ecommons University of Dayton Monish Ranjan Chatterjee University of Dayton, Tarig A. Algadey University of Dayton

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1 University of Dayton ecommons Electrical and Comuter Engineering Faculty Publications Deartment of Electrical and Comuter Engineering 3-05 Investigation of Electromagnetic Velocities and Negative Refraction in a Chiral Metamaterial with Second-order Material Disersion using Sectral Analyses and Disersive Models Monish Ranjan Chatterjee University of Dayton, mchatterjee@udayton.edu Tarig A. Algadey University of Dayton Follow this and additional works at: htt://ecommons.udayton.edu/ece_fac_ub Part of the Comuter Engineering Commons, Electrical and Electronics Commons, Electromagnetics and Photonics Commons, Otics Commons, Other Electrical and Comuter Engineering Commons, and the Systems and Communications Commons ecommons Citation Chatterjee, Monish Ranjan and Algadey, Tarig A., "Investigation of Electromagnetic Velocities and Negative Refraction in a Chiral Metamaterial with Second-order Material Disersion using Sectral Analyses and Disersive Models" (05). Electrical and Comuter Engineering Faculty Publications. Paer 335. htt://ecommons.udayton.edu/ece_fac_ub/335 This Article is brought to you for free and oen access by the Deartment of Electrical and Comuter Engineering at ecommons. It has been acceted for inclusion in Electrical and Comuter Engineering Faculty Publications by an authorized administrator of ecommons. For more information, lease contact frice@udayton.edu, mschlangen@udayton.edu.

2 Investigation of electromagnetic velocities and negative refraction in a chiral metamaterial with second-order material disersion using sectral analyses and disersive models Monish R. Chatterjee Tarig A. Algadey

3 Otical Engineering 54(3), (March 05) Investigation of electromagnetic velocities and negative refraction in a chiral metamaterial with second-order material disersion using sectral analyses and disersive models Monish R. Chatterjee* and Tarig A. Algadey University of Dayton, Deartment of Electrical and Comuter Engineering, 3 College Park, Dayton, Ohio 45469, United States Abstract. In recent years, considerable research has been carried out relative to the electromagnetic (EM) roagation and refraction characteristics in metamaterials with emhasis on the origins of negative refractive index. Negative refractive index may be introduced in metamaterials via different methods; one such is the condition whereby the Poynting vector of the EM wave is in oosition to the grou velocity in the material. Alternatively, negative refractive index also occurs when the grou and hase velocities in the medium are in oosition. The latter henomenon has been extensively investigated in the literature, including recent work involving chiral metamaterials with material disersion u to the first order. This aer examines the ossible emergence of negative refractive index in disersive chiral metamaterials with material disersion u to the second order. The motivation is to determine if using second- as oosed to first-order disersion may lead to more ractical negative index behavior. A sectral aroach combined with a slowly time-varying hasor analysis is alied, leading to the analytic derivation of EM hase and grou velocities, and the resulting hase and grou velocities and the corresonding hase and grou indices are evaluated by selecting somewhat arbitrary disersive arameters. The results indicate the emergence of negative index (via negative hase indices along with ositive grou indices, as reorted in the literature) or negative index material (NIM) behavior over information bandwidths in the low RF range. The second-order results are not significantly better than those for first-order results based on the theoretical analysis; however, greater arametric flexibility exists for the second-order system leading to the higher likelihood of achieving NIM over ractical frequency bands. The velocities and indices comuted using the Lorentzian and Condon models yield an NIM bandwidth around 4 Mrad sec, about orders of magnitude higher than that for the arametric aroach; more imortantly, NIM is found not to occur in the first order when using ractical models. 05 Society of Photo-Otical Instrumentation Engineers (SPIE) [DOI: 0.7/.OE ] Keywords: hase velocity; energy velocity; grou velocity; circular olarization; higher order disersion; Poynting vector; chiral constitutive relations; chirality; negative index; contra-roagation; material disersion. Paer 573P received Jan. 6, 05; acceted for ublication Mar. 6, 05; ublished online Mar. 4, 05. Introduction It is known that counter roagation between such standard electromagnetic (EM) entities as the Poynting vector, the roagation vector, the grou velocity and the hase velocity (in aired combinations) leads to the onset of negative refractive index behavior in the material. 6 Over the ast 5 or more years, several research studies have focused on the emergence of, and consequent behavior due to negative index in such (meta)materials corresonding to several different domains of oeration and their otential alications. 7 9 Since refractive index in a material is tyically exressed via the hase and grou indices, it is natural to determine which of these (erhas one or both) would become negative in the negative index material (NIM) regime. It has been shown that in the NIM regime, the grou index (n g ) remains ositive, while it is the hase index (n ) which becomes negative. 