**Abstract**

This work presents a methodology to estimate the pumping power required for the first Stokes to reach its maximum stored energy level, before it generates the next Stokes. These estimates are achieved by experimentally measuring the critical power and the relationship between the pumping power (PP0) and the small signal of the stimulated Raman spread (*PF0*). For our study we used 1 km of 1060-XP fiber, experimentally obtaining *Pcr* = 6.693 W, *PF0*/*PP0* = 6.759x10<sup>6</sup> . With these experimental data, the pump power required for the first Stokes to reach its maximum stored energy level was 13.39 W, and the stored energy in the first Stokes was 9.88 W. It is important to note that the Raman threshold ln(*PP0*/*PF0*) = 11.9 is smaller than the initially reported 16.

**Keywords:** fiber optics, Raman scattering, Raman fiber lasers

### **1. Introduction**

Stimulated Raman Scattering (SRS) in optical fibers has been an intensified subject of studies during the last two decades; at the beginning it was investigated in non-doped optical fibers but during the last decade the hybrid rare-earth-doped combined with the pure Raman effect has been developed and attracted much attention [1]. The SRS is a phenomenon responsible of a new generation of fiber lasers and Raman amplifiers because the high brightness of the confined powers traveling and interacting along optical fibers greatly facilitate the Raman scattering (RS) based amplification. Raman scattering occurs when a monochromatic light beam propagates along an optical fiber, whereas most of the beam power is transmitted without change, a small quantity in the order of 10<sup>6</sup> per kilometer of fiber is scattered isotropically with a new frequency [2]. At high coupled pump powers, the stimulated version of this phenomenon, the SRS, occurs and the portion of this frequency-shifted scattered light that travels through the fiber core becomes amplified and demands energy to feed itself from the pump power in order be amplified. This amplification is very efficient such that most of the traveling pump power can be transferred to this Stokes component. Therefore, the SRS is the platform for the development of Raman fiber lasers and amplifiers. As the Stokes component grows very remarkably, the pump power decreases until it is unable to continue transferring energy. In a sufficiently pumped optical fiber such that the pump is practically extinguished during this energy transfer, now this Stokes can be high enough to produce the following Stokes by the same SRS-process; again, a strong enough second Stokes can produce a third one and so on. Using the historically well developed laser operating around 1064 nm as a pump source for this type of Ramanbased lasers it is possible to obtain Stokes components that cover the 1.1–1.7 μm region that is of great importance for several applications such as: optical fiber communications (the internet platform), material processing (cutting, soldering, ablating, etc.), laser spectroscopy and medicine [3]. Equations for the behavior of the Raman amplification have been described in [4, 5] and their solutions have been proposed in [4, 6]. Despite these advances with the propagation equations describing the energy transfer among stokes components in Raman amplification, there is a lack of simple methods that allow the exact estimation of some important parameters such as: the Raman threshold power, the maximum pumping power necessary for the power stored in the Stokes signal to be maximized, the Raman gain coefficient and the power required to obtain any N-order of Stokes.

Solving the differential equation by the separable equation method, we obtain

*υS νP*

Where the parameters represent: *PPL* output pump power, *PP0* input pump power, *PFL* output power Stokes and *PF0* initial power Stokes, and evaluated in

First, we will multiply the Eq. (2) by (νp/νs), and consequently, the algebraic

Performing algebraic operation and integrating by separable variable, we obtain:

where PP0 and PPL are the input and output powers, PF0 and PFL are initial and

On the other hand, for a 1-Km of any telecom optical fiber, approximately a few millionths of the input pump power (*PP0*) becomes available at the output as RS (spontaneous, linear) Stokes signal (*PF0*), this is practically the signal that initiates the amplification process once the stimulated process takes place. Therefore, the term ((*νS*/*νP*)*PP0* + *PF0*) ≈ (*νS*/*νP*)*PP0* [2]. Substituting Eq. (7) in Eq. (5), it is obtained a general solution to differential Eqs. (1) and (2), given by [6]:

> <sup>¼</sup> *gr Aeff*

where *L*eff = [(1-exp(�*αL*))/*α*] represents the interaction length of the pumping and the Stokes signal. Eq. (5) provides information on parameters such as the Raman gain, the powers of the Pump and Stokes signals at the output end of the fiber.

