Effective negative refractive index in ferromagnet-semiconductor superlattices Roland H. Tarkhanyan IRPhE/NAS, 378410 Ashtarack-2, Armenia [email protected]

Dimitris G. Niarchos IMS/NCSR “Demokritos”, 15310 Athens, Greece [email protected]

Abstract: Problem of anomalous refraction of electromagnetic waves is analyzed in a superlattice which consists of alternating layers of ferromagnetic insulator and nonmagnetic semiconductor. Effective permittivity and permeability tensors are derived in the presence of an external magnetic field parallel to the plane of the layers. It is shown that in the case of the Voigt configuration, the structure behaves as a left-handed medium with respect to TE-type polarized wave, in the low-frequency region of propagation. The relative orientation of the Poynting vector and the refractive wave vector is examined in different frequency ranges. It is shown that the frequency region of existence for the backward mode can be changed using external magnetic field as tuning parameter. © 2006 Optical Society of America OCIS Codes: (260.0260) Physical optics : Physical optics; (160.0160) Materials : Materials ; (999.9999) Left-handed media; Superlattice; Semiconductor; Negative Index material, Backward Waves, Ferromagnet.

___________________________________________________________________________ References and Links 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12.

R.A.Shelby, D.R.Smith, S.Schultz., “Experimental Verification of a Negative Index of Refraction,” Science, 292, 77, 2001. P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002). D.R. Smith, S. Schultz, P. Markos and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002). S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003). V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and μ,” Sov. Phys. Usp. 10, 509 (1968) J.B. Pendry, “Negative refractions makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005). A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004). A.S. Raspopin, A.A.Zharov, H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005). A. Pimenov, A. Loidl, P. Przyslupski, “Negative refraction in Ferromagnet –Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005). R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975). D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).

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13.

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985). 14. R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech. 3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech 4, 1085 (2004). 15. D.J.Bergman, X.Li, Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one- or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005). 16. R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

1. Introduction Recently, there have been many studies of a novel class of artificial materials which exhibit negative refraction of electromagnetic waves (EMW) in certain frequency ranges (see, for example, Refs. [1-4] and citations therein). These materials, termed as “left-handed”, are characterized by a negative permittivity and simultaneously a negative permeability, and lead to a variety of unusual optical phenomena [5]. In addition, in such materials the subwavelength focusing is possible which can be used to fabricate a perfect lens [6]. More recently, the predictions of Veselago and Pendry stimulated strong theoretical and experimental efforts to study negative refraction in various systems such as two-dimensional photonic-gap crystals [7], multilayer structures involving anisotropic materials [8], semiconductor superlattices in the presence of Bloch oscillations [9]. Experimental success has been demonstrated in ferromagnet-superconductor superlattices for millimeter waves [10]. In this paper, the problem of anomalous refraction is studied in a periodic ferromagnetsemiconductor multilayer composite, and the differences between the phase and group refracted indices are discussed. We show that in certain frequency range, left-handed behavior in such a structure is possible. Analytical expressions for both phase and group refractive indices are obtained. It is shown that the group index can be both positive and negative depending on the wave frequency and material parameters while the phase index is always positive.In section 2, we describe the physical picture of the problem and find the effective permittivity and permeability tensors in the presence of an external magnetic field. In section 3, the regions of propagation and dispersion curves of TE-wave are studied and the conditions of existence for the backward mode are obtained. In section 4, the pecularities of the phase and group refractive indices are examined and the problem of reflection at negative refraction is considered.Section 5 contains a summary of the principal results obtained in this paper. 2. Effective permittivity and permeability tensors Consider a superlaffice (SL) or a periodic multilayer composite which consists of alternating layers of a ferromagnetic insulator (for example, ZnMnO) and nonmagnetic semiconductor (ZnO), in the presence of an external magnetic field B 0 parallel to the plane of the layers. Let’s choose a Cartesian coordinate system with z-axis along B 0 and y-axis along the optical axis of SL, so as the layers are parallel to the xz-plane; the space region y0; the space y ω s 3 , if the effective plasma frequency

if

ω p1 < ω s 3 , and ω r < ω < ω s3 ,

ω > ω p1 ,

(11a)

(11b)

ω p1 > ω s 3 . Here ω r = (ω 02 + ω 0ω s + ω s1ω s 2 sin 2 β )1 / 2

is the resonance frequency at which the wave number k → ∞ , wave vector k and the +y-axis;

ω = ω p1

and

ω = ω s3

β

(12)

is the angle between the

are the cutoff frequencies at which

k =0. The corresponding dispersion curves are shown schematically in Fig. 2, for three different values of the effective plasma frequency: ω p1 < ω r [Fig. 2(a)], ω r < ω p1 < ω s 3 [Fig. (2b)] and

ω p1 > ω s 3

[Fig. (2c)].

