Analysis of all-optical temporal integrator employing phased-shifted DFB-SOA

Xin-Hong Jia; Xiao-Ling Ji; Cong Xu; Zi-Nan Wang; Wei-Li Zhang

doi:10.1364/OE.22.028530

1. Introduction

The time integration is a basic signal processing function with considerably wide applications in the fields of computing and communication, etc. The typical bandwidth of state-of-the-art time integrator based on electronics technologies is restricted within a few GHz range due to the electronic speed bottle-lock. In recent years, all-optical temporal integrator was paid special attentions, since it has operation speed that is well beyond the range of electronic integrator. Several building blocks of performing all-optical first-order and/or high-order integrations have been proposed and demonstrated [1–11].

The proposed passive all-optical integrators mainly includes single or cascade phase-shifted fiber Bragg grating (FBG) operated in transmission, the specially designed FBG with appropriate refraction variation distribution operated in reflection, time-spectrum convolution system, and micro-ring resonator based on complementary metal-oxide-semiconductor (CMOS) compatible doped glass technology, etc. In particular, operation bandwidths as large as 200GHz/400GHz over 800ps/12.5ps time windows have been experimentally demonstrated by resonator structure [9].

Although the quite high bandwidth can be reached using passive all-optical integrator, it still suffers from the limited integration time and energy transmittance caused by the inevitable cavity loss. To overcome this limit, active all-optical integrators with gain compensation have been proposed. R. Slavik et al. reported the first experimental demonstration of all-optical integrator implemented by superimposed FBG made in Er-Yb co-doped optical fiber, and the function of this device was tested by integrating the optical pulses with time duration down to 60ps [3]. Very recently, an active Fabry-Perot cavity based all-optical integrator was proposed by N. Huang et al. The operation bandwidth of 180GHz over integration time window of 160ns was predicted theoretically [10].

On the other hand, distributed-feedback semiconductor laser (DFB-LD) has been widely used in fiber-optic communication system, due to its superior characteristics in terms of narrow line-width, single-mode output, good stability and integrable capability, etc. If the working point is below lasing threshold, it behaves as a distributed-feedback semiconductor optical amplifier (DFB-SOA). The DFB-SOA has been explored as an active optical filter, and all-optical flip-flop using its bi-stable feature [12].Here, all-optical active temporal integrator employing a phase-shifted distributed-feedback semiconductor optical amplifier (DFB-SOA) is investigated. The influences of system parameters on its energy transmittance and integration error are analyzed.

2. Theoretical model and operation principle

It is well known that the impulse response h(t) of an integrator is proportional to the unit step function u(t), expressed by [9]

h (t) \propto u (t) = {\begin{matrix} 1 & f o r t \geq 0 \\ 0 & f o r t < 0 \end{matrix}

In frequency domain, the transfer function H(ϖ) of the integrator can be described by the Fourier transform of the Eq. (1):

H (ϖ) \propto \frac{1}{i (ϖ - ϖ_{0})} + π δ (ϖ)

where ω is the optical frequency variable, ω₀ is the carrier frequency.

In the following, we will prove that, for the sufficiently high gain near the lasing threshold, the transfer function of the DFB-SOA can be well approximated by Eq. (2).

The dynamic evolution of forward and backward amplitudes is described by the coupled-mode equations [12]

\frac{\partial A_{f}}{\partial z} + \frac{1}{v_{g}} \frac{\partial A_{f}}{\partial t} = i [σ A_{f} + κ A_{b}]

\frac{\partial A_{b}}{\partial z} - \frac{1}{v_{g}} \frac{\partial A_{b}}{\partial t} = - i [σ A_{b} + κ A_{f}]

where A_f and A_b are the slowly varying amplitudes of forward and backward waves, respectively. Besides, v_g is the group velocity in the active layer, κ is the coupling coefficient, σ is the complex detuning, expressed by

σ = Δ - i g^{'} / 2

where Δ and g^’ are the detuning and net gain coefficient, respectively, defined as

Δ = δ - \frac{Γ g}{2} α = (n_{e f f} \frac{ϖ}{c} - \frac{π}{Λ}) - \frac{Γ g}{2} α, g^{'} = Γ g - α_{int},

where δ is the initial detuning when gain reaches transparency, π/Λ is the Bragg wave number, Λ is the grating period, n_eff is the effective mode refractive index, α is the line-width enhancement factor of active layer, Γ is the confinement factor, g and α_int are used to characterize the gain and loss coefficients, respectively.

