James R. Suckling, Alastair P. Hibbins, Matthew J. Lockyear, T. W. Preist, J. Roy Sambles, and Christopher R. Lawrence, Physical Review Letters, Vol. 92, article 147401 (2004). "Copyright © 2004 by the American Physical Society."Fabry-Perot–like resonant transmission of microwave radiation through a single subwavelength slit in a thick aluminum plate is quantified for a range of slit widths. Surprisingly, and in contrast to previous studies [e.g., Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001)], the resonant frequency exhibits a maximum as a function of slit width, decreasing as the slit width is reduced to less than 2% of the incident wavelength. This result accords with a new model based on coupled surface plasmon theory taking into account the finite conductivity, and hence permittivity, of the metal. This is contrary to a common assumption that metals can be treated as infinitely conducting in this regime

A. P. Hibbins

C. R. Lawrence

J. R. Sambles

J. R. Suckling

J. R. Reitz

M. J. Lockyear

S. Astilean

S. Collin

T. W. Preist

Physical Review Letters

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Finite Conductance Governs the Resonance Transmission of Thin Metal Slits at Microwave Frequencies

Fabry-Perot-like resonant transmission of microwave radiation through a single subwavelength slit in a thick aluminum plate is quantified for a range of slit widths. Surprisingly, and in contrast to previous studies [e.g., Phys. Rev. Lett. 86, 5601 (2001)]], the resonant frequency exhibits a maximum as a function of slit width, decreasing as the slit width is reduced to less than 2% of the incident wavelength. This result accords with a new model based on coupled surface plasmon theory taking into account the finite conductivity, and hence permittivity, of the metal. This is contrary to a common assumption that metals can be treated as infinitely conducting in this regime

Suckling, JR

Hibbins, AP

Lockyear, MJ

Preist, TW

Sambles, JR

Lawrence, CR

Surrey Research Insight

Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies.

James R. Suckling, Alastair P. Hibbins, Matthew J. Lockyear, T. W. Preist, J. Roy Sambles, and Christopher R. Lawrence, Physical Review Letters, Vol. 92, article 147401 (2004). Copyright © 2004 by the American Physical Society.Fabry-Perot–like resonant transmission of microwave radiation through a single subwavelength slit in a thick aluminum plate is quantified for a range of slit widths. Surprisingly, and in contrast to previous studies [e.g., Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001)], the resonant frequency exhibits a maximum as a function of slit width, decreasing as the slit width is reduced to less than 2% of the incident wavelength. This result accords with a new model based on coupled surface plasmon theory taking into account the finite conductivity, and hence permittivity, of the metal. This is contrary to a common assumption that metals can be treated as infinitely conducting in this regime

Suckling, James R.

Hibbins, Alastair P.

Lockyear, Matthew J.

Preist, T. W.

Sambles, J. Roy

Lawrence, Christopher R.

Open Research Exeter

P H Y S I C A L R E V I E W L E T T E R S week ending9 APRIL 2004VOLUME 92, NUMBER 14Finite Conductance Governs the Resonance Transmissionof Thin Metal Slits at Microwave FrequenciesJ. R. Suckling, A. P. Hibbins, M. J. Lockyear, T.W. Preist, and J. R. SamblesThin Film Photonics Group, School of Physics, University of Exeter, Devon EX4 4QL, United KingdomC. R. LawrenceQinetiQ, Cody Technology Park, Farnborough GU14 0LX, United Kingdom(Received 18 August 2003; published 7 April 2004)147401-1Fabry-Perot –like resonant transmission of microwave radiation through a single subwavelength slitin a thick aluminum plate is quantified for a range of slit widths. Surprisingly, and in contrast toprevious studies [e.g., Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001)], the resonant frequency exhibits amaximum as a function of slit width, decreasing as the slit width is reduced to less than 2% of theincident wavelength. This result accords with a new model based on coupled surface plasmon theorytaking into account the finite conductivity, and hence permittivity, of the metal. This is contrary to acommon assumption that metals can be treated as infinitely conducting in this regime.DOI: 10.1103/PhysRevLett.92.147401 PACS numbers: 78.20.Ci, 41.20.Jbf  cN2t: (1) to zero (wavelength to infinity) if the metal has finiteconductivity.There is much current interest in the remarkable trans-mission of subwavelength apertures in otherwise opaquescreens following the work of Ebbesen et al. [1].While theinitial study was for an array of holes in an opaque screen,further theoretical work has been undertaken for deepgratings with narrow slits in metal screens [2–6], andtheir Fabry-Perot –like nature has been demonstrated. Thesingle slit case has also been studied both theoretically [7]and experimentally [8,9], and although [8] was not anexact test of [7] it indicated the correct functional form ofresonant frequency on gap width.In this Letter the resonant response of a single, deepmetallic slit is studied for a range of slit widths andfrequencies. It is shown that while the resonances areFabry-Perot –like in nature they always occur at frequen-cies lower than that predicted from a simple Fabry-Perotmodel [7]. This reduction in resonant frequency withincreasing slit width is anticipated simply from the differ-ent boundary conditions at the open ends of the thin slitscompared to those of a Fabry-Perot interferometer. Bycontrast, a further reduction in resonant frequency atvery narrow widths ( < 70 m), illustrated in the presentstudy, is entirely associated with the finite conductivity ofthe aluminum. Because the resonant mode in the slit is acoupled surface mode, as the slit gap reduces so theinfluence of the finite skin depth in the metal becomessignificant. This results in a rapidly decreasing resonantfrequency and broadening resonance with reducing slitwidth.Simple Fabry-Perot theory for the resonant frequenciesof a cavity consisting of two nearly perfectly reflectingmirrors separated by distance t is given by Eq. (1), wherec is the speed of light and N the mode number.0031-9007=04=92(14)=147401(4)$22.50 However, a single, narrow metallic slit has differentboundary conditions that cause the frequency to drop asthe slit width is increased. For a perfect conductor thedegree of shift in wavelength is predicted by Takakura [7]to beshiftFP 2w=tlnw=FP  3=22w=tlnw=FP  1=2   ; (2)where w is the slit width, t the depth, and FP the Fabry-Perot wavelength. However, for narrow slits, where thecorrection due to end effects may be quite small, a sec-ond, and more important effect, arises. The resonantmodes, which are in effect long-range coupled surfaceplasmons with wave vector just beyond the light line,shift down in frequency and broaden. Sobnack et al.[10] give a formal expression for the wave vector forcoupled surface plasmons in the gap between twosemi-infinite, metal layers in the visible regime. For themicrowave domain, when the metal permittivity is domi-nated by the imaginary component, their expression isfound [11] simply to reduce tof  fFP  c22p j"ij1=2w; (3)where fFP is the simple Fabry-Perot frequency of thecavity, w is the width of the slit, and "i is the imagi-nary part of the permittivity of the metal. A slit ofinfinitesimal width in a perfect conductor is predictedto have a resonant frequency equal to the Fabry-Perotcavity of the same depth [7]. By complete contrast Eq. (3)predicts a turn down in the resonant frequency as the slitapproaches zero width, with the resonant frequency going2004 The American Physical Society 147401-1(a)P H Y S I C A L R E V I E W L E T T E R S week ending9 APRIL 2004VOLUME 