**2. Theoretical analysis**

whose operation relies on nonlinear optical phenomena, has raised great interests in recent years as an alternative to electronic switching in optical communication systems [1, 2]. These devices have a strong potential and can be very useful for a number of applications in diverse areas such as ultrahigh‐speed optical telecommunications, wavelength conversion, pulse regeneration, optical monitoring, optical computing, etc. Nonlinear effects will be particularly important in the next generation of optical networks, which will rely on all‐optical functions for higher speed and greater capacity. Many principles and devices have been reported; of them a great part is performed by devices based on the Kerr effect in optical fibers, such as the nonlinear optical loop mirror (NOLM) first proposed by Doran and Wood [3], devices based on cross‐phase modulation (XPM) proposed by Mamyshev and four‐wave mixing (FWM) proposed by Caramella and Stefano [4, 5], logic gates [6], and wavelength conversion [7]. First reports were followed by numerous investigations proving high performance of all‐fiber devices. The NOLM was investigated for soliton switch [8], wavelength demultiplexing [9], mode locking [10], etc. Some interferometric devices such as NOLM present an oscillating power transfer function but the ideal characteristic is the step‐like function. The power trans‐ fer function of the FWM‐based devices approaches the step‐like function; nevertheless, for an only stage scheme, a flat response is obtained only at spaces, whereas at marks an oscillating dependence was measured [11]. A flat response was shown in double stage scheme at both

Recently, there has been a lot of interest in stimulated Raman scattering (SRS) and is con‐ sidered one of the most important nonlinear effects in optical fibers. SRS is one of the oldest and most well‐studied optical phenomena and can anticipate great advantages for optical signal processing circuits because of high amplification of the signal and naturally existing possibility for wavelength conversion. In spite of that, SRS has just obtained some attention in the context of Raman amplifiers and only a few works were published on the use of SRS for designing optical signal processing circuits and optical switches. Furthermore, SRS can be hoped to present many advantages due to high amplification of the signal and intrinsi‐ cal compatibility with communications systems using Raman amplification of signals [13]. The strong dependence of the Raman amplification on pump power that was considered for wavelength conversion with high extinction ratio of the output signal and can be base for the design of Raman circuits [14]. In this work, the output signal pulses at Stokes wavelength are generated as result of the Raman amplification caused by input signal used as the pump. Because of the strong dependence of Raman amplification on pump power, the extinction ratio of the output signal can be much higher than that of the input signal. The exploitation of pump saturation in presence of the signal was shown in [15]. In this work, the pulses at Stokes wavelength are used as input signal, and the pump pulses are considered as an output signal. In the absence of the Stokes pulse (spaces), the pump pulse travel in the fiber without saturation and has high power at the fiber output (marks), whereas in presence of the Stokes signal (marks) the pump pulse is depleted and has a low power at the fiber output (spaces). Disadvantage of this approach is the low extinction ration of the output signal. The most interesting work was suggested by Belotiskii [16] where authors used a two‐stage setup. In this work, we consider experimentally, the Raman circuit using the approach suggested in [16] and show that it allows the step‐like power transfer function with high differential gain

spaces and marks [12].

182 Raman Spectroscopy and Applications

and low input signals.

In this section, we present simple numerical calculations to show the basic principles and poten‐ tial of the approach [17]. **Figure 1** presents the diagram of the Raman circuit, it consists of two stages. In stage 1, the pump pulses are saturated if the signal is at "ON" or passes through stage 1 without saturation if the signal is at "OFF," this means that stage 1 woks as saturated amplifier. The amplification in stage 2 depends on the pump power entering from stage 1; that is, stage 2 works as Raman amplifier. When the pump pulse enters stage 2 without saturation as a result of that the input signal is at "OFF," then this will result in the generation of the high output pulses at the Stokes wavelength, this mean that the output signal is at "ON." For the opposite case when the input signal is at "ON," then the pump pulse is depleted in stage 1, the input signal is at "ONs" and therefore the output signal is at "OFF," so in this way the Raman circuit works as an inverter. For this operation of the Raman circuit one should take into account the following considerations: (a) the pump power has to be high enough to provide strong amplification of the signal and (b) at the same time the pump power has to be lower than the SRS threshold at which strong Stokes pulses and pump depletion appears as a result of the amplification of the initial spontaneous Stokes waves. The wavelength of the continuous wave (CW) seeding laser wave defines the wavelength of the output signal. The walk‐off effect between pump and Stokes pulses is inevitable because of the big difference between pump and signal wavelengths. To avoid the degradation of the operation of the Raman circuit because of walk‐off effect we pro‐ pose to use the dispersion management using the special fibers, which they are formed by con‐ necting the fibers in which the signal travels faster than pump with the fibers in which the signal travels slower than pump. We must consider that if we use the fibers with anomalous, the modu‐ lation instability (MI) effect can be expected, which will complicate circuit operation drastically.

