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Generation of NLFM microwave waveforms based on controlled period-one dynamics of semiconductor lasers

Pei Zhou, Renheng Zhang, Kunxi Li, Zhidong Jiang, Penghua Mu, Hualong Bao, and Nianqiang Li

Open Access Open Access

Abstract

We propose an approach to generating nonlinear frequency-modulated (NLFM) microwave waveforms, which is based on controlled period-one (P1) dynamics of an optically injected semiconductor laser (OISL). When the optical injection is modulated, the OISL, which originally operates at a P1 oscillation state, acts as a microwave voltage-controlled oscillator (VCO). In the proposed system, the microwave frequency output depends closely on the optical injection strength controlled by the modulation voltage input, while the electrical modulation signal required to generate a desired NLFM microwave waveform can be calculated on the basis of the “voltage-to-frequency” transfer function of the established VCO system. Our simulations and experiments demonstrate that both single-chirp and dual-chirp NLFM microwave waveforms can be readily generated with a bandwidth up to 9 GHz. Considering peak-to-sidelobe ratio (PSLR) of the compressed pulses, the NLFM signals generated by the VCO exhibit a practical improvement of ∼13 dB when compared with LFM signals with the same bandwidth, and the tunability of the generated NLFM signals is also experimentally demonstrated.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Microwave arbitrary waveforms have extensive applications in the fields of radar, communications, electronic warfare and modern instrumentation [1,2]. Due to the advantages of the high speed, large bandwidth, low loss, fast tunability as well as reconfigurability, photonic-assisted microwave waveform generation has become a hot research topic in recent years [3]. In radar systems, linear frequency-modulated (LFM) microwave waveforms with a large time-bandwidth product (TBWP) have been widely applied, since they can achieve an improved range resolution through pulse compression [4–9]. However, the compressed pulse of an LFM signal after matched filtering has a low peak-to-sidelobe ratio (PSLR), which might mask the returning echo of a target adjacent to the main target. To deal with this problem, nonlinear frequency-modulated (NLFM) microwave waveform has been introduced [10,11]. A NLFM signal has a faster frequency sweep rate at the pulse edges and a slower frequency sweep rate at the pulse center, leading to a better PSLR after pulse compression. Thus, a NLFM signal is very promising in increasing the signal-to-noise ratio (SNR) and the dynamic range of radar systems [12]. Up to now, photonic generation of NLFM microwave waveforms has rarely been reported [13]. In [13], NLFM microwave waveform generation is demonstrated based on current modulation of a distributed feedback laser diode (DFB-LD) in a self-heterodyne scheme. Nevertheless, both bandwidth and tuning ability of the generated NLFM signal are limited.

Photonic microwave generation based on period-one (P1) nonlinear dynamics of semiconductor lasers has received great attention in microwave photonics [14–29]. After proper optical injection, the P1 oscillation state can be excited through undamping the relaxation resonance, and the P1 oscillation frequency of an optically injected semiconductor laser (OISL) can be tuned from a few GHz to over 100 GHz through simply adjusting the injection parameters. Specifically, for a fixed master-slave detuning frequency, the P1 frequency would increase approximately linearly with the injection strength over a large range. Various types of microwave signals have been successfully generated by using P1 oscillation states, including single-frequency microwave signals [14–20], microwave frequency combs [21,22], triangular pulses [23], frequency-hopping sequences [24] and frequency-modulated continuous-wave (FMCW) signals [25–29]. In [29], generation of LFM signal with a large TBWP has been realized based on an OISL.

In this paper, photonic generation of NLFM microwave waveforms based on controlled P1 dynamics of an OISL is demonstrated theoretically and experimentally. When operating at a P1 oscillation state, the semiconductor laser subjected to modulated optical injection is functioned as a microwave voltage-controlled oscillator (VCO), that is, the microwave frequency output is related to the optical injection strength, which can easily be controlled by the modulation voltage input. The electrical modulation signal required to generate a desired NLFM microwave waveform can be calculated according to the “voltage-to-frequency” transfer function of the established VCO system. Both single-chirp and dual-chirp NLFM microwave waveforms are generated with a bandwidth up to 9 GHz. Compared with LFM signals with the same bandwidth, the generated NLFM signals have a ∼13-dB improvement in PSLR of the compressed pulses. In addition, the tunability of the generated NLFM signals is also experimentally demonstrated. The proposed photonic NLFM signal generator may find wide applications in radar systems, such as weather radar and long-range early warning radar.

