Polarization of Light
This part continues from the linear optical susceptibility tutorial.
Consider a monochromatic plane optical wave that has a complex field
\[\tag{58}\mathbf{E}(\mathbf{r},t)=\pmb{\mathcal{E}}\exp(\text{i}\mathbf{k}\cdot\mathbf{r}-\text{i}\omega t)=\hat{e}\mathcal{E}\exp(\text{i}\mathbf{k}\cdot\mathbf{r}-\text{i}\omega t)\]
where \(\pmb{\mathcal{E}}\) is a constant independent of r and t, and \(\hat{e}\) is its unit vector.
The polarization of the optical field is characterized by the unit vector \(\hat{e}\).
The wave is linearly polarized, also called plane polarized, if \(\hat{e}\) can be expressed as a constant, real vector. Otherwise, the wave is elliptically polarized in general, and is circularly polarized in some special cases.
For the convenience of discussion, we take the direction of the wave propagation to be the z direction so that \(\mathbf{k}=k\hat{z}\) and assume that both E and H lie in the xy plane.
Note: this assumption is generally true if the medium is isotropic. It is not necessarily true if the medium is anisotropic. Propagation and polarization in isotropic and anisotropic media are discussed in the following two parts. However, the general concept discussed here does not depend on the validity of this assumption.
Then, we have
\[\tag{59}\pmb{\mathcal{E}}=\hat{x}\mathcal{E}_x+\hat{y}\mathcal{E}_y=\hat{x}|\mathcal{E}_x|e^{\text{i}\varphi_x}+\hat{y}|\mathcal{E}_y|e^{\text{i}\varphi_y}\]
where \(\mathcal{E}_x\) and \(\mathcal{E}_y\) are space- and time-independent complex amplitudes, with phases φx and φy, respectively.
The polarization of the wave depends only on the phase difference and the magnitude ratio between the two field components \(\mathcal{E}_x\) and \(\mathcal{E}_y\). It can be completely characterized by the following two parameters:
\[\tag{60}\varphi=\varphi_y-\varphi_x,\qquad\qquad-\pi\lt\varphi\le\pi\]
and
\[\tag{61}\alpha=\tan^{-1}\frac{|\mathcal{E}_y|}{|\mathcal{E}_x|},\qquad\qquad0\le\alpha\le\frac{\pi}{2}\]
Because only the relative phase φ matters, we can set φx = 0 and take \(\mathcal{E}\) to be real in the following discussions. Then \(\pmb{\mathcal{E}}\) from (59) can be written as
\[\tag{62}\pmb{\mathcal{E}}=\mathcal{E}\hat{e},\qquad\text{with}\qquad\hat{e}=\hat{x}\cos\alpha+\hat{y}e^{\text{i}\varphi}\sin\alpha\]
Using (39) from the harmonic fields tutorial, the space- and time-dependent real field is
\[\tag{63}\pmb{E}(z,t)=2\mathcal{E}[\hat{x}\cos\alpha\cos(kz-\omega t)+\hat{y}\sin\alpha\cos(kz-\omega t+\varphi)]\]
At a fixed z location, say z = 0, we see that the electric field varies with time as
\[\tag{64}\pmb{E}(t)=2\mathcal{E}[\hat{x}\cos\alpha\cos\omega t+\hat{y}\sin\alpha\cos(\omega t-\varphi)]\]
In general, \(\mathcal{E}_x\) and \(\mathcal{E}_y\) have different phases and different magnitudes. Therefore, the values of φ and α can be any combination. At a fixed point in space, both the direction and the magnitude of the field vector E in (64) can vary with time. Except when the values of φ and α fall into one of the special cases discussed below, the tip of the this vector generally describes an ellipse, and the wave is said to be elliptically polarized.
Note that we have assumed that the wave propagates in the positive z direction. When we view the ellipse by facing against this direction of wave propagation, we see that the tip of the field vector rotates counterclockwise, or left handedly, if φ > 0, and clockwise, or right handedly, if φ < 0.
Figure (4) below shows the ellipse traced by the tip of the rotating field vector at a fixed point in space. Also shown in the figure are the relevant parameters that characterize elliptic polarization.
In the description of the polarization characteristics of an optical wave, it is sometimes convenient to use, in place of α and φ, a set of two other parameters, θ and ε, which specify the orientation and ellipticity of the ellipse, respectively.
The orientationally parameter θ is the directional angle measured from the x axis to the major axis of the ellipse. Its range is taken to be 0 ≤ θ < π for convenience. Ellipticity ε is defined as
\[\tag{65}\epsilon=\pm\tan^{-1}\frac{b}{a},\qquad-\frac{\pi}{4}\le\epsilon\le\frac{\pi}{4}\]
where a and b are the major and minor semiaxes, respectively, of the ellipse.
