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6.6 Waveforms

Global variable: group Waveforms

In Angora, a time waveform is defined as a self-contained structure that can be used by other structures; such as a Hertzian dipole source or a plane-wave injector. The library of available time waveforms will be expanded in the future. Please send any comments, suggestions, and requests to help@angorafdtd.org.

An example usage of Waveforms:

 
Waveforms:
{
    GaussianWaveforms:
    (
        {
            …
        }
    );

    DifferentiatedGaussianWaveforms:
    (
        {
            …
        }
    );
    …
    …
};

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6.6.1 Gaussian Waveforms

Sub-variable of Waveforms: list GaussianWaveforms

This variable is used to define Gaussian time waveforms given by the formula

\begin{displaymath}
f(t)=A\ \exp\left({{-(t-n_{\tau}\tau)^{2}}\over{2\tau^{2}}}\right)
\end{displaymath}

The peak, 10%-amplitude (-20 dB power), and 1%-amplitude (-40 dB power) frequencies in the spectrum of the Gaussian are $\omega=0,\ \omega=2.15/\tau,\ \omega=3.035/\tau$ , respectively.

Gaussian waveforms are defined as follows:

 
Waveforms:
{
    GaussianWaveforms:
    (
        {
            waveform_tag = "my_waveform";
            amplitude = 1.0;
            tau = 2.1291e-15;
            delay = 3;
        },
        {
            …
            …
        }
    );
};
Sub-variable of GaussianWaveforms: string waveform_tag

This is the string tag by which the waveform will later be referred to by another structure that requires a time waveform in its definition.

Sub-variable of GaussianWaveforms: floating-point amplitude (default: 1.0)

This specifies the variable $A$ in the above equation defining the Gaussian waveform.

Sub-variable of GaussianWaveforms: floating-point tau (units: sec)

This specifies the variable $\tau$ in the above equation defining the Gaussian waveform.

Sub-variable of GaussianWaveforms: floating-point delay (default: 0.0)

This specifies the variable $n_{\tau}$ in the above equation defining the Gaussian waveform.


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6.6.2 Differentiated-Gaussian Waveforms

Sub-variable of Waveforms: list DifferentiatedGaussianWaveforms

This variable is used to define differentiated Gaussian time waveforms, given by the formula

\begin{displaymath}
f(t)=A\ {{d^n}\over {dt^n}}\left[\exp\left({{-(t-n_{\tau}\ta...
...ght)\exp\left({{-(t-n_{\tau}\tau)^{2}}\over{2\tau^{2}}}\right)
\end{displaymath}

where $H_n(x)$ are the (physicists’) Hermite polynomials.

The peak frequency in the spectrum of the differentiated-Gaussian is $\omega=1/\tau$ , the 10%-amplitude (-20 dB power) frequencies are $\omega=0.06/\tau$ and $\omega=2.76/\tau$ ; and the 1%-amplitude (-40 dB power) frequencies are $\omega=0.006/\tau$ and $\omega=3.57/\tau$ .

Differentiated Gaussian waveforms are defined as follows:

 
Waveforms:
{
    DifferentiatedGaussianWaveforms:
    (
        {
            waveform_tag = "my_waveform";
            amplitude = 1.0;
            tau = 2.1291e-15;
            delay = 3;
            n_diff = 3;
        },
        {
            …
            …
        }
    );
};
Sub-variable of DifferentiatedGaussianWaveforms: string waveform_tag

This is the string tag by which the waveform will later be referred to by another structure that requires a time waveform in its definition.

Sub-variable of DifferentiatedGaussianWaveforms: floating-point amplitude (default: 1.0)

This specifies the variable $A$ in the above equation defining the differentiated Gaussian waveform.

Sub-variable of DifferentiatedGaussianWaveforms: floating-point tau (units: sec)

This specifies the variable $\tau$ in the above equation defining the differentiated Gaussian waveform.

