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Abstract
We present a simple system for realizing single-shot ultrafast sequential imaging based on spatial multiplexing in-line holography. In this system, we propose to combine a specially designed mini-reflector delay-line array with digital in-line holography. The former including a group of adjustable mini-reflectors can easily generate an array of probe sub-pulses that can be controlled independently in the propagation direction and time delays. The object beams formed by the different sub-pulses will propagate and fall on different recording regions of the image sensor to generate a single-shot spatial-multiplexing in-line hologram. The geometry of the digital in-line holography can simplify the complexity of the system and enable complex amplitude imaging. In addition, the time resolution of this system is limited only by the pulse duration, which allows this system to study the dynamic processes with the femtosecond order. In an experiment about the laser-induced air plasma, our proposed system achieves nine frames sequential holographic images with the frame rate of 7.5 trillion frames per second (Tfps).
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultrafast optical imaging with pico- or femtosecond temporal resolution is of great importance in research of dynamics of transient events such as ultrafast laser induced plasmas, filaments and ablations [1–3]. In the early stage, time-resolved repetitive pump-probe imaging (RPI) techniques were often adopted for the purpose [4–7]. However, in the RPI methods, multiple repeat measurements with repeated pump pulses and different probe pulse delays must be carried out to obtain the dynamics of a transient process, which made the methods inapplicable to the situations where the probed ultrafast phenomena are either non-repeatable or difficult to reproduce [8].
To overcome the shortcomings of RPI methods, many single-shot sequential imaging (SSI) techniques without repetitive operations [9–11] have been developed, which can be broadly divided into two categories: intensity SSI methods [12–28] and complex amplitude SSI methods [29–35]. For example, the methods based on polarization multiplexing [12], chirped-pulse temporal wavelength division [13–15], and structured illumination [16,17] belong to the former. As the vanguard of intensity SSI methods in the last decade, compressed ultrafast photography (CUP) [18–24] based on compressed sensing and streak camera has been great success in the SSI with a large sequence depth and high framer rates. However, it is a challenge for these CUPs to improve the spatial resolution owing to the spatial-temporal mixture of streak camera and the sparsity of scenes. Another approach for realizing intensity SSI is to apply spatiotemporal division to ultrafast laser pulses [25,26]. In a recent research in intensity SSI methods [27], it utilized a diffractive optical element and a transparent echelon to produce an array of time-delayed beamlets and further realize single-shot non-synchronous array photography. However, its delay mechanism means that the frames interval cannot be adjusted flexibly. The same concern about delay line with adjustment flexibility is present in the latest study [28]. In addition, these intensity SSI methods are unavailable or limited in some applications such as quantitative phase characterization in physics [36] or biomedicine [37] owing to that they are not able to capture phase information of object wave within transient processes.
To circumvent the limitations of intensity SSI methods, complex amplitude SSI methods are set up based on the digital holography [29–35], which can image the amplitude and phase information of tested ultrafast phenomenon in one-shot camera exposure, simultaneously. For example, Wang et al. [29] earlier proposed a pulsed off-axis holographic SSI system that associates the different instants with an interference fringes at different orientation. Three sets of amplitude and phase images of air plasma with frames interval of 300 fs were reconstructed from recorded single spatial frequency multiplexing hologram (SFMH). This off-axis holographic recording scheme based on the SFMH can ensure that each sub-hologram extracted from the same SFMH has a large field of view by overlapping the sub-holograms in recording plane. Then, Chen et al. [30] also raised an off-axis holographic SSI method by utilizing a group of parallel coherence shutters, and four frames spatial separating holograms with a frame interval of 34 ps were captured. In our previous work [31], we designed a compact off-axis holographic SSI system, in which a specially designed sequence pulse train generator with a group of diffractive gratings is adopted to synchronously generate the probe pulse train (PPT) and the reference pulse train (RPT) required for a single SFMH recording. The proposed system can provide a large field of view because tilted front of the RPT can perfectly overcome the disadvantage of the walk-off effect in off-axis holographic recording [32,38]. However, these off-axis holographic SSI methods need two groups of pulse trains including the coaxial PPT and the off-axis RPT, which leads to the considerable complexity of the system. Most recently, Godin’s group [33] reported a versatile complex amplitude SSI technique combining the sequentially timed all-optical mapping photography with acousto-optics programmable dispersive filtering (AOPDF) and the spatial-multiplexing in-line holography. The former allows the exposure time and frame interval can be independently adjusted. The latter achieves spatial-multiplexing SSI by separating those sub-holograms in the recording plane and the application of in-line holography enables the system to obtain sample’s phase images without the reference arm. However, the exposure time of each sub-pulse was expanded owing to the accumulated dispersion of AOPDF and the spectral bandwidth of band-pass spectral filter. This can cause a significant temporal overlap between the close sub-pulses as the frame interval was up to femtosecond scale because the time resolution of the system determined by the probe pulse duration reduces.
