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added analytic formulas (#46)
* added analytic formulas
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‎.ipynb_checkpoints/faq_categories-checkpoint.json‎

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‎_faqs/.ipynb_checkpoints/what-does-the-FluxMonitor-record-checkpoint.md‎

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---
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_schema: default
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title: What does the FluxMonitor record?
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date: 2025-09-08 17:29:25
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enabled: true
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category: Monitors
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_inputs:
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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hidden: true
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date:
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type: datetime
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label: DATE
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instance_value: NOW
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category:
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type: select
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The FluxMonitor records field data tangential $E$ and $H$ fields colocated to the cell boundaries in the monitor's 2D plane grid. It then computes and integrates the Poynting vector, returning the real part of this integral as the flux.
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<div> </div>
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<center>
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Real$(\sum\limits_{\text{cells in monitor}}\frac{1}{2}(E_1\cdot H_2^*-E_2\cdot H_1^*)dA_{\text{cell}})$
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</center><br>
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where $dA$ is the area of each cell in the monitor, and the field components 1 and 2 are given depending on the monitor geometry: for $x$-normal monitors, 1 is $y$ and 2 is $z$; for $y$-normal monitors, 1 is $x$ and 2 is $z$; for $z$-normal monitors, 1 is $x$ and 2 is $y$. See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/data/monitor_data.html#ElectromagneticFieldData).

‎_faqs/.ipynb_checkpoints/what-is-the-formula-for-ContinuousWave-checkpoint.md‎

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---
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_schema: default
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title: What is the formula for ContinuousWave?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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type: switch
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date:
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label: DATE
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instance_value: NOW
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values: data.faq_categories
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value_key: key
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text:
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- key: category_name
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---
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The ContinuousWave sourcetime, if the DC component is zeroed out, has the following formula:
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<div> </div>
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<center>
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$A(1+e^{-\frac{(t-t_o*t_w)}{t_w}})^{-1}e^{i\phi}e^{-i\omega_0 t}$
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</center><br>
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<div>where $A$ is the amplitude, $t_o$ is the time offset, $t_w=\frac{1}{2\pi f_{width}}$ is the width of the pulse in seconds, $\phi$ is the phase shift, and $\omega_0=2\pi f_0$.</div>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/source/time.html#ContinuousWave).</div>

‎_faqs/.ipynb_checkpoints/what-is-the-formula-for-GaussianBeam-checkpoint.md‎

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---
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_schema: default
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title: What is the formula for GaussianBeam?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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_inputs:
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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type: switch
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hidden: true
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date:
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type: datetime
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label: DATE
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instance_value: NOW
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category:
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type: select
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The GaussianBeam source has the following scalar field amplitude, in cylindrical coordinates:
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<div> </div>
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<center>
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$u(r,z)=\frac{w_0}{w(z)}e^{-\frac{r^2}{w(z)^2}}e^{i(zk_0 + \frac{r^2k_0}{2R(z)} - \psi_g)}$
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</center><br>
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where:
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<ul>
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<li>$z$ is the propagation direction</li>
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<li>$k_0=\frac{2\pi nf}{c}$ are the wavenumbers of the frequencies $f$ where the beam is sampled</li>
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<li>$w_0$ is the beam waist</li>
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<li>$w(z)=w_0\sqrt{1 + \frac{(z + z_0)^2}{z_r}}$, where $z_0$ is the waist distance and $z_r=\frac{1}{2}w_0^2k_0$ is the Rayleigh range</li>
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<li>$R(z)=z(1 + (\frac{z_r}{z})^2)$ is the radius of curvature of the wavefront at $z$</li>
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<li>$\psi_g=\arctan(\frac{z+z_0}{z_r})-\arctan(\frac{z_0}{z_r})$ is the Gouy phase</li>
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</ul>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/beam.html#GaussianBeamProfile).</div>

‎_faqs/.ipynb_checkpoints/what-is-the-formula-for-GaussianPulse-checkpoint.md‎

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---
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_schema: default
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title: What is the formula for GaussianPulse?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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instance_value: NOW
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The GaussianPulse sourcetime, if the DC component is zeroed out, has the following formula:
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<div> </div>
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<center>
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$(i+\frac{t}{t_w^2\omega_0})Ae^{i\phi}e^{-i\omega_0 t}e^{-\frac{(t - t_o*t_w)^2}{2t_w^2}}$
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</center><br>
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<div> </div>
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<div>If the DC component is not zeroed out, the formula is</div><br>
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<center>
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$iAe^{i\phi}e^{-i\omega_0 t}e^{-\frac{(t - t_o*t_w)^2}{2t_w^2}}$
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</center><br>
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<div>where $A$ is the amplitude, $t_o$ is the time offset, $t_w=\frac{1}{2\pi f_{width}}$ is the width of the pulse in seconds, $\phi$ is the phase shift, and $\omega_0=2\pi f_0$.</div>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/source/time.html#GaussianPulse).</div>

‎_faqs/.ipynb_checkpoints/what-is-the-formula-for-PlaneWave-checkpoint.md‎

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---
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title: What is the formula for PlaneWave?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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title:
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label: QUESTION TITLE
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label: DATE
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instance_value: NOW
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values: data.faq_categories
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value_key: key
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- key: category_name
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---
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The PlaneWave source has the following scalar field amplitude:
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<div> </div>
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<center>
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$u(z)=e^{ik_z z}$
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</center><br>
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where $z$ is the propagation direction and $k_0=\frac{2\pi nf}{c}$ are the wavenumbers of the frequencies $f$ where the beam is sampled. If the source is specified as a fixed angle source, $k_0$ is multiplied by $\cos\theta$.
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/beam.html#PlaneWaveBeamProfile).</div>

‎_faqs/what-does-the-FluxMonitor-record.md‎

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---
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_schema: default
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title: What does the FluxMonitor record?
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date: 2025-09-08 17:29:25
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enabled: true
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category: Monitors
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_inputs:
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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label: DATE
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instance_value: NOW
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category:
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The FluxMonitor records field data tangential $E$ and $H$ fields colocated to the cell boundaries in the monitor's 2D plane grid. It then computes and integrates the Poynting vector, returning the real part of this integral as the flux.
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<div> </div>
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<center>
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Real$(\sum\limits_{\text{cells in monitor}}\frac{1}{2}(E_1\cdot H_2^*-E_2\cdot H_1^*)dA_{\text{cell}})$
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</center><br>
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where $dA$ is the area of each cell in the monitor, and the field components 1 and 2 are given depending on the monitor geometry: for $x$-normal monitors, 1 is $y$ and 2 is $z$; for $y$-normal monitors, 1 is $x$ and 2 is $z$; for $z$-normal monitors, 1 is $x$ and 2 is $y$. See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/data/monitor_data.html#ElectromagneticFieldData).

‎_faqs/what-is-the-formula-for-ContinuousWave.md‎

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---
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title: What is the formula for ContinuousWave?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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type: switch
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hidden: true
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date:
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type: datetime
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label: DATE
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instance_value: NOW
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category:
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The ContinuousWave sourcetime, if the DC component is zeroed out, has the following formula:
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<div> </div>
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<center>
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$A(1+e^{-\frac{(t-t_o*t_w)}{t_w}})^{-1}e^{i\phi}e^{-i\omega_0 t}$
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</center><br>
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<div>where $A$ is the amplitude, $t_o$ is the time offset, $t_w=\frac{1}{2\pi f_{width}}$ is the width of the pulse in seconds, $\phi$ is the phase shift, and $\omega_0=2\pi f_0$.</div>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/source/time.html#ContinuousWave).</div>

‎_faqs/what-is-the-formula-for-GaussianBeam.md‎

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---
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_schema: default
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title: What is the formula for GaussianBeam?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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_inputs:
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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type: switch
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hidden: true
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date:
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type: datetime
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label: DATE
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instance_value: NOW
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category:
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type: select
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The GaussianBeam source has the following scalar field amplitude, in cylindrical coordinates:
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<div> </div>
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<center>
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$u(r,z)=\frac{w_0}{w(z)}e^{-\frac{r^2}{w(z)^2}}e^{i(zk_0 + \frac{r^2k_0}{2R(z)} - \psi_g)}$
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</center><br>
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where:
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<ul>
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<li>$z$ is the propagation direction</li>
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<li>$k_0=\frac{2\pi nf}{c}$ are the wavenumbers of the frequencies $f$ where the beam is sampled</li>
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<li>$w_0$ is the beam waist</li>
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<li>$w(z)=w_0\sqrt{1 + \frac{(z + z_0)^2}{z_r}}$, where $z_0$ is the waist distance and $z_r=\frac{1}{2}w_0^2k_0$ is the Rayleigh range</li>
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<li>$R(z)=z(1 + (\frac{z_r}{z})^2)$ is the radius of curvature of the wavefront at $z$</li>
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<li>$\psi_g=\arctan(\frac{z+z_0}{z_r})-\arctan(\frac{z_0}{z_r})$ is the Gouy phase</li>
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</ul>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/beam.html#GaussianBeamProfile).</div>

‎_faqs/what-is-the-formula-for-GaussianPulse.md‎

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---
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_schema: default
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title: What is the formula for GaussianPulse?
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date: 2025-09-05 17:29:25
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enabled: true
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category: Sources
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_inputs:
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title:
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type: text
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label: QUESTION TITLE
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enabled:
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type: switch
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hidden: true
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date:
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type: datetime
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label: DATE
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instance_value: NOW
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category:
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type: select
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options:
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values: data.faq_categories
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value_key: key
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preview:
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text:
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- key: category_name
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---
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The GaussianPulse sourcetime, if the DC component is zeroed out, has the following formula:
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<div> </div>
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<center>
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$(i+\frac{t}{t_w^2\omega_0})Ae^{i\phi}e^{-i\omega_0 t}e^{-\frac{(t - t_o*t_w)^2}{2t_w^2}}$
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</center><br>
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<div> </div>
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<div>If the DC component is not zeroed out, the formula is</div><br>
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<center>
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$iAe^{i\phi}e^{-i\omega_0 t}e^{-\frac{(t - t_o*t_w)^2}{2t_w^2}}$
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</center><br>
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<div>where $A$ is the amplitude, $t_o$ is the time offset, $t_w=\frac{1}{2\pi f_{width}}$ is the width of the pulse in seconds, $\phi$ is the phase shift, and $\omega_0=2\pi f_0$.</div>
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<div>See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/source/time.html#GaussianPulse).</div>

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