ω ω

ω

p1

S3

ω

p1

ω ω

r p1

a

b

c

ck

ε∞

Fig. 2. Dispersion curves of TE-wave for the cases:

a.

ω p1 < ω r ,

b.

ω r < ω p1 < ω s 3 , c. ω p1 > ω s 3 .

One can see that the dispersion of the high-frequency branch is always normal: dω / dk >0 and the angle ϕ between the phase and group velocities is acute while the

δ = ω p1 / ω r . For δ < 1, the δ > 1, it is anomal: dω / dk π / 2 . To be convinced of such a behavior of the wave, let us examine the sign of the

scalar product kS, where S is the time averaged Poynting vector. For TE-wave under consideration, vector S has the following form:

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2

E μ S={ 1 k x , k y ,0 } μ2 2ωμ 0 μ v

.

(13)

Using Eqs. (10) and (13), we obtain kS = ( μ k + μ 2 k ) 2 1 x

2 y

E

2

1 = ωε 0 ε 3 | E | 2 , 2ωμ 0 μ 2 μν 2

according to which the wave is only forward (kS>0,

ϕ < π / 2 ) for ε 3 > 0,

(14)

that is, in the

ω > ω p1 . Thus, in the high-frequency regions of the wave propagation in

frequency region

Figs. 2(a,b,c), the SL behaves as a regular right-handed medium. On the other side, the power flow in the wave is in the backward direction with respect to the phase velocity, that is, kS π / 2 , only if the following conditions are fulfilled:

ε 3 < 0 , μ v < 0 , 1 + μ1 μ 2−1 tan 2 β > 0 .

(15)

After a simple analysis we conclude that conditions (15) can only be fulfilled simultaneously in the frequency range

ω r < ω < min{ω s 3 , ω p1 } ,

(16)

that is, for the low-frequency branchs in Figs. 2(b,c). Note that conditions (15) are quite different from those in an isotropic case when scalar ε and μ should be negative simultaneously [5]. Indeed, in (15) only one of the components of effective permittivity tensor is to be negative, namely, the diagonal component ε 3 along the direction of the external magnetic field. Also, instead of the scalar μ , the Voigt permeability

μv

must be negative. Moreover, an additional condition restricting the direction

of the backward mode propagation appears. For instance, if condition (15) gives tan

2

μ1 0,

the third

β < μ 2 / | μ1 | .

4. Negative refraction and problem of reflection. Phase and group refractive indices Consider now a TE-type polarized EMW which is incident from the vacuum on the plane surface of the medium, at the angle of incidence α . In this case, the refracted wave is TE polarized too and is described by Eq.(10). According to the causality principle, the energy flux of the wave is always directed away from the interface: S y > 0 . Then, using Eq.(13) we conclude that in the region (16), the normal component of the wave vector is negative, that is, the wave vector of the refracted wave is directed to the interface. In other words, with respect to the low-frequency branch, the medium behaves as a left – handed (LH) one. Using Eq. (14), for the angle ϕ between the vectors k and S we obtain

cos ϕ =

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ε 3μ 2 μv n p n g μ1

,

(17)

