## Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper

Optics Express, Vol. 14, Issue 13, pp. 6035-6054 (2006)

http://dx.doi.org/10.1364/OE.14.006035

Acrobat PDF (773 KB)

### Abstract

In this paper, the mathematical model of the modified step-theory is derived based on the platform of two-dimensional photonic crystal structure that is infinitely long in third dimension. The mode coupling mechanism of photonic crystal tapers is theoretically studied using the modified step-theory. The model is verified by comparing the transmission spectrum obtained for the input/output defect coupler where it shows a good match of less than 5% discrepancy. The modified step-theory is applied to different taper structures to investigate the power loss during the transmission. The power loss at the relative position of the taper provides an explanation as to which taper designs give the highest coupling efficiency.

© 2006 Optical Society of America

## 1. Introduction

1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” *Nature* , **386**, 143 (1997) [CrossRef]

6. S. G. Johnson, P. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Linear waveguide in photonic crystal slab,” Phys. Rev. B **62**, 8212–8222 (2000). [CrossRef]

1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” *Nature* , **386**, 143 (1997) [CrossRef]

7. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through sharp bends in Photonic Crystal Waveguide,” Phys. Rev. Lett. **77**, 3787–3790 (1996). [CrossRef] [PubMed]

8. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. **26**, 1102–1104 (2001). [CrossRef]

14. M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic mode conversion,” Appl. Phys. Lett. **78**, 1466–1468 (2001). [CrossRef]

8. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. **26**, 1102–1104 (2001). [CrossRef]

12. E. H. Khoo, A. Q. Liu, and J. H. Wu, “Nonuniform photonic crystal taper for high efficient mode coupling,” Opt. Express **13**, 7748–7759 (2005). [CrossRef] [PubMed]

13. Ph. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguide,” Opt. Express **10**, 354–359 (2002). [PubMed]

14. M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic mode conversion,” Appl. Phys. Lett. **78**, 1466–1468 (2001). [CrossRef]

15. M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, “Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions,” Phys. Rev. E **67**, 046613 (2003). [CrossRef]

20. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic Crystal circuits,” J. Lightwave Technol. **22**, 684–691 (2004). [CrossRef]

15. M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, “Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions,” Phys. Rev. E **67**, 046613 (2003). [CrossRef]

16. M. Skorobogatiy, “Modeling the impact of imperfections in high-index-contrast photonic crystal waveguides,” Phys. Rev. E **70**, 046609 (2004). [CrossRef]

17. A. A Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. Martin de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E **60**, 6118 (1999). [CrossRef]

18. A. A Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. Martin de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E **62**, 5711 (2000). [CrossRef]

19. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technology **18**, 102–110 (2000). [CrossRef]

20. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic Crystal circuits,” J. Lightwave Technol. **22**, 684–691 (2004). [CrossRef]

21. A. F. Milton and W. K. Burns, “Mode Conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. **11**, 32–39 (1975). [CrossRef]

24. O. Mitomi, K. Kasaya, and H. Miyazawa, “Design of a single-mode tapered waveguide for low-loss chip-to-fiber coupling,” IEEE J. Quantum Electron. **30**, 1787–1793 (1994). [CrossRef]

24. O. Mitomi, K. Kasaya, and H. Miyazawa, “Design of a single-mode tapered waveguide for low-loss chip-to-fiber coupling,” IEEE J. Quantum Electron. **30**, 1787–1793 (1994). [CrossRef]

9. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos “Adiabatic theorem and continuous coupled mode theory for efficient taper transition,” Phys. Rev. E **66**, 066608 (2002). [CrossRef]

## 2. Fundamentals of light propagation in photonic crystal waveguide

*E*is the real amplitude of the propagating

_{0}*E*-field at a position

*x. ξ*(

*z*) is the z dependent field distribution of the guided mode. ξ is dependent on the width and the mode number of the waveguide.

*β*is defined as the propagation constant of the guided mode and the electromagnetic wave is propagating in the x direction.

