A. Agrawal, T. Matsui, W. Zhu, A. Nahata, Z.V. Vardeny
S1. Experimental Methods and Sample Fabrication
We used a conventional time-domain THz spectroscopy setup [S1] to characterize the fractal aperture arrays. Photoconductive devices were used for both emission and coherent detection. An off-axis paraboloidal mirror was used to collect and collimate the
THz radiation from the emitter to the samples. The THz beam was directed onto the plasmonic fractals at normal incidence. Samples were attached to a solid metal plate with a 5 cm x 5 cm opening that was placed in the path of the collimated THz beam. The 1/e
THz beam diameter was smaller than the aperture opening in the metal holder and the spatial extent of the aperture array structures, thereby minimizing edge effects due to the finite size of the array sample. Reference transmission spectra were taken with just the metal holder in the THz beam path using the same setup.
The fractal and the random aperture arrays used in our study were fabricated on
75μm thick free standing stainless steel metal foils. Through apertures were milled on metal foils using a commercial laser micromachining system. The metallic foils were completely opaque prior to fabrication of the arrays; therefore transmission through the perforated metallic films was uniquely due to the apertures. The detected transient photocurrent was then Fourier transformed and normalized to the reference transmission, yielding the electric field transmission spectrum, t (
ν
) in the range ~ 0.05 to 0.6 THz. THz time-domain spectroscopy allows for the direct measurement of the THz electric field, yielding both amplitude and phase information. By transforming the time-domain data to the frequency domain, we are able to determine independently both the magnitude and phase of the amplitude transmission coefficient, t(ν) using the relation: t (

)
 t (

) exp[ i

(

)]

E transmitte d
E incident
(

(

)
)
(S1)

In this expression E incident
and E transmitted
are the incident and transmitted THz fields, respectively, | t(ν) | and

( ν ) are the magnitude and phase of the amplitude transmission coefficient, respectively, and
ν
is the THz frequency.
S2. Aggregation Procedure for the BA and the DLA Fractal Structures
As mentioned in the text, the ballistically aggregated (BA) and diffusion-limited aggregated (DLA) fractals were designed using an aggregation process where only the nearest neighbor distance ( d ) was preserved. The stepwise aggregation process for these two aggregated fractal structures is described below and shown schematically in Fig. S1:
1.
The aggregation process starts by placing one monomer at the origin and another monomer at a random position far from the origin.
2.
For the BA fractal, the second monomer is moved along a straight line path towards the origin until it adheres to the first monomer, at a fixed d .
3.
For the DLA fractal, the second monomer is moved along a 2D trajectory in a random walk manner until it adheres to the first monomer at a fixed d .
4.
The same aggregation procedure is repeated for all other monomers in order to form the fractal clusters shown in Fig. 1(b) and 1(c) of the manuscript.
(a) BA fractal

(b) DLA fractal
Fig. S1 : Schematic stepwise design procedure for (a) BA and (b) DLA plasmonic fractals. For the BA fractal, individual monomers are designed to travel along a straight line path toward the origin, where they adhere with other monomers. For the DLA fractal the monomers move along a 2D trajectory in a “random walk” manner until they adhere with the monomers at the origin. The procedure is repeated for each subsequent monomer to get the fractal clusters shown in Fig. 1(b) and 1(c) of the manuscript.
S3. Dimensionality of Plasmonic Fractals
Using THz transmission measurements, we obtained the fractal dimensionality ( D ) of the three plasmonic fractals, as shown in Fig. 3(a) of the manuscript. In order to compare our measurements with theory, we calculated D of these fractal structures using Eq. (1) directly from the engineered structures. In Fig. S2 we plot the number of monomers (N) vs. the gyration radius (R g
) for the three fractal structures; the slope of this curve is directly proportional to D . The values that we obtained by applying this procedure are:
D
HF
= 1.77; D
BA
= 1.88; and D
DLA
= 1.61. These values agree with the theoretical values in the literature [S2-S5], and are also in excellent agreement with the measured D values from the THz transmission as shown in Fig. 3(a) of the manuscript.

3.5
3.0
2.5
2.0
1.5
D
BA
=1.88
1.0
1.2
1.6
Log (R g
/a)
(a)
2.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
0.6
D
DLA
=1.61
0.8
1.0
1.2
Log (R g
/a)
(b)
1.4
2.5
2.0
1.5
1.0
4.0
3.5
3.0
D
HF
=1.77
0.0
0.5
1.0
Log(R g
/a)
1.5
2.0
(c)
Fig. S2 : Calculated fractal dimensionality ( D ) of the three fractal structures discussed in the text. (a) ballistic aggregated (BA) (b) diffusion-limited aggregated (DLA) and (c) hexa-flake (HF) fractals shown in Fig. 1. D is calculated directly from the engineered structure by simply counting the apertures using Eq. (1), which is equivalent to the slope of the log of the number of apertures (N) vs. the gyration radius ( R g
). The solid lines correspond to a straight line fit through the data points. The values of D obtained are given and in excellent agreement with theory [S2-S5].

