Russian Physics Journal, Vol. 57, No. 5, September, 2014 (Russian Original No. 5, May, 2014)
ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY REFRACTION INTO AN ISOTROPIC MEDIUM WITH NEGATIVE PERMITTIVITY AND PERMEABILITY V. V. Fisanov
UDC 537.874.33
Generalized representations of the refraction law are obtained for electromagnetic waves at the interface of transparent isotropic media with positive and/or negative permittivity and permeability. Keywords: magnetodielectric, Snell’s law, negative refraction, refractive index, refraction vector.
The phenomenon of reversed (negative) refraction has been well known among physicists for a long time. Sir Horace Lamb [1], responding to the model proposed by Lord Rayleigh (John Strutt) of a device in the form of a clamped wire subjected to tension and torsion [2], conjectured that in a cylindrical wire under certain conditions vibrations can also arise having group velocity that is negative with respect to the phase velocity. He considered the passage of a one-dimensional wave from a segment of the wire with the usual vibrational regime to a segment with the anomalous regime and was the first to observe that the transmitted wave should have a phase velocity oppositely directed relative to the phase velocity of the wave incident on the inhomogeneity. That same year, A. Schuster noticed and graphically elucidated [3, p. 317, Fig. 179] the curious result of the negative angle of the refracted light wave in a medium with opposed phase and group velocities, which was necessary to ensure unidirectional translation of the intersection lines of the wave fronts of the incident and refracted wave with the boundary of the two media. In optics and electrodynamics the simplest isotropic medium, at the boundary with which negative refraction can take place, is a magnetodielectric with negative values of its electric permittivity ( 0 ) and magnetic permeability ( 0 ), which was first pointed out by D. V. Sivukhin [4] and V. G. Veselago [5–7]. Negative refraction is most graphically manifested, for example, for oblique incidence of a plane wave from vacuum ( 0 and 0 ) into anti-vacuum (this term was proposed in [8], 0 and 0 ), where 0 and 0 are the electric and magnetic constants. In this
case, reflection of a wave from the interface does not take place, but the initial direction of propagation as the wave propagates into the second medium is changed to the direction of the wave vector of the absent reflected wave. Here the refracted wave, being a backward wave, transfers energy in accordance with the radiation condition from the interface into the periphery of the second medium. References [9–11] are dedicated to historical and other aspects of the electrodynamics of media with backward waves and their discussion in the literature. A modification of Snell’s refraction law was proposed in [5–7] relevant to the interface with a hypothetical lefthanded medium with both and negative, presaging the introduction of the new concept of a negative refractive index. After the first experimental confirmation of the phenomenon of negative refraction for microwaves [12] and the appearance of the concept of electromagnetic metamaterials, the English-language version [7] of Veselago’s paper [6] acquired some fame and became widely cited and much in demand. Despite some doubts and objections [13–16], the majority of researchers got on board with the term negative refractive index.
V. D. Kuznetsov Siberian Physical-Technical Institute at Tomsk State University, Tomsk, Russia; Institute of Physical Material Science of the Siberian Branch of the Russian Academy of Sciences, Ulan-Ude, Russia, e-mail:
[email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 111–116, May, 2014. Original article submitted January 9, 2014. 1064-8887/14/5705-0691 2014 Springer Science+Business Media New York
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Unfortunately, there are some unclarities in the works of Veselago which hinder an understanding of negative refraction and can lead to invalid representations as in [17]. Shen et al. [15] remarked that in the explanatory figure (Figure 3) in [7] the direction of the wave vector of the refracted wave in the left-handed medium is incorrectly indicated. This same figure was also reproduced in [5, 6, 18–20]. In the figure in [21] corresponding to Figure 3, the directions of both the wave vector and the Umov–Poynting vector are indicated, but it is not indicated for which of them Snell’s law formula applies. In addition to this, sometimes [6, 7, 20] the negative refraction angle has been denoted in figures as , but in other publications the minus sign has been left out without clarification. The present work sets itself as its goal to obtain a representation of the refraction law in the form of a generalization of Snell’s formula suitable for all possible variants of a combination of transparent isotropic media with both and positive and/or both negative. The problem is addressed in terms of plane waves and boundary conditions represented in vector form using the concept of vector refraction introduced by F. I. Fëdorov [22], and the covariant method based on it for the phenomenological theory of wave propagation in homogeneous media in which no reference coordinate system is introduced [23]. We consider two isotropic media, separated by a planar surface S . In each of the two media homogeneous harmonic plane waves with angular frequency propagate, having scalar components of the form f t A exp i A exp i k t ,
(1)
where A is the complex amplitude. The phase of the wave t k t depends on the spatial variable and time t . Let the unit vector kˆ be normal to the wave front. Then the distance in this direction, reckoned from some starting point, is equal to = kˆ r , where r is the radius vector, and the function f t has a constant value at any fixed moment in time at points in the plane kˆ r = const . The wave vector k = kkˆ , pointing in the direction of increasing distance , has length k k 0 , i.e., the wave number k in formula (1) is positive by definition. To describe a counterpropagating wave with wave vector k we must replace by = kˆ r in formula (1).
