Chapter 1 Introduction
1.3 Surface Plasmon Polaritons
Surface plasmon is a surface wave due to collective oscillation of carriers (electrons) in conducting materials such as metals or doped semiconductors at optical frequency[5]. This surface mode is confined at the interface between materials with positive and negative dielectric constants respectively. Furthermore, if electromagnetic wave is coupled with the carriers at the surface, we call it surface plasmon polariton (SPP). To get a simple physical picture, we can consider the following situation: a stimulating electric field creates two opposite electric displacements in phase with each other across the interface. From Maxwell’s equations, we can see that these two opposite displacements act to attract and confine an AC current to the interface, and thus generate the collective oscillation of electrons.
The mathematical description of the phenomenon of SPP can be referred to Ref. [5].
Then a dispersion relation of this non-radiative electromagnetic modecan be derived.
We don’t derive the dispersion relation here because in fact it is not the mechanism responsible for the phenomenon in our system.
1.4 Extraordinary Optical Transmission (EOT) through Sub-Wavelength Metallic Hole Arrays
In 1998, Ebbesen et al. reported the surprising property of optical transmission on metallic gratings [1]. They drilled cylindrical holes (150nm for the diameter) in optically thick (200nm) metallic films in fashion of 2D lattice (900nm for the lattice constant) on a glass. Although bi-dimensional metallic gratings have been studied over many years before 1998, the most attractive characteristic of their findings was the distinct spectrum of transmission, as shown in the Fig. 1.2. In Fig. 1.2, the peculiar part of the spectrum is the transmission intensity at the wavelengths above
the periodicity, a0. We have already known that the minimum at a0 is the result of
Rayleigh type of Wood’s anomalies, and the peak right after a0 can, in general, be explained by the Fano type. However, the peculiarity is that another peak occurs at even longer wavelength (1370nm) which is nearly ten times the hole diameter.
Furthermore, if focusing on the transmission efficiency, one can find that the absolute transmission efficiency obtained by taking the ratio of total transmittance (zero-order) to the fraction of surface area occupied by the holes is larger than 2. That is, more than twice as much as energy can be transmitted through the holes when the light illuminates directly on hole area. This new phenomenon cannot be explained by Bethe’s theory which states that the transmission efficiency of a single sub-wavelength aperture can be described as ( / )r λ 4 [6]. Apparently, the existence of grating does change the whole situation. Ebbesen attributed this phenomenon to the resonant excitation of surface plasmon polaritons (SPP). After Ebbesen, many researchers backed up this explanation by investigating this SPP-enhanced phenomenon both theoretically [7] [8] and experimentally [9]. However, there are other researchers questioning this SPP explanation. Theoretically they found that even a structure such as perfect electrical conductor (PEC) which cannot support surface plasmon on it also has a bounded surface wave on its surface [10][11], and hence can causes an extraordinary optical transmission [12]. Additionally, even both matter waves [13] and sound waves [14][15] through holey slabs show extraordinary transmission phenomenon.
In this thesis, we focus on the EOT phenomenon with nearly perfect electrical
the skin depths of those metals can be calculated to be several tens of nanometers, and can be negligible when compared to the incident wavelength.
Fig. 1.2 Zero-order transmission spectrum of hoe array on Ag [1].
1.5 Spoof Surface Plasmon
In 2004, J. B. Pendry, et al. reported an original work showing that even a perfect conductor cam support confined surface wave as long as the surface are not purely flat [10]. The authors call this surface mode “Spoof Surface Plasmon”. Because such spoof surface plasmon (SSP) involves no carriers in the metal, they concluded that it is simply the geometry of the structure responsible for this surface mode. They also suggested that there will be a hybrid surface mode, which is the mixture of surface plasmon and spoof surface plasmon, on real metals. In their derivation, the long-wavelength approximation is assumed, i.e., the characteristic length of the structure are much smaller than the wavelength, and therefore the structured metal can be described as a homogeneous medium with effective dielectric constant and permeability. If a structure with characteristic length comparable to incoming wavelength is considered, the spoof surface plasmon still exists. In this case, the diffracted modes have also to be considered, and the main effect of the diffraction is to couple the confined spoof surface plasmon to free space. Therefore, an anomaly optical transmission can also occur when the incident light resonantly excites this surface mode. In the following part of the thesis, our theoretical ground will base on this result.
