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BRAC University Journal, Vol. X, No. 1 & 2, 2013, pp. 13-29 ON TERAHERTZ LASER PULSE PROPAGATION IN NEGATIVE INDEX METAMATERIALS U. A. Mofiz1 Plasma Astrophysics Laboratory Department of Mathematics and Natural Science BRAC University, 66 Mohakhali Dhaka-1212, Bangladesh email: [email protected] ABSTRACT The nonlinear propagation of a terahertz laser pulse propagation in the resonant region of a negative index meta-material is investigated. The resonant frequency of a split ring resonator (SRR), comprosing the negative index metamaterial, is calculated to be in the order of terahertz. The analysis shows that the negative refractive index in SRR is maintained within a narrow band of the frequency, which is more narrowed down with the intensity of the laser radiation. From Maxwell’s system of equations, a system of coupled equations descring the nonlinear propagation of laser pulse in negative index metamaterials is obtained. The system of the coupled equations are solved analytically in the high frequency and in the low frequency responses. In the high frequency response, a nonlinear field dependent dispersion relation is obtained, and the corresponding group velocity is determined. In the low frequency response, introducing the Lorentz invariant stretched coordinates, a system of coupled nonlinear Schrödinger equations (NLSE) are obtained. Considering the equal intensities of electric and magnetic fields of the laser pulse, the coupled NLSE equations are solved , which shows that the laser radiation creates bright and dark optical solitons, respectively, before and after a critical frequency, within the narrow band, determined by the intensity of the laser radiation. The instability of the bright soliton is investigated while the dark soliton is modulationally stable. Key words: meta-material; negative refractive index; bright and dark solitons. I. INTRODUCTION I.1. History of Negative Index Metamaterials Vaselago [1], a Russian physicist in 1967 hypothetically studied the optical properties of an isotropic medium with simultaneously negative electric permittivity ( ) and magnetic permeability ( ), which he named a left- handed material (LHM). He showed LHMs display unique "reversed" electromagnetic (EM) properties having the triad k (wave vector), E (electric field), H (magnetic field) left handed, hence exhibiting phase and energy velocities of opposite directions. A LHM is characterized by negative refractive index n , therefore, their alternative name, negative 1. For all correspondence index materials (NIMs). Such materials demonstrate a number of peculiar properties: reversal of Snell’s law of refraction, reversal of the Doppler shift, counterdirected Cherenkov radiation cone, the refocusing of EM waves from a point source, etc. The remarkable property of a LHM is that it possees a negative refractive index due to which a thin negative index film will behave as a “superlens”, providing image detail with a resolution beyond the diffraction limit. Conventional positive index lenses require curved surface to bent the rays coming from an object to form an image but in the case of negative refractive index a planar slab can produce image within the slab and also outside the slab (Fig.1). U. A. Mofiz Negative refraction in photonic crystal is also feasible [5] in doped graphene metamaterials . I.2. Negative Refraction in Photonic Crystals Photonic crystals (PCs) are made only from dielectrics, thus smaller losses than metalic LHMs, especially at high frequencies. In PCs, to achieve negative refraction the size and periodicity of the “atoms” (the elimentary units) are of the order of the wave length. In PCs, no effective and can be defined, although the phase and energy velocity can be opposite as in the case in normal LHMs. Both negative refraction and superlensing have observed in PCs [6]. An alternative approach for fabricating metamaterial is to use PCs composed of polartonic materials [7]. Fig.1. A negative n medium bends light to a negative angle relative to the surface normal. Light formerly diverging from a point source, converged back within the metamaterial as well as second time in the