Performance Analysis of Incoherent Multi- Wavelength Ocdma Systems under the Impact of Four-wave Mixing References and Links Performance Enhancement of the 2-d Wavelength-hopping/time-spreading Synchronous Ocdm System Using a Heterodyne Detection Receiver and Ppm Signaling, "

In this paper, we comprehensively analyze the impact of four wave mixing (FWM) on the performance of incoherent multi-wavelength optical code-division multiple-access (MW-OCDMA) systems. We also consider many other interferences and noises, including multiple access interference, optical beating interference, and receiver noise, in the analysis. From the numerical results, we can find the power ranges of different MW-OCDMA systems, in which the impact of FWM is dominant and consequently results in an increase in the bit-error rate of the systems. We also find that the impact of FWM becomes more severe when the frequency spacing is small and/or dispersion-shifted fiber is used. In addition, we quantitatively discuss the impact of FWM on the number of supportable users and power penalty in the MW-OCDMA systems. Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security, " J. Noise limits of optical spectral-encoding CDMA systems, " Electron. Performance enhancement of spectral-amplitude-coding optical CDMA using pulse-position modulation, " IEEE Trans. Modified quadratic congruence codes for fiber Bragg-grating-based spectral-amplitude-coding optical CDMA systems, " IEEE/OSA, " J. Spectral-amplitude-encoding optical-code-division-multiplexing system with a heterodyne detection receiver for broadband optical multiple-access networks, " J. Impact of GVD on the performance of 2-D WH/TS OCDMA systems using heterodyne detection receiver, " IEICE Trans. The impact of group velocity on frequency-hopping optical code division multiple access system, " J. Theoretical evaluation of error rate degradation due to fiber four-wave mixing in multichannel FSK heterodyne envelope detection transmission, " J. Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions, " J. Impact of four-wave mixing on 2-D optical code-division multiplexing systems " in Bragg grating based spectral encoder/decoder for lightwave CDMA, " Electron. Experimental demonstration of frequency-encoded optical CDMA using superimposed fiber Bragg gratings, " in Proceeding of OSA Top. Experimental study on time-spread/wavelength-hop optical code division multiplexing with group delay compensating en/decoder, " IEEE Photon. Characteristics of a single-mode optical fibre and cable, (ITU-T 2005). 25. Recommendation G.653, Characteristics of a dispersion-shifted single-mode optical fibre and cable, (ITU-T 2006). 26. Recommendation G.655, Characteristics of a non-zero dispersion-shifted single-mode optical fibre and cable, (ITU-T 2006).


Introduction
Optical communication systems have been playing an important role in telecommunication networks in general and access networks in particular.In order to make use of the vast bandwidth provided by optical fiber, multiple-access techniques have been proposed, such as time-division multiple-access (TDMA), wavelength-division multiple-access (WDMA), and optical code-division multiple-access (OCDMA).In the area of optical access networks, OCDMA has been regarded as a promising candidate thanks to its advantages over conventional techniques.First and the most important one is that OCDMA allows users to access the network asynchronously and simultaneously without strict timing synchronizations and wavelength managements as in the cases of TDMA and WDMA, respectively.In addition, OCDMA network can support a large number of users, which has a soft limit with a graceful degradation of performance with increasing number of users.Finally, OCDMA's ability of offering service differentiation at the physical layer and its inherent security are advantages that no other techniques can provide [1].
In OCDMA systems, transmitted signal can be encoded using either time or frequency domain.The disadvantage of time-domain encoding OCDMA is its spectral inefficiency.Incoherent multi-wavelength OCDMA (MW-OCDMA) is the class of OCDMA technique that uses the spectral amplitudes of carrier signal to encode data hence offers better spectral efficiency than that of the time-domain counterpart [2,3].Signal can also be additionally encoded in time domain (using two-dimensional (2-D) codes) so that even more users can be supported [3].Furthermore, MW-OCDMA is especially favorable in the access network environment, as it does not require an expensive coherent source [4].
For design and deployment of MW-OCDMA in access networks, it is necessary to investigate the effects of physical layer impairments on the performance of MW-OCDMAbased systems.Though there have been many works devoted to this endeavor, including system performance analysis considering various noise, interferences such as receiver noise, multiple access interference (MAI), optical beating interference (OBI), and dispersion [5][6][7][8][9][10][11][12], the impact of four wave mixing (FWM), a nonlinear effect related to multi-wavelength transmission, has not been considered.

FWM in MW-OCDMA systems
Four-wave mixing (FWM) is a nonlinear phenomenon originating from the third order nonlinear susceptibility of optical fiber.Whenever three waves, assuming with frequencies of f p , f q and f r , respectively (r ≠ p, q), co-propagate inside the fiber and are overlapped in time, they interact with each other through the third order nonlinear susceptibility of the optical fiber and generate a new wave at the frequency f i = f p + f qf r [13].
FWM is a well-studied issue in multi-wavelength optical networks, e.g.WDMA networks, with many studies devoted to the analysis of the impact of FWM on the system performance [13][14][15][16][17].It is seen that FWM results in performance degradation [14,15] and its impact becomes especially critical when the number of wavelengths is large and/or the launched power is high [16].

Desired signal
Transmitter side Considering Fig. 1, for a MW-OCDMA system using N frequencies, p, q, and r can vary from 1 to N resulting in a very large combination of new frequencies generated by FWM.In the case that existing frequencies are equally spaced, the new frequencies coincide with the existing ones, leading to crosstalk that causes incorrectness in the decoding process.Furthermore, the launched power at each frequency is relatively high because it is the total transmitted power from all users.FWM is therefore potentially a serious performance degradation factor in MW-OCDMA systems and hence should be well understood.
It is necessary to note that as each user in MW-OCDMA systems is allocated a set of multiple frequencies according to its signature code, the FWM power depends on not only conventional parameters such as the launched power and the number of frequencies, but also the parameters of signature code set.Combined with the decoding process at the receiving end, the impact of FWM on the performance of MW-OCDMA systems becomes more complex than that of wavelength-independent systems, such as WDM.

Main contributions
This paper, which is comprehensively extended from our previous works [18,19], presents the first quantitative analysis and detailed discussion on the FWM impairment in MW-OCDMA systems.In addition, many other interferences and noises including MAI, OBI, and receiver noise are also included in the analysis.The impact of FWM on various aspects of MW-OCDMA system performance such as bit-error rate (BER), number of users, and power penalty, is quantitatively discussed.To obtain the useful information for practical system design, we investigate the system performance with several types of optical fibers, including single mode fiber (SMF), dispersion-shifted fiber (DSF), and non-zero DSF (NZ-DSF).In addition, different values of frequency spacing are taken into account.
The rest of the paper is organized as follows.Section 2 presents the descriptions of the MW-OCDMA systems.FWM impact model is the content of Section 3. The performance analysis and numerical results are presented in Section 4 and 5, respectively.Finally, Section 6 concludes the paper.Figure 2 shows the schematic diagram of a MW-OCDMA system with K users.A set of a combiner and a splitter is used to combine the signals from all transmitters and broadcast them to all receivers.For illustrative purposes, one transmitter and one receiver are depicted in detail in this figure.

