The Electromagnetic Spectrum and Superposition

So far, we have discussed harmonic plane waves with a specific given wavelength.

Such waves are called monochromatic. The wavelength can be nearly anything, ranging for instance from 10⁻¹³m for gamma rays to 10⁵ m for radio frequen-cies [447]. This range of wavelengths is known as the electromagnetic spectrum and is presented in Figure 2.4.

The range of wavelengths that the human eye is sensitive to is called the visible spectrum and ranges from roughly 380 nm to 780 nm. This range constitutes only a very small portion of the electromagnetic spectrum and coincides with the range of wavelengths that interacts sensibly with materials. Longer wavelengths have lower energies associated with them, and their interaction with materials is usually limited to the formation of heat. At wavelengths shorter than the visible range, the energy is strong enough to ionize atoms and permanently destroy molecules [815].

As such, only waves with wavelengths in the visible range interact with the atomic structure of matter in such a way that it produces color.

So far, our discussion has centered on monochromatic light; the shape of the wave was considered to be sinusoidal. In practice, this is rarely the case. Ac-cording to the Fourier theorem, more complicated waves may be constructed by a superposition of sinusoidal waves of different wavelengths (or, equivalently, of different frequencies). This leads to the superposition principle, which is given here for a dielectric material (compare with (2.59c)):

Ex(z,t) =^{ ∞}

ω=0

E_m,ω⁺ cos(ω^t−βz) + E_m,ω⁻ cos(ω^t+βz)

dω. (2.73) In addition, radiators tend to emit light at many different wavelengths. In Fig-ure 2.5, an example is shown of the wavelength composition of the light emitted by a tungsten radiator heated to 2000 K.

300 350 400 450 500 550 600 650 700 750 800 0.41

0.42 0.43 0.44 0.45 0.46 0.47 0.48

Wavelength λ (nm)

Emissivity

Emissivity of tungsten at 2000 (K)

Figure 2.5.The relative contribution of each wavelength to the light emitted by a tungsten radiator at a temperature of 2000 K [653].

2.3 Polarization

We have already shown that harmonic waves are transversal: both E and H lie in a plane perpendicular to the direction of propagation. This still leaves some degrees of freedom. First, both vectors may be oriented in any direction in this plane (albeit with the caveat that they are orthogonal to one another). Further, the orientation of these vectors may change with time. Third, their magnitude may vary with time. In all, the time-dependent variation of E and H leads to polarization, as we will discuss in this section.

We continue to assume without loss of generality that a harmonic plane wave is traveling along the positive z-axis. This means that the vectors E and H may be decomposed into constituent components in the x- and y-directions:

E=

E_xex+ Eyey

e^{−iβ z}. (2.74)

Here, ex and eyare unit normal vectors along the x- and y-axes. The complex amplitudes E_xand E_yare defined as

E_x= |Ex|e^iϕx, (2.75a)

E_y= |Ey|e^iϕy. (2.75b)

The phase angles are therefore given byϕ^xandϕ^y. For a given point in space z= r·s, as time progresses the orientation and magnitude of the electric field intensity vector E will generally vary. This can be seen by writing Equations (2.75) in their

2.3. Polarization 39 It is now possible to eliminate the component of the phase that is common to both of these equations, i.e.,ω^t−βz, by rewriting them in the following form (using identity (B.7a); see Appendix B):

Ex Repeating this, but now solving for sin(ω^t−βz) and equating the results, we find

Ex

|Ex|sin(ϕy) − Ey

|Ey|sin(ϕx) = cos(ω^t−βz)sin(ϕy−ϕx). (2.79) By squaring and adding these equations, we obtain

 Ex This equation shows that the vector E rotates around the z-axis describing an ellipse. The wave is therefore elliptically polarized. The axes of the ellipse do not need to be aligned with the x- and y-axes, but could be oriented at an angle.

Two special cases exist; the first is when the phase anglesϕxandϕyare sepa-rated by multiples ofπ^:

ϕy−ϕx= mπ (m = 0,±1,±2,...). (2.81)

For integer values of m, the sine operator is 0 and the cosine operator is either+1 or−1 dependent on whether m is even or odd. Therefore, Equation (2.80) reduces

to 

E

B

λ

Wave propagation

Figure 2.6. An electromagnetic wave with wavelengthλ is defined by electric vector E and magnetic induction B, which are both orthogonal to the direction of propagation. In this case, both vectors are linearly polarized.

