The propagation of surface plasmons on metal hole arrays with elliptical holes in various configurations

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The propagation of surface plasmons on metal hole

arrays with elliptical holes in various configurations

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THESIS

submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE in

PHYSICS

Author: Michel Hubert

Student ID: 1608908

Supervisor: Martin van Exter

2nd_corrector: _{Sense Jan van der Molen}

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Abstract

In this report are the results of measurements done on surface plasmons interacting with metal hole arrays. The holes are elliptical, and they are orientated on the sample in two different ways. The first way is the same orientation everywhere on the sample, the second way is in a spiral shape. The samples consist of layers InP and InGaAs with on top a layer of gold with the hole arrays in it. The sample is illuminated at the InP side with a 1064 nm laser. This creates surface plasmons between the InGaAs and the gold. The interactions with the metal hole array cause backscattering and sidescattering. The surface plasmons also scatter to photons, which can be measured. We measure this light both in the near field and in the far field.

In the near field we look at the polarization of the emitted light. This polarization coincides more or less with the orientation of the ellipses, specifically along the short axis of the ellipse. This also means that for the spiral samples, the spiral pattern can be seen when measuring the polarization of the emitted light on different locations on the sample.

In the far field we measure the band structure of the sample. For the uniformly oriented ellipses we have a theory that predicts the bands based on the scattering rates into the different directions, γ for backscattering, κ for scattering along the ellipse and µ for scattering against the ellipse. The found values are γ/ω0=0.006±0.002, κ/ω0=0.004±0.002 and µ/ω0=0.020±0.002. For the spiral samples we have

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1 Introduction

In this report we will investigate the propagation of surface plasmons on metal hole arrays. These hole arrays have elliptical holes in a square lattice. We have samples on which the orientation of the ellipses is uniform on the sample, and we have samples on which the ellipses form a spiral.

How surface plasmons propagate on the sample is well explained by W.L. Barnes et al.1_{. They also}

explain how a periodic structure leads to a band structure. That periodic structures enhance the optical transmission is shown by H. F. Ghaemi et al.2_{and by T.W. Ebbesen et al.}3_.

To get emitted light from the sample we pump it with a 1064 nm laser on the InP side of the sample. The emission generates surface plasmons that travel on the sample and interact with the metal hole array. This interaction is scattering, in particular backscattering, scattering along the ellipse and scattering against the ellipse. To predict the band structure we will develop a theory involving the scattering rates. This band structure is measured by looking at the far field of the emitted light. We have measured band structures for both types of samples. We will make a fit of the band structure of the sample with the uniformly orientated ellipses. For the band structure of the spiral samples we do not have a theory. Besides far field measurements we will also present near field measurements for both samples.

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2 Theory

In this chapter we will present theory about band structures. V.T. Tenner et al.4_{designed a coupled}

mode model with which they can predict the band structure of a sample with round holes on a square lattice. If we adapt this theory slightly, we can also predict the band structure for the sample with elliptical holes with uniform orientation. We will first succinctly explain their model and then show how we have adapted it.

The starting point of the coupled mode model is the notion that photons emitted with parallel momentum kıı couple only to surface plasmons with momenta 𝒌_𝒔𝒑= 𝒌_||+ Gi, where 𝑮_𝒊 is a lattice

vector of the array, with|𝑮𝒊| =2𝜋_𝑎₀ . There are the four fundamental lattice modes in a square lattice. Ref

4 starts with the following expression for the out of plane electric field.

𝐸(𝑟, 𝑡) = (𝐸𝑥(𝑡)𝑒𝑖𝐺𝑥+ 𝐸−𝑥(𝑡)𝑒−𝑖𝐺𝑥+ 𝐸𝑦(𝑡)𝑒𝑖𝐺𝑦+ 𝐸−𝑦(𝑡)𝑒−𝑖𝐺𝑦)𝑒𝑖𝑘𝑥𝑥+i𝑘𝑦𝑦 (2.1)

If the four E-field components are combined in a vector |𝐸 > , its time evolution is described as 𝑑|𝐸 >/𝑑𝑡 = -iH|𝐸 >, where H is a matrix describing time evolution and the scattering.

For circular holes, ref 4 uses the following matrix H,

𝐻 = [ 𝜔0+ 𝑐2𝜃2 𝛾 𝜅 𝜅 𝛾 𝜔₀+ 𝑐₂𝜃2 _𝜅 _𝜅 𝜅 𝜅 𝜔₀+ 𝑐₁𝜃 𝛾 𝜅 𝜅 𝛾 𝜔₀− 𝑐₁𝜃] (2.2)

with 𝜔0=_𝑎₀2𝜋𝑐_𝑛_𝑒𝑓𝑓, c1=𝜔0/𝑛𝑒𝑓𝑓, 𝑐2= 𝜔0/2𝑛𝑒𝑓𝑓2 , γ is the backscattering rate and κ the left/right scattering

rate. The forward scattering rate is absorbed into 𝜔0.

To further improve this model we have to account for the frequency dependence of the refractive index. We presume that this dependence is linear. This is a little bit tricky though, because the frequency is on its turn dependent on the refractive index. The employed algorithm works as follows. First the frequency is calculated without the correction 𝜔0=_𝑎2𝜋𝑐

0𝑛𝑒𝑓𝑓 . This frequency is then used to calculate the correction on the diagonal terms of H, 𝛥𝑛 = (𝜔0+𝑐_𝜔2𝜃2

0 − 1) 𝑛𝑒𝑓𝑓1 or 𝛥𝑛 = (

𝜔0±𝑐1𝜃

𝜔0 − 1) 𝑛𝑒𝑓𝑓1 . This correction is then used to calculate the new frequency 𝜔0=_𝑎 2𝜋𝑐

0(𝑛𝑒𝑓𝑓+𝛥𝑛). So 𝑛𝑒𝑓𝑓1 can be seen as 𝑛𝑒𝑓𝑓1= 𝜔0(

𝑑𝑛 𝑑𝜔)_𝜔₀

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Figure 1: A sketch of an incoming surface plasmon on an elliptical hole. The symmetry between scattering to the left and the right is broken. We now have three scattering rates, γ for backscattering, κ for scattering along the ellipse and µ for scattering against the ellipse. The forward scattering/transmission is absorbed into the 𝜔0.

The main difference between circular and elliptical holes is the left/right scattering. For round holes, the situation is symmetric, and the right and left scattering rate are equal. For elliptical holes this is no longer the case, the symmetry between left and right is broken. To represent this broken symmetry we replace some scattering rates by µ. κ represents the scattering along the ellipse and µ the scattering against the ellipse. This is shown in figure 1. This gives the following time evolution matrix H.

𝐻 = [ 𝜔0+ 𝑐2𝜃2 𝛾 µ 𝜅 𝛾 𝜔₀+ 𝑐₂𝜃2 _𝜅 _µ µ 𝜅 𝜔₀+ 𝑐₁𝜃 𝛾 𝜅 µ 𝛾 𝜔₀− 𝑐₁𝜃] (2.3)

To calculate the band structure as a function of θ we calculate the eigenvalues of this matrix. This gives four bands. Please note that the matrix remains Hermitian, which is necessary for energy conservation, since we consider a lossless scattering process here.

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3 Experimental setup

Figure 2: A sketch of our experimental setup

The used experimental setup is shown in figure 2. All the way to the left is the pump laser, with wavelength 1064 nm and maximum power 2 W. The setup also contains some devices with which the laser power can be attenuated, these are not included in figure 2. The laser beam is focused on the sample by lens 1 (L1). The sample is mounted into the cryostat, which is kept at T=100 K with liquid nitrogen.

The image is formed by an objective with a 20x magnification and a numerical aperture of 0.4. After this, the bundle can be sent into two directions using a flip mirror.

The first way is down in figure 2, where the beam goes through lens 2 (f=200 mm) and lens 3 (f=200 mm if looking at the far field or f=100 mm if looking at the near field). The bundle then falls onto the camera, with which we take an image. If the intensity of the bundle is too high, we can insert ND-filters, these are not shown in figure 2.

The second way is straight through, so when the mirror is down. The beam then goes through lens 4 (f=200 mm) and lens 5 (f=50 mm if looking at the far field or f=25 mm if looking at the near field). In between these two lenses, there is a polarizer which can be taken out if desired. The bundle then falls onto a fiber with a diameter of 10 µm. The fiber can be moved through the bundle using a computer controlled translation stage. The fiber then leads to a grating spectrometer, with a CCD with 512 pixels in the detection plane. With the grating we have used, every pixel corresponds to 0.5 nm.

All the data we have taken was processed using Python. All the figures have been made with the Holoviews package for Python.

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4 Samples used in experiment

4.1 Introduction

In this chapter, we present an overview of the used samples. We will first present overview pictures of the two wafers. Then detailed pictures of the samples will be presented, made with a SEM. At the end of the chapter we will present a table that lists all the properties of each sample.

4.2 Overview pictures

We have two overview pictures, one of each wafer. One of them was made using a regular microscope, the other one using a SEM.

Figure 3: An overview of the samples with a spiralized lattice. This picture was taken using a regular microscope, with a 200x magnification. The size of the samples is 50x50 µm.

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Figure 4: An overview of the samples with elliptical holes. The picture was taken using a SEM, with a 80x magnification. The samples on the left are 25x25 µm, the ones on the right 50x50 µm.

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4.3 Example pictures

We will now provide some example pictures of the samples.

Figure 5: An overview picture of sample Ad4. This was taken using a SEM, with a 4000x magnification. The size of the sample is 50x50 µm.

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Figure 7: An overview picture of sample k2d4, which was taken using a SEM, with a 8.000x magnification. The vortex in the corner can clearly be seen. The size of the sample is 25x25 µm.

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Figure 8: A zoomed in picture of sample wk1d3. This was taken using a SEM, with a 40.000x magnification. The spiralized lattice can clearly be seen.

Figure 9: A zoomed in picture of sample wk2d1. This was taken using a SEM, with a 40.000x magnification. The spiralized lattice can clearly be seen.

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4.4 Analysis of the samples

We will now provide some analysis of the samples. In the table below all kinds of properties of the samples are given. All these properties were measured from the SEM pictures.

Sample name Length holes (nm) Width holes (nm) Aspect ratio Lattice constant (nm) Sample size (µm) Topological charge θ (degrees) Standard deviation of θ Filling fraction wk1d4 Failed sample wk1d3 280 133 2.1:1 467 52 1 - 2.8 1 wk1d2 Failed sample wk2d1 280 147 1.9:1 467 52 2 - 3.0 1 wk1d1 Failed sample Sd1 130 130 1.0:1 455 48 0 - - 0.15 Ad1 293 190 1.5:1 445 51 0 -69 35 0.90 Sd2 198 198 1.0:1 453 49 0 - - 0.48 Ad2 293 190 1.5:1 441 51 0 -53 6.6 >0.99 Sd3 198 198 1.0:1 441 50 0 - - 0.94 Ad3 293 198 1.5:1 441 52 0 -53 2.4 >0.99 Sd4 198 198 1.0:1 441 51 0 - - >0.99 Ad4 345 250 1.4:1 441 52 0 -55 3.6 1 Sd5 215 215 1.0:1 453 52 0 - - >0.99

Ad5 Failed sample

k2d1 Failed sample k1d1 Failed sample k2d2 333 158 2.1:1 451 26 0.5 - >0.95 k1d2 333 158 2.1:1 451 26 0.25 - >0.95 k2d3 333 158 2.1:1 451 26 0.5 - >0.99 k1d3 333 158 2.1:1 451 26 0.25 - >0.99 k2d4 316 211 1.5:1 451 26 0.5 - >0.99 k1d4 316 211 1.5:1 451 26 0.25 - 1 k2d5 Failed sample k1d5 Failed sample

The names of the samples work as follows. The (quarter) spirals have a k in their name. The number after the k is the topological charge (if it would be a full spiral). The uniformly orientated samples start either with an A or an S. The A stands for Angled and means that the holes are elliptical and are oriented at an angle. The S stands for Straight and means that the holes are round. All the samples have a d in their name, this stands for depth. d1 means that the illumination with which the samples were written was the lowest, d5 means that it was the highest.

The length and the width of the holes, the lattice constant and the sample size are measured by pixel counting (i.e. comparing the amount of pixels of the scale bar with the amount of pixels of the hole/ hole spacing).

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θ is the angle the ellipses have compared to the sample. θ=0˚ in this case means a horizontally orientated ellipse, θ=±90˚ means a vertically orientated ellipse. Θ and its standard deviation are measured using ImageJ, a program for processing photos.

The filling fraction is measured by manually counting the amount of filled holes in a part of the sample. We have declared some samples to be failed. This was either because the illumination had been too high, which caused the holes to merge, or because the illumination had been too low, which caused a large lack of holes.