Received 23 February 2006; revised 14 April 2006; accepted 29 April 2006

12 June 2006 / Vol. 14, No. 12 / OPTICS EXPRESS 5439

where

np =

ck

ω

and

ng = are the phase and group refractive indices,

β

=

sin α sin β

(18)

sin α sin γ

(19)

is the angle of refraction and

γ

is the angle

between the group velocity of the refracted wave V g = ∂ω / ∂ k (or the Poynting vector S) and the +y-axis. Both n p and n g depend on the angle of incidence and are given by

n p = [ε 3 μ v + (1 − μ1 / μ 2 ) sin 2 α ]1 / 2 ,

(20a)

n g = [ε 3 μ v ( μ 2 / μ1 ) 2 + (1 − μ 2 / μ1 ) sin 2 α ]1 / 2 Sgn( μ1 μ v ) . (20b) Note that in Eq.(20a) n p is positive while after Veselago[5] it is accepted to use negative sign for it. However, a careful examination of the wave refraction problem shows (for the details see Ref.[16]) that the phase refractive index determined as the ratio of the light speed in vacuum to the modulus of the phase velocity ( ck / ω ) is always positive while the group refractive index can be both positive and negative. The confusion in the sign of n p appears because of in an isotropic LHM, one has n p =| n g | . For the backward wave under consideration, n g changes its sign at

ω = ω σ , where ω σ

is given in Eq.(5c) and corresponds to zero of μ1 (ω ) . Thus, the frequency region of existence for the backward mode (16) must be divided in two parts: I.

ω r < ω < ωσ

(21a)

II.

ω σ < ω < min{ω p1 , ω s 3 } ,

(21b)

and

where the symbols

ω σ , ω r , ω p1

and

ω s3

are key model parameters the meaning of which

is explained above [see also Eqs. (4c), (9a) and (12)]. In part I, μ1 < 0, S x > 0 and consequently, n g is positive; in other words, SL behaves as a positive index LH-medium [see Fig. 3(a)]. Note that in this case, the refraction occurs at an arbitrary value of the angle of incidence.

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y

y

k

S

S

k

φ

φ x

x

α

k0

α

kR

k0 kR

(a)

(b)

Fig. 3. Refraction of TE-wave in the case of positive index (a) and negative index (b) LHM. k 0 , k R and k are wave vectors for incident, reflected and refractive waves, respectively. (a)

μ1 < 0 , S x > 0 , n g > 0 .

In part ΙΙ, we have

(b)

μ1 > 0 , S x < 0 , n g < 0 . In both cases ε 3 , μ v < 0 .

μ1 > 0, S x < 0

and n g < 0 , that is, SL behaves as negative index

LH (or absolute left-handed) medium. The corresponding picture of refraction is shown in Fig. 3(b). In this case, the refraction is only possible for α < α c , where the critical angle of incidence is given by

sin α c = ε 3 μ 2 μ v / μ1 .

(22)

We have to note again that for both cases shown in Figs. 3(a) and 3(b), the medium is left-handed: the refracted wave vector is directed to the interface, and the angle between the vectors k and S is obtuse: π / 2 < ϕ < π . These vectors can only be exactly opposite in part II, and only for

α = α c . However, in this case k y = 0 , the wave is propagated along the

interface in the direction parallel to the +x-axis. The main difference between Figs. 3(a) and 3(b) is the different orientation of the Poynting vector with respect to the normal to the interface: for the case n g < 0, it is in the same side of the normal as in the incident wave while for the case n g > 0 it is in the opposite side, as in a regular medium. It is important to note the possibility of tuneability of both regions of anomalous refraction (21a) and (21b) by means of external magnetic field variation. Using Eqs. (5c), (9a) and (12) for the characteristic frequencies ω σ , ω s 3 and ω r , it is easy to see that with increasing magnetic field the low frequency region (21a) shifts toward the high frequency side without any change of its width. The tuneability of the negative index region (21b) is more

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interesting and promising from experimental point of view: for a given value of

ω p1 > ω s 3 ,

the lower limit of the region increases monotonically with increasing magnetic field while the interval of the region increases initially, reaches a maximum value at ω s 3 = ω p1 , then decreases rapidly and vanishes at

ω σ = ω p1 .

Finally, let us consider briefly the problem of reflection from the surface of LHM. Using the standard continuity conditions for the tangential field components at the interface, we obtain the reflection coefficient for our TE-polarization case in the form

R=

( μ v cos α + n p2 − sin 2 α ) 2 + ( μ a / μ 2 ) 2 sin 2 α ( μ v cos α − n 2p − sin 2 α ) 2 + ( μ a / μ 2 ) 2 sin 2 α

,

(23)

where n p is given by Eq.(18a). It is easily to see that the function R (α ) has a minimum value Rm at

α = α m , where Rm = tan 2 α m =

BC − A BC + A

,

μ 2 (C − ε 3 μ v B ) AB

(23a)

,

A = ε 3 μ 2 μ v − μ1 , B = 1 − ε 3 μ 2 , C = − Aμ v + ε 3 ( μ1 − μ v ) .

