# 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

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.

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