Plane wave decomposition of the fields of a dipole antenna

[Spinnor]

A dipole antenna will have near fields and far fields. Can both the near and far fields can be decomposed into an infinite sum of plane waves?

If so, are the plane waves for far fields and near fields of different type or class? Near fields must die off at infinity but far fields do not.

Thanks.

 

[tech99]

I can maybe imagine using a sum of spherical waves but not plane waves.
The near field you mention is more accurately called the reactive near field. It stores energy each half cycle and returns it to the antenna the next. So in my mind I cannot visualise the field extending to infinity, or the energy would not be able to return in time for the next half cycle. It is therefore a localised field and would seem to contain standing waves rather than the travelling waves of the far field.
Regarding the field strength of the far field at infinity, the F/S is proportional to 1/distance, so if distance is infinity then the field strength is zero.

 

[DaveE]

No, I don't think so. Certainly not in the near field.

Huygens principle allows use to use an infinite number of point sources (spatially infinite). But point sources have essentially spherical radiation. In summing the waves the key term is the exponential $e^{ikr}$ where r is the radial distance. This equates to $e^{ik\sqrt{x^2+y^2+z^2}}$. That $\sqrt{}$ is a PITA, so Fresnel approximates it with the quadratic terms of a Taylor expansion, like $e^{\frac{ik}{2}(x^2+y^2+z^2)}$, which is pretty good in the near field. Fraunhofer makes a further approximation that only works in the far field, but still uses quadratic terms in the exponent.

My description is terribly sloppy, because I'm too lazy to write it correctly. You can see the correct version here.

In any case, plane waves will have the form $e^{ikx}$ with only linear (1st order) terms of distance. OTOH, this is really just a very complex version of approximations a la Taylors theory. So I suppose you could model it with 1st order terms, but it wouldn't be a great model except in the very far field.

edit: Wait, I take it back! Yes, you can use a basis of plane waves to approximate the wavefront. This is Fourier optics which is a powerful technique. The math is nasty in the near fields, but easy (relatively) in the far fields.

 

[tech99]

I think that when you refer to the near field you are referring to the near radiation field (the Fresnel Region), and not the reactive near field. It is important to use exact terms for this or we will be at cross purposes.
[see for instance the following link https://www.iala-aism.org/wiki/dictionary/index.php/Fresnel_region]

 

[Spinnor]

This paper may also help, buy allowing the wave vector to be complex we can get near fields? Thanks for your help.

http://kirkmcd.princeton.edu/examples/virtual.pdf

Section 2.5 covers the Oscillating Electric Dipole.

 

[Baluncore]

The near field is particularly complex to calculate because it includes local currents and differential voltages, all influencing and approaching each analysis point from many directions.

The transition from the near field, to the far field, involves a change of assumptions with a change of equations. Crossing that bridge is a particularly complex, expensive, and unnecessary exercise.

The computation of the far field can be very much simpler, since it can be based on the current moments in the antenna elements alone. The magnetic far field is a plane wave, being summed from one direction. The associated electric far field is simply computed through the intrinsic impedance of free space, which is a real constant.

 

[marcusl]

Propagating waves in the far field have azimuthal (circular) symmetry about the dipole and may be described by spherical waves. Far from the source, curvature of the wave over distances of interest (such as across a receiving antenna) is negligible and the wave may be considered planar to a good approximation.

 

[Spinnor]

The near field is particularly complex to calculate because it includes local currents and differential voltages, all influencing and approaching each analysis point from many directions.

The transition from the near field, to the far field, involves a change of assumptions with a change of equations. Crossing that bridge is a particularly complex, expensive, and unnecessary exercise.

The computation of the far field can be very much simpler, since it can be based on the current moments in the antenna elements alone. The magnetic far field is a plane wave, being summed from one direction. The associated electric far field is simply computed through the intrinsic impedance of free space, which is a real constant.

I am sure you would understand the paper I linked. I think my question is answered? The fields of a Hertzian antenna can be decomposed into an infinite sum of plane wave that give both the near and far fields? Is it hard, looks like it. Why Professor Kirk T. McDonald would solve this problem I guess only he knows. I am glad he did though.

