1. Introduction

Digital holography [1,2], which digitally records a hologram and numerically reconstructs three-dimensional (3D) information of an object, has been actively researched in many fields [3–7]. The technique can record and reconstruct both amplitude and phase distributions of an object. Because the phase distribution contains 3D information of an object and is helpful for estimation of the thickness and the refractive index of a transparent object, the technique is capable of 3D measurement of an object. In particular, parallel phase-shifting digital holography (PPSDH) [8–16] is a promising technique for 3D measurement of fast moving objects with high accuracy and large field of view. In previous work, we proposed parallel phase-shifting digital holographic microscopy for instantaneous measurement of the 3D structures of living specimens [12]. We also achieved high-speed 3D imaging of gas jet by PPSDH at frame rate of 180,000 frames per second [15]. However, the temporal resolution in the previous PPSDH system for high-speed 3D imaging is limited by the shutter speed or exposure time of the high-speed camera of the system. The shutter speed of a high-speed camera is tens of nanosecond at the highest. Therefore, it is quite difficult to measure ultrafast phenomena, such as a plasma filament [17,18], structural changes of protein and living cells [19,20] and material processing by a femtosecond light pulse [21,22], by a high-speed camera. The elucidation of the mechanism of a plasma filament, for example, is helpful and significant to apply a filament to terahertz emission [23] and measurement and sensing of electric field [24]. Therefore, 3D measurement, observation, and visualization of ultrafast phenomena are quite important for elucidation of their dynamics and mechanism, and require ultra-high temporal resolution which exceeds the shutter speed of a high-speed camera. If we use an ultrashort pulsed laser as an optical source, we can obtain the ultra-high temporal resolution which corresponds to the duration of an ultrashort light pulse. However, it is quite difficult to carry out conventional sequential phase-shifting interferometry within the temporal duration of an ultrashort light pulse. Phase-shifting interferometry is powerful for high-precision 3D measurement and has been widely used in various applications [3,4,25–30]. Indeed PPSDH is feasible to carry out phase-shifting interferometry within femtosecond-order temporal duration in theory, but no PPSDH which uses a femtosecond pulsed laser as an optical source has been demonstrated yet.

In this letter, we proposed and demonstrated a single-shot femtosecond-pulsed phase-shifting digital holography to aim at achieving ultra-high temporal resolution. The proposed technique is based on PPSDH using a single femtosecond light pulse. Spark discharge in atmospheric pressure air induced by applying a high voltage to between two electrodes was recorded as a dynamically moving object.

2. Parallel phase-shifting digital holography

Figure 1 shows the image reconstruction process of PPSDH. PPSDH records a single hologram in which the information of multiple phase-shifted interference fringe images is combined by space-division multiplexing by an image sensor. In phase-shifting interferometry, three or more step phase-shifting method is frequently employed taking account of obtaining the robustness against the changes in environmental conditions. Oh the other hand, in PPSDH, the sampling interval of each phase-shifted hologram is narrowed as the reduction of the number of phase shifts [13]. As a result, the recordable space-bandwidth product of each phase-shifted interference fringe image increases. Therefore, the quality of obtained images of an object improves and the field of view increases. Thus, we adopted two-step phase-shifting method [29] in our study. Figure 1 shows the schematic flow of parallel two-step phase-shifting digital holography [10]. Pixels with each phase shift are extracted from the recorded single hologram in which the information of two phase-shifted interference fringe images is combined by space-division multiplexing. Two images that have vacant pixels are obtained after the extraction. By interpolating the vacant pixels in the obtained two images, we can generate two phase-shifted holograms I(0) and I(-π/2). On the other hand, the intensity distribution of a reference wave is recorded because it is required for two-step phase-shifting method. We can calculate the complex amplitude distribution of an object wave on the image sensor plane by applying two-step phase-shifting method to the generated holograms and the intensity distribution. Therefore, we can reconstruct only the image of the object by applying diffraction integral to the complex amplitude distribution.

 

Fig. 1 Schematic flow of parallel two-step phase-shifting digital holography.