2. Experimental setup

Figure 1 provides the schematic diagram of the proposed NFLM microwave signal generator based on controlled P1 dynamics of semiconductor lasers. A commercial single-mode DFB-LD (Wuhan69 BF14) driven by a low noise current-temperature controller (Throlabs ITC4001) is served as the slave laser (SL). Under a bias current of 24 mA (∼3 times of its threshold) and a stabilized temperature of 25°C, the free-running wavelength and output power of the SL are 1551.432 nm and 5.7 mW, respectively. A tunable laser (Agilent N7714A) is applied as the master laser (ML). Its output power and wavelength are set to 20 mW and 1551.416 nm. A continuous-wave (CW) light from the ML is injected to the SL after passing through a variable attenuator (VA), a 10-Gb/s Mach–Zehnder modulator (MZM, Sumitomo Co.), a polarization controller (PC) and an optical circulator (CIR). Here, the VA is used to achieve the suitable injection strength to excite P1 dynamics, and the PC is inserted before the CIR to align the polarization of the injection light with that of the slave laser to maximize the injection efficiency. An electrical modulation signal V(t) from a 120-MHz arbitrary waveform generator (AWG, Agilent 81150A) is used to drive the MZM for rapid variation of the optical injection strength. The output of the injected SL is exported through port 3 of the CIR. A 90/10 fiber coupler (FC) is inserted to tap 10% of the signal power to monitor the optical properties in an optical spectrum analyzer (OSA, Ando AQ6317B). The other 90% of the SL output is sent to a 30-GHz photodetector (PD, Optilab PD-30) before measurement in a 20-GHz real-time oscilloscope (OSC, LeCroy 820Zi-B).

 figure: Fig. 1.

Fig. 1. Schematic diagram of the proposed NFLM microwave signal generator. ML: master laser; SL: slave laser; VA: variable attenuator; V(t): modulation voltage input; MZM: Mach–Zehnder modulator; PC: polarization controller; CIR: optical circulator; FC: fiber coupler; PD: photodetector; OSC: oscilloscope; OSA: optical spectrum analyzer.

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When operating at P1 oscillation state, the semiconductor laser subjected to modulated optical injection is modeled as a microwave VCO. In the system, the microwave frequency output is related to the optical injection strength, which can easily be controlled by the modulation voltage input. First, the “voltage-to-frequency” transfer function of the established VCO system can be characterized. Then, a specially shaped modulation signal V(t) is calculated according to the transfer function. When the suitable V(t) is applied, a desired NLFM microwave waveform can be generated.

3. Numerical simulations

3.1 Simulation model

Our proposed scheme is verified by analyzing the dynamical behavior of the OISL consisting of an SL subjected to optical injection from an ML, which can be modeled by the modified Lang-Kobayashi rate equations [20,25]:

