One-dimensional photonic crystals: equivalent systemsto single layers with a classical oscillator like dielectric function

Francisco Villaa,*, J.A. Gaspar-Armentaa,b, F. Ramos-Mendietab

a Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, Le�oon Gto. 37150, M�eexicob Centro de Investigaci�oon en F�ıısica de la Universidad de Sonora. Apdo. Post. 5-088, Hermosillo Sonora 83190, M�eexico

Received 2 July 2002; received in revised form 27 September 2002; accepted 26 November 2002

Abstract

One-dimensional photonic crystals are analyzed by using symmetric periodic multilayers to determine an equivalent

dielectric function associated to these systems. This equivalent dielectric function has the characteristic shape of those

dielectric materials composed by Lorentz oscillators in some of their bands. It is found that the equivalent longitudinal

and resonance frequencies delimit the band gaps of the crystal where the dielectric function is negative. In this way a

one-dimensional photonic crystal can be considered as a single thin film, and the dispersion curves of surface modes can

be determined by applying the boundary conditions to this equivalent medium. In the case of non-symmetric truncated

one-dimensional photonic crystals, the concept of total admittance is applied to determine the surface modes under the

more general conditions.

� 2003 Elsevier Science B.V. All rights reserved.

PACS: 42.25.Bs; 42.70.Qs; 42.79.Wc; 78.20.Ci

1. Introduction

Surface electromagnetic excitations in photonic

crystals are being the subject of research during

last years [1–7]. It is known from recent results that

these surface waves (SW) can exist in truncated

photonic crystals [1–3,7]. These non-radiative

waves propagate along the crystal–air interface

with evanescent fields in the perpendicular direc-

tion away from the surface plane constituting a

mechanism for energy loss – via tunneling – of a

localized bulk excitation [2]. Quite recently it hasbeen found experimentally [4,5] that SW can be

excited on the surface of a truncated one-dimen-

sional photonic crystal (1D-PC) by optical tech-

niques identical to those used for excitation of

surface plasma waves on metals by attenuated

total reflectance [8], and the application in a sensor

device has been proposed [9].

In the theoretical study of photonic crystals, it isclaimed that a variety of effects are potentially

applicable to manage the flow of light by taking

advantage of periodicity in these systems [10]. It

will be found in this work that it is possible to

Optics Communications 216 (2003) 361–367

www.elsevier.com/locate/optcom

*Corresponding author. Fax: +52-477-717-50-00.

E-mail address: fvilla@cio.mx (F. Villa).

0030-4018/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0030-4018(02)02279-4

concentrate on the material properties from a point

of view of an equivalent medium, which is a rather

simple point of view: that of a simple material with

a given dielectric function. Although the concept of

effective media is not new [1,2,11–13], these ideas

will provide us a new method to determine theband structure and SW on 1D-PC to get a new

insight in the knowledge of these systems.

2. Equivalent media: the dielectric function of a

symmetric photonic crystal

In the formalism of the characteristic matrixmethod of propagation of light through multilay-

ers [11], any single thin film has associated a 2� 2matrix that is only function of the parameters of

the corresponding jth film

mðgj; djÞ ¼cosðdjÞ

i

gjsinðdjÞ

igj sinðdjÞ cosðdjÞ

24

35: ð1Þ

This characteristic matrix relates the total elec-tric and magnetic fields from one medium to the

next. Here,

dj ¼2pK

�kkzjdj ð2Þ

represents the phase thickness of the layer, dj itsphysical thickness,

gj ¼y�kkzj= �xx; TE polarization;yn2j �xx=�kkzj; TM polarization;

(ð3Þ

stands for its optical admittance (the ratio of the

total magnetic to electric H=E fields), and

�kkzj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2j �xx2 � �bb2

qð4Þ

is the reduced perpendicular component (to inter-

faces) of the wave vector. The constant y repre-

sents the admittance of vacuum. Here �xx ¼ xK=2pc, and �bb ¼ bK=2p represent the reduced fre-

quency and the reduced parallel component of the

wave vector, respectively. The constant K hasthe same dimension as the thickness of layers, c is

the speed of light in vacuum, and nj the refractiveindex of the medium.

