High-quality amplitude jitter reduction and extinction enhancement ...

High-quality amplitude jitter reduction and extinction enhancement using a powersymmetric NOLM and a polarizer O. Pottiez1, B. Ibarra-Escamilla2 and E. A. Kuzin2 1

Centro de Investigaciones en Óptica, Loma del Bosque 115 Col. Lomas del Campestre, León Gto 37150, Mexico [email protected] 2

Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Departamento de Óptica, Puebla Pue 72000, Mexico

Abstract: Four procedures for simultaneous high-quality amplitude jitter reduction and extinction ratio enhancement of optical data streams are presented and studied using numerical simulations. They all rely on the use of a power-balanced NOLM, optionally followed by a polarizer. The setup can be operated in various regimes, leading to several switching characteristics with different merits in the frame of the proposed application. These are discussed and compared with the results obtained using other NOLM configurations. ©2007 Optical Society of America OCIS codes: (230.4320) Nonlinear optical devices; (060.4370) Nonlinear optics, fibers; (230.5440) Polarization-sensitive devices; (060.2330) Fiber optics communications.

References and links 1. 2. 3.

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S. Yamashita and M. Shahed, “Optical 2R regeneration using cascaded fiber four-wave mixing with suppressed spectral spread,” IEEE Photon. Technol. Lett. 18, 1064-1066 (2006). E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142-144 (2001). S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957-959 (2003). P. V. Mamyshev, “All-optical data regeneration based on self-phase modulation effect,” in Proceedings of European Conference on Optical Communications (1998), pp. 475-476. T.-H. Her, G. Raybon, and C. Headley, “Optimization of pulse regeneration at 40 Gb/s based on spectral filtering of self-phase modulation in fiber,” IEEE Photon. Technol. Lett. 16, 200-202 (2004). M. Matsumoto, “Performance analysis and comparison of optical 3R regenerators utilizing self-phase modulation in fibers,” J. Lightwave Technol. 22, 1472-1482 (2004). M. Matsumoto, “Efficient all-optical 2R regeneration using self-phase modulation in bidirectional fiber configuration,” Opt. Express 14, 11018-11023 (2006). N. J. Doran and D. Wood, “Nonlinear optical loop mirror,” Opt. Lett. 13, 56-58 (1988). M. Attygalle, A. Nirmalathas, and H. F. Liu, “Novel technique for reduction of amplitude modulation of pulse trains generated by subharmonic synchronous mode-locked laser,” IEEE Photon. Technol. Lett. 14, 543-545 (2002). M. D. Pelusi, Y. Matsui, and A. Suzuki, “Pedestal suppression from compressed femtosecond pulses using a nonlinear fiber loop mirror,” IEEE J. Quantum Electron. 35, 867-874 (1999). M. Meissner, R. Rösch, B. Schmauss, and G. Leuchs, “12 dB of noise reduction by a NOLM-based 2-R regenerator,” IEEE Photon. Technol. Lett. 15, 1297-1299 (2003). O. Pottiez, E. A. Kuzin, B. Ibarra-Escamilla, F. Gutiérrez-Zainos, U. Ruiz-Corona, and J. T. CamasAnzueto, “High-order amplitude regularization of an optical pulse train using a power-symmetric NOLM with adjustable contrast,” IEEE Photon. Technol. Lett. 17, 154-156 (2005). A. Bogoni, P. Ghelfi, M. Scaffardi, and L. Potì, “All-optical regeneration and demultiplexing for 160-Gb/s transmission systems using a NOLM-based three-stage scheme,” IEEE J. Sel. Top. Quantum Electron. 10, 192-196 (2004). E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, “Theory of nonlinear loop mirrors with twisted low-birefringence fiber,” J. Opt. Soc. Am. B 18, 919-925 (2001).

