Light diffraction by a subwavelength circular aperture

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1 Early Vew publcaton on ssue and page numbers not yet assgned; ctable usng Dgtal Object Identfer DOI) Laser Phys. Lett ) / DOI 1.12/lapl Abstract: Dffracton of normally ncdent lght by a subwavelength crcular aperture s calculated analytcally. The aperture s opened on a perfectly conductng planar screen wth nfntesmal thckness. In our model, the aperture s replaced by unform magnetc currents and charges. The model allows one to obtan the normalzed cross secton for the aperture radus up to half of the wavelength, whch exceeds the.2 wavelength lmt of the Bethe-Bouwkamp s dpole model [1 3]. Also, n addton to reproducng the ka) 4 dependence, whch s characterstc of the dpole mode, our unform feld model explans the transmsson enhancement obtaned n Abajo s numercal smulaton [4]. Normalzed cross secton , 2.35).59,1) Abajo Bethe Boukamp Unform feld radus / wavelength Normalzed transmsson cross secton as a functon of the normalzed radus c 25 by Astro Ltd. Lght dffracton by a subwavelength crcular aperture Che-We Chang, A.K. Sarychev, and V.M. Shalaev School of Electrcal & Computer Engneerng, Purdue Unversty, West Lafayette, IN 4797, Unted States Receved: 25 January 25, Accepted: 29 January 25 Publshed onlne: 17 February 25 Key words: apertures; dffracton theory; scannng mcroscopy; superresoluton PACS: Fx, 42.5.St, 42.7.Ln 1. Introducton The non-geometrc behavor of lght passng through an aperture has been studed snce the ffteenth century. It was observed that the transmtted lght dverges from the orgnal path f the aperture shrnks to approxmately ten wavelengths. Ths dffracton phenomenon becomes much more complex when the aperture decreases further to the subwavelength range. For a crcular aperture on a planar conductng screen, the dpole model [1 3] predcts that the normalzed cross secton, whch s defned as the rato of the total transmtted power to the total ncdent power over the aperture area, s proportonal to ka) 4 a s the radus of the aperture, and k s the wavenumber of the ncdent wave). However, recent numercal smulatons [4] show that ths model works only for rad smaller than.2 wavelengths. Recently, the enhanced transmsson was observed for a subwavelength aperture array [5], C-shaped [6], and H- shaped apertures [7]. The radaton ntensty wth focused pattern was observed for an aperture wth a perodc pattern on the ext sde [8]. Currently, there s no a sngle analytcal model that could explan all these new phenomena. The dpole model assumes that the radus of the aperture s much smaller than the wavelength and thus has ts lmtatons. In ths paper, we assume that the magnetc current s unform wthn the aperture. Ths smple model s capable of descrbng the normalzed cross secton for larger apertures. 2. Mathematcal formulaton 2.1. Problem defnton The free space s separated nto two regons by a planar, perfectly conductng screen located at z =. A crcular aperture wth radus a s opened at the orgn. The thckness of the screen s set to be nfntesmal. We focus our consderaton on relaton between the aperture radus and Correspondng author: e-mal: chang22@purdue.edu c 25 by Astro Ltd.

2 2 Che-We Chang, A.K. Sarychev, et al.: Lght dffracted by a subwavelength crcular aperture H E k Fgure 1 onlne color at A crcular aperture wth radus a s centered on a perfectly conductng screen. A lnearly polarzed plane wave wth wavelength λ s ncdent normally from the left-hand sde of the screen normalzed cross secton. A plane wave wth wavelength λ s ncdent normally from the left-hand sde of the screen. The electrc feld of the ncdent wave s lnearly polarzed n the x drecton. The confguraton s sketched n Fg. 1. In the followng dscusson, ndces 1 and 2 refer to the left- and the rght-hand sdes, respectvely Felds radated by magnetc currents and charges From the boundary condtons: J s = n 12 H 2 H 1 ), 1) ρ s = D n1 D n2, 2) H 2 and D n2 can be defned by electrc currents and electrc charges whle the felds n regon 1 are replaced by the null felds. Ths represents the equvalence prncple [9 13]. However, the boundary condtons for the tangental electrc feld and the normal magnetc flux densty do not nclude feld sources. Therefore, we must add some fcttous terms, magnetc currents and magnetc charges, n Maxwell s equatons to descrbe the feld dscontnutes. The modfed versons of Maxwell s equatons are as follows E = 1 B K, 3) H = 1 D + J, 4) D = ρ e, 5) y x z B = ρ m, 6) J = 1 ρ e, 7) K = 1 ρ m, 8) ɛ =1, 9) µ =1. 1) Snce K and ρ m are only symbols representng the tangental electrc feld and the normal magnetc flux densty, these addtonal terms do not change the felds n regon 2 when we apply the equvalence prncple. The reason for settng the felds n regon 1 as null s that the aperture can be closed by a perfect electrc conductor, wthout changng the felds n both sdes. In such case, J s short-crcuted, and only magnetc currents and magnetc charges contrbute to the felds n regon 2. The magnetc charges are dstrbuted along the crcumference of the aperture to termnate the magnetc currents. In the followng steps, the flled screen s removed by applyng the method of mages, and thus the magnetc currents are doubled. K s = 2n E). 11) To calculate the felds radated by K and ρ m, t s useful to ntroduce the electrc vector potental A E n ths electrc-sources-free problem. The electrc vector potental s defned as E = A E. 12) By substtutng ths defnton n Eq. 4) and movng the curl operator out, we fnd the magnetc feld as H = 1 A E φ H. 13) The magnetc scalar potental φ H s added because φ H =. Next, wth Eq. 3) and the Lorentz gauge, A E = 1 φ H, 14) we derve the wave equaton n terms of A E : 2 A E + k 2 A E = K. 15) Smlarly, the wave equaton for φ H can be derved from Eq. 6) and Eq. 13) 2 φ H + k 2 φ H = ρ m. 16) The solutons to wave Eqs. 15) and 16) are gven by the convoluton of the source terms n the rght-hand sde wth the Green s functon, G: A E r) = 1 G r r )K dv, 17) 4π φ H r) = 1 4π G r r )ρ mdv, 18) c 25 by Astro Ltd.

