Units and normalization conventions in Lumerical solvers

FDTD MODE DGTD CHARGE HEAT FEEM

Units for electrical and thermal solvers

Unless otherwise stated, Lumerical uses SI units.

Quantity	Description	Units	Unit description
L	Length units in semiconductor models This is reflected in the semiconductor device literature, and in the parameter coefficients for the material models.	cm	Centimeter
E	Energy The electron energy E is related to the local electrostatic potential (voltage) as E = -qV. All energies (and voltages) are referenced from the (equilibrium) Fermi level of an electrical contact in the system.	eV	Electron volt
n	Electron density	1/cm3	Per centimeter cube
p	Hole density	1/cm3	Per centimeter cube
Jn	Electron current density	A/cm2	Ampere per centimeter square
Jp	Hole current density	A/cm2	Ampere per centimeter square
N	Net doping density The doping density is negative for p-type (acceptor) dopants and positive for n-type (donor) dopants	1/cm3	Per centimeter cube
E	Electric field	V/m	Volt per meter
V	Electrostatic potential (voltage)	V	Volt

Units for optical solvers

Unless otherwise stated, Lumerical's optical solvers used SI units at all times.

General

Quantity	Description	Units	Unit description
f=w/2p	Frequency	Hz	Hertz
x,y,z	Position	m	Meter
t	Time	s	Seconds

Time domain electromagnetic fields

Quantity	Description	Units	Unit description
E(t)	Electric field as function of time	V/m	Volts per meter
\|E(t)\|2	Electric field intensity as a function of time	(V/m)2	Volts squared per meter squared
H(t)	Magnetic field as a function	A/m	Amperes per meter
\|H(t)\|2	Magnetic field intensity as a function of time	(A/m)2	Amperes squared per meter squared
P(t)	Poynting vector as a function of time	W/m2	Watts per meter squared
Power(t)	Power as a function of time	W	Watts

Dipole moments

Quantity	Description	Units	Unit description
p	Electric dipole in 3D	Cm	Coulomb meters
m	Magnetic dipole in 3D	Am2	Ampere meters squared
p	Electric field in 2D	Cm/m	Coulomb meters per meter
m	Magnetic dipole in 2D	Am2/m	Ampere meters squared per meter

Frequency domain electromagnetic fields - Steady state, single frequency, cwnorm data

Quantity	Description	Units	Unit description
E(w)	Electric field as a function of angular frequency	V/m	Volts per meter
\|E(w)\|2	Electric field intensity as a function of angular frequency	(V/m)2	Volts squared per meter squared
H(w)	Magnetic field as a function of angular frequency	A/m	Amperes per meter
\|H(w)\|2	Magnetic field intensity as a function of angular frequency	(A/m)2	Amperes squared per meter squared
P(w)	Poynting vector as a function of angular frequency	W/m2	Watts per meter squared
Power(w)	Power as a function of angular frequency	W	Watts
Power(w)	2D Power as a function of angular frequency	W/m	Watts per meter

Frequency domain electromagnetic fields - nonorm data

Quantity	Description	Units	Unit description
E(w)	Electric field as a function of angular frequency	V/m/Hz	Volts per meter per Hertz
\|E(w)\|2	Electric field intensity as a function of angular frequency	(V/m/Hz)2	Volts squared per meter squared per Hertz squared
H(w)	Magnetic field as a function of angular frequency	A/m/Hz	Amperes per meter per Hertz
\|H(w)\|2	Magnetic field intensity as a function of angular frequency	(A/m/Hz)2	Amperes squared per meter squared per Hertz squared
P(w)	Poynting vector as a function of angular frequency	W/m2/Hz2	Watts per meter squared per Hertz squared
Power(w)	Power as a function of angular frequency	W/Hz2	Watts per Hertz squared
Power(w)	2D Power as a function of angular frequency	W/Hz2/m	Watts per Hertz squared per meter

Source amplitudes

Beam sources

When specifying the amplitude for beam sources, the "amplitude" refers to the peak electric field amplitude in units of V/m. For example, if a Gaussian beam has the following electric field distribution in time and space:

$$
E(x, y, z, t)=E_{0} \sin \left(\omega_{0}\left(t-t_{0}\right)\right) \exp \left(-\frac{\left(t-t_{0}\right)^{2}}{2(\Delta t)^{2}}\right) \exp \left(-\frac{\left(x^{2}+y^{2}\right)}{w_{0}^{2}}\right)
$$

Then the "amplitude" refers to the value of E0 and has units of V/m. It is worth noting that different beams will inject different amounts of power for a given source amplitude.

Dipole sources

For dipole sources, amplitude refers to the amplitude of the point source whose units are listed below. Base amplitude refers to the amplitude that will generate a radiated CW power of 10 nW/m in 2D simulations and 1 fW in 3D simulations, and total amplitude refers to the amplitude actually used in the simulations which is the product of the amplitude and the base amplitude.

Dipole source amplitude units are

Cm for 3D electric dipole sources
Am2 for 3D magnetic dipole sources
Cm/m for 2D electric dipole sources
Am2/m for 2D magnetic dipole sources

Field region

The field region object is used for inverse design with the lumopt Python module. This object acts as a frequency-domain monitor without spatial interpolation during the forward simulation, retaining data when switching back to the layout mode. During the adjoint simulation, lumopt activates “source mode”, which turns this object into a volumetric current source.

While in source mode, the field region converts the recorded field into an array of dipole moments, located at each grid point $x_i, y_j, z_k$, with phase and orientation determined by the electric field. The base amplitude of the dipole is set such that it radiates $|E(x_i, y_j, z_k)|$ fW of CW power. This means that for a single-point field region object that recorded electric field with an amplitude of V/m, the radiated CW power is 1fW.