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## Wave Equations

### Electromagnetic Wave Polarization and Photon Spin  For coherent EM wave propagating along the +z axis (Figure 07a), the two transversal electric fields can be expressed as :
Ex = Ex0 cos( t - kz + x) ---------- (53a)
Ey = Ey0 cos( t - kz + y) ---------- (53b)
where Ex0, Ey0 are amplitudes, and x, y are the phase angles of the x, y components respectively (Figure 07b). These equations can be simplified somewhat if the view is facing the x-y plane fixed at z=0 (such as the view in Figure 07c), and by using the relative phase angle = x - y :

#### Figure 07b Phase Angles [view large image]

Ex = Ex0 cos( t) ---------- (53c)
Ey = Ey0 cos( t - ) ---------- (53d)
By defining an angle in term of the ratio of the two components (Figure 07b) : = tan-1(Ey/Ex) = tan-1{(Ey0/Ex0)[cos( ) + tan( t)sin( )]} ---------- (53e)
the type of polarization can be readily discerned by the values of Ex0, Ey0, and as shown in Table 01 below.

Phase Angle tan( ) Polarization
0o 0 0o Linear along x-axis
0o 90o Linear along y-axis
0o 1 45o Linear at 45o
45o (1/2)1/2(Ey0/Ex0)[1 + tan( t)] Right-handed rotation Elliptical or circular
90o (Ey0/Ex0)[tan( t)] Right-handed rotation Elliptical
90o tan( t) t Right-handed Circular for Ey0=Ex0
180o 0 0o Linear along x-axis
180o - -90o Linear along y-axis
180o -1 -45o Linear at -45o
-90o -(Ey0/Ex0)[tan( t)] Left-handed rotation Elliptical
-90o -tan( t) - t Left-handed Circular for Ey0=Ex0

#### Table 01 Types of Polarization (from formula) Table 01 shows that for linear polarization the angle is constant with Ex and Ey varying in such a way that its ratio is still a constant. For circular polarization the length of the field |E| is constant, but changes with a frequency determined by . For elliptical polarization even |E| is not a constant. Another way to check out the polarization is to view the animation provided by Amanogawa in Figure 07c. Table 02 shows the types of polarization associated with different sets of parameters in the application.

#### Figure 07c Polarization of EM Wave [see animation]

Amplitude x Phase x Amplitude y Phase y Polarization
1.0 0o 1.0 0o Linear at 45o
1.0 0o 0.0 0o Linear along x-axis
0.0 0o 1.0 0o Linear along y-axis
1.0 45o 0.0 0o Linear along x-axis
1.0 45o 1.0 0o Right-handed elliptical
0.5 0o 1.0 0o Linear at 63.4o
0.5 30o 1.0 30o Linear at 63.4o
1.0 0o 1.0 90o Left-handed circular
0.2 0o 1.0 90o Left-handed elliptical

#### Table 02 Types of Polarization (from animation)

Linear polarization is characterized by the relative phase angle = 0, the direction of oscillation depends on the ratio Ey0/Ex0. Let's take Ey0=Ex0 to simplify the formulas, then the polarization vector can be expressed as :
|e> = cos |ex> + sin |ey> ---------- (53f)
where |ex> = and |ey> = are the unit vector in the x (upper component) and y (lower component) directions respectively. It is similar to the Jones vector in classical electrodynamics and the polarization vectors in Eq.(52). Since the unit vectors satisfies the orthogonality relation : < ei|ej> = (i-j) for (i, j) = (x or y), < e|e> = 1. If it is interpreted as the total probability, then cos2 and sin2 can be interpreted as the probability in polarization state |ex> or |ey> respectively.

For = 90o or -90o, = + t (right-handed circular polarization) or - t (left-handed circular polarization). The polarization vector in Eq.(53f) can be expressed in the form :
|e> = (e-i t/ ) |eL> + (ei t/ ) |eR> ---------- (53g)
where
|eL> = (|ex> - i|ey>)/ = / ---------- (53h)
|eR> = (|ex> + i|ey>)/ = / ---------- (53i)
are the unit vectors for circular polarization to the left and right respectively. Thus, circular polarization can be expressed in terms of the combination of linear polarizations and vice versa as shown below :
|ex> = (|eL> + |eR>)/ ---------- (53j)
|ey> = i(|eL> - |eR>)/ ---------- (53k)
Since the circular unit vectors again satisfies the orthogonality relation : < ei|ej> = (i-j) for (i, j) = (L or R), < e|e> = 1. If it is interpreted as the total probability, then there are 50% probability for each of the circular polarization state in Eq.(53g). In order to obtain a definite circular polarization, we have to pick either + t or - t but not both. A linear polarization will be produced if we keep both + t and - t in Eq.(53g). For example, the purely right-handed rotating polarization vector is described by :
|e> = ei t |eR> ---------- (53l)

The angular momentum density of classical electromagnetic waves is :
S = r [E(r,t) B(r,t)]/(4 c) ---------- (53m)
where r is a positional vector pointing out from the origin of the coordinate system. It can be shown that a circularly polarized plane wave moving in the z direction has a finite extent of the field in the x and y directions (Problem 6.11, Classical Electrodynamics, by J. D. Jackson, 1967). Such spatial extension has a direction associated with either |eR> or |eL>. Averaging over space and time of Eq. (53m) by one wavelength yields :
S = U/| | ---------- (53n)
where U=|E|2/(8 ) is energy of the wave per unit volume, the + sign is for right-handed circular polarization, the - sign for the left-handed one. An experiment back in 1936 had demonstrated conclusively that circular electromagnetic wave does carry angular momentum.

Transition to quantum theory is accomplished by expressing U=N  /V, where N is the number of photons in volume V, and the energy has been quantized to  . Thus, the angular momentum of a photon (N = 1 in unit volume V = 1) can be parallel or anti-parallel to the z-axis :
sz =  ---------- (53o) This formula shows that angular momentum of electromagnetic wave is related to the right- or left-handed polarization, which becomes photon spin of + or - respectively when goes over to quantum theory, i.e., the photon is spin 1 particle. It is said to have right-handed helicity if the spin aligns with the direction of motion (+ ), or left-handed helicity in opposite direction (- ) as shown in Figure 07d. As usual in quantum theory, the photon would be in a superposition of both the right- and left-handed helicity states, it is only when a measurement is performed that it assumes a definite right- or left-handed helicity state (see Copenhagen Interpretation).

#### Figure 07d Photon Helicity [view large image]

For linear polarization, although the average of large number of photons shows zero angular momentum, individual photon does carry either right-handed or left-handed polarization with 50% probability for each as shown in Eq.(53g).

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