1. Introduction

Time Division Multiplexed (TDM) Raman pumping scheme is an attractive solution to avoid pump-to-pump nonlinear Four Wave Mixing (FWM) and pump-to-pump interaction by temporally separating the pump forming train of pump pulses [1]. Pump wavelengths are time division multiplexed in counter-propagation direction so that pump-signal RIN transfer can be minimized at relatively lower modulation frequency. In order to reduce the receiver penalty due to the temporal gain variation, pulse repetition rate higher than 100 kHz is required [2]. Higher instantaneous gain in backward direction, especially at small pulse duty cycle imposes higher backward ASE noise, which ultimately increases the forward ASE through Rayleigh backscattering. Noisy pump also affects the amplifier noise figure significantly especially when there is group velocity matching between backward ASE and pump [3]. All these effects can be reduced by having higher pulse repetition rate and/or bigger pulse duty cycle. Pulse repetition up to 1GHz was recommended to reduce the backward ASE enhancement. Nevertheless, high repetition rate may cause pump pulses start to overlap. This interaction causes pump-to-pump interaction and possibly noise due to four-wave mixing effect. Therefore in this letter, we investigate the upper limit of pulse repetition rate and the impact of pump-to-pump interaction when the repetition rate is beyond this limit. In section two, we present the analytical model of the TDM gain property and employ this model into multi-wavelength case. We define the upper limit of the pulse repetition rate and discuss its impact to the flat Raman gain in section three followed by conclusion.

2. Analytical model

Consider Raman pump pulses propagate in backward direction with repetition rate f, and duty cycle ρ. Signals and pumps propagate with group velocities V_s(n) and V_p(m) respectively, where n and m denote the number of signal and pump wavelengths. The interaction between counter-propagating pump and signal is governed by:

where P(z), S(z), α_p, α_s, υ_p, υ_s and g_R denote pump power, signal power, fiber attenuation at pump and signal wavelength, pump and signal wave-number and Raman gain coefficient (W^-1km^-1) respectively.

The solution of (1) and (2) can simply be

where Δz denotes segmented interaction length.

Raman gain is generated by pump pulses interacting with the signal pulse within period of

Assuming initial number of pump pulses exist in the amplification medium at time t=0 is $N_0=f\frac{L}{V_p}$, thus total number of pump pulses interacting with signal pulse is

Interaction time between two counter-propagating pulses is given by

where T_p and T_s are the active period of pump and signal pulses respectively. Assume that signal bit rate is much higher than f such that T_s→0 for very small segmentation of signal pulses. Interaction length between two thus can be defined as

The interaction between counter-propagating pulses occurs at regular interval with total number of interactions equals to N_tot. The interval distance L_n can thus be approximated as:

Under no pump depletion condition, the pump power distribution is given by P_i(z_i,0)=P_L.exp{-α_p(L-(i-1)L_n}, where P_i(z,0) denotes the pump power at location points z_i=i.L_n and at instantaneous time t=0. P_L denotes the pump power at z=L. Temporal gain fluctuation is obtained by temporally shifting the pump power distribution with a regular interval Δt, i.e. P_ij(z, j. t).

From (4), (8) and (9), on-off Raman gain is defined as:

Inserting the pump power distribution into (10), the on-off gain becomes

At any amplification length much longer than the effective length L_eff(L≫L_eff) and pulse repetition rate is sufficiently high, τ approaches a constant value as shown in Fig. 1 below.

 

Fig. 1. Pump and signal wavelengths are 1450 nm and 1550 nm respectively with attenuation α_p=0.26dB/km. When pump pulse repetition rate is sufficiently high f>200 kHz and L≫L_eff, the factor τ≈L_eff≈1/α_p

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Raman on-off gain can thus be approximated as:

3. Results and discussion

We plot flat Raman gains by using multi-wavelength Raman pump on 100 km Non Zero Dispersion Shifted Fiber (NZDF) with fiber attenuation coefficient α_p=0.22 dB/km, 2 ps/nm-km dispersion at 1550 nm, zero dispersion slope S₀=0.078 ps/nm²-km and peak Raman gain coefficient g_R=0.598W^-1km^-1. Figure 2 shows the on-off Raman gain in C-band for two different pump wavelength combinations. Raman pumps were RIN-free and square modulated with 500 kHz repetition rate, 50% duty cycle, and infinite extinction ratio. Pump power of each pump source is within a range of 0.8W–1.2W to generate 12 dB average on-off gains. From our observation, when pump wavelengths are far away separated, i.e pump configuration in (A), one or two pump source will dominate (i.e. higher power) the others in order to maintain similar flat gain within similar band. Peak pump power as high as 1.5W is required by the pump source with wavelength λ_p=1444nm in (A). Closer pump wavelength separation makes almost-uniform pump power levels possible.

