Reflection and transmission of obliquely incident graphene plasmons by discontinuities in

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1 Home Search Collection Journal About Contact u My IOPcience Reflection and tranmiion of obliquely incident graphene plamon by dicontinuitie in urface conductivity: obervation of the Brewter-like effect Thi content ha been downloaded from IOPcience. Pleae croll down to ee the full text J. Opt ( View the table of content for thi iue, or go to the journal homepage for more Download detail: IP Addre: Thi content wa downloaded on 20/05/2016 at 10:10 Pleae note that term and condition apply.

2 Journal of Optic J. Opt. 18 (2016) (9pp) doi: / /18/7/ Reflection and tranmiion of obliquely incident graphene plamon by dicontinuitie in urface conductivity: obervation of the Brewter-like effect Saeed Farajollahi, Behzad Rejaei and Amin Khavai Department of Electrical Engineering, Sharif Univerity of Technology, PO Box , Tehran, Iran Received 30 January 2016, revied 3 April 2016 Accepted for publication 18 April 2016 Publihed 18 May 2016 Abtract Scattering of graphene urface plamon that are obliquely incident on a line dicontinuity in graphene urface conductivity i invetigated. The analyi i baed on a olution of the quaitatic integral equation for urface charge denity. It i hown that the reflection coefficient of the graphene plamon reache a minimum at a pecific angle of incidence that depend on the ratio of conductivitie of the two region urrounding the dicontinuity. Thi effect, which i imilar to the well-known Brewter effect, i pronounced for abrupt dicontinuitie, but become weaker a the width of the tranition region increae. The reult obtained can be ued for the deign and analyi of device baed on graphene layer with non-uniform conductivity pattern. A an example, an approximate method i preented for obtaining the diperion relation of waveguide mode in a graphene ribbon. Keyword: graphene plamon, cattering, oblique incidence, reflection and tranmiion coefficient, brewter effect, graphene ribbon waveguide (Some figure may appear in colour only in the online journal) 1. Introduction Surface plamon propagating on graphene layer have attracted much attention in recent year due to their trong confinement, hort wavelength, and relatively low lo [1 3]. Thee propertie have led to variou device being propoed for application at terahertz and optical frequencie [4 7]. But the mot intereting property of graphene plamon (GP) i their high tunability by mean of adjuting the chemical potential of the graphene layer [8]. The urface conductivity of graphene i mainly dependent on it chemical potential, which can be eaily tuned locally via gating or chemical doping. Thi property can be ued for controlling GP propagation in graphene-baed tructure. A a reult, variou component uch a ub-wavelength plamonic witche [9], planar lene [10] and tranformation optical device [11] can be realized in a ingle flake of graphene with a non-uniform urface conductivity pattern. To thi end, it i vital to know how GP catter at dicontinuitie in urface conductivity. The cattering of GP normally incident on line defect [12] and waveguide edge [13] ha already been tudied numerically. In [14] analytical reult were derived for the reflection and tranmiion coefficient of GP normally incident on an abrupt line dicontinuity in graphene urface conductivity. However, in practical cae an abrupt dicontinuity i not viable and the conductivity of graphene change gradually in a tranition region between two region with contant, but different, conductivitie. For thee non-abrupt dicontinuitie the cattering of normally incident GP have been invetigated uing full-wave imulation [15]. Recently a olution baed on olving an integral equation for urface current denity ha been preented for thi problem [16]. In thi paper, the problem of cattering of GP by dicontinuitie in graphene conductivity i tudied in the mot /16/ $ IOP Publihing Ltd Printed in the UK

