Optics Communications 223 (2003) 109–115

www.elsevier.com/locate/optcom

Electromagnetic surface waves: photoniccrystal–photonic crystal interface

Francisco Villa*, J.A. Gaspar-Armenta1

Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, Le�oon Gto 37150, Mexico

Received 13 February 2003; received in revised form 12 June 2003; accepted 12 June 2003

Abstract

It is demonstrated the existence of electromagnetic surface modes at the interface of two different one-dimensional

photonic crystals. Contrary to common behavior of modes present at the surfaces of truncated photonic crystals

embedded in dielectric media, these modes can reside above and below the light line for vacuum within the overlapping

of band gaps of both photonic crystals. This behavior implies that they can be excited and observed without prism or

grating configurations even under normal incidence from vacuum if proper conditions are fulfilled.

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

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

Keywords: Thin Films; Surface modes; Photonic crystals

1. Introduction

The properties of photonic crystals have been

the subject of intense research during last years invirtue of their potential to develop a new tech-

nology of pure optical integrated circuits [1–4],

among other interesting applications [5,6]. These

systems constitute periodical arrays of one, two,

* Corresponding author. Tel.: +52-477-7175823; fax: +52-

477-7175850.

E-mail address: fvilla@cio.mx (F. Villa).1 On leave from Centro de Investigaci�oon en F�ıısica de la

Universidad de Sonora. Apdo. Post. 5-088, Hermosillo Sonora

83190, Mexico.

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

doi:10.1016/S0030-4018(03)01644-4

and three dimensions of dielectric materials that

present some properties for the propagation of

electromagnetic waves that are analogous to those

of real crystals for electronic charges. The moreremarkable similarity is the characteristic band

structure with spectral regions where light can

(bulk bands) or cannot propagate (band gaps).

It is well known that when an ideal infinite

photonic crystal is truncated, surface waves (SEW)

appear on the truncation surfaces [7–9]. These

modes are electromagnetic fields that decay expo-

nentially in normal direction going away from theinterface between two optically different media (a

photonic crystal, and a dielectric homogeneous

medium). From this research, it is known that

SEW can exist in the case of finite one-dimensional

ights reserved.

Fig. 2. Band structure for the TE polarization of an infinite

110 F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115

photonic crystals (F1DPC) below the vacuum light

line in the dispersion diagram (frequency vs. par-

allel component of the wave vector). This means

that photonic surface modes can be excited ex-

perimentally using attenuated total reflectance

configurations [5,10,11].We will demonstrate in this work that when two

different F1DPCs are placed in contact, it is pos-

sible that new SEW appear on the boundary

between them. We call these modes photonic

crystal–photonic crystal surface waves (PCPCSW)

and we will show that these surface waves appear

whenever certain conditions are satisfied, that are

indeed closely related to the conditions for theexistence of SEW in F1DPC.

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

shed line. Surface photonic crystal–photonic crystal modes are

indicated by dotted lines. The inset shows the magnitude of the

electric field for the surface mode indicated by a triangle in the

same figure.

2. Band structure of one-dimensional photonic

crystals

Let us consider a symmetric 1DPC (Fig. 1)

which is composed of periods of three layers withtwo different materials with refractive indices

nq1 ¼ 2:22, np1 ¼ 1:46, and a thickness relation

nq1dq1 ¼ 2np1dp1 ¼ k1=4, for an arbitrary wave-

Fig. 1. Configuration of symmetric one-dimensional photonic

crystal embedded in dielectric media.

length k1 (in our case we stated k1 ¼ 1200 nm).The physical thicknesses of the constituent layers

are dp1, dq1. This election guarantees the widest

band gaps achievable with the chosen materials

[12], besides that the first order band gap should

appear at wavelength k1 [13]. The band structure

of this system (1DPC1) is given in Figs. 2 and 3

(gray and white regions) for both polarizations, in

Fig. 3. Band structure for the TM polarization of the same

infinite crystal of previous figure. The light line for vacuum is

given in dot-dashed line. Surface photonic crystal–photonic

crystal modes are given in dotted lines. The inset shows the

magnitude of the electric field for the surface mode indicated by

a diamond in the same figure.

F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115 111

the scale of reduced units of frequency �xx ¼xK1=2pc, and the parallel component of the

wave vector �bb ¼ bK1=2p. In the case of the system

we are considering K1 ¼ 2dp1 þ dq1 represents the

physical thickness of one period.

