Single-mode fibers

This post answers the question: “What are features of single mode fiber?”. Single-mode fibers are fibers that support only

$H{E}_{11}$

mode, fundamental mode.

Single-mode fiber determined by the value of

$V$

where

$T{E}_{01}$

and

$T{M}_{01}$

are cutoff.

Eigenwave equations there can be obtained from the cutoff condition we discussed before where

$m=0$

. The cutoff condition is

$pa=V$

and

$q=0$

.

$p{J}_0\left(pa\right){K_0}^{‘}\left(qa\right)+q{J_0}^{‘}\left(pa\right)K_0\left(qa\right)=0$ $p{n_2}^2{J}_0\left(pa\right){K_0}^{‘}\left(qa\right)+q{n_1}^2{J_0}^{‘}\left(pa\right)K_0\left(qa\right)=0$

For

$J_0\left(V\right)=0$

the smallest

$V=2.405$

. In this condition fiber supports only fundamental , which is condition of the single-mode fiber.

Cutoff condition can also help to obtain core radius of a fiber using the equation

$V\approx \frac{2\pi }{\lambda }a{n}_1\sqrt{2∆}$

, here

$a$

is a core radius. The mode index for the operating wavelength is

$\overline{n}=n_2+b∆$

. The

$H{E}_{11}$

is linearly polarised fiber mode. Axial components

$E_z$

and

$H_z$

are very small. If we take

$E_y=0$

so

$E_x=E_0\left\{\begin{array}{l}\left[\frac{J_0\left(p\rho \right)}{J_0\left(q\rho \right)}\right]exp\left(i\beta z\right), \rho \le a\\ \left[\frac{K_0\left(q\rho \right)}{K_0\left(qa\right)}\right]exp\left(i\beta z\right), \rho >a\end{array}\right.$

. Electric field

$E_x$

is related to magnetic field

$H_y$

via relationship

$H_y=n_2\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}{E}_x$

.

If we will assume that

$E_x=0$

, we can obtain magnetic field

$H_x$

via similar relationship with

$E_y$

.

Real fibers can have variation of the shape of the core along the fiber length, or experience nonuniform stress. For this reason modal  birefringence of the fiber is

$B_m=\left|{\overline{n}}_x–{\overline{n}}_y\right|$

, here

$\overline{n_x}$

and

$\overline{n_y}$

are mode indices for the orthogonally polarised modes.

Due to birefringence two polarisation components are periodically exchange power between them with period

$T_B=\frac{\lambda }{B_m}$

.

Lineraly polarised light is lineraly polarised only when it is polarised along one of the axes

$x, y, z$

. In other case the polarisation periodically changes from linear to eliptical.

One of the features of single mode fiber is birefringence. In single mode fibers birefringence changes randomly along the fiber because of anisotropic stress variations in the core shape. The linear polirised light reaches the mode with arbitrary polarisation. Different frequency components of a pulse are characterised with different polarisation states, the pulse start to broaden. This phenomena is called polarisation-mode dispersion (PMD). Polarisation-maintaining fibers, are fibers that are not influenced by core shape and size ununiformity.

Another feature of single mode fiber is spot size. Field distribution in the fiber is often described by Gaussian distribution 

$E_x=C{e}^{–\frac{{\rho }^2}{w^2}}{e}^{i\beta z}$

, here

$w$

describes the spot size and called as field radius.

Optical fibers are characterised with effective core area, parameter, describing how tightly light contains in the core and defined by formula

$A_{eff}=\pi w^2$

. The power that contains in the core is defined by formula

$\Gamma =\frac{P_{core}}{P_{total}}=1–e^{–\frac{2a^2}{w^2}}$

and called confinement factor.   Using confinement factor

$\Gamma$

for various

$V$

values we can show  that most telecommunication single-mode fibers are designed to work  with the

$V$

values from 2 till 2.4.

Source: “Fiber-optic communication systems”, Govind P.Agrawal, 2002

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