3 Figure 1. (a) The tructure of the problem: an infinite heet of graphene with non-uniform conductivity i placed on a dielectric ubtrate whoe permittivity i given by ε. (b) An example of the urface conductivity profile in the tranition region. general cae where the angle of incidence and the profile of conductivity in the tranition region are arbitrarily choen. To that end, a one-dimenional integral equation i formulated that govern the behavior of urface charge denity in the quai-tatic approximation. Since well away from the dicontinuity the olution mut conit of imple incoming and outgoing wave, the integral equation i only olved in a limited region around the dicontinuity uing the well-known method of moment. In order to find the unknown reflection and tranmiion coefficient, two additional boundary condition are impoed. The reult obtained from thi integral equation are verified by comparion with full-wave imulation. For an abrupt dicontinuity, it i hown that the reflection coefficient exhibit a minimum for a pecific angle of incidence that depend on the ratio of conductivitie. Thi effect i imilar to zero reflection of an obliquely incident TM wave on the boundary between two dielectric at the Brewter angle. However, it fade away a the length of the tranition region increae. Moreover, uing our method, we calculate the phae of the reflection coefficient for GP that are obliquely incident on the edge of a emi-infinite heet of graphene. Thi reult i then ued to derive an approximate diperion relation for waveguide mode of a graphene ribbon waveguide [17, 18]. Thu, thee waveguide mode could be interpreted a interference of GP propagating in a graphene patch where multiple reflection from edge occur. 2. Integral equation for charge denity Figure 1(a) how the tructure of the problem: an infinite heet of graphene with non-uniform conductivity i placed on an infinitely thick dielectric ubtrate whoe permittivity i given by ε. The optical conductivity of graphene can be modeled by uing Kubo formalim: m = - je2k BT c + ( -m 2ln e c/ kbt g + 1) p 2 ( w - j2g) kbt m ( w ) + - je2 - - j G ln 2 c 2 4p 2 m - ( w + j2g) c ( 1) where μ c i the chemical potential, Γ i the phenomenological cattering rate, T i the temperature, e i the charge of an electron, i the reduced Plank contant and k B i the Boltzmann contant. A time-dependence of exp( jωt) i aumed where ω i the radial frequency. To tudy the cattering of GP, we aume a onedimenional conductivity profile where g i a function of x and change from L for x 0 to R for x 0 in a tranition region around x = 0. Figure 1(b) how an example of the urface conductivity profile in the tranition region. Take note that the propagation contant of a GP on a uniform graphene layer with the urface conductivity LR, i given by: -2jwea klr, ~ ( 2) LR, within the quai-tatic approximation in which klr, w c where c i the peed of light [9]. Here ea = ( e0 + e) 2 with ε 0 the vacuum permittivity. Since a time-dependence of exp ( jωt) i aumed for the field, the imaginary part of the urface conductivity of graphene hould be negative for upporting GP. Conider now a GP with the propagation contant k L incident on the dicontinuity from the left with an angle of incidence q i. One expect reflected and tranmitted GP to be induced on the left and right of the junction with the propagation contant k L and k R, repectively. Note, however, that thi picture i only valid ufficiently far away from the dicontinuity [13], in contrat to the cattering of a conventional electromagnetic plane wave by the interface between two dielectric. Since the tructure i uniform along the 2

4 Figure 2. Magnitude of reflection (a) and tranmiion (b) coefficient of urface charge denity wave veru angle of incidence for different value of L R for R < L. Solid line are reult obtained from olving equation (14) and circle how the reult from full-wave imulation. dicontinuity (y-direction), the y-component of the wave vector of the reflected and tranmitted wave are preerved o that the uual relationhip k q = q ( 3) i r in ( q) = k in ( q) ( 4) L i R t will till hold where q R, q t denote the angle of reflection and tranmiion, repectively. Therefore, for L < R total reflection occur at the critical angle defined by: q = in - 1 ( ) ( 5) c L R the following equation for urface charge denity - jwr () r = [ () r E ()] r 1 - () r 4pe T g T ex a T g T r ( ) r d S. ( 7) r - r Although the incident wave may be aumed to be generated by external ource, we neglect thee ource and, intead, introduce the incident wave by mean of extra boundary condition, a will be explained later. In addition, becaue we take g to depend on x alone, the tructure i homogeneou along y. Therefore, the olution of (7) may everywhere be written a: where (2) ha been ued. Our aim i next to calculate the complex coefficient of reflection and tranmiion for a given conductivity profile and angle of incidence q i. To that end, we employ the quai-tatic integro-differential equation for a graphene urface current J = ( J, x, J, y), which i given by: where k r () r = r () x e-jk y y () 8 we = ( q ) = - 2j a k in in ( qi) ( 9) y L i L J () r () r g 1 = ET ex () r - 4pe a T r ( r ) ds ( 6) r - r where r = ( x, y), T = ( / x, / y), E = ( E, E ) denote the tangential component of the electric field generated by external ource in the abence of the graphene layer, and r i the local denity of urface charge on the graphene layer. Equation (6) tate that at any point on the urface of graphene, the total electric field ( J () r ()) r conit of an external electric field and an electrotatic field due to the urface charge denity r. Subtitution of the above relation into the continuity equation J = -jwr lead to T ex x ex y ex T i the y-component of the incident wave vector. Subtitution of (8) into (7) yield, after neglecting E ex T and carrying out the integration over y, where 1 + r () x = G( x, x ; ky) r ( x ) dx ( 10) 2jpew - d d G( x, x ; ky) = g( x) ( - ) x x K 0 k y d d x x 2 - k () x K ( k x - x ) ( 11) y g 0 y 3

