# Wave Optics

Definition: the description of optical phenomena based on wave models

Alternative term: physical optics

More general term: optics

German: Wellenoptik

Author: Dr. Rüdiger Paschotta

Early attempts to scientifically describe light and optical phenomena were based on corpuscular theories (developed by René Descartes, Isaac Newton and others), assuming light to consist of small particles which at least in free space travel along straight lines with a velocity. That fits to geometrical optics, where light is represented by geometrical rays. From the 17^{th} century on, scientists like Robert Hooke and Christiaan Huygens followed more closely the evidence for the wave nature of light, and this finally led to the wave theory of light (*wave optics*), which was mathematically worked out by Huygens (with publication in 1690, see Ref. [1]) and later on in more detail by the civil engineer Augustin-Jean Fresnel. Wave optics was not immediately accepted as an appropriate model for describing the nature of light; only in the early 19^{th} century, it became really widely accepted in the scientific community, particularly after observation of the so-called Arago spot by Dominique-François-Jean Arago, with important contributions from Augustin-Jean Fresnel. Based on wave theory, one could now well describe the following important optical phenomena:

- effects of diffraction of light e.g. at narrow optical slits (later studied in more detail, e.g. by Thomas Young)
- interference phenomena
- polarization of light (with pioneering contributions from Augustin-Jean Fresnel)

In the 1860s, optical waves were identified with electromagnetic waves by James Clerk Maxwell. However, a lot of productive research in wave optics was already possible before that, i.e., without understanding the detailed physical nature of light waves.

Modern applications of wave optics can be mathematically based on Maxwell's equations as the most fundamental basis; one can directly derive a *wave equation* from it, which is a second-order differential equation in time and space. For monochromatic light, one obtains the Helmholtz equation. In many cases one uses simplified equations which are approximate but fairly accurate in certain limited domains. For example, one can often neglect the second-order derivative with respect to the <$z$> coordinate for fields propagating largely in <$z$> direction. One also often uses the paraxial approximation. Scalar wave models, ignoring the transverse nature of electromagnetic waves, are also widely used in optics, e.g. for calculating fiber modes. For some applications, one requires more sophisticated models for a full description of electromagnetic wave propagation.

Wave optics is usually understood to be a completely classical approach, not taking into account any quantum effects. Quantum optics works with extended theories, where the electromagnetic field is treated on a new basis. Wave properties of light continue to play a central role, although some kind of particle character (→ photons) also became apparent.

A very important concept in wave optics is Fourier optics, which essentially means the application of transverse spatial Fourier transforms. This allows both for an intuitive qualitative explanation of various phenomena and design techniques and for quantitative calculations. Such calculations can partly be done with analytical means only.

Frequently, numerical software is used for simulating light propagation based on some kind of wave equation. While this approach can in principle be fairly general, computation time and memory requirements can be excessive when not using various restricting assumptions – for example, that light propagates essentially only in one direction. As far as geometrical optics are sufficient for realistic descriptions, that approach is often preferred over wave optics because it is computationally much less demanding.

A related term is *physical optics*, which may be interpreted as the same as wave optics, or in a more restricted sense when certain approximations are applied. The term emphasizes that such wave-based models are physically more realistic than geometrical optics, even if they are not based on full Maxwell's equations.

### Bibliography

[1] | C. Huygens, “Traité de la Lumière”, Leiden: Pieter van der Aa (1690), available at archive.org |

See also: optics, geometrical optics, Fourier optics, diffraction, interference, polarization of light

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