 

Fig. 2. Flat Raman on-off gain in 1525–1560 nm wavelength range with pump wavelengths: (A) λ_p=1420nm, 1444nm, 1477nm, and 1490nm, (B). λ_p=1421nm, 1442nm, 1453nm, and 1479nm. Gain ripple=±0.8 dB.

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High instantaneous gain in backward direction enhances the backward ASE, which ultimately enhances the forward ASE. This affects the amplifier noise figure severely especially when pump noise is high. To reduce this detrimental effect, higher duty cycle and higher repetition rate are required. Pulse repetition rate up to 1GHz is required to introduce sufficient walk-off that averages the backward ASE [3]. However, multi-wavelength pumps set the upper limit of the repetition rate before two neighboring pump pulses start to interact. The maximum pulse repetition rate required before two pump pulses with group velocity V_p1 and V_p2 interact is defined as:

where ḺC̱ denotes the location where two neighboring pump pulses start to collide and after which interact with each other. Based on the linear dispersion profile, ${\underline{f}}_{\underline{max}}$ can be estimated. Obviously, two neighboring pump pulses having wider wavelength separation has lower ${\underline{f}}_{\underline{max}}$ if zero-dispersion wavelength is not located in between the two pump wavelengths. Table 1 shows the various f_max for two pump wavelength combinations as used in Fig. 2 above and for various L_C. We set zero-dispersion wavelength λ ₀=1400nm and 50% pulse duty cycle. In our simulation, maximum pulse repetition rates f_max are measured based on the worst condition, which is when two neighboring pump pulses are having the biggest group-velocity difference, i.e. pump pulses with λ_p=1421nm and 1479nm are adjacent. When pump wavelength separations are smaller, as shown in column 2, the biggest group velocity difference is smaller, which then increases f_max∸

Table 1. Maximum pulse repetition rates f_max for two pump wavelength combinations (A) and (B) at various length of collision L_C.

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Figure 3(a) above shows that under group velocity matching condition, i.e. zero-dispersion wavelength is located in the mid of two pump wavelengths, f_max is dramatically increased, i.e. when pump wavelengths are λ_p=1421nm, 1442nm, 1453nm,1479nm and λ₀=1450nm, f_max can be higher than 10 GHz. Under this condition, slight change of pulse duty cycle will significantly change the allowable f_max as shown in Fig. 3(b).

 

Fig. 3. (a) f_max with respect to zero-dispersion wavelengths for pump wavelengths combination (a) λ_p=1421nm, 1442nm, 1453nm, and 1479nm; (b) λ_p=1420nm, 1444nm, 1477nm, and 1490nm; (b) f_max with respect to pulse duty cycles for pump wavelengths λ_p=1421nm and 1479nm with respect to zero-dispersion wavelengths (a) λ₀=1450nm, (b) λ₀=1445nm, (c) λ₀=1400nm and (d) λ₀=1500nm

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When pulse repetition rate is set higher than f_max, pump interaction starts to occur. We quantify the impact of pump-to-pump interaction to the flatness of Raman gain for the two pump wavelength combinations in Fig. 4 below based on similar condition as in Fig. 2. Pulse repetition rate as high as 92 MHz does not significantly affect the flatness of the gain in Fig. 4(a) as compared to Fig. 4(b). This is because the location of collision L_C in Fig. 4(a) is further away from the one in Fig. 4(b), such that smaller pump powers are involved in the power transfer process. A 92 MHz pulse repetition rate causes gain ripple degradation about ±0.2dB in Fig. 4(a) and ±0.7dB in Fig. 4(b). Furthermore, pulse repetition rate as high as 122.5 MHz can cause gain ripple degradation up to ±3dB in Fig. 4(a) and ±5.5dB in Fig. 4(b). Higher instantaneous gain at smaller duty cycle may induce much more pump power transfer, which makes the gain profile more sensitive to the pump-to-pump interaction effect. Further work needs to be done to investigate the noises generated by four-wave mixing during the pump-interaction process.

 

Fig. 4. Flat Raman gain is affected by pump-to-pump interaction when pulse repetition rates are 73.5 MHz (line with circles), 92 MHz (line with stars) and 122.5 MHz (line with squares) respectively. Pump wavelength combination are (a) λ_p=1421nm, 1442nm, 1453nm, and 1479n), (b) λ_p=1420nm, 1444nm, 1477nm, and 1490nm).

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4. Conclusions

Multi-wavelength TDM Raman pump requires more stringent requirement to the pulse repetition rate and duty cycle to avoid the pump-to-pump interaction effect. Smaller pulse duty cycle induces more severe pump-to-pump interaction effect due to high instantaneous pump power. Having zero-dispersion wavelength in the mid of pump wavelengths and/or having shorter pump wavelength separation help to increase the maximum repetition rate. Further investigation needs to be done to observe the significance of maximum pulse repetition rate to the noise performance of TDM multi-wavelength Raman amplifier.