## Numerically-assisted coupled-mode theory for silicon waveguide couplers and arrayed waveguides

Optics Express, Vol. 17, Issue 3, pp. 1583-1599 (2009)

http://dx.doi.org/10.1364/OE.17.001583

Acrobat PDF (860 KB)

### 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

## 1. Introduction

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-*μ*m 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 sub-micrometer aperture waveguides,” Opt. Express **15**, 2299–2306 (2007). [CrossRef] [PubMed]

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]

*n*(

*x*,

*y*) in the cross-sectional plane [11

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]

*ab initio*numerical simulations in every case. Secondly, our method of solution of the inverse problem can be easily applied to design couplers from a mode solver rather than a time-consuming propagation simulation.

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]

## 2. Coupled-mode theory (CMT) of the modes of multi-slot waveguides

16. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. **10**, 125–127 (1984). [CrossRef]

*ab initio*numerical algorithm we have encoded in MATLAB to calculate the modes accurately. The next part of this section presents the analysis of the modes based on supermode theory. In the following sections, we will compare the predictions of CMT with the FDFD calculations.

### 2.1. Finite-difference frequency-domain (FDFD) algorithm

*et al*. [20

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]

**E**

_{⊥}(

*x*,

*y*) is the transverse electric field vector, is written in matrix form as

*n*

^{2}

*E*and

_{x}*n*

^{2}

*E*are both continuous across any dielectric discontinuity, and with the graded-index approximation, the central difference equations can be applied directly without any special treatment at the boundaries. Fully discretized versions of these operators can be found in Ref. [21

_{y}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]

### 2.2. Coupled mode theory and its predictions

#### 2.2.1. TE Polarization

*n*

^{2}(

*x*,

*y*) can be written as a sum of individual waveguide contributions, so that

*n*

^{2}

_{s}(

*x*,

*y*) corresponds to the cladding. Thus,

*n*

^{2}

_{s}(

*x*,

*y*) + Δ

*n*

^{2}

_{l}(

*x*,

*y*) would yield the dielectric coefficient profile of the

*l*th waveguide in the absence of the others. Substituting the above two equations into the wave equation, we have,

*N*equations are formed by multiplying Eq. (8) by

*𝓔*

_{j}^{*}(

*j*= 1,2,…,

*N*), and integrating each of these equations over

*x*and

*y*,

*I*is the overlap integral of the modes of two waveguides which are not orthogonal to each other (particularly in the case of small waveguide separation), and

_{jl}*κ*are the self-coupling and cross-coupling (exchange coupling) coefficients familiar from coupled-mode theory [19, p. 362].

*M*.)

*l*=

*j*-1,

*j*,

*j*+1.

*M*takes the tridiagonal form,

*M*can be further simplified by setting

*β*

^{2}

_{1}=

*β*

^{2}

_{2}… =

*β*

^{N}

_{1}≡

*β*

^{2}

_{0}and also,

*I*

_{l,l+1}=

*I*

_{l-1,l}≡

*I,κ*

_{l,l+1}=

*κ*

_{l-1,l}≡

*κ*. However, even if the waveguides are identical and equally spaced,

*κ*

_{11}and

*κ*are not equal to

_{NN}*κ*

_{22},

*κ*

_{33},…,

*κ*

_{N-1N-1}. In fact, Eq. (10b) shows that for those waveguides at the edges (

*l*= 1 and

*I*=

*N*) there are approximately only half as many contributing terms as the other waveguides: there are no waveguides to the left of the

*l*= 1 waveguide, and there are no waveguides to the right of the

*l*=

*N*waveguide, whereas all the other waveguides have contribution terms from both the left and right halves of their modal profiles.

*κ*

_{self}≡

*κ*

_{22},

*κ*

_{33},…,

*κ*

_{N-1N-1},

*κ*

_{self,edge}≡

*κ*

_{11}and

*κ*, and

_{NN}*δκ*

_{self}=

*κ*

_{self}-

*κ*

_{self,edge}.

