Temporal cloak based on tunable optical delay and advance

Mingyang Zhou; Hongjun Liu; Qibing Sun; Nan Huang; Zhaolu Wang

doi:10.1364/OE.23.006543

1. Introduction

In recent years, the invisibility cloak has been widely studied both theoretically and experimentally [1–18]. By manipulating electromagnetic fields, probe light can flow around an area to create a ‘hole’ or ‘gap’ to hide objects. Inspired by the space-time cloak [10], the concept of temporal cloaking has been developed to conceal events rather than objects, and different temporal cloak schemes have been proposed in fiber-optics [11–13], accelerating wave packets [14], as well as atomic vapors [15]. The first temporal cloak has been demonstrated experimentally by implementing a specially designed split time-lens (STL) to imprint a suitable nonlinear frequency chirp on the probe beam [11]. After propagating through a dispersive medium, the probe beam separates into two parts, producing a controllable temporal gap to hide events. The second temporal cloak, which operates at telecommunication data rate, has been realized by exploiting temporal self-imaging through the time domain Talbot effect [12]. Using time-lenses based on electro-optic phase modulators, a series of temporal gaps were created to hide events from a receiver. The fundamental theory of these two kinds of cloaks is space-time duality, which represents the mathematical equivalence between the equations describing paraxial diffraction and narrowband dispersion [19, 20]. Another kind of cloak takes into account the state of polarization (SOP) of the incident beam, which has been manipulated using Omnipolarizers to generate a polarization gap [13].

In the previous two space-time schemes, the temporal gap created in the probe beam contains negligible optical energy. Any interaction which happens in the gap period would be invisible to receivers of the probe beam. However, when the cloak is operational, the event which occurs in the gap cannot be transmitted as a useful message to the receiver. Considering the temporal cloak in a different way, we use an optical data stream rather than continuous wave (CW) light to create the temporal gap, which can be regarded as a ‘zero’ added in an amplitude modulated optical stream [16]. Furthermore, if the input signal is modulated by adding several gaps, it can be considered as an encoding process. At the receiving end, an inverse modulation will be done to remove these gaps, leaving an output signal the same as the input, which can be considered as a decoding process. Optical signals transmitted in fiber with several temporal gaps will contain different information.

In this paper, we propose a temporal cloak scheme based on tunable optical delay and advance, which are generated by using all-optical switches, linearly chirped fiber gratings and phase modulators. The front part of an input optical signal is advanced while the rear part is delayed, which opens a temporal gap containing negligible optical energy. Different from the temporal cloak schemes demonstrated in [11, 12] where a CW probe is modified to hide the event from observers, we use an optical data stream as the probe beam. We analyze the performance of the temporal cloak by simulations, which can be optimized by tuning the parameters of the phase modulators and linearly chirped fiber gratings. In addition, further application of optical data encoding and decoding is demonstrated.

2. Theory

While the Fourier transform theory is widely used in the studies of electrical and optical signal processing, it can also be applied to analyze spatial and temporal cloaks [17, 18].There is a well-known Fourier shift theorem that adding a linear phase shift in the frequency domain introduces a translation in the time domain, which can be written as [21]

\begin{array}{l} G (ω) = F {g (t)} = \int_{- \infty}^{+ \infty} g (t) \exp (- j ω t) d t, \\ F {g (t \pm Δ t)} = G (ω) \exp (\pm j ω Δ t), \end{array}

where F denotes the Fourier transformation. Therefore, in order to open a temporal gap, we firstly divide an input signal g(t) into two parts at a time point t₀. The front part g₁(t) (t>t₀) and the rear part g₂(t) (t<t₀) of the signal can be expressed as

g_{1} (t) = {\begin{cases} g (_{t)} {_{}}_{} {_{}}_{} t > t_{0} \\ 0_{} {_{}}_{} {_{}}_{} {_{}}_{} {_{}}_{} t < t_{0} \end{cases}, g_{2} (t) = {\begin{cases} 0_{} {_{}}_{} {_{}}_{} {_{}}_{} {_{}}_{} t > t_{0} \\ g (_{t)} {_{}}_{} {_{}}_{} t < t_{0} \end{cases} .

