## Local density of states analysis of surface wave modes on truncated photonic crystal surfaces with nonlinear material

Optics Express, Vol. 12, Issue 20, pp. 4855-4863 (2004)

http://dx.doi.org/10.1364/OPEX.12.004855

Acrobat PDF (1270 KB)

### Abstract

The local density of states and response to an incident plane wave of a finite sized photonic crystal (PC) with nonlinear material (NLM) is analyzed. Of particular interest is the excitation of surface wave modes at the truncated surface of the PC, which is collocated with the NLM material. We compute the 2D Green function of the PC with linear material and then include the Kerr NLM in a self-consistent manner. The 2D PC consists of a square array of circular rods where one row of the rods is semi-circular in order to move the surface wave defect mode frequency into the band gap. Since the surface modes are resonant at the interface, the NLM should experience at least an order of magnitude increase in field intensity. This is a possible means of increasing the efficiency of the PC as a frequency conversion device.

© 2004 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

2. W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. **18**, 528 (1993). [CrossRef] [PubMed]

3. R. Hillebrand, St. Senz, W. Hergert, and U. Gösele, “Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap,” J. Appl. Phys. **94**, 2758 (2003). [CrossRef]

4. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K.M. Ho, “Photonic crystal-based resonant antenna with a very high directivity”, J. Appl. Phys. **87**, 603 (2000). [CrossRef]

5. I. V. Konoplev, A. D. R. Phelps, A. W. Cross, and K. Ronald, “Experimental studies of the influence of distributed power losses on the transparency of two-dimensional surface photonic band-gap structures,” Phys. Rev. E **68**, 066613 (2003). [CrossRef]

6. Y. Wang, B. Cheng, and D. Zhang, “Distribution of density of photonic states in amorphous photonic materials,” Appl. Phys. Lett. **83**, 2100 (2003) ; Y.W. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, “Localized modes in defect-free dodecagonal quasiperiodic photonic crystals,” Phys. Rev. B **68**, 165106 (2003). [CrossRef]

7. D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,“Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E **67**, 045601 (2003) [CrossRef]

8. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E **63**, 046612 (2001). [CrossRef]

4. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K.M. Ho, “Photonic crystal-based resonant antenna with a very high directivity”, J. Appl. Phys. **87**, 603 (2000). [CrossRef]

2. W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. **18**, 528 (1993). [CrossRef] [PubMed]

6. Y. Wang, B. Cheng, and D. Zhang, “Distribution of density of photonic states in amorphous photonic materials,” Appl. Phys. Lett. **83**, 2100 (2003) ; Y.W. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, “Localized modes in defect-free dodecagonal quasiperiodic photonic crystals,” Phys. Rev. B **68**, 165106 (2003). [CrossRef]

7. D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,“Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E **67**, 045601 (2003) [CrossRef]

8. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E **63**, 046612 (2001). [CrossRef]

9. F. Villa, L. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. Lopez-Rios, “Photonic crystal sensor based on surface waves for thin-film characterization,” Opt. Lett. **27**646 (2002) [CrossRef]

10. J. Gaspar-Armenta and F. Villa, “Photonic surface-wave excitation: photonic crystal-metal interface,” J. Opt. Soc. Am. B **20**2349 (2003) [CrossRef]

## 2. Theory

### 2.1. Dyson’s equation and iterative solution

12. O. J. Martin, A. Dereux, and C. Girard, “Generalized field propagator for arbitrary finites-size photonic band gap structures,” J. Opt. Soc. Am. A **11**1073 (1994) [CrossRef]

13. O. J. Martin, C. Girard, D. R. Smith, and S. Schultz, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” Phys. Rev. Lett. **82**315 (1999) [CrossRef]

14. O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E **58**, 3903 (1998) [CrossRef]

*iωt*). Here we consider an especially simple case where the electric vector is polarized parallel to the axis of the dielectric cylinders. In this case, the Green dyadic

**G**=

*ẑẑG*

_{zz}effectively becomes a scalar that satisfies Dyson’s equation

**E**=

*ẑE*

_{z}satisfies either of the following forms of the Lippmann-Schwinger equation,

*k*

_{0}=

*ω/c*and

*ρ*= (

*x,y*). The integration in Eqs. (1)–(3) is only over the cross-sectional area of the cylinders. The

*G*

_{zz}is the Green function with the effects of the PC included. The

*ρ*) is the incident electric field. Note that when

*ρ*=

*ρ́*in Eqs. (1)–(3) the singularity associated with the Green function would normally require special treatment, but in the simple case treated here, the singularity is more easily handled.[15

15. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE **68**, 248 (1980). [CrossRef]

*z*subscripts henceforth.

