V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of and

SOVIET PHYSICS USPEKHI 538.30 VOLUME 10, NUMBER 4 JANUARY-FEBRUARY 1968 THE ELECTRODYNAMICS OF SUBSTANCES WITH SIMULTANEOUSLY NEGATIVE VALUES OF e AND M V. G. VESELAGO P. N. Lebedev Physics Institute, Academy of Sciences, U.S.S.R. Usp. Fiz. Nauk 92, 517-526 (July, 1964) 1. INTRODUCTION J. HE dielectric constant e and the magnetic permea- bility n are the fundamental characteristic quantities which determine the propagation of electromagnetic waves in matter. This is due to the fact that they are the only parameters of the substance that appear in the dispersion equation -kikj = 0, (1) which gives the connection between the frequency OJ of a monochromatic wave and its wave vector k. In the case of an isotropic substance, Eq. (1) takes a simpler form: (2) Here n2 is the square of the index of refraction of the substance, and is given by ra^ep.. (3) If we do not take losses into account and regard n, e, and M as real numbers, it can be seen from (2) and (3) that a simultaneous change of the signs of € and ix has no effect on these relations. This situation can be interpreted in various ways. First , we may admit that the properties of a substance are actually not affected by a simultaneous change of the signs of e and ix. Second, it might be that for e and y. to be simultaneously negative contradicts some funda- mental laws of nature, and therefore no substance with e < 0 and ix < 0 can exist. Finally, it could be admitted that substances with negative e and \x have some properties different from those of substances with positive e and /x. As we shall see in what fol- lows, the third case is the one that is realized. It must be emphasized that there has not so far been any experiment in which a substance with e < 0 and ix < 0 could be observed. We can, however, at once give a number of arguments as to where and how one should look for such substances. Since in our opinion the electrodynamics of substances with e < 0 and ix < 0 is undoubtedly of interest, independently of our now having such substances available, we shall at first consider the matter purely formally. There- after in the second part of this article we shall con- sider questions connected with the physical realiza- tion of substances with e < 0 and ix < 0. II. THE PROPAGATION OF WAVES IN A SUBSTANCE WITH e < 0 AND \x < 0. "RIGHT-HANDED" AND "LEFT-HANDED" SUBSTANCES To ascertain the electromagnetic laws essentially connected with the sign of e and M» we must turn to those relations in which e and \x appear separately, and not in the form of their product, as in (1)—(3). These relations are primarily the Maxwell equations and the constitutive relations t, 1 dB rot E — c dl , „ 1 dB rot H = - , c dt• = eE. (4)" (4') For a plane monochromatic wave, in which all quantities are proportional to e^kz-wt^ t n e expres- sions (4) and (4') reduce to (5)t[ kHJ- - -2 - It c a n be s e e n a t o n c e f r o m t h e s e e q u a t i o n s t h a t if e > 0 a n d n > 0 t h e n E, H, a n d k f o r m a r i g h t - h a n d e d t r i p l e t of v e c t o r s , a n d if e < 0 a n d n < 0 t h e y a r e a l e f t - h a n d e d s e t . ' 1 - ' If w e i n t r o d u c e d i r e c t i o n c o s i n e s f o r t h e v e c t o r s E, H, a n d k a n d d e n o t e t h e m b y cq , fi{, a n d yi, r e s p e c t i v e l y , t h e n a w a v e p r o p a - g a t e d in a g i v e n m e d i u m w i l l b e c h a r a c t e r i z e d b y t h e m a t r i x ^ /<%! a2 a3 \ G= Pi P2 P 3J. (6) \Yi Y2 Y.3/ T h e d e t e r m i n a n t of t h i s m a t r i x i s e q u a l t o +1 if t h e v e c t o r s E, H, a n d k a r e a r i g h t - h a n d e d s e t , a n d - 1 if t h i s s e t i s l e f t - h a n d e d . D e n o t i n g t h i s d e t e r m i n a n t b y p , w e c a n