## Abstract

We propose a simple technique to optimize a multi-wavelength backward-pumped fiber Raman amplifier. Based on the geometric characteristics of Raman gain profile, we approximate it using several straight lines and utilize slope compensation technique to achieve flat and wideband gain profile. Good simulation results are obtained.

©2004 Optical Society of America

## 1. Introduction

Optimization of wavelength and power of pump lasers to achieve flat gain spectrum over the amplifier bandwidth is an important issue in distributed Raman amplifier design. Due to pump-to-pump and pump-to-signal Raman interaction, a set of coupled nonlinear equations have to be made to model the Raman amplification process. No simple relationship between Raman gain and pump power of multi-wavelength pumps can be found unless major simplifications are used which will compromise the accuracy [1]. To solve this problem a new method was proposed in [2], where genetic algorithm (GA) was used to find the pump wavelengths and the pump power integrals along the fiber before an iterative algorithm was applied to determine the required pump powers. The approach allows the Raman amplifier design problem to be finally solved irrespective of time-consuming searches of combinations of pump powers and wavelengths. However, its convergence can be slow due to the complexity of GA.

Although no simple relationship between Raman gain and pump power of multi-wavelength pump lasers can be found, it was shown in [2] that the logarithmic Raman gain expressed in dB can be given as a linear addition of pump integrals (integration of pump power distribution of a given pump wavelength along the fiber). As a result, a much simpler method may be used to find the pump wavelengths and the pump integrals first, before an iterative method is used to obtain the pump power. In this paper, we show that through approximating a given Raman gain profile and fiber loss profile using several asymptotic lines, the flat gain spectrum design problem is converted to a simple geometry equalization problem and it is also possible to separate the problem of compensating the gain curve profiles (involving the determination of the pump wavelengths and power integrals) from the problem of calculating the pump power values. This method greatly simplifies the design procedure and enables fast amplifier performance optimization to be carried out.

## 2. Theoretical model and design method

As shown in [2] the logarithmic net Raman gain of the k_{th} signal channel in dB can be described as a linear superposition of the gain spectra of individual pump wavelengths with respective weighting factors given by the correspending pump integrals, as shown in Eq. (1).

$$k=n+1,n+2,\dots n+m;j=1,2,\dots n$$

Where g_{jk}=g_{υj}(υ_{j}-υ_{k})/(K_{eff}A_{eff}) for υ_{j}>υ_{k} and g_{jk}=-g_{υk}(υ_{k}-υ_{j}) υ_{k}/(υ_{j} K_{eff}A_{eff}) for υ_{j}<υ_{k}, the gain coefficient at pump frequency υ_{i} are given by g_{υi}(Δυ)=gR(Δυ). υ_{j}/υ_{0}, where g_{R}(Δυ) is the Raman gain spectrum measured at a reference pump frequency υ_{0}, A_{eff} is the effective area of the fibre and K_{eff} is the polarization factor. k represents one of the m signal channels, j represents one of the n pump wavelengths. g_{jk} and I_{j} correspond to the gain profile and the power integral (the area below the power distribution curve of pump lights along the fiber) of different pump and signal waves. α and L are the loss coefficient of fiber at the signal wavelength and the length of the fiber respectively.

In the proposed design approach, we describe the gain profile of a given pump wavelength j using multi-segment straight lines, a_{j0}a_{j1}a_{j2}a_{j3}a_{j4}a_{j5}a_{j6}a_{j7}a_{j8} as shown in Fig. 1. For simplicity, frequency rather than wavelength is used for the gain spectrum representation. For silica fiber the frequency intervals are a_{j1}a_{j2}=a_{j3}a_{j4}=a_{j4}a_{j5}=a_{j5}a_{j6}=a_{j6}a_{j7}=a_{j7}a_{j8}=1.7THz and a_{j2}a_{j3}=3.4THz respectively. In Fig. 2, S_{1} is used to represent the gain spectrum of the longest wavelength pump, S_{2}, S_{3}, S_{4} … and S_{n} are used to represent the gain spectra of the other pump wavelengths. Typically the integral of the longest wavelength pump light S_{1} is much larger than those from the other pump wavelengths due to strong stimulated Raman interaction of pumps. As a result, we may neglect the contribution of a_{j0}a_{j1} and represent the gain spectra of all the other pump wavelengths with a_{j1}a_{j2}a_{j3}a_{j4}a_{j5}a_{j6}a_{j7}a_{j8} (j=2,3…n). Since the portion of gain spectrum of a_{12}a_{13} can be considered flat, it is possible to equalize the rest of the gain spectrum of S_{1}, using contributions from the a_{j1}a_{j2} (j=2…n) portion of the gain spectra of all the other pump sources to achieve flat gain spectrum. The range of the gain spectrum to be equalized will decide the number of pump wavelengths required.

