Ghost imaging constructs an image by correlating two signals: one that interacts with an object but possesses no spatial information, and the other that contains spatial information but does not interact with the object. Ghost imaging can be extended into the time domain by using laser intensity fluctuations to reconstruct an unknown time-varying pattern, but this requires the measurement of laser fluctuations on ultrafast timescales, a significant limitation at wavelengths where ultrafast detectors are not available. We overcome this by using wavelength conversion to shift the probe laser into a spectral region where ultrafast detectors are available, and we apply this technique to image a temporal object at 2 μm. Our results demonstrate that temporal correlation information can be transferred to an arbitrary spectral region, opening possibilities for ultrafast ghost imaging at new wavelengths.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Ghost imaging is an indirect measurement technique based on the correlation of multiple probing patterns with the integrated signal measured after transmission through (or reflection from) an object [1–8]. The probing patterns may be random, in which case, they must be measured in a reference arm, or they can be pre-programmed prior to illuminating the object [9–15]. Originally demonstrated in the spatial domain to reconstruct images of real physical objects [1,3,5], ghost imaging can in fact be applied to retrieve the characteristics of a given object in any measurement domain, provided that the probe patterns exhibit corresponding parameter fluctuations. Indeed, ghost imaging has been recently demonstrated in the spectral domain for spectroscopic applications , including greenhouse gas absorption measurement , and to recover a hidden state of polarization .
There has been particular interest in applying the ghost imaging concept in the time domain, transposing the principles of spatial ghost imaging to measure ultrafast “temporal objects” in the form of modulated high-speed bit pattern sequences [14,19–22]. In addition to showing how the random-intensity fluctuations of a laser source can be used to probe an ultrafast temporal modulation without directly detecting it, these results have stimulated interest into new potential applications of ghost imaging in various areas, including the dynamical characterization of free-electron lasers , secure and quantum communications [24–26], optical cryptography , quantum device characterization , and focal-plane three-dimensional imaging .
Irrespective of the domain in which ghost imaging is applied, the ultimate measurement resolution is determined by the characteristic scale of the probe patterns fluctuations and/or the corresponding resolution with which they can be measured (or pre-programmed). In the particular case of time-domain ghost imaging, retrieving an ultrafast temporal object or modulation with a high resolution requires high-speed detectors at the wavelength illuminating the temporal object. However, since such detectors are not available at all wavelengths, this is an experimental constraint which can significantly limit applications of temporal ghost imaging, particularly for measurements extending to longer wavelengths in the infrared spectral region.
In this paper, we transpose the approach of two-color ghost imaging developed in the spatial domain [30–32] to perform a time-domain correlation at two very distinct wavelengths, and thus remove the need for a fast detector at the wavelength that illuminates the temporal object. In particular, we use the random intensity fluctuations from a quasi-continuous wave laser source in the infrared at 2 μm to image a temporal object synthesized from the on/off keying transmission of a fast intensity modulator. The light transmitted through the modulator is directly detected with a slow integrating detector without resolution of any temporal structure, while the source intensity fluctuations are frequency doubled to 1 μm and measured with a high-speed detector. By correlating the unresolved integrated signal at 2 μm with the random-intensity fluctuations measured at 1 μm, the bit pattern sequence is retrieved. In this proof-of-concept demonstration, we compare the retrieved sequence with a direct measurement and find excellent agreement. These results demonstrate that temporal ghost imaging can be performed without the need for a particularly fast or sensitive detector at the specific wavelength at which the integrated signal is measured after the object. With suitable wavelength conversion, ghost imaging can be performed with an illumination wavelength optimally chosen to image the desired temporal object, while allowing the temporal object characteristics to be measured at a wavelength where a suitable detector is available. Our experiments open the door to ultrafast imaging in spectral regions where fast detectors are not available, including, e.g., pump-probe experiments to characterize materials in wavelength regimes that are difficult to access, such as the mid-infrared and THz regimes.
