## Abstract

We study the scattering of surface plasmons from sub-wavelength holes and find that it exhibits a stronger wavelength dependence than the traditional *λ*^{−4} scaling found for Rayleigh scattering of light from small particles. This experimental
observation is consistent with recent theoretical work and linked to the two-dimensional nature of the surface plasmon and the wavelength dependence of its spatial extent in the third dimension. The scattering cross sections are obtained with a frequency-correlation technique, which
compares intensity speckle patterns observed behind various random structures of holes and recorded at different wavelengths. This powerful technique even allows us to distinguish between scattering of surface plasmons into photons and scattering into other surface plasmons.

© 2014 Optical Society of America

## 1. Introduction

Sub-wavelength holes are important building blocks for novel photonic structures, given that these holes are used in metamaterials [1, 2], photonic crystal slabs [3], sensors [4] and possibly thin film solar cells [5]. In the context of the extraordinary optical transmission [6], the transmission of light through single sub-wavelength holes in metal films has attracted much interest and its physics is surprisingly rich [7–9]. To measure this single hole transmission, random patterns of sub-wavelength holes can be used [10–14]

The excitation [15, 16] and scattering [17, 18] of surface plasmons by single sub-wavelength holes has
been studied both theoretically and experimentally. The wavelength dependence of these scattering processes, which play a central role in recently developed microscopic models [19, 20], can reveal
the underlying physics of surface plasmon scattering. So far, this wavelength dependence has only been studied using metal hole arrays. One study reports the traditional [21] 1/*λ*^{4} dependence
[22], while another study reports a *λ*^{−n} wavelength dependence where the power *n* depends strongly on hole size [23]. Both experimental observations contradict theories on surface plasmon scattering [17, 18].

For surface plasmons scattered at a single hole, the scattering cross section has unit length instead of an area [17,18]. This is because the cross section is the scattered power divided by the
incident power in the plasmon mode *per unit width*, measured along the surface and perpendicular to the propagation direction [17, 18, 24].
Because the scattering cross section for surface plasmons has unit length, the traditional expression [21] of *λ*^{−4} times a volume squared can not be correct.

In this letter we extract the Rayleigh scattering cross sections of surface plasmons from single sub-wavelength holes by studying the optical transmission of random patterns of holes. An advantage of using random patterns is that most interference effects can be averaged, in contrast to the transmission of arrays which is entirely dominated by interference effects. Another important advantage of random patterns is that they enable us to separate (ohmic) absorption from (radiative) scattering loss, by comparing samples of different hole densities [25].

Figure 1 (a–c) show the three physical processes that we probe: (a) scattering of surface plasmons to free space, (b) surface-plasmon-mediated optical transmission, and (c) direct transmission. Figure 1(d)
shows a sketch of the experiment. This experiment yields three quantities: the surface plasmon absorption length *L*_{abs}, the scattering cross section *σ*, and a new concept that we name the intensity-ratio cross section *A*. The
surface plasmon absorption length *L*_{abs} quantifies the ohmic loss of the surface plasmons. The scattering cross section *σ* characterizes the radiative loss of a surface plasmon at a single hole (Fig.
1(a)); it singles out the scattering to free space and is insensitive to scattering within the surface plasmon manifold. The intensity-ratio cross section *A* describes the transmission of light via a surface plasmon, where a surface plasmon is first excited at one
hole and then transmitted at another hole (Fig. 1(b)). This cross section is approximately equal to, but slightly smaller than, the scattering cross section from surface plasmons to surface plasmons, such that *A* =
*ησ*_{spp} with *η* ≈ 1 (see below). Before presenting the wavelength dependence of *L*_{abs}, *σ* and *A*, we will first show how we extract
these quantities from the optical transmission of random patterns of holes.

