**4. Raman gain**

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 depends on the loss coefficient and the fiber length.

An analysis of the behavior below threshold, at threshold and above threshold is presented by the corresponding spectra on **Figure 2**. These spectra exactly correspond to the same experiment corresponding to **Figure 1**.

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 pump [9].

At any pump power, at the fiber output the relation between the output coupled pump power PP and the produced Raman signal (PF0) may be quantified by:

$$\frac{P\_P}{P\_{FO}} = \mathbf{10^{\frac{\Delta P\_{uor}(dRu)}{10}}} \tag{11}$$

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

**Figure 2.** *Output signals of the 1-km 1060-XP optical fiber.*

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

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

<sup>¼</sup> *gr Aeff*

Taking into account that for standard fibers of 1-km length PF0 is in the order of

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

*PPcrLeff* (9)

**Max [W]** *P*Sto

**<sup>1</sup> [W]**

*PcrLeff* (10)

without forming a laser cavity; i.e., in a free-running configuration.

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

; under this assumption, Eq. (9) usually approximates:

<sup>16</sup> <sup>≈</sup> *gr Aeff*

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

**Fiber span [km] Leff [km] P1st**

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

*Application of Optical Fiber in Engineering*

1 0.846 13.39 9.88 2 1.443 7.48 3.9 5 2.371 4.77 0.88

PP0x10�<sup>6</sup>

**84**

**Figure 1.**

**Table 1.**

*coupled at the input.*

*Application of Optical Fiber in Engineering*

$$P\_P = P\_{P\_0} e^{-aL} \tag{12}$$

Eq. (15) represents the pumping power that must be used for the first Stokes to reach its maximum energy level. Substituting Eq. (15) in Eq. (7), and neglecting the terms PF0 and PPL, because their values are very small, consequently, an equation is

> 2 ln ð Þ*<sup>ε</sup> Aeff gr*

Taking previous data for the 1060-XP fiber, ln(ε) = ln(PP0/PF0) = 11.9, gR = 2.1

] and 1.5 dB/Km. On the other hand, using λs = 1120 nm and λp = 1064 nm, we can estimate the maximum pump power and the maximum

In general, the pumping power for storing the maximum energy in N-order

1 þ *λP λs* þ *λs λs*2

With this equation, we obtain approximately the maximum power stored in the N-order Stokes before generating the N + 1-order Stokes. This equation is versatile because the parameters involved are simple to calculate. To date, the Eq. (17) at

And finally, the ratio between the maximum power stored and coupled pump

¼ *λP λs e*

This equation depends on the attenuation and the fiber length, and describes the optical conversion efficiency for a particular fiber length (*L*) and fiber attenuation (*α*). If *L* = 0, the Eq. (18) gives the maximum energy conversion efficiency or

In this sense, for a longer optical fiber the energy conversion efficiency is lower. For example, for 5 km of 1060-XP fiber this value can be �17%. This implies that the fibers for the design of Raman lasers and Raman amplifiers must be of a length

An analytical equation was developed to predict the pumping power necessary to store the maximum energy in the first Stokes, this is a powerful tool for numerical simulations, which can provide specific data for the design of Raman fiber lasers. Consequently, an analytical equation is deduced that can predict the stored power in

the following Stokes orders, that is, progressively, the second, the third, etc.

For a 1 km fiber span, 13.39 W of pumping power is required to get the first Stokes to reach its maximum stored energy level of 9.88 W, while for 5-km of fiber span a pump power of 4.77 W is required to obtain 0.88 W. It is important to mention that the experimental calculation of ln(PP0/PF0) let us conclude that the fiber is not intended for telecommunications as the �16 number is not satisfied. Finally, it is important to note that commercial single-mode fiber lasers normally operate at pump powers of around 10 W, therefore, an estimate such as the one

1 *Leff e*

þ … þ

*λ<sup>N</sup>*�<sup>1</sup> *λN* (17)

�*α<sup>L</sup>* (18)

�*α<sup>L</sup>* (16)

obtained for the power stored in the first Stokes, given by:

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

*PSto* <sup>1</sup> <sup>¼</sup> *<sup>υ</sup><sup>S</sup> νP*

power stored in the first Stokes, for different fiber span.

Max <sup>¼</sup> 2 ln ð Þ*<sup>ε</sup> Aeff*

that allows an acceptable conversion efficiency [10, 11].

least of our knowledge has not been reported.

*gr*

1 *Leff*

*PSto* 1 P1st Max

[W�<sup>1</sup> km�<sup>1</sup>

Stokes is given by;

quantum efficiency.

**6. Conclusions**

**87**

made in this work is fundamental.

PN

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

power for the first Stokes is given by;

Now, substituting Eq. (7) in (6), it is possible to numerically establish a relationship for *PP0*/*PF0* given by:

$$\frac{P\_{PO}}{P\_{FO}} = 10^{\frac{\Delta P\_{water}(dBn)}{10}} e^{aL} \tag{13}$$

Note that the term on the left side of Eq. (13) is constant, and it depends on the physical characteristics of the optical fiber and its response to spontaneous RS [8]. And it can be evaluated just when the spontaneous Raman Scattering becomes stimulated Raman scattering, then this relationship can be obtained experimentally from **Figure 2**.

