## New insight into quasi leaky mode approximations for unified coupled-mode analysis |

Optics Express, Vol. 18, Issue 20, pp. 20595-20609 (2010)

http://dx.doi.org/10.1364/OE.18.020595

Acrobat PDF (1484 KB)

### Abstract

To facilitate the analysis of radiation mode couplings, quasi leaky mode approximations were utilized in coupled-mode analysis. The key to effectively and accurately apply this approach is how to well approximate radiation modes by the quasi leaky modes in an equivalent closed waveguide model. In this paper, the principle, applicability and accuracy of the approximations are demonstrated, and the detailed implementation is also suggested by applying a unified coupled-mode analysis to fiber gratings. First of all, based on a thorough study on the characteristics of the complex modes, for the first time, quasi leaky modes are classified into guided-mode-like inner-cladding and radiation-mode-like outer-cladding leaky modes so as to explicitly establish equivalence relationships between the discrete leaky modes and the continuous radiation modes. With this new insight, the whole analysis process especially for some of the practically tricky issues such as the criteria for developing the proper equivalent waveguide model and the subsequent mode expansion basis are better understood and easier to be dealt with for different problems where radiation modes come into play. Moreover, as essential preconditions to extend the conventional coupled mode analysis to the present unified one, the couplings between the guided core mode and a leaky mode are studied in a systematic and consistent manner. An intuitive and then a deep understanding on the roles of complex modes on mode coupling and power exchanging are thus gained for further simulations. Lastly, the transmission spectra of fiber gratings with different surrounding indices are simulated. The simulated results agree well with those obtained theoretically and experimentally in the literatures, which strongly validate the principle of quasi leaky mode approximations and its implementation on the unified coupled-mode analysis expounded in this paper.

© 2010 OSA

## 1. Introduction

2. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. **31**(10), 1790–1802 (1995). [CrossRef]

3. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

4. F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. **7**(11), 371–373 (1997). [CrossRef]

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. **27**(11), 1461–1468 (2009). [CrossRef]

10. D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. **28**(11), 1661–1669 (2010). [CrossRef]

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**(21), 19134–19152 (2009). [CrossRef]

12. Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express **18**(2), 713–726 (2010). [CrossRef] [PubMed]

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. **28**(5), 761–767 (2010). [CrossRef]

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**(21), 19134–19152 (2009). [CrossRef]

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. **28**(5), 761–767 (2010). [CrossRef]

15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A **14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. **16**(9), 1606–1612 (1998). [CrossRef]

22. Y. Y. Shevchenko and J. Albert, “Plasmon resonances in gold-coated tilted fiber Bragg gratings,” Opt. Lett. **32**(3), 211–213 (2007). [CrossRef] [PubMed]

15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. **16**(9), 1606–1612 (1998). [CrossRef]

19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. **13**(4), 308–310 (2001). [CrossRef]

20. Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. **18**(9), 1220–1225 (2000). [CrossRef]

6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. **49**(2), 185–195 (2001). [CrossRef]

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A **14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

21. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science **305**(5680), 74–75 (2004). [CrossRef] [PubMed]

## 2. Theoretical model of fiber grating

_{oc}. Here the surround is seen as an outer-cladding in the following discussions. The refractive indices and radii of the core and inner-cladding are n

_{co}and n

_{ic}, r

_{co}and r

_{ic}, respectively. The periodically modulated refractive index in the core along the longitudinal direction of the grating is described as usual,where Λ is the grating period, Δn is the index change spatially averaged over a period, and ν is the fringe visibility of the index change.

3. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

23. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. **26**(13), 1868–1876 (2008). [CrossRef]

_{oc}, the thickness d

_{PML}and the reflection coefficient R

_{PML}, the PML cannot only perfectly match with the outer-cladding, but also rapidly attenuate the out-going field in the PML to effectively eliminate the additional field reflected by PEC. This model is thus at most time equivalent to the original open waveguide. The principle of how to set PML parameters to obtain an equivalent waveguide model will be discussed in the following sections. Meanwhile, the modes in the equivalent waveguide model will be examined before being selected as the expansion basis in the unified coupled-mode analysis.