*u*(

_{kx,kz}*x,z*) is a periodic function in

*x*and

*z*direction. The periodicity in

*x*and

*z*direction leads to a

*xz*dependent for

*E*-field, which is a product of plane wave with a

*xz*-periodic function. This is known as Bloch’s theorem.

26. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B **58**, 4809–4817 (1998). [CrossRef]

27. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B **64**, 155113 (2001). [CrossRef]

*b*, on which an artificial periodicity of

*d*is imposed. The equation for the mode field of the guided modes is given as [26

26. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B **58**, 4809–4817 (1998). [CrossRef]

*m*and the width of the waveguide,

*b*. For

*m*=1, this refers to the fundamental even mode where larger values imply higher order modes. When m=

*1, 3, 5*, …, (2

*υ*+1), …, it corresponds to even modes with a cosine mode profile that is symmetrical about the x direction. When

*m*=

*2, 4, 6*, …, (2

*υ*), …, it corresponds to odd modes with a sine profile that is asymmetrical about the

*x*direction. The constant

*b*is dependent on the structure of the lattice arrangement. It corresponds to the width of the periodicity imposed waveguide, which is usually in multiples of the lattice constant.

*ψ*is the phase coefficient, which is dependent on the artificial periodicity of

*d*given as

## 3. Mathematical model of Modified step-theory

*i*and

*j*traveling in a two-dimensional photonic crystal waveguide of width

*b*and artificial periodicity of

*d*. Both modes travel to a narrower waveguide of width

*b*with the same periodicity. The wider waveguide is labeled as the

^{’}*n*th step with subscript

*n*, while the narrower waveguide is labeled as the (

*n*+1)th step with subscript

*n*+1. Using Eq. (3) for the description of the mode field for

*i*and

*j*, the incident, reflected and transmitted components for the

*E*and

*H*field at the boundary between steps are matched and the amplitude of transmitted mode

*j*in the (

*n*+1)th step is given by

*A*refers to the ratio of the modal field amplitude of the ith mode at the

_{in}*n*th step, in the presence of mode conversion to the initial incident amplitude that corresponds to the fundamental mode and has a unit power. Similar assignments are given for

*A*+1 and

_{jn}*A*. The coupling constant

_{jn}*c*between mode

_{ij}*i*and

*j*is given as

*γ*is to account for the Bloch modes in periodic structure for convergence in high index contrast material. When

*γ*is set to zero, Eq. (6b) is simplified to the case of the conventional waveguide. The field overlap integral

*I*is given by

_{in,jn+1}*b*change. Therefore, the integral varies and is evaluated for every transmission between the steps. For the case of multiple modes propagating in the waveguide, the general expression for the transmitted

*j*th mode is given as the sum of all the field profile of the incident mode which is given as

*j*mode is given by

^{th}*ψ*is the phase coefficient that is defined as

*c*and transmission amplitudes,

_{ij}*A*are again derived for the three-dimensional photonic crystal tapered waveguide structures. The three-dimensional model of the modified step-theory will be discussed in other papers.

_{jn+1}11. P. Pottier, I. Ntakis, and R. M. De La Rue, “Photonic crystal continuous taper for low-loss direct coupling into photonic crystal channel waveguides and further device functionality,” Opt. Commun. **223**, 339–347 (2003). [CrossRef]

28. S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5 µm light in photonic crystals based on dielectric rods,” Appl. Phys. Lett. **85**, 6110–6112 (2004). [CrossRef]

11. P. Pottier, I. Ntakis, and R. M. De La Rue, “Photonic crystal continuous taper for low-loss direct coupling into photonic crystal channel waveguides and further device functionality,” Opt. Commun. **223**, 339–347 (2003). [CrossRef]

9. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos “Adiabatic theorem and continuous coupled mode theory for efficient taper transition,” Phys. Rev. E **66**, 066608 (2002). [CrossRef]

## 4. Verification of the Modified step-theory

9. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos “Adiabatic theorem and continuous coupled mode theory for efficient taper transition,” Phys. Rev. E **66**, 066608 (2002). [CrossRef]

*z*direction. There is also a row of linear defects in the input tapering area which acts as a linear chain of waveguide resonators for the propagation of lightwave. The input taper is then joined by an identical taper but with increasing width towards the other end. A twin head input/output photonic crystal taper is then formed.

