Motivation
The dipole antenna is one of the simplest antennas you have modeled. We will now study an array of such antenna. To obtain a useful model, you can merge the void cell of the grating section and the dipole antenna of the antenna section. Although this seems trivial, there are some aspects to consider.
Model
Assume that you have an array of dipole antennas along the x axis. The distance between each neighboring antenna shall be 1m. Thus, you can use the void cell of the grating section with 1m side length in the x direction. Theoretically, you can put the dipole antenna anywhere in the cell, but when the antenna is close to one of the boundaries, you obtain strongly varying fields at that boundary. Therefore, it is advisable to put the antenna in the center of the cell. Just as when you modeled a single dipole antenna, you should introduce a small, circular auxiliary boundary around the antenna to avoid problems with the singularity of the field. The weight of this boundary should be zero because no boundary conditions should be fulfilled on it.
Excitation
When you have loaded the void cell model of the grating section, replace the plane wave excitation in the top layer by the dipole antenna in the center of the cell. Remember that the complex constant Cx in the Project dialog was evaluated by MMP from the incident plane wave. Now, you have to define Cx manually. Start with Cx=0. This means that the complex amplitudes of all antennas in the array are equal in magnitude and phase. As soon as you have done that, you can run MMP and view the results.
Validate the results
To validate the results, you can proceed as you did in the previous sections. Note that there is a symmetry with respect to the line parallel to the x axis on which the antennas are located. Moreover, there is a symmetry with respect to several lines parallel to the y axis. Can you observe these symmetries? Since the field in the top and bottom layers is modeled with Rayleigh expansions only, the symmetry of the problem causes certain relations between the amplitudes of the different Rayleigh terms. Can you discover these relations by watching the parameters in the Expansion dialog?
Expansions
The best expansion for the void cell was a Bessel expansion. This expansion could be replaced, for example, by four multipole expansions located outside the cell. Two of these multipoles are located in the centers of the two neighboring cells and the other ones are in the top and bottom layers. Since no energy is radiated from the top and bottom layers into the cell, you can omit the multipoles in these layers. Compare the results obtained with a Bessel expansion with those obtained with two and four multipole expansions. Can you explain why the amplitudes of the multipoles in the top and bottom layers are not zero and are not even small? Which of the models leads to the most accurate results for a fixed number of unknown parameters? To answer this question, compare the results with a sufficiently fine model with high multipole orders or with high Bessel expansion orders.
Far field
Since the antenna array extends to infinity in the x direction, the far field computation is different from the far field computation of usual antennas. First of all, the observation point can only be far away from the array in the +y or -y direction in the top and bottom layers. Therefore, the far field is entirely described by the Rayleigh terms. Since the field of evanescent Rayleigh terms decays exponentially, these terms can be omitted from the far field computation. All other Rayleigh terms describe simple plane waves propagating in different directions, i.e., you obtain a discrete directional spectrum for the radiated far field. The radiation directions are obtained from the wave number and from the periodicity constants. The orientations of the dipole antennas have no influence on these directions.
Rotate the antenna
You can verify this by rotating the antenna in the original cell. Note that the discrete direction spectrum varies with the rotation angle, but the directions remain unchanged. This is not obvious when you observe, for example, the Poynting vector field near the antenna. The power radiated in the different directions depends on the orientation of the antenna and on the phase shift between the antenna in the original cell and its neighbors.
Phase shifts
You should also observe the effects caused by phase shifts. The phase shift is defined by the real part of the constant Cx that is defined in the project dialog. You can also change the phase shift in degrees with the directive "INCrease PERiod Phase".
Frequency and wavelength
If you studied the field obtained with different orientations of the dipole antenna and with different phase shifts, you probably observed some surprising effects. These effects are caused by interference of the waves generated by the different antennas in the array. Obviously, the distance of the antenna with respect to the wavelength plays an important role. Thus, you should also vary the frequency. Note that you obtain essentially the same results when you increase the frequency as when you decrease the distance between the antennas.
DIPP100.PRO
On your CD-ROM you can find the project DIPP100. This project contains all the necessary files for exploring an array of dipole antennas. Adapt the files according to your interests.