## 2. Experiment

Our experiments are performed on a series of random patterns of sub-wavelength holes in a metal film. The series contains 8 patterns of which only the hole density was varied. We choose the area per hole (= inverse hole density) to be $q{a}_{0}^{2}$, with *a*_{0} = 0.45 *μ*m and *q* ∈ [1, 2, 3, 4, 9, 16, 25, 36]. The circular holes (diameter of 140
± 8 nm) perforate a 150 nm thick gold film, which is deposited directly on glass omitting the commonly used adhesion layer. A subsequently deposited 20 nm chromium layer damps the surface plasmons on the gold-air interface, allowing us to selectively study surface plasmons on the
gold-glass interface (see Fig. 1(c)). The random pattern is generated using a random number generator that generates the coordinates of the holes one by one. When new holes almost (< 50 nm) overlap existing holes, they are placed at new random
positions instead.

We illuminate these random patterns of sub-wavelength holes with monochromatic light and record the far-field speckle intensity *I*(*θ⃗*, *λ*) (see Fig. 1(d)). The change of the
speckle pattern with wavelength can be quantified by calculating the correlation between the measured speckle intensity at wavelengths *λ*_{0} and *λ*_{1} = *λ*_{0} +
Δ*λ*, resulting in a correlation function *C*(Δ*λ*) [25–31]. More precisely, we compare intensities at the same
transverse momentum *k⃗*_{||} = 2*π/λ* (sin*θ _{x}*, sin

*θ*), which is achieved experimentally by rescaling the recorded speckle patterns [25]. We perform these measurements in a large wavelength range using a supercontinuum laser source (Fianum Whitelase 400SC) of which we select a narrow line (∼ 1 nm) with a spectrometer.

_{y}Using a simple model, which assumes that only surface plasmons (SP) are excited at the holes and thus neglects the quasi-cylindrical wave contribution [32–34], we find an analytic expression for the correlation function [25]:

*L*

_{tot}, the propagation distance of the surface plasmons, which includes both radiative and non-radiative losses, and the intensity ratio 〈

*I*〉/〈

_{s}*I*〉 between the intensities of the SP-mediated and direct transmission. The term Re[Δ

_{d}*k*

_{spp}] is the difference between the surface plasmon momenta at wavelengths

*λ*

_{0}and

*λ*

_{1}. It can be approximated by Re[Δ

*k*

_{spp}] ≈ 2

*π*Re

*n*

_{eff}Δ

*λ/λ̄*

^{2}, with

*n*

_{eff}the effective refractive index of the surface plasmon mode and

*λ̄*the average wavelength in vacuum. Equation (1) is a Lorentzian with an almost wavelength-independent background correlation 〈

*I*〉

_{d}^{2}/〈

*I*+

_{d}*I*〉

_{s}^{2}.

Figure 2 shows three examples of measured correlation functions (on a log-linear scale) for three different hole densities. The scans in this plot are performed from *λ* = 690 nm (Δ*λ*
= 0 nm) to 790 nm (Δ*λ* = 100 nm). With increasing hole density the background correlation, visible at large Δ*λ*, decreases while the spectral width of the correlation increases. The observation that the
background correlation decreases shows that the efficiency of transmission via surface plasmons increases with density, as a larger fraction of the excited surface plasmons is coupled out instead of being absorbed. This increase in outcoupling is also evidenced by the increasing spectral
width, which is directly related to the losses of the surface plasmons.

The three fits in Fig. 2 are based on Eq. (1) and in good correspondence with the data. From each fit two density-dependent parameters can be extracted: *L*_{tot} and
〈*I _{s}*〉/〈

*I*〉. Figure 3(a) shows the density dependence of

_{d}*L*

_{tot}, not only for the correlation measurements starting from

*λ*

_{0}= 690 nm, which is labeled with the average wavelength

*λ̄*= 705 ± 15 nm, but also for

*λ̄*= 803 ± 13 nm and 881 ± 9 nm. Each of these sets of results obeys the expected relation:

*L*

_{abs}is the surface plasmon absorption length in the absence of the holes and

*σ*is a scattering cross section that describes the radiative loss of a surface plasmon at a single hole. For

*λ̄*= 705 ± 15 nm we thus obtain the (density-independent) inverse absorption length

*L*

_{abs}= 0.20 ± 0.02

*μ*m

^{−1}and the scattering cross section

*σ*= 80 ± 15 nm. For the two other wavelengths, we find a similar linear dependence ${L}_{\text{tot}}^{-1}(\rho )={L}_{\text{abs}}^{-1}+\rho \sigma $, but with different slopes and different axis cutoffs. The slope

*σ*decreases by as much as a factor four between

*λ̄*= 705 ± 15 nm and 803 ± 13 nm, and almost vanishes at

*λ̄*= 881 ± 9 nm. The axis cutoff ${L}_{\text{abs}}^{-1}$ decreases with wavelength by more than a factor two over this wavelength range. Both observations are consistent with the theoretically expected dependence. They will be discussed in the next section, where we will combine them with similar results obtained at other wavelengths.

The second parameter that we obtain from the correlation functions is the intensity ratio 〈*I _{s}*〉/〈

*I*〉. In Fig. 3(b) we show the density dependence of the obtained intensity ratio and compares it with the expected dependence [25]:

_{d}*A*is the third density-independent parameter, which we name the intensity-ratio cross section. Equation (3) provides a good fit of the experimental data, using only

*A*as a free parameter, in combination with the values of

*L*

_{abs}and

*σ*obtained from the fit of Fig. 3(a). Figure 3(b) also shows a signature of the quasi-cylindrical wave, as the intensity ratio for the largest density is consistently larger than predicted by our model. For this reason we limited our analysis to hole densities smaller than 2.5

*μ*m

^{2}. A comparison of the fitted values of

*A*at the three wavelengths (

*λ̄*= 705, 803, 881 nm) shows that the wavelength dependence of this parameter is even slightly stronger than that of the scattering cross section

*σ*. It will be discussed in the next section.

To summarize this section, we have measured the correlation functions *C*(Δ*λ*) of samples with different hole densities and fitted these with Eq. 1, using two *density-dependent*
parameter: the SP propagation length *L*_{tot}(*ρ*) and the intensity ratio 〈*I _{s}*〉/〈

*I*〉 (

_{d}*ρ*). Next, we analyze the

*ρ*-dependence of these parameter in order to extract three

*density-independent*parameters: the absorption length

*L*

_{abs}, the scattering cross section

*σ*, and the intensity-ratio cross section

*A*. By performing this analysis for different values of the reference wavelength

*λ*

_{0}, we also find the wavelength dependence of these parameters. In the rest of the paper, we try to understand the wavelength dependence of

*L*

_{abs}and the scattering parameters

*σ*and

*A*, using Rayleigh scattering of surface plasmons at single holes as microscopic model.

## 3. Results

In Fig. 4 we show the measured wavelength dependence of the absorption length *L*_{abs} to the power −1. This length increases by approximately a factor four from *L*_{abs} ≈ 5
*μ*m to *L*_{abs} ≈ 20 *μ*m, when the wavelength is increased from 650 nm to 950 nm. The data matches very well with the theory for which we use literature values of the refractive index of gold [35, 36]. This correspondence is very important as it demonstrates the validity of our approach, both qualitative and quantitative.

In Fig. 5(a) we plot the extracted value for the scattering cross section *σ* as a function of wavelength. This cross section shows a steep decline from slightly more than 100 nm at a wavelength of 675 nm to around 15 nm at 875
nm. This decline is significantly steeper than the traditional expression for Rayleigh scattering (*σ* ∝ *λ*^{−4}) indicated by the dashed line.