## 3. Characteristics of complex modes in equivalent reference waveguide model

_{co}= 2.5μm, the inner-cladding radius r

_{ic}= 62.5μm, the core refractive index n

_{co}= 1.458 and the inner-cladding refractive index n

_{ic}= 1.45, which are chosen to be same as those in [5

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. **27**(11), 1461–1468 (2009). [CrossRef]

_{oc}will be chosen to be higher than or equal to n

_{ic}for different cases. The main model parameters of PML are respectively r

_{oc}= 80.0μm, d

_{PML}= 10.0μm, and R

_{PML}= 10

^{−12}, which will be used in the following simulations if no special declarations. The working wavelength is set to be 1.55μm.

23. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. **26**(13), 1868–1876 (2008). [CrossRef]

4. F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. **7**(11), 371–373 (1997). [CrossRef]

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. **27**(11), 1461–1468 (2009). [CrossRef]

6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. **49**(2), 185–195 (2001). [CrossRef]

_{oc}will be successively discussed as follows.

_{oc}is higher than n

_{ic}

_{oc}= 1.46 are shown in Fig. 2 , where the horizontal and vertical abscissas represent the real and imaginary parts of the effective refractive indices, respectively. There are several obviously divided branches which can be well defined by simultaneously looking into their field patterns.

_{11}mode is denoted by a diamond. Its amplitude and phase are shown in Fig. 3(a) . As seen from Fig. 3(a), its amplitude is the same as that in the open waveguide only in the core and inner-cladding regions. In the outer-cladding and PML region, the magnitude of the guided mode is negligible, and the phase increases rapidly.

**27**(11), 1461–1468 (2009). [CrossRef]

6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. **49**(2), 185–195 (2001). [CrossRef]

_{PML}= 1 to R

_{PML}= 10

^{−30}, and intercepted a part of mode spectrum interested in the following simulations to clearly show the convergent process of effective refractive indices with the variation of R

_{PML}, as illustrated in Fig. 4 . It is observed that the results become convergent when R

_{PML}= 10

^{−12}. In addition, we also plotted field pattern of a typical inner-cladding leaky mode, namely, HE

_{16}as shown in Fig. 3(b). Due to partial reflection at the inner- and outer-cladding interface, the leaky modes are guided-mode-like in the inner-cladding region with sufficient thickness as compared to the wavelength. Meanwhile, the exponentially growing field in the outer-cladding is effectively attenuated by PML, and smoothly transit to vanish before reaching PEC.

_{PML}= 10

^{−12}and R

_{PML}= 10

^{−60}. It is noted that, despite the fact that the PML reflection is extremely small, the residual reflection effect from the PEC still plays some role in confining the field in the outer-cladding region, leading to a quasi-standing wave pattern, which is especially obvious for a higher order outer-cladding leaky mode, as illustrated in Fig. 3(d).

_{oc}, more and more outer-cladding leaky modes emerge in the right branch. Compared with the guide-mode-like inner-cladding leaky modes, outer-cladding leaky modes are more radiation-mode-like. Meanwhile, they move left and upside, even to the left of the left branch if further enlarging r

_{oc}, but distinguished from inner-cladding ones, as shown in Fig. 5 . Actually, in the case of n

_{oc}<n

_{ic}, the leaky modes can also be classified as the inner- and outer-cladding modes in a similar way, though the distributions of the two kinds of leaky modes are different from those in the present case of n

_{oc}>n

_{ic}. In both cases, the difference between the inner- and outer-cladding modes decreases with the decrease of the thickness of the inner-cladding or the index difference between the inner- and outer-cladding until degenerates to the next case of n

_{oc}= n

_{ic}.

_{oc}is equal to n

_{ic}

_{oc}= 80.0μm, there are only a few leaky modes denoted by stars in the mode spectrum, as shown in Fig. 7 which is also intercepted from the mode spectra. With the location of PML moving to be far away from the core, much more leaky modes appear and tend to be a continuum; meanwhile, they move left and upside to approach to the radiation modes in open waveguide, as shown by circles in Fig. 7, where r

_{oc}= 500.0μm. Intuitively, the latter will be a better approximation to the radiation modes in the original waveguide with infinite cladding, which will be validated in the following simulations for the transmission spectra of fiber gratings with index-matched outer-claddings.

