## All-frequency effective medium theory of a photonic crystal

Optics Express, Vol. 11, Issue 13, pp. 1590-1595 (2003)

http://dx.doi.org/10.1364/OE.11.001590

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### Abstract

We consider light propagation in a finite photonic crystal. The transmission and reflection from a one-dimensional system are described in an effective medium theory, which reproduces exactly the results of transfer matrix calculations.We derive simple formulas for the reflection from a semi-infinite crystal, the local density of states in absorbing crystals, and discuss defect modes and negative refraction.

© 2003 Optical Society of America

## 1. Introduction

3. D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. **61**, 1118 (1993). [CrossRef]

## 2. Transfer matrix approach

*N*of unit cells.We use units with

*c*=1 and denote

*ω*the wavevector in the background medium with permittivity

*ε*=1. The transfer matrix

*T*connects the field

*E*

_{n}at the left edge of the

*n*th unit cell to the field

*E*

_{n+1}at the next cell. Expanding the fields in plane waves propagating to the left and right

*a*is the crystal period, we have (see chap. 6 of [1])

*T*=1, its eigenvalues λ

_{±}can be written in the form λ

_{±}=

*e*

^{±ika}where

*k*is the Bloch quasimomentum. An eigenvalue on the unit circle (real

*k*) corresponds to a propagating (extended) Bloch mode whereas real eigenvalues (

*k*imaginary) are found in the band gaps. An example is shown in Fig. 1 for the Kronig-Penney model detailed below.

*N*th power of the primitive transfer matrix, e.g.

*r*

_{N}=-

*T*

^{N}is particularly simple to compute in the Bloch basis:

*k*are

*N*

_{±}chosen for the Bloch eigenstates, the reflection coefficient is given by (in agreement with Refs. [3

3. D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. **61**, 1118 (1993). [CrossRef]

7. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107 (1996). [CrossRef]

*N*, the reflectance |

*r*

_{N}| shows oscillations as a function of frequency due to the formation of standing waves between the end faces of the crystal, with a fringe spacing scalingwith 1/

*N*. If these are not resolved due to some finite frequency resolution (or fluctuations in the crystal thickness), the envelope of the reflectance is a useful generalization. Maximizing |

*r*

_{N}| with respect to

*N*for each fringe period, we find from Eq. (5) in allowed bands (see Fig. 2)

3. D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. **61**, 1118 (1993). [CrossRef]

9. M Wubs and A. Lagendijk, “Local optical density of states in finite crystals of plane scatterers,” Phys. Rev. E **65**, 046612 (2002). [CrossRef]

*α*

_{n}characterizes the strength of the nth scatterer. We focus on

*α*

_{n}=

*α*to get a periodic crystal. The continuity of the electric field

*E*and its first derivative on the scatterers leads to the transfer matrix

_{+}is located on or inside the unit circle (provided

*α*has an infinitesimally positive imaginary part).

## 3. Half-space-approximation

*r*for a semi-infinite crystal. The key idea is to use the transformation matrices

*M*and

*M*

^{-1}occurring in Eq. (3) as transfer matrices, linking plane waves in the vacuum outside the crystal to Bloch modes inside the crystal. For a unit incident amplitude from the left, transmission and reflection amplitudes are thus given by

*r*≈(1-

*n*

_{eff})/(1+

*n*

_{eff}) with the effective index

*n*

_{eff}=(1+

*α*/

*a*)

^{1/2}=lim

_{ω→0}

*k*(

*ω*)/

*ω*. Figure 2 shows that Eq. (11) gives, for all frequencies, a good approximation to the reflectance from a finite crystal, when |

*r*

_{N}| is averaged over the standing wave fringes. In the band gaps, we recover the intuitively expected perfect reflector, |

*r*|=1.

