6.4: Rayleigh-Sommerfeld Diffraction Integral
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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 \[r=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+z^{2}} . \nonumber \]
Remarks.
- 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.
- 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.
- 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.