Working with units in MEEP is easy. The quantities which we are measure like transmitted/reflected power, etc are all unitless. Before diving into the methods, lets look into key concepts.

**Scalar Invariance:**

Scalar invariant laws do not change on multiplying or dividing by a common factor. For example consider a homogenous function f(x). If we are scaling 'x' to 'sx' ,

The function retains its form and is scaled according to the scaling factor.

Maxwell's equations are scale invariant.

In the absence of currents the charges, the wave equation is

Using scaling:

Substituting the value of scaled variables in the wave equations, it is evident that E' is also a solution.

The entire task of computational electromagnetism is to solve the wave equation. When we are scaling a quantity, we multiply all the quantities (except **frequency **)with the scaling factor.Frequency is divided by the scaling factor.

**Assigning Units ^{1}**

Let

**unit length**(eg. nm, cm) be 'a'. Note: in MEEP c=1.

**Unit time**= (Unit length)/c . Hence unit of time is "a/c" or simply "a".

**Frequency** =1/(Unit time). Hence frequency is expressed as "c/a" units.

Also note, frequency=c/(wavelength).

is expressed as units

Optical time period (a/c)units

Consider the following equation^{2} :

--- (1)

Upon scaling,

equation (1) becomes

---(2)

The following are some useful cases, to extend our analysis from one scale to another.

Case 1: In order to find mode profile of a system where the dielectric constant is either compressed or expanded, i.e

rewriting the equation:

---(3)

The mode profile in this configuration H'(r')=H(r'/s) is obtained by rescaling the old profile and frequency.

case 2: Analysing a medium in which Dielectric configuration differs everywhere by constant factor of a known profile, i.e,

we can use (1) and substitute . This yields

---(4)

Hence, the same mode profile is obtained by scaling epsilon and frequency. ( )

The equation (4) can also be rewritten in the following form:

---(5)

The above equation also yields the similar profile by scaling and coordinates (r'=sr).

**To Summarize:**

Step 1: Decide the regime in which you are working, for example micron range and set.

Step 2: Define source parameters : wavelength or frequency.

Step 3: Set the desired run time of the program

**Example: **

Consider solving a fiber optic cable with parameters wavelength 1.55µm, Core 8µm diameter and cladding 125µm diameter. Since the domain is in micrometer scale, set a=1µm. Wavelength is 1.55a or simply 1.55. Frequency is 1/1.55. In order to run your simulation for 200 time periods we need to run it for 200/f or 200*1.55 time units. (1 Time period= (1/f)time units)

**References:**

1) MEEP website