The Vacuum Impedance and Unit Systems

IEICE TRANS. ELECTRON., VOL.E92–C, NO.1 JANUARY 2009 3

PAPER Special Section on Recent Progress in Electromagnetic Theory and its Application The Vacuum Impedance and Unit Systems

Masao KITANO†,††a), Member

SUMMARY In the electromagnetic theory, the vacuum impedance Z0 cases. However conventionally the asymmetric pairs (c0,μ0) is a universal constant, which is as important as the velocity of light c0 or (c0,ε0) are used and SI equations become less memo- in vacuum. Unfortunately, however, its significance is not appreciated so rable. well and sometimes the presence itself is ignored. It is partly because in the In this paper, we reexamine the structure of electro- Gaussian system of units, which has widely been used for long time, Z0 is a dimensionless constant and of unit magnitude. In this paper, we clarify that magnetism in view of the SI (The International System of Z0 is a fundamental parameter in electromagnetism and plays major roles Units) system and find that Z0 plays very important roles in the following scenes: reorganizing the structure of the electromagnetic as a universal constant. In this process we also find that a formula in reference to the relativity; renormalizing the quantities toward natural unit systems starting from the SI unit system; and defining the mag- wide-spread belief that the Gaussian system of units is akin nitudes of electromagnetic units. to natural unit systems and suitable for theoretical studies is key words: vacuum impedance, wave impedance, Maxwell’s equation, SI just a myth. unit system, Gaussian unit system, metamaterial The recent development of a new type of media called metamaterials [3] demands the reconsideration of 1. Introduction wave impedance. In metamaterials, both permittivity  and permeability μ can be varied from values√ for vacuum = The notion of vacuum impedance was introduced in late and thereby the phase velocity vph 1/ μ and wave 1930’s by Schelkunoff [1] in the study of wave propagation. impedance Z = μ/ can be adjusted independently. With It is defined as the amplitude ratio of the electric and mag- the control of wave impedance the reflection can be reduced netic fields of plane waves in vacuum, Z0 = E/H, which has or suppressed [5]. the dimension of electrical resistance. It is also called the characteristic impedance of vacuum or the wave resistance 2. Relativistic Pairs of Variables of vacuum. Due to the historical reasons, it has been rec- ognized as a special parameter for engineers rather than a In relativity, the space variable x and the time variable t are universal physical constant. Compared with the famous for- combined to form a 4-vector (c t, x). The constant c ,which mula [2] for the velocity of light c in terms of the vacuum 0 0 0 has the dimension of velocity, serves as a factor matching permittivity ε and the vacuum permeability μ , 0 0 the dimensions of time and space. When we introduce a ≡ 1 normalized variable τ c0t, the 4-vector is simplified as = √ c0 , (1) (τ, x). In this form, space and time are represented with the μ0ε0 same dimension. the expression for the vacuum impedance Normally this is done offhandedly by setting c0 = 1.  However, this procedure is irreversible; c0 becomes dimen- μ0 sionless and the dimensional distinction between space and Z0 = , (2) ε0 time is lost. It is desirable to introduce normalized quantities such as τ when we compare the different systems of units. is used far less often. It is obvious when we look up index Here we introduce notation for dimensional analysis. pages of textbooks on electromagnetism. A possible reason When the ratio of two quantities X and Y is dimensionless is perhaps that the Gaussian system of units, in which Z0 is (just a pure number), we write X ∼D Y and read “X and Y are a dimensionless constant and of unit magnitude, has been D dimensionally equivalent.” For example, we have c0t ∼ x. used for long time. If a quantity X can be measured in a unit u, we can write As we will see, the pair of constants (c , Z ) can be D D 0 0 X ∼ u. For example, for l = 2.5mwehavel ∼ m. conveniently used in stead of the pair (ε ,μ )formany 0 0 With this notation, we can repeat the above discussion. D D Manuscript received April 18, 2008. For c0t ∼ x, instead of recasting t ∼ x by forcibly setting † The author is with the Department of Electronic Science and c0 = 1, we introduce a new normalized quantity τ = c0t and Engineering, Kyoto University, Kyoto-shi, 615-8510 Japan. D †† have τ ∼ x. Accordingly velocity v and c0 is normalized as The author is with CREST, Japan Science and Technology = = = Agency, Kawaguchi-shi, 332-0012 Japan. v˜ v/c0 andc ˜0 c0/c0 1, respectively. a) E-mail: [email protected] From the relativistic point of view, the scalar potential DOI: 10.1587/transele.E92.C.3 φ and the vector potential A are respectively a part of a uni-

Copyright c 2009 The Institute of Electronics, Information and Communication Engineers IEICE TRANS. ELECTRON., VOL.E92–C, NO.1 JANUARY 2009 4

we will see below, the vacuum impedance Z0 (or the vac- uum admittance Y0 = 1/Z0) plays the role to connect the quantities in the F and S series via the constitutive relations.