$${\kern 1pt} \frac{{dA}}{{dt}} = \frac{{1 - ib}}{2}(g - {\gamma _\textrm{c}})A + {\gamma _\textrm{c}}\xi (t)|{A_0}|exp ( - i2\pi {f_\textrm{i}}t)\textrm{ + }\chi $$
$${\kern 1pt} {\kern 1pt} \frac{{dN}}{{dt}} ={-} {\gamma _\textrm{s}}(N - {N_0}) - {\gamma _\textrm{s}}J{N_0}(\frac{{g|A{|^2}}}{{{\gamma _\textrm{c}}|{A_0}{|^2}}} - 1)$$
$$g = {\gamma _\textrm{c}} + \frac{{{\gamma _\textrm{c}}{\gamma _\textrm{n}}(N - {N_0})}}{{{\gamma _\textrm{s}}J{N_0}}} - {\gamma _\textrm{p}}(\frac{{|A{|^2}}}{{|{A_0}{|^2}}} - 1)$$
where A(t) represents the slowly varying complex electric field of the SL, N(t) stands for the carrier density of the SL, and g denotes the optical gain. A0 and N0 are the free-running values of A(t) and N(t), respectively. The linewidth enhancement factor b, cavity decay rate γc, spontaneous carrier relaxation γs, differential carrier relaxation rate γn, and nonlinear carrier relaxation rate γp are the laser intrinsic parameters [15]. The dimensionless injection current parameter J is the ratio between the bias current and the threshold current level for the SL. fi is the frequency detuning of the ML with respect to the free-running SL. χ means complex Langevin fluctuating force, which is used to characterize the spontaneous emission noise of SL. Its real part and imaginary part are mutually independent and are also independent at different times. The spontaneous emission noise of the SL is given by following averages:
$$< \chi (t){\chi ^ \ast }(t^{\prime}) > = {\beta _{\textrm{sp}}}\delta (t - t^{\prime}), $$
$$< \chi (t)\chi (t^{\prime}) > {\kern 1pt} {\kern 1pt} = 0, $$
$$< \chi (t) > {\kern 1pt} = 0. $$
Here, βsp = 5.99 × 1019 V2m-1s-1 is the strength of χ [20], and ξ(t) is the dimensionless injection strength, which can be written as [30]:
$$\xi (t) = {\xi _0}\sqrt {{T_{\textrm{MZM}}}[V(t)]} = {\xi _0}\sqrt {\frac{\textrm{1}}{2}[1 + \cos (\Delta \varphi + \frac{{\mathrm{\pi }V(t)}}{{{V_\mathrm{\pi }}}})} ], $$
where ξ0 is the injection strength without modulation, which is proportional to the amplitude ratio between the injected light and the free-running SL. TMZM[V(t)] is the optical power transmission function of the MZM, V(t) is the input modulation signal, Δφ is the phase shift difference induced by the DC bias, and Vπ = 5 V is the half-wave voltage.

In the numerical simulations, a fourth-order Runge-Kutta algorithm is used to solve Eqs. (1)-(3) with a time step of 1 ps. The parameter values of the SL used here are listed as follows [20]: γc = 5.36 × 1011 s-1, γs = 5.96 × 109 s-1, γn = 7.53 × 109 s-1, γp = 1.91 × 1010 s-1, b = 3.2, and J = 1.222.

3.2 Simulated results

By setting an optical injection with (fi, ξ0) = (6 GHz, 0.15) and Δφ = 0, the SL would operate at the P1 oscillation state. It has been proved that, for certain fixed master-slave detuned frequencies, the P1 oscillation frequency fm would almost monotonically increase with the injection strength ξ0 [29,30]. When the input modulation signal V(t) is applied, the output microwave frequency, equal to P1 oscillation frequency fm(t), can be expressed as:

$${f_\textrm{m}}(t) = H[\xi (t)] = H\{ {\xi _0}\sqrt {{T_{\textrm{MZM}}}[V(t)]} \} = G[V(t)]{\kern 1pt} $$
where H(*) is a function that relates injection strength ξ(t) with P1 oscillation frequency fm(t). Here, a “voltage-to-frequency” transfer function termed G(*) is introduced to replace H(*), which relates the microwave frequency output fm(t) and modulation voltage input V(t). Equation (8) indicates that, when operating at P1 oscillation state, the SL subjected to modulated optical injection is functioned as a microwave VCO. In order to generate a desired NLFM microwave waveform, the required modulation voltage input V(t) can be calculated as:
$$V(t) = {G^{ - 1}}[{f_\textrm{m}}(t)]{\kern 1pt} $$
where G-1(*) is the inverse function of the G(*). Figure 2 depicts the simulation results for the LFM and NLFM signal generation. We assume that V(t) is a sawtooth-shaped voltage signal with an amplitude of ∼2.4 V and a period of 1 μs, as shown in Fig. 2(a-i). Figures 2(b-i) and 2(c-i) are the corresponding temporal waveform and instantaneous frequency-time diagram (based on short-time Fourier transform). By using Eqs. (8) and (9), the “voltage-to-frequency” transfer function G(*) and its inverse function G-1(*) are calculated based on voltage input V(t) and microwave frequency output fm(t). Likewise, we can obtain the required modulation voltage input V(t) for generating LFM and NLFM microwave waveforms, and the results are shown in Figs. 2(a-ii) and 2(a-iii), respectively. The temporal waveform and the calculated instantaneous frequency of LFM signal are shown in Figs. 2(b-ii) and 2(c-ii), where the instantaneous frequency increases linearly from 15.7 GHz to 22.7 GHz in the time period of 1 μs. When V(t) is set to have a specially-designed shape displayed in Fig. 2(a-iii), the desired NLFM microwave waveform based on our scheme is also obtained numerically, and the corresponding temporal waveform is shown in Fig. 2(b-iii). The calculated instantaneous frequency of NLFM signal is given in Fig. 2(c-iii). The comparison between Figs. 2(c-ii) and 2(c-iii) indicates that the generated NLFM signal has a faster frequency sweep rate at the edges and a slower sweep rate throughout the middle. It should be noted that the bandwidth of the NLFM signal is almost identical to that of the LFM case, estimated as 7 GHz (from 15.7 GHz to 22.7 GHz).

