# Dispersion in Single-Mode Fibers

We have seen that intermodal dispersion in multimode fibers leads to considerable broadening of short optical pulses (- 10 ns/km). In the geometrical-optics description such a broadening was attributed to different paths followed by different rays. In the modal description it is related to the different mode indices (or group velocities) associated with different modes.

The main advantage of single-mode fibers is that intermodal dispersion is absent simply because the energy of the injected pulse is transported by a single mode. However, pulse broadening does not disappear altogether. The group velocity associated with the fundamental mode is frequency dependent because of chromatic dispersion. As a result, different spectral components of the pulse travel at slightly different group velocities, a phenomenon referred to as *group-velocity dispersion* (*GVD*), *intramodal dispersion*, or simply fiber dispersion.

Intramodal dispersion has two contributions known as material dispersion and waveguide dispersion. This tutorial considers both of them and discusses how GVD limits the performance of lightwave systems employing single-mode fibers.

#### 1. Group-Velocity Dispersion

Consider a single-mode fiber of length *L*. A specific spectral component at the frequency *ω* would arrive at the output end of the fiber after a time delay *T* = *L/v _{g}*, where

*v*is the group velocity defined as

_{g}*v _{g}^{-1}* =

*dβ/dω*

By using , one can show that , where is the group index given by

The frequency dependence of the group velocity (or GVD) leads to pulse broadening simply because different spectral components disperse during propagation and become desynchronized at the fiber output. If *Δω *is the spectral width of the pulse, the extent of the pulse broadening is governed by

The parameter *β _{2}* =

*d*is known as the GVD parameter. It determines how much an impulse would broaden on propagation inside the fiber.

^{2}β/dω^{2}In optical communication systems the frequency spread *Δ**ω* is often determined by the range of wavelengths *Δλ* emitted by the optical source. It is customary to use *Δλ *in place of *Δ**ω*. By using *ω = 2πc/λ* and *Δ**ω = (-2πc/λ ^{2})Δλ, *we can rewrite the equation as

where

*D* is called the *dispersion parameter* and is expressed in units of ps/(km-nm).

The effect of dispersion on the bit rate *B* can be estimated by using the criterion *BΔT* < 1 in a manner similar to that used previously. By using *ΔT *from last equation this condition becomes

*BL*|*D*|*Δλ < *1

This equation provides an order-of-magnitude estimate of the bit rate-distance product *BL* offered by single-mode fibers. In the next two sections the wavelength dependence of *D* is studied. It is found that D ~ 1 ps/(km-nm) can be obtained in the wavelength region near 1.3 μm. For a semiconductor laser the spectral width *Δλ *is 2-4 nm even when the laser operates in several longitudinal modes. The *BL* product of such lightwave systems can exceed 100 (Gb/s)-km. Indeed, a second-generation commercial telecommunication system in 1987 was capable of operating 1.7 Gb/s with a repeater spacing of 40-50 km, demonstrating the potential of single-mode fibers for optical communications. The *BL* product of single-mode fibers should be compared with multimode step-index fibers for which *BL* < 100 (Mb/s)-km because of intermodal dispersion.

The dispersion parameter *D* can vary considerably when the operating wavelength is shifted from 1.3 μm, near which it takes the minimum value. The wavelength dependence of *D* is governed by the frequency dependence of the mode index , *D* can be written as

If we substitute and use other equations, *D* can be written as the sum of two terms

*D* = *D _{M}* +

*D*

_{W}where the material dispersion *D _{M}* and the waveguide dispersion

*D*are given by

_{W}Here n_{2g} is the group index of the cladding material and the parameter *V* and *b* are given previously. *Δ* was assumed to be frequency independent. A third term known as differential material dispersion can be added when *d**Δ/dω* ≠ 0. Its contribution is, however, negligible in most cases of practical interest.

