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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..

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  • $\begingroup$ see section 4.2 The Fresnel Approximation $\endgroup$
    – hyportnex
    Nov 22, 2023 at 23:23

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