A Theoretical Approach to analyze impact of FWM on DWDM Systems

Abstract

Increase in the data rate of fiber optic communication system is limited due to the nonlinear effects like Self phase modulation (SPM) Cross phase modulation (CPM) Four wave mixing ( FWM ) Stimulated Raman Scattering(SRS) and Stimulated Brillion scattering (SBS). Many investigations are carried to mitigate these effects and in a Light wave Multiplexed systems the dominating effect is Four wave mixing(FWM). In multi-channel systems, four-wave mixing (FWM) in optical fibers [l] induces channel crosstalk and possibly degrades system performance.When signal wavelengths are positioned around the zero-dispersion wavelength of the transmission fibers, the phase-matching condition is nearly satisfied and the FWM light is efficiently generated. Especially in multi-repeater systems using optical in-line amplifiers, the generated power of FWM light is accumulated during the signal transmission [2].In this paper an analytical model has been developed and Four wave mixing power has been derived in the first part of this paper. Using this expression probability function of FWM noise is simulated and the performance of the crosstalk and channel spacing is plotted.

Keywords:Four wave mixing (FWM),.Crosstalk, channel spacing.

INTRODUCTION

The performance of WDM networks are strongly influenced by both linear and nonlinear phenomena that determines the signal propagation inside the fiber. However linear propagation effects are compensated and there is a class of nonlinear effects that pose additional limitations in DWDM systems. Both CPM and FWM cause interference between channels of different wavelengths resulting in an upper power limit for each WDM channel .The most severe problems are imposed by Four-wave mixing (FWM), also known as four-photon mixing, is a parametric interaction among optical waves, which is analogous to inter modulation distortion in electrical systems. In a multi-channel system, the beating between two or more channels causes generation of one or more new frequencies at the expense of power depletion of the original channels. When three waves at frequencies f_P, f_q,and f_r are put into a fiber, new frequency components are generated at f_FWM= f_P+f_q– f_r. In a simpler case where two continuous waves (cw) at the frequencies f₁ and f₂ are put into the fiber, the generation of side bands due to FWM is illustrated in Fig 1 The number of side bands due to FWM increases geometrically, and is given by

M= (1)

where N is the number of channels and M is the number of newly generated sidebands For example, eight channels can produce 224 side bands. Since these mixing products can fall directly on signal channels, proper FWM suppression is required to avoid significant interference between signal channels and FWM frequency components. The power of FWM product is inversely proportional to the square of the channel spacing .when all the channels have the same input power FWM efficiency is give by

(2)

Where A_eff is the effective area of the fiber and D is the dispersion parameter . section 1.Discusses the analytical treatment of FWM expression in the multiplexed systems.section2. briefs about cross talk performance, section 3.gives FWM noise distribution, section 4 presents the discussion and section 5 presents the summary of the paper.

Fig 1. Four wave mixing effect

ANALYTICAL TREATMENT

In this section, an analytical expression is derived for fiber FWM in multi-amplifier systems with non uniform chromatic dispersion. The system model considered here is illustrated in Fig.2. The total transmission line consists of M sections and ( M - 1) in-line amplifiers. The repeater span is equal throughout the transmission. The gain of each amplifier is adjusted to compensate the transmission loss of the section just before the amplifier. Thus, the signal power output from an inline amplifier, i.e., fiber input power, is equal for each section. One section consists of N fiber lengths with different zero dispersion wavelength, which is assumed to be uniform along one length. The length of each fiber is equal. For this system configuration, FWM light at f_F = f_P+ f_q– f_r is evaluated, where. f_{P ,} f_{q ,} f_r are the light frequencies of transmitted signals. Since the main concern here is to investigate the effect of the non uniformity of chromatic dispersion, the polarization states of each light are assumed to be matched throughout the transmission and the effect of polarization mode dispersion is ignored for simplicity. In actual systems, the FWM influence is relieved by this effect, compared with results In our model we assume M denotes number of sections., N number of fibers in one section section, L Length of one fiber, L _oLength of one section. α fiber loss coefficient, κ constant representing FWM efficiency at the phase-matched condition. E_P^(mn) light field at the beginning of fiber n in section m.E_q^(mn)light field at the beginning of fiber n in section m.E_r^(mn)light field at the beginning of fiber n in section m E_F^(mn)FWM light generated in Fiber n in section m.β_p^(mn)Propagation constant for f_p, in fiber n in section m .β_q^(mn)Propagation constant for f_q, in Fiber n in section m.β_r^(mn) Propagation constantfor f_r in Fiber n in section m.β_F(mn)Propagation constant for FWM light in Fiber.