image plane outside. I.3. Graphene Doped Metamaterials Pendry[2] carefully rexamined the plannar lense and found that it may recover the evanescent waves coming from an object. In a planar negative index lense, an evanesciting wave decaying away from an object grows exponentially in the lens and deacys agin until it reaches the image plane (Fig.2). Graphene is a two-dimensional , one atom thick allotrope of carbon holds the promise for building advanced nano-electronic devices since its discovery in 2004 [8]. It exhibits very unique optical properties, especially in the terahertz (THz) frequency range. To date, novel photonic devices such as THz devices, optical modulators, photodetectors, and polarizers were successfully realised. The physics of graphene can be considered as a unifying bridge between lowenergy condensed matter physics and quantum field theory , as its two-dimensional quasi-electrons behaves like massless “relativistic” Dirac fermions, very similarly to electrically charged “neutrinos”[9]. Currently, the huge unexplored potential of graphene for nonlinear optics has been outlined. Strong nonlinear optical responses have investigated in papers [10,11]. Preliminary experimental include ultrafast saturable absorption and the observation of strong four wave mixing [12], which are the building blocks of nonlinear optics. This discoveries and the advancement in optical communications are the motivations to investigate the nonlinear propagation of intense terahertz laser pulse in meta-materials in the resonant region. Fig.2. A decaying wave grows exponentially within the metamaterial and then decays again utside. Vaselago’s idea remains hypothetical for a long time, untill a breakthrough was announced in 2000 by Smith and co-workers [3] presented evidence for a composite material of conducting rings and wires – called split ring resonator (SRR)-displaying the negative value of and . The SRR structure has proven a remarkably efficient means of producing a magnetic response by scaling down the size thus upwards in frequency to produce metamaterials in the terahertz frequencies [4]. In this paper, we have investigated the nonlinear propagation of terahertz laser pulse in the resonant region of a negative index meta-material. The laser pulse is considered as the circularly polarized electromagnetic (EM) wave. In section II, we have 14 On Terahetrz Laser Pulse Propagation studied the optical properties of negative index meta-materials under EM fields and analyzed the nonlinear resonant frequency of the SRR. In section III, following Maxwell system equations, Meta materials based on strong artificial resonant elements can also be described efficiently with the Drude-Lorentz formulas. These structures are the metallic wire structure which provides a predominently free-elctron respone to EM fields and the SRR structure provides a predominently magnetic response to EM fields. ~ ~ and the complex magnetic field H are obtained, coupled equations for the complex electric field E which are solved analytically in the high frequency as well as the low frequency responses. In the high frequency response, a nonlinear field dependent dispersion relation is obtained, which is analyzed with the increased field intensities. In the low frequency response, coupled nonlinear Schrödinger equations are obtained, the analysis of which shows that bright and dark optical solitons may propagate through NIMs within the frequency band, below and after some critical frequency, which is defined by the intensities of the wave fields. The bright soliton is modulationally unstable, its growth rate is calculated, while the dark soliton is modulationally stable. Results and discussions are given in section IV, while section V concludes the paper. II.2. Nonlinear Form of Dielectric Permittivity and Magnetic Permeability The dielectric and the magnetic nonlinear responses of NIMs under EM fields without losses are given by the dispersive (frequency dependent) and the nonlinear (field dependent) expressions as [14]: = L ( ) NL ; | E |2 , = L ( ) NL ; | H |2 , L = 0 1 p2 / 2 , L = 0 1 F 2 / 2 02 , . (6) NL = 0 | E |2 /Ec2 , (7) 2 2 NL = 0 | H | /Ec . (8) Here, p is the plasma frequency, F << 1 is the filling factor, 0 is the linear resonant frequency, Ec is the critical electric field, = 1 corresponds to a focusing dielectric and = 1 corresponds to a defocusing dielectric, > 0 is a fitting The Drude-Lorentz forms for dielectric permittivity and magnetic permeability describe the EM response of materials over the high ranges of frequencies which are given by [13]: 2 pe , (1) = 0 1 2 02e ie 2 pm , (2) = 0 1 2 02m im where, 0 and 0 are the vacuum permittivity parameter. II.3. Analysis of the Linear Response The eigenfrequency of the system of SRRs in the linear limit is defined as [15]: 0 0 = 1/c 2 ; 0 = (c/as ) d g /h D 0 , pm are the electron /magnetic plasma frequency, 0 e and 0 m are the electron / and magnetic resonant frequency, (5) Under weak fields approximation: II.1. Drude-Lorentz Form for Linear Dielectric Permittivity and Magnetic Permeability pe (4) where, II. OPTICAL PROPERTIES OF NEGATIVEINDEX MATERIALS UNDER ELECTROMAGNETIC FIELDS and permeability, respectively, with (3) where, d g is the size of the SRR gap, h is the width of the ring, e and m are the speed of light. electron/ magnetic damping coefficient and is the frequency of the EM wave. The free electron contribution in metals corresponds to 0 e 0 . (9) a s is the radius and c is the D0 is the linear part of the dielectric. As per Ref. [15] for D 0 = 12.8 , h = 0.003 cm, d g = 0.1 cm, a s = 0.003 cm, 15 U. A. Mofiz c = 3 1010 cm s 1 , we find 0 = 9.1 1012 s 1 i.e the linear 0 frequency 5 f 0 = 0 /2 = 1.45 10 Hz = 1.45 THz. For simultaneous L < 0 , L < 0 , and n = L L / 0 0 , , f . f . p p M M , we find which corresponds Here, 0 < < n 12 min 15 f 0 < f < min 20 0.15 M = 0 / 1 F . For f M = 1.87 THz; then, 1.45 2 , and n band as functions of frequency 4 0 6 8 10 12 14 0.8 1.0 1.2 1.4 < 1.6 Figure 3: The linear dielectric permittivity as a function of frequency , which remains negative in the 0 < < p 0 < < M . | H |2 >= 0.5 , we plot Ec2 , and n as functions of frequency within the frequency band, which are shown in FIG.6-8, respectively. The figures show that the frequency band for NIMs become more narrower with the applied EM fields. p frequency band 0.19 Now, we consider the nonlinear response on the dielectric permittivity and magnetic permeability, and on the corresponding refractive index due to applied EM fields. In this case, for weak fields we have 2 | E |2 (10) = 0 1 p2 2 , E c F 2 | H |2 (11) = 0 1 2 2 . 2 El 0 2 Considering an average value of < | E | >= 0.5 and Ec2 2 0.6 0.18 II.4. Analysis of the Nonlinear Response 0 0.4 0.17 Figure 5: The linear refractive index n as a function of frequency , which remains negative in the frequency within the frequency band 1.45THz < f < 1.87 THz , which are shown in FIG.3-5, respectively. 0.2 0.16 p F = 0.4 ; we find THz < f < 1.87 THz. Considering p = 2 10 THz and 0 = 2 1.45 THz, we plot 10 . 0 2 0.0 4 0.5 0 6 0 8 1.0 10 1.5 12 14 2.0 0.2 0.16 0.17 0.18 0.19 0.20 0.21 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.22 p Figure 6: The nonlinear dielectric permittivity as a function of frequency under EM fields, which remains negative in the frequency band 0 < < P . Solid line (– )line for = 0 , p Figure 4: The linear magnetic permeability as a , which remains negative in the frequency band 0 < < M . function of frequency broken line (- - -) for = 1 (focusing case), dotted line (...) for = 1 (defocusing case) . 16 On Terahetrz Laser Pulse Propagation | H |2 Ec2 2 M = . | H |2 1 2 F Ec H Introducing H = , we have Ec 02 1 0.0 0 0.5 1.0 1.5 (13) 1/2 2.0 0.16 0.18 0.20 0.22 0.24 For (14) as a function of magnetic field H , which is shown FIG.9. The figure shows that the magnetic frequency shifts towards the linear resonant frequency with the increase of field intensity. It indicates that the narrow frequency band is more narrowed down with the increased magnetic field. p Figure 7: The nonlinear magnetic permitivity M 1 | H |2 = . 0 1 | H |2 F F = .4 and = 1 , we plot M as a function of frequency under EM fields, which remains negative in the frequency band 0 < < M . Solid line (– )line for = 0 ,broken line (- - -) for = 1 . 