Some remarks regarding electrodynamics of materials with negative refraction

Transcription

1 Appl. Phys. B 81, (2005) DOI: /s Applied Physics B Lasers and Optics V.G. VESELAGO Some remarks regarding electrodynamics of materials with negative refraction Moscow Inst. of Physics and Technology, Dolgopzudny, Russia A.M. Prokhorov Inst. of General Physics, Moscow, Russia Received: 20 March 2005 / Revised version: 7 May 2005 Published online: 15 July 2005 Springer-Verlag 2005 ABSTRACT In the following article some electrodynamical problems of materials with negative refraction are considered. In contrast to the usual case when the index of refraction is positive, for these sorts of materials many laws and equations must be recorded differently. Special note is taken of the fact that most of the books and textbooks are written with the so-called nonmagnetic approach, which is only valid for nonmagnetic materials (µ = 1). This approach is undoubtedly unfit for material with a negative index of refraction. It is shown that materials with simultaneously negative dielectric and magnetic permeabilities undoubtedly must possess the frequency dispersion. Correlation is brought between phase and group velocities for these sorts of materials. The question is considered in detail about the so-called overcoming of the diffraction limit by means of plates from materials with a negative factor of refraction. It is shown that this effect is indeed reduced to the problem of matching between source and receiver of radiation. Such matching is possible by spreading the so-called evanescent modes, for which a diffraction limit does not exist. These modes fade within a distance of the order of the wavelength, and only at such a distance is the transfer of picture details that are smaller than the wavelength possible. PACS Lr; Gy In the last several years, a new chapter of classical electrodynamics the electrodynamics of materials with negative index of refraction has begun to develop. The first experiments in this area [1, 2] were carried out by a group of physicists from San Diego University (USA). In their work unusual electrodynamic properties of some composite materials were demonstrated. These characteristics can be explained formally if one assumes that the given materials possess the negative index of refraction n [3]. It has been shown [3] that the negative value of n can be realized in materials which possess simultaneously negative values of permeabilities ε and µ. The inverse statement is also correct if an isotropic material possesses a negative value of refraction index n, then Fax: , v.veselago@relcom.ru it must have simultaneously negative values of ε and µ, and its phase and group velocity will have the opposite direction. However, the term negative refraction presently constitutes a much broader class of phenomena in which waves with opposite directions of phase and group velocities take place. Moreover, many authors understand the term opposite direction broadly enough, considering that this case always exists when phase ν ph and group ν gr velocity are connected with the vector equation (ν ph ν gr ) < 0 (1) This correlation is often valid, in particular, for electromagnetic waves in photonic crystals [4]. However photonic crystals could not be described by one scalar parameter, for example, by an index of refraction n. It is necessary to show the opposite direction of phase and group velocities is nothing new. This question, in particular, was discussed many years ago by L.I. Mandelstam [5]. Besides, electronic devices of the past (for instance, lamps of backward wave, LBW), in which the phase velocity had an opposite direction of flow of energy, were relatively common. It is worth mentioning that in the case of simultaneously negative values ε and µ, the phase and group velocity have opposite directions [6]. The appearance of materials with a negative value of n brings up a very important question: In cases where n < 0, for which values of the refraction index n are all the laws and formulas of electrodynamics, optics and adjacent technical sciences still valid? Can we always reckon the correct result under direct change n n as, for example, in the case of Snellius law [3]? Generally, the answer to this question is negative. It is given that most laws and equations of electrodynamics and optics correspond to the case when one or more materials is nonmagnetic and has magnetic permeability µ = 1. Using such a nonmagnetic approach leads to the conclusion that many formulas in which µ enters after substitution of µ 1 are drastically changed and do not give correct results. Table 1 explains this situation. From the table, it can be seen that three groups of physical laws and effects, wordings which have different outcomes when turning from a nonmagnetic approach equation to exact expressions, exist.

