## Gap-edge asymptotics of defect modes in two-dimensional photonic crystals

Optics Express, Vol. 15, Issue 8, pp. 4753-4762 (2007)

http://dx.doi.org/10.1364/OE.15.004753

Acrobat PDF (616 KB)

### Abstract

We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps.

© 2007 Optical Society of America

## 1. Introduction

2. M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. **92**, 063903 (2004). [CrossRef] [PubMed]

3. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves in Random and Complex Media **16**, 293–382 (2006). [CrossRef]

*E*

_{∥}or TM polarization), one might be confident that the behaviour for electrons should carry over to photons. However, for an electric field polarized perpendicular to the axes (i.e,

*H*

_{∥}or TE polarization), the boundary conditions for electrons and photons differ, and so it is an open issue whether the asymptotics for TE polarization are similar to, or quite different from, that for electrons.

5. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E **69**, 016609 (2004). [CrossRef]

6. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E **71**, 056606 (2005). [CrossRef]

## 2. Analytic method

*n*(

**r**) and

*ε*(

**r**) =

*n*

^{2}(

**r**), respectively giving the refractive index and dielectric constant (permittivity) in the array. In our treatment, we assume that

*ε*(

**) is a real function of position so that the system is lossless. The Green function for the PC is expanded as a superposition of quasiperiodic Green functions. In turn, each of these is expanded in the basis of Bloch functions {**

*r**ψ*(

_{m}

*k*_{0},

*)} which obey the Helmholtz equation, the Bloch condition associated with the Bloch vector*

**r**

**k**_{0}, and polarization dependent boundary conditions on the interfaces between cylindrical inclusions and the matrix material in which they are placed.

*ψ*

_{∥}) in the PC: for TM (

*E*

_{∥}) polarization, we define

*ψ*=

*E*,

_{z}*p*(

*) = 1,*

**r***s*(

*) =*

**r***ε*(

*), and, for TE (*

**r***H*

_{∥}) polarization, we introduce

*ψ*=

*H*,

_{z}*p*(

*) = 1/*

**r***ε*(

*),*

**r***s*(

*) = 1. In terms of these, the Helmholtz equation, for either polarization, can be written as*

**r***ψ*} are orthogonal with respect to this inner product and satisfy

_{m}*M*= 〈

_{n}*ψ*,

_{n}*ψ*〉.

_{n}*G*that we require represents the electromagnetic field corresponding to a single source at a point

**′, and satisfies the differential equation**

*r*5. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E **69**, 016609 (2004). [CrossRef]

*ω*lies in a complete band gap, and close to a gap edge which has the frequency

*ω*, using

_{L}*L*to denote the closest band to the frequency ω. We begin with the simplest (nondegenerate) case for which the band edge occurs when the Bloch vector

**k**

_{0}=

**k**

_{L}. To lowest order, we will assume that the

*L*

^{th}band surface is parabolic near its edge, i.e.,

*C*characterizes the band curvature and is equivalent to the effective mass of the electron in semiconductor theory. Using this parabolic form, we can approximate the required integral in Eq. (6) to leading order. When the band edge corresponds to a single, nondegenerate Bloch vector

_{L}

**k**_{0}that is contained completely within the Brillouin zone (BZ), we have

5. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E **69**, 016609 (2004). [CrossRef]

8. D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Two-dimensional treatment of the level shift and decay rate in photonic crystals,” Phys. Rev. E **72**, 046605 (2005). [CrossRef]

*L*to the Green function (6) dominates all other contributions, provided that

*ω*is very close to

*ω*. We thus deduce

_{L}

*k*_{0}, each of which can contribute to the Green function. For example, for a hexagonal lattice, the lower edge of the first band gap (see Fig. 1) is characterized by maxima in the frequency surface at the six equivalent

*K*points of the BZ, with each necessitating an integration over an internal angle of

*ϕ*= 2

_{L,j}*π*/3. Correspondingly, the upper edge of this band gap is characterized by minima at the six equivalent

*M*points, each with internal angle

*ϕ*=

_{L,j}*π*. Accordingly, the expression on the right hand side of Eq. (9), corresponding to the single Bloch vector

*, must thus be replaced by a sum over those*

**k**_{L}*, values that correspond to the band edge, with each contribution weighted by*

**k**_{L,j}*θ*=

_{L,j}*ϕ*/(2

_{L,j}*π*).