(23b)

(23c)

For a given value of ω in the range (16), all the parameters A, B, C are positive. The minimum exists only if the condition

2 μ a2 μv > 1+ ε3 Aμ 2

(24)

is fulfilled. Normal component of the refracted wave vector at α = α m is given by

k y = − k 0 CB −1 cos α m .

(25)

Note that in different frequency ranges (21a,b), angle dependence of the function R (α ) is different [see Figs. 4(a,b)].

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R 1 R0

R0

0 a. 0

αm

b.

π/2

(a)

0

αm

αc

π/2

α

(b)

Fig. 4. Angle dependence of the reflection coefficient from the surface of LHM with positive (a) and negative (b) group refractive index.

In these figures, the coefficient of reflection at the normal incidence is given by

R0

and the angle

αc

⎛ =⎜ ⎜ ⎝

| ε3 | − | μv | ⎞

2

⎟ ,

| ε 3 | + | μ v | ⎟⎠

(26)

at which the total reflection occurs, is given by Eq.(22).

5. Conclusions The obtained results may be summarized as follows. The dispersion, refraction and reflection of electromagnetic waves are studied analytically in ferromagnet-semiconductor multilayer in the presence of an external magnetic field. It is shown in terms of effective medium approach and Voigt configuration that such a multilayer or a superlattice can behave as a left-handed medium for a TE-polarized wave. General necessary conditions are found for anomalous refraction [see Eq.(15)] which are quite different from those in an isotropic case considered in [5]. It is shown that a backward mode exists only if the effective plasma frequency, at which the wave has a cutoff, exeeds the resonance frequency given in Eq.(12). Unlike the isotropic case, only one of the diagonal components of the effective permittivity tensor, namely, the component along the external magnetic field direction should be negative. Also, instead of the scalar μ (ω ) in an isotropic case, the Voigt permeability μ v (ω ) given in Eq.(9) must be negative. Finally, an additional condition restricting the direction of the refracted wave propagation must be fulfilled. A frequency range is found in which all the necessary conditions are fulfilled simultaneously. It is shown that the range can be devided in two parts, in one of which (low frequency part) the group refractive index is positive while in the second one it is negative. These parts are separated by the characteristic frequency ω σ given in Eq.(5c). The main difference between these parts is the different orientation of the Poynting vector of the refracted wave with respect to the normal to the interface [see Figs. 3(a) and 3(b)]. In both

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these regions, the medium is left-handed: the refracted wave vector is directed to the interface and makes an obtuse angle ϕ > π / 2 with the Poynting vector. It is shown that with increasing magnetic field, the positive index region shifts toward the high-frequency side without any change of its width while the negative index region becomes wider initially, reaches a maximum value and then decreases monotonically. Application of this result is promising from experimental point of view, taking into account the possibility of tuneability of the regions by means of external magnetic field variation.

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Dimitris G. Niarchos IMS/NCSR “Demokritos”, 15310 Athens, Greece [email protected]

Abstract: Problem of anomalous refraction of electromagnetic waves is analyzed in a superlattice which consists of alternating layers of ferromagnetic insulator and nonmagnetic semiconductor. Effective permittivity and permeability tensors are derived in the presence of an external magnetic field parallel to the plane of the layers. It is shown that in the case of the Voigt configuration, the structure behaves as a left-handed medium with respect to TE-type polarized wave, in the low-frequency region of propagation. The relative orientation of the Poynting vector and the refractive wave vector is examined in different frequency ranges. It is shown that the frequency region of existence for the backward mode can be changed using external magnetic field as tuning parameter. © 2006 Optical Society of America OCIS Codes: (260.0260) Physical optics : Physical optics; (160.0160) Materials : Materials ; (999.9999) Left-handed media; Superlattice; Semiconductor; Negative Index material, Backward Waves, Ferromagnet.

___________________________________________________________________________ References and Links 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12.

