Biophotonics_Concepts_to_Applications

2.2.1 Linear Polarization

Using Eq. (2.2), a train of electric or magneticfield waves designated byAcan be
represented in the general form

ArðÞ¼;t eiA 0 exp½iðxtkrފ ð 2 : 7 Þ

where r=xex+yey+zez represents a general position vector, k=kxex+
kyey+kzezis the wave propagation vector,ejis a unit vector lying parallel to an
axis designated by j (where j = x, y, or z), and kjis the magnitude of the wave
vector along the j axis. The parameter A 0 is the maximum amplitude of the wave
andω=2πν, whereνis the frequency of the light.
The components of the actual (measurable) electromagneticfield are obtained by
taking the real part of Eq. (2.7). For example, ifk=kez, and ifAdenotes the
electricfieldEwith the coordinate axes chosen such thatei=ex, then the mea-
surable electricfield is

ExðÞ¼z;t ReðÞ¼E exE0xcosðxtkzÞ¼exExðÞz ð 2 : 8 Þ

which represents a plane wave that varies harmonically as it travels in the z
direction. Here E0xis the maximum wave amplitude along the x axis and ExðÞ¼z
E0xcosðxtkzÞis the amplitude at a given value of z in the xz plane. The reason
for using the exponential form is that it is more easily handled mathematically than
equivalent expressions given in terms of sine and cosine. In addition, the rationale
for using harmonic functions is that any waveform can be expressed in terms of
sinusoidal waves using Fourier techniques.
The plane wave example given by Eq. (2.8) has its electricfield vector always
pointing in theexdirection, so it is linearly polarized with polarization vectorex.
A general state of polarization is described by considering another linearly polar-
ized wave that is independent of thefirst wave and orthogonal to it. Let this wave be

EyðÞ¼z;t eyE0ycosðxtkzþdÞ¼eyEyðÞz ð 2 : 9 Þ

whereδis the relative phase difference between the waves. Similar to Eq. (2.8), E0y
is the maximum amplitude of the wave along the y axis and EyðÞ¼z E0ycosðxt
kzþdÞis the amplitude at a given value of z in the yz plane. The resultant wave is

EðÞ¼z;t ExðÞþz;t EyðÞz;t ð 2 : 10 Þ

Ifδis zero or an integer multiple of 2π, the waves are in phase. Equation (2.10)
is then also a linearly polarized wave with a polarization vector making an angle

h¼arc sin

E0y

E0x

ð 2 : 11 Þ

2.2 Polarization 31