## Abstract

The method for designing the incoherent pump spectrum for the distributed fiber Raman amplifiers (DFRAs) of low gain ripple is studied, in which the wavelength of the maximum power spectral density can be assigned. The assigned wavelength is called the main pump wavelength. Incoherent pump spectrum is described with the power spectral density function (PSDF) that comprises a set of piece-wise continuous functions. PSDF is optimized for the minimum gain ripple with the least-square minimization method. An extremum pump wavelength condition is applied to PSDF. A proper initial trial PSDF is given so that the optimized PSDF converges to the desired result and the power spectral density at the extremum pump wavelength is the maximum. With this design method, we show the optimized PSDFs for the DFRAs using backward pumping and bidirectional pumping. The gain ripples of considered DFRAs are less than 0.1 dB for 20-dB ON-OFF Raman gain over 70-nm bandwidth. The reduction of average effective noise figure with shorter main pump wavelength is shown and investigated.

© 2007 Optical Society of America

## 1. Introduction

Fiber Raman amplifiers (FRAs) using incoherent pumping are interested recently [1–4]. FRAs have the advantages of polarization insensitivity and reducing the nonlinear effects such as the stimulated Brillouin scattering (SBS) of pumps [5] and the four-wave mixings (FWMs) of pump-pump, pump-signal, and pump-noise [6–9]. The reduction of nonlinear effects is owing to the broadband, low power spectral density (PSD), and random phase of incoherent pumps. For the wavelength-division-multiplexing system, the optical amplifier of broadband equalized gain is required. Gain-equalized FRAs can be achieved by the use of multiple coherent pumping [10–13], multiple incoherent pumping [1, 2, 4], and composite coherent-incoherent pumping [3]. Because the spectral width of an incoherent pump is large, it was shown that the gain ripple of the distributed FRA (DFRA) using two incoherent counter-pumps is less than that of the DFRA using six coherent counter-pumps [2]. It was also shown that the uses of four incoherent pumps and five incoherent pumps are enough to obtain the gain ripples of less than 0.05 dB for the DFRAs using backward pumping and bidirectional pumping, respectively, in which their gain bandwidths are 70 nm [4].

Reference [4] introduced a method for designing the pump spectra of incoherent pumps so that the gain ripple of an FRA can be minimized. A pump power spectrum is divided into a number of sub-bands, in which each sub-band is described with a polynomial and the piece-wise continuous boundary condition is applied to neighboring sub-bands. Thus the power spectral density function (PSDF) of a pump spectrum can be represented with a set of piece-wise continuous functions (PWCFs). The polynomial coefficients of the pump PSDF are optimized with the least-square minimization method for reducing the gain ripple. It was shown that the ultra-low gain ripple of 0.02 dB can be achieved. For realizing the pump PSDF, it was also shown that the optimized PSDF can be approximately synthesized with multiple Gaussian incoherent pumps at the expense of slightly higher gain ripple. The increased gain ripple can be decreased to be less than 0.05 dB by further optimizing the parameters of the multiple Gaussian incoherent pumps.

The method presented in Ref. [4] has the characteristic that the optimized pump spectrum depends on the number of sub-bands because the minimization process only freezes to a local minimum of the objective function representing gain ripple, i.e., there exists a number of different optimized pump spectra for the ultra-low gain ripple. From system design considerations, it is desired that the characteristics of optimized pump spectrum can be assigned. In this paper we propose a method for designing the optimized pump spectrum with the assigned wavelength of the maximum PSD, which is called the main pump wavelength. The desired main pump wavelength may depend on available pump sources or the other practical reasons. The reduction of average effective noise figure with the shorter main pump wavelength is also shown and investigated for the DFRAs designed with this method.

## 2. Amplifier model and design method

The steady-state power evolutions of the pumps, signals, and amplified spontaneous emission noise (ASEN) in a DFRA can be described with a set of coupled differential Eq. [14], which can be numerically solved by iteration [15]. As in Refs. [2, 4], an 100-km TW-Reach fiber DFRA is used as example. 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 spectra of incoherent pumps, forward ASEN, and backward ASEN are broadband, they are respectively sliced into a number of beams. The bandwidth of each beam is 100 GHz.

