# Paraxial Spherical wave emanating from point source

I'm reading through chapter 5.3.1(Impulse response of a Positive lens), in Goodman's "Fourier Optics"(p.109). An object is placed a distance $$z_1$$ in front of a lens.

If we place a point source at the object plane, meaning a $$\delta$$ function at coordinates $$(x,y)=(\xi,\eta)$$ of that plane,i.e. $$U_0(x,y)=\delta(x-\xi,y-\eta)$$.

If we look a distance $$z_1$$ away from this point source and stay close to the z-axis, then according to the book, the incident wave on the lens will appear a spherical wave diverging from that point, which in the paraxial approx. has the following form: $$U_l(x,y)=\frac{1}{j\lambda z_1} \cdot exp(\frac{jk}{2z_1}\cdot [(x-\xi)^2+(y-\eta)^2])$$

Could someone elaborate how this is derived?

I know that a unit-amplitude,diverging spherical wave emanating from $$(\xi,\eta)$$ is generally given by: $$\frac{exp(jk(r-r_0))}{r-r_0}$$ Where $$r_0=(\xi,\eta,0)$$. Furthermore, I know from the paraxial approximation that this leads to approximating $$r=z+\frac{(x-\xi)^2+(y-\eta)^2}{2z}$$, for which in the exponent we take all of the terms and for the denominator only the first term. Furthermore, we usually ignore the constant phase factor $$exp(jkz)$$, but this would lead to:

$$U(x,y)=\frac{exp(\frac{jk}{2z_1}\cdot [(x-\xi)^2+(y-\eta)^2])}{z}$$ which misses the $$\frac{1}{j\lambda}$$ term..

• see section 4.2 The Fresnel Approximation Nov 22, 2023 at 23:23