Raman threshold is a very important parameter since above this power the energy transfer from pump to the first Stokes becomes highly efficient (**Table 1**). In optical amplification, it is important to operate above threshold; on the contrary, not reaching threshold becomes crucial in multichannel communications systems

**Figure 1** presents the output powers of: (a) Residual pump (black line with squares) and (b) Stokes signal at 1117 nm (red line with circles) at the output of

as non-linearly produced Stokes-copies of any channel can interfere with

*νP*

� �*PP*<sup>0</sup> <sup>þ</sup> *PF*<sup>0</sup> � �*<sup>e</sup>*

¼ �*α PP* þ

*vp vs PF*

� �*PP*<sup>0</sup> <sup>þ</sup> *PF*<sup>0</sup> � �

� *<sup>υ</sup><sup>S</sup> νP*

� � (6)

*PP*0*Leff* (8)

�∝*<sup>L</sup>* (7)

(5)

� �*PPL* <sup>þ</sup> *PFL* � � � �

the following expression:

*DOI: http://dx.doi.org/10.5772/intechopen.94350*

*PPL* � � � � <sup>¼</sup> *gr*

all the fiber from *z* = 0 to *z* = *L*.

*d dz PP* <sup>þ</sup>

*νS νP*

output Stokes signal. *L* is the fiber length.

B. Adding the Eqs. (1) and (2).

operation results:

**3. Raman threshold**

neighboring channels. [2].

**83**

*Aeff* ! *<sup>υ</sup><sup>P</sup>*

*νS* � � 1 *α*

*Analysis of Energy Transfer among Stokes Components in Raman Fiber Lasers*

*vp vs PF* � �

� �*PPL* <sup>þ</sup> *PFL* � � <sup>¼</sup> *<sup>ν</sup><sup>S</sup>*

ln *PF Pp PP*<sup>0</sup> *PF*<sup>0</sup> � �

ln *PFL PF*<sup>0</sup> � � *PP*<sup>0</sup>

This chapter describes the maximum energy that can be stored in a given Stokes line as after such maximum storage it starts producing the next Stokes. From the existing physical–mathematical theories, a formula has been obtained by us and has been verified with experimental results.

### **2. Theory**

In order to find a mathematical relationship that predicts the maximum power stored in the Stokes signal is necessary to solve the equations that describe the forward propagation of the pumping power and Stokes line in a single mode fiber given by [4, 7, 8];

$$dP\_P/dz = -a\_P P\_P - (\upsilon\_P/\nu\_S) \left(\mathbf{g}\_r/\mathbf{A}\_{\text{eff}}\right) P\_P P\_F \tag{1}$$

$$dP\_F/dz = -a\_S P\_F + (\mathbf{g}\_r/A\_{\rm eff})P\_P P\_F \tag{2}$$

where *PP* and *PF*, *ν<sup>S</sup>* and *νP*, *α<sup>P</sup>* and *α<sup>S</sup>* are powers, frequencies and loss coefficients of pump power and Stokes signal, respectively. *Aeff* is effective area, and *gr* is the Raman gain coefficient of the Stokes signal.

Assuming that α<sup>p</sup> ≈ α<sup>s</sup> it is possible to minimize the errors by establishing *α = (α<sup>p</sup> + αs)/2* given that the attenuation curve of the silica fiber is approximately constant in the region covering pump and first Stokes signals. An analytical solution of the equations that govern the SRS can be obtained by the following procedure:

A. First dividing the Eq. (1) between Eq. (2),

$$\frac{d\mathbf{z}d\mathbf{P}\_P}{d\mathbf{z}d\mathbf{P}\_F} = \frac{\left(-a - \frac{v\_P}{v\_r}\frac{gr}{A\_{\text{eff}}}P\_F\right)P\_P}{\left(-a + \frac{gr}{A\_{\text{eff}}}P\_P\right)P\_F} \tag{3}$$

Where the term *dz* is canceled, and consequently we proceed to the separation of variables, to obtain the following mathematical expression:

$$\left(-a + \frac{gr}{A\_{\rm eff}} P\_P\right) \frac{dP\_P}{P\_P} = \left(-a - \frac{v\_p}{v\_s} \frac{gr}{A\_{\rm eff}} P\_F\right) \frac{dP\_F}{P\_F} \tag{4}$$

Solving the differential equation by the separable equation method, we obtain the following expression:

$$\ln\left[\left(\frac{P\_{FL}}{P\_{F0}}\right)\left(\frac{P\_{P0}}{P\_{PL}}\right)\right] = \left(\frac{\mathcal{g}\_r}{A\_{\sharp\mathcal{f}}}\right)\left(\frac{\nu\rho}{\nu\_\mathcal{S}}\right)\frac{1}{a}\left[\left(\left(\frac{\nu\_\mathcal{S}}{\nu\_P}\right)P\_{P0} + P\_{F0}\right) - \left(\left(\frac{\nu\_\mathcal{S}}{\nu\_P}\right)P\_{PL} + P\_{FL}\right)\right] \tag{5}$$

Where the parameters represent: *PPL* output pump power, *PP0* input pump power, *PFL* output power Stokes and *PF0* initial power Stokes, and evaluated in all the fiber from *z* = 0 to *z* = *L*.