An incoherent pump PSDF is taken as a function of the optical frequency *v*. Its spectrum is divided into *N* sub-bands. A PWCF for the *i*-th sub-band is defined as [4]

where *i* = 1,2,…,*N*-1, and

In Eq. (1), *M* is the polynomial order of a sub-band; *a _{ij}*’s are the polynomial coefficients;

*v*

_{i-1}and

*v*are the boundary optical frequencies of the

_{i}*i*-th sub-band. As there is no negative PSD, the PSDF of the

*i*-th sub-band is taken as

The piece-wise continuous boundary condition is applied to neighboring sub-bands, which can be written as

where *i* = 1,2,…, *N* -1; the prime represents the first derivative with respect to *v*. Note that the pump band lies within the end-point frequencies *v _{0}* and

*v*. The function and its first derivative are set to be zeros at

_{N}*v*and

_{0}*v*, i.e.

_{N}The end-point pump wavelengths *λ*
_{ 0} = *c*/ *v*
_{0} and *λ _{ N}* =

*c*/

*v*, where

_{N}*c*is the speed of light in vacuum. The spectral range within the end-point pump frequencies or wavelengths is called the pump band. For the assigned wavelength

*λ*that corresponds to a local extremum of PSDF, we have the extremum pump wavelength condition

_{ e}where *v _{e}* =

*c*/

*λ*. When the PSD at

_{ e}*λ*is the maximum,

_{ e}*λ*becomes the main pump wavelength of PSDF. From the results shown in Ref. [4], the main pump wavelength is usually close to

_{ e}*λ*. Therefore, we set

_{ N}*λ*to be near

_{ e}*λ*. Multiple extremum wavelengths can be assigned for tailoring a PSDF but that may increase the gain ripple of optimized result. We therefore assign only one extremum wavelength for the DFRAs considered in this paper.

_{ N}It requires an initial trial PSDF (IT-PSDF) for the minimization routine that is used to find the optimized PSDF for the minimal gain ripple. The null IT-PSDF takes all the coefficients of PSDF to be zero [4]. The use of the null IT-PSDF may result in the desired solution when the assigned *λ _{ e}* is close to the main pump wavelength of the optimized PSDF that is not constrained by Eq. (5). Otherwise the use of the null IT-PSDF may result in the following undesirable results: (i) there exists the other extremum wavelength that its PSD is comparable to or even larger than the PSD at

*λ*, (ii) the PSD at

_{ e}*λ*is a local minimum, and (iii) the gain ripple is large. From Eq. (5), in fact

_{ e}*λ*is only the wavelength corresponding to a local extremum that may be either the local maximum or the local minimum. These undesirable results can be avoided by the use of a proper non-null IT-PSDF. A non-null IT-PSDF can be obtained by fitting the coefficients of a PSDF to the following function

_{ e}where *p*
_{0} is the maximum PSD at *λ _{ e}*; Δ

*v*is the parameter related to the pump bandwidth in long-wavelength region. The pump bandwidth in unit of wavelength is Δ

_{ w}*λ*=

_{ w}*c*Δ

*v*/

_{ w}*λ*

_{ e}^{2}. The values of

*p*

_{0}and Δ

*λ*can be chosen according to the optimized PSDF using the null IT-PSDF. Note that the other proper function can be used in stead of Eq. (6) for specific design preference. However, from our experience, the use of Eq. (6) gives satisfactory results for the DFRAs of low gain ripple considered in this paper. Because

_{ w}*λ*is set to be near

_{ e}*λ*, we take the function given in Eq. (6) in short-wavelength region to be a linear function of frequency. We may use a nonlinear function in stead of the linear function in short-wavelength region but that will introduce the additional parameters that are needed to be adjusted. Because the optimization routine will further adjust the coefficients of PSDF for reducing gain ripple, the use of the nonlinear function in short-wavelength region is not necessary.

