Field Function Examples

As described in the 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 Scheme 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 arbitrary example, suppose we are interested in the strange function:

We would define this function, in Scheme, by:

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

Note that the r argument is a vector3, and can be manipulated by the functions defined in the Libctl User Reference.

Now, suppose we want to compute the integral of this function, over the whole computational cell. We can do this by calling the function integrate-field-function, as follows:

(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 is a list (a standard Scheme type) of component constants, specifying in order the list of field components our function f expects to be passed. Meep will then call f for every point in the computational 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, specifying an integration volume in case you don't want the integral over the whole computational 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: "
       (integrate-field-function (list Ex Hz Dielectric) f (volume (size 1 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: "
       (max-abs-field-function (list Ex Hz Dielectric) f (volume (size 1 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 computational cell:

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

Here, 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 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 run-until instead, using at-every to print or output at periodic time intervals. A common mistake is to do something like the following:

(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 run-until must be functions. For example, if you call (run-until 200 output-hfield), output-hfield is the name of a function which run-until will call to output the field. The incorrect code above, however, first calls the function output-field-function to output an HDF5 file, and then passes the result of this function to run-until. Instead, you must write a new function which you can pass to run-until, like the following:

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

Here, we have defined a function my-weird-output of no arguments that, when called, outputs our function f. We then pass this function to run-until. In contrast, our incorrect code above corresponds to passing (my-weird-output), the result of calling my-weird-output, to run-until.

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:

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

See also the section writing your own step functions.