3 Laser Phys. Lett. 25) / 3 Gr) = ekr 4πr. 19) Fnally, by substtutng Eqs. 11) and 8) nto Eqs. 17) and 18), we fnd the felds n regon 2 as A E = 1 G r r )n E )ds, 2) φ H = G r r ) n E )ds. 21) E', a.u , 4.616) 2.3. Boundary condtons 1 The tangental electrc feld s the only unknown n Eqs. 2) and 21), and ths feld can be related to the ncdent feld n regon 1 va the boundary condtons. The total feld n regon 1 contans three parts: the ncdent feld E, H ), the reflected feld from the screen wthout the aperture E r, H r ), and the scattered feld from the magnetc currents and magnetc charges E 1, H 1 ). The total feld n regon 2 comes only from the magnetc currents and magnetc charges, E 2 and H 2. The boundary condtons requre that E t + E rt + E 1t = E 2t, 22) H t + H rt + H 1t = H 2t. 23) Now, by applyng E t = E rt and H t = H rt, Eqs. 22) and 23) can be reduced to the followng equatons E 1t = E 2t, 24) 2H t + H 1t = H 2t. 25) Snce the drectons of the scattered felds n regons 1 and 2 are opposte, k 1 = k 2, we obtan that H 1t = k 1 E 1t, 26) H 2t = k 2 E 2t = H 1t. 27) By combnng Eqs. 27) and 25), we fnd that H 2t = H t. 28) 2.4. Problem soluton When we use for A E, φ H n Eq. 13) formulas 2) and 21), the dffracted magnetc feld becomes a functon of the tangental electrc feld n the hole. Thus, n E can be solved wth the boundary condton gven by Eq. 28). The calculaton can be smplfed by computng the feld at the orgn. Then, the assumpton that K s constant n the aperture s appled. Detals for ths calculaton are gven n the Appendx. The result s E = H t x. 29) 1+ expka) 2 ka 1) Fgure 2 onlne color at Electrc feld n the center of the aperture The E feld s plotted n Fg. 2. For the felds n the far zone r λ), we have E = ka2 expka) ˆr [n E ], 3) 2r H = ka2 expka) ˆr ˆr [n E ]. 31) 2r Hereafter, the subscrpts for the felds, referrng to the regon of space, are omtted snce only the felds n the rghthand sde regon wll be dscussed. a/λ 2.5. Normalzed cross secton In order to compare our results wth the Bethe formula, the normalzed cross secton s calculated. Frst, the Poyntng vector s gven by S = c 8π ReE H )= 32) = ck2 a 4 3r 2 E 2 cos 2 θ + sn 2 θ cos 2 ϕ)ˆr. Here θ denotes the angle between ẑ and ˆr, and ϕ gves the angle between the plane of ẑ and ˆx and the plane of ẑ and ˆr. The power densty s not unform, nether n θ drecton nor n the ϕ drecton. Thus, the total power through the aperture should be ntegrated over the sem-sphere. S tot = π 2 S r 2 sn θdθdϕ = ck2 a 4 24 E 2. 33) Fnally, the normalzed cross secton s found as σ = S tot S nc = 34) c 25 by Astro Ltd.

