Dispersion surfaces and light propagation in homogeneous dielectric-magnetic uniaxial medium

Transcription

1 Journa of Phsics: Conference Series Dispersion surfaces and ight propagation in hoogeneous dieectric-agnetic uniaia ediu To cite this artice: M S Rafaean and A H Gevorgan 0 J. Phs.: Conf. Ser Reated content - Singe refection quarter-wave device design G Raffa A Bianchetti M T Garea et a. - Obique tota transissions through epsion-near-zero etaaterias with hperboic dispersions Jie Luo Yadong Xu Huanang Chen et a. View the artice onine for updates and enhanceents. - A birefringent refector fro a D anisotropic photonic crsta N Ouchani D Bria B Djafari Rouhani et a. This content was downoaded fro IP address on 7/0/08 at 9:0

2 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00 Dispersion surfaces and ight propagation in hoogeneous dieectric-agnetic uniaia ediu M. S. Rafaean and A. H. Gevorgan Yerevan State Universit Ae Manoogan St. Yerevan 005 Arenia E-ai: Abstract. We investigated ight propagation in a hoogeneous ediu having doube anisotrop (i.e. with anisotrop of both dieectric and agnetic perittivities) and an arbitrar orientation of its optica ais in the pane of incidence. We investigated wave surfaces for the considered ediu. We showed that on soe groups of surfaces can arise. We investigated the conditions for tota independent of poarization refection and the conditions for tota transission as we as possibiities of the subject sste of serving as: an onidirectiona refector a bea spitter and a phase retarder.. Introduction Metaaterias are artificia coposites containing sub-ong-wave structures and anifesting such inear and non-inear optica properties as: negative refraction reverse Dopper effect eectroagnetic wave energ propagation in the direction opposite to the wave vector etc. [ ]. The aso have other etraordinar appications such as: perfect enses [] invisibe coaks [4 5] perfect absorbers [6] etc. Though the easiest wa of having negative refraction is the appication of the isotropic etaaterias this refraction can aso be observed in anisotropic etaaterias. Moreover in the ast genera case it is not necessar to require that a the eeents of the dieectric and agnetic perittivities be negative [7]. Recent the investigation of such anisotropic etaaterias has been of great interest [8-4]. However the iterature ain considers the cases when the dieectric and agnetic tensor principa eeents are either parae or perpendicuar to the boundaries of the sste. In [0] the pecuiarities of super-ight propagation in an anisotropic etaaterias were investigated for an arbitrar orientation of the optica ais in the incidence pane. In [] the case of the onidirectiona tota transission and possibiit of eistence of a negative Brewster s ange at the boundar isotropic ediu-anisotropic ediu for an arbitrar orientation of the optica ais when ˆ Iˆ ( ˆ is the agnetic perittivit tensor Î is the unit atri) are investigated. In [9] the possibiities of tota refection at the boundar isotropic ediu-anisotropic ediu are investigated and a condition for tota refection is obtained. In [0] the possibiities of tota negative refection at the boundar isotropic ediu-anisotropic ediu are investigated for an arbitrar orientation of the optica ais again for ˆ Iˆ. In [6] the dispersion equations for anisotropic etaaterias are cassified. In the present paper the dispersion surface pecuiarities and dependences of the dispersion curves on the optica ais orientation are investigated. We aso investigated the conditions for independent of poarization tota refection and the conditions of the tota transission as we as the possibiities of use of the subject sste as: an onidirectiona refector a bea spitter and a phase retarder. Pubished under icence b IOP Pubishing Ltd

3 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00. Dispersion Surfaces Let us consider pecuiarities of the dispersion surfaces of an anisotropic ediu having arbitrar orientated optica ais in the incidence pane. We assue that the dieectric and agnetic perittivit oca tensors of the ediu can both be diagonaized and the tensors ˆ 0 and ˆ 0 have the foowing for in the corresponding frae: ˆ ˆ () i.e. it is assued that the principa aes of the dieectric and agnetic perittivit tensors coincide. Beow we aso assue that the ediu is uniais i.e. = and =. In the aborator frae ˆ and ˆ have the foowing for: where ˆ ˆ Tˆ [ ] ˆ Tˆ [ ] ˆ Tˆ [ ] ˆ Tˆ [ ] () 0 0 is the rotation atri for the ais at ange. A pane eectroagnetic wave of frequenc and k wave vector propagates in the entioned ediu. Fro Mawe s equations we obtain the foowing dispersion equations for refractive inde: n ( ) - n ( ) - 0 () where n cos n sin n n n n z z / / / / n k n k nz kz k k kz - are the wave vector coponents and is the waveength in the ediu. The dispersion equation for pane waves in such a ateria can be factorized into two ters: One dispersion equation for eectric odes and the other dispersion equation for the agnetic odes. Let us reduce the dispersion equation of eectric odes to the canonic for. To do it we represent n n and n z in the foowing fors: n a n n n n n b n n n n c n z where sin cos sin (4) a b c cos sin Then the dispersion equation for eectric odes takes the for: n n n (5) where sin sin. sin sin Dispersion equation for agnetic odes aso has the sae for but in this case n n n are obtained b the interchanges: and in (4) and (6). Dispersion surfaces characterize the dependence of the eectroagnetic wave refraction in the ediu on the ight propagation direction. Eectroagnetic pane waves propagating inside the ateria depending on the vaues of and can ehibit dispersion surfaces in the for of eipsoids of revoution hperbooids of one sheet or hperbooids of two sheets. Furtherore depending on the optica ais orientation the intersections of these surfaces with the propagation pane can be circes eipses hperboas or straight ines. Now et us go into the detais of the probe. I. if in (5) are positive that is when: f 0 g sin 0 and h ( sin ) 0 (6)

4 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00 the dispersion surface of eectric odes is an eipsoids of revoution with seiaes aong the directions: n n and n i.e. aong the directions: n n ˆ ˆ sin ˆ ˆ cos ˆ nzz n nzz n (7) n n ˆ n zˆ n ˆ sin n ˆ n zˆ cos n n ˆ n zˆ z z z where ˆ ŷ and ẑ are the unit vectors of the and z aes. II. If <0 <0 и <0 i.e. for f<0 g>0 and h<0 the ode is evanescent. III. If one of is negative and the others are positive i.e. for f<0 and g<0 or f<0 и h>0 then the dispersion surface is a hperbooids of one sheet with seiaes aong the directions n n and n. IV. If one of is positive and the others are negative i.e. for f > 0 and g<0 or for f>0 and h>0 then the dispersion surface is a hperbooids of two sheet with seiaes aong the directions n n and n. V. If the dispersion surface is a pane and for we have for eectric odes: n cos n sin 0 i.e. the dispersion surface becoes a pane. It shoud be noted that for z the dispersion surface has the for: n n n cos sin 0. Fro this foows that for n 0 z the dispersion surface is the straight ine nz ntg. In the opposite case the ode is evanescent. Let us note that the pane arises on for and the straight ine for. On figure we present (for various paraeters of the ediu) the possibe (in the genera case) pairs of dispersion surfaces (one for the eectric odes the other for the agnetic odes). The can be defined fro dispersion equation (). Figure. The dispersion surfaces for various paraeters of the sste b: d: a: c: e: Let us note that in the genera case the foowing pairs are ipossibe: an eipsoid of revoution with a hperbooid of one sheet a hperbooid of one sheet with a hperbooid of two sheets one evanescent ode with a hperbooid of two sheets. It is natura for one can show that that is on even nubers of negative i can eist. Here 4 5 and 6 are the corresponding coefficients for the canonica dispersion equation of agnetic odes. The are obtained b the interchanges and in the. Let us aso note that if the dispersion surface of one of the odes is a pane then the other is either a conica surface (turning either into a pane or a straight ine in particuar cases) presented in figure е or an evanescent. And if the

5 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00 dispersion surface of one of the odes is a straight ine then the other can be: a revoution eipsoid; a hperbooid of one sheet; a hperbooid of two sheets; a straight ine (figure ). Figure. The dispersion surfaces for the case when one of the is a straight ine b: d: a: c: Now we pass on to detaied anasis of the dispersion equation () for n 0 that is for fied incident pane. Let s investigate the dependences of dispersion curves on the optica ais orientation. Figure a presents the dependences of dispersion curves of eectric odes on the paraeter for the sae paraeters of the probe for which the dispersion curve is an eipse. If the rotation ange of the optica ais is equa to k then the eipse seiaes are directed aong the directions: n ˆ and n ˆz. For the other vaues of this ange the eipse seiaes are shifted fro the directions n ˆ and n ˆz. Figure b presents the dependences of the dispersion curves of eectric odes on the paraeter for the sae paraeters of the probe for which the dispersion curve is a hperboa. Figure. The dependences of dispersion curves on the optica ais orientation. a: b: As it is seen fro that picture if the ange changes the dispersion curves which are hperboas rotate in the pane nn z and (for soe specific vaue of that ange) these directions becoe asptotes. At the end of this section et us note that the above considerations reain true aso for the agnetic odes.. Onidirectiona refection and tota transission Let us consider ight refection and refraction on the border of an anisotropic ediu uniaia anisotropic etaateria. The ediu is uniaia and it occupies the haf-space z 0 i.e. the ediu border is parae to the pane and the incident pane coincides with the pane z (z is the aborator sste). An eectroagnetic wave of frequenc is incident at the incidence ange 4