0 2 1.25 6 0 M 4 n 1.20 8 10 12 1.15 14 0.0 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.2 0.4 0.6 0.8 0.185 H Figure 9: Magnetic frequency p magnetic field intensity | H | Figure 8: The nonlinear refractive index n as a function of frequency under EM fields, which remains negative in the frequency band = 0 and = 0 , broken line (- - -) and dotted line (….) for = 1 ; = 1 . M as Following Ref. [15], the nonlinear magnetic permeability can be written as : ; | H | = 1 2 , then from Eq. (11), we get FM2 | H |2 = 0, M2 02 Ec2 M 0 II.6. Analysis of the Nonlinear Resonant Frequency = M , the magnetic permeability /0 = 0 1 . Here, as a function of | H |2 1. Ec2 line (– )line for At M Ec2 0 < < M . Solid II.5. Analysis of Magnetic Frequency 2 where, (12) 0 NL F 2 , (15) | H |2 2 i 02NL is the nonlinear resonant frequency and is the loss coefficient. We also have the which yields 17 U. A. Mofiz expression for the nonlinear dielectric permittivity as D0 Here, |E| . Ec2 (16) is the linear dielectric. Then, the | H | = A 2 1 X X 2 2 2 X6 2 2 1.15 1.10 relation between the macroscopic magnetic field and the dimensionless nonlinear resonant frequency can be obtained as : 2 2 0 1.20 oNL D | E |2 = D0 1.25 2 1.05 1.00 , 0.0 0.2 0.4 0.6 0.8 1.0 H (17) c = /0 , A2 = 16 D3 002 h 2 /c 2 , X = 0 NL /0 , and = /0 . where, Figure 11: Nonlinear resonant frequency function of magnetic field intensity (defocusing nonlinearity). |H | 2 0 NL for as a = 1 Near the resonant region without loss, we consider = 1, = 0 . These figures (Fig. 10 and Fig. 11) show that the nonlinear resonant frequency 0 NL goes below the Then, from Eq.(17), we find linear resonant frequency 0 NL 0 1 = 2 | H | 2 A 2 2/3 | H | 4 2 A 2 1/3 III. COUPLED EVOLUTION EQUATIONS III.1. Maxwell’s Equations We start with the Maxwell equations B E = , t D H = , t D = 0, B = 0, 1.00 0.95 0.90 0 in the case of focusing ( = 1 ) nonlinearity and it goes up in the case of defocusing ( = 1 ) nonlinearity. , (18) which is shown graphically in FIG.8 and FIG. 9, for = 1 (focusing nonlinearity) and = 1 (defocusing nonlinearity, respectively. From the available data, we find the value of the resonator 2 parameter A = 2.78 . oNL 0 0.85 0.80 (19) (20) (21) (22) where, E and H are the electric and magnetic 0.75 0.70 0.0 0.2 0.4 0.6 0.8 1.0 H c Figure 10: Nonlinear resonant frequency function of magnetic field intensity (focusing nonlinearity). |H | 2 0 NL for fields, respectively, D and B are the electric and magnetic flux densities which arise in response to the electric and magnetic fields inside the medium and are related to them through the constitutive relations as a given B = ( L NL ) H . =1 18 by D = ( L NL ) E , On Terahetrz Laser Pulse Propagation By using the Maxwell’s systems of equations, we have the following coupled wave equations | E |2 0 2 t Ec 2E 2 E L L 2 L 2 NL E ( E ) t t 2 | H | 2 H | H |2 0 2 H 0 2 t Ec Ec t NL H L E t t L NL H , ( 23) NL t t = H L H L t t L zˆ E . (26) t z 2H 2 H L L 2 L 2 NL H ( H ) Now, considering that the laser radiation is t t circularly polarized, i. e. E E x , E y ,0 , H H x , H y ,0 and introducing the complex = NL E L H t t ~ ~ fields: E E x iE y , H H x iH y , from NL L NL E. (24) Eqns. (25) and (26), we find the following coupled t t equations for the complex fields: 2 Substituting: NL = 0 | H |2 /Ec2 NL = 0 | E |2 /Ec2 , and taking ~ ~ 2E 2E 2 | E |2 ~ E L L 0 L z 2 t 2 t 2 Ec2 2 | H | ~ = i 0 2 H t z Ec ~ | H | 2 E L ~ 0 2 L E t Ec t t zˆ/z , Ez = H z = 0 , the above coupled equations take the following forms: 2E 2E 2 | E |2 E L L 0 L z 2 t 2 t 2 Ec2 | H |2 = 0 2 zˆ H t z Ec | H | 2 E L 0 2 L E t Ec t t | E | 2 E | E |2 0 2 E 0 2 t Ec Ec t L zˆ H , (25) t z ~ | E | 2 E | E |2 ~ 0 2 E 0 2 t Ec Ec t i L ~ H , t z (27) ~ ~ 2H 2H 2 | H |2 ~ H L L L 0 z 2 t 2 t 2 Ec2 | E |2 ~ = i 0 2 E t z Ec ~ | E | 2 H L ~ 0 2 L H t Ec t t 2H 2H 2 | H |2 H L L L 0 z 2 t 2 t 2 Ec2 | E |2 = 0 2 zˆ E t z Ec 19 U. A. Mofiz ~ | H | 2 H | H |2 ~ 0 2 H 0 2 t Ec Ec t ~ i L E, t z ( ) = 1 (28) 1/2 ~ H = H(z, t) exp[i(kz t )] with (1/ | E |) | E | /t << , (1/ | E |) | E | /z << k , L L and considering 0, 0 , L 0, z t t L 0, from Eqn. (27) or Eqn.