An automatic step adjustment method for average power analysis technique used in fiber amplifiers

Xue-Ming Liu

doi:10.1364/OE.14.002589

1. Introduction

Erbium-doped fiber amplifiers (EDFAs) are key components for wavelength-division-multiplexed (WDM) transmission systems [1]-[4]. To reduce costs and optimize system performance, the effective methods for designing EDFAs are necessary [4]-[6]. If the traditional numerical tools such as Runge-Kutta methods and Adams methods are used to solve the propagation equations of EDFAs, the required computation time makes this task be a serious problem. To overcome this difficulty, Hodgkinson had proposed an average power analysis (APA) technique [7] and its improved modification [8] on the basis of the semiconductor laser analysis technique [9]. Hereafter, the simulation tools for the system model of fiber amplifiers are originated from the APA technique [6], [10]-[14]. However, this technique employed a fixed constant step-size, i.e., the total amplifier length was divided into a number of fixed elemental lengths. To overcome this limitation, an automatic step adjustment (ASA) method for the APA technique has been proposed in this paper for the first time to author’s best knowledge.

2. Theoretical modeling and APA technique

The spectral gain and the spectral noise figure of EDFAs can be completely described by the propagation and rate equations modeling the interaction of the optical field with erbium ions [15]-[21]. Usually, EDFAs are modeled by a two-level system [22]-[24]. When there are M input wavelengths, the equations, which describe the power evolution and govern the amplifier behavior, can be expressed as [8]

\frac{d P_{k}^{\pm}}{dz} = \pm {[(σ_{21}^{k} + σ_{12}^{k}) N_{2} (z) - σ_{12}^{k} N_{t}] Γ_{k} - α_{k}} P_{k}^{\pm} (z),

\frac{d P_{i}^{\pm}}{dz} = \pm ({[(σ_{21}^{i} + σ_{12}^{i}) N_{2} (z) - σ_{12}^{i} N_{t}] Γ_{i} - α_{i}} + 2 σ_{12}^{i} N_{2} (z) \frac{Γ_{i} h v_{i} Δ v}{P_{i}^{\pm} (z)}) P_{i}^{\pm} (z) .

Here, + means forward propagating (+) and backward propagating (-), k is the wavelength identifier, i is the ASE wavelength identifier, P is optical power, α is the fiber background loss coefficient, Γ is the overlap integral factor between the dopant and the optical mode, N_t is the local erbium ion density, N ₂ is the metastable level population density, h is Planck’s constant, σ ₂₁ and σ ₁₂ are, respectively, stimulated emission and absorption cross-sections, and P_i represents ASE power in the frequency interval Δv at the center of v_i.

Since the power evolution of light along the erbium-doped fiber resembles exponential pattern, it is intuitive that the exponential change of a variable may lead to faster convergence and higher speed than some linear or polygonal approximation. Taking advantage of this property, the APA technique is constructed to increase the computing speed [7], [8].

P_{k}^{\pm out} = P_{k}^{\pm in} G_{k} (z), P_{i}^{\pm out} = 2 h v_{i} n_{sp} Δ v (G_{i} (z) - 1),

where the power gain G_k,i (z) and the population inversion factor n_sp are defined as

G_{k, i} (z) = \exp ({[(σ_{21}^{k, i} + σ_{12}^{k, i}) N_{2} - σ_{12}^{k, i} N_{t}] Γ_{k, i} - α_{k, i}} z),

n_{sp} = \frac{N_{2}}{[(1 + \frac{σ_{12}^{k, i}}{σ_{21}^{k, i}}) N_{2} - N_{t} \frac{σ_{12}^{k, i}}{σ_{21}^{k, i}} - \frac{α_{k, i}}{Γ_{k, i} σ_{21}^{k, i}}]} .

After averaging Eq. (2), the length-averaged power in each elemental amplifier section can be written as the following expression

〈 P_{k} 〉 = P_{k}^{\pm in} (\frac{G_{k} - 1}{In (G_{k})}), 〈 P_{i} 〉 = 2 h v_{i} n_{sp} Δ v (\frac{G_{i} - 1}{In (G_{i})} - 1) .

Therefore, the relatively simple algebraic calculations replace the time-consuming numerical integrations so that the computation time is decreased.

