## Abstract

We show that it is possible to find analytic expressions for characterizing the evolution of signal and noise photon numbers along the active fiber of a forward-pumped Raman amplifier with unequal signal and pump loss coefficients. We confirm the validity of the result by comparing the analytical solutions with numerical calculations and by analytically deriving the well-known 3 dB noise figure limit for high Raman gain. Apart from aiding the analysis and design of forward pumped Raman amplifiers, these results also enable one to find approximate analytical solutions for bidirectional Raman amplifiers and backward pumped Raman amplifiers with Rayleigh backscattering and Brillouin scattering.

©2004 Optical Society of America

## 1. Introduction

The transmission of many wavelength division multiplexed (WDM) channels over highspeed/long-haul systems can be limited by variety of impairments including the limited bandwidth of optical amplifiers and their noise performance [1]. One of the most promising technologies that can significantly overcome these impairments is optical amplification by stimulated Raman scattering [2]. This is mainly attributed to the fact that Raman gain is non-resonant, broadband and reasonably spectrally flat over a wide wavelength range [2, 3]. As standard installed fiber has a reasonbly strong Raman scattering response at standard telecommunicaitons windows, it is also possible to use already installed fiber as a gain medium for stimulated Raman scattering based distributed amplification. Since stimulated Raman scattering process is non-resonant, it has been shown that the bandwidth of Raman amplifiers can be further extended and arbitrary gain profile can be synthesized using multiple pumping wavelengths [4, 5]. Moreover, as a result of low noise performance of Raman amplifiers compared with conventional Erbium doped fiber amplifiers (EDFAs), average optical channel power required to achieve an acceptable signal to noise ratio along installed system fiber can be reduced considerably [6]. This reduction in channel power directly leads to the reduction in nonlinear impairments, thus improving the overall system performance. Therefore, much research effort has been spent on designing efficient Raman amplifiers for long-haul WDM systems.

There are several publications on analytical characterization of forward and backward pumped Raman amplifiers [7, 8, 9]. Dakss and Melman [7] have derived analytical expressions for signal and noise power evolution along the fiber of a forward pumped Raman amplifier. However, their results are applicable only for fibers with equal signal and pump loss coefficients. Chinn [8] has derived analytical expressions for signal and noise-figure evolution along the fiber of backward-pumped Raman amplifiers with un-equal signal and pump loss coefficients. However, similar study has not been done for forward-pumped Raman amplifiers.

In this paper, We show that it is possible to find analytic expressions for characterizing the evolution of signal and noise photon numbers along the active fiber of a forward-pumped Raman amplifier with unequal signal and pump loss coefficients using two different methods.

The first method is based on expanding signal and noise photon numbers along the fiber as a power series of relative difference of signal and pump attenuation coefficients. The main attractiveness of this method is, even though approximate, it allow us to derive approximate solutions for multiple-wavelength-pumping configurations. Apart from giving insight into the interaction between pumps in multiple-wavelength-pumping configurations, such analytical results can also be used to seed more esotoric methods of configuring wavelengths and powers [10].

The second method generates an exact solution for unequal signal and pump loss coefficients using hypergeometric special functions [11, 12]. However, unlike the power series solution, the exact solution is valid only for a single pump wavelength. Approximate analytical result for multiple pumping can be obtained from this method by using effective values for pump power, pump loss coefficient and pump wavelength. This analytical solution can be used to derive compact analytical results for backward pumped Raman amplifiers with Rayleigh backscattering [14, 15] and/or Brillouing scattering [16].

For both methods, we confirm the validity of the result by comparing the analytical solutions with numerical results obtained by direct integration of signal evolution equations and by theoretically demonstrating the derivation of the well-known 3 dB noise figure limit for high Raman gain [3, 6].

