MB1.4 11:45-12:00
Transformation of nonlinear phase noise statistics in a phase-sensitive amplifier Adonis Bogris, Thomas Kamalakis, Dimitris Syvridis, Thomas Sphicopoulos Optical Communications Laboratory, Department of Informatics and Telecommunications, National and Kapodestrian University of Athens, Panepistimiopolis, Ilissia, Athens, GR-15784, Greece. e-mail:
[email protected]
Abstract— The nonlinear phase noise statistics of a phase modulated signal coming through a phase-sensitive amplifier are semi-analytically analyzed.
the phase sensitive relation of the output signal with respect to the input can be expressed as [12] (1) BS ( z ) µ( z ) BS (0) v( z ) BS* (0) where the functions µ, v are expressed as
Index Terms – phase-sensitive amplification, regeneration, four-wave mixing, phase-shift keying.
all-optical
µ( z ) v( z )
D
URING the last decade, differential phase-shift keying (DPSK) has attracted the interest of numerous groups on one hand due to the 3dB receiver sensitivity enhancement compared to the on-off (OOK) approach [1] and on the other hand thanks to the high tolerance to nonlinear effects [2]. Despite these advanced characteristics, the performance of DPSK signals can be severely degraded due the linear phase noise (PN) added by the optical amplifiers and the nonlinear PN caused by the fiber non-linearity effects respectively [3]. Appropriate devices for simultaneous amplitude and phase noise treatment seem to be the phase-sensitive amplifiers (PSAs). Recently, both interferometric and noninterferometric techniques have been considered as DPSK signal regenerators [4, 5]. A DPSK regenerator based on a non-degenerate four-wave mixing (FWM) process in nonlinear fibers was theoretically proposed in [5]. The specific scheme offers both amplitude and phase regeneration. The former is accomplished when the amplifier operates in the pump depletion regime, while the latter is attributed to the phase-sensitive nature of the optical gain. In this work, the transformation of the nonlinear phase noise statistics in this phase sensitive amplifier is semi-analytically studied. Before describing the methodology of this paper, the basic characteristics of the specific PSA are highlighted. The phase-sensitivity is achieved by placing the pumps at frequencies that provide a non-frequency shifted image of the signal. The frequency condition for phase-sensitive interaction is ȦP1+ȦP2=2ȦS. Under this condition, the idler coincides with the signal, and their interaction becomes dependent on the input signal phase. Assuming the small-signal approximation,
978-1-4244-1595-3/08/$25.00 ©2008 IEEE
cosh( gz ) i i
ț sinh( gz ) 2g
2ȖAP1 (0) A P 2 (0) sinh( gz ) g
(2) (3)
and ț=ǻȕ+Ȗ(PP2+PP1), g 4Ȗ 2 PP1 PP 2 (ț / 2) 2 1 / 2 are the phasematching factor and the parametric gain coefficient respectively. Moreover, Ȗ is the fiber nonlinear parameter, PP1, PP2 are the two pump powers AP1(0), AP2(0) are the input pump fields and ǻȕ=2ȕS-ȕP2-ȕP1 is the linear phase mismatch factor (wave-vector mismatch). Considering all the propagation effects, the output signal is finally given by the expression. AS ( z )
B S ( z ) exp[iǻȕz / 2 i3Ȗ( PP 2 PP1 ) z / 2]
(4)
The joint probability density function (PDF)of phase ij and normalized amplitude Į in an optical communication system mainly impaired by the interaction between amplified spontaneous emission of the in-line amplifiers and the Kerr nonlinearity of the transmission fiber is given by the expression [6]
§ Cn2 a 2 1 · Į ¨ ¸ exp ¦ ¨ P TC 2¸ n f ʌPASE Tn C n © ASE n n ¹ f
P(a, ij)
(5)
§ 2a · ¸¸ u exp^ in>ij ijNL 1 Tn @` u I n ¨¨ © PASETnCn ¹ where PASE is the inverse of the optical signal-to-noise ratio, ijNL is the nonlinear phase shift, In is the nth order of the Bessel function, and Cn, Tn functions of n, PASE, ijNL [6]. Considering that the amplifier is working in the linear regime, eq. (1), (2), (3) can be straightforwardly utilized in order to determine the new joint PDF P’(A, ȥ) (A, ȥ the normalized amplitude and the phase at the output of the amplifier respectively) that corresponds to the statistics of a signal at the
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Fig. 2. Probability density function of the nonlinear phase noise before and after the PSA considering two pump power cases. The reduction of noise build up is obvious.
