- #1
chiraganand
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Hi, I have read about the rayleigh sommerfeld integral and its a surface integral. Why is it difficult to calculate the integral directly?
Why Is the Rayleigh Sommerfeld Integral Challenging to Compute Directly?
In summary, the Rayleigh Sommerfeld integral is a surface integral and it is difficult to calculate directly because of the presence of an oscillating complex factor. This can be addressed by using asymptotic expansions such as the stationary phase method or expressing it in terms of special functions, but finding an exact solution is not simple.
- #2
Ssnow
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Can you give a reference for this question ?
Ssnow
Ssnow
- #3
berkeman
Mentor
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@chiraganand was last at PF in Sept. 2020, but may still be receiving e-mail notifications of updates to his threads. Hopefully he will be able to respond. 
- #4
chiraganand
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Oh wow I had moved on from this one, but thanks for replying! This is the the reference. so there are a lot of different ways to make it easier to calcualte but my main question was how would one go about solving it without any simplifications and at which step would one get stuckSsnow said:
- #5
Ssnow
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Ok, now I understand the question, the principal problem is that the surface integral is not a simple surface integral but it has an oscillating part given by the oscillatory complex factor ##e^{ik|\vec{R}_{2}-\vec{r}_{1}|}##. This is the reason you can try to have an asymptotic expansion using different ways as the stationary phase expansion or others ... In addition you have the variable up to the exponent and also in the denominator- You can try to express it in terms of special functions as ##\int_{0}^{t}\frac{e^{x}}{x^{\alpha}}dx## but to have an exact primitive I don't think it is simple ...
Ssnow
Ssnow
Related to Why Is the Rayleigh Sommerfeld Integral Challenging to Compute Directly?
1. What is the Rayleigh-Sommerfeld integral?
The Rayleigh-Sommerfeld integral, also known as the Fresnel diffraction integral, is a mathematical formula used to calculate the diffraction pattern of a wave passing through an aperture or diffracting object. It was developed by British physicist Lord Rayleigh and German physicist Arnold Sommerfeld in the late 19th and early 20th centuries.
2. How is the Rayleigh-Sommerfeld integral derived?
The Rayleigh-Sommerfeld integral is derived from the Huygens-Fresnel principle, which states that every point on a wavefront can be considered as a point source for a new wave. By integrating the contributions of all these point sources, the resulting formula describes the diffraction pattern of the wave.
3. What is the significance of the Rayleigh-Sommerfeld integral in optics?
The Rayleigh-Sommerfeld integral is an important tool in the field of optics, as it allows scientists and engineers to predict and analyze the diffraction patterns of light passing through various diffracting objects. This is crucial for understanding and designing optical systems, such as microscopes and telescopes.
4. Are there any limitations to the Rayleigh-Sommerfeld integral?
Yes, the Rayleigh-Sommerfeld integral has some limitations. It assumes that the diffracting object is infinitely thin and that the diffracted waves propagate in a vacuum. It also does not take into account the effects of polarization or multiple scattering, which may be important in certain scenarios.
5. How is the Rayleigh-Sommerfeld integral used in practical applications?
The Rayleigh-Sommerfeld integral is used in various practical applications, such as designing optical systems for imaging and lithography, analyzing the diffraction patterns of laser beams, and studying the behavior of waves in different media. It is also used in fields like acoustics, where it can be applied to predict the diffraction of sound waves.
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