4.5 Mathematical description of the spirals

In the spiral samples the ellipses have a different orientation all over the sample. This can be described by the following two equations.

𝜃 = arctan𝑥+𝑦_𝑥−𝑦= 𝜑 + 45˚ (4.1)

𝜃 = arctan𝑥2−𝑦2+2𝑥𝑦

𝑥2_−𝑦2_−2𝑥𝑦= 2𝜑 + 45˚ (4.2)

θ=0˚ in this case means a horizontally orientated ellipse, θ=±90˚ means a vertically orientated ellipse. 𝜑 is here the polar coordinate, 𝜑 = 0 at the positive x-axis. The number in front of 𝜑 is the topological charge.

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5 Results for spiral samples

5.1 Introduction

In this chapter we will analyse the data taken from the spiral samples. We have analysed four samples, wk1d3, wk2d1, k1d3 and k2d3. That means both the full and the quarter spirals (with the vortex in the corner), and both topological charge 1 and 2. For all cases we have near field and far field

measurements.

5.2 Near Field

We have taken several types of near field images of different samples. We first took some pictures with the Xenics camera, using colour filters. These pictures showed interesting behaviour in the near field, and therefore we took near field pictures with the spectrometer. This allows for detailed analyse of the spectrum of the near field. We also put a polarizer in front of the spectrometer, to check the

polarization dependence of the spectrum of the near field.

5.2.1 Camera pictures

Using the Xenics camera, we took three types of pictures of every sample. One without colour filter, one with a 1450±12 nm colour filter and one with a 1480±12 nm colour filter. We used these filters to find out how the shape of the near field depends on the wavelength. One example of each of these pictures is given below. All the pictures have been taken at T=100 K and with an illumination power of 100 mW. The exposure time of the Xenics camera was 1 ms. To ensure that all the pictures have more or less the same brightness, ND-filters were used.

Figure 10: Near field pictures of sample k2d3 with (A) no colour filter, (B) a 1480±12 nm filter, (C) a 1450±12 nm filter. The size of the sample is 25x25 µm. Because of the used ND-filters, figure 10(A) is actually 316 times brighter.

In picture H can be seen that the near field light from sample k2d3 is certainly not Gaussian. Therefore it is certainly interesting to make near field pictures with the spectrometer, which we subsequently did.

5.2.2 Spectrometer pictures with polarizer

Figure 10 shows that the near field pictures look different for different wavelengths. This effect gets stronger if polarization is also included. We measured this in the following way. For all the polarizations between 0˚ and 180˚, in steps of 5˚ or 10˚, we took a 2D-scan. This means measuring a spectrum for every x and y position. This results into 4D data: Intensity as a function of x, y, λ and 𝜑 (polarization angle). We have six data sets, namely for wk1d3, wk2d1, k1d3 and k2d3. We will now present different cuts of these data sets. All data was taken at a temperature of 100 K and pumping power of 50 mW (the full spirals) or 100 mW (the quarter spirals).

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5.2.2.1 k1d3

Here we will present different cuts of the data of sample k1d3. First we take x and y as free parameters, while fixing λ at 1622 nm and φ at 0˚, 45˚, 90˚ and 135˚.

Figure 11: Near field pictures of k1d3, with x and y as free parameters and the x- and y-axis. λ=1622 nm and (A) φ=0˚, (B) 45˚, (C) 90˚, (D) 135˚. The vortex is situated in the top left corner.

Figure 11 shows that the near field picture strongly depends on the polarization. To better see how exactly, we will now fix x, y and λ, keeping only 𝜑 as a free parameter.

Figure 12: The intensity in arbitrary units plotted versus the polarization angle (blue line), and a fitted function of the form 𝑎 +

𝑏 𝑠𝑖𝑛 2𝜑 + 𝑐 𝑐𝑜𝑠 2𝜑. The wavelength is 1547 nm, x and y are fixed, the exact values are irrelevant.

As can be seen in figure 12, the intensity behaves, as expected, as a trigonometric function. We

therefore have fitted it to 𝑎 + 𝑏 sin 2𝜑 + 𝑐 cos 2𝜑. This works out nicely. 𝑎 is not zero because there is light with a circular polarization. This light does not change intensity when the angle of the polarizer is changed. Incoherent light will also affect 𝑎.

