Example · Double-Slit Diffraction by FDTD
Problem statement
Solve the 2-D scalar wave equation on a \(256\times256\) grid containing a barrier with two slits; drive a sinusoidal wave from one wall and observe the interference pattern.
Given information
- Wave equation: \(\partial_t^2 U = c^2\nabla^2 U\), \(c = 1\), unit box
- Update rule (FDTD): \(U^{n+1}_{i,j} = 2U^n_{i,j} - U^{n-1}_{i,j} + \frac{c^2\Delta t^2}{\Delta x^2}\text{Lap}^n_{i,j}\)
- CFL: \(\Delta t = \frac{\sqrt2}{2}\Delta x/c\)
- A boolean mask marks walls and the barrier (\(U = 0\) there); two gaps in the barrier are the slits
Solution strategy
- Build the mask: outer walls + a barrier row, then carve two slit openings.
- Keep two time levels (\(U\), \(U_\text{prev}\)); compute the
5-point Laplacian with
np.roll. - Each step: update → zero the masked cells → drive the source wall \(U(0, :) = \sin\omega t\) → rotate time levels.
lap = (np.roll(U,1,0) + np.roll(U,-1,0)
+ np.roll(U,1,1) + np.roll(U,-1,1) - 4*U)
Unew = 2*U - Uprev + fac*lap
Unew[mask] = 0
Unew[0,:] = np.sin(omega*t)
Expected results
Beyond the barrier, the two slit-waves overlap into alternating bright/dark fringes — constructive where path lengths differ by \(m\lambda\), destructive at \((m + \frac12)\lambda\). Experiments to run:
- Wavelength vs slit width: \(\lambda \sim\) slit → strong diffraction; \(\lambda \ll\) slit → sharp geometric shadows
- Slit separation sets fringe spacing (\(\Delta y \approx \lambda D/d\))
- One slit closed → single-slit envelope
Key takeaways
- No optics was programmed — interference emerges from the discretized wave equation. Huygens' principle is a theorem of the PDE, not an extra assumption.
- The same code structure (mask + stencil + march) handles waveguides, cavities, and scattering obstacles: change the mask, not the method.
- Hard walls reflect — visible artifacts motivate absorbing boundaries (PML) in serious electromagnetic work.