**66**, 066608 (2002). [CrossRef]

## 5. Numerical results and discussions

### 5.1 Abrupt Step Waveguide and Step Taper Waveguide

*a*is the lattice constant,

*r*is the radius of the rods and the

*r/a*ratio is 0.2. Using plane wave calculation for high refractive index rods, a photonic bandgap exists in normalized frequency range of 0.274–0.429 for TM polarization but there is no bandgap for TE polarization. The transverse magnetic (TM) polarization is defined to have magnetic field in

*x-z*plane and the electric field is perpendicular to the plane while vice visa for the transverse electric (TE) polarization.

*k*is the fundamental even mode while the other is the higher order odd mode. For the higher frequency of 0.4, the wider waveguide can support all the three guided modes.

*q*rows ×1 column rods as shown in Fig. 2. Only one column of rods is needed because the wave vector is conserved in the direction of propagation.

*q*is the number of rows of rods to be considered in the supercell such that the eigenvalues converge and accurate band diagram is obtained. For this paper, the

*q*value of 15 is sufficient to serve for this purpose.

*k*for the fundamental mode and 0.707

_{0}*k*for the odd mode where

_{0}*k*is the free space wave vector from Fig. 3(a) and (b). The narrower waveguide has a propagation constant of 0.7334

_{0}*k*. Using the modified step-theory, the power transmitted from the input power of unity with the even mode is 0.6559. The result is obtained using the exact scattering method. The different between the two values differ by less than 2 %. This shows a good match between the modified step-theory and the scattering method. When the input is excited by the odd mode, the transmission is zero. The scattering method gives a value of 3×10

_{0}^{-5}. The reason for this small value is due to scattered stray light that is calculated using the scattering method. However, the value detected by the scattering matrix is close to zero. Therefore, it can be concluded that the odd and even mode do not couple to each other.

8. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. **26**, 1102–1104 (2001). [CrossRef]

*a*which gives a width of approximately 10 µm. This corresponds to the modal diameter of single mode optical fibre source. The taper is known as a “step” taper because the side-wall of taper resembles that of a step staircase. Each step has a fixed length but different width as shown in Fig. 5. In Fig. 5, each step is fixed at a length of 3

*a*with the largest width of 18

*a*at the input and 2

*a*at the output of the taper. By varying the length of the step, the total length of the taper is changed. The change in the taper length can result in a more gradual or abrupt mode transition which affects the coupling efficiency of the taper.

^{th}step is actually the single mode photonic crystal waveguide. The difference is that to calculate the transmission from the wider to the narrower end of the taper, the value of

*b*changes as the width changes. For the step taper waveguide, the complexity increases as greater number of modes are involved compared to the step waveguide.

*a*and 40

*a*using the modified step-theory. For the step structure, the transmitted power has a discrete step function as the power coupling only occurs at the boundary between steps. When the light propagates through the taper, the transmitted power begins to reduce because of the uneven side-wall and scattering at the corner of the steps. The higher order even modes are excited and some power is transferred to them. Due to the abrupt change in the width of the taper, the power carried by the higher order mode is reflected rather than transferred to the lower order mode. Lower power is received in the photonic crystal waveguide. For the shorter taper length, the wave propagation is abrupt and fast tapering of the tapered waveguide induces a lot of intermodal coupling. However, when the taper length increases, the transmitted power increases because of more gradual change in the width of the taper.

### 6.2 Smooth linear and nonuniform taper design

12. E. H. Khoo, A. Q. Liu, and J. H. Wu, “Nonuniform photonic crystal taper for high efficient mode coupling,” Opt. Express **13**, 7748–7759 (2005). [CrossRef] [PubMed]

*α*parameter determines the different curvature of the taper waveguide.

*d*and

_{i}*d*are the input and output width of the taper respectively.