0 In Valanju et al., 0 these roerties are verified both exerimentally and via simulations. Dong et al. have reorted that in the NIM regime, *Address all corresondence to: Monish R. Chatterjee, mchatterjee@ udayton.edu both the medium ermittivity and ermeability are negative (as is well known), and the corresonding EM behavior (whereuon the hase velocity inside the metamaterial is negative) is described as left handed. It is commonly found that NIM behavior is more readily realizable at microwave frequencies, and in articular in some comosites. 8, Realizing NIM at otical frequencies continues to be more challenging, articularly because samles tend to be thin and exhibit high degrees of dissiation. 3 Since under counter roagation of hase and grou velocities in a metamaterial (one of the rerequisites for NIM behavior) both the ermeability and ermittivity assume negative values, it turns out that the hase index in such a situation is to be regarded as the negative root of the roduct of the relative ermeability and ermittivity; similarly, with both ermeability and ermittivity being negative, a real hase velocity emerges with a negative sign. This real velocity ensures EM roagation in the medium; however, since the regime is NIM, the grou velocity (and consequently the grou index) is ositive here. Since n is negative in NIM, the resulting effects for /05/$5. 05 SPIE Otical Engineering March 05 Vol. 54(3)

4 refraction under Snell s law are very different than customarily haens in ordinary (ositive) index materials. One consequence is that at the NIM interface, one generates a backward wave for which the hase moves in a direction oosite to the direction of the energy flow. 4 It has already been noted that achieving a negative index in the otical frequency range is much more comlicated for several reasons. However, it turns out that the ermittivities of a variety of materials (esecially metals) tend to be very large at RF or microwave frequencies, and are dramatically reduced in the otical band. Hence, nominally, NIM is difficult to achieve at otical frequencies. Fortunately, it also turns out that at otical frequencies the EM resonse of metals is vastly different from those at lower (RF or microwave) frequencies, where the ermittivity is extremely large and metals behave as nearly erfect conductors. At otical frequencies, the ermittivity of metals may become comarable to the dielectric ermittivity of a host material, allowing the excitation of a surface lasmon resonance that leads to another means of achieving negative ermittivity and ermeability, thereby allowing negative index behavior at otical frequencies. 5 In recent work, Banerjee and Chatterjee investigated the EM roagational velocities in a chiral metamaterial under material disersion due to disersive ermittivity, ermeability and chirality u to the first order (assuming a lossless material). The analysis involved alying relevant constitutive relations to Maxwell s equations combined with slowly time-varying hasors and Fourier transforms, thereby develoing lane wave solutions for the electric and magnetic fields. This aroach led to the usual finding of three ossible wavenumbers of roagation as functions of frequency and the different material arameters. Using one of these wavenumbers, it was ossible to derive the hase, grou, and energy velocities for these lane waves by assuming first-order disersion in each of the three arameters in terms of the modulation/sideband frequency. The oerational/carrier frequency could be in any chosen band, including RF or otical. Normalizing the velocities and alying ractical disersive models led to frequency-deendent hase and grou indices, and it was shown that the material may oerate in the negative index (NIM) region within arbitrary (modulation) frequency bands. Overall, this first-order analysis demonstrated that that NIM behavior occurs under counter-roagation of either the Poynting vector and the roagation vector, or alternatively the hase and grou velocities. The first-order analysis indicated that oosition of hase and grou velocities could be achieved over certain sideband frequency windows for secific choices of firstorder disersive arameters. Furthermore, by assuming standard disersive models, it was ossible to redict the velocity counter-roagation windows and the corresonding negative index windows by adating the models to the sectral analysis carried out for EM roagation in a disersive chiral material. A rimary motivation for conducting the secondorder analysis resented in this work is to incororate disersive effects that may be tailored u to the second order in each of the rimary arameters (ermittivity, ermeability, and chirality) used in the sectral analysis. Since the NIM windows in a sideband frequency are determined by the choices of the disersive arameters, it is reasonable to assume that one would have greater flexibility in achieving desired NIM bands and negative index amlitudes when working u to the second order. In reality, as we show later, a set of first- and second-order arameters actually roduces a wider NIM window in the first-order comared to the second order. However, such an occurrence is only due to the secific choice of arameters examined in this analysis. Obviously, for other choices of the arameters, one may exect widening of the NIM band in the second order. Adatation of the disersive models to the second-order roblem likewise yields ractical and robably realizable values of both the negative indices and the NIM bands. Uon further examination, it is found that NIM behavior using the model-based analyses occurs only under secondorder disersion. When examined under first-order with chosen ractical arameter values comatible with those used for the second order, it is shown that NIM does not occur. Furthermore, it is found that while the hysical values