A simple photonic generation of linearly chirped microwave pulse with large time-bandwidth product and high compression ratio

Hongbiao Gao; Cheng Lei; Minghua Chen; Fangjian Xing; Hongwei Chen; Shizhong Xie

doi:10.1364/OE.21.023107

1. Introduction

Microwave or millimeter radar has attracted great attention due to its wide applications in medical healthcare, physical chemistry, radar system, etc. Continuous wave (CW) Doppler radar has been used for non-contact remote detection of vital signs, such as searching for survivors after earthquake and monitoring sleeping infants or adults to detect abnormal breathing condition [1]. The frequency modulated continuous wave (FMCW) radar has also shown its advantage over the CW Doppler radar in the application of contactless sensing, because it can monitor the user motor activity as well as the cardiac activity and respiration [2]. In biomedical imaging, chirp-pulse microwave computed tomography (CP-MCT) in which the linearly chirped microwave pulse is used to extract the straight path from multiple paths has been demonstrated in noninvasive thermometry [3] and shown great promise in breast tumor detection [4]. In physical chemistry, broadband chirped pulse Fourier transform microwave (CP-FTMW) spectrometer is also widely used for measuring the rotational spectrum for molecular structure determination of gas phase samples [5]. Based on pulse compression technology, frequency-modulated or phase-coded signal source is extensively employed in modern radar system to increase range resolution and detecting distance [6]. Moreover, as stated in [6], linearly chirped or phase-coded microwave pulse with time-bandwidth product (TBWP) of 100 or more is extensively required for further application. Conventionally, linearly chirped or phase-coded microwave pulse can be generated by electronic circuit [7], but the center frequency and bandwidth which are required to be up to tens of gigahertz for many applications are greatly limited by the low speed electronic devices.

Owing to the inherent advantages of ultra-high speed and ultra-wide band offered by optics, various photonics-based methods to generate chirped or phase-modulated microwave waveform have been proposed in recent years. One popular method is based on photoelectric hybrid implementation in which active electrical devices, such as digital signal generator, microwave source and modulator, are used for phase modulation [8–14]. An obvious disadvantage of the method is that the usage of the active electrical devices greatly complicates the system configuration and limits the bandwidth of the generated microwave pulse. Method based on spatial light processing devices, such as spatial light modulator (SLM) and virtually-imaged phased-array (VIPA) has been proposed [15–18], which shows high flexibility in terms of pulse shape and center frequency of the generated chirped microwave pulse. However, the complex alignment and inevitable coupling loss between optical fiber and free space put the system at a disadvantage. Generation of chirped microwave pulse based on all-optical devices, such as especially fabricated fiber Bragg grating (FBG), has attracted extensive research interest [19–24]. Nevertheless, the limited bandwidth and chirp rate of the fabricated FBG used for imparting initial chirp limit the time duration, bandwidth and finally TBWP and compression ratio of the generated chirped microwave pulse. Moreover, the nonuniform reflection and group delay ripple of the FBG also lead to microwave pulse deterioration. Although the approach proposed in [25] shows reconfigurability of the center frequency and the chirp rate of the achieved microwave pulse, the undesired fact that the chirp rate will change when the center frequency is reconfigured greatly limits its practical application. In [26], chirped microwave pulse has been generated using a wavelength sweeping distributed-feedback (DFB) laser. While, there is a significant trade-off between sweeping rate and scanning bandwidth in the system.

In this paper, a simple generation of linearly chirped microwave pulse based on all-optical fiber using heterodyne beating of pre-dispersed optical pulse and CW light is demonstrated and linearly chirped microwave pulse with bandwidth of 33GHz, TBWP of 106 and compression ratio of 160 is obtained. The TBWP and compression ratio are at least two times higher than that of the pulses generated in [8–25], among which the achieved largest TBWP and compression ratio is 45.56 and 62.5 respectively. Compared with the fabricated FBG, wavelength sweeping laser and other designed configuration, the DCF used for imparting initial linear chirp in our method is commonly accessible and can be modeled as an all-pass filter with uniform frequency and identical dispersion response. Our approach also shows the promise to generate linearly chirped microwave pulse with bandwidth up to THz which is verified by our theory and simulation results. The center frequency of the generated linearly chirped microwave pulse can also be adjusted by tuning the frequency of the tunable CW. The time-frequency character of the chirped microwave pulse is presented by the short time Fourier transform (STFT) analysis which also indicates the stability and repeatability of the chirp rate of the microwave pulse achieved in our experiment. Our generated chirped microwave pulse also shows a vacuum range resolution of about 12mm which can be demonstrated by off-line processing.

