# Interference at the focus of a convex lens or a concave mirror

Now consider parallel rays travelling towards the convex lens. After emerging they get converged at the focus. Won't this lead to interference between light rays? Practically it should but I would like to know why this does not happen.

Look at the focus in the above image. The rays get converged and shouldn't that lead to interference. Similar phenomenon should occur with a concave mirror at its focus.

## 2 Answers

Yes you get interference - but a well constructed lens has introduced just enough phase shift in the wave front at every point that the interference is constructive. Because the lens is finite in size there will be some interference pattern observed at the focus - something known as "Airy's rings"

In fact - the way a lens works is precisely by creating a phase shift between light rays traveling along different paths, and after the phase shift the ray changes direction because that is the direction in which the interference is constructive.

The following diagram tries to explain this - I am using the conventional Huyghens construction to show that every point on a wave front can be considered a source of a wavefront that travels in all directions - with the final wave front continuing in the direction where all of these interfere constructively. The blue wedge is a prism - a very small piece of a lens. Inside the prism, the wavelength of the light is shorter (because of the refractive index of the lens), so the wave fronts (little circles) that represent a wavelength are closer together. You can think of a spherical lens as being made up of many prisms - each acting in the same manner (although the phase difference will change depending on the thickness of the lens). Note that in my drawing, the upper ray has exactly one wavelength inside the prism and two outside, while the lower ray has two wavelengths inside and only one outside. In both cases, the line connecting the wave fronts corresponds to exactly three wavelengths after the entrance plane of the prism. There are of course infinitely many rays between these two - if there were not, you would have something akin to a Young's Slits experiment setup, and would see interference patterns (several directions in which constructive interference can occur).

Incidentally - the picture you show in your question is very misleading. The rays don't "magically change direction" at the center of the lens - instead, they are refracted both at the entrance face and exit face of the lens. The following shows what I mean (in reality the angles are not quite as drawn - there is a thing called "spherical aberration" that is ignored here - but I hope you get the idea. I drew just the top few rays inside the lens in red; obviously the same thing is true for the bottom half):

UPDATE to explain how this works for a concave mirror:

If you take an arbitrary ray traveling parallel to the horizontal axis in this image:

You can compute its length as

$$length = D - y + \sqrt{h^2 + (f-y)^2}$$

Now if we want to set the length to a constant value regardless of $h$, we can say

\begin{align}\\ y + length - D &= \sqrt{h^2 + (f-y)^2}\\ y + C &= \sqrt{h^2 + (f-y)^2}\\ (y+C)^2 &= h^2 + (f-y)^2\\ y^2 + 2Cy + C^2 &= h^2 + f^2 -2fy + y^2\\ 2(C+f)y +C^2 + f^2 &= h^2\\ y &= \frac{h^2}{2(C+f)}-C^2-f^2\\ \end{align}

Which describes $y$ as a parabolic function of $h$. In other words - in a parabolic (convex) mirror, the path length for all rays to the focal point is the same. So once again, there will be constructive interference at the focal point.

• Well so in a convex lens we have spherical aberration. What about a concave mirror then? How do we explain the interference in that? And just one last thing, how exactly does a lens accomplish phase shift? – rahulgarg12342 Jun 8 '14 at 16:26
• The phase shift in the lens is really a result of the fact that the light traverses a different distance through glass (where it has a shorter wavelength) depending on where it is. A slightly different principle holds for a mirror: if you measure the total path length (for a parabolic mirror) you find that the total path length (reference to mirror + mirror to focal point) is the same for each parallel ray (with the reference being a plane normal to incident direction). – Floris Jun 8 '14 at 17:01
• Hey man, thanks for getting into the math of it but I just really wanted a yes/no question. So I conclude that it does lead to interference. Please correct me if I am wrong. I just wanted to ask one thing. Why isn't this phenomenon listed anywhere on the internet? I searched for it and seems like no one has touched upon this subject. What if you put a screen at the focus. What kind of an image would you get. I know it would be a point image but won't the color change due to interference? – rahulgarg12342 Jun 8 '14 at 18:51
• Why don't you get a different looking image then, as all the light rays pass through the focus undergoing interference. shouldn't the image look different from the actual image on a screen? – rahulgarg12342 Jun 8 '14 at 18:58
• To get "global" constructive interference the relative phase shifts of individual rays should be within a quarter wavelength for monochromatic light. For visible light that would be around 150nm or better. I have difficulty believing such surface and material control for a 1" diameter piece of glass. Is that really possible? – hyportnex Jun 8 '14 at 20:25

We don't have to assume that light goes in straight lines when it is in a uniform material like air or water; even that is explainable by the general principle of quantum theory. It appears that light goes in a straight line.$_1$

Light doesn't really travel only in a straight line; it 'smells' the neighboring paths around it, and uses a small core of nearby space. (In the same way, a mirror has to have enough size to reflect normally: if the mirror is too small for the core of neighboring paths, the light scatters in many directions, no matter where you put the mirror.)$_2$

In the source, each excited atoms emits light in a time of the order of $10^{-8}\text{s}$.$_3$ They could emit light in different direction and not necessarily the photons should form straight line.$_4$

So, in my opinion I feel, it is not better for me to discuss the parallel rays passing through the lens. In this context, for sure laser is not considered.

Quote to be remembered always. "Perfect clarity would profit the intellect but damage the will"- Pascal.

Credits:QED, the strange theory of light and matter $_1$Page No.53 $_2$Page No.54-Moderns ABC of Physics $_3$Page No.974. $_4$Reference needed. Data is subjected to modification, and page no's are subjected to change depending on editions.