Then using this numerical result, one obtains the Raman gain efficiency by rewriting Eq. (9) as:

$$\mathbf{g}\_R = \frac{\mathbf{g}\_r}{A\_{\epsilon f}} = \ln\left(\frac{P\_{P0}}{P\_{F0}}\right) \frac{\mathbf{1}}{P\_{cr}L\_{\epsilon f}}\tag{14}$$

**Figure 2** shows the spectral composition of the signal delivered by te 1-km 1060- XP optical fiber. Note that, �18.3 dBm level corresponds to the residual pump *PP*(*L*). At the lowest pump power *PP0*, only spontaneous RS occurs with signal from 1090 to 1130 nm, its maximum level is around �68.5 � 1 dBm. Taking this level as *PF0*, the difference is ΔPower = 50.2 � 1 dBm. The fiber loss is 1.5 dB/Km at 1064 nm according with the technical specification provided by de manufacturer. Substituting these data on the Eq. (13) yields *PP0*/*PF0* = 147,945.506 and ln(*PP0*/*PF0*) = 11.9, hence *PF0* = 6.759x10�<sup>6</sup> *PP0*. These numbers are of great importance, instead of obtaining the 16-number of Eq. (10), we have obtained 11.9, meaning that this is not considered a standard telecommunications fiber, this is true since it was designed for operating in the 1060-nm region; also note that the RS signal produced by 1-km of this fiber was 6.76x10�<sup>6</sup> multiplied by the pump power. Everything is within the values expected.

Using the critical power *Pcr* = 6.693 W from **Figure 1**, and calculating the effective length *Leff* = 0.846 km one may use the Eq. (14) to obtain the Raman gain efficiency *gR* = 2.1[W�<sup>1</sup> km�<sup>1</sup> ]; very important value to be used to model the propagation of pump and stokes in a fiber Raman laser system using Eqs. (1) and (2).

### **5. Maximum power stored in first stokes**

Once the Raman threshold is reached, first Stokes grows quickly until the pump power is unable to continue transferring power, and this energy transfer is described by Eq. (8). When the energy transfer ceases, the first Stokes reaches its maximum value, denoted by *P*<sup>1</sup>*st Max*, and precisely the first Stokes is capable of generating the second Stokes. We have previously known that approximately six millionth of pump power is transferred in spontaneous RS, that is, *PF0*/*PP0* = 6.759x10�<sup>6</sup> . We propose that the pump power ceases to transfer energy to the first Stokes when it reaches a value of six millionth multiplied by the threshold power. In this sense, and to estimate an approximate value to produce the next Stokes, we propose this be similar to PF0/PP0 = PP/PF = 1/ε, in such a way that Eq. (8) becomes;

$$P\_{\text{Max}}^{\text{1st}} = 2 \ln \left( \varepsilon \right) \frac{A\_{\text{eff}}}{\text{g}\_r} \frac{\mathbf{1}}{L\_{\text{eff}}} \tag{15}$$

*Analysis of Energy Transfer among Stokes Components in Raman Fiber Lasers DOI: http://dx.doi.org/10.5772/intechopen.94350*

Eq. (15) represents the pumping power that must be used for the first Stokes to reach its maximum energy level. Substituting Eq. (15) in Eq. (7), and neglecting the terms PF0 and PPL, because their values are very small, consequently, an equation is obtained for the power stored in the first Stokes, given by:

$$P\_1^{\text{Sto}} = \frac{\nu\_\text{S}}{\nu\_P} 2 \ln \left( e \right) \frac{A\_{\text{eff}}}{\text{g}\_r} \frac{\mathbf{1}}{L\_{\text{eff}}} e^{-aL} \tag{16}$$

Taking previous data for the 1060-XP fiber, ln(ε) = ln(PP0/PF0) = 11.9, gR = 2.1 [W�<sup>1</sup> km�<sup>1</sup> ] and 1.5 dB/Km. On the other hand, using λs = 1120 nm and λp = 1064 nm, we can estimate the maximum pump power and the maximum power stored in the first Stokes, for different fiber span.

In general, the pumping power for storing the maximum energy in N-order Stokes is given by;

$$\mathbf{P}\_{\text{Max}}^{\text{N}} = 2 \ln \left( \varepsilon \right) \frac{A\_{\text{eff}}}{\mathbf{g}\_r} \frac{\mathbf{1}}{L\_{\text{eff}}} \left[ \mathbf{1} + \frac{\lambda\_P}{\lambda\_s} + \frac{\lambda\_s}{\lambda\_{s2}} + \dots + \frac{\lambda\_{N-1}}{\lambda\_N} \right] \tag{17}$$

With this equation, we obtain approximately the maximum power stored in the N-order Stokes before generating the N + 1-order Stokes. This equation is versatile because the parameters involved are simple to calculate. To date, the Eq. (17) at least of our knowledge has not been reported.