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Received 30 August 2006; revised 23 January 2007; accepted 29 January 2007

5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2564

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O. Pottiez, E. A. Kuzin, B. Ibarra-Escamilla, and F. Méndez Martínez, “Theoretical investigation of the NOLM with highly twisted fibre and a λ/4 birefringence bias,” Opt. Commun. 254, 152-167 (2005). A. Striegler and B. Schmauss, ‘‘Extinction ratio improvement by an advanced NOLM setup,’’ IEEE Photon. Technol. Lett. 18, 1058-1060 (2006). P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing,” Opt. Lett. 24, 1454-1456 (1999). T. Sakamoto and K. Kikuchi, “Nonlinear optical loop mirror with an optical bias controller for achieving full-swing operation of gate switching,” IEEE Photon. Technol. Lett. 16, 545-547 (2004). I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, “A receiver model for optical fiber communication systems with arbitrarily polarized noise,” J. Lightwave Technol. 23, 14781490 (2005). A. Bogoni, L. Potì, R. Proietti, G. Meloni, F. Ponzini, and P. Ghelfi, ‘‘Regenerative and reconfigurable alloptical logic gates for ultra-fast applications,’’ Electron. Lett. 41, 435-436 (2005). S. Bigo, O. Leclerc, and E. Desurvire, “All-optical fiber signal processing and regeneration for soliton communications,” IEEE J. Sel. Top. Quantum Electron. 3, 1208-1223 (1997).

1. Introduction In order to ensure high capacity and long transmission distances in future ultrahigh-speed communication systems, high-quality all-optical signal regeneration techniques will be required. Of particular importance are the improvement of signal-to-noise ratio (SNR, reduction of amplitude jitter on logical ones) and of extinction ratio (ER, compression of noise on the logical zeros, reduction of ghost pulses). Two fiber-based techniques allowing simultaneous SNR and ER enhancement recently emerged in the literature, one based on cascaded four-wave mixing in fiber [1-3], and the other relying on self-phase modulation followed by offset filtering [4-6]. The first technique allows realizing near-ideal step-like transfer functions, at the price however of a rather complicated setup, including one (or multiple) intense continuous-wave pump, and a phase or frequency modulation scheme for both signal and pump in order to suppress stimulated Brillouin scattering. Another inconvenient of the technique is that the wavelength of the regenerated signal differs from the initial signal wavelength. The latter technique also allows high-quality regeneration, and usually leads to simpler regenerator structures, as no pump source is needed. However it also imposes a wavelength shift to the regenerated signal. Two stages are needed for wavelengthshift-free regeneration, which complicates the setup [7]. An alternative approach consists in the use of a fiber Nonlinear Optical Loop Mirror (NOLM) [8], due to its simplicity, flexibility and ultrafast switching operation. This device typically exhibits a sinusoidal nonlinear transmission characteristic T(Pin) = Pout/Pin. Regions were the slope of the output power characteristic Pout = TPin is smaller than 1 in modules (ideally zero) can be used to reduce amplitude noise on the logical ones [9], whereas low transmission at low power is needed to enhance the ER [10]. Large amplitude fluctuations were suppressed using a strongly power imbalanced setup, in which a 0.9/0.1 coupler is employed, as for this particular coupling ratio a large plateau appears in the Pout characteristic [11]. We showed that a similar characteristic could be obtained using a power-symmetric architecture [12]. In both cases however, low-power transmission is high, yielding poor ER improvement. On the other hand, a NOLM designed for ER improvement is not proper for improving poor SNR, as the plateau is short [10]. In general, a NOLM-based setup for signal regeneration includes several devices [13]. In this Paper, we show through numerical simulations that it is possible to enhance both SNR and ER of a degraded data stream using a single NOLM. Four regimes are considered, referred to as A, B, C and D. 2. Description of the device operation The proposed device is shown in Fig. 1. It consists of a symmetric coupler, a piece of highly twisted fiber and a quarter wave retarder (QW) inserted after one of the coupler output ports. Although the power is balanced between the two arms, the QW and fiber twist allow the device to switch through nonlinear polarization rotation [14]. The transmission characteristic of this device strongly depends on the input polarization. Zero transmission at low power is #74572 - $15.00 USD

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ensured by setting the QW angle at αc = Ω/2 + kπ/2, where k is an integer (for simplicity and without loss of generality we consider here k even), and Ω = ρL is the total optical activity of the fiber, corresponding to about 5% of the total twist in silica fiber (ρ is the rotatory power and L is the fiber length) [15]. An optional polarizer P is also represented at the NOLM output. y

Twist

αc

QW

50/50

y

ψ −αc

y

x

ψout

z

Pin

χ αc

αc

z x

y

x

x

Pout

Polarizer P

NOLM

Fig. 1. Proposed setup. Input and output polarizations angles, ψ and ψ out, as well as the polarizer orientation χ are defined with respect to the QW orientation αc.

