Surface Plasmon Polaritons in One-Dimensional Surface Relief Gratings
Electromagnetic Theory
Jimmy Zhan
Dr. Thomas Krause
December, 2013
Abstract
Surface plasmons (SP) are quanta of plasma oscillations that are restricted to and propagate along the interface between two media (usually a metal and a dielectric). Surface plasmons polaritons (SPP) are those surface plasmons that are additionally coupled with some external electromagnetic wave. The propagation of the SP waves, its dispersion relations, and its coupling properties were studied in this project. A brief overview of the underlying concepts of surface wave propagation, dispersion relations, and surface plasmonic crystals were first given. Then a rigorous derivation of key quantities of interests were conducted. The characteristic SP tangential propagation length and normal penetration depths were found. The dispersion relation of SPs were derived using the Drude model for the permittivity of metals. Finally the coupling conditions for a SP with an incident radiation that has its tangential wave vector boosted by a surface relief grating was determined. Common applications involving surface plasmon polaritons were also given, and some applications were described further in detail.
Table of Contents
Surface Plasmon Polariton Excitation Conditions
Introduction
A plasmon is a quasi-particle representing a quantum of plasma, or in other words they are electron density oscillations. When plasmons are coupled to an electromagnetic wave, they are known as plasmon polaritons (PP). Another term that is often used is surface plasmons (SP). The prefix “surface” signifies that they are plasmons that travel on the surface of a medium. The lack of the suffix “polariton” signifies that they are not coupled with an electromagnetic wave.
Combining the prefix and the suffix, surface plasmon polaritons (SPP) are quasi-particles representing quanta of plasma oscillations, that are confined to - and propagate along - a two-dimensional surface, and that they are coupled with an external electromagnetic wave. In most cases, they are quasi two-dimensional electromagnetic waves propagating along the interface between a metal and a dielectric. However the two media need not be metal and dielectric, any two media such that the real part of the dielectric function changes sign across the interface would support surface plasmon propagation [1][2].
Overview of Theories
Notation on Polarization
To understand the definition of polarizations, one must understand the plane of incidence. The plane of incidence is defined as the plane formed by the normal of the surface of incidence, and the incident wave vector. Note that the wave vector is perpendicular to both the E and B fields, as it is the direction of wave propagation. 
. A transverse magnetic (TM) wave is an EM wave such that its magnetic field is perpendicular (transverse) to the plane of incidence. Likewise, a TE wave is when the E-field is perpendicular to the plane of the incidence. Note since that an EM wave can be described using either one of its two fields since the other field is simply a phase shift. Often people choose to describe the EM wave using its E field, this gives rise to the term p (parallel) and s (senkrecht) polarizations. Note that if the E-field is parallel to the POI, then the B-field is necessarily transverse to it. Similarly, if the E-field is senkrecht (perpendicular) to the POI, then the B-field is necessarily parallel to it. Therefore, under the assumption that one uses the E-field to represent the EM wave, then s polarization is equivalent to TE, and p polarization is equivalent to TM.
Propagation and Skin Depth
Consider a dielectric-metal interface, and an incident TM polarized light, shown in Figure 1.
Figure 1. A TM polarized light incident upon a metal-dielectric interface at 
, showing surface plasmon propagating along the 
axis, and evanescently decaying along the 
direction [1].
The incident light is TM polarized, with wave vector 
. (The plane of incidence is the plane formed by the normal of the interface and the incident wave vector). Due to the evanescent nature of surface plasmons, the SP waves are strong only near the surface, in this case the plane, and decays exponentially into either adjoining media. The SP waves penetrate much further into the dielectric than the metallic side (as shown in Figure 1 and Figure 2). On the metallic side, at low frequencies, the penetration depth is comparable to that of the characteristic skin depth [2]. Therefore, the surface plasmon waves have their maximum at the surface. This is shown in Fig 2.
Figure 2. Surface plasmon waves penetration depth into the dielectric (top) and metallic (bottom) media. Note that the axis here is not the same as Fig. 1 [1].
As the surface plasmon polariton propagates along the surface, it loses energy to the metal due to absorption, and loses energy to free-space radiation via defects on the dielectric side (scattering). This is the reason that surface plasmon polaritons are evanescent. This unique property makes surface plasmons very sensitive to perturbations on the surface, and thus SPP are often used to probe inhomogeneities of a surface.