of the indices tend to be imractical under the assumed theoretical/arametric models, the corresonding values found under model-based analyses are actually quite comatible with that seen in the literature (thus, the negative hase index ranges in the neighborhood of tyically to 5). Finally, the absence of NIM under first-order model-based analyses indicates a significant difference from the corresonding second-order result, highlighting the imortance of second-order analysis as ursued in this aer. In Sec., we resent a quick review of EM field solutions under first-order material disersion and associated roagation velocities reviously derived by Banerjee and Chatterjee. We then examine the effect on the field comonents and velocities when disersion via material arameters is considered u to the second order, i.e., u to order OðΩ Þ in sideband frequency, as will be exlained in Sec. 3 in some detail. We show that once again the fields exhibit circular olarization, a roerty that manifests itself in chiral materials indeendent of the order of disersion. 6 It is shown that the derived results are consistent with the cases of zerothand first-order disersions. Additionally, we find that in order to reduce the disersive material roblem to the nonchiral limit, one cannot simly set the chirality arameter to zero and exect the usual results to follow. The couling effect of chirality-induced constitutive relations obviously induces fundamental changes in the hysical equations governing EM roagation. The motivation for analysis u to the second order arises from not only the desire to find the differences that occur under grou-velocity-tye disersion (which tyically occurs in second order), but also to see if additional arametric flexibilities might exist in terms of achieving negative index behavior in the material in realizable frequency ranges and numerical magnitudes of the hysical quantities involved. As will be shown, such flexibilities likely emerge under the assumtions made, and to emhasize the more novel features of this analysis, a section in the text is devoted to direct comarisons between the first- and second-order comutations. Incidentally, normalized exressions for the energy velocity (ṽ e3 ) based on the Poynting vector and the stored energy density inside the medium have been derived and certain interesting conditions observed that secifically enable the velocity to be written in an amlitude-indeendent form. Energy velocity derivations, however, will not be discussed in this aer. Normalized hase and grou velocities (ṽ 3N and ṽ g3n ) Otical Engineering March 05 Vol. 54(3)

5 are obtained after considerable algebra in Sec. 3, and it turns out that these are obtainable without the arametric balances that become necessary when deriving the energy velocity. It is evident from these results that the resulting deendence on Ω is much more extensive than in the first-order case. 6 Direct comarisons for hase and grou velocities, sideband resonances, negative index domains, and actual realized hase and grou indices are also carried out in Sec. 4 in some detail. Section 5 concludes this aer with a summary of the derived second-order fields, velocities, and indices and their similarities and differences with resect to the standard disersionless and first-order results. Brief Overview of Electromagnetic EM Analysis Under First-Order Disersion The negative index in the resence of chirality and material disersion u to the first order has been studied by Banerjee and Chatterjee. The basic methodology consists of using Maxwell s equations, constitutive relations, sectral analyses, and lane wave solutions to eventually obtain the grou, hase, and energy velocities in the medium, and thereafter finding conditions for the negative index. 6 A tyical schematic scenario leading to negative index is shown in Fig.. Starting with exansions of the material arameters u to the first order, one may exress the frequency-deendent chirality arameter, electric ermittivity, and the magnetic ermeability in first-order Taylor exansions. Next, the EM constitutive relations for a recirocal chiral medium in the frequency domain are incororated into the standard Maxwell s curl equations. Using Fourier transforms and the constitutive relations, a set of nontrivial lane-wave field solutions are derived, from where four ossible solutions for the wavenumber in the medium comatible with the lane wave under investigation are obtained. 7 Using the wavenumber k z3 (for illustration uroses) given by ffiffiffiffiffiffiffiffiffi k z3 ¼ ω κ P μ 0 ε 0 þ ω ffiffiffiffiffiffiffiffiffi μ ε ; () Fig. Conditions leading to ossible onset of negative index behavior (after Ref. 6). the field solutions are obtained in terms of a free amlitude Ẽ x as Ẽ x ; arbitrary; () Ẽ y ¼ ω μ ε þ α þ k z j α k z Ẽ x ; (3) H x ¼ k z þ ω μ ε þ α j α ω μ Ẽ x ; (4) H y ¼ α þ k z þ ω μ ε ω μ k z Ẽ x : (5) By using the second relation (3), it is shown that in the resence of chirality the electric field is indeed circularly olarized Ẽ y ¼ jẽ x, as is the magnetic field. To find the hase and grou velocities, defining ṽ ðωþ ¼ ðk z ωþ and ṽ g ðωþ ¼ ð k z3 ΩÞ ¼ ð k z3 ωþ, one finally obtains the velocities as follows: ṽ 3 ðωþ ¼ ffiffiffiffiffiffiffiffiffi κ 0 μ 0 ε 0 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi μ 0 ε 0 þ Ω ð ε 0 μ 0 0 þ ε 0 0 μ 0Þ ffiffiffiffiffiffiffiffiffi μ 0 ε 0 μ 0 ε 0 ffiffiffiffiffiffiffiffiffi μ 0 ε 0 κ 0 0 ^a z ; (6) and ṽ g3 ðωþ ¼8 h ffiffiffiffiffiffiffiffiffi κ 0 μ 0 ε 0 ω 0 κ 0 ffiffiffiffiffiffiffiffiffi 0 μ 0 ε 0 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi ω μ 0 ð ε 0 μ 0 0 ε 0 þ ε 0 0 μ 0Þ 0 μ 0 ε 0 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi i μ 0 ε 0 >< þω κ 0 ffiffiffiffiffiffiffiffiffi 0 μ 0 ε 0 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi μ 0 ε 0 ffiffiffiffiffiffiffiffiffi μ 0 ε 0 ðω 0 Þð ε 0 þ 0 μ 0 0 þ ε 0þ μ 0 Þ μ 0 ε 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi ðω >: 0 Þð ε 0 μ 0 μ 0 ε 0 þ ε 0 0 μ 0Þ 0 4 μ 0 ε 0 ð ε 0 μ 0 0 þ ε 0 0 μ 0Þ μ 0 ε 0 9 ^a z : >= >; (7) 3 Sectral Analysis Leading to Negative Index Under Second-Order Disersion In this section, we begin by exanding the material arameters as defined earlier u to the second order in frequency using the Taylor series, so that 3 ε ðωþ ε 0 ðωþþðωþ ε 0 0 þðω Þ ε μ ðωþ ¼ 4 μ 0 ðωþþðωþ μ 0 0 þðω Þ μ 0 α ðωþ ω ffiffiffiffiffiffiffiffiffi μ 0 ε 0 ð κ 0 þðωþ κ 0 0 þðω Þ κ 0 Þ 3 7 5: (8) We next exress k z3 in terms of the chiral wave number α as k z3 ¼ð α þ ω ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi μ ε Þ ¼ α ω α μ ε þ ω μ ε. (9) Otical Engineering March 05 Vol. 54(3)

6 Substituting the various frequency-deendent terms into Eq. (3), and after considerable algebra, we may exress the field comonents to the second order in terms of arametric coefficients as follows: Ẽ x ðωþ; arbitrary (0) Ẽ y ðωþ ¼jðA 3 þ ΩB 3 þ Ω C 3 ÞẼ x ðωþ; () H x ðωþ ¼jðA 3 þ ΩB 3 þ Ω C 3 ÞẼ x ðωþ; () H y ðωþ ¼jðA 33 þ ΩB 33 þ Ω C 33 ÞẼ x ðωþ: (3) Carrying out the necessary algebra, first these coefficients are found to be A 3 ¼ ; B 3 ¼ 0; C 3 ¼ 0: (4) Substituting the above in Eq. () immediately indicates that once again Ẽ y ¼ jẽ x. This result imlies that even under second-order disersion, the EM field in the resence of chirality usually exhibits circular olarization, as was also seen in the first-order case. Thus, the resence of the chiral constitutive relations in the EM analysis automatically leads to circular olarization, indeendent of disersion. The resulting Y-comonent of the electric field leads the X-comonent by 90 deg. Substituting k z3 and α into Eq. (4), the magnetic field comonent H x may be written as ffiffiffiffiffi ε H x ¼ j ffiffiffiffiffi Ẽ x ; (5) μ where, using Eqs. (8) and (), the corresonding coefficients are found to be sffiffiffiffiffiffiffi ε 0 A 3 ¼ ; (6) B 3 ¼ μ 0 sffiffiffiffiffiffiffi ε 0 μ 0 ε 0 o ε 0 μ 0 o μ 0 and sffiffiffiffiffiffiffi ε 0 ε o C 3 ¼ 4 ε 0 8 μ 0 ε o 0 ε 0 ; (7) μ o 0 ε o 0 μ o þ 3 μ 0 ε 0 4 μ 0 8 μ o 0 μ 0 : (8) Finally, using Eqs. (8) and(3), we obtain the last series of coefficients as sffiffiffiffiffiffiffi ε 0 A 33 ¼ A 3 ¼ ; (9) B 33 ¼ B 3 ¼ μ 0 sffiffiffiffiffiffiffi ε 0 μ 0 ε 0 o ε 0 μ 0 o μ 0 ; (0) and C 33 ¼ C 3 sffiffiffiffiffiffiffi ε 0 ε o ¼ ε o 0 μ 0 4 ε 0 8 ε μ o 0 0 μ 0 ε 0 o ε 0 4 μ o μ 0 þ 3 8 μ o 0 μ 0 : () Once again, consistent with chiral behavior, we find that H y ¼ j H x, indicating circular olarization. In what follows, we rimarily use the frequency deendence of the wavenumber to find the hase and grou velocities under disersion. For these calculations, field amlitude information is not needed. However, if calculating the energy velocity (ṽ e ), the field amlitudes are necessary in order to comute the Poynting vector and the stored energy. In the resent work, energy velocity is not ursued. 3. Derivation of Phase Velocity from Sectral Analysis Using the relation ṽ ðωþ ¼ ðk z3 ωþ, we may calculate the hase velocity by using k z3 and retaining u to the second order in Ω, as described below. Hence, beginning with ṽ 3 ðωþ ¼ ffiffiffiffiffiffiffiffiffi κ μ 0 ε 0 þ ffiffiffiffiffiffiffiffiffi ; () μ ε and using aroriate exansions, we finally get ṽ 3 ðωþ ¼ 8 ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ½ κ 0 μ 0 ε 0 þð μ 0 ε 0 ÞŠ ffiffiffiffiffiffiffiffiffi ð ε 0 μ 0 þω 0 þ ε 0 0 μ 0Þ μ 0 ε 0 μ >< 0 ε 0 ffiffiffiffiffiffiffiffiffi μ 0 ε 0 þω ffiffiffiffiffiffiffiffiffi μ 0 ε 0 κ 0 þ ð ε 0 0 μ 0 0 þ ε 0 μ þ ε 0 4 μ 0 ε 0 ffiffiffiffiffiffiffiffiffi ð ε >: 0 μ 0 0 þ ε 0 0 μ 0Þ μ 0 ε 0 8 μ 0 ε 0 κ μ 0Þ 9 ; >= ffiffiffiffiffiffiffiffiffi μ 0 ε 0 >; (3) It turns out that by alying the arametric ratio condition, ratios ε o ε 0 0 ¼ μ o μ 0 0 and ε o ε 0 ¼ μ o μ 0, the hase velocity in Eq. () simlifies to ð ffiffiffiffiffiffiffiffiffi μ 0 ε 0 Þ ṽ 3 ðωþ ¼8 >< κ 0 þ ffiffiffiffiffiffiffiffi h ε 0 ε r μ r þ Ω o >: þω κ 0 þ 4 ε 0 ffiffiffiffiffiffiffiffi ε o ε r μ r ε 0 ffiffiffiffiffiffiffiffi ε r μ r i 9 : κ 0 0 >= >; (4) We note that the condition stated above is not necessary for evaluating the velocity; it merely makes the algebra slightly simler. Also, the result in Eq. (4) reverts to the first-order case uon eliminating the second-order terms. Otical Engineering March 05 Vol. 54(3)

7 3. Derivation of Grou Velocity from Sectral Analysis We next calculate grou velocity using the definition, ṽ g ðωþ ¼ ð k z3 ΩÞ ¼ ð k z3 ωþ : (5) Alying the same arametric ratio condition as for the hase velocity, we rewrite the grou velocity in a relatively simlified form as follows: ð ffiffiffiffiffiffiffiffiffi μ 0 ε 0 Þ ṽ g3 ðωþ ¼nh κ 0 ω 0 κ 0 0 þ ffiffiffiffiffiffiffiffi ω 0 ε 0 0 þ ffiffiffiffiffiffiffiffi io þ Ω ð Þ κ 0 0 ω 0 κ þ ffiffiffiffiffiffiffiffi ε 0 o ε r μ rω0 þ ε o þ Ω 3 ε 0 3 ffiffiffiffiffiffiffiffi ε r μ r o ε o ε o ε o κ 0 þ 3 ε r μ r 0 þ ffiffiffiffiffiffiffiffi ε r μ r ε r μ r ε 0 0 ε 0 ε 0 ffiffiffiffiffiffiffiffi ε 0 o ε r μ rω0 ε 0 o ε o ε þ o ε o ε o ffiffiffiffiffiffiffiffi ε r μ r : (6) Note that the grou velocity deends on Ω, Ω (sideband frequency) and ω 0 (carrier frequency). As with ṽ 3 ðωþ, the result in Eq. (6) reverts to the first order uon eliminating all second-order terms. 