Reduction of Nonlinear Optical Self Phase Modulation in Optical Fiber at 40 Gb/s

ABSTRACT A nonlinear Schrodinger equation has been solved in order to establish the nonlinear optical self phase modulation (SPM) actual magnitude in fiber (within optical domain), the SPM magnitude was simulated at 40 GHz in relation to normalized time and actual distance within the fiber. When the SPM signal comes out of the fiber and enters a pre-designed transimpedance amplifier (TIA) (electronic domain), and according to simulation, the SPM signal corresponds to an input noise spectral density of 320.3 pA/Hz (electronic domain) that enters the TIA. A significant reduction in S21 scattering parameter, nonlinear optical power and nonlinear noise figure were registered when the 320.3 pA/Hz is entered at the TIA input. That effectively means a reduction in SPM value using the TIA at 40 GHz. A Microwave Office software was used for simulation.


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INTRODUCTION Experimental performance evaluation of SPM compensation and its application in optical transport network was introduced by (Yamazaki et al., 2010). Experimental results showed transmission distances of 100 Gb/s signal reaches 16.340 and 12.015 km in single channel 10 channel wavelength division multiplexing (WDM) conditions respectively (Yamazaki et al., 2010).
A double cell, self-phase modulation based optical regenerator was designed and optimized to reduce signal noise and distortion. Investigation on multiple passes through the proposed optical regenerator showed significant amplitude jitter suppression compared single pass alternatives. An improvement in Q-factor became negligible after only five loops for this system and a double Q-factor is achieved demonstrating the efficiency of the proposed scheme (Bolt and Ensser, 2010).
A multilevel coded modulation system in the presence of nonlinear phase noise for fiber optical communication was introduced. The scheme exploited a 16-point ring constellation with nonlinear post compensation of the self-phase modulation via the Kerr effect. A new set partitioning based on the Ungerboeck approach was introduced to maintain unequal error protection in amplitude and phase direction. Simulation results showed that the proposed system provides up to 2 dB gain over a forward error correcting scheme for a block error rates around 10 -9 , with the same overhead (7%) and complexity (Beygi et al., 2010).
The effect of dispersion slope of highly nonlinear fiber on the performance of self-phase modulation based 2R-optical regenerator was investigated numerically. This study showed that the dispersion slope has a significant impact on the shape of the transfer function of the regenerator. The dispersion slope can therefore be used together with the dispersion value and the filter offset as an additional parameter to control the performance of the regenerator. (Ghafoor and Petropoulos, 2010).
An analytical study of self phase modulation (SPM) and cross-phase modulation (XPM) was performed for coherent Differential Phase Shift Keying DPSK and ON-OFF (OOK) signal in Non-Return to Zero (NRZ) and Return-to Zero (RZ) modulation format for wavelength division multiplexing (WDM) system employing distributed Raman amplifier (DRA). The study showed that 40 Gb/s RZ-DPSK signal with 33.3% duty cycle experiences minimum XPM and SPM induced crosstalk. The results also revealed that minimum crosstalk was induced in backward pumped DRA among the three pumping schemes i.e forward, backward and bi-directional. Results assumed importance for minimizing deleterious XPM and SPM effects in optical communication system (Anamika, 2013).

Self Phase Modulation in Optical Domain
For pulse evolution in an optical fiber, the nonlinear Schrodinger (NLS) equation governs that as follows (Agrawal, 2001): is the pulse optical field within the fiber, z is distance that the pulse moves in fiber, T is pulse periodic time, is the dispersion coefficient and is the attenuation coefficient. The actual optical field envelope in fiber is given as (Agrawal, 2001): The chirping frequency is set at 1.
is considered to be the pulse width.
To start deriving the value equivalent to the self phase modulation (SPM) magnitude which is (given in equation 1), equation 2 can be used in which the following and parameters can be identified: equation is derived as: To satisfy equation (3), the values of and can be obtained: Therefore, from equation (3), After defining which is part of the left hand side of equation (1), the right hand side terms of equation (1) can be defined. Beginning with , the value is derived first as follows: Hence, The other term part of equation (1) which is can be obtained as follows: By multiplying by , the magnitude of SPM can be obtained. To simulate the magnitude against and , the following typical values (at 40 Gb/s) can be set. is set to 15 ps 2 /km, the pulse width is set to 25 ps (@40 GHz), the pulse period is considered to range from 25 ps to 100 ps, = 0.02 at wavelength of 1.54 µm. The value of can be up to 0.4 meter. Fig. 1 illustrates the magnitude (SPM value) against pulse time ratio and .