3. Adaptive step-size control algorithm

When the APA technique is implemented to solve the propagation equations of EDFAs, its two disadvantages, i.e., the fixed step-size and the two-order accuracy, obstruct the enhancement of the computing speed. Here, the n-order accuracy is defined as ε ∝ hⁿ, i.e., the local truncation error ε is proportional to n power of the step size h. In order to overcome this limitation, an adaptive step-size control algorithm has been proposed to increase the computing speed and improve the accuracy.

By introducing vectors P and f, Eq. (1) can be recast in a concise notation, i.e.,

\frac{d P}{dz} = f (z, P) ∙ P .

By substituting y to ln(P) [i.e., y =ln(P)], Eq. (6) is changed to

\frac{dy}{dz} = f (z) .

We denote y(z) to be the exact solution of Eq. (7), and the initial condition is assumed as yn=y(z_n). The solution of the next step (i.e., z _n+1=z_n+h) can be approximated as [25]

y_{n + 1,1} = y_{n} + h ∙ f (z_{n}, y_{n}),

y_{n + 1,2} = y_{n} + 0.5 h f (z_{n}, y_{n}) + 0.5 h f (z_{n} + 0.5 h, y_{n} + 0.5 h f (z_{n}, y_{n})),

y_{n + 1,3} = 2 y_{n + 1,2} - y_{n + 1,1} .

After Taylor expansion, the corresponding local truncation errors ε of Eqs. (8a)-(8c) can be expressed as, respectively,

ε_{1} = y (z_{n} + h) - y_{n + 1,1} = K h^{2} + O_{1} (h^{3}),

ε_{2} = y (z_{n} + h) - y_{n + 1,2} = 0.5 K h^{2} + O_{2} (h^{3}),

ε_{3} = y (z_{n} + h) - y_{n + 1,3} = 2 O_{2} {(h^{3})}^{} - O_{1} (h^{3}),

where $K = \frac{1}{2} \frac{d^{2} y (x)}{d z^{2}} ∣_{z = z_{0}}, O_{1} (h^{3}) = \frac{1}{6} h^{3} \frac{d^{3} y (z)}{d z^{3}} ∣_{z = ξ_{1}} and O_{2} (h^{3}) = \frac{1}{6} h^{3} \frac{d^{3} y (z)}{d z^{3}} ∣_{z = ξ_{2}} .$ .The corresponding error difference rate between Eq. (9a) and Eq. (9b) is about

ε_{r} = \frac{∣ y_{n + 1,2} - y_{n + 1,1} ∣}{h} \approx 0.5 ∣ K ∣ h .

If ε_r is less than a required error ε ₀ (i.e., ε_r > ε ₀), we can accept y _n+1,3 as an approximate value of y(z_n+h). In contrast, if ε_r > ε ₀, we reject y _n+1,3 and repeat the current step with a new trial step-size h´. The old and new step sizes have the following relationship

h' = \frac{ε_{0} h}{ε_{r}} .

By substituting P to exp(y) [i.e., P =exp(y)], Eqs. (8a)-(8c) can be rewritten as

P_{n + 1,1} = P_{n} . \exp (h f (z_{n}, P_{n})),

P_{n + 1,2} = P_{n} ∙ \exp [0.5 h f (z_{n}, P_{n})] ∙ \exp {0.5 h f (z_{n} + 0.5 h, P_{n} ∙ \exp [0.5 h f (z_{n}, P_{n})])},

P_{n + 1,3} = \frac{P_{n + 1,2}^{2}}{P_{n + 1,1}} .

The corresponding ε_r is changed as

ε_{r} = \frac{∣ In (P_{n + 1,2}) - In (P_{n + 1,1}) ∣}{h} \approx 0.5 ∣ K ∣ h .

Therefore, on the basis of Eqs. (11)-(13), an adaptive step-size control algorithm can be constructed. The ASA method is shown as follow:

4. Test, results and discussion

In comparison with the APA technique, the proposed method has suggested two key merits:

The traditional APA technique is originated from Eq. (12a) with the error shown in Eq. (9a), but the proposed method is based on Eq. (12c) with the error given by Eq. (9c). The error is the three-order accuracy in the latter rather than the two-order accuracy in the former, so that the proposed method improves the accuracy of the numerical solutions.
Based on Eqs. (11) and (13), the adaptive step-size control algorithm can be implemented in this paper. Unfortunately, the APA technique used so far was lacking an ASA mechanism. Therefore, the combination of the ASA method and the APA technique can increase the computing speed and improve the solution accuracy.

