Gaussian beam propagation through a metamaterial lens

Transcription

1 Calhoun: The NPS Institutional Achive Faculty and Reseache Publications Faculty and Reseache Publications 4 Gaussian beam popagation though a metamateial lens Zhou, Hong Gaussian beam popagation though a metamateial lens, Univesal Jounal of Applied Mathematics, (4): 84-94, 4

2 Univesal Jounal of Applied Mathematics (4): 84-94, 4 DOI: 89/ujam444 Gaussian Beam Popagation though a Metamateial Lens Hong Zhou Depatment of Applied Mathematics, Naval Postgaduate School, Monteey, CA , USA *Coesponding Autho: hzhou@npsedu Copyight 4 Hoizon Reseach Publishing All ights eseved Abstact We study a Gaussian beam popagation though a metamateial lens by diect numeical simulations using COMSOL We find that a metamateial lens can deflect the beam significantly by eithe adjusting the shape of the lens o inceasing the dielectic pemittivity of the metamateials Keywods Gaussian Beam, Metamateial Lens AMS Subject Classification: 5Q6; 78A99; 78-4 Intoduction Metamateials ae atificially stuctued mateials with unusual and exciting electomagnetic and optical popeties that ae not geneally found in natue (Cai and Shalaev ; Chen, Chan and Sheng ; Kildishev and Shalaev ; Noginov and Podolskiy ) These mico-stuctued metamateials have povided a wide aay of potential inteesting applications Some examples include invisibility cloaks, diective and omnidiectional antennas, Lunebeg and Eaton lenses, waveguide tapes, photonic band gap stuctues, double negative media (ie media having both negative pemittivity and negative pemeability), and shielding stuctues fom eathquake Undestanding the popagation of a Gaussian beam though a metamateial lens is impotant since it has many potential applications In paticula, by tuning the lens shapes and adjusting mateial popeties one can hope to popagate the beam as desied In this pape we diectly apply numeical techniques to undestand the popagation of a Gaussian beam though a metamateial lens and exploe the effects of vaious paametes in the design of the metamateial lens Ou goal is to find some effective way to deflect the Gaussian beam by using a metamateial lens This pape is oganized as follows Afte a bief discussion of the poblem setup and the elevant mathematical fomulations in Section, we cay out numeical calculations fo the Gaussian beam popagation in a metamateial lens in Section Section 4 is devoted to conclusions and some futue wok Poblem Setup and Mathematical Fomulations Following (COMSOL ), we conside a two-dimensional metamateial lens enclosed in a squae ai domain which is suounded by a pefectly matched laye (PML) on each side, as illustated in Figue A PML is an atificial absobing laye used to tuncate unbounded computational egions in numeical methods fo wave equations so that waves incident upon the PML ae absobed without eflection at the inteface Assume a Gaussian beam entes the domain fom the left side, via a suface cuent excitation at an inteio bounday given by J y a = Ae, whee a is the beam waist size and A measues the beam powe The excitation is at the bounday between PML and the modeling domain, and poduces a wave that popagates in both diections into the PML and into the modeling domain The wave taveling into the PML is completely absobed by the PML wheeas the wave taveling into the modeling domain is diffacted by the lens Ou main focus hee is to study how to contol the popagation of the beam by adjusting the metamateial lens Figue A metamateial lens enclosed in an ai domain The eal shape of the metamateial lens is defined as functions of the Catesian coodinates (, ) undefomed ectangula fame: u u () x y of the