92, NUMBER 14The microwave regime was chosen for this experimentfor two main reasons. First, the available analytic theoryfor single slit response [7] assumes that the metal isperfectly conducting. While this is certainly not true atoptical frequencies, it has been taken by many to be agood assumption in the microwave regime. However, ourresults forcefully demonstrate that this is not correct.Second, it is clearly far easier to manufacture the struc-ture and alter its parameters in the microwave regimethan in the visible.The experimental setup is shown in the inset of Fig. 1.The slit is made up of two large, polished aluminumplates of face area approximately 400	 200 mm2 andthickness 19.58 mm. The machining of the plates is tosuch an accuracy that the slit width varies by less than25 m. The plates are separated by smooth, dielectricspacers placed outside the incident beam spot, and thetransmission through the structure for a range of slitwidths (15  w  1000 m) is characterized. The slitis placed directly between two microwave horns, oneacting as source and the other as detector. The detectorhorns are placed at 10 cm, far enough away not to beconsidered near field, yet as close as possible to giveacceptable signals from the very narrow slits. A typicalset of data is shown in Fig. 1 for a 250 m slit width.An infinite plane wave source induces no momentumalong the slit length (for normally incident radiation);however, experimentally this is impossible to attain.The momentum associated with the Fourier transformof a finite-width beam width results in a correspondingincrease in resonant frequency. It is essential that this iswell quantified to allow very accurate determination ofthe influence of slit width on the resonant frequencies tobe extracted. To characterize this frequency shift effectthe slit length was constrained with parallel aluminumspacers, replacing the dielectric, and the change in reso-FIG. 1. Resonant transmission of a 250 m slit. Inset: theslit labeled with dimensions and polarization of incidentmicrowaves.147401-2nant frequency as a function of slit length characterized.This change in frequency is given, to first order, byf / 1fFPl2 ; (4)where l is the slit length. As expected from this simpletheory, a graph of frequency against inverse slit lengthsquared gave a straight line (see inset of Fig. 2). Theintersection of this line with the frequency axis givesthe ‘‘infinite’’ beam resonant frequency, and comparingthis with the frequency actually measured for the sampleof infinite slit length gave the frequency shift due to thefinite beam width. These shifts, which were subtractedfrom all the experimental data to allow comparison with(b)FIG. 2. (a) Mode 2: Experimental data, slit widths20–990 m (points), resonant frequency corrected by0:019 0:005 GHz due to finite beam spot size, width cor-rected by 10 4 m due to plate bowing and surface rough-ness. HFSS [12] modeling (solid line) and Takakura prediction(dashed line). Inset: frequency against inverse slit lengthsquared for slit width of 80 m (points), linear fit (solid line),frequency at intercept 15:05 GHz. (b) Mode 9: Experimentaldata for slit widths 24–994 m (points), resonant frequencycorrected by 0:035 0:008 GHz, width corrected by 133 m, HFSS modeling (solid line) and Takakura predic-tion (dashed line).147401-2FIG. 3. Mode shifts from the simple Fabry-Perot model, asgiven by Takakura predictions and the data for a 500 m slit.P H Y S I C A L R E V I E W L E T T E R S week ending9 APRIL 2004VOLUME 92, NUMBER 14plane wave modeling, were found to be 0:035 GHz forthe N  9 mode at 68.6 GHz, corresponding to a beamwidth of 68 mm diameter, and 0:019 GHz for theN  2 mode at 15.1 GHz, corresponding to a beam widthof 190 mm. In addition, the plates, separated by spacersat the ends, bowed slightly to close the slit, requiring acorrection of slit width when fitting to theory, stated incaptions to Figs. 2(a) and 2(b).(Note also from Fig. 1 that the peak transmission is oforder 8	 103 for a :w ratio of about 20:1. For similarratios Takakura [7] predicts a transmission of 2:5	 103.This is in reasonable agreement considering the strongdiffraction of the microwaves at the slit exit.)Figure 2 shows the frequency dependence of twomodes as a function of slit width for the N  2 mode(15:1 GHz) and the N  9 mode (68:1 GHz). Alsoshown for comparison is the Takakura [7] predictionand that from a finite element computer model [12].While Takakura’s prediction has the correct form, start-ing at the Fabry-Perot frequency and dropping monotoni-cally in frequency with increasing slit width, it is clearthat the magnitude does not accord with the data. Thefinite element model