## Abstract

The method to design the incoherent pump power spectrum described with a set of piece-wise continuous functions (PWCFs) for the distributed fiber Raman amplifier (DFRA) is presented. The pump power spectrum is divided into a number of sub-bands, in which each sub-band is described with a polynomial. The power spectral density function (PSDF) is the absolute value of the set of PWCFs, in which the polynomial coefficients are optimized with the least-square minimization method for reducing the signal gain ripple. Two 100-km TW-Reach DFRAs using backward pumping and bidirectional pumping respectively are taken as examples. The numerical results show that the gain ripple of less than 0.02 dB over 70-nm bandwidth can be achieved. The spectral characteristics of the optimized PSDF for the ultra-low gain ripple are investigated. The optimized PSDF can be synthesized with multiple incoherent pumps. The synthesis examples using the multiple Gaussian incoherent pumps are shown, in which the gain ripples are increased to 0.3 dB due to the discrepancy between the optimized PSDF and the synthesized PSDF. The gain ripples can be reduced to 0.05 dB by further optimizing the parameters of the multiple Gaussian incoherent pumps.

© 2006 Optical Society of America

## 1. Introduction

Distributed fiber Raman amplifiers (DFRAs) have the advantages of improving optical-signal-to-noise ratio (OSNR) and reducing fiber nonlinear impairments [1–5]. Recently the DFRA using incoherent pumping is interested and experimentally demonstrated [6–8]. Comparing to coherent pumping, incoherent pumping has the advantages of polarization insensitivity and reducing the nonlinear effects, such as the stimulated Brillouin scattering (SBS) of pumps [9] and the four-wave mixings (FWMs) of pump-pump, pump-signal, and pump-noise [10–13]. Both the SBS and FWMs can be reduced owing to the broadband and low power spectral density (PSD) of the incoherent pumps. The random phase of the incoherent pumps further reduces the FWMs.

An incoherent pump provides wider gain bandwidth than a coherent pump because of its larger spectral width. Broadband equalized gain is required for the applications of the optical amplifiers to the wavelength-division-multiplexing (WDM) systems, which can be achieved by the use of the multiple coherent pumping [2–5], multiple incoherent pumping [6, 7], and composite coherent-incoherent pumping [8]. Using the multiple coherent pumping, the gain ripple of a DFRA can be reduced by optimizing the wavelengths and powers of the pumps. Using the multiple incoherent pumping, the gain ripple of the DFRA can be reduced by optimizing the bandwidths of the incoherent pumps in addition to their wavelengths and powers. Reference [7] shows that the gain ripple of the DFRA using two incoherent counter-pumps is less than that of the DFRA using six coherent counter-pumps.

In this paper a method to design the pump spectra for minimizing the gain ripple of the DFRAs using the incoherent pumping is presented. The spectral characteristics of the optimized pumps are studied. By the theory of Taylor series expansion, an incoherent pump power spectral density function (PSDF) can be represented by a polynomial as a function of optical wavelength or frequency in general. The PSDF for low gain ripple can be found by optimizing the coefficients of the polynomial in principle. However, as the required polynomial order is high, the oscillating nature of the high order terms leads to poor optimization result. In this paper, we take the other strategy to design the incoherent pumps by optimizing a set of piece-wise continuous functions (PWCFs) representing the pump PSDF. The pump spectrum is divided into a number of sub-bands, in which each sub-band is described with a polynomial of low order so that the pump spectrum can be effectively optimized. The results show that the gain ripple can be less than 0.02 dB over 70-nm bandwidth with this optimization method. The spectral characteristics of the optimized PSDFs for the ultra-low gain ripple are investigated. The implementations of the optimized pumps are also discussed. Examples of synthesizing the optimized pump power spectra with the multiple Gaussian incoherent pumps are shown.