Substitution of wave vectors (1) into the Maxwell equations with allowance for the material equations D = 0E and B = 0H reduces them to the form
k E H , k H E , k E 0 , k H 0 .
(2)
System of equations (2) for the complex amplitudes has a solution in the form of harmonic vector plane waves under the condition that the dispersion relation k 2 2 0 0 k02
(3)
is fulfilled, where k0 00 is the wave number for vacuum. We introduce the refraction vector m , based on the relation k = k0 m = k0 m mˆ , where mˆ = kˆ . The length of the refraction vector m m n is the refractive index n (relative to vacuum) [23]. Dispersion relation (3) takes the form m 2 n 2 .
(4)
The permittivity and permeability, and , are real quantities if the medium is transparent (i.e., if conductivity is absent in the medium). In this case, we must take the positive value of the square root n
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(5)
as the value of the refractive index even if both and are negative. In order to find the vector solution of Eq. (4), we make use of the equality
mˆ 2 mˆ mˆ mˆ mˆ 1 ,
(6)
by introducing it on the right-hand side of Eq. (4). The minus sign in Eq. (6) belongs to the unit vector of the phase front of the counterpropagating wave, which corresponds to negative values of . Both solutions of quadratic equation (4) can be written as m n mˆ , where the refraction vectors of the counterpropagating waves m and m have the same refractive index (5). If there is absorption in the medium, then the permittivity and permeability become complex quantities with positive imaginary parts. It follows from the form of formula (4) that the refraction vector is also a complex quantity and we correspondingly represent it as m = m + im , where the real vectors m and m are in general not parallel: m m 0 . The vector m determines the variation of amplitude of the wave and is called the extinction vector; the physical meaning of the vector m is that it determines the variation of the phase of the wave, and its length m n 0 is the refractive index n in a conducting medium. The complex refractive index n n in arises only in the case when the vectors m and m are parallel, which corresponds to a homogeneous decaying wave in an absorbing medium. However, a wave with such a field structure cannot always exist. In the presence of the interface with a transparent medium from which the electromagnetic wave penetrates into the absorbing medium, the extinction vector of the refracted wave m is directed across the boundary of the media. Consequently, the refraction vector m should be oriented likewise, which takes place only for normal incidence from the non-conducting medium [23, p. 84]. For this reason, the statement of the problem for oblique incidence of the plane wave into an absorbing medium with complex refractive index is impossible. Let the planar surface S of the interface of two half-spaces filled with homogeneous isotropic magnetodielectric media with permittivity and permeability 1 , 1 and 2 , 2 , respectively, be oriented such that the normal unit vector to it qˆ points from medium I into medium II. The boundary conditions on the electric ( E ) and magnetic ( H ) fields at the interface S are written in vector form:
EI qˆ = EII qˆ , H I qˆ = H II qˆ .