5 Figure 3. Magnitude of reflection (a) and tranmiion (b) coefficient of urface charge denity veru angle of incidence for different value of L R for R > L. Figure 4. Reflection (a) and tranmiion (b) coefficient for power tranferred in the x-direction veru angle of incidence for different value of L R for R < L. where we have ued + - jk yy - e dy ( x - x ) + ( y ) 2 2 = 2K ( k x - x ) ( 12) with K 0 the 0th order modified Beel function of the econd kind. Equation (10) mut, in principle, be olved over the entire x-axi. But, ince well away from the dicontinuity the olution conit of incoming and outgoing wave, the 0 y unknown function r () x in equation (10) i expreed a: r ( e-jklx, x + rejklx, x 0 ) x < -DL r () x = r te -jkrx, x x >D. ( 13) 0 R r () x -D < x <D tr L R Here r 0 i the amplitude of the incident charge denity wave with the wave vector ki = ( kl, x, ky), with klx, = klco ( qi), krx, = krco ( qt), and r, t deignate the reflection and tranmiion coefficient, repectively. The 4

6 Figure 5. Reflection (a) and tranmiion (b) coefficient for power tranferred in the x-direction veru angle of incidence for different value of R Lfor R > L. Figure 6. The angle of minimum reflection a a function of kr kl for (a) R < L and (b) R > L. parameter DL, DR determine the interval -D L < x < DR where the olution rtr () x cannot be written a the uperpoition of plane wave and mut be treated numerically. Subtitution of (13) into (10) lead to an integral equation for rtr () x that mut be olved over the interval -D L < x < DR, where 1 DR r tr () x - G( x, x ; ky) r ( x ) dx 2jpew -DL = f () x + rf () x + tf () x ( 14) DL Lx, f () x = e G( x, x ; k ) dx ( 15) jk x y -DL Lx, f () x = e G( x, x ; k ) dx ( 16) 2 - jk x - jk x Rx, f () x = e G( x, x ; k ) d x. ( 17) 3 DR The integral equation in (14) i then olved by applying the method of moment baed on pule function and point matching. Since the reflection and tranmiion coefficient appearing in (14) are unknown, we need two more equation which are obtained by impoing the continuity boundary condition: r (-D ) = r ( ejklx, DL + re-jklx, DL) ( 18) tr L 0 y y 5

7 Figure 7. The phae of the reflection coefficient of urface charge denity a a function of incident angle for (a) R < L and (b) R > L. Solid line are reult obtained from (14). Circle how the reult from full-wave imulation. Figure 8. The effect of increaing the tranition length on (a) the reflection and (b) the tranmiion coefficient for L R = 2. r ( D ) = r te -jkrx, DR. ( 19) tr R 0 Although the method outlined above yield the reflection and tranmiion coefficient of wave of charge denity, it can be hown that the ame parameter relate the amplitude of the tangential component of the electric field accompanying thoe wave. The GP traveling in the uniform region far from the junction are TM wave. The tangential component of the electric field accompanying each wave i directed along it direction of propagation, and can be written a E = E e- Lx, - y ( co q, in q) ( 20) T i T i jk x jk y i i E = E e Lx, - y (-co q, in q) ( 21) T r T r jk x jk y i i E = E e- Rx, - y ( co q, in q). ( 22) T t T t jk x jk y t t By uing equation (20) (22) and (2), the continuity equation for urface current denity, and the relation 6

