1. Gratings and the Plane Wave Spectrum

3. Lecture Outline • Review • Wave vectors
• Phase matching at an interface
• Electromagnetic behavior at an interface
• Waves in periodic media
• The plane wave spectrum

4. Typical 2D FDTD Grid Layouts

5. The Total‐Field/Scattered‐Field Framework

6. Corrections to Finite‐Difference Equations
at the Problem Cells (Ez Mode)
On the scattered‐field side of the TF/SF interface, the curl equation contains a term from the total‐field side. The source must be subtracted from this term to make it look like a scattered‐field quantity.
On the total‐field side of the TF/SF interface, the curl
equation contains a term from the scattered‐field side.
The source must be added to this term to make it look
like a total‐field quantity.

7. Calculation of the Source Functions (Ez Mode)
We calculate the electric field as
We calculate the magnetic field as

8. TF/SF Block Diagram for Ez Mode

9. TF/SF Block Diagram for Hz Mode

10. Wave Vectors

11. Wave Vector 
A wave vector conveys two pieces of information at the same time. First, its direction is normal to the wave fronts. Second, its magnitude is 2π divided by the spatial period of the wave (wavelength).

12. The Complex Wave Vector
A wave travelling the +z direction can be written in terms of the wave number k as
The wave vector can be complex. We can write it in terms of a real and an imaginary part.
Substituting this into or uniform plane wave formula let’s us interpret the real and imaginary parts of k.

13. 1D Waves with Complex k Decaying amplitude Oscillations move energy
Considered to be a propagating wave (not
evanescent)
Decaying amplitude
No oscillations, no flow of energy
Considered to be evanescent
Uniform amplitude
Oscillations move energy
Considered to be a propagating wave

14. 2D Waves with Doubly Complex k

15. Phase Matching at an Interface

16. Dispersion Relation and Index Ellipsoids
The dispersion relation for a material relates the wave vector to frequency.
Essentially, it tells us the refractive index as a function of direction through the material.
For isotropic materials, this is
The dispersion relation defines a surface called an “index ellipsoid.” The vector connecting the origin to a point on the sphere is the k‐vector for that direction from which refractive index can be calculated.
This surface is a sphere for isotropic materials.

17. Geometrical Interpretation of the Dispersion Relation
The dispersion relation for isotropic materials is essentially just the Pythagorean theorem.
It says a waves sees the same refractive index no matter what direction the wave is travelling.

18. Index Ellipsoid in Two Different Materials

19. Phase Matching at the Interface Between Two Materials

20. Phase Matching at the Interface Between Two Materials

21. Electromagnetic
Behavior at an Interface

22. Longitudinal Component of the Wave Vector
1. Boundary conditions require that the tangential component of the wave vector is continuous across the interface. Assuming kx is purely real in material 1, kx must be purely real in material 2.
We have oscillations and energy flow in the x direction.
Knowing that the dispersion relation must be satisfied, the longitudinal component of the wave vector in material 2 is calculated from the dispersion relation in material 2.

23. Field at an Interface Above and Below the
Critical Angle (Ignoring Reflections)
1. The field always penetrates material 2, but it may not propagate.
2. Above the critical angle, penetration is greatest near the critical angle.
3. Very high spatial frequencies are supported despite the dispersion relation.
4. In material 2, energy always flows along x, but not necessarily along y.

24. Field Visualization for θC=45°

25. Waves in Periodic Media

26. What is a Periodic Structure?
Periodicity at the Atomic Scale
Larger‐Scale Periodicity
Pure materials are periodic at the atomic scale. Our devices are periodic at a much larger scale, but smaller than a wavelength. The math describing how things are periodic is the same for both atomic scale and larger scale.

27. Examples of Periodic Electromagnetic Devices

28. Fields are Perturbed by Objects
A portion of the wave front is delayed after travelling through the dielectric object.

29. Fields in Periodic Structures
Waves in periodic structures take on
the same periodicity as their host.

30. Diffraction from Gratings

31. Grating Equation
The angles of the diffracted modes are related to the wavelength and grating through the grating equation.
The grating equation only predicts the directions of the modes, not how much power is in them.

32. Summary of Effect of Grating Periodicity

33. The Plane Wave Spectrum

34. 1D Complex Fourier Series
If a function f(x) is periodic with period Λx, it can be expanded into a
complex Fourier series.
Typically, we retain only a finite number of terms in the expansion.

35. Periodic Functions Can Be Expanded into a Fourier Series
Waves in periodic structures obey Bloch’s equation
The envelop A(x) is periodic with period Λx so it can be expanded into a Fourier series.

36. A periodic field can be expanded into a Fourier series.
This has the form of a sum of
plane waves all at different angles

37. The Plane Wave Spectrum
We rearranged terms and saw that a periodic field can also be thought of as an infinite sum of plane waves at different angles. This is the “plane wave spectrum” of a field.

38. Longitudinal Wave Vector Components of the Plane Wave Spectrum
The wave incident on a grating can be written as
Phase matching into the grating leads to
Note: kx(m) is always real.
Each wave must satisfy the dispersion relation.

39. Visualizing Phase Matching into the Grating
The wave vector expansion for the first 11 modes can be visualized as…
Each of these is phase matched into material 2. The longitudinal component of the wave vector is calculated using the dispersion relation in material 2.
ky is imaginary.
The field in material 2
is evanescent.
ky is real.
A wave propagates into material 2.
ky is imaginary. The field in material 2 is evanescent.
Note: The “evanescent” fields in material 2 are not completely
evanescent. They have a purely real kx so they do flow energy
in the transverse direction.