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).

Where g_jk=g_υj(υ_j-υ_k)/(K_effA_eff) for υ_j>υ_k and g_jk=-g_υk(υ_k-υ_j) υ_k/(υ_j K_effA_eff) for υ_j<υ_k, the gain coefficient at pump frequency υ_i are given by g_υi(Δυ)=gR(Δυ). υ_j/υ₀, where g_R(Δυ) is the Raman gain spectrum measured at a reference pump frequency υ₀, 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.

 

Fig. 1. Asymptotic linear approximation of a typical Raman gain spectrum g_R(Δυ) of a silica fiber at pump wavelength λ₀=1µm

Download Full Size | PDF

In the proposed design approach, we describe the gain profile of a given pump wavelength j using multi-segment straight lines, a_j0a_j1a_j2a_j3a_j4a_j5a_j6a_j7a_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_j1a_j2=a_j3a_j4=a_j4a_j5=a_j5a_j6=a_j6a_j7=a_j7a_j8=1.7THz and a_j2a_j3=3.4THz respectively. In Fig. 2, S₁ is used to represent the gain spectrum of the longest wavelength pump, S₂, S₃, S₄ … and S_n are used to represent the gain spectra of the other pump wavelengths. Typically the integral of the longest wavelength pump light S₁ 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_j0a_j1 and represent the gain spectra of all the other pump wavelengths with a_j1a_j2a_j3a_j4a_j5a_j6a_j7a_j8 (j=2,3…n). Since the portion of gain spectrum of a₁₂a₁₃ can be considered flat, it is possible to equalize the rest of the gain spectrum of S₁, using contributions from the a_j1a_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.

 

Fig. 2. Schematic diagram of the geometric compensation scheme (i=2,3,4), with wavelength independent loss considered. S₁, S₂ and S₃ show the partial gain spectral; Δf shows the target amplification band; υ_a1,1, υ_a21, υ_a31 and υ_a32 related to frequency allocation of gain spectra from individual pumps

Download Full Size | PDF

We consider a case of three pump wavelengths and assume the pump laser with the longest wavelength is S₁ and the two other pumps are S₂ and S₃ 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_j1a_j2(j=2,3) in the gain profile of the two other pump lights to compensate a portion of the gain profile of S₁ to obtain a flat gain spectrum. To compensate the slope of a₁₃a₁₄ in S₁ with the slope of a₂₁a₂₂ in S₂, we should let the frequency of point a₂₁ be equal to that of point a₁₃, 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 I₂ of S₂ should be such that the slope of a₂₁a₂₂ is the same as the slope of a₁₃a₁₄ but with opposite sign. Similarly to compensate a₁₄a₁₅ using a₃₁a₃₂ of S₃, we should let the frequency of point a₃₁ be equal to that of point a₂₂. Thus the frequency of the third pump light should be shifted 1.7Thz(about 12.75nm) from the frequency of S₂. The pump integral I₃ of S₃ should be such that the slope of a₃₁a₃₂ is the same as the slope of a₁₄a₁₅ but with opposite sign. When wider amplifier bandwidth is needed, we can use more pump wavelengths and compensate the other part of S₁ 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 S₁ from S₂ and S₃ should be considered. From the above discussions and with wavelength independent considered, we can obtain following relationship

Where G is the targeted Raman net gain in dB. I₁ is the pump integral of S₁ 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.

 

Fig. 3. Schematic diagram of the geometric compensation scheme (i=2,3,4), with wavelength dependent loss considered. Triangle: wavelength dependent loss of the fiber; Circle: individual gain spectral

Download Full Size | PDF

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 S₁ is also represented by multi-segment lines α₃ α₄ α₅ 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 S₁. Eq. (2) can thus be modified as:

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 S₁. 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.

 

Fig. 4. example of a C-band Raman amplifier design

Download Full Size | PDF

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 S₁ 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 S₁ 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.