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904 J. Opt. Soc. Am. B/Vol. 23, No. 5 /May 2006 Shen et al.

Total transmission of electromagnetic wavesat interfaces associated with an

indefinite medium

Nian-Hai Shen, Qin Wang, Jing Chen, Ya-Xian Fan, Jianping Ding, and Hui-Tian Wang

National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University,Nanjing 210093, China

Yongjun Tian

Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China

Received August 3, 2005; revised October 20, 2005; accepted December 12, 2005; posted December 22, 2005 (Doc. ID 63791)

We investigate the problem of total transmission at the interface separating an isotropic regular material andan indefinite medium [Phys. Rev. Lett. 90, 077405 (2003)] in which not all of the principal elements of thepermeability and permittivity tensors have the same sign, for TE- and TM-polarized electromagnetic waves.We make a detailed investigation on the existence conditions of total transmission and the correspondingBrewster’s angles when an electromagnetic wave is incident on the interface from an isotropic regular mate-rial. We show that the propagation characteristics of electromagnetic waves at such interfaces are quite dif-ferent from those at regular interfaces. For both TE and TM waves, total transmission is possible at the in-terfaces containing an indefinite medium; in particular, normally incident total transmission andomnidirectional total transmission are also allowed, provided that suitable physical parameters for the twomaterials across the interface are chosen. © 2006 Optical Society of America

OCIS codes: 260.2110, 120.7000, 160.1190.

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. INTRODUCTIONn 1968, Veselago1 first suggested hypothetical materialsith negative electric permittivity � and negative mag-etic permeability �, which did not rivet scientists’ atten-ion at that time. These kinds of materials are called left-anded materials (LHMs), owing to their left-handedroperty (i.e., the electric and magnetic fields form a left-anded orthogonal vector triplet with the wave vector). It

s well known that materials with negative �, such as aonducting metal below its plasma frequency, can be eas-ly found. However, it is quite difficult to find materialsith negative �. This situation has changed since 1999,hen artificially structured nonmagnetic materials con-

isting of arrays of split-ring resonators were introducedo provide the possibility of achieving magnetic responserom inherently nonmagnetic materials (conductors).2

HM-related topics have then become rapidly growingelds, owing to the realization of Veselago’s hypothesis inoth theory3 and experiment.4 Such composite LHMsould be constructed by combining two types of arraysi.e., split-ring resonators and thin metal wires), whichould exhibit simultaneously ��0 and ��0 within a fre-uency region that is determined by the lattice param-ters of the two arrays. Some peculiar electrodynamicroperties (such as negative refraction and reversal Dop-ler effect) have been theoretically predicted or experi-entally demonstrated or both, and a potential applica-

ion in subwavelength imaging5–7 has been found. Someropagation characteristics of electromagnetic waves as-

0740-3224/06/050904-9/$15.00 © 2

ociated with an isotropic LHM, even an indefiniteedium8 in which not all the principal elements of the

ermeability and permittivity tensors have the same sign,ave been discussed in Refs. 8–10. Hu and Chui10 pre-icted the existence of an anomalous total reflection phe-omenon at the interface separating a regular materialnd an indefinite medium under suitable conditions andven the appearance of omnidirectional total reflection,hich could also be realized in one-dimensional photonic

rystals using photonic heterostructures.11

In the present paper, we investigate the problem of to-al transmission of electromagnetic waves at the inter-aces formed by an isotropic regular material and an in-efinite medium. We present a detailed analysis on thexistence conditions of total transmission for all the pos-ible physical parameters. This total transmission phe-omenon has significant differences from that at the in-erfaces formed by regular materials. This paper isrganized as follows. For universality, we first give a briefeview on the Fresnel coefficients at the interface betweenn isotropic regular material and an indefinite medium inection 2. Then we consider total transmission at the in-erface formed by an isotropic regular material and an in-efinite medium in Section 3. In Section 4 we considerhree special cases of the indefinite medium to explore theotal transmission phenomenon: (i) the indefinite mediumegenerates to an isotropic LHM, (ii) the symmetric prin-ipal axes of the permittivity and permeability tensorsre coincident normal to the interface, and (iii) the sym-

006 Optical Society of America

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Shen et al. Vol. 23, No. 5 /May 2006/J. Opt. Soc. Am. B 905

etric principal axes of the permittivity and permeabilityensors are coincident parallel to the interface. In the lastection, we summarize our main results.

. FRESNEL COEFFICIENTS AT ANNTERFACE OF AN ISOTROPIC REGULARATERIAL AND AN INDEFINITEEDIUM

he interface problem treated here is illustrated in Fig. 1.he interface is parallel to the xy plane, that is to say, theormal of the interface is along the z direction. The inter-ace is formed by two media: medium 1 is always an iso-ropic regular material, which has a positive permittivity1 and a positive permeability �1, whereas medium 2 is anndefinite medium, in which the principal components ofhe permittivity and permeability tensors may be of dif-erent signs. For universality, we first consider such aeneral case. Then for some special cases such as the in-efinite medium degenerating to an isotropic LHM, theesults can be easily obtained.

For the sake of simplicity, here we consider only thease that the permittivity and permeability tensors of me-ium 2 are simultaneously diagonalizable in the same ba-is, which have the forms as follows:

�2 = ��2x 0 0

0 �2y 0

0 0 �2z�, �2 = �

�2x 0 0

0 �2y 0

0 0 �2z� . �1�

Without loss of generality, we can choose the plane ofncidence (defined as a plane formed by the wave vectornd the normal of the interface) to be in the xz plane.lectromagnetic waves can be classified into two types:M and TE waves. For a TM wave, its magnetic field iserpendicular to the plane of incidence (the xz plane), andhere are field components Ex, Ez, and Hy only. A TE waveits electric field is normal to the plane of incidence) haseld components Hx, Hz, and Ey only. Considering an elec-romagnetic wave incident on the interface from medium, this wave should undergo reflection and transmissiont the interface. According to Maxwell’s equations, the in-ident, reflected, and transmitted fields for TE waves cane written as

Ei = E0ey exp�jk1zz�, �2�

Hi =E0

��0�1�− k1zex + kxez�exp�jk1zz�, �3�

ig. 1. (Color online) Geometry of the interface between media 1nd 2 and the coordinate system.

Er = rE0ey exp�− jk1zz�, �4�

Hr =rE0

��0�1�k1zex + kxez�exp�− jk1zz�, �5�

Et = tE0ey exp�jk2zz�, �6�

Ht =tE0

��0�−

k2z

�2xex +

kx

�2zez�exp�jk2zz�, �7�

here ex, ey, and ez are unit vectors along the x, y, and zxes and r and t represent the Fresnel amplitude reflec-ance and transmittance, respectively. Three subscripts i,, and t represent the quantities related to the incident,eflected, and transmitted waves, respectively. Note thathe common factor exp�jkxx− j�t� has been omitted in allhe above expressions and kx has the following form:

kx2 = ��2/c2��1�1 sin2 �1, �8�

here �1 is the incident angle in medium 1 (as shown inig. 1), � is the angular frequency of the incident electro-agnetic wave, and c is the velocity of light in vacuum.he normal component of a wave vector in medium 2, k2z,

s determined by

kx2

�2y�2z+

k2z2

�2y�2x=

�2

c2 �9�

or TE waves and

kx2

�2z�2y+

k2z2

�2x�2y=

�2

c2 �10�

or TM waves. However, the normal component in me-ium 1, k1z, can always be written for both TE and TMaves as

k1z2 = ��2/c2��1�1 − kx

2. �11�

Since kx represents the wave-vector component parallelo the interface and is conserved across the interface, inhe absence of losses, the sign of k2z

2 can be used to distin-uish the nature of the plane-wave solutions. k2z

2 �0 cor-esponds to real-valued k2z and propagating solutionscorresponding to so-called volume waves). k2z

2 �0 corre-ponds to imaginary k2z and exponentially growing or de-aying solutions (corresponding to so-called evanescentaves).From the boundary conditions, we can get the expres-

ions of the Fresnel amplitude reflectance and transmit-ance for TE waves:

rTE =�2xk1z − �1k2z

�2xk1z + �1k2z, �12�

tTE =2�2xk1z

�2xk1z + �1k2z. �13�

imilarly, we also easily obtain the Fresnel amplitude re-ectance and transmittance for TM waves:

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906 J. Opt. Soc. Am. B/Vol. 23, No. 5 /May 2006 Shen et al.

rTM =�2xk1z − �1k2z

�2xk1z + �1k2z, �14�

tTM =2�2xk1z

�2xk1z + �1k2z. �15�

t is easily found from Eqs. (12)–(15) that total transmis-ion (i.e., zero reflectance; the corresponding incidentngle is a so-called Brewster’s angle and is labeled as �1B)ccurs when

