Optics Communications 254 (2005) 10–18

www.elsevier.com/locate/optcom

Isotropic-uniaxial crystal interfaces: Negative refractionand backward wave phenomena

Liliana I. Perez a,b,c,*, Marıa T. Garea a,b, Rodolfo M. Echarri a,b,c

a Grupo Optica de Interfaces, Dpto. de Fısica, Facultad de Ingenierıa, Universidad de Buenos Aires (UBA), Av. Paseo Colon

850, C1063 ACV Ciudad Autonoma de Buenos Aires, Argentinab Laboratorio de Optica, Dpto. de Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (UBA),

Intendente Guiraldes 2160, Pabellon I, Ciudad Universitaria C1428 EGA, Ciudad Autonoma de Buenos Airesc Consejo de Investigaciones Cientıficas y Tecnicas, Buenos Aires, Argentina

Received 31 December 2004; received in revised form 6 May 2005; accepted 9 May 2005

Abstract

We analyse the conditions necessary for negative refraction and backward wave phenomena to be produced on real

dielectric non-magnetic isotropic-uniaxial crystal interfaces. The analyses are based on the geometric description of the

extraordinary ray and its relationship with the extraordinary wave number vector. We also make an analytical study of

the wave number and the time-averaged Poynting vectors under these conditions. We show that the proposed vectorial

method allows making the calculi for both phenomena for every orientation of the optical axis with regard to the plane

of incidence and, consequently, it could be applied in order to examine these effects and their consequences.

� 2005 Elsevier B.V. All rights reserved.

PACS: 260-1180; 260-1440; 160-1190

Keywords: Uniaxial crystals; Birefringence; Refraction

1. Introduction

Historically, the study of birefringent media has been developed through two methods: the method of

Fresnel�s ellipsoids and the analytical one. The first one began with Huygens and made it possible to deter-

mine the essential characteristics of propagation of plane waves in uniaxial and biaxial media. Since

the establishment of Maxwell�s equations for propagation, reflection and transmission of light, combined

0030-4018/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2005.05.010

* Corresponding author. Tel./fax: +54 11 4343 1418.

E-mail address: lperez@fi.uba.ar (L.I. Perez).

L.I. Perez et al. / Optics Communications 254 (2005) 10–18 11

methods were developed making use of Fresnel�s ellipsoids [1–6]. Since 1970 pure analytical methods have

been formulated in order to describe the properties of systems composed by at least one anisotropic med-

ium [7–10]. Among these analytical methods, the vectorial one has been shown as the most adequate for the

qualitative and quantitative study of the characteristics or revealing phenomena in birefringent media be-

cause it allows obtaining explicit formulas even when considering geometries without symmetries.It is well known that birefringent media and interfaces that involve these kinds of media present peculiar

characteristics as double refraction, double angle of total reflection, inhibited reflection, phase shift in

refraction, etc. Other two phenomena that can take place when real crystals (i.e. with positive principal

dielectric constants and permittivities) are those called backward wave and negative refraction. When we

are dealing with the backward wave phenomenon we mean that the component of the wave number vector

that is perpendicular to the interface has opposite sense to the inner interface normal after refraction [11].

Nevertheless, this fact is only related to the backward of the wave number vector and not to the backward

of the direction of propagation of energy. In previous works, we demonstrated that the backward wavephenomenon could take place either in reflected [12,13] or refracted waves [14] when at least one of the med-

ia is anisotropic and extraordinary waves are considered. This phenomenon gives rise to situations that are

not possible in isotropic media, as seen in Fig. 1(a). In this figure the effect that the backward wave phe-

nomenon produces is observed on the uniaxial-isotropic interface: the incident ray �R00and its associated

refracted one are included in the same half-plane. This way, the phenomenon can be visualized.

On the other hand, in negative refraction the incident and refracted rays are contained in the same half-

plane with regard to the normal to the interface. It was demonstrated theoretically and experimentally [15–

18] that this behaviour is possible in interfaces with real crystals for extraordinary rays when the optical axisis contained in the plane of incidence (Fig. 1(b)).

Fig. 1. Backward wave and negative refraction effects in a uniaxial plate. (a) Directions of the refracted rays when the backward wave

phenomenon is present for the extraordinary wave ~k00when the optical axis is in the plane of incidence. ~k

0and �R

0correspond to the

ordinary wave number vector and ray. (b) Directions of the rays when the refracted ray is extraordinary (in the negative refraction

range).

12 L.I. Perez et al. / Optics Communications 254 (2005) 10–18

Manufacturing these kinds of materials is of great interest because of the possibility of negative refrac-

tion and the consequential focusing properties of plane-parallel plates, sub wavelength focusing and other

associated phenomena [19]. In last years the experimental and theoretical studies of materials with negative

phase velocity (LHM) have awaken a great attention [11,20–22]. Nevertheless, media with negative phase

velocity are artificial structures and it would be desirable to determine if there are conditions in which it ispossible to substitute an isotropic LHM by an anisotropic RHM under conditions of negative refraction.

Nevertheless, it has not to be forgotten that when dealing with real birefringent media the causes that pro-

duce negative refraction are different.

The existence of these two phenomena can be easily understood through the relationship that exists

between the direction of the extraordinary wave front and the direction of propagation of energy (ray

direction). This treatment, as far as we know, is not in bibliography and has the advantage of giving

explicit and general formulas for both phenomena and not using the geometric construction of the in-

dex ellipsoids. The relationship normal to the wave front-ray as a function of the constitutive param-eters of the media allowed not only to obtain in a simple way the images formed by any crystalline

device but also their associated interference figures [23–27]. Having a quite knowledge of the behaviour

of rays inside and outside the anisotropic media will also allow resolving the inverse problem, i.e. opti-

cal components design.

2. Waves and rays in uniaxial media

When the wave equation in an anisotropic uniaxial medium is resolved, we obtain that for a given wave

propagation direction~k there exist two possible phase velocities u 0 and u00 [3,4]. The first one does not de-

pend on the advance direction of the wave, while the second one does. The most appreciable characteristic

of the electric field correspondent to the wave with phase velocity u 0 (ordinary wave) is that it doesn�t con-tain a component in the direction of the optical axis, while for the one that corresponds to the wave that has

a velocity u00 (extraordinary wave) is that it is not perpendicular to the direction of the wave advance. This

leads to the fact that the direction of propagation of energy does not coincide with the direction of the

extraordinary normal to the wave front.When considering a plane wave that impinges on an interface formed by an isotropic medium and a uni-

axial crystal with an arbitrary direction of the optical axis, the boundary conditions lead to the existence of

two waves in the anisotropic medium with different propagation directions~k0and ~k

00, because of there exist

two possible propagation velocities (Fig. 2). The directions of the fields associated to the ordinary and

extraordinary waves will depend on the direction of the incidence, on the orientation of the optical axis

and on the dielectric constants of the media e, eo and ee (that correspond to the isotropic media, the prin-

cipal ordinary and principal extraordinary permittivities). The electric field associated to the extraordinary

wave is given by [28]

~E00 ¼ gðkx; ky ; kzÞ ~k

00 � ~k00 � �z3

� �þ ee � eo

eolx2eo � ~k

00 � �z3� �2

� ��z3

� �expðið~k00 �~r � xtÞÞ; ð1Þ

where �z3 indicates the direction of the optical axis, g(kx,ky,kz) is a function that depends on the incidence

direction, the media characteristics and the polarization of the incident wave, and [29]

k002x ½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� þ 2k00xkzðee � eoÞð�z � �z3Þð�x � �z3Þ þ eok2y þ k2z ½eoð�x � �z3Þ

2 þ eeð�z � �z3Þ2� ¼ lx2eoee;

ð2Þ

Fig. 2. Co-ordinate systems. The vectors �k;~k�;~k

0and ~k

00correspond to the incident, the reflected and the ordinary and the

extraordinary refracted wave number vectors, respectively. The angle # (positive in the figure) determines the direction of the optical

axis z3 of the crystal; the angle d indicates the plane of incidence, e is the dielectric constant of the isotropic medium, eo the ordinary

principal dielectric constant and ee the extraordinary one.