 figure: Fig. 2.

Fig. 2. Simulation results for LFM and NLFM signals generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency. (i) “voltage-to-frequency” transfer function calculation, (ii) LFM signal generation, and (iii) NLFM signal generation.

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The proposed scheme is then extended to generate dual-chirp LFM and NLFM waveforms, which are useful in reducing the range-Doppler coupling [30]. As plotted in Fig. 3(a-i), a modulation voltage input with an amplitude of ∼2.4 V and a time period of 1 μs is applied. The corresponding temporal waveform and instantaneous frequency-time diagram are shown in Figs. 3(b-i) and 3(c-i), respectively. It is composed of two complementary linear chirp waveforms (with up- and down-chirp alternately) and has a bandwidth of 7 GHz. Likewise, one can also generate a dual-chirp NLFM microwave waveform with nonlinear up- and down-chirp alternately. Figure 3(a-ii) shows the required modulation signal with an amplitude of ∼2.4 V, and Fig. 3(b-ii) illustrates the generated dual-chirp NLFM microwave waveform. As can be seen from the instantaneous frequency-time diagram in Fig. 3(c-ii), it contains both NLFM up-chirp and down-chirp in the range of 15.7-22.7 GHz.

 figure: Fig. 3.

Fig. 3. Simulation results for (i) dual-chirp LFM signal and (ii) dual-chirp NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency.

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The pulse compression performance of the NLFM signal can be quantitatively evaluated by calculating its autocorrelation function and measuring its PSLR [29]. Figure 4 shows the autocorrelation function of both the dual-chirp LFM (shown in Fig. 3(b-i)) and the dual-chirp NLFM (shown in Fig. 3(b-ii)) signals. Generally, PSLR is defined as the ratio of the main lobe to the maximum sidelobe [10]. Here, we compare the remaining maximum side lobes of LFM and NLFM signals while keeping the same main lobe width. It is interesting to find from Fig. 4 that the generated NLFM signal demonstrates a ∼15-dB enhancement in PSLR, which is very promising in increasing the SNR as well as the dynamic range of radar systems. It is worth noting that the autocorrelation function of the dual-chirp LFM is similar to that of the single-chirp LFM, which is not shown here.

 figure: Fig. 4.

Fig. 4. Pulse compression performance of the simulated dual-chirp NLFM (red) and dual-chirp LFM (blue) signals.

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4. Experimental results

We carry out experiments for NLFM microwave waveform generation based on the above setup. Firstly, the modulation voltage input V(t) is not applied, and an optical injection of (fi, ξ0) = (2 GHz, 0.50) is introduced. Similar to the simulations, the injection strength ξ0 is defined as the amplitude ratio between the injected light and the free-running SL [30]. The blue dotted and green dashed curves in Fig. 5(a) describe the optical spectra of the ML and free-running SL, respectively. As shown by the red solid curve in Fig. 5(a), a typical P1 oscillation state with fm = 19.7 GHz is excited. Then, the OISL-based VCO system is characterized by measuring the relationship between the generated microwave frequency and the offset voltage of the modulation input, as shown in Fig. 5(b). It is obvious that the “voltage-to-frequency” relationship is not ideally linear, which is induced by the nonlinear transfer function of the injected SL and MZM [29,30].

 figure: Fig. 5.

