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Is it possible to make some separation of amplitudes and phase for a general vector-wave field?

For example, like a paraxial approximation of a complex scalar field of the form $$\Phi(x,y,z) = A(x,y,z)e^{i\theta(x,y,z)}.$$ If this is possible, what are some common methods?

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  • $\begingroup$ Expanding in Fourier series? That's what comes of the top of my head atm... $\endgroup$ – dingo_d Oct 14 '13 at 16:51
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It certainly can be done and the commonest method is simply to shove the Ansatz you cite into whatever the relevant wave equation is. When we do this with Maxwell's equations, this leads, with the slowly varying envelope approximation (I'll say exactly what this is below), to the Eikonal equation, which is equivalent to raytracing with Snell's law and Fermat's principle of least time in a variable refractive index medium, if the refractive index variation is spatially slow, i.e. $|\nabla n(\mathbf{r})| \ll k\, n(\mathbf{r})$ where $n(\mathbf{r})$ is the refractive index as a function of position $\mathbf{r}$ and $k$ the freespace wavenumber for the light in question (or the freespace wavenumber of the light with the longest important wavelength in a spectrum of light of different wavelengths).

This kind of procedure with the Schrödinger equation shows that the Hamilton Jacobi equation (see Wiki page with this name) can be thought of as an Eikonal approximation (see Wiki page with this name), or, equivalently, the Hamilton-Jacobi equation of classical mechanics can be thought of as the limit of the Schrödinger equation as $\hbar\to0$. Thus your ansatz with the Eikonal equation yields insight into how classical mechanics is an approximation embedded in the more general quantum theory and how we can continuously deform one into the other with $\hbar$ continuously running between nought and its experimentally witnessed value.

You can see the Wiki pages on how the ansatz plays out in quantum mechanics. Here's how it happens with the Maxwell equations. We write the electomagnetic fields in the form $\mathbf{E}\left(\mathbf{r}\right) = \mathbf{e}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$, $\mathbf{H}\left(\mathbf{r}\right) = \mathbf{h}\left(\mathbf{r}\right) e^{i\,\varphi\left(\mathbf{r}\right)}$ where $\mathbf{e}$ and $\mathbf{h}$ are slowly varying vectors and the phase $\varphi\left(\mathbf{r}\right)$ is real valued. Then Faraday's and Ampère's laws become:

$$\begin{array}{lcl} \nabla \wedge \mathbf{e} + i\, \nabla \varphi \wedge \mathbf{e} &=& -\mu\,\partial_t \mathbf{h}\\ \nabla \wedge \mathbf{h} + i\, \nabla \varphi \wedge \mathbf{h} &=& \epsilon\,\partial_t \mathbf{e} \end{array}\quad\quad\quad\quad(1) $$

So far there is no approximation; one then makes the slowly varying envelope approximation: that the envelopes $ \mathbf{e}$ and $ \mathbf{h}$ vary much more slowly with position than does the phase, {\it i.e.} $\left|\mathbf{e}\right|^{-1} \left|\nabla \wedge \mathbf{e}\right| \ll \left|\nabla \varphi_e\right| \approx \left|k\right|$ and $\left|\mathbf{h}\right|^{-1} \left|\nabla \wedge \mathbf{h}\right| \ll \left|\nabla \varphi_h\right| \approx \left|k\right|$ and we also assume a monochromatic (time-harmonic) field. The equations then become:

$$\begin{array}{lcl} \nabla \varphi \wedge \mathbf{e} &\approx& \omega\,\mu\,\mathbf{h}\\ \nabla \varphi \wedge \mathbf{h} &\approx& -\omega\,\epsilon\, \mathbf{e} \end{array}\quad\quad\quad\quad(2) $$

Both of these equations individually say that $\mathbf{e}$ and $\mathbf{h}$ are orthogonal. On forming $-\omega^2\,\mu\,\epsilon\, \mathbf{e}\wedge\mathbf{h} = \left(\nabla \varphi \wedge \mathbf{h}\right)\wedge\left(\nabla \varphi \wedge \mathbf{e}\right)$ and simplifying, one gets:

$$\omega^2\,\mu\,\epsilon\, \mathbf{e}\wedge\mathbf{h} = \nabla \varphi \cdot \left(\mathbf{e}\wedge\mathbf{h}\right) \nabla \varphi \quad\quad\quad\quad(3)$$

so that $\nabla \varphi$ is orthogonal to both $\mathbf{e}$ and $\mathbf{h}$ and aligned with $\mathbf{e}\wedge\mathbf{h}$, whence $\omega^2\,\mu\,\epsilon\, \left|\mathbf{e}\wedge\mathbf{h}\right| = \left|\nabla \varphi\right|^2 \left|\mathbf{e}\wedge\mathbf{h}\right| $, therefore $\mathbf{e}$, $\mathbf{h}$ and $\nabla \varphi$ in that order are a mutually orthogonal, right-handed triple and:

$$ \left|\nabla \varphi\right|^2 = \omega^2\,\mu\,\epsilon = \left|\mathbf{k}\right|^2 \quad\quad\quad\quad(4)$$

which is the Eikonal Equation: the fundamental equation defining raytracing. More fully, we can summarise everything inferred from Eq.(2) as the "Eikonal Equations" as follows:

$$\begin{array}{lcl} \mathbf{k}\left(\mathbf{r}\right) &\stackrel{def}{=}& \nabla \varphi\left(\mathbf{r}\right)\\ n\left(\mathbf{r}\right) &\stackrel{def}{=}& \sqrt{\frac{\mu\left(\mathbf{r}\right)\,\epsilon\left(\mathbf{r}\right)}{\mu_0\,\epsilon_0}} = \sqrt{\mu\left(\mathbf{r}\right) \epsilon\left(\mathbf{r}\right)} \;c\\ \mathcal{Z}\left(\mathbf{r}\right) &\stackrel{def}{=}& \sqrt{\frac{\mu\left(\mathbf{r}\right)}{\epsilon\left(\mathbf{r}\right)}}\\ \mathcal{Y}\left(\mathbf{r}\right) &\stackrel{def}{=}& \mathcal{Z}\left(\mathbf{r}\right)^{-1} = \sqrt{\frac{\epsilon\left(\mathbf{r}\right)}{\mu\left(\mathbf{r}\right)}}\\ k\left(\mathbf{r}\right) &=& \left|\mathbf{k}\left(\mathbf{r}\right)\right| = \frac{\omega}{c} \, n\left(\mathbf{r}\right) \\ \mathbf{\hat{k}}\left(\mathbf{r}\right) &\stackrel{def}{=}& \frac{\mathbf{k}\left(\mathbf{r}\right)}{k\left(\mathbf{r}\right)} = \frac{ \nabla \varphi\left(\mathbf{r}\right) }{\left|\nabla \varphi\left(\mathbf{r}\right)\right|}\\ \mathbf{e}\wedge\mathbf{h} &=& \mathcal{Y} \left|\mathbf{e}\right|^2 \mathbf{\hat{k}}= \mathcal{Z} \left|\mathbf{h}\right|^2 \mathbf{\hat{k}} \\ \mathbf{e} &=& -\mathcal{Z}\, \mathbf{\hat{k}} \wedge\mathbf{h} \\ \mathbf{h} &=& \mathcal{Y}\, \mathbf{\hat{k}} \wedge\mathbf{e} \\ \end{array}\quad\quad\quad\quad(5)$$

These equations are exactly fulfilled when $\mathbf{e}$ and $\mathbf{h}$ are constant (independent of $\mathbf{r}$) vectors and $\varphi\left(\mathbf{r}\right) = \mathbf{k} \cdot \mathbf{r}$ for some constant wavevector $\mathbf{k}$, when they define a plane wave, and plane waves are the only exact solution of the Eikonal equations. The uniqueness of plane waves in fulfilling the Eikonal equations exactly is another way of stating the strict contradictory nature of a ray - a true, exact ray represents a wholly delocalised photon and only the axial component of $\mathbf{k} \cdot \mathbf{r}$ is important for setting its phase. The assertion that the Eikonal equations approximately describe more general waves is the intuitive one that $\mathbf{e}\left(\mathbf{r}\right)$ and $\mathbf{h}\left(\mathbf{r}\right)$ vary slowly enough with $\mathbf{r}$ that they locally differ only slightly from plane waves.

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  • $\begingroup$ Hello @wetsavannaanimal-aka-rod-vance, I really appreciate your very thorough answer, it has enlightened me about some issues with which I am dealing now. I have not the email notifications set up, so I noticed your reply only with a considerable delay. I still am wondering, whether a case of strongly focused laser field (lets say with a microscope objective) can be treated in the eikonal approximation as the field variation can be quite large. Anyway, I have enjoyed following (+pen+paper) your argumentation. $\endgroup$ – LukeWasinahurry Mar 27 '16 at 11:56
  • $\begingroup$ @LukeWasinahurry Yes many microcope objective fields will be well modelled by the eikonal equation, although as the lens gets very fast, say for an NA of greater than 0.5 you'll begin to see significant error, as the fields vary very swiftly with position. For NA >0.7 you really need to turn to full vector solutions of Maxwell's equations for accurate models for many tasks. BTW I am talking numerical aperture as a geometric quantity here, $\arcsin(k_\perp/k)$ without heed to the refractive index. Most often NA is defined with a refractive index, with the weird result that you can have $NA > 1$ $\endgroup$ – WetSavannaAnimal Mar 28 '16 at 9:12

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