�2xk1z = �1k2z �16�

or TE waves and when

�2xk1z = �1k2z �17�

or TM waves.At the interface formed by two regular media (i.e., �2x

�2y=�2z=�1=1 and �2x ,�2y ,�2z��1), comparing Eq. (9)ith Eq. (11) we can find that k1z�k2z. Hence, total trans-ission never occurs for TE waves, as one can see fromq. (12). In contrast, if the interface is associated with an

ndefinite medium, the situation should be quite differentecause the permeabilities of the two media across the in-erface also could be different. We can find from Eqs. (9)nd (11) that total transmission could be allowed for TEaves, provided that the appropriate conditions, Eqs. (12)nd (16), are satisfied.

. GENERAL CASE OF TOTALRANSMISSION AT AN INTERFACE OF AN

SOTROPIC REGULAR MEDIUM ANDN INDEFINITE MEDIUM

n this section, we consider a general case to explore theroblem of total transmission in which the electromag-etic wave is incident on the interface formed by an iso-ropic regular medium 1 (with �1�0 and �1�0) and anndefinite medium 2 with its permittivity and permeabil-ty tensors as shown in Eqs. (1).

We first deal with the case of TE waves. The require-ent for the existence of total transmission has been

iven in Eq. (16). Now we discuss the existence possibilityf the Brewster’s angle for different signs of �2x. As shownn Fig. 1, since the incident wave vector K1 and Poyntingector S1 point forward top right, the energy flow of theefracted wave in medium 2 should propagate upwardway from the interface (that is to say, the Poynting vec-or S2 in medium 2 must have the normal componentlong the +z direction), implying that k2z should have theame sign as �2x.

7,8 Owing to both positive �1 and k1z, ones allowed to hold the requirement of Eq. (16) for the totalransmission. From Eq. (9) and referencing Eq. (8), we get

k2z2 = ��/c�2��2y�2x − ��2x/�2z��1�1 sin2 �1�. �18�

fter some algebra, we find the Brewster’s angle satisfieshe relation as follows:

sin2 �1BTE =

��1�2x − �2y�1��2z

��2x�2z − �12��1

. �19�

We now need to explore the existence conditions of therewster’s angle for different physical parameters. Therst restriction comes from the span of sin2 �1B

TE; thus theollowing inequality set must be satisfied:

0 ���1�2x − �2y�1��2z

��2x�2z − �12��1

� 1. �20�

he above inequality is, in fact, equivalent to two inequal-ty sets as

��2x�2z − �12���1�2x − �2y�1� � 0,

��2x�2z − �12���2y�2z − �1�1� � 0, �21�

or the case of �2z�0, and

��2x�2z − �12���1�2x − �2y�1� � 0,

��2x�2z − �12���2y�2z − �1�1� � 0, �22�

or the case of �2z�0.On the other hand, the transmitted wave in our discus-

ion is a volume wave or a propagation wave (i.e., threeomponents of its wave vector must be real simulta-eously); therefore, the following restrictions should alsoe required:

�2y�2z � �1�1, �23�

or �2x�2z�0, and

��2x�2z − �12���2y�2z − �1�1� � 0, �24�

or �2x�2z�0.We note that for TE waves the existence of total trans-ission is determined only by the three elements of the

ensors, i.e., �2y, �2x, and �2z, while it is independent ofhe others. Table 1 summarizes the existence conditionsf the Brewster’s angle (i.e., total transmission) for TEaves at the interface separating an isotropic regular ma-

erial and an indefinite medium with the form of Eqs. (1).In addition, we need to point out three special cases for

E waves. (i) When �2y /�2x=�1 /�1��2x�2z��12, the

rewster’s angle is zero (i.e., �1BTE=0), meaning that total

ransmission occurs only in the case of normal incidenceor TE waves. (ii) If �2y /�2x��1 /�1��2x�2z=�1

2, therehall never exist any incident angle for zero reflectance.

Table 1. Existence Conditions of Brewster’s Anglesfor TE Waves at Interface of Isotropic Regular

Medium 1 and Indefinite Medium 2

2z �2x �2y Existence Conditions

� � �2y�2z��1�1

� � �2y /�2x��1 /�1

� � �2y /�2x��1 /�1��2y�2z��1�1

�2y /�2x��1 /�1��2y�2z��1�1

� �

� �

� � �2y /�2x��1 /�1��2y�2z��1�1

�2y /�2x��1 /�1��2y�2z��1�1

� � �2y /�2x��1 /�1

� � �2y�2z��1�1

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Shen et al. Vol. 23, No. 5 /May 2006/J. Opt. Soc. Am. B 907

iii) For �2y /�2x=�1 /�1��2x�2z=�12, it is significant that

here occurs the phenomenon of omnidirectional totalransmission.

