## Fiber propagation loss definition

The total fiber loss can be divided into material losses and fiber induced losses.

Material losses include Rayleigh scattering, ultraviolet (UV), infrared (IR) absorption,

and hydroxyl (OH) absorption losses. Material losses are the limiting losses in fibers.

Fiber loss is defined as the ratio of the optical output power *P _{out}* from a fiber of length

L to the optical input power

*P*. The symbol á is commonly used to express loss in

_{in}decibels per kilometer:

## Rayleigh scattering model

Because of the granular appearance of atoms or molecules of the glass fiber, light

transmitted through the fiber suffers scattering loss. This is known as Rayleigh

scattering loss.

Rayleigh scattering losses in a fiber are typically determined through experimental

measurement. The loss is expressed in decibels per kilometer through [Buck,

1995[4]]:

For a single-component glass such as *Si0 _{2}*

where *n0* is the refractive index, p is the photoelastic coefficient, β is the thermal

compressibility, k is the Boltzmann coefficient, and T is the absolute temperature of

the sample.

## Ultraviolet absorption model

Ultraviolet or UV absorption results from electronic absorption bands in the ultraviolet

region. The electronic absorption bands are associated with the band gaps of

amorphous glass materials. Absorption occurs when a photon interacts with an

electron in the valence band and excites it to a higher energy level. The UV absorption at any wavelength can be expressed as a function of the mole fraction x of *Ge0 _{2}*

[Buck, 1995[4]]:

UV loss is small compared to scattering loss in the near infrared region.

## OH-radical absorption model

The OH radical of H2O molecule vibrates at a fundamental frequency corresponding

to IR light wavelength of λ = 2.8μ*m*. Since the OH radical is slightly anharmonic,

“overtones” can occur. These cause OH absorption lines to occur at λ = 1.39,

0.95,and 0.725μ*m*, the second, third, and fourth harmonics of fundamental

frequencies, respectively. Broad peaks can appear [see, for example, Marcuse,

1981[7]]. OH absorption can be characterized by fitting the absorption lines by

Lorentzian or Gaussian method.

**Lorentzian fit method:**

**Gaussian fit method:**

In the above two equations, *A _{i}* is amplitude,

*A*is absorption peak position, and σ

_{i}_{i}

is the width of the i-th absorption line. Using up to seven absorption lines fits the OH

absorption spectrum here.

In contemporary state-of-the-art fibers the hydroxyl-group absorption is greatly

reduced, and only the peak at λ = 1.38 – 1.39μ*m* still retains some practical

importance.

## Infrared absorption model

Infrared or IR absorption is associated with the characteristic vibration frequency of

the particular chemical bond between the atoms of which the fiber is composed. An

interaction between the vibrating bond and the electromagnetic field of the optical

signal results in a transfer of energy from the field to bond, thereby causing

absorption.

The IR absorption expression for *Ge0 _{2} – *Si0

_{2}glass is [Buck, 1995[4]]:

## Macrobending loss model

The macrobending loss *Y *is a radiative loss when the fiber bend radius is large

compared to the fiber diameter. It is defined as usual by P(z)=P(0) exp (-*yz*)where

P(0) is the input power and P(z) is the output power at distance z respectively.

Currently OptiFiber implements two different macrobending loss models:

1. The first model uses the closed-form integral formula, published by J. Sakai and T. Kimura [J. Sakai and T. Kimura, 1978[26]]. It is appropriate for calculating the macrobending loss of any LP mode, both fundamental and higher-order, in arbitrary-index profile optical fibers. Using this formula the macrobending power loss coefficient is expressed as a function of the bending radius *Rb* in the form:

The parameters appearing above are given by:

(the normalized dimensionless frequency)

where *r _{c} *denotes the fibre core radius,

*N*is the maximum value of the

_{max}refractive index and

*N*is the cladding index, β is the propagation

_{clad}constant of the mode,

*k*is the prop. constant in vacuum,

_{0}*V*is the azimuthal

mode number,

*s*= 2 if

*v*= 0 or

*s*= 1 for

*v*≠ 0 and

*K*is the modified

_{v}Bessel function of the second kind of order

*v*.

2. Using the second macrobending loss model the coeficient *y *can be expressed

as [Snyder and Love, 1983[20]]:

where *F _{0}* is the radial field of the fundamental mode:

and N(R) is the refractive index profile of the fiber. The other parameters are

given above.

The two models give similar results for step-index fibers.

The loss coefficient *y* can be converted to loss in decibels per kilometer units as

follows:

## Microbending loss model

Microbending loss is a radiative loss in fiber resulting from mode coupling caused by

random microbends, which are repetitive small-scale fluctuations in the radius of the curvature of the fiber axis.

An approximate expression for the attenuation coefficient is given by [Petermann,

1976[12]]:

where A is a constant, *d _{n}* is the near field diameter,

*n*is the core refractive index

_{1}of fiber, k is the free space wavenumber, and p is the exponent in the power law.

## Splice loss model

A splice is the dielectric interface between two optical fibers. Any index-of-refraction

mismatch at any point in this interface will produce reflection and refraction of the light

incident at that point. For splicing calculations, we assume that the mode field of

single-mode fibers is nearly Gaussian. The coupling losses for the splicing connectors

can be calculated by evaluating the coupling between two misaligned Gaussian

beams.

Based on the above model, the coupling loss between two single mode fibers is given

by [Miller and Kaminow, 1988[10]]:

where

The symbols above are:

*n _{1} – core refractive index of the fiber*

*n2 – *refractive index of the medium between the two fibers

*λ *– wavelength

*W _{1} *– near field mode field radius of transmitting fiber

*W _{2 }*– near field mode field radius of receiving fiber

*x *– lateral offset

*z *– longitudinal offset

θ – angular misalignment