Sub-variable of DifferentiatedGaussianWaveforms: floating-point delay (default: 0.0)

This specifies the variable $n_{\tau}$ in the above equation defining the differentiated Gaussian waveform.

Sub-variable of DifferentiatedGaussianWaveforms: integer n_diff

This specifies the order of differentiation $n$ in the above equation defining the differentiated Gaussian waveform.


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6.6.3 Modulated-Gaussian Waveforms

Sub-variable of Waveforms: list ModulatedGaussianWaveforms

This variable is used to define sinusoidally-modulated Gaussian time waveforms, given by the formula

\begin{displaymath}
f(t)=A\ g\big(2\pi f_0 (t-n_{\tau}\tau)+\phi\big)\exp\left({{-(t-n_{\tau}\tau)^{2}}\over{2\tau^{2}}}\right)
\end{displaymath}

where the function $g(t)$ is a sinusoidal function, being either $\sin(t)$ or $\cos(t)$ .

The peak frequency in the spectrum of the modulated-Gaussian is $\omega=\omega_0=2\pi f_0$ , the 10%-amplitude (-20 dB power) frequencies are $\omega=\omega_0\pm2.15/\tau$ ; and the 1%-amplitude (-40 dB power) frequencies are $\omega=\omega_0\pm3.035/\tau$ . A MATLAB script named ‘mod_gaussian_wf.m’ is distributed as part of the Angora package, which calculates the center frequency $f_0$ and the time constant $\tau$ of a modulated-Gaussian waveform that has the desired lower and upper cutoff wavelengths, and the desired amount of attenuation at these wavelengths. It also outputs the -40 dB wavelength and the suggested maximum spatial time step in the simulation (which is the -40 dB wavelength divided by 15). This script is installed in the directory ‘$(prefix)/share/angora/’ (see Compilation and Installation). If Angora was installed without any $(prefix) configuration option, the default location is ‘/usr/local/share/angora/’. This script can also be downloaded directly from the Angora website (link here).

Modulated Gaussian waveforms are defined as follows:

 
Waveforms:
{
    ModulatedGaussianWaveforms:
    (
        {
            waveform_tag = "my_waveform";
            modulation_type = "sine";
            amplitude = 1.0;
            tau = 2.1291e-15;
            f_0 = 5.8929e14;
            delay = 3;
            phase = 90;
            differentiated = false;
        },
        {
            …
            …
        }
    );
};
Sub-variable of ModulatedGaussianWaveforms: string waveform_tag

This is the string tag by which the waveform will later be referred to by another structure that requires a time waveform in its definition.

Sub-variable of ModulatedGaussianWaveforms: string modulation_type

If assigned "sine", the modulation function $g(t)$ in the above equation becomes a sine. If assigned "cosine", it becomes a cosine.

Sub-variable of ModulatedGaussianWaveforms: floating-point amplitude (default: 1.0)

This specifies the variable $A$ in the above equation defining the modulated Gaussian waveform.

Sub-variable of ModulatedGaussianWaveforms: floating-point tau (units: sec)

This specifies the variable $\tau$ in the above equation defining the modulated Gaussian waveform.

Sub-variable of ModulatedGaussianWaveforms: floating-point f_0 (units: Hz)

This specifies the modulation frequency $f_0$ in the above equation defining the modulated Gaussian waveform.

Sub-variable of ModulatedGaussianWaveforms: floating-point delay (default: 0.0)

This specifies the variable $n_{\tau}$ in the above equation defining the modulated Gaussian waveform.

Sub-variable of ModulatedGaussianWaveforms: floating-point phase (units: degrees, default: 0.0)

This specifies the extra phase $\phi$ in the above equation defining the modulated Gaussian waveform. This phase should be specified in degrees, which is then converted internally to radians, which are the actual units of $\phi$ .

Sub-variable of ModulatedGaussianWaveforms: boolean differentiated (default: false)

If set to true, the waveform in the above equation is differentiated once with respect to time.


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