4 Numerical Results and Interretations In the following, we resent numerical results for the hase and grou velocities ertinent to first- and second-order disersions uon making certain simlifying arametric assumtions. We first examine the frequency-deendent (normalized) hase velocity. This is followed by similar grahical examination of the (normalized) grou velocity. The lots are carried out for both first- and second-order disersions for comarison uroses. The velocity lots are followed by lots of the hase and grou refractive indices, again under first- and second-order aroximations. In each case, it is found that there exists a region or band within which the velocities are in oosition (the so-called negative index material or NIM region). Fortuitously, we also find that it is the hase index which is negative in the NIM region, while the grou index remains ositive. Likewise, the hase velocity in the NIM region is negative, while the grou velocity is ositive. These results are found to be consistently true in the analyses resented here, and are in agreement with known results in the literature. 5,7,8,0 The overall results are then comared and the trends in first- and second-order analyses are assessed. 4. Phase Velocity and Phase Index Under First- and Second-Order Disersions We examine the frequency behavior of the hase velocity normalized relative to the free-sace value of c. The lot of ṽ 3N ðωþ versus frequency is carried out by assuming μ r ¼ ε r ¼ 4, κ 0 ¼ 5. Rewriting Eq. (4) as follows: ṽ 3N ðωþ ¼ i h ε 0 þ Ω o ε 0 κ 0 0 þ Ω or equivalently ṽ 3N ðωþ ¼ and κ 0 þ ε o ε 0 ; ; in the second order (7) c þ bω þ aω ṽ 3N ðωþ ¼ ; in the first order (8) c þ bω where a, b, and c are the arameter-deendent constants derived from Eq. (7). 4.. Phase velocity under first-order disersion We choose κ 0 0 ¼ 0.ð ε o ε 0 0 Þ and ε o 0 ¼ in order to lace the velocity resonance or transition in the neighborhood of a sideband around a few Mrad sec. With the chosen values, the hase velocity transitions under first-order material disersion around (Ω t ) aroximately rad sec. If Ω < Ω t, denominator <0, then ṽ 3N ðωþ < 0. On the other hand, if Ω > Ω t, and denominator >0, then ṽ 3N ðωþ > 0. Figure (a) shows the hase velocity over the frequency interval 0 Ω < πf max, where f max ¼ 0 6 Hz. From the lot we can see that the transition frequency is around rad sec. This reasonably matches the numerical value. We note that in this instance, ṽ transitions from negative to ositive around Ω t. Figure (b) shows the hase index n 3 which is calculated by using n 3 v 3N and drawn over the interval 0 Ω < πf max. The hase index lot shows that n 3 asses through zero and the transition frequency Ω t, and becomes negative for Ω < Ω t and ositive for Ω > Ω t. Thus, the NIM region exists in the range 0 Ω Ω t. We also observe that the numerical value of n 3 ranges aroximately from. to in the NIM. These numbers matched u well with ractical values seen in the theocratical and exerimental literature. 8,0, 4.. Phase velocity under second-order disersion Using Eq. (8) with the aroriate coefficients and arameters, the normalized hase velocity is lotted and the results are shown in Fig. (c). For this case, we additionally choose κ 0 ¼ 0.ð ε o ε 0 Þ and ε 0 ¼ We note that these arameters along with those for the first order are again selected in order to roduce sideband resonances around a few Mrad sec. From the lot we note that the (ositive) resonant frequency Ω r is around rad sec. We see that in this instance, ṽ 3 again transitions from negative to ositive around the resonance. Proceeding as in the first-order case, we define the hase index n 3 v 3N ¼ c þ bω þ aω, and obtain the hase index shown in Fig. (d). As before, n 3 transitions from zero around the resonance to negative below Ω r and ositive above. Therefore, the NIM is exected in the range 0 Ω Ω r. The hysical range of n 3 values in second order are.6 to, in fair agreement with the literature. 0,3 Otical Engineering March 05 Vol. 54(3)

8 Fig. Phase velocity and hase index (a) ṽ 3 under first order; (b) n 3 under first order; (c) ṽ 3 under second order; (d) n 3 under second order. 4. Grou Velocity and Grou Index Under Firstand Second-Order Disersions To determine and lot the (normalized) grou velocity disersion, we substitute again the material arameters μ r ¼ ε r ¼ 4, κ 0 ¼ 5, and ε 0 ¼ ε r ε 0 in Eq. (6). This leads to ṽ g3n ðωþ ¼ ða þ bω þ cω Þ ; (9) with arametric-deendent constants a ¼ ω 0 κ 0 0 þ ω 0 ð ε 0 0 ε 0Þ, b ¼ κ 0 0 ω 0 κ 0 þ ω 0ð ε 0 ε 0Þþð ε 0 0 ε 0Þ, c ¼ ð3 Þ κ 0 þð3 Þð ε 0 ε 0Þ, ω 0 ¼ 0 4 rad sec. We observe here that in the lots that follow, some of the chosen arameters (such as the carrier ω 0 ) are selected in accordance with values used in the literature Grou velocity under first-order disersion For the first order, we chose κ 0 0 ¼ 0.ð ε o ε 0 0 Þ, and again ε o 0 ¼ , as were chosen for hase velocity. Also, note that the transition frequency is now negative (around aroximately rad sec). The grou velocity transitions from negative to ositive for frequencies higher than rad sec. ForΩ > 0, we observe that ṽ g3 is consistently ositive. A last observation regarding the hysical values of ṽ 3N and ṽ g3n is in order here. For the arametric chosen, ṽ 3N tends to be u to orders of magnitude higher than c, while ṽ g3n is u to 6 orders of magnitude above c. These somewhat unexected values, however, are still comatible with other numerical findings 7,,4 as in Fig. 3(a). The grou index calculated by using n g3 v g3n is shown in Fig. 3(b). We observe that n g3 once again asses Fig. 3 Grou velocity and grou index (a) ṽ 3 under first order; (b) n 3 under first order; (c) ṽ 3 under second order; (d) n 3 under second order. Otical Engineering March 05 Vol. 54(3)

