# Focal Curve of an Achromatic Doublet

1. Feb 29, 2012

### joaocosta23

I have a problem with the determination of the correct focal curve of an achromatic double.

I'm considering here a doublet formed by a biconvex Flint Glass lens attached to a plano-concave crown glass lens. ( I know that the typical design of the doublet is a biconvex crown lens attach to a plano-concave flint glass lens. but here I'm interested in the opposite.)

I'm also assuming that I aim to achromatize the C and F Fraunhofer lines.

The optical powers of the two lenses are:
P1 = n_flint * (1/R1-1/R2)
P2 = n_crown*(1/R3) [ I assume that R4=∞, since this is a plano-concave lens ]

I also know the optical power (Pt) of my system at the D Fraunhofer line.

The classic textbook formulas say that the conditions required to design a doublet are:

P1 = Pt * Δ2/(Δ2-Δ1)
P2 = -Pt * Δ1/(Δ2-Δ1)

Where Δi = (ni_F - ni_C) / (ni_D-1) ( i=1,2 stands for the first or second lens, respectively), and n_F, n_C and n_D are the refractive indexes of the respective glass.
We have 2 equations and 3 unknowns ( R1,R2 and R3), so in order to solve this problem I must choose an arbitrary radius R1 for example.

My problem is that after I solve the system and find the required radius, when I plot the focal shift of my system it yields a focal shift of 1*10^-15 in the whole visible spectrum, when in theory the focal shift should only be zero valued for the C and F lines..

Can anyone Explain me what I'm doing wrong?

2. Feb 29, 2012

### Andy Resnick

I'm having trouble following your 'textbook formula'- where is the lens spacing (and lens thickness) specified? Kingslake's book even works out a flint-in-front doublet.

3. Mar 1, 2012

### joaocosta23

You are right, but I forgot to mention that the lenses are thin lenses and that they are in contact with each other. In this case, the formulas I mentioned are the correct ones.

4. Mar 1, 2012

### joaocosta23

You can find the formulas here: http://pt.scribd.com/doc/58102523/138/Achromatic-Doublet-Lens-Design [Broken]

but they can also be found for example in the classical book Principles of Optics of Max Born and Wolf..

Last edited by a moderator: May 5, 2017
5. Mar 1, 2012

### Andy Resnick

Thanks- the reference helped me a lot.

Ok, so I got a different solution from you. Using the numbers from the referenced book, the total optical power $\phi$ = 1/254 = 0.0039 and the Abbe numbers for the front flint and back crown are 36.3 and 64.5, respectively.

I can interpret your design in two ways: either the front element is a (hopefully biconvex) flint, or the front element is a plano-convex crown.

For a flint-first, I get that the flint must have negative power, because Va < Vb. You can make it a biconcave, but it cannot be biconvex or even plano-convex

For a plano-convex front crown, the optical powers $\phi_{a}$ and $\phi_{b}$ of the two elements are the same as 'normal' (0.009 and -0.0051), but again the flint element must have a negative power and cannot be biconvex.

Does this help? Or am I missing something?

6. Mar 1, 2012

### joaocosta23

Thank you so much for your quick answer. My problem is actually related with a correct determination of the focal curve of an achromatic doublet.

In my problem I am using a standard solution, which consists of a biconvex Crown glass lens attached to a plano-concave Flint Glass Lens.

In order to model the refractive indices of the two glasses in the visible domain I am using the well-known Cauchy's equations:
(http://en.wikipedia.org/wiki/Cauchy's_equation)

n(λ) = A + B/λ2 ( with λ in μm).

n1(λ) - Flint Glass
n2(λ) - Crown Glass

For the flint glass I am considering:
A=1.7280
B=0.01342

For the crown glass I am considering:
A=1.5220
B=0.00459

Now lets say I want my lens to have a total focal lenght f-1=100mm

For Simplicity I assume that I have a equiconvex Crown glass lens ( like in the example I showed you) such that r2=-r1.

So basically I want to find two radiis, r2 and r3. When I do so and I try to plot the total focal shift of the systems, it yields a curve that is practically zero valued in the whole visible spectrum, but it should only be zero for the C and F lines..

7. Mar 1, 2012

### Andy Resnick

I'm afraid I'm really confused now: