Open Access
Add to BibSonomy
Abstract
Measurement of fast signal is getting more and more important in many fields. In this paper, we propose to detect a temporal signal based on the idea of computational ghost imaging (GI), which can greatly reduce requirements on bandwidth of detectors. In experiments, we implement retrieving of a temporal signal with time scale of 50ns using a detector of 1kHz bandwidth, which is much lower than the requirement on bandwidth of detector according to information theory. The performance of our technique are also investigated under different detection bandwidths.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the development of science and technology, people are more interested and have to deal with faster and faster signals, such as in the field of ultrafast physics [1]. According to information theory, the bandwidth of the detector is expected to match the highest frequency of the signal spectrum and the sample rate is also expected to be higher than twice of that frequency, to record a signal without distortion. Since existing fastest detectors can only respond to signals of tens of picoseconds, different methods were proposed to detect signals indirectly.
Ghost imaging (GI), especially computational ghost imaging, is one of indirect measurement methods, which realizes spatial-resolved measurement via a single-pixel detector, therefore provides a way for reconstructing a signal beyond bandwidth limitation of the detector. Since the first demonstration, more and more attention has been attracted to GI in the past 20 years [2–21]. In a typical GI system, a light beam with random fluctuations is illuminated onto the object, with the transmission or reflection light being collected by a non-spatial-resolved detector (single pixel, called bucket detector). At the same time, the illumination beam also propagates directly towards a spatial-resolved detector, with which the fluctuations of the source is recorded. The second-order correlation between results of these two detectors yields an image of the object. By actively controlling the wavefront of the illumination field thus calculating the intensity distribution on the object plane, the spatial-resolved detector can be omitted. Therefore the simplified GI system contains only a single-pixel detector [8–10]. This is the main idea of computational GI [8], in which image of the object can be reconstructed with a detector of much lower spatial resolution.
Recently, the concept of GI has been extended from space domain to time domain [13, 14, 16, 17]. Ryczkowski et al demonstrated a temporal analogue of GI [14]. However, a fast photodiode is still needed to record the temporal fluctuation of the source, with the bandwidth higher than that of the signal.
In this paper, we propose to reconstruct temporal signal with a much slower detector, based on the idea of computational GI, releasing the requirements on detection bandwidth. Experimentally, a signal with time scale of 50ns is well reconstructed, using a photodetector with bandwidth of only 1kHz.
2. Results
2.1. The scheme and experiment setup
To obtain the image of an object with computational GI, many measurements with different random illumination patterns are required, with a corresponding sequence of detection results from the bucket detector recorded. The second-order correlation between those patterns and the detection results provides the image. For each run, the random illumination pattern can be taken as a random modulation on the signal. According to this idea, we propose to measure a temporal signal S(t). Assume S(t) is of defined length T and repeatable. N realizations are considered and known random modulations Rk(t)(k = 1, 2, …, N) are performed on the signal for those realizations, with the modulated signal recorded by a slow detector (similarly, we call it bucket detector). The result of the bucket detector for the kth run reads
According to the computational GI theory, the signal can be reconstructed by calculating the second-order correlation, Here 〈·〉 denotes averaging over N realizations.Under this scheme, the time resolution of the retrieved signal is determined by that of R(t). Namely, it is determined by the bandwidth of the modulator, instead of the detector. At the same time, fast detector is no more required since the modulations can be actively controlled and recorded. We experimentally demonstrated measurement of a temporal signal with a much slower detector.
A digram of our experimental setup is shown in Fig. 1. A repeatable optical signal is generated via a laser diode, with the output waveform controlled by a pattern generator (Tektronix AWG7082C). Then the signal passes through an intensity modulator (Photline MXER-LN-10, 10GHz bandwidth), with the modulated signal detected with a photoelectric receiver (Newport 2051-FC-M). The bandwidth of the detector is adjustable from 10MHz to 10Hz discretely. Then the output of the detector is recorded with an oscilloscope (Agilent DSO9104A, 1GHz bandwidth, sampling rate being 200MHz). Without loss of generality, out experiments are performed with discrete signals and modulations. The time resolution of the interested signal is set to be 50ns, with the length of the signal being T = 15µs. Therefore the signal contains M = 300 time resolutions. Note that the sampling rate of the oscilloscope is 200MHz, thus there are 3000 sampled points in the signal. In each realization, the signal is randomly modulated by the intensity modulator with a known 0–1 random pattern.