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**(21), 19134–19152 (2009). [CrossRef]

*m*=

*n*, we defined the integral coefficient for the

*m*-th mode aswhich will be used in the unified coupled-mode equations.

## 4. Unified coupled-mode equations for fiber gratings

**17**(21), 19134–19152 (2009). [CrossRef]

**17**(21), 19134–19152 (2009). [CrossRef]

*a*and

_{n}*b*are respectively the expansion coefficients or the complex amplitudes of the fields of the forward and backward propagating modes.

_{n}*m*-th complex mode. The coupling coefficients between co-propagating modes and contra-propagating modes

*κ*and

_{mn}*χ*are respectively expressed as follows,where

_{mn}**e**

*and*

_{tn}**h**

*,*

_{tn}**e**

*and*

_{zn}**h**

*are the transverse and longitudinal mode functions of complex modes in the reference waveguide, respectively. n*

_{zn}_{0}and n are respectively index profiles of the reference waveguide and the structure under consideration which is fiber grating in the following discussions. N

*is defined in Eq. (3b).*

_{m}16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A **14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

*β*is the phase detuning factor and defined as

_{n}*β*is defined as

_{n}*δ*vanishes, that is,

*c*usually equals 2, except for the coupling to the counter-propagating core mode, where

*c*= 1 [16

**14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. **13**(4), 134–141 (1977). [CrossRef]

_{1n}and even leaky modes of EH

_{1n}in the case of a higher-index outer-cladding. As seen from the figure, when the mode order is lower, the coupling strength of the core mode to even leaky modes is very weak compared with that to odd leaky modes of the same order. However, it increases with the increase of the radial order, and becomes comparable with that to the same order odd one when the radial order is larger than 40 in this case. The variation trends are common for waveguides with different structure and PML model parameters, except that the variation curves change with the PML model parameters in the case of infinite cladding. Meanwhile, the variation trends shown in Fig. 8 are similar to those observed in the couplings between the core and guided cladding modes in [16

**14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

## 5. Numerical results and discussions for fiber gratings

^{−4}, and L = 2.5cm, which are also chosen to be same as those in [5

**27**(11), 1461–1468 (2009). [CrossRef]

*δ*, the power percentage of the core mode taking part in the power exchanging decreases, meanwhile, the period of the power exchanging decreases. The envelops of the periodically oscillating powers of the two modes both decrease at first, and then gradually change to monotonic attenuations, though the power of the leaky mode is not comparable with that of the core mode. From the power evolutions in Fig. 9, the roles of complex propagation constants and complex coupling coefficients on mode coupling and power exchanging can be intuitively understood. The differences from those of non-complex ones can be attributed to the leakage, which can be further understood by analogy to a simple harmonic vibration with damping. For the applications in fiber gratings, the roles will result in a larger power loss of the core mode at a certain coupling wavelength and length. The rules of power exchanging in the coupling between complex modes can be used in the design and optimization of devices, for example, properly setting parameters such as grating length to realize required performances such as coupling ratio or properly setting leakage loss to effectively shorten the coupling length. Generally, the roles of complex modes in the mode couplings have relation to multiple factors, which will not be further discussed in this paper due to the limited space.

_{oc}is higher than n

_{ic}

17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. **16**(9), 1606–1612 (1998). [CrossRef]

19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. **13**(4), 308–310 (2001). [CrossRef]

**27**(11), 1461–1468 (2009). [CrossRef]

**17**(21), 19134–19152 (2009). [CrossRef]

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. **28**(5), 761–767 (2010). [CrossRef]

_{oc}= 1.46 is taken as example from the three cases discussed successively in Fig. 10(a)–10(c), as shown in Fig. 10(a). As seen from the figure, convergent results are obtained with only three leaky modes used in the mode expansion, and the results obtained by the reduced two-mode model can be taken as a good approximation. Therefore, it is obvious that in this case of n

_{oc}>n

_{ic}, the present coupled-mode analysis is greatly simplified compared with the conventional expansion in terms of the guided core mode and radiation modes.