For the purpose of intuition, we summarize total trans-ission (concluding with the special cases) for TE waves

n Table 2.Following the case of TE waves, we shall briefly discuss

he case of TM waves. In fact, we can easily get the rel-vant results for TM waves, provided that � and � are re-laced by � and �, respectively. Therefore, referencing Eq.19), we find that the Brewster’s angle for TM waves isetermined by

sin2 �1BTM =

��2x�1 − �1�2y��2z

��2x�2z − �12��1

. �25�

The existence conditions of the Brewster’s angle for TMaves are determined by the same two restrictions as for

he case of TE waves. We have given a summary (in Table

Table 2. Total Transmission (Brewster’s Angle) forTE Waves at Interface of Isotropic Regular

Medium 1 and Indefinite Medium 2

Conditions Brewster’s Angle for TE Waves

�2y /�2x=�1 /�1��2x�2z=�12 �1B

TE� �0, /2��2y /�2x��1 /�1��2x�2z=�1

2

�2y /�2x=�1 /�1��2x�2z��12 �1B

TE=0

�2y /�2x��1 /�1��2x�2z��12 sin2 �1B

TE =��1�2x − �2y�1��2z

��2x�2z − �12��1

Table 3. Existence Conditions of Brewster’s Anglesfor TM Waves at Interface of Isotropic Regular

Medium 1 and Indefinite Medium 2

2z �2x �2y Existence Conditions

� � �2z�2y��1�1

� � �2x /�2y��1 /�1

� � �2x /�2y��1 /�1��2z�2y��1�1

�2x /�2y��1 /�1��2z�2y��1�1

� �

� �

� � �2x /�2y��1 /�1��2z�2y��1�1

�2x /�2y��1 /�1��2z�2y��1�1

� � �2x�2y��1�1

� � �2z�2y��1�1

Table 4. Total Transmission (Brewster’s Angle) forTM Waves at Interface of Isotropic Regular

Medium 1 and Indefinite Medium 2

Conditions Brewster’s Angle for TM Waves

�2x /�2y=�1 /�1��2x�2z=�12 �1B

TM� �0, /2��2x /�2y��1 /�1��2x�2z=�1

2

�2x /�2y=�1 /�1��2x�2z��12 �1B

TM=0

�2x /�2y��1 /�1��2x�2x��12 sin2 �1B

TM =��2x�1 − �1�2y��2z

��2x�2z − �12��1

) regarding the general existence conditions of the Brew-ter’s angle for TM waves at such an interface as men-ioned above. Actually, we find that for the case of TMaves the total transmission characteristic also dependsnly on the three tensor elements, which are �2y, �2x, and2z, while it is independent of the others.

Similarly, for the case of TM waves, there are also threepecial cases that need to be discussed expressly. (i) If2x /�2y=�1 /�1��2x�2z��1

2, the total transmission phe-omenon occurs only in the normal incidence case (i.e.,

1BTM=0). (ii) If �2x /�2y��1 /�1��2x�2z=�1

2, total transmis-ion can never be found regardless of any incident angle.iii) For �2x /�2y=�1 /�1��2x�2z=�1

2, the omnidirectional to-al transmission is allowed.

We also give a summary in Table 4 for the total trans-ission of TM waves (the special cases are included).With the above discussions, considering the total trans-ission of electromagnetic waves incident at the interface

ormed by an isotropic regular material and an indefiniteedium (we discuss only the case in which ��2 and �� 2 are

iagonalizable in the same basis), we find some peculiarssues different from the regular case (both sides of the in-erface are regular materials). (i) The phenomenon of to-al transmission may occur not only for TE waves but alsoor TM waves incident at such an interface. (ii) If the ap-ropriate elements of tensors are chosen, normal incidentaves may be totally transmitted; i.e., �2y /�2x�1 /�1��2x�2z��1

2 leads to zero reflectance of normal in-ident TE waves, and �2x /�2y=�1 /�1��2x�2z��1

2 corre-ponds to total transmission of normal incident TMaves. (iii) The most attractive and significant issue is

hat for both TE and TM waves the phenomenon of omni-irectional total transmission may occur. (iv) The deter-inants of total transmission for the two polarized waves

re generally independent of each other so that we canreely realize zero reflectance of TE and TM waves. There-ore, we can obtain either of the two polarized waves ac-ording to our need in experiments, and we also can makeE and TM waves be totally transmitted simultaneously

f necessary; for example, when �2y /�2x=�2x /�2y�1 /�1��2x�2z=�1

2��2x�2z=�12, both of the two polarized

aves will have omnidirectional total transmission (like alackbody), which may have beneficial applications in im-ge optics.

