1. Introduction

All-optical signal processing is crucial to avoid the electronic bottleneck. A number of ultrahigh-speed all-optical switches have been demonstrated. Important role play devices based on nonlinear carrier dynamics in semiconductor optical amplifiers (SOA) due to their low switching energies and short latencies. Generally, all-optical gates can be divided into two major groups. Non-interferometric gates can be based on cross-gain saturation, cross-phase modulation, four-wave mixing. Interferometric gates are based on Sagnac [1], Mach-Zehnder [2], or single-arm interferometer [3]. Usually, differential scheme is used to eliminate the slow carrier dynamics of SOA. Advantages of the first group of gates are simplicity and robustness. The main advantage of the second group is better extinction ratio.

Ultrafast nonlinear interferometer (UNI) based on single-arm interferometer was first proposed in [3] and its scheme is shown in Fig. 1. The concept of operation of the UNI relies on polarization rotation of data pulse in the presence of the control pulse. The incoming data pulse is split into two orthogonally polarized components due to a walk-off between the fast-and slow-axis of a highly birefringent fiber (HBF). On exiting the SOA, the relative delay between the two replicas is compensated for by another length of a highly birefringent fiber and the pulses are allowed to interfere destructively on a polarizer. When a strong control pulse is brought into the SOA just before the delayed replica of data pulse, a time-dependent refractive index change occurs due to carrier depletion, which in turn imparts a phase change of this replica. This effect changes the destructive interference to constructive at the output polarizer and data pulse emerges at the output of UNI. Long-lived transients of amplitude and phase are balanced out to first order, as they are perceived equally by both the polarization components of data pulse. For the analysis of UNI performance, it is usually assumed that the SOA is an isotropic device. Nonlinear polarization rotation due to dynamic birefringence has been studied both theoretically and experimentally by Dorren et.al. [4]. It was employed in all-optical switches [5], wavelength converters [6], passively mode-locked lasers [7], flip-flop memories [8], payload-header separators [9], 2R-regenerators [10].

 

Fig. 1. Scheme of UNI gate.

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The full vectorial model of the SOA was developed by Dorren [4] and it was used to study the nonlinear polarization rotation. A simplified model suitable for picosecond pulses was presented in [8]. Dynamic birefringence of linear optical amplifier was taken into account in the study of nonlinear polarization rotation in work of Baets et al. [10]. The dynamics of UNI gate was studied with scalar models, however. In this paper, we present a theory of the UNI gate that takes into account the time dependence of polarization dependent gain (PDG) and birefringence of the SOA. Adverse effects of dynamic birefringence (DB) and PDG in the SOA on the performance of the UNI add-drop multiplexers are discussed. The theory is formulated independently of the particular model of the SOA making it independent of the actual bit rate as long as chromatic dispersion of pulses can be neglected in fibers connecting the optical elements.

29 Theory

We develop the Jones matrix formalism for the UNI gate shown in Fig. 1. Let the first polarization beam splitter (PBS) polarizes the light along the x-axis. A highly birefringent fiber is oriented in such a way that both the slow- and fast-axis are equally excited. It means that its main axes are rotated by 45° with respect to PBS and its Jones matrix is R _-BR ₊, where R _± are the matrices that describe the rotation by ±45°. The propagation of the light through the highly birefringent fiber is accounted for by the generalized Jones matrix B

where coordinate frame moving with the group velocity of fast axis is used. The operator $\widehat{\Delta }$ describes the time delay Δ caused by walk-off between the fast and slow axis, $\widehat{\Delta }$ E_y(t) = E_y(t -Δ). The matrix B can be rewritten as

where matrices P _x,P _y describe the polarizer oriented along the x and y axis, respectively

and time delay matrix D is defined as

where E is unit matrix. Its commutation relation with time-dependent matrix can be written as DA(t) = A(t - Δ)D. Matrix B is usual Jones matrix of birefringent element

The SOA is a nonstationary birefringent element and its nonunitary matrix can be written as

The scalar function T(t) describes the gain saturated by the control pulse and ψ(t) is the common phase influenced by time-dependent refractive index. The polarization dependent gain is accounted for by the matrix

The matrix reflects the influence of data and control pulses on the PDG variations in time. These variations are caused by interaction of TE and TM modes with different hole reservoirs [8]. Therefore not only the gain varies after the arrival of the control pulse but also the PDG. Matrix

accounts for the dynamic birefringence (DB) of the SOA. Dynamic birefringence and PDG are closely related through the bandwidth enhancement factor [8]. The walk-off between the TE-and TM-modes of the SOA is neglected. Matrices R ^M _± are rotation matrices

and account for the general orientation of the SOA with respect to the laboratory system. The SOA is surrounded by polarization controllers that are described by unitary matrices ℓ_1,2 of the following structure