The fitting function can also be written as 𝑎 + 𝑏 cos(2(𝜑 − 𝜑0)). 𝜑0 is a parameter that indicates for

which 𝜑 the intensity of the emitted light has its maximum, being the polarization angle of the emitted light. In further analysis, φ0 is going to play a big role. First we will plot 𝜑₀ versus λ, for a fixed x and y.

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Figure 13: The fit parameter φ0 plotted versus the wavelength for a fixed x and y near the centre of the sample. The average angle θ the ellipses make at these x and y is -9.5˚. Please note that θ and φ0 are the other way around, because the bundle is mirrored at the location of the polarizer.

Figure 13 shows that 𝜑0 is relatively constant as a function of the wavelength. It becomes even more

constant for higher wavelengths, especially closer to the rim of the sample. To make sure we are always in a stable interval, we take the average of 𝜑0 over the wavelengths from 1600 nm till 1650 nm.

We have calculated this average 𝜑0for every measured pixel on the sample. This is a 10x10 data set,

because the pixels on the rim are clearly not a part of the sample as can be seen in figure 11. In figure 14 𝜑0is plotted in a false colour plot.

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Figure 14: A false colour picture of k1d3, with the colour representing the polarization angle of emission φ0 averaged over the wavelength from 1600 nm till 1650 nm. The vortex is situated in the top left corner.

This can now be compared to the angles the ellipses make on the sample. These angles are described by formula 4.1. Since the vortex is in the top left corner, x goes from 0 to 9 and y from 0 to -9. This results in a 10x10 matrix, which we can point-plot against the measured average 𝜑0. This is shown in figure 15.

Figure 15: The polarization angle of the emitted light φ0 averaged over the wavelengths from 1600 nm till 1650 nm plotted versus the corresponding angle of the ellipses on the sample k1d3. Every dot represents a location on the sample. The correlation between these two quantities is clear. The red line is the line y=-x.

The correlation between these two quantities is clear. This means that the polarization angle of the emitted radiation is determined by the local orientation of the ellipses. The red line in figure 15 is the line y=-x. We have no good explanation why this line does not exactly fit the data. The correlation is

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negative because at the location of the polarizer the bundle is mirrored. They are also 90˚ out of phase, which means that the ellipses radiate the strongest along their short axis. This is in agreement with what Gordon et al.6_{have found.}

5.2.2.2 k2d3

We can make the same analysis for sample k2d3, the quarter spiral with topological charge 2. The found results are very similar to those of sample k1d3. Figure 16 shows the false colour plot representing the polarization angle of emission 𝜑0. Comparing this angle with the angle the ellipses make on the sample

gives figure 17.

Figure 16: A false colour plot of sample k2d3. The false colour represents the polarization angle of emission φ0 in degrees. Please note that the colour scale is continuous.

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Figure 17: The polarization angle of emission φ0 averaged over the wavelengths from 1600 nm till 1650 nm plotted versus the corresponding angle of the ellipses on the sample k2d3. Every dot represents a location on the sample. The correlation between these two quantities is clear. The red line is the line y=-x.

The correlation between these two quantities in figure 17 is clear. This means that the polarization angle of the emitted radiation is determined by the local orientation of the ellipses. The red line in figure 17 is the line y=-x. Again, we have no good explanation why it does not exactly fit the data. The relatively large deviations from the fit around 45˚ is caused by some deviations around the vortex of the sample.

5.2.2.3 wk1d3

For sample wk1d3, which is the full spiral with topological charge 1, we have similar figures. The false colour plot of the polarization angle of emission 𝜑0 is again shown in figure 18, while figure 19 shows 𝜑0

compared with the angle the ellipses make. In this figure the line of dots appears to be broken at ±45˚. This is caused by how the average orientation of the ellipses on the sample is sampled. To avoid an ill-defined ellipse at the vortex, the points are taken around the vortex, so not on the ‘x-axis’ and ‘y-axis’. As can be seen in figure 18, on these axis the orientation if ±45˚. Therefore it seems as if these

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Figure 18: A false colour plot of sample wk1d3. The false colour represents the polarization angle of emission φ0 in degrees. Please note that the colour scale is continuous.

What is interesting about the samples with the vortex in the centre, is that we can examine the vortex more closely. This is possible because the bundle of the pump laser can be focused on the vortex. Figure 20 is a false colour plot of the intensity of the polarized emitted laser light in arbitrary units. An

interesting feature that can be seen here is the lower intensity of light in and near the vortex. The explanation for this effect is that in the vortex the orientation of the ellipse is ill-defined. Therefore the amplitude of the cosine of figure 12 has to go to zero, to prevent a maximum from existing. A different way of explaining this effect is concluding that the ellipses around the vortex have a rapidly variating orientation, and therefore the emitted light of these various ellipses cancels itself.