. THREE SPECIAL CASES OF TOTALRANSMISSION

n this section, we focus on the total transmission forhree special cases. For the first case, the indefinite me-ium degenerates to an isotropic LHM. For the secondase, the symmetric principal axes of the permittivity andermeability tensors are coincident yet normal to the in-erface, as presented in Refs. 8–10. For the third case, theymmetric principal axes of the permittivity and perme-bility tensors are coincident yet parallel to the interface,s occurred in Ref. 4.

. First Casen this subsection, we study the total transmission of anlectromagnetic wave incident on the interface separatingn isotropic regular medium 1 and an isotropic LHM. For

tms=c

a

tc

o

L�c(tin

s

a

o

w�ac

Fttrctional total transmission, respectively.

908 J. Opt. Soc. Am. B/Vol. 23, No. 5 /May 2006 Shen et al.

he isotropic LHM (medium 2), both permittivity and per-eability tensors described by Eqs. (1) degenerate into

calar quantities, i.e., �2x=�2y=�2z=�2�0 and �2x=�2y�2z=�2�0. For this special case, the relevant resultsan be obtained easily on the basis of Section 3.

According to Eq. (19), the relation of the Brewster’sngle for TE waves is given as

sin2 �1BTE =

��1�2 − �2�1��2

��22 − �1

2��1. �26�

Considering the possibility of the existence of totalransmission for TE waves, we simplify the constraintonditions of expressions (21)–(24) to

�2�2 � �1�1 � �2/�2 � �1/�1 �27�

r

�2�2 � �1�1 � �2/�2 � �1/�1. �28�

For the case of the indefinite medium being an isotropicHM, the three special cases are as follows. (i) When

2/�1=�2 /�1�−1, total transmission occurs when the in-ident wave is normal to the interface only (i.e., �1B

TE=0).ii) If �2 /�1��2 /�1=−1, there is no incident angle for totalransmission. (iii) When �2=−�1��2=−�1, the reflectances always zero for arbitrary incoming directions (i.e., om-idirectional total transmission, �1B

TE� �0, /2�).On the other hand, for TM waves, the Brewster’s angle

atisfies

sin2 �1BTM =

��2�1 − �1�2��2

��22 − �1

2��1, �29�

nd the existence conditions are

�2�2 � �1�1 � �2/�2 � �1/�1 �30�

r

�2�2 � �1�1 � �2/�2 � �1/�1. �31�

Specially, if �2 /�1=�2 /�1�−1, there is no reflection justhen the TM wave is of normal incidence; if �2 /�1�2 /�1=−1, the Brewster’s angle never exists; if �2=−�1

nd �2=−�1, omnidirectional total transmission also oc-urs as the TE wave case.

Table 5. Total Transmission (Brewster’s Anand Isotropic L

Conditions Brewster’s Angle

�2 /�1=�2 /�1=−1 �1BTE� �0,

�2 /�1=�2 /�1�−1 �1BTE=

�2 /�1��2 /�1=−1

−1=�2 /�1��2 /�1sin2 �1B

TE =��1�

��

−1��2 /�1��2 /�1�−1 sin2 �1BTE =

��1�

��

gle) at Interface of Isotropic Regular Materialeft-handed Material

for TE Waves Brewster’s Angle for TM Waves

/2� �1BTM� �0, /2�

0 �1BTM=0

sin2 �1BTM =

��2�1 − �1�2��2

��22 − �1

2��1

2 − �2�1��2

22 − �1

2��1

2 − �2�1��2

22 − �1

2��1sin2 �1B

TM =��2�1 − �1�2��2

��22 − �1

2��1

Table 6. Existence Conditions of TotalTransmission for TE Waves at Interface of

Isotropic Regular Material and Indefinite Mediumwith Symmetric Principal Axes of �2 and �2

Coincident Normal to Interface

�2 �2� �2� Existence ConditonsCorrespondingRegions (Fig.)