R.A.Shelby, D.R.Smith, S.Schultz., “Experimental Verification of a Negative Index of Refraction,” Science, 292, 77, 2001. P. Markos and C.M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401 (2002). D.R. Smith, S. Schultz, P. Markos and C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterilas from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002). S.G. Parazzoli, R.B. Gregor, K. Li, B.E. Koltenbah, M. Tanielian, “Experimental verification and simulation of Negative Index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003). V.G. Veselago, “The electrodynamics of substances with simultaneously negative ε and μ,” Sov. Phys. Usp. 10, 509 (1968) J.B. Pendry, “Negative refractions makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in 2D photonic crystals:A study of anomalous refractive effects,” Phys. Rev. B72, 165112 (2005). A. Mandatory, C. Sibilia, M. Bertolotti, S. Zhukovsky, J.W. Haus, M. Scalora, “Anomalous phase on onedimensional, multilayer, structures with birefringent materials,” Phys. Rev. B70, 165107 (2004). A.S. Raspopin, A.A.Zharov, H.L. Cui, “Spectrum of electromagnetic excitations in a dc-biased semiconductor superlattice,” J. Appl. Phys. 98, 103517 (2005). A. Pimenov, A. Loidl, P. Przyslupski, “Negative refraction in Ferromagnet –Fuperconductor Superlattices,” Phys. Rev. Lett. 95, 247009 (2005). R.H. Tarkhanyan, “On the Theory of Surface Waves in a Uniaxial Semiconductor Slab,” Phys. Status Sol.(b) 72, 111 (1975). D. Polder, “Theory of electromagnetic Resonance,” Philos. Mag. 40, N 300, 99 (1949).

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(C) 2006 OSA

Received 23 February 2006; revised 14 April 2006; accepted 29 April 2006

12 June 2006 / Vol. 14, No. 12 / OPTICS EXPRESS 5433

13.

V.M. Agranovich and V.E. Kravtsov, “Notes on crystal optics of superlattices,” Sol. St. Commun. 55, 85 (1985). 14. R.H. Tarkhanyan and A.G. Nassiopoulou, “Electromagnetic instability of surface waves in semiconductor superlattices,” J. Nanosci. Nanotech. 3, 549 (2003), and “Influence of magnetic field on electromagnetic instabilities in semiconductor superlattices,” J. Nanosci. Nanotech 4, 1085 (2004). 15. D.J.Bergman, X.Li, Y.M. Strelniker, “Macroscopic conductivity tensor of a three-dimensional composite with a one- or two-dimensional microstructure,”Phys. Rev. B71, 035120 (2005). 16. R.H. Tarkhanyan and D.G. Niarchos, “Wave refraction and backward magnon-polaritons in left-handed antiferomagnetic/semiconductor superlattices,” JMMM (to be published).

1. Introduction Recently, there have been many studies of a novel class of artificial materials which exhibit negative refraction of electromagnetic waves (EMW) in certain frequency ranges (see, for example, Refs. [1-4] and citations therein). These materials, termed as “left-handed”, are characterized by a negative permittivity and simultaneously a negative permeability, and lead to a variety of unusual optical phenomena [5]. In addition, in such materials the subwavelength focusing is possible which can be used to fabricate a perfect lens [6]. More recently, the predictions of Veselago and Pendry stimulated strong theoretical and experimental efforts to study negative refraction in various systems such as two-dimensional photonic-gap crystals [7], multilayer structures involving anisotropic materials [8], semiconductor superlattices in the presence of Bloch oscillations [9]. Experimental success has been demonstrated in ferromagnet-superconductor superlattices for millimeter waves [10]. In this paper, the problem of anomalous refraction is studied in a periodic ferromagnetsemiconductor multilayer composite, and the differences between the phase and group refracted indices are discussed. We show that in certain frequency range, left-handed behavior in such a structure is possible. Analytical expressions for both phase and group refractive indices are obtained. It is shown that the group index can be both positive and negative depending on the wave frequency and material parameters while the phase index is always positive.In section 2, we describe the physical picture of the problem and find the effective permittivity and permeability tensors in the presence of an external magnetic field. In section 3, the regions of propagation and dispersion curves of TE-wave are studied and the conditions of existence for the backward mode are obtained. In section 4, the pecularities of the phase and group refractive indices are examined and the problem of reflection at negative refraction is considered.Section 5 contains a summary of the principal results obtained in this paper. 2. Effective permittivity and permeability tensors Consider a superlaffice (SL) or a periodic multilayer composite which consists of alternating layers of a ferromagnetic insulator (for example, ZnMnO) and nonmagnetic semiconductor (ZnO), in the presence of an external magnetic field B 0 parallel to the plane of the layers. Let’s choose a Cartesian coordinate system with z-axis along B 0 and y-axis along the optical axis of SL, so as the layers are parallel to the xz-plane; the space region y0; the space y ω s 3 , if the effective plasma frequency