## 2. Amplifier model and optimization method

The Raman amplification in a DFRA can be described with a set of coupled differential equations that represent the steady-state power evolutions of the pumps, signals, and amplified spontaneous emission noise (ASEN) [14]. The coupled equations are numerically solved by iteration [15]. In this paper, an 100-km TW-Reach fiber DFRA is taken as an example, in which the numerical parameters of the fiber are the same as in [7,8]. Eighty six signal channels following ITU grid are considered, in which their wavelengths are from 1530.61 nm to 1600 nm. The signal power of 0.5 mW for every channel is assumed. As the spectrum of an incoherent pump is broadband, an incoherent pump is sliced into a number of beams. The bandwidth of each beam is 100 GHz.

In this paper, an incoherent pump PSDF is taken as a function of the optical frequency *v*. The spectrum is divided into N sub-bands. A PWCF for the i-th sub-band is defined as

where *i* = 1, 2, …, N-1, and

In the Eq. (1a), *M* is the polynomial order of a sub-band; *a*_{ij}
’s are the polynomial coefficients; *v*_{i}
-1 and *v*_{i}
are the boundary optical frequencies of the *i*-th sub-band. The entire pump spectrum lies within the frequencies *v*
_{0} and *v*_{N}
which are called the end-point pump frequencies. The corresponding end-point pump wavelengths λ_{0} = *c*/*v*
_{0} and λ_{N} = *c*/*v*_{N}
, where *c* is the speed of light in vacuum. The spectral range within the end-point pump frequencies or wavelengths is called the pump band. Piece-wise continuous boundary condition is applied to the neighboring sub-bands, which can be written as

where *i* = 1, 2,…,N -1; the prime represents the first derivative with respect to v. We take the function and its first derivative to be zeros at the end-point frequencies, i.e.,

so that the function smoothly changes from the end-point frequencies to the frequencies outside the pump band. As there is no negative PSD, the PSDF of the i-th sub-band is taken as

From the Eq. (1a), there are *N*(*M*+1) polynomial coefficients. Eqs. (2) and (3) give 2(*N*-1)+4 conditions. Therefore, the total number of the polynomial coefficients that are not constrained by the Eqs. (1b) and (3) is

These unconstrained coefficients are represented by a *N*_{A}
-dimension vector *A*. For the case of unidirectional pumping, *N*_{A}
= *U*. The optimization can be carried out with the least-square minimization methods. The objective function to be minimized is defined as

where * A* is the vector to be optimized;

*N*

_{S}is the number of signal channels which is also the number of gain requirements for the minimization problem;

*G*

_{k}(

*) is the signal gain of the*

**A***k*-th channel using the pump PSDF specified by

*;*

**A***G*

_{kT}is the target signal gain of the

*k*-th channel.

*conventional least-square minimization routine starts with a trial solution of*

**A***. In searching for the minimum objective function,*

**A***is changed according to the minimization algorithm. The modified Levenberg-Marquardt method is used to minimize the objective function [16]. In every searching step, the signal gains are obtained by solving the coupled differential equations of the DFRA.*

**A**There are numerous methods to choose the components of * A*. We use the method with the simplest formulas relating the components of

*and the coefficients constrained by the Eqs. (2) and (3). The components of*

**A***are the coefficients*

**A***a*

_{i2},

*a*

_{i3}, …,

*a*

_{iM}of the sub-bands labeled with

*i*= 1, 2, ‥, N-2 in the Eq. (1a), and the coefficients

*a*

_{i3}, …,

*a*

_{iM}of the sub-bands labeled with

*i*=

*N*-1 and

*N*in the Eq. (1a). That is

As to the other polynomials coefficients in the Eq. (1a), for the first sub-band, *a*_{10}
= *a*_{11}
= 0 from the Eqs. (3a) and (3c); for the *N*-th sub-band, *a*_{N0}
= *a*_{N1}
= 0 from the Eqs. (3b) and (3d); the coefficients *a*_{i0}
and *a*_{i1}
of the i-th sub-band (*i*= 2, 3,‥, *N*-1), and the coefficients *a*
_{(N-1)2} and *a*
_{N2} can be easily solved from the Eqs. (2a) and (2b) for a given * A*. For the