2.1 Regime A For the first regime (A), we consider the NOLM without output polarizer, and a linear input polarization (1st Stokes parameter As = 0) making an angle ψ = π/4 with the QW axes. This particular orientation yields the smallest switching power Pπ = 4π/βL that can be obtained from a power-balanced NOLM with fiber length L and nonlinear coefficient β [14, 15]. As an example, for 1 km of high nonlinearity fiber (β = 10 rad/W/km), Pπ = 1.26 W. In this case, the transmission T is sinusoidal, and Pout = TPin only exhibits a short 1st-order maximum (dnPout/dPinn ≠ 0 for n > 1) between two consecutive zeros (red curves in Fig. 2). However we showed previously [15] that, if |As| is increased (i.e., input polarization is made elliptic, while maintaining ψ = π/4), in the high-twist limit, the transmission writes as: ⎛

T = 12 − 12 cos⎜⎜ πAs ⎝

Pin ⎞ ⎛ 2 P ⎟ cos⎜ π 1 − As in ⎟ ⎜ Pπ ⎠ ⎝ Pπ

⎞ ⎟. ⎟ ⎠

(1)

Equation (1) shows that T is now the beating between two sinusoidal functions of Pin. As a consequence, the second transmission minimum is raised. For high enough values of |As|, this tends to generate a wide (2nd-order) plateau in the curve of Pout. The slope of this plateau can not be canceled, however, its minimal value ≈ -0.2 for As ≈ 0.34 (Fig. 2, green bold curves). The curves of T and Pout presented in this work were simulated using a matrix approach [15], based on the continuous-wave and low-nonlinearity approximations [14]. We used a fiber twist rate of 3 turns/m for a beat length LB = 10 m, yielding no substantial difference with the high-twist limit. The QW angle for zero low-power transmission was αc =Ω/2 = -0.71. Our approach to assess the device performances is similar to [16]. We consider a return-tozero on-off-keying (RZ-OOK) signal degraded by amplitude noise, like the amplified spontaneous emission (ASE) produced by Erbium-doped fiber amplifiers in a transmission link, and presenting in the spaces ghost pulses [17] or a continuous optical power level which may result from demultiplexing in optical time-division multiplexed (OTDM) systems [18]. Timing jitter introduced by the link is assumed to have a negligible effect on the regenerator performances (which do not include retiming), and is not considered in the calculations. #74572 - $15.00 USD

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1

(a)

(b) 1

-αc x

0.6

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Transmission

0.8

y

ψ

0.4

0.5

0.2 0

0 0

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Pin/Pπ

2

0

1

Pin/Pπ

2

Fig. 2. Transmission (a) and output power (b) of the NOLM versus input power for As = 0, 0.28, 0.34, 0.47 (arrow), and ψ = π/4. The angle and shape of the input ellipse in each case is shown in Fig. 2(b), inset.

