Characterization of Left-Handed Materials

Characterization of Left-Handed Materials Massachusetts Institute of Technology 6.635 lecture notes 1 Introduction 1. How are they realized? 2. Why the denomination “Left-Handed”? 3. What are their properties? 4. Does it really work? It has already been shown (see previous classes) that rings, or split-rings, can realize a negative permeability (µ < 0) over a certain frequency band. In addition to this, we need to realize a negative permittivity (² < 0). It has also been shown (see previous classes) that: • lossless: ²metal = 1 − ωp2 , ω2 where ωp = ne2 ²0 me (n: electron density, e: electron charge, me : effective mass of electrons). • lossy: ²metal ωp2 =1− . ω(ω + iγ) A typical transmission curve looks like shown in Fig. 1. T 1 PSfrag replacements no transmission (² < 0) transmission (² > 0) ωp ω Figure 1: Transmission curve for a plasma-like medium. . 1 2 Section 2. Why “left-handed”? With these characteristics, ² < 0 has been realized already at infrared frequencies (where metals behave like plasmas). Problem: how to realize it at GHz frequencies? Solution: by reducing n, the electron density. One way of doing this is to confine the electrons in space. This can be achieved by an array of rods for example, as shown in Fig. 2. a r PSfrag replacements Figure 2: Array of rods confining the electrons in space. Note: it is important that the wires are thin, so as to reduce the radiation interaction and allow penetration into the structure. Effect of the wires: to reduce n to neff = n πr2 . a2 (1) Finally, note also that the rods have to be parallel to the electric field. This, plus the (known already) fact that rings have to be perpendicular to the magnetic field, gives an idea on how to realize physically LH metamaterials (see Fig. 3). 2 Why “left-handed”? At this point, we have a metamaterial which can realize ² < 0, µ < 0 . (2) We shall now see what does it imply on the electromagnetic fields. Let us write Maxwell’s curl equations for plane wave solutions and time harmonic notations: k̄ × Ē(r̄) = ωµH̄(r̄) , (3a) k̄ × H̄(r̄) = − ω²Ē(r̄) . (3b) 3 0.5 mm 5 mm z 1.3 cm x εs FRONT BACK (a) 1 mm y εs x (b) Figure 3: A realization for LH material. In standard materials, Eq. (3) implies that the tryad (Ē, H̄, k̄) forms a right-handed system. However, under Eq. (2), we will have: (Ē, H̄, k̄) form a left-handed (LH) tryad. However, the time average Poynting power is still < S̄(r̄) >= 1 <{Ē(r̄) × H̄ ? (r̄)} 2 and remains in the same direction so that we have the set up shown in Fig. 4. Characteristics: • k̄ is in the phase velocity direction. • phase velocity and energy flux are in opposite directions. 3 Properties of LH media Some know characteristics are: • Reversed Doppler effect (track the phase), • Reversed Čerenkov radiation (cf. 6.632), (4) 4 3.1 Reversed Snell’s law Ē k̄ S̄ PSfrag replacements H̄ Figure 4: Electric field (Ē), magnetic field (H̄), wave-vector (k̄) and Poynting power (S̄) in an LH medium. • Negative index of refraction. This last item is very significant, and we shall spend some time discussing it. The index of refraction of a medium is defined as n= √ ² r µr , (5) or, writing explicitly the frequency dependence (cf. later), n(ω) = p ²r (ω) µr (ω) . (6) For those frequencies inside the left-handed band (i.e. in the band where ² < 0 and µ < 0), we can write: ²(ω) < 0 ⇒ µ(ω) < 0 ⇒ ²(ω) = |²(ω)| eiπ , µ(ω) = |µ(ω)| eiπ , (7a) (7b) Eventually, we write n from Eq. (6): n= 3.1 p |²(ω)µ(ω)| eiπ = − p |²(ω)µ(ω)| . Reversed Snell’s law An important consequence of this fact is the reversal of Snell’s law. • Ray diagram: (8) 5 n>0 PSfrag replacements θr −θr θi n<0 • k̄ diagram for an LH medium: PSfrag replacements kx k̄2 k̄1 kx S̄ k1z 3.2 k2z Energy Traditionally, the energy is given by W = ²E 2 + µH 2 . (9) What happens if ² < 0 and µ < 0? Is W < 0? Actually no, but this direct conclusion from Eq. (9) shows that this equation is not valid as is. In fact, these materials have to be modeled by frequency dispersive permittivity and permeability. In that case, the relation of Eq. (9) becomes (from Poynting’s theorem): W = ∂(²ω) 2 ∂(µω) 2 E + H ∂ω ∂ω (10) and we must have: ∂(²ω) > 0, ∂ω ∂(µω) > 0. ∂ω (11a) (11b) 6 3.3 Properties of an LHM slab When LH materials are studies as bulk materials, two models are commonly used for the permittivity/permeability: 1. Drude model: ²r = 1 − 2 ωep , ω(ω + iγe ) (12a) µr = 1 − 2 ωmp , ω(ω + iγm ) (12b) which is schematically represented in Fig. 5. Real(eps/eps0) Imag(eps/eps0) 10 0.02 0.015 0 0.01 −10 0.005 −20 0 −0.005 −30 −0.01 −40 wp = 100 GHz wp = 266.5 GHz wp = 500 GHz wp = 1000 GHz −50 0 0.2 0.4 0.6 0.8 1 w/w0 1.2 1.4 1.6 1.8 wp = 100 GHz wp = 266.5 GHz wp = 500 GHz wp = 1000 GHz −0.015 2 (a) <(²). −0.02 0 0.2 0.4 0.6 0.8 1 w/w0 1.2 1.4 1.6 1.8 2 (b) =(²). Figure 5: Permittivity for various values of ωep in the Drude model (f0 = 30 GHz, ωep = ωp ). 