# Understanding frequency domain CW normalization

The frequency domain field monitors in the FDTD and varFDTD solvers record the electric and magnetic fields at a series of user-defined frequencies. These can be returned in either the Continuous Wave Normalization state (cwnorm), or the No Normalization state (nonorm). For most applications, the default cwnorm state is the best choice.

In the nonorm state, the returned fields are simply the Fourier transform of the simulated time domain fields, and we use the subscript sim to refer to these fields in the table below. In the cwnorm state, the fields are normalized by the Fourier transform of the source pulse, thereby yielding the impulse response of the system, and we use a subscript imp to refer to these fields in the table below.

Quantity | Definition | Normalization state |
---|---|---|

Esim(w) |
$$\overrightarrow{E}_{s i m}(\omega)=\int \exp (i \omega t) \overrightarrow{E}(t) d t$$ |
nonorm |

Hsim(w) |
$$\overrightarrow{H}_{s i m}(\omega)=\int \exp (i \omega t) \overrightarrow{H}(t) d t$$ |
nonorm |

Psim(w) |
$$\overrightarrow{P}_{\operatorname{sim}}(\omega)=\overrightarrow{E}_{\operatorname{sim}}(\omega) \times \overrightarrow{H}_{\operatorname{sim}}^{*}(\omega)$$ |
nonorm |

Eimp(w) |
$$\overrightarrow{E}_{i m p}(\omega)=\frac{1}{s(\omega)} \int \exp (i \omega t) \overrightarrow{E}(t) d t=\frac{\overrightarrow{E}_{s i m}(\omega)}{s(\omega)}$$ |
cwnorm |

Himp(w) |
$$\overrightarrow{H}_{i m p}(\omega)=\frac{1}{s(\omega)} \int \exp (i \omega t) \overrightarrow{H}(t) d t=\frac{\overrightarrow{H}_{s i m}(\omega)}{s(\omega)}$$ |
cwnorm |

Pimp(w) |
$$\overrightarrow{P}_{i m p}(\omega)=\overrightarrow{E}_{i m p}(\omega) \times \overrightarrow{H}_{i m p}^{*}(\omega)=\frac{\overrightarrow{E}_{s i m}(\omega) \times \overrightarrow{H}_{s i m}^{*}(\omega)}{|s(\omega)|^{2}}$$ |
cwnorm |

s(w) |
$$ s(\omega)=\frac{1}{N} \sum_{s o u r c e s} \int \exp (i \omega t) s_{j}(t) d t $$ where sj(t) is the source time signal of the jth source and N is the number of active sources in the simulation volume |
N/A |

## Understanding Continuous Wave Normalization (cwnorm)

### Impulse response

FDTD is a time domain method, ie. the electromagnetic fields are calculated as a function of time. Here, the system being simulated is excited by a dipole, beam, mode or imported source, and the time signal of the source, s(t), is a pulse. For example, this could be

$$s(t)=\sin \left(\omega_{0}\left(t-t_{0}\right)\right) \exp \left(-\frac{\left(t-t_{0}\right)^{2}}{2(\Delta t)^{2}}\right)$$

and the Fourier transform of s(t) is s(w)

$$s(\omega)=\int \exp (i \omega t) s(t) d t$$

Ideally, s(t) would be a dirac delta function (in which case s(w) = 1). This would allow us to obtain the response of the system at all frequencies from a single simulation. For a variety of reasons, it is more efficient and numerically accurate to excite the system with a short pulse such that the spectrum, |s(w)|2, has a reasonably large value over all frequencies of interest.

In the nonorm state, power and profile monitors return the response of the system to the simulated input pulse s(t):

$$\overrightarrow{E}_{s i m}(\omega)=\int \exp (i \omega t) \overrightarrow{E}(t) d t$$

The simulated electric field as a function of angular frequency, Esim(w), depends on both the source pulse used, s(t), and the system under study.

In the default cwnorm state, power and profile monitors return the impulse response of the system.

$$\overrightarrow{E}_{i m p}(\omega)=\frac{\overrightarrow{E}_{s i m}(\omega)}{s(\omega)}$$

The impulse response of the system is a much more useful quantity because it is completely independent of the source pulse used to excite the system.

### Example

Consider a beam source injected into free space at z=z0. The source signal is

$$s(t)=\sin \left(\omega_{0}\left(t-t_{0}\right)\right) \exp \left(-\frac{\left(t-t_{0}\right)^{2}}{2(\Delta t)^{2}}\right)$$

The electric field at the source injection plane has the following form:

$$E\left(x, y, z_{0}, t\right)=E_{0}\left(x, y, z_{0}\right) s(t)$$

In the cwnorm state,

$$E(x, y, z, \omega)=E_{0}(x, y, z)$$

In other words, the field returned in the cwnorm state is the field that would exist if a CW source of amplitude E0 had been used at the angular frequency w. It removes any frequency dependence due to the finite pulse length of the source, and the units of the returned fields are the same as time domain fields.