3 Univesal Jounal of Applied Mathematics (4): 84-94, 4 85 ( ) (, 4 ) x= x a+ ay y= y a + ax () u u u u whee the typical values fo the paametes ae a =, a = 5, a =, a4 = 5 The dielectic distibution of the metamateial lens is defined on the oiginal Catesian domain by the elationship ( b ) by u ε = + () which leads to a spatial vaiation in the dielectic distibution on the defomed lens The typical values fo the paametes ae b =, b = 5 The special case whee b = coesponds to the situation without a metamateial lens The left bounday of the lens (shown in Figue ) whee the beam entes the lens is specified by the cuve x= a+ ay, y= y a+ a whose cuvatue is ( u ) u( 4) 4 xy yx ( a+ a4)( a) x + y ( ay u ) + ( a+ a4) a( a+ a4) / ( ay) + ( a+ a) κ= = = / / (4) u 4 whee the pimes indicate the deivatives with espect to y u So the cuvatue depends on the spatial vaiable yu and the paametes a and a + a4 Figue The left bounday of a metamateial lens whee the beam entes the lens Fom () and (4) we see that the dielectic distibution of the metamateial lens is invesely popotional to the cuvatue of the inteface of the lens In a flat thee-dimensional Euclidean space, the beam popagation though the lens is descibed by the macoscopic Maxwell s equations (Yeh, 5): B E + = D H = J D = ρ B = The fist two equations ae also called Faaday s law and Maxwell-Ampee s law, espectively Equation thee and fou ae the electic and magnetic fom fom of Gauss s law, espectively In these equations, E denotes the electic field vecto while H epesents the magnetic field vecto The quantities D and B ae the electic displacement (o electic flux density) and the magnetic induction (o magnetic flux density), espectively The emaining quantities ae the electic cuent density J and the electic chage density ρ Maxwell s equations contain eight scala equations that involve twelve vaiables To obtain a closed system, Maxwell s equations ae supplemented by the constitutive elations that descibe the macoscopic popeties of the medium: D= εe = εe+ P (6) B= µ H = µ H + M Hee ε is the dielectic o pemittivity tenso, and µ is the pemeability tenso of the mateial The constant ε is the pemittivity of a vacuum and has a value of 8854 F / m, µ is the pemeability of a vacuum 7 and has a value of 4π H / m The electic polaization vecto P descibes how the mateial is polaized when an electic field is pesent in matte Similaly, the magnetization vecto M descibes how the mateial is magnetized when a magnetic field is pesent Fo linea mateials, the polaization is diectly popotional to the electic field and the magnetization is diectly popotional to the magnetic field Fo such mateials, the constitutive equations become D= εε E (7) B= µµ H whee the paamete ε is the elative pemittivity and µ is the elative pemeability of the mateial These ae usually scalas if the mateial medium is isotopic, and tensos fo anisotopic mateial If we substitute the constitutive elation (6) fo B into Faaday s law in (4), divide both sides by µµ, and apply the cul opeato, we get ( µ µ E) + H = (8) Now diffeentiating Maxwell-Ampee s law in (4) with espect to time and combining it with (7), we find (5)

4 86 Gaussian Beam Popagation though a Metamateial Lens D ( µ µ E) + + J = (9) Using the mateial equation (6) and assuming J = σ E with σ being the electic conductivity, we obtain E E ( µ µ E) + εε + σ = () We conside the time-hamonic electomagnetic fields whee the electic field can be witten as E(, t) = Re( E( ) e iωt ) () Hee E( ) is a phase vecto, o phaso whose amplitude and phase ae time-invaiant, and ω is the angula fequency Note that the time deivative of () coesponds to a multiplication by a simple facto iω, namely, iωt E(, t) = Re( iωe( ) e ) () Employing the popety (), we can eplace the time-dependent equation (9) by a time-independent equation fo the phaso: µ E + ε µε iω E + µσ iω E = () ( ) ( ) ( ) o equivalently, iσ ( µ E) k ε E = (4) εω whee the fee space wave numbe k is given by k = ω / c= ωε µ and the speed of light is c = / εµ in Figue we plot the nom of the electic field passing though a metamateial lens Figue shows the diffeence between the two cases in Figue and Figue In Figue (d) we illustate the distibution of the electic field nom along a vetical line whee x = 5 Without a lens the maximum value of the electic field nom is about 4 V/m and it is educed to 9 V/m by the metamateial lens, which is about 5% eduction Numeical Results In this section we investigate the effects of vaious paametes on the popagation of a Gaussian beam though a metamateial lens We apply the commecial softwae COMSOL to solve the patial diffeential equation (4) with the pescibed initial and bounday conditions COMSOL is a multi-physics commecial softwae based on finite element methods which can be used to solve ou model equations efficiently Beam Effects Fist we fix the popeties of a metamateial lens and study the popagation of a Gaussian beam and how it is affected by vaious beam chaacteistics In Figue we compae the numeical solutions with and without a metamateial lens The paametes used hee ae A =, f ( ) = GHz, a = c / f whee f epesents the opeating fequency and a is the Gaussian beam waist size Hee the beam waist size is invesely popotionally to the opeating fequency All the othe paametes take the typical values specified ealie In Figue we plot the nom of the electic field though a medium of ai In contast,

