Far-field projection¶

Substrate projections: substantial measured error

An open plane or box over a substrate is an approximation. In this guide's dipole cases at substrate index 1.45, the pattern error is 13.5 to 21.6 % of the peak within 30 degrees, and the cone-power error is about 3 to 17 %. The range covers the open box and planes with and without a window, not one plane configuration. Higher substrate indices are outside what was measured.

Settings: vacuum wavelength 1 µm; dipole 0.48 µm above the interface; uniform mesh at 20 / 30 cells per substrate wavelength; flat interface on a grid line; subpixel averaging on, subpixel_method="contour"; no dispersion; PML one substrate wavelength thick; at most 100 periods with the default shutoff. See the surface comparisons and cone angles before using a substrate result.

A monitor records the field close to a device. An antenna, a grating coupler or a metalens is judged by the light far away. far_field carries a recorded surface to the far zone. It turns the tangential E and H on the surface into equivalent surface currents and integrates what they radiate. The result holds the far-field amplitudes, the intensity, the radiated power and the directivity.

This guide declares a surface, projects it, reads the labelled results and puts a recorded plane's E and H on common points. It then says when an open surface holds, what the warnings mean and how accurate the projection is. Every call is documented in Far field and diffraction.

The subpixel setting subpixel_method="contour" has an exact geometric fill only for boxes and vertical polygons. Cylinders and spheres use the supersampled fill.

Declare the surface¶

Declare the surface in the simulation with ph.FarFieldMonitor. A box encloses the sources. A plane looks into one half space.

import numpy as np

import photonhub as ph
from photonhub.analysis import far_field

ff = ph.FarFieldMonitor.box("ff", center_um=(0, 0, 0), size_um=(4, 4, 3),
                            wlens_um=[1.50, 1.55, 1.60])
up = ph.FarFieldMonitor.plane("up", axis="z", position_um=1.2, direction="+",
                              wlens_um=1.55, pts_per_wlen=10)
sim = ph.Simulation(..., monitors=[ff, up])
data = ph.run_local(sim)

The simulation records the surface the way far_field reads it best:

A box becomes six field monitors, ff.x_minus to ff.z_plus. Each records the four tangential components at every cell, one cell thick along its normal and one cell past the box in its plane. The solver stores E and H half a cell apart; the two samples along the normal let far_field bring them onto one surface.
A plane becomes one monitor, up.z_plus. It spans the domain inside the PML unless you give size_um with center_um. pts_per_wlen sets an in-plane stride that keeps that many samples per wavelength in the plane's medium.
Each face sits on the grid line nearest its declared plane.
The run keeps the recipe in client.json beside its results: the faces, their outward normals, the medium they sit in, and any symmetry planes, faces left out and edge window.

Building the simulation refuses a surface the projection cannot read, and names the face. It refuses a face in a PML or absorber layer, a face across a structure or a medium boundary, and faces in different media. It refuses a medium that is anisotropic, custom, PEC or lossy (a loss tangent above 1e-3). The build also warns: see Warnings.

Symmetry planes. Declare the box across a symmetry plane, centred on it or starting on it. The simulation records only the kept part, and far_field mirrors it back. A box or a plane that does not reach a symmetry plane, or crosses it off centre, is refused.

Open boxes and windows. exclude=("-z",) leaves out the box face whose outward normal is -z, for example above a substrate. window=(0.2, 0.2) on a plane scales the currents by sin² over the outer 20 % of the plane at each end, along both in-plane axes. The window is off by default, and a box refuses it. Read When an open surface holds before you use either.

Project¶

theta = np.linspace(0.0, np.pi, 91)
phi = np.linspace(0.0, 2 * np.pi, 72, endpoint=False)
pat = far_field(data, "ff", theta_rad=theta, phi_rad=phi)

far_field finds the faces by the monitor's name. It projects into the medium the faces sit in. Pass n_medium to override it: an index, not a permittivity. The medium must be homogeneous and lossless; a loss tangent above 1e-3 is refused. A layered half space, such as a cladding over a substrate, is outside the model. Record a plane in each half space instead.

