# Meep Tutorial/Near-to-far-field spectra

### From AbInitio

Revision as of 06:17, 18 April 2017 (edit)Ardavan (Talk | contribs) ← Previous diff |
Revision as of 15:36, 18 April 2017 (edit)Ardavan (Talk | contribs) (→Radiation Pattern of an Antenna) Next diff → |
||

Line 36: |
Line 36: | ||

==Radiation Pattern of an Antenna== | ==Radiation Pattern of an Antenna== | ||

- | We can also use the near-to-far-field transformation feature to compute the radiation pattern of an antenna. This example involves placing a single point-dipole source in the middle of a 2d computational cell surrounded by perfectly-matched layers (PMLs). The near-field surface is along the inner boundary of the PML. The far-field surface is a circle having a radius many times the source wavelength and thus lying outside of the computational cell. The far fields are computed at several equally-spaced points along the circumference of this circle. The simulation geometry is shown in the image below. | + | We can also use the near-to-far-field transformation feature to compute the [[w:Radiation pattern|radiation pattern]] of an antenna. This example involves an electric-current point source as the emitter in 2d vacuum. We will investigate the radiation pattern for different polarizations of the input source. The source is placed in the middle of the 2d computational cell which is surrounded by perfectly-matched layers (PMLs). The near-field surface, used to compute the far fields as described above, is along the inner boundary of the PML. The far fields are computed at several equally-spaced points along the circumference of a circle having a radius many times the source wavelength and thus lying outside of the computational cell. The simulation geometry is shown in the image below. |

- | [[Image:Near2far_simulation_geometry.png|center|Simulation geometry used to compute the far-field radiation pattern of a point-dipole source in 2d.]] | + | [[Image:Near2far_simulation_geometry.png|center|Simulation geometry used to compute the radiation pattern of a point source in 2d.]] |

- | We use the <code>get-farfield</code> routine to compute the far fields. The control file is shown below. | + | We use the <code>get-farfield</code> routine to compute the far fields by looping over a set of points along the circumference of a circle. The control file is shown below. |

(set-param! resolution 50) | (set-param! resolution 50) | ||

Line 47: |
Line 47: | ||

(set! geometry-lattice (make lattice (size (+ sxy (* 2 dpml)) (+ sxy (* 2 dpml)) no-size))) | (set! geometry-lattice (make lattice (size (+ sxy (* 2 dpml)) (+ sxy (* 2 dpml)) no-size))) | ||

(set! pml-layers (list (make pml (thickness dpml)))) | (set! pml-layers (list (make pml (thickness dpml)))) | ||

+ | |||

(define-param fcen 1.0) | (define-param fcen 1.0) | ||

(define-param df 0.4) | (define-param df 0.4) | ||

(define-param src-cmpt Ez) | (define-param src-cmpt Ez) | ||

(set! sources (list (make source (src (make gaussian-src (frequency fcen) (fwidth df))) (center 0) (component src-cmpt)))) | (set! sources (list (make source (src (make gaussian-src (frequency fcen) (fwidth df))) (center 0) (component src-cmpt)))) | ||

+ | |||

(if (= src-cmpt Ex) | (if (= src-cmpt Ex) | ||

(set! symmetries (list (make mirror-sym (direction Y))))) | (set! symmetries (list (make mirror-sym (direction Y))))) | ||

Line 57: |
Line 59: | ||

(if (= src-cmpt Ez) | (if (= src-cmpt Ez) | ||

(set! symmetries (list (make mirror-sym (direction X)) (make mirror-sym (direction Y))))) | (set! symmetries (list (make mirror-sym (direction X)) (make mirror-sym (direction Y))))) | ||

+ | |||

(define nearfield | (define nearfield | ||

(add-near2far fcen 0 1 | (add-near2far fcen 0 1 | ||

Line 63: |
Line 66: | ||

(make near2far-region (center (* 0.5 sxy) 0) (size 0 sxy)) | (make near2far-region (center (* 0.5 sxy) 0) (size 0 sxy)) | ||

(make near2far-region (center (* -0.5 sxy) 0) (size 0 sxy) (weight -1.0)))) | (make near2far-region (center (* -0.5 sxy) 0) (size 0 sxy) (weight -1.0)))) | ||

+ | |||

(run-sources+ (stop-when-fields-decayed 50 src-cmpt (vector3 0 0) 1e-8)) | (run-sources+ (stop-when-fields-decayed 50 src-cmpt (vector3 0 0) 1e-8)) | ||

+ | |||

(define-param r (* 1000 (/ fcen))) ; 1000 wavelengths out from the source | (define-param r (* 1000 (/ fcen))) ; 1000 wavelengths out from the source | ||