SPM Reduction at Electronic Domain
When the signal comes out of the fiber (optical domain carrying SPM noise) and enters a transimpedance (TIA) amplifier (electronic domain), the following work deals with the reduction of that noise. A schematic diagram of a transimpedance (TIA) amplifier electronic circuit is shown in Fig. (2). The RLC FET transistor circuit is commonly used for several purposes, however, it has not been used before as an SPM reducing mechanism. Furthermore, the transistor type used in this work is a high-speed STATZ GaAs FET transistor (Statz et al., 1987) which is used in this work as a new application of TIA amplifier. It is useful to say that R F = 148.2 Ohm (feedback resistor value), L F = 0.3 nH (feedback inductance), C F = 91 pF (feedback capacitance), and C i = 1 pF (input capacitance).
STATZ GaAs FET transistor characteristics can be described in which under normal operating conditions the gate-to-source capacitance is much larger than the gate-to-drain capacitance. At zero drain-to-source voltage, both capacitances are about equal. For negative drain-to-source voltages, the original source acts like a drain and vice versa. Consequently, the normally large gate-to-source capacitance becomes small and acts like a gate-to-drain capacitance. In order to model these effects, it is necessary to realize that there are no separate gate-to-source and gate-to-drain charges, but that there is only one gate charge which is a function of gate-to-source and gate-to-drain voltages (Statz et al., 1987).

Fig. 2: Transimpedance amplifier schematic design.
In the TIA circuit, the simulated input noise current i i is entered to correspond to a value of the actual magnitude of SPM which is (from optical domain). The best value that worked well (in simulation within electronic domain) for an input noise spectral density was 320.3 pA 2 /Hz. We can consider that is the input capacitance, V i is the input noise voltage (transformed SPM). The output voltage is denoted as . The transimpedance inverting amplifier voltage gain is given as (Razavi, 2006 From electronic circuits principles, the sum of currents at the V i node is: From equations (9) and (10), we have: Multiplying by , Obtaining : Hence, We replace by as in equation (9):

Multiply by
Further simplification indicates that the transfer function of the TIA amplifier is: For which: As mentioned earlier, the best value that worked well (in simulation within electronic domain) for an input noise spectral density was 320.3 pA 2 /Hz. Therefore, the value of SPM (from optical domain) should correspond to 320.3 pA 2 /Hz of input noise spectral density (electronic domain) in order to have an efficient reduction in SPM. In order to study the reduction in SPM using the TIA configuration, two aspects must be taken into account. The first is how does the S 21 scattering parameter behave and secondly, what would be the nonlinear noise output power actual magnitude. The S 21 noise spectrum shows the noise gain reduction for SPM corresponding to 320.3 pA 2 /Hz of input current spectral density (electronic domain) at 40 GHz. Figure 3 illustrates the reduction of S 21 noise spectrum (corresponding to SPM noise at optical domain) at 40 GHz, while at lower frequencies, S 21 noise spectrum is much higher. At the same time, Fig. (3) includes the TIA gain (transfer function given in equation 14) versus frequency.  Fig. (3), several comments can be made. To start with, the negative sign of the TIA gain means that the TIA amplifier is an inverting one, and that is clear from the minus sign in equation 14. Furthermore, the TIA gain maintains its strength beyond 40 GHz which is a good efficiency indication. It is important to notice that along side the rise in TIA gain of main signal, there is a downward direction of the noise S 21 parameter which is highly useful in terms of noise reduction capability. At 40 GHz, both TIA gain and noise S 21 scattering parameter are the subject of study in this work, although data shown at 50 GHz have shown equally important results. Main fiber optic communication systems focus on 40 Gb/s data bit rate and that explains the reason behind the 40 GHz frequency point. However, the range of up to 50 GHz covers a wide area of fiber optic high data bit rate operating conditions.
The nonlinear noise output power of the transimpedance amplifier when SPM noise is applied that corresponds to 320.3 pA 2 /Hz (electronic domain) is shown in Figure 4. A reduction in the nonlinear noise output power is clearly shown at 40 GHz (up to 50 GHz) compared to lower frequencies. The nonlinear noise figure (NF) is reduced as in Fig. (5). That implicitly means a real reduction in SPM value, while the transfer function magnitude (equation 14 as TIA gain of main signal) is also embedded in which the TIA gain rise in an opposite direction of the reduced nonlinear noise figure (NF). There is a significant and important gap between the TIA gain and NF at 40 GHz. Further analysis of Fig. (5) indicates that it is possible to make use of data not only at 40 Gb/s but it can also be useful to do that at 10 Gb/s. Although that analysis below 5 GHz may not be that useful due to the fact that the gap between the TIA gain and the nonlinear noise figure (NF) gets narrow. Finally, from Figs. (3 & 5), an inverse relationship between noise (in terms of SM) and TIA gain develops most importantly at 40 GHz and even beyond that. The concept of noise reducing transimpedance amplifier is therefore implemented in this work.

CONCLUSION
The actual magnitude of self phase modulation (SPM) can be established for 40 GHz signal in an optical fiber. It was confirmed that a significant reduction in SPM can occur when the signal is applied to a STATZ GaAs FET based transistor transimpedance amplifier (TIA). Furthermore, this SPM reduction was demonstrated through the actual reduction in S 21 scattering parameter, nonlinear optical power and nonlinear noise figure. Another advantage of the TIA system is that alongside the noise reduction, there was a TIA gain (main signal) derived from the TIA transfer function. Although that the input noise spectral density of 320.3 pA 2 /Hz is proved to be a good indication of the TIA system capability for noise reduction, however, future research may be possible in improving the system sensitivity in terms of input noise.