System descriptions
At the transmitter, an optical pulse coming from a broadband light source, whose spectrum is assumed flat, is first on-off keying (OOK) modulated by binary data.In OOK systems, an optical pulse is transmitted when a data bit is "1", otherwise no pulse is sent if the data bit is "0".The optical pulse corresponding to the data bit "1" is fed into a MW-OCDMA encoder, which can be implemented using the well-studied fiber Bragg gratings (FBGs) array structure [20,21].
In case of 1-D encoding, or spectral amplitude encoding (SAE), frequency components of optical pulse representing a bit "1" are selectively reflected at different FBGs, where each FBG is centered at a frequency (wavelength) defined by the corresponding signature code.More specifically, only frequency components corresponding to chip "1s" of the signature code can pass through the MW-OCDMA encoder.In case of 2-D encoding, reflected frequency components are additionally time-delayed in accordance with the second dimension of the signature code.In the FBG-based encoder, the time-delays are determined by the positions of FBGs in the array.The encoded signal in the case of 1-D and 2-D encoding are illustrated in Fig. 3(a) and Fig. 3(b), respectively.
The encoded signal is then combined with signals from other transmitters, transmitted over an optical fiber, and broadcasted to all receivers.When the multi-wavelength signal propagates along the optical fiber, FWM light is generated due to the interaction among the signals, which are overlapped in time but at different frequencies.
At the receiver, the received signal is first decoded by a MW-OCDMA decoder, which is also constructed by a FBGs array.In case of 1-D decoding, the desired signal from the targeted transmitter, whose frequencies are matched to FBGs tuned in the decoder, will be collected.The decoding process in case of 2-D also requires the frequency-matched pulses to be temporally coincided with the positions of the decoder's FBGs.For the receiver to decode signal from a specific transmitter, its FBGs are tuned to the same frequencies as those in the transmitter.Additionally in case of 2-D decoding, the positions of FBGs are in the inverse order of those in the transmitter [10].
The output of the decoder consists of an autocorrelation peak of height w (w is the code weight of the signature code) as shown in Fig. 3. MAI and FWM light, whose frequencies are coincided to desired ones, affect on the autocorrelation peak and may cause bit error.Decoded signal is then converted into photocurrent by a photodetector (PD).It is at the PD, OBI will occur due to the beating between signals with nearly the same frequency.Finally, the binary data is recovered by a threshold detector.

FWM impact model
Through FWM process, three waves of frequency f p , f q and f r (r ≠ p, q) interact and generate a new frequency f i = f p + f qf r .The power of FWM light at frequency f i can be expressed as [13] ( where d F is the degeneracy factor defined such that its value is 1 when p = q but 2 when p ≠ q.L is the transmission length, γ is the nonlinearity coefficient, and α is the attenuation coefficient of the optical fiber.P c is the transmitted power per chip pulse (i.e.chip "1"), which is assumed to be the same for all users and frequencies.n p , n q , and n r are the number of pulses appearing at the frequencies f p , f q , and f r in one chip period (1 ≤ p, q, r ≤ N). n p P c , n q P c and n r P c are therefore the launched power (i.e. the power at T-point in Fig. 2) at the frequencies f p , f q , and f r , respectively.η F is the FWM efficiency defined as [13] ( ) ( ) The FWM efficiency η F depends on the frequency spacing through the phase match governed by where β 2 is the group velocity dispersion parameter at ω c = (ω p + ω q )/2 (ω p,q,r = 2πf p,q,r ).As mentioned in Section 1.1, when p, q, and r vary from 1 to N, there are different combinations of f p , f q , and f r that generate the frequencies coinciding with the existing ones.For example, in a system with four equally spaced frequencies (N = 4; f 1 , f 2 , f 3 and f 4 ), the number of combinations generating FWM light at the frequency f 1 or f 4 is three, while it is five at the frequency f 2 or f 3 , as shown in Table 1.
Table 1.Frequency combinations caused by FWM in a 4-equally-spaced frequency system (N = 4) Frequency combination Denote P i F as the total power of FWM light generated at frequency f i , it can be calculated as , where the summation term in Eq. ( 4) contains all combinations satisfying the condition f p + f q f r = f i .The power of each combination (P pqr ) is obtained from Eq. (1).