We simulated the SNR and ER enhancement capabilities of the NOLM for As = 0.34, and an average input peak power matching the center of the plateau. The ER is defined as the power ratio P(1)/P(0) between the logical ones (mean pulse peak power) and zeros (mean power on spaces), and the SNR as the ratio between mean pulse peak power and standard deviation of amplitude jitter on ones, modeled as a Gaussian distribution, and which was added to the input signal. Although it does not accurately model true signal impairments, Gaussian statistics is often preferred for its simplicity, and it yields a good estimate of system performance in some cases [19]. To calculate output ER and SNR, we determine output power levels from input power values using the output power characteristic of the regenerator. The results are presented in Fig. 3 (curves labeled A). For comparison are also shown the curves corresponding to a transmission of the form T = 12 − 12 cos(π Pin Pπ ) (curves labeled “sin”). This characteristic can be obtained from a conventional NOLM with balanced coupler, where internal loss is used to provide power imbalance (instead of an asymmetric coupler, which does not allow zero low-power transmission). The same T can be obtained with the NOLM of Fig. 1, for As = 0 (see Eq. (1) and red curves in Fig. 2), without loss. Curves for the NOLM with large power imbalance (coupling ratio 0.91/0.09) are also included in Fig. 3 (“lpi”). Curves “sin” and “lpi” shown in Fig. 3 were computed in the same way as it was done for A, B, C, D, and other schemes, from the corresponding transmission characteristics. Figure 3(a) shows that, for highly degraded input SNR (< 10), the improvement using the A configuration is substantial, comparable to “lpi” and better than “sin”. The growth rate is 3 dB/dB (2 dB/dB for “sin”). For input SNR < 6, SNR enhancement can be slightly improved by slightly modifying As: as the slope of the plateau grows, it is widened, too (black curve in Fig. 2(b)). For input SNR > 10 however, curve A only exhibits a linear growth (1 dB/dB) due to the nonzero slope of the plateau that limits the reduction of low amplitude jitter. Figure 3(b) shows for the A configuration a very good ER improvement comparable with the “sin” case (3 dB/dB slope). Note that for “lpi” there is practically no ER improvement (output ER ≈ input ER), due to the large value of the low-power transmission. For the following regimes, the NOLM is followed by a polarizer P whose angle χ can be adjusted (Fig. 1). As the NOLM output ellipse orientation ψout, and thus the adjustment of χ, depend on the value of the QW orientation, these angles are measured with respect to αc. 2.2 Regime B Regime B consists in the use of the NOLM+P with linear input polarization (As = 0). Adjustment parameters in this case are the orientation ψ of the input polarization, and the angle χ of P. We showed earlier that, for linear input polarization, the NOLM transmission is given by TN = 12 − 12 cos(π Pin Pπψ ) , where Pπψ = Pπ sin (2ψ ) [15]. Hence the switching power #74572 - $15.00 USD

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Pπψ increases as ψ deviates from π/4. We also showed that in this case the Stokes parameter at the NOLM output is given by Asout = sin (2ψ ) , and is thus fixed by ψ and does not depend on power, while the ellipse rotates with power, as ψ out = πPin 2 Pπψ . From these results, one can show that the transmission of the NOLM output beam through the polarizer is given by TP = 12 − 12 cos(2ψ ) cos(π Pin Pπψ − 2 χ ) . The amplitude of TP can be adjusted through ψ, and its phase through χ. As a result, the global transmission of the system, TNP = TNTP, is quite adjustable. More specifically, TP can be employed to smoothen the peak power transmission.

Output SNR

103

(a)

lpi

B C

nps

sin

102

101

A D 101

Output ER

106

(b)

104

sin A

D C

B

102

100 100

102

Input SNR

nps 101

lpi 102

Input ER Fig. 3. SNR (a) and ER (b) enhancement for various NOLM designs and operational modes. Curves are identified by arrows using labels whose meaning is given in the text. Curves labeled D: solid As = 0.83, dashed As = 0.92, dotted As = 0.98. Black dashed curves correspond to equal output and input SNR (a) or ER (b), thus marking the border between enhancement and degradation.

Bold curves in Fig. 4 show that a wide 3rd-order plateau (dnPout/dPinn ≠ 0 for n > 3) can be obtained by adjusting ψ (0.73π/4) and χ (-1.3), without substantially increasing the switching power from Pπ. SNR and ER improvement performances of this transfer characteristic when the input peak power is matched to the center of the plateau (~1.3Pπ) are displayed in Fig. 3 (curves labeled B). The best SNR improvement of all is obtained in this case, with a growth #74572 - $15.00 USD

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rate of 4 dB/dB at large input SNR. In particular, substantial improvement is obtained with respect to “sin” (1st-order maximum) and “lpi” (2nd-order plateau), over the whole input SNR range. Again, for the case of very large amplitude jitter, we investigated the possibility of further widening the plateau. It is possible, if ψ deviates further from π/4, although a ripple appears on the plateau [Fig. 4(b)]. Unfortunately, the power of the plateau center is also increased, as a consequence of the increase of Pπψ , and no improvement was found in terms of SNR enhancement at small input SNR. As to the ER enhancement, the curve is similar to the A and “sin” ones, with a 3 dB/dB slope.

