Module II: Part B Optical Fibers: Dispersion
Dispersion We had already seen that that intermodal dispersion can be, eliminated, in principle, using graded-index fibers. We had also seen that single-mode, step-index fibers do not inherently have intermodal dispersion. However, dispersion is not eliminated overall even in a single-mode fiber because such a single mode is not monochromatic (single wavelength or color). This is due to the mere fact that a pulse is finite in duration---think of the Fourier transform and the uncertainty principle. A pulse can be thought of a a weighted superposition of plane waves at difference wavelengths. Each one will have its own velocity (called phase velocity), which is due to the dispersive (i.e., frequency dependent) nature of the refractive index.
Other Limitations: Group velocity dispersion Recall that for a plane wave, E(r,t)=E 0 exp{i(k.r ωt), and along the direction of propagation, viz., r in the direction of k, E(r,t) = E 0 exp{i( k r ωt) and the propagation velocity is v p = ω/ k (also, λ = π/ k ). v p is called the phase velocity. Aside from plane waves, as in optical pulse propagation in a fiber, the time it takes the pulse (or its energy) to propagate is given by the group velocity, v g : v g = 1/ dβ/dω
Group velocity and group index Recall from the previous set of notes that β = n k = n / 0 ω where n bar is the mode index. We can differentiate and apply the product rule to obtain dβ dn = c 1 ( ω dω dω + Thus, if we define we obtain v = c / g n g n) dn n g = n + ω d ω which has the looks of the ordinary (phase) velocity of a plane wave in a material whose refractive index is n g It therefore makes sense to call n g the group index c
Pulse broadening Now, the pulse propagation time (over distance L) is T = L/v g Write T (dt/dω) ω = d(l/v g )/dω ω= Ld β/dω ω Lβ ω β d β/dω is called the group-velocity-dispersion (GVD) parameter Similarly, we can consider T dt/dλ λand obtain T dt/dλ DL λ where D = d/dλ (1/v g ) = -πcβ /λ and it is called the dispersion parameter
Bandwidth limitation As before, let B = 1/T b, where T b is the bit duration To avoid intersymbol interference, we want T < T b, which yields BL D λ< 1 or BL < 1/ D λ Example: For silica fiber, D 1 (ps/km).nm at 1.3 µm For a semiconductor laser, λ -4 nm Thus, BL< 100 (Gb/s).km (if we take a conservative estimate of λ 10 nm) Also, for a Gb/s system utilizing a single-mode laser (conservatively, λ 1nm), and if L=50 km, then BL 1(Tb/s).km
Further insight into D It turns out that D has two components: one that is related only to the material properties of the fiber, and the other that has to do with the geometry (as well as the material) of the fiber. The former is called the material-dispersion component, W m, and latter is called the waveguidedispersion component, W w. This decomposition is shown below: We start by recalling that D = -(πc/λ ) β = (πc/λ ) (d/dω)(dβ/dω) = (πc/λ ) (d/dω)(1/v g ). But v g = c/ n g So πc dn d n D = + ω λ dω dω We now need to do some further substitutions using: n n 1 ( + b ) π an 1 λ V [b and where introduced In earlier notes]
Components of D After some substitutions and algebra, we obtain D = D M + D W where material-dispersion component waveguide-dispersion component D M 1 dng = c dλ D W = π λ ng Vd ( Vb) n ω dv + dn g Vd( Vb) dω dv and dn n g = n + ω d ω Next, we discuss these in more detail
Material Dispersion It occurs because the refractive index changes with optical frequency ω. Its origin is due to the nature of resonance frequencies (of the material) at which the material interacts with light. Refractive index can be modeled as: ( ω) = 1 + K j = ω where ω j is the jth resonance frequency and B j s are constants to be determined For pure silicon and using K=3, B 1 = 0.6961663; B = 0.407946 ; B 3 = 0.8974794 λ 1 = 0.0684043 µm; λ = 0.116414 µm; λ 3 = 9.896161 µm n B j ω j ω 1 j
Material Dispersion From this model we can determine Note that dn n g = n + ω d ω D M = (-1/c)dn g /dλ = 0 at λ = 1.76 µm λ ZD Also, D M is negative below λ ZD and positive above λ ZD λ ZD
Material dispersion Useful approximation: D M = 1(1- λ ZD / λ) λ ZD can be varied in the range 1.7-1.9 µm by doping of the cladding and core
Waveguide dispersion Recall that D W = π λ ng Vd ( Vb) n ω dv + dn g Vd( Vb) dω dv D W is always negative since both terms in the bracket are positive, but D M is negative before λ ZD and positive after λ ZD. Therefore, after adding D W to D M, D M is shifted down (i.e., D M increases) before λ ZD and shifted up to the after λ ZD (i.e., D M decreases) as shown in the previous plot. As a result, the new zero-dispersion wavelength λ ZD moves to the right to 1.31 µm.
Waveguide Dispersion Fortunately, in the range λ ZD to 1.6 µm, where optical communication is used, waveguide dispersion compensates for some of the material dispersion! This opens up the door for a field known as dispersion management or compensation, where fiber technology can be exploited to compensate for the material limitations. In step-index fibers, this can be achieved by manipulating and the core radius a. For example, λ ZD can be shifted to near 1.55 µm. See next figure.
Dispersion compensation
Dispersion flattening We can also design the fiber to have a slow flat variation about the operating wavelength. This is important in wavelengthdivision multiplexing (WDM) where we want to avoid any nonuniformity in dispersion between channels. Such fibers are called dispersion-flattened fibers. See previous plot.
Higher-order dispersion Recall that BL < 1/ D λ What would happen when λ is very small, as in single-mode, distributed lasers? Would the BL increase indefinitely? The answer is NO. This is because higher-order effects will kick in. To see this, first recall that in deriving the above limit, we used the approximation T dt/dλ λ, whose accuracy deteriorates as λ increases. In actuality, we have a Tailor-series approximation for T, and dt/dλ λis only the first order term for in the expansion. It is because the second-order term was neglected that the approximation in the LB bound fails to serve us when λ is very small.
Higher-order dispersion A more careful dispersion analysis yields (omit derivation): D = D M + D W + S λ where 3 3 4 β λ π β λ π + = c c S and 3 ω β ω β β d d d d = = Clearly, at λ = λ ZD, D S λ and BL < 1/ S ( λ)
Examples
Next topics So far we haven t considered what happens to the shape of a pulse in the presence of dispersion as it propagates through the fiber. Nonetheless, we were able to develop some rules of thumb for BL. These rules of thumb, however, do not take into account the fact that a narrow pulse has many wavelengths (think in terms of Fourier transforms), and since different wavelengths see different refractive indices, their collective dispersive effect is complicated. In the next Part, we will use wave theory to have further insight into dispersion. This will lead to the so-called polarization-mode dispersion (PMD), which is extremely important in today s lightwave systems. Please read the text, pages 37-43.