Unified Theory of Surface-Plasmonic Enhancement and Extinction of Light Transmission Through

Unified Theory of Surface-Plasmonic Enhancement and Extinction of Light Transmission Through Metallic Nanoslit Arrays

(Supplementary Information)

Jae Woong Yoon1, Jun Hyung Lee2, Seok Ho Song2, and Robert Magnusson1

1Department of Electrical Engineering, University of Texas at Arlington, Box 19016, Arlington, TX 76019, USA

2Department of Physics, Hanyang University, Seoul 133-791, KOREA

I. Analytic expressions of the SPP resonance wavelength and bandwidth

In this section, we derive analytic expressions ofthesurface plasmon-polariton(SPP) resonance wavelength SP and bandwidthSPbased on the microscopic pure-SPP model developed by Liu and Lalanne [R1]. We use elementary scattering coefficients tS, rS, a, and b() as defined in Fig. S1(a).Note that we use symbols different from those in [R1] because some symbols therein are identical to ones in our main text but have different definitions.In Fig. S1(b), amplitudes of SPPs excited at the n-th slit are denoted by Un and Vn for SPPs propagating toward the +x and –x directions, respectively.Using the Bloch-Floquet condition (Un+1, Vn+1) = (Un, Vn)exp(ikx) with kx = (2/)sin [R1],the SPP amplitudes under surface-normal incidence of transverse-magnetic polarized planewave is reducedto

,(A1)

where u() = exp(inSP/c) with an effective propagation constant of an SPP nSP = [M/(1+M)]1/2, angular frequency of the incident wave , and speed of light in vacuum c. Equation (A1) describes the SPP resonance by its pole at the complex frequency SP=SP–itot. Therefore, the SPP resonance condition can be written by

. (A2)

Figure S1| (a) Elementary scattering of an SPP at a single silt. The elementary scattering coefficients are SPP transmission coefficient tS, SPP reflection coefficient rS, SPP-CM coupling coefficient a, and SPP-radiation coupling coefficient b(). (b) Excitation of SPPs on a periodic array of slits by an external planewave. Un and Vn represent the amplitudes of SPPs propagating along the +x and –x directions from the n-th slit, respectively.

In a slowly decaying regime such that totSP, Eq. (A2) yields

, (A3)

, (A4)

where nSP′ and nSP″ are real and imaginary parts of nSP, respectively; we assume first-order coupling. On the right-hand side of Eq. (A4) for the SPP decay rate, the first term represents the radiation decay rate raddue to scattering at slits while the second term is the non-radiative decay rate nrdue to ohmic damping of free electrons. Finally, analytic expressions for theSPP resonance wavelengthSP and bandwidth SP(full-width at half-maximum) are immediatelyderived from Eqs. (A3) and (A4) asfollows.

, (A5)

, (A6)

where SPF = nSP′ is the SPP resonance wavelength on a flat, unpatterned metal surface.

II. The SPP resonance wavelength and bandwidth in our case

We estimate SP and SPfor our case with a metal dielectric constant M = –5 and a slit width w = 0.05. The SPP transmission tS and reflection rS coefficients are obtained by FEM calculation. Figure S2(a) shows the FEM calculation result (magnetic field, Hy) of an SPP scattered by a single isolated slit. The SPP source located at x = 0 launches a unit-amplitude SPP at  = 1.062 to a 0.05-wide slit centered at x = 5.025. The SPP from the source is transmitted (tS), reflected (rS), and also scattered to the external radiation and cavity mode.In Fig. S2(a), we observe the reflected SPP for x < 0, coexisting launched and reflected SPPs for 0 x 5, transmitted SPP for x 5.05, cylindrical pattern of emitted external radiation, and the cavity mode inside the slit.The fields associated with the cavity mode and external radiation are negligibly small at the air-metal interface when compared to the SPP amplitude. We therefore assume that the calculated field at the interface can be expressed solely by the pure SPP amplitude as

(A7)

