# Near-to-far field tranformation

1. Jul 14, 2009

### krindik

Hi,
Can somebody point me to a good explaination and theoretical background on near-to-far fields tranformation. Google gives so many publications which are pretty hard to understand.

Thanks

Krindik

2. Jul 14, 2009

### Born2bwire

Do you have a specific application in mind? Usually it is a bit of a vague definition. The transition between the the various regions is not well defined, we can easily say when we have near or far field but it gets a bit sticky in between. Take a look at stuff like the Fraunhofer region, Fraunhofer distance, Fresnel zone. Those are common terms when discussing near and far field regions in electromagnetics. Fraunhofer may also be used in other waves like acoustic waves.

3. Jul 15, 2009

### krindik

Thank you.

What I would like to know is how the far-field can be obtained from the near-field data. (I saw that Green's function? is used). Can u point me somewhere that decribes the steps gradually?
You can assume that I'm comfortable with Maxwell equation and EM wave propagation etc.

Thanks again

4. Jul 15, 2009

### Born2bwire

What kind of near field data are we talking about? If somebody gives you the equations of the fields in the near field then the far field results are done in a limiting case in terms of kr >> 1.

5. Jul 15, 2009

### krindik

Lets say, we somehow manage to get all the electric field data very close to an antenna. Then how do u extrapolate the field values at some far point from the antenna?

Thanks

6. Jul 15, 2009

### Born2bwire

You really can't, unless you know the exact equations of the waves. The problem with near-field is that a portion of the fields/energy are trapped, non-propagating. I do not think that you can differentiate between the portions of the fields that will ultimately propagate and those that will remain trapped via measurement. This is especially true because the non-propagating modes will generally be more and more dominant the closer you are to the antenna. The best you can do I think is to automatically ignore any longitudinal components, that is the r-hat vector since far-field waves are polarized along the phi hat and theta hat directions in spherical coordinates (provided you place the source at the origin of your coordinate system). This is because the antenna will look like a point source at sufficient distance and the k-vector of any wave coming off of it will be along the r hat direction.

7. Aug 12, 2009

### aro

I would recommend Principles of Nano-Optics, by Novotny and Hecht.

I'll try to give an explanation to give you at least an idea of how it works:

In near field optical microscopy a probe of subwavelenth dimensions (which usually can be modelled as a dipole) is placed in very close proximity to a sample so that it couples to the sample via the near field, including all k-vectors and not just those that satisfy the propagating photon dispersion relation.
The high k-vector modes that have interacted with the sample now include information about small ('subwavelength') scale structure. The key is to read this information from the outgoing field at the detector, here your detection window consists only of propagating wave k-vectors, so not the higher ones which correspond to evanescent modes that don't reach the detector. The key is that in fact you can read the information from interaction of high-k modes from the photons arriving at the detector. The general idea is:

i)You have a source field which propagates freely until it is at the sample. Sample and source are very close so the evanescent modes reach the sample.

ii)The interaction is taken into account by multiplying a transfer/transmission function with the source field at the position of the sample, this gives you the field just after interaction. The transfer function contains info about the sample and this you want to obtain, call this function T for now.

iii)Now the field propagates to the detector. And you detect only propagating photons, so the part of the outgoing field that has wavevector k' with -k<k'<+k, where k=\omega/c

Now consider the fourierspectrum of the field at ii). The product turns into convolution. If you consider a single frequency component of the source field this introduces a deltafunction in the convolution integral and the consequence is a shift. The result is that you have performed a translation on the sample spectrum (FT of the transfer function) and parts of this spectrum which would otherwise be outside of the detection window are now translated to within the detection window.
In theory, if you do this for all frequency components, you 'scan' the entire sample spectrum. After you have summed all single frequency contributions you know the response of the sample at all(!) wavevectors (meaning: you can construct T)
In practice I think you will have to do difficult data analysis, because while all the information is in principle at hand you really have to extract it from your measurements. And of course there are probably practical limitations and approximations involved.

I only know this global outline, but I am sure that Novotny and Hecht treat specific experimental setups and they have all the equations included.

Cheers,