(a)

TP

(b)

TN

1.5

Pout/Pπ

Transmission

1

0.5

1 y x

0.5 χ

TNP

0 0

αc

2

1

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0

0

2

1

Pin/Pπ

Fig. 4. (a). Transmissions TN of the NOLM, TP of the P and TNP of the NOLM+P, for χ = -1.0, 1.3 and -1.6 (gray arrows) and ψ = 0.73π/4 in all cases; (b) output power of the NOLM (orange dashed) and of the NOLM+P (green solid) for ψ = 0.73π/4, 0.65π/4 and 0.55π/4 (gray arrows), and χ = -1.3. Blue and red dashed lines are NOLM+P output power for χ = -1.0 and -1.6, respectively, and ψ = 0.73π/4 in both cases. The evolution of output polarization is also represented in Fig. 4(b) for the case ψ = 0.73π/4 (violet line materializes polarizer direction).

2.3 Regime C For regime C, we again start with linear input polarization (As = 0) and ψ = π/4 (corresponding to minimal switching power), and we then adjust As. This is similar to case A, except that now we use the polarizer P, whose orientation χ is the second adjustment parameter. Contrary to case B, here the NOLM output ellipse changes both orientation and ellipticity as power is increased, however it is still possible to generate a flat 3rd-order plateau by selecting one particular polarization orientation across the output ellipse (Fig. 5, green bold). SNR and ER enhancement performances are shown in Fig. 3 (curves labeled C). The SNR curve C roughly presents the same evolution as curve B (with a 4 dB/dB slope for large input SNR), although it remains below, as the plateau appears for larger input power (~1.6Pπ) than in case B. The advantage of the 3rd-order plateau over the “sin” case is obvious in Fig. 3(a), however the improvement with respect to the “lpi” case (2nd-order plateau) is less evident, in particular at low input SNR. The SNR improvement in this region can be slightly increased, however, by modifying As. In this case, indeed, the plateau is widened further, at the price of some ripple, but without substantial increase in the power of the plateau center (see red curves in Fig. 5). Hence, contrary to case B, the output SNR can be further improved for small input SNR ( 10-20, general performances are lower than in the other cases, including the “sin”. This is due to the high power value of the plateau regions (~2.3Pπ). For As = 0.83, although a 2nd-order plateau appears in the output power characteristic (Fig. 6), SNR enhancement [orange solid in Fig. 3(a)] barely exceeds the “sin” characteristic. On the basis of this analysis, we will now compare the proposed schemes and give some guidelines to determine the best choice for regenerating degraded signals from a particular transmission system (with specific degrees of degradation of SNR and ER). Signals with SNR #74572 - $15.00 USD

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and ER below ~10 can be considered as seriously degraded, yielding unacceptable bit-error rate levels (typically > 10-9). Figure 3(a) shows that all the proposed schemes except D yield SNR improvement for input SNR higher than 4 (or even smaller in the cases A and C, if As is slightly modified), and also improve the ER for input ER higher than 3. When the input SNR is severely degraded (< 10), configuration A can be advantageously used, as the absence of output polarizer will simplify the adjustment procedure. However an arbitrary elliptical polarization state at the NOLM input may be difficult to obtain in practice, or require longterm stabilization, which would complicate the setup. In this case, configuration B appears as a good alternative, as it operates with linear input polarization. Compared to cases A and B, the C configuration presents the inconvenient of a larger plateau power, which increases substantially the required input peak power (~1.6Pπ instead of ~1.3Pπ). Moreover, it also requires elliptical input polarization. These requirements may be a high price to pay for the better ER enhancement observed in Fig. 3(b). For input SNR > 10, configurations B or C may be used for SNR enhancement, the best results being obtained with B, but scheme A should be discarded, as it appears clearly from Fig. 3(a). As to ER enhancement, A, B and C remain quite equivalent on a wide range of input ER, as shown by the closely packed curves in Fig. 3(b). Finally, the D configuration should only be used when a very strong enhancement of a very degraded ER (as low as 2) is expected, and will degrade the SNR if its initial value is below ~10-20, Transmission

0.6

(a)

