write-up

Surface plasmon resonance is sensitive mechanism for observing slight changes near the surface of a dielectric-metal interface. It is commonly used today for discovering the ways in which proteins interact with their environment, and are valuable in the production of new pharmaceuticals. The phenomenon of surface plasmon resonance has been known for the last century, but has matured to become a state-of-the-art technology within the last 30 years. We will discuss them simplest configuration supporting surface plasmons, the interface of a semiinfinite metal with a semi-infinite dielectric. By taking the curl of the curl of the electric field, (or similarly, of the H field) one can obtain the wave equation describing the propagation of light: ∇ E− 2

ε∂ 2 E c 2 ∂t 2

=0

(1)

By assuming a harmonic time dependence, we can rewrite the E field as E=E(r)e-iwt. Substituting this back into (1), we obtain the Helmholtz equation: ∇ 2 E(r ) + ko 2 εE(r ) = 0

(2)

We now want to investigate how this oscillating electric field can behave at a metaldielectric interface. In order to do this, we first define our propagation geometry, as shown in Figure 1.

Figure 1

It is then apparent that ε is a function of z, with z = 0 defining the surface at the interface. We set the direction of propagation on the surface in the x direction and assume there is no variation along the y direction. On the surface, the wave can now be written as: E( r ) = E( z ) e iβ x

(3)

β is then the wave vector of the surface wave. We will later show that β is related to the dielectric constants of the surrounding media and thus to the properties of the free electrons of the metal. Substituting (3) back into the Helmholtz equation yields the following expression, which gives us the general form of our surface wave: ∂ 2 E( z ) + (ko 2 ε − β 2 )E( z )e iβx = 0 2 ∂z

(4)

In order to discover the manner in which the field depends on z, we need to find expressions for each component of the electric (and H) field. This is done by taking the curl of E and H as given by Maxwell’s equations, leaving a total of six equations which can be reduced and separated into two sets of solutions corresponding to TE and TM polarization. For TM polarization, only the Hy, Ex, and Ez components of the fields remain, resulting in the following defining equations:

Ex = −i Ez = −

∂Hy ωεoε ∂z

(5)

β Hy ωεoε

(6)

1

∂ 2 Hy 2 + (k o ε − β 2 ) Hy = 0 2 ∂z

(7)

For TE polarization, only the Ey, Hx, and Hz components of the fields remain and the following equations are obtained:

Hx = i

1 ∂Ey ωμo ∂z

(8)

Hz =

β Ey ωμo

(9)

∂ 2 Ey 2 + (k o ε − β 2 ) Ey = 0 2 ∂z

(10)

W can now assume an exponentially decaying solution, so that the field is confined to the surface. For TM modes, for example, we have that: Hy ( z ) = A2e iβx e − k 2 z

(11)

Hy ( z ) = A1e iβx e k 1 z

(12)

for z > 0 and:

for z < 0. The corresponding electric fields can be found by substituting (11) and (12) into (5) and (6). In order to solve for Ai, β, and ki, we apply boundary conditions. From Maxwell’s equations, we know that the tangential components of the H and E field must be continuous. By equating the Hy in (11) and (12), we find that A1=A2. Now equating Ex in both regions leads to the following relation: k2 ε2 =− k1 ε1

(13)

It is important to note here the negative sign on the right side of the expression. Since the wave vectors in the two media have the same sign, it is clear that the dielectric constant of media 1 and media 2 must be opposite. Thus, surface plasmons can only exist at the interface of a conductor and insulator. Now, we would like to find an equation relating β to ε1 and ε2. Plugging equations (11) and (12) into (7) yields the following expressions:

k2 k1

2

2

= β 2 − ko ε 2

(14)

= β 2 − ko ε1

(15)

2

2

Putting (14) and (15) into (13) leads to our expression for β:

β = ko

ε 1ε 2 ε1 + ε 2

(16)

Having discovered the possibility of a surface wave excited by TM polarized light, we now consider the possibility of surface waves excited by TE polarization. If we assume that Ey = A1eiβxe-k1z for z > 0 and Ey = A1eiβxe-k2z for z < 0, as before, and equate Ey in both regions and Hx in both regions, we find that we must have: k 2 = − k1

(17)

As noted before, k1 and k2 must have the same sign, and therefore we see that it is impossible to satisfy the boundary conditions and thus surface plasmons cannot be excited using TE polarized light. We can now take a closer look at the properties of surface plasmons by examining (16). From the Drude model, we know that we can write the dielectric constant of our metal (assuming negligible damping) as:

ε1 = 1 −

ωp2 ω2

(18)

Where ωp is the plasma frequency of the free electrons and ω is the frequency of the incident light (and thus of the surface wave). Plotting ω as a function of β allows us to see explicitly how the surface plasmon frequency depends on the wavevector, as shown in Figure 2.

Figure 2 We can see that as β becomes larger, the frequency asymptotically approaches a maximum value. In order to find this maximum frequency, we set β = ∞, and thus ε1 + ε2 = 0. Using (18), we find:

ωsp =

ωp 1+ ε 2

(19)

Where ωsp is the cutoff frequency. It is thus impossible to excite surface plasmon resonance with a frequency higher than ωsp will be futile. We can discover another interesting property of surface plasmons from Figure 2. Examination of the light line of air and the dispersion of a surface plasmon excited at a metal-air interface, we see that the two never cross. Thus it is impossible to excite a plasmon by simply shining light incident from air on a smooth interface. One alternative is to use prism coupling. Since glass has a higher refractive index than air, its light line will have a smaller slope, and thus will intersect with the plot of the surface Plasmon in air at some β. However, prism coupling requires the use of a thin metallic film rather than a semi-infinite metal, leading to a more complex situation than discussed here. Grating coupling is another possibility, in which the x component of ko can be made to match β by patterning the surface of the metal with shallow gratings.

In summary, we have shown that a surface wave of oscillating free electrons driven by an incident electromagnetic wave can exist at the interface between a metal and a dielectric. Practically, either prism coupling to a thin metallic film or the use of a grating is required to achieve resonance; however, the ideas introduced here provide a basis for further calculations.

Bibliography Wangsness, Roald K. Electromagnetic Fields. John Wiley and Sons Inc, 1979, 1986. Print Maier, Stefan A. Plasmonics Fundamentals and Applications. Springer Science+Buisness Media LLC, 2007. Print

Surface Plasmon Polaritons at Metal-Dielectric Interfaces Courtney Byard Physics 545