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Power Flow and PMLPlacement in FDTD

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2Lecture Outline

• Review

• Total Power by Integrating the Poynting Vector

• Total Power by Plane Wave Spectrum

• Example of Grating Diffraction

• PML Placement

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3Complex Wave Vectors

• Uniform amplitude• Oscillations move en-

ergy• Considered to be a prop-

agating wave

• Decaying amplitude• No oscillations, no flow

of energy• Considered to be

evanescent

• Decaying amplitude• Oscillations move energy• Considered to be a propa-

gating wave (not evanes-cent)

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Evanescent Fields in 2D Simulations

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5Fields in Periodic Structures

Waves in periodic structures take on the same periodicity as their host.

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6The 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 spec-trum” of a field.

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7The 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 n2.

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Total Power by Integrating Poynting Vector

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9Concept 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.

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10Power 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.

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11Vector 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 direc-tion.

This must be considered when calculating transmitted vs. reflected power. We re-verse the sign to calculate reflected power.

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We defined our 2D grid to be in the x-y plane. To be most consistent with conven-tion, 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 cal-culate the Poynting vector!

The Ez mode does not contain Ex or Hz so the Poynting vector is simply

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Total Power by Plane Wave Spectrum

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14Electromagnetic 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

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15Power 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

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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

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

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18RMS 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

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19Power of the Incident Wave

From the previous equation, the power flow of the incident wave into the grating is

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20Diffraction Efficiency

Diffraction efficiency is defined as the power in a specific diffracted order di-vided 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.

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So far, we have derived expressions for the incident power and power in the spa-tial 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

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22Diffraction 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 dif-fraction efficiency equations are

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Calculating Power Flow in FDTD

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24Process of Calculating Transmittance and Reflectance

1. Perform FDTD simulation

① Calculate the steady‐state field in the reflected and transmit-

ted record planes.

2. For each frequency of interest…

① Calculate the wave vector components of the spatial harmon-

ics

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.

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25Step 1: Perform FDTD Simulation

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Step 1: Perform FDTD Simulation

This same philosophy for constructing the problem holds for three dimensions as well.

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Step 2: Calculate Steady‐State Fields

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28Step 3: Calculate Wave Vector Components

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29Step 4: Calculate the Amplitudes of the Spatial Harmonics

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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

incS

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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. There-fore, it is equal to the sum of all the transmitted modes.

Note: This operation is performed for every frequency of interest.

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32Step 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 en-ergy. 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.

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33Visualizing the Data Arrays

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Procedure for FDTD

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35MATLAB Code for Calculating Power

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Example of GratingDiffraction

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37Binary Grating (Use as a Benchmark)

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PML Placement

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39Electromagnetic Tunneling

Evanescent fields do not oscillate so they cannot “push” power.

Usually, an evanescent field is just a “tempo-rary” configuration field power is stored. Even-tually, 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 mate-rial cuts through the evanescent field. The field may then become propagating in the high-index material. This is analogous to electron tunneling in semiconductors.

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• 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

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When a model incorporates waves at angles, fieldscan become evanescent.

It is good practice to place the PML well outside ofthe evanescent field.

For non‐resonant devices, the space between thedevice and PML is typically λ/4. For resonantdevices, this is more commonly λ.

Some structures have evanescent fields thatextend many wavelengths. You can identify thissituation by visualizing your fields during thesimulation.

To be sure, run a simulation and visualize the field.

Evanescent Fields in 2D Simulations

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

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