Dispersion Compensation using High-Positive Dispersive Optical Fibers
11 Dispersion Compensation using High-PositiveDispersive Optical Fibers
Mohammad Hadi, Farokh Marvasti, Mohammad Reza Pakravan
Abstract —The common and traditional method for dispersioncompensation in optical domain is concatenating the transmitoptical fiber by a compensating optical fiber having high-negativedispersion coefficient. In this paper, we take an opposite directionand show how an optical fiber with high-positive dispersion coef-ficient can also be used for dispersion compensation. Our opticaldispersion compensating structure is the optical implementationof an iterative algorithm in signal processing. The proposeddispersion compensating system is constructed by cascadinga number of compensating sub-systems and its compensationcapability is improved by increasing the number of embeddedsub-systems. We also show that the compensation capability isa trade-off between transmission length and bandwidth. We usesimulation results to validate the performance of the introduceddispersion compensating module. Photonic crystal fibers withhigh-positive dispersion coefficient can be used for constructingthe proposed optical dispersion compensating module.
Keywords — Fiber Optics Communications, Dispersion, Disper-sion Compensation Devices, Photonic Crystal Fibers, IterativeMethod
I. I
NTRODUCTION S INCE the revolution of fiber optics communication indecade , millions of kilometers of optical fibers havebeen laid all over the world to convey the ever growingdata streams primarily driven by the exponential increase incommunicated video and data traffic. This exponential growthin data traffic is met through increasing per-channel bit rateor number of accommodated sub-channels using techniquessuch as Wavelength Division Multiplexing (WDM) or OpticalOrthogonal Frequency Division Multiplexing (O-OFDM) [1],[2]. Increasing per-channel bit rate needs transmission of nar-rower optical pulses which multiplies signal degradation dueto high amount of accumulated chromatic dispersion duringpulse propagation through the optical fiber. This high amountof dispersion may result in Inter-Symbol Interference (ISI),information loss and increased Bit Error Rate (BER) value.on the other hand, increasing the number of accommodatedsub-channels results in reduced space between adjacent sub-channels, more transmitted optical power and more sensitivityto destructive fiber nonlinear effects. Consequently, both ofincreasing per-channel bit rate and number of accommodatedsub-channels lead to excessive sensitivity to certain fiberimpairments that degrade clean transmission of the optical M. Hadi is a PhD Candidate at Electrical Engineering Department, SharifUniversity of Technology, E-mil: [email protected]. Marvasti is director of ACRI and fauclty member of Electrical Engineer-ing Department, Sharif University of Technology, E-mil: [email protected]. R. Pakravan is faculty member of Electrical Engineering Department,Sharif University of Technology, E-mil: [email protected] signal [3]. Practically, the increased optical power along withtighter sub-channel spacing and longer transmission distanceare translated to a trade-off between nonlinear propagation ef-fects and chromatic dispersion in a fiber optics communicationsystem. Today’s, advanced optical fibers are designed suchthat they exhibit finite dispersion in the transmission band.This finite amount of dispersion reduces the growth of non-linear effects such as Four-Wave Mixing (FWM) and Cross-Phase Modulation (XPM) which are particularly deleterious inDWM and O-OFDM communication systems [4], [5]. In orderto resolve the ISI and BER forced by the mentioned finiteamount of dispersion, proper dispersion compensating tech-niques have been proposed to compensate for the accumulateddispersion in the propagated pulse through the optical fiber.Compensation can be achieved in the optical domain by the useof different dispersion compensation devices such as Disper-sion Compensating Gratings (DCG), Dispersion CompensatingFibers (DCF), Dispersion Compensating Arrayed Waveguides(DCAW), etc [3], [6], [7]. Electronic Dispersion Compensators(EDC) are also proposed to compensate for the accumulateddispersion in the electrical domain [8], [9].Conventional DCF are suitably constructed optical fiberswith an appropriate refractive index profile such that theyexhibit the desired dispersion value at the wavelength ofoperation. Since the dispersion coefficient of the transmissionoptical fiber is usually positive, the conventional DCF shouldexhibit negative dispersion coefficient. They should also havedispersion slope matching, low bend loss, low propagationloss, and relatively large mode effective area [3], [10]. Designtrade-offs to meet some of these requirements are necessary,e.g., small dispersion