After the Fourier transformation of each part, a negative phase shift exp(-jωΔt) is multiplied with G₁(ω) and a positive phase shift exp(jωΔt) is multiplied with G₂(ω), where G_1,2(ω) are the Fourier transformations of g_1,2(t), respectively. According to the shift theorem, after the inverse Fourier transformation g_1.2(t) can be written as

\begin{array}{l} g_{1} (t - Δ t) = F^{- 1} [G_{1} (ω) e x p (- j ω Δ t)] = {\begin{cases} g (_{t - Δ t)} {_{}}_{} {_{}}_{} t > t_{0} + Δ t \\ {_{}}_{} {_{}}_{}_{} 0_{} {_{}}_{} {_{}}_{} {_{}}_{} {_{}}_{}_{} t < t_{0} + Δ t \end{cases}, \\ g_{2} (t + Δ t) = F^{- 1} [G_{2} (ω) e x p (j ω Δ t)] = {\begin{cases} {_{}}_{} {_{}}_{} {_{}}_{} 0_{} {_{}}_{} {_{}}_{} {_{}}_{} {_{}}_{} t > t_{0} - Δ t \\ g (_{t + Δ t)} {_{}}_{} {_{}}_{} t < t_{0} - Δ t \end{cases}, \end{array}

combination of g₁(t-Δt) and g₂(t + Δt) is

g' (t) = g_{1} (t - Δ t) + g_{2} (t + Δ t) = {\begin{cases} g (_{t - Δ t)} {_{}}_{} {_{}}_{}_{} t > t_{0} + Δ t \\ {_{}}_{} {_{}}_{}_{} 0_{} {_{}}_{} {_{}}_{} {_{}}_{} {_{}}_{}_{} t_{0} - Δ t < t < t_{0} + Δ t \\ g (_{t + Δ t)} {_{}}_{} {_{}}_{}_{} t < t_{0} - Δ t \end{cases} .

From Eq. (4) we can get that a temporal gap with the width of 2Δt is opened around the center time point t₀. An inverse process can be done by adding an opposite phase shift on each part to close this gap, and output signal will be identical with input.

Based on the time-space duality, it has been proved that placing a phase modulator between two linearly chirped fiber gratings (LCFGs) with opposite dispersive coefficients could be used to generate tunable optical delay and advance [22–24]. After propagating through the LCFG operating in reflection mode, spectral components of the input signal undergo a realignment process in the time domain [25,26]. However, the time domain Fraunhofer condition cannot be satisfied on both the first and second dispersive elements in this condition. If an optical signal ɑ_i(t) is injected into the first LCFG, the output signal can be written as [23]

a_{r} (t) = a_{i} (t) * h (t) = C \int_{- \infty}^{+ \infty} a_{i} (τ) \exp [\frac{j π}{K} {(t - τ)}^{2}] d τ,

where h(t) is the temporal impulse response of the LCFG, K is the dispersive coefficient. In order to generate the temporal delay or advance, a ramp-type phase modulation is introduced after the first LCFG, and the phase modulated signal can be expressed as [26]

a_{m} (t) = a_{r} (t) \exp (\pm \frac{j π V}{V_{π}}) = a_{r} (t) \exp (\pm j \frac{α π t}{V_{π}}),

where V = α(t-nT₀) is the periodic voltage signal imposed on the phase modulator (PM), nT₀<t<(n + 1)T₀ and n is an integer. α = V_max/T₀ = 2V_π/T₀ is the slope of the voltage signal, V_π is the half-wave voltage of the PM and T₀ is the period of the voltage signal. Finally, after the second LCFG which has opposite dispersive coefficient -K, the output signal can be written as

\begin{array}{l} a_{o} (t_{D}) = C \int_{- \infty}^{+ \infty} a_{m} (t) \exp [- j \frac{π}{K} {(t_{D} - t)}^{2}] d t \\ = C \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} a_{i} (τ) \exp [j \frac{π}{K} {(t - τ)}^{2}] \exp (\pm j \frac{α π t}{V_{π}}) \exp [- j \frac{π}{K} {(t_{D} - t)}^{2}] d τ d t \\ = C \exp (- j \frac{π}{K} t_{D}^{2}) \exp [j \frac{π}{K} {(t_{D} \pm Δ t)}^{2}] a_{i} (t_{D} \pm Δ t), \end{array}

where Δt = Kα/2V_π = K/T₀ is the time delay or advance (determined by the sign of Δt). From Eq. (7) we can get that |ɑ_o(t_D)|²∝|ɑ_i(t_D ± Δt)|², which indicates that the output signal can be regarded as delay or advance of the input signal with a phase factor.