(define-param npts 100) ; number of points in [0,2*pi) range of angles | (define-param npts 100) ; number of points in [0,2*pi) range of angles | ||

Line 76: |
Line 81: | ||

(arith-sequence 0 1 npts)) | (arith-sequence 0 1 npts)) | ||

- | We compute the far fields for 3 different point sources: Ex, Ey, and Ez. The output consists of a set of columns containing the far-field point's index number, angle (in radians), followed by the magnitudes of each of the six fields. Since this is a 2d simulation, we plot the scalar field in each case normalized by its maximum value to produce a range of values between 0 and 1. This is shown in polar coordinates below. | + | We compute the far fields for 3 different point sources: <i>Ex</i>, <i>Ey</i>, and <i>Ez</i>. The output consists of a set of columns containing the far-field points' index (integer), angle (radians), followed by the magnitudes of each of the six fields. Since this is a 2d simulation, we plot the scalar field in each case normalized by its maximum value to produce a range of values between 0 and 1. This is shown in polar-coordinates plots below. |

[[Image:Source_radiation_pattern.png|center|Far-field radiation patterns of 3 point-dipole sources in 2d.]] | [[Image:Source_radiation_pattern.png|center|Far-field radiation patterns of 3 point-dipole sources in 2d.]] | ||

+ | |||

+ | The <i>Ex</i> and <i>Ey</i> sources produce dipole radiation patterns while <i>Ez</i> has a monopole pattern, as expected. |

## Revision as of 15:36, 18 April 2017

In this example, we demonstrate Meep's near-to-far-field transformation feature, which requires version 1.3+. This feature uses the fields from a "near" bounding surface *inside* the computational cell to compute the resulting "far" fields *outside* the computational cell via an analytical transformation. Note that this only works if the "near" surface and the "far" region lie in a single, homogeneous, non-periodic 2d or 3d medium. The analytical transformation is based on the principle of equivalence: given the Fourier-transformed tangential fields on the "near" surface, Meep computes equivalent currents and convolves them with the analytical Green's functions in order to compute the fields at any desired point in the "far" region. The use of the Fourier-transformed fields for this operation is similar to that for the flux and force spectra: we specify a set of desired frequencies, Meep accumulates the Fourier transforms, and then Meep computes the fields at each frequency for the desired far-field points.

There are three steps to using the near-to-far-field feature: first, we need to define the "near" surface(s) as a set of surfaces capturing all outgoing radiation in the desired direction(s); second, we run the simulation, typically with a pulsed source, to allow Meep to accumulate the Fourier transforms on the near surface(s); third, we have Meep compute the far fields at any desired points with the option to save the far fields from a grid of points to an HDF5 file.

For this demonstration, we will compute the far-field spectra of a resonant cavity mode in a holey waveguide; a structure we had explored in a separate tutorial. To do this, we simply modify the last portion of that control file beginning with the following:

(define-param d1 0.2) (define nearfield (add-near2far fcen 0 1 (make near2far-region (center 0 (+ (* 0.5 w) d1)) (size (- sx (* 2 dpml)) 0)) (make near2far-region (center (+ (* -0.5 sx) dpml) (+ (* 0.5 w) (* 0.5 d1))) (size 0 d1) (weight -1.0)) (make near2far-region (center (- (* 0.5 sx) dpml) (+ (* 0.5 w) (* 0.5 d1))) (size 0 d1))))

Here, we are creating a "near" bounding surface, consisting of three separate regions surrounding the cavity, that captures *all* outgoing waves in the top-half of the computational cell. Note that the *x*-normal surface on the left has a `weight`

of -1 (corresponding to the direction of the *outward normal* vector relative to the *x* direction) so that the far-field spectra is correctly computed from the outgoing fields, similar to the flux and force features. The parameter `d1`

is the distance between the edge of the waveguide and the bounding surface, as shown in the schematic above, and we will demonstrate that changing this parameter does not change the far-field spectra which we compute at a single frequency corresponding to the cavity mode.

We then time step the fields until, at a random point, they have sufficiently decayed away (since the computational cell is surrounded by PMLs) and output the far-field spectra over a rectangular area that lies *outside* of the computational cell:

(run-sources+ (stop-when-fields-decayed 50 Hz (vector3 0.12 -0.37) 1e-8)) (define-param d2 20) (define-param h 4) (output-farfields nearfield (string-append "spectra-" (number->string d1) "-" (number->string d2) "-" (number->string h)) (volume (center 0 (+ (* 0.5 w) d2 (* 0.5 h))) (size (- sx (* 2 dpml)) h)) resolution)

The first item to note is that the far-field region is located *outside* of the computational cell, although in principle it can be located anywhere. The second is that the far-field spectra can be interpolated onto a spatial grid that has any given resolution but in this example we used the same resolution as the simulation. Note that the simulation itself used purely real fields but the output, given its analytical nature, contains complex fields. Finally, given that the far-field spectra is derived from the Fourier-transformed fields which includes an arbitrary constant factor, we should expect an overall scale and phase difference in the results obtained using the near-to-far-field feature with those from a corresponding simulation involving the full computational volume. The key point is that the results will be qualitatively but not quantitatively identical. The data will be written out to an HDF5 file (having a filename prefix with the values of the three main parameters) that will automatically include the far-field spectra for all six field components, including real and imaginary parts.