*ε̂*(

*ρ*) that is non-zero only within the dielectric rods. Initially, we assume the rods are homogeneous with permittivity

*ε*

_{c}embedded in a background medium of unit permittivity. This yields

*ρ*is within any rod and zero otherwise. Numerical solution to Eqs. (1) and (3) can be performed by direct inversion, but this would typically be very large in terms of memory requirements, especially in Eq. (1). Here we provide only a brief description of the iterative method. The computational space is divided into an array of grids where the

*i*th grid has area Δ

*A*. The center of the

*i*th and

*j*th grid is located at

*ρ*

_{i}and

*ρ*

_{j}. The space is initially empty and the PC is then constructed by adding one “infinitesimal”

*ε̂*element at a time. When the

*n*th element

*ε̂*

*k*

_{n}is added we have

*A*=

*Â*. To initiate the iteration algorithm, set

*n*= 1 and there is a single perturbation

*ε*

_{k1}at grid

*k*

_{1}. The

*M*

^{0}

_{k1}term represents the integration term when

*i*=

*j*=

*k*

_{1}as

*R*. The electric field solution can be built up in a similar iterative fashion or calculated as

*i*th coordinate and

*N*is the number of perturbation elements

*ε̂*

*k*

_{n}in the structure. We have given a very brief outline of the method since details [12

12. O. J. Martin, A. Dereux, and C. Girard, “Generalized field propagator for arbitrary finites-size photonic band gap structures,” J. Opt. Soc. Am. A **11**1073 (1994) [CrossRef]

13. O. J. Martin, C. Girard, D. R. Smith, and S. Schultz, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” Phys. Rev. Lett. **82**315 (1999) [CrossRef]

14. O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E **58**, 3903 (1998) [CrossRef]

### 2.2. Local density of states

*G*

^{0}is the free space Green function as given in Eq. (1), and following Eq. (3), and ℑ indicates the imaginary part. In the case of linear media, the LDOS as calculated from Eq. (8) is only dependent on the material and structural parameters of the PC. If nonlinear material is introduced, then obviously an excitation field is required to activate the nonlinearity. In this case, we still use Eq. (8) to calculate the LDOS with the PC illuminated by an incident plane wave of unit amplitude and at a given angle of incidence. This is done self-consistently yielding changes in the permittivity as described in the next section. The resulting changes in permittivity yields different values for

*G*(

*ρ*,

*ρ*) and consequently the LDOS(

*ρ*). The modeling of the nonlinearity is discussed in the following section.

7. D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,“Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E **67**, 045601 (2003) [CrossRef]

8. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E **63**, 046612 (2001). [CrossRef]

*G*

^{0}(

*ρ*,

*ρ*) = 0.25. It follows that the normalized LDOS given in numerical results below can easily be compared to free space relative to 0.25.

### 2.3. Non-linear material

16. P. Tran, “Photonic-band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B **52**, 10673 (1995) ; [CrossRef]

18. V. Lousse and P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E **63**027602 (2001) [CrossRef]

19. M. Soljačič, D. Luo, J.D. Joannopoulos, and S. Fan “Non-linear photonic crystal microdevices for optical integration,” Opt. Lett. **28**637 (2003) [CrossRef]

20. M. Bahl, N. Panoiu, and R. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E **67**056604 (2003) [CrossRef]

21. S. Mingaleev and Y. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B **19**2241 (2002) [CrossRef]

12. O. J. Martin, A. Dereux, and C. Girard, “Generalized field propagator for arbitrary finites-size photonic band gap structures,” J. Opt. Soc. Am. A **11**1073 (1994) [CrossRef]

13. O. J. Martin, C. Girard, D. R. Smith, and S. Schultz, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” Phys. Rev. Lett. **82**315 (1999) [CrossRef]

14. O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E **58**, 3903 (1998) [CrossRef]

18. V. Lousse and P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E **63**027602 (2001) [CrossRef]

*ρ*), is obtained when

*ε̂*

^{0}(

*ρ*) =

*ε*

_{c}- 1. Using

*ρ*), the permittivity is then changed to

*ε̂*

^{1}(

*ρ*) =

*ε*

_{c}- 1 +

*χ*|

*ρ*)|

^{2}. Using

*ε̂*

^{1}(

*ρ*), a new Green function

*G*

_{ij}and field distribution

*ρ*) are calculated. This process is repeated until succeeding electric fields throughout the PC converge to a stable result. This yields not only the electric fields, but also any changes in the LDOS.