2 404 Applied Physics B Lasers and Optics Physical law Equation for nonmagnetic approach Correct equation Snellius, Doppler, Cherenkov n = ε n = εµ sin ϕ/sin ψ = n 21 = ε 2 /ε 1 sin ϕ/sin ψ = n 21 = ε 2 µ 2 /ε 1 µ 1 if ε, µ < 0, than n < 0 Fresnel n = ε 1/Z = ε/µ r = n1 cos ϕ n2 cos ψ n 1 cos ϕ+n 2 cos ψ r = Z2 cos ϕ Z1 cos ψ Z 2 cos ϕ+z 1 cos ψ Reflection coefficient for normal fall of light on the border between two media r = (n 1 n 2 )/(n 1 + n 2 ) r = (Z 2 Z 1 )(Z 2 + Z 1 ) Condition for full matching n 1 = n 2 Z 1 = Z 2 Brewster angle tgϕ = n tgϕ = q ε2 ε 1 ε 2µ 1 ε 1µ 2 ε 2µ 2 ε 1µ 1 TABLE 1 Nonmagnetic and exact forms for some equations of electrodynamics The Snellius law and the effects of the Doppler and Cherenkov laws are in the first group. In its formulas, usually applicable in a nonmagnetic approach, an expression n = ε simply must be replaced on n = εµ, and if ε and µ are both negative, then there must be a minus sign before n, too. The laws of reflection and refractions of light, and, in particular, Fresnel s formulas, pertain to the second group. In these formulas when turning from the nonmagnetic approach approximation to exact formulas a value n = ε should be changed not to n = εµ, butto ε µ value Z is the wave impedance Z = µ ε =1/z, where of the media. The wave impedance has a dimensionality of resistance (Ohm) and is a unique feature of each media, like the speed of light in it. From the table it can be seen that the nonmagnetic approach gives an incorrect condition for the absence of reflection of light on a flat border between two media. This condition consists not in the equality of indexes of refraction for two media, but in the equality of their wave impedances. It is important to show that under negative values of ε and µ, wave impedance Z, unlike value n, remains positive. And, finally to the third group of equations, which strongly depends on n and is highly variable when turning from the nonmagnetic approach to exact formulas, pertains, in particular, to the formula for Brewster angle tgϕ = n. The exact expression for the Brewster angle is provided in the last line of the table. It is important to note that the expression under the square root in this exact formula is not altered under simultaneous change of the signs for ε and µ of one media. It is necessary to remember that the formula for the Brewster angle provided in the table corresponds to one polarization of light. For another, perpendicular polarization, the formula can be found in the table by changing ε µ and µ ε in the expression under the square root. Thereby, the reflection under the Brewster angle always exists, under any values of permeabilities, but only for one of the two possible polarizations of falling light. The introduction to the scientific notion negative index of refraction also changes the wording of fundamental principles, such as the Fermate principle. This question is considered in detail in publication [7], where it is shown that correct wording of the Fermate principle suitable to the spreading of the electromagnetic wave through materials with any sign index of refraction n is a requirement of local extremum for total length of the optical path δl = δ n l = 0 (2) Integration in this expression (which is, in essence, eikonal) is produced on real optical path. Such an approach demonstrates that the length of the optical path, passable by electromagnetic wave in media with a negative value of n, is also negative. Therefore, it follows that in some cases the full total length of the optical path can be negative or even zero; however, the geometric length of the path, on which the light spreads, and the time of spreading are certainly not equal to zero. Exactly such a situation exists when the light is spread through a flat plate from a material with ε = µ = n = 1. Such a plate, as is seen in Fig. 1, is capable of focusing in a point a radiation coming out of the point source and disposed on the other side of the plate. From Fig. 1 it can be seen that path Am, the passable light from the source in front of the plate, and path nb,fromthe plate to the image, are together equal to path mn, whose light passes inside the plates Am + nb = mn (3) FIGURE 1 Penetration of light from source A to image B through a flat slab of media with ε = µ = n = 1. Outside of the slab is a vacuum with ε = µ = n = 1

3 VESELAGO Some remarks regarding electrodynamics of materials with negative refraction 405 Such a correlation is valid for any other possible path of spreading the light from A to B, for instance AcgB or Ad f B. But since inside the slab the index of refraction is n = 1, but outside n =+1, the full optical length for light going from point A to point B will, in accordance with the expression (2), be zero for any possible penetration path. At the same time, as was already mentioned, the time it took for the light to spread from point A to point B greatly differs from zero. The fact of focusing of the point source of light in a point on the other side of plate does not mean that this plate is a lens. Such a plate is an ideal optical instrument, which carries the space of the subject to the space of scenes without any distortion. But such carrying is possible only for subjects for which the distance to the plate is not greater than the thickness of the plate. The plate undoubtedly cannot focus into a point a parallel cone of rays coming from infinity. Also, this plate lacks an optical axis. However characteristics of such a plate are undoubtedly interesting and can have practical value. Under the general estimation of characteristics of materials with a negative index of refraction, it is necessary to note that these materials must possess the frequency dispersion. If ε and µ are both negative, in the absence of dispersion full energy materials following equation W = εe 2 + µh 2 (4) will be