The output HBF and PBS are rotated by 90°. The generalized Jones vector at the output of the UNI can be written as

This can be rewritten using the Eq. (2) as

Now we can apply the matrix D to time-dependent matrices and vectors and we get

We can immediately see that up to three pulses can appear at the output of the UNI gate. These are the central pulse resulting from the interference and two satellite pulses propagating all the time either in the fast or slow mode of PM-fibers.

3. Discussion

To achieve a proper function of the UNI it is necessary to null the interferometer in the absence of control pulses. First we will assume that control pulses are not present and data pulses are sufficiently weak so they do not disturb appreciably the SOA. Therefore the gain matrix M(t) in Eq. (6) is stationary

and we will refer to T₀ as static PDG. It is given by Eq. (7) with ε(t) replaced by ε ₀. Analogously, we will refer to F ₀ as static birefringence with δψ(t) = δψ ₀ in equation (8). We will discuss implications of static PDG and static birefringence on the settings of polarization controllers ℓ_1,2 that null the interferometer in the absence of the control pulses. Some authors use a scheme with only one polarization controller placed either in front of the SOA or behind it, others use scheme with the SOA surrounded by two polarization controllers. We will discuss the latter scheme first.

The polarization controller ℓ₁ should be nominally set to R ^M _- R ₊, or R ^M _- R _-. In the first case the pulse from the fast axis of PM fiber is coupled to TE-mode of the SOA and the pulse from the slow axis propagates in TM-mode. The modes are interchanged for the latter case. These are the well studied cases and we briefly recapitulate one of them for completeness. For this purpose we introduce the real-valued analogy of Pauli matrices

and the following identities

We will also use the fact that any pair of diagonal Jones matrices A,B commutate, AB = BA.

Assume that the polarization controller ℓ₁ is set to R ^M _- R ₊. In the absence of control pulses, the unperturbed SOA matrix is M ₀ and the other compensator ℓ₂ should be set to R _- M ^-1 ₀ R ^M ₊ to null the interferometer. Using the commutation rules of diagonal matrices, it can be easily verified that the first and the last terms of Eq. (13) are zero. The same is true for the sum of two middle terms, as can be verified using the identities (16).

In the absence of static PDG, ε ₀ = 0, the element ℓ₂ = R _- M ^-1 ₀ R ^M ₊ = R _- F ^-1 ₀ R ^M ₊ can be realized, in accordance with Eq. (10), by conventional polarization controller that compensates for the static birefringence of the SOA. When the SOA exhibits the PDG, the matrix ℓ₂ = R _- M ^-1 ₀ R ^M ₊ is nonunitary and it can not be realized by polarization controller. We must resort to a more general compensating element that is able to compensate for the PDG as well as the static birefringence. Otherwise the interferometer cannot be nulled completely in the absence of control pulses.

When conventional polarization controller is used in the presence of static PDG, it can be set to compensate just for the static birefringence of the SOA ℓ₂ = R _- F ^-1 ₀ R ^M ₊. The first and the last terms of Eq. (13) are still zero, since matrices F, F ₀, T are diagonal and commutate with matrices P _x,y. The sum of two middle terms gives

or

where Δψ(t) =ψ(t -Δ)-ψ(t)+[ζ(t -Δ)+ζ(t)], ζ(t) = (δψ(t) -δψ ₀)/2,

and T _x,y(t) = T(t)(1±ε(t)). This is a well known formula up to the phase correction ζ(t -Δ)+ ζ(t) originating from the DB. The last effect is of minor influence and can be easily eliminated with proper adjustment of control pulse energy.

In gain transparent regime [11], only the refractive index of the SOA is influenced by control pulses while the gain (in fact loss) and eventually PDG are constant, T(t) = T ₀ and ε(t) = ε ₀. The maximum extinction ratio of this interferometer is then E.R. = 1/ε ₀ ².

Since the control of the input state of polarization is difficult for the fiber-pigtailed SOA we consider the limit case when the power of every data pulse is split equally between the TE-and TM-mode, i.e. the input polarization controller ℓ₁ is set to R ^M _-. For normal functioning of the UNI gate with high extinction ratio we need to null the device in the absence of control pulses using the polarization controller ℓ₂. We should distinguish two cases, one with the time-dependent input J ₀(t) (multiplexers, logic gates), and the second with the time-independent input J ₀(t) = J ₀ (wavelength converters).