In this paper, we propose a simple system for realizing ultrafast SSI with adjustable delay interval. The key component of the system is a specially designed mini-reflector delay-line array (MDA), in which each mini-reflector is mounted on a micrometer head with a mini cardan joint, so the direction and the time delay of the beam reflected by each mini-reflector can be controlled respectively according to the requirement of the optical configuration. Based on this MDA, we have designed a spatial-multiplexing in-line holographic SSI (SI-HSSI) system, and demonstrated its feasibility in realizing ultrafast HSSI by utilizing it to record the dynamic evolution of air plasma induced by a femtosecond laser pulse in experiments.
2. Method
Figure 1(a) shows an example of the schematic of our SI-HSSI system based on the specially designed MDA. As shown in Fig. 1(a), the system is mainly composed by a beam extender lens (EL), a splitter (BS), a collimating lens (CL), an MDA and an image sensor (IS). An input laser pulse, after being expanded and collimated by the lenses EL and CL, directly illuminates on the mini-reflectors of the MDA. The reflected light is divided into an array of sub-pulses. These back-reflected sub-pulses are further focused into an array of spherical sub-beams by the CL and reflected to the object plane by the BS inserted between the EL and CL, forming a sequence of probe sub-pulses. Because each mini-reflector of the MDA is mounted on a specially designed micrometer head with a mini cardan joint, the propagation direction as well as the time delay of the probe sub-pulses can be adjusted by requirement. If the object plane is set between the focal plane of the CL and the recording plane of the IS as shown in Fig. 1(a), and all the sub-pulse spots are adjusted to be completely overlapped each other on the sample (S), the transmitted object beams formed by different sub-pulses will propagate and fall on different regions of the recording plane. Thus a single-shot spatial-multiplexing in-line hologram (SSIH) can be formed and recorded by the IS, in which an array of point source in-line sub-holograms is included. From one SSIH a sequence of high time-resolved holographic images can be reconstructed by using the algorithm described below, in which the imaging frame rate can be controlled by the time delay of the probe sub-pulses generated from the MDA.
Fig. 1. (a) Schematic of the proposed SI-HSSI system. (b) The coordinate geometry of the digital in-line holography under a spherical beam illumination with any angle.
Suppose $\lambda $ is the wavelength of the input pulse, ${z_f}$ is the distance from the object plane to the focal plane of the CL, and ${z_r}$ is the distance from the object plane to the recording plane of the IS. Under the coordinate geometry as shown in Fig. 1(b), the ith input spherical sub-beam illuminating to the object plane can be written as [39]:
Thus, the intensity forming the ith sub-hologram of the SSIH recorded on the recording plane can be expressed as:
From Eq. (7), it can be seen that this intensity formed by the recording geometry shown in Fig. 1 is the same as that formed by the following equivalent recording geometry: the sample on the object plane is pre-magnified by a factor of $M$ given in Eq. (4) and is illuminated by a plane beam; at the same time, the distance between the object plane and the recording plane is changed into ${z_t}$ determined by Eq. (5). Although the ith sub-hologram could be superimposed with some high frequency components belonging to other sub-holograms on the recording plane, they are incoherent with each other because the sub-pulses forming corresponding sub-holograms will arrive on the recording plane at different moments.
