Hi all,
I'd posted this email on a different thread and now that the units
thread has explicitly surfaced, I'm pasting my take on it so that others
could comment on it and correct me where I'm wrong.
Length is specified as the "number of" 'a' units where 'a' is your
length-scale, usually a wavelength.
Velocities are specified as the "number" of c, where c is the speed
of light in vacuum.
These two scales therefore fix the time-scale - time is specified as
the "number of" a/c. So if a=1um, then time is specified as the "number
of" a/c=(1e-6/3e8) second intervals.
If you have a monochromatic source with time-period T expressed in
the Meep scale, i.e., T*a/c is the actual period in seconds, then 1/T is
the frequency in Meep units
Meep internally multiplies this by 2\pi to convert it to angular
frequency. We as users never need to explicitly provide this 2\pi value
because the actual frequency and angular frequency are equivalent
through this scaling transform. I believe the Meep documentation would
be much clearer if it avoids mentioning angular frequency when requiring
the real frequency as input.
The resolution is the number of intervals (Yee-cells) you want each
spatial unit 'a' to be divided into. Suppose you choose 'n' for your
resolution and your simulation "volume" to be v spatial units long. Then
you actually have n*v intervals in your simulation - the E and H fields
are distributed on the boundaries according to Yee's scheme.
Each Yee cell is therefore 1/n Meep units long which corresponds to
real length of a/n. In FDTD, no signal travels faster than speed of
light in vacuum, c. Therefore signals from one point can reach a
neighbouring point only at or after (a/n)/c = (a/cn) seconds. In Meep
units, divide by (a/c) to get 1/n to be this minimum time interval. So
you see how spatial resolution affects the time-resolution also.
Neighbouring points must be spaced so that this causality is respected.
Now, in FDTD, E fields affect H and H fields affect E. So the signal
relationships are actually between field values that are displaced by
half the extent of the Yee-cell (in each dim). So the dt now becomes
0.5/n instead of 1/n. Each FDTD iteration updates the H fields and then
updates the E fields. You could refer to this as two time steps with one
for H and one for E, or as one time-step with half for H and half for E.
I'm not sure which convention Meep adopts.
For numerical stability a "Courant factor" multiples the 0.5/n. This
takes into account the deviation of the speed of light from its vaccuum
value of 'c' due to refractive index effects. So your time step now is
0.5S/n where S is the Courant factor.
My extrapolation of Meep units: absolute permittivity and
permeability are specified in units of \eps_0 and \mu_0, the values for
free-space. These quantities involve dimensions of mass and charge in
addition to length and time. Therefore, these units presumably fix the
mass-scale and charge-scale as well but since we don't explicitly deal
with these in usual FDTD problems, we don't "see" these scales.
cheers!
Manoj
shiv chawla wrote:
> Hello Everyone
>
>
> The thing which is puzzling me is units
> in meep. I have set the the meep parameter 'a' ,say to
> be,1 micrometer(1 um). Now, I donot understand about
> the base time units and frequency units. I think that
> a=1 um, sets, time = 1e-15 s. Am I interpreting it
> correctly? Also how resolution affects the time step
> and spatial step of yee's algroithm? Can you help me
> with this by an example.
>
> Thanks
>
> Regards
> Shiv
>
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