How to Determine the Validity of the Fresnel Approximation?

[the_godfather]

Homework Statement

A friend of mine asked me on the Fresnel approximations earlier, and I couldn't really remember many of the details other than it was an approximation for spherical waves based on the Taylor series. So basically I had to look it up in a textbook.
One of the exercises (2.2-1) in the book fundamentals of photonics [b. saleh] was a question on the validity of the Fresnel approximation:
Determine the radius of a circle within which a spherical wave of wavelength λ = 633nm, originating a a distance 1m away, may be approximated by a paraboloidal wave. Determine the maximum angle θ and the Fresnel number N_f.

Homework Equations

a^4 << 4z^3λ

(N_f*θ^2) / 4 << 1

N_F = a^2 / λ

The Attempt at a Solution

I'm not really sure where to start as I don't really understand.
My initial thought was to calculate the Fresnel number using N_F = a^2 / λ. If I take the a (circle radius) as 1m. It's simple.
N_F = 1/633nm
I can then use N_F to find the maximum angle when the radius is 1m.
Now that seems way too simple.

 

[NateD]

I'm guessing I need to use the two equations given to find the radius and the angle. So I set up two simultaneous equations:a^4 << 4z^3λand(N_f*θ^2) / 4 << 1I can then solve for a and θ. The first equation givesa^4 = 4z^3λa = (4z^3λ)^1/4 where z = 1mNow substituting this into the second equation gives θ = (4/N_f)^1/2Substituting N_f for 1/633nm gives θ = (4/1/633nm)^1/2θ = 0.068rad or 3.9°So the radius is 0.04m and the maximum angle is 3.9°Do I have this right? Any advice on where I went wrong or how I could improve my answer would be greatly appreciated.