2. EXPERIMENTAL SETUP
A schematic of the experimental setup is shown in Fig. 1. The light source is a custom-made quasi-CW cladding-pumped thulium-doped fiber laser operating at 1970 nm. It consists of a 793 nm laser diode pump, a 4-m-long thulium-doped fiber, and a pair of high-reflective and low-reflective fiber Bragg gratings. The laser can generate up to 1.5 W (unpolarized) output power, with a linewidth of 0.1 nm corresponding to random intensity fluctuations with a characteristic timescale of . The laser light is collimated and divided between the test and reference arms with an 8:92 beam splitter. In the reference arm, the frequency-doubled laser temporal fluctuations at 1 μm are generated in a -barium borate crystal via Type I phase matching and detected by a fast InGaAs avalanche photodiode (Thorlabs, APD310, 1 GHz bandwidth) with an operating wavelength range from 850 to 1650 nm. To eliminate any potential remaining light at 2 μm, a combination of long-pass and short-pass filters (Thorlabs, FELH1500 and DMSP 1180) is inserted before the detector. The resolution of the ghost image is determined by the convolution of the intensity fluctuation time scale at the fundamental wavelength with the fast detector bandwidth. Here, the intensity fluctuations time scale is nearly an order of magnitude shorter than the fast detector response time. It is therefore the detector bandwidth that defines the resolution of the ghost temporal image. In the test arm, light is linearly polarized with a polarizer along the same direction as that of the phase-matched second harmonic generation component used in the reference arm, driven by an arbitrary waveform generator (ADVANTEST D3186). The driving signal consists of a repeated on/off keying sequence of 0 and 1 bits. The transmission of the modulator follows the driving bit sequence and plays the role of the temporal object to be retrieved. The light transmitted through the modulator is detected by an integrating photodiode with an operating wavelength range extending beyond 2 μm (Thorlabs, DET05D, 20.6 MHz bandwidth). The photodiode speed is far too slow to resolve the structure of the bit sequence. The signals from the reference and test arms are recorded simultaneously by a real-time oscilloscope (Tektronix DSA72004) triggered by the bit pattern generator.
The ghost image of the temporal object is obtained by calculating the normalized intensity correlation function defined by20,33].
A. Frequency Conversion of Intensity Fluctuations
A central prerequisite for the wavelength-conversion ghost-imaging scheme is that the intensity fluctuations of the light source are preserved during the nonlinear frequency conversion process. To ensure that it is indeed the case, a set of control measurements was first performed without any temporal object in the test arm (i.e., when the modulator is removed) and replacing the slow-integrating detector by another fast InGaAs photodetector operating at 2 μm (EOT, ET-5000, 12.5 GHz bandwidth). This allows for recording simultaneously in real time the intensity fluctuations before and after the nonlinear crystal and measure their correlations. From second-harmonic generation theory, one expects that the frequency-doubled intensity fluctuations are proportional to the square of the intensity fluctuations at the fundamental frequency such that . An example of simultaneously recorded intensity fluctuations at 2 μm (red solid line) and the square root of intensity fluctuations at 1 μm (blue solid line) over a 30 ns time window is illustrated in Fig. 2(a). Note that avalanche photodiodes have a DC-blocked amplifier, which results in zero-mean fluctuations when measured by the oscilloscope. We can see how the fluctuations of the fundamental light at 2 μm follow nearly perfectly the square root of the fluctuations measured after the nonlinear crystal at 1 μm. This excellent correspondence is confirmed over a much larger number of temporal windows in Fig. 2(b), where we plot the time-to-time cross correlation between the intensity fluctuations of the fundamental light at 2 μm and that of the second-harmonic square root intensity at 1 μm, calculated over a recorded set of 8000 consecutive temporal windows with a 30 ns duration. The cross-correlation matrix clearly displays a near one-to-one correspondence, but we do note a small residual negative correlation at short times induced by the detector response. However, as we shall see below, this has only a minor effect on the ghost-imaging measurements.
Two additional pre-requisites to temporal ghost imaging using random intensity fluctuations are that (i) the fluctuations within the measurement window of the temporal object are uncorrelated and (ii) that the fluctuations from one realization to another are independent. This is shown in Fig. 3, where we plot the time-to-time correlation of the frequency-doubled light at 1 μm calculated over an ensemble of 8000 consecutive realizations [Fig. 3(a)] and the realization-to-realization correlation map calculated over the 30 ns duration of the temporal window of a single realization [Fig. 3(b)]. Both correlation maps show only non-zero values on the diagonal, indicating that the intensity fluctuations within the measurement window of the object are indeed uncorrelated, but also that the different realizations are independent from one another. Finally, for completeness, we show in Fig. 3(c) the experimentally measured probability density function of the laser temporal intensity fluctuations relative to the calculated mean. One can see how this distribution is well fitted by a generalized extreme value distribution, in agreement with previous observations .
