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Hi.

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

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

Optical path equation:

[tex]L = \displaystyle\int^d_0 n(\rho) dl[/tex]

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

The trajectory equation is:

[tex]\rho(z) = \displaystyle\frac{\sin\gamma_0}{\alpha} \sin\left(\displaystyle\frac{\alpha z}{\cos \gamma_0}\right)[/tex]

[tex]dl = \sqrt{d\rho^2+dz^2}[/tex] (infinitesimal calculus).

I've tried to write optical path equation in function of variable z. So, the resultant integral is:

[tex]\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[/tex]

How can I solve this integral?

Thanks!

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. [tex]\gamma_0[/tex] is the launch angle (with z-axis).

## Homework Equations

Optical path equation:

[tex]L = \displaystyle\int^d_0 n(\rho) dl[/tex]

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

The trajectory equation is:

[tex]\rho(z) = \displaystyle\frac{\sin\gamma_0}{\alpha} \sin\left(\displaystyle\frac{\alpha z}{\cos \gamma_0}\right)[/tex]

[tex]dl = \sqrt{d\rho^2+dz^2}[/tex] (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:

[tex]\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[/tex]

How can I solve this integral?

Thanks!

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