摘要

摘要

本論文對於使用非同調光幫浦的分散式拉曼光纖放大器，

提出一種設計最佳化光功率頻譜的方法，使得其尖峰功率之波 長可以預先指定。此波長稱之為極值波長。我們以應用於 100km 的 TW-Reach 分散式拉曼光纖放大器為設計範例，其中 考慮逆向幫浦及雙向幫浦兩種情形。所設計的放大器皆能滿足 在 70-nm 信號頻寬中有低於 0.1dB 增益差的要求。對於使用 逆向幫浦的分散式拉曼光纖放大器，較短的極值波長可以有較 低的等效雜訊指數，但是需要較高的幫浦功率。對於使用雙向 幫浦的分散式拉曼光纖放大器，我們將極值波長設定在順向幫 浦頻譜中，以降低等效雜訊指數。最佳化的逆向幫浦具有雙峰 的結構。我們發現將順向幫浦光的極值波長，設計為對逆向幫 浦光的短波長頻帶之尖峰波長有最大的拉曼增益，可以得到最 低的等效雜訊指數。

Abstract

The method to design the incoherent pump power spectrum with pre-assigned extremum pump wavelength for the distributed fiber Raman amplifier (DFRA) is presented. Two 100-km TW- Reach DFRAs using backward pumping and bidirectional pumping respectively are taken as examples, in which the gain ripple is less than 0.1 dB over 70-nm bandwidth. For the DFRA using backward pumping, the use of shorter extremum pump wavelength results in lower effective noise figure but larger pump power. For the DFRA using bidirectional pumping, the extremum wavelength is set to the co-pump. The optimized spectrum of counter-pump has the double-lobe structure. It is found that the effective noise figure is the minimum for the case using the extremum pump wavelength that has the maximum Raman gain for the wavelength of the short-wavelength lobe of the counter- pump.

致謝

承蒙溫盛發老師的悉心指導、教誨與鼓勵，使我得以順利

完成碩士論文與學業，謹致上我最誠摯的感謝與敬意。同時也 要感謝祁甡博士、吳俊傑博士、洪端佑博士及高川原博士在口 試時對於本論文提供寶貴的意見。

在此，特別感謝黃明同同學、池振村學弟和林耿伊學妹，

無論在學業上或生活上給予我很大的鼓勵與幫助。

最後感謝我的家人，在這幾年中給我很大的空間，讓我無 後顧之憂可以做自己該做的事。很高興可以完成這份論文，感 謝。

Contents

1. Introduction………5 2. Amplifier model and optimization method…7 3. Pump band and initial trial solution ………13 4. Numerical results and discussions…………..15 5. Conclusion……….20 Figure captions ………22 References……….38

1. Introduction

Fiber Raman amplifier (FRA) using incoherent pumping have

the advantages of polarization insensitivity and reducing the nonlinear effects [1-4], 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 the nonlinear effects is owing to the broadband, low power spectral density (PSD), and random phase of the incoherent pumps. For the wavelength-division-multiplexing (WDM) system, the optical amplifier of broadband equalized gain is required. Gain-equalized FRA can be achieved by the use of the multiple coherent pumping [10-13], multiple incoherent pumping [1,2,4], and composite coherent-incoherent pumping [3]. Because of the large pump spectral width, 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 four incoherent pumps and five incoherent pumps are enough to reduce the gain ripples to be less than 0.05 dB for the

DFRA using backward pumping and bi-directional pumping, respectively, in which the gain band is 70 nm [4].

Reference [4] introduced a method to design the pump spectra of the incoherent pumps for minimizing the gain ripple of the FRA. 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 the neighboring sub-bands. Thus the pump power spectral density function (PSDF) can be represented with a set of piece-wise continuous functions (PWCFs). The polynomial coefficients of 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. It was also showed that the optimized pump power spectra can be approximately synthesized with the multiple Gaussian incoherent pumps at the expense of higher gain ripple.

