How close is a 2D Gaussian to an Airy disk?

In summary, the central peak of an Airy disk can probably be approximated as a Gaussian function, but the secondary maxima are not visible if the diffraction pattern is indeed an Airy disc.
  • #1
omoplata
327
2
So we were taking measurements for an experiment in our radio astronomy lab. For the first part of the experiment, we recorded the intensity of a far away point source ( the signal from a TV satellite was used for the point source ) detected by commercial satellite dish and receiver.

When we plotted the intensity against the angle ( we traversed the dish across the satellite, so the curve we had was a 1D cross section of the 2D intensity distribution ), it looked like a Gaussian (http://en.wikipedia.org/wiki/Gaussian_function ). I was actually able to fit a Gaussian to it.

But then I read about what we were supposed to get, and found that we were supposed to get an Airy disk from a point source ( http://en.wikipedia.org/wiki/Airy_disk ).

So I'm curious. How close is an Airy disk to a Gaussian?

Can I get a Gaussian from an Airy disk by making some parameter close to zero or infinty or something?

Thanks.
 
Physics news on Phys.org
  • #2
The central peak of an Airy function can probably be approximated as a Gaussian function, so the issue is why didn't you measure the secondary maxima? I can think of a few reasons, but without knowing any details it's hard to say. How big is your receiver, in terms of wavelength?
 
  • #3
The detector ( LNBF ) that we are using has a bandpass centered at around 2.7 cm ( 11 GHz ). It's used with an off-axis parabolic satellite dish ( the width of the dish is about 1 m, if that matters ).

we didn't measure the secondary maxima ( if what you mean are the secondary peaks of the Airy disk ) specifically.

We just used a an interface instrument ( Vernier Lab Pro ) with a PC to record the voltage coming off the detector 10 times a second, and moved the satellite dish with constant speed, using a motor, so that the direction it's pointing towards traced a line across the position of the satellite in the sky.

So we were taking data of Voltage vs. time, but the dish was moving with constant angular velocity that we know, so we can convert the data to Voltage vs. angle.

If there were supposed to be secondary peaks in the plot of Voltage vs. angle ( which there should be if it was the cross section of an Airy disk ), they are not visible.
 
  • #4
Here are a few things to consider:

0) if the diffraction pattern is indeed an Airy disc, how large should it be at your receiver (assuming none of the below effects)?
1) the satellite dish/detector is not sensitive to a singe direction- there is an angular acceptance angle.
2) Propagation of light through the atmosphere introduces scattering (and attenuation), blurring the diffraction pattern
3) your source could be moving during signal acquisition
4) Your measured data is the convolution of the 'ideal' signal and all these effects, similar to blurring in any other optical imaging system.

What do other members of your group think?
 
  • #5
0) If by "how large" you mean the angular diameter of the Airy disk, looking at the wikipedia article, the intensity of the Airy disk [itex]I(\theta)[/itex] is related to the radius of the aperture [itex]a[/itex] ( in this case the radius of the satellite dish? ), using the same notation as the wikipedia article. ( I'm using the term Airy "disk" and not "function" here, because it seems different from an Airy function )
1) If detector (LBNF) has full sensitivity to an angle larger than the angle of the dish edge, I think we can take ignore the angular sensitivity distribution of the detector, because then the aperture is going to be the whole dish ( if the background behind the dish emits little radio signals of the relevant frequency) . If the sensitivity of the detector falls off with increasing angle from the central axis with such a rapid rate that it drops off before it reaches the dish edge, then I guess we have to take that into account.
2) This is certainly an issue.
3) We were detecting a geocentric satellite. So it did not move in relation to our position.
4) So I guess considering 1) and 2), it is possible that our plot can NOT be the cross section of an Airy disk.

When I told the professor in charge of the experiment that I fitted a Gaussian, he said "Why did you fit a Gaussian? It's supposed to be an Airy disk." The reason why did not fit one yet is because the fitting software I am using right now does not have Bessel functions ( the wikipedia Airy disk equation is given in terms of Bessel functions ). I'm going to find another fitting software which has Bessel functions and fit it. So this observation might be premature, but I can't see any secondary peaks.

The other two members in our group have the same problem. How to find a fitting software that can fit an Airy disk. But we think we found one ( IDL ), and we are going to try it out.

This link lists the functions in the NASA IDL Astronomy User's Library for IDL (which is very widely used in the Astronomy community) , and even they seem to be fitting Point Spread Functions with Gaussians and not Airy disks ( functions FIND and GETPSF ).
 
  • #6
Ok- I think you are on the right track. However, answer (0) should be more quantitative- I mean you should calculate how large a diameter (in meters/cm/feet/cubits..) you expect the Airy disk to be at your detector, and compare that to the size of your detector. Yes, the dish is the aperture.

And don't forget, since you are detecting the intensity and not the field, the fit function is actually [J_0(kr)/kr]^2.
 

FAQ: How close is a 2D Gaussian to an Airy disk?

Question 1: What is a 2D Gaussian distribution?

A 2D Gaussian distribution is a type of probability distribution that describes the spread of values for a two-dimensional variable. It is characterized by a bell-shaped curve and is commonly used to model natural phenomena such as the distribution of stars in a galaxy or the distribution of particles in a fluid.

Question 2: What is an Airy disk?

An Airy disk is a diffraction pattern that is produced when light passes through a circular aperture. It is named after British astronomer George Airy and is characterized by a bright central spot surrounded by concentric rings of decreasing intensity. It is commonly observed in telescopes and microscopes.

Question 3: How are 2D Gaussians and Airy disks related?

A 2D Gaussian can be thought of as a simplified version of an Airy disk, where the concentric rings of the Airy disk are collapsed into a single bell-shaped curve. This is because the distribution of light in an Airy disk follows a 2D Gaussian distribution.

Question 4: How close is a 2D Gaussian to an Airy disk?

The closeness of a 2D Gaussian to an Airy disk depends on the specific parameters of the Gaussian distribution, such as its standard deviation and mean. In general, the resemblance between the two will be greater for Gaussians with smaller standard deviations and closer means to the center of the Airy disk.

Question 5: Why is it important to understand the relationship between 2D Gaussians and Airy disks?

Understanding the relationship between 2D Gaussians and Airy disks is important in various scientific fields, such as astronomy and microscopy. It allows scientists to accurately model and analyze natural phenomena and to optimize the performance of instruments that rely on the diffraction pattern of Airy disks.

Back
Top