We consider a case of three pump wavelengths and assume the pump laser with the longest wavelength is S_{1} and the two other pumps are S_{2} and S_{3} as indicated in Fig. 2. The loss of the fibre is considered to be wavelength independent at first. The flat gain profile can be achieved for the signal channels over a frequency region Δf in the gain profile produced by the longest wavelength pump light. After the gain bandwidth region for the amplifier has been decided, the frequency of the longest wavelenght pump light is decided accordingly. We can then use a_{j1}a_{j2}(j=2,3) in the gain profile of the two other pump lights to compensate a portion of the gain profile of S_{1} to obtain a flat gain spectrum. To compensate the slope of a_{13}a_{14} in S_{1} with the slope of a_{21}a_{22} in S_{2}, we should let the frequency of point a_{21} be equal to that of point a_{13}, thus the frequency of the second pump light should be shifted 5.1Thz (about 37.5nm) from that of the longest wavelenghth pump light. The pump integral I_{2} of S_{2} should be such that the slope of a_{21}a_{22} is the same as the slope of a_{13}a_{14} but with opposite sign. Similarly to compensate a_{14}a_{15} using a_{31}a_{32} of S_{3}, we should let the frequency of point a_{31} be equal to that of point a_{22}. Thus the frequency of the third pump light should be shifted 1.7Thz(about 12.75nm) from the frequency of S_{2}. The pump integral I_{3} of S_{3} should be such that the slope of a_{31}a_{32} is the same as the slope of a_{14}a_{15} but with opposite sign. When wider amplifier bandwidth is needed, we can use more pump wavelengths and compensate the other part of S_{1} in the same way. The frequencies of all the pump lights can be determined accordingly. However, when deciding the pump integrals, the contributions to the slope of S_{1} from S_{2} and S_{3} should be considered. From the above discussions and with wavelength independent considered, we can obtain following relationship

$${I}_{j}={I}_{1}\frac{(g({\nu}_{{a}_{1,j+1}})-g({\nu}_{{a}_{1,j+2}}))}{{\nu}_{{a}_{1,j+1}}-{\nu}_{{a}_{1,j+2}}}\frac{{\nu}_{{a}_{j,1}}-{\nu}_{{a}_{j,2}}}{g({\nu}_{{a}_{j,1}})-g({\nu}_{{a}_{j,2}})}$$

$$\text{j}=2\dots \text{n}$$

Where G is the targeted Raman net gain in dB. I_{1} is the pump integral of S_{1} and I_{j} is pump integral of S_{j}. ν_{ai,j} is the frequency corresponding to point a_{i,j} and g(ν_{ai,j}) is the Raman gain at frequency ν_{ai,j}.

