A METHOD OF VIRTUAL IMAGE FOR DETERMINATION OF NEGATIVE REFRACTION INDEX OF A NANOMEDIUM

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1 Roanian Repots in Physics 69, 44 (217) A METHOD OF VIRTUAL IMAGE FOR DETERMINATION OF NEGATIVE REFRACTION INDEX OF A NANOMEDIUM V. SERGENTU 1, V. URSAKI 2 1 Institute of Applied Physics of the Acadey of Sciences of Moldova, 5 Acadey Steet, MD-228 Chisinau, Moldova, E-ail: vsegentu@yahoo.co 2 Institute of Electonic Engineeing and Nanotechnology of the Acadey of Sciences of Moldova, Acadey st. 3/3, MD-228 Chisinau, Moldova, E-ail: usaki@yahoo.co Received May 15, 216 Abstact. In this pape we popose a ethod fo easuing the negative efaction index of fils of optically tanspaent ateials. The ethod is based on ecoding the diection of popagation and the shifting of the ay eflected fo the fil. It is shown that the ethod can be applied fo both acoscopic and nanoetic saples. Howeve, a pecise contol of paaetes of the adiation souce is needed fo using this ethod in the case of nanoetic saples. The conditions fo a coposite ediu to acquie a negative efactive index ae discussed. Key wods: optical nanoediu, etaateial, negative efaction index, wave font, quasi-plane wave, optical ipedance. 1. INTRODUCTION New nanoedia with unusual popeties ae nowadays in the focus of attention of any eseach goups. Optical nanoedia with negative efaction index ae of especial inteest [1, 2, 3]. Howeve, all the used etaateials epesent systes of heteogeneous eleents with substantially diffeent popeties, and thee is a contadiction when dealing with such systes. On the one hand, one has to use coponents with daatically diffeent popeties. On the othe hand, one has to get a substantially hoogeneous optical etaateial fo wavelengths of adiation copaable with the ean distance between the eleents. Actually, a heteogeneous syste consisting of individual paticles can behave like a vitually hoogeneous optical etaateial independently on the wavelength of adiation, if soe specific conditions ae satisfied. Howeve, one needs to have a siple and eliable ethod fo the deteination of optical paaetes of etaateials in ode to be able to contol the fulfillent of these conditions. The usual schee fo easuing the negative efaction index of a plate of a tanspaent ateial iplies easuing the paaetes of the eal iage of a point souce of adiation. Howeve, the expeiental ethods fo testing such popeties equie

Aticle no. 44 V. Segentu, V. Usaki 2 sophisticated tools of nanoeasueents [4, 5]. Apat fo this, such a schee is actually applicable only fo the case of n = 1, when ε = µ= 1 [6].

2 Aticle no. 44 V. Segentu, V. Usaki 2 sophisticated tools of nanoeasueents [4, 5]. Apat fo this, such a schee is actually applicable only fo the case of n = 1, when ε = µ= 1 [6]. In this pape we popose to use a ethod based on the well known expeiental technique fo the investigation of Goos-Hänchen ect, which allows one to evaluate the degee of hoogeneity and efaction coicient of the ediu [7]. 2. A SIMPLE MODEL OF OPTICAL NANOMEDIUM WITH NEGATIVE REFRACTION INDEX As entioned above, usually the negative efaction index of a plate of a tanspaent ateial is easued on the basis of a schee with the eal iage of a point souce of adiation (Fig. 1). Fig. 1 A usual schee fo the deteination of the negative efaction index (the ethod of eal iage). In contast to this, we will use a ethod with a vitual iage fo the deteination of negative efaction index of a nanoediu. The poposed ethod is suitable fo vaious etaateials [8, 9], including those with a subsyste of cylindes with negative efaction index [1]. Let us fist conside a siple odel of nanoateial in the fo of a syste of nanocylindes iesed in vacuu. The nanocylindes with the axis oiented along the Z axis ae aanged in a peiodic two-diensional aay in the two-diensional space ρ = [ X,Y ] with the lattice constant equal to a. The elative dielectic peittivity and agnetic peeability of such a syste ae changing in the space accoding to a peiodic law µ ε = ε (ρ). In the vacuu ε µ = 1, while inside cylindes µ ε = 1. Let us assue that in a vey thin bounday laye with thickness δ between the cylinde and the vacuu we have ε ( ρ). The polaization vecto of the incident adiation is diected along the cylinde axis.

