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Abstract
In this study, we proposed a hologram calculation method for light-in-flight recording by holography (LIF holography). First, we simulated the behavior of ultrashort pulsed light on the diffuser plate using the two-dimensional finite-difference time-domain method. Second, we calculated the light propagation from the diffuser plate to the recording material and generated a hologram based on the calculation model of LIF holography. We reconstructed moving pictures of pulsed-light propagation from the calculated hologram. Because the behavior of the pulsed light in the moving pictures agreed well with that obtained from the reported optical experiment, the proposed method was successfully validated.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The imaging, analysis, and evaluation of the behavior of the ultrashort pulsed light are critical for innovative ultrafast-photonics fields and applications [1–6]. Researchers have reported studies and techniques enabling us to record, observe, and visualize the behavior of the ultrashort pulsed light and the ultrafast phenomena induced by the ultrashort pulsed light [7–17]. Femtosecond time-resolved optical polarigraphy was proposed for imaging the ultrashort pulsed light propagating in a nonlinear optical medium [8,12]. The streak camera and the streak-camera-based techniques can visualize fluorescence and bioluminescence without a specialized active illumination [9,11]. Single-shot frequency domain tomography was proposed for imaging the transient perturbation of the refractive index in a medium [10]. Sequentially timed all-optical mapping photography can visualize light-induced plasma and phonon polariton waves [13]. Single-photon avalanche diode detectors can measure the dynamic quantum phenomena [14,17]. Time-resolved holographic polarization microscopy was proposed for imaging the laser-induced damage in a polarization-sensitive material [15]. We focused on light-in-flight recording by holography (LIF holography) [7,18–35] because of its many attractive features compared with other techniques. LIF holography can record and reconstruct the behavior of pulsed light as spatially and temporally continuous moving pictures. LIF holography records spatiotemporally continuous moving pictures using a single pulsed light, negating the need for a pulsed-light train and multiple exposures [19]. Furthermore, because this technique is based on holography that can record and reconstruct both amplitude (or intensity) and phase information of light, LIF holography enables us to obtain both two-dimensional (2D) and three-dimensional (3D) moving pictures of the behavior of pulsed light [27]. Therefore, LIF holography can potentially realize the 3D measurements of ultrafast phenomena as moving pictures, which are difficult to acquire using other techniques. For example, spatiotemporal observations of pulsed-light propagation with multiple polarization states [34] and ultrafast behavior of polarized pulsed light [35] have been recently realized using LIF holography, elucidating the polarization properties of ultrashort pulsed light. Spatiotemporal observations of polarization properties are useful for analyzing and understanding certain phenomena. In particular, some ultrafast phenomena and their behaviors induced by ultrashort pulsed light depend on the polarization state of the pulsed light [36–38], and occur in polarization-sensitive materials [39,40]. However, LIF holography has a major problem of spatiotemporal distortions of the moving pictures reconstructed from the recorded hologram [26–29,31], causing the moving pictures to incorrectly express the behavior of pulsed light. Therefore, numerical simulation models of LIF holography are useful for compensating the spatiotemporal distortions and analyzing the spatiotemporal behaviors of light obtained by experimental results. However, conventional simulation models [31] are based only on ray tracing, which is unsuitable for treating not only the temporal information of light, but also the complex behaviors of light in materials. Moreover, because no hologram calculation of LIF holography has been performed based on conventional simulation models, it is unclear which moving pictures could be obtained by holographic reconstruction. Therefore, LIF holography needs a numerical simulation model that can treat the holographic recording and the spatiotemporal behavior reconstruction of light in complex situations and various materials.
In the present study, we propose a hologram calculation method for LIF holography. We use the finite-difference time-domain (FDTD) method to treat both the spatial and temporal information of light easier than the conventional simulation models of LIF holography [41–48]. The FDTD method can simulate the light behavior considering both the spatial and temporal information in a scattering medium [44]. We can obtain the polarization state of light by computing the xyz components of the electromagnetic field using the FDTD method [46]. Moreover, the FDTD method can simulate nonlinear phenomena by incorporating dispersive second-order nonlinear susceptibility tensor elements [47,48]. Therefore, the proposed method can treat complex situations such as light propagation in volumetric scattering medium and photonic crystals and observe polarization-state changes, which are challenging for conventional simulation models to treat. We generate holograms using the proposed method and reconstruct moving pictures from them to validate the proposed method. The proposed method is the first step toward simulating ultrashort pulsed light behavior and compensating spatiotemporal distortions in moving pictures obtained using LIF holography.