Fig. 1 Experimental setup for detecting a temporal signal via computational GI. The repeatable signal is generated from a laser diode, with the waveform of which is being controlled by a pattern generator. For direct measurements, the optical signal goes through the modulator without modulation. For computational GI, the optical signal is modulated by the modulator with the known patterns. In both cases the signal is detected by the photodetector of adjustable bandwidth.
2.2. Experiment results
A repeating binary optical signal is generated, with three peaks at different time included, of the same height. For comparison, the signal is both directly detected and reconstructed with computational GI. For direct detection, the modulator is set to leave the signal going through directly. For computational GI, the results recorded on the oscilloscope within the signal period are summed up to offer bucket results. To verify the influence of detector’s bandwidth, the photodetector is set to work at different 3-dB bandwidths, from 10MHz to 1kHz. For each setting of the detector, a reconstructed signal is obtained from detection results of N = 133333 runs.
Results form direct detection are shown as blue dots in Fig. 2. For a fair comparison, the direct detection results are averaged over 133333/3000≈44 measurements. With the bandwidth declining, the response time of the detector becomes longer, thus the rising and falling edge of the detection results become prolonged. And there will be distortion or even crosstalk when the bandwidth is low. The heights of the three peaks are the same in the prepared signal. However, as shown in Figs. 2(c)–2(e), the latter peak is influenced by the former ones, so the heights of three peaks increase in turn. At the same time, due to the frequency response of the photodetector getting worse, the height of the detection results declines as the bandwidth being lower and lower. Thus the noise increases relatively, which is more apparent when the bandwidth of detector is very low, as shown in Figs. 2(f). By contrast, the bandwidth of the detector has less influence on the reconstructed results of computational GI. In each run, the results of direct detection are summed together as one bucket result. The reconstructed results from computational GI are shown as pink asterisks in Fig. 2. The distortion in direct detection does not show much influence in this case. That is because we only need an overall intensity for the bucket detection and information of the prepared signal is obtained by correlation. Even through the waveform of the direct detection results distorts due to bad response of the photodetector, the bucket result values maintain relatively unchange and only suffers from attenuation. Such attenuations will contribute nothing to the second-order correlation, therefore the results of GI shows better robustness. It should be noticed that, when the bandwidth of the detector is very low, the fidelity of the reconstructed signal is also reduced even using GI, especially obvious in Figs. 2(e)–2(f). The reason lies in that the percentages of the three peaks in the bucket results becomes different due to the crosstalk in the direct detection, which were expected to be the same. The time length of detection and sampling is defined to be the same as the length of the prepared signal. As shown in Figs. 2, for low detection bandwidth, there is a long tail before the detection values reach the minimum or the level of background noise. Therefore, a fraction of the tail can be casted if it does not reach the minimum at the end of sampling. Under this situation, the later the peak occurs, the more contribution will be casted. Therefore the early peak contributes larger weight in the bucket values, resulting in the calculated correlation for which is also higher. The results of direct measurement before average are also shown in Figs. 2(g), 2(h) and 2(i). The signal-to-noise ratio (SNR) of direct measurement thus that of the bucket results is extremely low. That is the reason why high number of measurements are obtained to improve the SNR of the reconstructed results.
Fig. 2 Results of direct detection and computational GI, with the bandwidth of detector set as 10MHz, 1MHz, 300kHz, 100kHz, 10kHz and 1kHz, from (a) to (f). The blue dots represents direct measurement results and the pink asterisks represents GI reconstruction results. Results of direct measurement without average are shown from (g) to (i) with 100kHz, 10kHz and 1kHz bandwidth.
To quantitatively analyze the results, the peak signal-to-noise ratio (PSNR) is employed [23], which is defined as
where L is the number of pixels in the signal and W(i) describes the prepared waveform of the signal. G(i) is the reconstructed result. MAX is the maximum value within the measurement result. Normalization is adopted to calculate the PSNR. Since we did normalization, MAX = 1. From this definition, the bigger PSNR, the better quality of result will be, which means the result is more close to the original waveform.For different settings of detection bandwidth, the calculated PSNRs are shown in Fig. 3. Every point in the plot is obtained over statistic of 5 separate experiments. Red circles show the results of GI and blue squares for that of direct measurement. The results of PSNR also infers that GI provides stronger robustness against detection bandwidth.
Fig. 3 PSNR of the results achieved from two techniques, computational GI and direct measurement (DM), varying with the bandwidth of the photodetector. Each point is obtained by a statistic over 5 times of measurements. It shows that the quality of direct measurement declines with the bandwidth, while that of GI appears more robust.
2.3. Partition reconstruction strategy
When the bandwidth of the detector is extremely low, the GI reconstruction results still have distortions. In order to improve the quality of reconstructed signal, we employ partition reconstruction strategy, since SNR of ghost image is inversely proportional to the size of the object (in number of pixels) according to GI theory [21]. The signal is considered as M sequential pieces and reconstructed piece by piece. In each realization, the detection result of the photodetector is divided into F sequential pieces. For each piece, the detection results are summed together as the bucket result. The random pattern is also divided into F sequential pieces accordingly. The signal is reconstructed piece by piece, and the combination of those pieces is takes as the final result. For our signal containing 3000 points, we separate it into F = 300 pieces, with the results shown in Fig. 4.
Fig. 4 Results of computational GI with partition strategy. Three typical cases with bandwidth 10MHz, 10kHz and 1kHz are shown from (a) to (c). The background noise can be suppressed and the height of three reconstructed peaks turns out to be the same. From (a) to (c) the PSNR of the results are 24.5dB, 20.1dB and 18.6dB, which are higher than that of unimproved results, being 22.3dB, 18.7dB and 16.9dB, respectively.
Comparing with the results shown in Fig. 2, the partition reconstruction strategy can improve the quality of reconstructed results effectively. The results also shows that PSNRs of the reconstructed result for each piece are approximatively equal, equal to that of the final signal. For the case of 10MHz detection bandwidth, PSNR of the final result is 24.5dB, which is higher than that of the unimproved reconstruction result, 22.3dB. About 2dB improvement in PSNRs are also achieved for the cases that the detection bandwidth are set as 10kHz and 1kHz. The distortion of GI reconstructed results is also suppressed effectively. When the partition strategy is adopted, although the bucket result for each piece might contains crosstalk from other pieces, the waveform for the piece can be correctly reconstructed since those crosstalk can be taken as uncorrelated noise. GI appears robust against such noises, as has been verified via GI in the spatial domain [12].
2.4. About the sampling rate
In spatial GI, a lens is usually used to collect photons into a single-pixel detector, serving as the bucket detector. Due to the lack of “time-lens”, the electrical signals of the photodetector are sampled by the oscilloscope, and summed up as bucket results. Despite of the bucket detection method in the experiment, only one pixel value is necessary for the bucket detection, if it contains the whole information about the bucket result according to GI theory. Namely, only once sampling should be able to reconstruct the prepared signal.
Firstly, we examine the situation that only one pixel is sampled. We choose result of the 950th pixel from direct detection as the bucket result and try to reconstruct the signal via second-order correlation. With detection bandwidth being 10MHz, 300kHz, and 10kHz, the results are shown in Fig. 5. When the detection bandwidth is 10kHz, only one pixel sampling can reconstruct the whole signal, while for higher bandwidth it can not work well.
Fig. 5 The results of using only one pixel (the 950th) of direct detection as the bucket value. This pixel is apart from three peaks and the detection value contains no information about them for the case of 10MHz bandwidth. Due to the distortion and crosstalk for the case of 300kHz and 10kHz bandwidth, the information of the three peaks is (partially) included in the detection value of the 950th pixel. Therefore the signal can be (partially) reconstructed.
Results of direct measurement shown in Fig. 2 can be explained as following. For the case of 10MHz, direct measurement can characterize the signal well. Therefore each point contains information only about the very time pixel. Since the 950th pixel is apart from three peaks thus containing no information about those peaks, the reconstructed result can not reflect information about the whole signal. While the bandwidth is lower, certain degree of distortion and crosstalk among the whole signal occur, due to the slow response of the detector. For the 950th point departs from three peaks at different distances, the rates of included information differ. The lower the detection bandwidth, the closer rates for three peaks. That explains why the reconstructed result is better for the case of 10kHz.
