# Field Functions

As described in Python Interface, Meep provides several routines to integrate, analyze, and output arbitrary user-defined functions of the field components. See the functions whose names end with _field_function. This facility, while powerful, requires a bit more programming than most Meep usage, and is best illustrated by a few examples.

Note: field functions can be applied to time-domain and frequency-domain fields.

## Arbitrary Functions

Every field-function that can be passed to these routines is of the form f(r,components...), where r is a position vector and "components..." are zero or more field components that the function depends on. The set of desired components is user-specified. As an example, suppose we are interested in the arbitrary function:

We would define this function by:

Python

def f(r, ex, hz, eps):
return (r.x * r.norm() + ex) - (eps * hz)


Scheme

(define (f r ex hz eps)
(- (+ (* (vector3-x r) (vector3-norm r)) ex) (* eps hz)))


Note that the (mandatory) first argument r is a Vector3 (Python) or vector3 (Scheme) object.

Now, suppose we want to compute the integral of this function, over the whole cell. We can do this by calling the function integrate_field_function (Python) or integrate-field-function (Scheme), as follows:

Python

print("The integral of our weird function is: {}"
.format(meep.Simulation.integrate_field_function([meep.Ex, meep.Hz, meep.Dielectric], f)))


Scheme

(print "The integral of our weird function is: "
(integrate-field-function (list Ex Hz Dielectric) f) "\n")


Note that the first argument to integrate_field_function (Python) or integrate-field-function (Scheme) is a list, which is a standard type, of component constants, specifying in order the list of field components the function f expects to be passed. Meep will then call f for every point in the cell in parallel on a parallel machine, and return the integral approximated by a trapezoidal rule.

You can also specify an optional third argument to integrate_field_function (Python) or integrate-field-function (Scheme), specifying an integration volume in case you don't want the integral over the whole cell. For example, the following code computes the integral of f along a line from (-1,0,0) to (1,0,0):

Python

print("The integral of our weird function from (-1,0,0) to (1,0,0) is: {}"
.format(meep.Simulation.integrate_field_function([meep.Ex, meep.Hz, meep.Dielectric], f, meep.Volume(size=meep.Vector3(2,0,0), center=meep.Vector3(0,0,0)))))


Scheme

(print "The integral of our weird function from (-1,0,0) to (1,0,0) is: "
(integrate-field-function (list Ex Hz Dielectric) f (volume (size 2 0 0) (center 0 0 0))) "\n")


## Maximum Absolute Value

Instead of computing the integral, Meep also provides a function to compute the maximum absolute value of our given function:

Python

print("The maximum absolute value of our weird function from (-1,0,0) to (1,0,0) is: {}"
.format(meep.Simulation.max_abs_field_function([meep.Ex, meep.Hz, meep.Dielectric], f, meep.Volume(size=meep.Vector3(2,0,0), center=meep.Vector3(0,0,0)))))


Scheme

(print "The maximum absolute value of our weird function from (-1,0,0) to (1,0,0) is: "
(max-abs-field-function (list Ex Hz Dielectric) f (volume (size 2 0 0) (center 0 0 0))) "\n")


## Outputting to an HDF5 File

We can also output our function to an HDF5 file, similar to the built-in functions to output selected field components, and so on. The following outputs an HDF5 file consisting of our function f evaluated at every point in the cell:

Python

meep.Simulation.output_field_function("weird-function", [meep.Ex, meep.Hz, meep.Dielectric], f)


Scheme

(output-field-function "weird-function" (list Ex Hz Dielectric) f)


The first argument is used for the name of the dataset within the HDF5, and is also used for the name of the HDF5 file itself plus a .h5 suffix and a time stamp, unless you have specified the output file via to_appended (Python) or to-appended (Scheme) or other means.

The above example calls the integration, maximum, and output routines only once, at the current time. Often, you will want to pass them to meep.Simulation.run(..., until=...) (Python) or run-until (Scheme) instead, using at_every (Python) or at-every (Scheme) to print or output at periodic time intervals. A common mistake is to do something like the following:

Python

meep.Simulation.run(
mp.at_every(1, meep.Simulation.output_field_function("weird-function", [meep.Ex, meep.Hz, meep.Dielectric], f)),
until=200)


Scheme

(run-until 200
(at-every 1 (output-field-function "weird-function" (list Ex Hz Dielectric) f)))


This is wrong, and will cause Meep to exit with a strange error message. The reason is that the step functions you pass to meep.Simulation.run (Python) or run-until (Scheme) must be functions. For example, if you call meep.Simulation.run(meep.output_hfield, until=200) (Python) or (run-until 200 output-hfield) (Scheme),output_hfield (Python) or output-hfield (Scheme) is the name of a function which meep.Simulation.run (Python) or run-until (Scheme) will call to output the field. The incorrect code above, however, first calls the function output_field_function (Python) or output-field-function (Scheme) to output an HDF5 file, and then passes the result of this function to meep.Simulation.run (Python) or run-until (Scheme). Instead, you must write a new function which you can pass to meep.Simulation.run (Python) or run-until (Scheme), like the following:

Python

def my_weird_output(sim):
meep.Simulation.output_field_function("weird-function", [meep.Ex, meep.Hz, meep.Dielectric], f)

meep.Simulation.run(meep.at_every(1,my_weird_output), until=200)


Scheme

(define (my-weird-output)
(output-field-function "weird-function" (list Ex Hz Dielectric) f))

(run-until 200 (at-every 1 my-weird-output))


We have defined a function my_weird_output (Python) of one argument (the simulation instance) and my-weird-output (Scheme) of no arguments that, when called, outputs our function f. We then pass this function to meep.Simulation.run (Python) or run-until (Scheme). In contrast, our incorrect code above corresponds to passing my_weird_output(t) (Python) or (my-weird-output) (Scheme), the result of calling my_weird_output to meep.Simulation.run (Python) or my-weird-output to run-until (Scheme).

As described in Synchronizing the Magnetic and Electric Fields, because this example function combines electric and magnetic fields, we may want to synchronize them in time in order to compute this function more accurately, by wrapping it with synchronized_magnetic (Python) or synchronized-magnetic (Scheme):

Python

meep.Simulation.run(meep.synchronized_magnetic(meep.at_every(1,my_weird_output)), until=200)


Scheme

(run-until 200 (synchronized-magnetic (at-every 1 my-weird-output)))


## Coordinates of the Yee Grid

As a final example, the function integrate_field_function (Python) or integrate-field-function (Scheme) can be used to obtain the coordinates of the Yee grid. As long as the field arguments are on the same grid (e.g., $E_x$ and $D_x$, $E_y$ and $D_y$, etc.), the integral is computed over the exact Yee grid coordinates rather than being interpolated to the center of each grid point if fields from different grids are used (consistent with Meep's paradigm of pervasive interpolation).

Python

def f(r,fc):
print("({:.5f}, {:.5f}, {:.5f})".format(r.x,r.y,r.z))
return 0

meep.Simulation.integrate_field_function([mp.Ex],f)


Scheme

(define (f r fc)
(begin
(print "(" (vector3-x r) ", " (vector3-y r) ", " (vector3-z r) ")\n")
0))
(integrate-field-function (list Ex) f)


This function prints the $(x,y,z)$ Yee grid coordinates of all $E_x$ fields and returns a value of 0 which is never used. In contrast, the output functions output_field_function (Python) or output-field-function (Scheme) (as well as output-real-field-function) interpolate all fields onto the center of each grid point.