FEATool Multiphysics  v1.10
Finite Element Analysis Toolbox
Interference and Diffraction

The classic double-slit experiment considers a planar and periodic oscillating wave which hits and passes two narrow slits. Assuming the slits are narrow enough, the passing waves will bend and cause an interference pattern, while diffraction will attenuate the off axis resulting amplitude.

In the example the Helmholtz equation is used to model the wave phenomena

\[ -( \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} ) - k^2 A = 0 \]

where A is the amplitude of the wave,and k the wave number (k = 2·π/λ).

Tutorial

This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Classic PDE > Interference and Diffraction from the File menu. Or alternatively, follow the step-by-step instructions below.

  1. To start a new model click the New Model toolbar button, or select New Model... from the File menu.
  2. Select the Custom Equation physics mode from the Select Physics drop-down menu.
  3. Enter A into the Dependent Variable Names edit field.

    interference_diffraction1_03_50.png
  4. Press OK to finish the physics mode selection.
  5. Select Circle from the Geometry menu.
  6. Enter 0.8 into the radius edit field.
  7. Press OK to finish and close the dialog box.
  8. Select Rectangle from the Geometry menu.
  9. Enter -0.8 into the xmin edit field.
  10. Enter 0.8 into the xmax edit field.
  11. Enter -0.8 into the ymin edit field.
  12. Enter 0 into the ymax edit field.
  13. Press OK to finish and close the dialog box.
  14. Select C1 and R1 in the geometry object Selection list box.
  15. Press the - / Subtract geometry objects Toolbar button.

    interference_diffraction1_15_50.png
  16. Select Rectangle from the Geometry menu.
  17. Enter -0.08-0.02 into the xmin edit field.
  18. Enter -0.08+0.02 into the xmax edit field.
  19. Enter -0.2 into the ymin edit field.
  20. Enter 0 into the ymax edit field.
  21. Press OK to finish and close the dialog box.
  22. Select Rectangle from the Geometry menu.
  23. Enter 0.08-0.02 into the xmin edit field.
  24. Enter 0.08+0.02 into the xmax edit field.
  25. Enter -0.2 into the ymin edit field.
  26. Enter 0 into the ymax edit field.
  27. Press OK to finish and close the dialog box.
  28. Select CS1, R2, and R3 in the geometry object Selection list box.
  29. Press the + / Add geometry objects Toolbar button.

    interference_diffraction1_29_50.png
  30. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
  31. Enter 0.015 into the Grid Size edit field, and press the Generate button to call the grid generation algorithm.

    interference_diffraction1_32_50.png
  32. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.
  33. Press the edit button.
  34. Enter -(Ax_x + Ay_y) - k^2*A_t = 0 into the Equation for A edit field.

    interference_diffraction1_35_50.png
  35. Press OK to finish and close the dialog box.
  36. Press OK to finish the equation and subdomain settings specification.
  37. Press the Constants Toolbar button, or select the corresponding entry from the Equation menu, and add the following modeling constants for the wave length wl, and wave number k in the Model Constants and Expressions dialog box.
NameExpression
wl0.08
kpi*2/wl


interference_diffraction1_38_50.png
  1. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.

First set homogenous Neumann conditions for all boundaries.

  1. Select all the boundaries (1 - 11) in the Boundaries list box.
  2. Select the Neumann, g_A radio button.
  3. Enter 0 into the Dirichlet/Neumann coefficient edit field.

    interference_diffraction1_42_50.png

An incoming planar wave is featured at the inlet with the complex boundary condition n·∇(A) + k·i·A = 2·k·i which can be implemented as a Neumann boundary condition.

  1. Select 4 and 8 in the Boundaries list box.
  2. Enter -k*i*A + 2*k*i into the Dirichlet/Neumann coefficient edit field.

    interference_diffraction1_44_50.png

The outlet is assumed non-reflective and n·∇(A) + k·i·A = 0.

  1. Select 1 and 2 in the Boundaries list box.
  2. Enter -k*i*A into the Dirichlet/Neumann coefficient edit field.

    interference_diffraction1_46_50.png
  3. Press OK to finish the boundary condition specification.
  4. Switch to Solve mode by clicking on the corresponding Mode Toolbar button.
  5. Press the = Toolbar button to call the solver. After the problem has been solved FEATool will automatically switch to postprocessing mode and plot the computed solution.

After the problem has been solved FEATool will automatically switch to postprocessing mode and here display the computed wave amplitude A. The interference pattern can be clearly seen with four lines where the waves have been canceled out completely.

interference_diffraction1_49_50.png

The Point/Line Evaluation tool can be used to visualize the interference and diffraction pattern at the boundary.

  1. Select Point/Line Evaluation... from the Post menu.
  2. Select 1 and 2 in the Boundaries list box.
  3. Press OK to finish and plot the amplitude curve.

    interference_diffraction1_52_50.png

The interference and diffraction custom equation model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.