It has been shown elsewhere [11,12] that a basic

multilayer of three layers that is symmetric with

respect to a plane passing through its center of

symmetry can be represented mathematically by a

single equivalent layer, that has associated also an

equivalent characteristic matrix mðge; deÞ, with agiven equivalent phase thickness de and opticaladmittance ge. It is well known also that anysymmetric system can be represented by a repeti-

tion of the basic symmetric three-layer system pqp

repeated r-times (truncated 1D-PC of r periods orequivalently 2r þ 1 layers, Fig. 1. With this kind ofsystems K ¼ 2dp þ dq represents the thickness of aperiod if we assume that the layers of the sym-

metric period have the thicknesses dp, and dq re-spectively.In general any arbitrary symmetric system can

be considered as a single layer with an equivalent

phase thickness rde and the same equivalent op-tical admittance of the basic three-layer period.

According to the equivalence theorem [11], the

equivalent functions de and ge are given by therelations

cosðdeÞ ¼ cosð2dpÞ cosðdqÞ � qþ sinð2dpÞ sinðdqÞ;ð5Þ

Fig. 1. Periodic multilayer of three symmetric periods. g0 and gsstand for the optical admittance of the incidence and trans-

mission media respectively.

362 F. Villa et al. / Optics Communications 216 (2003) 361–367

ge ¼gp

sinðdeÞ½sinð2dpÞ cosðdqÞ þ qþ cosð2dpÞ sinðdqÞ

� q� sinðdqÞ�; ð6Þwhere gp; gq represent the optical admittance of thelayers of the period, dp; dq their respective phase

thicknesses, and

qþ ¼ 12

gp

gq

þ

gq

gp

!; ð7Þ

q� ¼ 12

gp

gq

�

gq

gp

!: ð8Þ

If we define an equivalent dielectric function eeassociated to the equivalent optical admittance as

eeð �xx; �bbÞ ¼ g2eð �xx; �bbÞ=y2, and plot its real part for theTE case as a function of the reduced frequency,

while keeping �bb constant (Fig. 2, solid line), it ispossible to appreciate in some spectral regions the

familiar Lorentzian curve of a classical oscillator

with its characteristic longitudinal xL and reso-

nance xT frequencies. In the case of classical os-

cillators xL represents the frequency where the

dielectric function is zero, and xT the resonance

frequency where the real part of the dielectric

function has a discontinuity with a change of sign.In the case of a 1D-PC these frequencies determine

the band gap boundaries since within these regions

(xT < x < xL) the effective dielectric function ee is

negative. Similar conditions apply to the TM po-

larization as can be appreciated from Fig. 3.

It is worth to mention that the imaginary part

of the dielectric function is zero anywhere when

materials are perfectly non-absorbing dielectric

except on the band edges where we have Diracdelta functions. If we consider some absorption in

one of the constituent layers, delta functions

transform to observable Lorentzian curves. This

fact can be appreciated from Fig. 4 where we

Fig. 2. Equivalent dielectric function of the 1D-PC under TE

polarization (solid line), and the imaginary part of the phase

thickness multiplied by a scaling factor of 30 (dot–dot-dashed

line). Both functions were determined by considering �bb ¼ 1.

Fig. 3. Equivalent dielectric function of the same 1D-PC under

TM polarization (solid line), and the imaginary part of the

phase thickness multiplied by a scaling factor of 40 (dot–dot-

dashed line). Both functions were determined by considering�bb ¼ 1.

Fig. 4. Real (dot–dot-dashed line) and imaginary (solid line)

parts of the equivalent dielectric function for TM polarization

when all the p layers have an intrinsic absorption index of

kp ¼ 9� 10�3. In this case also �bb ¼ 1.

F. Villa et al. / Optics Communications 216 (2003) 361–367 363

assumed an absorption index of kp ¼ 9� 10�3 inthe p layers of the symmetric period.