To be clearer, we also simulated the case of different sampling rates. Part of the recorded pixels are chosen as sampled value, at regular intervals. For different lengths of the intervals, we can simulate different effective sampling rates. For example, if the length of the intervals is 120, the effective sampling rate is 200MHz/120 = 1.67MHz.
For different effective sampling rates, PSNR of reconstructed waveform are shown in Fig. 6. For the case of 1kHz bandwidth, PSNR becomes stable soon with increasing effective sampling rate. An effective sampling rate of 300kHz (around the knee point) is sufficient to recover the waveform of the signal. Note that the time scale of the signal is 50ns, with bandwidth of which is much greater than the effective sampling rate. Namely, our detection method can also reduce requirements on sampling rate. The effective bandwidth around the knee points are 1.3MHz and 25MHz for the case of bandwidth 100kHz and 10MHz. Due to the distortion and crosstalk, the original bucket result can be replaced by the sum of a few sampling values, thus low sampling rate can be sufficient to recover the signal.
Fig. 6 PSNR of reconstructed results under different effective sampling rates. The effective bandwidth around the knee points are 300kHz, 1.3MHz and 25MHz for the case of bandwidth 1kHz, 100kHz and 10MHz, respectively. Each point here is obtained by a statistic over 5 times of measurement.
2.5. Comparison with scanning detection
An alternative way for such a signal detection is to scan by applying a short time gate and detecting the transmitted energy, which can be realized with the same experimental configuration. The results of scanning method are shown in Fig 7. For each point, the result is also averaged over 44 measurements. From left to right, the bandwidth of the detector are 10MHz, 1MHz, and 300kHz. When the bandwidth is 300kHz, single pixel transmission detection does not work.
With scanning method, only a small fraction of the interested signal could reach the detector, which results in low SNR or even no response from the detector, due to low transmitted energy. For a given signal, the transmitted energy decrease with increasing number of pixels. Thus the time resolution could be limited. By contrast, our scheme can work better. Under random modulation applied, the transmitted energy will be a combination of all transmitted pixels, which is averagely half of the total energy. Namely, there is more likely response in the detector with our method under the same detection condition. The time resolution of our technique is independent to energy response of the detector. At the same time, our scheme is more robust against uncorrelated noises compared to scanning detection, such as environmental noise and detection noise. In the scanning detection or direct detection the noises will be added into the final results directly. However, our method can filter out these uncorrelated noises, since we are using second-order correlation between the modulation and the bucket results. This feature has also been verified in spatial domain [12]. It has been shown that the quality of ghost imaging does not change much when the noise is getting stronger, while that of direct imaging (corresponding to scanning detection) declines dramatically.
3. Discussion and conclusion
Similar to spatial GI, we also need results from many runs to reconstruct the final result. This limited temporal GI useful only for repeatable signal. However, single-shot acquisition of non-reproducible time object is also possible using temporal computational GI, based on spatial multiplexed measurement, as has been discussed by Devaux et al [15].
The time resolution is determined by the bandwidth of the modulator. If bandwidth of the modulator is limited, sub-pixel-shift method [22, 24, 25] can be used to improve the resolution. That is, introducing controlled sub-pixel time shift between the interested signal and the modulator can help to improve time resolution. Since optical delay line with high precision can be easily achieved, sub-pixel-shift method combined with temporal computational GI can get high time resolution.
We are introducing idea of spatial computational imaging and computational ghost imaging into time domain. Therefore, there is no intrinsic difference between spatial computational GI and temporal GI. Features of spatial GI can be maitained in our scheme. Many related discussions, such as designing different illumination patterns [19, 26, 27] or employing different algorithms [28, 29], to improve performance of GI can also be applied in our scheme. We are here demonstrating proof-of-principle experiments for our scheme. Other kinds of time signal or dynamic process can also be investigated with this scheme, and the modulation can be realized without such a modulator of high bandwidth. Besieds, modulation of high bandwidth can be easier to obtain than detection of high bandwidth, since combination of different modulations or transferring between modulations on different degrees of freedom can help to improve the performance of the overall modulation. For example, our scheme can be used for detecting THz wave or some dynamic processes in the field of ultrafast physics, where pump-probe technique is widely employed [30, 31]. Using the technique of pulse shaping, modulate the probe laser into different time patterns, then detect the signal with an ordinary detector to obtain the bucket signal, the dynamic process can be reconstructed according to our scheme.