And finally, the ratio between the maximum power stored and coupled pump power for the first Stokes is given by;

$$\frac{P\_1^{\text{Sto}}}{P\_{\text{Max}}^{\text{1st}}} = \frac{\lambda\_P}{\lambda\_s} e^{-aL} \tag{18}$$

This equation depends on the attenuation and the fiber length, and describes the optical conversion efficiency for a particular fiber length (*L*) and fiber attenuation (*α*). If *L* = 0, the Eq. (18) gives the maximum energy conversion efficiency or quantum efficiency.

In this sense, for a longer optical fiber the energy conversion efficiency is lower. For example, for 5 km of 1060-XP fiber this value can be �17%. This implies that the fibers for the design of Raman lasers and Raman amplifiers must be of a length that allows an acceptable conversion efficiency [10, 11].

### **6. Conclusions**

*PP* ¼ *PP*<sup>0</sup> *e*

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

Note that the term on the left side of Eq. (13) is constant, and it depends on the physical characteristics of the optical fiber and its response to spontaneous RS [8]. And it can be evaluated just when the spontaneous Raman Scattering becomes stimulated Raman scattering, then this relationship can be obtained experimentally

Then using this numerical result, one obtains the Raman gain efficiency by

<sup>¼</sup> *ln PP*<sup>0</sup> *PF*<sup>0</sup> 1

**Figure 2** shows the spectral composition of the signal delivered by te 1-km 1060-

of obtaining the 16-number of Eq. (10), we have obtained 11.9, meaning that this is

designed for operating in the 1060-nm region; also note that the RS signal produced by 1-km of this fiber was 6.76x10�<sup>6</sup> multiplied by the pump power. Everything is

Using the critical power *Pcr* = 6.693 W from **Figure 1**, and calculating the effective length *Leff* = 0.846 km one may use the Eq. (14) to obtain the Raman gain efficiency

Once the Raman threshold is reached, first Stokes grows quickly until the pump power is unable to continue transferring power, and this energy transfer is described by Eq. (8). When the energy transfer ceases, the first Stokes reaches its maximum

second Stokes. We have previously known that approximately six millionth of pump

that the pump power ceases to transfer energy to the first Stokes when it reaches a value of six millionth multiplied by the threshold power. In this sense, and to estimate an approximate value to produce the next Stokes, we propose this be similar to

Max <sup>¼</sup> 2 ln ð Þ*<sup>ε</sup> Aeff*

*gr*

1 *Leff*

]; very important value to be used to model the propagation of

*Max*, and precisely the first Stokes is capable of generating the

not considered a standard telecommunications fiber, this is true since it was

pump and stokes in a fiber Raman laser system using Eqs. (1) and (2).

power is transferred in spontaneous RS, that is, *PF0*/*PP0* = 6.759x10�<sup>6</sup>

P1st

PF0/PP0 = PP/PF = 1/ε, in such a way that Eq. (8) becomes;

**5. Maximum power stored in first stokes**

XP optical fiber. Note that, �18.3 dBm level corresponds to the residual pump *PP*(*L*). At the lowest pump power *PP0*, only spontaneous RS occurs with signal from 1090 to 1130 nm, its maximum level is around �68.5 � 1 dBm. Taking this level as *PF0*, the difference is ΔPower = 50.2 � 1 dBm. The fiber loss is 1.5 dB/Km at 1064 nm according with the technical specification provided by de manufacturer. Substituting these data on the Eq. (13) yields *PP0*/*PF0* = 147,945.506 and ln(*PP0*/*PF0*)

*PcrLeff*

*PP0*. These numbers are of great importance, instead

Now, substituting Eq. (7) in (6), it is possible to numerically establish a

*PPO PFO*

*gR* <sup>¼</sup> *gr Aeff*

relationship for *PP0*/*PF0* given by:

*Application of Optical Fiber in Engineering*

from **Figure 2**.

rewriting Eq. (9) as:

= 11.9, hence *PF0* = 6.759x10�<sup>6</sup>

within the values expected.

*gR* = 2.1[W�<sup>1</sup> km�<sup>1</sup>

value, denoted by *P*<sup>1</sup>*st*

**86**

�*α<sup>L</sup>* (12)

*<sup>α</sup><sup>L</sup>* (13)

(14)

. We propose

(15)

An analytical equation was developed to predict the pumping power necessary to store the maximum energy in the first Stokes, this is a powerful tool for numerical simulations, which can provide specific data for the design of Raman fiber lasers. Consequently, an analytical equation is deduced that can predict the stored power in the following Stokes orders, that is, progressively, the second, the third, etc.

For a 1 km fiber span, 13.39 W of pumping power is required to get the first Stokes to reach its maximum stored energy level of 9.88 W, while for 5-km of fiber span a pump power of 4.77 W is required to obtain 0.88 W. It is important to mention that the experimental calculation of ln(PP0/PF0) let us conclude that the fiber is not intended for telecommunications as the �16 number is not satisfied.

Finally, it is important to note that commercial single-mode fiber lasers normally operate at pump powers of around 10 W, therefore, an estimate such as the one made in this work is fundamental.

*Application of Optical Fiber in Engineering*