Give one target grid per call:

Target grid	Arguments	Result dims
Directions	`theta_rad`, `phi_rad`	`(f, theta, phi)`
Direction cosines	`ux`, `uy`	`(f, ux, uy)`
Points on a plane	`plane_axis`, `plane_um`, `x_um`, `y_um`	`(f, x, y)` on a plane normal to z
Paired points	any of the above with `mesh=False`	`(f, point)`

theta_rad is the polar angle from +z; phi_rad the azimuth from +x.
ux and uy are direction cosines along the two in-plane axes of plane_axis, in xyz order. A plane uses its own normal and side. A box needs plane_axis=, and direction="-" for the half space below it. A point with ux² + uy² > 1 has no direction: its fields are nan and its intensity 0.
On a plane over one period of a periodic structure, the point u = mλ/(nL) is the direction of grating order m. The order's power is U (λ/n)² / (A cos θ) in W, with U the intensity there, A the period's area in m² and θ the angle from the plane's normal. To compare it with diffraction_orders, mind two conventions. That function's (k1, k2) run along its in_plane_axes in cyclic order, so on a plane normal to y its k1 is uy and its k2 is ux, times the wavenumber. Its order powers carry the period's area in µm², 1e12 times the watts of the formula.
x_um and y_um run along the plane's two in-plane axes, in xyz order. Each point reads the direction it lies in from origin_um, the surface's centre by default. Pass origin_um to compare two runs.

The field at a finite distance. The far-field approximation drops the terms that fall faster than 1/r. exact=True evaluates the field the currents radiate at each point with the full Green's function of the medium:

xs = ys = np.linspace(-10.0, 10.0, 81)       # um
spot = far_field(data, "ff", plane_axis="z", plane_um=20.0,
                 x_um=xs, y_um=ys, exact=True)
sphere = far_field(data, "ff", theta_rad=theta, phi_rad=phi,
                   exact=True, r_um=5.0)

A meshed grid of points on a plane parallel to an evenly sampled recorded plane uses that plane's angular spectrum. Anything else, and a spectrum that does not settle within its padding, sums over every recorded sample, so its cost grows as the number of points times the number of samples.

Read the results¶

far_field returns a FarField. Its dataset is an xarray.Dataset labelled by frequency and by the grid's own axes. Along f every array carries the vacuum wavelength wlen_um and the index n_medium.

ds = pat.dataset
f = float(ds["f"][1])                    # Hz, the middle wavelength, 1.55 um
U = ds["intensity"].sel(f=f)             # W/sr, dims (theta, phi)
D = ds["directivity"].sel(f=f, phi=0.0)  # 4 pi U / P, the phi = 0 cut
P = ds["radiated_power"]                 # W, one per frequency

e_theta and e_phi (V) are the far-field amplitudes, with the e^{ikr}/r spreading removed. An exact projection adds e_r, h_r, h_theta and h_phi.
intensity (W/sr) is the power per unit solid angle.
radiated_power (W) and directivity come on a meshed angle grid. The power integrates over the grid's directions, so it is the total only when the grid covers the whole sphere, or the whole half space a plane faces.

The methods return labelled results too:

Call	What it gives
`pat.gain(p_in)`	`4πU / P_in`. Pass the power the source delivers.
`pat.cross_section()`	`dσ/dΩ = U / I_inc` in µm²/sr. A run driven by one `TFSFBox` supplies `I_inc`; otherwise pass it.
`pat.polarization("ludwig3")`	Co- and cross-polar amplitudes, Ludwig's third definition, about +z by default on an angle grid; undefined opposite the boresight.
`pat.polarization("circular")`	Right- and left-hand amplitudes, IEEE handedness with `e^{-iωt}`.
`pat.renormalize(distance_um)`	A copy read at that distance, for `power_density()` (W/m²) and `cartesian()` (`Ex` ... `Hz`).
`ph.viz.plot_far_field_cut(pat, phi_rad=0.0, freq_hz=f)`	A polar cut through `phi` and `phi + π`.
`ph.viz.plot_far_field_map(pat, freq_hz=f)`	A map over direction cosines, or over a Cartesian plane.

On an angle grid, Ludwig-3 defaults to the +z boresight even when the recorded plane faces downward. Its basis is undefined opposite the boresight. For numerical stability, co and cross are nan throughout the small antipodal neighborhood 1 + r_hat.dot(boresight_hat) <= 1e-12, where both vectors are unit vectors. With boresight="+z", this includes theta = pi and directions within approximately 1.4e-6 radians of it. The SDK emits a UserWarning identifying the selected boresight when requested directions enter this neighborhood. A NumPy sum that includes those samples is also nan; the finite circular amplitudes at the same directions do not indicate a field-projection failure. For a downward hemisphere, choose boresight="-z", reference="x" explicitly. This puts the masked neighborhood around theta = 0, outside that hemisphere. For a full sphere, partition the directions and choose a boresight for each hemisphere. The full-sphere grid above includes an undefined pole for either choice. Changing the boresight changes the co/cross basis.

