Sub-THz-range linearly chirped signals characterized using linear optical sampling technique to enable sub-millimeter resolution for optical sensing applications

Shuai Wang; Xinyu Fan; Bin Wang; Guangyao Yang; Zuyuan He

doi:10.1364/OE.25.010224

1. Introduction

Pulse compression techniques based on matched filtering method using linearly chirped pulses or phase coded pulses are widely used in modern radar systems and distributed fiber sensing systems for high spatial resolution and large dynamic range [1–6]. In some other applications such as biomedical imaging, linearly chirped pulses are used as optical sources for extracting depth of the organization sample, which show a great prospect in tumor detection [7]. In particular for long distance applications such as coherent radar systems and distributed fiber sensing systems, pulse compression techniques allow to transmit long pulses minimizing the transmitted peak power while maintaining a high spatial resolution. Conventionally, linearly chirped pulses are generated by using direct digital synthesizers (DDSs) or microwave generators. However, state-of-the-art DDSs have a maximum frequency limited to few GHz, which results in a small time-bandwidth product (TBWP). To deal with this problem, a couple of photoelectric techniques have been proposed to generate chirped or phase-modulated pulses taking advantage of the high speed and broad bandwidth of optical signals [8–11]. Phase modulation using conventional modulators based on photoelectric processing is a popular method, whereas the necessity of using modulators greatly limits the bandwidth of the generated signals and makes the system complicated, and the TBWP is usually less than 100 due to the limited modulation index [11]. Another promising method is to use the quadratic phase modulation effects of chromatic dispersion for generating linearly chirped pulses. In this system, ultrashort optical pulses launching from a mode-locked laser are sent into a dispersion compensating fiber (DCF) or a linearly chirped fiber Bragg grating (LCFBG), and are stretched to be several nanoseconds. Microwave signals may be generated based on heterodyning a linearly chirped pulse with a continuous wave (CW) laser or two linear pulses with different chirp rates [12–14]. However, mainly limited by the bandwidth of the photodetector, photonic-assisted linearly chirped pulses are usually restricted to ~60 GHz [12]. For applications such as frequency modulated continuous wave (FMCW) LIDAR and optical frequency domain reflectometry (OFDR), they directly deduce target range from the beat frequency generated from the pulse reflected by the target and the reference pulse. In this case, the detection bandwidth may be compressed greatly. However, the nonlinearity of the frequency sweep deforms the range resolution or the spatial resolution [15, 16]. Since the pulse compression technique does not require a very strict linear relationship of the frequency sweep, it enables a high spatial resolution for optical sensing applications if we realize a large frequency sweep and characterize it with a receiving technique having a high detection-bandwidth.

Linear optical sampling (LOS) technique [17–19], known for its capability of observing the complex amplitude response of large bandwidth optical signals using slow electronics with low bandwidth and shot-noise limited sensitivity, has been used in many fields for monitoring the waveform in high speed transmission systems. To achieve a large TBWP, LCFBG with a large dispersion coefficient is a good option. Therefore, to use LCFBG for generating linearly chirped pulse signal, and to detect in combination with linear optical sampling method, make it possible to achieve linearly chirped pulse with a large bandwidth and TBWP.

In this paper, we generate a sub-THz-range linearly chirped signal by stretching ultrashort optical pulses, showing that a sub-millimeter spatial resolution can be achieved with the help of linear optical sampling technique. The key distinctive feature of this system is that the employment of linear optical sampling technique enables the possibility to characterize THz-bandwidth signals. By using an LCFBG with a large dispersion coefficient, we generate a sub-THz-range linearly chirped signal with TBWP of 4500 and compression ratio of 4167. Also, a correlation gating scheme based on phase modulation is implemented to extend the time aperture and enhance the signal to noise ratio (SNR).

2. Operation principle

The schematic diagram of our proposed method is shown in Fig. 1. An ultrashort optical pulse train with tens of nanometers wavelength range from a femtosecond laser is rectangular shaped through an optical bandpass filter. An LCFBG is utilized as a dispersive element to implement a wavelength-to-time mapping effect. Let τ_p be the duration time of optical pulse and $Φ^{'}$ be the first order dispersion coefficient, the pulse duration and chromatic dispersion coefficient meet the spatial Fraunhofer condition | τ_p²/(2π) $Φ^{″}$ |<<1 according to the real time Fourier transform [20]. Dispersive element can be modeled as linear time invariant systems by means of a transfer function $H (ω) = | H (ω) | exp [jΦ (ω)]$ , while the spectral phase of the dispersive element can be expanded by Taylor series:

Fig. 1 Experimental scheme to generate and detect linearly chirped optical signal.

