S. Blair February 15, 2012 23 2.2. Pulse dispersion Pulse dispersion is the spreading of a pulse as it propagates down an optical fiber. Pulse spreading is an obvious detrimental effect that limits the minimum bit time T B, and thereby limits the maximum bit rate B = 1/T B. During system design, there is typically a maximum amount of pulse spreading allowable. The dominant source of pulse dispersion in multi-mode fiber is due to modal dispersion - the fact that each mode propagates at a slightly different velocity. Single-mode fiber has no modal dispersion. Both types of fiber do exhibit chromatic dispersion. Due to the spectral bandwidth of any information-carrying pulse, data transmission down fiber incurs intramodal dispersion because the material and fiber parameters vary with frequency (or wavelength). Polarization mode dispersion can also result due to the difference in fiber response to the state of polarization. 2.2.1. Multipath dispersion (intermodal dispersion) Angles within the fiber are quantized to modulo 2π phase shifts in a transverse round trip. Each angle corresponds to a transverse mode. Each mode travels with a different velocity and gives rise to modal dispersion. The intermodal time delay can be written τ mod = T modal = longest ray path shortest ray path ( ) n 1 L L = n ( ) 1L 1 1 = L n 2 1 Ln 1 }{{} c sinφ c c sinφ c c n 2 c. 1/phase vel. The NA increases with, but modal dispersion also increases. Note that the text book uses τ for the pulse spread. We will use T, since it is in more common use. For a desired bit rate B = 1/T B, we need to have T modal < T B /4. Note that, at this point, this condition is at best a rule of thumb. Some designers will use the condition T modal < T B /2 (which is what the text book uses), but we will stick with the factor 1/4 to be consistent with the rest of the world. Later on, we ll talk about system design and use bandwidth and rise time budgets. Using this condition, B T < 1 4 or BL < n 2 c 4n 2 1 c 4n 1 which is the fundamental bit-rate distance product for multi-mode fiber (MMF). Using typical MMF values = 2 10 3 n 1.5, gives the value BL <100 Mb-km/s, which is suitable for a local area network. 2.2.2. Graded-index fiber Graded-index fiber can reduce, or even eliminate, modal dispersion, and is in common use today for low performance networks. The radial refractive index profile can be written { n1 [1 (ρ/a) n(ρ) = α ] ρ < a n 1 (1 ) = n 2 ρ a
S. Blair February 15, 2012 24 a = core radius step index profile when α With this profile, all ray velocities are nearly the same. Larger angle sees longer distance, but nearly the same optical density due to local decrease in refrative index. Lower index Higher index The minimum modal dispersion occurs when α = 2(1 ), with intermodal time delay T = n 1 L 2 /8c and BL < 8c/n 1 2 10 Gb km/s using typical values. Single-mode fiber (SMF) can do considerably better, but is more expensive and more difficult to use. Graded-index plastic fiber is used for data link applications. The core size is about 1 mm (to give large area for efficient coupling from an LED), and the loss 50 db/km. However, this fiber still achieves BL > 1 Gb-km/s, which is suitable for gigabit ethernet. 2.2.3. Chromatic Dispersion Chromatic dispersion results from the fact that different frequencies (or wavelengths) propagate down an optical fiber at different velocities. We will first consider chromatic dispersion due to the variation in the refractive index of glass with wavelength. Group delay dispersion (GDD) is the lowest-order form of chromatic dispersion and describes the variation in group delay (inverse velocity) with frequency or wavelength. For a given frequency component of a pulse, the time delay through the length L of a fiber is T(ω) = L υ g (ω), where υ g (ω) is the group velocity at frequency ω. Note that 1/υ g is called the group delay. The group velocity is defined as ( ) 1 dβ υ g (ω) = = c. dω n g Here, n g is called the group index. The mode eigenvalue can be written β = nω/c, allowing us to evalute the group index dβ dω = ω dn c dω + 1 c n 1 c n g. The following figure shows the variation of n and n g with wavelength for fused silica.