(28) we find the z | E |2 | H |2 vg = c 2 2 Ec Ec 0 < < M L and L 1/2 (29) frequency band for both the linear and nonlinear 0.4 as 0.2 (30) 0.0 0.15 0.16 0.17 0.18 p (31) Figure 12: Group velocity where , 2 the c = c, 1 vg as a function of the 0.8 vg 0 0 1 0 , 2 2 F L = 0 1 2 0 , 02 ( ) = 1 in (35) 0.6 1 L = 0 1 . cases, which are shown in FIG.12. The figure shows that the frequency band for the wave propagation is more narrowed down ( M decreases) in the nonlinear case with the increase of field intensities. Now, introducing the dimensionless parmeters: and writing frequency 1/2 We plot the group velocity | E |2 | H |2 = k L L 0 L 2 0 L 2 Ec Ec with | E |2 | H |2 2 2 Ec Ec the following field dependent nonlinear dispersion relation: = , 0 = 0 p p | E |2 | H |2 2 2 Ec Ec | E |2 | H |2 2 2 . (34) Ec Ec The group velocity vg = /k is found to be ~ E = E( z, t ) exp[i(kz t )] and . = kc III.2. High Frequency Response and Nonlinear Dispersion Relation | E |2 | H |2 0 0 2 2 Ec Ec (33) the above dispersion relation (Eq.(29)) can also be written in the following form: which we study below for the high frequency and the low frequency responses to the fields. Taking F 2 2 02 in the frequency band vg as a function of frequency 0 < < M . Solid line (—-) is the linear case, dash and dot lines(- - -) for (32) nonlinear case for = 1 , respectively. 20 = 1, | E |2 | H |2 = .5, 2 = .5 2 Ec Ec and On Terahetrz Laser Pulse Propagation ~ ~ 2H 2H Z 2 T 2 ~ ~ 2 | H |2 ~ ~ 2 2H H | H | T 2 T 2 ~ ~ ~ 2 | E |2 | E |2 E = i 0 Ex 0 TZ Z T ~ 2 ~ ~ | E | | H |2 ~ ~ 2 2 | H |2 ~ H| E | H T T 2 T ~ ~ ~ | H |2 H | E |2 T T ~ 2 ~ ~ | E | H ~ 2 2H |E | T 2 T T ~ 2 ~ ~ ~ | E | H ~ 2 ~ 2 | H |2 H |H | |E | T T T T ~ ~ ~ 2H | E |2 | H |2 T 2 ~ 0 2 ~ E i E . (37) 0 TZ Z T III.3. Low Frequency Response: Coupled NLS Equations Normalizing the field variables E and H by Ec and introducing dimensionless space-time coordinates: Z = z/ p , and T = t/t p , where p is the pulse length, and p ct p , t p is the pulse time with the coupled equations, Eqs. (27) and (28) in dimensionless forms are: ~ ~ 2E 2E Z 2 T 2 ~ ~ 2 | E |2 ~ ~ 2 2E E | E | T 2 T 2 ~ 2 2 ~ ~ 0 | H | ~ | H | 2 H = i H 0 TZ Z T ~ ~ ~ | H |2 | E |2 ~ ~ 2 2 | E |2 ~ E | H | E T T 2 T Now, we consider the following Lorentz invariant stretched coordinates [16]: = Z vg T and 1 v ~ ~ ~ 2 | E | 2 E | H | T T ~ ~ ~ | H | 2 E ~ 2 2E |H | T 2 T T = 2 T vg Z , where < 1 , vg vg /c , and we expand 2 1/2 , g the quantities ~ ~ E and H as: ~ E = 2E0 l El eil ( kZT ) E*l e il ( kZT ) , (38) ~ ~ ( ) | H | 2 ~ ~ 2 E E 2| H | T T T 2 ( ) ~ 2 ~ |H| E T 2 ~ ~ 2 ~ ~ | H | E ~ 2 ~ 2 | E | 2 E |E| |H | T T T T ~ ~ ~ 2E | H |2 | E |2 T 2 ~ 0 2 ~ H i H , (36) 0 TZ Z T l =1 ~ H = 2H0 l Hl eil ( kZT ) H*l e il ( kZT ) . (39) l =1 Here, (E0 , H0 , El , Hl ) A(1) A(2) 2 A(3) ... are functions of stretched coordinates ( , ) . Then in the order 0( ) , we find = k/ , and in the order 0( 2 ) , we find vg = 1/ approving the compatibility condition. Thus, following the standard technique [17] in the order 0( ) , we arrive at the following coupled NLS equations: 3 and 21 U. A. Mofiz a 2a P 2 Qa | a |2 a Qb | b |2 a = 0, (40) b 2b i P 2 Qb | b |2 b Qa | a |2 b = 0, (41) 0.000 i Qa 0.002 a E1(1) , b H1(1) and P , Qa , Qb are where 0.006 given by the expressions 0.008 1 vg2 P= , 2 kvg / , 2 kvg / Qa = 0.150 0.175 0.180 0.185 nonlinearity) 0.008 0.006 Qa are the nonlinear coefficients, the nature of which on the frequency in the frequency band 0 < < M , we study below. 0.004 0.002 P , Qa , and Qb 0.000 0.150 0.155 0.160 0.165 F = 0.4 , 0 = 0.145 , 0.170 0.175 0.180 0.185 p <| a | >= 0.5 , <| b | >= 0.5 , = 1 , and = 1 , the plots of P , Qa , Qb , and Q = Qa Qb are depicted in FIGs. 13-17, 2 2 0.170 in the frequency band 0 < < M . Qa is always negative but changes its value abruptly at a critical frequency cr 0.1742 for = 1 (focusing frequency (43) P is the dispersion coefficient, Qa and Qb Taking the parameters 0.165 Figure 14: The nonlinear coefficient Qa as a function of III.4. Analysis of 0.160 p Qb = . (44) 2 kvg / Here, 0.155 (42) and 0.004 Figure 15: The nonlinear coefficient Qa as a function of frequency in the frequency band 0 < < M . Qa is always negative but changes its value abruptly at a critical frequency cr 0.1742 for = 1 respectively. (defocusing nonlinearity) 0.0 4 0.5 P Qb 2 1.0 0 1.5 2 2.0 0.150 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.155 0.160 0.165 0.170 0.175 0.180 0.185 0.185 p p Figure 16: The nonlinear coefficient Qb as a function of Figure 13: Dispersion coefficient P as a function of frequency in the frequency band 0 < < M . P in the frequency band 0 < < M . Qb is always negative but changes its value abruptly at a critical frequency cr 0.1742 . frequency changes its sign from negative to positive at a critical frequency cr 0.1742 . 