1. Electromagnetic waves

The electromagnetic spectrum

EM-waves are all around us in our everyday life. Only a small fraction of the wide variety of EM-waves surrounding us is visible to our eyes, which is known as “light” or the visible part of the EM spectrum (figure 1). This visible part of the spectrum consists of photons (the carrier of light, or more general of EM-waves) with a wavelength between 380 and 760 nm. Photons with an arbitrary wavelength outside of this range are not visible to the human eye, but they are very important in technological applications surrounding us. Some examples include the MHz and GHz bands for our mobile phones, Radio frequencies (RF) used for radio and TV broadcasts and X-rays used for medical imaging.

EMspectrum
Figure 1: the electromagnetic spectrum

This wide variety of different EM-waves originates from the same theory, their main difference is the fact that they have different wavelengths (or frequencies) and very different propagation properties in different media. In this thesis we will focus on visible and near infrared (NIR) EM waves for plasmonic metamaterial biosensors.

Basic properties

The most fundamental form of EM-waves are transverse plane waves, which can be easily derived starting from the Maxwell equations (1-4) in an infinite medium in the absence of sources. Here we show the formal derivation of such waves, which are essential in the understanding of the propagation behavior for any other polarization state of an EM-wave.

(1)  \nabla \cdot {\bf B} = 0
(2) \nabla \times \mathbf{{\bf E}} + \frac{\partial \mathbf{{\bf B}}} {\partial t} = 0
(3) \nabla \cdot {\bf D} = 0
(4) \nabla \times \mathbf{{\bf H}} - \frac{\partial \mathbf{D}} {\partial t} = 0

If we assume solutions with a harmonic time dependence e^{-i \omega t}, these equations can be rewritten in order to obtain the field amplitudes.

(5)  \nabla \cdot {\bf B} = 0
(6) \nabla \times \mathbf{{\bf E}} - i \omega {\bf B} = 0
(7) \nabla \cdot {\bf D} = 0
(8) \nabla \times \mathbf{{\bf H}} + i \omega {\bf D} = 0

In uniform isotropic media, these expressions can be reformulated in terms of the electric permittivity \epsilon and the magnetic permeability \mu which describe the relationship between the electric displacement {\bf D}, the magnetic field {\bf H}, the electric field {\bf E} and the magnetic induction {\bf B} (9-10). Both \epsilon and \mu are in general complex valued functions of the angular frequency \omega, but for now we assume that they are real and positive (i.e. no losses).

(9) {\bf D} = \epsilon {\bf E}
(10) {\bf B} = \mu {\bf H}

Plugging equations 9-10 into 5-8 we obtain the equations for {\bf E} and {\bf B}

(11) \nabla \times \mathbf{{\bf E}} - i \omega {\bf B} = 0
(12) \nabla \times \mathbf{{\bf B}} + i \omega \mu \epsilon {\bf E} = 0

By calculating the divergence of equations 11-12, we obtain the Helmholtz wave equation

(13) ( \nabla^{2} + \mu \epsilon \omega^{2}) \begin{Bmatrix} {\bf E} \\ {\bf B} \end{Bmatrix} = 0

If we look for plane waves propagating in the z-direction as possible solutions (~ e^{i k z - i \omega t}), we obtain the relationship between the wave number k and the frequency \omega.

(14) k = \sqrt{\mu \epsilon} \omega

This expression gives rise to the definition of the phase velocity \nu and the refractive index n, which are related by equation 15. The phase velocity describes how any frequency component in a wave propagates, which is determined by the refractive index, that in general is a complex function of the frequency.

(15) \nu = \frac{\omega}{k}=\frac{1}{\sqrt{\mu \epsilon}}=\frac{c}{n}

The refractive index itself can therefore be described in its most general shape

(16) n = n' + i n'' = \sqrt{\frac{\mu}{\mu_{0}}\frac{\epsilon}{\epsilon_{0}}}

For a nondispersive medium (\mu and \epsilon independent of \omega), we can rewrite the possible plane wave solutions of the Maxwell equations (1-4) in their most general shape:

(17) u(z,t) = a e^{i k z - i \omega t} + b e^{-i k z -i \omega t} = a e^{ik ( z - \nu t)} + b e^{-i k ( z + \nu t)}

from which it is clear that we are looking at waves traveling in the positive and negative z-direction with phase velocity \nu. The derivations above also hold in case of dispersive materials, but in that case the shape of the propagating wave changes as it propagates. In order to formally derive the behavior of a plane wave with frequency \omega and wave vector {\bf k}=k {\bf q} we have to verify that the solution in equation 17 is a solution of equations 1-4 and 13. If we look at the real parts of the complex fields {\bf E} and {\bf B}, the solutions can be rewritten to

(18) {\bf E}({\bf r},t) = {\bf E_0} e^{i k {\bf q} \cdot {\bf r} - i \omega t}
(19) {\bf B}({\bf r},t) = {\bf B_0} e^{i k {\bf q} \cdot {\bf r} - i \omega t}

in which {\bf E_0}, {\bf B_0} and {\bf q} are constant vectors. By plugging in these values into equation 13, it follows that {\bf q} should be a unit vector, and the only thing that remains is to fix the vector parameters such that equations 1-4 are satisfied. In that way, we obtain the following conditions from the divergence equations

(20) {\bf q} \cdot {\bf E_0} = 0
(21) {\bf q} \cdot {\bf B_0} = 0

which implies that {\bf E} and {\bf B} are both perpendicular to the propagation direction {\bf q}. From the curl equations we can derive the other constraints that apply to {\bf E} and {\bf B}, which result in the following condition

(22) {\bf B_0} = \sqrt{\mu \epsilon} {\bf q} \times {\bf E_0} = \frac{n}{c} {\bf q} \times {\bf E_0}

from which we can see that c {\bf B} and {\bf E} have the same dimensions and the same magnitude (in free space). Moreover, it is also clear that the electric and magnetic fields are perpendicular to the propagation direction and to each other. Plane waves are often expressed in terms of {\bf E} and {\bf H} and in that case equation 2.22 can be rewritten as follows

(23) {\bf H_0} = {\bf q} \times \frac{{\bf E_0}}{Z}

with Z (equation 24) being the impedance of the medium.

(24) Z = \sqrt{\frac{\mu}{\epsilon}}

In the case n is a real number, the electric and magnetic fields have the same phase and the resulting transversal plane wave propagates as illustrated in figure 2.

EMwave
Figure 2: Schematic representation of a transverse electromagnetic plane wave propagating to the right.

A propagating plane wave transports energy and the time-averaged energy flux can be extracted from the real part of the Poynting vector {\bf S}

(25)  S = \frac{1}{2} {\bf E} \times {\bf H^*}

The resulting time-averaged energy density u is given by equation 26. Note that both electric and magnetic fields contribute to the energy flow, but that these are the same in magnitude for a homogeneous plane wave, such that we write the energy flow as function of electric fields only.

(26) u = \frac{\epsilon}{2} |E_0|^2

So far we only considered the refractive index to be real, but in its most general shape (equation 17), it also has an imaginary part. If we also take this into account, we are looking at inhomogeneous plane waves, which decay (positive values) or increase (negative values) in amplitude as they propagate.

Polarization states

In the previous section we derived the properties of a transverse plane wave, the most fundamental type of EM-wave. Now we want to take a look at other polarization states, which are a superposition of different plane wave states. We start from 2 transverse plane waves which are polarized perpendicular with respect to each other.

(27) {\bf E_1} = {\bf \mathcal{E}_1} E_{1,0} e^{i {\bf k} \cdot {\bf r} - i \omega t}
(28) {\bf B_1} = \sqrt{\mu \epsilon} \frac{{\bf k} \times {\bf E_{1,0}}}{k}
(29) {\bf E_2} = {\bf \mathcal{E}_2} E_{2,0} e^{i {\bf k} \cdot {\bf r} - i \omega t}
(30) {\bf B_2} = \sqrt{\mu \epsilon} \frac{{\bf k} \times {\bf E_{2,0}}}{k}

A linear combination of these two plane waves with electric field direction (polarization) {\bf \mathcal{E}_1} and {\bf \mathcal{E}_2} and complex electric field amplitude E_{1,0} and E_{2,0} is the most general homogeneous plane wave propagating wave vector {\bf k}.