Chapter 2 Method of Measurement
We used Fourier Transform Infrared Spectroscopy (FTIR) as our measurement method to analyze the transmission spectra of the devices under study in THz region.
In this chapter, we will briefly introduce the fundamentals of FTIR method and the details of this measurement instrument. Fig. 2.1 is the basic schematics of a Michelson interferometer. A mercury lamp is used as the far-infrared light source.
When light impinges on the beam splitter (50% transmitted and 50% reflected), the differences of light path can be adjusted by moving the mirrors, M1 and M2. In our instrument, M1 is held fixed while M2 is varied. As Fig. 2.1 shows, the reflected part of the light that goes to the fixed M1 in a distance L is reflected there and impinges on the beam splitter again after a total path of 2L. The same action happens to the transmitted part of the beam. Nevertheless, since the reflecting mirror M2 is not held fixed but can be moved very precisely back and forth around L by a distance x, the total path length of this light is consequently 2(L+x) . Then, when the two halves of the light recombine again on the beam splitter, they possess a path length difference of 2x and thus show a interference pattern. The light leaving the interferometer is then passed through the sample under test and is finally focused on the detector. In fact, the quantity measured by the detector is the intensity I(x) which is a function of moving mirror displacement x, the so-called interferogram. Here we use the zero crossings of the interferogram of He-Ne laser to sample that of the sample under measurement.
One of the advantages of FTIR is its measurement accuracy. The accuracy of the sampling spacing between two zero crossings is only determined by the precision of the laser wavelength itself. And the common FTIR spectrometers have a built-in
wavenumber calibration of high precision of about 0.01cm . Besides its high -1 accuracy, FTIR has others prior features to conventional IR grating spectrometers: the signal intensity. Because the circular apertures used in FTIR spectrometers have a larger area than the linear slit used in grating spectrometers, the throughput of light can be enhanced considerably. It is especially useful to the far-infrared measurement since the power density of general far-infrared light source is very weak. After data acquisition, we cannot directly read the spectrum information. The digitized, discrete and equidistant interferogram ( )I x must be converted to a spectrum S kv( ) by discrete Fourier transformation (DFT):
1
0
( ) ( ) exp( 2 / )
N
n
S k v − I n x i πnk N
=
⋅Δ =∑ ⋅Δ ,
where (S k⋅ Δ is the magnitude of the spectrum, (v) I n⋅ Δ is the magnitude of x)
Fig. 2.1 Schematics of a Michelson interferometer. S: light source. D: detector. M1:
fixed mirror. M2: movable mirror. X: mirror displacement.
(2.1)
interferogram, Δx is the sampling distance, and Δv is the interval of the
The interferogram (I n⋅ Δ can be reconstructed by inverse discrete Fourier x) transformation (IDFT):
The above is the mathematical fundamental of DFT. Fig. 2.2 shows some examples of Fourier Transform. The final transmittance spectrum can be obtained by three steps: a) an interferogram measured without sample in the optical path is Fourier transformed and generates the single channel reference spectrum R v (referred to Fig. 2.3(a)); b) ( ) an interferogram with a sample in the optical path is measured and Fourier transformed and generate the single channel sample spectrum S v (referred to Fig. ( ) 2.3(b)); c) the final transmittace spectrum T v is defined as ( ) ( )
( ) ( ) T v S v
= R v (referred to Fig. 2.3(c)). To further eliminate the H O2 and CO2 absorptions in THz region of the optical path, we vacuum the chamber for every measurement. Some typical spectrums are shown in Fig. 2.3.
The type of FTIR instrument in our lab is “Bruker IFs66vs”, and the measurement wavenumber range of liquid-He-cooled bolometer is from 50cm to 700-1 cm which -1 is equal to 14μm to 200μm in wavelength.
(2.2)
(2.3)
Fig. 2.2 Examples of spectra (left side) and their corresponding interferograms (right side).
Fig. 2.3 Three kinds of transmission spectra: (a) reference spectrum, (b) spectrum of absorbing sample, (c) transmittance spectrum obtained by dividing (b) by (a).
