Is there any limit to distance between two waves when used in Huygens principle?

Question

Apologising for not being clear.

When I used Huygens principle to determine the spherical wavuefront of radius $R/2$ created by reflection from a spherical concave lens of radius $R$, I got amazingly different results for different size of wave front created. I used a hemispherical lens. When I used a wavefront of width $2R$ I really got a spherical wavefront of radius $\frac{R}{2}$. But in this case all rays were meeting at different position and not at $\frac{R}{2}$. Also laws of reflection was violated.

When I used a wavefront of much smaller width I got most of rays meeting at focus and obeying law of reflection. But the wavefront was not spherical.

Can anyone tell me what I did wrong and what I should do to get correct results?

Here a ray falls at pt. A. Consider a big and vertical wavefront here touching other side of mirror. After central ray of this wavefront, reach at pole of mirror, a new secondary wavefront is created at A of radius AB.

Now I drew a perpendicular from O on arc of this wavefront, intersecting at B. This created a ray AB. So, light should pass from A to B.

1)Law of reflection state , It should reflect along OD.

This is a contradiction.

2) Then, I used a new wavefront(look at point near D). Made required drawings and found ray reflecting near focus.

All this helped me to draw a circular wavefront created due to all such secondary wavelets.

This circle tells, rays should focus at focus, without any spherical abberation. How?

Answer 1

I think that you have found from first principles that a hemispherical lens suffers from spherical aberration?

Later

Here is my attempt.

It might look a mess but I think that it does illustrate the fact that the wavefronts converge towards a region which is about half the radius of the mirror.

The radius of the mirror is 90 mm and the wavelets have a radius of 10 mm.
You start with a wavefront and draw wavelets centred on that wavefront.
You then construct the new wavefront and repeat the process.
The smaller you make the wavelets the more accurate will be the construction.

When you "know" what the new wavefront is going to look like then the size of the secondary wavelets does not matter as is shown in the right hand diagram.

Answer 2

You can apply the Huygen's construction to any distance over which the local environment is optically homogeneous. Encountering a reflective or refractive surface is encountering a non-homogeneity, so you can't draw big circles that include a mirror or a lens.

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You can modify the Huygen's principle so that you can use near such boundaries, but the description of those modifications in words is not easy.

Let's take lenses first. The new rule is that you draw around each point on the original wave-front the convex hull for which the sum $d_1 n_1 + d_2 n_2$ is some chosen constant. Here the $d$s are distances traveled in a particular medium and the $n$s are the indicies of refraction.

In a single medium this reduces to the usual Huygen's rules, but near an edge the portion of the circle penetrating into the other medium is magnified or shrunk.

Mirrors are similar in the sense that you modify the arc that would have originally penetrated the mirror surface, but this time instead of scaling that arc you reflect it across the surface so that it is once again on the front side of the mirror.