2. Operation principle and simulation

The schematic diagram of our proposed method is shown in Fig. 1. According to the real time Fourier transform [27], if the pulse width $Δ t_{0}$ of the input optical pulse $x (t)$ and the dispersion coefficient $\ddot{Φ}$ of the dispersive element meet the relation $| Δ t_{0}^{2} / (2 π \ddot{Φ}) | < < 1$ , the output optical pulse from the dispersive element can be written as

y (t) = C \exp (j \frac{t^{2}}{2 \ddot{Φ}}) {F [x (t)]}_{ω = t / \ddot{Φ}},

where C is constant and

F [x (t)]

denotes the Fourier transform of

x (t)

. The

ω

in above equation is the angular frequency deviation relative to the center frequency

ω_{0}

of the optical pulse. If we rewrite the equation with absolute angular frequency

ω

, then we get

Fig. 1 Schematic diagram of the proposed method to generate linearly chirped microwave pulse.

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y (t) = C \exp (j ω_{0} t) \exp (j \frac{t^{2}}{2 \ddot{Φ}}) {F [x (t)]}_{ω - ω_{0} = t / \ddot{Φ}} .

Hence, the envelope of the output optical signal is proportional to the spectrum of the input signal. An additional quadratic phase is imparted on the optical pulse providing a linear chirp which is inversely proportional to the dispersion coefficient. A CW light $x_{C W} (t)$ with center frequency $ω_{C W}$ and initial phase $φ_{C W}$ is coupled with the linearly chirped optical pulse before being detected by a PD which is assumed to be without bandwidth limit. The coupled light can then be written as $y_{1} (t) = y (t) + \exp (j ω_{C W} t + j φ_{C W})$ and the output current of the PD is

\begin{matrix} i (t) = ℜ \cdot [y_{1} (t) \cdot y_{1}^{*} (t)] \\ = ℜ \cdot {1 + C^{2} A^{2} (t) + 2 \cos [ψ (t)] A (t)}, \end{matrix}

where

A (t) ​ ​ ​ ​ ​ ​ ​ = | {F [x (t)]}_{ω - ω_{0} = t / \ddot{Φ}} |

and

ψ (t) = (ω_{0} - ω_{C W}) t + t^{2} / (2 \ddot{Φ}) + Δ φ

with

Δ φ

the initial phase difference between the optical pulse and CW light,

ℜ

is the response factor of the PD. The quadric phase and

A (t)

indicate that the obtained electronic pulse is linearly chirped and its envelope is also proportional to that of the spectrum of the optical pulse. The chirp rate

R

_, center frequency

f_{c e n t e r}

, bandwidth

B

and TBWP of the generated microwave pulse can be respectively written as

R = \frac{\partial^{2} ψ (t)}{2 π \partial t^{2}} = \frac{1}{2 π \ddot{Φ}}, ​ f_{c e n t e r} = \frac{ω_{0} - ω_{C W}}{2 π}, B \approx \frac{c}{λ_{0}^{2}} Δ λ, ​ ​ T B W P \approx \frac{4 π c^{2}}{λ_{0}^{4}} \ddot{Φ} {(Δ λ)}^{2},

where

λ_{0}

and

Δ λ

is the center wavelength and the 3dB bandwidth of the optical pulse out of the dispersive element. The chirp rate is determined by the dispersion coefficient and the center frequency can be adjusted by changing the center frequency of the CW light. The bandwidth and TBWP are only decided by the bandwidth of the dispersed optical pulse, which indicates the promise to greatly increase the carrier frequency and bandwidth of the generated chirped microwave pulse, even up to THz.