In these figures, the shaded regions representthe bulk bands and non-shaded ones represent the

band gaps, except for the lower region that cor-

responds to electromagnetic waves with a parallel

wave vector that is beyond the light line of any of

the constituent materials of the multilayer. This

condition implies a perpendicular wave vector that

is imaginary, as a consequence we have purely

evanescent fields inside the multilayer.If we consider another 1DPC (1DPC2) com-

posed with materials whose refractive indices are

nq2 ¼ 2:0, np2 ¼ 3:0 while keeping the same optical

thickness relation nq2dq2 ¼ 2np2dp2 ¼ k1=4. In this

case the band structures are superimposed to those

of 1DPC1 (solid thick lines). The peculiarity of

these two different systems is that their band gaps

are quite similar under normal incidence in spitethat materials are completely different for both

structures.

The band structures of these 1DPCs were de-

termined by following the method of equivalent

media based on the characteristic matrix [4]. It is

well known that a basic multilayer of three layers

that is symmetric with respect to a plane passing

through its center of symmetry is equivalent to asingle layer with phase thickness de, and optical

admittance ge [5,13,14] given by:

cosðdeÞ ¼ cosð2dpÞ cosðdqÞ � qþ sinð2dpÞ sinðdqÞ;ð1Þ

ge ¼gp

sinðdeÞsinð2dpÞ cosðdqÞ�

þ qþ cosð2dpÞsinðdqÞ � q�sinðdqÞ�; ð2Þ

where,

dj ¼2pK

�kkzjdj; ð3Þ

stands for the phase thickness of the layer, the

subscript j denotes the corresponding material p1,

p2 or q1, q2. dj represents the physical thickness,

and the optical admittance (the ratio of the total

magnetic to electric H=E fields) of different media

are given by:

gj ¼y�kkzj= �xx TE polarization;

yn2j �xx=�kkzj TM polarization;

(ð4Þ

where

�kkzj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2j �xx2 � �bb2

q; ð5Þ

represents the perpendicular (to interfaces) com-

ponent of the reduced wave vector �kkzj ¼ K=2pð Þkzj,y is the optical admittance of vacuum (in the MKSsystem), and

qþ ¼ 1

2

gp

gq

þ

gq

gp

!; ð6Þ

q� ¼ 1

2

gp

gq

�

gq

gp

!: ð7Þ

In general a symmetric system with complete pand q layers of any number of periods can be

represented by a repetition of the basic symmetric

three layer system pqp (truncated 1DPC of r pe-

riods or equivalently 2r þ 1 layers, Fig. 1). As aconsequence, such symmetric systems can be con-

sidered as single layers with an equivalent phase

thickness rde and the same equivalent optical ad-

mittance of the basic three layer period.

The criteria to determine the band structure of

any 1DPC is that band gaps are present wherever

the equivalent optical admittance ge is imaginary

in the ð�bb; �xxÞ space except for the lower region ofthe light line of the highest refractive index of

constituent materials.

3. Surface modes in the interface: bulk dielectric-

photonic crystal

From the equivalent parameters given in Eqs.(1) and (2) it becomes an easy task to derive an

analytic expression to determine SEW, by consid-

ering the Fresnel equations for the system dia-

gramed in Fig. 1. It can be demonstrated that a

dispersion relation that determines the existence of

modes on the surfaces of finite 1DPC is given

by [4]:

112 F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115

g0 þ gs þ ig0gs

ge

�þ ge

tgðrdeÞ ¼ 0 ð8Þ

which results from assuming a null incident wave

in the Fresnel reflection coefficient of the system.

This expression that is valid only within the band

gaps, allows us to determine SEW on truncatedand infinite 1DPC embedded on a given incidence

and transmission media as indicated in Fig. 1.

This formalism allows us to treat a rather in-

volved problem of determining surface modes in a

symmetric system of many layers in a simple way

by considering a single equivalent layer embedded

in the incident (g0) and transmission (gs) media.

Fig. 4. Imaginary parts of optical admittance (normalized di-

viding by admittance of vacuum y) of 1DPC1 (dot-dot-dashed)

and the 1DPC2 (solid line) for �bb ¼ 0. The imaginary part of the

equivalent phase thickness de1 is indicated on the same figure

with dashed line.