_{oc}is equal to n

_{ic}

**16**(9), 1606–1612 (1998). [CrossRef]

18. D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. **11**(3), 343–345 (1999). [CrossRef]

19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. **13**(4), 308–310 (2001). [CrossRef]

_{oc}, the radial order from which the coupling strength to the odd and even one becomes comparable also greatly increases, thus the even modes can still be neglected to simplify the simulations.

_{oc}>n

_{ic}, the radiation modes in couplings are well approximated by at most 3 inner-cladding leaky modes. It is thus suggested that a moderate PML with which inner-cladding leaky modes can be convergent is enough for mode solving, such as R = 10

^{−12}as seen in Fig. 4, while an even stronger PML will induce much more useless outer-cladding leaky modes and PML modes, and then unnecessary complication in the whole analysis. In the case of n

_{oc}= n

_{ic}, the convergent spectra owe to the contribution of about 11 radiation-mode-like and nearly phase-matching outer-cladding modes obtained by using an enlarged r

_{oc}instead of a stronger PML with a less R

_{PML}. It is thus evident the establishing of equivalence relations between guided-mode-like inner- or radiation-mode-like outer-cladding leaky modes and radiation modes for different cases is also helpful for suitably setting PML parameters of the equivalent waveguide model.

^{−4}, and L = 5.0mm. The Bragg resonant wavelength between two contra-propagating core modes is designed at λ

_{B}= 1.55μm.

_{oc}and distributed more concentrated. In this case, when r

_{oc}is increased to 200.0μm, well convergent results are obtained with 11 outer-cladding leaky modes taken into mode expansion, as seen in Fig. 11(b). Figure 11(c) and 11(d) show the cases that the refractive indices of the surrounds are higher than those of the claddings. Seen from the transmission spectra, the dips correspond to the couplings between the guided core mode and the inner-cladding leaky modes. The variation trend of a certain dip with the change of n

_{oc}is similar to that in LPGs of the same case.

15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

**14**(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

20. Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. **18**(9), 1220–1225 (2000). [CrossRef]

_{oc}= n

_{ic}and n

_{oc}>n

_{ic}, respectively. The reduced two-mode model may be an acceptable approximation in the case of n

_{oc}>n

_{ic}, however, in the case of n

_{oc}= n

_{ic}, at least 9 leaky modes need to be included to approximate radiation modes.

## 6. Conclusion

## Acknowledgments

## References and links

1. | J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China. |

2. | S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. |

3. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

4. | F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. |

5. | Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. |

6. | H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. |

7. | H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. |

8. | K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. |

9. | J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express |

10. | D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. |

11. | W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express |

12. | Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express |

13. | N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. |

14. | L. Yang and L. L. Xue, “Simplified treatment for radiation mode in coupled-mode analysis,” ICMMT 2010, Chengdu, May, 2010. |

15. | T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. |

16. | T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A |

17. | H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. |

18. | D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. |

19. | Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. |

20. | Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. |

21. | V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science |

22. | Y. Y. Shevchenko and J. Albert, “Plasmon resonances in gold-coated tilted fiber Bragg gratings,” Opt. Lett. |

23. | Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. |

24. | R. E. Collin, |

25. | W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2310) Fiber optics and optical communications : Fiber optics

(230.7370) Optical devices : Waveguides

(350.2770) Other areas of optics : Gratings

(350.5610) Other areas of optics : Radiation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 22, 2010

Revised Manuscript: September 5, 2010

Manuscript Accepted: September 7, 2010

Published: September 14, 2010

**Citation**

Li Yang, Lin-Lin Xue, Yu-Chun Lu, and Wei-Ping Huang, "New insight into quasi leaky mode approximations for unified coupled-mode analysis," Opt. Express **18**, 20595-20609 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20595

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### References

- J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.
- S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995). [CrossRef]
- J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
- F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997). [CrossRef]
- Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]
- H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001). [CrossRef]
- H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20(7), 1141–1148 (2002). [CrossRef]
- K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005). [CrossRef]
- J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16(22), 18152–18163 (2008). [CrossRef] [PubMed]
- D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28(11), 1661–1669 (2010). [CrossRef]
- W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]
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