if

ω p1 < ω s 3 , and ω r < ω < ω s3 ,

ω > ω p1 ,

(11a)

(11b)

ω p1 > ω s 3 . Here ω r = (ω 02 + ω 0ω s + ω s1ω s 2 sin 2 β )1 / 2

is the resonance frequency at which the wave number k → ∞ , wave vector k and the +y-axis;

ω = ω p1

and

ω = ω s3

β

(12)

is the angle between the

are the cutoff frequencies at which

k =0. The corresponding dispersion curves are shown schematically in Fig. 2, for three different values of the effective plasma frequency: ω p1 < ω r [Fig. 2(a)], ω r < ω p1 < ω s 3 [Fig. (2b)] and

ω p1 > ω s 3

[Fig. (2c)].

ω ω

ω

p1

S3

ω

p1

ω ω

r p1

a

b

c

ck

ε∞

Fig. 2. Dispersion curves of TE-wave for the cases:

a.

ω p1 < ω r ,

b.

ω r < ω p1 < ω s 3 , c. ω p1 > ω s 3 .

One can see that the dispersion of the high-frequency branch is always normal: dω / dk >0 and the angle ϕ between the phase and group velocities is acute while the

δ = ω p1 / ω r . For δ < 1, the δ > 1, it is anomal: dω / dk π / 2 . To be convinced of such a behavior of the wave, let us examine the sign of the

scalar product kS, where S is the time averaged Poynting vector. For TE-wave under consideration, vector S has the following form:

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2

E μ S={ 1 k x , k y ,0 } μ2 2ωμ 0 μ v

.

(13)

Using Eqs. (10) and (13), we obtain kS = ( μ k + μ 2 k ) 2 1 x

2 y

E

2

1 = ωε 0 ε 3 | E | 2 , 2ωμ 0 μ 2 μν 2

according to which the wave is only forward (kS>0,

ϕ < π / 2 ) for ε 3 > 0,

(14)

that is, in the

ω > ω p1 . Thus, in the high-frequency regions of the wave propagation in

frequency region

Figs. 2(a,b,c), the SL behaves as a regular right-handed medium. On the other side, the power flow in the wave is in the backward direction with respect to the phase velocity, that is, kS π / 2 , only if the following conditions are fulfilled:

ε 3 < 0 , μ v < 0 , 1 + μ1 μ 2−1 tan 2 β > 0 .

(15)

After a simple analysis we conclude that conditions (15) can only be fulfilled simultaneously in the frequency range

ω r < ω < min{ω s 3 , ω p1 } ,

(16)

that is, for the low-frequency branchs in Figs. 2(b,c). Note that conditions (15) are quite different from those in an isotropic case when scalar ε and μ should be negative simultaneously [5]. Indeed, in (15) only one of the components of effective permittivity tensor is to be negative, namely, the diagonal component ε 3 along the direction of the external magnetic field. Also, instead of the scalar μ , the Voigt permeability

μv

must be negative. Moreover, an additional condition restricting the direction

of the backward mode propagation appears. For instance, if condition (15) gives tan

2

μ1 0,

the third

β < μ 2 / | μ1 | .