This paper is organized as follows: In Section 2, we describe the photon number evolution equations for signal and noise photons along a forward pumped optial fiber. In Section 3, without loss of generality, we describe the approximate power series approach for single pumping case. As stated before, direct extension of this method allows us to derive approximate analytical results for forward multiple-wavelength-pumped Raman amplifiers. Moreover, the method of solution also enable us to highlight all of the previously known analytical solutions for forward pumped Raman amplifiers. In Section 4, we show the derivation of the exact analytical result for forward pumped Raman amplifiers using hyper-geometric special functions. In Section 5, we derive analytical expressions for noise figure of forward pumped Raman amplifier using power series and hypergeometric function methods. We show theoretically that the noise figure expressions of both methods tend to the well-known 3 dB value for high Raman gain. In Section 6, a comprehensive numerical analysis is given to compare the power series and hyper-geometric function results against numerical results generated by direct integration of photon number evolution equtions in Section 2. Section 7 will gives a qualitative directions for applying the derived analytical result for two different amplifier configurations: 1. backward pumped Raman amplifiers with Rayleigh backscattering and Brillouing scattering and 2. bi-directional Raman amplifiers. This sections also shows steps required to extend the power series method discussed in Section 3 for multiple pump wavelenghths. Section 9 will concludes this paper after summarizing the major results.

## 2. Theoretical model

Figure 1 shows a schematic diagram of forward pumped fiber Raman amplifier considered in this paper. The signal propagation length of the fiber, *z* , is measured from the end of the pump coupler to the right as shown in Fig. 1.

Withouth loss of generality, we consider the single pumping case throughout this paper. Extension of the results to multiple pumping case is qualtitatively considered in the Section 7. In the case of single pumping scheme, the evolution of signal and noise photon number, *n*(*z*), and pump photon number, *nP*(*z*), along the fiber can be characterized by following differential equations [7, 8]:

where *γ* is the Raman gain coefficient, *α _{S}* is the signal loss coefficient and

*α*is the pump loss coefficient. The experimental evidence for the adequacy and validity of Eq. (1) and Eq. (2) for describing Raman amplification in optical fibers can be found in [17]. Under normal small signal operating conditions of Raman amplifiers, depletion of pump can be ignored with high accuracy, leading to following expression for pump evolution along the fiber [7, 8].

_{P}Substitution of Eq.(3) to Eq.(1) gives following differential equation for signal and noise photon number, *n*(*z*):

Equation (4) represents the signal and noise photon number evolution along the fiber. We solve this approximately and exactly in Section 3 and Section 4, respectively.

## 3. Approximate analytical solution using power series

In this section we analytically solve Eq. (4) using a power series of relative difference of signal and pump attenuation coefficients. The main attractiveness of this method is, even though approximate, it allow us to derive approximate solutions for multiple-wavelength-pumping configurations. Apart from giving insight into the interaction between pumps in multiple-wavelength-pumping configurations, such analytical results can also be used as initial guesses for iterative numerical solution techniques for configuring multiple pumping schemes. Dakss and Melman [7] have shown that an analytical result for Eq. (4) exists when *α _{P}* =

*α*. To exploit this feature, we seek a solution for Eq. (4) as a power series of the parameter,

_{S}*α*= 1 -

*α*/

_{S}*α*,which represents the relative difference of signal and pump attenuation coefficients.

_{P}where *n _{i}*(

*z*) :

*i*= 0,1,⋯,∞ are some auxiliary functions that need to be determined recursively by using lower order solutions. It is interesting note that the parameter

*α*tends to zero as

*α*tends to

_{P}*α*. Thefore, when signal and pump loss coefficients are equal, the above power series reduces to a single term

_{S}*n*

_{0}(

*z*) with known analytical result. Substitution of Eq. (5) to Eq. (4) results in following set of coupled system of differential equations:

Eq. (6) can be solved analytically by using an integrating factor [11] to get the following closed form solution

where *u*
_{0} =*γn _{P}*(0)/

*α*,

_{P}*u*(

*z*) =

*u*

_{0}exp(-

*α*) and

_{P}z*H*(

_{P}*z*) = exp(-

*α*-

_{P}z*u*(

*z*)) . In Eq. (8), Ei(

*x*) denotes the exponential integral [12]

where P.V. denotes the principle value [13] of the integral of Eq. (9). Substitution of explicit expression for *n*
_{0}(*z*) given in Eq. (8) to Eq.(7) when *i* = 1 gives the following analytic expression for *n*
_{1}(*z*) :

The integral in Eq. (10) is uniformly convergent. Therefore, it can be evaluated approximately by using term-by-term integration of the series expansion of the exponential integral to get the following expression:

where parameter *C* = 0.5772156649⋯ is called the Euler’s constant [12]. Substitution of this expression to Eq. (10) results in following approximate expression:

Continuing this process and ignoring higher order terms, it is possible to calculate approximate expressions for functions *n _{i}*(

*z*) :

*i*= 2,3,⋯,∞ . Substituting these obtained values to Eq. (5) gives the following expression for signal and noise power evolution along the Raman amplifier:

$$\phantom{\rule{2em}{0ex}}+\alpha {\alpha}_{P}\times \left(\genfrac{}{}{0ex}{}{\mathrm{exp}\left({u}_{0}\right)n\left(0\right)z+\frac{{\alpha}_{P}}{2}{u}_{0}{z}^{2}+(\mathrm{Ei}\left({u}_{0}\right)}{-C-\mathrm{ln}\left({u}_{0}\right)\left)\right){u}_{0}z-\frac{{u}_{0}^{1+\mathrm{ln}\left({u}_{0}\right)}}{{\alpha}_{P}}\left(1-\frac{{u}^{2}\left(z\right)}{{u}_{0}^{2}}\right)}\right)$$

$$\phantom{\rule{2em}{0ex}}+\sum _{i=2}^{\infty}\frac{{\left({\alpha}_{P}z\right)}^{i}}{i!}\left(\mathrm{exp}\left({u}_{0}\right)n\left(0\right)+{u}_{0}\mathrm{Ei}\left({u}_{0}\right)\right){\alpha}^{i}$$

Even though this expression looks bit complicated, it can be written in more intuitive compact form after considering underlying physical processes:

$$\phantom{\rule{2em}{0ex}}+{H}_{S}\left(z\right){u}_{0}\mathrm{Ei}\left({u}_{0}\right)-{H}_{P}\left(z\right){u}_{0}\mathrm{Ei}\left(u\left(z\right)\right)$$

where *H _{P}*(

*z*) = exp(-

*α*-

_{P}z*u*(

*z*)),

*H*(

_{S}*z*) = exp(-

*α*-

_{S}z*u*(

*z*)) and Raman signal gain at distance

*z*from left end of fiber (c.f., Fig. 1) is given by

*G*(

_{S}*z*) = exp(

*u*

_{0})

*H*(

_{S}*z*). The term ℘(

*z*) represent a scattering noise contribution when

*α*=

_{P}*α*and is given by

_{S}It is interesting to consider value of Eq. (14) when *α* → 0 (i.e.*α _{P}* →

*α*. Then

_{S}*H*(

_{P}*z*) →

*H*(

_{S}*z*) and we get the following result:

This result has already been reported for this special case in [7].

## 4. Exact analytical solution

In this section we solve Eq. (4) exactly using hypergeometric special functions. This is done by first looking at the solution of Eq. (4) without noise terms and then seeking a general solution having a similar form. The solution for Eq. (4) without the noise term can be written as:

Using this, we seek a solution for Eq. (4) in the following form:

where *n̂*(*z*) is an unknown function which assumes input photon number (i.e. *n*(0)) when Raman amplifier is free of noise. Substitution of Eq. (18) to Eq. (4) gives a differential equation for *n̂*(*z*):

where *u*
_{0} = *γn _{P}*(0)/

*α*,

_{P}*u*(

*z*) =

*u*

_{0}exp(-

*α*) and

_{P}z*α*= 1-

*α*/

_{S}*α*. Integrating along the fiber and using

_{P}*n̂*(0) =

*n*(0) results in

where the integral *I*(*z*)is defined as

Using the Taylor series expansion for exp(*v*) and integrating term by term (note: this operation is valid given that the resulting series is uniformly convergent for positive real numbers), *I*(*z*) can be written in the following form:

where we utilized the definition of Pochhammers symbol, (*α*)_{k} = *α*(*α* + 1) ⋯ (*α* + *k* - 1), [12]. But noting that the confluent hypergeometric function is defined by [12]

we can write the following closed form solution for *n*(*z*) along the fiber

This shows that total noise photon number generated by the forward pumped Raman amplifier can be reduced to an equivalent noise photon injection, *n _{noise}*(0) at input with following value:

As done in the end of previous Section, it is interesting to study the behavior of Eq. (24) when *α _{P}* →

*α*because exact analytical solution for this case has already been reported in literature [3, 5]. When

_{S}*α*→

_{P}*α*,α → 0 and hence second bracketed term in Eq.(24) assumes 0/0 indeterminate form. Therefore, limit of Eq. (24) as

_{S}*α*→ 0 needs to be calculated. Applying LHospitals rule [12], Eq. (24) can be written in the limit

*α*→ 0 as:

$${u}_{0}{H}_{S}\left(z\right)\times \underset{\alpha \to 0}{lim}\frac{d}{\mathit{d\alpha}}\left(\Phi (\alpha ;\alpha +1;{u}_{0})-\Phi (\alpha ;\alpha ;+1;u\left(z\right)){\left(\frac{u\left(z\right)}{{u}_{0}}\right)}^{\alpha}\right)$$

Term by term differential of the definition given in Eq. (23) and letting *α* → 0 results in

But noting that the exponential integral Ei(*x*) is defined using Euler’s constant *C* = 0.5772156649⋯ as [12]

we can obtain the result given in Eq. (16), confirming the equality of the two approches in the limit of equal pump and signal loss coefficients.

## 5. Noise Figure

The noise figure of Raman amplifier, NF(*z*), at distance *z* from input can then be written as [8]

By substituting Eq. (14) to Eq. (29), we get the following expression for the noise figure NF_{a}(*z*) for power series solution Eq. (14)

and following expression for the noise figure NF_{e}(*z*) for exact analytical solution using Eq. (24)

We show that both NF_{a}(*z*) and NF_{e}(*z*) approach 2.0 (ie. 3dB) for large Raman gain, *G _{S}*(

*z*) Large Raman gain amounts to large pump powers (ie. large

*u*

_{0}) and long amplifiers (ie.

*z*≈ 1/

*α*). Under these conditions, first term of Eq. (30) approaches zero rapidly and hence, Eq. (30) can be written in following approximate form

_{P}The asymptotic limit of this expression for large *u*
_{0} can be calculated by using an asymptotic expansion of exponential integral for large *x* [12].

where O is a Landue symbol [12] . Substituion of Eq. (33) to Eq. (32) shows that NF_{a} → 2 for large *u*
_{0} and *z* ≈ 1/*α _{P}* (ie. large Raman gain). Similarly taking limit of Eq. (31) for large Raman gain (ie. large

*u*

_{0}) and long amplifiers (ie.

*z*≈ 1/

*α*), it can be shown that NF

_{P}_{e}→ 2.

## 6. Comparison with simulations

We use the numerical solution of Eq. (1) and Eq. (2) under un-depleted pump approximation to test the validity of analytical solutions derived in Sections 3 and 4. Photon numbers were converted into optical powers by using the mapping *γn _{P}*(0) →

*g*where

_{Rm}P_{P}*g*is the modal Raman gain with and

_{Rm}*P*is the input pump power [8]. Following parameters were used in all the numerical calculations unless explicitly specified otherwise:

_{P}*g*= 0.67W

_{Rm}^{-1}km-1 ,

*α*= 0.22dB/km and

_{S}*P*= 400mW.

_{P}Figure 2 shows the photon number, *n*(*z*) of approximate power series solution in Eq. (13) against fiber length for two different pump loss coefficients *α _{P}* = 0.30dB/km and

*α*= 0.40dB/km. The solid lines in Fig. 2 shows the approximate power series solution while dashed lines represent the numerical solution of Eq. (1) and Eq. (2). Figure 2 clearly shows that the analytical solution is in very good agreement with typical usage limits of Raman amplifiers. It also shows that the accuracy of the analytical solution increases as the Raman gain increases. This is expected because the mismatch of signal and pump loss coefficients influence mainly the noise added by the Raman amplifier

_{P}To compare the accuracy of exact analytical solution, we look at the behavior of Eq. (21). Figure 3 shows the exact analytical solution of integral *I*(*z*) (c.f. Eq. (21))against fiber length. The solid-lines in Fig. 3 represent the hyper-geometrical functions based solution, while triangles represent the results of direct numerical integration. For comparison, *I*(*z*) for *α _{S}* =

*α*with

_{P}*α*= 0.30dB/km and

_{P}*α*= 0.40dB/km are shown using dashed lines in Fig. 3. These results clearly confirm the validity of the current derivation within the typical usage limits of Raman amplifiers. It also shows that commonly used approximation

_{P}*α*=

_{S}*α*is not accurate enough for characterizing forward-pumped Raman amplifiers.

