Lecture 4 Fiber Optical Communication Lecture 4, Slide 1

Transcription

1 ecture 4 Dispersion in single-mode fibers Material dispersion Waveguide dispersion imitations from dispersion Propagation equations Gaussian pulse broadening Bit-rate limitations Fiber losses Fiber Optical Communication ecture 4, Slide 1

2 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 ecture 4, Slide

3 Fiber Optical Communication ecture 4, Slide 3 Each spectral component of a pulse has a specific group velocity The group delay after a distance 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 d d d dt T d dn n c d dn n c n c v g g d d v T g d n d n n g

4 The dispersion parameter Measuring the spectral width in units of wavelength rather than frequency, we can write the broadening as ΔT = D Δλ, 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 ecture 4, Slide 4

5 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 ecture 4, Slide 5

6 Waveguide dispersion.3.3 The waveguide dispersion arises from the modes dependence on frequency D W n g 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 ecture 4, Slide 6

7 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 < 1 where B is the bit-rate, Δλ the spectral width, and the fiber length Fiber Optical Communication ecture 4, Slide 7

8 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 ecture 4, Slide 8

9 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 ecture 4, Slide 9

10 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 ecture 4, Slide 1

11 Index profiles of different fiber types Standard single-mode fiber SMF Dispersion-shifted fiber DSF Dispersion-flattened fiber DFF Fiber Optical Communication ecture 4, Slide 11

12 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 ecture 4, Slide 1

13 Polarization-mode dispersion PMD.3.5 An optical fiber is always slightly elliptical Different index of refraction for x and y polarization, difference is Δn This leads to: Changes along the fiber Changes with frequency Changes with time The state-of-polarization SOP of the signal in the fiber will be random Different delay for different polarizations, differential group delay DGD 1x 1y 1 vgx vgy DGD acting along varying axes leads to polarization-mode dispersion PMD T 1 l c z D p D p is the PMD parameter typical value is.5 1 ps/ km Fiber Optical Communication ecture 4, Slide 13

14 Fiber Optical Communication ecture 4, Slide 14 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 Fx, y describes the mode in the transverse directions Az, t is the complex field envelope β is the propagation constant corresponding to ω Only Az, 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, ~, ~

15 Fiber Optical Communication ecture 4, Slide 15 The propagation constant The propagation constant is in general complex α is the attenuation δn N is a small nonlinear = power dependent change of the refractive index Dispersion arises from β ω The frequency dependence of β N and α is small We now expand β ω in a Taylor series around ω = ω Δω = ω ω 1/v p 1/v g GVDrel. to D dispersion sloperelated to S , 6 m m m d d / / / ] [ i i c n n N N

16 Basic propagation equation.4.1 Substitute β with the Taylor expansion in the expression for the evolution of Az, 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 N A A The nonlinearity is quantified by using δn N = n I where n [m /W] is a measure of the strength of the nonlinearity, and I is the light intensity β N = 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 ecture 4, Slide 16

17 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 NSE 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 ecture 4, Slide 17

18 Chirped Gaussian pulses.4. To study dispersion, we neglect nonlinearity and loss The formal solution is A i z Note: Dispersion acts like an all-pass filter We study chirped Gaussian pulses A t 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 ecture 4, Slide 18

19 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 expiβ z iω t = A expiφ, 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 ecture 4, Slide 19

20 Frequency increases with time A linearly chirped pulse Frequency decreases with time ω c ω c t t Fiber Optical Communication ecture 4, Slide

21 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 ecture 4, Slide 1

22 Fiber Optical Communication ecture 4, Slide We introduce ξ = z/ D where the dispersion length 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 / s C s C C sc b f 1/ 1 1 T z T z C T z T z b f

23 Fiber Optical Communication ecture 4, Slide 3 Broadening of chirp-free Gaussian pulses Short pulses broaden more quickly than longer pulses Compare with diffraction of beams 1 1 T z z z b D f

24 Broadening of linearly chirped Gaussian pulses For C β <, pulses initially compress and reaches a minimum at z = C /1+C min T 1 D at which C 1 = and T1 1 C Chirped pulses eventually broaden more quickly than unchirped pulses Fiber Optical Communication ecture 4, Slide 4

25 Fiber Optical Communication ecture 4, Slide 5 Non-Gaussian pulses Only Gaussian pulses remain Gaussian upon propagation Not even Gaussian pulses remain Gaussian if β 3 cannot be ignored In these cases, the RMS-pulse width can be used σ p can be calculated when the initial pulse is known Example 1: An unchirped rectangular pulse will, in presence of only β, broaden as Example : An unchirped Gaussian pulse will, in presence of only β, broaden as t t p dt t z A dt t z A t t m m,, 3 1 D p 1 D p

26 Fiber Optical Communication ecture 4, Slide 6 Chirped Gaussian pulses in the presence of β 3 Higher order dispersion gives rise to oscillations and pulse shape changes C C / T

27 Fiber Optical Communication ecture 4, Slide 7 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 V C V C p V ω << 1 when the source spectral width << the signal spectral width

28 imitations on bit rate, incoherent source.4.3 If, as for an ED light source, V ω >> 1 we obtain approximately A common criteria for the bit rate is that D 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 4B D 1 σ λ is the source RMS width in wavelength units Example: D = 17 ps/km nm, σ λ = 15 nm B max 1 Gbit/s km Fiber Optical Communication ecture 4, Slide 8

29 imitations on bit rate, incoherent source In the case of operation at λ = λ ZD, β = we have S With the same condition on the pulse broadening, we obtain 8B S The dispersion slope, S, will determine the bit rate-distance product 1 Example: D =, S =.8 ps/km nm, σ λ = 15 nm B max Gbit/s km Fiber Optical Communication ecture 4, Slide 9

30 imitations on bit rate, coherent source.4.3 For most lasers V ω << 1 and can be neglected and the criteria become Neglecting β 3 : / D The output pulse width is minimized for a certain input pulse width giving 4B 1 Example: β = ps /km B max 3 Gbit/s km 5 Gbit/s, 3 1 Gbit/s If β = close to λ : 3 / 4 / D For an optimal input pulse width, we get 1/3 B 3.34 Fiber Optical Communication ecture 4, Slide 3

31 imitations 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 ecture 4, Slide 31

32 Super-Gaussian pulses Super-Gaussian pulses are flat-top and can be used to model NRZ...but more accurate modeling is preferred If m = 1 We recover the Gaussian A, t A If m > 1 The shape is more rectangular 1 ic exp Numerical simulation shows: Super-Gaussian has: Smaller bit rate-distance product...due to more problems with dispersion......due to the sharper edges A small chirp can be beneficial: C 1 is optimal Fiber Optical Communication ecture 4, Slide 3 t T m

33 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 ecture 4, Slide 33

34 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 1 e 1 log e log1 1 log Typical value in SMF at 155 nm α db =. db/km α =.46 km -1 = 1/1.7 km Fiber Optical Communication ecture 4, Slide 34

35 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 ecture 4, Slide 35

36 Total attenuation Minimum theoretical loss is.15 db/km at 155 nm Main contributions: Rayleigh scattering and IR absorption eft figure: Theoretical curves and measured loss for typical fiber Right figure: oss for sophisticated fiber with negligible loss peak Fiber Optical Communication ecture 4, Slide 36