8 Figure 9. The effect of increaing the tranition length on (a) the reflection and (b) the tranmiion coefficient for R L = 2. Figure 10. The propagation of waveguide mode in a graphene ribbon waveguide. The waveguide mode can be thought of a a conventional GP that i multiply reflected by the edge of the ribbon. The interference of multiply reflected wave i contructive when GP are in phae at point A and C. Figure 12. The diperion relation of the firt two waveguide mode of a graphene ribbon waveguide with width of 5 μm. The chemical potential and carrier mobility of graphene are taken a 0.2 ev and cm 2 V 1 ), repectively. The olid line are calculated from equation (28) and (29). The circle are reult from full-wave imulation. Figure 11. The phae of the reflection coefficient for an obliquely incident GP on the edge of a emi-infinite waveguide veru the angle of incidence. J = get for each wave, one arrive at E E E E T t T i T r T i = t ( 23) = r. ( 24) 7

9 3. Reult In order to numerically invetigate the cattering of GP, we firt have to chooe a model for the tranition region. We aume that the graphene conductivity i given by: R + L R - L = + ( ) tanh x l ( 25) 2 2 where l i the length of the tranition region (for an abrupt tranition we aign a very mall number to l ). The reult for an abrupt tranition are hown in figure 2 for R < L and figure 3 for R > L. The propagation contant of GP on the left of the junction i k = 2 mm- L 1 and the reflection and tranmiion coefficient are plotted veru q i for different value of kr kl = L R in the quai-tatic approximation. Loe are neglected and graphene i aumed to be freetanding ( t, e = e0). For comparion, reult obtained from full-wave imulation carried out uing Anoft HFSS oftware are alo hown. Take note that the coefficient for the reflected and tranmitted power are different from r and t. If the loe are negligible (if conductivitie have negligible real part), thee quantitie are repectively given by = = - = R r 2 R P, TP 1 RP t L 2 co( qt) co( q ) i ( 26) and are plotted in figure 4 and 5 for R < L and R > L, repectively. A hown in figure 3 and 5, for angle of incidence larger than the critical angle defined by equation (5), total reflection occur. It i alo oberved from the above figure that for a given ratio of conductivitie, the reflection coefficient ha a minimum for a pecific angle of incidence. Thi angle i hown in figure 6 a a function of kr k L. Although the abolute of reflection at thi angle i not zero, thi effect i imilar to zero reflection of an obliquely incident TM wave on the boundary between two dielectric at the Brewter angle. Figure 7 how the reflection phae F R of the charge denity wave defined by r = r e-f j R ( 27) a a function of incident angle and for different value of R L. Thi unuual phae dependence of reflection coefficient i a reult of highly evanecent mode that are excited cloe to the dicontinuity [13]. A in the cae of the normal incidence of GP [14], the phae of the tranmiion coefficient i mall and i not hown here. Figure 8 and 9 how the effect of increaing the length of the tranition region for L R = 2 and R L = 2. A illutrated in thee figure, the reflection coefficient i reduced a the length of the tranition region increae. The Brewter effect een for the abrupt tranition alo diappear a the tranition become moother. 4. The diperion relation for waveguide mode in ribbon waveguide The method decribed in the previou ection can be ued to obtain an approximate diperion relation for the waveguide mode propagating along a graphene patch [17, 18]. The electromagnetic field of the waveguide mode of a graphene ribbon are characterized by a more or le inuoidal behavior acro it width. By contrat, the edge mode of a graphene ribbon have electromagnetic field that drop exponentially a one move away from the two edge of the ribbon (figure 10). The waveguide mode can thu be thought of a a conventional GP that i multiply reflected by the edge of the ribbon. The propagation contant of thee mode can be written a: k = k in ( q) ( 28) mode g where k mode i the propagation contant of the waveguide mode in the graphene ribbon waveguide, k g i the propagation contant of GP in the infinite graphene heet given in equation (2), and θ i the angle of incidence of GP to the edge of waveguide hown in figure 10. Thi interference of multiply reflected wave i contructive when GP are in phae at point A and C in figure 10. Thi condition could be written a: 2dk co( q) + 2F ( q) = 2Np ( 29) g where fr ( q ), which i plotted in figure 11, i the reflection phae of a GP that i obliquely incident on the edge of a emiinfinite graphene layer. Thi phae wa calculated by olving equation (14) for an abrupt tranition in which R 0. Take note that for q 0 (normal incidence) F R i expected to equal π/4 ince, at normal incidence, F R differ from the current denity reflection phae ( 3π/4 [14]) by π due to the continuity equation Jx x= -jwr. Solving equation (29) give the angle of incidence for a waveguide mode at a pecific frequency. By obtaining thee angle, the propagation contant of waveguide mode are calculated uing equation (28). The diperion relation of the firt two waveguide mode of a graphene ribbon waveguide with a width of 5 μm i hown in figure 12. The chemical potential and carrier mobility of graphene are taken to be 0.2 ev and cm 2 V 1 1 ), repectively. The olid line are calculated uing the mentioned procedure and the circle are reult from full-wave imulation. Although the loe are ignored in the calculation of the reflection phae, good agreement can be een from thi figure. 5. Concluion The problem of the cattering of GP by dicontinuitie in graphene conductivity wa tudied in the mot general cae where the angle of incidence and the profile of conductivity in the tranition region are arbitrarily choen. A one-dimenional integral equation governing the behavior of the urface charge denity wa formulated in the quai-tatic approximation, and the numerical olution of thi integral equation wa hown to R 8