Recently, an analytic expression was derived for the scattering cross section of surface plasmons scattered at a sub-wavelength hole [18]. This theory treats the hole as a polarizable object, relative to its surroundings, and distinguished between scattering to other surface plasmons and to photons. For surface plasmons scattered to the photon field this expression is (see Appendix A):

where*a*is the hole radius,

*k*is the wave vector in air,

*d*

_{spp}is the mode size of the surface plasmon. The dimensionless proportionality constant

*ξ*is radius independent in the Rayleigh limit, i.e. for

*ka*≪ 1. Hence, the expression for surface-plasmon scattering resembles the expression

*σ*

_{3}

*∝*

_{D}*k*

^{4}

*a*

^{6}for the scattering of light by three dimensional particles, but the surface-plasmon mode size

*d*

_{spp}enters as a proportionality factor. This factor indicates that the hole is polarized more effectively when the surface plasmon mode is more compact. For our experiments

*d*

_{spp}is well approximated by the 1/e width of the intensity tail in the dielectric. For surface plasmons at a metal-air interface we find ${d}_{\text{s}pp}\approx \sqrt{\left|\epsilon \right|}/(2k)$, with

*ε*the dielectric constant of the metal (assuming |

*ε*| ≫ 1). At sufficiently large wavelength, where the Drude result |

*ε*| = |

*ε*

_{∞}−

*b/k*

^{2}| ∝

*k*

^{−2}applies, we thus expect

*d*

_{spp}∝

*k*

^{−2}and

*σ*∝

*k*

^{6}. For accurate fitting, we use the literature values of

*ε*[36], rather than the Drude approximation.

The solid curve in Fig. 5(a) shows that Eq. (4) fits the data much better than the ordinary Rayleigh scaling *σ* ∝ *λ*^{−4}. This is a
very important result, as it shows that the wavelength dependence of surface-plasmon scattering differs from that of photon scattering and that it can still be understood and described well with a simple expression.

The scattering cross sections that we measure are surprisingly large. The data presented in Fig. 5(a) correspond to *ξ* = 36 ± 13, whereas theory predicts *ξ* = 0.24 for a simplified
geometry [18]. There are several reasons for this discrepancy. First of all, the mentioned theoretical value was derived for a metal-air interface. By adapting the theory to a metal-glass interface, we predict that Eq. (4) should be multiplied by *n*^{6} = 11.9, with *n* the refractive index of glass (see appendix A), thereby increasing the theoretical expectation to
*ξ* = 2.8. The equations presented above automatically include this scaling when *k* = 2*π/λ* = 2*πn/λ*_{0} in interpreted as the *wave number in the
medium* and use ${d}_{\text{s}pp}\approx \sqrt{\epsilon}/(2{n}^{2}{k}_{0})$. Secondly, this value was derived for a perfect-electrical-conductor slab of zero thickness. The field penetration into the metal can increase the effective hole radius by ∼ 15 nm, thereby
increasing *a*^{6} by another factor ∼ 3, making *ξ* ∼ 9. But even this number is only a rough approximation. There is no real theory for our glass-metal-air geometry, which includes two dielectrics and a lossy metal of finite
thickness, and the mentioned *n*^{6} scaling only applies to a metal film that is fully embedded by a single dielectric. Hence, the quantitative difference between experiment and theory does not worry us too much. For now, we are only interested in the wavelength
dependence of the cross section. The factor *ξ* is just a constant in the Rayleigh limit, albeit a complicated constant that depends crucially on the geometry and material composition of the hole and its surroundings.

Our technique is sensitive enough to observe a gradual changes in the structure. For this, we compare the results presented in this paper with the single-wavelength results presented in [25]. The later results were obtained with the
same method and on the same sample, but one year before the current measurements. During that year, the scattering cross section at *λ* = 740 nm has almost doubled from the value *σ* ≈ 36 nm (= intensity cross section
= 2× amplitude cross section reported in [25]) to the value *σ* ≈ 60 nm that we now find. At the same time, the average transmission of the random-hole patterns increased by a factor
∼ 3×, while SEM images show that the average hole size has increased from the original radius *a* = 60 nm to a new radius *a* = 70 nm. Our observation that the absorption loss remains unchanged, and in agreement with literature
values, indicates that the changes occur in the geometry rather than in the quality of the metal-glass interface. Metal films are known to change in time, but aging is typically observed at elevated temperatures. Our measurements indicate that phenomena such as diffusion of chromium in gold
[37] and grain growth and grain boundary migration [38] could also be important at room temperature, at least if one waits long enough. A systematic study of these dynamic
processes is difficult though, as they typically depend crucially on growth conditions, such as deposition rate and substrate temperature [37]. Fortunately, the precise composition of the film only shows up in the pre-factor
*ξ* and has no effect on the studied wavelength dependence of the scattering cross sections.