In the first case, for the arbitrary J ₀(t), it is necessary to null separately the first and last terms in the Eq. (13), and also the sum of middle two terms, i.e. the terms that deal with input signal in different times. It can be seen that nulling all the three terms is achieved for

This corresponds to an element that is able to compensate for both the static PDG and static birefringence. Now we discuss the consequences of using the conventional polarization controller ℓ₂ in the presence of PDG. Let this controller is set in such a way that it compensates for static birefringence of the 9OA,

Using the identities BS ₂ B = exp(-iωΔ)S ₂, P _x S ₃ P _y = P _y S ₃ P _x = 0, and writing

where a(t) = cosζ(t) +iε(t)sinζ(t), b(t) = isinζ(t)+ε(t)cosζ(t), it can be shown that

If the SOA has no DB, ζ(t) = 0, and no PDG, ε(t) = 0, we have a(t) = 1 and b(t) = 0 and the Eq. (22) transforms to (17) with T_x = T_y as should be expected for an isotropic device. In the SOA with the PDG, but without DB, the middle two terms remain intact, since a(t) = 1, but the first and the last term give rise to daughter pulses, since b(t) = ε(t). In a device with the DB only, a(t) = cosδ(t), and b(t) = isinδ(t). Now two daughter pulses appear at the expense of the middle peak. For the given amount of static birefringence and PDG, the polarization controller ℓ₂ allows to minimize the signal at the output at the absence of control pulses, but it does not allow to null the interferometer completely.

We can consider a particular case of gain transparent UNI (GT-UNI). Then gain (loss) and PDG are time independent even in the presence of control pulses, T(t) = T ₀ and ε(t) = ε ₀. If this device is used as the gate or in 3R-regenerator, the switching window is typically wider then the data pulse time-width, and J _0x(t)J _0x ^*(t - Δ) ≈ 0. Then the intensity of the field at the output of the GT-UNI gate is

In the absence of DB, the achievable extinction ratio is 4/ε ₀ ². It is higher than for the “properly” aligned interferometer with ℓ₁ = R ^M _- R ₊, because the PDG leads to appearance of the satellite pulses but does not impair the main interference term. However, presence of the DB leads to rapid degradation of the extinction ratio because it influences the amplitude of satellite pulses as well as the main peak.

Somewhat different situation occurs when the UNI is used as a wavelength converter or 2R-regenerator. Then J ₀ is a time-independent vector and it is not necessary to null terms of Eq. (13) individually. Hence the UNI can be nulled with a simple polarization controller even in the presence of static PDG. It is important to note, however, that matrix ℓ₂ now generally depends on matrix B, i.e. on the birefringence of PM-fiber. The temperature dependence of the stress-induced birefringence is very strong for PM-fibers and it would be impractical to follow these changes using the polarization controller. A practical solution to this problem is represented by the UNI in folded geometry where the static birefringence is compensated for in the same PM-fiber but single pass through the SOA is maintained [12].

The UNI gate is often operated with one polarization controller only [12, 13]. In this case, the optimum setting of the polarization controller is ℓ_i = R ^M _- F ^-1 ₀ R ^M ₊, the other polarization controller is replaced by the unit matrix. The interferometer can be nulled completely in the absence of PDG. In the presence of PDG, the appearance of the satellite pulses and their magnitude will depend on the actual orientation of the SOA described by the matrix R^M _± and we have no further control of it.

4. Conclusions

We have analyzed an ultrafast nonlinear interferometer (UNI) based on a semiconductor optical amplifier (SOA) exhibiting both polarization dependent gain (PDG) and dynamical birefringence (DB). We have found that misalignment of main axes of SOA and input PM fiber leads to appearance of satellite pulses in the presence of PDG or DB. The performance of the gate could be seriously impaired if the satellites of neighbor pulses overlap. A practical solution would be keeping the walk-off Δ different from the half of bit-period.

The misalignment of the axes of input PM fiber and the SOA can be precluded with the counter-propagating control pulse geometry and the SOA pigtailed at the input side with PM fiber of appropriate length defined by necessary walk-off. This configuration guarantees that satellite pulses are avoided. The achievable extinction ratio is then limited by the PDG and could be improved when PDG is compensated for. The effect of DB is restrained by a proper choice of control pulse energy.