The dynamic evolution of gain coefficient g can be obtained from rate equation theory for carrier density, expressed by

\frac{\partial g}{\partial t} = \frac{g_{0} - g}{τ_{c}} - \frac{g P}{τ_{c} P_{s a t}}

where τ_c is the carrier lifetime (typically ~100ps-1ns), P = |A_f|² + |A_b|² is the total optical power, g₀ and P_sat are the small signal gain coefficient and saturated power, respectively, defined as

g_{0} = a N_{0} (\frac{I}{I_{0}} - 1), P_{s a t} = \frac{A_{c r o s s} h ν}{Γ a τ_{c}},

where a is the differential gain, N₀ is the carrier density at transparency, I is the injected current, I₀ = eN₀V/τ_c is the current required to reach transparency(V is the volume of active layer). Besides, A_cross is the cross-sectional area of active-layer, hν is the photon energy.

In the case of signal-signal input, the gain saturation can be omitted. From steady-state solution of Eqs. (5) and (6), we infer that g≈g₀. By setting the derivatives relative to t to zeros, the steady-state output of forward and backward waves can be obtained by the following transfer matrix:

[\begin{matrix} A_{f} (L) \\ A_{b} (L) \end{matrix}] = T_{_{Σ}} [\begin{matrix} A_{f} (0) \\ A_{b} (0) \end{matrix}] = T_{2} T_{ϕ} T_{1} [\begin{matrix} A_{f} (0) \\ A_{b} (0) \end{matrix}]

where T₁,T_Ф, and T₂ are the transfer matrixes for propagation and phase-shifted sub-sections, calculated by [12]

T_{1} = T_{2} = [\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}] = [\begin{matrix} \cos h (γ l) + i \frac{σ}{γ} \sin h (γ l) & i \frac{k}{γ} \sin h (γ l) \\ - i \frac{k}{γ} \sin h (γ l) & \cos h (γ l) - i \frac{σ}{γ} \sin h (γ l) \end{matrix}]

T_{ϕ} = [\begin{matrix} ϕ_{11} & 0 \\ 0 & ϕ_{11}^{*} \end{matrix}] = [\begin{matrix} \exp (i ϕ_{s h} / 2) & 0 \\ 0 & \exp (- i ϕ_{s h} / 2) \end{matrix}]

where l = L/2(L is the length of active region), Ф_sh is the phase shift value,

γ = \sqrt{k^{2} - σ^{2}}

.

From Eqs. (7a)-(7c),the total transfer matrix is obtained as follows

T_{_{Σ}} = [\begin{matrix} T_{Σ, 1, 1} & T_{Σ, 1, 2} \\ T_{Σ, 2, 1} & T_{Σ, 2, 2} \end{matrix}] = [\begin{matrix} T_{_{11}}^{2} ϕ_{11} + T_{12} T_{21} ϕ_{_{11}}^{*} & T_{_{11}}^{} T_{_{12}}^{} ϕ_{11} + T_{12} T_{22} ϕ_{_{11}}^{*} \\ T_{11} T_{21} ϕ_{11} + T_{21} T_{22} ϕ_{_{11}}^{*} & T_{21}^{} T_{_{12}}^{} ϕ_{11} + T_{_{22}}^{2} ϕ_{_{11}}^{*} \end{matrix}]

For the DFB-SOA with π phase-shift located at the middle of the cavity, the resulting transfer function (i.e., transmittance of forward wave amplitude) is

H (ϖ) = \frac{1}{T_{Σ 2, 2} (ϖ)} = \frac{1}{\frac{i k^{2}}{γ^{2}} \sin h^{2} (γ l) - i {[\cos h (γ l) - i \frac{σ}{γ} \sin h (γ l)]}^{2}}

