Progress In Electromagnetics Research M, Vol. 18, 31 42, 2011

Transcription

1 Progre In Electromagnetic Reearch M, Vol. 18, 31 42, 2011 UNIFORM PLANE WAVE REFLECTION FROM PEC PLANE EMBEDDED IN A NONLINEAR MEDIUM M. M. Hunain and M. J. Mughal Faculty of Electronic Engineering GIK Intitute of Engineering Science and Technology Topi, Swabi, Khyber Pakhtunkhwa 23640, Pakitan Abtract Reflection from a perfect electric conductor (PEC plane of infinite dimenion embedded in a econd order nonlinear medium i tudied. The reflected wave ha two part, one due to linear behavior, the other due to nonlinear behavior of the medium. The expreion of the reflection coefficient for parallel and perpendicular polarization cae are obtained. The reduction of reflection coefficient to a linear medium cae i alo reported. Dependence of the aid coefficient on incident electric field intenity and the angle of incidence i alo plotted. 1. INTRODUCTION All medium are baically nonlinear. For example, a nonlinear phenomena like photon-photon cattering in a magnetized vacuum ha been reported [1]. The experiment of Franken et al. to generate econd harmonic i marked a the birth of nonlinear optic [2]. Electromagnetic phenomena are nonlinear in the ene that they occur when the repone of a material ytem to an applied electromagnetic field depend in a nonlinear manner on the trength of the aid field. For example, an applied field produce nonlinear polarization in the medium which caue parametric or nonparametric phenomena [4]. The former i governed by the contitutive formulation and the latter by the Maxwell potulate [3]. Many problem related to diffraction, cloaking, inviibility, focuing ytem and cattering problem etc. involve the PEC boundarie [4 8]. Some author have alo worked on reflection of plane wave from a PEC planar interface embedded in a medium [7 11]. Ellipometry i baed on the reflection and i ued for Received 13 January 2011, Accepted 3 April 2011, Scheduled 11 April 2011 Correponding author: Mohammad Mazhar Hunain (mazhar.hunain@gmail.com.

2 32 Hunain and Mughal the invetigation of the dielectric propertie (complex refractive index or dielectric function of thin film [12 15]. Reflectance meaurement are alo ued to find optical propertie uch a optic axi, complex refractive index, optical contant etc. [16 19]. In the preent work the behavior of a uniform plane wave i tudied when it trike an infinite PEC plane which i embedded in a econd order nonlinear optical medium. It i aumed that the aid medium i iotropic, lole and ha no diperion. Thi work i carried out to tudy PEC reflector placed in a nonlinear medium to find high frequency field at the cautic uing Malov method. Thi method require the initial value of the reflected field [7 9]. There are two canonical cae of thi problem, and the electric field i either parallel polarized or perpendicularly polarized. The reflection coefficient for both cae have been derived which can be ued to tudy the PEC reflector and catterer. Perpendicular polarization cae i dicued in Section 2 and parallel polarization in Section 3. The reult are hown and dicued in Section 4. Some application are alo mentioned in Section 4.1. The paper i concluded in Section PERPENDICULAR POLARIZATION Before coming to the original problem, the form of the wave equation i conidered for the propagation of light through a nonlinear optical medium. The Maxwell equation in SI unit can be written a [4] D = ρ, (1 B = 0, (2 B Ẽ = t, (3 H = D t + J. (4 We ue a tilde ( to denote a quantity that varie rapidly in time. It i aumed that the region of interet ha no free charge and current, o that ρ = 0, (5 J = 0. The material i alo nonmagnetic, o that B = µ 0 H (6 The electric flux denity i related to electric field and polarization in a nonlinear medium a D = ɛ 0 ɛ r (ω Ẽ + P, (7

3 Progre In Electromagnetic Reearch M, Vol. 18, where the polarization vector P depend nonlinearly upon the electric field trength Ẽ, and ɛ r(ω i the nonlinear relative permeability of the medium. To derive the optical wave equation, we take the curl of the curl-ẽ Maxwell equation (3, wapping the order of pace and time derivative on the right-hand ide of the reulting equation, then uing Equation (4, (5 and (6 to replace B by µ D 0 and obtain the equation Ẽ + µ 2 D 0 t 2 = 0, Now uing the definition of D given in Equation (7 to eliminate it from the above equation. We alo replace µ 0 by 1 ɛ 0 on the left ide c 2 of the aid equation and obtain the following expreion 2 Ẽ ɛ(ω 2 Ẽ c 2 t 2 t = 1 µ 0 2 P t 2 (8 Now conider an infinite PEC plane which i embedded in a medium having econd order nonlinear optical propertie. The PEC plane i placed on z = 0 in a yz plane a hown in Figure 1. The medium i nonlinear for z < 0 and z > 0. The incident electric field i perpendicular to xz plane, which can be defined a [4] Ẽ i (r, t = E i (ω i e iω it Figure 1. Oblique incidence on a PEC plane embedded in a econd order nonlinear medium.

4 34 Hunain and Mughal where and E i (ω i = A i (ω i ŷe ik i(ω i (x in θ i +z co θ i k i (ω i = ɛ(ω i ω i /c A the medium i a econd order nonlinear optical type, the reflected fundamental wave will create a nonlinear polarization at frequency ω = 2ω i which can be repreented a where and P (r, t = P e iω t P = pe ik (ω r p = ˆpɛ 0 χ (2 eff Γ2 (ω A 2 i (ω i Where χ (2 eff i the nonlinear uceptibility of the medium. Thi nonlinear polarization will caue a radiation of the econd-harmonic frequency ω. The propagation of reflected electric field E R in the medium can be expreed a follow uing Equation (8 and putting the expreion for the nonlinear polarization 2 E R (ω + [ ɛ(ω ω 2 /c 2] E R (ω = ( ω 2 /µ 0 p e ik r Where p i the component of P perpendicular to the plane of incidence. The olution of the above equation conit of a particular olution plu a general olution. The general olution i obtained by etting it right-hand ide equal to zero. All of the appropriate boundary condition can be met by auming that the homogeneou olution i an infinite plane wave of amplitude Γ(ω A i (ω i and wavevector k R (ω. A PEC plane will offer a dicontinuity to the incident wave o the amplitude of reflected electric field will differ from incident field by a factor Γ(ω the reflection coefficient which will be found after applying the boundary condition. The reflected field will ha a nonlinear part alo becaue of nonlinear polarization of the medium. Thu the olution to the above nonlinear wave equation can be written a [4] E R (ω = Γ(ω A i (ω i ŷe ik R(ω (x in θ R z co θ R ŷ ( ω 2 + /µ 0 k 2 k R (ω 2 ɛ 0χ (2 eff Γ2 (ω A 2 i (ω i e ik (x in θ z co θ where k R (ω = ɛ(ω ω /c

5 Progre In Electromagnetic Reearch M, Vol. 18, and k (ω = 2k R (ω (9 By applying the following boundary condition at z = 0 to the tangential component (E i + E R tangential z=0 = 0 (10 For the baic frequency A i (ω i ŷe ik i(ω i (x in θ i + Γ(ω A i (ω i ŷe ik R(ω (x in θ i = 0 (11 and for double frequency A i (ω i ŷe ik i(ω i (x in θ i + Γ(ω A i (ω i ŷe ik R(ω (x in θ i ŷ ( ω 2 + /µ 0 k 2 k R (ω 2 ɛ 0χ (2 eff Γ2 (ω A 2 i (ω i e ik (x in θ =0 (12 To meet the boundary condition at each point along the interface, it i neceary that the wavevector of the incident wave, the reflected wave and reflected econd-harmonic wave have identical component along the plane of the interface. We thu require that let k i x(ω i in θ i = k R x (ω in θ R = k x in θ (13 B = ( ω 2 /µ 0 k 2 k R (ω 2 ɛ 0χ (2 eff By examining Equation (11 and (13, it can be een that for the baic frequency the reflection coefficient Γ(ω i 1. From Equation (12 and (13 By implifying A i (ω i + Γ(ω A i (ω i + BΓ 2 (ω A 2 i (ω i = Γ(ω + BΓ 2 (ω A i (ω i = 0 (14 Solution of the above quadratic equation give the reflection coefficient a Γ 1,2 (ω = 1 ± 1 4BA i (ω i 2BA i (ω i By examining Equation (14 it can bee een that for a linear medium aumption the lat term in the above equation will be zero a there will be no contribution from nonlinear polarization in reflected field and thi will give the reflection coefficient Γ(ω equal to 1 a earlier which i true for a PEC plane.

6 36 Hunain and Mughal 3. PARALLEL POLARIZATION For parallel polarization cae, the incident electric field vector lie in the xz plane, and can be written a E i (ω i = A i (ω i (ˆx co θ i ẑ in θ i e ik i(ω i (x in θ i +z co θ i The reflected field will be E R (ω = Γ(ω A i (ω i (ˆx co θ R + ẑ in θ R e ik R(ω (x in θ R z co θ R + ˆx co θ (ω 2 /µ 0 k 2 k R (ω 2 ɛ 0 χ (2 eff Γ2 (ω A 2 i (ω i + ẑ in θ(ω2 /µ 0 ɛ k 2 k R (ω 2 0 χ (2 eff Γ2 (ω A 2 i (ω i After applying the boundary condition given in Equation (10 ik(x in θ z co θ e ˆxA i (ω i co θ i e ik i(ω i (x in θ i +ˆxΓ(ω A i (ω i (co θ R e ik R(ω (x in θ i ( ω 2 +ˆx co θ /µ 0 k 2 k R (ω 2 ɛ 0χ (2 eff Γ2 (ω A 2 i (ω i e ik(x in θ = 0 A θ i = θ R, from Equation (9 and the condition given in Equation (13 we can write and co θ = 1 in2 θ i 4 A i (ω i co θ i + Γ(ω A i (ω i (co θ i + BΓ 2 (ω A 2 i (ω i B i ame a defined in Section 2, after implification co θ i + Γ(ω (co θ i + BΓ 2 (ω A i (ω i Solving for Γ(ω give co θ i ± Γ 1,2 (ω = 1 in2 θ i 4 co 2 θ i 4BA i (ω i 2BA i (ω i 1 in2 θ i 4 1 in2 θ i 4 = 0 = 0 (15 1 in2 θ i 4 co θ i In the cae of normal incidence co θ i will be one and in θ i will be zero then Γ 1,2 (ω = 1 ± 1 4BA i (ω i 2BA i (ω i Thi i the ame reult a obtained for the perpendicular polarization cae. Similarly, a dicued in previou ection for the cae of a linear medium, the lat term in Equation (15 will be zero, and the reflection coefficient Γ(ω will be 1.

7 Progre In Electromagnetic Reearch M, Vol. 18, Γ 1 (ω A (ω i x Figure 2. Reflection coefficient Γ 1 (ω for perpendicular polarization a a function of incident electric field A i (ω i. Γ 2 (ω x A i (ω x Figure 3. Reflection coefficient Γ 2 (ω for perpendicular polarization a a function of incident electric field A i (ω i. 4. RESULT AND DISCUSSION By uing the value of χ (2 eff given in [4], the reflection coefficient Γ 1,2 (ω i plotted againt the incident electric field A i (ω i, for perpendicular cae in Figure 2 and 3 and for parallel cae in Figure 4 and 5. It i aumed that the wave i incident at an angle of 30, ω i i equal to rad/ and the permeability of the medium ɛ doe not depend upon frequency of the incident or reflected wave, i.e., the medium ha no diperion. The ɛ i taken a a calar and i given by ( ɛ = ɛ χ (2 eff

8 38 Hunain and Mughal Γ 1 (ω A (ω i x Figure 4. Reflection coefficient Γ 1 (ω for parallel polarization a a function of incident electric field A i (ω i. 0 x Γ 2 (ω A i (w x Figure 5. Reflection coefficient Γ 2 (ω for parallel polarization a a function of incident electric field A i (ω i. A the angle of incidence i fixed the behavior of Γ 1,2 (ω are imilar in both perpendicular and parallel polarization cae. Γ 1 (ω firt decreae then increae. Γ 2 (ω increae a A i (ω i increae in both cae. In our cae, the boundary i a PEC plane; the incident field i reflected back in the medium; and ome of the incident energy i ued to give birth to nonlinear part of the reflected field. The reflection coefficient in the parallel polarization cae alo depend on the angle of incidence. In Figure 6 and 7 the dependence of Γ 1,2 (ω on angle of incidence ha been hown graphically. The amplitude of incident electric field A i (ω i i aumed a

9 Progre In Electromagnetic Reearch M, Vol. 18, Γ 1 (ω q i Figure 6. Reflection coefficient Γ 1 (ω for parallel polarization a a function of incident angle θ i Γ 2 (ω q i Figure 7. Reflection coefficient Γ 2 (ω for parallel polarization a a function of incident angle θ i. for thee plot. Γ 1 (ω decreae from 1 to 2 then goe to zero at the angle 90 and Γ 2 (ω increae and goe to zero at the angle 90 becaue there i no reflection at that angle and urface wave will generate [26] Application Waveguide i baically two plane of PEC or any other dielectric media which are facing each other. The reult can be ued to olve the waveguide filled with a nonlinear medium. Our work i a preliminary one to find the field at a focal point of reflector (embedded in a nonlinear medium uch a circular, pherical or parabolic reflector

10 40 Hunain and Mughal uing Malov Method or any other method a geometrical or phyical optic. Many author have explored the reflection and tranmiion of wave at the interface of linear and nonlinear medium [20 25]. The main difference in our formulim i that the wave reflected completely due to PEC plane, i.e., no tranmiion. 5. CONCLUSIONS The problem of oblique incidence on a PEC plane embedded in a econd order nonlinear optical medium ha been analyzed. The effect of the medium nonlinearity on the reflection coefficient have been revealed. A the nonlinear polarization depend on the incident electric field, the reflection coefficient alo varie with the amplitude of the aid field. Thi dependence i hown graphically. The change in the amplitude of reflection coefficient with repect to angle of incidence i alo plotted. Thi change ha to be incorporated in finding the field at the focal point of a reflector. It i neceary to um all the reflected field to find the intenity of the field at the cautic. Thi work i carried out to find the aid field. Our reult can be ued to find field in nonlinear dielectric filled planar waveguide, cavitie etc. It i aumed that the medium i iotropic, lole and nondiperive. The reflection coefficient have negative value becaue of a PEC boundary. A the governing equation of the reflection coefficient are quadratic in nature o the amplitude of incident electric field hould (ω be le than 1/(4 2/µ 0 k 2 k R (ω ɛ 2 0 χ 2 eff otherwie the reflection coefficient will become imaginary and reflected wave will be nonuniform [26]. REFERENCES 1. Ding, Y. J. and A. E. Kaplan, Second harmonic generation due to photon-photon cattering in magnetized vacuum, Nonlinear Optic: Material, Phenomena and Device, Diget. NLO 90, Franken, P. A., A. E. Hill, C. W. Peter, and G. Weinreich, Generation of optical harmonic, Phy. Rev. Lett., Vol. 7, 118, Shen, Y. R., The Principle of Nonlinear Optic, Wiley- Intercience, New York, Boyd, R. W., Nonlinear Optic, 3rd edition, Academic Pre, Elevier, Burlington, Gennarelli, G. and G. Riccio, Diffraction by a double-negative