For eliminating the noises such as the air agitation, pulse intensity difference and phase inclination factor in the recording process, it is necessary to take a background hologram B including an array of ${B_i}({{x_2},{y_2}} )$ under the same experimental conditions as the SSIH except without the sample. The ${B_i}({{x_2},{y_2}} )$ can be expressed as based on Eq. (7):
In the following imaging reconstruction, the reconstruction of these sub-holograms are mainly performed independently, taking into account the computer configuration requirements and post image calibration. When the extracted sub-hologram and background sub-hologram are illuminated by a plane beam in the reconstruction, the imaging reconstruction will be achieved by an iteration algorithm based on the propagation back and forth of light filed between the recording plane and the imaging plane with a distance ${z_t}$ determined by Eq. (5). The iteration procedure consists of the following steps:
- (I) The initial reconstructed complex object field $t_i^0\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)$ and complex background field $t_i^B\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)$ are first obtained by calculating the inverse diffraction operation from the ith sub-hologram and the corresponding background sub-hologram according to Eqs. (7) and (9), respectively. Then we can simply divide the complex object field over the complex background field to capture an iteration initial complex object field $t_i^1\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)$.
- (II) An object support S is applied to the reconstructed image on the object plane. Specifically, the S is a region established based on the pre-knowledge of the shape and size of the test sample. In the iteration procedure, the part of the obtained complex object field inside this region will be retained, while the part of the field outside this region will be assigned to 1, which can be expressed as: where N is the number of the iteration.
- (III) For the region inside the object support, a constraint is adopted so that the amplitude must not exceed 1. In order to avoid stagnation during the iteration, a relaxation parameter $\varepsilon $ is applied in the constraint [40], and the update complex object field can be expressed as:$$\left|{t_i^{N^{\prime}}\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)} \right|= \left\{ {\begin{array}{{cc}} {\left|{t_i^N\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)} \right|- \varepsilon \times \left|{t_i^N\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)} \right|}&{N = 1}\\ {\left|{t_i^{N - 1}\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)} \right|- \varepsilon \times \left|{t_i^N\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)} \right|}&{N \ge 2} \end{array}} \right.$$in which, the relaxation parameter $\varepsilon $ is a real with ranges in 0 to 1.
- (IV) When $t_i^{N^{\prime}}\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)$ propagates forward to the recording plane, it gives rise a new complex field $U_i^N({{x_2},{y_2}} )$. A normalized hologram ${{{H_{Ni}}({{x_2},{y_2}} )= {H_i}({{x_2},{y_2}} )} / {{B_i}({{x_2},{y_2}} )}}$ is introduced [41], and the square root of this normalized hologram $\sqrt {{H_{Ni}}({{x_2},{y_2}} )} $ is used as an amplitude constraint on the recording plane [42]. In other words, the phase information obtained from $U_i^N({{x_2},{y_2}} )$ is retained, and the amplitude information is replaced by $\sqrt {{H_{Ni}}({{x_2},{y_2}} )} $. The updated complex field $U_i^{N + 1}({{x_2},{y_2}} )$ propagates backward to the imaging plane, then the forming new complex object field $t_i^{N + 1}\left( {\frac{{{{x^{\prime}}_1}}}{M},\frac{{{{y^{\prime}}_1}}}{M}} \right)$ as the input for the next iteration starting at step (II).
The iteration operation will run until all pixel values within the object support are not exceed 1. If the constraint is satisfied at the (N + 1)th iteration, the amplitude and phase distribution of the tested sample can be obtained:
3. Experiment and results
3.1. Experiment configuration
An experimental setup was constructed according to the principal schematic shown in Fig. 1(a). Figure 2(a) shows a photo of the experimental setup. In the experiments, the input beam is a single pulse emitted from Ti:sapphire regenerative laser amplifier system with the center wavelength of 800 nm, the repetition frequency of 1 kHz and the pulse duration of 30 fs. The focal lengths of the EL and CL are -15 mm and 100 mm, respectively. The MDA is composed of nine mini-reflectors; each is mounted on a micrometer head with a displacement range of 6 mm and a smallest division of 10 µm. The SSIH containing nine sub-holograms is recorded by a CMOS camera (HIKVISION, MV-CH120-10UM) with global shutter. The camera has a large the pixel number of 4096 × 3000 and small pixel size of 3.45 µm × 3.45 µm. During the formation of this SSIH, each mini-reflector of the MDA is slightly angled inward to ensure that all the sub-pulses spots overlap completely each other on the sample. The tilted angle of the ith mini-reflector is about $\theta = \arctan ({{{{\Delta_i}} / {{z_r}}}} )$, where ${\varDelta _i}$ is the distance between the sub-hologram corresponding to the ith mini-reflector and the sub-hologram located at the center of the SSIH.
