Lecture 3 Dispersion in single-mode fibers Material dispersion Waveguide dispersion Limitations from dispersion Propagation equations Gaussian pulse broadening Bit-rate limitations Fiber losses Fiber Optical Communication Lecture 3, Slide 1
Dispersion, qualitatively Different wavelengths (frequency components propagate differently A pulse has a certain spectral width and will broaden during propagation The index of refraction as a function of wavelength The dispersion in SMF (red and different dispersion-shifted fibers Fiber Optical Communication Lecture 3, Slide
Fiber Optical Communication Lecture 3, Slide 3 Each spectral component of a pulse has a specific group velocity The group delay after a distance L is The group velocity is related to the mode group index given by Assuming that Δω is the spectral width, the pulse broadening is governed by where β is known as the GVD parameter (unit is s /m or ps /km Group delay, group index, and GVD parameter (.3.1 L d d L d dt T d dn n c d dn n c n c v g g d d L v L T g d n d n n g
The dispersion parameter Measuring the spectral width in units of wavelength (rather than frequency, we can write the broadening as ΔT = D Δλ L, where D [ps/(nm km] is called the dispersion parameter D is related to β and the effective mode index according to D c d, v d d d 1 c d 1 d vg g dn d n d d The dispersion parameter has two contributions: material dispersion, D M : The index of refraction of the fiber material depends on the frequency waveguide dispersion, D W : The guided mode has a frequency dependence Fiber Optical Communication Lecture 3, Slide 4
The material dispersion is related to the dependence of the cladding material s group index on the frequency D M dn g d An approximate relation for the material dispersion in silica is D M M 1 1 where D M is given in ps/(nm km Material dispersion (.3. Fiber Optical Communication Lecture 3, Slide 5
Waveguide dispersion (.3.3 The waveguide dispersion arises from the modes dependence on frequency D W ng Vd n dv V Vb dn g dvb d dv n g : the cladding group index V: the normalized frequency V a n1 n an1 c b: the normalized waveguide index n n b n n 1 Fiber Optical Communication Lecture 3, Slide 6
Total dispersion The total dispersion D is the sum of the waveguide and material contributions D = D W + D M Note: D W increases the net zero dispersion wavelength The zero-dispersion wavelength is denoted either λ or λ ZD An estimate of the dispersionlimited bit-rate is D B Δλ L < 1 where B is the bit-rate, Δλ the spectral width, and L the fiber length Fiber Optical Communication Lecture 3, Slide 7
Anomalous and normal dispersion The dispersion can have different signs in a standard single-mode fiber (SMF D > for λ > 1.31 μm: anomalous dispersion, the group velocity of higher frequencies is higher than for lower frequencies D < for λ < 1.31 μm: normal dispersion, the group velocity of higher frequencies is lower than for lower frequency components Pulses are affected differently by nonlinear effects in these two cases Fiber Optical Communication Lecture 3, Slide 8
Different fiber types The fiber parameters can be tailored to shift the λ -wavelength from 1.3 μm to 1.55 μm, dispersion-shifted fiber (DSF A fiber with small D over a wide spectral range (typically with two λ - wavelengths, dispersion-flattened fiber (DFF A short fiber with large normal dispersion can compensate the dispersion in a long SMF, dispersion compensating fibers (DCF Dispersion compensating fiber Fiber Optical Communication Lecture 3, Slide 9
This dispersion compensating module contains 4 km of DCF... Fibers in the lab...and it compensates the dispersion in this 5 km roll of SMF Fiber Optical Communication Lecture 3, Slide 1
Index profiles of different fiber types Standard single-mode fiber (SMF Dispersion-shifted fiber (DSF Dispersion-flattened fiber (DFF Fiber Optical Communication Lecture 3, Slide 11
Higher order dispersion (.3.4 Near the zero-dispersion wavelength D The variation of D with the wavelength must be accounted for S dd d c We have used β = S [ps/(nm km] is called the dispersion slope Typical value in SMF is.7 ps/(nm km 3 d 3 d Fiber Optical Communication Lecture 3, Slide 1
Fiber Optical Communication Lecture 3, Slide 13 Basic propagation equation We will now develop the theory for signal propagation in fibers The electric field is written as The field is polarized in the x-direction F(x, y describes the mode in the transverse directions A(z, t is the complex field envelope β is the propagation constant corresponding to ω Only A(z, t changes upon propagation (described in the Fourier domain Each spectral component of a pulse propagates differently exp(, (, ( ˆ Re, ( t i z i t z A y x F t x E r d t i t z A z A z i z i A z A exp(, (, ( ~ ( exp (, ~, ( ~
Fiber Optical Communication Lecture 3, Slide 14 The propagation constant The propagation constant is in general complex α is the attenuation δn NL is a small nonlinear (= power dependent change of the refractive index Dispersion arises from β L (ω The frequency dependence of β NL and α is small We now expand β L (ω in a Taylor series around ω = ω (Δω = ω ω 1/v p 1/v g GVD(rel. to D dispersion slope(related to S 3 3 1..., ( 6 ( ( ( m m m L d d / ( ( ( / ( / ]( ( ( [ ( i i c n n NL L NL
Basic propagation equation (.4.1 Substitute β with the Taylor expansion in the expression for the evolution of A(z, ω, calculate A/ z, and write in time domain by using Δω i / t A A 1 i z t A 3 t 6 3 A 3 t i NL A A The nonlinearity is quantified by using δn NL = n I where n [m /W] is a measure of the strength of the nonlinearity, and I is the light intensity β NL = γ A, where γ = πn /(λ A eff is the nonlinear coefficient A eff is the effective mode area and A is normalized to represent the power γ is typically 1 W 1 km 1 Fiber Optical Communication Lecture 3, Slide 15
Basic propagation equation Use a coordinate system that moves with the pulse group velocity! This is called retarded time, t = t β 1 z We neglect β 3 to get A i z This is the nonlinear Schrödinger equation (NLSE The primes are implicit A t i A A The loss reduces the power reduces the impact from the nonlinearity The average power of the signal during propagation in the fiber is P T / 1 av ( z lim A( z, t dt Pav( T T T / Note: α is in m -1 while loss is often expressed in db/km A e z Fiber Optical Communication Lecture 3, Slide 16
Chirped Gaussian pulses (.4. To study dispersion, we neglect nonlinearity and loss The formal solution is A A i z t Note: Dispersion acts like an all-pass filter We study chirped Gaussian pulses A ~ z A ~ (, (, expi z A(, t A exp (1 ic( t / T 1 A is the peak amplitude C is the chirp parameter T is the 1/e half width (power T (ln T 1. T 1/ FWHM 665 Fiber Optical Communication Lecture 3, Slide 17
For a chirped pulse, the frequency of the pulse changes with time What does this mean??? Study a CW (continuous wave A is a constant Chirp frequency Writing A exp(iβ z iω t = A exp(iφ, we see that ω = φ/ t We define the chirp frequency to be We allow φ to have a time dependence We get φ from the complex amplitude In this way, the chirp frequency can depend on time For the Gaussian pulse we get ω c = Ct/T E r, t Re xˆ F( x, y A( z, texp( i z i ( t ( t / t c Fiber Optical Communication Lecture 3, Slide 18
Frequency increases with time A linearly chirped pulse Frequency decreases with time ω c ω c t t Fiber Optical Communication Lecture 3, Slide 19
Time-bandwidth product The Fourier transform of the input Gaussian pulse is A ~ (, T A 1 ic 1/ exp T (1 ic The 1/e spectral half width (intensity is The product of the spectral and temporal widths is C / T 1 C 1 T If C = then the pulses are chirp-free and said to be transform-limited as they occupy the smallest possible spectral width Using the full width at half maximum (FWHM, we get T ln 1 C.44 FWHM FWHM 1 C Fiber Optical Communication Lecture 3, Slide
Fiber Optical Communication Lecture 3, Slide 1 We introduce ξ = z/l D where the dispersion length L D = T / β In the time domain the dispersed pulse is The output width (1/e-intensity point broadens as Chirped Gaussian pulses (.4. A Gaussian pulse remains Gaussian during propagation The chirp, C 1 (ξ, evolves as the pulse propagates If (C β is negative, the pulse will initially be compressed C i b T t ic b A t A f f 1 arctan (1 exp, ( 1 sign ( (1 ( (1 ( 1 1/ s C s C C sc b f 1/ 1 1 ( ( T z T z C T z T z b f
Fiber Optical Communication Lecture 3, Slide Broadening of chirp-free Gaussian pulses Short pulses broaden more quickly than longer pulses (Compare with diffraction of beams 1 1 ( T z L z z b D f
Broadening of linearly chirped Gaussian pulses For (C β <, pulses initially compress and reaches a minimum at z = C /(1+C min T 1 L D at which C 1 = and T1 1 C Chirped pulses eventually broaden more quickly than unchirped pulses Fiber Optical Communication Lecture 3, Slide 3