4. Negative refraction and problem of reflection. Phase and group refractive indices Consider now a TE-type polarized EMW which is incident from the vacuum on the plane surface of the medium, at the angle of incidence α . In this case, the refracted wave is TE polarized too and is described by Eq.(10). According to the causality principle, the energy flux of the wave is always directed away from the interface: S y > 0 . Then, using Eq.(13) we conclude that in the region (16), the normal component of the wave vector is negative, that is, the wave vector of the refracted wave is directed to the interface. In other words, with respect to the low-frequency branch, the medium behaves as a left – handed (LH) one. Using Eq. (14), for the angle ϕ between the vectors k and S we obtain

cos ϕ =

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ε 3μ 2 μv n p n g μ1

,

(17)

Received 23 February 2006; revised 14 April 2006; accepted 29 April 2006

12 June 2006 / Vol. 14, No. 12 / OPTICS EXPRESS 5439

where

np =

ck

ω

and

ng = are the phase and group refractive indices,

β

=

sin α sin β

(18)

sin α sin γ

(19)

is the angle of refraction and

γ

is the angle

between the group velocity of the refracted wave V g = ∂ω / ∂ k (or the Poynting vector S) and the +y-axis. Both n p and n g depend on the angle of incidence and are given by

n p = [ε 3 μ v + (1 − μ1 / μ 2 ) sin 2 α ]1 / 2 ,

(20a)

n g = [ε 3 μ v ( μ 2 / μ1 ) 2 + (1 − μ 2 / μ1 ) sin 2 α ]1 / 2 Sgn( μ1 μ v ) . (20b) Note that in Eq.(20a) n p is positive while after Veselago[5] it is accepted to use negative sign for it. However, a careful examination of the wave refraction problem shows (for the details see Ref.[16]) that the phase refractive index determined as the ratio of the light speed in vacuum to the modulus of the phase velocity ( ck / ω ) is always positive while the group refractive index can be both positive and negative. The confusion in the sign of n p appears because of in an isotropic LHM, one has n p =| n g | . For the backward wave under consideration, n g changes its sign at

ω = ω σ , where ω σ

is given in Eq.(5c) and corresponds to zero of μ1 (ω ) . Thus, the frequency region of existence for the backward mode (16) must be divided in two parts: I.

ω r < ω < ωσ

(21a)

II.

ω σ < ω < min{ω p1 , ω s 3 } ,

(21b)

and

where the symbols

ω σ , ω r , ω p1

and

ω s3

are key model parameters the meaning of which

is explained above [see also Eqs. (4c), (9a) and (12)]. In part I, μ1 < 0, S x > 0 and consequently, n g is positive; in other words, SL behaves as a positive index LH-medium [see Fig. 3(a)]. Note that in this case, the refraction occurs at an arbitrary value of the angle of incidence.

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y

y

k

S

S

k

φ

φ x

x

α

k0

α

kR

k0 kR

(a)

(b)

Fig. 3. Refraction of TE-wave in the case of positive index (a) and negative index (b) LHM. k 0 , k R and k are wave vectors for incident, reflected and refractive waves, respectively. (a)

μ1 < 0 , S x > 0 , n g > 0 .

In part ΙΙ, we have

(b)

μ1 > 0 , S x < 0 , n g < 0 . In both cases ε 3 , μ v < 0 .

μ1 > 0, S x < 0

and n g < 0 , that is, SL behaves as negative index

LH (or absolute left-handed) medium. The corresponding picture of refraction is shown in Fig. 3(b). In this case, the refraction is only possible for α < α c , where the critical angle of incidence is given by

sin α c = ε 3 μ 2 μ v / μ1 .

(22)

We have to note again that for both cases shown in Figs. 3(a) and 3(b), the medium is left-handed: the refracted wave vector is directed to the interface, and the angle between the vectors k and S is obtuse: π / 2 < ϕ < π . These vectors can only be exactly opposite in part II, and only for

α = α c . However, in this case k y = 0 , the wave is propagated along the

interface in the direction parallel to the +x-axis. The main difference between Figs. 3(a) and 3(b) is the different orientation of the Poynting vector with respect to the normal to the interface: for the case n g < 0, it is in the same side of the normal as in the incident wave while for the case n g > 0 it is in the opposite side, as in a regular medium. It is important to note the possibility of tuneability of both regions of anomalous refraction (21a) and (21b) by means of external magnetic field variation. Using Eqs. (5c), (9a) and (12) for the characteristic frequencies ω σ , ω s 3 and ω r , it is easy to see that with increasing magnetic field the low frequency region (21a) shifts toward the high frequency side without any change of its width. The tuneability of the negative index region (21b) is more

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interesting and promising from experimental point of view: for a given value of

ω p1 > ω s 3 ,

the lower limit of the region increases monotonically with increasing magnetic field while the interval of the region increases initially, reaches a maximum value at ω s 3 = ω p1 , then decreases rapidly and vanishes at

ω σ = ω p1 .

