# Eikonal equation and trajectory of a ray of light

• PhilDSP
In summary, on page 122 of Born and Wolf's "Principles of Optics", the equation for the trajectory of a ray of light is derived in association with the eikonal equation. This equation was later developed by J.J. Thomson into an equation of motion for the electron, but details in its derivation are missing. The equation is only approximately correct for small wavelengths and breaks down when the wavelength approaches zero. The eikonal equation can be derived using Fermat's principle and the calculus of variations or by starting with Huyghen's wavelets. For a detailed derivation, one must do the work from scratch. J.J. Thomson also specified a relationship between the group velocity of a particle and its acceleration, derived
PhilDSP
On page 122 of Born and Wolf's "Principles of Optics" the following equation for the trajectory of a ray of light is glibly derived in association with the eikonal equation.

$$\frac{d}{d \bf s} (n \frac{d \bf r}{d \bf s}) = \nabla n$$
where n is the index of refraction and r is the displacement vector

This equation is extremely interesting because much earlier J. J. Thomson developed it into an equation of motion for the electron. But details in its derivation are sadly missing in both places.

What are the limitations? Does the equation degenerate as the wavelength approaches zero? Does anyone have references to a more detailed derivation?

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It is only approximately correct if the wavelength is much smaller than the typical variation length scale. In other words,
##|k^{-2}||\nabla k| \ll 1##
where k is the wavenumber.

If k goes to zero somewhere, then the eikonal approximation breaks down.

1 person
You can derive the eikonal equation by using Fermat's principle and the calculus of variations.

Thanks Khashishi and UltrafastPED. That equation stands a bit apart from the eikonal equation and so doesn't necessarily inherit all of the eikonal equation's limitations. So it seems that the only way to know the details is to do the work of deriving that equation from scratch.

I give an informal derivation of both forms of the eikonal equation, along with some discussion in lecture 10 of Notes on Analytical Mechanics: "Connection to Optics".

1 person
Excellent! Thanks again

J. J. Thomson specified, by the way, that ${\bf v}$ will represent the group velocity of a particle from which we can get the relationships

$d {\bf s} = {\bf v} dt$ and ${\bf v} = nc$

Hence
$$\frac{d}{d \bf s} (\frac{\bf v}{c} \frac{d \bf r}{d \bf s}) = \nabla n$$
and
$$\frac{1}{\bf v} \frac{d}{dt} (\frac{1}{\bf v} \frac{d \bf r}{dt}) = \frac{c}{\bf v} \nabla n$$
or
$$\frac{d^2 \bf r}{dt^2} = nc^2 \nabla n$$
Equating $\frac{d^2 \bf r}{dt^2}$ with the acceleration of the particle and allowing the particle to have mass we get
$${\bf F} = m{\bf a} = mnc^2 \nabla n$$
This applies to a [STRIKE]charged[/STRIKE] elementary particle only of course which is not close to any boundary

P.S. I've striked through the word charged as evidently the equation before inserting mass applies to a photon.

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## 1. What is the Eikonal equation?

The Eikonal equation is a partial differential equation that describes the trajectory of a ray of light in a medium with varying refractive index. It is derived from Fermat's principle, which states that light travels in a path that minimizes the travel time.

## 2. How is the Eikonal equation used in optics?

The Eikonal equation is used to calculate the path of a ray of light through a medium with varying refractive index. It is commonly used in the design of optical systems such as lenses and mirrors, as well as in the simulation of light propagation through complex media.

## 3. What is the relationship between the Eikonal equation and Snell's law?

The Eikonal equation is a generalization of Snell's law, which describes the relationship between the angle of incidence and the angle of refraction of a ray of light passing through a boundary between two media. Snell's law can be derived from the Eikonal equation by assuming a constant refractive index within each medium.

## 4. How is the Eikonal equation solved?

The Eikonal equation can be solved using various numerical methods such as the finite difference method or the finite element method. These methods involve discretizing the equation and solving it iteratively to obtain a numerical solution. Analytical solutions are also possible for certain simplified cases.

## 5. Can the Eikonal equation be applied to other wave phenomena?

Yes, the Eikonal equation can be applied to other wave phenomena such as sound waves or electromagnetic waves. It is a general equation that describes the propagation of any type of wave through a medium with varying properties. However, the specific form of the equation may differ depending on the type of wave being studied.

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