6 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00 fro isotropic and hoogenous ediu having the paraeters a n d (the dieectric and agnetic perittivities of the ediu) on the subject haf-space. Now we consider the possibiit of obtaining tota refection on the base of etaaterias such that it does not depend on the incidence ange and poarization and tota transission independent of poarization for certain incidence anges. Tota refection condition for p- poarization p (refection coefficient for p- poarization) has the foowing for: n cos 0. (8) The condition of tota refection for p- poarization for an arbitrar incident ange has the foowing for: 0 0 or 0 cos (9) Doing the interchanges and in the (8) and (9) we wi obtain anaogous conditions for s- poarization consequent requiring the conditions for s- and p- poarizations siutaneous we can obtain onidirectiona refection. Our cacuations show that in particuar for / 4 onidirectiona refection takes pace. For the condition: n cos cos (0) we have p (transission coefficient for p- poarization). Fro (0) foows that tota transission for p- poarization is possibe for certain incidence anges which are defined fro (0) (the Brewster ange for p- poarization): ( cos sin ) p B sin. () ( ) It is to be noted that in contrast to the refection on the border of two isotropic edia when there is on one Brewster ange here we have two of the (for p- and s- poarizations). Having sae s p conditions for s- poarization we can cacuate the condition of : B B B ( )( ) sin ( ) ( ) ( ) Consequent there is tota transission at the incident ange B regardess of the poarization. Now et us consider possibiities of anisotropic etaaterias as bea spitters. The refraction anges of the two forward eigen waves ( Pz 0 and Pz 0 P i is the Ponting vector of the i-th eigen wave) and spit ange are defined as foows: P P tan tan and P z P. () z Figure 4 presents the dependence of on the. It is to be noted that for each B and the ange is chosen in such a wa that satisfies () i.e. here is the bea spitting ange for tota transission. As it is seen fro the figure 4 for the certain paraeters. Consequent it is possibe to design iniature bea spitters without an intensit oss on the base of etaaterias. At 0 both refraction anges are the sae. At this condition the sste can work as a phase retarder. Fro figure 4 we can see that for certain paraeters 0 therefore on this paraeters of the probe the sste can work as an idea phase retarder again without an intensit oss. () 5

7 Internationa Sposiu on Optics and its Appications (OPTICS0) Journa of Phsics: Conference Series 50 (0) 00 IOP Pubishing doi:0.088/ /50//00 Figure 4. The dependence of spit ange on the dieectric perittivit. The probe paraeters are:..5.4 B B 4. Concusion Concuding et us to note that we investigated dispersion surfaces for the anisotropic etaaterias with dieectric and agnetic anisotropies. We showed that for an arbitrar orientation of the optica ais in the incidence pane soe groups of surfaces can arise. The conditions for independent of poarization tota refection and the conditions for tota transission are obtained as we as obtained the conditions of the subject sste of serving as: an onidirectiona refector a bea spitter and a phase retarder. 5. References [] Veseago V G 00 Sov.Phs.Usp [] Lee S H Park C M Seo Y M and Ki C K 00 Phs. Rev. B 8 40 [] Pendr J B 000 Phs. Rev. Let [4] Au A and Engheta N 005 Phs. Rev. E [5] Leonhardt U 006 Science 777 [6] Land N I Sajuigbe S Mock J J Sith D R.and Padiia W J 008 Phs. Rev. Lett [7] Linde I V Tretakov S A Nikoskinen K I and Ivonen S 00 Microw. Opt. Techno. Lett. 9 [8] Xiang Y Dai X Wen S 007 Opt. Coun [9] Shen N H Wang H T Tian Y 008 Europhs. Lett [0] Luo H Hu W Shu W Li F Ren Z 006 Europhs. Lett [] Gevorgan A H 00 Mo. Crst. Liq. Crst. 8-9 [] Gevorgan A H 00 Opt. Spectrosc [] Chen H Xu Sh Li J 009 Opt. Epress [4] Rafaean M S and Gevorgan A H 00 Proc. SPIE K; doi:0.7/.895 [5] Rafaean M S and Gevorgan A H 00 Modern Probes in Optics & Photonics. Proc. of Int. Adv. Res. Workshop [6] Depine R A Inchaussandague M E and Lakhtakia 006 A J. Opt. Soc. Aer. A 949 [7] Sith D R and Schurig D 00 Phs. Rev. Lett [8] Pishnak O P and Lavrentovich O D 006 App. Phs. Lett [9] Xiang Y Dai X and Wen S 007 Opt. Coun [0] Yonghua L Pei W Peijun Y Jianping X and Hai M 005 Opt. Coun [] Luo H Ren Z Shu W and Li F 007 App. Phs. A [] Lu Y Pei W Yao P Xie J and Ming H 005 Opt. Coun [] Marke V A Schotand J C 00 J. Opt. 050 [4] Hu L Chui S T 00 Phs. Rev. B