Figure S2| (a) FEM calculation result of SPP scattered by a single slit. An SPP is launched from the SPP source at x = 0 and scattered by a single slit centered at x =x0 = 5.025 with width w = 0.05. (b) Magnetic field profile at the metal surface. Red circles (●) are obtained by FEM calculation in (a) while the black solid curve (─) is due to the unmixed pureSPP model in Eq. (A8)with tS = 0.925896–0.14069iand rS = –0.073074–0.150357i. The phase factor  = 1.08.

where the in-plane wavevector of the SPP kSP = nSP/c, the slit center position x0 = 5.025 and the is arbitrary phase constant. Fitting the pure SPP amplitude in Eq. (A7) to the field at the interface due to the FEM calculation yields tS = 0.925896–0.14069i and rS = –0.073074–0.150357i as confirmed in Fig. S2(b).Applying these values to Eqs. (A5) and (A6), we finally obtain the SPP resonance wavelength and bandwidth

SP = 1.0624 andSP = 0.03346 .(A8)

These values quantitatively agree with SP = 1.062 and SP = 0.03211 from the surface excitation spectrum in Fig. 2c of the main text.

III.Formal consistency of our theory with the microscopic theory of EOT

The microscopic theory of EOT developed by Liu and Lalanne yields the single-interface transmission coefficient [R1]

(A9)

for surface-normal incidence. On the right-hand side of Eq. (A9), the first and second terms represent the non-resonant and resonant contributions, respectively. Near the SPP resonance condition = SP+, where SP, SPcan be rewritten by a Lorentzian function.For the reinterpretation of SP in terms of Lorentzian resonance parameters, we apply the Taylor series expansion of u–1() as

(A10)

In this expansion, Eqs. (A3) and (A4) for SP were used. For a slowly decaying resonance that satisfies SP, SP has a significant amplitude only within a narrow frequency range around SP; therefore, we can take the first order of the expansion in Eq. (A10) as a reasonable approximation.Finally, we obtain

, (A11)

where reduced frequency  = /tot. Formal consistency of Eq. (A11) with

, (A12)

i.e., Eq. (2) in the main text, is obtained by directly comparing these two expressions and including a relation in/ex = |a/b|2 according to the definitions of in, ex, a, and b. We finally obtain

and ,(A13)

. (A14)

For these expressions to have practical meaning in a quantitative manner, normalization of modes should be identical in two different approaches.In defining the elementary scattering coefficients tS, rS, a, and b, the external planewave and slit-guided mode should be normalized so that they carry unit surface-normal power within a period.The SPP mode should be normalized so that it has unit energy when its energy density is integrated over a period.

IV.The surface-plasmonic Fano resonance theory vs.previous interpretations

The excitation of an SPP induces various features such as antiresonant extinction, a null-field at the aperture opening, an abrupt change in the cavity resonance condition, and resonance peak narrowing. As elaborated in detail throughout the main text, these features are all rooted in a single resonance interaction caused bythe surface-plasmonic Fano resonance at the interface.In Table S1, we summarize the main issues under debate regarding the role of SPPs, various interpretations describing such issues, and our unified explanations based on the surface-plasmonic Fano resonance theory.

Table 1.SPP issues in previous debates and the unified interpretations due to our theory.

Issues / Previous interpretations / The unified interpretation
based on the surface-plasmonicFano resonance
Description / Refs.
Transmission peak extinction / Enhanced absorption at the SPP resonance condition on a flat metal surface. / R2, R3. / Closed-cavity (high-Q) regime

Associated resonance processes
The Fano-type interference between the SPP-resonant and direct coupling at the interface is
-constructive in the internal reflection,
-totally destructive in the transmission.
Consequences
-The slits act as closed cavities.
-Very narrow, high-Q cavity resonance.
-The transmission peaks are highly sensitive to the dissipative losses.
-A null-field at the slit opening is observed under the interface excitation by an external planewave as a result of destructive interference in the transmission to the cavity mode.
Destructive interference between the surface and cavity resonances. / R4.
Surface-plasmonic bandgap effect. / R5, R6.
Zero of the interface transmittance associated with the SPP excitation. / R7-R9.
Null-field at the aperture opening / Excitation of the nonresonant SPP. / R8, R9.
Destructive interference between multiply scattered SPPs by the slit array. / R10, R11.
Peak narrowing effect / Transition of the operative resonance mode between the SPP and cavity modes. / R12, R13.
Guided-mode-like resonance at the surface-plasmonic band edge. / R5.
Abrupt shift of the cavity (slit) resonance condition / Effect of the Rayleigh anomaly. / R14. / Open-cavity (low-Q) regime