Polarization methods return amplitudes, so a sum of their squared magnitudes is not an integrated power share. Integrate with the solid-angle weight sin(theta) dtheta dphi, then normalize. For a downward pattern down on an increasing theta grid from pi/2 to pi and a uniformly spaced full azimuth grid with endpoint=False:

pol = down.polarization("ludwig3", boresight="-z", reference="x")
angular_power = (
    (abs(pol)**2 * np.sin(pol["theta"])).mean("phi") * (2 * np.pi)
).integrate("theta")
total = angular_power["co"] + angular_power["cross"]
co_share = angular_power["co"] / total
cross_share = angular_power["cross"] / total

This gives one share per frequency. The common factor 1 / (2 eta) cancels in the shares; divide angular_power by 2 eta for watts, using the projected medium's impedance at each frequency. The azimuth mean above covers the periodic interval without duplicating its endpoint.

The plots draw one frequency. With several, pick it with freq_hz=; the plot takes the projected frequency nearest it.

The projection cannot tell how much power the source delivers, so gain requires it. Measure it with PowerMonitor planes that close a small box centred on the source. Off centre, the box's total is off by an amount first order in the cell size times the offset. A dipole's own analytic power holds only when it sits alone in a homogeneous lossless medium.

Earlier releases returned flat arrays indexed [frequency, point]. They still work for one release, with a FutureWarning that names the replacement. For example, pat.reshape(theta.size, phi.size)["intensity"][0] becomes pat.dataset["intensity"].isel(f=0).values, the same numbers.

Near fields on a recorded plane¶

The solver records each field component at its own point in the cell. data[name] labels them all with the cell's corner. A product such as E × H* read at those labels is wrong at first order in the cell. colocated_fields puts every component on the same points:

from photonhub.analysis import colocated_fields

nf = colocated_fields(data, "ff.z_plus")                # dims (f, x, y)
sz = 0.5 * np.real(nf["Ex"] * np.conj(nf["Hy"]) - nf["Ey"] * np.conj(nf["Hx"]))

It reads a ProfileMonitor plane, a FarFieldMonitor face such as "ff.z_plus" (every cell, one cell thick), or a FarFieldMonitor plane by its own name. In the plane, a component half a cell off a grid point is interpolated linearly from the two samples either side. A plane recorded with a stride (pts_per_wlen, every s cells) is interpolated across the stride, about 2s - 1 times less accurately: for a product of E and H, record every cell. Along the normal, a plane recorded one cell thick has E and H on one surface. Every FarFieldMonitor face is. A plane with one sample keeps H half a cell above E, and warns when it holds both.

Record a plane one cell thick for any product of E and H. On a uniform mesh, give it the size dl along its normal, centred at (k + 3/4) dl. Keep it clear of sources: the interpolation's error grows fast near one. On a dipole's exact field (λ = 1 µm), a plane recorded one cell thick reads each component within 2.2 % of its peak at 20 cells per wavelength and 0.59 % at 40 when it lies 0.3 to 0.6 µm from the dipole, but 32 to 62 % and 9.5 to 13 % when it lies 0.09 µm away. For the power alone, a PowerMonitor is exact on the solver's grid. To project a surface, use far_field, which reads each component where it was recorded.

When an open surface holds¶

A closed box in one medium sees all the light. An open box and a single plane are truncated surfaces. Each reads the pattern well only where the field it cuts off is small, such as a beam that has died out well inside the plane's edges. A point dipole's field is not small there, and the error shows even straight ahead.