The gain ripples can be reduced to 0.05 dB by further optimizing the parameters of the multiple Gaussian incoherent pumps. The method presented in the Reference [4] has the characteristic that the optimized pump power spectra depends on the number of sub-

bands because the minimization process only freezes to a local minimum of the objective function, i.e. there exist a number of different optimized pump power spectra for the ultra-low gain ripple. From system design considerations, it is preferred that the characteristics of the optimized pump power spectrum can be pre- assigned. In this paper we propose a method to set the wavelength of the peak spectral power for the optimized pump spectrum. The preferred wavelength of the peak spectral power may depend on the available pump sources or the other practical reasons. In this paper, we will show the dependences of the requirements pump power and effective noise figure on the wavelength of the peak spectral power.

2. Amplifier model and optimization 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 equations [14], which can be numerically solved by iteration [15]. As in [2,4], an 100-km TW-Reach fiber DFRA is taken as an 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 an incoherent pump and 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]

( ) (

)

0

M j

i ij i

j

f ν a ν ν ₋

=

=

∑

ν

≤ ≤ ν ν

where i =1, 2,...,N −1_{, and}

( ) ( )

0

M j

N Nj N

j

f ν a ν ν

=

=

∑

ν

≤ ≤ ν ν

In the Eq. (1), M is the polynomial order of a sub-band; aij’s are the polynomial coefficients; vi-1 and vi are the boundary optical frequencies of the i-th sub-band. As there is no negative PSD, the PSDF of the i-th sub-band is taken as

( ) ( )

i i

P ν = f ν

ν

≤ ≤ ν ν

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

( )

( )

i i i i

f ν = f

ν

( )

( )

i i i i

f ^′ ν = f

^′ ν

where i =1, 2,...,N − ; the prime represents the first derivative 1 with respect to v. Note that the pump band lies within the end-point frequencies v0 and vN. The function and its first derivative is set to be zeros at v0 and vN, i.e.,

( )

0

f

ν =

( ) ⁰

N N

f ν =

( )

0

f

^′ ν =

( ) ⁰

N N

f ^′ ν =

λ _e corresponding to a local extremum of PSDF, we have the condition

( ) ⁰

o e

f ^′ ν =

where ve = c/^λ e. λ _e is called the extremum wavelength. The local extremum may be either the local maximum or minimum. Because of the conditions given by Eqs.(4), the PSDF increases from the end-point pump wavelengths and the wavelength corresponding to a local maximum of the PSDF usually closes to one of the end- point pump wavelengths [4]. Therefore, by properly choosing the end-point pump wavelengths, we can enforce the extremum wavelength correspond to a local maximum of the PSDF. Multiple extremum wavelengths can be assigned to tailor the PSDF but that may increase the gain ripple of the optimized result. We therefore set only one extremum wavelength for the results shown in this paper. However sometimes the use of the constraint Eq.(5) does not lead to the preferred result. For an example, there maybe exists the other extremum wavelength of higher spectral power in the optimized pump spectrum in addition to the pre-assigned

extremum wavelength λ e. For such a case, a proper initial trial solution for the pump spectrum should be taken that will be discussed in the next section.

There are N(M+1) polynomial coefficients in the Eq. (1). Eqs.

(3)-(5) give 2(N-1)+3 conditions. Therefore, the total number of the polynomial coefficients that are not constrained by the Eqs. (3)- (5) is U=N(M-1)-3. These unconstrained coefficients are optimized for the minimum gain ripple, in which the objective function representing gain ripple is given in [4]. The 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 the DFRA. 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. We set the same N and M for the two PSDFs for simplicity. Because the characteristics of the DFRAs mainly depend on the co-pump, we set an extremum wavelength in the co-pump while no extremum wavelengths are pre-assigned to

the counter-pump for simplifying the analysis of the results. In the next section we show the numerical results for the two DFRAs using backward pumping and bidirectional pumping respectively.

N= 9 and M= 4 are taken for all the shown results.