s a y t h a t p c h a r a c t e r i z e s t h e " r i g h t - n e s s " of t h e g i v e n m e d i u m . T h e m e d i u m i s " r i g h t - h a n d e d " if p = + 1 a n d " l e f t - h a n d e d " if p = - 1 . T h e e l e m e n t s of t h e m a t r i x (6) s a t i s f y t h e r e l a t i o n Gih = pAih. (7) H e r e Ajj j i s t h e a l g e b r a i c c o m p l e m e n t of t h e e l e m e n t F u r t h e r m o r e t h e e l e m e n t s of G a r e o r t h o - *rot = curl. t [kE] .kxE. 509 510 V. G. V E S E L A G O b) FIG. 1. a) Doppler effect in a right-handed substance; b) Doppler effect in a left-handed substance. The letter A represents the source of the radiation, the letter B the receiver. normal. The energy flux carried by the wave is de- termined by the Poynting vector S, which is given by S rpu l / Q\ According to (8) the vector S always forms a right- handed set with the vectors E and H. Accordingly, for right-handed substances S and k are in the same direction, and for left-handed substances they are in opposite directions.^3-' Since the vector k is in the direction of the phase velocity, it is clear that left- handed substances are substances with a so-called negative group velocity, which occurs in particular in anisotropic substances or when there is spatial dispersion.'-4-' In what follows we shall for brevity use the term "left-handed substance," keeping in mind that this term is equivalent to the term " sub - stance with negative group velocity." Let us now consider the consequences of the fact that in left- handed substances the phase velocity is opposite to the energy flux. First, in left-handed substances there will be a reversed Doppler effect.[1'3-' Indeed, suppose for example that a detector of radiation which is in a left-handed medium moves relative to a source which emits a frequency COQ- In its motion the detector will pursue points of the wave which correspond to some definite phase, as is shown in Fig. 1. The frequency received by the de- tector will be smaller than a>oi not larger as it would be in an ordinary (right-handed) medium. Us- ing the quantity p for the medium in question, we can write the formula for the Doppler shift in the form (0 -=(O0(l—/?-£-) . (9) Here the velocity v of the detector is regarded as positive when it is receding from the source. The velocity u of the energy flux is regarded as always positive. The Vavilov-Cerenkov effect will also be r e - versed, just like the Doppler effect.[1 '3] If a particle FIG. 2. a) The Vavilov-Cerenkov effect in a right-handed sub- stance; b) The same effect in a left-handed substance. moves in a medium with speed v in a straight line (Fig. 2), it will emit according to the law e i (k z z+krr -o ; t ) ) a n ( i ^e w a v e vector of the radiation will be given by k = k z / co s 6 and is in the general direction of the velocity v. The quantity k r will be different in different media, in accordance with the expression (10) This choice of the sign for the square root in (10) will assure that the energy moves away from the radiating particle to infinity. It is then clear that for left- handed media the vector k r will be directed toward the trajectory of the particle, and the cone of the radiation will be directed backward relative to the motion of the particle. This corresponds to an obtuse angle 6 between v and S. For a medium of either "Tightness" this angle can be found from the expres- sion cos 9 = p (11) III. THE REFRACTION OF A RAY AT THE BOUND- ARY BETWEEN TWO MEDIA WITH DIFFERENT RIGHTNESSES In the passage of a ray of light from one medium into another the boundary conditions Eh = Eh, Htl = HH, (12) ni V ni n2 (13) must be satisfied, independently of whether or not the media have the same Tightness. It follows from (12) that the x and y components of the fields E and H in the refracted ray maintain their directions, inde- E L E C T R O D Y N A M I C S O F S U B S T A N C E S WITH N E G A T I V E e AND 511 FIG. 3. Passage of a ray through the boundary between two media. 