2. Light-in-flight recording by holography (LIF holography)
2.1 Recording process
The schematic diagram of the top view of a basic optical setup for the recording process of LIF holography is shown in Fig. 1(a). The beam splitter divides the pulsed light emitted from the ultrashort pulsed laser into two. The pulsed light reflected by the beam splitter is collimated by the concave and collimator lenses and obliquely introduced into the diffuser plate against the normal of the diffuser plate. The diffuser plate then scatters the pulsed light generated from the cross-section between the ultrashort pulsed light and the diffuser plate. Because the scattered pulsed light is used as the object light, this setup records the behavior of the ultrashort pulsed light propagating on the diffuser plate [29]. The pulsed light passing through the beam splitter is used as the reference light, illuminating the recording material such as a photosensitive material or an image sensor after being collimated by the concave and collimator lenses. Here the incident angle of the reference light is inclined from the normal of the recording material. Because of the inclination of the reference light, the moment when the reference light illuminates the recording material is different, depending on the horizontal position of the recording material. An ultrashort pulsed light has low coherence, forming interference fringes on the recording material only when the object and reference pulsed lights simultaneously arrive at the same position of the recording material. Therefore, LIF holography can record temporal information of the object light by converting it into spatial information of the hologram.
Fig. 1. Schematic diagram of top view of basic optical setup for light-in-flight recording by holography. (a) Recording process. (b) Reconstruction process. BS: beam splitter; M: mirror; NL: concave lens, and CL: collimator lens.
2.2 Reconstruction process
The schematic diagram of the top view of a basic optical setup for the reconstruction process of LIF holography is shown in Fig. 1(b). Holography needs the reconstruction process to obtain holographic images from the hologram recorded in the recording process; thus, LIF holography also requires the reconstruction process to obtain moving pictures. Instead of an ultrashort pulsed laser, a continuous-wave laser is used as the optical source. The light emitted from the laser illuminates the hologram at the same angle of the reference light in the recording process. The illumination light is diffracted by the hologram, and the holographic images are then reconstructed from the hologram. Here, different temporal information of the object light is recorded at different positions horizontally along the hologram. We can obtain moving pictures from the behavior of the object light from the hologram by positioning the observation point close to the hologram and moving the gazing point horizontally along the hologram [29]. The moving pictures to be reconstructed are spatially and temporally continuous, which is an attractive feature of LIF holography.
3. Hologram calculation for LIF holography using the 2D-FDTD method
Figure 2 shows a flowchart of the procedure of the proposed method for hologram calculation of LIF holography. Each parameter used in Fig. 2 will be presented in detail later. The procedure is roughly divided into two parts: the 2D-FDTD and ray-tracing calculations. Because the FDTD method can simulate the behavior of light considering both spatial and temporal information, it is the most desired solution to generate a hologram for LIF holography. However, because the FDTD method is both time-consuming and a memory-hogging approach, we cannot directly calculate a hologram for LIF holography using commercial computers. For example, in the situation that will be described below, the computational amount for the hologram calculation using only the FDTD method is at least $10^4$ times larger than that using the ray-tracing-based method. Therefore, we applied the 2D-FDTD method to only simulate the behavior of pulsed light on the diffuser plate and calculated light propagation from the diffuser plate to the recording material using the ray-tracing-based calculation model [31].
Fig. 2. Flowchart of the procedure of the proposed method for the hologram calculation of light-in-flight recording by holography. $t_R$ and $\tau _R$ denote the times at which the reference and object lights, respectively, arrive at an arbitrary point $\mathrm {R}$ on the recording material. Each parameter will be presented in detail.
