New Physics of Metamaterials

Physica B 443 (2014) 114–119

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Physica B journal homepage: www.elsevier.com/locate/physb

New Physics of Metamaterials Zhong-Yue Wang Engineering Department, SBSC, No. 580 Songhuang Road, Shanghai 201703, China

art ic l e i nf o

a b s t r a c t

Available online 12 March 2014

Einstein utilized Lorentz invariance from Maxwell's equations to modify mechanical laws and establish the special theory of relativity. Similarly, we may have a different theory if there exists another covariance of Maxwell's equations. In this paper, we find such a new transformation where Maxwell's equations are still unchanged. Consequently, Veselago's metamaterial and other systems have negative phase velocities without double negative permittivity and permeability can be described by a unified theory. People are interested in the application of metamaterials and negative phase velocities but do not appreciate the magnitude and significance to the spacetime conception of modern physics and philosophy. & 2014 Elsevier B.V. All rights reserved.

Keywords: Metamaterial Maxwell's equations Negative momentum Advanced potential Faster than light

E2 ¼ p2 C 2 þ m20 C 4

1. Introduction Consider an inertial reference frame K 0 moves at a constant velocity V with respect to another inertial system K as shown in Fig. 1. For convenience, the three sets of axes are parallel and their relative motion is along the common x–x0 axis. The form of Maxwell's equations in K 0 and K does not change under the following Lorentz transformation: x  Vt x0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð1Þ

y0 ¼ y

ð2Þ

z0 ¼ z

ð3Þ

C2 ¼

1

ð4Þ

ð5Þ

εμ

Einstein assumed mechanical laws to satisfy this transformation and get 0 1 p ¼ mV

V ⪡C

-

m0 B C m0 V@m ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1  ðV 2 =C 2 Þ

E ¼ mC 2

E-mail address: [email protected] http://dx.doi.org/10.1016/j.physb.2014.03.002 0921-4526/& 2014 Elsevier B.V. All rights reserved.

Although the Lorentz transformation is seemingly strange and complex, it can reduce to the Galilean transformation at low speeds (V⪡C) x0 ¼ x V t

ð9Þ

y0 ¼ y

ð10Þ

z0 ¼ z

ð11Þ

t0 ¼ t

ð12Þ

which agrees with our classical intuition. Addition of velocities is

2

t  ðV=C Þx t 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð8Þ

ð6Þ

ð7Þ

dx0 dx ¼ V dt 0 dt

ð13Þ

However, nobody had ever thought of the following space-time transformation to Fig. 1 x þ Vt x0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð14Þ

y0 ¼ y

ð15Þ

0

z ¼z

ð16Þ

t þ ðV=C 2 Þx t 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð17Þ

because the consequence is inconsistent with common senses even in the classical limit. For example, x þ Vt x0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

V ⪡C

-

x þ Vt

ð18Þ

Z.-Y. Wang / Physica B 443 (2014) 114–119

115

ρ þ ðV =C 2 Þjx ρ0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2

2

1  ðV =C Þ

jx þ ρV j0x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 ðV 2 =C 2 Þ

j0y ¼ jy ;

j0z ¼ jz

ð30Þ

In K,

Fig. 1. Two inertial frames with a common axis.

dx0 ðdx=dtÞ þ V ¼ dt 0 1 þ ðV =C 2 Þðdx=dtÞ

dx þV dt

V⪡C

-

ð19Þ

Hereinafter, we show that Maxwell's equations are invariant under the incredible transformation (14)–(17) and develop a tentative mechanical theory to compare with experiments.

2. Anomalous invariance and Maxwell's equations Maxwell's equations in K are ∂Ex ∂Ey ∂Ez ρ þ þ ¼ ε ∂x ∂y ∂z

ð20Þ

∂Ez ∂Ey ∂Bx  ¼ ∂y ∂z ∂t

ð21  xÞ

∂Ex ∂Ez ∂By  ¼ ∂z ∂x ∂t

ð21  yÞ

∂Ey ∂Ex ∂Bz  ¼ ∂x ∂y ∂t

ð21  zÞ

∂Bx ∂By ∂Bz þ þ ¼0 ∂x ∂y ∂z

ð22Þ

∂Bz ∂By 1 ∂Ex  ¼ μjx þ 2 ∂y ∂z C ∂t

ð23  xÞ

∂Bx ∂Bz 1 ∂Ey  ¼ μjy þ 2 ∂z ∂x C ∂t

ð23  yÞ

∂By ∂Bx 1 ∂Ez  ¼ μjz þ 2 ∂x ∂y C ∂t

ð23  zÞ

ð24Þ

∂ ∂ ¼ ∂y ∂y0

ð25Þ

∂ ∂ ¼ ∂z ∂z0

ð26Þ

∂ 1 ∂ ∂ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þV 0 ∂t ∂t 0 ∂x 2 2 1  ðV =C Þ

ð31Þ

E0y  VB0z Ey ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð32Þ

E0z þ VB0y Ez ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðV 2 =C 2 Þ

ð33Þ

Bx ¼ B0x

ð34Þ

B0y þ ðV=C 2 ÞE0z By ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð35Þ

B0z ðV=C 2 ÞE0y Bz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðV 2 =C 2 Þ

ð36Þ

ρ0 ðV=C 2 Þj0x ρ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð37Þ

j0x  ρ0 V jx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð38Þ

jy ¼ j0y

ð39Þ

jz ¼ j0z

ð40Þ

1  ðV 2 =C 2 Þ

Substituting Eqs. (24)–(26),(31)–(33),(37) and (5) into (20),   E0y  VB0z E0z þV B0y 1 ∂ V ∂ ∂ ∂ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 0 E0x þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ∂x C ∂t ∂y ∂z 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ

ρ0  ðV=C 2 Þj0x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 1  ðV 2 =C 2 Þ

ð41Þ

0 0 ∂E0x ∂Ey ∂E0z ∂B0z ∂By 1 ∂E0x þ 0 þ 0 V  0 2 0 0 0 ∂x ∂y ∂z ∂y ∂z C ∂t

According to (14)–(17),   ∂ 1 ∂ V ∂ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 ∂x ∂x' C ∂t' 1  ðV 2 =C 2 Þ



Ex ¼ E0x

 ð27Þ

! ¼

ρ0  V μj0x ε

ð42Þ

i.e. 0 ∂E0x ∂Ey ∂E0z ρ0 þ 0þ 0 ¼ 0 ∂x ∂y ∂z ε

ð200 Þ

0 ∂B0z ∂By 1 ∂E0  0 ¼ μjx ' þ 2 0x 0 ∂y ∂z C ∂t

ð23  x0 Þ

Likewise, (21-x) is now   E0z þ VB0y E0y  VB0z ∂ ∂ 1 ∂ ∂ q q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0  ¼  þ V ∂y0 ∂z0 ∂t 0 ∂x' x 2 2 2 2 2 2 1  ðV =C Þ 1  ðV =C Þ 1  ðV =C Þ ð43Þ

Physical quantities in K 0 , E0x ¼ Ex ;

Ey þV Bz E0y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1  ðV 2 =C 2 Þ

Ez  VBy E0z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð28Þ

B0x ¼ Bx ;

By ðV=C 2 ÞEz B0y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1  ðV 2 =C 2 Þ

Bz þ ðV=C 2 ÞEy B0z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðV 2 =C 2 Þ

ð29Þ

0 0 ∂E0z ∂Ey ∂B0x ∂B0x ∂By ∂B0z  þ þ V þ þ 0 ∂y0 ∂z0 ∂t ∂x0 ∂y0 ∂z0

! ¼0

ð44Þ

i.e. 0 ∂E0z ∂Ey ∂B0  ¼  0x ∂y0 ∂z0 ∂t

ð21  x0 Þ

116

Z.-Y. Wang / Physica B 443 (2014) 114–119

0 ∂B0x ∂By ∂B0z þ 0 þ 0 ¼0 0 ∂x ∂y ∂z

ð220 Þ

Eq. (21-y):   E0z þ VB0y ∂E0x 1 ∂ V ∂  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ∂x C ∂t ∂z 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ   0 2 0 1 ∂ ∂ By þ ðV=C ÞEz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þV ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ∂t ∂x' 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ

ð45Þ

 ∂E0  ∂B ∂E0x 1 1 y z  ¼ 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ 2 2 ∂z0 1 ðV 2 =C 2 Þ ∂x0 ∂t 0 1  ðV =C Þ ð46Þ

Eq. (23-y):   0 2 0 ∂B0x 1 ∂ V ∂ Bz  ðV=C ÞEy  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ∂x C ∂t ∂z 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ   E0y  VB0z 1 ∂ ∂ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ V ¼ μj0y þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ∂t ∂x 1  ðV 2 =C 2 Þ C 2 1 ðV 2 =C 2 Þ

! ! 0 ∂B0x 1 V 2 ∂B0z 1 V 2 ∂Ey 0   1   ¼ μ j þ 1  y 0 ∂z0 1  ðV 2 =C 2 Þ C 2 ∂x0 C 2 ∂t C 2 1  ðV 2 =C 2 Þ ð54Þ

0

∂B0y ∂E0x ∂E0z  0¼ 0 0 ∂z ∂x ∂t

ð21  y0 Þ

Eq. (21-z):   E0y  VB0z 1 ∂ V ∂ ∂ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 E0x þ 0 2 ∂t 0 ∂x ∂y 2 2 2 2 C 1  ðV =C Þ 1  ðV =C Þ   0 2 0 1 ∂ ∂ Bz  ðV=C ÞEy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þV 0 ∂t ∂x 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ ! 0 ! V 2 ∂Ey ∂E0x 1 V 2 ∂B0z  ¼ 1 2 1 2 ∂x0 ∂y0 ∂t 0 C C 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ 1

∂E0y ∂E0x  ∂x0 ∂y0

¼

ð47Þ

ð48Þ

0

∂B0x ∂B0z 1 ∂Ey  ¼ μj0y þ 2 0 ∂z0 ∂x0 C ∂t

ð21  z0 Þ

ð23  y0 Þ

Eq. (23-z):   0 2 0 1 ∂ V ∂ By þ ðV=C ÞEz ∂B0x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 þ 0 2 ∂t 0 ∂x ∂y C 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ   E0z þ VB0y 1 ∂ ∂ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ μj0z þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 þV ∂t ∂x0 C 2 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ

ð55Þ

! 0 ! V 2 ∂By ∂B0x 1 V 2 ∂E0z 0   1 2  ¼ μjz þ 1 2 ∂x0 ∂y0 ∂t 0 C C 1  ðV 2 =C 2 Þ C 2 1  ðV 2 =C 2 Þ 1

ð56Þ ∂B0y ∂x0

∂B0  0z

ð53Þ



∂B0x 1 ∂E0 ¼ μj0z þ 2 0z ∂y0 C ∂t

ð23  z0 Þ

Eq. (22):

Maxwell's equations remain in the same form under two transformations (1)–(4) and (14)–(17). It is a pity that the latter had never been studied.

  2 0 2 0 0 0 1 ∂ V ∂ ∂ By þ ðV=C ÞEz ∂ Bz  ðV=C ÞEy 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B þ þ þ ¼0 x 0 ∂x0 C 2 ∂t ∂y0 ∂z0 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ

3. New mechanics and measurable effects

∂t

ð49Þ 0 ∂B0x ∂By ∂B0z þ 0þ 0þ 0 ∂x ∂y ∂z

V C2

0 ∂E0z ∂Ey ∂B0x  0þ 0 0 ∂y ∂z ∂t

! ¼0

ð50Þ

i.e. 0 ∂B0x ∂By ∂B0z þ þ ¼0 ∂x0 ∂y0 ∂z0 0 ∂E0z ∂Ey ∂B0  ¼  0x ∂y0 ∂z0 ∂t

0

ð22 Þ

ð21  x0 Þ

Under this new transformation, the product of a physical quantity times the velocity V is reversed, e.g. j ¼  ρV;

Vector potential A ¼ 

ϕ C2

V

ð57Þ

Introduce a mechanical theory which is consistent with (14)–(17), as Einstein had done to special relativity and the Lorentz transformation (1)–(4). Hence, the momentum and total energy should be 0 1 p ¼  mV

V ⪡C

-

m0 B C  m0 V@m ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 1  ðV =C Þ

ð58Þ

Eq. (23-x): 2 0 2 0 0 0 ∂ Bz  ðV=C ÞEy ∂ By þðV=C ÞEz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ∂y ∂z' 1  ðV 2 =C 2 Þ 1  ðV 2 =C 2 Þ

  ρV 1 ∂ ∂ ¼ μqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0x 0 þV 0 ∂t ∂x 1  ðV 2 =C 2 Þ C 2 1  ðV 2 =C 2 Þ j0x 

0

ð51Þ

! 0 0 ∂B0z ∂By 1 ∂E0x V ∂E0x ∂Ey ∂E0z 0 0  ¼ μjx  μρ V þ 2 0 þ 2 þ þ ð52Þ ∂y0 ∂z0 C ∂t C ∂x0 ∂y0 ∂z0       Owing to ð1=C 2 Þ ¼ εμ (5) and ∂E0x =∂x0 þ ∂E0y =∂y0 þ ∂E0z =∂z0 ¼ ρ0 =ε (200 ), Eq. (52) is 0 ∂B0z ∂By 1 ∂E0  ¼ μj0x þ 2 0x ∂y0 ∂z0 C ∂t

ð23  x0 Þ

E ¼ mC 2

ð7Þ

E2 ¼ p2 C 2 þ m20 C 4

ð8Þ

It is surprising that the momentum p is in a direction opposite to the velocity V. That is to say, this new theory is symmetric to relativistic mechanics and Newtonian. In view of de Broglie's relation p ¼ ℏk, the wave vector k of a photon should be antiparallel to the arrival of the mass m ¼ ðℏω=C 2 Þ and energy ℏω (mass–energy equivalence). As electromagnetic waves consist of a 2 stream of photons, the phase velocity ðω=k Þk [1] is also negative. We can obtain Snell's law of refraction from conservation of the momentum component parallel to the interface for a single photon [2] (Fig. 2). ℏk1 sin γ 1 ¼ ℏk2 sin γ 2

ð59Þ

Z.-Y. Wang / Physica B 443 (2014) 114–119

117

Fig. 6. Reversed Cherenkov radiation.

Fig. 2. Snell's law.

Fig. 7. Doppler effect (the receiver is moving away from the source).

Fig. 8. Doppler effect (one approaches the other).

Fig. 3. Negative refraction.

Fig. 9. Inverse Doppler effect (they move away from each other).

Fig. 4. Momentum conservation.

Fig. 10. Inverse Doppler effect (one is moving towards the other).

ω  kV ω0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ω ðFig: 10Þ 1  ðV 2 =C 2 Þ

ð63Þ

Fig. 5. Cherenkov radiation.

4. Application: Veselago's materials and others If the momentum of a photon is opposite to the arrival of energy (red arrow), the light ray should be refracted on the same side of the incident beam otherwise the horizontal momentum of the photon is non-conservational (Fig. 3). Moreover, the Cherenkov effect can be deduced from the photon theory [3,4] and energy–momentum conservation (Fig. 4). For normal photons see Fig. 5. As to photons have a negative momentum see Fig. 6. Regular Doppler effect derived by Lorentz transformation:

ω  kV ω0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ω ðFig: 7Þ

ð60Þ

ω þ kV ω0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ω ðFig: 8Þ

ð61Þ

1 ðV 2 =C 2 Þ

1 ðV 2 =C 2 Þ

5. Physical meaning of advanced potentials

Inverse Doppler effect:

ω þ kV ω0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ω ðFig: 9Þ 1 ðV 2 =C 2 Þ

A method to realize (58) is to generate an even number of imaginary momenta so the final effective momentum is a negative quantity. For instance, single negative materials with εef f o 0 or μef f o 0 cause an imaginary wave vector(momentum) respectively and the combination results in a negative wave vector(momentum). This is just the mechanism of Veselago's material [5] to have negative refraction [6], reversed Cherenkov effect [7] and inverse Doppler effect [8]. But εef f o 0 and μef f o 0 are not necessary conditions. Actually, negative phase velocities sporadically appeared in other studies [9–12]. A practical structure is the slow wave device in vacuum electronics where the phase constant β and phase velocity ω=β can be negative [13]. They have nothing to do with Veselago's proposal εef f o 0 and μef f o 0.