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Φ (ω) = Φ + Φ^{'} ω + \frac{1}{2} Φ^{″} ω^{2} + \frac{1}{6} Φ^{‴} ω^{3} .

In this experiment, the dispersion coefficient of the LCFBG is $Φ^{″}$ = −2568 ps² and $Φ^{‴}$ = −9.27 ps³ respectively. Compared with the second order dispersion coefficient, $Φ^{‴}$ and higher order dispersion can be neglected and the impulse response of dispersive element is simplified to:

h (t) \propto \exp (j \frac{π}{2 Φ^{″}} t^{2}) .

Let the input ultrashort pulse be a Sinc-shaped one, $E_{i n} (t) = \frac{sin (π t / τ_{p})}{π t / τ_{p}} exp (j ω_{0} t)$ , the output optical pulse from the dispersive element can be written as:

\begin{matrix} E_{o u t} (t) = E_{i n} (t) * h (t) = C \exp [j (ω_{0} t + \frac{1}{2 Φ^{″}} t^{2})] F F T [\frac{\sin (π t / τ_{p})}{π t / τ_{p}}] \\ = C_{1} \exp [j (ω_{0} t + \frac{1}{2 Φ^{″}} t^{2})] r e c t (t / T), \end{matrix}

where ω₀ is the center frequency of the optical pulse, C and C₁ are constants, T is the pulse duration time and rect() is a rectangular function. Therefore, the envelope of the output signal is a rectangular pulse. It is worth noting that the phase of the output signal has a new quadratic term, providing a linear chirp which is inversely proportional to the dispersion coefficient. Ignoring the higher order chromatic dispersion, the pulse duration time T, sweeping bandwidth B, chirp rate R and TBWP can be respectively written as:

T \approx \frac{| 2 π c Δ λ Φ^{″} |}{λ_{0}^{2}}, B \approx \frac{c}{λ_{0}^{2}} Δ λ, R = \frac{1}{2 π Φ^{″}}, T B W P \approx \frac{4 π c^{2}}{λ_{0}^{4}} Φ^{″} Δ λ^{2},

where Δλ is the 3 dB bandwidth of the pulse and λ₀ is the center wavelength of the ultrashort optical pulse. Equation (4) indicates that the sweeping bandwidth is determined by the bandwidth of the pulses and the theoretical one is calculated to be 625 GHz (5 nm bandwidth centered in 1550 nm), which is far beyond the bandwidth of the state-of-the-art photodetector. This explains the reason why the bandwidth achieved in the previous works is limited to tens of GHz. To make the best of the large frequency-range linearly chirped signal and characterize these chirped pulses, another ultrashort pulse train is launched from another femtosecond laser with a period of T_S, and the electrical field of the pulse train is written as:

E_{S} (t) = \sum_{N} ε_{S} (t - N T_{S}) \exp (j ω_{S} t),

where

ε_{S} (t)

is the complex envelope of the sampling pulses and ω_S is the center frequency. It is required that the spectrum of the sampling pulses covers the linearly chirped signal and exhibits a flat shape. It is convenient to make the period of the sampling pulses slightly different from an integer multiple of the linearly chirped pulses, thus the sampling rate f_sampling is given as follows:

f_{s a m p l i n g} = 1 / (T_{S} - N / f_{p u l s e}),

where f_pulse is the repetition rate of the linearly chirped signal, N = mod(f_pulse, 1/T_S) and mod() is the modulus function. The sampling rate is limited by the timing jitter of the sampling system, which is usually restricted by the timing jitter of the femtosecond laser. Then the launched ultrashort pulse train interfere with the linearly chirped signals and detected by a photodetector. Assuming the impulse response of the photodetector is R(t), the output signal of the photodetector is:

S (t) = \int_{- \infty}^{+ \infty} E_{o u t} (t) E_{S}^{*} (t) R (τ - t) d t .

As we mentioned before, the spectrum of the sampling pulse covers which of the linearly chirped pulse and the intensity spectrum is flat over the whole wavelength range, thus the sampling pulse has a constant spectral density. Finally, what we obtain from the A/D card is a series of discrete sampling points:

S = \sum_{N} \exp [j (ω_{S} - ω_{0}) N T_{S}] {\tilde{E}}_{S}^{*} (ω) E_{o u t} (N T_{S}),

where E_S*(ω) is the spectral density of the sampling pulse and can be regarded as a constant value. For the purpose of simplification, we lock the center frequencies of the two lasers together as

ω_{S} = ω_{0}

. Considering Eq. (3) and (8), the sampled signal becomes:

S (t) = C^{'} E_{S}^{*} (ω) \exp (j \frac{1}{2 Φ^{″}} t^{2}) r e c t (t / T),

while the impulse response of the matched filter is determined by:

h (t) = S^{*} (- t) = C^{'} E_{S}^{*} (ω) \exp (- j \frac{1}{2 Φ^{″}} t^{2}) r e c t (- t / T) .

Then the final demodulation process is a convolution between the sampled signal and the matched filter shown as:

C (t) = S (t) * h (t) = {(C^{'} E_{S}^{*} (ω))}^{2} r e c t (- t / 2 T) \frac{T \sin [\frac{π}{2 Φ^{″}} (T - | t |) t]}{\frac{π}{2 Φ^{″}} T t},

where C(t) denotes the waveform of the compressed pulse. It has a maximum value at t = T, and the full width at half maximum (FWHM) of C(t) is Δt = 2π

Φ^{″}

/T. The spatial resolution Δz, defined here as the FWHM of the compressed pulse, can be derived from Eq. (4) and is given as:

Δ z \approx \frac{λ_{0}^{2}}{2 n Δ λ},

where n is the refractive index of the transmission medium. The result shows that the spatial resolution is determined by the bandwidth of linearly chirped pulses.

From the discussion above, we may understand that taking advantage of the capability to characterize the ultra-wide frequency-range signals using the linear optical sampling technique, we are able to generate and detect sub-THz-range linearly chirped signal, and achieve a sub-millimeter spatial resolution with a low-bandwidth photodetector and an A/D card, breaking the limitation of the electronic bottleneck.

3. Experiment and results

Figure 2 illustrate the experimental configuration to generate and detect the linearly chirped signals. The lower femtosecond pulse laser (FSPL) which launches pulses with 50-MHz repetition rate acts as the sampling laser. The upper FSPL which has a wavelength range centered at 1550 nm and has a repetition rate of 250 MHz is used as the seed pulse source. By using an external clock to synchronize both FSPLs and a pulse pattern generator (PPG), an intensity modulator (IM) serves as a pulse picker. As a result, the repetition rate of the upper FSPL is reduced to be 50 MHz with −3dBm average power. We set the sampling rate to be 150 TS/s according to Eq. (6). The dispersive element we use is an LCFBG operating from 1528 nm to 1565 nm incorporated into an optical circulator to operate in reflection mode. The signal is detected by a balanced photodetector (Thorlabs PDB450C) and captured by a 10-bit real time oscilloscope (Keysight DSOS204A) with a sampling rate of 1 GS/s.

Fig. 2 Experimental configuration to generate and detect linearly chirped optical signal. FSPL: femtosecond pulsed laser; BFP: optical bandpass filter; PPG: pulse pattern generator; IM: intensity modulator; LCFBG: linearly chirped fiber Bragg grating; CIR: optical circulator; PD: photodetector.

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3.1 Signal generation and pulse compression

To achieve a moderate pulse duration, the ultrashort pulses from the femtosecond laser (Menlo Systems C-Fiber Sync 250MHz) is filtered to be 5 nm centered at 1550 nm as the blue lines shows in Fig. 3(a), the red line represents the optical spectrum of the sampling laser. Figure 3(b) shows the measured trace of the original pulse with linear optical sampling technique before launched into the dispersed LCFBG. With a rectangular spectrum the pulse should be Sinc-shaped, but deformed because of the uneven optical spectrum of the sampling laser. By choosing a 10-m long LCFBG with 1981 ps/nm dispersion coefficient, the spectrally shaped pulse is then stretched to a temporal duration of 10 ns. Figure 4 illustrates the measured linearly chirped pulse with linear optical sampling method. The generated linearly chirped pulse characterized by using linear optical sampling technique is shown in Fig. 4(a), which shows that a linearly chirped pulse with 10 ns duration is obtained. The dynamic range of linear optical sampling technique is measured to be 27 dB without any averages. The red box of Fig. 4(a) is the zoom-in picture of the measured signal, however the extinction ratio ofpulse picker (Mach-Zehnder modulator, Photoline) limits the dynamic range of this system, which reflects in the black box in Fig. 4(a). The signals in the black box is the sidelobes caused by the unwanted adjacent pulses which are supposed to be suppressed by the pulse picker. The extinction ratio, which defined as the ratio between the amplitude of the linearly chirped pulses to the amplitude of the sidelobes, is measured to be 12 dB. The sampling jitter of this system is several femto-seconds, and is also shown in Fig. 4(a) in red box which gives detailed information of generated waveform. Here, the sampling interval shown in the Figure is 6.7 fs. The theoretical sweeping range is 625 GHz as mentioned before, but filtered to be 450 GHz to maintain a good SNR. To obtain the instantaneous frequency of the signal, we take the short time Fourier transform (STFT) analysis of the linearly chirped pulse, and the time-frequency characteristic is shown as the color maps in Fig. 4(b). Here, the close-to-linear down-slope illustrates the decreasing instantaneous frequency with a chirp rate of about 45 GHz/ns along the whole 10 ns pulse duration. As a result, a TBWP of ~4500 (450 GHz × 10ns) is achieved in this experiment. In order to evaluate the pulse compression of the generated linearly chirped pulse, we calculate the autocorrelation of the pulse depicted in Fig. 4(c). The main lobe of the autocorrelation has an FWHM of 2.4 ps. Since the linearly chirped pulse has a temporal duration of 10 ns, the pulse compression ratio is calculated to be 4167. Moreover, an even larger compression ratio can be achieved when using a much shorter original optical pulse, which is mainly limited by the wavelength range of the sampling laser. We also note that the chirp rate of the linearly chirped pulse is not a constant, as shown in Fig. 4(b). As shown in Eq. (3), the chirp rate is given as R(t) = 1/2π $Φ^{″}$ , which means that it is determined by the dispersion coefficient of the LCFBG, not having a constant dispersion coefficient over a wide wavelength range in this experiment. In the next section, we are going to compare the nonlinearity effect of using pulse compression method with that of using the FMCW method.