In our scheme, the input signal is divided into two parts at the time point t₀. The front part (t> t₀) is advanced by imposing a negative voltage signal upon the phase modulator (PM), while the rear part (t< t₀) is delayed by imposing a positive voltage signal. Combination of two parts opens a temporal gap at t₀. The width of the gap can be tuned by changing dispersion coefficient K and voltage signal’s period T₀. In order to divide the input signal, ultra-high speed optical switches (OS) are used in our scheme. Among different types of optical switches, we choose the quantum dot semiconductor optical amplifier based Mach-Zehnder interferometer (QD-SOA-MZI) logic gate, because it has a great potential in high speed optical signal processing and well developed theoretical models feasible in numerical simulations [27–33]. Rate equations to describe the carrier density of QD-SOA are as follows [27–30]

\begin{array}{l} \frac{\partial N_{w}}{\partial t} = \frac{J}{e L_{w}} - \frac{N_{w} (1 - h)}{τ_{w 2}} + \frac{N_{w} h}{τ_{2 w}} - \frac{N_{w}}{τ_{w R}}, \\ \frac{\partial h}{\partial t} = \frac{N_{w} L_{w} (1 - h)}{N_{Q} τ_{w 2}} - \frac{N_{w} L_{w} h}{N_{Q} τ_{2 w}} - \frac{(1 - f) h}{τ_{21}} + \frac{f (1 - h)}{τ_{12}}, \\ \frac{\partial f}{\partial t} = \frac{(1 - f) h}{τ_{21}} - \frac{f (1 - h)}{τ_{12}} - \frac{f^{2}}{τ_{1 R}} - \frac{g L}{N_{Q}} S \frac{c}{\sqrt{ε_{r}}}, \end{array}

where N_w is the electron density in the wetting layer (WL); h and f are the electron occupation probability of the excited state (ES) and ground state (GS), respectively; J is the injection current density; L_w is the effective thickness of active layer; N_Q is the surface density; ε_r is the SOA material permittivity and c is the velocity of light in free space. τ_w2 is the electron relaxation time from the WL to the ES; τ_2w is the electron escape time from the ES to the WL; τ_wR is the spontaneous radiative lifetime in the WL, τ₂₁ is the electron relaxation time from the ES to the GS; τ₁₂ the electron escape time from the GS to the ES; τ_1R is the spontaneous radiative lifetime in the QD; L is the length of the SOA; g is the modal gain; S is the photon density. The equation of S is,

\frac{\partial S}{\partial z} = g S - α_{int} S,

where g = g_max(2f-1) and S = P/(A_effV_gh_pv); h_p is the Planck’s constant; α_int is the absorption coefficient of the material; g_max is the maximum modal gain; P is the input signal power; A_eff is the effective cross section; z is the distance in the longitudinal direction that z = 0 represents the input facet of the QD-SOA and z = L represents the output facet. Time-dependence of the integral gain and the phase-shift of the QD-SOA can be expressed as

\begin{array}{l} G (t) = \exp (\int_{0}^{L} g (z, t) d z), \\ φ (t) = - \frac{α_{L}}{2} (\int_{0}^{L} g (z, t) d z), \end{array}

where α_L is the linewidth enhancement factor.

3. System and simulation results

The Schematic of our temporal cloak is illustrated in Fig. 1. The input optical signal is separated into two parts equally using a 3 dB coupler, and then the divided signal is injected into the optical switch in each branch as the optical data stream. The output signal of the optical switch is the result of the logic AND operation of the data stream and the control stream. In each branch, the first LCFG (color in red), the electro-optic phase modulator (EOPM) and the second LCFG (color in blue) are used to add a time shift. The arbitrary waveform generator (AWG) is used to provide the voltage signal for driving the EOPM. After the second LCFG in each branch, optical signals combine together to form the transmitted signal with a temporal gap. The following structures simply remove the temporal gap and leave output signals the same as the input.

Fig. 1 Schematic of the temporal cloak, OS stands for optical switch; LCFG stands for linearly chirped fiber grating; AWG stands for arbitrary waveform generator; PM stands for phase modulator.