The general form of this equation is either x²+y²= 2xy or x²+y²= −2xy. We are interested in the ratio between x and y, as this determines the level of eccentricity of the ellipse. We find this as follows:

x²+ y²= 2xy. (2.83a)

By dividing both sides by y²we get x²

y²+ 1 = 2x

y. (2.83b)

Solving for the ratio x/y yields

x

y= 1. (2.83c)

Solving x²+ y²= −2xy in a similar manner yields x/y = −1. We therefore find that the ratio between Exand Eyis

Ex

Ey= (−1)^mE_y

E_x. (2.84)

As such, the ratio between x- and y-components of E are constant for fixed m.

This means that instead of inscribing an ellipse, this vector oscillates along a line.

Thus, when the phase anglesϕ^{x and}ϕ^y are in phase, the electric field intensity vector is linearly polarized, as shown in Figure 2.6. The same is then true for the magnetic vector H.

2.3. Polarization 41

y

x Wave

propagation E

Figure 2.7.For a circularly polarized wave, the electric vector E rotates around the Poynt-ing vector while propagatPoynt-ing. Not shown is the magnetic vector, which also rotates around the Poynting vector while remaining orthogonal to E.

The second special case occurs when the amplitudes|E_x| and |E_y| are equal and the phase angles differ by eitherπ/2 ± 2mπ^or−π/2 ± 2mπ. In this case, (2.80) reduces to

E_x²+ Ey²= ±|Ex|². (2.85)

This is the equation of a circle, and this type of polarization is therefore called circular polarization. Ifϕy−ϕx=π/2 ± 2mπthe wave is called a right-handed circularly polarized wave. Conversely, ifϕ^y−ϕ^x= −π/2 ± 2mπ the wave is called left-handed circularly polarized. In either case, the field vectors inscribe a circle, as shown for E in Figure 2.7.

The causes of polarization include reflection of waves off surfaces or scat-tering by particles suspended in a medium. For instance, sunlight enscat-tering the Earth’s atmosphere undergoes scattering by small particles, which causes the sky to be polarized.

Polarization can also be induced by employing polarization filters. These fil-ters are frequently used in photography to reduce glare from reflecting surfaces.

Such filters create linearly polarized light. As a consequence, a pair of such filters can be stacked such that together they block all light.

This is achieved if the two filters polarize light in orthogonal directions, as shown in the overlapping region of the two sheets in Figure 2.8. If the two filters are aligned, then linearly polarized light will emerge, as if only one filter were present. The amount of light Eetransmitted through the pair of polarizing filters

Figure 2.8. Two sheets of polarizing material are oriented such that together they block light, whereas each single sheet transmits light.

Figure 2.9.A polarizing sheet in front of an LCD screen can be oriented such that all light is blocked.

2.3. Polarization 43

Figure 2.10.A polarizing sheet is oriented such that polarized laser light is transmitted.

is a function of the angleθbetween the two polarizers and the amount of incident light E_e,0:

Ee= Ee,0cos²(θ). (2.86)

This relation is known as the Law of Malus [447, 730, 1128].

The same effect is achieved by placing a single polarizing filter in front of an LCD screen, as shown in Figure 2.9 (see also Section 14.2). Here, a backlight emits non-polarized light, which is first linearly polarized in one direction. Then the intensity of each pixel is adjusted by means of a second variable polarization in the orthogonal direction. Thus, the light that is transmitted through this sequence of filters is linearly polarized. As demonstrated in Figure 2.9, placing one further polarizing filter in front of the screen thus blocks the remainder of the light.

In addition, laser light is polarized. This can be shown by using a single polarization filter to block laser light. In Figure 2.10, a sheet of polarizing material is placed in the path of a laser. The sheet is oriented such that most of the light is transmitted. Some of the light is also reflected. By changing the orientation of the sheet, the light can be blocked, as shown in Figure 2.11. Here, only the reflecting component remains.

Polarization is extensively used in photography in the form of filters that can be attached to the camera. This procedure allows unwanted reflections to be

re-Figure 2.11.A polarizing sheet is oriented such that polarized laser light is blocked.

Figure 2.12.The LCD screen emits polarized white light, which undergoes further polar-ization upon reflection dependent on the amount of stress in the reflective object. Thus, the colorization of the polarized reflected light follows the stress patterns in the material.