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

1 1 ^ASE

on off

ENF P

G hν ν

+

−

⎛ ⎞

= ⎜⎝ + Δ ⎟⎠, (6)

where PASE⁺ is the output power of the forward ASEN; Δv 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 Gon-off = 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.

3. Pump band and initial trial solution

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. For a given extremum wavelength ^λe, we set λ_{N =} λ _e -20nm so that the optimized pump PSDF of the pump band smoothly increases from λ _{N to} λ _e for avoiding the oscillation of the set of PWCFs that cross zeros. The oscillation of PSDF will result in poor gain ripple. The minimization routine is to find the optimal solution of the unconstrained coefficients of the PSDF that minimize the gain ripple. It is found that, if the null

initial trial solutions for the unconstrained coefficients are taken as in Reference [4], there maybe exists the other extremum wavelength for the optimized result. Fig. 1 shows the example of the optimized pump PSDF for the DFRA using backward pumping and λ _e = 1360 nm. One can see that there appears an additional extremum wavelength at 1378 nm and its PSD is higher than the PSD at 1360 nm. Therefore we set a set of initial trail solutions for the unconstrained coefficients. The initial trail solutions is obtained by fitting the PSDF

( )

( )

2 0

0 0

0 0

( ) exp ,0

,

e

e e

t

e e N

N e

P f

P P

ν ν ν ν ν ν

ν ν ν

ν

ν ν ν ν ν ν ν

⎧ − ⎡−⎛ − ⎞ ⎤ ≤ ≤

⎪ − ⎢ ⎜ Δ ⎟ ⎥

⎪ ⎢⎣ ⎝ ⎠ ⎥⎦

= ⎨⎪⎪⎩ − − − ≤ ≤

, (7)

where P0 is the maximum PSD and Δν is a parameter relating to the bandwidth of the pump band. The values of P0 and Δν are trialed until satisfactory optimized result is obtained. Usually P0

can be set to be mW. Therefore only Δν is required to be trialed.

An example of the trial initial solution is given in Fig. 1, in which P0= 8 mW and Δν= 310 nm. The optimized PSDF is also shown in

Fig.1, in which the maximum PSD is at the pre-assigned extremum wavelength.

4. Numerical results and discussions

For the DFRAs 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 the long-wavelength signals. The short-wavelength pump lobe not only amplifies the short-wavelength signals but also the long-wavelength pump lobe. Therefore, the power of the short- wavelength lobe is higher than the long-wavelength lobe and it is preferred to set the extremum wavelength as the wavelength of the short-wavelength pump lobe. Fig.2 shows the optimized pump PSDFs for several extremum wavelengths, in which their corresponding gain ripples, effective noise figures, and total pump powers are shown in Figs. 3, 4, and 5 respectively. From Fig.3, the corresponding gain ripples for the all cases shown in Fig.2 are less than 0.1 dB. From Fig.4, the effective noise figure is larger for short-wavelength signals because the maximum ASEN PSD is near

the short-wavelength region of the signal band that is amplified by the short-wavelength pump lobe [2,4]. The use of shorter extremum wavelength shifts the maximum ASEN away from the signal band and decreases the effective noise figure but at the expense of higher pump power. Compared with the case using 1420-nm extremum wavelength, the noise figure of the case using 1380-nm extremum wavelength is reduced about 0.5 dB while the increment of total pump power is 214 mW.

For the DFRA using bidirectional pumping, the pump bands of the co-pump and counter-pump are respectively specified. We set

λ _{N =} λ _e -20 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.

There are two lobes for the counter-pump, in which the short- wavelength and long-wavelength pump lobes respectively amplify

the signals near 1565 nm and 1600 nm. Because the wavelength of the co-pump is the shortest, it also amplifies the counter-pump and requires higher pump power. We set the extremum wavelength as the wavelength of the co-pump. Figs.6(a) and 6(b) show the optimized pump PSDFs for the co-pump and counter-pump respectively, in which several extremum wavelengths for the co- pumps are shown. The corresponding gain ripples, effective noise figures, and total pump powers are shown in Figs. 3, 7, 5, respectively. From Fig.3, the corresponding gain ripples for all the cases shown in Fig.6 are less than 0.1 dB. From the consideration of pumping efficiency for the short-wavelength signals, the use of shorter extremum wavelength requires higher pump power because it shifts away form the effective Raman gain bandwidth. The use of the extremum wavelength near 1370 nm provides the best pumping efficiency for the amplification of the short-wavelength lobe of the counter-pump near 1460 nm. In addition, there is the water absorption peak near 1390 nm for the transmission fiber. The three factors effect required total pump power. From Fig. 5, the largest

required pump power is the case using 1350-nm extremum wavelength.

