Abstract

In this paper we demonstrate that rigorous high-order perturbation of surfaces (HOPS) methods coupled with analytic continuation mechanisms are particularly well-suited for the assessment and design of nanoscale devices (e.g., biosensors) that operate based on surface plasmon resonances generated through the interaction of light with a periodic (metallic) grating. In this connection we explain that the characteristics of the latter are perfectly aligned with the optimal domain of applicability of HOPS schemes, as these procedures can be shown to be the methods of choice for low to moderate wavelengths of radiation and grating roughness that is representable by a few (e.g., tens of) Fourier coefficients. We argue that, in this context, the method can, for instance, produce full and precise reflectivity maps in computational times that are orders of magnitude faster than those of alternative numerical schemes (e.g., the popular “C-method,” finite differences, integral equations or finite elements). In this initial study we concentrate on the description of the basic principles that underlie the solution scheme, including those that relate to analytic continuation procedures. Within this framework, we explain how, in spite of conventional wisdom to the contrary, the resulting perturbative techniques can provide a most valuable tool for practical investigations in plasmonics. We demonstrate this with some examples that have been previously discussed in the literature (including treatments of the reflectivity and band gap structure of some simple geometries) and extend this to demonstrate the wider applicability of the proposed approach.

© 2013 Optical Society of America

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2012

T. W. Johnson, Z. J. Lapin, R. Beams, N. C. Lindquist, S. G. Rodrigo, L. Novotny, and S.-H. Oh, “Highly reproducible near-field optical imaging with sub-20-nm resolution based on template-stripped gold pyramids,” ACS Nano 6, 9168–9174 (2012).
[CrossRef]

N. C. Lindquist, P. Nagpal, K. M. McPeak, D. J. Norris, and S.-H. Oh, “Engineering metallic nanostructures for plasmonics and nanophotonics,” Rep. Prog. Phys. 75, 036501 (2012).
[CrossRef]

A. D. Baczewski, N. C. Miller, and B. Shanker, “Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions,” J. Opt. Soc. Am. A 29, 531–540 (2012).
[CrossRef]

2011

L. Li and G. Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A 28, 738–746 (2011).
[CrossRef]

I. S. Spevak, M. A. Timchenko, and A. V. Kats, “Design of specific gratings operating under surface plasmon–polariton resonance,” Opt. Lett. 36, 1419–1421 (2011).
[CrossRef]

J. Bischoff and K. Hehl, “Perturbation approach applied to modal diffraction methods,” J. Opt. Soc. Am. A 28, 859–867 (2011).
[CrossRef]

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
[CrossRef]

A. Shahmansouri and B. Rashidian, “Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation,” J. Opt. Soc. Am. B 28, 2690–2700 (2011).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. Tasolamprou, T. Yioultsis, and E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

M. Wang, C. Engstrom, K. Schmidt, and C. Hafner, “On high-order FEM applied to canonical scattering problems in plasmonics,” J. Comput. Theor. Nanosci. 8, 1564–1572 (2011).
[CrossRef]

N. C. Lindquist, T. W. Johnson, D. J. Norris, and S.-H. Oh, “Monolithic integration of continuously tunable plasmonic nanostructures,” Nano Lett. 11, 3526–3530 (2011).
[CrossRef]

A. Trügler, J.-C. Tinguely, J. R. Krenn, A. Hohenau, and U. Hohenester, “Influence of surface roughness on the optical properties of plasmonic nanoparticles,” Phys. Rev. B 83, 081412(R) (2011).
[CrossRef]

A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
[CrossRef]

2010

C.-C. Chao, S.-H. Tu, C.-M. Wang, H.-I. Huang, C.-C. Chen, and J.-Y. Chang, “Impedance-matching surface plasmon absorber for FDTD simulations,” Plasmonics 5, 51–55 (2010).
[CrossRef]

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[CrossRef]

B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
[CrossRef]

H.-Y. Xie, M.-Y. Ng, and Y.-C. Chang, “Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect,” J. Opt. Soc. Am. A 27, 2411–2422 (2010).
[CrossRef]

2009

M. Huber, J. Schöberl, A. Sinwel, and S. Zaglmayr, “Simulation of diffraction in periodic media with a coupled finite element and plane wave approach,” SIAM J. Sci. Comput. 31, 1500–1517 (2009).
[CrossRef]

J. Hoffmann, C. Hafner, P. Leidenberger, J. Hesselbarth, and S. Burger, “Comparison of electromagnetic field solvers for the 3D analysis of plasmonic nano antennas,” Proc. SPIE  7390, 73900J (2009).