(7)
Additional relations of continuity of the normal components of the electric and magnetic inductions must be provided together with conditions (7) and are not presented here. Since relations (7) are valid only on the surface S , we form a radius vector originating from some point on this surface, r S , and lying entirely within it, such that qˆ = 0 . As a result of incidence of a plane wave from medium I into medium II there arise a reflected wave (in medium I) and a refracted wave (in medium II). In analogy with expression (1), we represent the electric fields of these waves in the form
j E j E0 exp i j , j k j r t k0 m j r t ,
where the indices j 0, 1, 2 are for the incident, reflected, and refracted waves, respectively. The magnetic fields have the same structure. Substitution into boundary conditions (7) leads to the relation
0 1 2 E0 qˆ exp i 0 E0 qˆ exp i1 E0 qˆ exp i 2 0 , where r .
Since the vector amplitudes are not equal to zero, all of the phases should coincide; therefore, m0 m1 m2 and m1 m0 0 , m2 m0 0 . As a consequence of the fact that the vector is perpendicular to both the unit vector qˆ and the differences of the refraction vectors, the latter are parallel; therefore, the following relations are valid:
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m0 qˆ m1 qˆ m2 qˆ .
(8)
Taking into account that m j = n j mˆ j , we rewrite formula (8), explicitly singling out the refractive indices:
n0 mˆ 0 qˆ n1mˆ 1 qˆ n2 mˆ 2 qˆ .
(9)
Formula (9) is the general vector form of the reflection law and the refraction law for waves at a planar interface of two media. It is valid for and both positive and for and both negative for each of the two media. Since the reflected and incident waves propagate in the same medium (the index n1 n0 ), the reflection law takes the form
mˆ 0 mˆ 1 qˆ 0 . It follows from this formula that the vector difference mˆ 0 mˆ 1 is parallel to the unit vector qˆ ; therefore, mˆ 1 mˆ 0 Aqˆ . Taking the square of this relation, solving the algebraic equation for the constant A and discarding the value A 0 , we obtain mˆ 1 mˆ 0 2 mˆ 0 qˆ qˆ .
(10)
We define the angle of incidence with the help of the formula 0 arccos mˆ 0 qˆ ; it lies in the first quadrant 0 2 . It follows from formula (10) that the angle of reflection 1 belongs to the second quadrant since mˆ 1 qˆ mˆ 0 qˆ cos 0 0 . This means that 1 0 , and with respect to the negative normal direction the angle of reflection is equal to the angle of incidence: 1 arccos mˆ 1 qˆ 0 .
In accordance with relations (9), we represent the refraction law in the form
mˆ 0 n20 mˆ 2 qˆ 0 ,
(11)
where n20 n2 n0 is the relative refractive index. It follows from Eq. (11) that n20 mˆ 2 mˆ 0 Вqˆ , where the scalar В is to be determined. Taking the square of this latter equality, we find two solutions of the resulting quadratic equation which differ by the sign on the square root. The refraction vector of the refracted wave reduces to n20 mˆ 2 mˆ 0 mˆ 0 qˆ
2 1 qˆ . mˆ 0 qˆ 2 n20
(12)
Taking the scalar product of relation (12) and the unit normal vector qˆ , we find the slope of the refraction vector of the 2 refracted wave n20 mˆ 2 qˆ mˆ 0 qˆ 2 n20 1 . For the upper sign in front of the radical the refraction vector of the
refracted wave takes the limiting value of the vector of the incident wave if we set n20 1 . For this reason, the angle of refraction 2 arccos mˆ 2 qˆ lies in the first quadrant and is calculated according to the formula 2 2 2 arccos 1 sin 2 0 n20 ,
(13)
which corresponds to the standard form of Snell’s law n2 sin 2 n0 sin 0 for both media having positive and . For the lower sign in front of the radical and taking n20 1 the refraction vector of the refracted wave will be oriented in the direction of the reflected wave (in the second quadrant), i.e., refraction into the anti-vacuum will take place. Therefore, for n20 1 the refraction angle is now also located in the second quadrant and is calculated according to the formula 2 2 . Snell’s law changes its form to n2 sin 2 n0 sin 0 . This case is realized only if the refracted 694
wave satisfies the damping principle in the positive direction of the unit normal qˆ to the surface S , i.e., if it is a backward wave [24, 25], and medium II is characterized by negative values of and . We introduce into consideration the unit vector of the ray that indicates the direction of motion of the energy flow of an electromagnetic plane wave pˆ E H E H . For a wave propagating in a right-handed medium with positive and , the unit vectors mˆ and pˆ are parallel, so that their scalar product mˆ pˆ 1 , and in a left-handed medium with negative and they will be antiparallel: mˆ pˆ 1 . Employing this result and introducing the symbolic Heaviside function , we can write Snell’s law for the wave normals in the generalized form n2 sin mˆ 2 pˆ 2 2 mˆ 2 pˆ 2 n0 sin 0 ,
(14)
where x 1 for x 0 and x 0 for x 0 . Employing the equality mˆ mˆ pˆ pˆ , we arrive at the refraction law
pˆ 0 n20 mˆ 2 pˆ 2 pˆ 2 qˆ 0 ,
(15)
setting pˆ 0 mˆ 0 in medium I. Relation (15) is equivalent to the equation n20 pˆ 2 mˆ 2 pˆ 2 pˆ 0 Cqˆ . After finding the scalar C , the expression for the refraction vector of the refracted ray takes the form n20 pˆ 2 mˆ 2 pˆ 2 pˆ 0 mˆ 2 pˆ 2 pˆ 0 qˆ
2 1 qˆ . pˆ 0 qˆ 2 n20
(16)
The sign in front of the square root ensures fulfillment of the radiation condition. Indeed, the projection of the refraction 2 vector onto the normal n20 pˆ 2 qˆ pˆ 0 qˆ 2 n20 1 is a positive quantity, which ensures transfer of wave energy
from the surface of the interface into the depth of medium II. If medium II is optically less dense (the index n20 1 ) and the condition of total reflection is fulfilled, then the square root takes a positive imaginary value and the amplitude of the refracted wave decays exponentially with increasing distance from the surface S in medium II. The projection onto the tangent direction (with unit vector ˆ in the plane of incidence) n20 pˆ 2 ˆ mˆ 2 pˆ 2 pˆ 0 ˆ allows us to identify the orientation of the vector pˆ 2 : it lies in the first quadrant if the permittivity and permeability of medium II are positive and in the fourth quadrant if they are negative. Introducing the notation pˆ 0 ˆ sin 0 and p p p p p pˆ 2 ˆ sin 2 , we obtain n2 sin 2 n0 sin 0 and n2 sin 2 n0 sin 0 , where 2 2 and 2 2 or p p 2 2 depending on how we reckon the angle 2 (reckoning counterclockwise starting from the direction of the
unit vector qˆ corresponds to the positive value). Snell’s law for rays is now written in general form as
n2 sin 2 mˆ 2 pˆ 2 n0 sin 0 .
(17)
As is obvious by comparing formulas (14) and (17), in the presence of a medium with negative values of and it is necessary to distinguish between Snell’s law for the wave vectors and Snell’s law for the ray vectors. If we allow that medium I can also be a left-handed medium, then formulas (14) and (17) must be supplemented. The most general formulas for the refraction law at the interface of two transparent magnetodielectrics allow for all variants of the combination of two media with two pairs of positive and negative values of the permittivity and permeability, specifically: the law for the wave normals has the form n0 sin mˆ 0 pˆ 0 0 mˆ 0 pˆ 0 n2 sin mˆ 2 pˆ 2 2 mˆ 2 pˆ 2 ,
(18)
whereas the formula for the rays looks simpler:
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n0 sin 0 mˆ 0 pˆ 0 n2 sin 2 mˆ 2 pˆ 2 .
(19)
Here n0 and n2 are the refractive indices of the media (positive quantities), 0 and 2 are the angles of incidence and refraction, respectively, corresponding to the canonical form of Snell’s law for two right-handed media. It follow from formulas (18) and (19), in particular, that at the interface of two left-handed media refraction takes place in the same way as in the case of two right-handed media. This property was already noted earlier in [26]. REFERENCES
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