negative. However if the frequency dispersion exists, the expression (4) must be written in the form: W = (εω) ω E 2 + (µω) ω H 2 (5) It is clear that values (µω) and (εω) are positive if the law ω ω of frequency dispersion for ε and µ is chosen in a general enough form µ = 1 A2 m ω 2 (6) ε = 1 A2 e (7) ω 2 If we assume A 2 e = A2 m = A2 >ω 2 (8) the index of refraction becomes negative, and for phase V ph = C and the group V 1 A2 gr = C velocities will be valid, giving ω 2 1+ A2 ω equation: 2 C + C = 2 (9) V ph V gr For the waves in media with negative n, we must choose a sign minus before wave vector k. However in media with absorption, vector k has not only a real, but also an imaginary part. The appearance of this imaginary part is a result of the presence of imaginary parts in expressions for ε and µ. So, the question arises does the change of the sign before the imaginary part of the wave vector follow if the sign is changed before its real part? Let us write expressions for ε and µ as ε = ε + jε,µ= µ + jµ (10) It is not difficult to see that for the case of a small dissipation an expression for k will be k = k + jk = ω C (ε + jε )(µ + jµ ) = ω C [ ε µ 1 + j ( ε )] + µ 2 ε µ (11) From (11), it is easy to see that the change of the sign itself beside the real parts does not entail the automatic change of the sign beside the imaginary part of the wave vector. To change the sign of the imaginary part of the wave vector, it is necessary to change the sign beside the imaginary parts of ε and µ that corresponds to the transition from a material with the positive absorption to a material with the negative absorption as in the case of quantum amplifiers, for example. Such a transition, in general, is not connected with a possible transition from the usual materials with positive refraction to materials with negative refraction. The estimated importance of this notion greatly depends on the fact that we can really hold materials with negative refraction in our hands. In addition to our efforts, this question arises in other publications [3]. We have spent observable efforts in preparing material with negative refraction on the basis of the magnetic semiconductor CdCr 2 Se 4 ; however, these efforts ended in failure because of essential technological difficulties, which characterize the syntheses of this material. At this point, it is inappropriate to discuss an exotic mixture of electric and magnetic charges, which was considered in [8]. The appearance of a new class of material a material with negative refraction, which shows many unusual electrodynamic properties has borne an intensive search for new characteristics and possible practical applications. Herewith some statements in the literature prompted forceful objections. So, in work [9], it became firmly established that negative refraction exists for phase velocity only, while group velocity follows the usual law of refraction with the positive value of refraction index n. The authors of this work are not embarrassed by the fact that the difference in directions of phase and group velocity are typical properties of optical anisotropic media, which cannot be characterized by a scalar value of n. The mistake of the authors [9] can be explained by the fact that the authors muddled a the direction of the group velocity with the direction of the perpendicular to surfaces of constant amplitude for modulated waves. This mistake is considered in detail and explained in the paper [10]. There is one more problem, which is in close relation to the appearance of materials with negative refraction. This is the problem known as the overcoming of diffraction limit, which consists of the problem of the increasing of so-called evanescent modes. For the first time this problem was discussed by J.B. Pendry in his work [11], where it was shown that the material with a negative refraction can successfully

4 406 Applied Physics B Lasers and Optics spread waves, for which the component k z of the wave vector along the direction of spreading has an imaginary value ω 2 k z = i C 2 k2 x (12) This equality is valid for very large k x, that is to say for very short waves. In material with positive values of n, the amplitude of such waves (the evanescent modes), in accordance with (12), will exponentially decrease along z axis, and exactly this fact explains the impossibility of the image by optical systems of objects, with sizes noticeably smaller than the wavelength. However in work [11] and in many articles that followed, the authors suggest that in material with a negative refraction index, waves with large values of k x do not decrease, but increase. Hereunder it is suggested that it is possible to transfer the images with sizes much smaller than the wavelength from one point of space to another. This suggestion was motivated by the possibility of resonance increasing of the surface modes in material with negative refraction. The work [11] proposes the notion of superlenses for devices like those shown in Fig. 1, confirming that for this sort of device the classical restriction on the diffraction limit is not valid. The authors of the work [11], and many others, have more recently drawn a veil over the fact that the overcoming of the diffraction limit automatically means the breaching of the uncertainty principle. For our case the uncertainty equation can be written as follows k x d 2π (13) Here k x is a component of the wave vector orthogonal to the z axis, along which the wave spreads and d is the transverse size of the focused spot of light. Value k x cannot be more than wave vector k 0 in free space: k x < k 0 = ω c = 2π (14) λ From (13) and (14) immediately follows: d λ (15) The possibility of a breach of the diffraction limit is equivalent to the statement about the unacceptability of (13), or, more exactly, to render its execution unnecessary. Such a statement