4. Surface modes in the boundary of two different

photonic crystals

Following the method briefly outlined in pre-

vious sections, the matter of analyzing different

configurations involving multiple 1DPCs is greatly

simplified. Let us consider an air–1DPC1–1DPC2system. In this case the dispersion relation will be

given by:

g0 þ ge2 þ ig0ge2

ge1

�þ ge1

tgðr1de1Þ ¼ 0; ð9Þ

where ge1 and ge2 stand for the equivalent opticaladmittance of each photonic crystal. In this case

we consider that the 1DPC2 is of infinite thickness

and the 1DPC1 has the finite phase thickness r1de1.

Being r1 the number of periods of 1DPC1. When

the phase thickness of the finite crystal goes to

infinity (r1 ! 1), Eq. (9) simplifies to:

ge1 ¼ �g0; ð10Þge1 ¼ �ge2: ð11Þ

Eq. (10) gives the condition for the existence of

modes between the incidence medium and the

1DPC1, so that, if the incidence medium is a

dielectric this is a condition for existence of thewell-known SEW. However, Eq. (11) is more

interesting since it establishes the condition for

existence of surface modes in the boundary

1DPC1–1DPC2. According to the analysis given

elsewhere [4] the regions where optical admittance

of the 1DPCs satisfy this condition lie within their

corresponding band gaps and in these regions the

optical admittance can be positive or negative de-

pending on the order of the band gap, besides the

combination of the materials composing the crys-

tal pqp or qpq. This becomes evident in Fig. 4,where we show the imaginary part of the equiva-

lent optical admittance (normalized dividing by

the admittance of vacuum y) of both crystals under

normal incidence (�bb ¼ 0) as a function of the

reduced frequency. In this figure the equivalent

admittance of 1DPC2 is graphed with negative

sign (solid line) to appreciate easily the intersection

with the curve corresponding to the admittance of1DPC1 (dot-dot-dashed line) which is the condi-

tion for existence of modes. The imaginary part of

the equivalent phase thickness is given in dashed

line. All these functions are zero outside the band

gaps.

These modes are new to our knowledge, and

contrary to the well known surface modes they can

be above or below the vacuum light line. They areillustrated in the dispersion diagrams (Figs. 2 and

3) with dotted lines. It can be appreciated that

these modes lie in the overlapping regions of the

band gaps of both crystals and are present for both

polarizations.

It is worth mentioning that Eq. (11) establishes

the proper conditions for the existence of our

Fig. 5. Transmittance and reflectance vs. wavelength, illus-

trating the mode in the first band gap under normal incidence.

System: air–ðp1–q1–p1Þ8ðp2–q2–p2Þ8–glass.

Fig. 6. Absorptance and reflectance vs. wavelength, illustrating

the mode in the first band gap under normal incidence. System:

air–ðp1–q1–p1Þ8ðp2–q2–p2Þ28–glass. In this case transmittance is

almost zero in the considered spectral range.

F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115 113

modes in the ideal interface between two semi-in-

finite photonic crystals, and Eq. (9) is involved

with necessary conditions for the excitation of

these modes.

As one example of these modes, we show in the

inset of Fig. 3 (TM case) the magnitude of theelectric field profile in both media for the point

indicated by a diamond in the same figure. These

results were obtained for a system air–1DPC1–

1DPC2 (6 periods each crystal). All these modes

are of the same nature except those that reside

below the light line (dot-dot-dashed line, Figs. 2

and 3) for the highest refractive index material of

1DPC1 (np1), which are evanescent inside thiscrystal. One example is illustrated in the inset of

Fig. 2, where the amplitude of the electric field is

indicated for a system air–1DPC1–1DPC2 of two

periods each one. In this case the mode is located

at �bb ¼ 1:4 and �xx ¼ 0:627 and indicated by a tri-

angle. The difference in the decaying length (or the

confinement) between the modes of both examples

is due to the relative position of each mode insidetheir corresponding band gap. It has been shown

elsewhere [6] that the decaying length is minimum

in the middle of the band gap and goes to infinity

at the edge of the band gaps, since it is inversely

proportional to the imaginary part of the equiva-

lent phase thickness (see Fig. 4).