*i*-th sub-band (

*i*= 2, 3,‥,

*N*-1),

where Δ*v*_{i-1}
=*v*_{i-1}
- *v*_{i-2}
. For the (*N*-1)-th and *N*-th sub-bands,

where for *i* = *N*-1, *k* = *N*; for *i* = *N*, *k* = *N*-1; Δ*v*_{N-1}
= *v*_{N-1}
- *v*_{N-2}
and Δ*v*_{N}
= *v*_{N-1}
- *v*_{N}
. From the Eqs. (7)-(10), it requires *M* ≥ 3. It is found that the optimized results slightly depend on *M* for *M* ≥ 4. Therefore we take *M*= 4 for all the examples shown in the next section.

It requires two sets of PWCFs to describe the PSDFs of co-pump and counter-pump for the DFRAs using bidirectional pumping. The polynomial coefficients of the two PSDFs are optimized simultaneously, i.e., the components of * A* are the unconstrained coefficients of the two PSDFs. We set the same

*N*and

*M*for the two PSDFs for simplicity. In this case, the dimension of

**is 2**

*A**U*, where

*U*is given by the Eq. (5).

Note that, for the minimization problem of a large number of unknowns, it is difficult to find the global minimum. For an example, from the Eq. (5), *U*= 25 for *N*= 9 and *M*= 4. Therefore, the optimized pump PSDF depends on the initial trial solution because the minimization process only freezes to a local minimum of the objective function, Eq. (6). A proper non-null initial trial solution can be applied for the minimization routine. In this paper, we take a null initial trial solution so that the optimized pump PSDF is not pre-biased by the initial trial solution, i.e., the components of the initial trial solution of * A* are all taken to be zeros.

In the next section we show the numerical results for the two DFRAs using backward pumping and bidirectional pumping respectively. It is found that the optimized pump PSDFs depend on the number of sub-bands that is due to the boundary conditions set by the Eqs. (1b) and (2a) in addition to the output of the minimization routine only corresponding to a local minimum of the objective function. It is also found that the use of larger *N* may not result in lower gain ripple because of the less effective optimization resulting from too large dimension of the vector * A*. Several design examples of low gain ripple using the different numbers of sub-bands are shown in the next section. From the results, though the spectral shapes of the optimized pump PSDFs change with the number of sub-bands, their characteristics are similar.

## 3. Numerical results and discussion

As distributed amplifiers are considered, the ON-OFF Raman gain (*G*_{on-off}
) and effective noise figure (ENF) are taken to evaluate the gain and noise performance of the DFRAs. *G*_{on-off}
is the ratio of the signal power with pumps ON over the signal power with pumps OFF. ENF is defined as

where ${P}_{\mathit{\text{ASE}}}^{+}$
is the output power of the forward ASEN; Δ*v* is the bandwidth of the ASEN power; *hv* is the photon energy at the optical frequency *v*. The power spectrum of either the forward or backward ASEN is sliced into a number of beams. The bandwidth of every ASEN beam is 100 GHz. The target gains can be chosen to just compensate for the fiber loss and they vary with the signal wavelength. However, in this paper, we take *G*_{on-off}
= 20 dB for all the signal channels, i.e., *G*_{kT}
= 20 dB for *k*= 1, 2, …, *N*_{S}
in the Eq. (6) as is usually taken in the literatures. The comparison of ENFs relates to the comparison of the forward ASEN photon-number spectral densities for the same gain.