**Figure 1.** Setup of Raman circuit.

Because of its speed and good results split‐step Fourier method (SSFM) is the most commonly used method for numerical analysis of the nonlinear equation of Schrödinger (NLSE), which uses the algorithm's finite Fourier transforms. We use this method to evaluate the operation of the Raman circuit based on the coupled equations for pump *A*<sup>P</sup> , and Stokes *A*<sup>S</sup> , pulses:

$$\frac{\partial A\_s(t)}{\partial z} + \left(\frac{1}{V\_p} - \frac{1}{V\_s}\right) \frac{\partial A\_s(t)}{\partial t} + \mathrm{i}\frac{\beta\_z}{2} \frac{\partial^2 A\_s(T, z)}{\partial t^2} = \frac{\mathcal{g}}{2} A\_s(t) \left|A\_p(t)\right|^2 \tag{1}$$

$$\frac{\partial A\_p(t)}{\partial z} + i \frac{\beta\_2}{2} \frac{\partial^2 A\_p(T, z)}{\partial t^2} = -\frac{g}{2} A\_p(t) \left| A\_p(t) \right|^2 \tag{2}$$

where *β*<sup>2</sup> is the group velocity dispersion (GVD) parameter considered equal for pump and Stokes wavelengths; *g* is the Raman gain coefficient equal to 10‐13 m/W for silica glass at 1550 nm wavelength; *V*P and *V*<sup>S</sup> are the group velocity of pump and Stokes pulses, respectively. Here we do not consider the effects connected to the Kerr effect and widening of the pulses due to GVD that is possible if the fiber length is less than the scattering length. The param‐ eters of the fibers used in calculations correspond to the fibers used in experiments. Fiber 1, Corning SMF‐LS dispersion shifted fiber with normal dispersion; Fiber 2, SMF‐28 fiber with anomalous dispersion; and Fiber 3, OFS True Wave fiber with anomalous dispersion. **Figure 2** shows examples of the pump pulses at the output of Fiber 1 (**Figure 2a**) and at the output of Fiber 2 (**Figure 2b**). Input pulse is shown by the solid line and the output by dashed line.

**Figure 2.** Input (solid line) and output (dashed line) waveforms of the pump at the end of (a) fiber 1 and (b) fiber 2.

The parameters used for calculations were as follow: Fiber 1 SMF‐LS with 100 m length, Fiber 2 SMF‐28 with 25 m length. The pump pulse presents the Gaussian shape of 100 ps and 25 W of power. The input Stokes pulse has a super Gaussian shape with 300 ps and a peak power of 1 mW and effective area of fibers of 50 μm<sup>2</sup> . Fiber 1 has normal dispersion, so Stokes pulse has high speed and the first half of pulse manifests stronger depletion than the second half of pulse. Depletion can be made large when normal dispersion fiber is spliced with a fiber with anomalous dispersion, see **Figure 2(b)**, this happens because in fiber with normal dispersion the signal travels faster than the pump and the fibers with anomalous dispersion the signal travels slower than the pump, therefore pumping depletion becomes stronger in this case [18].

where *β*<sup>2</sup>

dashed line.

nm wavelength; *V*P and *V*<sup>S</sup>

184 Raman Spectroscopy and Applications

is the group velocity dispersion (GVD) parameter considered equal for pump and

are the group velocity of pump and Stokes pulses, respectively.