Finally, let us consider briefly the problem of reflection from the surface of LHM. Using the standard continuity conditions for the tangential field components at the interface, we obtain the reflection coefficient for our TE-polarization case in the form

R=

( μ v cos α + n p2 − sin 2 α ) 2 + ( μ a / μ 2 ) 2 sin 2 α ( μ v cos α − n 2p − sin 2 α ) 2 + ( μ a / μ 2 ) 2 sin 2 α

,

(23)

where n p is given by Eq.(18a). It is easily to see that the function R (α ) has a minimum value Rm at

α = α m , where Rm = tan 2 α m =

BC − A BC + A

,

μ 2 (C − ε 3 μ v B ) AB

(23a)

,

A = ε 3 μ 2 μ v − μ1 , B = 1 − ε 3 μ 2 , C = − Aμ v + ε 3 ( μ1 − μ v ) .

(23b)

(23c)

For a given value of ω in the range (16), all the parameters A, B, C are positive. The minimum exists only if the condition

2 μ a2 μv > 1+ ε3 Aμ 2

(24)

is fulfilled. Normal component of the refracted wave vector at α = α m is given by

k y = − k 0 CB −1 cos α m .

(25)

Note that in different frequency ranges (21a,b), angle dependence of the function R (α ) is different [see Figs. 4(a,b)].

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R 1 R0

R0

0 a. 0

αm

b.

π/2

(a)

0

αm

αc

π/2

α

(b)

Fig. 4. Angle dependence of the reflection coefficient from the surface of LHM with positive (a) and negative (b) group refractive index.

In these figures, the coefficient of reflection at the normal incidence is given by

R0

and the angle

αc

⎛ =⎜ ⎜ ⎝

| ε3 | − | μv | ⎞

2

⎟ ,

| ε 3 | + | μ v | ⎟⎠

(26)

at which the total reflection occurs, is given by Eq.(22).

5. Conclusions The obtained results may be summarized as follows. The dispersion, refraction and reflection of electromagnetic waves are studied analytically in ferromagnet-semiconductor multilayer in the presence of an external magnetic field. It is shown in terms of effective medium approach and Voigt configuration that such a multilayer or a superlattice can behave as a left-handed medium for a TE-polarized wave. General necessary conditions are found for anomalous refraction [see Eq.(15)] which are quite different from those in an isotropic case considered in [5]. It is shown that a backward mode exists only if the effective plasma frequency, at which the wave has a cutoff, exeeds the resonance frequency given in Eq.(12). Unlike the isotropic case, only one of the diagonal components of the effective permittivity tensor, namely, the component along the external magnetic field direction should be negative. Also, instead of the scalar μ (ω ) in an isotropic case, the Voigt permeability μ v (ω ) given in Eq.(9) must be negative. Finally, an additional condition restricting the direction of the refracted wave propagation must be fulfilled. A frequency range is found in which all the necessary conditions are fulfilled simultaneously. It is shown that the range can be devided in two parts, in one of which (low frequency part) the group refractive index is positive while in the second one it is negative. These parts are separated by the characteristic frequency ω σ given in Eq.(5c). The main difference between these parts is the different orientation of the Poynting vector of the refracted wave with respect to the normal to the interface [see Figs. 3(a) and 3(b)]. In both

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12 June 2006 / Vol. 14, No. 12 / OPTICS EXPRESS 5443

these regions, the medium is left-handed: the refracted wave vector is directed to the interface and makes an obtuse angle ϕ > π / 2 with the Poynting vector. It is shown that with increasing magnetic field, the positive index region shifts toward the high-frequency side without any change of its width while the negative index region becomes wider initially, reaches a maximum value and then decreases monotonically. Application of this result is promising from experimental point of view, taking into account the possibility of tuneability of the regions by means of external magnetic field variation.

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Received 23 February 2006; revised 14 April 2006; accepted 29 April 2006

12 June 2006 / Vol. 14, No. 12 / OPTICS EXPRESS 5444