coefficient is usually characterized bysmall mode effective area and consequently large nonlineareffects and vice versa [3], [6], [10]. In this letter, we take anopposite direction and show how an optical fiber with highpositive dispersion coefficient can also be used to compensatefor the dispersion. In fact, our compensating procedure isan adoption of an iterative method in signal processing thatuses a given system to implements its inverse system [11].The proposed dispersion compensating structure is a cascaderepetition of a sub-system and its compensation capabilityis improved by higher number of cascaded sub-systems inthe structure. Furthermore, the capability of the proposeddispersion compensating technique is a trade-off betweenthe transmission length and bandwidth. It is noteworthy thatthe introduced structure can simultaneously compensate fordispersion, dispersion slope, and other high order dispersions.Photonic Crystal Fibers (PCF) with high positive dispersionand low attenuation coefficients can be considered as a maincandidate for constructing the offered compensating module a r X i v : . [ c s . I T ] S e p [7], [12], [13]. High positive dispersion and low attenuationcoefficients of these PCFs improve the compensation abilityof our structure by providing dispersion compensation withlower latency, attenuation, and sensitivity to nonlinear effectsin comparison with the conventional DCFs.The rest of the paper is organized as follows. In SectionII, we present a brief survey of chromatic dispersion andits mathematical model. Following a concise review of thementioned iterative method, we introduce and analyze ournew dispersion compensating system in Section III. In SectionIV, the performance of the proposed dispersion compensat-ing structure is verified using simulation results. Finally, weconclude the paper in Section V.II. C HROMATIC D ISPERSION
One of the most significant impairments in fiber opticscommunication systems is chromatic dispersion. The lightpulses propagating along the optical fiber become distortedbecause different spectral components of the signal travel withdifferent speed and hence experience different propagationdelay during transmission. This means those parts of the signalwill reach the receiver at different time instants which resultsin a temporal pulse distortion and broadening. Dispersion ischaracterized by the parameter D which describes how thepulse is broadened. There are two physical issues accountingfor the chromatic dispersion in optical fibers. The first oneis material dispersion which is due to the fact that the coreand cladding are made of dispersive materials and this meansthat the refractive index is frequency dependent. The secondone is waveguide dispersion which is caused by the frequencydependence of the propagation constant to the geometry anddesign of the optical fiber. Assuming that the fiber is acylindrical dielectric waveguide along the z-axis, the wavepropagation in the frequency domain along the positive z-coordinate is given by [6]: E ( z, ω ) = E (0 , ω ) e − jβ ( ω ) z (1)where E (0 , ω ) and E ( z, ω ) are pulse fields at frequency ω anddistances and z with respect to the origin, respectively and β ( ω ) is the fiber frequency-dependent propagation constant.Assuming that the spectral width ∆ ω = ω − ω is muchsmaller than the carrier frequency ω = πcλ , the Fourier seriesexpansion of the propagation constant around ω is: β ( ω ) = n ( ω ) ωc = β + β (∆ ω ) + β ω ) + · · · (2)where the series coefficients are written as β n = ∂ n β∂ω n | ω = ω .The group delay per unit of length is τ ( ω ) = ∂β∂ω . Thefirst term of (2) represents a frequency independent phaserotation that can be disregarded for the propagation of thepulse. The second parameter β is equal to the group delayper unit of length at the carrier frequency and also equals tothe inverse of the group velocity at the carrier frequency i.e. β = V g ( ω ) , V g ( ω ) = cn ( ω ) ( n ( ω ) is the frequency dependentrefractive index). The third term of (2) describes first-orderchromatic dispersion and sometimes it is called the group-velocity dispersion. Chromatic dispersion parameter equals to {} E x x {} E K x K x Fig. 1: Direct implementation of the iterative method which is acascade repetition of the highlighted part. Each highlighted part isan error operator followed by an add operation. β = dτdω | ω = ω and it is responsible for linear variation ofgroup delay with frequency. Commonly, chromatic dispersionparameter is characterized in terms of ∆ λ instead of ∆ ω : D = dτdλ = − πcλ β (3)where D is the mentioned dispersion coefficient and its unitis usually expressed as ps/nm/km . Typical values of themain dispersion parameters in standard single mode fiberat the wavelength λ = 1550 nm are β = − ps /km and D = 17 ps/nm/km . As previously mentioned, the totaldispersion parameter is given by the sum of both the ma-terial dispersion D m and the waveguide dispersion D w i.e. D = D m + D w . Other parameters β n , n = 3 , , · · · arerelated to high order dispersions and usually neglected incomparison with the main dispersion parameter β . Clearly, theeffect of dispersion on pulse propagation through the opticalfiber is totally characterized by the dispersion transfer function H D ( ω ) : H D ( ω ) = exp( − jz ∞ (cid:88) i =2 β i i ! (∆ ω ) i ) (4)III. P ROPOSED D ISPERSION C OMPENSATING S YSTEM