The plus sign for ε > 0 is taken to correspond to φ > 0 for left handed polarization, whereas the minus sign for ε < 0 is taken to correspond to φ < 0 for right-handed polarization. The two sets of parameters (α, φ) and (θ, ε) have the following relations:
\[\tag{66}\tan2\theta=\tan2\alpha\cos\varphi\]\[\tag{67}\sin2\epsilon=\sin2\alpha\sin\varphi\]
Either set is sufficient to characterize the polarization state of an optical wave completely.
The following special cases are of particular interest.
1. Linear Polarization.
This happens when φ = 0 or π for any value of α. It is also characterized by ε = 0, and θ = α if φ = 0 or θ = π - α if φ = π.
Clearly, the ratio \(\frac{\mathcal{E}_x}{\mathcal{E}_y}\) is real in this case; therefore, linear polarization is described by a constant, real unit vector as
\[\tag{68}\hat{e}=\hat{x}\cos\theta+\hat{y}\sin\theta\]
It follows that E(t) described by (64) reduces to
\[\tag{69}\pmb{E}(t)=2\mathcal{E}\hat{e}\cos\omega{t}\]
The tip of this vector traces a line in space at an angle θ with respect to the x axis, as shown in figure (5) below.
2. Circular Polarization.
This happens when φ = π/2 or - π/2, and α = π/4. It is also characterized by ε = π/4 or - π/4, and θ = 0. Because α = π/4, we have \(|\mathcal{E}_x|=|\mathcal{E}_y|=\mathcal{E}/\sqrt2\).
There are two different circular polarization states:
a. Left-circular polarization. For φ = π/2, also ε = π/4, the wave is left-circularly polarized if it propagates in the positive z direction. The complex field amplitude in (62) becomes
\[\tag{70}\pmb{\mathcal{E}}=\mathcal{E}\frac{\hat{x}+\text{i}\hat{y}}{\sqrt2}=\mathcal{E}\hat{e}_+\]
and E(t) described by (64) reduces to
\[\tag{71}\pmb{E}(t)=\sqrt2\mathcal{E}(\hat{x}\cos\omega t+\hat{y}\sin\omega t)\]
As we view against the direction of propagation \(\hat{z}\), we see that the field vector E(t) rotates counterclockwise with an angular frequency ω. The tip of this vector describes a circle. This is shown in figure 6(a) below. This left-circular polarization is also called positive helicity. Its eigenvector is
\[\tag{72}\hat{e}_+\equiv\frac{\hat{x}+\text{i}\hat{y}}{\sqrt2}\]
b. Right-circular polarization. For φ = - π/2, also ε = - π/4, the wave is right-circularly polarized if it propagates in the positive z direction. We then have
\[\tag{73}\pmb{\mathcal{E}}=\mathcal{E}\frac{\hat{x}-\text{i}\hat{y}}{\sqrt2}=\mathcal{E}\hat{e}_-\]
and
\[\tag{74}\pmb{E}(t)=\sqrt2\mathcal{E}(\hat{x}\cos\omega t-\hat{y}\sin\omega t)\]
The tip of this field vector rotates clockwise in a circle, as shown in figure 6(b) below. This right-circular polarization is also called negative helicity. Its eigenvector is
\[\tag{75}\hat{e}_-\equiv\frac{\hat{x}-\text{i}\hat{y}}{\sqrt2}\]
As can be seen, neither \(\hat{e}_+\) nor \(\hat{e}_-\) is a real vector. Note that the identification of \(\hat{e}_+\), defined in (72), with left-circular polarization and that of \(\hat{e}_-\), defined in (75), with right-circular polarization are based on the assumption that the wave propagates in the positive z direction.
For a wave that propagates in the negative z direction, the handedness of these unit vectors changes: \(\hat{e}_+\) becomes right-circular polarization, while \(\hat{e}_-\) becomes left-circular polarization.
Linearly polarized light can be produced from unpolarized light using a polarizer. A polarizer can be of transmission type, which often utilizes the phenomenon of double refraction in an anisotropic crystal, or of reflection type, which takes advantage of the polarization-sensitive reflectivity of a surface.
A very convenient transmission-type polarizer is the Polaroid film, which utilizes a material with linear dichroism, having low absorption for light linearly polarized in a particular direction and high absorption for light polarized orthogonally to this direction. The output is linearly polarized in the direction defined by the polarizer irrespective of the polarization state of the input optical wave.
A polarizer can also be used to analyze the polarization of a particular optical wave. When so used, a polarizer is also called an analyzer.
The next part continues with the Propagation in an Isotropic Medium tutorial.