Fiber Optical Communication Lecture 3, Slide 4 Chirped Gaussian pulses in the presence of β 3 Higher order dispersion gives rise to oscillations and pulse shape changes 3 3 4 (1 1 C L L L C / T
Fiber Optical Communication Lecture 3, Slide 5 Effect from source spectrum width Using a light source with a broad spectrum leads to strong dispersive broadening of the signal pulses In practice, this only needs to be considered when the source spectral width approaches the symbol rate For a Gaussian-shaped source spectrum with RMS-width σ ω and with Gaussian pulses, we have where V ω = σ ω σ 3 3 4 (1 (1 1 L V C L V L C p V ω << 1 when the source spectral width << the signal spectral width
Limitations on bit rate, incoherent source (.4.3 If, as for an LED light source, V ω >> 1 we obtain approximately A common criteria for the bit rate is that ( L ( DL T B / 4 1/(4B For the Gaussian pulse, this means that 95% of the pulse energy remains within the bit slot In the limit of large broadening 4BL D 1 σ λ is the source RMS width in wavelength units Example: D = 17 ps/(km nm, σ λ = 15 nm (BL max 1 (Gbit/s km Fiber Optical Communication Lecture 3, Slide 6
Limitations on bit rate, incoherent source In the case of operation at λ = λ ZD, β = we have 1 1 ( 3L ( SL With the same condition on the pulse broadening, we obtain 8BL S The dispersion slope, S, will determine the bit rate-distance product 1 Example: D =, S =.8 ps/(km nm, σ λ = 15 nm (BL max (Gbit/s km Fiber Optical Communication Lecture 3, Slide 7
Limitations on bit rate, coherent source (.4.3 For most lasers V ω << 1 and can be neglected and the criteria become Neglecting β 3 : ( L / D The output pulse width is minimized for a certain input pulse width giving 4B L 1 Example: β = ps /km (B L max 3 (Gbit/s km 5 km @.5 Gbit/s, 3 km @ 1 Gbit/s If β = (close to λ : ( 3L / 4 / D For an optimal input pulse width, we get 1/3 B( 3 L.34 Fiber Optical Communication Lecture 3, Slide 8
Limitations on bit rate, summary A coherent source improves the bit rate-distance product Operation near the zero-dispersion wavelength also is beneficial...but may lead to problems with nonlinear signal distortion Fiber Optical Communication Lecture 3, Slide 9
Dispersion compensation Dispersion is a key limiting factor for an optical transmission system Several ways to compensate for the dispersion exist More about this in a later lecture... One way is to periodically insert fiber with opposite sign of D This is called dispersion-compensating fiber (DCF Figure shows a system with both SMF and DCF The GVD parameters are β 1 and β Group-velocity dispersion is perfectly compensated when β 1 l 1 + β l =, which is equivalent to D 1 l 1 + D l = GVD and PMD can also be compensated in digital signal processing (DSP Fiber Optical Communication Lecture 3, Slide 3
Fiber losses (.5 Fiber have low loss but the loss grows exponentially with distance Approx. 5 db loss over 1 km Optical receivers add noise......and the input power may be too low to obtain sufficient SNR The optical power in a fiber decreases exponentially with the propagation distance as P out = P in exp( αz α is the attenuation coefficient (unit m -1 Often, attenuation is given in db/km and its relation to α is db 1 1log L 1 e L 1 L log e log1 L 1 log1 4.343 Typical value in SMF at 155 nm α db =. db/km α =.46 km -1 = 1/(1.7 km Fiber Optical Communication Lecture 3, Slide 31
Material absorption Attenuation mechanisms Intrinsic absorption: In the SiO material Electronic transitions (UV absorption Vibrational transitions (IR absorption Extrinsic: Due to impurity atoms Metal and OH ions, dopants Rayleigh scattering Occurs when waves scatter off small, randomly oriented particles (Makes the sky blue! Proportional to λ -4 Waveguide imperfections Core-cladding imperfections on > λ length scales (Mie scattering Micro-bending (bending curvature λ Macro-bending (negligible unless bending curvature < 1 5 mm Fiber Optical Communication Lecture 3, Slide 3
Total attenuation Minimum theoretical loss is.15 db/km at 155 nm Main contributions: Rayleigh scattering and IR absorption Left figure: Theoretical curves and measured loss for typical fiber Right figure: Loss for sophisticated fiber with negligible loss peak Fiber Optical Communication Lecture 3, Slide 33
Lecture Why/when are nonlinear phenomena important? Different types of fiber nonlinearities The Kerr effect: SPM, XPM, FWM Fiber Optical Communication Lecture 3, Slide 34
