Math problem in Huygens-Fresnel principle

[mensa]

in Huygens-Fresnel principle, U(P)=K*∫∫U(Q)*F(θ0，θ0)*exp{ikr}/r*d∑
K=-i/λ; F(θ0，θ0)=0.5(cosθ0+cosθ) is inclination factor; d∑ is a small part of any close surfaces; these are all include in any Beam Optics book

I want to demonstrate the spherical wave point source S gives out, all the d∑ in the spherical surface as Secondary Source, whose light come to point P, equals to S only

I don't know how to integrate,coz it seems to hard to me.
Please show the process. Any book involves is available too.Thanks

 

[AndyW]

Thank you for your interest in the Huygens-Fresnel principle. This principle is a fundamental concept in optics that helps explain the propagation of light as a wave.

To demonstrate the spherical wave point source S giving out light, we can use the Huygens-Fresnel principle to calculate the light intensity at point P. Let's break down the equation you provided and explain each component.

U(P) represents the amplitude of the light at point P. This amplitude is determined by the integral of the secondary sources U(Q) over the entire spherical surface. This means that the light from point S is spreading out in all directions, and each small part of the spherical surface is acting as a secondary source.

The constant K is equal to -i/λ, where λ is the wavelength of the light. This constant helps us calculate the amplitude of the secondary sources.

The function F(θ0, θ0) is called the inclination factor. This factor takes into account the direction of the light as it propagates from the secondary sources to point P. It is a function of the angles θ0 and θ, which represent the angles between the direction of the light and the normal to the spherical surface at point S and point P, respectively.

The exponential term represents the phase difference between the light from point S and the secondary sources at point Q. This phase difference depends on the distance r between points S and Q, as well as the wavelength of the light.

Finally, the term d∑ represents a small part of any closed surface. This means that we are integrating over the entire spherical surface, taking into account the contributions of all the secondary sources.

To calculate the integral, we can use the Huygens-Fresnel diffraction formula, which is a simplified version of the Huygens-Fresnel principle. This formula assumes that the secondary sources are very close together, so we can approximate the integral as a sum. This makes the integration much easier.

I recommend consulting any standard textbook on optics, such as "Fundamentals of Optics" by Francis Jenkins and Harvey White, or "Optics" by Eugene Hecht. These books will provide a more detailed explanation of the Huygens-Fresnel principle and its applications.

I hope this explanation helps you better understand the Huygens-Fresnel principle and how to apply it in your research. Best of luck with your studies!

 

[sir-pinski]

The Huygens-Fresnel principle is a fundamental concept in wave optics that explains the propagation of light as a series of secondary wavefronts. In this principle, the secondary wavefronts are generated by the interference of the primary wavefront at each point on the surface of a spherical wave source. This principle is widely used in the field of Beam Optics, and it is important to understand how to use it to solve problems.

To solve the math problem given in this content, we need to apply the Huygens-Fresnel principle and use the given equations to integrate the expression U(P)=K*∫∫U(Q)*F(θ0，θ0)*exp{ikr}/r*d∑, where K=-i/λ, F(θ0，θ0)=0.5(cosθ0+cosθ) is the inclination factor, and d∑ is a small part of any closed surface.

First, we need to understand that the integration is performed over all the secondary sources on the spherical surface. These secondary sources are generated by the primary wavefront from the point source S. The aim is to show that the contribution from all the secondary sources is equivalent to the primary wavefront at point P.

To solve this, we can divide the spherical surface into small patches of area d∑. Each of these patches acts as a secondary source for the point P. We can then rewrite the integral expression as a summation over all these small patches, as follows:

U(P)=K*∑U(Q)*F(θ0，θ0)*exp{ikr}/r*d∑

Next, we can use the given equations to simplify the expression. We know that K=-i/λ, and F(θ0，θ0)=0.5(cosθ0+cosθ). We can also substitute the expression for r, which is the distance between the secondary source and the point P, as r=|PQ|.

U(P)=-i/λ*∑U(Q)*0.5(cosθ0+cosθ)*exp{ik|PQ|}/|PQ|*d∑

Now, we can use the Huygens-Fresnel principle to show that the contribution from each secondary source is equivalent to the primary wavefront at point P. This is because the primary wavefront at point P is the sum of all the secondary wavefronts from