1 — incident ray; 2 — reflected ray; 3 — re- flected ray if the second medium is left- handed; 4 — refracted ray if the second medium is right-handed. pendently of the rightnesses of the two media. As for the z component, it keeps the same direction only if the two media are of the same Tightness. If the right- nesses are different, the z components change sign. This corresponds to the fact that in passage into a medium of different Tightness the vectors E and H not only change in magnitude owing to the difference in e and M but also undergo a reflection relative to the interface of the two media. The same thing hap- pens to the vector k also. The simultaneous reflec- tion of all three vectors corresponds precisely to a change of sign of the determinant G in (6). The path of the refracted ray produced as the result of such reflections is shown in Fig. 3. As we see, when the second medium is left-handed the refracted ray lies on the opposite side of the z axis from its position in the case of a right-handed second medium J5-' It must be noted that the direction of the reflected ray is always the same, independent of the rightnesses of the two media. It can be seen from Fig. 3 that the usual Snell's law sin

, ij>. In order not to make mistakes one must always use the absolute values of these quantities in Fresnel 's formulas. An interesting case is that of a ray passing from a medium with e4 > 0, ^ > 0 into one with e2 = —e1( ;U2 = — Mi- In this case the ray undergoes refraction at the interface between the two media, but there is no reflected ray. The use of left-handed substances would in principle allow the design of very unusual refracting systems. An example of such a system is a simple plate of thickness d made of a left-handed substance with n = - 1 and situated in vacuum. It is shown in Fig. 4 that such a plate can focus at a point the radiation from a point source located at a dis- tance I < d from the plate. This is not a lens in the usual sense of the word, however, since it will not focus at a point a bundle of rays coming from infinity. As for actual lenses, the paths of rays through lenses made of a left-handed substance are shown in Fig. 5. It is seen that the convex and concave lenses have "changed places ," since the convex lens has a diverging effect and the concave lens a converging effect. FIG. 6. Reflection of a ray propagated in a medium with ( < 0 and fi < 0 from an ideally reflecting body. The source of radiation is denoted by a heavy black point. 512 V. G. V E S E L A G O M FIG. 7. e — fi diagram. A monochromatic wave in a left-handed medium can be regarded as a stream of photons, each having a momentum p = hk, with the vector k directed toward the source of radiation, not away from it as is the case in a right-handed medium. Therefore a beam of light propagated in a left-handed medium and incident on a reflecting body imparts to it a momen- tum p = 2Nhk (N is the number of incident photons) directed toward the source of the radiation, as shown in Fig. 6. Owing to this the light pressure character- istic for ordinary (right-handed) substances is r e - placed in left-handed substances by a light tension or attraction. These are some of the features of the electrody- namics of left-handed substances. Let us now con- sider the question of their physical realization. For this purpose we first examine what the values of e and n are that various substances may have. IV. WHAT SORT OF VALUES OF e AND n ARE IN PRINCIPLE POSSIBLE? Figure 7 shows a coordinate system in which values of e and n are marked off on the axes. We shall try to locate in it all known substances, at first confining ourselves to the case in which e and M are isotropic. Then the first quadrant contains the majority of isotropic dielectrics, for which e and /x are positive. In the second quadrant (e < 0, M > 0) there will be plasmas, both gaseous plasmas [6] and solid-state plasmas.