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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 13 — Jun. 30, 2003
  • pp: 1590–1595
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All-frequency effective medium theory of a photonic crystal

Geesche Boedecker and Carsten Henkel  »View Author Affiliations


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

Periodic crystals do not exist in the real world, they are all finite. It turns out that in a finite crystal one has to consider the reflection of the Bloch waves at the endfaces. This leads to the formation of standing waves, as mentioned, e.g., by [3

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

]. Reflection amplitudes are easy to find in the long wavelength limit using effective medium theory.We construct here an effective medium theory that describes at all frequencies the light propagation through a finite crystal. Its key ingredients are the reflection and transmission coefficients for a half-space crystal, calculated using an expansion in Bloch waves. With our approach, standard formulas for a homogeneous layer can be used for a finite crystal.We consider a few important applications: the local density of states for finite or infinite systems, even with absorption, and frequencies of defect modes.

Fig. 1. (Left panel) complex eigenvalue λ+=eika vs. frequency for the Kronig-Penney model, Eq. (9), with point scatterers of polarizability α=(0.2+0i)a. (Right panel) band structure ω(k) of the infinite crystal. The thick dispersion curves correspond to the physical Bloch momentum fixed by the requirements of causality and energy conservation. Even bands exhibit negative refraction (k<0).

We discuss negative refraction. Our results are illustrated by exact formulas for a 1D Kronig-Penney (planar scatterer) model.

2. Transfer matrix approach

En(x)=aneiω(xna)+bneiω(xna),
(1)

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

1. E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).

])

(an+1bn+1)=T(anbn).
(2)

Since detT=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.

The reflection and transmission amplitudes from a finite crystal are given by the Nth power of the primitive transfer matrix, e.g. rN =-T21N/T22N. TN is particularly simple to compute in the Bloch basis:

TN=M(eikaN00eikaN)M1,M=(N+NN+c+Nc).
(3)

The ratios of the plane wave amplitudes for the Bloch states ±k are

c±=e±ikaT11T12=T21e±ikaT22.
(4)

Independent of the normalization factors 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

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]

])

rN=c+c(eikaNeikaN)c+eikaNceikaN.
(5)
Fig. 2. (Left) reflectance |rN | for the Kronig-Penney model with α=(0.2+0i)a, N=4 (blue line). The envelope, Eq. (6), is shown as well. Solid green line: half-space approximation |r|, Eq. (11). Dashed dark green line: |r| for absorbing scatterers (α=(0.2+0.02i)a). (Right) reflection coefficient |r i| for Bloch waves reflected from the end face of a semi-infinite crystal. Solid line: our result, Eq. (13); dashed line: proposed by Sakoda [4]. Kronig-Penney model with α=(0.2+0i)a.

For a finite number of periods N, the reflectance |rN | 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 |rN | with respect to N for each fringe period, we find from Eq. (5) in allowed bands (see Fig. 2)

0rN2c+cc++c,
(6)

provided energy conservation holds. This envelope function is consistent with [3

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

] and overestimates the reflectance in the band gap.

In the preceding plots, a Kronig-Penney type model has been used for definiteness. We consider a wave equation with point scatterers (see Ref. [9

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

] for a 3D generalization)

d2E(x)dx2+ω2(1+n=1Nαnδ(x(n12)a))E(x)=0
(7)

where the polarizability αn characterizes the strength of the nth scatterer. We focus on αn =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

T=((1+i2ωα)eiωai2ωαi2ωα(1i2ωα)eiωa)
(8)

whose eigenvalues are given by

λ±=cos(ωa)ωα2sin(ωa)±isin(ωa)1+ωαcot(ωa)14ω2α2
(9)

We have chosen the sign of the square root such that λ+ is located on or inside the unit circle (provided α has an infinitesimally positive imaginary part).

3. Half-space-approximation

We show here that the Bloch modes for the infinite crystal can be used to define a reflection coefficient 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

(t0)=M1(1r).
(10)

Independent of the eigenmode normalizations, this yields

r=c+=eiωaeikaeiωaeika1,
(11)

where the second equation is valid for the Kronig-Penney model. At low frequency, we recover the standard effective medium result r≈(1-n eff)/(1+n eff) with the effective index n eff=(1+α/a)1/2=limω→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 |rN | is averaged over the standing wave fringes. In the band gaps, we recover the intuitively expected perfect reflector, |r|=1.

In a similar manner, we define a coefficient r i for the ‘interior’ reflection of a Bloch wave from a crystal end face, say the left one:

(ri1)=M1(0t)
(12)

Reflection from both end faces can be described by the same coefficient. This fixes the normalization of the Bloch eigenmodes:

ri=NN+=N+c+Nc=c+.
(13)

For the last equality, we have used the relation 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.