3 3 A ethod fo deteination of negative efaction index Aticle no. 44 We assue that the dispesion law is the ain chaacteistic of the tanspaent optical ediu. By using the standad ethod fo the deteination of the dispesion law fo two-diensional photonic cystals [11] we obtain ε ( ρ) ( ε ( ρ) Ε ( ρ)) = ( ω / c) Ε( ρ), (1) whee Ε( ) is the electic field stength, and ω is the fequency of the wave. The equation (1) can be tansfoed into ω Ε ( ρ) + ( ln( ε ( ρ))) Ε ( ρ) = ε ( ρ) Ε ( ρ) 2. (2) Since in the standad ethod of plane waves the atix eleents of the second te in the left-hand side of the equation (2) tend to zeo δ, one can keep only the fist te in the left-hand side, and obtain 2 2 ( ω / c ) Ε ( ρ) = ε ( ρ) Ε ( ρ). (3) 2 The equation (3) fo deteination of eigenvalues of ω is actually equivalent to a siila equation fo ef. [11]. We see that the atheatical poble of finding the spectu of electoagnetic oscillations fo ou syste pactically does not diffe fo a siila poble fo a syste of dielectic cylindes, but with substitution of ε ( ) by ε ( ) 1. The spectu of such a syste does not diffe fo the spectu of vacuu with the efaction index n = n v = ±1 (which is equivalent to µ 1/ε ). Howeve, the syste ipedance Z = μ /ε is diffeent fo the case of vacuu Z v 1. It is known [12, 13] that a usual coposite ediu behaves like a hoogeneous ediu with a definite efaction index, but only fo λ 2π / k >> a. Howeve, the consideed in this pape syste will behave like a = hoogeneous ediu with a definite efaction index up to values of λ ~ a. The eason fo this phenoenon was specified in ef. [14]. Each of tanspaent cylindes scattes the plane wave, ainly in the diection paallel to the diection of incident wave popagation [14] (Fig. 2). It can be agued that the inclusions fo a conventional optically tanspaent ateial behave like scattees. Howeve, the inclusions fo a etaateial behave like scattee antipodes. The value of ε fo λ > a can be estiated fo the foula ε ( 1 2c) [13], whee c is the shae of ateial in the volue of the etaateial. One can see fo this foula that µ, ε 1, Z 1 fo c 1 (lage adii of the cylindes R a / 2 ). A siple case of vacuu ( ε = 1, µ

4 Aticle no. 44 V. Segentu, V. Usaki 4 Z = 1) is obtained with deceasing the adius of cylindes ( R ), while a case with ε, µ = 1/ ε, Z is obtained fo c 1/ 2. Fig. 2 The scatteing indicatix of vaious types of tanspaent cylindes fo the wavelength of λ = 4R. 1 is fo a usual tanspaent ateial with ε = 1, and 2 is fo a etaateial with ε = µ = 1. Siila esults with ino diffeences ae obtained fo the case when the polaization vecto of the incident adiation is pependicula to the cylindes axis. The lage is the adiation wavelength as copaed to the lattice constant, the oe applicable ae the above entioned conclusions. 3. DESCRIPTION OF THE METHOD ON THE EXAMPLE OF A SIMPLE MODEL OF OPTICAL NANOMEDIUM The ethod is based on deteination of paaetes of the vitual iage occuing at the incidence of a light bea on a plate of optically tanspaent ateial with paallel sufaces and efaction index n. The eflected ay shifts to diffeent diections depending on the sign of the efaction index (Fig. 3). The obseved vitual iage of the light souce also shifts in diffeent diections elative to the io iage at the inteface ediu/vacuu.

5 A ethod fo deteination of negative efaction index Aticle no. 44 Fig. 3 The eflected ay shifts to diffeent diections depending on the sign of the efaction index.

5 5 A ethod fo deteination of negative efaction index Aticle no. 44 Fig. 3 The eflected ay shifts to diffeent diections depending on the sign of the efaction index. The obseved vitual iage of the light souce also shifts in diffeent diections. The ay shift is actually an analog of the Goos-Hänchen shift, but fo ou case. We will assue in ou futhe calculations that the plate with paallel sufaces is located at an ideally eflecting suface (Fig. 4). Pactically any substate with peittivity ε >> 1 satisfies this condition, since we will futhe deal with stongly etched saples with a shae of ateial in the volue of the etaateial of the ode of seveal pecents [15]. Fig. 4 The phenoenon of efaction/eflection of the incident adiation fo the case of a plate with paallel sufaces with ε, µ located at an ideally eflecting suface.

