasers and Electro-Optics (CLEO, 2008), paper CTuNN1.
  8. P. Cheben, J. H. Schmid, A. Delage, A. Densmore, S. Jannz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D. C. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
    [CrossRef] [PubMed]
  9. K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70x60 μm2 size based on Si photonic wire waveguides," Electron. Lett.  41, 801-802 (2005).
  10. P. Dumon, W. Bogaerts, D. V. Thourhout, D. Taillaert, and R. Baets, "Compact wavelength router based on a Silicon-on-insulator arrayed waveguide grating pigtailed to a fiber array," Opt. Express 14, 664-669 (2006).
    [CrossRef] [PubMed]
  11. H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972).
    [CrossRef]
  12. A. Hardy and W. Streifer, "Coupled-mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985).
    [CrossRef]
  13. W. P. Huang, "Coupled-mode theory for optical waveguides: An overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
    [CrossRef]
  14. K. S. Chiang, "Coupled-zigzag-wave theory for guided waves in slab waveguide arrays," J. Lightwave Technol. 10, 1380-1387 (1992).
    [CrossRef]
  15. F. P. Payne, "An analytical model for the coupling between the array waveguides in AWGs and star couplers," Opt. Quantum Electron. 38, 237-248 (2006).
    [CrossRef]
  16. E. Kapon, J. Katz, and A. Yariv, "Supermode analysis of phase-locked arrays of semiconductor lasers," Opt. Lett. 10, 125-127 (1984).
    [CrossRef]
  17. A. Klekamp and R. Munzner, "Calculation of imaging errors of AWG," J. Lightwave Technol. 21, 1978-1986 (2003).
    [CrossRef]
  18. S. H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
    [CrossRef] [PubMed]
  19. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 2005).
  20. C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
    [CrossRef]
  21. W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
    [CrossRef]
  22. M. Kuznetsov, "Expressions for the coupling coefficient of a rectangular waveguide directional coupler," Opt. Lett. 8, 499-501 (1983).
    [CrossRef] [PubMed]
  23. S. Mookherjea, "Spectral characteristics of coupled resonators," J. Opt. Soc. Am. B 23, 1137-1145 (2006).
    [CrossRef]
  24. G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990).
    [CrossRef]
  25. M. Popovic, C. Manolatou, and M. Watts, "Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters," Opt. Express 14, 1208-1222 (2006).
    [CrossRef] [PubMed]
  26. E. Marcatili, "Improved coupled-mode equations for dielectric guides," IEEE J. Quantum Electron. QE-22, 988-993 (1986).
    [CrossRef]
  27. H. A. Haus, W. P. Huang, S. Kawakami and N. A. Whitaker, "Coupled-mode theory of optical waveguides," J. Lightwave Technol. LT-5, 16-23 (1987).
    [CrossRef]

2008

2007

2006

2005

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

2004

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

2003

1994

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang, "Coupled-mode theory for optical waveguides: An overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
[CrossRef]

1993

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

1992

K. S. Chiang, "Coupled-zigzag-wave theory for guided waves in slab waveguide arrays," J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

1990

G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990).
[CrossRef]

1987

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

1986

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

1985

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

1984

1983

1972

H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

Baets, R.

Bandaru, P. R.

Beausoleil, R. G.

Bogaerts, W.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

Cheben, P.

Chiang, K. S.

K. S. Chiang, "Coupled-zigzag-wave theory for guided waves in slab waveguide arrays," J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

Cohen, O.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Cooper, M. L.

Delage, A.

Densmore, A.

Dumon, P.

Fattal, D.

Green, W. M. J.

Hardy, A.

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

Haus, H. A.

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

Huang, W. P.

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang, "Coupled-mode theory for optical waveguides: An overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

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

Jannz, S.

Jones, R.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Kapon, E.

Katz, J.

Kawakami, S.

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

Klekamp, A.

Kogelnik, H.

H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

Kuznetsov, M.

Lamontagne, B.

Lapointe, J.

Lenz, G.

G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990).
[CrossRef]

Liao, L.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Liu, A. S.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Manolatou, C.

Marcatili, E.

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

Mookherjea, S.

Munzner, R.

Nicolaescu, R.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Paniccia, M.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Payne, F. P.

F. P. Payne, "An analytical model for the coupling between the array waveguides in AWGs and star couplers," Opt. Quantum Electron. 38, 237-248 (2006).
[CrossRef]

Popovic, M.

Post, E.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Rooks, M.

Rooks, M. J.

Rubin, D.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Salzman, J.

G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990).
[CrossRef]

Samara-Rubio, D.

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Schmid, J. H.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Sekaric, L.

Shank, C. V.

H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

Stern, M. S.

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

Streifer, W.

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

Taillaert, D.

Thourhout, D. V.

Vlasov, Y.

Vlasov, Y. A.

Waldron, P.

Watts, M.

Whitaker, N. A.

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

Xia, F.

Xu, C. L.

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

Xu, D. C.

Xu, Q.

Q. Xu, D. Fattal, and R. G. Beausoleil, "Silicon microring resonators with 1.5-μm radius," Opt. Express 16, 4309-4315 (2008).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Yang, S. H.

Yariv, A.

Appl. Phys.

H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

IEEE J. Quantum Electron.

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

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

J. Lightwave Technol.

K. S. Chiang, "Coupled-zigzag-wave theory for guided waves in slab waveguide arrays," J. Lightwave Technol. 10, 1380-1387 (1992).
[CrossRef]

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

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

G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990).
[CrossRef]

A. Klekamp and R. Munzner, "Calculation of imaging errors of AWG," J. Lightwave Technol. 21, 1978-1986 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Photonics

F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Nature

A. S. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, "A highspeed silicon optical modulator based on a metal oxide-semiconductor capacitor," Nature 427, 615-618 (2004).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Opt. Express

P. Dumon, W. Bogaerts, D. V. Thourhout, D. Taillaert, and R. Baets, "Compact wavelength router based on a Silicon-on-insulator arrayed waveguide grating pigtailed to a fiber array," Opt. Express 14, 664-669 (2006).
[CrossRef] [PubMed]

M. Popovic, C. Manolatou, and M. Watts, "Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters," Opt. Express 14, 1208-1222 (2006).
[CrossRef] [PubMed]

P. Cheben, J. H. Schmid, A. Delage, A. Densmore, S. Jannz, B. Lamontagne, J. Lapointe, E. Post, P. Waldron, and D. C. Xu, "A high-resolution silicon-on-insulator arrayed waveguide grating microspectrometer with submicrometer aperture waveguides," Opt. Express 15, 2299-2306 (2007).
[CrossRef] [PubMed]

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, "Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects," Opt. Express 15, 11934-11941 (2007).
[CrossRef] [PubMed]

W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, "Optical modulation using anti-crossing between paired amplitude and phase resonators," Opt. Express 15, 17264-17272 (2007).
[CrossRef] [PubMed]

Q. Xu, D. Fattal, and R. G. Beausoleil, "Silicon microring resonators with 1.5-μm radius," Opt. Express 16, 4309-4315 (2008).
[CrossRef] [PubMed]

S. H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Quantum Electron.

F. P. Payne, "An analytical model for the coupling between the array waveguides in AWGs and star couplers," Opt. Quantum Electron. 38, 237-248 (2006).
[CrossRef]

Optoelectron.

C. L. Xu, W. P. Huang, M .S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc.-Optoelectron. 141, 281-286 (1994).
[CrossRef]

Other

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 2005).

X. Liu, I. Hsieh, X. Chen, M. Takekoshi, J. I. Dadap, N. C. Panoiu, R. M. Osgood,W. M. Green, F. Xia, and Y. A. Vlasov, "Design and fabrication of an ultra-compact silicon on insulator demultiplexer based on arrayed waveguide gratings," in Proceedings of the Conference on Lasers and Electro-Optics (CLEO, 2008), paper CTuNN1.

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70x60 μm2 size based on Si photonic wire waveguides," Electron. Lett.  41, 801-802 (2005).

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Figures (9)


                  Fig. 1.
Fig. 1.

(a) Refractive index profile in the transverse plane n(x,y) for an N arrayed-waveguide structure. (b) The same refractive index profile may be decomposed, mathematically, into the sum of parts, Δn2 i = n2 i -n2 s , each of which appears in integrals equation for the coupling coefficients. For the structures considered in this paper, h = 500 nm, w = 200 nm, and s varies over the range 50 nm to 1μm. For these waveguide widths and heights (similar to those in Ref. [18]), the polarization direction of the principal transverse component of the electric field is indicated for the (quasi) TE and TM modes.

Fig. 2.
Fig. 2.

TE Polarization Ey : The modes of an N = 5 coupled waveguide array for λ = 1550 nm, calculated using coupled-mode theory (blue solid lines), and a finite-difference frequency-domain algorithm (black crosses). The coupled-mode theory calculations were done by using the effective index method, calculating the overlap integrals, solving Eq. (11), and reassembling the field. Waveguide height = 500 nm, width = 200 nm, separation = 200 nm, ncore = 3.47, and nclad =1.46. Under nearest neighbor coupling, the scaling relationship predicted by Eq. (13) adequately predicts the field amplitudes within each waveguide.

Fig. 3.
Fig. 3.

TM Polarization Ex : The modes of an N = 5 coupled waveguide array, calculated using coupled-mode theory (blue solid lines), and a finite-difference frequency-domain algorithm (black crosses). The coupled-mode theory calculations were done by using the effective index method, calculating the overlap integrals, solving Eq. (11), and reassembling the field. Waveguide height = 500 nm, width = 200 nm, separation = 1μm, ncore = 3.47, and nclad =1.46. Under nearest neighbor coupling, the scaling relationship predicted by Eq. (13) adequately predicts the field amplitudes within each waveguide.

Fig. 4.
Fig. 4.

Error versus N: Exact eigenvalues of a tridiagonal symmetric matrix of size N were perturbed by values chosen from a uniform random distribution with variance chosen to be ten percent of the first eigenvalue. The variance and mean of the reconstructed nearest-neighbor coupling and next-to-nearest-neighbor coupling coefficients are plotted, calculated from a distribution of coupling matrices generated by 105 iterations, showing that Eq. (20) is a good predictor of the reconstruction accuracy.

Fig. 5.
Fig. 5.

Ratio of coupling coefficients for different separation distances extracted from Eq. (11), which was reconstructed using an algorithm described in the text. (a) TE Polarization An exponential fit expected from a simple nearest-neighbor-coupling theory holds throughout this regime. (b) TM Polarization At a separations less than 450 nm, the ratio deviates significantly from the predicted behavior. (c) TE Polarization The ratio of cross coupling coefficients show that the reconstructed coupling matrix M becomes asymmetric as the waveguide separation is reduced. (d) TM Polarization The asymmetry of the coupling matrix begins at a larger separation.

Fig. 6.
Fig. 6.

Left column: TE Polarization, and Right column: TM polarization. Effective index of the five supermodes for different separation distances with ncore = 3.47, and (a,b) nclad =1.46, (c,d) nclad = 1 (e,f) nclad = 2.05. For each case as the separation between the waveguides increases, the effective indexes of the modes converge to that of the single waveguide. These values are (a) neff = 2.36 and (b) neff = 1.66 for oxide cladding, (c) neff = 2.24 and (d) neff = 1.07 for air cladding, (e) neff = 2.56 and (f) neff = 2.26 for nitride cladding. The shaded regions indicate > 5% deviation of neff for the m=3 supermode from its theoretical value, which as discussed in the text, is predicted by CMT to be independent of the separation distance.