The specification of the pump band can be determined from practical considerations, such as the signal wavelengths and the available pump wavelengths. If the pump band is limited within the 14xx-nm region, in which λ_{N} = 1400 nm and λ_{0} = 1500 nm, it is found that the minimum gain ripple is as high as 0.6 dB because the long-wavelength signal gains near 1600 nm are not enough. Therefore we take λ_{0} = 1520 nm to provide higher gains for the long-wavelength signals. Figure 1(a) shows the optimized pump PSDFs of the pump band λ_{N} = 1400 nm and λ_{0} = 1520 nm for the cases of *N* = 6, 9, and 12. The corresponding gains, effective noise figures, and output forward ASEN PSDFs for the three cases are shown in Figs. 1(b)–1(d), respectively. The gain ripples are 0.042 dB, 0.051 dB, and 0.014 dB for the cases of *N*= 6, 9, and 12, respectively. The total pump powers are 910 mW, 912 mW, and 930 mW for the cases of *N*= 6, 9, and 12, respectively. One can see that the spectral characteristics of the three cases are similar. There are a high-power lobe and a low-power lobe in the pump spectrum near 1420 nm and 1490 nm respectively. The low-power lobe is used to amplify the
long-wavelength signals. The high-power lobe is used to amplify the short-wavelength signals and the low-power lobe. The result agrees to the optimized PSDF with two counter-incoherent 20-nm bandwidth pumps in [7], in which the gain ripple is 0.57 dB and the wavelengths of the
high-power pump and low-power pump are at 1429 nm and 1491 nm respectively. From the Fig. 1(a), it requires the continuous spectrum near 1450 nm for reducing the gain ripple to ultra-low level. This indicates that the use of the two incoherent pumps of few tens of nanometers bandwidth is not enough to achieve the ultra-low gain ripple.

From the Fig. 1(c), the effective noise figures of the three cases of *N*= 6, 9, and 12 are about the same. The case of the largest difference is that the effective noise figure of the case
of *N*= 6 is 0.15-dB lower than the case of *N*= 9 at the shortest signal wavelength. Comparing the Figs. 1(a) and 1(d), the peak ASEN near 1500 nm comes from the Rayleigh back-scattering of the low-power pump lobe and the peak ASEN near 1515 nm results from the amplification of the ASEN by the high-power pump lobe. For reducing the effective noise figure, the wavelength of the high-power pump lobe should be shorter so that the wavelength of the peak ASEN amplified by the high-power pump lobe can be shifted away from the
signal band. For the high-power pump lobe of the case of *N*= 6, its wavelength is shorter than the case of *N*= 9 and its peak power is less than the case of *N*= 12. Such characteristics of the high-power pump lobe results in the lowest effective noise figures for the case of *N*= 6.

For the DFRA using bidirectional pumping, the pump bands of the co-pump and counter-pump are respectively specified. We set λ_{N} = 1400 nm and λ_{0} = 1460 nm for the co-pump; λ_{N} = 1450 nm and λ_{0} = 1520 nm for the counter-pump. The pump band of the co-pump is chosen to be in shorter wavelength region so that the amplification of the ASEN near the shortest signal wavelength by the co-pump can be attenuated in the middle section of the transmission fiber and the effective noise figures can be reduced. Figure 2(a) shows the optimized pump PSDFs for the cases of *N*= 6, 9, and 12. The corresponding gains, effective noise figures, and output forward ASEN PSDFs for the three cases are shown in Figs. 2(b)– 2(d), respectively. The gain ripples are 0.0033 dB, 0.019 dB, and 0.011 dB for the cases of *N* = 6, 9, and 12, respectively. The pump powers of co-pumps are 607 mW, 652 mW, and 598 mW for the cases of *N*= 6, 9, and 12, respectively. The pump powers of counter-pumps are 505
mW, 501 mW, and 511 mW for the cases of *N*= 6, 9, and 12, respectively. One can see that the spectral characteristics of the three cases are also similar. For the co-pumps, their wavelengths of the maximum PSDF are near 1410 nm. In addition, there are the double-knee structures in their power spectra near 1427 nm and 1450 nm. For the counter-pumps, there are the double-lobe structures in their power spectra near 1465 nm and 1500 nm. These results indicate that, to achieve the ultra-low gain ripple, there requires three and two incoherent
pumps for the co-pump and counter pump respectively, in which the bandwidth of the
individual incoherent pump is about 10 nm.

