Fresnel Equations for a beam instead of a plane wave

[montealeku]

In many books we can find the derivation of Fresnel Equations (TE and the TM reflection and transmission coefficients), but always for the case of a plane wave. How can we go about obtaining the reflection coefficient if the incident light took the form of a beam rather than a plane wave?

We may use the paraxial approximation, or we may consider the particular case of a Gaussian beam at its waist, but I cannot figure out the solution for the general case. Any help is most appreciated!

 

[montealeku]

Note: The question is related to Problem 6.2-3 in the Fundamentals of Photonics book, by Saleh and Teich.

 

[montealeku]

Could a spatial Fourier transform be used? I'd really appreciate if anyone could share his/her thoughts and perhaps provide more elaboration

 

[sgd37]

The starting point of the fresnel equations is

\frac{1}{\mu_i v_i } (E_i-E_r) cos(\theta_i) = \frac{1}{\mu_t v_t } E_t cos(\theta_t)

So you can derive it for any electric field you want

 

[montealeku]

sgd37,

You can go for a rigorous treatment of the topic (please check this reference:
http://webee.technion.ac.il/courses/046244/files/ComplexPointSource.pdf), which as you can see is not straighforward, but in this case I need just a few sentences explaining how to tackle the problem... Any ideas?

Regards

 

[sgd37]

well what is the difference between a plane wave and a beam, isn't it the electric field

And could you maybe explain your problem in more detail

 

[montealeku]

sgd37,

First of all, thank you for your interest in the topic and your previous replies. The question, as I mentioned above, is taken from Problem 6.2-3 in Fundamentals of Photonics, 2nd Edition, by Saleh and Teich (and no further information is provided). My undestanding is that the book author is not asking for a formal and rigorous solution (as that proposed in the article that I was suggesting), but a short explanation of how could we deal with calculating the reflection coefficient if instead of the idealized case of a plane wave we had a beam (which does not need to be Gaussian). I was thinking of a spatial Fourier transform, but I am not sure... if anyone can think of a different approach, or detail a bit how to do it with the spatial Fourier transform (I am not keen on the topic), it would be most appreciated.