From Fig. 7, the wavelengths of their maximum effective noise figures are near 1565 nm. Fig. 8 shows the output forward ASEN PSDF for the cases of 1340 nm, 1370 nm, and 1400 nm.

The evolutions of the forward ASEN PSDs at 1530 nm, 1565 nm, and 1600 nm for the cases shown in Fig. 8 are shown in Fig. 9. As is explained above the ASEN near the shortest signal wavelength amplified by the co-pump is attenuated in the middle section of the transmission fiber and the corresponding effective noise figure is reduced. This results in the lowest effective noise figure near 1530 nm. The maximum ASEN near 1565 nm is due to the amplification of the ASEN by the short-wavelength lobe of the counter-pump which can be effectively amplified by the co-pump in the middle section of the transmission fiber. As to the ASEN near 1600 nm, it is mainly amplified by the long-wavelength of the counter-pump because its wavelength differences from the co-pump and the short-wavelength lobe of the counter-pump are far beyond the 13.2-THz Raman gain bandwidth. Near the output end of the

transmission fiber, as the power of the long-wavelength lobe of the counter-pump is less than the short-wavelength lobe of the counter- pump, the ASEN near 1600 nm is less amplified than the ASEN near 1565 nm. Therefore the effective noise figure near 1600 nm is lower than near the effective noise figure near 1565 nm. For the cases shown in Fig.7, the best extremum wavelength for the minimum effective noise figure is 1370 nm. From Fig. 6, one can see that the power of the short-wavelength lobe of the counter- pump near 1460 nm is the minimum for the case of 1370 nm because the wavelength difference of 1460 nm and 1370 nm corresponds to the maximum Raman gain. Therefore the ASEN near 1565 nm at the output end of the transmission fiber for this case is less amplified than the other cases and the corresponding effective noise figure is also the least. Compared with the case using 1420-nm extremum wavelength, the noise figure of the case using 1370-nm extremum wavelength is reduced about 1.75 dB while the increment of total pump power is 500 mW.

5. Conclusion

We present the method for designing the incoherent pump power spectra for the DFRAs with a pre-assigned extremum pump wavelength to meet the preference of the applications. 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. In case there appears the additional extremum wavelength that the corresponding PSD is higher than the PSD of the pre-assigned extremum wavelength, proper initial trial solutions for the coefficients of pump PSDF can be taken to enhance the PSD at the pre-assigned extremum wavelength. Two DFRAs using backward pumping and bidirectional pumping respectively are taken as examples. The gain ripples of the considered DFRAs are less than 0.1 dB for 20 dB signal gain over 70-nm bandwidth. The dependences of the pump power and effective noise figure on the extremum pump

wavelength are studied. For the DFRA using backward pumping, the use of shorter extremum pump wavelength results in lower effective noise figure but larger pump power. For the DFRA using bidirectional pumping, the extremum wavelength is set to the co- pump. The optimal spectrum of counter-pump has the double-lobe structure. It is found that the effective noise figure is the minimum for the case using the extremum pump wavelength that has the maximum Raman gain for the wavelength of the short-wavelength lobe of the counter-pump.

FIGURE CAPTIONS

Fig.1 Optimized pump power spectral density functions (PSDFs) for the DFRAs using backward pumping and 1360-nm extremeum wavelength. The cases (a) with null initial trail solutions and (b) with the non-null initial trail solutions are shown. In the figure, the non-null initial trail solutions given by Eq.(7) is also shown.