For a simple proof of our proposed method, a simulation is firstly carried out. The optical pulse with a 3dB bandwidth of 8nm is firstly dispersed by DCF with dispersion coefficient of −6ns/nm and its pulse width is broadened to be about 48ns. The optical frequency deviation $ω_{0} - ω_{C W}$ is set to be zero to generate linearly chirped microwave pulse with the same envelope of the broadened optical pulse as shown in Fig. 2(a). The lower part in Fig. 2(a) shows the detailed profile of the chirped microwave pulse center marked by a red dotted rectangle frame, which indicates an obvious linearly and centrally symmetrical instantaneous frequency change along the whole pulse duration. The STFT analysis of the chirped microwave pulse shown in Fig. 2(b) indicates a 20GHz/ns chirp rate and a 500GHz frequency distribution symmetrical with respect to the microwave pulse center, which coincide exactly with our theoretical results predicted by Eq. (4). The autocorrelation shown in Fig. 2(c) represents the compressed profile of the chirped microwave pulse with a full width at half maximum (FWHM) of about 2ps leading to a compression ratio of about 24000.

Fig. 2 Simulation results. (a) Temporal profile of the simulated linearly chirped microwave pulse. (b) STFT analysis of the microwave pulse in (a). (c) Autocorrelation of the microwave pulse in (a).

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The ultra-wide bandwidth of the simulated linearly chirped microwave pulse benefits from the broadband optical pulse, which can be up to THz. Despite of the practical bottleneck that it is impossible to capture chirped microwave pulse with such wide bandwidth owing to the bandwidth limit of the equipment used in our experiment, our method shows advantages over other methods in terms of increasing the bandwidth, TBWP and compression ratio of the generated linearly chirped microwave pulse.

3. Experiment configuration and results

The experiment setup of our proposed method is presented in Fig. 3. The ultra-short optical pulse from a femtosecond pulsed laser (FSPL) is firstly filtered by an optical band-pass filter (OBPF). After being amplified by the erbium doped fiber amplifier (EDFA), the optical pulse is linearly chirped or dispersed by the DCF. The linearly chirped optical pulse is then coupled with a CW light in the optical coupler (OC). After the heterodyne beating in the PD, the linear chirp of the optical pulse is converted to the generated microwave pulse.

Fig. 3 Experiment configuration to generate linearly chirped microwave pulse. FSPL: Femtosecond Pulsed Laser; CW: Continuous Wave; OBPF: Optical Band-Pass Filter; EDFA: Erbium Doped Fiber Amplifier; DCF: Dispersion Compensation Fiber; OC: Optical Coupler; PD: Photodetector; DSO: Digital Sampling Oscilloscope; OSA: Optical Spectrum Analyzer.

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The FSPL used in our experiment is a lab-made laser with 3dB bandwidth of 8nm, output power of 0.2dBm and center wavelength of 1560nm as the red line shows in Fig. 4(a) and the black line represents the filtered optical spectrum after the OBPF. The DCF with a chromatic dispersion of −6ns/nm broadens the time duration of the optical pulse to be about 4ns as shown in Fig. 4(b). In order to generate linearly chirped microwave pulse with different center frequency, the frequency deviation between the optical pulse and CW light emitted from the tunable CW source, whose output power keeps 6dBm constant, is set to be 0, −33 and 33GHz respectively. Here, the largest frequency deviation, i.e. 33GHz, is determined by the hardware bandwidth of the DSO. The bandwidth of the PD is 40GHz which enables us to take full use of the bandwidth of the DSO.

Fig. 4 Measured results before heterodyne beating. (a) Spectrum of the FSPL (red line), the filtered optical spectrum (black line). (b) Temporal profile of the dispersed optical pulse.