L.I. Perez et al. / Optics Communications 254 (2005) 10–18 13

kx ¼ ½lx2e� k2y � k2z �1=2 ¼

ffiffiffiffiffiffiffiffiffiffilx2e

pcos a; ð3Þ

ky ¼ffiffiffiffiffiffiffiffiffiffilx2e

psin a sin d; ð4Þ

kz ¼ffiffiffiffiffiffiffiffiffiffilx2e

psin a cos d. ð5Þ

The second term of Eq. (1) shows that the advance direction of the extraordinary wave front is not per-pendicular to the extraordinary electric field.

From the results obtained in [6] we can obtain the relationship between the extraordinary wave number

vector ~k00and the associated ray �R

00is [9,13]

�R00 ¼ 1

fn½eo~k

00 þ ðee � eoÞð~k00 � �z3Þ�z3�; ð6Þ

where fn is a normalization factor, i.e. the ray direction is given by the extraordinary wave number vector

plus a vector in the direction of the optical axis �z3. The equal-phase condition on the separation surface

leads to the fact that, generally, �R00is not contained in the plane of incidence, except in high symmetry sit-

uations, e.g. when the incidence is produced on a principal plane of the crystal. Besides, the incidence on

one of these planes shows the advantages of the vectorial method for the description of the extraordinary

negative refraction and backward wave phenomena.

3. Optical axis in the plane of incidence

When the incidence of a plane wave is produced on the principal plane that contains the optical axis and

the normal to the interface (Fig. 3), from Eq. (2) one can obtain that the relation between the components

of the extraordinary wave number vector is

Fig. 3. Extraordinary wave number vectors and rays in an isotropic-uniaxial interface (optical axis in the plane of incidence) (a)

corresponds to extraordinary negative refraction and (b) corresponds to extraordinary backward wave.

14 L.I. Perez et al. / Optics Communications 254 (2005) 10–18

k00x ¼kzðeo � eeÞð�z � �z3Þð�x � �z3Þ þ

ffiffiffiffiffiffiffiffieoee

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilx2½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� � k2z

q½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2�

. ð7Þ

Moreover, from Eq. (6) the components of the ray that are perpendicular and parallel to the interface are

given by

R00x ¼

1

fn

ffiffiffiffiffiffiffiffieoee

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilx2½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� � k2z

q; ð8Þ

R00z ¼

1

fn

ffiffiffiffiffiffiffiffieoee

p

½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2�ffiffiffiffiffiffiffiffieoee

pkz � ðeo � eeÞð�x � �z3Þð�z � �z3Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilx2½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� � k2z

q� �.

ð9Þ

The three simple expressions given by Eqs. (7)–(9) let us to describe completely (qualitatively and quanti-

tatively) both phenomena.

From Eq. (9) we can explain that it is possible that the components that are parallel to the interface of

the extraordinary time-averaged Poynting vector �R00z and of the incident ray kz have opposite senses if kz and

ðeo � eeÞð�x � �z3Þ are simultaneously positive or negative. That is to say, the fulfilment of this condition de-

pends on the incidence direction, kind of birefringence and orientation of the optical axis. If we considerthat both the angle of incidence a and ðeo � eeÞð�x � �z3Þ are positive, this condition will be fulfilled for every

angle of incidence a < a0N , where a0N is the limiting angle and is given by

sin a0N ¼ ðeo � eeÞð�x � �z3Þð�z � �z3Þffiffie

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieoð�z � �z3Þ2 þ eeð�x � �z3Þ2

q . ð10Þ

The obtained result corresponds to one of the situations posed in [17]. In Fig. 4 we draw the angles thatare formed by the normal to the extraordinary wave front b00 and the associated ray b00

R and the normal to

the interface as a function of the incidence angle, considering three different isotropic-negative uniaxial

interfaces (calcite with # = �45�). In Fig. 4(a), where the first medium is vacuum or air, negative refraction

is produced when the angle of incidence is included in the interval [0�, 10�] approximately, while for

other angles of incidence the incident and refracted rays are contained in different half-planes of incidence

with regard to z = 0. In Fig. 4(b), where the isotropic medium is glass with refraction index 1.50, negative

-90 -45 0 45 90

-60

-30

0

30

60

-90 -45 0 45 90

-90

-45

0

45

90

-90 -45 0 45 90

-90

-45

0

45

90

Fig. 4. Extraordinary refraction angle as a function of the incidence angle for three different isotropic-uniaxial interfaces (a) vacuum-

calcite, (b) glass-calcite, (c) dense glass-calcite with # = �45�. b00 and b00R are the angles that the wave number vector and the ray form

with the normal to the interface, respectively. The plane of incidence contains the optical axis.