In conclusion, we demonstrated computational temporal GI, reconstructing a time object with requirements on the bandwidth of detectors greatly reduced. A signal with a time scale of 50ns can be well reconstructed with detection bandwidth of only 1kHz. Performance of our method using detectors of different bandwidths was also investigated. In addition, our detection method can reduce the requirements on sampling rate. Out signal can be recovered with an effective sampling rate of 300kHz. Combining with the techniques of pulse shaping and pump-probe, applications of this method in the field of ultrafast physics and THz wave measurements are quite straightforward. We believe this method will find wide applications in many different fields, even beyond optics and physics.
Funding
National Natural Science Foundation of China under Grant Nos. 11374368, 11674397 and 61671015.
References and links
1. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163 (2009). [CrossRef]
2. T. B. Pittman, Y. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995). [CrossRef] [PubMed]
3. R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002). [CrossRef]
4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]
5. A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed]
6. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004). [CrossRef] [PubMed]
7. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). [CrossRef] [PubMed]
8. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]
9. Y. Bromberg, O. Katz, and Y. Silberberg,” Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]
10. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 34, 844 (2013). [CrossRef]
11. S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6, 37013 (2016). [CrossRef] [PubMed]
12. Y. K. Xu, W. T. Liu, E. F. Zhang, Q. Li, H. Y. Dai, and P. X. Chen, “Is ghost imaging intrinsically more powerful against scattering?” Opt. Express 23(26), 32993–33000 (2015). [CrossRef]
13. T. Setälä, T. Shirai, and A. T. Friberg, “Fractional Fourier transform in temporal ghost imaging with classical light,” Phys. Rev. A 82, 043813 (2010). [CrossRef]
14. P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics , 10, 167–170 (2016). [CrossRef]
15. F. Devaux, P. A. Moreau, S. Denis, and E. Lantz, “Computational temporal ghost imaging,” Optica , 3(7), 698–701 (2016). [CrossRef]
16. F. Devaux, K. P. Huy, S. Denis, and E. Lantz, “Temporal ghost imaging with pseudo-thermal speckle light,” J. Opt. 19(2) 024001 (2016). [CrossRef]
17. S. Denis, P. A. Moreau, F. Devaux, and E. Lantz, “Temporal ghost imaging with twin photons,” J. Opt. 19(3), 034002 (2017). [CrossRef]
18. M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016). [CrossRef]
19. M. J. Sun, L. T. Meng, M. P. Edgar, M. J. Padgett, and N. Radwell, “A Russian Dolls ordering of the Hadamard basis for compressive single-pixel imaging,” Sci. Rep. 7, 3464 (2017). [CrossRef] [PubMed]
20. D. B. Phillips, M. J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017). [CrossRef] [PubMed]
21. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34(21), 3343–3345 (2009). [CrossRef] [PubMed]
22. M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016). [CrossRef] [PubMed]
23. Q. Huynh-Thu and M Ghanbari, “Scope of validity of PSNR in image/video quality assessment,” Ele. Lett. 44, 800–801(2008). [CrossRef]
24. Z. Wang, J. Zhu, F. Yan, and H. Jia, “Superresolution imaging by dynamic single-pixel compressive sensing system,” Opt. Engineering 52(6), 063201 (2013). [CrossRef]
25. S. Tetsuno, K. Shibuya, and T. Iwata, “Subpixel-shift cyclic-Hadamard microscopic imaging using a pseudo-inverse-matrix procedure,” Opt. Express 25(4), 3420–3432 (2017). [CrossRef] [PubMed]
26. S. M. Khamoushi, Y. Nosrati, and S. H. Tavassoli, “Sinusoidal ghost imaging,” Opt. Lett. 40(15), 3452–3455 (2015). [CrossRef] [PubMed]
27. K. Shibuya, K. Nakae, Y. Mizutani, and T. Iwata, “Comparison of reconstructed images between ghost imaging and Hadamard transform imaging,” Opt. Rev. 22(6), 897–902 (2015). [CrossRef]
28. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010). [CrossRef] [PubMed]
29. B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express 20(15), 16892–16901(2012). [CrossRef]
30. Y. S. Lee, Principles of Terahertz Science and Technology (Academic, 2009).
31. X. C. Zhang and J. Xu, Introduction to THz Wave Photonics (Acandemic, 2010). [CrossRef]
