# Calculate optical path in SELFOC cylindrical fiber optic

Hi.

This is my first message in this forum. I'm not English, so sorry my spelling.

## Homework Statement

Calculate the optical path done by a meridional ray, supposing it covers a horizontal distance, d, in z-axis direction. $$\gamma_0$$ is the launch angle (with z-axis).

## Homework Equations

Optical path equation:
$$L = \displaystyle\int^d_0 n(\rho) dl$$

Because we work with SELFOC fiber optic, refraction index is: $$n^2(\rho) = n^2_0\left(1-\alpha^2\rho^2\right)$$

The trajectory equation is:
$$\rho(z) = \displaystyle\frac{\sin\gamma_0}{\alpha} \sin\left(\displaystyle\frac{\alpha z}{\cos \gamma_0}\right)$$

$$dl = \sqrt{d\rho^2+dz^2}$$ (infinitesimal calculus).

## The Attempt at a Solution

I've tried to write optical path equation in function of variable z. So, the resultant integral is:
$$\displaystyle\int_{0}^{d}n_0\sqrt[ ]{1-\sin^2\left(\gamma_0\right) \sin^2\left(\displaystyle\frac{\alpha z}{\cos \gamma_0}\right)}\sqrt[ ]{1+\displaystyle\frac{\sin^2\gamma_0}{\cos^2\gamma_0}\cos^2\left(\displaystyle\frac{\alpha z}{\cos\gamma_0}\right)}dz$$

How can I solve this integral?

Thanks!

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