4 4 Che-We Chang, A.K. Sarychev, et al.: Lght dffracted by a subwavelength crcular aperture = ka)2 4 sn ka 5 4 cos ka ka + 1 ka) ka)4 [1+3ka) ka)4 ], 35) whch s vald at ka Dscusson In order to compare the unform model wth other models, the normalzed cross secton s shown as a functon of the normalzed radus, a/λ, n Fg. 3. Note that the screen thckness n the Abajo s boundary element method [4] s.1λ nstead of zero as n other models. It s clear that the unform feld model reflects properly the mportant characterstcs suggested by the dpole model and the boundary element method. Frst, n the unform feld model, the normalzed cross secton for a small radus s proportonal to ka) 4 f the hgher-order terms n Eq. 35) are neglected. Ths s consstent wth the Bethe-Bouwkamp dpole model. The Bouwkamp s curve approaches the unform feld model f more correcton terms are added. Only two correcton terms are ncluded n Bouwkamp s model n Fg. 3: σ = 64 27π 2 ka)4 36) [ ka) ka)4 +...]. Bouwkamp Second, the unform feld model has one local maxmum as the boundary element method does. As seen n Fg. 3, the local maxmum occurs at.13λ; ths corresponds to the maxmum electrc feld n Fg. 2. Although n Fg. 3 the peak of the boundary element method s at.27λ, Abajo reported that the peak occurs at a smaller radus and the enhancement s further ncreased as the screen thckness approaches zero. The normalzed cross secton drops down below 1 as the radus exceeds.2λ. Then, the normalzed cross secton s equal to 1 agan at.6λ and then t goes to nfnty. The Abajo calculaton clams that the normalzed cross secton approaches 1 for rad larger than one wavelength. We suggest that the unform feld model provdes a reasonable descrpton for rad smaller than a half of the wavelength. 4. Concluson In concluson, the normalzed cross secton for a crcular aperture n a perfectly conductng planar screen s calculated analytcally. A lnearly polarzed plane wave s ncdent normally onto the screen and the thckness of the screen s set to be zero. In our calculatons, the magnetc Normalzed cross secton , 2.35) radus / wavelength.59,1) Abajo Bethe Boukamp Unform feld Fgure 3 onlne color at Ths fgure shows the normalzed transmsson cross secton as a functon of the normalzed radus. The red sold lne s the unform feld model proposed n ths paper. The blue dot-dash lne s Bethe s dpole model. The green plus lne s Boukamp s modfed dpole model wth two hgher-order terms, Eq. 36). The black dash lne s Abajo s boundary element method. In Abajo s smulaton, the thckness of the screen s.1λ current densty n the aperture s assumed to be constant. Ths unform feld model descrbes properly the most mportant features of the two other models. Frst, t reduces to the ka) 4 equaton as Bethe-Bouwkamp s dpole model does for rad smaller than.13λ. Second, ths analytc model demonstrates the transmsson enhancement smlar to Abajo s numercal result. In the future, the unform feld model wll be appled to the subwavelength slt problem wth TM polarzaton. A proper modfcaton of the unform feld assumpton could mprove the result so that the normalzed cross secton approaches one when radus s larger than a wavelength. The model can be also mproved by usng the Taylor expanson for the electrcal feld n the center of the hole. Then, the ntegraton of terms n the Taylor seres as was done here for the case of the constant feld) would gve a set of lnear equatons that fully solve ths problem for a sngle hole. Note that for each step of the procedure, one would obtan analytcal expressons for both the far felds and local felds. 5. Appendx Ths secton descrbes how to obtan A E and φ H at the orgn. The calculaton s smplfed by computng the feld at the orgn. Then, the assumpton that K s constant n the aperture s appled. [ ] 1 A E = G r r )n E )ds = 37) r= r= c 25 by Astro Ltd.

5 Laser Phys. Lett. 25) / 5 = 1 expkρ) E ŷ)ρdρdϕ = 38) ρ = 1 k [expka) 1]E ŷ, 39) φ H = G r r ) n E )ds = 4) = G r r )) n E )ds = 41) = G r r )n E ) ˆndl = 42) ρ =a = E a φ H = E a [ G r r ) φ H r= G r r ) k ρ=a sn ϕ dϕ, 43) sn ϕ 44) ) 1 r r dϕ r r r r ], ρ =a = E a 45) sn ϕ expka) a k 1 ) aˆρ a a dϕ = = 2k E expka) k 1 ) ŷ. 46) a References [1] H.A. Bethe, Phys. Rev. 66, ). [2] C.J. Bouwkamp, Phlps Research Reports 5, ). [3] C.J. Bouwkamp, Reports on Progress n Physcs XVIII, ). [4] F.J. García de Abajo, Opt. Express 1, ). [5] T.W. Ebbesen, H.J. Lezec, H.F. Ghaem, et al., Nature 391, ). [6] X. Sh and L. Hesselnk, Jpn. J. Appl. Phys. 41, ). [7] E.X. Jn and X. Xu, n: Proceedngs of IMECE 3 Washngton, D.C., 23), pp [8] T. Tho, K.M. Pellern, and R.A. Lnke, Opt. Lett. 26, ). [9] J.A. Stratton, Electormagnetc Theory McGraw-Hll, New York, 1941). [1] R.F. Harrngton, Tme-Harmonc Electromagnetc Felds McGraw-Hll, New York, 1961). [11] R.F. Harrngton, Introducton to Electromagnetc Engneerng McGraw-Hll, New York, 1958). [12] S. Ramo, Felds and Waves n Modern Rado Wley, New York, 1953). [13] R.E. Colln, Antennas and Radowave Propagaton McGraw-Hll, New York, 1985). c 25 by Astro Ltd.