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Contents of this posting
 Single emitters
 Arrays of emitters  directed beams
 Fraunhofer diffraction
 Beam strength very close to an emitter array
 Reflection via emitters
 Resonant cavities via emitters
 Overview
 The diffraction question
Games with arrays of wave emitters
Assumptions of the emitter method 


One approach to solving a problem of waves in the presence of reflecting and absorbing walls is to model the problem with arrays of emitters. That is, replace all walls, reflectors and absorbers with emitters, nothing but emitters, and let the waves be free to propagate from these emitters without any objects in the space. In this article we examine the possibilities in algebra and images of the wave patterns of various emitter arrays in two dimensions.
One of the motivations for using this method is the number of variables involved. Conventional computer modeling of waves in the 2D world divides up the space into a 2D array of cells, each with variables to be calculated. For a two dimensional problem, the emitter method involves linear arrays of emitters and no 2D arrays of variables. It is therefore computationally cheaper.
In 2D differential equation methods, a user needs to add features to absorb radiation that should disappear into space. In emitter methods, this issue does not occur.
Emitter methods are most appropriate for systems that are mostly made up of a simple wave medium.
Single emitters  types
A monopole emitter
The equation for the 2D wave field around a monopole emitter located at the origin is:
where H_{0}^{(1)} is a Hankel function of order 0 and of the first kind. The Hankel function is a complex sum of Bessel functions of the first and second kinds with a 90 degree phase shift, i.e. H_{0}^{(1)} = J_{0} + i Y_{0}.
The wave field is a function of the wavenumber κ = 2π/λ, where r (the distance between the emitter and point of field measurement which equals the polar coordinate radius for an emitter at the origin), and λ which is the wavelength of the wave being emitted (measured a distance away from the emitter where the waves approximate a plane wave).
The variable Φ in (1) above is the distance varying part of the potential of the wave. We have not included the time part of the potential which is understood to be e^{−iωt}. The actual or physical wave potential is the real part of the total complex potential (that is the real part of (1) after it is multiplied by the time part). Φ is the velocity potential in the case of water waves. The parameter A in (1) sets the amplitude of the wave.
Below we show a video of such waves from an emitter at the origin.
A dipole emitter
The equation describing the wave field coming from a dipole emitter in two dimensions (2D) is as given in (2):
where H_{1}^{(1)} is a Hankel function of order 1 and of the first kind.
Note that in Fig. 3 below, the wave field is not uniform in all directions as it is in Figs. 1 and 2 but varies with direction as given by the cos ϕ as given by (2). This wave field is strongest in the positive and negative x directions where ϕ = 0 and ϕ = π . The wave field is zero in the positive and negative ydirections where ϕ = π/2 and ϕ = 3π/2 .
A directional emitter
To get an emitter that sends waves mostly in one direction, we combine the two above emitters with a 90 degree phase shift as given by:
We see the results in the images Figs. 5 and 6 below.
Fig. 5. Video clip of the fields around a directional emitter as given by (3) above.
Figs. 6a and 6b. Snapshot and amplitude of a directional emitter located at the origin. Note that the waves are maximum along the positive x axis and zero along the negative x axis. Note that the origin is left of center as indicated by the axis numbering. The amplitude varies as (cos φ + 1) . In two dimensions the amplitude of waves from a directional emitter falls off as 1/√r . Note that this (cos φ + 1) dependence is consistent with the HuygensKirchhoff diffraction formula [Elmore and Heald, physics of waves, p.331]. More on this below.  
Arrays of emitters
Location of the linear array of emitters used to make field mappings in Figs. 7, 8 and9. 
The HuygensKirchhoff diffraction formula is based on the idea that the wavefield in any space can be calculated based solely on the wave entering the space to power the wavefield there. The HuygensKirchhoff formula is basically just setting us up to calculate any arbitrary wave field as though it were created entirely by arrays of directional emitters arranged along a mathematical surface feeding the wavefield in question.
In Figs. 7, 8 and 9 below we show 2D false color snap shots and mappings of the amplitude of the waves coming from linear arrays of 11 emitters of three kinds: monopole, dipole and directional. In all three cases the emitters are located along the y axis and spaced at 0.3 wavelength intervals. In all cases the resulting wave pattern is more focused than with single emitters along the x axis. The beams created by the array of monopole and dipole emitters in Figs. 7 and 8 are focused in two directions: in the positive x direction and in the negative x direction. The beam created by the directional emitters in Fig. 9 is directed mostly in the positive x direction. The HuygensKirchhoff diffraction formula has a cosine factor in it that is mathematically equivalent to the directional emitter case.
In making the mappings below (as well as those above) we have used the Hankel function formula, Equations (1), (2) and (3) above, for the fields instead of the simpler approximate e^{ikr} term in the HuygensKirchhoff diffraction formula.
Fraunhofer diffraction
Fraunhofer diffraction refers to interference patterns that occur in wave fields at large distances from their sources. The most commonly mentioned of these are the single slit diffraction pattern and double slit diffraction pattern. Historically these patterns were produced by shining a single color of light through a single or pair of slits in another otherwise opaque barrier as shown in Fig. 10. The patterns from the setups in Figs. 7, 8 and 9 are shown in Fig. 12 below (scroll down) and are the classic single slit Fraunhofer diffraction patterns. Fig. 12 shows the amplitude of the waves from each of the setups falling on a screen located very far to the right of the emitter arrays. We see that in all three cases, we get the same diffraction pattern.
Next we will algebraically derive these diffraction patterns for an array of monopole emitters. Since the wave of a monopole is made up of Hankel functions and Fraunhofer diffraction is all about the wave field at large distances, we look at the behavior of Hankel functions for very large arguments. It is also important for the approximation that the patterns be relatively close to the x axis.
At very large arguments (z >> α^{2}+ ¼ ) the Hankel functions can be approximated by an asymptotic sinusoid:
Using (4) the asymptotic wave field of a monopole emitter becomes:
whereis the distance from an emitter at (x,y) = (0,y_{1}) to a point (x_{2},y_{2}) on the observation screen.
The normal way to calculate Fraunhofer diffraction is to imagine a continuum of emitters located on the y axis and extending from y = −a/2 to y = +a/2. The parameter a is the width of the slit (or opening) in the opaque barrier. We set up an integral to sum up the waves from these emitters.
To get to a simple mathematical equation, we need to approximate the r in (6) and (7) consistent with the screen being at a large distance from the slit, large compared with the wavelength of the waves. This calculation involves replacing the r in the exponent in (7) with an average distance r_{0} plus small dr that varies with the angle, i.e. dr = −y_{1} sinθ where θ is the angle between the line from y_{1} to y_{2} and the x axis as shown in Fig. 11:
The r in the denominator of the square root in (7) is considered slowly varying and replaced by r_{0} ≅ x_{2}. This is in contrast with the r in the exp() function which, because of it sinusoidal nature, small changes in r cause large changes in the result of the exp() operation.
The C in (8) stands for the lumped together complex constants which do not play a role in the amplitude pattern of the waves striking the screen.
An alternative way to derive (8) is to approximate (6) with the first two terms of a Taylor series expansion.
Fig. 13a. Field strengths were measured at the locatioins shown in green, very close to the linear array of emitters (shown in red). 
Beam strength very close to an emitter array
For the next set of simulations it is important to know the ratio between the strength of the emitters in a linear array and the amplitude of waves in the beam once the waves from the various emitters have more or less merged. Below are graphs of the amplitude of the partially "merged" beams for emitter strengths of 1.0 .
Reflection via emitters
To illustration an application of arrays of emitters, we next simulate a mirror using a linear array of monopole emitters. The strengths of the emitters in the mirror are set to cancel out the incident wavefront using the (π/2)×A/emitterDensity factor mentioned in the caption for Fig. 13 above.
Resonant cavities via emitters
One form of resonators is a resonant cavity. This type of resonator consists of a chamber with walls that reflect all waves inside the chamber and keep them bottled up. The waves can be optical, microwave, acoustical, water waves, etc and the chamber walls made of material suitable for the wave type. The best cavity resonators have very low loss so that the waves will reflect on themselves many, many times. If a driving source of the waves is inserted into the chamber and its frequency is adjusted just so, very large standing waves will develop in time. The frequencies at which this occurs are called the resonant frequencies.
Above we have just explored the use of a linear array of emitters to form a reflector of waves. We will next investigate the use of such an array to form a cavity resonator.
Above we set the strength of emission of each wall emitter so that it canceled the waves on the far side of the mirror. We set the strength of emission based on the waves present from all other sources. (These are the waves we wish to cancel on the far side.) We should be able to repeat this method here.