*r*

_{i}for the ‘interior’ reflection of a Bloch wave from a crystal end face, say the left one:

*c*

_{+}

*c*

_{-}=1 which follows from the symmetry relation

*T*

_{12}=-

*T*

_{21}of the transfer matrix for an even scatterer. Note the

*π*phase shift between ‘exterior’ and ‘interior’ reflection amplitudes.

*r*

_{i}(Sakoda)=(

*n*

_{g}-1)/(

*n*

_{g}+1) where the group index

*n*

_{g}=

*dk*/

*dω*. When the standing waves inside a finite crystal are used for lasing, |

*r*

_{i}| determines the quality factor of this cavity [4

4. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express **4**, 481 (1999),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-12-481. [CrossRef] [PubMed]

*r*

_{i}|, except very close to the band edges.

## 4. Applications and discussion

### 4.1 Effective medium theory

*r*

_{N}, e.g. would be given by the well-known Fabry-Pérot expression [2]

*r*

_{N,FP}coincides with the transfer matrix result

*r*

_{N}, Eq. (5), by substituting the reflection amplitudes, Eqs. (11, 13), and the corresponding transmissions. In fact, this is not surprising because according to the definitions Eqs. (10, 12) of these amplitudes, the transformation matrices

*M*and

*M*

^{-1}can play the role of transfer matrices at the crystal endfaces. If we express them in terms of the

*r, r*

_{i}

*, t, t*′, the expression Eq. (3) for

*T*

^{N}becomes precisely the product of transfer matrices one would write down for a homogeneous layer, and leads to Eq. (14).

*k*is a useful quantity to describe the propagation inside a finite-size crystal [7

7. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107 (1996). [CrossRef]

9. M Wubs and A. Lagendijk, “Local optical density of states in finite crystals of plane scatterers,” Phys. Rev. E **65**, 046612 (2002). [CrossRef]

5. D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. **92**, 4194 (2002). [CrossRef]

*et al.*[5

5. D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. **92**, 4194 (2002). [CrossRef]

*n*

_{eff}=

*k*/

*ω*close to the band edges. Our analysis indicates that this is due to the contribution of the Fabry-Pérot denominator in the expression

*t*

_{N}=

*tt*′

*e*

^{ikaN}/(1-

*e*

^{2ikaN}).

### 4.2 Local density of states

*x,ω*) is the key quantity for the radiative decay of a two-level system in the crystal [6

6. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. **63**, 455 (2000). [CrossRef]

*G*(

*x,x;ω*) (the field radiated by a pointlike test source). If we put two (finite or semi-infinite) crystals at distances

*d*

_{L,R}from a test source, the Green function is easily obtained from the corresponding reflection coefficients

*r*

_{L,R}. Normalizing to the free space LDOS, we get in this way

*d*

_{L,R}are defined relative to the reference planes implicit in the reflection coefficients

*r*

_{L,R}(located

*a*/2 in front of the first scatterer in our example). This simple formula reproduces more involved expressions given, e.g., in [8

8. A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. **46**, 419 (1999). [CrossRef]

7. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107 (1996). [CrossRef]

10. A. Tip, A. Moroz, and J. M. Combes, “Band structure of absorptive photonic crystals,” J. Phys. A: Math. Gen. **33**, 6223 (2000). [CrossRef]

11. J. B. Pendry, “Photonic band structures,” J. mod. Optics **41**, 209 (1994). [CrossRef]

*r*(

*ω*) be nonsingular as a function of complex frequency in the upper half plane (as dictated by causality). For the Kronig-Penney model, we can show that it suffices to choose the eigenvalue

*e*

^{ika}located inside the unit circle. (This condition is consistent with the limit

*N*→∞ of Eq. (5) for finite absorption.) The reflection coefficient for finite absorption is shown in Fig. 2: it drops below unity in the band gaps. The LDOS (Fig. 3) exhibits a smoothing out of the dielectric band edge while the singularity at the upper gap edge persists. This is due to the approximation of point-like scatterers in our Kronig-Penney model, which makes the mode functions at the air band edge insensitive to the scattering strength

*α*.