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3. Experiment

Figures 2 and 3 show the schematic diagram of optical setup and the transitions of the polarizations and the phase shifts of the object and reference waves in the experiment. We used a mode-locked Ti:sapphire laser with a regeneration amplifier (Solstice, Spectra-Physics Inc.) to generate a single-shot femtosecond light pulse. The center wavelength and temporal duration of the light pulse were 800 nm and 96 fs, respectively. A perpendicularly polarized light pulse emitted by the laser was divided into two pulses by the first half mirror as shown in Fig. 2. The pulse reflected by the half mirror was collimated by a plano-concave lens and a collimator lens, and illuminated the object. A pulse passing thorough the object or diffracted by the object was called as an object light pulse. The object light pulse was introduced into the polarizer in order to detect only perpendicularly polarized light pulse in the object wave. On the other hand, the pulse passing through the first half mirror was introduced into the mirror with micro-moving stage, and was reflected by the mirror and introduced into the first half mirror again. After that, the pulse reflected by the half mirror was called as a reference wave. Also the reference wave was collimated by a plano-concave lens and a collimator lens, and was introduced into the quarter-wave plate. Because the fast axis of the quarter-wave plate was inclined at an angle of 45° relative to the polarization direction of the original reference pulse, the linear polarization of the reference pulse was converted to the circular polarization by the quarter-wave plate. In other words, the phase shift of the reference pulse in the fast-axis direction compared with that in the slow-axis direction was -π/2. The reference wave and the object wave were combined by the second half mirror and formed interference fringes on the image sensor of the polarization-imaging camera. The polarization-imaging camera has the micro-polarizer array on the image sensor plane as shown in Fig. 2. Each pixel of the array corresponds to each pixel of the image sensor. Because each phase shift of the reference pulse was selected by the array, a single hologram in which the information of the two phase-shifted hologram, I(0) and I(-π/2), was contained were recorded by the polarization-imaging camera as shown in Fig. 3. Because we used a pulse of 800-nm center wavelength to record a hologram, a micro-polarizer array optimized to near-infrared light was required. In addition, a micro-polarizer array which selected two polarization axes for 1 × 2 pixels was required because we adopted two-step phase-shifting method. Although Sato et al. reported a micro-polarizer array which was optimized to green light [31], no polarization-imaging camera satisfying our requirements has been developed yet. Then, we designed and developed a polarization imaging camera which has the desired micro-polarizer array made of photonic crystals as reported by Sato et al [31]. Figure 4 shows the polarization-imaging camera we developed. The number of pixels of the image sensor and the micro-polarizer array were 1360(H) × 1024(V) and 1120(H) × 868(V), respectively. Because the intensity distribution of the reference light pulse was required in two-step phase-shifting method, we recorded the intensity distribution before recording of a hologram.

 

Fig. 2 Schematic of optical setup in the experiment. M: mirror, MM: mirror with micro-moving stage, HM: half mirror, NL: plano-concave lens, CL: collimator lens, QWP: quarter-wave plate, P: polarizer.

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Fig. 3 Transitions of polarizations and phase shifts of object and reference waves in the experiment.

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Fig. 4 Photographs of polarization-imaging camera and its image sensor plane.

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We set two fine electrodes made of stainless steel facing each other. The diameter of the two electrodes and the distance between the electrodes were 1.2 mm and 1.8 mm, respectively. The electrodes were positioned 31 cm away from the camera. A 10 kV was applied to the electrodes and spark discharge was induced between the electrodes. We recorded the spark discharge in atmospheric pressure air as an object by the technique. We extracted and used 1024 × 868 pixels of the recorded hologram to obtain reconstructed images. Figure 5(a) shows a photograph of the spark discharge taken with a digital still camera, and Figs. 5(b) and 5(c) show the reconstructed phase images without and with phase-shifting method, respectively. The reconstructed images were represented by pseudocolor of 256 gradations, and the relations between pixel (or phase) values and colors were shown by the color bar in Fig. 5. The pixel values of two electrodes were set to 0. The image in Fig. 5(b) was significantly degraded and the phase changes were unclear because the zeroth-order diffraction image and the conjugate image were superimposed on the desired image of the object. On the other hand, we can observe the phases between the two electrodes changed by the spark discharge in Fig. 5(c). However, because the size of the hologram was small, the striated noises which resulted from diffraction caused by small aperture arose and the image in Fig. 5(c) was degraded. In addition, in Fig. 5(c), background noises by residual unwanted images appeared and the image quality was also degraded due to errors of calculation of phase-shifting method caused by interpolation of vacant pixels of holograms as shown in Fig. 1 and by fluctuation of the intensity distribution of the reference light pulse. Because the two-step phase-shifting method we used is sensitive to the fluctuation particularly, we acquired over 20 images of the intensity distribution of the reference light pulse before recording of a hologram in order to suppress the errors caused by the fluctuation. After the recording, we searched the most appropriate distribution among the acquired multiple images. Figure 5(d) shows the phase image reconstructed by using the most appropriate distribution. Because the errors of calculation of phase-shifting method were suppressed, we can more clearly observe the phases between the two electrodes changed by the spark discharge in Fig. 5(d) than in Fig. 5(c). Therefore, we showed the effectiveness of the PPSDH using a single femtosecond light pulse.

 

Fig. 5 (a) Photograph of object (spark discharge) taken with a digital still camera. [(b) and (c)] Reconstructed phase images without and with phase-shifting interferometry, respectively. (d) Reconstructed phase image with phase-shifting interferometry by using the most appropriate distribution of the intensity of the reference light pulse.

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Next, we aimed at calculating the change of the refractive index of air from the reconstructed phase image. The relationship between the phase change Δθ and the change of the refractive index Δn is given by the following equation:

4. Conclusion

In conclusion, we successfully demonstrated a single-shot femtosecond-pulsed parallel phase-shifting digital holography for attainment of ultra-high temporal resolution. A moment of spark discharge induced by applying a high voltage was captured and clearly imaged by the technique. The temporal resolution of obtained images depends on the temporal duration of the single femtosecond light pulse, and the temporal resolution of 96 fs is, to our knowledge, the fastest in not only phase-shifting digital holography but also in phase-shifting interferometry. The proposed PPSDH will contribute to 3D measurement of ultrafast phenomena such as a plasma filament induced by focusing femtosecond light pulses, structural changes of protein and living cells, photochemical reaction, and so on.