# The Fresnel Diffraction Integral

The Huygen-Fresnel principle can be stated as

$$U(P_{0})=\frac{1}{i \lambda}\int \int_{\Sigma}

U(P_{1}) \frac{e^{i k r_{01}}}{r_{01}}\cos \theta ds$$

Figure 1.

This equation describes how the light travels from one plane to another a distance \(z\) apart. All points from the aperture \(\Sigma\) contribute towards the intensity at point \(P_{0}\).

Typically this equation is too computationally intensive to compute so we

make some approximations.

We note

$$\cos \theta = \frac{z}{r_{01}}$$

So

$$U(x,y)=\frac{z}{i \lambda}\int \int_{\Sigma} U(\xi,\eta) \frac{e^{i k r_{01}}}{r_{01}^{2}} d \xi d \eta$$

We can rewrite \(r_{01}\) as

$$r_{01}=\sqrt{z^{2}+(x-\xi)^{2}+(y-\eta)^{2}}$$

$$=z \sqrt{1+\frac{(x-\xi)^{2}}{z^{2}}+\frac{(y-\eta)^{2}}{z^{2}}}$$

To simplify this further we note that the binomial expansion below

$$\sqrt{1+b}=1+\frac{1}{2}b-\frac{1}{8}b^{2}+\cdots$$

So

$$r_{0} \approx z [1+\frac{1}{2}[(x-\xi)^{2}+(y-\eta)^{2}]-\frac{1}{8}[(x-\xi)^{2}+(y-\eta)^{2}]^{2}]$$

The third term can be dropped providing it contributes only a small amount

usually take as less than one radian in the \(exp\) term, i.e.,

$$\frac{z}{8}\frac{2 \pi}{\lambda} \left[(x-\xi)^{2}+(y-\eta)^{2}\right]^{2} \ll 1$$

i.e.,

$$z \gg \sqrt[3]{\frac{\pi}{4 \lambda} [(x-\xi)^{2}+(y-\eta)^{2}]}$$

The error contribution from the denominator is much smaller so only in

this case is ok to put \(r_{0} \approx z\).

We now get

$$U(x,y)=\frac{ e^{ikz}}{i \lambda z}\int \int U(\xi,\eta)e^{\frac{ik}{2z}[(x-\xi)^{2}+(y-\eta)^{2}]} d \xi d \eta$$

If we take the \(e^{\frac{ik}{2z}(x^2+y^2)}\) outside of the integral we

have

which is the orignal object multiplied by a quadratic phase term and

Fourier transformed – **The Fresnel Diffraction Integral.**

It can also be expressed as a convolution.

with a kernel of

Either can be used, some times one is easier to solve than the other.

Thank you so much for this! My Bsc project is on holographic microscope and reading up on this has been a great start!

Concise great introdunction.