3.1 Constitutive Relation

The constitutive relations for vacuum, D = ε0E and H = −1B μ0 , can be simplified by using the relativistic pairs of Fig. 1 The relativistic pairs of quantities are arranged as a diagram, the variables as =     rows of which correspond to the orders of tensors (n 1, 2, 3, 4). In the E D left column, the quantities related to the electromagnetic forces (the F se- c0 = Z0 . (5) ries), and in the right column, the quantities related to the electromagnetic c0B H sources (the S series) are listed. The exterior derivative “d” connects a pair of quantities by increasing the tensor order by one. These differential The electric relation and magnetic relation are united under relations correspond to the definition of (scalar and vector) potentials, the the sole parameter Z0. More precisely, with Hodge’s star Maxwell’s four equations, and the charge conservation. Hodge’s star oper- operator “∗” (see Appendix) [8], [9], it can be written as ator “∗” connects (E, c0B)and(H, c0D) pairs. This corresponds to the constitutive relations for vacuum and here appears the vacuum impedance (E, c0B) = ∗Z0(H, c0D). (6) Z0 = 1/Y0 as the proportional factor.

3.2 Source-Field Relation fied quantity. Considering φ ∼D V, A ∼D Vs/m, we introduce a pair of quantities: We know that the scalar potential Δφ induced by a charge

D Δq = Δv is (φ, c0A) ∼ V. (3) 1 Δv Similarly, we introduce other pairs: Δφ = , (7) 4πε0 r D (E, c0B) ∼ V/m, where r is the distance between the source and the point of (H, c D) ∼D A/m, observation. The charge is presented as a product of charge 0 density  and a small volume Δv. Similarly a current mo- J ∼D 2 ( , c0) A/m , (4) ment (current times length) JΔv generates the vector poten- tial where E, B, H, D, J,and are the electric field, mag- μ JΔv netic flux density, magnetic field strength, electric flux den- ΔA = 0 . (8) sity, current density, and charge density, respectively. More 4π r precisely those pairs are anti-symmetric tensors in the 4- The relations (3) and (4) are united as dimensional space as defined in Appendix.     We have seen that the constant c0 appears when we Δφ Z c  = 0 0 Δv. (9) form a relativistic tensor from a pair of non-relativistic elec- Δ(c0A) 4πr J tromagnetic quantities. In Fig. 1, such relativistic pairs are listed according to their tensor orders. We will find that this We see that the vacuum impedance Z0 plays the role to relate table is very helpful to overview the structure of electromag- the source (J, c0)Δv and the resultant fields Δ(φ, c0A)ina netism [6], [7]. unified manner.

3. Roles of the Vacuum Impedance 3.3 Plane Waves

In this section we show some examples for which Z plays For linearly polarized plane waves propagating in one di- 0 = important roles. The impedance (resistance) is a physical rection in vacuum, a simple relation E cB holds. If we = −1 = quantity by which voltage and current are related. In the SI introduce H ( μ0 B) instead of B,wehaveE Z0H.This ff system, the unit for voltage is V(= J/C) (volt) and the unit relation was introduced by Schelkuno [1] in 1938. The for current is A(= C/s) (ampere). We should note that the reason why H is used instead of B is as follows. The bound- latter is proportional to and the former is inversely propor- ary conditions for magnetic fields at the interface of media = = tional to the unit of charge, C (coulomb). We also note from 1and2areH1t H2t (tangential) and B1n B2n (normal). Fig. 1 and Eqs. (3) and (4) that the units for quantities related For the case of normal incidence, which is most important to the electromagnetic forces (left column, labeled F series) practically, the latter condition becomes trivial and cannot are proportional to the volts. On the other hand, the units be used. Therefore H is used more conveniently. The mixed for quantities related to the electromagnetic sources (right use of the quantities (E and H) of the F and S series invite column, labeled S series) are proportional to the ampere. As Z0 unintentionally. KITANO: THE VACUUM IMPEDANCE AND UNIT SYSTEMS 5

2    Z0 = {Z0}Ω=k {Z0}Ω = {Z0} Ω . (14) 3.4 Magnetic Monopole and the Fine-Structure Constant The numerical value must be changed from {Z0} = 377 to  2 {Z0} = 377k . For example, we could choose a new√ ohm Let us compare the force between electric charges q and   D Ω so that Z = 1 Ω is satisfied by setting k ∼ 1/ 377. that between magnetic monopoles g (∼ Vs = Wb). If these 0 Conversely, fixing {Z } to a particular number implies the forces are the same for equal distances r, i.e., q2/(4πε r2) = 0 0 determination of the magnitude of units (Ω, V, A, and oth- g2/(4πμ r2), we have the relation g = Z q. With this rela- 0 0 ers) in electromagnetism. tion in mind, the Dirac monopole g [10], whose quantiza- 0 Once k,or{Z } is fixed, the numerical values for quan- tion condition is g e = h, can be beautifully expressed in 0 0 tities in the F series are multiplied by k and those in the S terms of the elementary charge e as series are divided by k. The sole parameter k or {Z0} deter- h h Z e mines the numerical relation between the F and S series. g = = (Z e) = 0 , (10) 0 2 0 Coulomb’s law for charges q and q can be rewritten e Z0e 2α 1 2 as h = π where 2 is Planck’s constant. The dimensionless pa- 1 q q = Z e2 h = e2 c ∼ F = 1 2 = q E = {q } ×{E} rameter α 0 /2 /4πε0 0 1/137 is called the 2 2 2 As V/m fine-structure constant, whose value is independent of unit 4πε0 r = { }  ×{ }  systems and characterizes the strength of the electromag- q2 A s E V /m, (15) netic interaction. The fine-structure constant itself can be − where E = (4πε ) 1(q /r2) is the electric field generated by represented more simply with the use of Z .Further,by 0 1 0 q .Wesee introducing the von Klitzing constant (the quantized Hall 1 = 2  −1  resistance) RK h/e , the fine-structure constant can be ex- {q2} = k {q2}, {E} = k{E}, (16) pressed as α = Z0/2RK [11]. We have learned that the use and find that the numerical value for the charge q and that of Z0 helps to keep SI-formulae in simple forms. for the electric field E will be changed reciprocally. We also { } = −2{ } { } = 2{ } 4. The Magnitude of the Unit of Resistance note ε0 k ε0 and μ0 k μ0 . In the SI, the ampere is defined in terms of the force F Here we consider a hypothetical situation where we are al- between the parallel two wires carrying the same amplitude = 2 lowed to redefine the magnitudes of units in electromag- of current I.WehaveF/l μ0I /(2πr), where r is the = netism. separation and l is the length of wires. Substituting F × −7 = = = = × −7 The product of the unit of voltage and that of current 2 10 N, r l 1m,I 1A,wegetμ0 4π 10 H/m. { } { } should yield the unit of power, 1 V × 1A = 1W = 1J/s, Thus the numerial value μ0 (or Z0 )isfixed. { } which is a fixed quantity determined mechanically, or out- We could determine ε0 by the force between charges side of electromagnetism. Thus, a new volt V and a new with the same magnitude. In Giorgi’s unit system (1901), ampere A must be defined so as to satisfy which is a predecessor of the MKSA unit system or the SI system, k was fixed by determining the magnitude of the   −1 A = kA, V = k V, (11) ohm. The way of determination {Z0} has been and will be changed according to the development of high precision in terms of the currently used V and A, where k ( 0) is a measurement technique. scaling factor. Accordingly a new ohm Ω must be redefined as 5. Toward Natural Unit Systems Ω = −2Ω k . (12) We have seen that normalizing quantities we can reduce the number of base units or the dimension. As shown in Ta- We denote the numerical value as {A}≡A/u, when we ble 1(a), first we introduce a new set of normalized quanti- measure a physical quantity A with a unit u. For example, ties, X˜ = c X, derived from SI quantities X. We can remove for l = 1.3 m we write {l} = l/m = 1.3. We can have another 0 the unit for time and we only need three base units; the am- numerical value {A} ≡ A/u, when we measure the same pere, volt, and meter. quantity A in a different unit u. Now we have the relation Further, as seen in Table 1(b), when we introduce a   ∗ = A = {A}u = {A} u . (13) set of normalized quantities, X Z0X, by multiplying Z0, only the volt and meter are required. By normalizing the It should be stressed that the physical quantity A itself is quantities with the universal constants, c0 and Z0,wehavea independent of the choice of units. What depends on the simplified set of Maxwell’s equations: choice is the numerical value {A}. ∗ ∗ ∗ ∗ ∂D˜ ∗ In the SI system, from the definition of c0 and μ0,the ∇ · D˜ = ˜ , ∇ × H˜ = + J , ∂τ vacuum impedance is represented as Z0 = μ0/ε0 = c0μ0 = × × −7 = × ∂B˜ (299 792 458 m/s) (4π 10 H/m) (119.916 983 2 ∇ · B˜ = 0, ∇ × E = − , (17) π) Ω ∼ 377 Ω. In our new system, with (12) we have ∂τ IEICE TRANS. ELECTRON., VOL.E92–C, NO.1 JANUARY 2009 6

Table 1 Pairs of quantities in electromagnetism and their units (dimension). Quantities X normalized ∗ with c and Z0 are accented as X˜ = c0X and X = Z0X, respectively. As seen in (a) and (b), the variety of units is reduced owing to the normalization. Gaussian unit√ system (c) and√ the modified√ Gaussian√ unit system (d) are presented with normalized quantities: Sˆ = S/ ε0, Fˆ = F ε0, S´ = S/ μ0, F´ = F μ0, where S (F) represents a quantity in the S (F) series. We notice an irregularity in the fourth row of the Gaussian system. √ √ (φ, A˜)V (φ, A˜)V (φ,ˆ A´) √N (φ,ˆ A´) √N (E, B˜ )V/m (E, B˜ )V/m (Eˆ , B´ ) √N/m (Eˆ , B´ ) √N/m ∗ ∗ (H, D˜ )A/m (H , D˜ )V/m (H´ , Dˆ ) N/m (H´ , Dˆ ) N/m ∗ ∗ √ √ (J, ˜)A/m2 (J , ˜ )V/m2 (c−1Jˆ, ˆ) N/m2 (J´, ˆ) N/m2 (a) c0-normalization (b) (c0, Z0)-normalization (c) Gaussian (d) modified Gaussian with the constitutive relation, D˜ ∗ = E and H∗ = B˜ . universal constant. Considering τ = ct, this set of equations resembles to the In the cgs electrostatic system of units (esu), Maxwell equations in the Gaussian system of units except Coulomb’s law is expressed as for the rationalizing factor 4π [See (29)]. However there is 1 q q 1 qˆ qˆ F = 1 2 = 1 2 . (18) a significant difference; the factor 1/c is missing in the cur- 2 2 0 4πε0 r 4π r rent density term. We will return to this point later. It should Thus, the normalized charge be stressed that a natural system of units can be reached q C √ √ from the SI system by normalizations without detouring via qˆ = √ ∼D √ = Jm= Nm, (19) ε the Gaussian system. 0 F/m The number of basic units has been reduced from four is a quantity expressed by mechanical dimensions only. (m, kg, s, A) to two (m, V) by introducing the quantities We can confirm that the numerical values for the esu and Heaviside-Lorentz systems are related as normalized with c0 and Z0. For further reduction toward a √  √  natural unit system [12], [13], and the gravitational con- Nm 5 {qˆ}esu = 4π{qˆ}HL  = 4π × 10 {qˆ}HL, stant G can be used for example. dyn cm where dyn = g · c/ms2 is the cgs unit of force. 6. The Gauss and Modified Heaviside-Lorentz Systems The quantities in the S series, each of which is propor- of Units √tional to the coulomb, C, can be normalized by division with ε0. On the other hand, the quantities in the F series, each ff The SI and the cgs (esu, emu, Gaussian) systems di er in of which is inversely proportional√ to C, can be normalized three respects. First, in the cgs unit systems, no fundamen- by multiplication with ε0. For example, E, D, B,andH, tal dimensions are supplemented to the three fundamental are normalized as √ √ dimensions for mechanics; length, mass, and time. On the √ N D N other hand in the SI (MKSA) system, a new fundamental Eˆ = E ε ∼D , Dˆ = √ ∼D , 0 m ε m dimension that for electric current is introduced. The cgs √ 0 √ √ systems contain three basic units, while the SI system con- D Ns H D N Bˆ = B ε ∼ , Hˆ = √ ∼ , 0 2 (20) tains four. m ε0 s Secondly, the cgs systems are irrational systems; the respectively. We have the constitutive relation factor (1/4π) is erased from Coulomb’s law but the factor Dˆ = Eˆ Hˆ = 2Bˆ 4π appears in the source terms of Maxwell’s equations in- , c0 , (21) stead. The SI is a rational system, which has the opposite and the normalized permittivityε ˆ0 = 1 and permeability appearance. The Heaviside-Lorentz system is a rationalized 2 μˆ0 = 1/c . The normalized vacuum impedance is version of the Gaussian system. 0  Thirdly, the base mechanical system for the cgs sys- μ ˆ = ˆ0 = 1 ∼D s tems is the cgs (centimeter, gram, and second) mechanical Z0 . (22) εˆ0 c0 m system. That for the SI system is the MKS (meter, kilogram, and second) system. For the cgs electromagnetic system√ of units (emu), S- In order to focus all our attention on the first respect, series quantities are multiplied√ by μ0 and F-series quan- i.e., the number of basic units, we will ignore the differences tities are divided by μ0. With this normalization, μ0 is in the last two respects. From now on, we pretend that all eliminated from the magnetic Coulomb law or the law of the cgs systems (esu, emu, and Gaussian) are constructed magnetic force between currents. The fields are normalized as rationally on the MKS mechanical system. Especially the √ √