Measured on a dipole in a medium of index 1.45 at λ0 = 1 µm, at 20 / 30 cells per wavelength in the medium (uniform mesh, PML one medium wavelength thick, no structures and so no subpixel averaging, no dispersion, at most 100 periods with the default shutoff), against a closed box 0.62 µm around the dipole, as a fraction of the peak:

Open surface	Within 30°	Within 60°
Plane 0.62 µm away, 2.41 µm half width, no window	36.8 / 36.1 %	41.4 / 42.2 %
The same plane, window 0.2	14.2 / 14.8 %	33.6 / 33.7 %
The same plane, window 0.4	7.8 / 7.5 %	57.2 / 58.1 %
Box 0.62 µm half width, one face left out (its closed side)	10.2 / 10.1 %	22.9 / 22.8 %

The dipole's exact fields on the same surfaces read about the same (35 %, 15 % and 7.3 % for the plane; 10.2 % and 22.9 % for the box): the differences belong to the open surfaces, not to the solver. The power within a cone reads closer in one medium: within 2 % for the open box, up to 7.4 % for the plane. Where the open edge is declared matters. Moving the open box's declared edge by one cell, at 20 cells per wavelength, moved its pattern by 8.5 % of the peak within 30°.

Over a substrate the cone power is off too. The scene: a dipole 0.48 µm above a half space of index 1.45, vacuum above, λ0 = 1 µm; 20 / 30 cells per substrate wavelength on a uniform mesh; the substrate's flat top on a grid line, with the default subpixel averaging (subpixel_method="contour"); no dispersion; PML one substrate wavelength thick; at most 100 periods with the default shutoff. Against the closed form of a dipole above a half space:

the box open toward the substrate, its open edge 0.14 µm above it, reads the upper half space 13.7 / 13.5 % of the peak off within 30°, and its cone power by +3.9 / +3.5 % within 30° and -10.8 / -10.8 % within 80°;
a plane at the box's top reads 21.6 / 21.3 % off within 30° (14.1 / 14.0 % with a window of 0.4), and the windowed plane's cone power is off by +17.1 / +16.6 % within 30°.

A wider plane with a window helps fast. On the dipole's exact fields in index 1.45, at 20 cells per medium wavelength, a plane 0.9 medium wavelengths from the dipole with a half width of 14 medium wavelengths and a window of 0.2 reads within 0.27 % of the peak within 30°. Toward the plane's edge the window costs accuracy: within 60°, 7.3 % with a window of 0.2 and 2.4 % with 0.4. A larger open box helps much less: on the same exact fields, 10.2 %, 10.8 %, 8.5 % and 6.9 % within 30° of its closed side at half widths of 0.9, 1.8, 3.6 and 7.2 medium wavelengths.

So:

Close the box in one medium wherever the scene allows.
Use an open surface for fields that fall off before its edges.
For a dipole-like emitter in one medium, quote the power in a cone, not the pattern. Over a substrate the cone power is off by 3 to 17 % as well, so check it against a wider surface, a window or a second open-edge position.
Declare an open box with exclude=. A face list given by hand is read without its open edge.

Warnings¶

Far zone. On a grid of points, the far-field approximation reads each point with the amplitude of its direction. It needs the point in the far zone. far_field warns for points nearer than 2D²/λ, with D twice the surface's farthest reach from origin_um and λ in the medium. There the approximation's phase error passes π/8. Use exact=True closer in.

Near the surface. The exact projection loses accuracy close to the currents. far_field warns for points within two sample spacings or a tenth of a wavelength of them, whichever is farther. On a dipole's exact fields on a box (an x dipole 0.14 µm off the centre of a box 1.2 µm wide, vacuum, λ = 1 µm, uniform mesh) at 20 cells per wavelength, the error is 3 to 6 % of the peak one sample spacing out. Over the box's edges it grows on finer meshes: up to 11 % at 40 cells and 21 % at 80, one spacing out. It refuses points inside a box and points behind a plane. Directions of a plane's grid that end at or behind it read nan, with a warning.

Near a face. Building the simulation warns when a source or a structure lies within 2 cells of a face's innermost sample, since the error grows sharply there. It measures each source where the solver runs it: a point dipole on the nearest grid point of its field component, which can be half a cell from where you placed it; a plane wave or mode source on the grid line it is injected on, with the row of its correction half a cell before it; and each face of a TF/SF box on its grid line, with that row half a cell outside the box. It warns when a face is closer than half a wavelength to a structure: the near field there is not radiation. On a graded mesh it warns when a box face is nearer than 0.35 wavelengths to a change of cell size, or a source nearer than 0.1 wavelengths to one (see On a graded mesh).

Medium and recipe. A result reloaded without its client.json projects from the face names, with a warning, into vacuum unless you pass n_medium. It reads the faces as recorded, with no open edge and no window.

Accuracy¶

Each figure below is the worst over the stated directions, as a fraction of the pattern's peak, unless it says otherwise.

A closed box in vacuum¶

The error is second order in the cell: about 4x smaller per halving. For a point dipole in vacuum at λ0 = 1 µm and 25 cells per wavelength (uniform mesh, no structures and so no subpixel averaging, no dispersion, 25 PML layers, a box 1.2 wavelengths wide, at most 200 periods with the default shutoff), the front/back asymmetry is 0.1 % with the dipole half a cell from the box centre, up to about 0.80 % with the dipole near a box corner (0.10 to 0.12 µm from each of the three faces there) and up to about 1.06 % near a box edge, two faces near and the third coordinate centred: the worst measured over the placements the build does not warn about. The bound is 2 % of the peak wherever the build does not warn. The radiated power stays within 0.21 % of the near-field flux at every placement measured.

In a dielectric¶

Count the cells per medium wavelength, λ0/n, and use at least 30, where the bound is 2 % of the peak wherever the build does not warn. The solver's time step follows the vacuum limit, so at a fixed count the error grows with the index. For the vacuum box above in a uniform medium, every length scaled to the medium wavelength and every run to its 200-period step cap, a dipole near a box edge (two faces near, the third coordinate centred) reads up to 0.96 %, 1.07 % and 1.11 % at index 1.5, 2.5 and 3.5 at 30 cells. The worst of 22 other placements (both corners, both signs along each axis, four held three cells from a face, and eight fixed in space) reads:

Index	25 cells	30 cells	35 cells
1.5	0.96 %	0.70 %	0.53 %
2.5	1.11 %	0.81 %	0.61 %
3.5	1.15 %	0.84 %	0.64 %

For a source fixed in space it falls as the square of the cell. The radiated power stays within 0.27 % of the near-field flux.

On a graded mesh¶

Count the cells per wavelength in the background and use at least 30. Keep every box face at least 0.35 wavelengths from the nearest grid line where the cell size changes, along its normal, on either side. Keep every source at least 0.1 wavelengths from one. End a refinement inside the box with that room to spare, or carry it that far past every face.

In that range, for the vacuum dipole box above (MeshOverride grading by up to 1.4 per cell, no structures, no dispersion, PML one wavelength thick, at most 100 periods), the front/back asymmetry stays within 2 % at 30 background cells per wavelength wherever the build does not warn. The worst measured reads (the carried-past figures ran on the GPU solver, which stops on energy decay alone and agrees with the CPU within 0.15 % of each value):

Refinement	Inside the box, near a box edge	Inside the box, at a box corner	Carried past every face
2 times as fine	0.90 %	0.67 %	0.13 %
3 times as fine	0.93 %	0.71 %	0.055 %

It falls at about second order for a source fixed in space, and the radiated power stays within 0.24 % of the near-field flux. Higher ratios are not covered. Outside the range no bound is given. A source on or just past the edge of a refined region radiates an asymmetric field by itself: up to 1.085 % at 3 times as fine and 0.92 % at 2 times, at 30 background cells. A refined region that ends just outside a box face reads 1.79 %, 1.25 % and 0.92 % at 25, 30 and 35 background cells (2 times as fine, the worst of 33 placements).

Directivity and gain¶

On an Ex dipole in vacuum at λ = 1 µm (a FarFieldMonitor box of half width 0.6 µm, uniform mesh, PML one wavelength thick, the default DFT shutoff, no structures, no dispersion), the peak directivity reads 1.5156, 1.5067 and 1.5037 at 20, 30 and 40 cells per wavelength. The analytic value is 1.5; the error falls at second order. The gain from six PowerMonitor planes closing a box of half width 0.3 µm centred on the dipole reads 1.5166, 1.5070 and 1.5038.

Symmetry planes¶

A closed box in a run with symmetry planes projects as the full run does, within the projection's own second-order error. On dipole boxes in vacuum at 16 cells per wavelength (three cases: three planes; two planes; a graded mesh), a closed FarFieldMonitor box reads 0.15 to 0.43 % of the peak intensity away from the same declaration on the full run, and 0.04 to 0.10 % at 32 cells. The full run's far-side faces sit a cell nearer the planes, since each face snaps to the grid line nearest its plane.