## 4. Intensity-ratio cross section *A* = *ησ*_{spp}

Next, we consider the intensity-ratio cross section *A* and its wavelength dependence. In Fig. 5(b) we plot the extracted value for *A* as a function of wavelength. *A* spans roughly an order of magnitude
and is of comparable magnitude as *σ*, suggesting that *A* and *σ* may be related. Similar to *σ*, *A* has a stronger wavelength dependence than *λ*^{−4}.
At sufficiently large wavelength, where *d*_{spp} ∝ *k*^{−2}, we expect *A* ∝ *σ*_{spp} ∝ *k*^{7}.

In appendices A and B, we will show that the intensity-ratio cross section *A* = *ησ*_{spp}, where
*σ*_{spp} is the cross section from scattering of surface plasmons into other surface plasmons, instead of photons. The efficiency *η* describes, for an incident surface plasmon, how much power is radiated into the
substrate relative to the total power scattered out at this hole. When these scattering processes are mediated via the same (magnetic) dipole moment, we also find *A/σ* = *ησ/σ*_{spp} =
*η*(3*λ*/16*d*_{spp}), where *λ* = *λ*_{0}/*n*.

In Fig. 5(b) we plot a fit of *A* = *ησ*(3*λ*/16*d*_{spp}), using the efficiency *η* as the only free parameter.
We used scattering cross sections *σ* calculated from the fit from Fig. 5(a) to limit the noise. We obtain a fitted value of *η* = 0.67 ± 0.19, which is reasonable as we expect this
efficiency to be close to, but smaller than, one. This demonstrates the consistency of the experimental data and the data analysis. We are able to relate two independent quantities (the intensity ratio and the spectral width) to the same scattering cross section *σ*,
using a simple efficiency factor *η*.

## 5. Results for square holes

The results presented so far were obtained for random patterns of circular hole, with a diameter of 2*a* = 140 ± 8 nm. We have also performed similar measurements on random patterns of square holes with side length 151 ± 6 nm. We are interested whether
the shape has any influence on the magnitude of the scattering cross section and its wavelength dependence.

In Fig. 6(a) we plot the results for the scattering cross section of the square holes, along with the results for the round holes presented earlier. The measured scattering cross section *σ* is larger for the square hole than
that of the round holes, but its wavelength dependence is very similar. The suggested wavelength dependence *σ* = *ξk*^{4}*a*^{6}/*d*_{spp} accurately fits the
experimental data, where we choose *a* the rib length divided by two. The pre-factor *ξ* is found to be 1.7 ± 1.3 larger for the square holes.

In Fig. 6(b) we plot the results for the intensity-ratio cross section, also with the results of the round holes. The value of *A* is larger for the square holes too. We fit the expected wavelength dependence of *A*
= *ησ*(3*λ*/16*d*_{s}* _{sp}*), using the value of

*ξ*just found and leaving only

*η*as a free parameter. We thus find

*η*= 0.60 ± 0.13, which is comparable to that of round holes.

In conclusion, the data for the square hole shows the same wavelength dependence of *σ* and *A*. The pre-factors *η* and *ξ* obtained for the square holes do not differ significantly from those found for
round holes.

## 6. Conclusions

The scattering cross section of surface plasmons scattered by a sub-wavelength hole is measured in the wavelength range of 650–900 nm. The reported wavelength dependence is stronger than Rayleigh scattering predicts, because a surface plasmon polarizes the hole less efficiently at larger wavelengths. Nonetheless, this behavior can be captured in a simple expression.

Additionally, the measured scattering cross section explains the ratio between surface plasmon-mediated transmission and direct transmission of random hole patterns. Our results therefore imply that it may be viable to model particular complex plasmonic structures, like metal hole arrays, using only physical parameters like the hole size, hole density and film thickness. The magnitude of the measured scattering cross section is surprisingly large in comparison with recent theoretical predictions.