Heaviside-Lorentz system thus modified or the MKS ver- B D N √ D N B´ = √ ∼ , H´ = H μ0 ∼ , sion of rationalized Gaussian system is a three-unit, rational μ0 m m system based on the MKS system. √ √ E N √ Ns To go from the SI system to the cgs systems, we have E´ = √ ∼D , D´ = D μ ∼D . 0 2 (23) to reduce the number of basic units by normalization with a μ0 s m KITANO: THE VACUUM IMPEDANCE AND UNIT SYSTEMS 7

The constitutive relations are unit system or because of the compatibility with relativity is almost pointless. We should remember that the Gaussian H´ = B´ D´ = −2E´ , c0 , (24) unit system was established in 1870s, when the relativity or the Lorentz transformation were not known yet. and we have the normalized permeabilityμ ´ = 1 and per- 0 The modified Gaussian system [14], in which J´ is mittivityε ´ = 1/c2. The normalized vacuum impedance is 0 0 adopted and the above seam is eliminated, has been pro-  posed but is rarely used. Table 1(d) contains quantities in μ´ 0 D m the modified Gaussian system. They differ uniformly by a Z´0 = = c0 ∼ . (25) √ ε´0 s factor ε0 from the (c0, Z0)-normalized quantities in col- umn (b). The Gaussian system of units is a combination of the esu and emu systems. For electrical quantities, the esu nor- 7. Summary and Discussion malization is used and for magnetic quantities, the emu nor- malization is used. Namely,√ we use Eˆ , Dˆ , B´ ,andH´ ,allof The important expression (1) for the velocity of light also which have the dimension N/m. The constitutive relations holds for the esu and emu systems: are simplified as 1 1 1 c = √ =  =  , (30) Dˆ = Eˆ , H´ = B´ , (26) 0 μ0ε0 μˆ 0εˆ0 μ´ 0ε´0  and we haveε ˆ0 = 1and´μ0 = 1. So far it looks nice be- but not for the Gaussian system; 1/ εˆ0μ´ 0 = 1  c0. cause electric and magnetic quantities are treated symmetri- The expression (2) for the vacuum impedance, is cally. Due to this apparent simplicity the Gaussian system rewritten as Zˆ0 = 1/c0 and Z´0 = c0 for the esu and emu sys- has been used so widely. However, there is an overlooked tems, respectively. Maxwell himself worked with the emu problem in the normalization√ of current density. It is nor- system when he found that light is an electromagnetic dis- Jˆ = J malized as / ε0 in the Gaussian system. The cur- turbance propagated according to electromagnetic laws [2]. rent density is the quantity primarily connected to magnetic√ For the emu system the dimensions of resistance and veloc- J´ = J fields and therefore it should be normalized as μ0 ity are degenerate. For the Gaussian system, the vacuum as for the emu system. Because of this miscasting, we have impedance reduces to unity, μ´ 0/εˆ0 = 1. an irregularity in the fourth row of Table 1(c). The Gaus- Thus there is no room for the vacuum impedance in the sian normalization happens to make the pairs of quantities cgs systems, which contains only three base units. However J relativistic except for the ( ,)pair. when we move to a unit system with four base units, the vac- The relativistic expression for the conservation of uum impedance Z0 should be recognized as a fundamental charge should be constant as important as the velocity of light, c0. ∗ As has been pointed out by Sommerfeld [6], the Gaus- ∂˜ ∗ + ∇ · J = 0, (27) sian system tends to veil the significance of D and H in vac- ∂(c t) 0 uum. Sometimes it is told that in vacuum only E and B have as for the cases of c0-or(c0, Z0)-normalization. In the Gaus- their significance and D and H lose their meaning. This ar- sian system, however, the non-relativistic expression gument is strongly misled by the use of Gaussian system of units. Considering the tensorial nature of quantities as ∂ˆ + ∇ · Jˆ = 0, (28) in (6), the constitutive relations for the Gaussian system are ∂t expressed as is adopted. As a practical system of units, it is a reasonable (Eˆ , B´ ) = ∗(H´ , Dˆ ). (31) (and perhaps unique) choice. This quirk can clearly be seen, when we compare the with Hodge’s star operator, “∗.” This relation represents im- Maxwell’s equations (17) in the natural system of units and portant geometrical relations [4] of electromagnetic fields, that for the Gaussian system: which can hardly be suggested by the simple vector rela- tions (26). 