Variation of eikonal (phase) set to zero

[neelakash]

We know, in optics Fermat's principle is written in analogy with principle of Maupertuis in Classical mechanics (given by $$\delta S=\delta\int\vec{p}\cdot d\vec{l}=0$$). In terms of the wave vector it is written as $$\delta\psi=\delta\int\vec{k}\cdot d\vec{l}=0$$. Here $$\psi$$ is known as eikonal (or, phase).

For a monochromatic wave of fixed frequency,$$|\vec{k}|=\frac{\omega}{v}=\frac{\omega}{c}\frac{c}{v}=|\vec{k_0}|\frac{c}{v}=|\vec{k_0}|n$$ ($$n$$ is refractive index). Using that the above reduces to
$$|\vec{k_0}|\delta\int\ n\ dl=0$$ or $$\delta\int\ n\ dl=0$$ which can be identified as more familiar form of Fermat's principle.[we have calculated the line integral along the direction of $$\vec{k}$$]

Notice that it is reached by setting the variation of $$\psi$$ to zero like in mechanics we have principle of Maupertuis by setting variation of action $$S$$ equal to zero. It is known that action is minimized in mechanics to get the classical path traced by the system. But is the case same with the phase eikonal also? Landau (section 53-54) identifies $$\vec{k}=\nabla\psi$$ and thus $$\psi$$ is a scalar potential of a conservative field. That means whatever path you choose from A to B,the integral $$\int_{A}^{B}\vec{k}\cdot\ dl=\int_{A}^{B}|\vec{k}|\ dl$$ will remain the same. Anyway, the minimization of optical path is evident from the familiar form of Fermat's principle: $$\delta\int\ n\ dl=0$$ what can be proved using geometry.

The only difference is that action $$S$$ is not constrained by any partial differential equation like $$\psi$$ is. $$\psi$$ is constrained the eikonal equation given by $$(\nabla\psi)^2=\ n^2$$-here $$n$$ is refractive index...Anyway, what I wonder is that whether it is possible to call the problem of optics a variational problem: we are not actually minimizing $$\psi$$ for $$\psi(B)-\psi(A)$$ is the same for all the paths...(unless you constrain it via eikonal equation).

Do people know the explanation for it? May be I am missing something...

 

[Petar Mali]

Maupertuis in Classical mechanics

$$\delta\int^{t_1}_{t_0}2Tdt=0$$

$$2Tdt=m\upsilon^2dt=m\upsilon\upsilondt=pds$$

so

$$\delta\int^{M_1}_{M_0}pds=0$$

No scalar product!

Eikonal is solution of equation

$$(\frac{\partial F}{\partial x})^2+(\frac{\partial F}{\partial y})^2+(\frac{\partial F}{\partial z})^2=\frac{\omega^2}{c^2}n^2(x,y,z)$$

So I think is good to go with Hamilton - Jacobi equation

$$(\frac{\partial W}{\partial x})^2+(\frac{\partial W}{\partial y})^2+(\frac{\partial W}{\partial z})^2=2m[E-U(\vec{r})]=p^2$$

 

[neelakash]

Yes...but I did not ask that...what I asked is following:-

In mechanics, the problem is posed like this: you are to find the desired path by minimizing action...not all paths will lead to the minimum value of action.

But in optics, the variational principle $$\delta\int\vec{k}\cdot\ d\vec{l}=0$$ does NOT minimize the eikonal (for the integral is the same along all the paths).Yet when you reduce this to $$\delta\int\ n\ dl=0$$,the minimization is apparent...

Why is this difference?

 

[neelakash]

I do not know why you say there would not be any scalar product...In many texts including Landau the equation is written in terms of scalar product. Only when you choose your contour along the direction of motion,you get cosine term to be unity.

 

[PJC01]

I would like to provide a response to the content presented above. It is clear that the concept of the eikonal, or phase, is closely related to Fermat's principle in optics. Just like in classical mechanics, where the principle of Maupertuis is derived from the variation of action, in optics we can derive Fermat's principle from the variation of the eikonal.

It is interesting to note that the eikonal can be seen as a scalar potential of a conservative field, with the wave vector being the gradient of the eikonal. This means that the value of the eikonal will be the same for all paths from point A to point B, and the only difference will be in the magnitude of the wave vector. This is similar to the classical mechanics concept, where the action is minimized along the classical path traced by the system.

However, it is important to note that the eikonal is constrained by the eikonal equation, which states that the square of the wave vector is equal to the square of the refractive index. This means that the eikonal cannot take on any arbitrary value, and is instead constrained by this equation. This is different from the action in classical mechanics, which is not constrained by any partial differential equation.

In terms of calling the problem of optics a variational problem, it is important to consider the constraints on the eikonal. While we are not actually minimizing the eikonal itself, the variation of the eikonal is still used to derive Fermat's principle. This can be seen as a constrained variational problem, where the eikonal is constrained by the eikonal equation.

In conclusion, while there are similarities between the eikonal in optics and the action in classical mechanics, there are also important differences to consider, such as the constraints on the eikonal. Understanding these concepts is crucial in furthering our understanding of the principles governing optics and classical mechanics.