Associated resonance processes
The Fano-type interference between the SPP-resonant and direct coupling at the interface is
-destructive in the internal reflection,
-constructive in the transmission.
Consequences
-The slits act as open cavities.
-Relatively broad, low-Q cavity resonance.
-The transmission peaks are insensitive to the dissipative losses.
-The SPP is maximally excited at this condition.
-Strong SPP excitation leads to resonant phase change in the internal reflection of the cavity mode, causing an abrupt shift of the cavity resonance condition.
Transition of the operative resonance mode between the SPP and cavity modes. / R6, R7,R13, R15.
Phase change in the internal reflection of the cavity mode. / R15, R16.
Peak broadening effect / Strong coupling to the cavity mode at the surface-plasmonic band edge. / R5.

References

[R1] Liu, H & Lalanne, P. Microscopic theory of the extraordinary optical transmission. Nature 452, 728-731 (2008).

[R2] Lochbihler, H. & Depine, R. A. Properties of TM resonances on metallic slit gratings. Appl. Opt.51, 1729-1741 (2012).

[R3] Cao, Q. & Lalanne, P. Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits. Phys. Rev. Lett. 88,art. no. 057403 (2002).

[R4] Catrysse, P. B. & Fan,S. H. Transmission enhancement and suppression by subwavelength hole arrays in polaritonic films. Proc. of SPIE 6480, art. no. 64800C (2007).

[R5] Ceglia, D. de, Vincenti, M. A., Scalora, M., Akozbek, N. & Bloemer, M. J. Plasmonic band edge effects on the transmission properties of metal gratings. AIP Advances 1,art. no. 032151 (2011).

[R6] D’Aguanno, G., Mattiucci, N.,Bloemer, M. J.,Ceglia, D. de, Vincenti, M. A. Alù, A. Transmission resonances in plasmonic metallic gratings. J. Opt. Soc. Am. B 28, 253-264 (2011).

[R7] Guillaumée, M., Dunbar, L. A. &Stanley, R. P. Description of the modes governing the optical transmission through metal gratings. Opt. Express 19, 4740-4755 (2011).

[R8] Lalanne, P., Sauvan, C., Hugonin, J. P., Rodier, J. C. & Chavel, P. Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures. Phys. Rev. B 68,art. no. 125404 (2003).

[R9] Maystre, D., Fehrembach, A.-L. & Popov, E. Plasmonic antiresonance through subwavelength hole arrays. J. Opt. Soc. Am. A 28, 342-355 (2011).

[R10] Weiner, J. The physics of light transmission through subwavelength apertures and aperture arrays. Rep. Prog. Phys. 72, art. no. 064401 (2009).

[R11] Weiner, J. & Nunes, F. D. High-frequency response of subwavelength-structured metals in the petahertz domain. Opt. Express 16, 21256-21270 (2008).

[R12] García-Vidal, F. J. L. Martín-Moreno, L. Transmission and focusing of light in one-dimensional periodically nanostructured metals. Phys. Rev. B 66,art. no. 155412 (2002).

[R13] Marquier, F., Greffet, J. J. & Collin, S. Resonant transmission through a metallic film due to coupled modes. Opt. Express 13, 70-76 (2005).

[R14] Søndergaard, T., Bozhevolni, S. I.,Beermann, J.,Novikov, S. M., Deraux, E. & Ebbesen, T. W. Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders. J. Opt. Soc. Am. B 29, 130-137 (2012).

[R15] Ding, Y., Yoon, J., Javed, M. H., Song, S. H. & Magnusson, R. Mapping surface-plasmon polaritons and cavity modes in extraordinary optical transmission. IEEE Photon. J. 3, 365-374 (2011).

[R16] Sturman, B & Podivilov, E. Theory of extraordinary light transmission through arrays of subwavelength slits. Phys. Rev. B 77,art. no. 075106 (2008).

1