# Fresnel-Kirchoff integral and Huygens principle

Consider a monochromatic, uniformly polarized, EM wave: $$E(x,y,z,t)=\tilde E(x,y,z)e^{iwt}$$ If on a certain plane $$z=z_1$$ the field $$\tilde E(x_1,y_1,z_1)$$ is known, we can find $$\tilde E(x,y,z)$$ on a plane $$z$$ using the Fresnel-Kirchoff integral: $$\tilde E(x,y,z)=\frac i \lambda \iint_S \tilde E(x_1,y_y,z_1) \frac {\exp^{-ikr}}r \cos(\theta) dx_1dy_1$$ Where $$S$$ is some general aperture located in the $$z_1$$ plane.

After this example (that I took from the book principle of lasers by O. Svelto) the author tells that Fresnel-Kirchoff integral is the mathematical expression of the Huygens principle. However he doesn't tell that the wavefront of the EM wave should coincide with the aperture $$S$$.

My question is: Is it necessary that the wavefront of the wave coincide with $$S$$ to apply Fresnel-Kirchoff integral?

Suppose we apply this integral to $$S$$ of size much smaller than wavelength of light and much smaller than $$z-z_1$$. In this case the wavefront will coincide with $$S$$, by construction. Now consider a set of such small apertures $$\{S_i\}_{i=1}^N$$ on an arbitrary surface $$T$$. By superposition principle, the final EM field originating from $$T$$ will be the sum of fields originating on each of these apertures $$S_i$$.
Now let the number $$N$$ of these apertures approach infinity (with their areas and distances between them approaching zero). In the limit you'll get the same integral for the total field as you'd get if you just integrated over the whole surface $$T$$ from the beginning.
Notice how we didn't actually use the fact of coincidence of the wavefront with the point-like apertures $$S_i$$ we started with. This points to the indifference of the Fresnel-Kirchhoff's formula to this. The complex value of $$\tilde E(x,y,z)$$ can be anything. Just don't forget to use the correct $$\theta$$ as a function of $$x_1$$ and $$y_1$$: it must be relative to $$T$$.