Fig. 1. Signal gain and output phase as a function of input signal phase calculated analytically for three different pump power values.
distribution according to [5]. The PSA redistributes the nonlinear phase noise so as to reduce the noise around ij=0. It is important to say that BER improvement is not achieved
output of the amplifier. This can be done applying the theorem of transformation of random variables:
P c( A, ȥ )
wa wA P (a, ij) wȥ wA
wa wȥ wij wȥ
3S / 2
after the PSA. Taking into account that BER
(6)
the integral for all cases is practically the same. This is somehow expected due to one main reason; the input phase also determines output power, hence input phase noise (ij) is translated to output power noise (ǹ) in the PSA as depicted in fig. 1. This problem could be solved operating the device in the depleted-pump regime, where signal power is almost constant for all input phase values with the exception of the completely out-of-phase components which are parametrically attenuated [5]. Moreover, the regenerator can be very useful if placed periodically in a long-haul transmission system. The periodic phase noise compression prevents a sudden noise buildup and its associated BER degradation. In summary, the transformation of nonlinear phase noise statistics in a PSA was theoretically presented. The specific theoretical analysis can be very useful in order to predict the evolution of phase noise in a long-haul transmission system utilizing PSAs. It can be also applied for other phase noise PDFs that take into account multi-channel transmission.
The calculations can be simplified if the general form of the amplifier’s function is taken into account.
Ae jȥ
a( z1e jij z 2 e jij )
(7) The next step is to invert expression (7) so as to determine the solutions of Į=g(A, ȥ) and ij=h(ǹ, ȥ) respectively. The exact formula of the new PDF is a very complicated task, thus the semi-analytical approach was followed instead. In the subsequent analysis a typical HNLF having length L=200m, nonlinear parameter Ȗ=12W-1Km-1, zero-dispersion wavelength Ȝ0=1560nm, dispersion slope dD/dȜ=0.03ps/nm2/km, and ȕ4=-2.48×10-4ps4/km is assumed. The input signal wavelength is 1559.7nm and the two pump waves are placed at 1540nm and 1580nm respectively for the degenerate FWM process. The two pumps are placed far apart from each other in order to safely ignore the pump-pump FWM process. The dependence of power gain and signal output phase on the relative input phase is estimated based on eq. (1)-(4) and depicted in fig. 1 for different pump power values. It is observed that the gain peaks occur at multiples of ʌ and the output phase varies as a step-like function of the input phase exhibiting two phase states with difference equal to ʌ. The approximation of step-like function is more accurate as the pump power increases, achieving almost ideal quantization of the phase at two phase states (Pp=0.4W). The PDF of the nonlinear phase noise before and after the PSA is depicted in fig. 2. The unconditioned PDF of the phase is P ij
f
³ daP (a, ij) 0
and P ȥ
f
³ dǹPc(ǹ, ȥ )
REFERENCES [1] A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, “Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity,” IEEE Photon. Technol. Lett., vol. 15, no. 1, pp. 99-101, Jan 2003. [2] J. K. Rhee, D. Chowdhury, K. S. Cheng, and U. Gliese, “DPSK 32x10 Gb/s Transmission Modeling on 5x90 km Terrestrial System,” IEEE Photon. Technol. Lett. vo. 12, no. 12, pp. 1627-1629, Dec 2000 [3] J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communication systems using linear amplifiers,” Opt. Lett. vol. 15, no. 23, pp. 1351-1354, Dec. 1990. [4] K. Croussore, I. Kim, C. Kim, Y. Han, and G. Li, “Phase-and-amplitude regeneration of differential phase-shift keyed signals using a phase sensitive amplifier,” Opt. Express, vol. 14, no. 6, pp. 2085–2094, Mar. 2006. [5] A. Bogris, D. Syvridis, “RZ-DPSK Signal Regeneration Based on DualPump Phase-Sensitive Amplification in Fibers,” IEEE Photon. Technol. Lett. vo. 18, no. 20, pp. 2144-2147, Oct. 2006 [6] A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett., vol. 29, pp. 673-675, 2004.
for the two
0
cases respectively. The integration is done numerically. A transmission system of 1000km with mean optical power P=80μW and PASE=0.075 is considered. The PDF of the phase before the parametric amplifier resembles the Gaussian 978-1-4244-1595-3/08/$25.00 ©2008 IEEE
³ P(I )dI ,
S /2
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