Fig. 3 Characteristics of lightwave before using dispersion element. (a) Spectrum of the sampling laser (red line), the filtered spectrum of ultrashort pulses for stretching (blue line). (b) Original ultrashort pulses with 2-ps pulse duration.

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Fig. 4 Experimental results. (a) A temporal frame of 10 ns linearly chirped pulse. (b) STFT analysis of the linearly chirped pulse. (c) Calculated autocorrelation of the linearly chirped pulse.

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3.2 Comparison results of FMCW method by using the same linearly chirped signals

As mentioned in Ref [1], the spatial resolution of pulse compression system is only decided by the FWHM of the compressed pulse. In this experiment, the FWHM of the compressed pulse given in Eq. (12) only depends on the bandwidth of chirped pulses. Therefore, different from the FMCW method, the nonlinearity of the chirp rate does not deteriorate both the spatial resolution and the SNR as the detection distance increases, although the FMCWmethod does not impose strict bandwidth requirements on the receiver. To compare the spatial resolution of FMCW method and pulse compression method using this linearly chirped pulse, off-line processing have been made referring to Ref [21]. as for FMCW method. To consider the nonlinear chirp situation, the pulse on the reference beam can be written as $S (t) = C^{'} exp [j (ω_{0} t + π R (t) t^{2})]$ with a time-varying chirp rate R(t). For the target point whose round-trip time is τ, when the reflected beam returns, the amplitude is given by $S_{r} (t) = C^{'} exp [j (ω_{0} (t - τ) + π R (t - τ) {(t - τ)}^{2})]$ . The reflected beam is mixed with the reference beam, to be detected by the photodetector, which yields a photocurrent represented by:

Γ (t) \propto [R (t) - R (t - τ)] π t^{2} + 2 t τ R (t - τ) - π τ^{2} R (t - τ) .

By Fourier-transforming of the photo-current signal, the spatial resolution can be represented by the 3 dB FWHM of the spectrum peak. For demonstration, we adopt the off-line processing which is similar to microwave time delay spectroscopy [14, 22]. As shown in Fig. 5(a), we calculate the spatial resolution from 1 mm to 20 mm with 1 mm step, and the spatial resolution starts from 260 μm, quite close to the theoretical spatial resolution, but it suffers a great loss on spatial resolution and has a degradation on SNR. When the measured distance comes to 20 mm, the spatial resolution deteriorated to 10.8 mm. By contrast, the spatial resolution of pulse compression method remains 240 μm (corresponding to 2.4-ps autocorrelation trace duration) along the whole detection range, which is only determined by the 3 dB FMHW of autocorrelation trace.

Fig. 5 (a) Spatial resolution of the FMCW method as a function of target distance. (b) Measured shape of reflection peak by FMCW method at 1 mm, 20 mm and by pulse compression.

