Field example

Output visualization

Usually a proper field plot gives the best idea about the guiding characteristics of a dielectric structure. Besides, profile visualizations are required e.g. for checking the correctness of results, and for purposes of demonstration.

Since graphics capabilities are not directly incorporated into the WMM programs, a plot requires to export suitable data sets, where the output format naturally depends on the numerical sofware for which the data is intended. The export routines are favourably implemented as member functions of Mode (WMM_Mode) or WMM_ModeArray objects; the methods

writeprofile Evaluate a mode component on a finite rectangular regular mesh in the x-y-plane, store the profile data in a file.
writesec Evaluate a mode component on equidistant points on a line segment in the x-y-plane, save the profile section data to a file (extension .xyf).

(defined in mode.h, mode.cpp) may serve as examples, and as candidates for adapting the WMM output to a new visualization tool. The WMM_Mode members field and wg provide the necessary information about mode profiles and the underlying structures. WMM_ModeArray.field gives access to propagating guided fields.

This section describes a collection of functions which export the field data to MATLAB script files (extension '.m'). The routines are defined in mode.h, mode.cpp for the mode profiles, and in wmmarr.h, wmmarr.cpp for mode interference fields, with helper functions found in wmmvis.h and wmmvis.cpp (the latter actually write the MATLAB commands).

Besides the field data, a proper figure requires further information, e.g. about axis annotations or scaling ratios. In the present case, a sketch of the structure under consideration is usually appropriate (helpful for immediate on-screen inspection as well as for publishing). This information is at hand when the export routine is executed, and at this moment one has usually a shape of the final plot already in mind. Therefore, the output m-files produce an entire figure when invoked in MATLAB, including plot commands, axis annotations, scaling, and, if appropriate, lines or patches to indicate the dielectric structure. In this way the path from the raw mode solver output to an acceptable plot is short and almost automatized.

Since 'acceptable' is a matter of taste (and of the actual problem), some ways to change the look of the final figure are to be mentioned:

The type of plot is determined by selecting the appropriate function, or its argument, respectively. Additionally, a few global options influence the output (possible settings are YES and NO):

Mlop_Colour Decides, whether the plot uses colours. Set this value to NO to enforce a gray scale figure.
Mlop_Print Most types of m-files contain print commands at the very end, which are commented, if this option is not set to YES. Doing so enforces MATLAB to write a graphics file (encapsulated postscript (.eps) or portable network graphics (.png)) at completing the figure setup.

These options may be useful, if an entire sequence of graphics is to be generated automatically. They are introduced in wmmvis.h, wmmvis.cpp.

In conclusion, for plotting a field profile you may adhere to the following steps:

Most of the the figures shown below are generated in this way, without any further editing.

Mode profiles

The following functions, defined in mode.h and mode.cpp as members of WMM_Mode objects, initiate mode profile plots. Comments on some of the common arguments can be found at the bottom of this page. See e.g. rib.cpp for examples on the use of the plot commands; the figures in this section correspond to the profile of the fundamental symmetrical vectorial mode of the rib waveguide, which is analysed in that program.

acmfile Surface plots of the six (all) electromagnetic components collected in one figure. The subplot's axes are adjusted such that the field amplitudes are comparable within each row.
The script file contains a line 's = 1;' at the beginning of the section with plot commands. Changing this to 's = -1;' inverts all field amplitudes.
mfile, 'N' Generate a m-file for a single mode component. The last argument specifies the type of plot. 'N' or a character which is not listed below yields a m-file that contains merely the field data, without any plot commands.
mfile, 'C' A contour plot of one field component with a sketch of lines and patches that indicates the guiding structure.
The gray scale levels of the patches are adjusted to the actual refractive indices (darker shading means higher refractive index). To enlarge the contrast (e.g. between substrate and film regions with, compared to the cover, very similar permittivity), change these values, such that the steps between the refractive index levels become equal. The waveguide data can be found at the beginning of the m-file.
mfile, 'S' A surface mesh plot of one field component. The waveguide boundaries are indicated on top of the axis box.

If the mode solver happens to have made the wrong choice of signs for a proper look of the selected component, you might wish to invert the plot: insert a minus sign in the line starting 'fld = ...' at the beginning of the plot section (end of the m-file) and reverse the definitions of minf and maxf at the beginning of the file.