We run the above modified control file and in post-processing create an image of the real and imaginary parts of *H*_{z} over the far-field region which is shown in insets (a) above. For comparison, we compute the steady-state fields with a continuous source (making sure to force the fields to be complex) over the same region in a separate simulation using a larger computational cell that contains within it the far-field region, from which is obtained the images shown in (b). The difference in the relative phases among any two points within each of the two field spectra is zero, which can be confirmed numerically. Also, as would be expected, it can be shown that increasing `d1`

does not change the far-field spectra as long as the results are sufficiently converged (i.e., the resolution is made sufficiently large to mitigate discretization effects).

In general, it is tricky to interpret the overall scale and phase of the far fields, because it is related to the scaling of the Fourier transforms of the near fields. It is simplest to use the `near2far`

feature in situations where the overall scaling is irrelevant, e.g. when you are computing a ratio of fields in two simulations, or a fraction of the far field in some region, etcetera.

## Radiation Pattern of an Antenna

We can also use the near-to-far-field transformation feature to compute the radiation pattern of an antenna. This example involves an electric-current point source as the emitter in 2d vacuum. We will investigate the radiation pattern for different polarizations of the input source. The source is placed in the middle of the 2d computational cell which is surrounded by perfectly-matched layers (PMLs). The near-field surface, used to compute the far fields as described above, is along the inner boundary of the PML. The far fields are computed at several equally-spaced points along the circumference of a circle having a radius many times the source wavelength and thus lying outside of the computational cell. The simulation geometry is shown in the image below.

We use the `get-farfield`

routine to compute the far fields by looping over a set of points along the circumference of a circle. The control file is shown below.

(set-param! resolution 50) (define-param sxy 10) (define-param dpml 1) (set! geometry-lattice (make lattice (size (+ sxy (* 2 dpml)) (+ sxy (* 2 dpml)) no-size))) (set! pml-layers (list (make pml (thickness dpml))))

(define-param fcen 1.0) (define-param df 0.4) (define-param src-cmpt Ez) (set! sources (list (make source (src (make gaussian-src (frequency fcen) (fwidth df))) (center 0) (component src-cmpt))))

(if (= src-cmpt Ex) (set! symmetries (list (make mirror-sym (direction Y))))) (if (= src-cmpt Ey) (set! symmetries (list (make mirror-sym (direction X))))) (if (= src-cmpt Ez) (set! symmetries (list (make mirror-sym (direction X)) (make mirror-sym (direction Y)))))

(define nearfield (add-near2far fcen 0 1 (make near2far-region (center 0 (* 0.5 sxy)) (size sxy 0)) (make near2far-region (center 0 (* -0.5 sxy)) (size sxy 0) (weight -1.0)) (make near2far-region (center (* 0.5 sxy) 0) (size 0 sxy)) (make near2far-region (center (* -0.5 sxy) 0) (size 0 sxy) (weight -1.0))))

(run-sources+ (stop-when-fields-decayed 50 src-cmpt (vector3 0 0) 1e-8))

(define-param r (* 1000 (/ fcen))) ; 1000 wavelengths out from the source (define-param npts 100) ; number of points in [0,2*pi) range of angles (map (lambda (n) (let ((ff (get-farfield nearfield (vector3 (* r (cos (* 2 pi (/ n npts)))) (* r (sin (* 2 pi (/ n npts)))) 0)))) (print "farfield:, " (number->string n) ", " (number->string (* 2 pi (/ n npts)))) (map (lambda (m) (let ((ffc (list-ref ff m))) (print ", " (number->string (* ffc (conj ffc)))))) (arith-sequence 0 1 6)) (print "\n"))) (arith-sequence 0 1 npts))

We compute the far fields for 3 different point sources: *Ex*, *Ey*, and *Ez*. The output consists of a set of columns containing the far-field points' index (integer), angle (radians), followed by the magnitudes of each of the six fields. Since this is a 2d simulation, we plot the scalar field in each case normalized by its maximum value to produce a range of values between 0 and 1. This is shown in polar-coordinates plots below.

The *Ex* and *Ey* sources produce dipole radiation patterns while *Ez* has a monopole pattern, as expected.