Field Functions

As described in Python User Interface, Meep provides several routines to integrate, analyze, and output arbitrary user-specified 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.

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:


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


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

Note that the 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:


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


(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):


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)))))


(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")

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


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)))))


(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")

Finally, 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:


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


(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, 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.output_field_function("weird-function", [meep.Ex, meep.Hz, meep.Dielectric], f)), until=200)


(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 (Python) or run-until (Scheme) must be functions. For example, if you call, until=200) (Python) or (run-until 200 output-hfield) (Scheme),output_hfield (Python) or output-hfield (Scheme) is the name of a function which (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 (Python) or run-until (Scheme). Instead, you must write a new function which you can pass to (Python) or run-until (Scheme), like the following:


def my_weird_output(sim):
    meep.Simulation.output_field_function("weird-function", [meep.Ex, meep.Hz, meep.Dielectric], f),my_weird_output), until=200)


(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 (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 (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,my_weird_output)), until=200)


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

As a final example, the Python interface routine get_array_metadata used to obtain the coordinates of grid points in a volume slice can be replicated in Scheme via e.g.:

(define (f r eps) (vector3-x r))
(output-real-field-function "x" Dielectric f)

For more information, see Python User Interface or Scheme User Interface.