# Calculating coefficients of spherical harmonic expansion of electric field

1. Dec 27, 2009

### sally_a

I have data for the radiation pattern of antenna, given as the theta and phi components of the electric field (E_theta, E_phi), with 0<theta<180 deg, 0<phi<360 deg.

I want to describe this data as a spherical harmonic expansion. So, my task is to find the spherical harmonic expansion coefficients.

I assumed the theta and phi components of the electric field can be individually expanded as a sum of spherical harmonics, and found the coefficients by multiplying each (normalized) spherical harmonic term with the data and integrating, since the spherical harmonics are orthonormal.

However, this seems incorrect, the error between the reconstructed and measured electric fields increases as the order of the expansion increases.

What could be going wrong? I am trying to figure out if the error is conceptual or computational in nature.

I saw a paper which says the measured data has to be converted from spherical to Cartesian coordinates (convert (E_theta, E_phi) to (E_x, E_y, E_z)), and then each of Cartesian components has to be expanded in terms of spherical harmonics. Is this necessary, and if so, why? Why doesn't the spherical harmonic expansion hold good in spherical (theta,phi) coordinates?

2. Dec 27, 2009

### sally_a

Spherical harmonic expansion coefficients for electric field

I have data for the radiation pattern of antenna, given as the theta and phi components of the electric field (E_theta, E_phi), with 0<theta<180 deg, 0<phi<360 deg.

I want to describe this data as a spherical harmonic expansion. So, my task is to find the spherical harmonic expansion coefficients.

I assumed the theta and phi components of the electric field can be individually expanded as a sum of spherical harmonics, and found the coefficients by multiplying each (normalized) spherical harmonic term with the data and integrating, since the spherical harmonics are orthonormal.

However, this seems incorrect, the error between the reconstructed and measured electric fields increases as the order of the expansion increases.

What could be going wrong? I am trying to figure out if the error is conceptual or computational in nature.

I saw a paper which says the measured data has to be converted from spherical to Cartesian coordinates (convert (E_theta, E_phi) to (E_x, E_y, E_z)), and then each of Cartesian components has to be expanded in terms of spherical harmonics. Is this necessary, and if so, why? Why doesn't the spherical harmonic expansion hold good in spherical (theta,phi) coordinates?
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3. Dec 27, 2009

### Longstreet

It should work fine with theta/phi, i'm not sure the motivation to convert to Cartesian it seems to me that would be an ambiguous conversion.

$$A_{lm} = \int Y^*_{lm}(\theta, \phi) g(\theta, \phi) d\Omega$$

$$g(\theta, \phi) = \sum_{l=0}^{\infty}\sum_{m=-l}^{l} A_{lm} Y_{lm}(\theta, \phi)$$

For expanding some function g.

4. Dec 27, 2009

### Bob S

Re: Spherical harmonic expansion coefficients for electric field

The radiation pattern of a simple dipole radiator in spherical coordinates is given in Panofsky and Phillips, "Classical Electricity and Magnetism" first edition, pags 222-225, including the near field, transition field, and far field. Multipole radiation is also covered beginning on page 225.
Bob S

5. Dec 27, 2009

### Staff: Mentor

(Duplicate posts merged into the Classical Physics thread)

6. Dec 27, 2009

### sally_a

What kind of functions admit a spherical harmonic expansion? I understand that spherical harmonics provide a set of orthonormal basis functions and any 2 dimensional function expressed in spherical coordinates can be expanded in terms of spherical harmonics.

7. Dec 28, 2009

### diazona

I believe any function of the two angular variables that parametrize a 2D sphere (that is, a spherical surface of the sort which would be embedded in 3D Euclidean space) can be expanded in terms of spherical harmonics. As far as I know, the harmonics are a complete basis.

8. Dec 28, 2009

### sally_a

Am I having a problem because the field I am trying to represent is a vector field (electric field)? From what I have read on the internet, it seems I have to either express it in terms of vector spherical harmonics with scalar coefficients, or scalar spherical harmonics with vector coefficients.

9. Dec 29, 2009

### marcusl

In general you need vector spherical harmonics (SH)--remember that antennas produce polarized radiation. Scalar SH may be used in some cases. The foundations were worked out in the 1980's, and by now there are many treatments and refinements of the SH expansions needed to extrapolate near field measurements to far field antenna patterns. This book can get you started:

Hansen, Spherical Near-Field Antenna Measurements, IEE/Perigrinus, 1988.