_{ N}There are *N*(*M*+1) polynomial coefficients in Eq. (1). Equations (3)–(5) give 2(*N*-1) +3 conditions. Therefore, the total number of un-constrained polynomial coefficients is *N*(*M*-1)-3. These unconstrained coefficients are optimized for the minimum gain ripple, in which the objective function representing gain ripple is defined in Ref. [4]. Modified Levenberg-Marquardt method is used to minimize the objective function [16]. In searching for the minimum gain ripple, the signal gains are obtained by solving the coupled differential equations of DFRA. We will show the numerical results for the DFRAs using backward pumping and bidirectional pumping in the next two sections. It requires two sets of PWCFs for describing the PSDFs of co-pump and counter-pump for the DFRA using bidirectional pumping. The polynomial coefficients of the two PSDFs are optimized simultaneously. We set the same *N* and *M* for the two PSDFs for simplicity. Because the characteristics of the DFRA using bidirectional pumping mainly depends on its co-pump, we assign the main pump wavelength to its co-pump and no extremum wavelengths are assigned to its counter-pump for simplifying the analysis of numerical results. The IT-PSDF of the counter-pump is set to be null. For all the considered DFRAs, *N*= 9 and *M*= 4 are taken.

As distributed amplifiers are considered, ON-OFF Raman gain (*G _{on-off}*) and effective noise figure (ENF) are taken to evaluate the gain and noise performance of DFRAs.

*G*is the ratio of the signal power with pumps ON over the signal power with pumps OFF. The gain variation of the amplifier is usually represented by the gain ripple that is defined as the difference of the maximum gain and the minimum gain of all signal channels. For the DFRAs designed in this paper, we require the gain ripple to be less than 0.1 dB. ENF is defined as

_{on-off}where *P*
^{+}
* _{ASE}* is the output power of forward ASEN; Δ

*v*is the bandwidth of the ASEN power;

*hv*is the photon energy at the optical frequency

*v*. The target gains can be chosen to just compensate for fiber loss and they vary with signal wavelength. However, in this paper, we take

*G*= 20 dB for all the signal channels 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. Because ENF varies with signal wavelength, we define the average ENF as the average value of the ENFs of all signal channels for comparing the noise performance of DFRAs. In addition, the ENF variation is defined as the difference of the maximum ENF and the minimum ENF of all signal channels.

_{on-off}## 3. DFRA using backward pumping

For the DFRA using backward pumping, the optimized pump spectrum comprises a short-wavelength pump lobe and a long-wavelength pump lobe [4]. The long-wavelength pump lobe amplifies long-wavelength signals. The short-wavelength pump lobe not only amplifies short-wavelength signals but also the long-wavelength pump lobe. Therefore, the power of the short-wavelength pump lobe is higher than that of the long-wavelength pump lobe and it is preferred to assign *λ _{ e}* to the short-wavelength pump lobe. We set

*λ*

_{ 0}= 1520 nm and

*λ*=

_{ N}*λ*-20nm.

_{ e}Figures 1(a) and 1(b) show the optimized pump PSDFs and ENFs, respectively, for the cases using null and non-null IT-PSDFs, in which *λ _{ e}*= 1420 nm. In the figures, the cases using the non-null IT-PSDFs with several

*p*

_{0}and Δ

*λ*are shown. The gain ripples of all the cases shown in Fig. 1(a) are less than 0.1 dB. From Fig. 1(a), one can see that the optimized pump PSDFs are insensitive to their IT-PSDFs. As is shown in Ref. [4], the main pump wavelength of the optimized PSDF that is not constrained by Eq. (5) is close to 1420 nm, in which the pump band is the same as the cases shown in Fig. 1(a). Therefore the optimized pump PSDFs converge to similar solutions under the requirement of low gain ripple. The total pump power of the case using the null IT-PSDF is 917 mW.