Performance analysis
In this section, we will theoretically analyze the performance of MW-OCDMA systems based on the FWM impact model discussed in Section 3. We consider both types of MW-OCDMA systems, including 1-D encoding (SAE) [2] and 2-D encoding OCDMA systems, in which transmitted signals are encoded using both frequency and time domains [3].Besides the impact of FWM, we also take into account MAI, OBI, and receiver noise.In order to focus on the impact of FWM, the group velocity dispersion (GVD), self-phase mixing, and cross-phase mixing on the system performance is neglected in the analysis.The discussion on the impact of GVD on 2-D OCDMA systems can be found in [11], and as a matter of fact, GVD can be effectively compensated by using DSF or GVD compensation techniques [12,22].
In the following analysis, we assume that the bit and chip streams from all users are to be synchronized, i.e., the worst performance since the FWM-induced power is the strongest in this case.The binary channel is also assumed equally likely, i.e. the probabilities that a transmitter sends bit "1" and bit "0" are the same and equal to 1/2.In addition, we denote w as the number of frequencies assigned to each user, which is also the code weight of the signature code.Without loss of generality, the w frequencies for targeted transmitter (desired frequencies) are denoted from 1 to w.The remaining ones (i.e., Nw frequencies) are considered as undesired frequencies and denoted from w + 1 to N.

BER in SAE/OCDMA systems
In SAE/OCDMA systems, a receiver structure with two decoders connecting in a balance fashion, as shown in the Fig. 4, is employed so that MAI can be completely removed [2].The desired frequencies are collected by the direct decoder while the undesired ones will pass through the complementary decoder.To completely cancel MAI, a ratio between the optical powers that arrive at the two photodetectors is set to ξ = λ/(wλ), where λ is the crosscorrelation value of the signature code [9].When a transmitter sends bit "1", its w pulses are visible at w frequencies, i.e., one at each frequency.The probability that a pulse appears at a specific frequency can be expressed as p m = w/N.Consider a whole system with K users, the total number of pulses at a specific frequency (n p , n q , or n r ) can be modeled as a binomial random variable with a probability p m /2 (the probability of sending bit "1" is 1/2).According to the property of the signature code set, when the number of users is large enough, optical pulses from all transmitters are equally distributed over N frequencies.As a result, we can estimate n p , n q , and n r using averaging approach and the product of n p , n q , and n r in the calculation of FWM light [Eq.( 1)] can be replaced with (K p m /2) 3 .
As the impact of GVD is ignored, MAI current is completely eliminated in SAE/OCDMA systems [3].The total signal current at the output of the balanced detector considering FWM impact therefore can be derived as ( ) where b = 1 (or 0) when the targeted transmitter sends bit "1" (or bit "0").ℜ is denoted as the PD responsivity and P s is the received power per chip pulse.The relation between P s and P c is P s = [P c exp(-αL)]/K.Note that the FWM-induced current is contributed from FWM light at both w desired frequencies and (Nw) undesired ones.
Besides the OBI currents caused by the beating between desired and interfering pulses, and the beating among interfering pulses [7], when FWM is taken into consideration, additional OBI currents are generated.These OBI currents are caused by the beating between FWM light and desired, interfering pulses.The total variance of OBIs therefore can be calculated as where B e is the PD electrical bandwidth.τ c is the coherent time of the broadband source, which can be approximated as τ c = 1/B 0 , where B 0 is the optical bandwidth.k i is the number of interfering pulses at the frequency f i .The interfering pulses are from interfering users sending bit "1" denoted as k.The event k out of possible K -1 interfering users sending bit "1" can be modeled as a binomial distribution with probability 1/2.When k is large enough, similarly, interfering pulses are equally distributed over N frequencies.As a result, the total variance of OBI can be calculated by replacing k i in Eq. ( 6) with its average value 〈k i 〉 = 0.5(K-1)w/N.The receiver noise includes the contribution from both shot noise i s 2 = 2eB e I b and thermal noise i th , where e is the electron charge, k B is Boltzman's constant, T n is the receiver noise temperature, and C is the receiver capacitance.The total noise variance, i nb 2 (b ∈{0,1}), is expressed as follows 2 .

