# How to calculate the Geometrical Extent/Etendue for an Optical Fiber?

I would like to calculate the theoretical Geometrical Extent (G) for an optical fibre, however I am struggling a bit with the calculation of the needed solid angle based on the acceptance angle $\theta_{max}$

As far as I know Geometrical Extent is calculated by $$G_{OF} = A_{OF} \cdot \Omega_{OF}$$

where A is the area of the optical fibre side, which is the area of the circle assuming a radius $r$ $$A_{OF} = \pi \cdot r_{OF}^2$$ and $\Omega_{OF}$ is the solid angle of the optical fibre.

Reaching this point, I make use of the expression for the NA to get my $\theta_{max}$ since it will impose the solid angle for the fibre: $$NA = n_o\cdot \sin(\theta_{max}) \to \theta_{max} = \arcsin(NA/n_o)$$ Knowing that, I would like to obtaining the solid angle,$\Omega_{OF}$, given by the angle $\theta_{max}$. It will be defined by a cone due to our $\theta_{max}$ angle , looking for the solid angle of a cone in literature is defined as $$\Omega_{cone} = 2\cdot\pi\cdot(1-\cos(\theta))$$ The acceptance angle $\theta_{max}$ only refers to half angle, but I need the whole angle to calculate the solid angle thus I have to multiply $\theta_{max}$ by two $$\Omega_{OF} = 2\cdot\pi\cdot(1-cos(2\cdot\theta_{max}) = 2\cdot\pi\cdot(1-\cos(2\cdot\arcsin(NA/n_o))$$ and finally $$G_{OF} = 2\cdot r_{OF}^2 \cdot\pi^2\cdot(1-cos(2\cdot\arcsin(NA/n_o))$$

Is this reasoning correct?

Regards

• This looks sound, although in some fibers - particularly holey (photonic crystal fibers) with extremely high NA, the direction cosine of the field becomes significant and so you should calculate the grasp using the differential $n^2 \,\cos\theta \,\mathrm{d} A\,\mathrm{d}\Omega$ (you look like you're dropping the $\cos\theta$ term. You'll get a reasonable answer, but it's easy enough to include the exact definition. Commented Jun 10, 2017 at 8:49
• @WetSavannaAnimalakaRodVance I drop the $cos\theta$ because I am more interested on having the realtionship between the $G_{OF}$ and the NA, is that not appropiate? Commented Jun 12, 2017 at 8:35
• It may well be perfectly OK as an approximation - it all depends on what you want to do with your calculation. It would be interesting to see what would happen if you began with a more first principles definition of differential étendue rather than your first equation and carried out the integral. Commented Jun 12, 2017 at 10:41
• @WetSavannaAnimalakaRodVance do you mean starting from $dG = n^2*dS*cos(\theta)d\Omega$ and carrying the integral over the solid angle of the cone defined by the fiber NA? Commented Jun 19, 2017 at 7:51
• Yes, you may also want to include any apodization factor in the integral, too, although the $\cos\theta$ is the main issue. Commented Jun 19, 2017 at 8:18