Supplementary information to
Making Metals Transparent for White Light by Structured Surface Plasmons
Xian-Rong Huang,1 Ru-Wen Peng,2 and Ren-Hao Fan2
Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA (xiahuang@aps.anl.gov)
2
National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China (rwpeng@nju.edu.cn)
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I. Verification of RCWA
The rigorous coupled-wave analysis (RCWA) method is
a simple and well-developed Fourier method for numerically
solving Maxwell's equations
  E  i(2  ) H,   H  i(2  )  E,
where the dielectric permittivity (r) is periodically modulated in space (for details, see Refs. [6,8,9] of the paper as well
as Ref. [A1] below). It is a first-principles method with no
assumptions or approximations (except for the truncation of
very high diffraction orders). Its validity for 1D gratings has
been well verified by finite-difference time-domain (FDTD)
computations (Ref. [A2] as well as our own FDTD calculations) and by analytical models (Ref. [12]). The C++ computing code we developed was based on the algorithm in Ref.
[8], which may not have as fast convergence as that in the
later developed algorithms in Ref. [9]. Still our program can
easily retain hundreds to thousands of diffraction orders (on
personal computers) to achieve complete convergence and
accuracy with no artifacts. Our source code is freely available upon request. Our program can reproduce all the correct
RCWA computations (including oblique incidence) reported
in the literature (e.g., Ref. [12]).
To further verify the accuracy of our computations, we
have used the RCWA program to successfully simulate the
experimental measurements in the literature. For example, in
Ref. [13], Went et al. constructed 1D periodic gratings by
stacking well-defined aluminum slats (3 mm  64.7 mm 
600 mm) and measured their normal-incidence transmission
at microwave frequencies. Figure A1(a) was taken from Fig.
3 of Ref. [13], which shows the measured Fabry-Perot-like
transmission pattern as well as the curve fit with the empirical Fabry-Perot equations (see Ref. [13] for details)
1 R  A 
T0  

 1 R 
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as an example to verify RCWA because at this macroscopic
scale, the fabrication and construction of the grating can be
nearly perfect such that the experimental measurements are
reliable. Since the RCWA method based on Maxwell's equations universally and non-discriminately applies to any incidence geometry at any frequencies for gratings of any scales,
our computations as well as the mechanisms in the paper
should be convincing. In fact, we have also used FDTD to
verify our results (see the example in Fig. A4 below).


1
4R
1  F sin 2 ( 2) , F  (1  R) 2 .