The iterative method is a systematic technique in whichsuccessive operations of a given operator is used to provide anestimate of its corresponding inverse operation [11]. Consideran arbitrary operator named H {} and let I {} be the identityoperator. Defining E {} = I {} − H {} as error operator, E k {} means k consecutive operations of the error operator. It canbe shown that H − {} , the inverse of the operator H {} , canbe calculated as follows: I + E + E + ... + E k + ... = H − (5)provided that E {} is linear and its operator norm || E || = || I − H || is less than [11]. For sufficiently large values of k ,(5) provides an approximation of the inverse operator H − {} .Fig. 1 shows a systematic realization for (5). As shown inFig. 2, the inverse operator can also iteratively be realized byfeedbacking the output of the highlighted part in Fig. 1 to itsinput. For speeding up or down the convergency or extendingthe convergence region, a scaling factor µ can be includedin the definition of the error operator E {} = I {} − µH {} .Increasing µ speeds up the convergency at the cost of reducingthe convergence region [11]. Now, consider the structureshown in Fig. 3. This structure is an optical adoption of theintroduced iterative method that implements the inverse ofthe chromatic dispersion transfer function using high-positive {} E K x K x x Fig. 2: Feedback implementation of the iterative method which isa closed loop system constructed of the highlighted part of Fig. 1.Input signal should cycle K times in the feedback loop to provide anoutput equivalent to the output of the direct implementation with K repetition of the highlighted part. E SMF + E
SMF +G Amplifier Adder AdderIn OutDivider Divider Divider -E SMFPCF Attenuator S ub t r a c t o r D i v i de r Fig. 3: Block diagram of the proposed dispersion compensatingstructure including cascaded sub-systems E . dispersive optical fibers. Sub-system E ( ω ) gets an input opticalsignal, divides it between two optical fibers and constructthe output optical signal by subtracting the outputs of theoptical fibers. We assume the optical fiber in up branch ofthe sub-system is an ordinary optical fiber with characterizingparameters β SMFi , i = 0 , , , ... and length L while the opticalfiber in down branch is a special optical fiber with charac-terizing parameters β P CFi , i = 0 , , , · · · and length L suchthat β P CFi ≈ β SMFi , i = 0 , and β PCFi β SMFi (cid:29) , i = 2 , , · · · .A well-designed high-positive dispersive PCF can satisfy therequirements of the optical fiber in the down brand of the sub-system. If the attenuator in the down branch has an attenuationcoefficient α , the transfer function of the sub-system is givenby: E ( ω ) = 1 √ e − jL ∞ (cid:80) i =0 βSMFi i ! (∆ ω ) i − √ α √ e − jL ∞ (cid:80) i =0 βPCFi i ! (∆ ω ) i ≈ √ e − jL ( β SMF + β SMF ∆ ω ) (1 − √ αe − jL ∞ (cid:80) i =2 βPCFi i ! (∆ ω ) i )= 1 √ e − jL ( β SMF + β SMF ∆ ω ) E D ( ω ) (6)In Fig. 3, the optical fiber below each embedded sub-systemhas the same characterizing parameters as the optical fiber inthe up branch of the sub-system. Referring to (6) and assuming K cascaded copies of the analyzed sub-system, one can easilycheck that the total transfer function of the system H − D ( ω ) is simplified as: H − D ( ω ) ≈ √ G K +12 e − jKL ( β SMF + β SMF ∆ ω ) K (cid:88) k =0 E kD ( ω ) (7)For sufficiently large number of cascaded sub-systems K andfor | E ( ω ) | < : H − D ( ω ) ≈ √ G √ α K +12 e − jKL ( β SMF + β SMF ∆ ω ) e jL ∞ (cid:80) i =2 βPCFi i ! (∆ ω ) i (8)which is the desired inverse transfer function of the chromaticdispersion with a causal delay coefficient. Assume we wantto compensate for the chromatic dispersion of an opticalfiber with characterizing parameters β F IBi , i = 0 , , , · · · and length z . If we set β P CFi L = β F IBi z, i = 2 , , · · · and G = α K +1 , the proposed structure can totally remove thedispersion if: | E D ( ω ) | = | − √ αe − jz ∞ (cid:80) i =2 βFIBii ! (∆ ω ) i ) | < (9)Neglecting high order dispersion coefficients β F IBi , i =3 , , · · · against the main dispersion coefficient β F IB , (9) im-plies that we have the following trade-off between transmissionlength and bandwidth for a stable dispersion compensation: | β F IB | ω ) z < cos − ( √ α (10)It is noteworthy that the optical amplifier of the proposed struc-ture can be merged with the receiver amplifier and its otherparts can totally be constructed using passive optical elements.Furthermore, the length of the required compensating fibers L ,the compensation delay, and the compensation attenuation arereduced for higher values of | β P CF | .IV. S IMULATION R ESULTS