3.1 2D-FDTD calculation
The brief situation of 2D-FDTD calculation for simulating the behavior of the object light on the diffuser plate is shown in Fig. 3. We defined the center of the diffuser plate as the origin of the $xyz$ coordinates. We simplify the 2D-FDTD calculation by not considering the thick $z$-axis and loss of light intensity by scattering on the diffuser plate. Therefore, the electromagnetic field to be calculated is uniform along the $z$-axis, and we examined the 2D-FDTD calculation in the $xy$ plane. We assumed that the electromagnetic field has only $H_x$, $H_y$, and $E_z$ components, where $H_x$ and $H_y$ denote the $x$ and $y$ components of the magnetic field and $E_z$ denotes the $z$ component of the electric field. We set the resolution of the FDTD calculation region as $X_D \times Y_D$ cells. The spatial interval of each cell along the $x$- and $y$-axes, $\Delta x$ and $\Delta y$, was defined as $\lambda /10$, where $\lambda$ is the center wavelength of the pulsed light. We defined the time interval, $\Delta t$, as $\Delta t = \Delta x/(\sqrt {2}c) = \Delta y/(\sqrt {2}c) = \lambda /(10\sqrt {2}c)$. Here, $c$ is the speed of light in the vacuum. We prevent light from reflecting on the boundary of the calculation region by applying the Mur’s absorbing boundary condition with a second-order approximation [49]. We set light sources near the edge of the calculation region to generate a pulsed light with a Gaussian distribution [50]. The light sources, which were uniform along the $z$-axis, were aligned (Fig. 3). The $z$ component of the electric field generated from each light source, $E_{pls}(x, y; N)$, was given by
Here, the $x$ and $y$ coordinates of the center of the aligned light sources denote $x_{pls}$ and $y_{pls}$. $N$ represents the time step in the FDTD calculation. $w_r$ is the spatial full width at half maximum (FWHM) of the pulsed light along the aligned direction. The temporal FWHM of the pulsed light denotes $w_t$. $\theta _{pls}$ defines the angle of the aligned direction from the $y$-axis. The counterclockwise direction was selected as the positive rotation angle of $\theta _{pls}$. Each light source can be regarded as the point-light source in the $xy$ plane; hence, the pulsed light to be generated from the light sources propagates to the $\theta _{pls} \pm 90^\circ$ directions according to the Huygens–Fresnel principle.
As a computational environment, we used Microsoft Windows 10 Enterprise as the operating system, Microsoft Visual Studio Community 2019, CUDA Toolkit 10.2, a CPU (Intel Core i7-8700K with 3.7 GHz), and two GPUs (GeForce RTX 2080 SUPER). We assumed the total internal reflection in the glass substrate as the target situation to be recorded by LIF holography according to [25]. The schematic diagram of the calculation region of the 2D-FDTD calculation is shown in Fig. 4(a). We set the parameters for the 2D-FDTD calculation as $X_D \times Y_D = 81,920 \times 10,240 \,\mathrm {cells}$, $\lambda = 522 \,\mathrm {nm}$, $w_r = 100 \;\mu \mathrm {m}$, $w_t = 200 \,\mathrm {fs}$, and $M_M = 10$ by referring to previous optical experiments in the literature using LIF holography [33–35]. The variable $M_M$, denoting the magnification factor of the calculation region, will be discussed in detail hereinafter. A medium with a refractive index of 1.5 was positioned in the calculation region as the glass substrate. The refractive index, except for the glass substrate, was 1.0. We set $\left ( x_{pls},y_{pls} \right ) = \left ( -38,192\Delta x, 3,072\Delta y \right )$ and $\theta _{pls} = 30^\circ$. The refractive index distribution of the calculation region is shown in Fig. 4(b). Nine examples of the results obtained from the 2D-FDTD calculation are shown in Figs. 4(c)–(k). Figures 4(l)–(t) show enlarged pictures of the part surrounded by the red frame in Figs. 4(c)–(k), respectively. The dashed line in Figs. 4(e) and (n) indicates the boundary between the glass substrate and the free space. These pictures show the intensity distributions of the electric field obtained from the 2D-FDTD calculation. First, two pulsed lights were generated from the aligned light sources. One propagated to the bottom left and disappeared because of the absorption at the edge of the calculation region. Another propagated to the upper right and was introduced into the glass substrate. The edge of the glass substrate refracted most of the pulsed light and reflected part of the pulsed light. The pulsed light to be reflected propagated to the upper-left and disappeared. The pulsed light to be refracted traveled inside the glass substrate with the zig-zag path because of the total internal reflection at the boundary between the air and the glass substrate.
Fig. 4. Results of the 2D-FDTD calculation of total internal reflection. (a) Schematic diagram of calculation region. (b) Refractive index distribution of calculation region. (c)–(k) Nine examples of intensity distributions of the $z$ component of the electric field (see Visualization 1). (l)–(t) Enlarged pictures of the part surrounded by the red frame in (c)–(k), respectively. The dashed line in (e) and (n) indicates the boundary between the glass substrate and the free space.
3.2 Hologram calculation
The complex amplitude distributions formed by the object light at the recording material are necessary for the hologram calculation; therefore, we calculated the complex amplitude distributions using the results of the 2D-FDTD calculation based on the ray-tracing-based calculation model [31]. The schematic diagram of the hologram calculation model is shown in Fig. 5. For simplicity, we assumed that the recording material was set parallel to the diffuser plate, the pulse front of the reference light was normal to the propagation direction, and the incident angle of the reference light was oblique along only the $x$-axis (Fig. 5(a)). By these assumptions, the cross-section between the recording material and the reference light was linear along the $y$-axis.
Fig. 5. Schematic diagram of hologram calculation model. (a) Top view. (b)–(d) Bird’s eye view of the temporal series scenes at (b) $t = \tau _{D}$, (c) $t = t_{R_O}$, and (d) $t = t_R = \tau _{R}$.
First, we consider the behavior of the object light (Fig. 5(b)). In the present study, we expressed the behavior using the results of the 2D-FDTD calculation. We assumed that the intensity of the object light generated from the diffuser plate was proportional to the square of the $z$ component of the electric field obtained from the 2D-FDTD calculation. Then, the complex amplitude distribution of the object light at $\mathrm {D}\left ( x_D, y_D, 0 \right )$ at $t = N\Delta t \equiv \tau _{D}$ was obtained from
We then consider the behavior of the reference light on the recording material that was positioned at $z = z_R$ (Fig. 5(a)). The number of pixels consisting of the recording material was set as $X_R \times Y_R$. We defined $t_{R_O}$ as the time when the reference light arrived at $\mathrm {R_O} \left ( 0, 0, z_R \right )$, corresponding to the center of the recording material (Fig. 5(c)). We needed to calculate $t_{R_O}$ in advance because $t_{R_O}$ means the standard time for hologram calculation. We also defined $t_R$ as the time when the reference light arrived at $\mathrm {R} \left ( x_R, y_R, z_R \right )$, which was an arbitrary point on the recording material (Fig. 5(d)). Because the cross-sectional pattern of the reference light was linear along the $y$-axis, the time when the reference light arrived at $(0, y_R, z_R)$ was $t_{R_O}$. Then, using $t_{R_O}$, $t_R$ was obtained from
Here, $\theta _{R}$ is the incident angle of the reference light along the $x$-axis from the normal of the recording material (Fig. 5(a)).
We again focused on the behavior of the object light. We considered $L_{DR}$, showing the optical path length between $\mathrm {D}$ and $\mathrm {R}$. $L_{DR}$ was obtained from
Then, $\tau _{R}$ is the time when the object light from $\mathrm {D}$ arrived at $\mathrm {R}$, obtained from
If $\tau _{R} = t_R$, the reference light and the object light from $\mathrm {D}$ reached $\mathrm {R}$ at $t = \tau _{R}$ simultaneously, and both lights formed interference fringes. Because the pulsed light had the temporal width or coherence length in fact, the object light satisfying the following inequality was also recordable.
where $T_{th}$ was calculated according to the temporal width or coherence length. The complex amplitude distribution, formed by the object light scattered from $\mathrm {D}$ at $\tau _{D}$, at $\mathrm {R}$ was expressed byBy accumulating $u \left ( x_R, y_R, z_R; \tau _{R} \right )$ satisfying Eq. (9) while changing $N$, we could calculate $U(x_R, y_R, z_R)$, showing that the aggregate of the object light was recordable at $\mathrm {R}$. Finally, we generated the hologram pattern at $\mathrm {R}$ as
Here, the operator $\arg [C]$ is the argument of a complex number $C$. The kinoform was adopted as the phase modulation-type hologram [51]. After computing $H(x_R, y_R, z_R)$ modulo $2\pi$, we quantized its results to 256 levels. Applying Eq. (10) to all pixels of the recording material, we could generate the hologram of LIF holography.