9 through zero around the (negative) transition frequency Ω t, and is negative for Ω < Ω t and ositive for Ω > Ω t. Since Ω t < 0, the negative grou index behavior is irrelevant to the NIM band (which must fall within a ositive frequency band). This shall be discussed further later. Also to be noted is the fact that the numerical value of n g3 varies from about to Both these are excetionally high, but are a consequence of the unusually low values of v g3n derived earlier. 4.. Grou velocity under second-order disersion Under second-order disersion, first-and second-order arameters are again selected identical to those used for hase velocity. Using the aroriate arameter, the lot shown in Fig. 3(c) is obtained. It turns out that for the chosen arameters, both resonances of ṽ g3n are negative, of which the less negative is shown in Fig. 3(c). Once again, the grou velocity transitions from negative to ositive around the resonance frequency (about rad sec). The nature of ṽ g3n at Ω > 0 is again consistently ositive. Figure 3(d) shows the grou index under second-order disersion with the chosen arameters. As in the first-order case, the grou index once again transitions from negative to ositive around the resonance. Since the NIM region discussed later falls in the ositive frequency band, ñ g3 will be consistently ositive within the NIM band. This result is in agreement with the literature. 0 The numerical values of ñ g3n in the second order are again unusually large, as exected. 4.3 Note on Direct Comarison between ñ 3 and ñ g3 For the urose of comaring the hase and grou velocities for a given set of material constants across a range of frequencies, we next illustrate the cases reresented by Figs. (a) and 3(a). We note from Fig. (a) that for the secific choice of material constants, the hase velocity transitions from negative to ositive around the resonance. Likewise, the grou velocity also transitions from negative to ositive around the resonance as seen from Fig. 3(a). These transitions are further indicated in the index lots of Figs. (b) and 3(b). In Fig. 4, which schematically grahs the signs of both indices under the first order together, we find that the indices n 3 and n g3 are in oosition within the frequency range of aroximately rad sec u to about 3.7 Mrad sec. Hence, it is exected that the material may exhibit negative index characteristics in this range. In ractice, of course, the roer NIM region exists for ositive frequencies only; this usable band is shown in Fig. 4 from 0toΩ t. A similar schematic band icture of n 3 and n g3 is shown in Fig. 5 for second-order disersion. The usable NIM in this case turns out to be slightly narrower. However, this is of no great significance, as will be discussed later. Overall, it is established from those analyses that n 3 is always negative and n g3 is always ositive in the NIM, as exected. 0 5 Alication of Practical Disersive Models to the Second-Order System To study the erformance of the chiral disersive materials in terms of the normalized hase and grou velocities derived reviously for roagating signals consisting of ulsed or modulated carriers, we take u the Lorentzian disersive models for relative ermittivity and ermeability and the Condon model for chirality, as follows:,6 ε r ðωþ ¼ þ ðω Þ ðω c ω Þ ; (30) μ r ðωþ ¼ þ ðω mþ ðω c ω Þ ; (3) ðωþ κ r ðωþ ¼α c ðω c ω Þ ; (3) where ε r ðωþ, μ r ðωþ, and κ r ðωþ are resectively the relative (sectral) ermittivity, ermeability, and chirality admittance of the material under consideration. The frequencies ω and ω m arise from electric olarization and magnetization, resectively, while ω c reresents a (single) resonance in the neighborhood of the alied signal. Also α c reresents a chiral frequency arameter. From Eq. (3), we have the Fig. 4 Index distribution showing the (ositive) frequency band (NIM region) where n 3 and n g3 are in oosition under first-order material disersion. Otical Engineering March 05 Vol. 54(3)

10 Fig. 5 Index distribution showing the (ositive) frequency band (NIM region) where n 3 and n g3 are in oosition under second-order material disersion. relativity ermittivity under second-order material disersion as follows: ε ðωþ ¼ ε 0 þðωþ ε 0 0 þðω Þ ε 0 ; (33) where the sideband frequency Ω is equal to Ω ¼ ω ω 0, and ω 0 is the carrier frequency. We may note that tyically Ω ω 0, therefore, while the carrier may be in the otical domain, the sideband may well be in the RF (thus in the MHz-GHz range). In the case of Ω ω c, the relative sectral ermittivity in Eq. (30) can be written as follows: ε r ðωþ ¼ þ ω ω c ω ω c : (34) Using Taylor exansion u to the second order for the equation above and assuming Ω ω c and ω 0 ω c, and comaring Eq. (30) with Eq. (33), we will have the following: ε r0 0 ¼ ε 0 0 ¼ ω 0ω ; (35) ε 0 ε r0 ¼ ε 0 ω 4 c ¼ ω ; (36) ε 0 ω 4 c ε r0 ¼ þ ω ω ; (37) c Similarly, from the relative (sectral) ermeability relation and comaring with the Lorentzian disersive model for relative ermeability in Eq. (3), we get the following: μ ðωþ μ 0 ðωþ ¼ þ Ω μ 0 0 þ ðω Þ μ 0 μ 0 μ ; (38) 0 μ r0 0 ¼ μ 0 0 ¼ ω 0ω m ; (39) μ 0 μ r0 ¼ μ 0 ω 4 c ¼ ω m ; (40) μ 0 ω 4 c μ r0 ¼ þ ω m ω ; (4) c Finally, based on the chirality exression u to secondorder material disersion, the chiral coefficients may be derived as follows: κ ðωþ ¼ κ 0 þðωþ κ 0 0 þðω Þ κ 0 : (4) Uon then alying the Condon model as in Eq. (3), we obtain the following: ω κ r0 ¼ α 0 c ω ; (43) c κ r0 0 ¼ κ 0 0 ¼ α κ c 0 ω ; (44) c κ r0 ¼ κ 0 ω ¼ 6α 0 κ c : (45) 0 ω 4 c We next substitute Eqs. (35) to(37), (3) to(4), and (43) to (45) into Eq. (7) to comute the normalized hase velocity under the Lorentzian disersive model for relative ermittivity and ermeability and the Condon model for chirality. After considerable algebra, we obtain the (normalized) hase velocity as in Eq. (34). 6 To lot hase velocity versus the sideband frequency we assume lausible values for ω, ω m and ω c 8,9 Thus, we begin with the hase velocity as above in Eq. (8) Otical Engineering March 05 Vol. 54(3)

