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This is from Optics (5th ed.) by Hecht, p. 209. I was studying fiberoptics and this parameter called $V$-number came up. Also there is another term called the normalized propagation constant, b which is a function of V. I would also appreciate it if someone shed some light on this as well. Thank you.


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Extensive discussion about V-number and $\beta$ mode propagation constants can be found in Fundamentals of Photonics by B.E.A. Saleh and M.C.Teich. Here I'll try to answer your question about where that parameter appears in the equations, following reasoning from the book.

From Maxwell equations, you can obtain Helmholtz differential equation describing your field propagating through the fiber: $\nabla^2 U + n^2k_0^2 U = 0$, where $n$ is your refraction index and $k_0$ is the wave number of your field in vacuum. When you consider that equation in the cylindrical coordinate system you can postulate solutions in a form: $$ U(r, \phi,z) = u(r) e^{-il\phi}e^{-i\beta z}, \space \text{where} \space l=0, \pm 1,\pm2, \dots $$ Over here you already can see the meaning of the propagation parameter $\beta$. It is an effective $k$ number for your solution (fiber mode).

Putting that into your equation you get:

$$ \frac{\text{d}^2u}{\text{d}r^2} + \frac{1}{r} \frac{\text{d}u}{\text{d}r} + \left( n^2(r)k_0^2 - \beta^2 - \frac{l^2}{r^2} \right) u = 0. $$

For step-index fibers with $n_1$ - core and $n_2$ - cladding your mode has to obey: $n_1k_0 > \beta > n_2k_0$ to be bound to your fiber. You can understand that as a requirement for the effective wavenumber being somewhere between the wavenumber of light propagating only through the core and only through the cladding. With that requirement, you can include two positive numbers to further simplify the equation above:

$$ k_T^2 = n_1^2k_0^2 - \beta ^2 $$ and $$ \gamma^2 = \beta^2 - n_2^2k_0^2.$$

Because you are considering step-index fiber the equation can be split in regard to the function $n(r)$:

$$ \frac{\text{d}^2u}{\text{d}r^2} + \frac{1}{r} \frac{\text{d}u}{\text{d}r} + \left(k_T^2 - \frac{l^2}{r^2} \right) u = 0, \qquad \text{for} \qquad r<a $$


$$ \frac{\text{d}^2u}{\text{d}r^2} + \frac{1}{r} \frac{\text{d}u}{\text{d}r} - \left(\gamma^2 + \frac{l^2}{r^2} \right) u = 0, \qquad \text{for} \qquad r>a, $$

where $a$ is the radius of your core.

The solutions for $u(r)$ are proportional to $J_l(k_T r)$ in the core and $K_l(\gamma r)$ in the cladding, where $J_l(x)$ is the Bessel function of the first kind and order $l$ and $K_l(x)$ is the modified Bessel function of the second kind and the order $l$.

One can notice that at this point you can introduce a dimensionless parameter containing information about your system - V number that is independent from your solutions:

$$ \text{V}^2 = k_T^2 a^2 + \gamma^2 a^2 = a^2 \left( n_1^2k_0^2 - \beta^2 + \beta^2 - n_2^2k_0^2 \right) = a^2 k_0^2(n_1^2-n_2^2), $$ thus: $$ V=ak_0\sqrt{n_1^2-n_2^2} = 2 \pi\frac{a}{\lambda_0}\cdot\text{NA}, $$

as for a step-index fiber $\text{NA}=\sqrt{n_1^2-n_2^2}.$


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