4. Results and discussion
Based on the results from the 2D-FDTD calculation, we generated a hologram. We set each parameter for hologram calculation as $X_R \times Y_R = 163,840 \times 8,192$ pixels, $t_{R_O} = 94,000 \Delta t$, $z_R = 150 \,\mathrm {mm}$, $\theta _{R} = 45^\circ$, $m_D = 1/64$, and $T_{th} = 2w_t$. The pixel pitch of the recording material (or hologram) was $0.3\;\mu \textrm {m}$. We used each 100th frame from the 2D-FDTD results for hologram calculation to reduce the computational amount required. The total computational time of the 2D-FDTD and hologram calculations was approximately 150 h. We validate the proposed method by numerically reconstructing the generated hologram. The reconstruction of the hologram of LIF holography to obtain the moving pictures of the behavior of the object light is shown in Fig. 6 [31]. First, we extracted sub-holograms along the $x$-axis. After the extraction, we applied zero-padding to the sub-hologram. The resolutions of the sub-hologram and zero-padding hologram were $1,024 \times 8,192$ and $16,384 \times 16,384$ pixels, respectively. Next, we reconstructed the zero-padding sub-holograms. We adopted shifted Fresnel diffraction [52,53] that can vary the pixel pitch of the reconstructed image plane as the diffraction calculation algorithm. We set the pixel pitch of the reconstructed image plane three times larger than the hologram plane. After the reconstruction, we arranged each reconstructed image while maintaining the spatial relationship among the reconstructed images because the position of each sub-hologram shifted along the $x$-axis. The entire resolution after the arrangement was $70,656 \times 16,384$ pixels. Nine pictures extracted from the moving pictures are shown in Figs. 7(a)–(i). The reconstruction process per sub-hologram, including the outputting of each reconstructed image as the image file, took several seconds. First, one pulsed light beam propagated to the bottom left, and the other beam propagated to the upper right. Because the latter beam arrived at the edge of the glass substrate, it was mostly refracted, and a minor part was reflected. The refracted light traveled along a zig-zag path. These behaviors well agreed with the results of the 2D-FDTD calculation, thereby validating the proposed method. Similar behaviors were also reported in [25]. The time shown in Fig. 7 indicates the relative value with that of the image reconstructed from the left-end sub-hologram as 0 s. Because we applied the magnification factor $M_M = 10$ to the results of the 2D-FDTD calculation, the time shown in Fig. 7 was larger than that in Fig. 4. However, we can also see the enlargement degree of time does not correspond to $M_M$. This resulted from the spatiotemporal distortions to be caused by the principle of LIF holography [26–29,31].
Fig. 6. Reconstruction manner for the hologram of LIF holography. First, sub-holograms were extracted from the original hologram along the $x$-axis. The extracted sub-holograms were padded with zeros, and the reconstructed images were obtained from the zero-padded sub-holograms. Each reconstructed image was arranged while maintaining the spatial relationship among the reconstructed images.
Fig. 7. Reconstructed moving pictures of pulsed-light propagation with total internal reflection (see Visualization 2). The time in each figure indicates the relative value with that of the image reconstructed from the left-end sub-hologram as 0 s.