Fig. 5. (a) Optical spectra of the ML (blue dotted curve), the free-running SL (green dashed curve), and the injected SL (red solid curve), (b) the relationship between the output P1 oscillation frequency and the input voltage amplitude.

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When the input modulation signal V(t) is applied, the output microwave frequency, which is equal to P1 oscillation frequency fm(t), can be expressed as:

$${f_\textrm{m}}(t) = H\{ {\xi _0}\sqrt {{T_{\textrm{MZM}}}[V(t)]} \} = H\{ \sqrt {\frac{{{P_{\textrm{inj}}}{T_{\textrm{MZM}}}[V(t)]}}{{{P_{\textrm{SL}}}}}} \} = G[V(t)]$$
where Pinj and PSL are the optical powers of injected light and free-running SL, respectively. Figure 6(a-i) plots a sawtooth-shaped modulation signal V(t) with a 2.2-V amplitude and 1-μs period. Figures 6(b-i) and 6(c-i) are the corresponding temporal waveform and instantaneous frequency-time diagram. Similar to the previous simulation process, the “voltage-to-frequency” transfer function G(*) and its inverse function G-1(*) of the VCO system are calculated. Based on the obtained G(*) and G-1(*) functions, the required voltage input V(t) for LFM and NLFM signals are calculated, which are shown in Figs. 6(a-ii) and 6(a-iii). As shown in Figs. 6(c-ii) and 6(c-iii), both LFM and NLFM signals with a same bandwidth of 7 GHz (from 10.3 GHz to 17.3 GHz) are experimentally generated.

 figure: Fig. 6.

Fig. 6. Experimental results for LFM and NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency. (i) “voltage-to-frequency” transfer function calculation, (ii) LFM signal generation, and (iii) NLFM signal generation.

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In order to compare with simulations, dual-chirp LFM and dual-chirp NLFM signals are also experimentally generated. Figures 7(a-c) are the corresponding V(t), the temporal waveform and the instantaneous frequency. As can be seen, both dual-chirp LFM and dual-chirp NLFM are successfully generated with a bandwidth of 7 GHz (from 10.3 to 17.3 GHz). In addition, the tunability of the bandwidth is also experimentally demonstrated. By setting the amplitude of the modulation voltage input V(t) to 2.0 or 2.6 V, the generated NLFM signals are shown in Fig. 8, whose bandwidth are 6 and 9 GHz, respectively, and the temporal period of 1 μs remains unchanged. It should be noted that the method can be used to generate NLFM waveforms with a higher frequency and/or a larger bandwidth, and the limitations mainly originates from the limited bandwidths of the oscilloscope and photodetector used in our experiment.

 figure: Fig. 7.

Fig. 7. Experimental results for (i) dual-chirp LFM signal and (ii) dual-chirp NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency.

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 figure: Fig. 8.

Fig. 8. Instantaneous frequency-time diagrams of the generated dual-chirp NLFM signals with a bandwidth of (a) 6 GHz and (b) 9 GHz, respectively.

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Finally, we evaluate the pulse compression performance of the experimentally generated dual-chirp NLFM signal (corresponding to Fig. 7 (b-ii)), whose autocorrelation function is calculated. To demonstrate the improvement of PSLR, the autocorrelation function of the dual-chirp LFM signal (corresponding to Fig. 7 (b-i)) is also computed for reference. As shown in Fig. 9, a 13-dB improvement of PSLR is found in the NLFM signal as compared to the LFM signal with the same bandwidth, which indicates a visibility more than twenty times better than the LFM case for a target adjacent to the main target is achieved. The experimental results coincide with the above simulations, confirming the feasibility of the proposed scheme for NLFM signal generation and thus for PSLR improvement.

 figure: Fig. 9.

Fig. 9. Pulse compression performance of the experimentally generated dual-chirp NLFM (red solid curve) and dual-chirp LFM (blue dotted curve) signals.