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The diagram of the optical switch is shown in Fig. 2. In this structure, two identical QD-SOAs are planted in the upper and the lower arms of a symmetric Mach-Zehnder interferometer. A clock stream (at λ₁) is injected into port A while a much weaker data stream (at λ₂) is injected into port B and equally split into two arms. In order to eliminate the background noise of the clock stream, low power CW light (at λ₁) is injected into port C [31]. The output intensity at port D is the interference of the two arms, which can be expressed as [31–33]

P_{D} (t) = \frac{1}{4} P_{B} (t) [G_{1} (t) + G_{2} (t) - 2 \sqrt{G_{1} (t) G_{2} (t)} \cos (φ_{1} (t) - φ_{2} (t))],

where P_B is the power of the data stream, G_1.2(t) is the total gain and φ_1.2(t) is the phase shift. A band-pass filter (BPF) is used to filter out the control signal A and C. The output stream at port D is the result of the logic AND operation of the control signal A and the data signal B. When B = 0, the output is ‘0’; when B = 1, if A = 0, the gain and phase shift of the two arms are the same, hence the result is ‘0’; if A = 1, the output at D is ‘1’.

Fig. 2 Schematic of optical switch based on the QD-SOA Mach-Zehnder interferometer.

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Equations (8)–(11) are solved numerically to study the performance of the QD-SOA-MZI. Parameters used in the simulation are listed as follows [27, 28]: τ_w2 = 3 ps, τ_2w = 1 ns, τ_wR = 1 ns, τ₂₁ = 0.16 ps, τ₁₂ = 1.2 ps, τ_1R = 0.4 ns, L = 4 mm, α_int = 2 cm⁻¹, g_max = 12 cm⁻¹, J = 2 kA/cm², L_w = 0.2 μm, N_Q = 5 × 10¹⁰ cm⁻² and α_L = 0.1. The signal data rate is 40 Gbit/s and the full pulse width is 25 ps. Optical powers of the control stream, signal stream and CW light are 30 mW, 0.01 mW and 0.001 mW, respectively. Simulation results are displayed in Fig. 3. Equations (5)–(7) are calculated to study the temporal gap generation performance of the scheme. In our simulation, the parameters K = −5400 ps², 1/T₀ = 3.32 GHz and V_π = 4 V [26]. As shown in Fig. 4, a temporal gap with the width of 39 ps is opened, and the total time shift is 36ps.

Fig. 3 (a) Normalized average powers of the signal, control and output streams of optical switch 1 on the upper branch, (b) normalized average powers of the streams of optical switch 2 on the lower branch. The output stream is the result of logic AND operation of the control and signal stream.

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Fig. 4 (a) Normalized average power of the 40 Gbit/s input signal, (b) normalized average power of the output signal with a temporal gap. The width of the gap is 39 ps.

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The width of the temporal gap can be changed by tuning the dispersion coefficient K of the LCFG and the repetition rate 1/T₀ of the ramp-type voltage signal imposed on PM. Firstly, we set 1/T₀ = 3.32 GHz and change the dispersion coefficient K. The simulation results shown in Figs. 5(a)–5(d) indicate that the width of the temporal gap is increasing with the increase of the coefficient K. When K = 7000 ps², the width of temporal gap is 49 ps, and when K = 13000 ps² the width increases to 81 ps. However, the large coefficient K will cause signal distortion, which means the temporal gap may not be able to close or hide the event. In Figs. 5(e)–5(f), we set 1/T₀ = 4 GHz, when K = 12000 ps² the width is 93ps and when K = 5400 ps² the width is 43ps, both of them are larger than the condition of 1/T₀ = 3.32 GHz. Moreover, the change of T₀ would not influence the pulse shape and amplitude. In practice, T₀ is confined by the sampling speed of the arbitrary waveform generator used to generate the ramp-type voltage signal imposed on the PM [26].

Fig. 5 Normalized average power of the transmitted signals, (a-d) K = 7000 ps², 9000 ps², 11000 ps² and 13000 ps² with 1/T₀ = 3.32 GHz, the width Δt = 49 ps, 61 ps, 73 ps and 81 ps, respectively, (e-f) K = 5400 ps² and 12000 ps² with 1/T₀ = 4 GHz, Δt = 43 ps and 93 ps, respectively.

