Wave Optics

Early scientific attempts to describe light and optical phenomena were based on corpuscular theories, developed by figures such as René Descartes and Isaac Newton. These theories treated light as a stream of particles traveling in straight lines with a certain velocity — an idea consistent with the framework of geometrical optics, where light is represented by rays.

From the late 17^th century onward, scientists including Robert Hooke and Christiaan Huygens found growing evidence for the wave nature of light. This led to the development of the wave theory of light (also called wave optics), first formulated mathematically by Huygens in his Traité de la Lumière (1690, see Ref. [1]), and later refined in detail by the civil engineer Augustin-Jean Fresnel.

Wave optics was not immediately accepted as a correct description of light. It gained widespread acceptance only in the early 19^th century — particularly after the observation of the Arago spot by Dominique François Jean Arago and the theoretical contributions of Fresnel.

Based on wave theory, several important optical phenomena could now be successfully explained:

• diffraction of light, for example at narrow optical slits (later investigated in detail by Thomas Young)
• interference effects
• polarization phenomena (pioneering work by Augustin-Jean Fresnel)

In the 1860s, James Clerk Maxwell identified optical waves as electromagnetic waves, providing a unified physical basis for light. However, extensive and productive research in wave optics had already been achieved long before understanding the detailed physical nature of light waves.

After the development of wave theory based on Maxwell's equations, physicists initially believed they had found the complete and final theory of light. However, later discoveries revealed quantum effects that could not be explained within this classical framework.

To account for these phenomena, quantum optics was developed, extending classical wave theory by quantizing the electromagnetic field. In this framework, the wave properties of light remain fundamental, but the theory also incorporates its particle-like behavior in terms of photons.

Modern applications of wave optics are mathematically grounded in Maxwell's equations, from which one can directly derive a wave equation — a second-order differential equation in both time and space. For monochromatic light, one finds the vector Helmholtz equation:

In many practical situations, simplified forms of this equation are employed. When the refractive index contrast is weak and polarization coupling can be neglected, the field can be treated as scalar, yielding the scalar Helmholtz equation:

Here, the scalar field ($E(x, y, z)$) represents one polarization component of the total field.

For fields propagating largely in ($z$) direction, one can often apply the paraxial approximation, neglecting the second-order derivative with respect to the ($z$) coordinate. One obtains the first-order paraxial scalar wave equation

where the complex amplitude ($A(x, y, z)$) is related to the electric field amplitude by

and thus does not include the rapid increase in optical phase.

Scalar wave models, ignoring the vector nature of electromagnetic waves, are widely used in optics, e.g. for calculating fiber modes as LP modes. For some applications, one requires more sophisticated models (full-vector methods) for an accurate description of electromagnetic wave propagation.

A central concept in wave optics is Fourier optics, which applies transverse spatial Fourier transforms to describe light fields. This approach provides both intuitive insights into optical phenomena and powerful tools for system design and quantitative analysis. In some cases, analytical solutions can be obtained directly through this framework.

Numerical simulations are frequently used to model light propagation based on various forms of the wave equation. Such methods are very versatile, allowing one to study a far wider range of situations, since analytical solutions are often not available. However, numerical methods can be computationally intensive in terms of both time and memory. Therefore, simplifying assumptions — such as predominantly unidirectional propagation — are often employed, e.g. in methods of numerical beam propagation.

When geometrical optics provides a sufficiently accurate description, it is often preferred because of its lower computational demands.

The term physical optics is closely related to wave optics. It can either refer to the same concept or to a restricted form involving certain approximations. The term emphasizes that wave-based models are physically more realistic than those of geometrical optics, even when they do not rely on the full set of Maxwell's equations and ignore quantum effects.