6.4: Rayleigh-Sommerfeld Diffraction Integral

Another method to propagate a wave field is by using the Rayleigh-Sommerfeld integral. A very good approximation of this integral states that each point in the plane $z=0$ emits spherical waves, and to find the field in a point $(x, y, z)$, we have to add the contributions from all these point sources together. This corresponds to the Huygens-Fresnel principle postulated earlier in Section 5.6. Because a more rigorous derivation starting from the Helmholtz equation would be complicated and lengthy, we will just give the final result: $$\begin{aligned} U(x, y, z) &=\frac{1}{i \lambda} \iint U\left(x^{\prime}, y^{\prime}, 0\right) \frac{z e^{i k \sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+z^{2}}}}{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+z^{2}} \mathrm{~d} x^{\prime} \mathrm{d} y^{\prime} \\ &=\frac{1}{i \lambda} \iint U\left(x^{\prime}, y^{\prime}, 0\right) \frac{z e^{i k r}}{r} \mathrm{~d} x^{\prime} \mathrm{d} y^{\prime} \end{aligned} \nonumber$$ where we defined $$r=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+z^{2}} . \nonumber$$

Figure $\PageIndex{1}$: The spatial frequencies $k_{x}, k_{y}$ of the plane waves in the angular spectrum of a time-harmonic field which propagates in the $z$-direction. There are two types of waves: the propagating waves with spatial frequencies inside the cricle: $\sqrt{k_{x}^{2}+k_{y}^{2}}<k=2 \pi / \lambda$ and which have phase depending on the propagation distance $z$ but constant amplitude, and the evanescent waves for which $\sqrt{k_{x}^{2}+k_{y}^{2}}>k$ and of which the amplitude decreases exponentially during propagation.

Remarks.

1. The formula ( $\PageIndex{1}$ ) is not completely rigorous: a term that is a factor $1 /(k r)$ smaller (and in practive is therefore is very much smaller) has been omitted.
2. In ( $\PageIndex{1}$ ) there is an additional factor $z / r$ compared to the expressions for a time-harmonic spherical wave as given in (1.53) and at the right-hand side of (5.44). This factor means that the spherical waves in the Rayleigh-Sommerfeld diffraction integral have amplitudes that depend on the angle of radiation (although their wave front is spherical), the amplitude being largest in the forward direction.
3. Equivalence of the two propagation methods. The angular spectrum method amounts to a multiplication by $\exp \left(i z k_{z}\right)$ in Fourier space, while the Rayleigh-Sommerfeld integral is a convolution. It is one of the properties of the Fourier transform that a multiplication in Fourier space corresponds to a convolution in real space and vice versa. Indeed a mathematical result called Weyl’s identity implies that the rigorous version of ( $\PageIndex{1}$ ) and the plane wave expansion (i.e. angular spectrum method) give identical results.