10 be in full agreement with the reult of full-wave imulation. It wa hown that the reflection coefficient of the urface charge denity ha a minimum at a pecific angle of incidence imilar to zero reflection of a TM wave at the boundary between two dielectric at the Brewter angle. Thi phenomenon could be ueful in deigning novel plamonic device at terahertz frequencie, e.g., to minimize the reflection of GP at the interface between two graphene region with different conductivitie. Finally, the calculated phae of reflection coefficient for obliquely incident GP at the edge of a emi-infinite graphene layer wa ued to preent a novel method for obtaining the diperion relation of waveguide mode in graphene ribbon. Reference [1] Jablan M, Hrvoje B and Marin S 2009 Plamonic in graphene at infrared frequencie Phy. Rev. B [2] Koppen F H L, Chang D E and de Abajo F J G 2011 Graphene plamonic: a platform for trong light matter interaction Nano Lett [3] Garcia de A F J 2014 Graphene plamonic: challenge and opportunitie Ac Photonic [4] Ju L et al 2011 Graphene plamonic for tunable terahertz metamaterial Nat. Nanotechnol [5] Low T and Avouri P 2014 Graphene plamonic for terahertz to mid-infrared application Ac Nano [6] Chen J et al 2012 Optical nano-imaging of gate-tunable graphene plamon Nature [7] Jornet J M and Ian F A 2010 Graphene-baed nano-antenna for electromagnetic nanocommunication in the terahertz band Antenna and Propagation (EuCAP), 2010 Proc. of the Fourth European Conf. on (IEEE) [8] Mak K F et al 2008 Meaurement of the optical conductivity of graphene Phy. Rev. Lett [9] Gómez-Díaz J-S and Perruieau-Carrier J 2013 Graphenebaed plamonic witche at near infrared frequencie Opt. Expre [10] Vakil A and Engheta N 2011 Tranformation optic uing graphene Science [11] Cheng B H et al 2014 Actively controlled uper-reolution uing graphene-baed tructure Opt. Expre [12] Garcia-Pomar J L, Nikitin A Y and Martin-Moreno L 2013 Scattering of graphene plamon by defect in the graphene heet ACS Nano [13] Nikitin A Y, Low T and Martin-Moreno L 2014 Anomalou reflection phae of graphene plamon and it influence on reonator Phy. Rev. B [14] Rejaei B and Khavai A 2015 Scattering of urface plamon on graphene by a dicontinuity in urface conductivity J. Opt [15] Roolen G and Mae B 2015 Nonuniform doping of graphene for plamonic taper J. Opt [16] Farajollahi S et al 2015 Circuit model for plamon on graphene with one dimenional conductivity profile Photonic Technology Letter, IEEE [17] Nikitin A Y et al 2011 Edge and waveguide terahertz urface plamon mode in graphene microribbon Phy. Rev. B [18] Chritenen J et al 2011 Graphene plamon waveguiding and hybridization in individual and paired nanoribbon ACS Nano