Dispersion Relation
At low frequencies, the surface plasmon polariton dispersion approaches that of free-space electromagnetic wave (photons), the 
light line. As the incident wave vector (and frequency) increases, the dispersion curve bends over and approaches an asymptotic limit, known as the surface plasma frequency. This is shown in Figure 3.
Figure 3. The dispersion relations of a one-dimensional surface plasmonic structure, showing the light line (blue) and the surface plasmon dispersion (red). The surface plasma frequency is the dashed line [3].
From this plot, one can make two conclusions. The surface plasmon frequency never exceeds the bulk plasma frequency [3],
(1)
and the dispersion curve of the surface plasmon never intersects with the light line. This implies that surface plasmons should never be able to couple to free-space radiation, which leads to the next topic, excitation.
Free-Space Radiation Coupling
In general, SPP can be excited by both electrons and photons, the latter of which will be the focus of this project. In the case of electron excitation, free electrons are incident on a bulk metal and as they scatter, some energy is transferred into the bulk plasma. The component of the scattering vector parallel to the surface results in surface plasmon polariton [3].
In the case of photons, excitation depends on the surface structures. As shown in Figure 3, free-space radiation cannot directly couple with surface plasmons on a smooth surface (radiation cannot excite SP, and SP cannot lose energy to radiation), because at any given wave vector, they have different frequencies. As seen in Figure 3, the free-space radiation wave vector is always smaller than the SP wave vector, therefore one must boost the incident radiation wave vector via some mechanism [2]. There are generally two methods to couple the incident wave vector to SP. One method is using a prism, proposed by Kretschmann and Otto (shown in Figure 4), and the second method is using a surface relief grating, or any periodic lattice structure on the surface. This is shown in Figure 5. The second method will be the focus of this project.
Figure 4. The prism coupling method, to couple incident radiation to SP [3].
Figure 5. The grating coupling method [3].
Surface Relief Gratings
Surface plasmonic crystals support the coupling and propagation of surface plasmon polaritons. In most cases, they are a periodic lattice. The periodicity can be in one or two dimensions [4]. This project will focus on one-dimensional surface plasmonic crystals, which are simply ordinary surface relief gratings.
Analytical Solutions
In all the following derivations, bold letters will be used to denote vectors, and normal letters scalars. Complex numbers will be denoted by a tilde above the letter.
Fields and Propagation
Knowing that the surface plasmons are evanescent waves [1][2] that propagate only along the surface, one can make the conclusion that the at least one component of the wave vector (the normal component) is imaginary and the other component (tangential or in-plane component) of the wave vector can be either real or complex. Using the same coordinate axis as shown in Figure 1, the wave will exponentially decay towards 
.
(2)
The fields in the dielectric are thus given by [1],
(3)
(4)
and the fields within the metal are given by [1],
(5)
(6)
Dispersion Relations
In mathematical terms, the dispersion relation is a function that relates the wave number (magnitude of the wave vector) to the angular frequency of the wave, such that 
. In the case of free-space radiation (photons), the relationship is simply a straight line [4],
(7)
And in general [5],
(8)
(9)
The dispersion relation for surface plasmons will be slightly more complicated. Firstly since the SP decays rapidly in the normal directions, only the tangential (in-plane) component of the wave vector are of concern. Following the same coordinate notations, the dispersion becomes,
(10)
Using Maxwell’s curl equation (Ampere’s Law), assuming no free current and no dipoles [5],
(11)
Where 
denotes either medium.