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Experimentally, linearly chirped microwave pulse is firstly generated under the condition that the optical frequency deviation is zero and the results are shown in Fig. 5. The two linearly chirped microwave pulses shown in Figs. 5(a) and 5(d) with linearly changing frequency symmetrical with respect to the pulse center are captured by DSO at two different time. The difference in the center of the two pulses is caused by time-dependent initial phase deviation between the optical pulse and CW light, which can be explained by Eq. (3). The two waveforms maintain almost the same rectangular envelope with a FWHM of about 3.2ns. The slight amplitude fluctuation of the pulse envelope is mainly caused by the amplitude noise of the initial pulse as shown in Fig. 4(b). By taking the STFT analysis of the two microwave pulses, the time-frequency character is respectively shown as the colormaps in Figs. 5(b) and 5(e). The black dotted lines in the colormaps illustrate the instantaneous frequency distribution of the microwave pulse and indicate almost a same chirp rate of about 20GHz/ns which agrees well with our theoretical result calculated by Eq. (4). The bandwidth reaches up to 33GHz in the 3.2ns pulse duration and so a TBWP as large as 106 is firstly experimentally achieved. The frequency ambiguous at certain time of the colormaps is caused by the fast frequency change, i.e. large chirp rate, and the limited frequency resolution owing to the small time window used in the STFT analysis, but this has no influence on analyzing the frequency distribution versus time. By calculating autocorrelation, the microwave pulses in Figs. 5(a) and 5(d) are compressed to be about 20ps as shown in Figs. 5(c) and 5(f), which leads to a compression ratio as high as 160. It is obvious that the initial phase deviation between the optical pulse and CW light only affects the temporal profile of the pulse center, but has no influence on the time-frequency character, TBWP and compression ratio of the generated chirped microwave pulse.

Fig. 5 Experiment results for optical frequency deviation of zero. (a) and (d) Temporal profiles of two microwave pulses captured at two different time. (b) STFT analysis of (a), black dotted line indicates a slope of 20.08GHz/ns. (c) Autocorrelation of (a) with FWHM of 20ps. (e) STFT analysis of (d), black dotted line indicates a slope of 19.93GHz/ns. (f) Autocorrelation of (d) with FWHM of 20ps.

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In order to demonstrate the tunability of the center frequency of the generated linearly chirped microwave pulse, the optical frequency deviation between the optical pulse and CW light is then changed to be −33 and 33GHz respectively to generate two different types of microwave pulses. The temporal profiles of the measured microwave pulses with the same FWHM of about 1.75ns are shown in Figs. 6(a) and 6(d). The STFT analysis of the two microwave pulses, as shown in Figs. 6(b) and 6(e), illustrate the monotonously linearly increasing and decreasing instantaneous frequency with a chirp rate of about 20GHz/ns across the whole pulse duration which leads to a single-side band of 33GHz and center frequency of 16.5GHz. Compared with the centrally symmetrical microwave pulses in Figs. 5(a) and 5(d), the single-side band leads to nearly a half decrease of pulse duration and consequently the TBWP is reduced to 57.8. Figures 6(c) and 6(f) represent the compressed microwave pulses with the same FWHM as that in Figs. 5(c) and 5(f), which means that the compression ratio decreases to 87.5. By comparing the different results for frequency deviation of −33GHz and 33GHz, it is concluded that the TBWP and compression ratio of the generated linearly chirped microwave pulse remain unchanged in spite of the opposite sign of the chirp rate.

Fig. 6 Experiment results for different optical frequency deviations. (a) and (d) The temporal profiles of the measured microwave pulses for frequency deviation of −33GHz and 33GHz respectively. (b) STFT analysis of (a), black dotted line with a slope of −19.95GHz/ns. (c) Autocorrelation of (a) with FWHM of 20ps. (e) STFT analysis of (d), black dotted line with a slope of 20.02GHz/ns. (f) Autocorrelation of (d) with FWHM of 20ps.

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Additionally, the largest chirp rate deviation among the pulses shown in Fig. 5 and Fig. 6 is only about 0.4% which is mainly caused by the calculation error. This shows the ultra-high stability and repeatability of our method to generate chirped microwave pulse with different center frequency while the chirp rate keeps nearly constant (~99.6%).