11 Fig. 6 The modeled hase velocity under second order. ṽ 3N ðωþ ¼ þ Ω 8ω 0 ω ω 4 c i : κ α 0 c þ Ω ω 5α 0 ω c c þ 8ω ω 4 c ω 4 c (46) The (ositive) hase velocity resonance is at aroximately rad sec. From the lot in Fig. 6 we note that in this instance, ṽ 3N transitions from negative to ositive. Incidentally, the above sequence of calculations involves the choice of relevant material constants as κ r0 ¼ 6 0 0, ε r0 ¼.70 0, κ r0 0 ¼ 0, ε r0 0 ¼ , and ε r0 ¼.6. We may note that the values of the second-order ermittivity and chirality material constants based on the Lorentzian and Condon models turn out to be about 4 to 5 orders of magnitude larger than those that were used in the theoretical lots discussed earlier. Corresondingly, the (resonant) hase velocity for the modelbased lot is also about 4 orders of magnitude higher than that for the theoretical case. Therefore, there is a consistent scaling difference between the theoretical/analytical results and those based on the ractical models using tyical model frequencies. This scaling difference should not be of articular concern because the analytical model simly determines the feasibility of whether a negative index is realizable or not; actual ractical values and numbers would be roerly determined and assessed by incororating roer arameters in a model-based derivation. Similarly, for the normalized grou velocity, by using Eq. (9) we have ṽ g3n ðωþ ¼ ða þ bω þ cω Þ ; (47) where, a ¼ 5ðω 0 α c ω cþþð8ω 0 ω ω 4 cþ, b¼ 30ω 0 α c ðω 0 ω 4 cþþ8ω 0 ðω ω 4 cþ 0ðα c ω cþþð6ω ω 4 cþ, c ¼ 45α c ðω 0 ω 4 cþþðω ω 4 cþ. The grou velocity in Fig. 7 again transitions from negative to ositive around the resonance. It is clear from the lot that the (ositive) resonant frequency is aroximately.0 0 rad sec. We note that the model-based material Fig. 7 The modeled grou velocity under second order. Otical Engineering March 05 Vol. 54(3)

12 arameters for the grou velocity are the same as those for the corresonding hase velocity. An imortant feature regarding the model-based grou velocity as comared with the theoretical values obtained earlier is that for the model-based case, the resonant v g is about 4 m sec (and hence about 3 orders of magnitude lower than the model-based v ), while the same for the theoretical case is dramatically lower (about m sec or less), and which is also about 6 orders of magnitude lower than the corresonding v. These findings are somewhat inconsistent and difficult to exlain as such; however, as discussed earlier, the theoretical results are essentially feasibility studies, and the corresonding numerical values are relatively unimortant. 5. Estimation of Negative Index in the Velocity Counter-Proagation Regime In this section, we resent the grahical lots of hase and grou indices under model analysis for both the first- and second-order cases. The indices are obtained by simly finding the inverse of the corresonding normalized velocities. One goal of this analysis is to ascertain if working in the second order offers any significant advantage relative to the first-order analysis. Some asects of this were already discussed earlier in general terms; finding more quantifiable differences for the model analysis would likely enable a stronger case to be made for examining second-order material disersion, as resented in this aer. Fig. 8 Modeled hase index (a) zoom-out view; (b) zoom-in view showing ositive to negative transition. 5.. Plot of Phase and Grou Indices Under First- and Second-Order Disersions En Route to Negative Index Material Noting that with n 3 v 3N, and n g3 v g3n, we may once again straightforwardly lot the hase and grou indices for the second-order disersive system based on the Lorentzian and Condon models. These are shown in Figs. 8(a), 8(b) and 9. Figure 8(a) deicts a zoomed version of Fig. 8(b), whereby the change of sign of the hase index is clearly visible along the frequency axis. The change of sign for the grou index is evident in Fig. 9. We observe that for the given value of the arameter β ðω ω c Þ¼ (for the simulations here), each of the indices exeriences negative values [at frequencies above rad sec for n 3 as shown in Fig. 8(a)], and above.4 0 rad sec for n g3. Hence, a negative index occurs for the model case in the frequency range rad sec to.4 0 rad sec. Additionally, we note that the numerical values of both n and n g under the model analysis are consistent with those discussed in the literature. 6 Thus, within the NIM region, we find that the grou index is ositive while the hase index is negative (hence, negative index behavior is more directly aligned with the sign of the hase index). Figures 0 and show lots for the hase and grou indices with firstorder disersion under the assumed models. It is seen that neither index undergoes any change of sign versus frequency. Thus, n 3 is ositive and increases with frequency, whereas n g3 is negative and decreases with frequency. These characteristics, whereby n g3 is negative and n 3 is ositive, are contrary to exected NIM behavior, and moreover do not show any secific NIM band. This finding agrees with revious results found in Ref. This last finding is significant in Phase index Fig. 9 Modeled grou index under second-order disersion. 3 x Frequency(rad/sec) x 0 Fig. 0 Modeled hase index under first order. Otical Engineering March 05 Vol. 54(3)