*C*

_{0}to create the defect mode. The defect mode, which is localized and tends to zero exponentially with increasing distance from

*C*

_{0}, satisfies the Helmholtz equation ▽ ∙ (

*p*̃(

*)▽*

**r***ψ*(

*)) + (*

**r***ω*

^{2}/

*c*

^{2})

*s*̃(

*)*

**r***ψ*(

*) = 0, in which the perturbation is characterized by the quantities*

**r***p*̃(

*) and*

**r***s*̃(

*).*

**r***H*

_{∥}) polarization, for which

*p*(

**′) = 1/**

*r**ε*(

*′),*

**r***p*̃(

*′) = 1/*

**r***ε*(

*′), and*

**r***s*(

*′) =*

**r***s*̃(

*′) = 1, and substituting into Eq. (10) the leading order estimate (9) for*

**r***G*, we obtain

*j*sums over all values of

**k**_{0}=

*that correspond to the edge of band*

**k**_{L,j}*L*. The evaluation of this expression requires some delicacy, given the dependence on both the geometry of the Wigner-Seitz cell and the spatial characteristics of the mode. However, expression (11) simplifies considerably in cases of high symmetry, since the bulk modes at the band edge can be obtained from one another by an appropriate rotation transformation.

*K*points of the BZ. The

*ψ*are almost circular symmetric near the cylinder and so may be taken to be effectively identical in the vicinity of

_{L,j}*C*

_{0}. Furthermore, for this case,

*θ*= 1/3,

_{L,j}*C*=

_{L,j}*C*, and

_{L}*M*=

_{L,j}*M*, for all

_{L}*j*.

*ψ*(

*) =*

**r***ψ*(

**k**_{L,1},

*), corresponding to a particular mode. The sum in Eq. (11) now contains 6 identical terms each of which are weighted by*

**r***θ*= 1/3. This leads us to define a geometry factor

_{L,j}*f*, for which here

_{L}*f*, = 6 × 1/3 = 2. Of course, for a singularity that is completely enclosed within the BZ,

_{L}*f*= 1.

_{L}*M*by noting that the transverse resolute of the electric field is

_{L}*= -*

**E**_{t}*i*(

*z*̃ × ▽ψ)/(ωε). Thus, for all

*j*, this leads to

*c*

^{2}/

*ω*

^{2}

_{L}, the result in Eq. (12) is the ratio of the change in the electric energy (∫

*∙*

**E**

**D***d*

^{2}

*) resulting from the defect, to the electric energy of the unperturbed mode within the Wigner-Seitz cell. Since the perturbed dielectric constant*

**r***) differs from ε(*

**r***) only in the region*

**r***C*

_{0}, it follows that the integral in the numerator can be calculated over the entire Wigner-Seitz cell. Thus,

*N*= |

_{L}*C*|

_{L}*f*(2

_{L}A_{WSC}*π*) is the Density of States at the band edge

*L*[4, 5

**69**, 016609 (2004). [CrossRef]

*ψ*=

*E*. We now have

_{z}*p*(

*′) =*

**r***p*̃(

*̃) = 1,*

**r***s*(

*′) = ε(*

**r***′) and*

**r***s*̃(

*′) =*

**r***′) and, as for TE polarization, the substitution of these relations into Eq. (10) leads to*

**r****= (0,0,**

*E**ψ*).

*n*, except for a circular inclusion which has the refractive index

_{b}*n*. For a defect caused by changing the refractive index of a single cylinder to

_{c}*n*, we have δ𝓔

_{d}_{C0}/𝓔

_{C0}= δε

_{C0}/𝓔

_{C0}and so we derive the result

## 3. Numerical details

**69**, 016609 (2004). [CrossRef]

9. K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. **219**, 120–143 (2006). [CrossRef]

*δε*, when both are small, is computationally difficult, since the defect mode becomes arbitrarily extended, the more closely we search for modes near to the band edge. While ultimately any numerical technique fails in such a search, the FSS method [6

_{c}6. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E **71**, 056606 (2005). [CrossRef]

6. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E **71**, 056606 (2005). [CrossRef]

## 4. Defect modes

_{asymp}from the asymptotic analysis (19) and then compare this with the estimate 𝓢

_{FSS}obtained by fitting the evolution curve obtained from the FSS calculations to the model ln |

_{L}| = ln 𝓐

_{FSS}- ε

_{C0}𝓢

_{FSS}/δε in the vicinity of the band edges (10

^{-6}<|

_{L}| < 10

^{-3}). From this least squares fit, we determine values for the parameters 𝓐

_{FSS}and 𝓢

_{FSS}.

*n*= 3.0 and

_{c}*a*/

*d*=0.3, and with a background refractive index of

*n*= 1. Fig. 2 (a) shows the band diagram which has band gaps for

_{b}*ω*̃ ≡

*ωd*/(2

*πc*) =

*d*/

*λ*∈[0.265265,0.334947] and

*ω*̃ ∈ [0.468027,0.562353]. In Fig. 2 (b) we show the evolution of a defect mode, in each of the first and second band gaps, generated by varying the refractive index (

*n*) of the single defect cylinder. For

_{d}*n*>

_{d}*n*in the second band gap, there are two additional defect modes indicated in blue. These originate at a higher band edge and are not considered in our analysis.