Nonlinear effects When is a phenomenon nonlinear? Superposition does not apply The phenomenon is changed by an amplitude (power change Which is the same, e.g., doubling the amplitude is equivalent to a superposition of a pulse on itself In nonlinear optics, light cannot be viewed as a superposition of independently propagating spectral components Spectral components interact New frequencies can be generated, existing components can lose power IR light can become visible (green Fibers nonlinearity is important for moderate powers because The fiber core is small, the electric field intensity is high A fiber is long, allowing nonlinear distortion to accumulate Fiber Optical Communication Lecture 3, Slide 35
Why study fiber nonlinearities? What transmitted power would you choose in a fiber optic link? Laser output power is sufficient The energy cost is small (typical input power is 1 mw The figure shows that the SNR is proportional to the input power Clearly, higher input power is always better!?! No, actually it is not... Fiber Optical Communication Lecture 3, Slide 36
Why study fiber nonlinearities? What limits the launch power? Before 199: Limited by laser output power to 1 mw After 199: EDFAs enable power levels up to > 1 mw Performance is limited by fiber nonlinearities Noise limitation Nonlinear limitation The nonlinear trade-off: Low power: System is limited by noise High power: System is limited by nonlinearities BER for a system without nonlinearities There exist an optimum launch power A higher power is not always better! Fiber Optical Communication Lecture 3, Slide 37
Nonlinearities in fibers Two types of important nonlinear effects in fibers: Electrostriction Intensity modulation in the fiber leads to pressure changes in the density of the medium, which leads to changes of the refractive index Responsible for Stimulated Brillouin Scattering (SBS The Kerr effect The refractive index is changed in proportion to the optical intensity This gives rise to Self-phase modulation (SPM Cross-phase modulation (XPM Four-wave mixing (FWM Modulation instability Solitons, which propagate without any change of the shape The delayed response of the Kerr effect gives rise to a nonlinear frequency downshift called Stimulated Raman Scattering (SRS Fiber Optical Communication Lecture 3, Slide 38
Nonlinearities in fibers, scattering processes Stimulated Brillouin scattering Occurs only in the backward direction Light will be backscattered and downshifted 1 GHz Remaining photon energy is absorbed as a vibration mode in the fiber Requires power levels 1 mw Stimulated Raman scattering Occurs both in the forward and backward direction Appears over a wide spectral range (15 THz, 1 nm Photons are downshifted in frequency Remaining photon energy is absorbed by the fiber Requires power levels of about.1 1 W Fiber Optical Communication Lecture 3, Slide 39
Nonlinearities in fibers, the Kerr effect The Kerr effect means that the refractive index is intensity dependent The propagation constant becomes β(ω = β lin (ω + γ A(t The Kerr-effect gives rise to Self-phase modulation (SPM Causes spectral broadening Can counteract anomalous dispersion Can give rise to soliton pulses Solitons do not broaden in time or frequency Cross-phase modulation (XPM Causes frequency shift of other WDM channels Limits WDM systems performance Four-wave mixing (FWM Causes power exchange between WDM channels Limits WDM system performance The fundamental phenomenon is SPM XPM and FWM appear when we interpret SPM in a WDM system Fiber Optical Communication Lecture 3, Slide 4
Self-phase modulation (.6. Start from the NLSE and eliminate loss term by A( z, t P p( z U( z, t U is the normalized amplitude The NLSE for U(z, t becomes U i z t The function p(z varies periodically between 1 and exp( αl A L A is the amplifier spacing U i P p z U ( Neglecting the impact from dispersion, the NLSE is U i z p( z L L NL = 1/(γ P is the nonlinear length NL U U U The nonlinear length is the propagation distance over which the nonlinear effects become important Fiber Optical Communication Lecture 3, Slide 41
The solution to the NLSE without dispersion is The signal phase is changed by the signal itself self-phase modulation We have introduced L eff and φ NL φ NL is the nonlinear phase shift L eff is the effective length We have Self-phase modulation U( L, t U(, texp eff iu(, t L / L U(, texp i ( L, t The power decreases during propagation, the nonlinearity becomes weaker Therefore, the effective length is shorter than the physical length L eff L p( z dz N L A A p( z dz where N A is the number of amplified sections of fiber (often called spans N A NL NL 1 exp( L / N / A A Fiber Optical Communication Lecture 3, Slide 4