11 Progre In Electromagnetic Reearch M, Vol. 18, metamaterial layer with PEC backing, PIERS Online, Vol. 6, No. 8, , Alu, A., D. Rainwater, and A, Kerkhoff, Plamonic cloaking of cylinder: Finite length, oblique illumination and cropolarization coupling, New J. Phy., Vol. 12, No. 10, Faryad, M. and Q. A. Naqvi, High frequency expreion for the field in the cautic region of a cylindrical reflector placed in chiral medium, Progre In Electromagnetic Reearch, Vol. 76, , Faryad, M. and Q. A. Naqvi, High frequency expreion for the field in the cautic region of a parabolic refector coated with iotropic chiral medium, Journal of Electromagnetic Wave and Application, Vol. 22, No. 7, , Faryad, M. and Q. A. Naqvi, Cylindrical refector in chiral medium upporting imultaneouly poitive phae velocity and negative phae velocity, Journal of Electromagnetic Wave and Application, Vol. 22, No. 4, , Byun, W. J., J. W. Yu, and N. H. Myung, TM cattering from hollow and dielectric-filled emielliptic channel with arbitrary eccentricity in a perfectly conducting plane, IEEE Tran. Microwave Theory and Technique, Vol. 46, No. 9, Sep Rahim, T., J. Mughal and M. Hunain, Focal region field of a two dimenional gregorian ytem coated with iotropic chiral medium, J. Electromagnetic Analyi and Application, , Aug. 2010, jemaa Lekner, J., Determination of complex refractive index and thickne of a homogeneou layer by combined reflection and tranmiion ellipometry, Journal of the Optical Society of America A: Optic, Image Science, and Viion, Vol. 11, No. 7, , Jul Dorf, M. C. and J. Lekner, Reflection and tranmiion ellipometry of a uniform layer, Journal of the Optical Society of America A: Optic, Image Science, and Viion, Vol. 4, No. 11, , Nov Peteren, K. and O. Keller, Photoelatic propertie of metal meaured by off-null ellipometry, Applied Optic, Vol. 25, No. 2, , Jan. 15, Azzam, R. M. A., Determination of the optic axi and optical propertie of aborbing uniaxial crytal by reflection perpendicular-incidence ellipometry on wedge ample, Applied Optic, Vol. 19, No. 18, , Sep. 15, 1980.

12 42 Hunain and Mughal 16. Haar, U. C., Permittivity meaurement of thin dielectric material from reflection-only meaurement uing one-port vector network analyzer, Progre In Electromagnetic Reearch, Vol. 95, , Liu, C.-C., Y.-H. Chang, and C.-J. Wu, Refractometric optical ening by uing a multilayer reflection and tranmiion narrowband filter, Journal of Electromagnetic Wave and Application, Vol. 24, No. 2 3, , Azzam, R. M. A., Explicit determination of the complex refractive index of an aborbing medium from reflectance meaurement at and near normal incidence, J. Opt. Soc. Am., Vol. 72, 1439, Wolf, R., H. G. Birken, C. Bleing, and C. Kunz, Optical contant of gold in the oft-x-ray region from reflection and tranmiion meaurement, Applied Optic, Vol. 33, No. 13, , May 1, Bloembergen, N. and P. S. Perhan, Light wave at the boundary of nonlinear media, Phy. Rev., Vol. 128, , Shaarawi, A. M., I. M. Beieri, A. M. Attiya, and E. El-Diwany, Reflection and tranmiion of an electromagnetic xwave incident on a planar air-dielectric interface: Spectral analyi, Progre In Electromagnetic Reearch, Vol. 30, , Boiko, B. B., I. Z. Dzhilavdari, and N. S. Petrov, Reflection of plane light from a nonlinear tranparent iotropic medium, Journal of Applied Spectrocopy, Vol. 23, No. 5, , Dec. 7, Dadoenkova, Y. S., I. L. Lyubchankiil, Y. P. Lee, and T. Raing, Light reflection from nonlinear optical dielectric film on a bigyrotropic magnetoelectric ubtrate at angle cloe to Brewter angle, Low Temperature Phyic, Vol. 36, 538, Martorell, J., R. Vilaeca, and R. Corbaln, Peudo-metal reflection at the interface between a linear and a nonlinear material, Optic Communication, Vol. 144, No. 1 3, 65 69, Dec. 1, Kaplan, A. E., Hyterei reflection and refraction by a nonlinear boundary-a new cla of effect in nonlinear optic, JETP Lett., Vol. 24, No. 3, , Aug. 5, Balani, C. A., Advanced Engineering Electromagnetic, John Wiley and Son, Inc. New York, 1989.