In the above discussion, we have assumed that the loss profile is wavelength independent. However, in a practical system, it is normally not the case. To consider wavelength dependent loss, the loss spectrum of fiber over the same wavelength range as S_{1} is also represented by multi-segment lines α_{3} α_{4} α_{5} with 1.7THz per segment as shown in Fig.3, where the solid lines highlight the parts in the geometrical compensation model. We can then use it to modify the slope of S_{1}. Eq. (2) can thus be modified as:

$$j=2\dots n$$

Where α(ν_{ai,j}) is the loss at frequency ν_{ai,j}. This will allow us to obtain the required pump integrals, taking into consideration the wavelength dependant loss. It can be seen in Fig. 3 that the dot lines ignored are the origin of gain ripple. Due to the non-linear interaction, i.e., pump/signal depletion, cross-pump depletion due to inter-channel Raman interaction between pump wavelengths etc., the pump powers cannot be derived directly from the pump integrals. Using the same approach as in Ref. [2], we can go back to the coupled non-linear Raman propagation equations and use an iterative method to obtain the pump powers. Typically a few iterations will enable us to obtain the required pump powers. Here Raman gain tilt due to signal-to-signal interaction was not considered since it is related to transmission system parameters. However, we can reasonably ignore the effect of Raman process in obtaining the inter-channel Raman interaction slope for a given input signal condition without amplification and then use it to rectify the slope of S_{1}. In this way the effect of the inter-channel Raman interaction slope can also be equalized in the Raman amplifier design.

More divisions for straight line approximation of a gain profile can result in lower ripple and calculation error, however, this in turn requires more pumps in practical implementation that may not be economical, and the design procedure tends to be more complex and inefficient when compared with GA.

## 3. Numerical example and discussions

The technique is applied to optimize a C band Raman amplifier design as an example. The amplifier is backward pumped by three pump wavelengths with 80km SMF and 0dB targeted net gain. It is designed for DWDM system with 40 signal channels from 1530nm to 1565nm with input signal level at 1mW/ch. Wavelength dependent loss spectrum of the fiber is considered in the design.

After applying the design procedure discussed above, the resultant theoretical gain curve is given in Fig. 4(a). Clearly a gain spectrum flatness of about 0.5dB can be obtained. The gain ripple is mainly at the two sides of the gain spectrum. Clearly, if more pump wavelengths are used for compensation, better flatness can be obtained. The wavelengths of the pump lights are 1422.5nm, 1434nm and 1469nm, while the corresponding pump powers are 654mW, 349mW and 275mW respectively. To verify the result, we use the complete numerical model given by Kidorf [3], which considers the ASE power and Rayleigh scattering but not inter-channel Raman interaction among signal channels. The result is shown in the same figure. Apart from a slight difference in net gain, good agreement with designed theoretical result is achieved. This very small difference is as expected since in our design procedure the effects of Rayleigh scattering and ASE were not considered. When considering inter-channel Raman interaction among signals in the Kidorf model, the net gain spectrum will tilt noticably as shown in Fig. 4(b). This can be rectified by considering the effect of inter-channel Raman interaction in the amplifier design. Here the slope of inter-channel Raman interaction with the given input signal condition is obtained first, it is then used to modify the slope of S_{1} in the design process before the compensation procedure is carried out. After the amplification condition is established, the newly obtained inter-channel Raman interaction value is then used to modify the slope of S_{1} before a new compensation procedure is carried out. Very few iterations are required since the Raman tilt hardly changes after the signal distribution in the presence of the pump is more or less established. The obtained new result is plotted in Fig. 4(b). The modified pump powers are 682mW, 370mW and 254mW respectively. Clearly, this allow flat net gain spectrum to be achieved even if inter-channel Raman interaction is taken into consideration.

## 4. Conclusions

We proposed a simple geometry simulation technique to optimize the design of wideband distributed Raman amplifier gain based on the geometric characteristics of the Raman spectrum. In comparison with traditional solutions using rigorous iterative simulation, the suggested method is simpler and more effective. It is also possible to separate the problem of compensating the gain curve profiles (involving the determination of the pump wavelengths and the power integrals) from the problem of calculating the pump power values. The design results have been verfied using the complete numerical model and good agreement has been achieved.

## References and links

**1. **X Zhou and C Lu, et al, “A simple model and optimal design of a multi-wavelength backward-pumped fiber Raman amplifier,” IEEE Photon. Technol. Lett. **13**, 945–947 (2001). [CrossRef]

**2. **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]

**3. **Howard Kidorf and Karsten Rottitt, et al, “Pump interaction in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–533 (1999). [CrossRef]