# Evaluate the outgoing radiation from an optical fiber on a surface

Frostman
TL;DR Summary
Hi there, I hope to get some guidance on calculating the intensity of outgoing radiation from an optical fiber that reaches a surface. I'm feeling uncertain about the accuracy of my estimate based on the geometry configuration and procedure I used. Can you advise me, compare my approach to others, and suggest any alternative methods or tricks that could be helpful?
The geometric configuration that I am adopting is the following, I hope you understand.

The optical fiber is positioned relative to the bottom surface at a height ##a## and an angle ##\alpha## with respect to the y-axis in the yz-plane with x = 0. ##b## is the distance between the origin and the center of the ellipse that is projected onto the surface. ##c## is the semimajor axis, while ##d## is the semiminor axis. Moving on to the angles, ##\theta## is the angle formed in the yz-plane and specifies the angular opening of the blue cone. While ##\varphi## specifies the opening of the blue cone in the inclined plane at an angle ##\alpha## with respect to the Cartesian axis system chosen. I hope I explained the geometry of the system well.

At this point my idea for evaluating the intensity of radiation that reaches the surface is as follows: I have the characteristic emission spectrum of the optical fiber available, in the following figure I have normalized the intensity of radiation with respect to its integral.

I want to evaluate the function I obtained in another integral in which the integration extremes are from the angle ##\varphi## minimum to ##\varphi## maximum, and from ##\theta## minimum to ##\theta## maximum.

In my case I get
$$\theta_m = \frac{\pi}{2} - \alpha - \sin^{-1}\left(\frac{b-\frac c2}{\sqrt{a^2+\left(b-\frac c2\right)^2}}\right)$$
$$\theta_M = - \frac{\pi}{2} + \alpha + \sin^{-1}\left(\frac{b+\frac c2}{\sqrt{a^2+\left(b+\frac c2\right)^2}}\right)$$
$$\varphi_m = - \tan^{-1}\frac{\frac{d}{2}}{b}$$
$$\varphi_M = \tan^{-1}\frac{\frac{d}{2}}{b}$$

The aspects that don't convince me are the angles I calculated, in this case they are not with respect to the adopted coordinate system. For ##\theta## it is quite straightforward to arrange the values since we are in the third quadrant. For ##\varphi## instead it's less trivial and honestly I don't know how to fix it.

The integral then that I'm going to evaluate is a surface integral, but I'm not very convinced of ##f_\text{norm}## since that function must be seen in 3D as a surface of rotation.

I hope you can give me a hand and sort out this apparently chaotic configuration. In the end, what I want to obtain is the intensity of radiation that arrives on that blue ellipse starting from the emission profile of the optical fiber.