S. Blair February 15, 2012 25 If ω is the pulse spectral width (which is due to the source spectrum and the modulation bandwidth B = 1/T B ), then T chrom = dt dω ω = d ( ) L ω = d ( L dβ ) ω = L d2 β dω υ g dω dω dω 2 ω = Lβ 2 ω gives the duration of the pulse after propagation over a length L. In order for this expression to be valid, the length must be greater than the dispersion length L D (defined later), or L L D. The group-delay dispersion (GDD) coefficient is defined β 2 = 2 β ω 2, and has units of ps 2 /km. Note that many incorrectly refer to this parameter as the group-velocity dispersion (GVD) parameter. This misconception is common in the optical communications community. Just for completeness, the GVD parameter is really defined as the variation in group velocity with frequency, or dυ g dω = d ( ) 1 = d2 β/dω 2 dω dβ/dω (dβ/dω) 2 = β 2 υ 2 g It is also useful to define the dispersion parameter D, which is the variation in group delay with wavelength. It s use is very common in the telecommunications industry: T chrom = D(λ f )L λ, where D is the dispersion parameter (in units of ps/nm km), L is the length of fiber (in km), and λ is the spectral width of the optical channel in nm. Note that the textbook uses M rather than D as the dispersion parameter. We ll use D because that is the industry standard notation.
S. Blair February 15, 2012 26 The spectral width may be dominated by the range of wavelengths produced by the source, or for narrow linewidth sources, may be dominated by the frequency spreading caused by modulation. In order to derive the dispersion parameter, it is useful to know that d dω = λ2 f 2πc If we determine the output pulse duration as a function of wavelength spread, then T chrom = d ( ) L λ = L d ( ) dβ λ dλ υ g dλ dω = L dω ( ) d dβ λ dλdω dω = L dω dλ β 2 λ }{{} D d dλ. where the dispersion parameter is defined D dω dλ β 2 = d ( ) 1 dλ υ g = 2πc λ 2 β 2. f The dispersion parameter has units ps/nm km. The dispersion parameter is zero at a particular wavelength, called the zero-dispersion wavelength λ 0,. The dispersion parameter can be estimated from the following formula [ ] D(λ f ) = S 0 4 λ f λ4 0 λ 3 f where S 0 is the zero-dispersion slope (in ps/nm 2 km). For a multi-mode fiber, the pulse spreading can be calculated from the following expression T = Tmodal 2 + T2 chrom. 2.2.4. Higher-order dispersion Higher-order dispersion can play an important role in pulse propagation either at the zero dispersion wavelength or for very short pulses (<10 ps). The differential dispersion parameter is defined S = ( ) 2 ) 2πc 4πc λ 2 β 3 +( f λ 3 β 2, f which describes the slope of the dispersion parameter D with wavelength. An effective dispersion parameter can then be written D eff = D+ λs.
S. Blair February 15, 2012 41 2.5.2. Material and waveguide dispersions Chromatic dispersion results from two phenomena: variation in the refractive index of the constituent materials of the fiber with frequency and variation in the waveguide properties of the fiber with frequency. These are chromatic dispersion as they depend on frequency, or wavelength, and are described in composite through the mode eigenvalue β = ωn(ω)/c, which depends on frequency. In terms of the mode index, the dispersion parameter can be written If we recall the fiber b parameter, D = 2πc = 2π dn g = 2π ( ) d 1 = 2πc d dω dω υ g dω = 2π d dω ( 2 dn n dω +ωd2 dω 2 ( ng ( n+ω dn dω ). b = n n 2 n 1 n 2 = n 1 n n 2, we can write the modal index as n = n 2 +b /n 1. Here, n 2 is the cladding index and is material property, b is a waveguide property, and is the normalized index, which is a differential material property. Now, we can rewrite the total dispersion parameter D = D M +D W +D We will assume that the differential dispersion is small (meaning that the changes in n 1 and n 2 with frequency are about the same), leaving D W = 2π D M = 2π dn 2g dω = 1 dn 2g c dλ [ V n2 2g d 2 (Vb) + dn 2g ωn 2 dv 2 dω c ) ) ] d(vb), dv where n 2g is the group index in the fiber cladding. Material dispersion in pulse propagation down optical fiber occurs because the refractive index of fused silica (which is the main constituent of optical fiber) varies with frequency. Using a simple resonance model called the Sellmeier equation, the refractive index can be written n 2 (ω) = 1+ M j=1 B j ω 2 j ω 2 j ω2, where ω j denotes the frequency location of the relevant material resonances and B j denotes the strength (called the oscillator strength) of that resonance. For fused silica, these parameters are: 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. Note that ω j = 2πc/λ j. With the knowledge of n(ω), the group index can be calculated from n g = n+ω n ω.