_{ w}From Fig. 1(b), ENF decreases as signal wavelength increases. The maximum ASEN PSD is near the short-wavelength region of signal band, which is amplified by the short-wavelength pump lobe. The use of shorter *λ _{ e}* shifts the maximum ASEN away from the signal band and decreases ENF but at the expense of higher pump power [2, 4]. Considering the DFRA with

*λ*= 1380 nm and using the null IT-PSDF, we show its optimized pump PSDF and ENF in Figs. 2(a) and 2(b), respectively. The gain ripple is 0.068 dB. For this case, there is the pump lobe at 1410 nm in addition to 1380 nm. The power of the pump lobe at 1410 nm is larger than that at 1380 nm. Compared with the DFRA with

_{ e}*λ*= 1420 nm, the ENF of the DFRA with

_{ e}*λ*= 1380 nm and using the null IT-PSDF is not appreciably decreased because its main pump lobe is at 1410 nm.

_{ e}The role of the pump lobe at 1380 nm is to amplify the long-wavelength pump in stead of directly amplifying the signals because its frequency difference from the shortest signal wavelength is far beyond the 13.2-THz Raman gain bandwidth. Thus the PSD at 1380 nm can be increased by decreasing the PSD in the long-wavelength region of pump band. Figures 2(a) and 2(b) also show the optimized pump PSDFs and ENFs, respectively, for the DFRAs using the non-null IT-PSDFs with Δ*λ _{ w}* = 40 nm and several

*p*

_{0}. The gain ripples are 0.073 dB, 0.026 dB, and 0.054 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The PSD at 1380 nm does not necessarily increase with

*p*

_{0}because the non-null IT-PSDF just gives a starting point for searching the PSDF that minimizes gain ripple by the minimization routine. Considering ENF, one can see that the case with Δ

*λ*= 40 nm and

_{ w}*p*

_{0}= 10 mW/100GHz is preferred for the cases shown in Fig. 2. The average ENF and total pump power of this preferred case are decreased and increased by 0.49 dB and 216 mW, respectively, compared with the DFRA with

*λ*= 1420 nm and using the null IT-PSDF.

_{ e}The PSD at 1380 nm can be further increased by the use of narrower pump bandwidth but the gain ripple may be increased. Figures 3(a) and 3(b) show the optimized pump PSDFs and ENFs, respectively, for the DFRAs with *λ _{ e}*= 1380 nm and Δ

*λ*= 30 nm. The gain ripples are 0.12 dB, 0.37 dB, and 0.35 dB for the cases with

_{ w}*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The total pump powers are 1247 mW, 1339 mW, and 1396 mW for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. Comparing Fig. 3(b) with Fig. 2(b), one can see that the ENF for the case with Δ

*λ*= 30 nm can be decreased at the expense of larger gain ripple and pump power.

_{ w}## 4. DFRA using bidirectional pumping

For the DFRA using bidirectional pumping, the pump band of its co-pump is chosen to be in shorter wavelength region. Thus 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 ENF can be decreased [4]. Because the wavelength of the co-pump is shorter than that of the counter-pump, the co-pump also amplifies the counter-pump and requires higher pump power. Therefore we set *λ _{ e}* to be the main pump wavelength of the co-pump.

Figures 4(a) and 4(b) show the optimized pump PSDFs and ENFs, respectively, for the cases using null and non-null IT-PSDFs, in which *λ _{ e}*= 1410 nm. In the figures, the cases using the non-null IT-PSDFs with several Δ

*λ*and

_{ w}*p*0 are shown. In Fig. 4(a), the pump bands are chosen to be:

*λ*

_{ 0}= 1460 nm and

*λ*= 1400 nm for co-pump; and

_{ N}*λ*

_{ 0}= 1520 nm and

*λ*= 1450 nm for counter-pump as in Ref. [4]. The gain ripples for all the cases shown in Fig. 4(a) are less than 0.1 dB. From Fig. 4(a), the optimized pump PSDFs are insensitive to their IT-PSDFs. As is shown in Ref. [4], the main pump wavelength of the optimized co-pump PSDF that is not constrained by Eq. (5) is close to 1410 nm. Therefore the optimized pump PSDFs converge to similar solutions under the requirement of low gain ripple. The total pump powers of the case using the null IT-PSDF are 650 mW and 491 mW for co-pump and counter-pump, respectively. Note that there is the double-knee structure in the power spectrum near 1427 nm and 1450 nm for the co-pump shown in Fig. 4(a). The short-wavelength signals from 1530 nm to 1560 nm are mainly amplified by the double-knee structure. There is the double-lobe structure for the counter-pump shown in Fig. 4(a). The long-wavelength signals from 1560 nm to 1600 nm are mainly amplified by the counter-pump.