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Then we illustrate the temporal cloaking process in Fig. 6. Optical signals with and without the temporal gap are shown in Fig. 6(a) and Fig. 6(b), respectively. By tuning the parameters of the LCFG and phase modulator, a temporal gap with the width of 59 ps is opened in Fig. 6(b). Here we consider the event as a sinusoidal amplitude modulation of the optical signal in the gap period, which is generated by using an electro-optic intensity modulator [12, 17]. Figure 6(e) shows the electrical modulation signal imposed on the intensity modulator, which occurs in the temporal gap and multiplies by the input signal’s intensity. When the temporal gap is closed, the output signal is shown in Fig. 6(c), which is disturbed by the event. In contrast, when the gap is open, the event which happens in it will not influence the signal since the optical amplitude of the temporal gap is nearly zero, as shown in Fig. 6(d). At the receiving end, the temporal gap will be closed to recover the original signal. In order to close the gap, a positive phase shift is added on the front part (t> t₀) and a negative phase shift is added on the rear part (t< t₀). The event affected signal shown in Fig. 6(d) is used to recover the original signal. Simulation results of the original signal and the recovered signal are shown in Fig. 7(a) and Fig. 7(b), respectively. It can be seen that the recovered signal is nearly the same as the original. Therefore, the event will not be detected by receivers.

Fig. 6 illustration of the temporal cloaking process; (a) the optical signal without the temporal gap; (b) the optical signal with the temporal gap; (c) the output signal when the cloak is off ; (d) the output signal when the cloak is on; (e) the electrical modulation signal imposed on the intensity modulator.

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Fig. 7 (a) Normalized average power of the input original signal, (b) normalized average power of the output recovered signal.

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4. Further application

In this section, the application of our scheme in fiber-optical data encoding and decoding is demonstrated. Opening a temporal gap in an amplitude modulated optical signal is similar to adding a 0 bit code into a binary data stream, which means the information represented by the optical bit pattern is also changed. The encoding process can be done by adding or removing some 0 bit codes. The simulation results are shown in Fig. 8. An input optical bit stream is divided into two parts and injected into the upper and the lower branches of the scheme. The optical signal in the upper branch is advanced while in the lower branch which is delayed. Combination of two parts will generate the encoding signal transmitted in optical fibers. By doing the inverse operation, these gaps can be removed to recover the original signal. In practice, more complicated encoding patterns can be realized by adjusting the voltage signal of the PM and adding more branches. In each branch the optical stream can be delayed, advanced or unchanged, which is determined by the PM. The simulation results of an improved scheme are shown in Fig. 9. The optical stream in the upper branch is delayed while in the lower branch is unchanged.

Fig. 8 Simulation results of optical data encoding and decoding. (a) normalized average power of the input original signal; (b) the encoded signal; (c) the output decoded signal.

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Fig. 9 Simulation results of optical data encoding and decoding with the improved scheme. (a) normalized average power of the input original signal; (b) the encoded signal; (c) the output decoded signal.

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However, there are several limitations of the proposed scheme in the applications of signal encoding. Firstly, overlap between adjacent slots should be taken into account, and secondly, synchronizing the optical switches and phase modulators is important to recover the original signal. In our simulation, the input signal is divided at the middle of some continuous ‘0’ bit codes. It is better to add a ‘0’ between two slots and remove a ‘0’ at the nearby continuous 0 bit codes in order to avoid overlaps. Locations of the temporal gap are determined by the control streams of optical switches. If the input signal contains several continuous 0 bit codes, overlaps could be avoided. Considering the synchronization, control streams of the optical switches can be transmitted together with the data stream due to their different wavelengths. At the receiver port, control streams will be extracted and recovered to control the relevant optical switches. Concepts from the optical wavelength-division multiplexing (OWDM) technique can be applied in this condition [34].

5. Conclusion

In this paper, a temporal cloak scheme based on tunable optical delay and advance has been proposed. The input signal is divided into two parts using optical switches. Tunable optical delay and advance generated by dispersion and phase modulation are added on each part to open the temporal gap, which can be used to hide events from observers. System performance and width of the temporal gap have been analyzed by simulation. Furthermore, application of the temporal cloak system in optical signal encoding and decoding has been proposed. It has been demonstrated that the developed scheme can be applied in optical signal processing and fiber-optical secure communications.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61178023 and 61275134.

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Temporal cloak based on tunable optical delay and advance

Abstract

1. Introduction

2. Theory

3. System and simulation results

4. Further application

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (11)

Optics Express