Instead we decided to change the method a little. We added a separate array of sensors located a distance outside the cavity and mathematically requiring that the complex amplitude of each wall emitter be adjusted so that the total amplitude detected at each of these external sensors was zero. This separated the functions of detecting and emitting.
circular cavity
Below we see the results of an Octave simulation using a circular array of monopole emitters forming a circular cavity as shown in Fig. 16. There is a single active monopole emitter at the center of the circle to drive the resonance. There is also a cicular array of sensors outside the walls.
We want the wall emitters to broadcast a wave that cancels the wave from the central driver and all other wall emitters at the sensors' locations. We can express this mathematically as:
where
 Φ_{jthsensor} is the wave at the j^{th} sensor's location due to waves sent out by the driver and by all the cavity wall emitters.
 r_{drive  jthsensor} is the distance between the driver and j^{th} sensor. We will refer to this distance as r_{0,j}.
 r_{ithwall  jthsensor} is the distance between the i^{th} wall emitter and j^{th} sensor. We will refer to this distance as r_{i,j}.
In Fig. 16 we show these points and distances.
In (8), the A_{i}'s, the complex amplitudes of the wall emitters, are the unknowns while everything else are the knowns.
We can write (8) as a linear matrix equation:
where
In standard matrix notation we write (9) as:
M A = S , (11)
where A and S are vectors and M is a matrix.
In Octave, we first calculate S and M based on the geometry of our setup. We then use the standard linear matrix equation solver, linsolve, to solve for A. Once we know A we know the strengths of all emitters in the setup and we can calculate the field anywhere to make our 2D false color maps.
where p is the point at which you want to know the wave fields. In figure 16 we show these query points as an "array of field points" which we will use to make a 2D color map of the fields.
Resonant cavity simulation via emitters  

Figs. 17a and 17b. Cavity made of 15 emitters. The left image is a 2D false color mapping of the magnitude of waves for the first resonance, the (0,1) mode.
The right image is a graph of the amplitude near the center versus the frequency of the driver (which affects κ, the wavenumber). We see the standard resonance curve. This resonance has a Q of about 1200. (For more on Q's of resonators see this reference and other links at the beginning of that reference.) Use of more wall emitters will result in much higher Q's.  
Figs. 18a, 18b and 18c. The first two images are similar to the above images except they are for the second mode. Also we have plotted the real part of the complex wave (which goes positive and negative) instead of its magnitude. Zero field is light green as indicated by the colorbar. While the previous mode has only a central peak, this mode has the central peak plus an outer ring (blue). (For more on modes of a circular resonator see this reference.)
The last image shows the real part of the complex wave versus the radius (or x when constrained to a radial line). We see with Dirichlet boundary conditions, the field peaks at r = 0 goes through zero at r = 0.42, peaks in the negative at r = 0.7 and returns to zero at r = 1 which is the cavity wall. The field is zero outside the cavity. The precise field shown at the cavity wall depends on the exact path we chose through the array of emitters there. We try to pass cleanly through a space between the emitters and avoid passing too close to an emitter whose fields can be very strong. Mathematically we expect circularly symmetric modes at wavelengths of 2.6 and 1.14 times the cavity resonance radius. The radius of our cavity is 1.0 . This resonance has a Q of about 750. This resonance has a lower Q than that above because it is of higher frequency and thus shorter wavelength which tends to "slip" through our picket fence of emitters a little better. 
We might point out that the above setup is very symmetric and could be solved without matrix solving methods. On the other hand, the method used above is general and directly applicable to nonsymmetrical and oddly shaped cavity resonators (such as the Lshaped resonator discussed below). Treating such resonators involves a somewhat more complicated routine to specify the coordinates of the cavity wall emitter x,y locations.
modeling a rotating wave resonator
In an earlier posting we have discussed rotating wave fields in depth.
In fig. 19 we show use of the above methods to make an animation of a rotating mode in our cavity.
Fig. 19. Video clip of the fields in the (2,1) rotating mode modeled in Octave using a circular array of emitters for the cavity wall.
Two driver monopole emitters were used, one on the xaxis at (x,y)=(0.5,0) and one above it at (x,y)=(3.5,3.5), both positioned to be near the maximum field regions of their related standing wave modes. Fields from the second one were multiplied by the imaginary constant i, to cause a 90 degree temporal phase shift between the two standing wave modes. Fields of the two modes were then added together on a pointbypoint basis and used to make the color video shown. 