_{ N}It is found that ENF can be decreased by the use of shorter *λ _{ e}* for the DFRA using bidirectional pumping. When

*λ*is decreased, we find that the pump band of counter-pump has to be adjusted for reducing gain ripple. For the case with

_{ e}*λ*= 1370 nm, we take the pump bands to be: λ

_{ e}_{ 0}= 1460 nm and

*λ*= 1350 nm for co-pump; and λ

_{ N}_{ 0}= 1520 nm and

*λ*= 1430 nm for counter-pump. Figures 5(a) and 5(b) show the optimized pump PSDF and ENF, respectively, for the case using the null IT-PSDF. The gain ripple is 0.02 dB. From Fig. 5(a), there is no double-knee structure for the co-pump. For this case, the total pump powers of co-pump and counter-pump are 989 mW and 519 mW, respectively. Compared with the case with

_{ N}*λ*= 1410 nm and using the null IT-PSDF, the total pump powers are increased but the average ENF is not decreased. The average ENFs for the cases with

_{ e}*λ*= 1410 nm and 1370 nm are -6.41 dB and -6.3 dB, respectively. The ENF variations are 4.86 dB and 3.26 dB for the cases with

_{ e}*λ*= 1410 nm and 1370 nm, respectively. The ENF variation of the case with

_{ e}*λ*= 1370 nm is decreased but its average ENF is slightly increased.

_{ e}The PSD at 1370 nm can be further increased by decreasing the PSD in the long-wavelength region of co-pump and the PSD of counter-pump. It is found that the use of narrow Δ*λ _{ w}* increases the PSD at 1370 nm and decreases ENF. Figures 5(a) and 5(b) also show the optimized pump PSDFs and ENFs, respectively, for the DFRAs using the non-null IT-PSDFs with Δ

*λ*= 25 nm and several

_{ w}*p*

_{0}. Note that the PSD at 1370 nm also does not necessarily increase with

*p*

_{0}as the cases shown in Fig. 2(a). The gain ripples are 0.032 dB, 0.099 dB, and 0.021 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The average ENFs are -6.51 dB, -6.11 dB, and -6.92 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The ENF variations are 2.87 dB, 2.64 dB, and 2.47 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. From Fig. 5(b), one can see the significant reduction of the ENF variation for the case with

*p*

_{0}= 20 mW/100GHz, in which the total pump powers of co-pump and counter-pump are 1396 mW and 489 mW, respectively.

From Fig. 5(b), the wavelength of the maximum ENF is around 1570 nm. Figure 6 shows the output forward ASEN PSDs for the cases with *λ _{ e}* = 1370 nm and

*p*

_{0}= 20 mW/100GHz shown in Fig. 5(a). In Fig. 6, the case with

*λ*= 1410 nm and using the null IT-PSDF is also shown for comparison. One can clearly see the low ASEN PSD in signal band for the case with

_{ e}*λ*= 1370 nm. The evolutions of the forward ASEN PSDs at 1530 nm, 1570 nm, and 1600 nm for the cases shown in Fig. 6 are shown in Fig. 7. Figure 8 shows the evolutions of the co-pump PSDs at 1370 and 1430 nm for the case with

_{ e}*λ*= 1370 nm shown in Fig. 6. From Fig. 7, the ASEN at 1530 nm is significantly amplified by the co-pump around 1430 nm near the input end of the transmission fiber. Although the input PSD of the co-pump around 1430 nm is low for the case with

_{ e}*λ*= 1370 nm, it can be amplified by the co-pump PSD around 1370 nm near the input end as is shown in Fig. 8. Because the ASEN at 1530 nm is attenuated in the middle section of the transmission fiber and is less amplified near the output end of the transmission fiber, the ENF at 1530 nm is the smallest in signal band.