Δβ^(mn)=β_p ^(mn)+ β_q^(mn)- β_r^(mn) - β_F^(mn) (3)

= - ( π⁴/ c ²) (d D_c/d ) { f _p– f ₀^(mn) + f _q– f ₀^(mn) }

x ( f _P - f_r) ( f _q - f _r)

f _o^(mn)zero dispersion frequency of fiber n in section m .

_N _{N N}

Φ_P^(m)=Σ β_P^(mn) L_{0 ;}Φ_q⁽^m)=Σ β_q^(mn)L_{o ;}Φr⁽^m) =Σβ_r^(mn)L_{o ;}

^{n-1
n-1 n-1}

These expressions are the propagation phase of f_{P ,}f _q and f_r through section m . and Φ_F^(m)= Σ β_F^(mn) L₀is the propagation phase of FWM light.

ΔΦ^(m)= Φ_P^(m)+ Φ _q^(m) - Φ _r^(m) - Φ _F^(m) (4)

Fig 2.Analytical model

In order to derive an analytical expression, we consider that FWM light at the end of the transmission line is composed of FWM lights that are generated in each fiber length and linearly propagate through the remaining part of the system. Based on this model, we derive an expression for each FWM light and sum them at the end of the transmission line. First, FWM light generated in Fiber n in Section m is evaluated. At the end of this fiber piece, that FWM light is

E _F^{(mn )}=κE_P^(mn) E _q^(mn)E_r^(mn) exp( -α / 2)+ i β_F^(mn)) L_o)

* (1-exp( -α + i Δβ^(mn) L₀ )

(5)

With κ = i (2π)² D χ / n₀λ) where n₀ is the refractive index λ is the wavelength ,c is the velocity of the light, d is the degeneracy factor of FWM and χ is the third order nonlinear susceptibility.

_n-1

E_P^(mn)= E_p^(m
1)exp( Σ ( – α / 2) + iβ_p^(mj)L_o)

^{j =1}

_m-1_n-1

E_P^{(1 1)} exp (i Σ Φ_P^(k) + Σ (-α /2) + I β_P^(m
j)) L_o

^{k=1
j=1}

^- (6)

_m-1_n-1

E_q^(mn)₌ E_q^(1
1) exp (i Σ Φ_q^(k) + Σ (-α /2) + I β_q^{(m j)}) L_o

^{k=1 j=1}

(7)

_m-1_n-1

E_r^(mn) = E_r^{(1 1)} exp (i Σ Φ_r^(k) + Σ (-α /2) + I β_r^{(m j)}) L_o

^{k=1 j=1}

(8)

Substituting (6) (7) & (8) in Eqn (5) we get

_m-1

E_F^(mn)= κ E_p^(1
1)E_q^{(1 1)}E_r^{(1 1)} exp[ ( i Σ (Φ_F^{( k)} +ΔΦ^(k))]

^k=1

*exp ( Σ [ ( - α / 2 ) +i β_F^{( m j)}) L₀]

^j=1

_n _-1

*exp ( Σ [ ( - α) + iΔβ^{( m j)}) L₀]