SAE nb
OBIb sb th i i i i = + + (7) Each bit is detected by comparing the total signal current with a threshold current I D .The photocurrent as well as noises and interferences can be modeled as Gaussian random variables and the total probability of error can be calculated as where I 1 and I 0 are the total signal current (I b , b = 1 or 0) obtained from Eq. ( 5).Also, i n1 and i n0 are i nb with b = 1 and b = 0. Q function is written by To minimize the BER, we consider the case that I D is optimum [13], I D = (i n0 I 1 + i n1 I 0 )/(i n0 + i n1 ).The total probability of error hence can be expressed as

BER in 2-D OCDMA systems
We now extend our discussion to 2-D OCDMA systems, whose 2-D signature codes consists of a time-spreading pattern and a wavelength-hopping pattern.Denote β as the maximum number of overlapping pulses between the time-spreading pattern of an interfering user and the desired one's, the probability that one interfering pulse belongs to a desired chip period (out of w ones) is β /w.Furthermore, this interfering pulse can be visible at one of N frequencies.As a result, the probability that one interfering pulse appears at one specific frequency in one specific chip period is calculated as p m = β /(wN).Using the same approach with SAE/OCDMA systems, we can estimate the FWM light in 2-D OCDMA systems.In 2-D OCDMA systems, MAI current will appear in the total signal current.Denote k i as the number of MAI pulses at frequency f i , the total signal current hence can be expressed as As only signal at w desired frequencies can pass through the decoder, the total variance of OBIs is thus derived as To calculate the total signal current and OBI, the number of MAI pulses at each frequency (i.e., k i ) needs to be obtained.Defining p MAI as the probability that one pulse from an interfering user matches with a desired one in both time and frequency (i.e., MAI pulse), p MAI depends on the signature code set and will be mentioned in detail in the numerical result section.The event that there are j MAI pulses generated from k interfering users sending bit "1" can also be modeled as a binomial distribution with probability p MAI .When the number of interfering users is large enough, MAI pulses can be assumed to be equally distributed over w desired frequencies, k i hence can be replaced by its average value 〈k i 〉 = j/w.
The receiver noise as well as the total noise variance is calculated in a similar way with SAE/OCDMA system.As a result, the total probability of error can be calculated as [7] ( ) ( )

Numerical results
In our analysis, the N frequencies used in the system are centered at 193.4 GHz, which corresponds to the wavelength 1550.12 nm.The frequency spacing is chosen as 25 GHz, 50 GHz, and 100 GHz in accordance with standard ITU for DWDM [23].As the OCDMA is proposed for broadband optical access networks, we are interested in the system that can support at least 32 users with bit rate of 1Gbit/s per user.Other system parameters and constants used in the analysis are shown in Table 2.In addition, three types of optical fiber are considered in this analysis including normal SMF [24], DSF [25], and NZ-DSF [26] with attenuation coefficient α = 0.2 dB/km at wavelength 1550 nm.The dispersion coefficient D, the dispersion slope coefficient S, and the nonlinear parameter γ of optical fibers are shown in Table 3.  [2,6,8].Of these codes, MQC offers the best performance when OBI is considered [8].In fact, it is not practical to use msequence, Hadamard codes in the considered system, which is expected to support at least 32 users with bit rate per user of 1 Gbit/s [9].In the following discussion, we therefore consider only SAE/OCDMA systems using MQC code.MQC codes are defined by an odd prime number p.The code weight and the code length are w = p + 1 and N = p 2 + p, respectively.The cross-correlation is one and the cardinality of a MQC code set is p 2 .For the system to support 32 × 1 Gbit/s users with adequate BER, MQC code with p = 11 is used, equivalently we have (N, w, λ) = (132, 12, 1).
We first investigate the BER versus transmitted power per chip P c when K = 32 users, the transmission length L = 30 km. Figure 5 shows that the transmitted power range can be divided into two regions: receiver noise-dominant and FWM-dominant regions.In the first one, the BER decreases when P c is increased.However, reverse phenomenon occurs in the second region.This is because FWM power strongly increases when P c is large enough and its impact becomes dominant compared with the receiver noise.At the threshold power, i.e., the boundary between two regions, BER reaches the smallest value.For example, the threshold power is −12 dBm for the case of using SMF.
The influence of frequency spacing is further investigated in Fig. 7 when the power penalty versus transmitted power per chip is shown.The number of active users is 32, the transmission length is also 30 km.Similarly, the increase of frequency spacing helps to reduce the power penalty.Additionally, it is seen that the power penalty due to the impact of FWM is significantly high when the optical power per chip is high, especially for the case of narrow frequency spacing, for example, 9.6 dB for the case that frequency spacing is 25 GHz at P c = −8 dBm.
Finally, in Fig. 8, we investigate power penalty versus the transmitted power per chip when K = 32 and 40 users.It is found that the higher K results in the higher power penalty under the impact of FWM.This is because, when there are more users, the total launch power at a wavelength becomes higher, which consequently results in higher FMW.