22 On Terahetrz Laser Pulse Propagation 0.0 2 0.5 Q 2 magnetic fields: X = Y , we plot the graph of cr as a function of field X = Y , which is shown in FIG.18. The figure shows that the critical frequency cr goes to the linear resonant 1.0 frequency 0 with the increase of intensities of the wave fields. 1.5 0.185 2.0 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.185 0.180 p cri 0.175 Figure 17: The nonlinear coefficient as a function of 0 < < M . 0.170 is always negative but changes its value abruptly at a 0.165 frequency Q Q in the frequency band critical frequency cr 0.1742 . 0.160 0.0 We see from the Figs. 13-17 that P changes its sign from negative to positive , Qa is negative for 0.4 0.6 0.8 1.0 X cr as a function of applied field X = Y . Note that cr 0 as X 1 . Figure 18: focusing ( = 1 ) nonlinearity while it is positive for defocusing( = 1 ) nonlinearity , Qb an Q are always negative but change their values abruptly at a certain critical frequency cr , which is defined by the condition at 0.2 III.5. Solution of the NLSE From the symmetric nature of Eqs. (40) and (41) and considering the same intensities of the electric and magnetic fields of the wave, we can take vg /c = 1 , and it is determined from the Eq.(35) as | a |2 =| b |2 [18]. In this situation, we obtain from cr = bˆ bˆ 2 2aˆcˆ , 2aˆ either of Eq. (40) or Eq. (41) a 2a P 2 Q | a |2 a = 0, (46) where, Q = Qa Qb . In the present situation, it is seen that, P can be positive as well as negative for (45) i where, aˆ = F Y 2 (1 F )X 2 X 2Y 2 , different range of the values of the frequency , whereas Q is always negative. Thus, for obtaining solution of the above Eq. (46), we may have two bˆ = (1 F ) 02 (X 2 Y 2 X 2Y 2 ) Y 2 , cˆ = 02 (1 Y 2 ). (37) |E| | H| Here, X = and Y = . Ec Ec cases: Case I: Q < 0 , and 0 < < cr ; cr < < M ; For PQ > 0 where where P < 0 , and Case II: P > 0 , Q < 0 , and = 1, X = 0.5, Y = 0.5, 0 = 0.145, F = 0.4 , we find cr = 0.174822 for = 1 , and cr = 0.174837 for = 1 , respectively. PQ < 0 . For the case I, we take a new variable such that and assume = V a( , ) = u( ) exp[i( K )] , where V is Considering the same intensities of electric and a constant, then from Eq. (46) we find: 23 U. A. Mofiz (2PK V ) P du = 0, d a0 , V and D are arbitrary constants. They define gray solitons . For V = 2PD , it is a kink Here, (47) d 2u PK 2 u Qu 3 = 0, d 2 type dark soliton. Below, we investigate the modulational instabilities of these solitons. (48) III.6. Modulations instabilities of optical solitons which gives motion ( K = V/2P and yields an integral of Coherent dispersion: Hasegawa Formalism du 2 PK 2 2 Q 4 u ) u = const. (49) d P 2P For a localized solution, we In order to investigate the modulational instability of optical solitons, we follow Hasegwawa formalism [20,21]. Considering the modulation of a constant amplitude coherent phase dispersion, we consider u, du/ 0 as , and so, we put const. = 0 . At the maximum value of u = u0 , which defines du/d = 0 , introduce a phase shift a aExp(iQ | a0 |2 ) , a 0 is the constant equilibrium amplitude of where the field. Then, Eq.(46) can be written as: = Qu02 /2 V 2 /4P . i Thus, we arrive at the following solution of Eq.(49) a 2a P 2 Q(| a |2 | a0 |2 )a = 0. (53) Q (50) u( ) = a0 sech a0 , 2P where a 0 is a constant. Going back to the ( , ) In a = Exp(i ) , where = ( , ) and = ( , ) .This means that =| a | with an equilibrium value 0 =| a0 | . Equation(53) has the polar representation, coordinates, we have the solution of Eq. (46) as obvious solution = 0 , = 0 . Q a( , ) = a0 secha0 ( V ) 2P e 1 V 2 V i Qa02 2 P 2 2 P i e we write = 0 1 , where | 1 |<< 0 and = 1 . Then by linearizing Eq.