Fig. 7.
Fig. 7.

TM Polarization Ex : The field profile of the fifth eigenmode in the first waveguide. When the separation is decreased below 450 nm, the peak of the field in the high-index rib indicated by the dotted red line in (a) is no longer centered, and the mode shape is considerably altered, thereby changing both κ and n eff. Consequently, CMT can no longer accurately predict the mode coupling.

Fig. 8.
Fig. 8.

TE Polarization Ey : Using the exact solution from a FDFD simulation of a single waveguide, the horizontal cross section is extracted and five copies are shifted from one another so that their separation corresponds to a waveguide separation of 80 nm. (a) These individual waveguide modes are scaled in accordance with Eq. (13) for the fundamental mode (m=1). (b) The sumation of the individual waveguide modes; superimposed is the FDFD solution of the entire five waveguide structure. (c) Zoomed in to just the first waveguide. CMT and FDFD show a shift of the mode towards the center of the waveguide structure. (d-e) The fifth mode, both CMT and FDFD show a shift towards the edge of the waveguide structure however FDFD shows a shift of greater magnitude.

Fig. 9.
Fig. 9.

TM Polarization: finite element simulations (COMSOL) of the transverse field amplitudes, Ex and Ey , and Power flow, Pz for a separation of 1 μm (first Row) and 100 nm (second row) for the fifth mode. For small separations the polarization becomes strongly hybridized and the power is carried at the outer edges.

Tables (1)

Tables Icon

Table 1. An example of a reconstructed coupling (M) matrix from FDFD calculations of eigenmodes and eigenvalues. Si/SiO2, TE polarization, separation s = 350 nm, β 0 = 2.26128 (2π/λ). Although the nearest-neighbor coupling coefficients dominate, the self-coupling and off-tridiagonal coupling terms are non-zero.

Equations (32)

