Diffraction

In wave physics, diffraction refers to a deviation from rectilinear propagation when a wave encounters an obstacle in its propagation path. In this event, the wavefronts are modified (either in amplitude and/or in phase). The segments of the modified wavefronts interfere beyond the obstacle, resulting in wave propagation “around” the obstacle. Diffraction is a function of the relative side of the obstacle compared to the wavelength of the incident wave.

Intuitively, consider people blocked by a tall fence. They can hear each other (diffraction of sound waves) but can’t see each other (because light’s wavelength is extremely small).

Basics

We’ll use a simplified version of diffraction theory, with a few simplifying assumptions:

• Scalar field — the electric field is represented by a scalar, i.e., no polarisation dependence. This is okay when the aperture is small and near normal incidence, and the angle subtended by the diffraction pattern is small.
• Coherent illumination — the incident light has perfect temporal and spatial coherence over the region of the aperture. This condition can be satisfied when the aperture is small and/or when laser radiation is used.
• Far-field approximation — the $\vec{E}$ distribution of interest (i.e., the observation plane) is located far away from the aperture.
• Only relative intensity is considered.

Then, the general approach we take is as follows, called the Huygens-Fresnel principle:

• The field at the aperture is treated as an ensemble of secondary point sources. Each of them emits spherical waves, the strength and phase of which are defined by the incident wave and the aperture function.
• The field at the screen is the coherent superposition of all $\vec{E}$ contributions from the secondary point sources at the aperture. i.e., the field at the screen is the result of interference of all secondary waves emitted at the aperture.

Steps

We first define an aperture function:

$$E_o(x_o,y_o)$$

Then, we take the Fourier transform to find the Fraunhofer $\vec{E}$ distribution:

$$E_i(x_i,y_i)\propto \mathcal{F}\{E_o(x_o,y_o)\}$$

Of course, we then need the actual intensity distribution:

$$I_i(x_i,y_i)\propto E_i^2(x_i,y_i)$$

Tips

Note that the Fourier transform of two multiplied functions of two independent variables (i.e., a function of $x$ and a function of $y$) does not result in the convolution in the frequency domain. It just results in a multiplication:

$$\mathcal{F}\{f(x)\cdot g(y)\}=F(f_x)\cdot G(f_y)$$

There are a few useful Fourier transforms to take note of.