The presented methodology of obtaining scattering cross sections from transmission measurements on samples of different hole densities is surprisingly powerful, and may prove to be fruitful outside plasmonics too. Moreover, we showed the advantage of using random patterns instead of arrays, as the randomness allows measurements at virtually any wavelength without changing the illumination angle and thus the character of the excited dipole moments.

## Appendix A: Relating model parameters to polarizability

In this Appendix, we will briefly discuss a recent calculation of the scattering cross section of surface plasmons from a single hole in a metal film, presented as supplementary material to [18]. This calculation starts from an
incident surface plasmon on a metal dielectric interface, of which the power per unit length *P/L*_{⊥} is calculated. Next, the hole in the metal is treated as polarizable object, which is polarized relative to its surroundings, with an induced (dominantly
vertical) electric dipole *p* = *α _{E}E* and horizontal magnetic dipole

*m*=

*α*, where

_{M}H*E*and

*H*are the electric and magnetic field component of the incident surface plasmon, respectively. Finally, the authors calculate the field emitted by these induced dipoles, assuming an otherwise smooth film, and thereby the power scattered to free space

*P*and to the surface plasmon field

_{out}*P*

_{spp}. The associated scattering cross sections

*σ*and

*σ*

_{spp}are found after division by the power per unit length

*P/L*

_{⊥}[18]:

The polarizabilities *α _{E}* and

*α*follow from a modal expansion of the EM field in cylindrical waves by imposing field continuity at the material boundaries, but this calculation is difficult. The theoretical results presented in [18] show that each polarizability scale as

_{M}*a*

^{3}, where

*a*is the hole radius, multiplied by a dimensionless factor that is constant for

*a/λ*≪ 1. The magnetic polarizability

*α*exhibits a shape resonance around

_{M}*a/λ*≈ 0.2 and decreases for larger

*a/λ*. The electric polarizability is almost a factor two smaller for

*a/λ*≪ 1 and exhibits no resonance but simply decreases for larger

*a/λ*. The polarizability of an infinitely thick film is predicted to be somewhat smaller than that of a film with a zero thickness film. For the calculations in the main text, we used the zero thickness values

*α*= 0.106

_{M}*a*

^{3}and

*α*= 0.054

_{E}*a*

^{3}, which yields

*ξ*≈ 0.24 in the expression

*σ*=

*ξk*

^{4}

*a*

^{6}/

*d*

_{spp}.

For completeness we note that the electric and magnetic response of single sub-wavelength holes have also been measured recently by Rotenberg et al. [39]. From the optical field observed close to an illuminated hole they were able to
deduce the strength and angle-dependence of the surface plasmon to surface plasmon scattering. The magnetic polarizability *α _{M}* that they find is approximately as expected, but the measured electric polarizability

*α*is larger than expected and about as large as

_{E}*α*.

_{M}In the main text, we have rewritten Eq. (5) in terms of the mode size of the surface plasmon ${d}_{\text{s}pp}\approx \sqrt{\left|\epsilon \right|}/(2{k}_{0})$ to stress that the induced dipole should be proportional to the incident field and scale as $\propto 1/\sqrt{{d}_{\text{s}pp}}$. This removes the factor $\sqrt{\left|\epsilon \right|}$ from the denominator and allows us to write *σ* =
*ξk*^{4}*a*^{6}/*d*_{spp} for the scattering cross section of surface plasmons to photons. The power radiated to the surface plasmon field contains another factor $\lambda /{d}_{\text{s}pp}\propto 1/\sqrt{\left|\epsilon \right|}$ to account for the ’width of the angular spectrum of the surface plasmon’ [40].

The theory in [18] assumes that the surface plasmon exists on a metal-air interface. In our experiment, however, it exists on a gold-glass interface. This modifies the expressions. For Rayleigh scattering, we expect that the factor ${k}_{0}^{4}$ should be replaced by *k*^{4}, with *k* = *nk*_{0} as the wave vector inside the medium [41]. As the expression for the mode size ${d}_{\text{s}pp}\approx \sqrt{\epsilon}/(2{n}^{2}{k}_{0})$ contains the refractive index squared, we predict that the scattering cross section *σ* ∝ *n*^{6}.