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2 E + ( n 2 k 2 β 2 ) E = [ · E 1 n 2 · ( n 2 E ) ]
( P xx P xy P yx P yy ) ( E x E y ) = β 2 ( E x E y )
P xx E x = x [ 1 n 2 ( n 2 E x ) x ] + 2 E x y 2 + n 2 k 2 E x , P xy E y = x [ 1 n 2 ( n 2 E y ) y ] + 2 E y x y ,
P yx E x = y [ 1 n 2 ( n 2 E x ) x ] + 2 E x y x , P yy E y = y [ 1 n 2 ( n 2 E x ) y ] + 2 E y x 2 + n 2 k 2 E y .
2 E + ω 2 c 2 n 2 ( x , y ) E = 0 ,
E ̂ = y ̂ E ( x , y ) e iβz = y ̂ [ Σ l = 1 N A l 𝓔 1 ( x , y ) ] e iβz .
n 2 ( x , y ) = n s 2 ( x , y ) + Σ l = 1 N Δ n l 2 ( x , y )
( 2 + ω 2 c 2 [ n s 2 ( x , y ) + Σ l = 1 N Δ n l 2 ( x , y ) ] β 2 ) [ Σ l = 1 N A l 𝓔 1 ( x , y ) ] = 0 .
( 2 + ω 2 c 2 [ n s 2 ( x , y ) + Δ n l 2 ( x , y ) ] β l 2 ) 𝓔 1 ( x , y ) = 0
Σ l = 1 N A 1 ( Δ l + ω 2 c 2 Σ m = 1 m l N Δ n m 2 ( x , y ) ) 𝓔 1 ( x , y ) = 0
Δ 1 β l 2 β 2 .
Σ l = 1 N A 1 ( Δ l ∫∫ 𝓔 j * 𝓔 1 dxdy + ω 2 c 2 Σ m = 1 m l N ∫∫ 𝓔 j * Δ n m 2 ( x , y ) 𝓔 1 dxdy ) = 0 , j = 1,2 , , N .
I jl = ∫∫ 𝓔 j * 𝓔 1 dxdy ,
κ jl = ω 2 c 2 Σ m = 1 m l N ∫∫ 𝓔 j * Δ n m 2 ( x , y ) 𝓔 1 dxdy ,
∫∫ 𝓔 j * 𝓔 1 dxdy = 1 .
( β 1 2 + κ 11 Δ 2 I 12 + κ 12 Δ N 1 I 1 , N 1 + κ 1 , N 1 Δ N I 1 N + κ 1 N Δ 1 I 21 + κ 21 β 2 2 + κ 22 Δ N 1 I 2 , N 1 + κ 2 , N 1 Δ N I 2 N + κ 2 N Δ 1 I 31 + κ 31 Δ 2 I 32 + κ 32 Δ N 1 I 3 , N 1 + κ 3 , N 1 Δ N I 3 N + κ 3 N Δ 2 I N 1 + κ N 1 Δ 2 I N 2 + κ N 2 Δ N 1 I 1 , N 1 + κ N , N 1 β N 2 + κ NN ) ( A 1 A 2 A 3 A N )
= ( β 2 0 0 0 0 β 2 0 0 0 0 β 2 0 0 0 0 β 2 ) ( A 1 A 2 A 3 A N ) .
M = ( β 1 2 + κ 11 Δ 2 I 12 + κ 12 0 0 Δ 2 I 21 + κ 21 β 2 2 + κ 22 0 0 0 Δ 2 I 32 + κ 32 0 0 0 Δ N 1 I N , N 1 + κ N , N 1 β N 2 + κ NN ) .
A l ( m ) = ( 2 N + 1 ) 1 / 2 sin lmπ N + 1 δ κ self ( 2 N + 1 ) 3 / 2
× Σ m = 1 n m N 1 β 2 ( m ) β 2 ( m ) [ sin N + 1 sin N + 1 + sin mNπ N + 1 sin nNπ N + 1 ] sin lnπ N + 1
H = y ̂ H ( x , y ) e iβz = y ̂ [ Σ l = 1 N A l H 1 ( x , y ) ] e iβz .
E x = β ε ( x , y ) ω H y ,
Δ M ij = Σ k = 1 N [ ( u ik + u jk ) Δ u k λ k + u ik u jk Δ λ k ] .
Δ M jj + 1 = Σ k = 1 N 2 N + 1 sin jkπ N + 1 sin ( j + 1 ) N + 1 Δ λ k ,
Δ M j 1 j + 1 = Σ k = 1 N 2 N + 1 sin ( j 1 ) N + 1 sin ( j + 1 ) N + 1 Δ λ k .
E [ Δλ M jj + 1 ] = E [ Δλ ] Σ k = 1 N 2 N + 1 sin jkπ N + 1 sin ( j + 1 ) N + 1 = 0
Var [ Δλ M jj + 1 ] = Var [ Δλ ] ( 1 N + 1 ) 2 Σ k = 1 N ( cos 2 N + 1 cos 2 jkπ N + 1 ) 2
= Var [ Δλ ] ( 1 N + 1 ) ,
Nearest neighbor : E [ Δλ M jj + 1 ] = 0 , Var [ Δλ M jj + 1 ] = Var [ Δλ ] / ( N + 1 ) ,
Next to nearest : E [ Δλ M j 1 j + 1 ] = 0 , Var [ Δλ M j 1 j + 1 ] = Var [ Δλ ] / ( N + 1 ) .
Δ 3 I 13 + κ 13 Δ 2 I 12 + κ 13 κ 13 κ 12 = e p ( 2 s + w ) e ps = e p ( s + w )
β 2 ( m ) = β 0 2 + κ self + 2 ( κ + Δ 0 I ) cos N + 1 2 δ κ self N + 1 ( sin 2 N + 1 + sin 2 Nmπ N + 1 ) ,

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