B. Adding the Eqs. (1) and (2).

extinguished during this energy transfer, now this Stokes can be high enough to produce the following Stokes by the same SRS-process; again, a strong enough second Stokes can produce a third one and so on. Using the historically well developed laser operating around 1064 nm as a pump source for this type of Ramanbased lasers it is possible to obtain Stokes components that cover the 1.1–1.7 μm region that is of great importance for several applications such as: optical fiber communications (the internet platform), material processing (cutting, soldering, ablating, etc.), laser spectroscopy and medicine [3]. Equations for the behavior of the Raman amplification have been described in [4, 5] and their solutions have been proposed in [4, 6]. Despite these advances with the propagation equations describing the energy transfer among stokes components in Raman amplification, there is a lack of simple methods that allow the exact estimation of some important parameters such as: the Raman threshold power, the maximum pumping power necessary for the power stored in the Stokes signal to be maximized, the Raman gain coeffi-

This chapter describes the maximum energy that can be stored in a given Stokes line as after such maximum storage it starts producing the next Stokes. From the existing physical–mathematical theories, a formula has been obtained by us and has

In order to find a mathematical relationship that predicts the maximum power stored in the Stokes signal is necessary to solve the equations that describe the forward propagation of the pumping power and Stokes line in a single mode fiber

� �*PPPF* (1)

(3)

(4)

� �*PPPF* (2)

*dPP=dz* ¼ �*αPPP* � ð Þ *υP=ν<sup>S</sup> gr=Aeff*

*dPF=dz* ¼ �*αSPF* þ *gr=Aeff*

where *PP* and *PF*, *ν<sup>S</sup>* and *νP*, *α<sup>P</sup>* and *α<sup>S</sup>* are powers, frequencies and loss coefficients of pump power and Stokes signal, respectively. *Aeff* is effective area, and *gr* is

> �*<sup>α</sup>* � *vP vs gr Aeff PF*

�*<sup>α</sup>* <sup>þ</sup> *gr Aeff PP* � �

Where the term *dz* is canceled, and consequently we proceed to the separation

¼ �*<sup>α</sup>* � *vp*

*vs gr Aeff PF*

!

*dPF PF*

� �

*PP*

*PF*

Assuming that α<sup>p</sup> ≈ α<sup>s</sup> it is possible to minimize the errors by establishing *α = (α<sup>p</sup> + αs)/2* given that the attenuation curve of the silica fiber is approximately constant in the region covering pump and first Stokes signals. An analytical solution of the equations that govern the SRS can be obtained by the following

cient and the power required to obtain any N-order of Stokes.

been verified with experimental results.

*Application of Optical Fiber in Engineering*

the Raman gain coefficient of the Stokes signal.

A. First dividing the Eq. (1) between Eq. (2),

�*<sup>α</sup>* <sup>þ</sup> *gr Aeff PP*

!

*dzdPP dzdPF*

¼

of variables, to obtain the following mathematical expression:

*dPP PP*

**2. Theory**

procedure:

**82**

given by [4, 7, 8];

First, we will multiply the Eq. (2) by (νp/νs), and consequently, the algebraic operation results:

$$\frac{d}{dz}\left(P\_P + \frac{v\_p}{v\_s}P\_F\right) = -a\left(P\_P + \frac{v\_p}{v\_s}P\_F\right) \tag{6}$$

Performing algebraic operation and integrating by separable variable, we obtain:

$$\left( \left( \frac{\nu\_S}{\nu\_P} \right) P\_{PL} + P\_{FL} \right) = \left( \left( \frac{\nu\_S}{\nu\_P} \right) P\_{P0} + P\_{F0} \right) e^{-\infty L} \tag{7}$$

where PP0 and PPL are the input and output powers, PF0 and PFL are initial and output Stokes signal. *L* is the fiber length.

On the other hand, for a 1-Km of any telecom optical fiber, approximately a few millionths of the input pump power (*PP0*) becomes available at the output as RS (spontaneous, linear) Stokes signal (*PF0*), this is practically the signal that initiates the amplification process once the stimulated process takes place. Therefore, the term ((*νS*/*νP*)*PP0* + *PF0*) ≈ (*νS*/*νP*)*PP0* [2]. Substituting Eq. (7) in Eq. (5), it is obtained a general solution to differential Eqs. (1) and (2), given by [6]:

$$\ln\left(\frac{P\_F}{P\_p}\frac{P\_{P0}}{P\_{F0}}\right) = \frac{\mathcal{g}\_r}{A\_{\epsilon\mathcal{f}}}P\_{P0}L\_{\epsilon\mathcal{f}}\tag{8}$$

where *L*eff = [(1-exp(�*αL*))/*α*] represents the interaction length of the pumping and the Stokes signal. Eq. (5) provides information on parameters such as the Raman gain, the powers of the Pump and Stokes signals at the output end of the fiber.

