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Limting the discourse of this question only for scalar waves in optics, we have $\nabla^2 \phi - \dfrac{n^2}{c^2} \dfrac{\partial^2 \phi}{\partial t^2} =0 $.

Now, when we investigate about geometric optics limit of it, the eikonal equation is taken into consideration. $\phi(\mathbf{r},t) = \phi_0(\mathbf{r},t) e^{f (\mathbf{r},n(\mathbf{r}))- \omega t}$.

Here, refractive index $n$ is assumed to be weakly changing over wavelength. This requirement is translated into condition $\left | \nabla n(\mathbf{r}) \right | \ll | \mathbf{k} n(\mathbf{r})|$, where $\mathbf{k}= \dfrac{2 \pi}{\lambda}$.

It is also considered that amplitude $\phi_0$ is very slowly varying over wavelength, like in this answer and therefore, $\left | \nabla \phi_0(\mathbf{r},t) \right | \ll | \mathbf{k} \phi_0(\mathbf{r},t) |$ .

I fail to understand "exactly" how this mathematical expressions come from. Can anyone explain intuitively how these expressions come ?

I undertsand that change in amplitude is given by $\nabla \phi_0$ but to indicate that change in amplitude is very small over wavelength - how it is $\left | \nabla \phi_0(\mathbf{r},t) \right | \ll | \mathbf{k} \phi_0(\mathbf{r},t) |$ ?

curious_mind
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1 Answers1

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I fail to understand "exactly" how this mathematical expressions come from. Can anyone explain intuitively how these expressions come ?

When changing position by $\Delta\mathbf{r}$, the amplitude $\phi_0$ changes by $$\Delta\phi_0 = \nabla\phi_0\cdot\Delta\mathbf{r} \tag{1}$$

Expanding the scalar product we get $$\Delta\phi_0 = |\nabla\phi_0|\ |\Delta\mathbf{r}|\ \cos\alpha \tag{2}$$ where $\alpha$ is the angle between $\nabla\phi_0$ and $\Delta\mathbf{r}$.

Saying "the amplitude is slowly varying over a wavelength" just means: For $|\Delta\mathbf{r}|=\lambda$ the amplitude changes only by a small percentage, i.e. $$\Delta\phi_0 \ll \phi_0 \tag{3}$$

Using this together with (2) we get $$|\nabla\phi_0|\ \lambda\ \cos\alpha \ll \phi_0 \tag{4}$$ and with $|\mathbf{k}|=\frac{2\pi}{\lambda}$ $$|\nabla\phi_0|\ \cos\alpha \ll \frac{|\mathbf{k}|}{2\pi} \phi_0$$

Since we are only interested in orders of magnitude, we can neglect the factors $\cos\alpha$ and $2\pi$ (their order of magnitude is $\approx 1$) $$|\nabla\phi_0| \ll |\mathbf{k}|\ \phi_0 \tag{5}$$

You can do the same reasoning like above for the amplitude $\phi_0$ also for the refractive index $n$.

Thomas Fritsch
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