The photonic crystal–photonic crystal modes

can be observed by measuring transmittance orreflectance of a system with a finite number of

periods. As one example we have graphed in Fig. 5

the transmittance and reflectance under normal

incidence ð�bb ¼ 0Þ for the mode in the first band

gap and the system air� ðp1� q1�p1Þ8ðp2�q2� p2Þ8 � glass, where the exponent in the pa-

renthesis denotes the number of times the period is

repeated. The refractive index (ns ¼ 1:52) of theglass considered here is that corresponding to the

typical BK7.The reason of having a transmitted signal of

these modes under normal incidence, can be ex-

plained by the fact that we are considering finite

both photonic crystals and absorption free mate-

rials. They are quite similar to the well known

Fabry–Perot modes in multilayers [13]. However ifwe increase gradually the number of periods for

example of the 1PDC2 (in contact with the trans-

mission medium) the transmittance peak will be-

come narrower until disappearing. In this case the

mode can be observed only under reflection always

that the system has a small absorption to dissipate

energy. This can be made evident from Fig. 6,

where the same mode is observed now with asystem: air� ðp1� q1� p1Þ8ðp2� q2� p2Þ28�glass. Here we assumed an imaginary part for the

refractive index of material p2 that is equal to

kp ¼ 3� 10�3. So in this case indeed the energy

lost under reflection is absorbed by the system.

An alternative explanation of having a trans-

mitted signal with this photonic crystal–photonic

114 F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115

crystal modes can also be given in terms of the

well-known photonic crystal-bulk surface modes

that are always below the light line for vacuum. As

we know, in order to excite them it is necessary to

use for example a Kretschmann configuration

which constitutes a half-cylinder prism with a fi-nite 1DPC deposited on the flat face that contains

the axis of the cylinder [5,10,11]. Then, if the

proper angle and wavelength are chosen, a surface

mode will appear at the interface of the finite

1DPC–air interface. In this system transmission

has no sense because, the wave in the transmission

medium is evanescent. However, if we add a half-

cylinder prism as indicated in Fig. 7 the light canbecome radiative when the distance between the

flat face of the prism and the 1DPC is properly

chosen and then these kind of modes can be ob-

served under transmission.

Fig. 7. Optical set up to observe electromagnetic photonic

crystal-bulk dielectric surface modes. In this scheme the thin

films of the finite 1DPC are represented by A and B, respec-

tively. The variable s represents a truncation parameter for

layer A in contact with air. The angle of incidence is given by h.

The matter of light guiding along the 1DPC1–

1DPC2 interface is again very similar to what

happens in the Fabry–Perot modes. When light is

incident normally to the interface, energy is

transmitted (Fig. 5) or absorbed (Fig. 6). In this

case, it is not possible to have guided waves sincethe parallel component of the wave vector is null.

However, under oblique incidence indeed we do

have a parallel component of the wave vector that

becomes coupled with the wave vector of the mode

in order to excite it and produce a guided wave.

Although Eq. (11) is valid only within the

overlap of the band gaps of both crystals that are

indeed symmetric, it is possible to demonstrate [4]that a similar analysis applies for the general case

of non-symmetric finite crystals and any arbitrary

truncation. In such case it is better to consider the

total admittance of the systems 1DPC1–air (with

admittance Y1) and 1DPC2–air (with admittance

Y2). In this case the condition for the existence of

pc–pc modes is similar to that given by Eq. (11):

Y1 ¼ �Y2: ð12Þ

It is worth to mention that the well known and

common bulk modes of both crystals are always

present in the regions of overlap of band gaps–

bulk bands and bulk bands–bulk bands. Although

not indicated in the figures, they constitute over-lapping of the bulk modes of both crystals.

5. Conclusions

We have proved the existence of new electro-

magnetic surface modes appearing in the bound-

ary of two photonic crystals that have the property

of lying above and below the light line for vacuum

and inside the overlap of band gaps. This fact

makes possible to observe these modes even under

normal incidence from vacuum whenever theproper conditions are satisfied.

Another contribution of this work is the deter-

mination of existence of surface waves in the

photonic crystal–photonic crystal interface that

can be evanescent in one side of the interface

1DPC1–1DPC2.

In spite that we are considering one-dimensional

photonic crystals we will show in future work that

F. Villa, J.A. Gaspar-Armenta / Optics Communications 223 (2003) 109–115 115

these results canbe extended to arbitrarily truncated

crystals of two and three more dimensions.

One aspect that deserves a detailed future study

is the possibility that PCPCSW can be applied to

build also channels to guide (or loss) energy that

result just by the combination of different photoniccrystals in the very same way to the well-known

guided modes produced by point and line defects

in photonic crystals used to manipulate the flow of

light in integrated optical circuits.

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