#### 2. Material Dispersion

Material dispersion occurs because the refractive index of silica, the material used for fiber fabrication, changes with the optical frequency *ω*. On a fundamental level, the origin of material dispersion is related to the characteristic resonance frequencies at which the material absorbs the electromagnetic radiation through oscillations of bound electrons. Far from the medium resonances, the refractive index *n*(*ω*) is well approximated by the Sellmeier equation

where *ω _{j}* is the resonance frequency and

*B*is the oscillator strength. Here

_{j}*n*stands for

*n*or

_{1}*n*depending on whether the disperse properties of the core or the cladding are considered.

_{2}The sum in the equation extends over all material resonances that contribute in the frequency range of interest. In the case of optical fibers the parameters *B _{j}* and

*ω*are obtained empirically by fitting the measured dispersion curves to this equation with

_{j}*M*= 3. They depend on the amount of dopants and have been tabulated for several kinds of fibers. For pure silica these parameters are found to be B

_{1}= 0.6961663, B

_{2}= 0.4079426, B

_{3}= 0.8974794, λ

_{1}= 0.0684043 μm, λ

_{2}= 0.1162414 μm, and λ

_{3}= 9.896161 μm, where λ

_{j}= 2πc/ω

_{j}(j = 1, 2, and 3). The group index n

_{g}= n + ω(dn/dω) can be obtained by using these parameter values.

The above figure shows the wavelength dependence of *n* and *n _{g}* in the range 0.5-1.6 μm for fused silica. Material dispersion

*D*is related to the slope of

_{M}*n*by the relation

_{g}*D*=

_{M}*c*

^{-1}(

*dn*). It turns out that

_{g}/dλ*dn*/

_{g}*dλ*= 0 at

*λ*= 1.276 μm. This wavelength is referred to as the zero-dispersion wavelength

*λ*, since

_{ZD}*D*= 0 at

_{M}*λ*=

*λ*. The dispersion parameter

_{ZD}*D*is negative below

_{M}*λ*and becomes positive above that. In the wavelength range 1.25-1.66 μm it can be approximated by an empirical relation

_{ZD}*D _{M}* ≈ 122(1 -

*λ*/

_{ZD}*λ*)

It should be stressed that *λ _{ZD}* = 1.276 μm only for pure silica. It can vary in the range 1.27-1.29 μm for optical fibers whose core and cladding are doped to vary the refractive index. The zero-dispersion wavelength of optical fibers also depends on the core radius

*a*and the index step

*Δ*through the waveguide contribution to the total dispersion. This issue is discussed in the following section.

#### 3. Waveguide Dispersion

The contribution of waveguide dispersion *D _{W}* to the dispersion parameter

*D*is given in the previous equations and depends on the

*V*parameter of the fiber. The following figure shows how

*d(Vb)*/

*dV*and

*Vd*(

^{2}*Vb*)/

*dV*change change with

^{2}*V*. Since both derivatives are positive,

*D*is negative in the entire wavelength range 0-1.6 μm.

_{W}On the other hand, *D _{M}* is negative for wavelengths below

*λ*and becomes positive above that. The following figure shows

_{ZD}*D*,

_{M}*D*, and their sum,

_{W}*D*=

*D*+

_{M}*D*, for a typical single-mode fiber.

_{W}The main effect of waveguide dispersion is to shift *λ _{ZD}* by an amount 30-40 nm so that the total dispersion is zero near 1.31 μm. It also reduces

*D*from its material value

*D*in the wavelength range 1.3-1.6 μm that is of interest for optical communication systems. Typical values of

_{M}*D*are in the range of 15-18 ps/(km-nm) near 1.55 μm. This wavelength region is of considerable interest for lightwave systems since the fiber loss is minimum near 1.55 μm. High values of

*D*can limit the performance of 1.55-μm lightwave systems.

