OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 18 — Sep. 6, 2004
  • pp: 4235–4245
« Show journal navigation

Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers

Malin Premaratne  »View Author Affiliations


Optics Express, Vol. 12, Issue 18, pp. 4235-4245 (2004)
http://dx.doi.org/10.1364/OPEX.12.004235


View Full Text Article

Acrobat PDF (230 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).

]. One of the most promising technologies that can significantly overcome these impairments is optical amplification by stimulated Raman scattering [2

2. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. 8, 548–559 (2002). [CrossRef]

]. This is mainly attributed to the fact that Raman gain is non-resonant, broadband and reasonably spectrally flat over a wide wavelength range [2

2. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. 8, 548–559 (2002). [CrossRef]

, 3

3. M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).

]. 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

4. V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” IEEE J. Lightwave Technol. 20, 250–254 (2002). [CrossRef]

, 5

5. P. C. Xiao, Q.J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003). [CrossRef]

]. 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

6. M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer-Verlag, New York, 2003).

]. 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

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

, 8

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

, 9

9. B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. 24, 660–664 (1988). [CrossRef]

]. Dakss and Melman [7

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

] 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

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

] 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

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

].

The second method generates an exact solution for unequal signal and pump loss coefficients using hypergeometric special functions [11

11. D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).

, 12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]. 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

14. P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998). [CrossRef]

, 15

15. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” IEEE J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

] and/or Brillouing scattering [16

16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002). [CrossRef]

].

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

3. M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).

, 6

6. M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer-Verlag, New York, 2003).

].

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.

Fig. 1. A schematic diagram of forward pumped Raman amplifier.

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

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

, 8

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

]:

dn(z)dz=γnp(z)(n(z)+1)αSn(z)
(1)
dnP(z)dz=γnp(z)(n(z)+1)αPnP(z)
(2)

where γ is the Raman gain coefficient, αS is the signal loss coefficient and αP 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

17. J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. 14, 347–352 (1978). [CrossRef]

]. 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

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

, 8

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

].

nP(z)=nP(0)exp(αpz)
(3)

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

dn(z)dz+(αSγnP(0)exp(αPz))n(z)=γnp(0)exp(αPz)
(4)

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

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

] have shown that an analytical result for Eq. (4) exists when αP = αS. To exploit this feature, we seek a solution for Eq. (4) as a power series of the parameter, α = 1 -αS/αP,which represents the relative difference of signal and pump attenuation coefficients.

n(z)=i=0ni(z)αi
(5)

where ni(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 αP tends to αS. Thefore, when signal and pump loss coefficients are equal, the above power series reduces to a single term n 0(z) with known analytical result. Substitution of Eq. (5) to Eq. (4) results in following set of coupled system of differential equations:

dn0(z)dz+(αPγnP(0)exp(αPz))n0(z)=γnp(0)exp(αPz)
(6)
dni(z)dz+(αPγnP(0)exp(αPz))ni(z)=αPni1(z),i0
(7)

Eq. (6) can be solved analytically by using an integrating factor [11

11. D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).

] to get the following closed form solution

n0(z)=HP(z)exp(u0)n(0)+HP(z)(Ei(u0)Ei(u(z)))u0
(8)

where u 0 =γnP(0)/αP, u(z) = u 0 exp(-αPz) and HP(z) = exp(-αPz-u(z)) . In Eq. (8), Ei(x) denotes the exponential integral [12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]

Ei(x)=P.V.xu1exp(u)du,x>0
(9)

where P.V. denotes the principle value [13

13. M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).

] 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) :

n1(z)=αPHP(z)(exp(u0)n(0)+u0Ei)u0))zαPHP(z)u00zEi(u(z))dz
(10)

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:

0zEi(u(z))dz(C+ln(u0))zαP2z2+u0ln(u0)αP(1u2(z)u02)
(11)

where parameter C = 0.5772156649⋯ is called the Euler’s constant [12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]. Substitution of this expression to Eq. (10) results in following approximate expression:

n1(z)=ααP×exp(u0)n(0)z+αP2u0z2+(Ei(u0)Cln(u0)))u0zu01+ln(u0)αP(1u2(z)u02)HP(z)
(12)

Continuing this process and ignoring higher order terms, it is possible to calculate approximate expressions for functions ni(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:

n(z)HP(z)(exp(u0)n(0)+u0(Ei(u0)Ei(u(z)))
+ααP×exp(u0)n(0)z+αP2u0z2+(Ei(u0)Cln(u0)))u0zu01+ln(u0)αP(1u2(z)u02)
+i=2(αPz)ii!(exp(u0)n(0)+u0Ei(u0))αi
(13)

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

n(z)=GS(z)n(0)+(αPαS)HP(z)HS(z)(z)
+HS(z)u0Ei(u0)HP(z)u0Ei(u(z))
(14)

where HP(z) = exp(-αPz-u(z)), HS(z) = exp(-αSz-u(z)) and Raman signal gain at distance z from left end of fiber (c.f., Fig. 1) is given by GS(z) = exp(u 0)HS(z). The term ℘(z) represent a scattering noise contribution when αP = αS and is given by

(z)=αP2u0z2u0(C+ln(u0))zu01+ln(u0)αP(1u2(z)u02)
(15)

It is interesting to consider value of Eq. (14) when α → 0 (i.e.αPαS. Then HP(z) → HS(z) and we get the following result:

n(z)α=0=GS(z)n(0)+u0HS(z)×(Ei(u0)Ei(u(z)))
(16)

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

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

].

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:

n(z)=GS(z)×n(0)
(17)

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

n(z)=GS(z)×n̂(z)
(18)

where (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 (z):

dn̂(z)dz=αPu0exp(αPαzu0(1exp(αPz)))
(19)

where u 0 = γnP(0)/αP, u(z) = u 0 exp(-αPz) and α= 1-αS/αP. Integrating along the fiber and using (0) = n(0) results in

n̂(0)=n(0)+u0exp(u0)×I(z)
(20)

where the integral I(z)is defined as

I(z)=1u0αu(z)u0vα1exp(v)dv
(21)

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:

I(z)=k=0(α)ku0k(α+1)kk!(u(z)u0)αk=0(α)kuk(z)(α+1)kk!
(22)

where we utilized the definition of Pochhammers symbol, (α)k = α(α + 1) ⋯ (α + k - 1), [12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]. But noting that the confluent hypergeometric function is defined by [12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]

Φ(a;b;z)=k=0(a)kzk(b)kk!
(23)

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

n(z)=GS(z)n(0)+u0HS(z)α(Φ(α;α+1;u0)Φ(α;α+1;u(z))(u(z)u0)α)
(24)

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

nnoise(0)=u0exp(u0)α(Φ(α;α+1;u0)Φ(α;α+1;u(z))(u(z)u0)α)
(25)

n(z)α=0=GS(z)n(0)+
u0HS(z)×limα0d(Φ(α;α+1;u0)Φ(α;α;+1;u(z))(u(z)u0)α)
(26)

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

n(z)α=0=GS(z)n(0)+u0HS(z)×(k=1u0kk·k!+ln(u0)k=1uk(z)k·k!ln(u(z)))
(27)

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

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]

Ei(x)=C+ln(x)+k=1xkk·k!
(28)

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

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

]

NF(z)=1+2(n(z)GS(z)n(0))GS(z)
(29)

By substituting Eq. (14) to Eq. (29), we get the following expression for the noise figure NFa(z) for power series solution Eq. (14)

NFa(z)=1+2(αPαS)HP(z)HS(z)(z)+2(HS(z)u0Ei(u0)HP(z)u(z)Ei(u(z)))exp(u0)HS(z)
(30)

and following expression for the noise figure NFe(z) for exact analytical solution using Eq. (24)

NFe(z)=1G(z)+2u0exp(u0)α(Φ(α;α+1,u0)Φ(α;α+1,u(z)))
(31)

We show that both NFa(z) and NFe(z) approach 2.0 (ie. 3dB) for large Raman gain, GS(z) Large Raman gain amounts to large pump powers (ie. large u 0 ) and long amplifiers (ie. z ≈ 1/αP ). Under these conditions, first term of Eq. (30) approaches zero rapidly and hence, Eq. (30) can be written in following approximate form

NFa(z)2(u0Ei(u0)u(z)Ei(u(z)))exp(u0)
(32)

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

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

].

Ei(x)=exp(x)x(1+O(1x))
(33)

where O is a Landue symbol [12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

] . Substituion of Eq. (33) to Eq. (32) shows that NFa → 2 for large u 0 and z ≈ 1/αP (ie. large Raman gain). Similarly taking limit of Eq. (31) for large Raman gain (ie. large u 0 ) and long amplifiers (ie. z ≈ 1/αP ), it can be shown that NFe → 2.

6. Comparison with simulations

Fig. 2. Photon number, n(z), against fiber length for two different pump powers PP and pump loss coefficients αP

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 αP = 0.30dB/km and αP = 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

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 αS =αP with αP = 0.30dB/km and αP = 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 αS = αP is not accurate enough for characterizing forward-pumped Raman amplifiers.

Figure 4 shows noise figure, Eq. (29) , against fiber length for two different pump loss coefficients (a) αP = 0.4dB and (b) αP = 0.3dB/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.

Fig. 3. I(z) against fiber length for two different pump loss coefficients,αP = 0.4 dB/km and αP = 0.3 dB/km.
Fig. 4. Noise Figure against fiber length for two different pump loss coefficients: (a) αP = 0.4 dB/km (b) αP = 0.3 dB/km

7. Applications

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

dn(z)dz=(k=1MγknPk(z))(n(z)+1)αSn(z)
(34)
dnPi(z)dz=γinPi(z)(k=1kiMnPk(z)+n(z)+1)αPinPi(z),i=1,2,,M
(35)

where γk : k = 1,⋯,M are the Raman gain coefficients and αPk : k = 1,⋯,M are the pump loss coefficient for M pump wavelengths with photon numbers, nPk(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

18. A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001). [CrossRef]

] for forward propagating pumps and seeking a solution for signal photon number, n(z) as

n(z)=i=0ni(z)(1αMk=1Mαk)i
(36)

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

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

]. 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

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

].

The second method generates an exact solution for unequal signal and pump loss coefficients using hypergeometric special functions [11

11. D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).

, 12

12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]. 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

16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002). [CrossRef]

].

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.

References and links

1.

G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).

2.

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. 8, 548–559 (2002). [CrossRef]

3.

M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).

4.

V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” IEEE J. Lightwave Technol. 20, 250–254 (2002). [CrossRef]

5.

P. C. Xiao, Q.J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003). [CrossRef]

6.

M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer-Verlag, New York, 2003).

7.

M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985). [CrossRef]

8.

R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997). [CrossRef]

9.

B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. 24, 660–664 (1988). [CrossRef]

10.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

11.

D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).

12.

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

13.

M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).

14.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998). [CrossRef]

15.

P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” IEEE J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

16.

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. 20, 1635–1643 (2002). [CrossRef]

17.

J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. 14, 347–352 (1978). [CrossRef]

18.

A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

ToC Category:
Research Papers

History
Original Manuscript: July 27, 2004
Revised Manuscript: August 25, 2004
Published: September 6, 2004

Citation
Malin Premaratne, "Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers," Opt. Express 12, 4235-4245 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-18-4235


Sort:  Journal  |  Reset  

References

  1. G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).
  2. M. N. Islam, �??Raman amplifiers for telecommunications,�?? IEEE J. Select. Topics in Quantum Electron. 8, 548 �?? 559 (2002). [CrossRef]
  3. M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).
  4. V. E. Perlin and H. G.Winful,�??Optimal design of flat-gain wide-band fiber Raman amplifiers,�?? IEEE J. Lightwave Technol.20, 250 �?? 254 (2002). [CrossRef]
  5. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, �??A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,�?? IEEE Photon. Technol. Lett. 15, 206 �?? 208 (2003). [CrossRef]
  6. M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer- Verlag, New York, 2003).
  7. M. L. Dakss, and P. Melman, �??Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,�?? J. Lightware Technol. 3, 806 �?? 813 (1985). [CrossRef]
  8. R. Chinn, �??Analysis of counter-pumped small-signal fiber Raman amplifiers,�?? Electron Lett. 33, 607 �?? 608 (1997). [CrossRef]
  9. B. Bobbs, C.Warner, �??Closed-form solution for parametric second Stokes generation in Raman amplifiers,�?? IEEE J. Quantum Electron. 24, 660 �?? 664 (1988). [CrossRef]
  10. Y. Yan, J. Chen; W. Jiang, J. Li, J. Chen, X. Li, �??Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,�?? IEEE Photon. Technol. Lett. 13, 948 �?? 950 (2001). [CrossRef]
  11. D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).
  12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).
  13. M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).
  14. P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, D. J. DiGiovanni �??Rayleigh scattering limitations in distributed Raman pre-amplifiers,�?? Photon. Technol. Lett. 10, 159 �?? 161 (1998). [CrossRef]
  15. P. Parolari, L. Marazzi, L. Bernardini, M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�??IEEE J. Lightwave Technol. 21, 2224 �?? 2228 (2003). [CrossRef]
  16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, A. F. Evans, �??Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,�?? IEEE J. Lightwave Technol. 20, 1635 �?? 1643 (2002). [CrossRef]
  17. J. Auyeung and A. Yariv, �??Spontaneous and stimulated Raman scattering in long low loss fibers,�?? IEEE J. Quantum Electron. 14, 347 �?? 352 (1978). [CrossRef]
  18. A. Bononi, M. Papararo, A. Vannucci, �??Impulsive pump depletion in saturated Raman amplifiers,�?? Electron. Lett. 37, 886 �?? 887 (2001). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited