**How to use Spherical Wave Expansion in Feko**

Do You know Spherical Wave Expansion (SWE), and would like to find out how you can use this technology in Feko? Then you should read on now. The following illustrative examples show how SWE can be defined and calculated in Feko and for what kind of problems this approach is beneficial.

**What is Spherical Wave Expansion? **

Spherical wave expansion (SWE) is a mathematical series expansion that represents a radiated electromagnetic field of an antenna as a summation of spherical waves. In Equation (1) *E*(r,q,f) describes the outward propagation electric field, k is the wavenumber and h is the wave impedance. *F*_{smmn}(r,q,f) is the normalized spherical wave function and Q_{smn} denotes the complex expansion coefficients [1]. Theoretically, the SWE is an infinite series, but for practical applications a finite number for the upper index N is sufficient [2]. With equation (2) the index N is related to a hypothetical sphere with radius r_{0} that includes all relevant radiating structures of the source, that should be represented with the SWE. The cumulative modal power (CPM) can be analyzed using equation (3) to control the approximation error due to a finite series index N.

The radiated fields *outside *the hypothetical sphere can be computed efficiently from the spherical wave expansion. Therefore, SWE is a powerful tool for replacing antenna models with equivalent sources. They can be defined in Feko as so-called **spherical mode sources**. In contrast to far-field sources, spherical mode sources can also be used to calculate accurate near fields as long as they are located outside the hypothetical sphere. This is visualized in the near field plot of the previous picture for an open quad-ridged horn antenna (simulated at 1 GHz). Both plots differ only in the interior of the hypothetic sphere.

The spherical mode source is, so to speak, something between a near-field source and a far-field source. Comparing the different approaches can help to evaluate at which distance to an antenna the far-field conditions occur. That allows a less conservative far-field characterization as the standard far field formular r_{ff}=2D²/l , e.g. for 5G mobile devices as described in [3] by Benoit Derat.

**How to compute Spherical Wave Expansion in Feko **

It is straight forward to compute the SWE of an antenna in Feko. In the advanced tab of the far-field request for a 3D pattern, the following spherical mode options can be activated with checkboxes:

- Calculate spherical expansion mode coefficients,
- Specify number of modes,
- Export spherical expansions coefficients to ASCII file.

If the number of modes is not specified, Feko will automatically choose an appropriate upper mode index N with sufficient approximation accuracy. The export is done in the standard TICRA sph-file format.

Note, that in the far field request an appropriate SWE center (possible near the antenna center) should be defined. If the SWE center has some distance to the antenna, the series can still be computed, but more spherical modes with higher series index N and larger radius r_{0} will appear.

**How to define a Spherical Mode Source in Feko **

In the source/load tab of CADFEKO different kind of equivalent sources can be defined. In the spherical mode source button, a TICRA sph-file can be imported, and the position and orientation of the equivalent source are defined. It is also possible to define spherical modes manually in the Construct-Field/Current-data menu of CADFEKO.

In this way the sph-file can act as interface to other antenna simulation software like GRASP, ESTEAM or QUPES from TICRA (available in the Altair partner program), for example to describe the radiation properties of satellite antennas with parabolic or quasi-periodic reflectors and use them as equivalent sources in a Feko simulation.

Compared to an equivalent far-field source, there are two major advantages:

- The spherical modes source can be used for accurate calculation of the field strength in the near field.
- No information on the phase center is required for positioning the spherical mode source. The source is positioned at the geometric location that was used as the center for SWE calculation.

Before we turn to the applications, I would like to cite a work about accuracy investigations of spherical modes: The validation of spherical mode results in Feko was part of the PhD research of Tomislaw Marinovic at KU Leuven (Belgium) and Chalmers University of Technology (Sweden). His numerical results (with own implementation in MATLAB) show excellent agreement with Feko results [4].

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**Example 1: Equivalent Model for Array Antenna (30 GHz) **

Let’s start with a large antenna array example (35 dBi gain) build from 400 patch antenna:

The model is composed by 329.556 triangle elements and the dimensions of the array are 110 mm x 110 mm x 0.29 mm. Memory requirements for the MLFMM simulation are around 315 GB. The center of the spherical wave expansion is placed at the center of the antenna and Feko indicates that sufficient accuracy is achieved with series index N = 46 (r_{0} = 73.2 mm). The same near-field and far-field evaluation can be done for a simulation with equivalent spherical mode source. Both gain patterns are identical and the near fields (in a cut-plane perpendicular to the array) differ only inside the hypothetical sphere. The memory requirements for the equivalent model are just 21 MB.

The center for the SWE could defined at other locations. To demo this, three variants with sphere center below the array are computed. Then the mode number and the sphere radius will change, but the near field result outside the hypothetical sphere will remain the same. This degree of freedom can be helpful, if a radom in front of the antenna needs to be analyzed, to avoid intersections of the hypothetical sphere with the radom.

**Example 2: Determine relevant antenna integration area on a vehicle **

In [5] another idea to use SWE for antenna integration analysis was proposed: The size of the hypothetical sphere can be used to determine which parts of a platform contribute to the far field pattern of an antenna that is placed on that platform. For example, a V2X antenna (5.9 GHz) is placed in a shark fin configuration on the roof of a (generic) car. For a full vehicle simulation, the SWE around the antenna center can be computed in Feko. The relevant parts for antenna pattern analysis are characterized with mode index N = 22 and sphere radius r_{0 }= 178 mm. This means that vehicle parts outside this sphere have no significant impact on the antenna far-field pattern and could be neglected for additional simulations to investigate antenna placement. For electrically very large objects this saves a lot of memory and computing time.

The situation would be completely different for an antenna location inside the car. For a hat rack antenna, the equivalent spherical mode source needs N = 380 modes and the corresponding sphere radius would be more than 3 meters. This is not surprising because the reflections at the car body are very relevant for the antenna pattern. A very large radius of the hypothetical sphere is therefore an indication that the position for antenna placement is poorly suited.

**Example 3: RL-GO simulation in EMC chamber **

Equivalent spherical mode sources of antennas can speed up ray-based simulation methods like the faceted UTD (Uniform Theory of Diffraction) or ray-launching Geometrical Optics (RL-GO) in Feko. For the full antenna model every triangle element would act as a source in the ray-tracing or ray-launching approach. The equivalent spherical mode source is reducing the number of rays and enables faster simulation with reasonable memory requirements.

For a semi-anechoic chamber (23 m x 16 m x 9 m) with metallic ground surface and more than 10.000 absorbers at walls and the ceiling the quite zone behavior shall be investigated. The electric field distribution at specified locations is computed and compared with a reference solution created in free space (with infinite PEC ground). At the frequency of 1 GHz the chamber model would be electrically too large for a classical full wave MLFMM simulation. But this is where asymptotic methods can prove their strengths: Absorber and ground plane are assigned for RL-GO and the Bilog antenna is replaced with an equivalent modes source. This approach requires only 8.5 GB memory. Note, that runtime scales linear with the number of near-field request points. In this case, approximately 1 second runtime per field request is required.

The results shows that the filed distribution in the chamber is very similar to the reference solution and in the quite zone differences are sufficient small. Near the absorber walls the behavior is less good.

I hope that these examples have shown that SWE and equivalent spherical mode sources are powerful tools for several antenna related simulation tasks, especially at high frequencies. Try this approach for yourself: The model of the antenna array is attached to this article.

**References: **

- [1] F. Jensen:
*On the Number of Modes in Spherical Wave Expansions*, 26^{th}AMTA Conference, 2004. - [2] J. E. Hansen,
*Spherical Near-Field Antenna Measurements*, London, UK: The Institution of Engineering and Technology, 2008. - [3] B. Derat:
*5G antenna characterization in the far-field: How close can far-field be?*IEEE International Symposium on Electromagnetic Compatibility and IEEE Asia-Pacific Symposium on Electromagnetic Compatibility (EMC/APEMC), 2018. - [4] T. Marinovic :
*Vector Spherical Harmonic Expansion*, technical paper available at Altair University 2020. - [5] N. D. Chaudhury, L. Berkelmann, B. Jansen, J. Robert:
*On Vehicular Model Reduction for Antenna Simulation Using Spherical Wave Expansion*, IEEE AP-S/URSI International Symposium on Antennas and Propagation, 2023.