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Figure 19: The polarization angle of emissionφ0 averaged over the wavelengths from 1600 nm till 1650 nm plotted versus the corresponding angle of the ellipses on the sample wk1d3. Every dot represents a location on the sample. The correlation between these two quantities is clear. The red line is the linear line y=-x.

Figure 20: A false colour plot of the intensity of the polarized emitted laser light of sample wk1d3. Every pixel represents an area of 2.8x2.8 µm on the sample. What is remarkable is the lower intensity of the laser light in and around the vortex.

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5.2.2.4 wk1d1

For the sample wk2d1, the full spiral with topological charge 2, we have the same figures.

Figure 21: A false colour plot of sample wk2d1. The false colour represents the polarization angle of emission φ0 in degrees.

Please note that the colour scale is continuous.

Figure 22: The polarization angle of emission φ0 averaged over all the wavelengths plotted versus the corresponding angle of the ellipses on the sample wk2d1. Every dot represents a location on the sample. The correlation between these two quantities is clear. The red line is the line y=-x.

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Figure 23: A false colour plot of the intensity of the polarized emitted laser light of sample wk1d3. Every pixel represents an area of 2.8x2.8 µm on the sample. What is remarkable is the lower intensity of the laser light in and around the vortex.

5.3 Far Field

In this paragraph we will show and discuss the far field measurements. These show the band structure of the sample. We have taken 2D scans of the far field with the spectrometer, which gives a 3D data set, namely (kx, ky, λ). Here we will present different cross sections of these data sets, namely equal energy

cross sections (fixed λ) and band structures (fixed kx , ky or (kx ± ky)/√2). For each sample we have six

3D data sets, measured with different polarizations, namely 0˚, 90˚, 45˚, -45˚, left-handed and right-handed.

5.3.1 Band structures

Figure 24: An equal energy cross section to demonstrate in which four directions the cross sections are taken for the band structures.

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In this paragraph we will present four different cross sections under four different angles. These angles are shown in figure 24. The angle of polarization of the emitted light is 0˚ in figure 25.

Figure 25: Four different cross sections of the band structure of sample wk1d3. They are taken under the four different angles as shown in figure 24. The angle of polarization of the emitted light is 0˚

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The four different cross sections are shown in figure 25. The thing that immediately catches the eye is how bright the lowest band is compared to the others. A possible explanation for this is that the

semiconductor layers, responsible for the gain, does not work that well. V.T. Tenner et al.4_{measured the}

fluorescence as a function of wavelength, and found that for the wavelengths in the range 1400 nm till 1500 nm the fluorescence is less strong.

Other polarizations we have measured are left-handed and right-handed light. Because the spirals have a handedness, we expected a difference between these two polarizations. The difference turned out to be very minimal though, as can be seen by comparing figures 26 and 27.

Figure 26: Four different cross sections of the band structure of sample wk1d3. They are taken under the four different angles as shown in figure 24. The polarization of the emitted light is left-handed.

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Figure 27: Four different cross sections of the band structure of sample wk1d3. They are taken under the four different angles as shown in figure 24. The polarization of the emitted light is right-handed.

Because every unit cell in these spiral lattices is different, it is not so easy to develop a good theory to predict the band structure. Therefore we cannot present a fit of the band structure.

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5.4 Conclusion

In the near field, the polarization angle of emission 𝜑0 is strongly correlated with the orientation of the

ellipses. To be more specific, the strongest emission is along the short axis of the ellipse. For the two full spirals the correlation between the polarization angle of emission and the orientation of the ellipses is perfect. This means that it can be fitted to a line with slope -1. For the two quarter spirals the slope is not perfect, namely -1.12 and -0.82 for respectively k1d3 and k2d3.

Another interesting thing that can be seen in the near field is the reduced intensity of emitted light in and around the vortex. This is probably due to the rapidly varying orientation of the ellipses around the vortex, which makes the light from around the vortex cancel itself.

In the far field we have measured the band structure, to find that the bands with a higher wavelength are almost not illuminated. Unfortunately we do not have a theory to fit the band structure.