The lasing threshold gain g_th and frequency ω_th are determined by the condition of $T_{Σ 2, 2} (ϖ_{t h})$ = 0. If the working gain g of DFB-SOA is quite close to g_th (g≈g_th), in the vicinity of zero detuning relative to ω_th, we can expand the matrix element $T_{Σ 2, 2}$ around ω_th:

H (ϖ) \approx \frac{1}{T_{Σ 2, 2} (ϖ_{t h}) + T_{Σ 2, 2}^{'} (ϖ_{t h}) (ϖ - ϖ_{t h})} \propto \frac{1}{ϖ - ϖ_{t h}}

Thus, the DFB-SOA can be employed as a type of active device of performing all-optical temporal integration.

In the following sections, the time-domain transfer matrix method (TD-TMM) [13] is utilized to simulate and analyze the influences of system parameters on integration characteristics. The used data is listed in Table 1. Our discussions would be focused on the energy transmittance and integration error.

Table 1. Device Structure and Material Parameters Used in Simulation

View Table

3. Results and discussions

3.1 Transmission spectrum and basic integration characteristics

Figure 1 shows the unsaturated gain (a) and phase (b) spectrums under different current injection. The calculated threshold current I_th is ~123.3mA. It is shown that, when the injected current is close to this threshold, the peak gain becomes very narrow, which is consistent with the theoretical prediction by Eq. (10). Note that the peak detuning is also increased with enhanced current injection, due to the refractive index change in active region from carrier injection. As shown in Fig. 1(a), the operation bandwidth of ~145GHz can be determined from the span of two minimum points, at which the normalized initial detuning δL are ~2 and 7 respectively. Besides, a π phase-shift variation is also seen at detuning point near peak gain.

Fig. 1 Unsaturated gain (a) and phase (b) spectrums under different current injection.

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To show the basic integration function for the phase-shifted DFB-SOA, Fig. 2 reports the input (a), (c), (e) and output (b), (d), (f) waveforms for different cases. The full-width at half-maximum (FWHM) of input Gaussian pulse is 14ps. The injection current satisfies the condition of I/I_th = 0.999. The detuning is adjusted to match its peak gain point (ΔL = 0). It is shown that, when a single pulse is injected, the phase-shifted DFB-SOA behaves as an integrator (see Fig. 2(a) and 2(b)). The integration time (defined as the duration time from peak value of rising-edge to 80% point) can be as long as ~25ns. If two pulses with in-phase are injected, it acts as an all-optical counter (see Fig. 2(c) and 2(d)). This configuration can also be used for all-optical flip-flop if two pulses with out-of-phase are used, as shown in Fig. 2(e) and 2(f). Note that, if the injected current is very close to its threshold value, the integration time can be far larger than 25ns shown in Fig. 2(b).

Fig. 2 Input (a),(c),(e) and output (b),(d),(f) waveforms of phase-shifted DFB-SOA for different cases.(a),(b): integrator; (c),(d):counter;(e),(f):flip-flop.

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If the input waveform is the derivative of Gaussian pulse, the output of phase-shifted DFB-SOA should be Gaussian shape. To verify this point, Fig. 3 gives the real (solid) and ideal (dotted) output waveforms for different input pulse-width. Because the input pulse is irregular, the normalized root-mean-square (RMS) width relative to the round-trip time over half length of DFB-SOA (~6.67ps, defined as L/v_g) is used in the inset. The used data is the same with that in Fig. 2. From this figure, the real outputs well agree with the ideal cases. On the other hand, a slight waveform deviation and distortion occurs, especially for narrow input pulses. This result is mainly attributed to its low-pass filtering characteristics for the all-optical integrator. Since the high frequency components of short pulse are filtered out, some oscillating distortion is seen for the falling-edge of output pulse.

Fig. 3 Real (solid) and ideal (dotted) output waveforms for different input pulse-width. The input waveform is the derivative of Gaussion pulse.

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3.2 Influences of system parameters on energy transmittance and integration error

Figure 4 shows the energy transmittance (left) and integration error (right) as a function of the normalized input pulse-width for different current injection. The detuning ΔL is set to 0. It is seen that, the energy transmittance is enhanced with the increased input width. For the small normalized input pulse-width (<3), the integration error of 5% can be reached. This is caused by the pulse distortion due to low-pass filter characteristics of all-optical integrator, as shown in Fig. 3. On the other hand, for the sufficiently large normalized input pulse-width, the energy transmittance tends to saturate, and the integration error is also worsened. This is caused by the gain saturation mechanism and detuning shift relative to ΔL = 0 due to carrier-induced refractive index change. Besides, with the increased current injection below lasing threshold, both of energy transmittance and pulse-width range with lower integration error are enhanced; however, the integration error for too large normalized input pulse-width(>11) is worsened accordingly.

Fig. 4 Energy transmittance (left) and integration error (right) as a function of normalized input pulse-width for different current injection.red: I/I_th = 0.999; green: I/I_th = 0.995; blue: I/Ith = 0.990.The detuning ΔL = 0.

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Figure 5 reports the energy transmittance (left) and integration error (right) as a function of normalized pulse-width for different input peak power (P_s). The detuning is ΔL = 0.The energy transmittance is not sensitive to input peak power. With the increased input power, the pulse-width range with lower integration error is extended considerably. This result is attributed to the lower gain saturation and refractive change for smaller input power.

Fig. 5 Energy transmittance (left) and integration error (right) as a function of normalized pulse-width for different input peak power (P_s).red: P_s = −30dBm; green: P_s = −32.5dBm; blue: P_s = −35dBm. The detuning ΔL = 0.

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Figure 6 displays the energy transmittance (left) and integration error (right) as a function of normalized input pulse-width for different current injection. The detuning ΔL is set to 0.1. By comparison with Fig. 4, it is found that the both of energy transmittance and integration error are worsened considerably for larger detuning.

Fig. 6 Energy transmittance (left) and integration error (right) as a function of normalized input pulse-width for different current injection.red: I/I_th = 0.999; green: I/I_th = 0.995; blue: I/Ith = 0.990.The detuning ΔL = 0.1.

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4. Conclusion

In summary, all-optical temporal integrator based on phase-shifted DFB-SOA is analyzed, which has potentials for realizing high-speed all-optical integration, counter, and flip-flop, etc. Enhanced energy transmittance and integration time window can be simultaneously achieved by increased injected current in the vicinity of lasing threshold. The range of input pulse-width with lower integration error is highly sensitive to the injected optical power, due to gain saturation and induced detuning deviation. The initial frequency detuning should also be carefully chosen to suppress the integration deviation with ideal waveform output. Different from FP-SOA structure, an additional advantage of DFB-SOA is that, higher-order temporal integration can be directly realized by introducing multiple-phase-shifted structure [1,14], thus removing the need of cascade of multiple optical devices, which would lead to the increased complexity, length and cost [1].

Acknowledgments

The authors thank the reviewers for their helpful comments. This work is supported by the National Nature Science Foundation of China (NSFC) under grants No. 61205079, 61475105, 61205048,61106045, the Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province under grant No. 12TD008, and the 251 Talents Program of Sichuan Normal University.

References and links

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Symbol	Parameter	Value
L	Active layer length	500μm
A _cross	Active layer cross-sectional area	2 × 10⁻¹³m²
V	Active layer volume	1 × 10⁻¹⁶m³
Γ	Confinement factor	0.3
λ_B	Bragg wavelength	1550nm
Λ	Grating period	234.8 nm
n_eff	Refractive index of active layer	3.3
v _g	Group velocity	7.5 × 10⁷m/s
N ₀	Carrier density at transparency	1.1 × 10²⁴m⁻³
τ _c	Carrier lifetime	200ps
α _int	Internal power loss	4 × 10³m⁻¹
a ₀	Material gain constant	2.5 × 10⁻²⁰m²
α	Line-width enhancement factor	5
kL	Normalized coupling coefficient	3
Ф _sh	Phase-shift	π

Analysis of all-optical temporal integrator employing phased-shifted DFB-SOA

Abstract

1. Introduction

2. Theoretical model and operation principle

3. Results and discussions

3.1 Transmission spectrum and basic integration characteristics

3.2 Influences of system parameters on energy transmittance and integration error

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (14)

Optics Express