### **3. Raman threshold**

Raman threshold is a very important parameter since above this power the energy transfer from pump to the first Stokes becomes highly efficient (**Table 1**). In optical amplification, it is important to operate above threshold; on the contrary, not reaching threshold becomes crucial in multichannel communications systems as non-linearly produced Stokes-copies of any channel can interfere with neighboring channels. [2].

**Figure 1** presents the output powers of: (a) Residual pump (black line with squares) and (b) Stokes signal at 1117 nm (red line with circles) at the output of

#### *Application of Optical Fiber in Engineering*


to improve some of their properties for different applications; for such special fibers the Eq. (10) is not a good approximation, in this sense, the relationship *PP0/PF0* must be determined experimentally, as it is the case in this work. This ratio is unique for each optical fiber, that is, each optical fiber has its own response to the

*Analysis of Energy Transfer among Stokes Components in Raman Fiber Lasers*

The Raman gain coefficient *gr* describes how the Stokes power grows as the pump power is transferred through the stimulated Raman scattering; in our case, given that the optical fiber under test has common characteristics, Eq. (9) is useful to calculate the Raman gain. Note that the effective fiber length is a constant that

An analysis of the behavior below threshold, at threshold and above threshold

Observe that at 2.9-W pump power, the Stokes power has �68.5 dBm, this is the maximum RS level *PF0* (the figure presents an error, instead of *PP0* it should be *PF0*). The evolution of this signal is presented until the SRS signal is evident at 3.3-W

At any pump power, at the fiber output the relation between the output coupled

<sup>¼</sup> <sup>10</sup>*<sup>Δ</sup>Power dBm* ð Þ

<sup>10</sup> (11)

is presented by the corresponding spectra on **Figure 2**. These spectra exactly

pump power PP and the produced Raman signal (PF0) may be quantified by:

where *ΔPower = PP-PF0*. When the spontaneous Raman just appears at the output, practically, the pump signal only suffers linear attenuation and thus the relationship between the residual pump power *PP(L)* and the coupled pump

*PP PFO*

stimulated Raman scattering [9].

*DOI: http://dx.doi.org/10.5772/intechopen.94350*

depends on the loss coefficient and the fiber length.

correspond to the same experiment corresponding to **Figure 1**.

**4. Raman gain**

pump [9].

power *PP*<sup>0</sup> is:

**Figure 2.**

**85**

*Output signals of the 1-km 1060-XP optical fiber.*

#### **Table 1.**

*Three estimates of maximum stokes power for 1060-XP fiber.*

#### **Figure 1.**

*Residual-pump and stokes powers delivered at the output of an optical fiber as functions of the pump power coupled at the input.*

1-km 1060-XP optical fiber as a functions of 1064-nm pump power. Observe that the aforementioned threshold pump was around 2.9 W. There is another important parameter in which the residual and Stokes powers are equal, this is called the critical (pump) power that occurs at 6.693 W. These curves were obtained by splicing the 1060-XP optical fiber to an Ytterbium-based 1064-nm fiber laser without forming a laser cavity; i.e., in a free-running configuration.

At critical power (*PP = PF*) from Eq. (8) we have:

$$\ln\left(\frac{P\_{P0}}{P\_{F0}}\right) = \frac{\mathcal{g}\_r}{A\_{\text{eff}}} P\_{\text{Per}} L\_{\text{eff}} \tag{9}$$

Taking into account that for standard fibers of 1-km length PF0 is in the order of PP0x10�<sup>6</sup> ; under this assumption, Eq. (9) usually approximates:

$$\mathbf{16} \approx \frac{\mathbf{g}\_r}{A\_{\epsilon\mathcal{f}}} P\_{cr} L\_{\epsilon\mathcal{f}} \tag{10}$$

where *PP0* is equal to the critical power *Pcr*. This relationship has been previously reported [4] and used in several research works presenting an acceptable accuracy.

Modern specialty optical fibers are designed for purposes other than optical communications and are usually doped with materials such as P2O5, GeO2 and B2O4 to improve some of their properties for different applications; for such special fibers the Eq. (10) is not a good approximation, in this sense, the relationship *PP0/PF0* must be determined experimentally, as it is the case in this work. This ratio is unique for each optical fiber, that is, each optical fiber has its own response to the stimulated Raman scattering [9].