[7~9-' In a plasma with no mag- netic field the value of e is given by where OJ2 = 47rNe2/m, N being the concentration of the carr iers , e their charge, and m their mass, and the summation is over all types of ca r r i e r s . It is not hard to see that at small frequencies e is smaller than zero. For e > 0 and n > 0 the value of n2 given by (3) is negative, which leads to reflection of waves from such a medium. This fact is well confirmed by experiment, for example, in the ionosphere. The third and fourth quadrants in Fig. 7 are un- occupied. So far there is not a single substance known with ji< 0. Aswe shall see in what follows, this is not accidental. Let us now go on to anisotropic substances. In this case the quantities e and M are tensors, and we can- not make use at once of a diagram like Fig. 7. In some substances, however, we can do this for waves propagated in particular directions. Gyrotropic sub- stances are especially interesting in this respect. For gyrotropic substances the tensors e^ and are of the forms (17) (18) A well known example of a gyrotropic substance is a plasma in a magnetic field, which is characterized by a tensor e ^ of the form (17) and a scalar value of M- H a plane circularly polarized transverse wave of the form e^z - wt) j s propagated in such a plasma, with k II z II Ho, then n2 is given by El te2 0 Hi i\i2 0 ie2 «i 0 *H2 Hi 0 0s 0 e3/ 0 0 H3 ±e2). (19) The sign ± corresponds to the two directions of polarization of the wave. If | e21 < I £i I and et > 0, two waves can be propagated in the plasma, but if I €21 > I e i I = - £ i i then only oae wave is propagated, that for which n2 > 0. In these cases the plasma must be placed in the first quadrant of Fig. 7 (/i is of the order of 1). As for the second wave in the case | e21 > I ei I = - e l f it cannot be propagated because for it e < 0, which by (19) leads to an imaginary value of n. In this case the plasma belongs in the second quadrant in Fig. 7. Another example of gyrotropic substances is various magnetic materials, in which, in contrast to the plasma, it is /x and not e that is a tensor. For these materials the analog of (19) is re2 = e(n, ± (20) Here also there can in principle be a situation in which I n21 > l^i I = -Mi, and this case corresponds to the fourth quadrant in Fig. 7. Quite recently there have begun to be intensive studies of gyrotropic substances in which both e and ^ are tens or s.'-10"15-' Examples of such substances are pure ferromagnetic metals and semiconductors. For such substances the index of refraction of a circu- larly polarized wave travelling along the field is given by w2 = (e, ±E 2 ) (HI ± (21) and it is easy to see that in this case the effective electric and magnetic permeabilities can both be less than zero, while n2 remains positive and the wave will be propagated.t3-13 '1^ Such substances occupy the third and last quadrant of Fig. 7. Accordingly we see that we must look for substances with e < 0 and E L E C T R O D Y N A M I C S O F S U B S T A N C E S WITH N E G A T I V E e AND 513 H < 0 p r i m a r i l y a m o n g g y r o t r o p i c m e d i a . F u r t h e r - m o r e i t i s o b v i o u s t h t a n e g a t i v e v a l u e s o f e a n d M i n g y r o t r o p i c s u b s t a n c e s c a n b e r e a l i z e d o n l y f o r t h o s e w a v e s t h a t a r e p r o p a g a t e d a l o n g t h e m a g n e t i c f i e l d . F o r o t h e r d i r e c t i o n s o f p r o p a g a t i o