From the Fig. 2(c), the effective noise figures of the three cases of *N*= 6, 9, and 12 are about the same except in the long signal wavelength region. The case of the largest difference is that the effective noise figure of the case of *N*= 9 is 0.29-dB lower than the case of *N*= 12 at 1591.51 nm. The wavelengths of their maximum effective noise figures are near 1573 nm. From the Figs. 2(a) and 2(d), the peak ASEN near 1500 nm comes from the Rayleigh back-scattering of the long-wavelength pump lobe of the counter-pump and the peak ASEN near 1573 nm is due to the amplification of the ASEN by the short-wavelength pump lobe of the counter-pump. Figure 3 shows the evolutions of the forward ASEN PSDs at 1530 nm and 1600 nm for the three cases of *N*= 6, 9, and 12 shown in Fig. 2(a). One can clearly observe that, the amplification of the ASEN at 1530 nm is more near the input end by co-pump but is less near the output end by counter-pump. This results in the low effective noise figures for the short-wavelength signals. For the ASEN at 1600 nm, it is mainly amplified by the counter-pump because the wavelength difference between the co-pump and the ASEN is far beyond the 13.2-THz Raman gain bandwidth. Although the counter-pump power of the case of *N*= 9
is lower than the other two cases, it is higher than the other two cases near the input end of the transmission fiber because it is amplified near the input end by the co-pump of the largest power among the three cases. From the Fig. 3, one can see that the amplification of the 1600- nm ASEN for the case of *N*= 9 is more than the other two cases near the input end and is less than the other two cases near the output end. This results in the lower effective noise figures for the case of *N*= 9.

## 4. Synthesis of pump PSDF

High-power incoherent pump source is achieved through coupling a low-power ASE seed into a single-spatial-mode long-cavity semiconductor laser amplifier (SLA) [6]. The optimized PSDFs shown in the last section can be synthesized with the multiple incoherent pumps. The required number of pumps can be reduced by properly tailoring the spectra of the ASE seeds launched into the SLAs. In this section, we take the examples of synthesizing the optimized PSDFs with the multiple Gaussian incoherent pumps.

The PSDF of a Gaussian incoherent pump is taken as

where *P*_{p}
, *v*_{c}
and Δ*v*_{W}
are the amplitude, central optical frequency and spectral width (FHWM) of the PSDF. The spectral width in wavelength is defined as Δλ
_{W}
= *c*Δ*v*_{W}
/${v}_{\mathrm{c}}^{2}$. The pump power of a Gaussian incoherent pump is defined as the integration of the Eq. (12) over frequency, i.e., the total power of the Gaussian incoherent pump. The optimized PSDF is synthesized with the multiple Gaussian incoherent pumps by optimizing the amplitudes, central optical frequencies, and spectral widths of the Gaussian incoherent pumps. For using
the Gaussian incoherent pumps of larger bandwidth, the slope of the pump PSDF with respect to wavelength should not be steep. From the Figs. 1(a) and 2(a), the wavelength region of the largest slope is near 1400 nm. We take the cases of the smallest slope near 1400 nm as the examples to show the synthesis results.

As is indicated in the last section, two incoherent pumps of few tens of nanometers are not enough to synthesize the pump PSDF of the DFRA using backward pumping for the ultra-low gain ripple. It is found that four Gaussian incoherent pumps are required to synthesize the optimized pump SPD. Figure 4 shows the example of synthesizing the optimized PSDF of the case of *N*= 9 shown in the Fig. 1(a), in which the DFRA is backwardly pumped. The optimized pump PSDF is well synthesized except near 1400 nm. The gain ripple is increased to 0.3 dB with this synthesized PSDF. We further optimize the parameters of the four Gaussian incoherent pumps for reducing the gain ripple. The optimized results are shown in the Table 1. The total spectrum of the four optimized Gaussian incoherent pumps is also shown in the Fig. 4, in which it slightly differs from the original synthesized pump PSDF but the corresponding gain ripple is reduced to 0.047 dB. The gains, effective noise figures, and output forward ASEN PSDFs are shown in the Figs. 1(b)–1(c), respectively, in which the effective noise figures are about the same as the original case of *N*= 6. We have also used three Gaussian incoherent pumps to synthesize the pump PSDF. The original PSDF is poorly synthesized. However, we further optimize the parameters of the three Gaussian incoherent pumps. It is found that the gain ripple after the optimization is about 0.3 dB.