2. Resonant model: ²r = 1 − µr = 1 − 2 − ω2 ωep eo , 2 + iγ ω ω 2 − ωeo e (13a) 2 − ω2 ωmp mo , 2 2 ω − ωmo + iγm ω (13b) where ω(em)o are the electric/magnetic resonant frequencies and ω(em)p are the electric/magnetic plasma frequencies. An illustration of this model is given in Fig. 6. 3.3 Properties of an LHM slab Let us consider the situation depicted in Fig. 7. 7 Imag(eps/eps0) Real(eps/eps0) 100 1 wp = 266.5 GHz wp = 500 GHz 80 60 0.6 40 0.4 20 0.2 0 0 −20 −0.2 −40 −0.4 −60 −0.6 −80 −0.8 −100 wp = 266.5 GHz wp = 500 GHz 0.8 −1 0 0.2 0.4 0.6 0.8 1 w/w0 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 (a) <(²). 0.8 1 w/w0 1.2 1.4 1.6 1.8 (b) =(²). Figure 6: Permittivity for various values of ωep in the resonant model of Eq. (13a) (f0 = 30 GHz, ωep = ωp , ω0 = 100GHz). x y z PSfrag replacements Figure 7: LH slab in free-space. 2 8 3.3 Properties of an LHM slab Let us consider the case for which ²r = µr = −1 (working at the right frequency). 1. From simple ray diagrams, using the reversed Snell’s law: PSfrag replacements x #0 #1 #2 I2 S z I1 d We see that is the source is close enough to the slab (distance< d), the slab will produce two images, one inside the slab and one outside. The distance from source to the second image is S − I2 = 2d . 2. Rigorous calculation: Let us consider a TE wave impinging on this slab. We write, for a single interface: Ē0 =ŷE0 eikx x−iωt [eikz z + re−ikz z ] , Ē1 =ŷE0 eikx x−iωt te ikz0 z (14a) (14b) . We need to match the boundary conditions and, for simplicity, we set the boundary to be at z = 0. We get: • Tangential Ē field: • Tangential H̄ field (H̄ = 0 eikz z + re−ikz z = teikz z . 1 iωµ ∇ × Ē): eikz z − re−ikz z = µ0 kz0 ikz0 z te . µ1 k z Upon solving, we get the reflection/transmission coefficient from free-space to the medium: 2µ1 kz , µ1 kz + µ0 kz0 µ1 kz − µ0 kz0 r= . µ1 kz + µ0 kz0 t= (15a) (15b) 9 In a similar way, the reflection/transmission coefficients from the medium to free-space are: 2µ0 kz0 , µ1 kz + µ0 kz0 µ0 kz0 − µ1 kz r0 = . µ1 kz + µ0 kz0 t0 = (16a) (16b) In order to obtain the field inside the slab, we shall compute the transmission coefficient as: 0 0 0 2 0 0 T = teikz d t0 + teikz d r0 eikz d r0 eikz d t0 + tt0 (r0 )2 e5ikz d + · · · 0 = tt0 eikz d ∞ X 0 2 0 (r0 )n e2inkz d = n=0 tt0 eikz d . 1 − r 0 2 e2ikz0 d (17) Using the previous expressions, we obtain 0 T = 4µµ1 kz kz0 eikz d . (µ1 kz + µ0 kz0 )2 − (µ0 kz0 − µ1 kz )e2ikz0 d (18) For the specific case of ²r = µr = −1: 0 T = −4kz kz0 eikz d . (−kz + kz0 )2 − (kz0 + kz )e2ikz0 d (19) At this point, we need to define kz and kz0 : Propagating waves (k > kρ ) q k 2 − kρ2 kz kz = kz0 kz0 = − Relation q k 2 − kρ2 Evanescent waves (k < kρ ) kz = i q kρ2 − k 2 kz0 = i q kρ2 − k 2 kz0 = −kz Performing the calculation, we get: • Propagating waves (kz0 = −kz ): T = 4kz2 e−ikz d = e−ikz d . 4kz2 T = −4kz2 eikz d = e−ikz d . −4kz2 e2ikz d • Evanescent waves (kz0 = kz ): kz0 = kz 10 Section 4. Does it really work? Therefore, for all waves, we get: T = e−ikz d . (20) Conclusions: • Evanescent waves are amplified by the medium. • Propagating waves are “backward waves”. • Taking an infinite amount of those creates a source. • Two images can be formed, like shown in the ray diagram of the previous subsection. 4 Does it really work? The theoretical predictions have been verified in experiments. To date, essentially two experiments have been done: 1. Prism, 2. Gaussian beam. 4.1 Prism experiment First experiment (2001). Later reproduced. LHM RHM PSfrag replacements 4.2 Gaussian beam experiment First publication was theoretical. Later experiments followed, but still need to be improved. 11 RHM LHM PSfrag replacements