� � � �2��2 ��1�1 3—A1� � � �2� /�2���1 /�1 3—A2� � � �2� /�2���1 /�1��2��2 ��1�1 3—A3

�2� /�2���1 /�1��2��2 ��1�1 3—B1� � �

� � �

� � � �2� /�2���1 /�1��2��2 ��1�1 3—A4�2� /�2���1 /�1��2��2 ��1�1 3—B2

� � � �2� /�2���1 /�1 3—A5� � � �2��2 ��1�1 3—A6

ig. 2. (Color online) Existence regions of total transmission athe interface between an isotropic regular medium 1 and an iso-ropic left-handed medium 2, for TE and TM waves, on the pa-ameter plane ��2 /�1 ,�2 /�1�; the solid straight line and the solidircle correspond to the zero Brewster’s angle and omnidirec-

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wp�rt

siRi

i

f

f

csdia

shtwF

tmtsfi=sT

Shen et al. Vol. 23, No. 5 /May 2006/J. Opt. Soc. Am. B 909

In conclusion, when a beam passes from one isotropicegular medium into another isotropic left-handed me-ium, there are several new features regarding totalransmission.12 First, total transmission occurs not onlyor TM waves but also for TE waves; then, if �2 /�1�2 /�1�−1, the normal incident waves will totally trans-it for both of the two polarized waves, and the most re-arkable phenomenon is omnidirectional total transmis-

ion at the condition of �2=−�1 and �2=−�1. Table 5 is aummary of this subsection, and Fig. 2 shows the exis-ence regions of total transmission for both TE and TMaves.

ig. 3. (Color online) Total transmission for TE waves at the in-erface between an isotropic regular medium 1 and an indefiniteedium 2 of the first special category. Existence regions of total

ransmission for TE waves (a) for the case �2��2 ��1�1, with theolid straight line corresponding to the zero Brewster’s angle, (b)or the case �2��2 ��1�1, with the solid straight line correspond-ng to the zero Brewster’s angle, and (c) for the case �2��2

�1�1 on the parameter plane ��2� /�1 ,�2� /�1�, with the solidtraight line representing omnidirectional total transmission.he hollow circle needs eliminating.

. Second Cases shown in Fig. 1, medium 1 is still an isotropic regularaterial and occupies the z�0 semi-infinite space, and

he permittivity and permeability tensors of the indefiniteedium 2 have the forms as below:

�2 = ��2� 0 0

0 �2� 0

0 0 �2

�, �2 = ��2� 0 0

0 �2� 0

0 0 �2

� , �32�

here the symbols of � and denote the directions per-endicular and parallel to the symmetric principal axis of

�2 or �� 2. It is evident that, in this case, both the symmet-ic principal axes of ��2 and �� 2 are coincident and normalo the interface.

Such a category was considered to be uniaxial LHMs inome references.8,10 However, we think this understand-ng may be inapposite and have given an explanation inef. 13. Therefore, we consider it a special category of an

ndefinite medium instead of uniaxial LHMs.On the basis of the discussion in Section 3, we can eas-

ly find that the Brewster’s angle satisfies

sin2 �1BTE =

��1�2� − �2��1��2

��2��2 − �12��1

�33�

or TE waves and

sin2 �1BTM =

��2��1 − �1�2���2

��2��2 − �12��1

�34�

or TM waves.In relation to the existence of total transmission, we

an obtain the relevant restrictions by referencing expres-ions (20)–(24). Table 6 and Fig. 3 show the existence con-itions of the Brewster’s angle and the corresponding ex-stence regions for TE waves, respectively, and Table 7nd Fig. 4 are for the case of TM waves.As in the general case in Section 3, there are also three

pecial cases for both TE and TM waves. In Table 8, weave presented the main results of total transmission inhis subsection for both of the two polarized waves, inhich the special cases are included.

Table 7. Existence Conditions of TotalTransmission for TM Waves at Interface of

Isotropic Regular Material and Indefinite Mediumwith Symmetric Principal Axes of �2 and �2

Coincident Normal to Interface

�2 �2� �2� Existence ConditionsCorrespondingRegions (Fig.)

