Sinusoidal plane-wave solutions of the electromagnetic wave equation

Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation.

The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of different frequencies and polarizations.

The treatment in this article is classical but, because of the generality of Maxwell's equations for electrodynamics, the treatment can be converted into the quantum mechanical treatment with only a reinterpretation of classical quantities (aside from the quantum mechanical treatment needed for charge and current densities).

The reinterpretation is based on the theories of Max Planck and the interpretations by Albert Einstein of those theories and of other experiments. The quantum generalization of the classical treatment can be found in the articles on Photon polarization and Photon dynamics in the double-slit experiment.

Contents

Explanation

Experimentally, every light signal can be decomposed into a spectrum of frequencies and wavelengths associated with sinusoidal solutions of the wave equation. Polarizing filters can be used to decompose light into its various polarization components. The polarization components can be linear, circular or elliptical.

Plane waves

The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units)

\mathbf{E} ( \mathbf{r} , t ) = \begin{pmatrix} E_x^0 \cos \left ( kz-\omega t %2B \alpha_x \right ) \\ E_y^0 \cos \left ( kz-\omega t %2B \alpha_y \right ) \\ 0 \end{pmatrix} = E_x^0 \cos \left ( kz-\omega t %2B \alpha_x \right ) \hat {\mathbf{x}} \; %2B \; E_y^0 \cos \left ( kz-\omega t %2B \alpha_y \right ) \hat {\mathbf{y}}

for the electric field and

c \, \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) = \begin{pmatrix} -E_y^0 \cos \left ( kz-\omega t %2B \alpha_y \right ) \\ E_x^0 \cos \left ( kz-\omega t %2B \alpha_x \right ) \\ 0 \end{pmatrix} = -E_y^0 \cos \left ( kz-\omega t %2B \alpha_y \right ) \hat {\mathbf{x}} \; %2B \; E_x^0 \cos \left ( kz-\omega t %2B \alpha_x \right ) \hat {\mathbf{y}}

for the magnetic field, where k is the wavenumber,

\omega_{ }^{ } = c k

is the angular frequency of the wave, and c is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions.

The plane wave is parameterized by the amplitudes

E_x^0 = \mid \mathbf{E} \mid \cos \theta
E_y^0 = \mid \mathbf{E} \mid \sin \theta

and phases

\alpha_x^{ } , \alpha_y

where

\theta \ \stackrel{\mathrm{def}}{=}\ \tan^{-1} \left ( { E_y^0 \over E_x^0 } \right ) .

and

\mid \mathbf{E} \mid^2 \ \stackrel{\mathrm{def}}{=}\ \left ( E_x^0 \right )^2 %2B \left ( E_y^0 \right )^2 .

Polarization state vector

Main article: Jones calculus

Jones vector

All the polarization information can be reduced to a single vector, called the Jones vector, in the x-y plane. This vector, while arising from a purely classical treatment of polarization, can be interpreted as a quantum state vector. The connection with quantum mechanics is made in the article on photon polarization.

The vector emerges from the plane-wave solution. The electric field solution can be rewritten in complex notation as

\mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \}

where

|\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}

is the Jones vector in the x-y plane. The notation for this vector is the bra-ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.

Dual Jones vector

The Jones vector has a dual given by

\langle \psi | \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} = \begin{pmatrix} \quad \cos\theta \exp \left ( -i \alpha_x \right ) & \sin\theta \exp \left ( -i \alpha_y \right ) \quad \end{pmatrix} .

Normalization of the Jones vector

The Jones vector is normalized. The inner product of the vector with itself is

\langle \psi | \psi\rangle = \begin{pmatrix} \psi_x^* & \psi_y^* \end{pmatrix} \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = 1 .

Polarization states

Main article: Polarization (waves)

Linear polarization

Main article: Linear polarization

In general, the wave is linearly polarized when the phase angles \alpha_x^{ } , \alpha_y are equal,

\alpha_x = \alpha_y \ \stackrel{\mathrm{def}}{=}\ \alpha .

This represents a wave polarized at an angle \theta with respect to the x axis. In that case the Jones vector can be written

|\psi\rangle = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} \exp \left ( i \alpha \right ) .

Circular polarization

Main article: Circular polarization

If \alpha_y is rotated by \pi / 2 radians with respect to \alpha_x the wave is circularly polarized. The Jones vector is

|\psi\rangle = \begin{pmatrix} \cos\theta \\ \pm i\sin\theta \end{pmatrix} \exp \left ( i \alpha_x \right )

where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If unit vectors are defined such that

|R\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}

and

|L\rangle \ \stackrel{\mathrm{def}}{=}\ {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}

then a circular polarization state can written in the "R-L basis" as

|c\rangle = \psi_R |R\rangle %2B \psi_L |L\rangle

where

\psi_R \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) = \left ( {\exp(-i\theta) \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right )

and

\psi_L \ \stackrel{\mathrm{def}}{=}\ \left ( {\cos\theta %2Bi\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) = \left ( {\exp(i\theta) \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) .

Any arbitrary state can be written in the R-L basis

|\psi\rangle = a_R \exp \left ( i \alpha_x -i \theta \right ) |R\rangle %2B a_L \exp \left ( i \alpha_x %2B i \theta \right ) |L\rangle

where

1 = \mid a_R \mid^2 %2B \mid a_L \mid^2 .

Elliptical polarization

Main article: Elliptical polarization

The general case in which the electric field rotates in the x-y plane and has variable magnitude is called elliptical polarization. The state vector is given by

|\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix} .

References

See also