SPM impact on pulses In the absence of dispersion, the pulse shape will not change SPM introduces chirp and continually broadens the spectrum The chirping depends on pulse shape Super-Gaussian different from Gaussian pulse Solid line: A Gaussian pulse Dashed line: A super- Gaussian pulse with m = 3 max Leff / LNL P Leff (Remember the chirp frequency from last lecture Fiber Optical Communication Lecture 3, Slide 43 L NL ( t t t L NL t eff U(,
Spectral broadening from SPM Figures show the spectra for chirped Gaussian pulses affected by SPM Dispersion and loss are neglected In this numerical example φ max = 4.5 π Spectral broadening will continue if more SPM is introduced Chirp on the pulse will change the effect from SPM significantly When φ max is large, the spectral broadening is strong Dispersion will change this result! SPM and GVD acting simultaneously leads to nontrivial phenomena Fiber Optical Communication Lecture 3, Slide 44
Linear dispersive effects A L = A arg(a L = A L = 1.5L D A arg(a L > time (bit slots In the time domain: Pulses broaden......and start to interfere A phase shift (chirp will become an amplitude change The length scale for dispersion is the dispersion length L D = T / β frequency (normalized In the frequency domain: The amplitude is not changed Quadratic phase modulation Fig. shows spectrum for single pulse Fiber Optical Communication Lecture 3, Slide 45
Nonlinear propagation, SPM A arg(a L = A L = A arg(a L > A L > time (bit slots In the time domain: The amplitude is not changed A pulse-shaped phase shift is introduced Self-phase modulation frequency (normalized In the frequency domain: The spectrum is broadened Energy is conserved Notice: Different y-scales The length scale for the nonlinearity is the nonlinear length L NL = 1/(γ P Fiber Optical Communication Lecture 3, Slide 46
Fiber Optical Communication Lecture 3, Slide 47 Cross-phase modulation Consider (again the case A = a exp( iω a t + b exp( iω b t, insert into the NLSE, neglect FWM, and split into a coupled system of equations The group velocities are different This causes walk-off and limits the impact of XPM The wave at ω a notices the presence of the wave at ω b through the additional nonlinear term And vice versa XPM is stronger than SPM by a factor of two, but walk-off limits the impact from XPM, i.e., dispersion reduces XPM The equation system can be used only for waves well separated in freq. b b a i t b i t b v z b a b a i t a i t a v z a b g a g,, 1 1
Cross-phase modulation in WDM systems XPM on channel b from channel a gives b b exp[iγp a (tz] This changes the absolute phase, but can also......introduce a chirp that shifts the pulse up or down in frequency Figure shows that the sign of the shift depends on the pulse position Blue, solid line is the a channel, affected by the red, dashed b channel Remember the chirp frequency, ω c = φ(t/ t frequency upshift no frequency change frequency downshift The frequency shift depends on the relative position of the pulses The frequency shift will, via dispersion, give rise to timing jitter Dispersive walk-off will decrease the impact of XPM Fiber Optical Communication Lecture 3, Slide 48
Four-wave mixing The waves at three frequencies generate a fourth The frequencies can be different or some may be the same With N different frequencies, FWM will generate N (N 1/ mixing products The strength of each mixing product depends on The degeneracy (how many terms that contribute How close the process is to being phase matched Phase matching is strongly dependent on the dispersion FWM is strong for low dispersion, e.g., near the zero-dispersion wavelength At symbol rates > 1 Gbaud, FWM is weak Figure: Non-degenerate FWM Left: Measured FWM Right: Original and generated frequencies (dispersion not accounted for Fiber Optical Communication Lecture 3, Slide 49
Four-wave mixing in WDM systems Equal channel spacing FWM components overlap with the data channels FWM can be a problem Solution: Decrease the dispersion length to reduce phase matching SMF/DCF better than DSF Only SMF is even better DSP dispersion compensation Use unequal channel spacing Not compliant with standard frequency assignment (ITU grid Increases optical bandwidth Original signal Equal spacing Unequal spacing Fiber Optical Communication Lecture 3, Slide 5