The imaginary part of the phase thickness de ofone period is indicated by a dash-dotted line in

Figs. 2 and 3. This function is zero in the bulk

bands and negative within the band gaps as couldbe expected since it is proportional to the magni-

tude of the Bloch wave number K in the crystal [3],

de ¼ KK. This fact denotes the evanescent charac-ter of the envelope of the electromagnetic field in

this region. Its real part takes positive values in the

bulk bands and is zero or amultiple of p in the gaps.Strictly speaking, a dielectric function of a real

material should be independent of the parallelcomponent of the wave vector. However, this de-

pendence in the equivalent function is rather con-

venient since the points where it changes the sign

(x ¼ xT, x ¼ xL) determine the boundaries of the

band gaps in a 1D-PC. In fact, the band structures

given in Figs. 5 and 6 for TE and TM polarization

were determined applying this criteria by consid-

ering a 1D-PC composed of a periodic multilayer(Fig. 1) of two different materials with refractive

indices np ¼ 2:22, nq ¼ 1:46, and a thickness rela-tion dp ¼ dq between constituent layers. In thesefigures, the shaded regions represent the bulk

bands and non-shaded ones represent the band

gaps, except for the lower region that corresponds

to electromagnetic waves with a parallel wave

vector that is beyond the light line of any of the

constituent materials of the multilayer. This con-

dition implies a perpendicular wave vector that isimaginary, as consequence we have purely eva-

nescent fields inside the multilayer.

Electromagnetic waves whose frequencies cor-

respond to a region inside the gaps are not phys-

ically allowed to propagate in the crystal.

In the TM case the band gaps narrow satisfying

the Brewster condition between the internal

boundaries. These points are in the line �xx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2p þ n3q

q�bb=npnq given in dot-dashed line in Fig. 6.

3. Equivalent media and the relation dispersion of

surface modes

An ideal crystal is considered as an infinite pe-riodic system. However, it has been found that

under truncation, surface modes can appear within

the band gaps when the outermost period is in-

complete [1,2]. As an example, surface modes are

shown in short dashed lines appearing in the first,

and third band gaps when the crystal is truncated

in a layer of material with refractive index np andthickness d ¼ dp as that given in Fig. 1. As anexample the squared amplitude of the electric field

is shown in the inset of Fig. 5. This SW corre-

sponds to the point indicated with a diamond in

the first band gap.

Fig. 5. Band structure for the TE polarization of an infinite

periodic system. The light line for vacuum is given in dot–dot-

dashed line. Surface modes are indicated in short dashed lines.

The inset shows the squared amplitude of the electric field for

the SW indicated by a diamond within the first band gap for�bb ¼ 0:4369, x ¼ 0:3175.

Fig. 6. Band structure of the 1D-PC for the TM polarization.

The light line for vacuum is given in dot–dot-dashed line and

the Brewster line is given in dot-dashed line. Surface modes for

the normal system are indicated in short dashed lines.

364 F. Villa et al. / Optics Communications 216 (2003) 361–367

Surface modes of the 1D-PC given in Fig. 1

satisfy the dispersion relation

g0 þ gs þ ig0gsge

þ ge

�tgðrdeÞ ¼ 0; ð9Þ

which results from assuming a null incident wave

in the Fresnel reflection coefficient of the system.

This relation can also be obtained from applying

the boundary conditions to the equivalent thinfilm. In this equation, g0 and gs represent the op-tical admittance of the incident and transmission

medium respectively. Eq. (9) is valid for any

number of periods (rP 2). In opposition to well

known methods to determine SW [1,2] Eq. (9) is

useful to determine the surface modes of a finite

truncated 1D-PC that may be present in each one

of its surfaces in spite of having only few periods.It is worth to mention that one condition for the

existence of surface modes for Eq. (9) to be satis-

fied is the fact the equivalent dielectric function

should be negative, corresponding to evanescent

fields, as happens in this case only within the band

gaps. It is well known that such condition deter-

mines the existence of surface plasma waves on

metals [8] with the advantage that SW in 1D-PCcan exist in both polarizations.