ð62Þ

pffiffiffiffiffiffi Potentials with the time dependence t ðr=ð1= εμÞÞ and pffiffiffiffiffiffi t þ ðr=ð1= εμÞÞ are called retarded potentials and advanced potentials. At present, advanced potentials are deemed to violate the

118

Z.-Y. Wang / Physica B 443 (2014) 114–119

Fig. 11. Energy(mass) transfer.

principle of causality and are discarded although they are entirely pffiffiffiffiffiffi consistent with Maxwell's equations [14]. In fact, t þ ðr=ð1= εμÞÞ pffiffiffiffiffiffi can be rewritten as t ðr= ð1= εμÞÞ which implies that the phase pffiffiffiffiffiffi velocity is V p ¼  ð1= εμÞ o 0. “Advanced potentials” as potentials of negative phase velocities are not in conflict with causality.

The dispersion relation !

εef f ¼ ε0 1 

ω2p ω2  ω20



μef f ¼ μ0 1 

F ω2 ω2  ω2m

6. Energy transport and Poynting vector The energy ℏω and mass m ¼ ℏω=C 2 propagates from the source to the receiver, while the momentum p ¼  mV is negative in this new theory (Fig. 11). The momentum density of an EM field g ¼ Np ¼  NmV ¼ Nℏk

ðN is the number density of photonsÞ ð64Þ

is in the direction of k. In addition, 2

ðw ¼ Nℏω 4 0 is the density of energyÞ

ð65Þ

Therefore, the energy flux density defined as S ¼  wV

ð66Þ

like (57) and (58) is equal to gC 2 ¼ NℏkC 2 p k

2

ð67Þ

The Poynting vector S should be in the same direction of momentum density g and wave vector k (Fig. 11).

The form of Eq. (7) is still tenable in a theory of superluminal bodies [15] and the criterion for a particle or wave to exceed c ¼ 299; 752; 498 m=s should be V 2 4 c2

ð68Þ

p2 4 c2 m2

ð69Þ

ω2p ¼ constant ω20

ð76Þ

ℏ2 β

2

ðℏω=C 2 Þ2

4 c2

c2 C4

ð77Þ

ð78Þ

β2 4

ω2 c2

ω2 2 oc β2

ð79Þ

ð80Þ

describes slow phase velocity waves [13].

2

ðℏω=C 2 Þ2 k 4 ω2 2

2

ð75Þ

In vacuum,

7. Faster than light

k 4

c2

!  ω2p F ω2 1 2 1  ω2  ω2m ω  ω20

ð74Þ

This behaves as if a dielectric medium where c is replaced by C [17,18] and Eq. (71) should be applied. It makes against our aim to break the light barrier. Sometimes the momentum of a photon is p ¼ ℏβ [4] and the criterion should now be

β 2 4 ω2

ℏ2 k

ω2



pffiffiffiffiffiffiffiffiffiffiffi can meet Eq. (72) in the range ω o ωm =ð 1 F Þ,ω0 o ω⪡ωp (double negative) or ω o ωm , ω o ω0 (double positive). We plan to design suitable materials to observe [16] The angular frequency ω in a double positive material cannot be much less than ω0 otherwise !

εef f  ε0 1 þ

gC ¼  NmC V ¼  NℏωV ¼ wV 2

k ¼ ω εef f μef f ¼ 2

ð73Þ

ω2 c2

4 c2

c2 C4 ðC ¼ cÞ

ð70Þ

ð71Þ

ð72Þ

8. Conclusion We construct a new system from first principles to replace Veselago’s phenomenological theory. It can interpret all kinds of negative phase velocities which essentially represent a momentum antiparallel to the velocity. Retarded potentials and advanced potentials correspond to Lorentz invariance and the anomalous space-time transformation respectively. Faster than light energy is possible to

Z.-Y. Wang / Physica B 443 (2014) 114–119

119

exist in a metamaterial.

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