6 Aticle no. 44 V. Segentu, V. Usaki 6 In the case of the electic field pependicula to the plane of incidence of the ay we obtain [12] Ε = exp is, (4) 1 Ε whee Ε1,Ε ae the aplitudes of the eflected and incident waves, d is the thickness of the plate. k2x S = 2actg ctg( k ) 2xd, (5) µ kx ( ) whee k = k k, 2 2 x y k = ε µ k k x y The foula fo the shift of the ay afte passing though the plate is δ = S k ), (6) ( y Fo the case of ε = µ = ±1 we have 2k yd δ =, (7) µ k k fo which it follows that δ when k y k, i.e. fo lage angles of incidence. Futhe we will discuss the pocedues fo testing nanoplates. We will use a diectional adiation souce with an apetue ensuing obtaining a liited aea of the plana wavefont nea the saple suface, which will peseve this popety at a distance L geate than the thickness of the laye d. Let us nae it quasi-plane wavefont. We will futhe use in ou calculations a light souce adiating in the diection of the angleθ = / 4. The wave field is descibed by the foula whee π M l= M 2 2 y (1) Ε ( ρ, θ ) = exp( il ( θ θ + π / 2)) H l ( k ), (8) ρ (1) H l is the Hankel function. A wave bea with Ε ( ρ, θ ) 2 2π / ρ exp( i π / 4) δ ( θ θ) is obtained at М. ρ This type of two-diensional diffactionless wave field (self-econstucting light bea) is analogous to Bessel beas [16]. The adiation apetue is θ 1/M in the case of a finite М >> 1. If the condition M kr/π (R is the distance fo the eitte to the cente of the plate) is satisfied fo a wave-egion of the adiation,

7 A ethod fo deteination of negative efaction index Aticle no. 44 then the width of the bea in this egion satisfies the condition of R θ > λ and, at the sae tie, R >> λ.

7 7 A ethod fo deteination of negative efaction index Aticle no. 44 then the width of the bea in this egion satisfies the condition of R θ > λ and, at the sae tie, R >> λ. The length of the used egion of the bea L is salle than R, but it should satisfy the condition of L >> d. The paaetes ae chosen so that M=1 2 >> 1, d > λ and the souce is oved to a distance of R = 5 λ >> d, λ. A coicient is intoduced to assess the degee of pefection of the wave * 1 K( ) = [ H ( ) H ( ) + H ( ) H ( ) ] [ 2H ( ) H ( ) ], (9) whee H, H ae the intensities of agnetic fields of the plane ight-hand wave and the quasi-plane wave, espectively, fo which the wave vectos and the electic field intensities coincide; is the distance fo the cente of the bea to the souce. The value of K is an indicative of the diffeence of the quasi-plane wave fo the ideally plana ight-hand wave in the vacuu. If K = 1 and the intensity I in the cente of the bea eains constant at the distance L, then these waves can be consideed as pactically indistinguishable at this distance (Fig. 5). Othewise, thee is a significant pat of adiation in the wave which behaves like a left-hand wave when it is incident at the plane suface bounday. The adiation is diffacted to an opposite diffaction angle as copaed to the expected one, which can copletely distot the easueent esults by using a quasi-plane wave. Fig. 5 (a) Wave field of the adiation souce nea the saple suface (M = 1). The adiation bea ensues obtaining a quasi-plane wave font with a liited aea. The bea length L is finite, but it is lage that the plate thickness and the wavelength; b) the dependence of the coicient K (cuve 1) and the intensity log(i) in the cente of the bea (cuve 2) on the noalized distance (2/λ) fo the souce. The aow indicates the egion whee the adiation bea can be consideed to be quasi-plane. Let us fist conside a siple odel of the atte as a pefectly hoogeneous ateial with known paaetes, and constuct the wave field Real(E(ρ)) nea the nanosaples with an incident quasi-plane wave (Fig. 6). The foula (4) and the Matlab fft and ifft opeatos ae used. The incident wave is not

Aticle no. 44 V. Segentu, V. Usaki 8 taken into consideation when the iage of the wave field is constucted, since it will distub the obsevation of ain egulaities of the investigated phenoenon.

