A circularly polarized plane wave moving in the z direction...

A circularly polarized plane wave moving in the $z$ direction has a finite extent in the $x$ and $y$ directions. Assuming that the amplitude modulation is slowly varying (the wave is many wavelengths broad), show that the electric and magnetic fields are given approximately by $$ \begin{aligned} \mathbf{E}(x, y, z, t) &=\left[E_{0}(x, y)\left(\mathbf{e}_{1} \pm i \mathbf{e}_{2}\right)+\frac{i}{k}\left(\frac{\partial F_{0}}{\partial x} \pm i \frac{\partial E_{0}}{\partial y}\right) e_{y}\right] e^{i t_{z}-i n t} \\ \mathbf{B} &=\mp i \sqrt{\mu \epsilon} \mathbf{E} \end{aligned} $$ where $\boldsymbol{e}_{1}, \boldsymbol{e}_{\mathrm{J}}, \boldsymbol{e}_{3}$ are unit vectors in the $x, y, z$ directions.

A circularly polarized plane wave moving in the $z$ direction has a finite extent in the $x$ and $y$ directions. Assuming that the amplitude modulation is slowly varying (the wave is many wavelengths broad), show that the electric and magnetic fields are given approximately by
$$
\begin{aligned}
\mathbf{E}(x, y, z, t) &=\left[E_{0}(x, y)\left(\mathbf{e}_{1} \pm i \mathbf{e}_{2}\right)+\frac{i}{k}\left(\frac{\partial F_{0}}{\partial x} \pm i \frac{\partial E_{0}}{\partial y}\right) e_{y}\right] e^{i t_{z}-i n t} \\
\mathbf{B} &=\mp i \sqrt{\mu \epsilon} \mathbf{E}
\end{aligned}
$$
where $\boldsymbol{e}_{1}, \boldsymbol{e}_{\mathrm{J}}, \boldsymbol{e}_{3}$ are unit vectors in the $x, y, z$ directions.

Key Concepts

Paraxial Approximation

The paraxial approximation assumes that variations in the transverse direction (relative to the main propagation direction) are small, allowing the neglect of certain second-order transverse derivatives. This simplifies the mathematical treatment of beams that are almost collimated, meaning they propagate primarily in one direction with small angular divergence. It is particularly useful in optics and wave mechanics for analyzing beams with finite transverse extent.

Slowly Varying Envelope Approximation

The slowly varying envelope approximation is a method used in wave analysis where the amplitude or envelope of the wave changes slowly compared to the rapid oscillations due to its carrier frequency. This approximation allows significant simplifications in the treatment of modulated waves by neglecting higher-order spatial derivatives, and is often employed in situations where the wave extends over many wavelengths as in the given problem.

Maxwell's Equations in Free Space

Maxwell's equations govern the behavior of electric and magnetic fields in space and time. In free space, these equations show that time-varying electric fields induce magnetic fields and vice versa, leading to the propagation of electromagnetic waves. The relationship between the electric and magnetic fields is particularly evident in plane wave solutions, where the magnetic field is proportional to the electric field with a phase relationship that depends on the polarization.

Circular Polarization

Circular polarization refers to a state of electromagnetic wave polarization in which the electric field rotates in a circle around the direction of propagation, with its magnitude remaining constant. This occurs when two orthogonal linear polarization components have equal amplitude and a phase difference of ±?/2. It is a fundamental concept in wave physics and optics, important for applications ranging from telecommunications to the study of light-matter interactions.

Plane Waves

A plane wave is an idealized electromagnetic wave whose surfaces of constant phase are infinite, parallel planes. In such waves, the electric and magnetic fields are uniform in planes perpendicular to the direction of propagation. This idealization simplifies analysis by providing a uniform model to study the basic properties of wave propagation according to Maxwell's equations.

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