### 4.3 Defect modes

*α*

_{d}≠

*α*. Expanding the field in plane waves around the defect, we find that the defect mode frequency is determined by

*d*

_{R,L}are again the distances between the crystals and the defect. For given real

*ω*,

*α*

_{d}is in general complex and can be absorbing or even active. In the latter case, the crystal backscatters more light than is incident when the defect resonance is hit (see Fig. 4). By varying

*α*

_{d}, the defect mode frequency can be tuned across the band gap.

### 4.4 Negative refraction

*r*|

^{2}≤1 leads to the condition sin(

*ωa*) sin(

*ka*)≥0 for the ‘physical’ Bloch momentum

*k*. This is fulfilled when an incident plane wave in an even frequency band injects a Bloch wave with negative

*k*into the crystal. Note that this choice of

*k*is consistent with the well-known rule that the Bloch wave should have a positive group velocity

*v*

_{g}=

*dω*/

*dk*(see, e.g. [12

12. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction at media with negative refractive index,” Phys. Rev. Lett. **90**, 107402 (2003). [CrossRef] [PubMed]

## Acknowledgements

*Deutsche Forschungsgemeinschaft*in the framework of the

*Schwerpunktprogramm*1113 “Photonic Crystals”.

## References and links

1. | E. Merzbacher, |

2. | M. Born and E. Wolf, |

3. | D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. |

4. | K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express |

5. | D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, “Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,” J. Appl. Phys. |

6. | P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. |

7. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

8. | A. Moroz, “Minima and maxima of the local density of states for one-dimensional periodic systems,” Europhys. Lett. |

9. | M Wubs and A. Lagendijk, “Local optical density of states in finite crystals of plane scatterers,” Phys. Rev. E |

10. | A. Tip, A. Moroz, and J. M. Combes, “Band structure of absorptive photonic crystals,” J. Phys. A: Math. Gen. |

11. | J. B. Pendry, “Photonic band structures,” J. mod. Optics |

12. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction at media with negative refractive index,” Phys. Rev. Lett. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(260.2110) Physical optics : Electromagnetic optics

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 28, 2003

Revised Manuscript: June 23, 2003

Published: June 30, 2003

**Citation**

Geesche Boedecker and Carsten Henkel, "All-frequency effective medium theory of a photonic crystal," Opt. Express **11**, 1590-1595 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1590

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### References

- E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
- D. W. L. Sprung, H. Wu, and J. Martorell, �??Scattering by a finite periodic potential,�?? Am. J. Phys. 61, 1118 (1993). [CrossRef]
- K. Sakoda, K. Ohtaka, and T. Ueta, �??Low-threshold laser oscillation due to group-velocity anomaly peculiar to two-and three-dimensional photonic crystals,�?? Opt. Express 4, 481 (1999), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-12-481.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-12-481</a>. [CrossRef] [PubMed]
- D. Y. Jeong, Y. H. Ye, and Q. M. Zhang, �??Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction,�?? J. Appl. Phys. 92, 4194 (2002). [CrossRef]
- P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, �??Fundamental quantum optics in structured reservoirs,�?? Rep. Prog. Phys. 63, 455 (2000). [CrossRef]
- J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107 (1996). [CrossRef]
- A. Moroz, �??Minima and maxima of the local density of states for one-dimensional periodic systems,�?? Europhys. Lett. 46, 419 (1999). [CrossRef]
- M. Wubs and A. Lagendijk, �??Local optical density of states in finite crystals of plane scatterers,�?? Phys. Rev. E 65, 046612 (2002). [CrossRef]
- A. Tip, A. Moroz, and J. M. Combes, �??Band structure of absorptive photonic crystals,�?? J. Phys. A: Math. Gen. 33, 6223 (2000). [CrossRef]
- J. B. Pendry, �??Photonic band structures,�?? J. Mod. Optics 41, 209 (1994). [CrossRef]
- S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, �??Refraction at media with negative refractive index,�?? Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]

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