2-D OCDMA systems
In this paper, we consider the 2-D OCDMA systems using 2-D signature codes constructed by the popular prime sequences.The 2-D signature code set based on prime sequences is characterized by two prime numbers, p s and p h (p h ≥ p s ), which are used, respectively, for generating time-spreading and wavelength-hopping patterns.This 2-D signature code has weight of p s (i.e., w = p s ); and p h is the number of available frequencies in the system (i.e., N = p h ).The cardinality of the 2-D prime code set is p s × (p h -1).The details about structure and properties of 2-D signature codes can be found in [4].When this 2-D signature codes are used, the probability p MAI in corresponding to a pair of p s and p h then can be expressed as p MAI = µ λ /p s 2 [7], where is the average number of wavelengths common to a pair of two prime signature code.
We are interested in finding the operational range of transmitted power per chip.Due to the nature of the 2-D signature code based on prime sequence, the minimum and maximum operation ranges can be found in two special cases: (1) all users have the same time spreading pattern, and (2) all users have different time spreading patterns.Accordingly, we have β = p s for the first case, and β = 2 for the second one [4].Figure 9 shows BER versus the transmitted power per chip when p s = 29 and p h = 31.The frequency spacing is 25 GHz and the transmission length is 60 km.The number of active users is 30.The operational range of the transmitted power per chip is the most narrow when all users use the same time-spreading pattern and it is about 15.7 dB.On the other hand, this range is extended in the case that different patterns are used.
Finally, we assume all users use different time-spreading patterns and investigate the power penalty when the number of active users increases.Several values of the transmitted power per chip, P c = −6, −4, and −2 dBm, are used with the frequency spacing is 25 GHz and the transmission length is 60 km. Figure 10 shows that the power penalty increases in proportion with the number of users, especially when the transmitted power per chip is high.Specifically, the power penalty is almost 0.5 dB for every ten-user more when the transmitted power per chip is −2 dBm.

Conclusion
We have theoretically analyzed the impact of FWM on the performance of MW-OCDMA systems considering many other noises and interferences.We presented a quantitative discussion about the impact of FWM on various aspects of MW-OCDMA system performance, including the BER, the number of users, and the power penalty.We also investigated the system performance with several types of optical fibers such as SMF, DSF, and NZ-DSF taking account different values of frequency spacing.The numerical results show that FWM causes an increase in the BER of the systems, especially when transmitted power is high, frequency spacing is small, or DSF is used.FWM also limit the operational power range of the system.For example, it is only 6 dB in the case of SAE/OCDMA system using MQC code of (132, 12, 1).In addition, the impact of FWM is quantified by power penalty and investigated with different number of users to provide useful information for practical system design.

Fig. 9 .
Fig. 9. BER versus the transmitted power per chip in the case of using SMF when K = 32 users, L = 60 km, and ∆f = 25 GHz.

Fig. 10 .
Fig. 10.Power penalty versus the number of users in the case of using SMF when L = 60 km and ∆f = 25 GHz.

Table 3 . Optical fiber parameters
Several signature code sets can be used in SAE/OCDMA systems, including m-sequence, Hadamard, and modified quadratic congruence (MQC) code