13 Grou index NIM under first-order model-based analyses demonstrates a significant difference from the corresonding second-order result, thereby highlighting the imortance of second-order analysis as ursued in this aer. Acknowledgments The authors (MRC and TA) would like to thank the ECE Deartment for roviding travel suort toward resentation of this work. TA would also like to acknowledge the financial suort rovided by the Libyan Government higher education rogram for this research Frequency(rad/sec) x Fig. Modeled grou index under first order. that under material disersion models, NIM is achievable based on this analysis only under second-order disersion. 6 Conclusion Starting from the hasor aroach and slowly varying the enveloe aroximation combined with Maxwell s equations and the constitutive relations in a recirocal chiral material, exressions for electromagnetic hase and grou velocities have been derived assuming second-order material disersion. It is seen that in the resence of chirality, the roagating EM field is always circularly olarized indeendent of disersion. As was the case for first-order disersion, it is found that once again a negative index may also be realized under second-order conditions. It is imortant to note that the NIM behavior as seen here for both first- and second-order cases occurs under the arametric analysis with arbitrarily chosen numerical values of disersive arameters. It is demonstrated that there are measurable ranges of frequency where the hase and grou velocities are in oosition for identical material roerties, leading to the emergence of NIM over a secific frequency band. Phase and grou velocities based on the alication of Lorentzian and Condon models are also derived and lotted, and thereby comared case by case for comatible choices of modelbased material arameters. We note that the sideband resonances for both v and v g range in the neighborhoods of Mrad sec to 0 Grad sec under arametric analysis. Actual carrier frequencies will, of course, be much higher, as described in the text. Also, the derived velocity resonances for the analytical cases aear to be imractical; however, the numerical values are relatively unimortant for the theoretical (i.e., arametric) analyses as long as the ossible emergence of a negative index is established as feasible. Phase and grou indices for the model-based analyses under the second-order have also been derived and lotted, and indicate regions of negative excursions. It is found that NIM behavior using the model-based analyses occurs only under second-order disersion. When the indices are examined under first order, we find that for the chosen ractical arameter values, NIM does not occur. Moreover, the hysical range of index values found under model-based analyses is comatible with that seen in the literature. The absence of References. P. P. Banerjee and M. R. Chatterjee, Negative index in the resence of chirality and material disersion, J. Ot. Soc. Am. B 6, 94 0 (9).. J. B. Pendry, Negative refraction, Contem. Phys. 45, 9 0 (4). 3. V. G. Veselago et al., Negative refractive index materials, Comut. Theor. Nanosci. 3, 30 (6). 4. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Sringer-Verlag, New York (989). 5. S. Zhang et al., Negative refractive index in chiral metamaterials, Phys. Rev. Lett. 3, 0390 (). 6. M. R. Chatterjee and T. A. Algadey, Investigation of negative index in disersive chiral materials via contra-roagating velocities under second-order disersion GVD, Proc. SPIE 8837, 88370W (03). 7. R. A. Shelby, D. R. Smith, and S. Shultz, Exerimental verification of a negative index of refraction, Science 9, (). 8. W. J. Padilla, D. N. Basov, and D. R. Smith, Negative refractive index metamaterials, Mater. Today 9, 7 8 (6). 9. V. G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. USP 0(4), (968). 0. P. M. Valanju, R. M. Walser, and A. P. Valanju, Wave refraction in negative-index media: always ositive and very inhomogenous, Am. Phys. Soc. 88, ().. Z. G. Dong, S. N. Zhu, and H. LIU, Numerical simulations of negative-index refraction in wedge-shaed metamaterials, Am. Phys. Soc. 7, (5).. D. R. Smith et al., Comosite medium with simultaneously negative ermeability and ermittivity, Phys. Rev. Lett. 84(8), (0). 3. J. Valentine et al., Three- dimensional otical metamaterial with a negative refractive index, Nature 455, (8). 4. H. Chen and M. Chen, Fliing hotons backward: reversed Cherenkov radiation, Mat. Today 4( ), 34 4 (0). 5. V. M. Shalaev, Otical negative-index metamaterials, Nat. Photonics, 4 48 (7). 6. J. B. Pendry and D. R. Smith, Reversing light with negative refraction, Phys. Today 57, (4). 7. P. Yeh, Otical Waves in Layered Media, Sringer-Verlag, Wiley, New York (988). 8. P. P. Banerjee and G. Nehmetallah, Linear and nonlinear roagation in negative index materials, Ot. Soc. Am. 3, (6). 9. P. G. Zablocky and N. Engheta, Transients in chiral media with single-resonance disersion, Ot. Soc. Am. 0, (993). Monish R. Chatterjee has been a rofessor of electrical and comuter engineering (ECE) at the University of Dayton since. He has authored over 60 aers in archival journals and conference roceedings, several book chaters, three literary books, numerous literary essays, and resented over aers at international conferences. He is a senior member of the IEEE and the OSA, and a member of SPIE and Sigma Xi. Tarig A. Algadey received his BSEE degree in communication engineering from the College of Industrial Technology, Misurata, Libya, in, and his MSEE degree from the Libya Academy of Graduate Studies, Trioli, Libya, in 8. Currently, he is comleting his research for his PhD degree at the University of Dayton, Dayton, Ohio. His research interests include negative index in comlex media, metamaterials, electromagnetics, and digital communications. Otical Engineering March 05 Vol. 54(3)