Since the SPs are evanescent waves, the fields do indeed extend some length into the adjacent media, one can make the assumption that the metal is a non-perfect conductor, since in a perfect conductor no field can exist [5]. The boundary conditions stating the continuity and discontinuity of the tangential and normal components of the fields at are given by [1],
(12)
(13)
(14)
Using equation 11 in equations 3 to 6, one obtains
(15)
Separating the equation into components, for both the dielectric and metal media,
(16)
(17)
(18)
(19)
Applying boundary conditions given by equations 12 and 13 to equations 16 and 17,
(20)
And applying the boundary conditions given by equations 13 and 14 to equations 18 and 19,
(21)
Therefore, the normal components of the wave vector are discontinuous by a factor of the permittivity of the media, whereas the tangential components of the wave vector are continuous. Noting the that overall wave number is,
(22)
Isolating for 
and plugging into (20), and assuming the media are non-magnetic, then solving for 
, one obtains (see Appendix A),
(23)
Where the subscript 
denotes the relative permittivity. Note that for magnetic materials, there is an extra 
in the numerator inside the square root. Note that this derivation for the dispersion relation assumes a flat surface. A flat surface will allow for the propagation of surface plasmons, but cannot couple SPs to electromagnetic waves. Therefore this derivation is only an approximation, which is good for small surface modulations (grating depth << SP propagation length along the interface)[2]. However, the dispersion relations for a non-flat surface can also be derived, commonly using the Rigorous Coupled Wave Analysis (Fourier Modal Method) or the Rayleigh Scattering Hypothesis. These topics are complicated and require separate projects to describe. For the normal components of the wave vector,
(24)
(25)
Since the permittivity of the metal is a complex number, such that,
(26)
The wave numbers will also have a both a real and imaginary part. The real part 
represents SP propagation, and it gives information regarding the wavelength and the speed of the wave [5]. The imaginary part 
represents attenuation, and will give the skin depth [5]. Writing out the real portion of the wave vectors explicitly,
(27)
(28)
(29)
As stated before, in order for an evanescent wave to exist, one component of the tangential wave vector must be real, and all components of the normal wave vector must be purely imaginary. The wave will decay in the normal direction, and propagate as well as attenuate in the tangential direction.
(30)
(31)
Therefore, 
must be purely real, and 
must be purely imaginary. Overall, is complex, and is imaginary. Looking at equations 27 and 28, one can make the conclusion that,
(32)
(33)
A negative real permittivity means that the material attenuates EM waves. (It reaffirms the hypothesis made earlier that one of the media is a metal). The SP wavelength 
can be found via , and the SP propagation length 
(SP penetration length in the tangential direction) can be found via 
. The SP propagation length (skin depth) into the adjacent media 
and 
can be found via , using the general equations [5],
(34)
(35-a)
(35-b)
The dispersion relation of SPs is given in terms of the permittivities of the dielectric and the metal. The permittivity of dielectric is a constant for all frequencies, however the permittivity of the metal is a function of frequency [5]. Therefore, Drude model for the permittivity of metal will be used.
At optical frequencies, the real portion of the relative permittivity of a metal, at frequency 
, given by the Drude model is [4],
(36)
Where 
is the characteristic plasma frequency of the metal.
A plot of the permittivity of both the dielectric and the metal (Drude model) is shown in Figure below.
Figure 6. The dielectric and metal relative permittivities, as functions of angular frequency [1].
Now, in the infrared and visible spectra, the optical frequency will generally be less than the plasma frequency . This means that the 
will be a negative number, satisfying equation 32. However, as the optical frequency increases (towards ultraviolet), there will come a point when
(37)
And going beyond this frequency, equation 33 will no longer be satisfied. Thus there is a maximum SP frequency. By rearranging equation 37 one obtains,
(38)
Using the Drude model in the dispersion relation, and solving for as a function of in equation 27, one can then obtain a plot of the dispersion relation,
Figure 7. The surface plasmon dispersion curve. Note that at lower frequencies, it approaches a light line, and at higher wave numbers the dispersion curve reaches an asymptote. Also note that the curve is symmetrical about the axis [1].
Looking at the upper and lower limits,
(39)
Or
(40)
Which is a straight line.
At higher frequencies,
(41)
Or
(42)
In words, at lower frequencies and wave numbers the SP dispersion relation approaches that of a free-space electromagnetic wave, while at higher wave numbers the dispersion curve bends over and asymptotically approaches a maximum frequency.