For synthesizing the pump PSDFs of the DFRA using bidirectional pumping, three
Gaussian incoherent pumps for the co-pump and two Gaussian incoherent pumps for the
counter-pump are used as is indicated from the Fig. 2(a). Figure 5 shows the examples of synthesizing the optimized PSDFs of the case of *N*= 6 shown in the Fig. 2(a) for the co-pump
and counter-pump, in which the DFRA is bidirectional pumped. One can see the optimized pump PSDFs are not well synthesized especially for the counter-pump. The gain ripple is increased to 0.28 dB using the two synthesized PSDFs. We also further optimize the parameters of the five Gaussian incoherent pumps for reducing the gain ripple. The optimized results are shown in the Table 2. The total spectrum of the three optimized Gaussian
incoherent co-pumps and the total spectrum of the two optimized Gaussian incoherent
counter-pumps are also shown in the Fig. 5, in which they are blue-shifted with respect to the original synthesized pump PSDFs. Using the pump PSDFs of the five optimized Gaussian incoherent pumps, the gain ripple is reduced to 0.015 dB. The gains, effective noise figures, and output forward ASEN PSDFs are shown in the Figs. 2(b)–2(d), respectively. One can see the effective noise figures are larger than the original case of *N*= 6 shown in the Fig. 2(c).

Because the pump PSDFs are blue-shifted, the counter-pump amplifies the forward ASEN near the output end of the transmission fiber more than the original case.

## 5. Conclusion

The method to design the incoherent pump power spectra for DFRAs is presented. The set of PWCFs representing a pump PSDF is defined. The pump power spectrum is divided into a number of sub-bands, in which each sub-band is described with a polynomial. The pump PSDF is the absolute value of the set of PWCFs. The polynomial coefficients of pump PSDF are optimized with the least-square minimization method for reducing the gain ripple. Two DFRAs using backward pumping and bidirectional pumping respectively are taken as examples. The results show that the optimized pump PSDFs depend on the number of sub-bands. It is found that the gain ripples of the two pumping schemes can be less than 0.02 dB over 70-nm bandwidth. The spectral characteristics of the optimized pump PSDFs for the ultra-low gain ripple are investigated. For the DFRA using backward pumping, the optimal pump spectrum has the double-lobe structure which comprises a high-power lobe and a low-power lobe in long- wavelength region and short-wavelength region respectively. For the DFRA using bidirectional pumping, the optimal spectrum of co-pump has the double-knee structure in short-wavelength region and the optimal spectrum of counter-pump has the double-lobe structure in long-wavelength region. The noise characteristics of the DFRAs using the optimized pump PSDFs are studied. As to the implementation of the optimized pump PSDF, it can be synthesized with the multiple incoherent pumps. The examples using the optimized pump PSDFs synthesized with multiple Gaussian incoherent pumps are shown.

Their gain ripples are increased to about 0.3 dB because of the discrepancy between the optimized pump PSDF and the synthesized pump PSDF. By further optimizing the parameters of the multiple Gaussian incoherent pumps, the gain ripples can be reduced to less than 0.05 dB. The optimization method introduced in this paper can be further modified by including the constraints of the spectral characteristics of the pump PSDF in addition to the end-point boundary conditions given by the Eq. (3). Although gain ripple may be increased due to the additional constraints, specific design preferences can be met. In addition, the pump PSDFs can be optimized for compromising the requirements of the gain ripple, effective noise figures, and pump powers by modifying the objective function given by the Eq. (6).

## Acknowledgments

This work is supported in part by National Science Council, R. O. C., under Contract No. 94-2215-E-216-002.

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