Fig. 2. (a) The picture of the constructed experimental setup for realizing the SI-HSSI. (b) The recorded spatial multiplexing in-line holograms of a transparency. (c) The images reconstructed from 9 sub-holograms. (d) The PSNR of the reconstructed images, in which the No. 1 reconstructed image is taken as the standard image.
In order to verify the quality and consistency of the images reconstructed from different sub-holograms based on our SI-HSSI system, a transparency with some static lines and patterns was used as a test sample for the first experimental demonstration. Figure 2(b) shows a recorded SSIH when the distance ${z_f}$ from the focal plane to object plane is equal to about 17.5 mm and the distance ${z_r}$ between the object plane and recording plane is about 41.5 mm. According to Eq. (4), the magnification M of the system is about 3.37. Although there was no any change taken place in the tested sample, the different sub-holograms still recorded the states of the sample at different moments. The reconstructed region of each sub-hologram has been marked by the square dashed box. Figure 2(c) shows the amplitude distributions of the sample reconstructed from 9 sub-holograms when the iteration times is 35. For checking the consistency of the images reconstructed from different sub-holograms, we measured the diameter of a same circle pattern on the reconstructed images. For example, the diameter of the circle No. 4 on the image reconstructed by the first sub-hologram is measured to be 80 µm, while the diameter of the same circle pattern reconstructed by the fifth sub-hologram is 82 µm. The verge relative error relative error between the two measured diameter is about 2.5%. By comparing the difference between the measured diameters taken from nine reconstructed images, the average relative error is about 3%. It is allowable for this difference that may be caused by the recording or position errors. In order to quantitatively demonstrate the quality and consistency of the reconstructed images, we calculated the peak signal-to-noise ratio (PSNR) of the corresponding amplitude of the reconstructed image, which has been shown in Fig. 2(d). In this calculation, the image reconstructed from the first sub-hologram is considered as a standard image because of its specificity in position. From Fig. 2(d) it can be seen that the reconstructed images from different sub-holograms have approximately the same fidelity. Thus, for the thin test sample, such as a transparency or an air plasma, the difference among these sub-holograms can be ignored.
3.2. Evolution of the phase and refractive index of a femtosecond laser-induced air plasma
Next, the SI-HSSI system was applied to capture the sequential holographic images of an air plasma induced by the femtosecond laser pulse. The energy of a single pump pulse is 0.87 mJ, and then the pump pulse is focused by the focused lens (FL: f = 15 mm) to gain an intensity of $5.91 \times {10^{15}}{\textrm{W} / {\textrm{c}{\textrm{m}^\textrm{2}}}}$ in the focal spot. Air molecules absorbed the laser energy in the focal area are ionized, which can generate an air plasma. At the same time, the nine sub-pulses generated by MDA sequentially pass through the air plasma region, and these sub-pulses will carry a string of complex images of the plasma with different moments. The time interval among the sub-pulses reaching the air plasma can be independently changed via adjusting the micrometer head. The time interval between the pump pulse and the first sub-pulse is determined by an optical delay in the pump unit. In Figs. 3(a) and 3(b), nine sets of two-dimension sequential complex amplitude images quantitatively characterize the amplitudes and phase distributions of the air plasma evolution reconstructed from the recorded SSIH. Here, the birth and growth of this air plasma can visually be observed. The frame interval is up to 133 fs, which corresponds to a frame rate of 7.5 Tfps. The exposure time of each sub-pulse is always equal to the duration time of the input pulse, i.e. 30 fs. The pump pulse is incident from the right-hand side and gradually focused. When more laser energy is deposited within a small volume, the laser intensity quickly exceeds the air breakdown threshold and then an air plasma is formed. Generally, an air plasma will undergo photoionization, avalanche ionization, diffusion and recombination. In the stage of the plasma birth, the initial free electrons in this region will be generated by a multi-photon ionization process, which lead to the successive cascaded ionizations of the air such as tunneling ionization in the growth of the plasma.
Fig. 3. The reconstructed sequential images of a single femtosecond-laser-induced air plasma dynamics extracted from the SI-HSSI system. (a) Amplitude. (b) Phase. The laser beam is incident from the right-hand side. scale bars, 50 µm
In the tunneling ionization, a large number of free electrons are produced inside the plasma, thus the phase distribution of the air in this region will be changed. While the electron density within the plasma does not increase endlessly. As shown in Fig. 4, the electron density saturated at delay time of 667 fs, which means the air has been completely ionized. The phase shift diagram of air plasma also reflects its internal refractive index variation. Under the assumption that the morphology of single plasma is cylindrical symmetry, an inverse Abel transformation based on the Fourier-Hankel transform can be applied to reconstruct the refractive index distribution of the air plasma [43]. The reconstruction results of refractive index distribution at different moments are shown in Fig. 5. It can be seen that the refractive index values within the plasma are below the values of the ambient atmosphere from Fig. 5(a). This phenomenon is the result of the increase in free electrons density within the plasma, because the link between the refractive index and the free electrons density is $n = \sqrt {1 - {{{N_e}} / {{N_c}}}} $, where ${N_e}$ is the free electrons density and ${N_c}$ is the critical free electrons density at the wavelength of $\lambda $. It is clear from the relationship that free electrons exhibit negative feedback with the refractive index of the plasma. The air is further ionized as the pump laser pulse passed through the focal point, thus Fig. 5(b) shows that the refractive index in plasma kernel decreased to lower values. In a previous study [44,45], they gave refractive index distribution within the air plasma at femtosecond order, which is similar to the reconstructed results shown in Fig. 5. Thus, the results shown are in line with expectations.
Fig. 4. The profiles of phase distribution along propagation path of air plasma at different moments [marked by the black dashed in Fig. 3(b)]. scale bars, 50 µm
Fig. 5. The index of refractive distribution as a false color. (a) and (b) corresponding to the delay time are 400 fs and 667 fs, respectively. scale bars, 50 µm
The spatial resolution limit of the image reconstructed from each sub-hologram is influenced by many factors, including the wavelength, numerical aperture of each spherical illumination spot, source-sensor distance, object–sensor distance, as well as sampling parameters of the sensor. Some analysis method and basic conclusions, intrinsic to an in-line hologram recorded with spherical waves, had been reported in some published papers [46,47].
4. Conclusion
In summary, we have demonstrated the feasibility of our proposed SI-HSSI system based on a MDA. The MDA designed by integrating an array of mini-reflector and an array of micrometer lead with the mini cardan joint was used to generate multiple back-reflected sub-pulses with adjustable propagation angles and delay intervals. These propagating sub-pulses will overlap on the sample and eventually fall on different regions of the recording plane. Now a single-shot spatial-multiplexing in-line hologram can be formed and recorded by the IS. We provided a group of visualized complex amplitude sequential images of a single laser-induced air plasma with the frame rete of 7.5 Tfps and without loss of the exposure time of the original laser pulse, furthermore the distribution of the refractive index within the air plasma was revealed. In comparison with the existing intensity SSI methods [11–28], our system can accomplish SSI of complex amplitude including the amplitude and phase information of the tested sample. Compared with the existing off-axis holographic SSI [29–31], our system reaches a higher level of simplicity by the application of the digital in-line holography configuration, which does not require a reference arm. In addition, due to no significant extension of the sub-pulse duration, the time resolution of this SI-HSSI system allows us to study the dynamic processes with the femtosecond order, which is beyond the ceiling of the in-line holographic SSI system proposed by Godin’s group [33]. Although the sequence depth of our system is limited by the size of the image sensor, we believe that our method can record more frames of holographic images in a single-shot measurement, with the development of image sensor technology.
Funding
National Natural Science Foundation of China (91750105).
Acknowledgments
The authors thank Professor Chuanfu Cheng from School of Physics and Electronics, Shandong Normal University for the support in providing Ti:sapphire regenerative laser amplifier system.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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