(31)  {\bf E}({\bf r},t) = ({\bf \mathcal{E}_1} E_{1,0} + {\bf \mathcal{E}_2} E_{2,0}) e^{i {\bf k} \cdot {\bf r} - i \omega t}

The real and complex parts of the electric field amplitudes determine the polarization state of the combined EM-wave, which depends both on the magnitude and phase of the field amplitudes of the constituent waves. In the simplest case both waves are in phase, and their superposition will be a homogeneous plane wave (figure 3) of which the polarization angle \theta (with respect to {\bf \mathcal{E}_1}) and the magnitude E are given by equations 32-33.

(32) \theta = \arctan{E_{2,0}/E_{1,0}}
(33) E = \sqrt{{E_{1,0}}^2+{E_{2,0}}^2}

EMlin-lin-lin
Figure 3: Schematic representation of a linearly polarized wave, composed of 2 transverse electromagnetic plane waves which are in phase

When the two constituent plane waves are out of phase, their superposition will in general be elliptically polarized. The simplest example of such a wave consists of the superposition of two orthogonal plane waves with the same magnitude which are out of phase by \pi/2, which results in a circularly polarized wave (figure 4), of which the mathematical description is given by equation 34.

(34) {\bf E}({\bf r},t) = E_0 ({\bf \mathcal{E}_1} \pm i {\bf \mathcal{E}_2}) e^{i {\bf k} \cdot {\bf r} - i \omega t}

EMlin-lin-circ
Figure 4: Schematic representation of a circularly polarized wave, composed of 2 transverse electromagnetic plane waves which are out of phase by \pi / 2

If we define the direction of the unit vectors \bf \mathcal{E}_1 and \bf \mathcal{E}_2 to be along the directions x and y, and the propagation direction to be the z-direction, then the components of the electric fields are obtained by taking the real part of equation 34:

(35) {\bf E_x}({\bf r},t) = E_0 \cos (k z - \omega t)
(36) {\bf E_y}({\bf r},t) = \mp E_0 \sin (k z - \omega t)

From this formula it is clear that at a fixed point in space, the electric field magnitude is constant and rotates at a frequency \omega around the propagation direction. In case we take the upper signs in equations 35 and 36 the wave is left circularly polarized whereas for the lower signs it is right circularly polarized. These two circular polarization states can also be used as a set of basic fields in order to describe any other polarization state. The corresponding unit vectors can be written as follows:

(37) {\bf \mathcal{E}_\pm} = \frac{1}{\sqrt{2}} ({\bf \mathcal{E}_1} \pm i {\bf \mathcal{E}_2})

Using these unit vectors, a homogeneous plane wave (figure 5) can be constructed based on equation 37:

(38) {\bf E}({\bf r},t) = (E_+ {\bf \mathcal{E}_+} + E_- {\bf \mathcal{E}_-}) e^{i {\bf k} \cdot {\bf z} - i \omega t}

in which \mathcal{E}_+ and \mathcal{E}_- are the complex amplitudes, which in this case have equal magnitudes. 

EMcic-circ-lin
Figure 5: Schematic representation of two circularly polarized waves which make up a transverse electromagnetic plane wave

In the more general case where the magnitudes (amplitude and/or phase) of the circularly polarized waves differ, we end up with an elliptically polarized wave (figure 6). The principal axes of the ellipse are given by the vectors {\bf \mathcal{E}_1} and {\bf \mathcal{E}_2}. The ratio between the two axes is |(1+r)/(1-r)| with E_-/E_+ = r. In case there is also a phase difference between the constituent waves, which can be expressed as

(39) E_-/E_+ = re^{i \alpha}

where the rotation angle of the main axis of the ellipse is given by \alpha/2.

EMcic-circ-ellipt
Figure 6: Schematic representation of two circularly polarized waves which make up an elliptically polarized wave

Any elliptical polarization state can also be described as a superposition of two linearly polarized waves (equations 27-30) with different magnitudes for E_{1,0} and E_{2,0} and a phase difference between them, as illustrated in figure 7.

EMlin-lin-ellipt
Figure 7: Schematic representation of an elliptically polarized wave, composed of 2 transverse electromagnetic plane waves which are out of phase by \pi / 4