(a) (b) (c)
Chapter 3 Sample Design and Fabrication
Fig. 3.1 depicts the general profile of the samples fabricated by standard microlithography process. We defined the pattern by photolithography after coating the substrate surface with photoresist and then deposited 20nm-thick titanium for adhesive layer on intrinsic 1cm 1cm× GaAs ( εGaAs =13.7 ) substrate and 200nm-thick gold successively. Finally, a 2D hole array was perforated on the metal by lift-off process. The pattern on the metal was indeed a 2D Bravais lattice. It is known that there are five types of 2D Bravais lattices. Here we chose four types of lattices, which are square, rectangular, oblique, and triangular, to investigate the EOT phenomenon. To investigate the influence of hole shape on the transmission spectra, we varied the hole shape of the array with fixed periodicity. Especially we focused on square array with different hole shapes. As Fig. 3.2(a) shows, for square arrays, we varied the hole widths from 18μm to 3μm and with the hole lengths unchanged. In Fig. 3.2(b) we did the same work but started the shrinking from 14μm. It is shown that both spectra in Fig. 3.2 have non-monotonous redshifts as the aspect ratios of holes are very large. On the other hand, we made the same pattern as Fig. 3.2(b) but with different metal, titanium, of 200nm thickness to see the influence of finite conductivity, as shown in Fig. 3.3. The finite conductivity effect can result in larger loss and enlarge the cutoff wavelength of the holes [26]. In Fig. 3.4, we kept the aspect ratio of the holes unchanged but shrank hole area gradually, and we found that the peak positions of the spectrum blueshift with decreasing full width at half maximum (FWHM). Moreover, we also studied the effect of symmetry difference between hole and lattice. It is known that any Bravais lattice has its unique primitive unit cell which is called Wigner-Seitz cell. A Wigner-Seitz cell has the full symmetry
of the Bravais lattice, i.e., the Wigner-Seitz cell is as symmetrical as the Bravais lattice. Therefore, for each lattice, we defined the hole by Wigner-Seitz cell of the lattice. We found that if we kept the same symmetry but shrank the hole area, the peak position would be unchanged. The results are shown in Fig. 3.5(a)-(d). There is one thing that has to be mentioned: in Fig. 3.5(b), Fig. 3.5(c) and Fig. 3.5(d) the substrate material is changed to intrinsic silicon and the metal we use is 200nm-thick aluminum for economic consideration. The detailed discussion about these measurement results can be postponed until Chapter 5.
Fig. 3.1(a) Top-view of the sample under measurement. The gray region represents the substrate while the yellow region is the metal. (colors)
Fig. 3.1(b) Side-view of the sample under measurement. (colors)
Fig. 3.2(a) Evolution of transmittance with various aspect ratios. d: lattice constant. a:
the length of rectangular hole. b: the width of rectangular hole. (colors)
60 80 100 120 140 160
Fig. 3.3 Evolution of transmittance with low-conductive metal. d: lattice constant. a:
the length of rectangular hole. b: the width of rectangular hole. (colors)
Fig. 3.4 Transmission spectra with fixed aspect ratio of the holes. d: lattice constant. a:
the length of rectangular hole. b: the width of rectangular hole. (colors)
60 80 100 120 140 160
Fig. 3.5 (a) Evolution of transmittance with same symmetry between hole and lattice.
(colors)
Fig. 3.5 (b) Evolution of transmittance with same symmetry between hole and lattice.
50 100 150 200 250
Measurement: Rectangular Lattice
Tr ansmittance
Measurement: Square Lattice
Transmittance
Wavenumber (cm-1)
d22a18b18 d22a14b14 d22a9b9
Fig. 3.5 (c) Evolution of transmittance with same symmetry between hole and lattice.
(colors)
Fig. 3.5 (d) Evolution of transmittance with same symmetry between hole and lattice..
“d” represents the lattice constant. “a” represents the side length of Wigner-Seitz cell of triangular lattice. (colors)
50 100 150 200 250
Measurement: Oblique Lattice
Trans m ittance
Measurement: Triangular Lattice
Wavenumber (cm-1)
Chapter 4 Theoretical Formalism
In order to analyze the experimental data and understand the physics involved in the 2D structure, we have to do the calculation based on modal expansion. The unit system we adapt here is SI units.