When we have a system with a big number of

periods (r ! 1), it is possible to establish a con-dition independent of this parameter. In such case

the condition given in Eq. (8) transforms to

geð �xx; �bbÞ ¼ �g0ð �xx; �bbÞ;geð �xx; �bbÞ ¼ �gsð �xx; �bbÞ:

ð10Þ

This is an interesting result which establishesthat whenever the equivalent optical admittance of

the 1D-PC is equal to the negative optical admit-

tance of the incident or transmission media there

exist a surface mode. This equation which is valid

in both polarizations, resembles the Brewster

condition but with negative sign.

Given the symmetry of the system discussed so

far, the resulting surface modes determined byEqs. (9) and (10), correspond only to one possible

case of truncated crystal, to mention that with the

outermost layers with thicknesses dp. To consider amore general case, let us analyze the characteristic

matrix [10] of a multilayer like that given in Fig. 1

but considering explicitly the outermost periods:

M ¼ mðgp; scdpÞmðgq; sbdqÞmðgp; sadpÞmðge; rdeÞ� mðgp; s

0adpÞmðgq; s

0bdqÞmðgp; s

0cdpÞ; ð11Þ

where the truncation parameters s0a; s0b; s

0c; sa; sb;

sc;2 ½0; 1� in the matrices of the outermost periodsof last equation, are different factors introduced to

vary the thicknesses of these layers from 0 to dp (ordq depending on the material) with the purpose oflocating the SW in different positions inside the

gaps. In this case the more general dispersion re-lation that determines the existence of SW is given

by:

g0ðM11 þ gsM12Þ þM21 þ gsM22 ¼ 0 ð12ÞConditions given by Eq. (12) for the determi-

nation of SW are similar to those given by Eq. (9)

for that particular case. If we consider, an arbi-

trary periodic multilayer including some defects or

truncated layers, we can assume that the multi-

layer plus its transmission medium has associated

a total admittance Y given by

Y ¼ M21 þ gsM22

M11 þ gsM12

: ð13Þ

This property (total admittance) allows us to

consider a complex multilayer as a single medium

(see Fig. 7) so that the problem of determining the

conditions for the existence of SW reduces to

finding the condition in the denominator of re-flection coefficient

Y ¼ �g0: ð14ÞThis condition should be satisfied for the exis-

tence of SW in the interface between the media of

optical admittance Y and g0. If we want to deter-

Fig. 7. A truncated 1D-PC supported by a substrate of ad-

mittance gs can be modeled by a single medium of admittance

Y. In a similar way, a truncated 1D-PC supported by a sub-

strate of admittance g0 can be modeled by a single material ofadmittance Y 0.

F. Villa et al. / Optics Communications 216 (2003) 361–367 365

mine the existence of SW in the interface between

the media Y 0 and gs (Fig. 4) we can demonstrate bya similar procedure that SW are determined by the

condition

Y 0 ¼ �gs: ð15ÞConditions given in Eqs. (15) and (16) for the

existence of SW in both boundaries of the trun-cated 1D-PC are given simultaneously by Eq. (12).

These two conditions reduce to Eq. (9) for a per-

fectly symmetric system.

If we define a normalized (to one period) trun-

cation parameter s 2 ½0; 1� in terms of the trunca-tion parameters of individual layers introduced in

Eq. (11) and given the fact that in the symmetric

period dp ¼ dq=2, we have

s ¼sa4

if ðsb ¼ 0 and sc ¼ 0Þ;sb2þ 14if ðsa ¼ 1 and sc ¼ 0Þ;

sc4þ 34if ðsa ¼ 1 and sc ¼ 1Þ:

8<: ð16Þ

This parameter allows us to truncate the

thickness of the outermost period (p–q–p) in con-

tact with the incidence medium of admittance g0.A similar rule applies with the unitary truncation

factor s0 of the outermost period in contact withthe transmission medium of admittance gs. Thisfactor will be defined in terms of s0a, s

0b, and s0c.

In Fig. 8 we have an example (TE polarization)of the varying position of modes within the band

gaps as a function of the truncation of the outer-

most periods of a 1D-PC constituted by many

periods (r > 20) and different incident and trans-mission media. In this case s and s0 were variedindependently. The modes indicated by spheres

correspond to the surface air-1D-PC and those

indicated by diamonds correspond to the surface

1D-PC-BK7 glass, with a refractive index

ns ¼ 1:52. It is worth to mention that SW in the last

case do not appear in the two uppermost bands,

since the light line of this glass is below them.

In Fig. 9 we show the surface modes for aperfectly symmetric system air-1D-PC-air for TM

polarization. In this case the truncation parame-

ters s and s0 were tied together to have equalvariation in order to truncate symmetrically the

1D-PC.

4. Conclusions

Summarizing, the extraordinary fact that a 1D-

PC can be considered as a single thin film of a

dielectric material allows us to determine in a

simple way its band structure and SW. This

method and the properties of the effective dielectric

function in the band gaps, resemble many aspects

of the surface plasma waves in single metallicsurfaces which are physically different in nature.

The procedure outlined in this letter besides

giving a good insight on the properties of surface

modes and the conditions for their existence,

Fig. 8. Surface modes position as a function of truncation of

the outermost periods of each termination surface of the 1D-

PC. Case TE. In this example the incidence and transmission

media are air and BK7 glass respectively. SW in the interface

air–crystal are indicated by spheres, and those modes corre-

sponding to the boundary crystal–glass are indicated by dia-

monds.

Fig. 9. Surface modes position as a function of truncation of

the outermost periods of each surface of the 1D-PC. Case TM.

In this case the incidence and transmission media were con-

sidered equal to air.

366 F. Villa et al. / Optics Communications 216 (2003) 361–367

provides us a powerful and simple tool for analysis

of SW in 1D-PC even in finite systems of at least

two periods. This opens the possibility to study the

physical effects of truncation to determine how far

the real truncated crystals can approximate ideal

infinite systems.Although we propose the application of the

characteristic matrix for the electromagnetic field

given its conceptual simplicity, the Airy recurrent

formulas or the transfer matrix are suitable

methods to be applied to determine the equivalent

functions.

Until now other methods to determine surface

modes have been focused on the properties of thewave propagating within the crystal based mainly

on the properties of the Bloch wave number. This

work takes into account both the properties of the

wave but also the properties of material from the

simplest point of view: an equivalent material. In

the case when the multilayer is not symmetric, the

simple idea of a basic property keeps our theory

also simple based on the optical admittance of thecomplete system.

References

[1] F. Ramos-Mendieta, P. Halevi, J. Soc. Am. B. 14 (1997)

370.

[2] Pochi Yeh, Amnon Yariv, Chi-Shian Hong, J. Opt. Soc.

Am. 67 (1977) 423.

[3] Pochi Yeh, Optical Waves in Layered Media, Wiley, New

York, 1988.

[4] W.M. Robertson, J. Lightwave Technol. 17 (1999) 2013.

[5] W.M. Robertson, Appl. Phys. Lett. 74 (1999) 1800.

[6] F. Ramos-Mendieta, P. Halevi, Phys. Rev. B 59 (1999)

15112.

[7] Xiamin Yi, Pochi Yeh, John Hong, J. Opt. Soc. Am. B 18

(2001) 352.

[8] H. Raether, Surface Plasmons on Smooth and Rough

Surfaces and on Gratings, Springer, New York, 1986.

[9] F. Villa, F. Ramos-Mendieta, T. L�oopez-Rios, J. Gaspar

Armenta, L.E. Regalado, Opt. Lett. 27 (2002) 646.

[10] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic

Crystals: Molding the Flow of Light, Princeton University

Press, Princeton, NJ, 1995.

[11] H.A. Macleod, Thin Film Optical Filters, McGraw-Hill,

New York, 1989.

[12] C.J. Van der Laan, H.J. Franquena, Appl. Opt. 34 (1995)

681.

[13] J.M. Bendickson, J.P. Bowling, M. Scalora, Phys. Rev. E

53 (1996) 4107.

F. Villa et al. / Optics Communications 216 (2003) 361–367 367