J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of numerical methods for the analysis of plasmonic structures,” J. Comput. Theor. Nanosci. 6, 763–774 (2009).
[CrossRef]

H. Kurkcu and F. Reitich, “Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies,” J. Comput. Phys. 228, 75–95 (2009).
[CrossRef]

J. Bischoff, “Prospects and limits of the Rayleigh Fourier approach for diffraction modelling in scatterometry and lithography,” Proc. SPIE  7390, 73901E (2009).

D. Grieser, H. Uecker, S.-A. Biehs, O. Huth, F. Rüting, and M. Holthaus, “Perturbation theory for plasmonic eigenvalues,” Phys. Rev. B 80, 245405 (2009).
[CrossRef]

O. P. Bruno and M. C. Haslam, “Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences,” J. Opt. Soc. Am. A 26, 658–668 (2009).
[CrossRef]

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Figures (16)

Fig. 1.
Fig. 1.

Generic grating configuration for SPR coupling.

Fig. 2.
Fig. 2.

(a) (Normalized) reflectivity [“NR” in Eq. (16)] map for a sinusoidal silver profile with period d=650nm at normal incidence in TM polarization; (b) poles (open markers) and zeros (filled markers) of (the [32/32] Padé approximant of) the coefficient B0(h) for complex h and different values of the wavelength of radiation λ (660λ700). Note that no poles are present in a complex neighborhood of the real line (shaded area). The zeros indicated with an arrow correspond to close-to-zero reflectivity, indicating the presence of a plasmon resonance.

Fig. 3.
Fig. 3.

(a) Square of the absolute value |B1+|2 of the 1st order Rayleigh coefficient B1+ in Eq. (10) (“reflectivity at order 1”) for a sinusoidal silver grating of pitch d=400nm and height h=4nm, illuminated with a plane wave of energy 2.26 eV, corresponding to a wavelength λ=548.60nm, as a function of incidence angle; (b) same as in (a) but with height h=6nm; (c) same as in (a) but with height h=10nm.

Fig. 4.
Fig. 4.

Logarithm (base 10) of the maximum relative error in |B1+|2 (circles) and best linear fit (dashed line) corresponding to incidences in the interval [19.2°,19.4°] for the case in Fig. 3(c) (h=10nm). The optical constants used in the simulations are from [87].

Fig. 5.
Fig. 5.

(a) Configuration corresponding to a deep silver grating of height 124 nm and period 258 nm illuminated with a plane wave with an incidence angle of 30°; (b) reflectivity at order 0, |B0|2 [Eq. (10)], as a function of wavelength for a polynomial permittivity [85] as computed from a [32/32] Padé approximant. The dashed vertical line marks the wavelength below which the 1st Rayleigh mode becomes propagating.

Fig. 6.
Fig. 6.

(a) Zeroth order reflectivity as a function of height h for the example in Fig. 5 at a wavelength λ=410nm; the solid lines display the values of the Padé approximations corresponding to each value of N for which the Taylor partial sums are shown. The vertical line at h=hp depicts the absolute value of the closest (noncanceling) pole to the origin as approximated by the corresponding pole of the [32/32] Padé approximant (N=65). (b) Poles and zeros in the complex h plane of the [32/32] Padé approximant for the configuration in (a); the circles and crosses (square and stars) mark the location of the zeros and poles, respectively, that do (do not) cancel out. (c) Same as (a) but for a wavelength λ=495nm. (d) Same as (b) for the configuration in (c).

Fig. 7.
Fig. 7.

(Total) Reflectivity for an elliptical grating configuration, as shown in (a) (period d=630nm, linewidth w=400nm), for different incidence angles. The vertical lines mark the wavelength below which the 1st order becomes propagating.

Fig. 8.
Fig. 8.

Left- (horizontal lines) and right- (curves) hand sides of Eq. (28). Their intersections, marked by vertical lines, define the resonant wavelengths that would be excited for a silver interface of zero height and period d=630, as depicted in (a), under the dispersion derived from (cubic-spline) interpolation of the results in [88].

Fig. 9.
Fig. 9.

Results for a sinusoidal profile of period d=258nm illuminated with an incidence angle θ=30°. (a) Total reflectivity as a function of height and wavelength; the thick dashed vertical line marks the wavelength below which the 1st order becomes propagating. (b) Spatial distribution of the (transverse) reflected magnetic field, |u+,scat|=|Hz+,scat| [Eq. (3)], corresponding to the intersection of the dotted–dashed lines in (a): height h=20nm and wavelength λ=410nm.

Fig. 10.
Fig. 10.

Same as Fig. 9 for the configuration of Fig. 7(a) with incidence angle θ=1° [Fig. 7(d)]. (a) Reflectivity map; (b) spatial (transverse) magnetic field distribution corresponding to the intersection of the dotted–dashed lines in (a): height h=30nm and wavelength λ=640nm.