8 Aticle no. 44 V. Segentu, V. Usaki 8 taken into consideation when the iage of the wave field is constucted, since it will distub the obsevation of ain egulaities of the investigated phenoenon. At the sae tie, the aows on dawings indicate the tajectoies of two ays (the incident ay and the ay eflected fo the cente of the ea suface of the nanoplate). Fig. 6 The wave field Real(E(ρ)) nea the nanosaples with an incident quasi-plane wave with wavelength λ =.4d fo: (a) a etaateial with n = 1, Z = 1; (b) a etaateial with n = 1, Z = 4; (c) a etaateial with n = 1, Z = 6; (d) a usual tanspaent ateial with n = 3, Z =.11. The dawings in Fig. 6 ae easily and siply intepeted. Fig. 6a illustates the shift of a single bea fo a ediu with the efaction index n = 1. Thee ae no eflections fo the ateial suface, and the optical ipedance of the nanoateial is Z = 1. The bea is shifted to the left. Fig. 6b illustates the shift of the bea fo a ediu with the efaction index n = 1 and the optical ipedance of the nanoateial Z >> 1. The fist bea is the ay eflected fo the suface of the nanoateial, since Z 1. Othe ays anifest theiselfs afte eflections fo the pefectly eflecting suface, and/o fo the suface of the

9 9 A ethod fo deteination of negative efaction index Aticle no. 44 nanoateial. The shift of the bea fo a ediu with the efaction index n = 1 and the optical ipedance of the nanoateial Z >> 1 is shown in Fig. 6c. The fist weak bea is the ay eflected fo the suface of the nanoateial, since Z 1. Anothe ay (shifted to the ight) eeges afte eflection fo the pefectly eflecting suface. Figue 6d shows the shift of the bea fo a ediu with the efaction index n = 3.5 and the optical ipedance Z =.11. The fist weak bea is the ay eflected fo the suface of the nanoateial. Anothe ay eeges afte eflection fo the pefectly eflecting suface. Theefoe, one can ague that the adiation souce and the poposed ethod ae consistent with the consideed poble. Siila esults with ino diffeences ae obtained fo the case when the polaization vecto of the incident adiation is pependicula to the cylindes axis. The lage is the adiation wavelength as copaed to the lattice constant, the oe applicable ae the above entioned conclusions. 4. A NANOMEDIUM IN A REALISTIC MODEL Futhe we conside a case of nanosaple testing in the fae of a oe coplex and ealistic odel fo the atte of the plate. All the used etaateials epesent systes fo individual eleents with diffeent popeties. The poposed ethod allows a detailed taking into consideation of the phenoenon of adiation scatteing on individual eleents of the ediu. The etaateial is odeled as a plate consisting of a syste of diffeent nanocylindes with known efaction indexes iesed in the vacuu. The cylindes ae aanged in a tigonal lattice. A ultiple scatteing appoach is used fo calculations [17]. Nea the j th cylinde the electoagnetic field can be epesented as (1) [ ( j) J ( k ρ) + B ( j) H ( k ρ ρ )] exp( iθ ) l E( ρ) = α = l whee ρ = ρ ρ = ρ, θ ). ρj ρ j ( ρj ρj j ρj, (1) The elationship between the α( j) and B ( j) coicients is given by D ( j) = B ( j) / α ( j), (11) whee D ( j) appeas fo solving the poble of a single cylinde. Thee is a syste of equations fo the self-consistent solution fo all the cylindes + ~ ~ (1) B ( i) = α D ( i) + B ( j) D ( i) exp( i( l )( θ + π )) H ( k ρ ), (12) j i l= l ij l ij