Originally, the extension of the field axis is adjusted to the maximum and minimum of the electric or magnetic field with respect to all three components. Thus plots of different components of a single mode are comparable. However, if interest is in a single component only, uncomment the two redefinitions of minf and maxf at the beginning of the plot section in the m-file.
mfile, 'I' Intensity plot of one field component. The field amplitude is indicated by blue scale shading, with the levels extending linearly from the minimum amplitude (blue) to the maximum field (white).
In case one is tired of blue, MATLAB offers several beautiful predefined colormaps. With 'colormap(blue);' replaced by 'colormap(hot)' and with 'ls = 'w-';' substituted for 'ls = 'k-';' (boundaries in white), the m-file yields the following output:
Here the colour scaling renders larger regions with small field amplitudes visible. A somewhat more objective method, i.e. plotting the modulus of the field instead of its square (mfile-argument MOD instead of SQR), has a similar effect. The statement 'Mlop_Colour = NO;' before the mfile command switches the plot to gray scale levels and white lines for the permittivity boundaries.
fancymfile fancy (adj.): ..., bred esp. for bizarre or ornamental qualities that lack practical utility, ...
(Merriam-Websters Collegiate Dictionary).
A lighted surface plot. Black lines at the surface level indicate the dielectric discontinuities:
At present, the function accepts no Afo argument. The plot always depicts the original field component.

Sections of mode profiles

For a strictly quantitative illustration of mode profiles, one should restrict the graphics to two dimensional data. There is one WMM_Mode member function, which stores mode profile section data into m-files (see mode.h, mode.cpp for the precise definition):

secmfile, 'L' With argument 'L', the function evaluates a mode component on equidistant points on a line segment in the x-y-plane and writes the data to a file *L.m, together with figure initialization commands, axis annotations, plot commands, and instructions for illustrating the relevant dielectric structure. The line segment is to be specified by the coordinates (x0, y0) and (x1, y1) of the end points. If the segment crosses a dielectric boundary, the boundary position enters the plot. To indicate other positions or to modify the shading, edit the variable assignments below the '% Crossed boundaries' comment at the beginning of the output file (nbd, bd, and ri are the number of boundaries, the positions, and the refractive indices inbetween).
secmfile, 'V' 'V' restricts the output to field data and plot commands. The output is named *V.m. These files are intended to add curves to an m-file produced by the 'L' version of secmfile. The plot command section of the *L.m files (end of the file) contains the comments
% Include further field section data
% _______V

Replacing the second line by the (uncommented) base filename of an *V.m file adds the curve to the figure.
In rib.cpp, these functions are employed to generate plots, where the dominant electric component of the fundamental mode in a rib waveguide is evaluated along several horizontal ...

... and vertical lines of the x-y-plane:


rib.cpp writes eight files, which contain mode profile section data. First the files *h?V.m and *v?V.m were included into *haL.m or *vaL.m, respectively. To indicate all horizontal boundaries in the bottom figure, vcsEyvaL.m was edited to read
  % Crossed boundaries
  nbd = 3;
  bd = [0.0 0.5 1.0];
  ri = [1.9 2.2 1.5 1.0];

(line 43). Then invoking the *L scripts in MATLAB resulted in the figures as shown.

The transverse components Ex and Ey of the mode profile may be discontinuous at dielectric boundaries. Therefore, if different media are involved, the m-files written by secmfile usually define more than one curve. While the points in each segment are connected, the segments are not joined, such that the discontinuities become explicitely apparent.

At present, secmfile is retricted to either horizontal (x0==x1) or vertical (y0==y1) profile sections. A profile on a tilted line must be processed manually, e.g. using the output of the writesec function.

Mode interference

wmmarr.h and wmmarr.cpp define several procedures for illustrating the interference of WMM_Modes. The functions are members of WMM_ModeArrays. Confer the section below for comments on some of the arguments, and the corresponding section on the preceding page for remarks on mode interference.

The application mmi.cpp exemplifies the use of the interference visualization procedures. All figures included in this section belong to that program. It simulates the guided wave propagation in a multimode interference (MMI) coupler, here a wide, multimode raised strip waveguide. Semivectorial analysis yields three symmetrical and two antisymmetrical TE modes:

tesqEy00C.m teaqEy00C.m tesqEy01C.m teaqEy01C.m tesqEy02C.m

The following functions evaluate superpositions of these fields.

Single waveguide segment:

Initial mode amplitudes are supplied as explicit arguments.

mfile The function generates a plot of the mode interference field versus the position on the waveguide cross section plane, at a certain propagation distance. Analogously to mfile for single mode profiles, the last character argument (one of 'C', 'S', 'I', or 'N') specifies the type of plot, a contour plot, surface mesh, an intensity image, or the raw MATLAB data definitions. See the section on mode profile visualization for comments on the plot types.

In mmi.cpp, mfile serves to illustrate the superposition of the three symmetrical modes after exciting the multimode waveguide by the single mode of a centered input wavguide (width=1.3µm) at z=0:


After propagating a certain distance, the power concentrates around two focus points (note that the contour levels of the two plots are not comparable):

Where appropriate, the figures contain the z-position as an annotation.
fancymfile A lighted surface plot of the interference field over the x-y-plane at a fixed z-position, cf. WMM_Mode.fancymfile.
movie An animated sequence of plots of the two former types, for a specified number of equidistant z-positions. movie first generates a separate m-file for each propagation distance, numbered by the ext? character arguments. Afterwards the function writes a script *M.m, which collects MATLAB commands to initialize the single figures, and statements to show them sequentially afterwards. *M.m stores the entire sequence of frames in a MATLAB variable 'M'. The command
  movie(M, 10)
repeats the animation 10 times.

This is what it looks like for the MMI example with 32 frames on a distance of one focusing length (click on the figures for a magnified version):

intmEyplM.m              intfSzplM.m
Note that this is a stationary process; time represents the propagation distance z.
prop A plot of the interference pattern in the y-z-plane ybeg < y < yend, zbeg < z < zend at a certain elevation x, i.e. a top view of the waveguide. The blue or gray scale levels correspond to the squareroot of the local intensity (Sz). Now the y-direction is the vertical plot axis; the propagation distance z extends over the horizontal plot axis.

For the MMI waveguide from above, the prop-plot shows a focus point at a distance of about 32µm after the centered illumination, with a two focus position at half this length (cf. the figures shown for mfile).

This almost periodic pattern justifies repeating the animations shown before.

Three segment couplers:

Overlap integrals determine the initial mode amplitudes.

ioprop A function similar to prop, which assumes the three segment coupler model. Instead of initial amplitudes, ioprop takes input and output modes (WMM_Modes or WMM_ModeArrays, respectively) as arguments, and computes the mode amplitudes for the central segment and for the output channel(s) by overlap integrals. The plot may include the central interference segment as well as the straight input and output channels, with black lines indicating the device boundaries.

Depending on the position z, the plot displays the following information:

z < 0: the intensity profile of the input modes, scaled by the supplied amplitudes.
0 < z < l: the interference pattern of the WMM_ModeArrays modes. Overlap integrals with the input modes at z=0 determine the mode amplitudes. The argument l specifies the device length.
z < 0: the intensity profile of the output modes. Scaling factors are determined by computing overlaps between the output modes and the interference field at z=l.

The function exist in three versions, which differ by their input arguments. The information about the input respectively output channels can be supplied either as a single WMM_Mode or as a pair of a WMM_ModeArray and a Waveguide. In the former case, the mode's waveguide is employed to indicate the channel geometry; in the latter the supplied waveguide (needed for displaying purposes only) should be defined to represent the actual physical geometry of the ports.

In the case of more than one input or output channel, the model requires spatially well separated input and output ports; coupling between the port waveguides is assumed to be negligible. Note that the model neglects reflections at the waveguide junctions completely. Parts of the electromagnetic fields which are radiated in forward direction do not enter the plot as well (this shows up in small discontinuities in the intensity at the junctions).

Applied to the MMI example, ioprop yields the following plots:

Centered input and output waveguide, device length equals one focusing length:


One input port, device length is half the focusing length, two properly spaced output channels, a TE power splitter:



Two input waveguides, symmetrical excitation, one centered output channel, constructive interference:



Antisymmetrical excitation, destructive interference:



Only one input channel of the combiner is excited; about half of the power passes the device:


Common arguments

Some of the plot output functions listed above take arguments of somewhat exotic types, which may need commenting:

Rect A rectangular region, defined in waveg.h, waveg.cpp. The class contains four double values x0, y0, x1, y1 to fix the lower left and the upper right corners (note the orientation of the coordinate system). In the present context, the argument usually specifies a plot window, where the two accompanying integer values define the numbers of points in a regular output mesh.
Fcomp The field component which is to be plotted, one of EX, EY, EZ, HX, HY, HZ, or SZ for the three components of the electric field, the three magnetic components, and the longitudinal component of the time averaged Poynting vector, as declared in maxweq.h. For mode profiles, note the convention of real components, as explained in the comment on the WMM_Mode.field function.
Afo An attribute for the field output, declared in maxweq.h. Possible values are
 MOD:  the absolute value,
 ORG:  the original field,
 SQR:  the square of the field,
 REP:  the real part, and
 IMP:  the imaginary part.
Note that usually not all choices and combinations of Fcomp and Afo arguments are permitted and/or meaningful.

Additionally, in most cases two character arguments (named ext0 and ext1) have to be supplied to fill two identifier places in the automatically composed filename base.

SiIO, Simulations in Integrated Optics