Since the waveguide contribution *D _{W}* depends on the fiber parameters such as the core radius

*a*and the index difference

*Δ*, it is possible to design the fiber such that

*λ*is shifted in the vicinity of 1.55 μm. Such fibers are called

_{ZD}*dispersion-shifted fibers*. It is also possible to tailor the waveguide contribution such that the total dispersion

*D*is relative small over a wide wavelength range extending from 1.3 to 1.6 μm. Such fibers are called

*dispersion-flattened fibers*. The following figure shows typical examples of the wavelength dependence of

*D*for standard (conventional), dispersion-shifted, and dispersion-flattened fibers. The design of dispersion-modified fibers involves the use of multiple cladding layers and a tailoring of the refractive-index profile. Design issues are discussed in another tutorial.

#### 4. Higher-Order Dispersion

It appears that the *BL* product of a single-mode fiber can be increased indefinitely by operating at the zero-dispersion wavelength *λ _{ZD}* where

*D*= 0. The dispersive effects, however, do not disappear completely at

*λ*= λ

_{ZD}. Optical pulses still experience broadening because of higher-order dispersive effects. This can be understood by noting that

*D*cannot be made zero at all wavelengths contained within the pulse spectrum centered at

*λ*. Clearly, the wavelength dependence of

_{ZD}*D*will play a role in pulse broadening. The higher-order dispersion is thus governed by the "dispersion slope"

*S*=

*dD*/

*dλ*. The parameter

*S*is also called differential-dispersion parameter or second-order dispersion parameter. It can be written as

*S* = (*2πc*/*λ ^{2}*)

^{2}

*β*+ (

_{3}*4*

*πc*/

*λ*)

^{3}*β*

_{2}where *β _{3}* =

*dβ*/

_{2}*dω*=

*d*/

^{3}β*dω*.

^{3}At *λ* = λ_{ZD}, *β _{2}* = 0, and

*S*is proportional to

*β*. Typical value of

_{3}*S*at

*λ*= λ

_{ZD}are 0.085 ps/(km-nm

^{2}) for standard single-mode fibers and 0.05 ps/(km-nm

^{2}) for dispersion-shifted fibers with

*λ*= 1.55 μm. For a source of spectral width

_{ZD}*Δλ*, the effective value of dispersion parameter becomes

*D*=

*SΔλ.*The limiting bit rate-distance product can be estimated by using previous equation with this value of

*D*or by using

*BL*|*S*|(*Δλ*)^{2} < 1

For a multimode semiconductor laser with *Δλ = *2 nm and a dispersion-shifted fiber with *S* = 0.05 ps/(km-nm^{2}) at *λ = *1.55 μm, the *BL* product can approach 5 (Tb/s)-km. Further improvement is possible only by using single-mode semiconductor lasers.

#### 5. Polarization Dispersion

A possible source of pulse broadening is related to fiber birefringence. As discussed previously, small departures from perfect cylindrical symmetry lead to birefringence because of different mode indices associated with the orthogonally polarized components of the fundamental fiber mode. If the input pulse excites both polarization components, it becomes broader at the fiber output as the two components disperse along the fiber because of their different group velocities. This phenomenon is referred to as *polarization dispersion*. The time delay *ΔT* for a fiber of length *L* is given by

where subscripts *x* and *y* stand for the direction of linear polarization, and previous equation was used to relate the group velocity *v _{g}* to the propagation constant

*β*. Similar to the case of intermodal dispersion discussed for multimode fibers, the quantity

*ΔT*/

*L*is called polarization dispersion. Its values for conventional single-mode fibers are typically below 0.1 ps/km and are negligible compared to GVD. Such low values result from a coupling between the two modes induced by random perturbations along the fiber. The coupling tends to equalize the propagation time for the two polarization components. For polarization-preserving fibers

*ΔT*/

*L*can be quite large ( ~ 1 ns/km) if the two components are equally excited at the fiber input but can be reduced to zero by launching light along one of the principle axes. Most lightwave systems currently use conventional (nonpolarization-perserving) fibers. Polarization dispersion is not a practical limitation for such fibers until the effects of GVD become negligible.