S. Blair February 15, 2012 42 The following figure shows the variation of n and n g with wavelength for fused silica. The material dispersion parameter is written D M = (1/c)(dn g /dλ) and can be easily evaluated. Theslopedn g /dλ = 0atthezerodispersionwavelengthλ zd = 1.276µm,whereD M = 0. Forlonger wavelengths, D M is positive, while for short wavelengths, D M is negative. The zero dispersion wavelength for glass used in optical fiber due to dopants (such as germanium). The contribution from waveguide dispersion can dramatically alter the total dispersion seen by the pulse. The main effect of D W on the total dispersion D is to shift the zero-dispersion wavelength λ zd by 30-40 nm so that the zero dispersion in fiber (rather than in bulk fused silica) occurs at λ zd 1.31 µm, which is one common wavelength range of operation. For the other common range about 1.55 µm, typical D values are 15-18 ps/nm km. Since the total dispersion depends on D W through the fiber parameters, we have some control the position of λ zd. Because the minimum fiber loss occurs near 1.55 µm, it is very useful to make this the zero dispersion wavelength as well. These types of fibers are called dispersion-shifted fibers and are common in new fiber installations. Another type of fiber is dispersion-flattened, such that the dispersion is held low over a large wavelength range. Representative refractive index
S. Blair February 15, 2012 43 profiles for these fiber types are also shown in the figure. SMF DSF DFF 2.5.3. Dispersion summary Due to group-delay dispersion (the lowest-order contribution to chromatic dispersion), the pulse spread can be written T chrom = Lβ 2 ω T chrom = LD λ where β 2 is the group delay dispersion and D is called the dispersion parameter. We can define the higher-order dispersion parameters β 3 = β 2 ω = 3 β ω 3 S = D λ, where β 3 is the third-order dispersion and S is the dispersion slope. With these parameters, the group-delay dispersion parameters can be approximated β 2 (ω) β 2 (ω o )+(ω ω o )β 3 D(λ) D(λ o )+(λ λ o )S, where ω o and λ o are fixed. At the zero-dispersion point, pulse spread can be written in terms of the higher-order dispersion parameters T chrom = Lβ 3 ( ω) 2 T chrom = LS( λ) 2. 2.5.4. Dispersion compensation Since the beginning of optical communications systems, over 75 million miles of standard singlemode optical fiber (SMF-28) has been installed. The dispersion parameter for this fiber is in the
S. Blair February 15, 2012 44 Tx DCF Rx T range D 17 18 ps/nm km at 1550 nm wavelength. For these systems, data-carrying capacity is limited by dispersion, as an upgrade in capacity requires either the use of shorter modulation or the use of multiple wavelength channels, or both. In some more recent systems, dispersion-shifted fiber(dsf) was used such that a single channel can operate at high bit rates with very little dispersion. However, when capacity was added using WDM techniques, another problem came up, that of four-wave mixing. The detrimental effects of four-wave mixing are maximized near the zero dispersion point, and this became the next limitation. Current systems use non-zero dispersion-shifted fiber (NDSF), where the dispersion zero is shifted out of the 1550 nm band (i.e. the C-band), leaving a small amount of residual dispersion, about 2-4 ps/nm km. The way to further increase capacity, for both the legacy SMF-28 systems and the newest NDSF systems is to use a technique called dispersion compensation. This technique relies on a special type of fiber - dispersion compensating fiber (DCF) - which allows for the compensation of pulse broadening in an installed system without replacing the buried fiber. DCF is a fiber with a negative dispersion parameter. Its use is illustrated in the following figure: The pulse broadening due to the first, installed, fiber is given by T chrom = D(λ) λl. We therefore need the DCF to provide T comp = T chrom. For the DCF, we need a dispersion parameter D DCF (λ) = D L L DCF. Note, that in WDM, we need to compensate dispersion for all wavelength channels. This requires the compensation of dispersion slope. DCF is made by modification of the refractive index profile of the fiber, and typically has larger attenuation of 0.4 db/km. Two differences are used - a larger core/cladding refractive index of = 2.5% versus = 0.37% (which occurs through increased Ge doping of the core and results in greater Rayleigh scattering), and a decreased mode size of 4.7 µm versus 10.5 µm. The additional loss of using DCF must be compensated by an optical amplifier. The splicing loss between standard fiber and DCF is minimized by using an interim fiber.