Thanks.

 

[Baluncore]

Yes, that paper makes sense, but I don't have a good reason to get that near. The complexity of so many near field spherical waves has always been a poor fit to my brain. Near field plane waves are similar, possibly a sidestep more difficult.

The near field has now become the domain of FEM modelling, since any real antenna, will be beyond the human application of algebra. Far field modelling appears unchanged by the paper.

When I stumbled into Cerenkov Radiation from travelling wave antennas, 35 years ago, I could find no practical use in communications. The antennas were too difficult to engineer for a band of frequencies, and too difficult to aim in our real world.

 

[marcusl]

I am sure you would understand the paper I linked. I think my question is answered? The fields of a Hertzian antenna can be decomposed into an infinite sum of plane wave that give both the near and far fields? Is it hard, looks like it. Why Professor Kirk T. McDonald would solve this problem I guess only he knows. I am glad he did though.

Thanks.

I'm not convinced. First, the author states on p. 8

"To this author, there is very limited physical reality to the inhomogeneous plane
waves [describing the near field] identified in the expansions (41)-(43)."

Second, the paper treats a "point dipole" of negligible size, which is the first term in a multipole series expansion. A multipole series is used as a mathematical model of a physical radiator whereby a radiator of finite size is bounded by a surface (usually spherical) and the whole replaced by a "black box" containing an effective multipole source at the center of the volume. Fields outside the surface may then be computed. Accordingly, a point dipole is not ideal for describing fields next to (in the near field of) the physical dipole antenna having finite extent that you have asked about, for this region is right next to the dipole wire which is inside of the bounding surface.

The first problem in solving for near fields is determining the current on the dipole antenna wire, which is non-trivial. This is typically solved by the Method of Moments, using sinusoidal functions as a basis for the expansion. One then proceeds to find the fields. As Baluncore states, this is the domain of numerical modeling software. The procedure and equations may be found in antenna texts such as Elliott, Antenna Theory and Design where some 35 pages are devoted to just this problem (see chapter 7 and, particularly, sections 7.3-7.8).

 

[Spinnor]

I'm not convinced. First, the author states on p. 8

"To this author, there is very limited physical reality to the inhomogeneous plane
waves [describing the near field] identified in the expansions (41)-(43)."

Second, the paper treats a "point dipole" of negligible size, which is the first term in a multipole series expansion. A multipole series is used as a mathematical model of a physical radiator whereby a radiator of finite size is bounded by a surface (usually spherical) and the whole replaced by a "black box" containing an effective multipole source at the center of the volume. Fields outside the surface may then be computed. Accordingly, a point dipole is not ideal for describing fields next to (in the near field of) the physical dipole antenna having finite extent that you have asked about, for this region is right next to the dipole wire which is inside of the bounding surface.

The first problem in solving for near fields is determining the current on the dipole antenna wire, which is non-trivial. This is typically solved by the Method of Moments, using sinusoidal functions as a basis for the expansion. One then proceeds to find the fields. As Baluncore states, this is the domain of numerical modeling software. The procedure and equations may be found in antenna texts such as Elliott, Antenna Theory and Design where some 35 pages are devoted to just this problem (see chapter 7 and, particularly, sections 7.3-7.8).

"To this author, there is very limited physical reality to the inhomogeneous plane
waves [describing the near field] identified in the expansions (41)-(43)."

I read that part. He comes up with an expansion for the Hertzian dipole in terms of plane waves that are equivalent to the fields of the original problem. That was my question, seems answered. Does the plane wave superposition that represents the near fields have some physical reality, he does not think so and I am ok with that. He did not state as fact just his opinion, he makes an interesting observation. One could ask what is the reality of a plane wave in physics. You might just say its an abstraction used in physics that works?

I did say dipole antenna in my question, I could have been more clear. That it can be done for the Hertzian antenna is cool, might his technique work for a more realistic antenna I don't know.

Thanks.

 

[marcusl]

Agreed, if you are after a Hertzian dipole then the case is settled.