Surface Plasmon Polariton Excitation Conditions
As mentioned before, since the dispersion curve of the surface plasmon never intersects the free-space photon dispersion, it is impossible to couple free-space radiation into surface plasmon polaritons on a smooth surface [6]. Therefore, a surface plasmonic crystal is required to boost the tangential component of the incident wave vector so that it can couple with SPs. In the simplest case, a one-dimensional surface relief grating would work [7]. Light diffracted from a grating will acquire a boost to its wave vector component along the direction of the grating vector [7]. The grating vector is more generally known as the reciprocal lattice vector [8][9]. Consider a one dimensional Bravais lattice (such as a grating) that is uniform in the other two directions. The reciprocal lattice vector is given by [4][8],
(43)
Where 
is the lattice period, and 
is any integer corresponding to an order of diffraction. Note that the RL and BL vectors point in the same direction [8]. Now consider an incident light and a grating oriented in such a way that they line up (for example, along the direction). The incident light makes an angle 
with respect to the normal to the surface. Therefore, the magnitude of the in-plane (tangential) component of the incident light wave vector has contribution from both the wave vector component from the reflected wave along the grating vector, as well as the boost acquired from the RL vector itself. Mathematically, this can be expressed as [1][2][5][7],
(44)
Where
(45)
Where 
is the grating period, and is an integer.
Equation 27 can be rewritten as,
(46)
Since [5],
(47)
(48)
To find the coupling conditions, one must equate 
with 
, therefore,
(49)
Which can be rearranged to obtain,
(50)
This equation gives the wavelength of free-space radiation (obviously only the positive wavelength makes sense) that can be coupled into surface plasmon polariton, for a given set of the dielectric permittivity, the grating period, the real part of the metallic permittivity, and the incident angle.
A plot showing the boost in incident wave vector, and subsequent coupling of the incident wave to surface plasmon polariton shown below,
Figure 8. The SPP excitation conditions [1].
Applications
Surface plasmon has a wide range of applications [3][10]. Plasmonic-based circuits have been proposed as an alternative to existing electronic and photonic circuits [11]. It is able to operate at a higher frequency than electronics, which are limited by Joule heating. They are also able to have much smaller device dimensions than photonics, which have a size limit due to diffraction [12]. This is schematically shown in Figure 9.
Figure 9. Motivation for plasmonic circuits [13].
Another application of surface plasmons is surface sensing [14]. As shown before, since the surface plasmon polaritons are evanescent waves that propagate along the interface between two media (often a metal and a dielectric), they are extremely sensitive to slight perturbations and inhomogeneities within the skin depth on the surface [15]. Specifically, the resonant condition (when the incident optical frequency matches the natural frequency of surface plasmons oscillating with the positive nuclei) are used for surface sensing. Typical samples that can be probed range from metallic surfaces, to proteins and organic molecules [15]. An incident light is directed towards a surface, any slight changes on the surface, for example from the adsorption of a type of protein, or a crack on the surface of a metal, changes the resonant peaks of the reflected beam. See Figure 10. It is also effective for measuring slight changes to thickness and density of a smooth metallic surface. SPR enhanced spectroscopy forms the basis of many biosensors.
Figure 10. Surface plasmon resonance enhanced spectroscopy, measuring molecular adsorption of proteins. The resonant peaks shift as different molecules are adsorbed onto the surface [3].
Yet another application of surface plasmons is in organic light emitting diodes (OLED), and analogously organic solar cells (OSC). In the OLED, surface plasmons at a metallic cathode layer are created by electrically excited excitons, and they radiate out as light via a grating that is attached to the metallic cathode. The working mechanism of the OSC is nearly opposite to that of the OLED. An incident light whose polarization direction matches that of a grating which is attached to a metallic cathode gains an additional tangential wave vector, and become coupled to a surface plasmon polariton. These SPPs in turn excite excitons, which are separated and collected at the electrodes [1][2][7].
In the case of surface plasmon enhanced OSC, one can take advantage of surface plasmon resonance. Using nanoparticles or nanostructures (such as surface relief gratings), localized surface plasmon oscillations give rise to intense absorption at particular frequencies (resonance) [1][2][7]. Nanoparticles of noble metals exhibit strong absorption in the ultraviolet and visible spectra that are not present in bulk metals. This strong absorption thus in turn increases the light collecting efficiencies of solar cells [16]. Experimental research has shown that SPR enhanced OSC (using 
thick layer of bulk heterojunction P3HT-PCBM as active layer, and aluminum and gold as electrodes) achieves more than 
increase in photocurrent, for a given polarization [7].
