Calculating Far Field Pattern with Born Approximation & Gaussian

In summary, to calculate the far field pattern u(xhat), you will need to substitute the given values into the Born approximation equation and solve the integral in R^3. This will give you an expression for u(xhat) in terms of the constants A, a, and k. You can then use this expression to plot the far field pattern u(xhat) as a function of xhat, using a computer program or software to help with the calculations.
  • #1
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Homework Statement


The Born approximation is u(xhat) = -k2/(4*pi) [tex]\int_R[/tex] exp(-i k xhat . y)m(y)ui(y) dy where R is R3.
Suppose that m is a Gaussian m(y) = Aexp(-a|y|2), where A > 0 and a > 0. Consider an incident plane wave ui(y) = exp(iky1). Calculate the far field pattern u(xhat).

Homework Equations





The Attempt at a Solution


The Fourier transform of a Gaussian is a Gaussian. So the Fourier transform of m(y) is A/(sqrt(2a)*exp(-[tex]\omega^2/4a[/tex]). I guess what I am asking is what do I do next. Am I just solving the integral (In R^3)?
 
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  • #2


Hello,

Based on the given information, it seems like you are on the right track. To calculate the far field pattern u(xhat), you will need to substitute the given values into the Born approximation equation and solve the integral in R^3. This will give you an expression for u(xhat) in terms of the constants A, a, and k. You can then use this expression to plot the far field pattern u(xhat) as a function of xhat.

As a tip, you can use a computer program or software such as Mathematica to help you solve the integral and plot the far field pattern. This will make the process faster and more accurate.

I hope this helps. Good luck with your calculations!
 

1. What is the Born Approximation method and how is it used to calculate far field patterns?

The Born Approximation method is a mathematical approach used in scattering problems, where a wave is scattered by an object. It assumes that the scattered wave is much weaker than the incident wave, and therefore, the scattered field can be approximated as a linear function of the incident field. This method is used to calculate far field patterns by solving the integral equations that describe the scattered field using the incident field as the source.

2. What is the role of Gaussian beams in calculating far field patterns using the Born Approximation method?

Gaussian beams are often used as incident fields in the Born Approximation method because they have a simple mathematical form and can be easily manipulated. They also have the property of having a finite width, which allows for a more accurate representation of the incident field in the near field. This makes them a suitable choice for calculating far field patterns using the Born Approximation method.

3. Can the Born Approximation method be used for any type of scattering problem?

No, the Born Approximation method is most suitable for problems where the scatterer is small compared to the wavelength of the incident field. It also assumes that the scatterer is weak, which means that the scattered field is much smaller than the incident field. If these assumptions are not met, the Born Approximation method may not provide accurate results.

4. How can the accuracy of the Born Approximation method be improved?

The accuracy of the Born Approximation method can be improved by using higher-order approximations, such as the Rytov approximation or the Kirchhoff approximation. These methods take into account higher-order scattering effects and can provide more accurate results for larger scatterers. Additionally, using a larger number of incident beams can also improve the accuracy of the method.

5. Are there any limitations or drawbacks to using the Born Approximation method?

Yes, there are a few limitations to using the Born Approximation method. As mentioned earlier, it is only applicable for small scatterers and weak scattering. It also assumes that the incident field is known and does not take into account any multiple scattering effects. Additionally, the method may not be accurate in the near field region. Therefore, it is important to carefully consider the assumptions and limitations of the Born Approximation method before using it for a scattering problem.

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