Our expression for the interior reflection can be compared to a conjecture by Sakoda, r i(Sakoda)=(n g-1)/(n g+1) where the group index n g=dk/. 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]

]. Sakoda’s result qualitatively agrees with ours (see Fig. 2), but underestimates |r i|, except very close to the band edges.

4. Applications and discussion

4.1 Effective medium theory

In the preceding section, we have seen that physically reasonable reflection and transmission amplitudes can be introduced for the endfaces of a photonic crystal, when the field inside the crystal is expanded in Bloch modes. This immediately suggests an effective medium description of a finite length crystal. The reflectance rN , e.g. would be given by the well-known Fabry-Pérot expression [2

2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).

]

rN,FP=r+trite2ikaN1r12e2ikaN,
(14)

where the standing wave oscillations arise by summing a multiple scattering series.

It is easy to check that r N,FP coincides with the transfer matrix result rN , 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, ri, t, t′, the expression Eq. (3) for TN becomes precisely the product of transfer matrices one would write down for a homogeneous layer, and leads to Eq. (14).

Fig. 3. LDOS, Eq. (15), in an infinite crystal with scatterers at x=…,-a/2,a/2,… (Left) no absorption (α=0.2a). (Right) nonzero absorption (α=(0.2+0.05i)a).

It has been previously noted that the Bloch momentum 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

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

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]

]. In particular, Jeong 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]

] interpret the phase of the transmission amplitude in terms of a frequency-dependent effective index, but find differences with respect to the conventional expression 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 tN =tteikaN /(1-ri2 e 2ikaN).

4.2 Local density of states

The LDOS ρ(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]

], and it is well known that it is obtained from the imaginary part of the Green function G(x,x;ω) (the field radiated by a pointlike test source). If we put two (finite or semi-infinite) crystals at distances dL,R from a test source, the Green function is easily obtained from the corresponding reflection coefficients rL,R . Normalizing to the free space LDOS, we get in this way

ρ(x,ω)=Re(rL(ω)e2idLω+rR(ω)e2idRω+21rL(ω)rR(ω)e2i(dL+dR)ω1),dL=dR=x.
(15)

Note that the distances dL,R are defined relative to the reference planes implicit in the reflection coefficients rL,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]

]. For an infinite crystal with point scatterers, the LDOS is plotted in Fig. 3 and shows the characteristic inverse square root singularities close to the band edges [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]

]. One also recovers the well-known dielectric and air bands at the gap edges.

Our approach immediately allows for a nonzero absorption in the sample where, at least for an infinite crystal, standard band theory breaks down because all Bloch vectors become complex (extended states do not exist any more; see [10

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

] for a detailed discussion). For finite absorbing systems, it is well-known that transfer matrix techniques still provide the required reflection and transmission spectra (see, e.g., [11

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

]). To define consistently a half-space reflection amplitude Eq. (11), we require that 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 eika 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.

Fig. 4. (Left) reflectance from two N=6 layer crystals (α=0.2a) with a defect in between (dR =dL =a/2). Green line with peak: active defect with scattering strength α d/a≈-0.62-0.04i, given by Eq. (16) for ωa/2πc≈0.462. The same structure with a passive defect (α d≈-0.62a) gives the green line with the transmission dip. Blue line: reflectance for α d=α. (Right) defect frequency in the first band gap vs. real part of scattering strength α d.

4.3 Defect modes

As a final application, we investigate the situation that two crystals surround a defect scatterer with α dα. Expanding the field in plane waves around the defect, we find that the defect mode frequency is determined by

iωαd=rR(ω)e2iωdR1rR(ω)e2iωdR+1+rL(ω)e2iωdL1rL(ω)e2iωdL+1,
(16)

where dR,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

To conclude, we point out that the simple one-dimensional Kronig-Penney model provides an exactly soluble example of a photonic crystal with negative refraction. For a non-absorbing, semi-infinite crystal, the requirement that |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=/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]

]), as is manifest from the thick lines in Fig. 1.

Acknowledgements

We thank V. Raab, J. J. Saenz and M. Wilkens for encouraging comments and discussion. This work has been supported by the Deutsche Forschungsgemeinschaft in the framework of the Schwerpunktprogramm 1113 “Photonic Crystals”.

References and links

1.

E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).

2.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).

3.

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

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]

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]

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]

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]

8.

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

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]

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]

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

  1. E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley & Sons, New York, 1998).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
  3. D. W. L. Sprung, H. Wu, and J. Martorell, �??Scattering by a finite periodic potential,�?? Am. J. Phys. 61, 1118 (1993). [CrossRef]
  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), <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]
  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]
  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]
  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]
  8. A. Moroz, �??Minima and maxima of the local density of states for one-dimensional periodic systems,�?? Europhys. Lett. 46, 419 (1999). [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]
  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]
  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]

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