Huygens–Fresnel principle





The Huygens–Fresnel principle (named for Dutch physicist Christiaan Huygens, and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation (both in the far field limit and in near field diffraction). It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. This view of wave propagation helps better understand a variety of wave phenomena, such as diffraction.

For example, if two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them, a person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound. The same is true of light passing the edge of an obstacle, but this is not as easily observed because of the short wavelength of visible light.

Huygens principle follows formally from the fundamental postulate of quantum electrodynamics – that wavefunctions of every object propagate over any and all allowed (unobstructed) paths from the source to the given point. It is then the result of interference (addition) of all path integrals that defines the amplitude and phase of the wavefunction of the object at this given point, and thus defines the probability of finding the object (say, a photon) at this point. Not only light quanta (photons), but electrons, neutrons, protons, atoms, molecules, and all other objects obey this simple principle.

Single slit diffraction
A common application of Huygens' principle is for the case of a plane wave (usually light, radio waves, x-rays or electrons) incident on an aperture of arbitrary shape. Huygens' principle states that each point in the hole acts as a point source. A point source generates waves that travel spherically in all directions (similar to circular waves caused by dropping small stone in a pond). The sum of the waves from all the point sources at any point beyond the aperture can be calculated by integration or numerical modelling.

Consider the case of single slit diffraction where we have one slit through which we shine light onto a distant screen. Suppose, we want to calculate at which point on the screen interference minima (dark stripes) occur. We then replace this relatively wide slit by an increasing number of narrow ones (subslits), and add waves produced by each. Obviously, two small slits interfere destructively when their path lengths differ by $$\lambda/2$$ (a 180 degrees phase difference). We can calculate (using phasors or a similar wave-addition math) that for three waves from three slits to cancel each other the phases of slits must differ 120 degrees, thus path difference from the screen point to slits must be $$\lambda/3$$, and so forth. In the limit of approximating the single wide slit with an infinite number of subslits the path length difference between edges of slit must be exactly $$\lambda$$ to get complete destructive interference (and so a dark stripe on the screen).

Diffraction by a general aperture
The qualitative argument we used to glean some understanding of single slit diffraction using only Huygens' principle is difficult to apply in general to apertures of truly arbitrary shape. The wave that emerges from a point source has amplitude $$\psi$$ at location r that is given by the solution of the frequency domain wave equation for a point source,


 * $$\nabla^2 \psi + k^2 \psi = \delta(\bold r)$$

where $$ \delta(\bold r)$$ is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (aka scalar Laplacian) in the spherical coordinate system simplifies to (see del in cylindrical and spherical coordinates)

$$\nabla ^2 = \frac{1}{r} \frac {\partial ^2}{\partial r^2} (r \psi) $$

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention $$e^{-i \omega t}$$) is:


 * $$\psi(r) = \frac{e^{ikr}}{4 \pi r}$$

This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector $$\bold r'$$ and the field point is located at the point $$\bold r$$, then we may represent the scalar Green's function (for arbitrary source location) as:


 * $$\psi(\bold r | \bold r') = \frac{e^{ik | \bold r - \bold r' | }}{4 \pi | \bold r - \bold r' |}$$

Therefore, if an electric field, Einc(x,y) is incident on the aperture, the field produced by this aperture distribution is given by the surface integral:


 * $$\Psi(r)\propto \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y')~ \frac{\exp(ik | \bold r - \bold r'|)}{4 \pi | \bold r - \bold r' |} \,dx'\, dy',$$

where the source point in the aperture is given by the vector

$$\bold r' = x' \bold \hat x + y' \bold \hat y$$

In the far field, wherein the parallel rays approximation can be employed, the Green's function,


 * $$\psi(\bold r | \bold r') = \frac{\exp(ik | \bold r - \bold r' |) }{4 \pi | \bold r - \bold r' |}$$

simplifies to


 * $$ \psi(\bold r | \bold r') = \frac{e^{ik r}}{4 \pi r} \exp(-ik ( \bold r' \cdot \bold \hat r))$$

as can be seen in the figure to the right (click to enlarge).

The expression for the far-zone (Fraunhofer region) field becomes


 * $$\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') \exp(-ik ( \bold r' \cdot \bold \hat r ) ) \, dx' \,dy',$$

Now, since

$$\bold r' = x' \bold \hat x + y' \bold \hat y$$

and

$$\bold \hat r = \sin \theta \cos \phi \bold \hat x + \sin \theta ~ \sin \phi ~ \bold \hat y +  \cos \theta \bold \hat z$$

the expression for the Fraunhofer region field from a planar aperture now becomes,


 * $$\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') \exp(-ik \sin \theta (\cos \phi x' + \sin \phi y')) \, dx'\, dy'$$

Letting,

$$k_x = k \sin \theta \cos \phi $$

and

$$k_y = k \sin \theta \sin \phi $$

the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform


 * $$\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') \exp(-i (k_x x' + k_y  y') ) \,dx'\, dy',$$

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics).

Other reading
Stratton, Julius Adams: Electromagnetic Theory, McGraw-Hill, 1941. (Reissued by Wiley - IEEE Press, ISBN 978-0-470-13153-4).