Fig. 2. Optimized pump power spectral density functions (PSDFs) for the DFRAs using backward pumping, in which the cases of several extremum wavelengths ranging from 1360 nm to 1420 nm are shown.

Fig. 3. Gain ripples of the DFRAs using backward pumping and bi-directional pumping with the optimized pump power spectral density functions given in Figs.2 and 6, respectively.

Fig. 4.Effective noise figures of the DFRAs using backward pumping with the optimized pump power spectral density functions given in Figs.2.

Fig. 5. Total pump powers of the DFRAs using backward pumping and bi-directional pumping with the optimized pump power spectral density functions given in Figs.2 and 6, respectively.

Fig. 6. Optimized pump power spectral density functions (PSDFs) for the DFRAs using bi-directional pumping, in which the cases of several extremum wavelengths ranging from 1320 nm to 1420 nm are shown. The PSDFs of co-pump and counter-pump are shown in (a) and (b) respectively.

Fig. 7. Effective noise figures of the DFRAs using bi-directional pumping with the optimized pump power spectral density functions given in Figs.6, in which the extremum wavelength are (a) from 1320 nm to 1370 nm and (b) from 1370 nm to 1420 nm.

Fig. 8.Output forward ASEN PSDFs for the DFRAs using bi- directional pumping with the cases of 1340-nm, 1370-nm, and 1400-nm extremum wavelengths shown in Fig.6.

Fig. 9. Evolutions of the forward ASEN PSDs at 1530 nm, 1565 nm, and 1600 nm for the DFRA using bi-directional

pumping with the cases of 1340-nm, 1370-nm, and 1400- nm extremum wavelengths shown in Fig.6.

Fig. 1.(a)

Fig. 1.(b)

Fig. 1. Optimized pump power spectral density functions (PSDFs) for the DFRAs using backward pumping and 1360-nm extremeum wavelength. The cases (a) with null initial trail solutions and (b) with the non-null initial trail solutions are shown. In the figure, the non-null initial trail solutions given by Eq.(7) is also shown.

Fig. 2.

Fig. 2. Optimized pump power spectral density functions (PSDFs) for the DFRAs using backward pumping, in which the cases of several extremum wavelengths ranging from 1380 nm to 1420 nm are shown.

Fig. 3.(a)

Fig. 3.(b)

Fig. 3. Gain ripples of the DFRAs using backward pumping and bi-directional pumping with the optimized pump power spectral density functions given in Figs.2 and 6, respectively.

Fig. 4.

Fig. 4. Effective noise figures of the DFRAs using backward pumping with the optimized pump power spectral density functions given in Figs.2.

Fig. 5.(a)

Fig. 5.(b)

Fig. 5. Total pump powers of the DFRAs using backward pumping and bi-directional pumping with the optimized pump power spectral density functions given in Figs.2 and 6, respectively.

Fig. 6

Fig. 6. Optimized pump power spectral density functions (PSDFs) for the DFRAs using bi-directional pumping, in which the cases of several extremum wavelengths ranging from 1320 nm to 1420 nm are shown. The PSDFs of co-pump and counter-pump are shown in (a) and (b) respectively.

Fig. 7.(a)

Fig. 7.(b)

Fig. 7. Effective noise figures of the DFRAs using bi-directional pumping with the optimized pump power spectral density functions given in Figs.6, in which the extremum wavelength are (a) from 1320 nm to 1370 nm and (b) from 1370 nm to 1420 nm.

Fig. 8

Fig. 8. Output forward ASEN PSDFs for the DFRAs using bi- directional pumping with the cases of 1340-nm, 1370-nm, and 1400-nm extremum wavelengths shown in Fig.6.

Fig. 9.

Fig. 9. Evolutions of the forward ASEN PSDs at 1530 nm, 1565 nm, and 1600 nm for the DFRA using bi-directional pumping with the cases of 1340-nm, 1370-nm, and 1400-nm extremum wavelengths shown in Fig.6.

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