1. Introduction

Semiconductor Optical Amplifier (SOA) is a key device for nonlinear sub-systems implementation, enabling all-optical signal processing functionalities in Wavelength Division Multiplexing (WDM) networks [2]. SOA-based sub-systems can give feasible alternatives for wavelength conversion [2], switching [3] and pulse regeneration [4], among others. Also, SOA’s are essential for the development of small form factor, low power, uncooled high performance hybrid integrated optical systems [5]. In all these applications, simple models allowing reduction in testing time and resources for chip or package characterization are desirable, since this step is a costly one in the optoelectronic production chain. Also, simple algorithms covering the spectral gain and spontaneous emission over a large range of wavelengths and pump currents can enable the fabrication of tunable components for uncooled operation relying on simple in-package correction logic circuit.

Several different approaches to SOA modeling have been presented in the literature. While sophisticated models can provide design and analysis tools for the active region of the amplifiers [6], some simplified semi-empirical models are versatile for practical analysis [7].

We have recently proposed a Black Box Model (BBM) for the characterization of the gain and noise behavior of Semiconductor Optical Amplifiers (SOA) operating under CW conditions [1], i.e., a model that does not use any intrinsic device parameter and needs few experimental data points to map an entire range of SOA’s spectral properties. In this BBM three spectral gain or noise curves for different pump (bias current) condition are used to predict the SOA spectral response under any other bias condition, by knowing only two spectral points of the desired curve. In other words, the whole gain or noise spectrum of a SOA is predicted if only two spectral points of the curve are known and for this prediction we use three spectral functions that can be calculated from three gain or noise spectra. In this work, the BBM for SOAs operating under CW conditions is presented and experimentally validated to describe the spectral gain and amplified spontaneous emission (ASE) behavior. The derivation of the BBM for gain predictions is straightforward, while its application in ASE spectra modeling is not so obvious and deserves special attention. We discuss the conditions under which this BBM works for SOA gain and noise characterization and experimentally validate our results using a commercially available SOA.

2. Theory

2.1Gain modeling

In many recent SOA models, the incremental material gain dG (for a spatial discrete step dz inside the amplifier active medium) as function of the wavelength λ is related to the electronic population density N(z) by using a direct linear term and indirect terms, second and a third order (spectral gain’s peak shift), being expressed as [8]:

where λ_sh = [λ₀ - a₄ (N - N_tr)] is the shift in the central frequency λ₀ , N_tr the transparency carrier density, Γ the confinement factor, and a_1-4 are gain parameters [8]. Now, it is straightforward that the total amplifier gain in logarithmic scale can be approximated in the same way, but now with N being an averaged value of the electron-hole population density along the amplifier cavity. Thus,

From Eq. (2) it is easy to show that G(λ) can be written in terms of N as:

where R, S, T and W depend only on the SOA internal and intrinsic parameters and, obviously, on the wavelength λ. Now, if the cubic term in Eq. (3) is discarded (i.e., T = 0), Eq. (3) can be rewritten for three different wavelengths (λ, λ ₁ and λ ₂) and combined in order to eliminate N and N ², hence obtaining

where the functions F ₁, F ₂ and F ₃ depend on R, S and W, which are evaluated at λ, λ₁ and λ₂.

Equation (4) is the BBM fundamental equation and shows that the gain at any wavelength can be expressed as a linear function of the gain at the reference wavelengths λ₁ and λ₂ if the F’s spectral functions are known. The main advantage in the BBM interpolation process is that these three spectral functions (Fs) can be easily determined from the amplifier as a whole, including all penalties inherent to engineering mounts (packaging, gain polarization dependence, optical interconnections, etc). In fact, F₁, F₂ and F₃ are obtained from three complete spectral gain curves, each one measured under different SOA’s bias currents (say A, B and C). Writing Eq. (4) three times for these three different pump conditions, a set of equations are obtained which in a matrix form are:

Solving the system above, F₁, F₂ and F₃ are obtained and so the gain spectra at any different pump condition can be determined by Eq. (4) by measuring the optical gain just at the two reference wavelengths λ₁ and λ₂. If a reduction in device testing complexity is desired, one needs to measure the gain spectra for three bias currents and then, under any other operating condition, the optical gain at two fixed wavelengths to obtain the entire spectrum. In Section 3, experimental validation is presented. Now, it becomes important to evaluate an extension of the same approach to treat the ASE noise.

2.2 Noise modeling

It is well-known that the ASE output power of a SOA, in a bandwidth B, is given by:

where the superscript L indicates linear scale are used, N_sp is the so called noise factor where the rightmost term in Eq. (6) is valid for G^L ≫1, as is usually the case in SO As. Now, one can write Eq. (6) in logarithmic scale and use Eq. (3) with T=0 to obtain:

where Seq^dBm is equal to N_sp(λ)B in logarithmic scale (dBm), called ‘equivalent input noise term’ in order to stress that the ASE output power could be expressed as the amplification of this equivalent input noise. Now, Seq^dBm depends on N, the carrier population density, and so it can be expanded as a power series of N. Assuming this expansion up to the quadratic term and rearranging the terms proportional to each power of N in Eq. (7), one can proceed as in the derivation of Eq. (4) and write Eq. (7) for three different wavelengths, combining them to eliminate N and N ²:

Therefore, a similar approach as described in Section 2.1 can be, in principle, employed for the ASE characterization in amplifiers. One should observe that the main approximation in the Eq. (8) construction is the dispose of third and higher order terms in both Seq^dBm(λ) and G(λ) in Eq. (7). Thus, Eq. (8) is an even more limited solution for the ASE output power than Eq. (4) is for the gain of the SOA. Nevertheless, as shown in the next Section, this equation provides excellent theoretical predictions for the ASE output power of a commercially available SOA.

3. Experimental validation

ASE and optical gain measurements were used to validate the BBM. The gain and ASE spectra of a commercially available SOA (Corning Inc.) were measured with an Optical Spectrum Analyzer (Anritsu, MS96A), using 400-point discretization for the acquired spectral span. The optical signals were directly collected from the SOA module with single-mode fiber cables with FC-APC (angled) connectors, avoiding spurious reflections. The SOA bias current was varied, in 50-mA steps, from 100mA to 450mA, and the typical ASE spectra are presented at Fig. 1(a).

 

Fig. 1. SOA ASE power spectra for eight bias current levels: a) experimental data and b) BBM method numerical results.

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Fig. 2. Relative error between experimental and BBM predicted ASE data, for the eight SOA bias levels.

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With the experimental data collected, the BBM proceeds as follow: first, three ASE spectra are used in Eq. (5) to calculate the Fs spectral functions. Then, at different pump conditions (bias currents), the ASE power is measured at the two chosen reference wavelengths, λ₁ and λ₂, to predict the whole spectra through Eq. (8). In this example the curves corresponding to bias currents of 100mA, 250mA and 400mA were chosen to calculate the Fs functions, and λ₁ = 1450 nm and λ₂ = 1550 nm as the reference wavelengths. The calculated curves are shown in Fig. 1(b). To better visualize the BBM accuracy, the relative error ((P_exp- P_BBM)/ P_exp) was calculated and is presented at Fig. 2, with good agreement between the BBM reconstruction and the experimental within 2% in all cases.

The same procedure applied to the ASE data was done to the SOA optical gain. In this case, however, due to the limited bandwidth of our CW tunable laser, the model was tested in a much narrower band, from 1520 to 1570 nm. The optical power injected in the SOA by the tunable laser was -10 dBm. The SOA bias current was varied in 50 mA steps and a computer controlled Optical Spectral Analyzer measured the output optical power. The net SOA optical gain, considering the back-to-back link losses, is presented at Fig. 3(a).

 

Fig. 3. SOA optical gain spectra for eight bias current levels: (a) experimental and (b) predicted by BBM.

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Fig. 4. Relative error between experimental and calculated SOA optical gain spectra data, for the eight bias current levels.

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In this case, the gain curves corresponding to the bias currents of 100mA, 200mA and 400mA were chosen to calculate the Fs functions, and the wavelengths 1530 nm and 1555 nm as the reference wavelengths. The BBM prediction for the SOA optical gain is presented at Fig. 3(b). A good accuracy was obtained as shown by the relative error presented in Fig. 4, as done before for the ASE case, where relative errors within 4% are shown. The 10% relative error point (out of the figure span) at the 150 mA curve was due to an experimental fluctuation in the laser optical power at 1520 nm. Therefore, the accuracy is similar for both ASE and gain.

Two observations are relevant here. First, in Figs. 2 and 4 the BBM interpolation mechanism gives the exact result when predicting the gain (ASE) curves used to calculate the Fs functions or when predicting the gain (ASE power) at the reference wavelengths. This happens by construction and is natural, since these measured curves are used to interpolate the other spectra. For instance, if in the right side of Eq.(5) λ=λ ₁ is used, one will obtain F ₁(λ ₁,λ ₁,λ ₂)=1 and F ₂(λ ₁,λ ₁,λ ₂)=F ₃(λ ₁,λ ₁,λ ₂)=0. Similarly, if λ= λ ₂ is used, one will obtain F ₂(λ ₂,λ ₁,λ ₂)=1 and F ₁(λ ₂,λ ₁,λ ₂)=F ₃(λ ₂,λ ₁,λ ₂)=0. In the same way, substituting in Eq. (4) the expression for the Fs curves as a function of the measured gains A,B and C, given by the solution of Eq. (5), it is straightforward to show that Eq. (4) gives the trivial relations G_A,B,C(λ)= G_A,B,C(λ).

The second observation is that the noise over experimental data propagate during BBM interpolation mechanism, as can be seen in Fig. 1 around the region of 1.40 μm. The noisy characteristic of this region in Fig. 1(b) is a consequence of the fluctuations also observed in the spectral curves used to calculate the Fs functions (Fig. 1(a)).

To guarantee that the probe signal (-10 dBm) used to measure the SOA gain is not saturating the amplifier, the optical gain versus optical input power response were measured for four bias currents (50 mA, 100 mA, 250 mA and 500 mA, not shown) and optical input powers from -30 dBm up to 3.8 dBm, with less than 2 dB of gain depletion. This guarantee that linear gain regime was used.

4. Discussion and conclusion

The validity of a simple BBM has been experimentally demonstrated for ASE and optical gain data. Since we have neglected all cubic and higher order dependences on average carrier density for the model, we can conclude that these terms do not significantly affect SOA gain and ASE behavior in the C-band.

Interestingly, the result presented here concerning the linear relation between the gain at three different wavelengths, has been first shown to correctly predict Thulium-Doped Fiber Amplifiers (TDFAs) with good experimental accuracy [9]. But, while in the case of SOAs the linear relation between gain at three different wavelengths arises from the fact that the cubic terms of the electronic population density can be neglected in its modeling, in the TDFA case it arises from the fact that three energy levels are involved in the amplification process. The BBM presented in [9] is an extension for a tree-level system of a model originally presented for erbium-doped amplifiers, i.e., for a two-level system [10].