_{P}Figure 4 shows noise figure, Eq. (29) , against fiber length for two different pump loss coefficients (a) *α _{P}* = 0.4

*dB*and (b)

*α*= 0.3

_{P}*dB*/

*km*. The solid-lines in Fig. 4 represent the analytical solution, Eq. (31), while triangles represent the numerical solution obtained by substituting the numerically integrated solution of Eq. (1) and Eq. (2) to Eq. (29). For comparison, widely used analytical expression of noise figure with equal pump and signal attenuation coefficients are shown using dashed lines. These results clearly confirm the validity of the current derivation within the typical usage limits of Raman amplifiers. It also shows that commonly used equal pump and loss coefficient approximation is not accurate enough for characterizing forward-pumped Raman amplifiers. In conclusion, the analytical results presented here can be used to get insight into the noise performance of forward pumped Raman amplifiers and hence to improve their design.

## 7. Applications

Apart from getting insight into the operation of forward pumped Raman amplifiers, one of the main applications of the current theory is that, it enable us to approximately solve bidirectional Raman amplifiers [3, 6] and backward pumped Raman amplifiers with Rayleigh [14, 15] or Brillouin scattering [16]. Noting that both problems highlighted above have a similar mathematical structure, without the loss of generality, we limit our qualitative discussion to the case for Rayleigh scattering. Introduction of Rayleigh scattering leads to the initiation of two counter propagating Stokes waves in the Raman fiber with a coupling coefficient related to the Rayleigh scattering strength [14]. This coupled differential system can then be solved using a form similar to the power series method discussed in Section 3, leading to a recursive set of differential equations. The 0-order forward and backward propagating Stokes wave equations can then be easily solved using the Chinns method [8] and the confluent hyper geometric function solution method of this paper, respectively. These analytical solutions can then be used to recursively refine the resulting solution to a very good approximation.

The power series method presented in the the Section 3 can be easily extended to derive an approximate analtycal method for multiple pumping wavelength case. Assume that there are *M* different laser beams pumps the Raman fiber. Then, Eq. (1) and Eq. (2) need to be modified to

where *γk* : *k* = 1,⋯,*M* are the Raman gain coefficients and *α _{Pk}* :

*k*= 1,⋯,

*M*are the pump loss coefficient for

*M*pump wavelengths with photon numbers,

*n*(

_{Pk}*z*). All the other parameters assume same meaning as in Eq. (1) and Eq. (2). Using an approximation similar to impulsive pump depletion approximation [18] for forward propagating pumps and seeking a solution for signal photon number,

*n*(

*z*) as

we could arrive at an approximate analytical solution for multiple pump wavelengths. Such solutions are especially useful for seeding more accurate numerical methods for flatterning gain spectrum of Raman amplifiers [10]. The exact details of the above two approaches are beyond the scope of this paper and published elsewhere.

## 8. Conclusion

Analytic expressions for optical power and noise figure of forward-pumped Raman amplifier were derived for unequal signal and pump loss coefficients using two different methods. The first method is based on expanding signal and noise photon numbers along the fiber as a power series of relative difference of signal and pump attenuation coefficients. The main attractiveness of this method is, even though approximate, it allow us to derive approximate solutions for multiple-wavelength-pumping configurations. Apart from giving insight into the interaction between pumps in multiple-wavelength-pumping configurations, such analytical results can also be used to seed more esotoric methods of configuring wavelengths and powers [10].

The second method generates an exact solution for unequal signal and pump loss coefficients using hypergeometric special functions [11, 12]. However, unlike the power series solution, the exact solution is valid only for a single pump wavelength. Approximate analytical result for multiple pumping can be obtained from this method by using effective values for pump power, pump loss coefficient and pump wavelength. This analytical solution can be used to derive compact analytical results for backward pumped Raman amplifiers with Rayleigh backscatte-ring [14, 15] and/or Brillouing scattering [16].

We confirmed the accuracy of the calculation by direct comparison with numerical simulation of the representative evolution equations. We also showed analytically and numerically that at high Raman gain values, the noise figure of forward pumped Raman amplifier approaches the well-known 3dB noise figure limit. The results presented here can be used to get insight into the operation of forward pumped Raman amplifiers and hence to improve their design.

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