_{o}*l*is the total length of taper. The different taper structures are obtained by shifting the rods in the crystal lattice using Eq. (11) with respect to the centerline. Based on [12

12. E. H. Khoo, A. Q. Liu, and J. H. Wu, “Nonuniform photonic crystal taper for high efficient mode coupling,” Opt. Express **13**, 7748–7759 (2005). [CrossRef] [PubMed]

*α*=0.5 gives the highest coupling efficiency of 97.5 % at a taper length of 20.52 µm. The linear taper with

*α*=1 and concave taper

*α*=2 give a coupling efficiency of 91.2 % and 90.5 % respectively due to higher reflection loss. The results of the coupling efficiency for different taper curvature are discussed in depth using the modified step-theory. The transmitted power for the three different taper curvatures at varied length is shown in Fig. 9.

**13**, 7748–7759 (2005). [CrossRef] [PubMed]

15. M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, “Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions,” Phys. Rev. E **67**, 046613 (2003). [CrossRef]

20. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic Crystal circuits,” J. Lightwave Technol. **22**, 684–691 (2004). [CrossRef]

29. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. and Quantum Electron. , **33**, 327–341 (2001). [CrossRef]

30. A. Lavrinenko, P. Borel, L. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. Chong, “Comprehensive FDTD modelling of photonic crystal waveguide components,” Opt. Express **12**, 234–248 (2003). [CrossRef]

## 6. Conclusions

**66**, 066608 (2002). [CrossRef]

**66**, 066608 (2002). [CrossRef]

## Appendix A

*µ*and

*ε*are the permeability and permittivity of the material in which the field exist. Assuming linear and lossless material, the complex field

**and**

*E***can be expanded to a set of harmonic modes with complex time exponential**

*H**e*. Rearranging Eqs. (A1) and (A2) and substituting the field harmonics, the equation for the electric field is given as

^{iωt}*ν*is the speed of electromagnetic waves in the material with refractive index,

*n*given by

**field, similar expression can be obtained. By applying vector properties for the E field and Coulomb’s law, the curl of the curl symbol can be removed and be replaced by ∇**

*H*^{2}to form the scalar wave equation given as

*x*and

*z*direction. The wave solution for the electric field is described by the normal modes given as [25]

*β*is the propagating constant of the waves traveling in the waveguide and

*ξ*(

*z*) is the z dependent field distribution of the guided mode.

*E*is the real amplitude of the propagating electric field. As mentioned in section 2, photonic crystal waveguide mode field resembles modes in conventional waveguide [26

_{0}26. A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B **58**, 4809–4817 (1998). [CrossRef]

27. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B **64**, 155113 (2001). [CrossRef]

*b*, on which an artificial periodicity of

*d*is imposed. The equation for the mode field of the guided modes is given as

*ψ*is given as

*i*and

*j*are propagating at the x direction. Both modes travel to a narrower waveguide of width

*b*

^{’}with the same periodicity. The wider waveguide is labeled as the nth step with subscript

*n*, while the narrower waveguide is labeled as the (

*n*+1)th step with subscript

*n*+1. Matching the boundary conditions between the

*n*th and (

*n*+1)th steps, the incident, reflected and transmitted field components are given as

*E*and the reflected radiation modes,

^{trans}*E*. These two modes will vanish during the course of derivation by applying orthogonal condition. For

^{ref}*H*-field in

*z*direction, similar expressions can be obtained as

*E*refer to the real amplitude of the reflected guided mode. By multiplying Eqs. (A9) and (A10) by a transmitted mode field distribution and integrating over

^{R}*x*, the unguided radiation modes are eliminated because of the orthogonality. The aim now is to obtain an iterative equation for field amplitude and phase of

*j*th mode,

*E*and

_{jn+1}*ψ*. To do this, the expression for the reflected field of the

_{jn+1}*i*th mode must be obtained. Together with the assumption made in section 3 and multiplying Eqs. (A9) and (A10) by transmitted mode field of the

*ζ*and integrating over

_{in+1}*x*, the expression for the reflected field of mode is given as

*ζ*and integrating over

_{jn+1}*x*. After some algebraic manipulation and substituting the reflected field for

*i*th mode with Eq. (A11), the transmitted field expression for the

*j*th mode is obtained as

*k*is the free space wave vector. All the terms of Eq. (A12a) is divided by appropriate factor of the unity mode field and simplifying, the amplitude and phase of the