_{c}_{FSS}and 𝓢 = 𝓢

_{FSS}while the red curves display the results of the FSS calculations. Note the excellent agreement between the numerical and analytic results for frequencies close to the band edge, where the analytic result is valid. Note also the excellent agreement between the analytic (𝓢

_{asymp}) and fitted (𝓢

_{FSS}) values of the sensitivity parameter, shown in Table 1, which confirm the validity of the asymptotic analytic method.

*E*

_{∥}) polarization, and hole type structures in TE (

*H*

_{∥}) polarization. The geometry for TM polarization comprises rods of refractive index

*n*= 3 and normalized radius

_{c}*a*/

*d*= 0.2 in a background of refractive index

*n*= 1, while for the TE polarization, we invert the lattice so that

_{b}*n*= 1,

_{c}*n*= 3, preserving the value of

_{b}*a*/

*d*= 0.2. Fig. 3 shows band diagrams for (a) TM and (b) TE polarizations, which respectively exhibit band gaps for

*n*>

_{d}*n*) on a logarithmic vertical scale: the curves overlap for a normalized frequency difference (off the band edge) of more than 4 orders of magnitude. Figs 2(c) and (d) are similar, but refer to the hole-type lattice with TE polarization. We only show data for

_{c}*n*>

_{d}*n*since for

_{c}*n*<

_{d}*n*we could not accurately find the defect mode numerically (see the discussion below). Note again the excellent agreement between the analytic and numerical results. This agreement is further illustrated in Table 2, in which the data is presented as in Table 1: in all cases the agreement between 𝓢

_{c}_{FSS}and 𝓢

_{asymp}is much better than 1% for all cases in Table 2.

*n*<

_{d}*n*using the FSS method: the computed value of the sensitivity ε

_{c}_{C0}𝓢

_{asymp}≈ 900, implying that the defect modes are extremely close to the band edge, and thus difficult to locate. In turn, the origin of this high sensitivity can be seen from Fig. 5 which shows the normalized electric energy distribution

*f*(

*) = ε∥*

**r**the bulk modes for the hexagonal lattice*E*(

*)∥*

**r**^{2}/

*𝓔*

_{WSC}, so that ∫

_{WSC}

*f*(

*)*

**r***d*

^{2}

*= 1. Figs 5(a) and 5(b) show contour maps of the normalized electric energy distribution (on a base 10 logarithmic scale) for the bulk mode, respectively at the lower and upper edge of the first band gap, while Figs 5(c) and 5(d) respectively show horizontal and vertical sections through the centre of the defect for modes at the lower (red) and upper (blue) edge of the band gap. Clearly evident from these is the very low energy density within the cylindrical inclusion for the mode at the lower edge of the band gap, thereby explaining the very high sensitivity factor (19).*

**r**## 5. Discussion and conclusion

*C*| small; “effective mass”

*m** small), defect levels tend to remain close to the band edge. This is consistent with well known behaviour in solid state physics: for example, a shallow defect in a semiconductor can be described by a hydrogen atom model [10

10. W. Kohn and J. Luttinger, “Theory of donor states in silicon,” Phys. Rev. **98**, 915 – 922 (1955). [CrossRef]

*m**.

*C*

_{0}where the perturbation is applied, is of finite spatial extent.

## Acknowledgments

## References and links

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

2. | M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. |

3. | A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves in Random and Complex Media |

4. | E. N. Economou, |

5. | R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E |

6. | S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E |

7. | L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method,” Int. J. Microwave and Optical Technology |

8. | D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Two-dimensional treatment of the level shift and decay rate in photonic crystals,” Phys. Rev. E |

9. | K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. |

10. | W. Kohn and J. Luttinger, “Theory of donor states in silicon,” Phys. Rev. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(050.1970) Diffraction and gratings : Diffractive optics

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 6, 2007

Revised Manuscript: March 30, 2007

Manuscript Accepted: March 30, 2007

Published: April 4, 2007

**Citation**

K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan, and C. M. de Sterke, "Gap-edge Asymptotics of defect modes in 2D Photonic Crystals," Opt. Express **15**, 4753-4762 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4753

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).
- M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92, 063903 (2004). [CrossRef] [PubMed]
- A. Figotin and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006). [CrossRef]
- E. N. Economou, Green’s functions in quantum physics, 2nd ed. (Springer-Verlag, Berlin, 1983).
- R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, "Density of states functions for photonic crystals," Phys. Rev. E 69, 016609 (2004). [CrossRef]
- S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005). [CrossRef]
- L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave and Optical Technol. 1, 133-145 (2006).
- D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, "Two-dimensional treatment of the level shift and decay rate in photonic crystals," Phys. Rev. E 72, 046605 (2005). [CrossRef]
- K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006). [CrossRef]
- W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915 - 922 (1955). [CrossRef]

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