S4. Dimensionality and Structure Factor of Random Aperture Arrays
In this section, we present the measured and calculated D of random hole (RH) arrays. In
Fig. S3 (a) and (b), we show the real and reciprocal space representation of one specific
RH array used in this study. The RH arrays were designed to have the same fractional aperture area and aperture radius, a as those of the corresponding fractal arrays. The value of D obtained using the THz transmission measurements is shown in Fig. S3 (c), and is D ~ 2.02. The theoretical D (= 2) calculated from Eq. (1) via the method of aperture counting as a function of R g
(discussed above in section S3) is shown in Fig. S3
(d), and agrees well with the measured value. We also note that the structure factor of this
RH aperture array is smooth and does not contain any discrete or diffused reciprocal vectors.
(a) (b)
5.0
4.5
4.0
3.5
D
RH
~2.02
0.4
0.6
0.8
1.0
Log(R g
/a)
(c)
1.2
1.4
2.0
1.8
1.6
1.4
2.8
2.6
2.4
2.2
0.6
D
RH
=2.0
0.8
1.0
Log(R g
/a)
1.2
(d)
1.4

Fig. S3 : (a) The real and (b) reciprocal space representations of the random hole (RH) array used in our study. The black dots in (a) correspond to the location of apertures in the stainless steel metal films with aperture radius, a
= 300 μm. (c) ‘Fractal dimensionality’ (
D ) of the RH array shown in (a) extracted from the THz transmittance value at the isosbestic frequency ( ν o
= 0.33 THz), plotted vs. the gyration radius ( R g
). (d)
Fractal dimensionality of the RH array using Eq. (1) as extracted from the log of the number of apertures (N) vs. the gyration radius ( R g
) (the method described in section S3).
The solid lines correspond to a straight line fit through the data points.
S5. Conservation of integrated transmission spectra through DLA aperture arrays
Similar to our finding regarding the conservation of integrated transmission spectra for the BA plasmonic fractal shown in Fig. 4 of the manuscript, we now present our results for the DLA plasmonic fractal. In Fig. S4 (a) we plot the ratio, f of area under the T (
ν
) spectra for the DLA and HF arrays. The ratio f ~ 1.0 is consistent with the measurements on the BA fractals and confirms our model that the ‘total transmitted power’ through isotropic aperture arrays is conserved. In order to further confirm our model, we analyzed the structure factor associated with the DLA fractal shown in Fig. 1(c) of the manuscript.
Similar to the analysis of the BA fractal, we obtained the relative scattering spectral density Σ rel
F (i) (

) by integrating the amplitude of the various Fourier components
( Σ
~
( i
F
)
) over the azimuthal angle on rings in k-space representation for each
 k

(and corresponding frequency,

= c k ) for the fractal array normalized by that of the RH array, as shown for the DLA fractal in Fig. S4 (b). Consistent with our observations for the BA fractal, we find that Σ rel
F (i) (

) for the DLA fractal aperture arrays is also modulated around that of RH arrays, thus keeping their integrals fixed. The deviation from the value of unity in Fig. S4 (a) indicates that the DLA and HF fractal morphologies are not perfectly isotropic and, since the THz beam is linearly polarized, it does not sample all the apertures with the same weight. In contrast to the deviation from unity found in the
THz transmission data of Fig. S4 (a), when integrating
Σ rel
F
(i)
(

) up to 1 THz (Fig. S4
(b)), we find that it is zero within ~3 %.

1.1
1.0
DLA
HF
0.9
10 20 30
R g
/ a
(a)
40 50
150
100
50
0
-50
-100
0.2
0.4
0.6

[THz]
0.8
1.0
(b)
Fig. S4 : (a) The ratio of ‘area under the curve’ of the transmission spectra, f = A
FRAC
/A
RH plotted vs. the gyration radius ( R g
) for the DLA (blue line) and HF (green line) fractal arrays; a straight line fit to the data points is also shown. (b) The relative scattering spectral density,
Σ rel
F
(i)
(

) of the DLA structure (blue line), compared to that of the corresponding RH array (red line).The existence of a sum-rule for their integrals is evident; indeed the obtained
Σ rel
F
(i)
(

) integration of the DLA amplitude relative to that of the RH amplitude up to 1 THz yields zero to within ~3 % .

Supplementary References
[S1] H. Cao and A. Nahata, Opt. Express 12 , 1004 (2004).
[S2] http://en.wikipedia.org/wiki/Fractal
[S3] T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1992).
[S4] P. Meakin, Ann. Rev. Phys. Chem. 39 , 237-267 (1988).
[S5] T. A. Witten Jr. and L. M. Sander, Phys. Rev. Lett. 47 , 1400 (1981).