_{0}*j*

^{th}mode at (

*n*+1)th step is given by

*c*is given by

_{ij}*β*and the Bloch mode of the periodicity. The coefficient

*c*can be obtained by substituting

_{jj}*i*by

*j*. For multiple modes coupling, the general expression is given as

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” |

2. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

3. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel add-drop filters in photonic crystal,” Opt. Express |

4. | M. Bayindir, B. Temelkuran, and E. Ozbar, “Photonic-crystal-based beam splitter,” Appl. Phys. Lett. |

5. | Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. |

6. | S. G. Johnson, P. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, “Linear waveguide in photonic crystal slab,” Phys. Rev. B |

7. | A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through sharp bends in Photonic Crystal Waveguide,” Phys. Rev. Lett. |

8. | T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. |

9. | S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos “Adiabatic theorem and continuous coupled mode theory for efficient taper transition,” Phys. Rev. E |

10. | P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguide,” Opt. Soc. Am. B |

11. | P. Pottier, I. Ntakis, and R. M. De La Rue, “Photonic crystal continuous taper for low-loss direct coupling into photonic crystal channel waveguides and further device functionality,” Opt. Commun. |

12. | E. H. Khoo, A. Q. Liu, and J. H. Wu, “Nonuniform photonic crystal taper for high efficient mode coupling,” Opt. Express |

13. | Ph. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguide,” Opt. Express |

14. | M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic mode conversion,” Appl. Phys. Lett. |

15. | M. Skorobogatiy, S. G. Johnson, S. A. Jacobs, and Y. Fink, “Dielectric profile variations in high-index-contrast waveguides, coupled mode theory, and perturbation expansions,” Phys. Rev. E |

16. | M. Skorobogatiy, “Modeling the impact of imperfections in high-index-contrast photonic crystal waveguides,” Phys. Rev. E |

17. | A. A Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. Martin de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E |

18. | A. A Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. Martin de Sterke, “Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E |

19. | M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technology |

20. | T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic Crystal circuits,” J. Lightwave Technol. |

21. | A. F. Milton and W. K. Burns, “Mode Conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. |

22. | A. R. Nelson, “Coupling optical waveguides by tapers,” Appl. Opt. |

23. | D. Marcuse, |

24. | O. Mitomi, K. Kasaya, and H. Miyazawa, “Design of a single-mode tapered waveguide for low-loss chip-to-fiber coupling,” IEEE J. Quantum Electron. |

25. | J. David Jackson, |

26. | A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B |

27. | M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B |

28. | S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5 µm light in photonic crystals based on dielectric rods,” Appl. Phys. Lett. |

29. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. and Quantum Electron. , |

30. | A. Lavrinenko, P. Borel, L. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. Chong, “Comprehensive FDTD modelling of photonic crystal waveguide components,” Opt. Express |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(230.7380) Optical devices : Waveguides, channeled

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: March 7, 2006

Revised Manuscript: May 19, 2006

Manuscript Accepted: June 4, 2006

Published: June 26, 2006

**Citation**

E. H. Khoo, A. Q. Liu, J. H. Wu, J. Li, and D. Pinjala, "Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper," Opt. Express **14**, 6035-6054 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6035

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### References

- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature, 386, 143 (1997) [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel add-drop filters in photonic crystal," Opt. Express 3, 4-11 (1998). [CrossRef] [PubMed]
- M. Bayindir, B. Temelkuran, and E. Ozbar, "Photonic-crystal-based beam splitter," Appl. Phys. Lett. 77, 3902-3904 (2000). [CrossRef]
- Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, "Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 82, 1341-1343 (2003). [CrossRef]
- S. G. Johnson, P. R. Villeneuve, S. H. Fan, and J. D. Joannopoulos, "Linear waveguide in photonic crystal slab," Phys. Rev. B 62, 8212-8222 (2000). [CrossRef]
- A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High Transmission through sharp bends in Photonic Crystal Waveguide," Phys. Rev. Lett. 77, 3787-3790 (1996). [CrossRef] [PubMed]
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