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The phase shift is a function of time SPM (t) and knowing that frequency is the derivative of phase shift with respect to time, Eq (735) can be expressed as Eq (736) = d SPM dP = L dt dt eff
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(736)
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where is the change in frequency caused by the change in phase shift at location L, s 1 Equation (736) shows the relationship between the change in signal power and frequency shift around the carrier frequency, see Fig 79 The reason for this is explained as follows As an optical pulse propagates in a fiber, the leading edge of the pulse s amplitude causes the refractive index of the fiber to increase, resulting in a shift to a lower frequency for the beginning of the pulse The falling edge of the pulse s amplitude causes the refractive index to decrease, resulting in
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FIGURE 79
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Pulse frequency change due to SPM
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Seven
a shift to a higher frequency for the end of the pulse Over the pulse width, in the time domain, its frequency changes as shown in Fig 19a and b This is referred to as positive frequency chirp Since different optical frequencies travel at different speeds in a fiber, the pulse width expands or compresses Pulse width expansion leads to intersymbol interference and worst BER Refer to Chap 1 for more details on chirped pulse propagation The pulse s frequency change is a modulation caused by a phase shift induced by itself and therefore the effect is called self-phase modulation This effect caused by high optical power limits the maximum transmission rate possible in the fiber It should be noted that this wavelength chirping is not linear in time as the pulse propagates in the fiber, which is different from laser frequency chirping Solving the nonlinear Schr dinger equation [Eq (140)] using only the Kerr nonlinear term and with the Gaussian pulse Eq (145) where chirp C0 = 0, Cvijetic14 shows that the pulse width broadening factor due to SPM and chromatic dispersion can be estimated with Eq (737) A + z 1 A t + j 2 2 A 3 3 A + 2 t 2 6 t 3
CD effect CD slope
A 2
Attenua tion
= j A A
SPM 1/2
Group velocity
SPM 2 Leff L 2 4 L2 L2 2 eff 2 = 1 + + 1 + 4 2 2 o 2LNL o 3 3 LNL 4 o
(737)
Here the nonlinear length LNL term is introduced and is defined as the fiber length required to produce one radian of nonlinear phase rotation at power Po, see Eq (738) LNL = Aeff 2 n2 Po
(738)
The group-velocity dispersion (GVD) parameter 2 is defined in Eq (417) and is shown below 2 = where 2 CD c 2 c
SPM = signal RMS pulse width (Gaussian pulse shape) at location L, s o = signal RMS pulse width (Gaussian pulse shape) at the beginning of fiber, s SPM/ o = pulse width broadening factor due to SPM L = fiber link length, m LNL = fiber nonlinear length, m Leff = fiber effective length, m Po = pulse peak power at launch, W 2 = group-velocity parameter, s2/m
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= center wavelength of the optical signal, m CDc = chromatic dispersion coefficient, s/(m m) n2 = nonlinear-index coefficient which varies for different fibers between 20 10 20 to 35 10 20 m2/W, (typical is 30 10 20 m2/W) Aeff = fiber core s effective area, m2 Equation (737) shows that SPM effect is dependent on 1 Signal wavelength (the higher the wavelength the lesser the SPM) 2 Transmission rate, 10 Gbps (the higher the rate the greater the SPM effect) 3 Fiber core effective area (the larger the area the lesser the SPM) 4 Fiber dispersion (the lower the dispersion the lesser the SPM) 5 Fiber nonlinear index coefficient For transmission rates of 25 Gbps and less, SPM is not a limiting factor for NRZ and other OOK modulation formats The below discussion concentrates on NRZ modulated 10 Gbps and higher rates For 10 Gbps and higher transmission rates, we assume that optical signal pulses have a Gaussian shape as they propagate in a fiber A commonly used fit criteria for Gaussian pulses is that the pulse s RMS width needs to be less than 1/4 of the bit time slot T for at least 95% of the optical pulse power to fit within the time slot, see Eq (429) and Eq (739) o = 1 4R
(739)
where R = transmission rate, bps o = Gaussian pulse initial launch RMS width, s As a Gaussian pulse propagates in a fiber, it undergoes compression and expansion in the presence of SPM For example, assume a 10 Gbps signal with Gaussian-shaped pulses is launched into a standard single-mode fiber (G652) Pulse width changes due to SPM occur as the signal propagates in the fiber as shown in Fig 710 When SPM is present, positive chirp occurs and the pulse initially undergoes compression, then expansion as it propagates further in the fiber This compression is helpful in extending the propagation limit for a distance up to ~100 km as shown in the example Comparing this effect to a pulse launched in a fiber with no SPM [Eq (160), C0 = 0], the pulse only expands as it propagates in the fiber Eventually, the SPM pulse width surpasses the non-SPM pulse width at ~100 km Therefore, in
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