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5. Conclusion

In conclusion, a scheme for generating NLFM microwave waveforms is experimentally and theoretically demonstrated based on controlled P1 dynamics of an OISL. The feasibility of this OISL system as a microwave VCO is proved by building the relationship between the microwave frequency output and modulation voltage input. According to the measured “voltage-to-frequency” transfer function of the established VCO system, the electrical modulation signal required to generate a desired NLFM microwave waveform is calculated. It is shown that both single-chirp and dual-chirp NLFM microwave waveforms are generated with a bandwidth up to 9 GHz. Compared with LFM signals with the same bandwidth, the generated NLFM signals have a ∼13-dB improvement in PSLR of the compressed pulses. Furthermore, the tunability of the generated NLFM signals is also experimentally demonstrated. To the best of our knowledge, this is the first demonstration of NLFM signal generation with an enhanced PSLR using OISLs, which is very promising in increasing the SNR as well as the dynamic range of radar systems.

Funding

Project of Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education (RIMP2020001); Startup Funding of Soochow University (Q415900119, Q415900220).

Disclosures

The authors declare no conflicts of interest.

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Figures (9)
Fig. 1.
Fig. 1. Schematic diagram of the proposed NFLM microwave signal generator. ML: master laser; SL: slave laser; VA: variable attenuator; V(t): modulation voltage input; MZM: Mach–Zehnder modulator; PC: polarization controller; CIR: optical circulator; FC: fiber coupler; PD: photodetector; OSC: oscilloscope; OSA: optical spectrum analyzer.

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Fig. 2.
Fig. 2. Simulation results for LFM and NLFM signals generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency. (i) “voltage-to-frequency” transfer function calculation, (ii) LFM signal generation, and (iii) NLFM signal generation.

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Fig. 3.
Fig. 3. Simulation results for (i) dual-chirp LFM signal and (ii) dual-chirp NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency.

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Fig. 4.
Fig. 4. Pulse compression performance of the simulated dual-chirp NLFM (red) and dual-chirp LFM (blue) signals.

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Fig. 5.
Fig. 5. (a) Optical spectra of the ML (blue dotted curve), the free-running SL (green dashed curve), and the injected SL (red solid curve), (b) the relationship between the output P1 oscillation frequency and the input voltage amplitude.

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Fig. 6.
Fig. 6. Experimental results for LFM and NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency. (i) “voltage-to-frequency” transfer function calculation, (ii) LFM signal generation, and (iii) NLFM signal generation.

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Fig. 7.
Fig. 7. Experimental results for (i) dual-chirp LFM signal and (ii) dual-chirp NLFM signal generation. (a) input voltage, (b) output microwave waveforms, and (c) output microwave frequency.

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Fig. 8.
Fig. 8. Instantaneous frequency-time diagrams of the generated dual-chirp NLFM signals with a bandwidth of (a) 6 GHz and (b) 9 GHz, respectively.

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Fig. 9.
Fig. 9. Pulse compression performance of the experimentally generated dual-chirp NLFM (red solid curve) and dual-chirp LFM (blue dotted curve) signals.

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Equations (10)

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d A d t = 1 − i b 2 ( g − γ c ) A + γ c ξ ( t ) | A 0 | e x p ( − i 2 π f i t )  +  χ
d N d t = − γ s ( N − N 0 ) − γ s J N 0 ( g | A | 2 γ c | A 0 | 2 − 1 )
g = γ c + γ c γ n ( N − N 0 ) γ s J N 0 − γ p ( | A | 2 | A 0 | 2 − 1 )
< χ ( t ) χ ∗ ( t ′ ) >= β sp δ ( t − t ′ ) ,
< χ ( t ) χ ( t ′ ) > = 0 ,
< χ ( t ) > = 0.
ξ ( t ) = ξ 0 T MZM [ V ( t ) ] = ξ 0 1 2 [ 1 + cos ⁡ ( Δ φ + π V ( t ) V π ) ] ,
f m ( t ) = H [ ξ ( t ) ] = H { ξ 0 T MZM [ V ( t ) ] } = G [ V ( t ) ]
V ( t ) = G − 1 [ f m ( t ) ]
f m ( t ) = H { ξ 0 T MZM [ V ( t ) ] } = H { P inj T MZM [ V ( t ) ] P SL } = G [ V ( t ) ]
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