(53) from the Let imaginary and real parts , we obtain , (51) which is a bump type solution, known as the bright soliton, and the wave amplitude may be modulationally unstable. Similarly, the solution for Case II ( PQ < 0 ) is [19]: a( , ) = a0 tanha0 a Q ( V ) 2P (54) 1 P 2 1 2Q0 1 = 0. 0 2 (55) which yield 4 2 1 2 1 P 2QP02 1 = 0. (56) 2 4 Considering, 1 = 1Exp(iK ) ,we 2P V D Q 2 P V 2 i D u02Q 3 PD 2 2 DV 2P 1 2 1 P0 = 0. 2 readily obtain from dispersion relation Eq.56), the nonlinear 2 = PK 2 ( PK 2 2Q 02 ) = P 2 K 2 ( K 2 , = ( PK 2 Q02 ) 2 Q 2 04 . (52) 24 2Q 2 0 ) P (57) On Terahetrz Laser Pulse Propagation We see from Eq.(57) that if Q/P > 0 , becomes negative for values of 2 A. Fast Mode: Bright Soliton K below We see from above, the fast mode has PQ > 0 , and thus modulationally unstable. Possible final state can lead to the formation of bright soliton, i.e. a localized envelope modulating a carrier wave in the form a = Exp(i ) , where 2Q K cr = | 0 | , so that there is a purely growing P mode and the wave is modulationally unstable. The growth rate = Im() attains a maximum max = Q | 0 |2 Q | 0 | . Therefore, in P dimensionless variables: = / max , K = K/K cr : for K = = 1 (2K 2 1) 2 . = (58) (59) and FIG.19 shows the variation of the growth rate 1 2P V sech . L Q L = as a function of the wave number K . On the other hand, for Q/P < 0 , the wave is modulationally stable. where, 2P V 2 1 V L 2 0 . (60) 2 P V is the envelope speed, L is the pulse = 0 and 0 is an arbitrary spatial width at It is to be noted that for a fixed amplitude 0 , the critical frequency K cr is a function of frequencies phase. The maximum amplitude in the frequency band, which is shown in FIG. 0 is inversely L , i.e. 0 L = 2P/Q . We do an analysis of pulse width L on the proportional to the width 20. 1.0 frequencies within the band, which is shown in Fig. 21. It shows that the width is narrower at the resonant frequency while it becomes wider at higher frequencies. 0.8 max 0.6 0.4 3.0 0.2 2.5 0.0 0.2 0.4 0.6 0.8 2.0 1.0 L 0.0 k Figure 19: The growth rate 1.5 k cr as a function of K . 1.0 0.5 6 0.150 5 0.155 0.160 0.165 0.170 k cri 4 p 3 Figure 21: Pulse width L as a function frequency within the frequency band for 0 = 1 . 2 1 At a fixed time = 0 , we have shown modulational envelopes of the carrieer wave and the corresponding bright solitons at different fixed frequencies within the frequency band in FIGs. 2223 for L = .5 and FIGs. 24-25 for L = 2 , respectively. 0 0.150 0.155 0.160 0.165 0.170 p Figure 20: K cr as a function frequency in the frequency band for 0 = 1 . 25 U. A. Mofiz 1.0 1.0 0.9 0.5 2 0 I I2 a a0 0.8 0.0 0.7 0.6 0.5 0.5 1.0 2 1 0 1 2 0.4 2 L 1 Figure 22: Modulational envelope of fast mode for L = .5 corresponding to / p = .155 , V/2P = 30 at 1 2 L =0. Figure 25: Bright soliton for / p = .17 , at = 0 . 1.0 L=2 corresponding to B. The slow mode: Dark Soliton 0.8 The slow mode is modulationally stable since PQ < 0 . It produces dark and gray solitons[22]. The dark has 2 0 0.6 I I2 0 0.4 = 0.2 0.0 2 1 0 1 1 2P V tanh . L Q L (61) 1 V2 V 0 , 2P 2 (62) 2 and 2. Figure 23: Bright soliton for L = .5 corresponding to / p = .155 , at = 0 . = 1.0 while the gray has 0.5 a a0 = | 0.0 2P 1 V | 1 d 2 sech2 Q Ld L 0.5 and 1.0 V02 1 = V0 2 P 2 2 1 0 1 2 . (63) V (64) d tanh L s arcsin 0 . 2 2 V 1 d sech L L Figure 24: Modulational envelope of fast mode for L = 2 , corresponding to / p = .17 , V/2P = 30 at =0. 26 On Terahetrz Laser Pulse Propagation Here, the parameter d, lying in the range 0 < d < 1, regulates the modulation depth, 0.6 0.4 s = 1 , and V0 = V s(2P/Ld ) 1 d 2 . Note that for d = 1 , one recovers the dark soliton. The a a0 0 finite equilibrium amplitude 0.2 is now inversely 0.2 L and the parameter d , i.e. 0 Ld = 2 | P/Q | . The proportional to both the width minimum amplitude min = 0 1 d 2 is given 0.0 0.4 0.6 by 2 1 0 1 2 , which is zero in the dark L case. FIGs 26-29 show the modulational envelopes of the carrieer wave and the corresponding dark solitons at different frequencies within the frequency band at a fixed time = 0 for L = .5 and L = 2 , respectively. Figure 28: Modulational envelope for slow mode corresponding L = 2, V/2P = 30 at = 0 . to / p = .17 , 0.5 1.0 I I2 2 0 0.4 0.5 0.3 a a0 0.2 0.0 0.1 0.0 0.5 2 1 1.0 0 1 2 L 2 1 0 1 2 Figure 29: Dark soliton for / p = .17 , at = 0 . L L=2 corresponding to Figure 26: Modulational envelope of slow mode for L = .5 corresponding / p = .155 , V/2P = 30 at = 0 . IV. RESULTS AND DISCUSSIONS Thus, we analyze the optical properties of the negative index materials (NIMs) under external electromagnetic (EM) fields of intense laser radiation, their linear and nonlinear resonant frequencies, and the propagation of EM fields through NIMs near the resonant region. With the data available, the linear resonant frequency (eigenfrequency) of SSR is found to be 1.45 THz and the magnetic frequency to be 1.87 THz. The resonant frequency and the magnetic frequency creates the frequency band within which the dielectric permittivity and the magnetic permeability simultaneously are negative, which is the main criteria of NIMs. Thus, the founded frequency band is a narrow band. Analyzing Eq.(10) and Eq.(11), it is found that the frequency band becomes more narrower with the increasing intensities of the EM fields of the propagating 1.0 0.8 2 0 I I2 0.6 0.4 0.2 0.0 2 1 0 1 2 L Figure 27: Dark soliton for / p = .155 , at = 0 . L = .5 corresponding to 27 U. A. Mofiz wave. Eq.(14) shows that that the magnetic frequency shifts towards the linear resonant frequency with the increase of field intensity. We solve Eq.(17) to find the relation between the nonlinear resonant frequency with the intensity of the wave’s magnetic field. With the increasing magnetic field intensity, Eq.(18) shows that the nonlinear resonant frequency goes below the linear resonant frequency in the case of focusing nonlinearity, while it goes up in the case of defocusing nonlinearity . A system of coupled equations [ Eq.(25) and Eq.(26)] describes the EM fields propagation through NIMS. From these equations, in the high frequency response, we find the the nonlinear field dependent dispersion relation defined by the Eq. (34). The group velocity of the corresponding wave is defined by the Eq. (35) which shows that the frequency band for wave propagation is more narrowed down with the increase of the field intensities. wave. The nonlinear resonant frequency of the split ring resonator depends on the intensity of the wave, and it goes below the linear resonant frequency in the case of focusing nonlinearity, while it goes up in the case of defocusing nonlinearity. The nonlinear field dependent dispersion relation and the corresponding group velocity determined in the high frequency response, show that the frequency band of the wave propagation is more narrowed down with the increase of field intensities. The coupled nonlinear Schrödinger equations describing the propagation of the wave in the slow response, predicts the propagation of bright solitons and dark solitons before and after a critical frequency determined by the field intensities within the frequency band. The bright soliton is a fast mode and it is modulationally unstable. The growth rate of the instability is calculated. The dark soliton is a slow mode and it is modulationally stable. Solitons pulse widths are more narrower at the resonant frequency and they become more wider away from the resonant region. Results obtained in this investigation may have some applications in meta-materials using SRR with the negative refractive index in the terahertz spectrum and in photonic crystals, which are to be investigated in near future. In the low frequency response, using the reductive pertubation method, we obtain a coupled [(Eq.(40) and Eq.(41)] nonlinear Schrödinger (NLS) equations . The dispersion coefficient and the nonlinear coefficients of the NLS equations are investigated within the frequency band of NIMs. Considering the same intensities of electric and magnetic fields of the wave, the coupled system is solved, which predicts the propagation of bright and dark solitons through NIMs before and after some critical frequency within the band. Eq.(45) defines the critical frequency, the analysis of which shows that it goes to the linear resonant frequency with the increase of field intensities. The bright soliton is a fast mode which is modulationally unstable. 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