10 Aticle no. 44 V. Segentu, V. Usaki 1 whee ρ ij = ρ j ρi = ( ρij, θij ). A poble of calculating the distibution of wave fields in the space aound the syste of cylindes aises afte solving the equations. A siple technique is used. A cylinde with an exteely sall adius, which pactically does not scatte the incident adiation, is inseted at the point in space whee the field aplitude has to be found. Then, the field aplitude is deteined fo the petubation theoy, and it is popotional to ~ ~ W θ ). (13) (1) ( i) = Bl( j) exp( il( ji + π )) H l ( kρ ji j i l= The nanoplate is placed on an infinitely conductive substate, which seves at the sae tie as a syety plane fo all the syste. Theefoe, instead of the syste (13) we have ~ B + ( + ) ( i) = α D ( i) D ( + ) ( i) l= ~ B ( + ) l exp(( i l )( θ ( i)exp( i( l + )( θ + π)) H ( +, ) ii ( k ρ + π)) H ( +, + ) (1) ( +, + ) + ~ ( + ) ij l ij D ( i) Bl ( j) j + + i l (, ) (1) (, ) = exp( i( l + )( θij + π)) H l ( kρ ij whee ρ (+) v ( ) ρ, ρ = i x + i y, and, fo instance, ρ x ρ y ρ ( = ( ρ (, θ ( + ) ( ) ( + ) + ) + ) ij = j i ij ij (1) l ) ) ). ( k ρ ( +, ) ii These foulas allows one to constuct the wave field Real(E(ρ)) nea the nanosaples upon the incidence of a quasi-plane wave. Figue 7 is siply intepeted when copaed with the siple odel applied in Fig. 6. The siilaity with the siple odel is oe ponounced with inceasing the adiation wavelength. Figue 7a is siila to Fig. 6a, but with a oe ealistic odel of the finite size of the syste. The lage is the wavelength, the bette is the siilaity. Figue 7b is siila to Fig. 6b. The fist bea is the ay eflected fo the nanoateial suface (the optical ipedance of the nanoateial Z >> 1). Anothe ay eeges afte eflection fo the ideally eflecting suface. A shift to the left is obseved, siilaly to a ediu with efaction index n = 1. In Fig. 7c, which coesponds to cylindes fo a ateial with efaction index n = 3.2 (ε = 1) and the ipedance Z =.1, a ay eflected exactly fo the place of entance into the nanoateial is obseved. In contast to Fig. 6d, the efacted ay is absent, since the ediu is not hoogeneous. The coheent bea in the ediu is attenuated, and is quickly scatteed in all the diections. Fig. 7d coesponds to a ediu coposed of two types of cylindes: 75% of the have the efaction index n = 3.2 (the optical ipedance of the nanoateial is Z < 1), and 25% of ) + (14)

11 11 A ethod fo deteination of negative efaction index Aticle no. 44 the have the efaction index n = 1 (the optical ipedance of the nanoateial is Z = 1). A patial hoogenization of the ateial occus in this case. The eflected ay is shifted fo the place of entance into the plate. It eans that it was efacted, it enteed into the ateial, and it was eflected fo the io-like suface. At the sae tie, it was patially scatteed. Fig. 7 The wave field Real(E(ρ)) nea the nanosaples with an incident quasi-plane wave fo: (a) cylindes fo a etaateial with µ = ε = 1, the adius R =.5a, the wavelength λ = 2a; (b) cylindes fo a etaateial with µ = ε = 1, the adius R =.4a, the wavelength λ = 4a; (c) cylindes fo a usual tanspaent ateial with ε = 1, the adius R =.5a, the wavelength λ = 4a; (d) cylindes with adius R =.5a ae fo two types of ateials: 25% ae fo a etaateial with µ = ε = 1 and 75% ae fo a usual tanspaent ateial with ε = 1, the wavelength λ = 4a. Siilaly to Fig. 7, Fig. 8 illustates the wave field fo a ediu coposed of two types of cylindes, but with an invese popotion of the constituents. One type of cylindes (25% of the) ae fo a ateial with efaction index n = 3.2 (the optical ipedance of the nanoateial is Z =.1). Anothe type of cylindes (75% of the) ae fo a etaateial with efaction index n = 1 (the optical ipedance of the nanoateial is Z = 1). A patial dehoogenization of the

This pevents the anifestation of the phenoenon of ay shifting (Fig. 7b). The scatteing is weakned in Fig.

12 Aticle no. 44 V. Segentu, V. Usaki 12 etaateial occus in this case, i.e. it stats to scatte that pat of adiation which shifted the ay to the left in Fig. 7b.The ay efacted on the suface of nanoplate is scatteed inside the ediu. The phenoenon of efaction in a left-hand ediu occus via tapping [18], which is sensitive to scatteing. This pevents the anifestation of the phenoenon of ay shifting (Fig. 7b). The scatteing is weakned in Fig. 8b, which leads to the appeaance of seveal weak scatteed ays. Fig. 8 The wave field Real(E(ρ)) nea the nanosaples with an incident quasi-plane wave with wavelength λ = 4a fo a ediu coposed of two types of cylindes: one type of cylindes with adius R 1 (25% of the) ae fo a tanspaent ateial with ε = 1, while anothe type of cylindes with adius R 2 (75% of the) ae fo a etaateial with µ = ε = 1. The adii of cylindes ae as follows: a) R 1 = R 2 =.4a; b) R 1 =.25a, R 2 =.5a. If the scatteing is educed alost to zeo as in the case illustated in Fig. 9, then one of the weak scatteed ays esebles the shifted ay fo Fig. 7b. Fig. 9 The wave field Real(E(ρ)) nea the nanosaples with an incident quasi-plane wave with wavelength λ = 4a fo a ediu with 25% of epty space and 75% filled with cylindes of adius R =.5a fo a etaateial with µ = ε = 1.

13 13 A ethod fo deteination of negative efaction index Aticle no CONCLUSIONS A ethod is poposed fo deteination of paaetes of edia with negative efaction index. The ethod can be applied fo both acoscopic and nanoetic saples. A pecise contol of paaetes of the adiation souce is needed fo using this ethod in the case of nanoetic saples. One can judge about the optical paaetes of the nanoediu by investigating the behavio of the eflected ay (the diection of popagation and the shifting of the eflected ay). When designing a etaateial ediu with negative efaction index, one needs to cobine a scatteing atix fo a ateial with usual efaction index with inclusion of a ateial with negative efaction index. With inceasing the concentation of the inclusions, the scatteing ediu is gadually hoogenized, it having a positive efaction index. Howeve, the scatteing is educed to zeo with a futhe incease of the inclusions concentation, and the ediu acquies a negative efactive index. Acknowledgents. This wok was suppoted by the Acadey of Sciences of Moldova unde Gants Nos A and A. REFERENCES 1. W.J. Padilla, D.N. Basov, D.R. Sith, Mateials Today 9, (26). 2. S. Havey, Negative-Index Metaateials, Univesity of Washington P., I.M. Tiginyanu, E. Foca, V.V. Segentu, V.V. Usaki, F. Daschne, R. Knochel, and H. Foell, J. Nanoelecton. Optoe. 4, 2-39 (29). 4. N. Fang, H. Lee, C. Sun, X. Zhang, Science 38, (25). 5. B. Bhushan (Ed.), Spinge Handbook of Nanotechnology, Spinge, Belin, Heidelbeg, K. Yu. Bliokh, Yu. P. Bliokh, Phys. Usp. 47, (24). 7. P.R. Began, Phys. Rev. E 66, 6763 (22); X. Yin, L. Hesselink, Z. Liu, N. Fang, X. Zhang, Appl. Phy. Lett. 85, (24). 8. J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett. 9, 8391 (23). 9. A.A. Asatyan, K.B. Dossou, L.C. Botten, Local Density of States of Metaateial Photonic Cystals, Austalian Institute of Physics, Poc. 18 th National Congess, Adelaide, Austalia, V.V. Segentu, V.V. Usaki, I.M. Tiginyanu, F. Foca, H. Föll, Robet W. Boyd, Appl. Phys. Lett. 91, 8113 (27). 11. J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Cystals: Modeling the Flow of Light, Pinceton Univ. Pess, Pinceton, NJ, L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electodynaics of Continuous Media, Buttewoth-Heineann, Oxfod, R. Landaue, AIP Confeence Poceedings 4, 2 45, Aeican Institute of Physics, A.E. Mioshnichenko, axiv: [physics.optics] (29). 15. S. Ya. Pislopski, E.K. Nauenko, I.M. Tiginyanu, L. Ghipu, E. Monaico, L. Sibu, S. V. Gaponenko. Optics Lett. 36, (211). 16. D. Mcgloin, K. Dholakia, Contep. Phys. 46, (25). 17. X. Zhang, L.-M. Li, Z.-Q. Zhang, C. T. Chan, Phys. Rev. B 63, (21). 18. S. Foteinopoulou, E. N. Econoou, C. M. Soukoulis, Phys. Rev. Lett. 9, 1742 (23).

8-3 Magnetic Materials

8-3 Magnetic Materials 11/28/24 section 8_3 Magnetic Mateials blank 1/2 8-3 Magnetic Mateials Reading Assignent: pp. 244-26 Recall in dielectics, electic dipoles wee ceated when and E-field was applied. Q: Theefoe, we defined