First of all, we divide the whole system into three regions which are I, II, and III respectively as shown in Fig. 4.1(b). RegionI is the region of reflection where the EM fields can be expanded by the eigenmodes of Helmholtz’s equations in free space and the incident light is given in this region. RegionII is the structure region where the EM fields inside the hole can be expanded by rectangular waveguide modes of perfect electrical conductor (PEC). RegionIII is the substrate region which can be seen as another kind of free space except the light velocity there has to be divided by refraction index of the substrate material. The 2D structure under study is an infinite array of holes drilled periodically in a metal film of thicknessh. Fig. 4.1(a) depicts the definition of the primitive unit cell. The primitive unit cell is defined as a rectangular with length and width being A, B respectively, while the length and width of the rectangular hole inside the unit cell is denoted by a, b respectively.
We start from the Maxwell’s equations for complex time-harmonic fields in source free case:
( ) iωμ0 ( ) 0
∇×H r − E r =
( ) iωε ε0 ( ) ( ) 0
∇×E r + r H r =
(4.1) (4.2)
0 ( ) ( ) 0 ε ε
∇⋅ r E r = ,
where the related coefficients are the same as the standard notations in any electromagnetic textbook. In the following derivation, the only approximation we make is that the metal is considered to be perfect electrical conductor (PEC). This is a good approximation because our system is operated in THz region, where the skin depth of the metal with good conductivity is about only several tens of nanometer.
Combine (4.1) and (4.2) , we obtain
After further manipulation we can obtain two Helmholtz’s equations,
2
Both electric and magnetic fields satisfy the above two Helmholtz’s equations in each region. At any boundary the EM fields in between the regions satisfy the following boundary conditions,
where the numbers of subscripts mean different regions, n is the unitary vector normal to the surface, and Js and ρs are surface current and surface charge respectively. In the following, we will first attack this problem at each region individually and then match the boundary conditions at the interface of each region.
All EM fields at each region are governed by (4.7) and (4.8) . To simplify the derivation, we learned from Garcia’s paper in 2008 [16] to use Dirac’s notation for representation of each field, or eigenfunction. For example, the electric field can be written as E r( )= r E// E z( ) . The reason why we deliberately separate the z-dependent function from Dirac’s notation will be apparent in our derivation soon. To further simplify the derivation, according to Garcia, for the same mode E-field and H-field have the following relation
mode Ymode mode
− ×z H = ± E ,
where Ymode is the modal admittance. The choice of + or – depends on the propagation direction +z or –z respectively. Consequently we can consider only the eigenmodes of electric field.
Basically, we can categorize the system into two types, the free space type and the inside hole type. In free space, the eigenmodes of (4.7) are plane waves obeying Bragg diffraction law, i.e., k=k0+G, where k0 and k are incident and reflected wavevectors respectively, and 2 (m n )
A B
π
= +
G x y . In the previous expression ( , )m n denoting the diffraction order is a pair of arbitrary integers. Moreover, the plane wave eigenmodes can be further decomposed to two orthogonal functions based on the directions of polarization, p and s. The definitions of p- and s-polarizations are shown
(4.13)
Fig. 4.1(a) Top view of unit cell of the rectangular lattice.
Fig. 4.1(b) Schematics of the system under study.
Fig. 4.1(c) Schematics of the system with incident light.
which can be denoted by TE,pq and TM,pq , where ( , )p q represents a certain order of waveguide mode.