with finite conductivity agrees verywell with the data.The turn down in resonant frequency for the verynarrow slits confirms the Preist-Sambles model [Eq. (3)]for coupled surface plasmons at microwave frequencies.This drop in frequency with decreasing slit width alsoprovides a method for determining the imaginary per-mittivity of aluminum. To generate the predictions thatcompare with the data the computer modeling code usesfinite conductivity in the aluminum. From this it gener-ates the relative imaginary permittivity using [13]"i  g2"0f ; (5)where g is the conductivity and f the frequency of in-cident radiation. This equation may be applied when thephoton frequency (here 109 s1) is much less thanthe damping constant of the metal,   1014 s1. In themicrowave regime the conductivity of the metal would beexpected to be close to that of the d.c. value for alumi-num, 3:8	 107 Sm1. The conductivities required to fitthe data, 3:5 0:3	 107 Sm1 and 1:7 0:1	107 Sm1 for modes N  2 and 9, respectively, giveimaginary permittivity values of 4:170:35 	 107 at15.1 GHz and 4:480:26 	 106 at 68.1 GHz, respec-tively. Although the conductivity values are close tothat expected, their variance suggests that the simpleDrude model may not be appropriate. It is worth notingthat as the conductivity is reduced, for any given mode,the frequency turn down begins at increasingly widerslits, as also predicted by Preist-Sambles [Eq. (3)].Figure 3 shows the fall in resonant frequency from afixed slit width (500 m) across the full range of frequen-cies measured (15–69 GHz) compared to the simple147401-3Fabry-Perot prediction for the same depth interferometer.Also plotted are Takakura’s [7] predictions. Once again, itis clear that [7] has the right form, but predicts too low afrequency for each mode. This trend shows that it is theratio of the wavelength to the slit width, and not thenumber of wavelengths inside the slit, that determinesthe shift in frequency away from that of the simple Fabry-Perot prediction. It might be expected that more modeswithin the slit would minimize boundary effects, instead,as the frequency increases, the slit width becomes a moresignificant fraction of the wavelength, and causes agreater shift in frequency.In summary, the frequency of resonant transmission ofa narrow metallic slit has been measured as a function ofslit width. It is shown that one approximate theory [7] isflawed but that the data are in accordance with predictionsof a finite element code when finite conductivity is alsotaken into account. This is in contradiction to the assump-tion that metals can be treated as perfect conductors in themicrowave regime. It is also shown that for small slitwidths a simple coupled surface plasmon model predictsthe observed reduction in resonant frequency and fromfitting to this gap-dependent change in frequency theimaginary relative permittivity of aluminum at micro-wave frequencies has also been determined. As expected,the value of the imaginary permittivity was found tochange by almost one order of magnitude across the rangeof frequencies measured.The authors thank the Engineering and PhysicalSciences Research Council and QinetiQ (Farnborough)for their financial support.[1] T.W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, Nature (London) 391, 667 (1998).[2] J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, Phys. Rev.Lett. 83, 2845 (1999).147401-3P H Y S I C A L R E V I E W L E T T E R S week ending9 APRIL 2004VOLUME 92, NUMBER 14[3] S. Astilean, Ph. Lalanne, and M. Palamaru, Opt.Commun. 175, 265 (2000).[4] S. Collin, F. Pardo, T. Teissier, and J-L. Pelouard,J. Opt. A, Pure Appl. Opt. 4, 154 (2002).[5] H. E. Went, A. P. Hibbins, J. R. Sambles, C. R. Lawrence,and A. P. Crick, Appl. Phys. Lett. 77, 2789 (2000).[6] A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and D. M.Robinson, Appl. Phys. Lett. 79, 2844 (2001).[7] Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001).[8] F. Yang and J. R.Sambles, Phys. Rev. Lett. 89, 063901(2002).147401-4[9] A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, Appl.Phys. Lett. 81, 4661 (2002).[10] M. B. Sobnack, W. C. Tan, N. P. Wanstall, T.W.Preist, and J. R. Sambles, Phys. Rev. Lett. 80, 5667(1998).[11] T.W. Preist and J. R. Sambles (private communication).[12] HFSS, Ansoft Corporation, Pittsburgh, PA, U.S.A.[13] J. R. Reitz, F. J. Milford, and R.W. Christy, Founda-tions of Electromagnetic Theory (Addison-Wesley,Wokingham, 1993), 4th ed.147401-4

Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies

Finite conductance governs the resonance transmission of thin metal slits at microwave frequencies

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