n e a n d M c a n n o l o n g e r b e r e g a r d e d a s s c a l a r s . N e v e r t h e l e s s , f o r a c e r t a i n r a n g e o f a n g l e s b e t w e e n H a n d k t h e v e c t o r s S a n d k w i l l m a k e a n a n g l e c l o s e t o 1 8 0 ° a n d w i l l q u a l i t a t i v e l y s a t i s f y a l l o f t h e l a w s w h i c h a r e c h a r a c - t e r i s t i c f o r l e f t - h a n d e d s u b s t a n c e s . I n c o n c l u d i n g t h i s s e c t i o n w e n o t e t h a t s i m u l t a n e - o u s n e g a t i v e v a l u e s o f e a n d n c a n b e r e a l i z e d o n l y w h e n t h e r e i s f r e q u e n c y d i s p e r s i o n . I n f a c t , i t c a n b e s e e n f r o m t h e r e l a t i o n W = &E2 + \iH2 ( 2 2 ) t h a t w h e n t h e r e i s n o f r e q u e n c y d i s p e r s i o n n o r a b - s o r p t i o n w e c a n n o t h a v e e < 0 a n d n < 0 , s i n c e i n t h a t c a s e t h e t o t a l e n e r g y w o u l d b e n e g a t i v e . W h e n t h e r e i s f r e q u e n c y d i s p e r s i o n , h o w e v e r , t h e r e l a t i o n ( 2 2 ) m u s t b e r e p l a c e d b y 3 (eco) p 2 , d (LICO) r,2 W' = • da ( 2 3 ) I n o r d e r f o r t h e e n e r g y W g i v e n b y ( 2 3 ) t o b e p o s i t i v e i t i s r e q u i r e d t h a t d (eco) ( 2 4 ) T h e s e i n e q u a l i t i e s d o n o t i n g e n e r a l m e a n t h a t e a n d H c a n n o t b e s i m u l t a n e o u s l y n e g a t i v e , b u t f o r t h e m t o h o l d i t i s n e c e s s a r y t h a t e a n d n d e p e n d o n t h e f r e - q u e n c y . I t i s a p p r o p r i a t e t o e m p h a s i z e h e r e t h a t t h e c o n - c l u s i o n t h a t t h e r e i s a l i g h t a t t r a c t i o n i n l e f t - h a n d e d s u b s t a n c e s , w h i c h w a s o b t a i n e d a t t h e e n d o f S e c . I l l f r o m q u a n t u m a r g u m e n t s , c a n a l s o b e o b t a i n e d i n a p u r e l y c l a s s i c a l w a y . T o d o t h i s w e m u s t u s e t h e c l a s s i c a l e x p r e s s i o n f o r t h e m o m e n t u m o f t h e f i e l d '-17-' t h e r e l a t i o n s ( 2 3 ) a n d ( 2 4 ) , a n d a l s o t h e c o n n e c t i o n b e t w e e n t h e P o y n t i n g v e c t o r S a n d t h e g r o u p v e l o c i t y v g = 3 w / a k , S = P F v g r . ( 2 6 ) C o m b i n i n g t h e e x p r e s s i o n s ( 2 3 ) — ( 2 6 ) , w e g e t p = i L = i L . k . ( 2 7 ) v p h <•> I t f o l l o w s f r o m t h i s t h a t i n l e f t - h a n d e d s u b s t a n c e s t h e f i e l d m o m e n t u m p i s d i r e c t e d o p p o s i t e t o t h e P o y n - t i n g v e c t o r S . V . G Y R O T R O P I C S U B S T A N C E S P O S S E S S I N G P L A S M A A N D M A G N E T I C P R O P E R T I E S I t i s c h a r a c t e r i s t i c o f g y r o t r o p i c m e d i a o f t h i s k i n d t h a t , f i r s t , t h e y c o n t a i n s u f f i c i e n t l y m o b i l e c a r - r i e r s f o r m i n g a n e l e c t r o n - h o l e p l a s m a , a n d , s e c o n d , t h a t t h e r e e x i s t s a s y s t e m o f i n t e r a c t i n g s p i n s w h i c h p r o v i d e a l a r g e m a g n e t i c s u s c e p t i b i l i t y . T h i s a s s u r e s t h e s i m u l t a n e o u s p r o p a g a t i o n o f s p i n a n d p l a s m a w a v e s , a n d n a t u r a l l y t h e r e i s a n i n t e r a c t i o n b e t w e e n t h e m . I f t h i s i n t e r a c t i o n i s s t r o n g e n o u g h , t h e w a v e s p r o p a g a t e d i n s u c h a s u b s t a n c e a r e o f a m i x e d , s p i n - p l a s m a , c h a r a c t e r . I n t h i s c a s e t h e v a l u e s o f e a n d M a r e o f t h e f o l l o w i n g f o r m [ 1 5 j : ( 2 8 ) ± Q) 1 - Q ' ± CO ' n* = e\x. ( 2 9 ) H e r e u> i s t h e f r e q u e n c y , u \ = 4 7 r N e 2 / m i s t h e s q u a r e o f t