After choosing the expansion basis with consideration of Bloch’s theorem, we can write down the EM fields in different regions:
z Region I, (z<0)
Given a normal incident wave, the EM fields can be expressed as follows,
{ }
(1),00 1mnp mnp mns mns
mn
( )
coefficients to be calculated.z Region III, (z>z2)
Now we write down the real space expression for each eigenmode explicitly:
(4.20)
2 2
There is one important thing that has to be mentioned. For TE waveguide modes,
// TE,pq
surface have to be continuous everywhere on the surface and magnetic fields parallel to the surface have to be continuous on the holes area. Therefore, as Garcia did, we project the matching equations onto plane wave eigenmodes for electric fields and rectangular waveguide eigenmodes for magnetic fields. Besides, we can match the boundary only on one Wigner-Seitz unit cell, i.e., R=0, because our structure has perfect periodicity and thus satisfies Bloch’s theorem. Therefore we can abbreviate TM,pq,R=0 and TE,pq,R=0 to TM,pq and TE,pq respectively. The boundary conditions are matched at two interfaces:
At z=z1 (z1− = −z2 h): area of unit cell to obtain
{ }
an area of a hole to obtain of unit cell to obtain
( )
With the four simultaneous equations (4.36), (4.40), (4.43), and (4.46) , we can (4.47)
[ ]
Thus (4.36), (4.40), (4.43), and (4.46) can be expressed as(4.58)
[ ] [ ] [ ][ ] [ ][ ][ ]
a1 + b1 = M a2 + M E b2[ ] [ ] [ ] [ ]
M † Y1{
a1 − b1}
=[ ] [ ] [ ][ ]
Y2{
a2 − E b2} [ ] [ ] [ ][ ] [ ]
a3 = M { E a2 + b2 }[ ] [ ][ ] [ ] [ ][ ] [ ]
M † Y3 a3 = Y2{
E a2 − b2}
.With (4.53) being given and the four matrix equations (4.59), (4.60), (4.61) and (4.62) , we can determine the four column matrices (4.54), (4.56), (4.57) and (4.58) . The last step is to calculate the transmittance of this system. Because the system is considered to be lossless, the transmittance in each region must be equal. We use Poynting’s theorem to calculate the energy flux through a unit cell at a given z,
( ) ( )
*( )
unit cell
1Re ˆ , , , , d d
J z 2 ⎧ x y z x y z x y⎫
⎡ ⎤
= ⎨ ⋅⎣ × ⎦ ⎬
⎩
∫∫
z Ev Hv ⎭.Finally, the transmittance can be obtained by dividing ( )J z by incoming energy flux
J0.
(4.63) (4.62) (4.61) (4.60) (4.59)
Chapter 5 Simulation Results and Discussions
5.1 Preamble
In Chapter 4, our formalism is based on rectangular (or square) Bravais lattice and rectangular (or square) lattice basis (hole) for simplicity. Therefore, in this chapter, we will compare our simulation results with measurement ones restricted only to rectangular (or square) case. The dielectric constants of the substrate will be chosen to be either εSi =11.9 or εGaAs =13.7, just for matching the measurement conditions.
The real dielectric constants of substrate imply that the substrate is assumed to be lossless material. The incident light in our simulation is set to be 45 -polarized for 0 including the two possible polarizations. Besides, let us recall the two assumptions of calculation presented in Chapter 4. The first assumption is that the metal is a perfect electric conductor, so there is no EM field penetrating into the metal and apparently no surface plasmon polariton effect is considered. This is a good assumption because in our case the light frequency is at THz regime and the skin depth of the field into the metal with good conductivity can be calculated as the following [17],
1
δ 2
= μσ ω,
where μ is the magnetic permeability of the metal, ω is the angular frequency, and σ1 is the real part of conductivity which can be related to imaginary part of dielectric constant ε2 by [18]
( )
21 2
4 σ ωε
= π .
Since ε2 of gold is a very large value (larger than 80000) in THz regime [19], the (5.1)
(5.2)
skin depth can be calculated to be approximately 35nm which is around 1/10000 of the incident wavelength. The second assumption is that the substrate thickness is assumed to be infinite, i.e., we don’t consider the dielectric waveguide effect and interference of three layer system caused by the substrate. This also can be neglected because the accuracy Δk in our measurement is set to be large enough (say, 4cm ) -1 to make the interference due to substrate thickness unresolved in the spectrum. As to dielectric waveguide effect, our definition of transmittance1 can minimize this effect.
1
5.2 Simulation Results Compared with Measurement Results:
EOT Phenomenon
Fig. 5.1 shows the (zero-order) transmittance spectra of both simulation and
Fig. 5.1 shows the (zero-order) transmittance spectra of both simulation and