Fig. 11.
Fig. 11.

(a) A purely sinusoidal profile with amplitude a=5nm; and (b) a modulated profile with a second harmonic component as in Eq. (29) (a=5nm, b=2nm).

Fig. 12.
Fig. 12.

Reflectivities as a function of incidence angle and wavelength for: (a) the purely sinusoidal profile in Fig. 11(a); and (b) for the modulated profile Fig. 11(b). In both cases the simulations correspond to the HOPS approach using a truncated Taylor series of order N=5. The white area in the middle of the latter figure (corresponding to values that exceed the maximum in the plotted scale) indicates the region where the five-term Taylor series is already inaccurate.

Fig. 13.
Fig. 13.

Reflectivities as a function of incidence angle and wavelength for: (a) the purely sinusoidal profile in Fig. 11(a); and (b) a modulated profile with a second harmonic component [Eq. (29)] as depicted in Fig. 11(b). For the latter, the line λ=λ¯ marks the approximate location of the center of the band gap as predicted in Eq. (3b) in [96]. In both cases the simulations correspond to the HOPS approach using [2/2] Padé approximants (N=5).

Fig. 14.
Fig. 14.

(a) Numerical convergence analysis (and best linear fit) for the examples in Figs. 12(a) and 13(a); and (b) same for those in Figs. 12(b) and 13(b). Logarithm of the relative maximum error in the range of wavelengths between 635 and 670 nm, and angles between 1.5° and 1.5°. Note the slow convergence of the Taylor series in (a), and its divergence in (b).

Fig. 15.
Fig. 15.

(a) Reflectivity for a sinusoidal profile [b=0 in Eq. (29)] with period d=634nm and height h=2a=60nm. (b) Reflectivities corresponding to the same profile for varying heights h=1060nm.

Fig. 16.
Fig. 16.

(a) Reflectivity for a Fourier grating [as in Eq. (29)] with period d=634nm, height h=60nm, and a/b=2.5. (b) Reflectivities corresponding to the same profile for varying heights h=1060nm.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

y=f(x),f(x+d)=f(x),
(E⃗(x,t),H⃗(x,t))=eiωt(E(x),H(x)),x=(x,y,z),
u={Ezfor TE polarization,Hzfor TM polarization,
Δu±,scat+(k±)2u±,scat=0inD±,
{u+,scatu,scat=uinconS,u+,scatnC02u,scatn=uincnonS,
C0={1for   TEk+/kfor   TM.
uinc(x)=u(x,y)=eiαxiβy,
uinc(x+d,y)=eiαdeiαxiβy,
u±,scat(x+d,y)=eiαdu±,scat(x,y),
u±,scat(x,y)=r=Br±eiαrx±iβr±y,valid on±y>maxxR{±f(x)},
αr=α+rK,K=2πd,βr±=(k±)2(αr)2,ris an integer,and Im(βr±)0.
kx,SPP=±k0ε+εε++ε,
|Re(kx,SPP)|>k+,
Re(kx,SPP)αr,
RrUβr+β|Br+|21,
NRrUβr+β|Br+|2|B0+,flat|21,
NR=|B0+|2|B0+,flat|21.
maxxR(f(x))minxR(f(x))=1,
Δxu±,scat(x;h)+(k±)2u±,scat(x;h)=0inDh±,
{u+,scat(·;h)u,scat(·;h)=uinc(·;h),u+,scatn(·;h)C02u,scatn(·;h)=uincn(·;h),
u±,scat(x,y;h)=r=Br±(h)eiαrx±iβr±y,valid on±y>maxxR{±hf(x)}.
Br±(h)=n=0dn,r±hn,
dn,r+dn,r=(iβ)nCn,rk=0n1q=max[kF,r(nk)F]min[kF,r+(nk)F]Cnk,rq[(iβq+)nkdk,q+(iβq)nkdk,q],
iβr+dn,r++C02iβrdn,r=Cn,r(iβ)n1[(iα)(iKr)(iβ)2]+k=0n1q=max[kF,r(nk)F]min[kF,r+(nk)F]Cnk,rq{[iK(rq)](iαq)×[(iβq+)nk1dk,q+C02(iβq)nkdk,q][(iβq+)nk1dk,q+C02(iβq)nkdk,q]},
f(x)=p=FFC1,pei2πpx/d,
f(x)!=p=FFC,pei2πpx/d.
y=h2sin(2πdx),
n=0.060+3.586i.
r=[±ε+εε++εsin(θ)]dλ,
f(x)=asin(Kx)+bsin(2Kx),(K=2πd),
|Kb|1,
(2πλ¯)2(2πλ0)2[12(Kb)2],

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