L.I. Perez et al. / Optics Communications 254 (2005) 10–18 15

refraction is produced when the angle of incidence is between 0� and 6� approximately. As the normal to the

incident wave front approaches the optical axis, the normal to the wave front and ray come closer, and they

coincide when the normal to the wave front has the direction of the optical axis. In grazing incidence, ray

approaches to the interface before the normal to the wave front does. When the index of the isotropic med-

ium is a little larger (although lesser than the minimum that is necessary for extraordinary total reflection),

negative refraction exists for positive angles of incidence (Fig. 4(c)). Unlike what happens when dealingwith interfaces that involve an isotropic left-handed medium and despite of the existence of negative refrac-

tion, a divergent (convergent) beam can never be transformed in a convergent (divergent) one if there is

only one isotropic-uniaxial interface. This is due to the fact that the refracting angle that corresponds to

the refracted ray increases with the incidence angle either when negative refraction corresponds to positive

incidence angles or to negative ones. Nevertheless, negative refraction in real anisotropic slabs allows

obtaining some equivalent results to those obtained in isotropic media with negative phase velocity for lim-

ited beams (see Fig. 1 of [30]).

In Fig. 4(c), it is also observed that for negative incidence angles, the normal to the wave front can makean angle b00 greater than 90� while the associated ray is not grazing. This behaviour, called backward wave

phenomenon, can be also explained through the extraordinary wave number vector-ray relation.

Effectively, from Eq. (7) it clearly follows that the perpendicular to the interface component k00x can be

negative if the first term of the second member is negative. Consequently, it will be necessary for

kzðeo � eeÞð�x � �z3Þ to be negative for the existence of the backward wave phenomenon. Thus, the existence

of negative refraction and backward wave may take place in the same interface (dielectric constants and

direction of the optical axis with regard to the interface) if the incidence directions are adequately chosen.

The possibility of the existence of a backward refracted wave front was shown in [14] for crystal-isotropicinterfaces, and considering extraordinary total reflection for isotropic-uniaxial interfaces (see Fig. 2 in [13]).

From Eq. (8), the condition of backward wave verifies for positive angles of incidence a > a0R where a0R is

the limiting angle of backward wave

sin a0R ¼ffiffiffiffiffiffiffiffieoee

p

ffiffie

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieeð�z � �z3Þ2 þ eoð�x � �z3Þ2

q ; ð11Þ

when ðeo � eeÞð�x � �z3Þ < 0. An analogous expression can be obtained when kz < 0 and ðeo � eeÞð�x � �z3Þ > 0. It

must be pointed out that the expression backward wave does not mean backward energy, because it is the

normal to the wave front that has an unexpected direction whereas the ray (which is the one with physical

sense) has the expected direction.

16 L.I. Perez et al. / Optics Communications 254 (2005) 10–18

4. Arbitrary orientation of the optical axis

In the case we are not considering a principal plane of the crystal, the ray that is associated to the

extraordinary wave is not in the plane that contains the incident and reflected ray and the normal to the

interface. Consequently, the condition of negative refraction corresponds to the fact that the componentof the refracted ray that is parallel to the interface is contained in the half-space that contains the incident

ray and that is determined by the plane that is perpendicular to the interface and to the incidence plane

(Fig. 2). Therefore, we will calculate the component of the ray parallel to the interface and contained in

the plane of incidence (characterised by the xf co-ordinates) where the angle between z and f is d. FromEq. (6) the component of the ray in the plane of incidence and parallel to the interface is (regardless a nor-

malisation factor)

R00g ¼ eokgbeoð�z � �z3Þ2sin2dþ eeb1� ð�z � �z3Þ2sin2dcc þ ffiffiffiffi

eop ðee � eoÞð�x � �z3Þð�z � �z3Þ

� cos dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilx2eo½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� � k2g½ee þ ðeo � eeÞð�z � �z3Þ2sin2d�

q. ð12Þ

This component will have the same sign that kg if the angle of incidence is smaller than the limiting angle

aN

sin aN ¼ffiffiffiffieee

rðeo � eeÞð�x � �z3Þð�z � �z3Þ cos dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðeo sin2dþ ee cos2dÞ2ð�z � �z3Þ2 þ ½eoee þ ðeo � eeÞ2ð�z � �z3Þ2 sin2dcos2d�ð�x � �z3Þ2q ; ð13Þ

if kgðee � eoÞð�x � �z3Þ > 0. The limiting angle of negative refraction is drawn in Fig. 4(a) for a dense glass (with

refraction index 1.70) – calcite (with # = �30�) interface as a function of d, i.e. for every possible plane of

incidence. When the plane of incidence is the xy plane (d = 90�) there is not negative refraction because,

although the direction of propagation of energy does not coincide with the normal to the wave front,the contribution due to the optical axis is in the xz plane. Negative refraction neither exists when the optical

axis is perpendicular or parallel to the interface because the components on the interface of �R00and ~k

00have

same sense (see Eq. (6)).

Moreover, from Eq. (2) we obtain that it is possible to have backward waves for different planes of inci-

dence when k00x < 0, i.e. when

�kzðee � eoÞð�z � �z3Þð�x � �z3Þ þffiffiffiffieo

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlx2ee � k2yÞ½eoð�z � �z3Þ

2 þ eeð�x � �z3Þ2� � eek2z

q< 0; ð14Þ

that is verified for angles greater than the incidence angle aR

sin aR ¼ffiffiffiffiffiffiffiffieoee

p

ffiffie

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieo þ ðee � eoÞð�z � �z3Þ2cos2d

q . ð15Þ

Comparing Eq. (15) with the expression for the limiting angle of extraordinary total reflection

sin a00T ¼ffiffiffiffiee

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieoð�z � �z3Þ2 þ eeð�x � �z3Þ2

qffiffie

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½eoð�z � �z3Þ2 þ eeð�x � �z3Þ2� þ ðee � eoÞð�z � �z3Þ2cos2d

q ; ð16Þ

it can be deduced that the existence of an angle of backward wave is independent of the existence of the

angle of total reflection. In Fig. 5(b) it is shown the limiting angle of backward wave and of extraordinary

total reflection for a dense glass (with refraction index 1.70) – calcite (with # = +30�) interface as a function

of d. In these conditions the backward wave phenomenon exists for every plane of incidence except whenthe plane of incidence is perpendicular to the plane that contains the optical axis and the normal to the

0 30 60 90 120 150 180

-6

-4

-2

0

2

4

6

0 30 60 90 120 150 18060

65

70

75

α''T

αR

δ

αN

δ

(a) (b)

Fig. 5. Limiting angles of (a) negative refraction (b) extraordinary backward wave and total reflection, for isotropic-uniaxial interfaces

as a function of the angle between the plane of incidence and the plane that contains the optical axis and the normal to the interface.

L.I. Perez et al. / Optics Communications 254 (2005) 10–18 17

interface (d = 90�) because in that case sin2a00T ¼ sin2aR ¼ ee=e. An equivalent situation arises when the opti-

cal axis is parallel or perpendicular to the interface.

5. Conclusions

We have seen that both phenomena are direct consequence of both the existence of two possible phasevelocities (birefringence) and a preferential direction (optical axis). Beginning with the equation that relates

the extraordinary wave number vector and the direction of propagation of energy (ray) we were able to

determine qualitative and quantitatively the negative refraction phenomenon considering anisotropic uni-

axial media with positive permittivity, permeability and phase velocity. This treatment also allowed us to

explain the backward wave phenomenon, which could be visualized when we are dealing with limited beams

and two interfaces (e.g. isotropic-uniaxial-isotropic). The proposed method is completely analytical and

was extended without any difficult to every plane of incidence with regard to the optical axis. This simple

treatment, if applied to anisotropic multilayers, will allow not only to study the possibility of steering beamswithout reflection but also to examine the existence of other phenomena as the ones proposed in [17].

Acknowledgements

This work has been done with the support of CONICET and two grants of UBA.

18 L.I. Perez et al. / Optics Communications 254 (2005) 10–18

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