The range resolution of our generated chirped microwave pulse can also be verified by off-line processing based on microwave time delay spectroscopy (MTDS) [28], as shown in Fig. 7. The input chirped pulse can be written as $f_{i n} (t) = \cos (π R t^{2})$ where $R$ is the chirp rate. After the two delayed pulses mix with the reference pulse and only the beat frequency terms are taken into account, the output of the mixer can be simplified as $f_{o u t} (t) = \cos [π R (2 τ_{1} t - τ_{1}^{2})] + \cos [π R (2 τ_{2} t - τ_{2}^{2})]$ . It clearly shows that two intermediate frequency terms emerge with frequency separation of $Δ F = R | τ_{1} - τ_{2} |$ corresponding to range separation of $Δ D = c | τ_{1} - τ_{2} |$ where $c$ is the velocity of light in vacuum. This indicates that the range resolution of the chirped microwave pulse can be verified by analyzing the magnitude spectrum of $f_{o u t} (t)$ .

Fig. 7 Conceptual algorithm of the off-line-MTDS for verifying range resolution of the generated chirped microwave pulse.

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For demonstration, the generated chirped microwave pulse shown in Fig. 5(a) is used as $f_{i n} (t)$ , $τ_{1}$ is set to 400ps and $τ_{2}$ is set to 500, 440, 420, 410ps respectively. The magnitude spectrums of $f_{o u t} (t)$ for different $τ_{2}$ are shown in Fig. 8. It is obvious that when the relative delay between path 1 and path 2 is larger than 40ps the two intermediate frequency terms can be clearly distinguished. However, on the condition that the relative delay is smaller than 40ps, the two intermediate frequency terms will confuse each other. This indicates a vacuum range resolution of 12mm which agrees well with the theory result proposed in [28]. Moreover, as predicted in [28], when our generated chirped microwave pulse is used in specified medium, such as saline solution with relative permittivity of about 74 in [28], the range resolution will be increased to 1.06mm in theory. The very high range resolution offer us alternative to make use of our generated chirped microwave pulse for further applications, such as CP-MCT.

Fig. 8 Magnitude spectrums of $f_{o u t} (t)$ in condition that (a) $τ_{2}$ = 500ps, (b) $τ_{2}$ = 440ps, (c) $τ_{2}$ = 420ps and (d) $τ_{2}$ = 410ps. Insets show the detailed spectrum.

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4. Conclusions

Based on the heterodyne beating between the pre-dispersed optical pulse and the CW light, a simple method to generate linearly chirped microwave pulse has been demonstrated. The center-symmetrical linearly chirped microwave pulses, of which the FWHM, bandwidth, TBWP and compression ratio is increased up to 3.2ns, 33GHz, 106 and 160 respectively, are experimentally obtained. The tunability of the center frequency of the linearly chirped microwave pulse is illustrated by the two different types of monotonously linearly chirped microwave pulse with TBWP and compression ratio of 57.8 and 87.5, which are achieved by changing the frequency deviation of FSPL and CW. It is also shown that the initial phase deviation of FSPL and CW has no effect on the performance, such as TBWP and compression ratio, of the generated linearly chirped microwave pulse. The vacuum range resolution of our generated chirped microwave pulse is about 12mm which is verified by off-line processing based on MTDS.

As the linear chirp is originally imparted on the optical pulse by the DCF which can be regarded as an all-pass optical filter with uniform frequency response compared with the widely used FBG, our proposed method shows the promise to generate linearly chirped microwave pulse with bandwidth up to THz which is very attractive for future applications, such as radar or biomedical imaging system with ultra-high range resolution. Although the inevitable hardware bandwidth limit of the measurement equipment is the main barrier for capturing such broadband microwave pulse, the ultra-large TBWP, ultra-high compression ratio and ultra-stable chirp rate show the advantages of our method. What’s more, compared with methods employing modulators, microwave sources or digital signal generator, our all-fiber based method greatly simplifies the system configuration and is much more cost effective and competitive for practical application.

Acknowledgments

This work is supported by National Program on Key Basic Research Project (973) under Contract 2012CB315703, and NSFC under Contract 61120106001, 61132004 and 60807026 and the Program for New Century Excellent Talents in University (NCET-10-0520).

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A simple photonic generation of linearly chirped microwave pulse with large time-bandwidth product and high compression ratio

Abstract

1. Introduction

2. Operation principle and simulation

3. Experiment configuration and results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (4)

Optics Express