Open surfaces differ more. On exact fields in vacuum at λ = 1 µm and 16 cells per wavelength, of an Ex dipole 3.5 cells from the symmetry plane and, in the full run, its mirror image, about 0.5 µm below a plane 2 µm wide:

without a window, the plane differs between the two runs by up to 7.5 % of the peak within 30°: its recorded block ends half a cell differently at one edge;
with a window of 0.2, it reads the same to round-off;
a box 1 µm wide, centred on the symmetry plane and open on one side, differs by 1.2 % within 30° of its closed side, 3.5 % over that hemisphere and 7.7 % on its open side.

Scattering cross sections¶

On a sphere of index 1.5 and radius 0.6 µm at λ = 1 µm (a 6 µm vacuum cube, PML 0.5 µm, uniform mesh, the default subpixel averaging, no dispersion, a TFSFBox of half width 1.25 µm, a FarFieldMonitor box of half width 1.75 µm), cross_section() reads within 0.96 %, 0.80 % and 0.68 % of the Mie series' peak at 20, 25 and 30 cells per wavelength.

A sphere near a sharp resonance needs a long run. A sphere of index 2 and radius 1 µm in the same scene sits beside one: a 0.5 % lower index moves the series by 37 % of its peak. Once the pattern settles (1200 periods at 20 and 30 cells, 2400 at 25), it reads 3.8 %, 1.95 % and 1.4 % at 20, 25 and 30 cells per wavelength. After 100 to 300 periods it read 2.3 to 2.7 % at 25 cells. One frequency does not stand for the band: the 2400-period run at 25 cells reads 4.6 to 19.7 % at 0.9 to 1.1 times that frequency, each beside a mode the grid shifts.

Direction cosines and diffraction orders¶

On a 2-D grating at λ = 1 µm (a block of index 2 in a 2.1 × 1.5 µm vacuum cell, a normal-incidence plane wave polarized along x, 11 orders leaving; uniform mesh, the default subpixel averaging, no dispersion, PML one wavelength thick, at most 100 periods with the default shutoff), the order powers read from a k-space grid agree with diffraction_orders within 1.1e-4 at 20 cells per wavelength and 3.8e-5 at 30. The order directions agree exactly.

The exact projection¶

On a dipole's exact fields on a box (an x dipole 0.14 µm off the centre of a box 1.2 µm wide, vacuum, λ = 1 µm, uniform mesh), the exact field matches the dipole's own from 5 to 320 µm to 0.52 to 0.57 % of the peak at 20 cells per wavelength and 0.12 to 0.13 % at 40, and at 1.2 µm, just past the box's corners, to 0.70 % and 0.16 %. On FDTD, the exact projection of a FarFieldMonitor box of half width 0.6 µm around an Ex dipole onto a plane 1.5 µm above the dipole, 0.9 µm beyond the box, reads the field the solver recorded there to 1.12 %, 0.51 % and 0.29 % of the peak at 20, 30 and 40 cells per wavelength, second order (vacuum, λ = 1 µm, uniform mesh, no structures, no dispersion, PML one wavelength thick, at most 100 periods with the default shutoff).

Near fields¶

On an Ex dipole in vacuum at λ = 1 µm (uniform mesh, PML 0.5 µm, no structures, no dispersion), the power through a plane 0.6 µm above it, whose edges reach the PML, read from colocated_fields against a PowerMonitor on the same plane:

Cells per wavelength	20	30	40	60
Recorded one cell thick	-0.46 %	-0.21 %	-0.12 %	-0.052 %
Recorded with one sample	-2.0 %	-1.3 %	-0.91 %	-0.59 %

The first row is second order. The second is first order, from where the plane meets the PML. A plane cut in the open loses about one cell's strip at its edges: -1.8 %, -1.1 %, -0.78 % and -0.50 % cut 1 µm from the dipole axis. Record such a plane a few cells wider than the region you integrate over.

On a dipole's exact field at λ = 1 µm, the interpolation reads each component of the plane, as a fraction of that component's peak:

Plane	Recorded one cell thick	Recorded with one sample
0.3 to 0.6 µm from the dipole, 20 cells per wavelength	2.2 %	1.9 %
0.3 to 0.6 µm from the dipole, 40 cells per wavelength	0.59 %	0.47 %
0.09 µm from the dipole, 20 cells per wavelength	32 to 62 %	24 to 34 %
0.09 µm from the dipole, 40 cells per wavelength	9.5 to 13 %	8.0 to 9.7 %

The near-source ranges cover the dipole's orientation and the plane's axis and side.