Stokes wavelengths; *g* is the Raman gain coefficient equal to 10‐13 m/W for silica glass at 1550

Here we do not consider the effects connected to the Kerr effect and widening of the pulses due to GVD that is possible if the fiber length is less than the scattering length. The param‐ eters of the fibers used in calculations correspond to the fibers used in experiments. Fiber 1, Corning SMF‐LS dispersion shifted fiber with normal dispersion; Fiber 2, SMF‐28 fiber with anomalous dispersion; and Fiber 3, OFS True Wave fiber with anomalous dispersion. **Figure 2** shows examples of the pump pulses at the output of Fiber 1 (**Figure 2a**) and at the output of Fiber 2 (**Figure 2b**). Input pulse is shown by the solid line and the output by

**Figure 2.** Input (solid line) and output (dashed line) waveforms of the pump at the end of (a) fiber 1 and (b) fiber 2.

To calculate the depletion of the pump pulse we use as a ratio between the pulse energies at the output and the input. **Figure 3** shows the dependencies of the depletion of the pump on the input Stokes signal peak power. In the figure, the solid line represents the depletion when only the SMF‐LS fiber is considered; the fiber length is equals to 100 m (pump power is equals to 30 W). For the dashed line we have 100 m of SMF‐LS fiber added with 40 m of SMF‐28 fiber (pump power equals to 24 W to have the same total Raman amplification as it was in the first case). Finally, we show the depletion for dotted line and for this case we used 100 m of fiber 1 with low GVD so that walk‐off length is much longer than the length of the fiber (pump power was 30 W). It can be seen that the best results obtained are for the fiber with low dispersion; however, in practice it is not easy to fulfill the condition that the walk‐off length is larger than the length of the fiber, a special case is when we have low potencies and for this case are required large fiber lengths. However, a simple dispersion management technique using fibers with normal and anomalous dispersion may provide switching with a contrast (the ratio between the energies of the signals at the second stage output) of about 20 dB at input power less than 10‐3 of pump power [17].

**Figure 3.** Depletion of the pump at first stage. Solid line is for fiber 1, dashed line is for fiber 1 + fiber 2, and dotted line is for fiber 1 with low dispersion.

We calculate the energy of Stokes pulse of the input in the output circuit depending on the power input Stokes in the first state. The calculations were made with SSFM with *g* = 0.6×10−<sup>13</sup> m/W corresponding to the maximum Raman gain for the pump wavelength 1550 nm, the pulse duration pump of input is 100 ps with a power of 15 W, and the effective area of the fibers is 50 μm<sup>2</sup> . Three types of fibers with different GVD were used: fiber 1 D = −6 ps/(nm‐km); fiber 2 D = 20 ps/(nm‐km); fiber 3 D = −0.01 ps/(nm‐km). Fiber 1 corresponds to dispersion shifted fiber, fiber 2 is the SMF‐28 fiber, and fiber 3 corresponds to the fiber without walk‐off between pulses pump and Stokes. **Figure 4** shows the results for three different configurations of stage 1. Stage 1 is only of fiber 1 with 295 m of length (solid line). Stage 1 is comprised of three fibers connected in series: 150 m of fiber 1, 45 m of fiber 2, and 100 m of fiber 3 (dashed line). Stage 1 is also comprised of 295 m of fiber 3 (the dotted line). For the three cases 350 m of fiber 3 was used for stage 2. The best result is provided by the fiber with low dispersion.

**Figure 4.** Energy of Stokes pulses at the output of the circuit.

The characteristics of a circuit of two stages depend on the attenuation between them, and two circuits connected in series provide the step‐like dependence. The dependence of two cir‐ cuits for the case when stage 1 is comprised of only fiber 3 and stage 2 is comprised of fiber 3 is shown in **Figure 5(a)**. The dependence for the case when stage 1 is comprised of three fibers connected in series, fiber 1, fiber 2, and fiber 3, and for stage 2 that is comprised of fiber 3 is shown in **Figure 5(b)**. As mentioned earlier, the dependence of two circuits depends on the attenuation between them, and for **Figure 5(a)** and **(b)** the attenuation is different and shows the dependence of the circuits connected in series.

We calculate the energy of Stokes pulse of the input in the output circuit depending on the power input Stokes in the first state. The calculations were made with SSFM with *g* = 0.6×10−<sup>13</sup> m/W corresponding to the maximum Raman gain for the pump wavelength 1550 nm, the pulse duration pump of input is 100 ps with a power of 15 W, and the effective area of the fibers is

The characteristics of a circuit of two stages depend on the attenuation between them, and two circuits connected in series provide the step‐like dependence. The dependence of two cir‐ cuits for the case when stage 1 is comprised of only fiber 3 and stage 2 is comprised of fiber 3 is shown in **Figure 5(a)**. The dependence for the case when stage 1 is comprised of three fibers connected in series, fiber 1, fiber 2, and fiber 3, and for stage 2 that is comprised of fiber 3 is shown in **Figure 5(b)**. As mentioned earlier, the dependence of two circuits depends on the attenuation between them, and for **Figure 5(a)** and **(b)** the attenuation is different and shows

the dependence of the circuits connected in series.

**Figure 4.** Energy of Stokes pulses at the output of the circuit.

used for stage 2. The best result is provided by the fiber with low dispersion.

. Three types of fibers with different GVD were used: fiber 1 D = −6 ps/(nm‐km); fiber 2 D = 20 ps/(nm‐km); fiber 3 D = −0.01 ps/(nm‐km). Fiber 1 corresponds to dispersion shifted fiber, fiber 2 is the SMF‐28 fiber, and fiber 3 corresponds to the fiber without walk‐off between pulses pump and Stokes. **Figure 4** shows the results for three different configurations of stage 1. Stage 1 is only of fiber 1 with 295 m of length (solid line). Stage 1 is comprised of three fibers connected in series: 150 m of fiber 1, 45 m of fiber 2, and 100 m of fiber 3 (dashed line). Stage 1 is also comprised of 295 m of fiber 3 (the dotted line). For the three cases 350 m of fiber 3 was

50 μm<sup>2</sup>

186 Raman Spectroscopy and Applications

**Figure 5.** Dependencies of the output Stokes energy on the input power for two circuits connected in series: (a) attenuation is 10 times and (b) attenuation is 3 times.

A strong dependence of the output signal energy on the input signal power allows for improved signal‐to‐noise ratio of the signal. To show it, we put Gaussian noise to input Stokes. The Gaussian noise added to input Stokes is shown in **Figure 6a)** and the result of the input Stokes with Gaussian noise is shown in **Figure 6(b)**. We can consider two possibili‐ ties: one possibility, when input signal is "OFF," Gaussian noise is launched for fiber 1, and two possibility when input signal is "ON," Stokes more Gaussian noise is launched for fiber 1. We applied random noise and the results are show in **Figure 7** for 10 Stokes pulses when input signal is "ON" and "OFF." The parameters of the setup used for calculations were as follows: 100 m of fiber 1, 40 m of fiber 2, and 300 m of fiber 3, input pump power of 24 W, and the power of the input signal of 1 mW. As we can see, the output Stokes pulses are well distinguished [18].

**Figure 6.** The input signal "off" (a) and the input signal "on" (b) in the presence of random noise.

**Figure 7.** Output Stokes pulses for input pulses "ON" and "OFF."

From the above results, we can say that the setup of Raman circuit allows effective optical switching, logic operation, and noise reduction with signal of low power of the pump power. As we show above, to improve the operation of the Raman circuit we proposed the connection in series of fibers with normal and anomalous dispersion. We must consider that the power required for strong Raman amplification is lower than that required for the effects of MI and pulse breakup in fibers with anomalous GVD [19]. The pulse breakup will affect the depletion of the pump pulse and amplification of the Stokes pulses. There are few works that are related to this problem, especially related to supercontinuum generation and Raman amplification [20].