5 Univesal Jounal of Applied Mathematics (4): 84-94, 4 87 (d) Figue The nom of the electic field with beam waist size a = ( c/ f ) = 9986m without a lens; with a metamateial lens; diffeence between the case with lens and the case without lens; (d) along the vetical line whee x = 5 Now we investigate the effect of the beam waist size We double the beam waist size to a = 4 ( c/ f ) and give ou esults in Figue 4 Figue 4 coesponds to the beam popagation in ai while Figue 4 is elated to the beam popagation though a metamateial lens Figue 4 shows thei diffeence The metamateial lens has deflected the beam significantly Figue 4(d) gives the distibution of the electic field nom along the vetical line x = 5 The maximum nom of the electic field is deceased fom 8 V/m without a lens to 5 V/m with a metamateial lens, which is about 7% eduction In Figue 5 we decease the beam waist size to a = ( c/ f ) and show the esults simila to Figues -4 As shown in Figue 5(d), the maximum value of the electic field nom deceases fom 76 V/m to 66 V/m with a metamateial lens This is about % eduction in magnitude Figues -5 also suggest that the maximum value of the diffeence between the case with a lens and the case without a lens is popotional to the beam waist size O said diffeently, the maximum value of the diffeence between the case with a lens and the case without a lens is invesely popotional to the opeating fequency (d) Figue 4 The nom of the electic field with beam waist size a = 4 ( c/ f ) = 997m without a lens; with a metamateial lens; diffeence between case and the case ; (d) along the vetical line whee x = 5 In Figue 6 we incease the beam powe fom A = to

6 88 Gaussian Beam Popagation though a Metamateial Lens A = and keep all the othe paametes the same as in Figue A diect compaison between Figue and Figue 6 shows that as the beam powe inceases, the electic field inceases and thei elationship is linea Fom Figue 6(d) we see that the maximum nom of the electic field is educed about 5% fom 8V/m to 8 V/m by the metamateial lens (d) Figue 5 The nom of the electic field with beam waist size a = ( c/ f ) = 999m without a lens; with a metamateial lens; diffeence between case and the case ; (d) along the vetical line whee x = 5 Fom all the above simulations, we can see that the metamateial lens can diffact the beam and effectively educe the maximum nom of the electic field by moe than % in all the cases we have investigated hee

7 Univesal Jounal of Applied Mathematics (4): 84-94, 4 89 numeical data can be fitted by a linea function E = 89794a max vey well fo the paamete ange 8 a (d) Figue 6 The nom of the electic field with A = whee the beam waist size a = ( c/ f ) = 9986m without a lens; with a metamateial lens; diffeence between the case with lens and the case without lens; (d) along the vetical line whee x = 5 Effects of a Metamateial Lens Shape Having investigated the effect of beam paametes, we now move to study the effects of vaious paametes in the design of a metamateial lens All the paametes ae the same as in Figue unless specified explicitly Fist, we vay the paamete a and obtain vaious shapes of lens in Figue 7 and plot the nom of the electic field along the vetical line x = 5 fo each case in Figue 7 The lage value of a defines a lage lens and consequently educes the maximum nom of the electic field moe Due to the change of the shape of the lens, the cente of the bell-shaped cuve in each case is gadually shifted to the ight as the value of a inceases In Figue 7 we show the dependence of the maximum nom of the electic field along the line x = 5 on the paamete a We find that the Figue 7 Vaious shapes of lens descibed by diffeent values of a The nom of the electic field along the vetical line x = 5 fo vaious values of a The dependence of the maximum nom of the electic field along x = 5 on Second, we vay the paamete a As we change the

8 9 Gaussian Beam Popagation though a Metamateial Lens values of a fom negative to positive values, the concavity of the lens changes, as illustated in Figue 8 Howeve, fom Figue 8 we see that the maximum nom of the electic field is not sensitive to the change of a that much, the maximum nom of the electic field does not change damatically, as shown in Figue Figue 8 Vaious shapes of lens descibed by diffeent values of a The nom of the electic field along the vetical line x = 5 fo vaious values of a Next, we vay the paamete a and plot the esults in Figue 9 As a deceases, the lens shape gets shote and the maximum nom of the electic field deceases as well Figue 9 indicates that the dependence of the maximum nom of the electic field along the line x = 5 on the paamete a can be well appoximated by a quadatic function E = 9799a max + 7a fo the paamete ange 8 a Lastly, we investigate the effect of the paamete a 4 Since the vaiation of a 4 does not change the shape of lens Figue 9 Vaious shapes of lens descibed by diffeent values of a The nom of the electic field along the vetical line x = 5 fo vaious values of a The dependence of the maximum nom of the electic field along x = 5 on a

9 Univesal Jounal of Applied Mathematics (4): 84-94, 4 9 To conclude, the design paametes a and a4 in Equation () do not affect the maximum nom of the electic field that much, wheeas vaying the design paametes a and a can lead to significant changes to the maximum nom of the electic field To educe the maximum nom of the electic field, one would incease a o educe a a the quadatic fom E b b = 756 max fo the paamete ange 5 b Figue Vaious shapes of lens descibed by diffeent values of a 4 The nom of the electic field along the vetical line x = 5 fo vaious values of a 4 Effects of the Dielectic Popety of a Metamateial Lens We fist exploe the effect of the paamete b in the dielectic distibution of the lens As shown in Figue, the maximum nom of the electic field deceases as the value of b deceases Figue eveals that the maximum nom of the electic field along the line x = 5 can be appoximated as a function of b in Figue The nom of the electic field along the vetical line x = 5 fo vaious values of b The dependence of the maximum nom of the electic field along x = 5 on b Similaly, the effect of the paamete b is shown in Figue, togethe with snapshots of the elative pemittivity on the lens with b = and b = 7 As b inceases, the maximum nom of the electic field deceases The maximum nom of the electic field along the line x = 5 can be fitted as a function of b in the quadatic fom E = 756b max b+ 746 fo b 7 Fom above simulations, we see that in ode to educe the maximum nom of the electic field, one would design a