^j=1

*1 - exp [( - α) + iΔβ^{( m n)}) L₀] / (( - α) –iΔβ^{( mn)}

( 9 )

Equation (9) describes the FWM light at the end of fiber n in section m. This light propagates linearly and reaches the end of section m

E_F^(mn)(end of #m )= E_F^(mn) exp( Σ [(-α/2)+iβ_F^(mj))L₀]

^{j = n+1}

(10)

Further more it propagates to the end of the whole transmission line as

E_F^(mn) (end) = E_F^(mn) (end of #m) exp[ i ( Σ Φ_F^(k)]

^k=m+1

_m-1

= K exp[iΣ ΔΦ^(k)

^K=1

_n-1

= exp[ Σ (-α + iβ^(mj)L_o]

^j=1

*( 1-exp[( -α + iΔβ^(mn) L_o)]

_(k)

With K = κ E _p⁽¹¹⁾E_q⁽¹¹⁾ E_r^{(11) * e x p [ ( - α L / 2 ) + I
Σ Φ}_F^)]

(11)

Where (9 and (10 ) are substituted .

The total FWM light at the end of the transmission line is obtained by summing for all lengths and is written as

_{M N
M m-1 N}

E_F^(total) = Σ Σ E _F^(mn)_(end) =KΣ exp [ i Σ ΔΦ ^(k) ] Σ

^m=1 ⁿ⁼¹ ^{m=1 k= 1 n=1}

_n-1

* exp[ ( Σ (-α + iΔβ^(mn)L_o]

^j=1

*(1– exp[(-α + iΔβ^(mn)L₀] / (α – iΔβ^(mn)

(12)

This equation expresses FWM light amplitude .

CROSS TALK PERFORMANCE

FWM cross talk can be examined by the multiplexing channels and its spacing ., which can be estimated using the Power expression for the FWM.

P_FWM= [1024 π⁶ ( D χ )²/ ( η₀⁴λ²c ² )] *(P_PP _qP_r) / A²_eff

_{M m-1
‌‌}‌‌‌_{N n-1}

e^-^α^LΣ exp[ iΣ ΔΦ^(k)] * Σ exp [ Σ )(-α + iΔβ^(mj)L_o]

^{m=1 k=1
n=1 j=1}

* (1- exp (-α + iΔβ^(mn)L₀) / α –iΔβ^{(mn) 2}

(13)

The parameters taken in the calculations are L₀ = 10 Kms the number of spans N=50 ,repeater span L= 100 Kms fibe loss=0.2 db/km, the value of the Dispersion slope=0.7ps/km.nm²,third order nonlinear susceptibility χ=4x10^-8esu.zero dispersion wavelength=1550nm.From these observations on substituting into equation (13) If we propagate this signal through a fiber span the FWM tones will be generated [6]. The Pdf of the FWM noise can be approximated with the symmetric double sided exponential distribution is proved in accurate. The output power of the FWM is given by

(14)

where P_i(i = p,q,r)represents the input peak power at frequencies f_i.is nonlinear coefficient of the fiber is the fiber loss coefficient ,L is the total fiber length ,d_pqr is the degeneracy factor where d_{p q r}=3 when p=q and d_{p q r}= 6 when p not equal to q and the propagation of signal through the cascade of steps has been studied. Since signal (10) is a comb of CW carriers, its propagation through both the linear and nonlinear operators has an analytical solution. Therefore, we were able to theoretically obtain the resulting signal at the output of a fiber span, propagated through the numerical SSM.we get the FWM efficiency altered by the SSM algorithm .In this paper an accurate statistical description of FWM noise is considered and its influence on system performance is investigated[7]. To compute the error probability it is necessary to calculate probability density function of the received current where the sample space is the set of all the possible bit pattern in all the channels The properties of FWM noise are investigated by use of Random noise experiments.