Note that this simple empirical fitting process is completely
independent of RCWA. Since the permittivity Al (mainly
imaginary) of aluminum at microwave frequencies is in the
order of 107, we may roughly set Al = 106  i107. Based on
this value, we have used our RCWA program to calculate the
transmission spectrum of the same structure. The result is
shown in Fig. A1(b), which surprisingly agrees well with the
experimental data. The slight discrepancies may result from
the inaccurate value of Al we used. Note that here the
RCWA method does not use any Fabry-Perot interference
theories at all. It only uses the above Maxwell's equations!
Therefore, the agreement between Figs. A1(a) and A1(b)
experimentally demonstrates the validity and accuracy of
RCWA. Here we chose Fig. A1(a) at microwave frequencies
Fig. A1. Experimental verification of RCWA. (a) Measured and
Fabry-Perot-fit microwave transmission curves from an aluminum
grating with d = 4 mm, W = 1 mm,  = 64.7 mm (taken from Fig. 3
of Ref. [13]). Normal incidence. (b) The transmission curve calculated with our RCWA program based on the same lattice parameters and a rough permittivity value Al = 106 i107 (slight frequency
dependency ignored).
In addition, the RCWA has been later developed to treat
more complicated structures (e.g., see Ref. [A3]), so its validity demonstrated here indicates that this technique is an
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invaluable and extremely reliable method to perform “numerical experiments” on various plasmonic designs. It can
also be used to evaluate the fabrication perfection and the
device performance.
red-shift measurements at microwaves in Ref. [11]).
Therefore, the extension of the transparence in Fig. A2 does
not conflict with the SSP based interpretations of the related
physics.
II. Extension of transparence to microwaves
The nearly flat transmission of 1D metallic gratings can
be extended to microwaves. Since the experimentally measured permittivities of gold are only available for  < 10 µm
in Ref. [10], we have used the following Lorentz model to
extend the permittivity Au up to milimeter EM waves:
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 ( )  1  
j 1
( sj  1) pj2
   2  2i j
2
0j
where  = 2 (with  in µm), 02 j = {0, 4.42306,
17.69224, 226.38473, 475.74175, 4556.54001},
= {1591.53367, 50.25896, 20.94123,
( sj  1) pj2
148.68275, 1258.56808, 9180.63634}, 2j =
{0.26859, 1.22132, 1.74837, 4.40892, 12.63892,
11.21995}. The transmissivity curves calculated with this
equation are similar to those in Fig. 1 in the paper. Figure A2
shows the extension of the three curves of Fig. 1(b) to the
wavelength range up to  = 1 mm, in which the gold gratings
are all transparent. The higher transmission in the longerwavelength range is due to the higher conductivity (and less
ohmic loss) for longer wavelengths.
Fig. A3. (a) Dependence of f on the ratio W/d. (b)-(d) RCWA calculations of the nearly flat transmission curves for different W/d
ratios, calculated with the f angles predicted in (a). Calculated with
d = 2 m (but similar results can be achieved for other lattice constants d) and gold grating thickness  = 10 m.
III. Predicting the optimized transparence angle
Fig. A2. The extension of the three transmission curves in Fig. 1(b)
of the paper to the longer-wavelength range. The dielectric permittivities of gold are based on the above Lorentz model. d = 2 m, W
= 0.2 m,  = 84°.
Note that classical SPs on planar metal surfaces do not exist
for extremely long wavelengths, including THz and microwave frequencies. In our current work, the SSPs (also called
“structured SPs” in Ref. [6]) are not classical SPs. They are
surface charge density waves/patterns with the characteristics
of subwavelength periods and surface-bound modes. They
exist for frequencies up to microwave frequencies as verified
by our charge density calculations (see Ref.[6], also see the
By assuming that the incident angle f for flat transparence corresponds to the condition that the total forces exerted
by the incident wave on the wall BP and the upper surface
AB are equal (see Fig. 3), we have derived Eq. (1) in the paper [i.e., tanf = (d  W)/W] for predicting f. Figure A3(a)
shows the dependence of f on the ratio W/d, calculated with
this equation. Although we have ignored many possible factors (e.g. electric resistance) in the derivation, this equation is
quite consistent with the detailed RCWA calculations for
highly conducting gratings at long wavelengths. In addition
to Fig. 1 of the paper that corresponds to W/d = 0.1, here
Figures A3(b)-A3(e) show the RCWA calculations of the
transmission curves for W/d = 0.2, 0.35, 0.5 and 0.75, respectively, based on the predicted angles f from Eq. (1). For
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W/d > 0.1, the Fabry-Perot oscillations usually do not disappear completely (for any incidence angle), but obviously they
are significantly suppressed (minimized) in Figs. A3(b)-A3(e)
so that nearly white-beam transparence is achieved in all
these figures for long wavelengths. This indicates that Eq. (1)
is a reliable guideline for predicting f (particularly for small
W/d ratios and highly conductive metal structures). In our
calculations, we have proved that this guideline is valid for
various lattice constants d. Note that white-beam transparence occurs in a finite angular range  around f. For W/d ~
0.1,  ~ 1°. For W/d > 0.5,  ~ 10°. Our RCWA calculations show that the center of  is nearly f predicted by the
equation tanf = (d  W)/W (or slightly above f for large
W/d values).
Figure 3 indicates that: (1) f decreases (becoming less
oblique incidence) with increasing W/d; and (2) the overall
transmission increases with W/d as the slits become more
open.
Finally, Fig. A4 shows the transmission curve computed
by FDTD (red curve) under the conditions of Fig. A3(b).
Note that FDTD is a completely independent and different
computation technique, but here it still correctly repeats the
main features revealed. This should make our RCWA results
work more convincing. The small differences may result
from some FDTD computing artifacts or limited computation
precision.
Fig. A4. (a) Comparison between the FDTD and RCWA computations under the conditions of Fig. A3(b).
Additional references:
[A1] M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of
the rigorous coupled-wave analysis of binary gratings,” J. Opt.
Soc. Am. A 12, 1068-1076 (1995).
[A2] N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. 9, 490-495 (2007).
[A3] E. Silberstein, P. Lalanne, J. -P. Hugonin, and Q. Cao, “Use of
grating theories in integrated optics,” J. Opt. Soc. Am. A 18,
2865-2875(2001).
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