1. EMLAB 1 Power Flow and PML Placement in FDTD

2. EMLAB 2 Lecture Outline Review Total Power by Integrating the Poynting Vector Total Power by Plane Wave Spectrum Example of Grating Diffraction PML Placement

3. EMLAB 3 Complex Wave Vectors Uniform amplitude Oscillations move energy Considered to be a propagating wave Decaying amplitude No oscillations, no flow of energy Considered to be evanescent Decaying amplitude Oscillations move energy Considered to be a propagating wave (not evanescent)

4. EMLAB 4 Evanescent Fields in 2D Simulations

5. EMLAB 5 Fields in Periodic Structures Waves in periodic structures take on the same periodicity as their host.

6. EMLAB 6 The Plane Wave Spectrum (1 of 2) 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.

7. EMLAB 7 The Plane Wave Spectrum (2 of 2) The plane wave spectrum can be calculated as follows Each wave must be separately phase matched into the medium with refractive index n 2.

8. EMLAB 8 Total Power by Integrating Poynting Vector

9. EMLAB 9 Concept of Integrating the Poynting Vector The Poynting vector is the instantaneous flow of power. We must integrate the Poynting vector to calculate total power flowing out of the grid at any instant. S is the cross section of the grid.

10. EMLAB 10 Power Flow Out of Devices To calculate the power flow away from a device, we are only interested in the normal component of the Poynting vector Pz. For the diagram below, it is the z ‐ component.

11. EMLAB 11 Vector Components of the Poynting Vector Expanding the E×H cross product into its vector components, we get For power flowing into z ‐ axis boundaries, we only care about power in the z direction. This must be considered when calculating transmitted vs. reflected power. We reverse the sign to calculate reflected power.

12. EMLAB 12 We defined our 2D grid to be in the x-y plane. To be most consistent with convention, power will leave a device when travelling in the y direction. Power Flow for the Ez Mode CAUTION: We must interpolate Ez and Hx at a common point in the grid to calculate the Poynting vector! The Ez mode does not contain Ex or Hz so the Poynting vector is simply

13. EMLAB 13 Total Power by Plane Wave Spectrum

14. EMLAB 14 Electromagnetic Power Flow The instantaneous direction and intensity of power flow at any point in space is given by the Poynting vector. The RMS power flow is then This is typically just written as

15. EMLAB 15 Power Flow in LHI Materials The regions outside a grating are almost always linear, homogeneous, and isotropic (LHI). In this case E, H, and k are all perpendicular. In addition, power flows in the same direction as k. Under these conditions, the expression for RMS power flow becomes

16. EMLAB 16 The field magnitudes in LHI materials are related through the material impedance. Given this relation, we can eliminate the H field from the expression for RMS power flow. Eliminate the Magnetic Field

17. EMLAB 17 Power Flow Away From Grating To calculate the power flowing away from the grating, we are only interested in the z ‐ component of the Poynting vector.

18. EMLAB 18 RMS Power of the Diffracted Modes Recall that the field scattered from a periodic structure can be decomposed into a Fourier series. The term Sm is the amplitude and polarization of the mth diffracted harmonic. Therefore, power flow away from the grating due to the mth diffracted order is

19. EMLAB 19 Power of the Incident Wave From the previous equation, the power flow of the incident wave into the grating is

20. EMLAB 20 Diffraction Efficiency Diffraction efficiency is defined as the power in a specific diffracted order divided by the applied incident power. Assuming the materials have no loss or gain, conservation of energy requires that Despite the title “efficiency,” we don’t always want this high. We often want to control how much power gets diffracted into each mode. So it is not good or bad to have high or low diffraction efficiency.

21. EMLAB 21 So far, we have derived expressions for the incident power and power in the spatial harmonics. We also defined the diffraction efficiency of the mth harmonic. We can now derive expressions for the diffraction efficiencies of the spatial harmonics by combining these expressions. Putting it All Together

22. EMLAB 22 Diffraction Efficiency for Magnetic Fields We just calculated the diffraction efficiency equations based on having calculated the electric fields only. Sometimes we solve Maxwell’s equations for the magnetic fields. In this case, the diffraction efficiency equations are

23. EMLAB 23 Calculating Power Flow in FDTD

24. EMLAB 24 Process of Calculating Transmittance and Reflectance 1.Perform FDTD simulation ① Calculate the steady ‐ state field in the reflected and transmitted record planes. 2.For each frequency of interest… ① Calculate the wave vector components of the spatial harmonics ② Calculate the complex amplitude of the spatial harmonics ③ Calculate the diffraction efficiency of the spatial harmonics ④ Calculate over all reflectance and transmittance ⑤ Calculate energy conservation.

25. EMLAB 25 Step 1: Perform FDTD Simulation

26. EMLAB 26 Step 1: Perform FDTD Simulation This same philosophy for constructing the problem holds for three dimensions as well.

27. EMLAB 27 Step 2: Calculate Steady ‐ State Fields

28. EMLAB 28 Step 3: Calculate Wave Vector Components

29. EMLAB 29 Step 4: Calculate the Amplitudes of the Spatial Harmonics

30. EMLAB 30 Note 1: is the amplitude of the source obtained by Fourier transforming the source function.  Note 2: This operation is performed for every frequency of interest. Step 5: Calculate Diffraction Efficiencies

31. EMLAB 31 Step 6: Reflectance and Transmittance Reflectance is the total fraction of power reflected from a device. Therefore, it is equal to the sum of all the reflected modes. Transmittance is the total fraction of power transmitted through a device. Therefore, it is equal to the sum of all the transmitted modes. Note: This operation is performed for every frequency of interest.

32. EMLAB 32 Step 7: Calculate Energy Conservation Assuming you have not included loss or gain into your simulation, the reflectance plus transmittance should equal 100%. It is ALWAYS good practice to calculate this total to check for conservation of energy. This may deviate from 100% when: Energy still remains on the grid and more iterations are needed. The boundary conditions are not working properly and need to be corrected. Rounding errors are two severe and greater grid resolution is needed. You have included loss or gain into you materials. Note: This operation is performed for every frequency of interest.

33. EMLAB 33 Visualizing the Data Arrays

34. EMLAB 34 Procedure for FDTD

35. EMLAB 35 MATLAB Code for Calculating Power

36. EMLAB 36 Example of Grating Diffraction

37. EMLAB 37 Binary Grating (Use as a Benchmark)

38. EMLAB 38 PML Placement

39. EMLAB 39 Electromagnetic Tunneling Evanescent fields do not oscillate so they cannot “push” power. Usually, an evanescent field is just a “temporary” configuration field power is stored. Eventually, the power leaks out as a propagating wave. There exists one exception (maybe more) where evanescent fields contribute to power transport. This happens when a high refractive index material cuts through the evanescent field. The field may then become propagating in the high- index material. This is analogous to electron tunneling in semiconductors.

40. EMLAB 40 Fields that are evanescent at the record plane will not be counted as transmitted power. Evanescent fields can become propagating waves inside the PML and tunnel power out of the model. This provides an unaccounted for escape path for power. PMLs Should Not Touch Evanescent Fields

41. EMLAB 41 When a model incorporates waves at angles, fields can become evanescent. It is good practice to place the PML well outside of the evanescent field. For non ‐ resonant devices, the space between the device and PML is typically λ/4. For resonant devices, this is more commonly λ. Some structures have evanescent fields that extend many wavelengths. You can identify this situation by visualizing your fields during the simulation. To be sure, run a simulation and visualize the field. Evanescent Fields in 2D Simulations

42. EMLAB 42 Typical 2D FDTD Grid Layout (Style #1)

43. EMLAB 43 Typical 2D FDTD Grid Layout (Style #2)