Diffraction Patterns

This example simulates a double-slit interference experiment with water waves or sound. The model mimics the incoming plane-wave excitation with two thin waveguides leading to slits in a screen and computes the diffraction pattern on the other side of the screen.

Model Definition

Figure 1 shows the model geometry.

Figure 1: Model geometry with inflow and outflow boundaries indicated. On all other boundaries, a zero flux condition applies.

Theory predicts amplitude minima along rays where the difference in travel distance is an odd multiple of half the wavelength, and maxima at even multiples. For n = 0, ±1, ±2, …:

In this example, the distance D between the slits is 2λ. Maxima should then be found at θ = 0° and 30°, while minima should appear at θ = 14.48° and 48.59°.

Equation

For time-harmonic propagation, the wave equation turns into the Helmholtz equation:

Boundary Conditions

On the inflow and outflow boundaries (see Figure 1), absorbing boundary conditions apply. Let us briefly show how such conditions in their simplest form can be derived. First, assume the solution at the boundaries to be the sum of an incident plane wave, uin, propagating in an arbitrary direction and a scattered wave, usc, propagating in the normal direction:

(1)

Here n is the outward boundary normal vector and

. At the boundary of the modeling domain, Γ, we then have

(2)

where Equation 1 was used in the second step. It follows that

(3)

There is no incident wave on the outflow boundary, which means that the second term on the right-hand side of Equation 3 vanishes. For the inflow boundary, make the further approximation that the incident wave propagates in the inward normal direction, so that k = −kn. We then arrive at the following boundary conditions:

(4)

In this model, these conditions are readily imposed using the Coefficient Form PDE interface’s Flux/Source condition. The default condition, Zero Flux, applies on the remaining boundaries.

Results and Discussion

The plot in Figure 2 shows the diffraction pattern clearly. The effect of discretization is that the numerical wavelength differs from λ, which results in a shift of the angles. You can correct for this effect by adjusting the value of k in the Helmholtz equation to the element size. These practices are important for modeling the interference effects of monochromatic waves.