10 9 Gaussian Beam Popagation though a Metamateial Lens metamateial with smalle values of b o lage values of b in Equation () Anothe possibility is to change the dielectic distibution fom Equation () to some othe foms Figue The nom of the electic field along the vetical line x = 5 fo vaious values of b The dependence of the maximum nom of the electic field along x = 5 on b The contou plot of the dielectic distibution, shown hee is the elative pemittivity whee b = on the left and b = 7 on the ight As a peliminay study, we change the fom of the dielectic distibution into ( b ) bx 4 u ε = + (5) whee the default values of the two coefficients ae b =, b4 = 5 Figue shows how the nom of the electic field along the vetical line x = 5 is affected by the vaiation of b As b deceases, the maximum value of the nom of the electic field deceases Moeove, the maximum nom of the electic field along the line x = 5 can be epeesnted as a quadatic function of b : fo 5 b E b b = 664 max

11 Univesal Jounal of Applied Mathematics (4): 84-94, 4 9 metamateial lens shape and dielectic popeties to futue wok b) Figue 4 The nom of the electic field along the vetical line x = 5 fo vaious values ofb 4 Acknowledgment and Disclaime The autho would like to thank the Office of Naval Reseach (ONR) fo suppoting this wok The views expessed in this document ae those of the autho and do not eflect the official policy o position of the Depatment of Defense o the US Govenment Figue The nom of the electic field along the vetical line x = 5 fo vaious values of b The dependence of the maximum nom of the electic field along x = 5 on b The contou plot of the dielectic distibution with b = In contast, the effect of vaying b 4 fom to 7 is almost negligible, as seen in Figue 4 This obsevation, togethe the esults shown in Figue, also indicates that the inhomogeneity of the dielectic popety in the x-diection contibutes vey little to the eduction of the maximum electic field; wheeas the inhomogeneity of the dielectic popety in the y-diection can lead to significant decease in the electic field 4 Conclusions and Futue Wok We have investigated a Gaussian beam popagation in a metamateial lens, focusing on the effects of the beam, lens shape and dielectic popety of the metamateials Ou findings eveal that it is possible to deflect the beam by eithe adjusting the shape of the lens o vaying the dielectic pemittivity We postpone the seach of optimal design of the REFERENCES [] Cai, W and Shalaev, V () Optical Metamateials, Spinge, New Yok [] Chen, H, Chan, CT & Sheng, P () Tansfomation optics and metamateials, Natue Mateials, 9, [] Justice, BJ, Mock, JJ, Guo, L, Degion, A, Schuig, D and Smith, DR (6) Spatial mapping of the intenal and extenal electomagnetic fields of negative index metamateials, Optics Expess, 4(9), [4] Kildisheve, AV and Shalaev, VM () Tansfomation optics and metamateials, Physics-Uspehi, 54(), 5-6 [5] Kundtz, N and Smith, D R () Exteme-angle boadband metamateial lens, Natue Mateials, 9, 9- [6] Leonhadt, U and Phibin, TG (9) Tansfomation optics and the geomety of light, Pogess in Optics, 5, 69-5 [7] Noginov, MA and Podolskiy, VA () Tutoials in Metamateials, CRC Pess, Boca Raton, Floida [8] COMSOL (), Defining a mapped dielectic distibution of a metamateial lens, COMSOL libay [9] Robets, DA, Kundtz, N and Smith, D R (9) Optical

12 94 Gaussian Beam Popagation though a Metamateial Lens lens compession via tansfomation optics, Optics Expess, 7(9), 864 [] Thompson, R T and Cumme, SA () Tansfomation optics, Advances in Imaging and Electon Physics, 7, [] Yan, M, Yan, W and Qiu, M (9) Invisibility cloaking by coodinate tansfomation, Pogess in Optics, 5, 6-4 [] Yan, M, Yan, W and Qiu, M (8) Cylindical supelens by a coodinate tansfomation, Physical Review B, 78(), 5 [] Yeh, P (5) Optical Waves in Layeed Media, Wiley, Hoboken, New Jesey [4] Ziolkowski, RW and Heyman, E () Wave popagation in media having negative pemittivity and pemeability, Physical Review E, 64, 5665