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3.3 Long time apertures signal generation and SNR enhancement using phase modulation

In the demonstration described above, our femtosecond laser has 4 ns pulse period, corresponding to 40 cm measurement range in optical fiber and 60 cm in free space. To extend the measurement range, we use an intensity modulator acting as a temporal gate. By increasing the length of the pulse pattern, we are able to increase the time apertures freely. However, the dynamic range is restricted by the extinction ratio of the intensity modulator, and the average power has been degraded due to the use of temporal gating technique. For long time apertures measurement, a scheme has been proposed using noise-like phase modulation by an arbitrary waveform generator (AWG) in optical correlation domain reflectometry (OCDR) [23]. By synchronizing the AWG to the repetition rate of the optical source, the pre-programmed phase modulation signals yield a correlation gate at arbitrary delay with an average power kept to be high. Taking advantage of the delta function property of its autocorrelation function, pseudo-random binary sequence (PRBS) is used in ourexperiment to generate similar correlation gate since it is simple.

In this demonstration, a DCF instead of the LCFBG is used as the dispersion element. The length of the DCF (dispersion coefficient of −130ps/nm/km) is carefully chosen to be 5 km so that each 6-nm bandwidth pulse is stretched to 4 ns temporal period. After stretched by the dispersion element, the chirped pulse train is modulated by the intensity modulator. The intensity modulator working at the NULL point driven by the PRBS impose a differential phase modulation on the chirped pulse train with 0 or π phase shift compared to the original pulse train at every pattern. To exemplify the time apertures extension, the PRBS length is set to be 200 while the PPG generates 10 Gb/s pulse patterns. As shown in Fig. 6(a) and 6(b), the original 4-ns period has been extended to be 20 ns without sacrificing any average powers. The 20-ns up-conversional chirped signal covers a frequency range up to 700 GHz as we can see from the STFT shown in Fig. 6(b). However, compared to the formerly generated linearly chirped pulses, the phase term has been distracted by differential phase modulation. For observation of the time apertures extension, we calculate the autocorrelation of the phase-modulated pulse train, which is plotted in Fig. 6(c). The pulse has a strong autocorrelation peak at 20 ns, which is 5 times of the original 4-ns period. Weak peaks can still be noticed at the original 4-ns period. The extinction ratio between the main peak and the weak peaks is measured to be 11.4 dB. For longer measurement range and larger extinction ratio, by setting the period to be 386 pulses with 15820 PRBS sequences, which corresponding to 1544 ns time apertures, the extinction ratio improved to 20.4 dB. As shown in the zoom-in insets of Fig. 6(d), the FWHM is 1.2 ps, corresponding to 700-GHz chirped frequency-range which enables a 120 μm spatial resolution in optical fiber and a 180 μm spatial resolution in free space.

Fig. 6 Experimental results. (a) 20 ns measurement of linearly chirped pulse with phase modulation. (b) STFT analysis of the linearly chirped pulse with phase modulation. (c) Calculated autocorrelation of the 20-ns linearly chirped pulse with phase modulation. (d) Autocorrelation of the 1544-ns time-aperture pulse.

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4. Conclusions

Based on the chromatic dispersion effect, a sub-THz-range linearly chirped signals has been generated and detected with the help of linear optical sampling technique. The sub-THz-range linearly chirped pulses of which the maximum TBWP, single pulse duration and compression ratio is increased to 4500, 10 ns and 4167 respectively, are successfully demonstrated experimentally with low bandwidth electronics, breaking the limitation of the electronic bottleneck. Only restricted by the wavelength range of the sampling laser, the maximum frequency-range achieved in this experiment is 700 GHz, corresponding to a 120 μm spatial resolution in optical fiber and a 180 μm spatial resolution in free space. In addition, long time apertures chirped pulse train is achieved with PRBS phase modulation, making it possible to enable long measurement range as well as ultra-high spatial resolution.

As a stabilized femtosecond laser with a coherence length of hundreds of kilometers has the potential to cover a frequency range of 100 THz, as well as with the dispersion element, this proposed technique has the opportunity to achieve several micrometers spatial resolution and ultra-long measurement range with the help of pulse compression technique. Our work features the generation and detection of large bandwidth linearly chirped signals and paves the way for new horizons in ultra-high spatial resolution and long measurement range LIDAR, distributed fiber sensing and biomedical imaging.

Funding

National Natural Science Foundation of China (NSFC) (61575001, 61327812); Shanghai STCSM Scientific and Technological Innovation Project (15511105401).

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Sub-THz-range linearly chirped signals characterized using linear optical sampling technique to enable sub-millimeter resolution for optical sensing applications

Abstract

1. Introduction

2. Operation principle

3. Experiment and results

3.1 Signal generation and pulse compression

3.2 Comparison results of FMCW method by using the same linearly chirped signals

3.3 Long time apertures signal generation and SNR enhancement using phase modulation

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (13)

Optics Express