� � � �2�2���1�1 4—A1� � � �2� /�2���1 /�1 4—A2� � � �2� /�2���1 /�1��2�2���1�1 4—A3

�2� /�2���1 /�1��2�2���1�1 4—B1� � �

� � �

� � � �2� /�2���1 /�1��2�2���1�1 4—A4�2� /�2���1 /�1��2�2���1�1 4—B2

� � � �2� /�2���1 /�1 4—A5� � � �2��2���1�1 4—A6

CNaot

ic

rs

s

f

f

tmFawc

ctfstw

5IpamcdgmcdtswcataantbBi

e9os

Ftmtsfi=sT

910 J. Opt. Soc. Am. B/Vol. 23, No. 5 /May 2006 Shen et al.

. Third Caseow we investigate the phenomenon of total transmissiont the interface associated with another special categoryf indefinite medium 2; the permittivity and permeabilityensors of which are given as

�2 = ��2� 0 0

0 �2 0

0 0 �2�

�, �2 = ��2� 0 0

0 �2 0

0 0 �2�

� , �35�

ndicating the symmetric principal axes of ��2 and �� 2 areoincident and parallel to the interface in the y direction.

ig. 4. (Color online) Total transmission for TM waves at the in-erface between an isotropic regular medium 1 and an indefiniteedium 2 of the first special category. Existence regions of total

ransmission for TM wave (a) for the case �2�2���1�1, with theolid straight line corresponding to the zero Brewster’s angle, (b)or the case �2�2���1�1, with the solid straight line correspond-ng to the zero Brewster’s angle, and (c) for the case �2�2�

�1�1 on the parameter plane ��2� /�1 ,�2� /�1�, with the solidtraight line representing omnidirectional total transmission.he hollow circle needs eliminating.

The first structured metamaterial exhibiting negativeefraction experimentally in Ref. 4 in fact corresponds touch a category of indefinite medium in the above.

For total transmission, one can easily obtain the Brew-ter’s angle by referencing the discussion in Section 3:

sin2 �1BTE =

��1�2� − �2�1��2�

��2�2 − �1

2��1�36�

or TE waves and

sin2 �1BTM =

��2��1 − �1�2��2�

��2�2 − �1

2��1�37�

or TM waves.Similarly, the restrictions for the existence of total

ransmission for both of the two different polarized wavesay be deduced from expressions (20)–(24). Table 9 andig. 5 show the existence conditions of the Brewster’sngle and the corresponding existence regions for TEaves, respectively, and Table 10 and Fig. 6 are for the

ase of TM waves.Note that, for both of the two polarized waves, the spe-

ial cases as mentioned in Section 3 also exist. Becausehe determined elements of the tensors are independentor TE and TM waves, we shall summarize the conclu-ions on total transmission in this subsection includinghree special cases in Tables 11 and 12 for TE and TMaves, respectively.

. CONCLUSIONn this paper, we present a detailed investigation on theroblem of total transmission at the interface separatingn isotropic regular medium (medium 1) and an indefiniteedium (medium 2) when an electromagnetic wave is in-

ident on the interface from the side of medium 1. We firsteal with the case of the indefinite medium 2 being in aeneral form in order to present an outline of total trans-ission associated with an indefinite medium. Then we

onsider three special cases: first, the indefinite mediumegenerates to an isotropic LHM, and then in the otherwo cases the medium 2 has a special form. In our discus-ion, both of the two types of polarized electromagneticave, TE and TM waves, are considered. The existence

onditions of total transmission (Brewster’s angle) arelso investigated for both of the two different polariza-ions in detail. Total transmission is possible for both TMnd TE waves, which is significantly different from thatt a regular interface. We have predicted the existences oformally incident total transmission and omnidirectionalotal transmission. Such exotic phenomena should haveeneficial applications in imaging and designing therewster’s window for simplifying the laser arrangement

n the future.

This research is supported by the National Natural Sci-nce Foundation of China under grants 10325417 and0501006, the Key Program of the Ministry of Educationf China under grant 305007, and the National Basic Re-earch Program of China under project 2004CB719801.