FWM NOISE

At the receiver the photocurrent is proportional to the optical power. Assuming a single fiber span without optical amplification, all other noises at the receiver except for FWM can be ignored .In practice to evaluate the performance of the system it is useful to have a relation between the bit error rate and the input power. For calculations of the BER, the optimum threshold that minimize the error probability was chosen Hence for low input powers the distribution of mark and space state will not overlap which implies error free transmission. However in practical systems the existence of other noises will force the two distributions to overlap preventing the BER from becoming zero The probability density function of the received current can be computed The optical phases of all the channels are assumed to be uniformly distributed with in 0 to 2pi. Because of phase noise [5]and the data bits are assumed to be in the mark and space states with equal probability P(Bi=0)=P(Bi=1)=1/2.Hence the statistics of the receiver current will depend on the total number of channels N and as power of the FWM noise is proportional to the mixing efficiency it is expected that noise power will increase and performance of the system will degrade .similar behavior is observed when the number of channel is reduced In Fig 3.to study the FWM components the frequency allotted are as follows

[193.0,193.1,193.2,193.3]THZ,Freq spacing=100 GHZ

Crosstalk components=36.Out of 36 crosstalk components .The four wave mixing noise components in the Center channel is five in number.

Fig 3. crosstalk performance of FWM noise.

A main concern in system design is the allowable fiber input power determined by fiber FWM rather than the crosstalk value[8]. How to use crosstalk distribution to evaluate allowable input power depends on one's viewpoint in designing systems. If we take a position that any system should successfully work for any situation, the crosstalk value for uniform dispersion should be adopted for evaluating allowable input power. In this case, a severe limitation is imposed on system design due to fiber FWM. On the other hand, if we allow a certain percentage of failure, that limitation is relieved[4]. This issue is addressed below. Error rate degradation due to fiber FWM is dependent on modulation demodulation schemes.

Fig 4.Power per channel versus FWM noise

Carrying out this procedure for various system parameters, we can find system conditions satisfying a given value of failure probability .Fig 4. illustrates the power per channel in milli watts and FWM power products. The parameters taken for simulation is as follows Fiber length=17.5 km, Nonlinear refractive index=2.68e-20,core effective area =50 e -6m2,Freq=[193.0,193.1,193.2,193.3]THZ and the Power per channel =5 mwatts .Fig 5 depicts the number of worst FWM components in the particular channel considered. In Fig 6. Crosstalk as a function of channel spacing is shown .Assumed system conditions are 10 channels, 1000 km transmission distance, and -18 dBm/ch fiber input power. The phase-matching is nearly satisfied for a narrow channel spacing .

RESULTS AND DISCUSSION

An analytical expression for Fiber FWM in multi-channel multi-repeater systems operated in the zero-dispersion wavelength region was studied. The transmission line was composed of short fiber lengths with different zero-dispersion. crosstalk was calculated for various combinations of fiber lengths that were selected according to a random function. From these calculations, crosstalk distribution was obtained and allowable fiber input power was evaluated. When the channel spacing is narrows there is more crosstalk power and the increase in the channel spacing will cause constant crosstalk power.

Fig 5.Number of channels versus FWM components

on worst affected channels .

Fig 6 crosstalk as a function of channel spacing with

Zero dispersion wavelength.

CONCLUSION

The performance of crosstalk power and the channel spacing was estimated for the four wave mixing power derived analytically .It as found that unequal channel spacing reduces the crosstalk power .

REFERENCES

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[4] F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “WDM system with unequally spaced channels,” J. Lightwave Technol., vol. 13, pp. 889–897, May 1995.

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[6] Jose Ewerton P. de Farias and Seemant Teotia,”four wave mixing reduction is a long haul amplified multi channel optical transmission system.”, SBMOlJEEE MlT-S IMOC'97 proceeding.

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[8] G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S.Benedetto, “Suppression of spurious tones induced by the split-stepmethod in fiber systems simulation,” IEEE Photon. Technol. Lett., vol.12, pp.

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