In order to compare the proposed method to the traditional APA technique, an ordinary differential equation is tested, i.e.,

\frac{dx}{dt} = x - t + 1 = (1 - \frac{t}{x} + \frac{1}{x}) ∙ x, and x (2) = \exp (2) + 2 .

Figure 1 illustrates the relative error between the numerical value x_n and exact value x(t_n), i.e., |(x(t_n)-x_n)/x(t_n)|. From Fig. 1, we can see that, in order to reach the relative error of <2.5×10^-4 at t=20, (1) only 38 steps are enough when the proposed method is used to solve Eq. (14); (2) there are 4287 steps with the step-size of 0.0042 for the fixed step method used in the APA technique; and (3) the proposed method shortens the CPU time by more than a hundredfold in comparison with the APA technique. Moreover, the step-size h used in the ASA method is larger and larger when t increases. The reasons why the computing speed of the proposed method is two orders of magnitude faster than that of the APA technique originate from two factors: an ASA mechanism and a higher order accuracy.

Fig. 1. Relationship of relative error with t. The circular symbols in figure represent the points that are calculated in the computing procedure.

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On the basis of the parameters used in Ref. [7], the model equations of EDFAs are computed by the traditional APA technique and the proposed method, respectively. The corresponding numerical results are shown in Fig. 2. This figure demonstrates the relationship between the relative error E_r and the elemental amplifying section number n, where the whole amplifier length L is partitioned into n sections and each section length h is L/n in the APA technique. Note that the relative error E_r is defined by |(P(Z_L)-P_L)/P(Z_L)|, where P(Z_L) and P_L are the exact value and the numerical value at the total amplifier length L, respectively. From Fig. 2, we can see that (1) when n>20, the relative error E_r of the proposed method is two orders of magnitude less than that of the APA technique; (2) at the same E_r of 2×10^-2, the elemental amplifying section number n used in the proposed method is ~7 instead of ~100 in the APA technique; (3) with the increase of n, the solution accuracy in the proposed method is better and better than that in the APA technique, i.e., the difference of their E_r is greater and greater.

Numerical results also show that, if the solution accuracy is higher, the speed of convergence is faster and the CPU time efficiency is more. For instance, the computational time of the proposed method is about 15 times less than that of the APA technique at the same relative error E_r of 10^-2, while the computational speed of the proposed method can be increased by more than two orders of magnitude in comparison with that of the APA technique at the same E_r of 10^-4. However, the improvement of the computational efficiency is at the cost of the stability. For example, when L>120 m, our method is unstable to solve Eq. (1) whereas the APA technique is still stable.

Fig. 2. Relationship of the relative error E_r with the elemental amplifying section number n. Each section length h is L/n in the APA technique, but the length h is adaptive in the proposed method.

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In additional, since the power of lightwave in the fiber Raman amplifiers exponentially evolves, the APA technique has been used for the Raman amplifier modeling [26]-[28]. Therefore, the proposed method has the capacity to rapidly and accurately solve the propagation equations of fiber Raman amplifiers and semiconductor lasers.

5. Conclusion

In this paper, we have proposed an adaptive step-size control algorithm for the traditional APA technique used in fiber amplifiers for the first time. Compared to the conventional APA technique, the proposed method has two unique merits, i.e., a higher order accuracy and an ASA mechanism. So the computing time is significantly reduced and the solution accuracy is effectively improved. A test example shows that, in comparison with the APA technique, the proposed method increases the computing speed by two orders of magnitude under the same relative error. The model equations of EDFAs are solved and compared by the proposed method and the APA technique, respectively, and the numerical results show that the solution accuracy in the former is about two orders of magnitude higher than that in the latter at the same elemental amplifying section number n. Moreover, our method can effectively be used to fast solve the propagation equations of fiber Raman amplifiers and semiconductor lasers.

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An automatic step adjustment method for average power analysis technique used in fiber amplifiers

Abstract

1. Introduction

2. Theoretical modeling and APA technique

3. Adaptive step-size control algorithm

4. Test, results and discussion

5. Conclusion

References and links

Cited By

Figures (2)

Equations (21)

Optics Express