*δκ*

_{self}, the eigenvectors are

*m*is the modal number and

*l*indicates which high-index rib waveguide (or low-index slot) is being described. [The expression for the eigenvalues is written later, Eq. (22).]

*N*, the second term in the above expression, is smaller than the first by (

*N*+ 1)

^{-1}and can be ignored, yielding a simpler expression. The progression of peak-amplitude values (in the high index regions) {

*A*

^{(m)}

_{l}},

*l*= 1,…,

*N*matches with the numerical calculations shown in Fig. 2. However, we shall see that the agreement is good only at large separation distances between the individual waveguides.

#### 2.2.2. TM Polarization

## 3. Numerically-assisted CMT: The “Inverse Problem”

*M*in Eq. (11) is revealing—but the quantitative predictions of CMT are in error in high-index-contrast SOI structures at short separation distances. To obtain a numerically-accurate picture of modal coupling, we propose a new extension of CMT, which we call “numerically assisted” CMT, to use the simulation results of the FDFD algorithm to back-calculate the elements of the coupling matrix

*M*. We can thereby check if the assumption of nearest-neighbor coupling is valid at short separation distances, and identify various other interesting coupling phenomena (e.g., non-Hermiticity of

*M*) which have not been pointed out earlier.

23. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B **23**, 1137–1145 (2006). [CrossRef]

- First, we solve for the supermodes using FDFD, which does not contain any of the limitations of nearest-neighbor CMT under investigation. The propagation constants of the supermodes are also obtained by this algorithm.
- Having obtained both the eigenvectors (peak amplitudes) and eigenvalues (propagation constants), we construct the (non-singular) matrix of eigenvectors
*A*(whose columns are the linearly-independent supermodes), and the diagonal matrix of eigenvalues, Λ = diag{_{mn}*β*^{2}_{m}}. - Next, we reassemble
*M*[see Eq. (11)] by using the matrix theorem cited in Ref. [23, Eq. (7), Lem. 1–2]: if the eigenvalues are distinct (which they are in this case),23. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B

**23**, 1137–1145 (2006). [CrossRef]*M*can be reconstructed as follows:*M*=*A*Λ*A*^{-1}. The matrix is unique to within a similarity transformation, which does not affect the following step. An example is shown in Table 1. (Notice that*κ*_{11}and*κ*_{55}are approximately one-half of*κ*_{22},*κ*_{33}, or*κ*_{44}, as discussed earlier.) The values of the reconstructed*M*matrix may be useful to design couplers in the strongly-coupled regime from the output of the FDFD mode-solver algorithm itself, without having to carry out time-consuming beam-propagation simulations.

## 4. Asymptotic accuracy of numerically-assisted CMT

*λ*) and eigenvectors (

_{k}*A*

^{(k)}) are obtained from a computer simulation. However, they may be obtained from measurements on fabricated structures, in order to test whether the intended coupling matrix was successfully obtained in practice. The experimental procedure to measure eigenvalues and eigenvectors could be similar to that used to image the modes of laser resonators.

*M*can be written to first order as

*u*is zero, i.e., the errors are only in the measured eigenvalues, since for identical arrayed waveguide structures, successive eigenvectors look quite different from each other and are easily distinguished [23

_{k}**23**, 1137–1145 (2006). [CrossRef]

**u**

^{(j)}.

**u**

^{(j+1)}= 0).

*M*

_{j-1j+1}].

## 5. Discussion

### 5.1. Next-to-nearest-neighbor coupling

*M*contains useful information about non nearest-neighbor coupling. We can read off whichever coupling coefficients are needed: in particular, we calculate the ratio ∣

*κ*

_{13}/

*κ*

_{12}∣, i.e., the ratio of next-to-nearest-neighbor coupling coefficient to the nearest-neighbor coupling coefficient.

*s*. Using Kuznetsov’s solution for the coupling coefficients of two slab waveguides [22

22. M. Kuznetsov, “Expressions for the coupling coefficient of a rectangular waveguide directional coupler,” Opt. Lett. **8**, 499–501 (1983). [CrossRef] [PubMed]

*κ*in both the TE and TM cases varies with

*s*as

*κ*~

*e*

^{-ps}where

*p*is the field decay length in the cladding. Therefore, the ratio

*κ*

_{13}/

*κ*

_{12}for both polarizations has the following expression (to leading order),

### 5.2. Eigenvalue fanout: effective index of the supermodes versus separation distance

*m*-th supermode is given by the equation

*β*

_{0}is the propagation constant of a single waveguide in isolation. Note that for

*N*= 5, the

*m*= 3 supermode has the special property that the right-hand-side of the above expression , i.e., the index of that supermode does not change with the coupling coefficient

*κ*. Hence,

*n*

^{(3)}

_{eff}is only weakly dependent on the separation distance (through the self-coupling coefficients,

*κ*

_{11},

*κ*

_{22},…,

*κ*

_{55}).

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]

*M*itself results from a numerical calculation of the supermodes (and their eigenfrequencies), rather than individual waveguide modes and the propagation constants of isolated waveguides.

24. G. Lenz and J. Salzman, “Eigenmodes of multiwaveguide structures,” J. Lightwave Technol. **8**, 1803–1809 (1990). [CrossRef]

*m*= 3 supermode: at a certain (small) waveguide separation,

*n*

^{(3)}

_{eff}is no longer independent of s and begins to deviate substantially from a straight line, contrary to the prediction of Eq. (22). This deviation is much more pronounced in the case of the TM polarization.

### 5.3. Field skewing and reshaping

*M*can become non-symmetric (non-Hermitian), although the eigenvalues remain stricly real as long as the mode is above cut-off. This can be seen in fact in the matrix written in Table 1 and Fig. 5(c,d):

*κ*

_{12}≠

*κ*

_{21}and

*κ*

_{13}≠

*κ*

_{31}, etc.

*A*

^{(m)}in Eq. (13)—doing so would result in asymmetric

*M*matrices.

*m*= 1 and

*m*= 5 supermode (plotted with continuous lines) and the supermode calculation of FDFD (with crosses). Note that in both cases the field is asymmetrically centered within the dielectric boundaries of the outer waveguides. Recall that CMT is based on writing the field as a summation of the scaled individual waveguide modes, Fig. 8(a,d), each of which is centered within its own core-cladding boundaries. At short separation distances, when, for example, there is a significant contribution of the (asymmetric) tail from the field in the second waveguide to the (symmetric) mode of the first waveguide, CMT itself predicts a lateral shift of the peak (of the sum) away from the exact center of the waveguide. The scaling relationships from Eq. (13) will enhance this effect for an multi-waveguide arrayed structure compared to a (twin-waveguide) directional coupler.

### 5.4. Polarization hybridization

*E*field components at large and small separation distances, along with the cross section of the Poynting vector, which indicates power flow. Notice that at small separation distances, the polarization component that was previously negligible has become, in fact, the dominant one. Furthermore, the power is actually carried above and below the waveguide structure at the outer edges, rather than within the inner slots, contrary to the original intention of slot waveguides.

## 6. Conclusion

*M*(a procedure we have called numerically-assisted coupled-mode theory, NA-CMT). The NA-CMT framework can be used to find out when the nearest-neighbor-coupling approximation breaks down.

## Acknowledgment

## References and links

1. | F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics |

2. | Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5- |

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 |

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 |

5. | Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometer-scale silicon electro-optic modulator,” Nature |

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 |

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 |

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 sub-micrometer aperture waveguides,” Opt. Express |

9. | K. Sasaki, F. Ohno, A. Motegi, and T. Baba,“Arrayed waveguide grating of 70×60 |

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 |

11. | H. Kogelnik and C. V. Shank, “Coupled-mode theory of distributed feedback lasers,” Appl. Phys. |

12. | A. Hardy and W. Streifer, “Coupled-mode theory of parallel waveguides,” J. Lightwave Technol. |

13. | W. P. Huang, “Coupled-mode theory for optical waveguides: An overview,” J. Opt. Soc. Am. A |

14. | K. S. Chiang, “Coupled-zigzag-wave theory for guided waves in slab waveguide arrays,” J. Lightwave Technol. |

15. | F. P. Payne, “An analytical model for the coupling between the array waveguides in AWGs and star couplers,” Opt. Quantum Electron. |

16. | E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. |

17. | A. Klekamp and R. Munzner, “Calculation of imaging errors of AWG,” J. Lightwave Technol. |

18. | S. H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, “Giant birefringence in multi-slotted silicon nanophotonic waveguides,” Opt. Express |

19. | P. Yeh, |

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. |

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. |

22. | M. Kuznetsov, “Expressions for the coupling coefficient of a rectangular waveguide directional coupler,” Opt. Lett. |

23. | S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B |

24. | G. Lenz and J. Salzman, “Eigenmodes of multiwaveguide structures,” J. Lightwave Technol. |

25. | M. Popovic, C. Manolatou, and M. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express |

26. | E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. |

27. | H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.7380) Optical devices : Waveguides, channeled

(230.4555) Optical devices : Coupled resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 8, 2008

Revised Manuscript: January 14, 2009

Manuscript Accepted: January 22, 2009

Published: January 27, 2009

**Citation**

Michael L. Cooper and Shayan Mookherjea, "Numerically-assisted coupled-mode
theory for silicon waveguide couplers
and arrayed waveguides," Opt. Express **17**, 1583-1599 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1583

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### References

- F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007). [CrossRef]
- Q. Xu, D. Fattal, and R. G. Beausoleil, "Silicon microring resonators with 1.5-μm radius," Opt. Express 16, 4309-4315 (2008). [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]
- 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]
- 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]
- 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.
- 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]
- 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).
- 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]
- H. Kogelnik and C. V. Shank, "Coupled-mode theory of distributed feedback lasers," Appl. Phys. 43, 2327-2335 (1972). [CrossRef]
- A. Hardy and W. Streifer, "Coupled-mode theory of parallel waveguides," J. Lightwave Technol. LT-3, 1135-1146 (1985). [CrossRef]
- W. P. Huang, "Coupled-mode theory for optical waveguides: An overview," J. Opt. Soc. Am. A 11, 963-983 (1994). [CrossRef]
- K. S. Chiang, "Coupled-zigzag-wave theory for guided waves in slab waveguide arrays," J. Lightwave Technol. 10, 1380-1387 (1992). [CrossRef]
- 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]
- E. Kapon, J. Katz, and A. Yariv, "Supermode analysis of phase-locked arrays of semiconductor lasers," Opt. Lett. 10, 125-127 (1984). [CrossRef]
- A. Klekamp and R. Munzner, "Calculation of imaging errors of AWG," J. Lightwave Technol. 21, 1978-1986 (2003). [CrossRef]
- 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]
- P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 2005).
- 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]
- M. Kuznetsov, "Expressions for the coupling coefficient of a rectangular waveguide directional coupler," Opt. Lett. 8, 499-501 (1983). [CrossRef] [PubMed]
- S. Mookherjea, "Spectral characteristics of coupled resonators," J. Opt. Soc. Am. B 23, 1137-1145 (2006). [CrossRef]
- G. Lenz and J. Salzman, "Eigenmodes of multiwaveguide structures," J. Lightwave Technol. 8, 1803-1809 (1990). [CrossRef]
- 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]
- E. Marcatili, "Improved coupled-mode equations for dielectric guides," IEEE J. Quantum Electron. QE-22, 988-993 (1986). [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]

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