Division of Equation (6) by (5) yields the elegant result:

*λ*≡

*λ*

_{0}/

*n*. If the magnetic polarizability dominates over the electric polarizability, the ratio between these two cross sections is

*σ*

_{s}

*≈ (3*

_{pp}/σ*λ*/16

*d*

_{spp}). For the more realistic case |

*α*| = 2,

_{M}/α_{E}*σ*

_{s}

*≈ 1.2 × (3*

_{pp}/σ*λ*/16

*d*

_{spp}). This ratio is equal to the power radiated to the surface plasmon field relative to that radiated to free space. It is also equal to the density of modes of the surface plasmon field relative to that of the free space modes. This ratio is approximately 0.5 for gold at 800 nm.

## Appendix B: Intensity-ratio cross section *A* = *ησ*_{spp}

In this Appendix, we relate the intensity-ratio cross section *A*, extracted from our measurements, to the scattering cross section *σ*_{spp}. We do this by considering the power flow depicted in Fig. 7, which is linked to the power flow in our experiment by the principle of reciprocity. We consider an incident plane wave with power *P _{in}*, which polarizes a hole on the glass side of the gold film. The induced dipole will radiate
power into three channels:

*P*into the substrate,

_{d}*P*

_{spp}into the surface plasmon field, and

*P′*

_{1}back into the waveguide (not shown). The surface plasmon field is then either absorbed or scattered as photons, into the substrate (power

*P*

_{2}) or into the waveguide (power

*P*

_{1}). The corresponding loss rates for these processes are the rates ${L}_{abs}^{-1}$ and

*σρ*mentioned in the main text, making $({P}_{1}+{P}_{2})/{P}_{3}=\sigma \rho /\left({L}_{abs}^{-1}+\sigma \rho \right)$, a ratio that approaches one in the high-density limit where radiative loss dominates. Combination of these expressions now yields the intensity ratio of the surface-plasmon-mediated transmission over the direct transmission

*η*=

*P*

_{2}/(

*P*

_{1}+

*P*

_{2}), to quantify how much of the out-coupling is to the substrate relative to all light scattered out, and used

*P*

_{s}

*=*

_{pp}/P_{d}*σ*

_{s}

*. A comparison with Eq. (3) from the main text immediately shows that the intensity-ratio cross section*

_{pp}/σ*A*=

*ησ*

_{spp}.

This is a very important result as it shows that the intensity-ratio cross section *A* is closely related to the scattering cross section of surface plasmons into other surface plasmons. We expect *η* to be smaller but close to one, making
*A* just a little bit smaller than *σ*_{spp}. The measured ratio

*σ*

_{s}

*and (ii) our assumption that the magnetic dipole dominates over the electric dipole. Under this assumption, theory and experiment provides a good match for the reasonable efficiency*

_{pp}/σ*η*≈ 0.69 ± 0.19 mentioned in the main text.

We finally return to the correlation function *C*, which can be written as

*σ*=

*ξk*

^{4}

*a*

^{6}/

*d*

_{spp}, the intensity-ratio cross section

*A*=

*ησ*

_{spp}=

*ησ*(3

*λ*/16

*d*

_{s}

*) (for dominant magnetic dipoles), and the absorption length*

_{sp}*L*. These can in principle be calculated from the hole size

_{abs}*a*(and its geometry), the optical wavelength

*λ*, and the metal properties. The ratio 〈

*I*〉

_{d}^{2}/〈

*I*+

_{d}*I*〉

_{s}^{2}simply normalizes the result. The presented description is an important step forward in understanding random patterns in terms of their relevant design parameters. The next challenge will be to better understand the dimensionless quantities

*ξ*and

*η*in the expressions for

*σ*and

*A*.

## Acknowledgments

We acknowledge M.J.A. de Dood and M. Orrit for discussions. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).

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