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6 Results for uniform ellipses

6.1 Introduction

We will present the measurements done on the ellipses with a uniform orientation. The near field measurements have been done on sample Ad3, the far field measurements on sample Ad4. All the measurements were done at T=100 K and P=100 mW.

6.2 Near Field

Figure 29: A false colour plot of sample Ad3. The false colour represents the polarization angle of emission φ0 in degrees.

Figure 30: A false colour plot of sample Ad3. The false colour

represents the polarized intensity of the emitted light in arbitrary units.

Just as we have done in chapter 5 we have performed near field measurements with a polarizer. This allows one to find the polarization angle of emission 𝜑0. This is the angle under which most light is

emitted, the maximum of the fitted cosine from figure 12.

Figure 28: A histogram of the polarization angle of emission φ0 in degrees. The average angle is -48˚

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For this sample it is to be expected that 𝜑0 is the same everywhere on the sample, because the

orientation of the ellipses is constant. This turns out to be the case, as can be seen in figure 28. It is clear that the spreading of 𝜑0is quite small. Around the centre of the beam, which can be found in figure 30,

𝜑₀ is more or less constant. To the left of it however, 𝜑0 is a few degrees higher, and below it 𝜑0 is a

few degrees lower. This can be explained by travelling and scattering surface plasmons. The local part of the laser beam creates surface plasmons, which will radiate mostly perpendicular to the direction of the ellipses. However there are also travelling surface plasmons, mainly coming from the centre of the beam. These slightly change the angle under which the surface plasmons radiate the most.

The average φ0 is -48˚. The average angle the ellipses make on the sample is -53˚. This again confirms

that the ellipses radiate the strongest along the short axis. The 5˚ difference is probably caused by several small deviations, including a not perfectly aligned cryostat and a not perfectly aligned polarizer.

6.3 Far Field

In this paragraph we will show and discuss the far field measurements. These show the band structure of the sample. We have taken 2D scans of the far field with the spectrometer, which gives a 3D data set, namely (kx, ky, λ). Here we will present different cross sections of these data sets, namely equal energy

cross sections (fixed λ) and band structures (fixed kx , ky or (kx ± ky)/√2). For each sample we have four

3D data sets, measured with different polarization angles, namely 0˚, 90˚, 45˚ and -45˚.

6.3.1 Band structures

We will present a cross section where λ is plotted against k. This shows the band structure of the sample the clearest. The four different types of cross sections that we will make are shown in figure 24, they are the same as in chapter 5. The polarization of the light in the measurements is 0˚as defined in figure 24. The four different cross sections are shown in figure 31. At least three bands are clearly visible. The bands that have a higher wavelength (and therefore a lower energy) contain more light than the bands with a lower wavelength. The low bands are therefore hard to distinguish. A possible explanation for this is that the semiconductor layers, responsible for the gain, does not work that well. V.T. Tenner et al.4_{measured the fluorescence as a function of wavelength, and found that for the wavelengths in the}

range 1400 nm till 1500 nm the fluorescence is less strong.

To get some quantitative information from the band structure, we will fit the pictures of figure 31 using the theory from chapter 2. This will give us information about the backward, left, right and forward scattering of the elliptical holes. To improve the fit, the refractive index is made linearly dependent on the wavelength with coefficient neff1. The fitting of the band structure has been done manually, by two

independent people. Their results were in agreement with each other. The results are shown in figure 32. The found values are γ/ω0=0.006±0.002, κ/ω0=0.004±0.002, µ/ω0=0.020±0.002, neff=3.45±0.01 and

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31

Figure 32: The four different cross sections of the band structure of sample Ad4, including a manual fit using the theory from chapter 2. The found scattering rates are γ/ω0=0.006±0.002, κ/ω0=0.004±0.002, µ/ω0=0.020±0.002 and the refractive index

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6.3.2 Equal energy cross sections

Figure 33: Various equal energy cross sections of the far field of sample Ad4. The polarization of the emitted and detected light is 0˚. The NA of the eyepiece is 0.4. The four cross sections have been taken at different wavelengths. Please note that the false colour scale is not the same for the different pictures.

In figure 33 four equal energy cross sections are shown. They are taken at four different wavelengths. The asymmetry is clearly visible. The structure only has a 180˚ rotation symmetry, whereas the results presented by M.P. van Exter et al.5_{for circular holes had a 90˚ rotation symmetry for the same type of}

pictures. This is a clear indication that the symmetry is broken by the ellipticity of the holes.