_{ e}The ASEN near 1570 nm is the maximum that is due to the amplification of the ASEN by the short-wavelength lobe of counter-pump. From the counter-pump PSDFs shown in Fig.
5(a), the power of the short-wavelength pump lobe is larger than that of the long-wavelength pump lobe, in which the long-wavelength pump lobe near 1500 nm is used to amplify the signals near 1600 nm. Therefore the ASEN at 1570 nm is amplified more than that at 1600 nm near the output end of the transmission fiber and the ENF at 1570 nm is larger than that at 1600 nm. Figure 8 also shows the evolutions of the counter-pump PSDs at 1456 nm and 1500 nm for the cases with *λ _{ e}* = 1370 nm shown in Fig. 6. Note that the frequency difference of 1456 nm and 1370 nm is 12.9 THz which is close to the frequency difference of the maximum Raman gain. From Fig. 8, the short-wavelength pump lobe of counter-pump near 1456 nm is effectively amplified by the co-pump around 1370 nm. Thus, the ENF around 1570 nm decreases with the input power of the short-wavelength lobe of counter-pump. Because the use of the larger co-pump PSD at 1370 nm can decrease the required input power of the short-wavelength lobe of counter-pump, the ENF around 1570 nm decreases as the co-pump PSD at 1370 nm increases.

Figures 9(a) and 9(b) show the optimized pump PSDFs for the co-pump and counter-pump, respectively, in which the cases with *λ _{ e}* ranging from 1350 nm to 1390 nm are shown. The corresponding effective noise figures are shown in Fig. 10. The parameters of the initial trial solutions for the cases shown in Fig. 9 are Δ

*λ*= 40 nm, 40 nm, 25 nm, 25 nm, and 40 nm for the cases with

_{ w}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively, and

_{ e}*p*

_{0}= 8 mW/100GHz, 8 mW/100GHz, 21 mW/100GHz, 10 mW/100GHz, and 5 mW/100GHz for the cases with

*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively, so that their PSDs at

_{ e}*λ*are the maximum and their gain ripples are less than 0.1 dB. Note that the required initial trial bandwidth is larger for the case with

_{ e}*λ*away from 1370 nm. The average ENFs are -5.99 dB, -5.95 dB, -7.16 dB, -6.0 dB, and -5.07 dB for the cases with

_{ e}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively. The ENF variations are 2.63 dB, 2.84 dB, 2.4 dB, 2.28 dB, and 2.23 dB for the cases with

_{ e}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively. One can see that the average ENF is the smallest for the case with

_{ e}*λ*= 1370 nm, because it is able to pump the short-wavelength lobe of counter-pump the most effectively among the cases shown in Fig. 9.

_{ e}## 5. Conclusion

We have presented the method for designing the incoherent pump power spectra for the DFRAs with an assigned main pump wavelength. The extremum pump wavelength condition is applied to the set of PWCFs representing a pump PSDF. The polynomial coefficients of the pump PSDF are optimized with the least-square minimization method for reducing the gain ripple. A proper initial trial PSDF is given so that the optimized PSDF converges to the desired result and the PSD at the extremum pump wavelength is the maximum. With this method, we show the design examples of the DFRAs using backward pumping and bidirectional pumping. The gain ripples of considered DFRAs are less than 0.1 dB for 20-dB ON-OFF Raman gain over 70-nm bandwidth. For the DFRA using backward pumping, the use of shorter main pump wavelength results in smaller effective noise figure at the expense of larger pump power. For the DFRA using bidirectional pumping, the main pump wavelength is assigned to its co-pump. The optimized PSDF of counter-pump has a double-lobe structure. It is found that the average effective noise figure and effective noise figure variation decrease with the input power of the short-wavelength lobe of counter-pump. The input power can be decreased by using the co-pump that its main pump wavelength is able to effectively pump the short-wavelength lobe of counter-pump. The amplifiers of low gain ripple and effective noise figure variation are preferred. Although the designed gain spectra over signal band are flat, it is desirable to study the algorithm for further flattening effective noise figure spectra by properly choosing the main pump wavelength, tailoring the initial trial solution, and modifying the objective function for minimization.

## Acknowledgment

This work is supported in part by National Science Council, R. O. C., under Contract No. 95-2221-E-216-044.

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