Fint�st

Shen et al. Vol. 23, No. 5 /May 2006/J. Opt. Soc. Am. B 911

Table 8. Total Transmission (Brewster’s Angle) at Interface of Isotropic Regular Material and IndefiniteMedium with Symmetric Principal Axes of �2 and �2 Coincident Normal to Interface

Conditons Brewster’s Angle for TE Waves Brewster’s Angle for TM Waves

�1 /�2=�2� /�1=�2� /�1=�1 /�2 �1BTE� �0, /2� �1B

TM� �0, /2��1 /�2=�2� /�1=�2� /�1��1 /�2 �1B

TE=0 �1BTM� �0, /2�

�1 /�2��2� /�1=�2� /�1=�1 /�2 �1BTE� �0, /2� �1B

TM=0�1 /�2��2� /�1=�2� /�1��1 /�2 �1B

TE=0 �1BTM=0

�1 /�2=�2� /�1��2� /�1=�1 /�2

�1 /�2��2� /�1��2� /�1=�1 /�2sin2 �1B

TE =��1�2� − �2��1��2

��2��2 − �12��1

�1 /�2=�2� /�1��2� /�1��1 /�2 sin2 �1BTM =

��2��1 − �1�2���2

��2��2 − �12��1

�1 /�2��2� /�1��2� /�1��1 /�2sin2 �1B

TE =��1�2� − �2��1��2

��2��2 − �12��1

sin2 �1BTM =

��2��1 − �1�2���2

��2��2 − �12��1

Table 9. Existence Conditions of TotalTransmission for TE Waves at Interface of

Isotropic Regular Material and Indefinite Mediumwith Symmetric Principal Axes of ��2 and �� 2

Coincident Parallel to Interface

�2� �2 Existence ConditionsCorrespondingRegions (Fig.)

� � �2 /�2���1 /�1��2�2���1�1 5—1�2 /�2���1 /�1��2�2���1�1 5—2

� �

� �

� � �2 /�2���1 /�1��2�2���1�1 5—3�2 /�2���1 /�1��2�2���1�1 5—4

Ftmt�st

Table 10. Existence Conditions of TotalTransmission for TM Waves at Interface of

Isotropic Regular Material and Indefinite Mediumwith Symmetric Principal Axes of ��2 and �� 2

Coincident Parallel to Interface

�2� �2 Existence ConditionsCorrespondingRegions (Fig.)

� � �2� /�2 ��1 /�1��2��2 ��1�1 6—1�2� /�2 ��1 /�1��2��2 ��1�1 6—2

� �

� �

� � �2� /�2 ��1 /�1��2��2 ��1�1 6—3�2� /�2 ��1 /�1��2��2 ��1�1 6—4

ig. 5. (Color online) Total transmission for the TE waves at thenterface between an isotropic regular medium 1 and an indefi-ite medium 2 of the second special category. Existence regions ofotal transmission for TE waves on the parameter plane�2 /�1 ,�2� /�1�. The solid straight line represents the zero Brew-ter’s angle. The solid circle represents omnidirectional totalransmission, and the hollow circles need eliminating.

ig. 6. (Color online) Total transmission for TM waves at the in-erface between an isotropic regular medium 1 and an indefiniteedium 2 of the second special category. Existence regions of to-

al transmission for TM waves on the parameter plane�2� /�1 ,�2 /�1�. The solid straight line represents the zero Brew-ter’s angle. The solid circle represents omnidirectional totalransmission, and the hollow circles need eliminating.

b

R

1

1

1

1

912 J. Opt. Soc. Am. B/Vol. 23, No. 5 /May 2006 Shen et al.

H.-T. Wang, the corresponding author, can be reachedy e-mail at htwang@nju.edu.cn.

Table 11. Total Transmission (Brewster’s Angle)for TE Waves at Interface of Isotropic Regular

Material and Indefinite Medium withSymmetric Principal Axes of ��2 and �� 2

Coincident Parallel to Interface

Conditions Brewster’s Angle for TE Waves

�2 /�2�=�1 /�1� �2�=�1 �1BTE� �0, /2�

�2 /�2���1 /�1� �2�=�1

�2 /�2�=�1 /�1� �2���1 �1BTE=0

�2 /�2���1 /�1� �2���1sin2 �1B

TE =��1�2� − �2�1��2�

��2�2 − �1

2��1

Table 12. Total Transmission (Brewster’s Angle)for TM Waves at Interface of Isotropic Regular

Material and Indefinite Medium withSymmetric Principal Axes of ��2 and �� 2

Coincident Parallel to Interface

Conditions Brewster’s Angle for TM Waves

�2� /�2=�1 /�1� �2�=�1 �1BTM� �0, /2�

�2� /�2��1 /�1� �2�=�1

�2� /�2=�1 /�1� �2���1 �1BTM=0

�2� /�2��1 /�1� �2���1sin2 �1B

TM =��2��1 − �1�2��2�

��2�2 − �1

2��1

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