Figure 11. Left: AFM view of a surface relief grating, with 
. Right: A cut-away view of the solar cell, showing various layers [7].
Figure 12. The TE and TM polarization photocurrent as a function of wavelength, with no grating [7].
Figure 13. The photocurrents, when there is a grating of . As shown, compared with the case without gratings, photocurrents in the TM incidence increased by nearly [7].
Evidently, only the polarization of light with parallel (to the grating) component of the electric field shows increased current. This is expected as it was shown earlier that the grating will only provide a boost to the tangential wave vector component that is in the same direction as the reciprocal lattice vector. Therefore, to reap the full benefit of plasmonic enhanced solar cells, two-dimensionally periodic surface plasmonic structures (such as crossed gratings) must be used, in order to couple with both TE and TM polarized light [2].
There are a myriad of other applications for surface plasmons, these range from surface plasmon resonance imaging (SPRI), SPR immunoassay, data interpretation, creating photonic metamaterials, and building blocks for electro-optic plasmonic modulators [10][12][14][17][18].
Conclusions
Surface plasmon polaritons are those plasmons (which are quanta of plasma, or more specifically, coherent and collective oscillations of electron densities) that are coupled with an incident electromagnetic wave, and that travel along the interface of a metal and a dielectric. This project explored the theories and concepts behind surface plasmons, their evanescent nature and propagation characteristics, dispersion relations, and methods of coupling with free-space electromagnetic waves, as well as various applications involving surface plasmons. Firstly due to its evanescent nature, the normal component of the SP wave vector is purely imaginary, and the tangential component is complex. Thus the SP will decay exponentially into both adjacent media while propagating and attenuating along the surface. Secondly, the dispersion relations of surface plasmons were derived following the Drude model. Noting that at lower frequencies the SP dispersion curve approaches that of free-space photons, while at higher wave numbers the dispersion curve bends over asymptotically approaches a maximum frequency. This frequency is known as the surface plasma frequency, which is always lower than the bulk plasma frequency. In order for surface plasmons to exist, the real part of the metallic permittivity must be less than the negative of the dielectric permittivity, and its magnitude must be greater than that of the dielectric permittivity. Also the SP dispersion curve never intersects the free-space electromagnetic wave dispersion (the SP wave vector is always bigger), leading to the third result. A surface plasmonic structure must be used in order to couple free-space radiation with SPs. Any periodic (in one or two dimensions) structure can be used. The structure functions to boost the tangential component of the incident wave vector by an amount equaling the reciprocal lattice vector. However, the direction (and thus polarization) of the incident radiation must be matched with the reciprocal lattice vector direction. Finally, there are many applications of surface plasmons. Several applications were discussed in depth, including plasmonic circuits, surface sensing, and surface plasmon resonance OSC.
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Appendix A
MATLAB code for derivation of KSP (23), as derived from 20 and 22.
%equation 22
%kx^2 + kzd^2 = ed * (w/c)^2
%kx^2 + kzm^2 = em * (w/c)^2
%equation 20
%kzd/ed +- kzm/em = 0
%isolating for kzd and kzm from eq (22) and putting into eq (20), then solve
%for kx
syms ed em kx w c
solve(((ed*(w/c)^2)-kx^2)^(1/2)/ed+((em*(w/c)^2)-kx^2)^(1/2)/em==0,kx)
%note that in eq (20) it can be either + or -. Same answer
solve(((ed*(w/c)^2)-kx^2)^(1/2)/ed-((em*(w/c)^2)-kx^2)^(1/2)/em==0,kx)
%for magnetic materials (assuming both materials have the same permeability)
%eq (22) becomes
%kx^2 + kzd^2 = mu*ed * (w/c)^2
%kx^2 + kzm^2 = mu* em * (w/c)^2
syms mu
solve(((mu*ed*(w/c)^2)-kx^2)^(1/2)/ed-((mu*em*(w/c)^2)-kx^2)^(1/2)/em==0,kx)
%if two materials have different permeabilities, mud and mum
syms mud mum
solve(((mud*ed*(w/c)^2)-kx^2)^(1/2)/ed-((mum*em*(w/c)^2)-kx^2)^(1/2)/em==0,kx)