Complex Amplitude Expressions for Plane and Spherical Waves

In wave optics, light is fundamentally an electromagnetic wave. To rigorously and quantitatively analyze phenomena such as propagation, interference, and diffraction, establishing robust mathematical models is essential. Directly manipulating the instantaneous electric field strength—which involves high-frequency oscillations—is computationally cumbersome. This gives rise to the powerful concept of the complex amplitude. By representing light waves in the complex domain, we can drastically simplify the process of solving wave equations. This article explores the complex amplitude expressions for plane and spherical waves, contrasting their characteristics and examining their roles in the broader landscape of optics.

The light field of a monochromatic wave can generally be treated as a real function of spatial position $\mathbf{r}$ and time $t$. For instance, a specific component of the electromagnetic field can be written as:

$$U(\mathbf{r}, t) = A(\mathbf{r}) \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi_0)$$

Here, $A(\mathbf{r})$ represents the amplitude, $\mathbf{k}$ is the wave vector, $\omega$ is the angular frequency, and $\phi_0$ is the initial phase.

Drawing upon Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, we can transition this expression into the complex domain as the product of a complex amplitude and a harmonic time factor:

$$u(\mathbf{r}, t) = \text{Re}\left{ U(\mathbf{r}) e^{-i\omega t} \right}$$

The spatial term $U(\mathbf{r}) = A(\mathbf{r}) e^{i\phi(\mathbf{r})}$ is defined as the complex amplitude. It encapsulates both the spatial amplitude distribution $A(\mathbf{r})$ and the phase information $\phi(\mathbf{r})$, while temporarily factoring out the rapid temporal oscillation $e^{-i\omega t}$. For steady-state optical calculations, computations are streamlined by focusing exclusively on this complex amplitude $U(\mathbf{r})$.
A plane wave is one of the simplest and most idealized models in wave optics. Its wavefronts—surfaces of constant phase—form a series of infinitely extended, mutually parallel planes.

Wave Vector Definition: Let the propagation direction of the plane wave be designated by a unit vector $\mathbf{s}$, with a wavelength of $\lambda$. The magnitude of the wave vector $\mathbf{k}$ is given by $k = \frac{2\pi}{\lambda}$, pointing directly along the propagation path.
Mathematical Expression: Assuming a constant amplitude $A_0$ and a zero initial phase, the complex amplitude of a monochromatic plane wave propagating along an arbitrary direction $\mathbf{k}$ is expressed as:

$$U(\mathbf{r}) = A_0 e^{i \mathbf{k} \cdot \mathbf{r}}$$

In a Cartesian coordinate system, if the wave vector components along the three axes are denoted as $k_x, k_y,$ and $k_z$ (satisfying $k_x^2 + k_y^2 + k_z^2 = k^2$), the complex amplitude expands to:

$$U(x, y, z) = A_0 e^{i (k_x x + k_y y + k_z z)}$$

When the light wave propagates strictly along the positive $z$-axis, $k_x = 0, k_y = 0,$ and $k_z = k$, simplifying the expression to:

$$U(z) = A_0 e^{i k z}$$

Plane wave models are frequently utilized to approximate collimated beams emerging from beam expanders or local optical fields situated far from the source.

Complex Amplitude of Spherical Waves

In practical physical scenarios, light radiating outward from a point source is better modeled as a spherical wave. The wavefronts of a spherical wave consist of concentric spheres centered around the source.

Geometric Attenuation: Due to conservation of energy, as a spherical wave propagates outward, the wavefront area expands proportionally to the square of the radius $r$ ($r^2$). Consequently, its amplitude is inversely proportional to the propagation distance $r$, yielding an amplitude term of $\frac{A_0}{r}$.
Mathematical Expression: Setting the origin of the coordinate system at the point source, the complex amplitude for a diverging spherical wave is formulated as:

$$U(r) = \frac{A_0}{r} e^{i k r}$$

Conversely, if the light wave converges toward the origin, it is termed a converging spherical wave, with a complex amplitude expressed as:

$$U(r) = \frac{A_0}{r} e^{-i k r}$$

Spherical waves hold a foundational position in optical design and lens diffraction theory, serving as the essential building blocks for constructing complex field distributions in optical systems.

Comparative Overview and Applications

Within the methodology of wave optics, plane waves and spherical waves form the dual cornerstones for analyzing intricate light fields. They can be compared across several key dimensions:

Wavefront Geometry: Plane waves feature infinite parallel planes, whereas spherical waves consist of concentric spheres.
Amplitude Variation with Distance: In an ideal lossless medium, a plane wave maintains a constant amplitude ($A_0$), while a spherical wave's amplitude decays proportionally to $\frac{A_0}{r}$.
Mathematical Complexity: Plane waves offer the simplest mathematical formulation, making them ideal for spatial frequency analysis in Fourier optics. Spherical waves incorporate a distance-inverse term, though under paraxial approximations, they can be approximated as parabolic waves to simplify Fresnel diffraction integrals.

Within the broader macroscopic landscape of optics, these fundamental wave forms connect deeply to other core principles:

Huygens-Fresnel Principle: This principle states that every point on a wavefront acts as a source of secondary wavelets. The superposition of these secondary spherical waves ultimately dictates the evolution of the subsequent wavefront.
Interference and Diffraction: Whether analyzing Young's double-slit experiment or Fraunhofer diffraction, the underlying theoretical derivation involves decomposing an incident light field into a superposition of plane or spherical waves, then calculating their vector interference at a given point in space.
Polarization and Dispersion: When incorporating polarization, the complex amplitude naturally extends into a vector field with directional properties. When addressing chromatic dispersion, integrations over the wavelength-dependent wave vector $k(\lambda)$ become necessary.

Mastering the complex amplitude expressions for plane and spherical waves bridges the gap between geometric and wave optics, providing an indispensable foundation for modern optical system design and laser physics.