0.1

As = .92

0.4

(b)

.98 0.05 .83

0.2 0

Pout/Pπ

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.83

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1

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Pin/Pπ

Fig. 6. (a, b). Transmission and (c, d) output power characteristics of the NOLM+P for various values of As (indicated as curve labels). χ = 0.2 (green), -0.1 (red), -0.5 (blue). ψ = π/4 in all cases. (b) and (d) are magnifications of Fig. 6(a) and (c), respectively (dashed boxes). The evolution of the output polarization is shown in Fig. 6(d) for the green curve. Values of the power PL of the second zero of transmission are 0.82Pπ (blue), 0.90Pπ (red), and 1.03Pπ (green).

As shown in Fig. 3 and discussed in this Paper, the performances of the proposed NOLMbased regeneration schemes generally exceed, in terms of SNR and/or ER, those of other 1step NOLM-based schemes (“sin”, “lpi” and “nps”). We will now analyze how these novel schemes compare with a multiple-step, NOLM-based regeneration scheme. Let us consider the 3-step scheme described in [13], for regeneration of 160 Gb/s data (including retiming). In this experimental Paper, the authors report as their best result a Q factor improvement from 3.2 without regeneration to 6.3 when their regeneration scheme is used prior to detection. Remembering that the average photocurrent is proportional to the average power on either marks or spaces if a square-law photodetector is used, and considering that electrical and

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optical bandwidths are comparable, the Q factor can be estimated by Q ≈ (P(1)-P(0))/(σ1+σ0), where σ1 and σ0 are the standard deviations of amplitude jitter on marks and spaces, respectively. Neglecting P(0) with respect to P(1) and assuming that σ1 ≈ σ0, we find that Q ≈ P(1)/2σ1 = SNR/2, or SNR ≈ 2Q. Hence the reported Q factor enhancement translates in terms of SNR into an improvement from 6.4 to 12.6. On the other hand, Fig. 2 in Ref. [13] allows estimating that the ER is enhanced from ~5 to ~50. Considering now in Fig. 3(a) an input SNR = 6.4, we find an output SNR = 17 for case A, 26 for B, 15 for C and < 4 for all cases D. With an input ER = 5, Fig. 3(b) yields an output ER = 20 for case A, 18 for B, 45 for C and 120 for the dotted curve D. These results show that SNR and ER enhancement performances of scheme C are quite similar to those of the 3-stage regenerator. Schemes A and B allow better SNR enhancement, but lead to lower ER improvement than the 3-stage scheme for these particular input parameters. Finally, the D case (dotted curves in Fig. 3) yields a very large ER enhancement, but at the price of a degradation of the SNR. Although the single-stage scheme C supports the comparison with the more complex scheme proposed in Ref. [13], it has to be noted that the latter was designed to operate well below the NOLM switching power, whereas more than the switching power must be reached for scheme C. In addition, fiber dispersion, which reduces NOLM performance for very high bit rates, was not accounted for in the present simulations. Let us mention finally that Stimulated Brillouin Scattering (SBS), if present, may decrease the NOLM efficiency, or even cause device instability [21]. In this case, a SBS reduction technique should be implemented, for example pulse compression (linewidth broadening) through careful fiber composition, which would increase the complexity of the setup. 3. Conclusion We proposed four procedures, A, B, C and D, to realize simultaneous enhancement of the SNR and ER of an optical data stream, all based on polarization effects in a power-balanced fiber NOLM. The A configuration offers a simple way to reduce efficiently large amplitude fluctuations. With the additional complexity of an output polarizer, which introduces another adjustment parameter, configurations B and C yield excellent reduction of the amplitude jitter, over a very wide range of powers. These three configurations are also optimized for improving relatively large ER values. With the D configuration, very high ER enhancement occurs at both small and large ER ends, at the price however of less efficient amplitude jitter reduction. We believe that this work will be useful for the elaboration of high-quality signal regeneration techniques needed by future high-speed transmission networks. Acknowledgments O. Pottiez was supported by CONCYTEG grant entitled “Estudio y Diseño de un láser de fibra óptica para generación de pulsos ultracortos.” The authors thank J. W. Haus for fruitful discussions and careful reading of the manuscript.

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