Lshaped resonator
As an example of applying these emitter methods to other shaped resonators we next model an Lshaped resonator. The method is the same as above except that calculation of the resonator wall emitter locations is more burdensome and the calculation of the sampling points is even more complicated.
In order to avoid spurious resonances, each of the exterior sampling points needs to be closest to its own special wall emitter. This requirement entails a somewhat "messy" specifying of the sampler locations. Enough so that while we wrote the code for the wall emitter placement quite generally for a polygon resonator, we custom fit the sample point locations to our particular Lresonator at hand.
Resonator with Neumann boundary conditions
Neumann boundary conditions can be simulated with emitters by substituting dipole wall emitters in place of the monopole emitters used above for Dirichlet boundaries. The dipole emitters are a little more complicated to use because their field strength varies with direction requiring each of them to be properly oriented. The correct orientation is with the maximum emission being perpendicular to the cavity wall so that their fields add to the fields inside the cavity and subtract from the fields outside.
Suppose we have a dipole that is oriented horizontally. The field from this, radiating at ϕ_{2} relative to the horizontal, is given by:
which we repeat from (2) above. But our dipoles are not all oriented horizontally. Instead they are arranged around the circular cavity walls, everywhere perpendicular to these walls, as shown in Fig. 22. So what is the field for a dipole in the wall of our circular resonator as a function of its position in the resonator? That is to say, what is ϕ_{2} for (13) as a function of the polar angle ϕ_{1} that an emitter happens to be located at? By examining Fig. 22 we see that these angles are related by:
where ϕ_{3} is the angle relative to the horizontal for which we want the emission.
Fig. 23 shows the results of an Octave simulation using an array of dipoles in place of the cavity walls.
Overview
Let's reflect on the work shown above to model reflectors. We have used two methods. To make the reflectors shown in Figs. 14 and 15 we required that the reflector emitters react to the wave fields they sense at their location and zero this out. This method could be viewed as how atoms in real wave reflectors respond to incident waves and is therefore very natural to use.
This method can be extended by replacing the monopole emitters with dipole emitters oriented perpendicular to the surface to allow modeling Neumann boundary conditions (most of the above examples assumed Dirichlet boundary conditions.) Use of directional emitters, also oriented perpendicular to the surface, would allow modeling wave absorbing surfaces.
The question that we worked on in Fig. 13 but did not totally resolve is:
"exactly what field should or would an elementary emitter or atom emit for a given applied field?"
Another way to state this question is: "how stong should a wall emitter's amplitude A be so that its emitted waves merge with that of nearby emitters to cancel the total wave on the far side of a mirror?". This brings up the (π/2)×A/emitterDensity factor we mentioned in Fig. 13 above. Does this factor hold for corners in a segmented reflector as in the Lshaped resonator above? Does it hold for curved reflectors such as the walls of the circular resonator in Figs. 16 through 19?
Method 2 bypasses that question by using external sensors. It works nicely in a case where we know what the field should be at the sensor locations, as in zero outside a resonator.
If we answered the above question, we could use the first method in situations where we do not have regions of zero field or known field strength and wish to find the field everywhere. It would also open up the emitter method for use where dielectriclike materials are present. In this case we would model a dielectric as an array of emitters set to react less vigorously to the applied field as would an emitter in a conductor.
The diffraction question
Question of the exact nature of diffraction often comes up. If we have a narrow, wellcollimated beam traversing an otherwise empty wave medium, will the beam diffract? Exactly when and where will this occur?
 Some spreading will take place from the instant the beam emerges from the collimating structure.
 From the HuygensKirchhoff diffraction formula we can pick the beam anywhere and computationally let it diffract at that point with a (1+cosθ) dependence. We might think that this would lead to immediate destructive spreading out of the beam. However a careful considering of the HK formula shows that the beam with a reasonable width will resist spreading out, except for an insignificant amount. When we add all the wavelets from the whole beam width, the diffracting part is almost entirely canceled out.
 Waves cannot escape their tendency to diffract and spread out, however the amount of wave energy that goes to really odd unintuitive trajectories is extremely small.
 It would be interesting to do a real time double slit modeling using the HK formula with Hankel functions. Launch a short pulse of waves and watch them do all sorts of unintuitive things as a movie or series of stills. To see this, the log of the amplitude will need to be displayed in a 2D false color map. Might make an entertaining movie to show at a conference.
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