is exceedingly strong, much stronger than all other possible statements about any other unusual characteristics of the material with negative refraction. In our opinion, such statements are a result of the not very exact use of some terms, first of all the main term lens. This word lens itself characterizes the optical instrument, shown in Fig. 1, which is based on geometric optics laws. However, a flat plate from material with a negative refraction can be considered as a lens, only if its transverse size a, wavelength of radiation λ and period of internal structure δ satisfy the inequality a >λ>δ (16) Besides, the correlation must be executed in such a lens a = b + c (17) b n=1 FIGURE 2 n = 1 FIGURE 3 δ n= - 1 a c n=1 a=b+c a,b,c>λ>δ λ- wavelength δ - lattice constant The flat lens produced by a material with refraction index Waveguide with detector and matching screws Only in this case, shown in Fig. 2, does the lens truly qualify as an optical instrument, complying with geometric optics laws. Exactly this situation was meant long ago in our article [3], though this was not directly indicated. However, in the basic article [11] and following works, the situation was considered when correlation (16) was not executed, since the thickness of the lens was the same order as the wavelength, and such order has distances from lens to object and image planes. Such a system is not a lens, but some matching device, which will not at all work on the basis of geometric optics laws. As is well known, it is possible by means of matching devices to concentrate the flow of energy into spots, undoubtedly smaller, than the wavelength. To make this problem clear, let us consider the spread and registration of the electromagnetic wave in the usual metallic waveguide as is shown in Fig. 3. It is well known that the electromagnetic wave could spread in a hollow rectangular waveguide with broad wall size a if the following relation is valid λ 2a (18) If this condition is satisfied, the wave spreads on the waveguide with a little fading, and the field of output of such a waveguide could be considered as some rectangle image, which is formed by the cross section of the waveguide. The sizes of this rectangle are on the order of the value of size a. For the registration of radiation spreading in the waveguide, one usually needs detectors with sizes much smaller than

5 VESELAGO Some remarks regarding electrodynamics of materials with negative refraction 407 the transverse size of the waveguide, as is shown in Fig. 3. If such a detector is situated beside the output end of waveguide, it will register only a small part of radiation; the mainstream of which will go by the detector. If one wants to enlarge the power falling on the detector, one can narrow the waveguide so that the wave in the waveguide will begin to fade strongly as soon as it has broken the correlation (18). However, it is wholly possible to greatly enlarge the power on the detector by using some sort of matching elements, having placed them in the waveguide in close proximity of the detector. Such matching elements are usually a different sort of screws and/or slots, by means of which it is possible to enlarge greatly the power hitting the detector. This increase of power on the detector is possible to consider as a focusing of radiation to a spot, whose transverse size is the same size as the detector, and, naturally, much less than the wavelength. Hereunder, it is possible to confirm that in the waveguide with the matching device an undoubtedly broken diffraction limit exists, and the wave is focused on the spot with a size that is noticeably less than a wavelength. This is only true if both flat waves and long waveguides fall on the detector. Except for these flat waves, the evanescent waves, which are generated by the elements of the matching devices, act on the detector. The generation of evanescent waves results from the reradiation of flat waves in the waveguide on small aspects of matching devices. In principle, the same process exists in superlenses, which are shown in Fig. 2. This lens will in essence match the source of the radiation with the receiver. Herewith the correlations (16) and (17) are not executed; however, the lens indeed can send without garbling the scene with small sizes, but, regrettably, only over small distances, comparable with the wavelength [12]. In the entire work dedicated to the problem of overcoming the diffraction limit, beginning with work [11], it was not directly established what exact thickness the plate made from material with negative refraction must or can have in order to transfer without distortion pictures of sizes much smaller than the wavelength. However in most of the work dedicated to this problem, plates where the thickness was comparable or less than that of the wavelength were considered. Accordingly, distances from the source of radiation in front of the plate and from the plate to the image were also comparable to or less than than the wavelength. This situation is considered in detail in [12]. So, the superlens can really exist, but only from the domain of geometric optics. ACKNOWLEDGEMENTS This work is supported by the Russian Foundation for Fundamental Research, project # a REFERENCES 1 D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Let. 84, 4184 (2000) 2 R.A. Shelby, D.R. Smith, S. Schultz, Science 292, 77 (2001) 3 V.G. Veselago, Sov. Phys. Uspekhi 10, 509 (1968) 4 M. Notomi, Opt. Quantum Electron. 34, 133 (2002) 5 L.I. Mandelstam, Zh. Eksp. Teor. Fiz. 15, 475 (1945) 6 D.V. Sivukhin, Opt. Spektroskop. 3, 308 (1957) 7 V.G. Veselago, Phys. Usp. 45, 1097 (2002) 8 V.G. Veselago, Sov. Phys. JETP 25, 80 (1966) 9 P.M. Valanju, R.M. Walser, A.P. Valanju, Phys. Rev. Let 88, (2002) 10 J.B. Pendry, D.R. Smith, cond-mat/ J.B. Pendry, Phys. Rev. Let. 85, 3966 (2000) 12 X.S. Rao, C.K. Ong, cond-mat/