*χ*≤ 0.01 approximately 10 iterations or less were needed to converge. If the discretized computational domain contains

*M*points (

*x,y*), we calculate the electric field

*E*

^{(n)}(

*x,y*) for all

*M*points. We assume convergence is satisfied when

*r*

^{(n)}(

*x,y*) = [

*E*

^{(n)}(

*x,y*) -

*E*

^{(n-1)}(

*x,y*)]/

*E*

^{(n-1)}(

*x,y*) and

*n*is the number of iterations. We did not do an extensive parametric evaluation of the self-consistent convergence conditions. We did note that when

*χ*= 0.1, the self-consistent algorithm failed to converge. In any case, we did not try to use experimentally obtained

*χ*values, rather we note that

*χ*≤ 0.01 is well within any realistic material values.

## 3. Numerical results

*a*and radius

*r*with physical parameters [11] chosen so that there is a complete band gap for the infinite crystal and this is shown in Fig. 2(a) and the defect mode surface wave dispersion is shown in Fig. 2(b). The computational space is divided into square grids of side dimension Δ and the PC period is

*a*=

*N*Δ. We choose a frequency

*ω*

_{0}within the band gap where (

*ω*

_{0}/

*c*)(

*a*/2

*π*) = 0.350. We set

*N*= 20 and this yields a free space resolution of Δ = 0.0175

*λ*

_{0}with

*ω*

_{0}/

*c*= 2

*π*/

*λ*

_{0}. The resolution in a cylinder is then degraded to

*N*points, Eq. (2) can be solved for

*E*

_{z}(

*ρ*) by direct inversion. Also,

*E*

_{z}(

*ρ*) can be obtained from Eq. (3) after first obtaining

*ρ*,

*ρ́*). We find that the electric fields computed by both methods are in excellent agreement and we conclude that the method of calculating

*ρ*,

*ρ́*) is also accurate.

*θ*= 45

*°*, the computed effect of including NLM is shown in Fig. 3(b). The NLM coefficient is chosen as

*χ*= 0.01. The results of the LDOS calculations indicate a large concentration of states available at the truncated-rod interface. When the NLM coefficient

*χ*is non-zero, there is noticeable decrease in the LDOS. This decrease is due to the change in the permittivity of the semi-circular rods and the surface wave mode is quite sensitive to any physical or material changes in the semi-circular interface.

*E*(

*x,y*)|

^{3}since this is the magnitude relevant to Kerr nonlinearity. For both angles of incidence, there is strong excitation of the surface mode. It is seen that for

*χ*= 0.0 and 0.01 for both angles of incidence

*θ*= 45

*°*and 135

*°*, the surface mode peaks are quite similar in both cases. This is likely due to the fact that very little of the PC actually contains nonlinear material. Nevertheless, there are some differences in the |

*E*(

*x,y*)|

^{3}results when

*χ*= 0.0 and 0.01. The addition of

*χ*= 0.01 causes a slight decrease in |

*E*(

*x,y*)|

^{3}and this is consistent with the LDOS data in Fig. 4(b).

## 4. Conclusions

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. |

3. | R. Hillebrand, St. Senz, W. Hergert, and U. Gösele, “Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap,” J. Appl. Phys. |

4. | B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K.M. Ho, “Photonic crystal-based resonant antenna with a very high directivity”, J. Appl. Phys. |

5. | I. V. Konoplev, A. D. R. Phelps, A. W. Cross, and K. Ronald, “Experimental studies of the influence of distributed power losses on the transparency of two-dimensional surface photonic band-gap structures,” Phys. Rev. E |

6. | Y. Wang, B. Cheng, and D. Zhang, “Distribution of density of photonic states in amorphous photonic materials,” Appl. Phys. Lett. |

7. | D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,“Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E |

8. | A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E |

9. | F. Villa, L. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. Lopez-Rios, “Photonic crystal sensor based on surface waves for thin-film characterization,” Opt. Lett. |

10. | J. Gaspar-Armenta and F. Villa, “Photonic surface-wave excitation: photonic crystal-metal interface,” J. Opt. Soc. Am. B |

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

12. | O. J. Martin, A. Dereux, and C. Girard, “Generalized field propagator for arbitrary finites-size photonic band gap structures,” J. Opt. Soc. Am. A |

13. | O. J. Martin, C. Girard, D. R. Smith, and S. Schultz, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” Phys. Rev. Lett. |

14. | O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E |

15. | A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE |

16. | P. Tran, “Photonic-band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B |

