Sky scientific do software for lens design. They do a free trial download. I haven't tried it, but it may be useful for your app.

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Call any one of a number of optics companies in Rochester NY. The Rochester Cluster has several world class optics houses. I have found that engineers love to teach.

I don't know what your intended use is or how accurately you need to know the radius. A computer program for optical design is the best approach.

However, if an approximate solution is sufficient, then using paraxial or first order equations will work.

The following are good references

Melles Griot has a good introduction to fundamental optics. Look over the "Sign conventions", "Basic Paraxial Equations", and ""Paraxial Formulaie for Lenses in Air".

wikipedia also has the paraxial equations.

Consider this plano-concave lens

First, determine how much divergence you want. From geometry, identify the location of S₁ and S₂.

The point light source is at the left (S₁) and after the rays are bent (diverged), they appear to come from S₂.

Then the focal length of the lens, f, is given by

1/f = 1/S₁ - 1/S₂

The minus sign comes from S₂ being on the left of the lens. And the focal lengh of the lens is negative.

Now, the thin lens formula for radii with one radius being flat (infinite radius) simplifies to

1/f = (n-1)*(1/R)

where n is the index of refraction of the lens material. For crown glass this is 1.52

I hope this answers your question.

You might want to contact OSCI Inc's President, Michael Pate. He is an expert in optical lens design and has helped us with some tough lens design issues. www.oscintl.com or 541.255.2165