1 ∂Dˆ 1 ∇ · Dˆ = ,ˆ ∇ × H´ = + Jˆ, Now we have understood that without the help of Gaus- c0 ∂t c0 sian system, we can reach natural systems of units directly 1 ∂B´ from the SI system. We believe it’s time to say goodbye to ∇ · B´ = 0, ∇ × Eˆ = − . (29) the Gaussian system of units. We will not miss its simple- c0 ∂t ness if we use the vacuum impedance Z0 as a fundamental The factor 1/c0 in the current density term is a seam intro- constant. duced when the esu and emu systems are joined into the Gaussian system. Acknowledgements The common belief that the Gaussian system is supe- rior to the SI system because of the similarity to a natural We thank K. Shimoda, F.W. Hehl, M. Ueda, and V. Veselago IEICE TRANS. ELECTRON., VOL.E92–C, NO.1 JANUARY 2009 8 for helpful discussions. This work is partly supported by the defined as global COE program for “Photonics and electronics science X = e0 + X and engineering.” (α; )1 α , 0 (X; Y )2 = e ∧ X + Y, References 0 (Y ; β)3 = e ∧ Y + βσ, (A· 2) [1] S.A. Schelkunoff, “The impedance concept and its application to 1 2 3 problems of reflection, refraction, shielding, and power absorption,” where σ = e ∧ e ∧ e is the 3-form representing the vol- Bell Syst. Tech. J., vol.17, pp.17–48, 1938. ume element and “∧” represents the antisymmetric tensor [2] J.C. Maxwell, “A dynamical theory of electromagnetic field,” Royal = 3 3 e ∧ e product. Y j=1 k=1 ijkYi j k is a 2-form (in three Society Transactions, vol.155, pp.526–597, 1864. dimensional space) derived from Y ,whereijk is the Levi- [3] D.R. Smith, J.B. Pendry, and M.C.K. Wiltshire, “Metamaterials and Civita symbol. With these, the pair quantities in Fig. 1 are negative refractive index,” Science, vol.305, pp.788–792, 2004. [4] U. Leonhardt, “Optical conformal mapping,” Science, vol.312, defined as pp.1777–1780, 2006. A = − A [5] Y. Tamayama, T. Nakanishi, K. Sugiyama, and M. Kitano, “Obser- (φ, c0 ) ( φ; c0 )1, ff vation of Brewster’s e ect for transverse-electric electromagnetic (E, c0B) = (−E; c0B)2, waves in metamaterials: Experiment and theory,” Phys. Rev. B, H D = H D vol.73, pp.193104(4), 2006. ( , c0 ) ( ; c0 )2, [6] A. Sommerfeld, Electrodynamics — Lectures on Theoretical Physics, (J, c0) = (−J; c0)3. (A· 3) pp.45–54, pp.212–222, Academic Press, New York, 1952. [7] G.A. Deschamps, “Electromagnetics and differential forms,” Proc. The differential operator d is defined as IEEE, vol.69, pp.676–696, 1981. 3 [8] T. Frankel, The Geometry of Physics: An Introduction, 2nd ed., ∂ ∂ pp.118–123, Cambridge, 2004. d = e0 ∧ + ei ∧ . (A· 4) ∂x0 ∂xi [9] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrody- i=1 namics, Charge, Flux, and Metric, pp.143–162, Birkhauser,¨ Boston, 2003. andtheapplicationtoann-form results in an (n + 1)-form. [10] J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Read- For example, we have d(φ, c0A) = (E, c0B), which is ing, Mass., 1995. equivalent to E = −∇φ − ∂A/∂t and B = ∇ × A.The [11] F.W. Hehl, “Dimensions and units in electromagnetism,” arXiv: = / successive applications of d always yield zero (dd 0), physics 0407022, 2004. E B = [12] M.J. Duff, L.B. Okun, and G. Vemeziano, “Trialogue on the number therefore we have d( , c0 ) 0, which corresponds to of fundamental constants,” J. High Energy Phys. PHEP03, 023 2002; ∂B/∂t + ∇ × E = 0and∇ · B = 0. Furthermore, / arXiv:physics 0110060, 2002. d(H, c0D) = (J, c0) yields −∂D/∂t + ∇ × H = J and [13] F. Wilczek, “On absolute units, I: Choices,” Phys. Today, vol.58, ∇ · D = ,andd(J, c0) = 0 yields the conservation of pp.12–13, 2005. charge: ∂/∂t + ∇ · J = 0. [14] J.D. Jackson, Classical Electrodynamics, 3rd ed., pp.782–784, See footnotes for Tables 3 and 4 in Appendix, John Wiley & Sons, New Another important notation is Hodge’s star operator York, 1998. “∗,” which converts an n-form into a (4 − n)-form; [15] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, chapters 3 and 4, W.H. Freeman, New York, 1973. ∗ (α; X)1 = (X; α)3,

∗ (X; Y )2 = (−Y ; X)2, ∗ Y = Y · Appendix: Tensor Notations ( ; β)3 (β; )1. (A 5) From the second relation and (A· 3), the constitutive rela- In this appendix, we will explain mathematically the enti- tions are represented as (E, c0B) = ∗Z0(H, c0D). ties of Fig. 1. A more detailed description on the relativistic forms of electromagnetic equations are givin in [15]. With the basis vectors {e0, e1, e2, e3}, a space-time vector can be represented as Masao Kitano was born in Kyoto, Japan, on August 5, 1952. He received the B.S., M.S., and 3 Ph.D. degrees in electronic engineering from i (ct, x) = (ct)e0 + x, x = x ei. (A· 1) Kyoto University, Kyoto, Japan, in 1975, 1977, i=1 and 1984, respectively. In 1977 he joined the e · e = = Department of Electronics, Kyoto University The basis vectors satisfy μ ν gμν (μ, ν 0, 1, 2, 3). The and since 1999, he is a Professor at the De- nonzero elements are g00 = −1, g11 = g22 = g33 = 1. We partment of Electronic Science and Engineer- also introduce the dual basis vectors: {e0, e1, e2, e3}, with ing, Kyoto University. He spent the academic 0 i years 1984–1986 on leave at the Department of e = −e0, e = ei (i = 1, 2, 3). The quantities in electromagnetism are expressed by Physics, Princeton University. His research in- terest includes quantum optics, nearfield optics, optical pumping, electro- antisymmetric tensors of rank n (n-forms) in the four- magnetism. He received the Matsuo Science Prize in 2007. Dr. Kitano is a dimensional space [7]–[9]. For scalar fields α, β and 3- member of American Physical Society, the Physical Society of Japan, the dimensional vector fields X, Y , n-forms (n = 1, 2, 3) are Laser Society of Japan, and the Japan Society of Applied Physics.