The fit parameters found in paragraph 6.3.1 can also be used to make a fit in these equal energy planes. This is shown in figure 34.

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Figure 34: Six equal energy cross sections of the band structure of sample Ad4 including the fit made with the same parameters as in paragraph 6.3.1. The colour of the bands corresponds to the colours used in figure 32 (i.e. A=blue, B1=red, B2=yellow and C=green). The cross sections are taken at (A) 1438 nm, (B) 1488 nm, (C) 1503 nm, (D) 1575 nm, (E) 1613 nm and (F) 1638 nm.

6.4 Conclusion

In the near field, the polarization angle of emission is constant over the sample within 4˚ fluctuations. This is as expected, because the angle of the ellipses is also constant over the sample. Most of the fluctuations can be explained by travelling surface plasmons. The average angle we have found is -48˚, whereas the average angle the ellipses make on the sample is -53˚. The 5˚ deviation is probably caused by several alignment issues. The ellipses radiate the strongest along their short axis.

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In the far field we find four bands in the band structure, as expected. We have fitted these bands with the model described in chapter 2. The found values are γ/ω0=0.006±0.002, κ/ω0=0.004±0.002,

µ/ω0=0.020±0.002, neff=3.45±0.01 and neff1=0.3±0.2. This means that scattering against the ellipse is

dominant over scattering along the ellipse. The hole size is 345x250 nm. V.T. Tenner et al.4_{found for round holes the following values: γ/ω}

0=0.012, κ/ω0=0.004, neff=3.268. From

this we can conclude that elliptical holes increase scattering to the side, but only against the ellipse, and that it decreases backscattering. The size of the holes used by ref 4 is 180 nm. This is significantly smaller than our elliptical holes, which explains our increased scattering rates.

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7 Final conclusion

In this research project we have performed far field and near field measurements on samples either with elliptical holes with a uniform orientation or with elliptical holes a varying orientation, where the holes form a spiral.

In the near field we have concluded that the polarization of the emitted light is along the short axis of the local orientation of the ellipses. For the uniformly orientated sample this means that the polarization angle of emission is almost constant over the sample. For the spiral samples this means that the

polarization angle of emission follows the local orientation of the spiral. Around the vortex of the spiral samples, the emitted light turned out to have less intensity. This is probably due to the rapidly varying orientation of the ellipses around the vortex, which makes the light from around the vortex cancel itself. In the far field for the sample with uniformly orientated ellipses (we measured sample Ad4) we have found four bands in the band structure. We have fitted these bands to the theory described in chapter 2. This theory can be used to calculate the band structure if you know the backscattering rate γ, the scattering rate along the ellipse κ, the scattering rate against the ellipse µ, the lattice constant a0, the

index of refraction neff and the frequency dependence of the index of refraction neff1. The found values

are γ/ω0=0.006±0.002, κ/ω0=0.004±0.002, µ/ω0=0.020±0.002, neff=3.45±0.01 and neff1=0.3±0.2. The

lattice constant is a0=441 nm.

In the far field for the spiral samples we have also found bands, but only the band with the lowest energy is clearly visible. Some other bands are visible, but they are only weakly illuminated. Unfortunately we do not have a theory to fit these bands to.

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8 References

1: W.L. Barnes, A. Dereux and T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 2003. 2: H. F. Ghaemi, T. Thio, D.E. Grupp, T.W. Ebbesen and H.J. Lezec, Surface plasmons enhance optical transmission through subwavelength holes, Physical Review Letters 1998.

3: T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature 1997.

4: V.T. Tenner, A.N. van Delft, M.J.A. de Dood and M.P. van Exter, Loss and scattering of surface plasmon polaritons on optically-pumped hole arrays, Journal of Optics 2014, Volume 16

5: M.P. van Exter, V.T. Tenner, F. van Beijum, M.J.A. de Dood, P.J. van Veldhoven, E.J. Geluk and G.W. ‘t Hooft, Surface plasmon dispersion in metal hole array lasers, Optics Express 2013, Volume 21

6: R. Gordon, A.G. Brolo, A. McKinnon, A. Rajora, B. Leathem and K.L. Kavanagh, Strong polarization in the optical transmission through elliptical nanohole arrays, Physical Review Letters 2004, volume 92.

The propagation of surface plasmons on metal hole arrays with elliptical holes in various configurations

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