Fig. 4 shows the 2D region of transmission length z andbandwidth B = max {| ∆ ω |} /π values for which the intro-duced system can stably compensate for the dispersion. Werefer to this region as convergence region. As illustrated, theboundary of this region is governed by the explicit trade-offbetween transmission length and bandwidth in (10) and itsarea is extended for lower values of dispersion coefficient D F IB or attenuation coefficient α . Now, consider a sinc-shaped optical pulse with a given zero-to-zero pulse widthand its corresponding bandwidth B . This pulse is conveyed bycarrier wavelength λ = 1550 nm over an optical fiber havingthe second order dispersion parameter β F IB and length z . Weuse the proposed system to compensate for the accumulateddispersion in sinc-shaped pulse propagated through this opticalfiber. Fig. 5 shows the simulated broadening factor of thecompensated pulse (i.e. the ratio of the received pulse widthto the transmitted pulse width) in terms of the number ofembedded sub-systems E in the compensating system forvarious values of β F IB z (2 πB ) and attenuation coefficient α . Clearly, the compensation performance is increased for B (GHz) z ( k m ) Region: α = 0.2, D FIB = 8 ps/nm/kmRegoin: α = 0.6, D FIB = 8 ps/nm/kmRegion: α = 1, D FIB = 8 ps/nm/kmRegion: α = 1, D FIB = 17 ps/nm/kmRegion: α = 1, D FIB = 34 ps/nm/kmBoundary: α = 0.2, D FIB = 8 ps/nm/kmBoundary: α = 0.6, D FIB = 8 ps/nm/kmBoundary: α = 1, D FIB = 8 ps/nm/kmBoundary: α = 1, D FIB = 17 ps/nm/kmBoundary: α = 1, D FIB = 34 ps/nm/km
Fig. 4: 2D region of transmission length z and bandwidth B valuesfor which the proposed system can stably compensate for dispersion. B r oaden i ng F a c t o r α = 1, | β |z (2 π B) = 0.5 α = 1, | β | z (2 π B) = 1 α = 0.6, | β | z (2 π B) = 1.5 α = 0.6, | β | z (2 π B) = 2 α = 1, | β | z (2 π B) = 2 α = 1, | β | z (2 π B) = 1.5 Fig. 5: Broadening factor for a sinc-shaped optical pulse conveyedby carrier wavelength λ = 1550 nm in terms of the number ofcascaded sub-systems E in the compensating structure for variousvalues of | β FIB | z (2 πB ) and attenuation coefficient α . higher number of cascaded sub-systems. Furthermore, a pulsewith lower bandwidth and transmission distance needs lowernumber of sub-systems to get a desired dispersion compen-sated level. The simulation results also show decreasing theattenuation coefficient α decreases the convergence speed butas interpreted from Fig. 4, this can extend the convergenceregion to include a desired pair of transmission length andbandwidth.Assume that we desire to compensate for the accumulateddispersion of a single channel optical signal with GHz bandwidth and nm carrier frequency that propagatesthrough a km standard single mode optical fiber such thata broadening factor of . is achieved at the receiver side. Since | β F IB | z (2 πB ) ≈ , Fig. 5 shows that our proposeddispersion compensating module including embedded sub-system can provide the desired compensated broadening factor.If D P CF ≈ ps/nm/km (or equivalently β P CF ≈− ps /km ), the signal propagation path of the proposedcompensating structure will be around Km . As simulationresults show, the favorite level of the compensated broadeningfactor can also be obtained using a sample DCF with D DCF = − ps/nm/km and approximated propagation path of km [10]. The propagation path in the proposed module is shorterthan its counterpart DCF and hence it has a potential to providelower attenuation and delay during the compensation process.V. C ONCLUSION
In this paper, we took an opposite direction with respectto the conventional DCFs and showed an optical fiber withhigh positive dispersion coefficient can also be used fordispersion compensation. We proposed an optical structurebased on a well-known iterative algorithm in signal processingin which dispersion inverse transfer function is implementedusing high-positive dispersive optical fibers. We showed thatthe dispersion compensation capability of the proposed moduleis a trade-off between transmission length and bandwidth andis enhanced for the compensating structure including moredispersion compensating sub-systems. We also specified howsystem parameters should change to stabilize or speed-upthe system performance. Generally, the concepts and ideasbehind the proposed structure can be used in developing otherinnovative optical modules such as optical filters and otheroptical impairment compensating structures.R
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