17. | P. Tran, “Photonic band structure calculation of system possessing Kerr nonlinearity,” in “Photonic Band Gap Materials” NATO ASI Series E: Applied Sciences |

18. | V. Lousse and P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E |

19. | M. Soljačič, D. Luo, J.D. Joannopoulos, and S. Fan “Non-linear photonic crystal microdevices for optical integration,” Opt. Lett. |

20. | M. Bahl, N. Panoiu, and R. Osgood, “Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E |

21. | S. Mingaleev and Y. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(240.6690) Optics at surfaces : Surface waves

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 26, 2004

Revised Manuscript: September 10, 2004

Published: October 4, 2004

**Citation**

J. Elson and Klaus Halterman, "Local density of states analysis of surface wave modes on truncated photonic crystal surfaces with nonlinear material," Opt. Express **12**, 4855-4863 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4855

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### References

- E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, �??Observation of surface photons on periodic dielectric arrays,�?? Opt. Lett. 18, 528 (1993). [CrossRef] [PubMed]
- R. Hillebrand, St. Senz, W. Hergert, and U. Gösele, �??Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap,�?? J. Appl. Phys. 94, 2758 (2003). [CrossRef]
- B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K.M. Ho, �??Photonic crystalbased resonant antenna with a very high directivity�??, J. Appl. Phys. 87, 603 (2000). [CrossRef]
- I. V. Konoplev, A. D. R. Phelps, A.W. Cross, and K. Ronald, �??Experimental studies of the influence of distributed power losses on the transparency of two-dimensional surface photonic band-gap structures,�?? Phys. Rev. E 68, 066613 (2003). [CrossRef]
- Y. Wang, B. Cheng, and D. Zhang, �??Distribution of density of photonic states in amorphous photonic materials,�?? Appl. Phys. Lett. 83, 2100 (2003) ; Y.W. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, �??Localized modes in defect-free dodecagonal quasiperiodic photonic crystals,�?? Phys. Rev. B 68, 165106 (2003). [CrossRef]
- D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,�??Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,�?? Phys. Rev. E 67, 045601 (2003) [CrossRef]
- A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, �??Twodimensional Green�??s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,�?? Phys. Rev. E 63, 046612 (2001) [CrossRef]
- F. Villa, L. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. Lopez-Rios, �??Photonic crystal sensor based on surface waves for thin-film characterization,�?? Opt. Lett. 27 646 (2002) [CrossRef]
- J. Gaspar-Armenta and F. Villa, �??Photonic surface-wave excitation: photonic crystal-metal interface,�?? J. Opt. Soc. Am. B 20 2349 (2003) [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton NJ, 1995), See page 75 for a diagram of the surface mode dispersion.
- O. J. Martin, A. Dereux, C. Girard, �??Generalized field propagator for arbitrary finites-size photonic band gap structures,�?? J. Opt. Soc. Am. A 11 1073 (1994 [CrossRef]
- O. J. Martin, C. Girard, D. R. Smith, S. Schultz, �??Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,�?? Phys. Rev. Lett. 82 315 (1999) [CrossRef]
- O. J. Martin and N. B. Piller, �??Electromagnetic scattering in polarizable backgrounds,�?? Phys. Rev. E 58, 3903 (1998) [CrossRef]
- A. D. Yaghjian, �??Electric dyadic Green�??s functions in the source region,�?? Proc. IEEE 68, 248 (1980). [CrossRef]
- P. Tran, �??Photonic-band-structure calculation of material possessing Kerr nonlinearity,�?? Phys. Rev. B 52, 10673 (1995) ; [CrossRef]
- P. Tran, �??Photonic band structure calculation of system possessing Kerr nonlinearity,�??�?? in �??Photonic Band Gap Materials�?? NATO ASI Series E: Applied Sciences 315, 555, Ed. C. M. Soukoulis (Kluwer Academic Publishers, Boston, 1996).
- V. Lousse and P. Vigneron, �??Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,�?? Phys. Rev. E 63 027602 (2001) [CrossRef]
- M. Solja¡ci¡c, D. Luo, J.D. Joannopoulos, and S. Fan �??Non-linear photonic crystal microdevices for optical integration,�?? Opt. Lett. 28 637 (2003) [CrossRef]
- M. Bahl, N. Panoiu, and R. Osgood, �??Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,�?? Phys. Rev. E 67 056604 (2003) [CrossRef]
- S. Mingaleev, Y. Kivshar, �??Nonlinear transmission and light localization in photonic-crystal waveguides,�?? J. Opt. Soc. Am. B 19 2241 (2002) [CrossRef]

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