Explanation of ray caustics in E&M

My understanding (now) of a real caustic is that it is envelope of curves or ray-paths that arise due to reflection or refraction from the medium/manifold.

My main question is, I am seeing the term "imaginary caustic" in literature (like here) and cannot find a straightforward answer on what an imaginary caustic is and when/where they arise. What are imaginary caustics?

• Why would the propagation of waves break down on a caustic? I don't believe caustics even have the requirement that there is anything even close to a focal point. A caustic is an envelope of many rays and that is perfectly well defined. -1 for effort (and lack thereof). May 18 '16 at 7:50
• I'm voting to close this question as off-topic because it shows insufficient effort. May 18 '16 at 7:50
• @CuriousOne "direction of propagation" breaks down--meaning the direction of propagation is no longer well-defined. Is this wrong? Forgive me for a lack of fundamentals--I'm not a physicist/physics major. For more context about where I'm coming from, I've been reading about the canonical transform as applied to radio occultations, as described here: onlinelibrary.wiley.com/doi/10.1029/2000RS002592/pdf May 18 '16 at 9:22
• That is 100% wrong. The problem is more general, though. You (and a non-trivial number of books of the past) are trying to understand a wave equation with an approximation in a regime in which the approximation does not hold. That's meaningless, even though plenty of attempts have been made to extend the validity of the approximation in special cases. The problem is: it's only in these special cases where one gets somewhat meaningful results. In general (e.g. for a hologram) the approximation is completely wrong to begin with. May 18 '16 at 17:10

In Wikipedia the caustic is defined as follows.

In optics, a caustic is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface.

You can think of the envelope of a family of curves as a curve that is a tangent to each of them.

Here is a diagram on page 60 of "A Treatise on Optics" by David Brewster published in 1831.

There is a source of light at $R$ and the rays are reflected a circular surface $MN$ although only the rays reflected from part $MB$ are shown.
The angle of incidence is equal to the angle of reflection for each of the rays.
None of the rays meet the principal axis at the same point although those that hit the reflecting surface at positions $B, 1, 2$ and $3$ do so approximately and this is where the image of the object $R$ is formed.
The concave mirror is suffering from a mirror defect called spherical aberration.

Now if you look at all the rays in the upper half they define an area with no light and an area where there is light and in particular there is an envelope, shown in red at the bot one in the top half, which defines places where the rays are at a tangent - this is the caustic curve and here is a photograph of one although you may well see one very frequently in you cup of coffee or tea?

The caustic curve is particularly noticeable because it is brighter because there are more rays passing through that region then in other regions around the reflector.

In the diagram above one half of the imaginary caustic I have coloured blue.
This is obtained by moving the object to the other side of the reflector $MBN$ so that it is acting like a convex mirror. The reflected rays diverge but when back produced form the blue imaginary caustic $Nf'M$.

In recent years caustics have been used to find exoplanets.

A star and its orbiting plant act as a gravitational lens and form a caustic in a plane at right angled to the plane of the Earth's orbit around the Sun which is shown in blue in the left-hand diagram.
As the orbit of the Earth (shown in red) crosses the caustic the light intensity resulting from the caustic changes as shown in the graph of intensity against time.
Although the shape of the caustic is very complex that very complexity means that from the caustic information about the star and its exoplanet can be obtained.
The edX course "ASTRO2x Astrophysics: Exploring Exoplanets" is a good introduction to this effect and many others which are used to discover exoplanets and their properties.

• How do you know there is really a plant orbiting the star? :) Seriously, this is really fascinating stuff. I have started reading Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations By J.F Nye. It will take quite some time for me to go all the way through it, but that's what good books are for!
– uhoh
Mar 19 '17 at 12:29

Your understanding of a real caustic (I presume you call it real as opposed to the imaginary caustic that you mention later) is correct.

First the easy part: an imaginary caustic is a caustic located on the extension of the light rays beyond the optical system from which they arrive. For instance, in the presence of a convex lens, imaginary caustics may form behind the lens itself.

Then the more difficult part. Geometrical optics can be developed under the assumption that the wavelength (for a monochromatic wave) $\lambda$ is much smaller than any other physical length in the process. In this case, all physical quantities can be shown to be proportional to

$$f \propto e^{\imath\psi/\lambda}$$

where the function $\psi$ is the wave phase. Surfaces with $\psi =$ constant are wave surfaces, and caustics are the regions (if any) which are reached by (at least) two distinct wave surfaces, say one with $\psi = \psi_1$ and one with $\psi=\psi_2$. This mathematical characterization is global, not local, because the bending of light rays may occur either locally or remotely, and then lead to photons crossing paths.

There is another characterization of caustics, by means of the angular eikonal: if you are interested, you can find it in Landau & Lifshitz's Theory of Fields. In that case, finding caustics is shown to be equivalent to finding multiple solutions a system of four equations.

Lastly, you are right that near caustics geometrical optics breaks down; as a matter of fact, as rays cross caustics, there is this cute little phenomenon whereby their phase is changed by exactly $-\pi/2$, which is of course inexplicable in geometrical optics. Again, see Landau and Lifshitz for a poignant explanation.

• Light rays don't bend (at least not unless the index of refraction changes) and photons don't interact with each other. May 18 '16 at 16:49
• @CuriousOne Neither statement is correct: the eikonal approximation (the equation above) has been developed exactly to deal with the case of a smoothly changing refraction index. As for photons interacting, I have no idea where you gathered that I said anything to that extent. May 18 '16 at 18:26
• I think I was pretty clear about "unless the index of refraction changes" (agreed, I should have added "slowly") and that's the only case where these things work. None of this is any meaningful for the case where the changes are happening at the length scale of the wavelength. I don't know what you mean by "photons crossing paths". For one thing, photons don't move at all, even if we assume, in terms of the path integral interpretation of quantum mechanics, that they do, they are not moving on straight paths, at all. Even if we falsely assume that they do, any two photons do not interact. May 18 '16 at 19:34
• @CuriousOne I don't know what you mean by "photons crossing paths".. And yet it is easy.. And I will be so kind to skip over the other errors you committed. May 19 '16 at 5:26
• Photons are local measurements on a quantum field... they are not balls moving along rays and even if they were, they would not interact, at least not at this energy. That's just what the theory says. May 19 '16 at 5:29