GaussianPulse

GaussianPulse (opens new window) defines a carrier-modulated Gaussian pulse.

Formula

The plotted example compares both values of remove_dc_component using Tidy3D's amp_time and amp_freq methods.

Let $A$ be amplitude, $\phi$ be phase, $f_0$ be freq0, $f_w$ be fwidth, $\omega_0 = 2\pi f_0$, $\sigma_t = 1 / (2\pi f_w)$, and $t_p = \mathrm{offset}\,\sigma_t$.

When remove_dc_component is False, the pulse uses the unmodified Gaussian:

$s(t) = i A e^{i\phi} e^{-i\omega_0 t} \exp\left[-\frac{(t - t_p)^2}{2\sigma_t^2}\right]$

Its frequency-domain amplitude is:

$\hat{s}(f) = \frac{iA}{f_w} e^{i\phi + i2\pi(f - f_0)t_p} \exp\left[-\frac{(f - f_0)^2}{2f_w^2}\right]$

When remove_dc_component is True, Tidy3D applies a derivative-style correction. Define:

$f_p = \frac{1}{2}\left(f_0 + \sqrt{f_0^2 + 4f_w^2}\right), \qquad \Delta t = \begin{cases} \sigma_t\sqrt{1 - f_0^2/f_w^2}, & f_w > f_0 \\ 0, & f_w \le f_0 \end{cases}$

and $t_s = t_p + \Delta t$. The time-domain pulse becomes:

$s(t) = A e^{i\phi} e^{-i\omega_0 t} \exp\left[-\frac{(t - t_s)^2}{2\sigma_t^2}\right] \frac{i\omega_0 + (t - t_s)/\sigma_t^2}{2\pi f_p}$

The corresponding frequency-domain amplitude is:

$\hat{s}(f) = \frac{f}{2\pi f_p} \frac{iA}{f_w} e^{i\phi + i2\pi(f - f_0)t_s} \exp\left[-\frac{(f - f_0)^2}{2f_w^2}\right]$

Because of the leading $f$ factor, $\hat{s}(0)=0$, so the DC component is removed.

Parameters

freq0: Central frequency of the pulse.

• Unit: Hz
• Constraint: greater than 0
• Required field

fwidth: Standard deviation of the frequency content of the pulse.

• Unit: Hz
• Constraint: greater than 0
• Required field

offset: Time delay of the pulse peak in units of $1 / (2\pi f_\mathrm{width})$.

• Default: 5

remove_dc_component: Toggle whether to remove the DC component from the Gaussian pulse spectrum.

• Default: True

amplitude: Real-valued maximum amplitude.

• Default: 1

phase: Phase shift of the time dependence.

• Unit: rad
• Default: 0