B. Temporal Ghost Imaging
We subsequently proceeded to the ghost-imaging measurements of a bit sequence of a 30 ns total duration, and the results are shown in Fig. 4. Specifically, Figs. 4(a) and 4(b) show the experimental ghost image retrieved for two different bit sequences by correlating the temporal intensity fluctuations at 1 μm from the reference arm with the integrated intensity at 2 μm from the test arm over 8000 realizations (solid blue lines). For comparison, we also performed a direct control measurement of the bit sequences without the ghost-imaging approach, but using the fast InGaAs photodetector operating at 2 μm directly after the electro-optic modulator and averaging over 8000 consecutively recorded temporal windows of 30 ns duration (dashed red lines). We can see very good agreement between the ghost image and the control measurement, indicating the success of the temporal ghost-imaging scheme in retrieving the temporal object by transferring the correlation information to a wavelength different from that of the light illuminating the temporal object. The residual negative correlation values in the ghost images arise from the detectors response, as manifested in the 2–1 μm intensity cross-correlation results shown in Fig. 2(b).
Finally, Fig. 5 shows the ghost image of the bit sequence shown in Fig. 4(b) for an increasing number of realizations. As expected, we can see that the signal-to-noise ratio increases with the number of realizations, but, rather remarkably, we also see that the bit sequence can be already resolved with only 500 realizations. The improvement in the retrieved ghost image with realization number can be quantified by calculating the least squares error between the directly measured (dashed red line) and retrieved temporal sequences (solid blue line) normalized relative to the number of points in the sequence . The misfit values corresponding to the results in Fig. 5 are given in the caption and show the quantitative improvement with increased number of iterations.
4. DISCUSSION AND CONCLUSION
We have demonstrated here an extension of temporal ghost imaging that allows the unresolved temporal measurements after interaction with the object and the ultrafast temporal measurements of the source intensity fluctuations to be performed at different wavelengths. The development of such a wavelength-versatile scheme significantly expands the usability and applications of time-domain ghost imaging as it allows a wider choice of illumination and detection options in the experimental design. Although the approach here is based on second-harmonic generation for the wavelength-conversion step, the technique is applicable to any nonlinear process that preserves the source intensity statistics during frequency conversion. Indeed, our results further highlight a flexibility inherent to ghost imaging in that one does not necessarily need to correlate the unresolved measurements after interaction with the object with the actual pattern that illuminates the object. Rather, the correlation step can use any derived or otherwise related pattern that is itself correlated with the illumination pattern, similar to the recently introduced blind ghost-imaging method in the spatial domain .
The achievable temporal resolution in the ghost image is determined by the convolution of the characteristic fluctuation time of the illuminating light source (at the fundamental wavelength) with the response time of the detector (at the wavelength of the converted light). Using a detector with a response time of the order of the fluctuations time is therefore in principle optimum. The use of random fluctuation patterns from a laser source requires of course a large number of distinct realizations (on the order of few thousands) to reconstruct the temporal signal with a high signal-to-noise ratio. Although the approach of computational temporal ghost imaging can reduce the required number of realizations , it requires the use of suitable instrumentation such as, e.g., ultrafast modulators to pre-program illumination patterns at the source wavelength, which are not necessarily available in all wavelength regimes. In contrast, our approach allows the use of virtually any noisy quasi-CW laser source. In addition, because the temporal profile of the signal interacting with the object does not need to be resolved, ghost imaging is intrinsically insensitive to temporal distortion effects that take place after the object. This implies that light collection is possible using multimode fibers and large-area (low-bandwidth) detectors, and when combined with the wavelength versatility shown here, they will be expected to enable imaging at significantly reduced power levels. We anticipate particular applications of the wavelength conversion scheme for ghost imaging applied to pump–probe experiments in areas such as semiconductor carrier lifetime measurements in the mid-infrared  and THz regimes , allowing the use of standard photodiodes rather than cryogenically cooled detectors. Applications are also expected for the more general case of pump–probe spectroscopy, where the pump and probes are at different wavelengths.
Academy of Finland (298463); Photonics Research and Innovation (PREIN) (320165); Agence Nationale de la Recherche (ANR) (ANR-15-IDEX-0003).
G. G. acknowledges support from the Academy of Finland. J. M. D. acknowledges support from the French Investissements d’Avenir programme, project ISITE-BFC. H.W. acknowledges support from the China Scholarship Council.
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