These pictures are the intensity of the complex amplitude distributions to be reconstructed. First, the pulse width was significantly enlarged in the reconstructed images compared with that of the 2D-FDTD results, mostly because of temporal blurring in LIF holography. As described in 2.1, LIF holography records the temporal information of the object light as spatial information on the recording material, showing that the temporal resolution in LIF holography depends on the size of the sub-hologram along the $x$-axis [31,32]. Therefore, it is ideal for the horizontal size of the sub-holograms to be the same as the cross-sectional width between the recording material and reference light. Because the temporal width of the reference light was 200 fs, the cross-sectional width was estimated as $85\;\mu \textrm {m}$. However, the horizontal size of the sub-hologram was 2.5 mm, which was much larger than the cross-sectional width of the reference light. Consequently, the pulse width in the reconstructed moving pictures was significantly enlarged. In other words, the reconstructed moving pictures were temporally blurred. We ensured the effect of temporal blurring by also reconstructing the moving pictures from the sub-holograms with different horizontal resolutions. The reconstructed images from the sub-holograms with $256 \times 8,192$, $512 \times 8,192$, $1,024 \times 8,192$, $2,048 \times 8,192$, $4,096 \times 8,192$, and $8,192 \times 8,192$ pixels are shown in Fig. 8. Here, the zero-padding hologram’s resolution was $16,384 \times 16,384$ pixels. The time of each figure illustrated in Fig. 8 correspond to that in Fig. 7(d). Although it was expected that the pulse width shortened inversely proportional to the horizontal size of the sub-hologram, the case of $1,024 \times 8,192$ pixels had the shortest width among the six conditions, resulting from blurring because of diffraction. The actual sizes of the sub-holograms with $256 \times 8,192$ and $512 \times 8,192$ pixels were 77 and $154\;\mu \textrm {m}$ along the $x$-axis, respectively. Because the hologram size corresponds to the numerical aperture, the spatial resolution increases proportionally to the hologram size in the reconstruction process. The blurring due to diffraction also reduces when the hologram size increases. In the case of $256 \times 8,192$ and $512 \times 8,192$ pixels, blurring due to diffraction was more dominant than the temporal blurring. Conversely, temporal blurring was dominant in the case of $2,048 \times 8,192$, $4,096 \times 8,192$, and $8,192 \times 8,192$ pixels. In other words, the spatial and temporal resolutions are trade-off in LIF holography. The relationship between them was numerically detailed in [31].
Fig. 8. Reconstructed images from sub-holograms with different horizontal resolutions. The time of each figure correspond to that in Fig. 7(d).
As another example of a more complex situation, we assumed a medium with a graded-index distribution in the calculation region (Fig. 9(a)). Its refractive index was linearly changed from 1.0 to 1.5 along the $y$-axis. It is challenging for the conventional simulation model [31] based on only ray tracing to treat this situation because LIF holography should consider the temporal information of the object light. Conversely, the proposed method could treat the situation because it applied the FDTD method to the calculation model. We set each parameter for the 2D-FDTD calculation as $X_D \times Y_D = 81,920 \times 10,240$ cells, $\lambda = 522 \,\mathrm {nm}$, $w_r = 100\;\mu \textrm {m}$ fs, and $M_M = 10$. We set $(x_{pls}, y_{pls}) = (-39,936 \Delta x, -2,560 \Delta y)$ and $\theta _{pls} = 0^\circ$. The refractive index distribution of the calculation region is shown in Fig. 9(b). Twelve examples of the results obtained from the 2D-FDTD calculation are shown in Figs. 9(c)–(n). The pulsed light first propagated along the $x$-axis before the incident into the graded-index medium. In the graded-index medium, the pulsed light gradually traveled to the bottom-right and was reflected by the bottom edge of the graded-index medium because of the total internal reflection. Although the pulsed light to be reflected once traveled to the upper right, its propagation direction gradually changed downward and is again reflected by the bottom edge. We set each parameter for hologram calculation as $X_R \times Y_R = 140,800 \times 8,192$ pixels, $t_{R_O} = 80,800\Delta t$, $z_R = 150$ mm, $\theta _R = 45^\circ$, $m_D = 1/64$, and $T_{th} = 2w_t$. The pixel pitch of the hologram was $0.3\;\mu \textrm {m}$. We used each 100th frame of the 2D-FDTD results for hologram calculation to reduce the computational amount required. Twelve examples extracted from the moving pictures to be obtained from the hologram are shown in Fig. 10. Here, the resolutions of the sub-hologram and zero-padding hologram were $1,024 \times 8,192$ and $16,384 \times 16,384$ pixels, respectively, and the pixel pitch of the reconstructed image plane was three times larger than the hologram plane. The time shown in Fig. 10 indicates the relative value with that of the image reconstructed from the left-end sub-hologram as 0 s. As with the case of total internal reflection, we could obtain the moving pictures of propagation of the pulsed light in the graded-index medium. The behavior of the pulsed light in the moving pictures to be reconstructed also agreed well with that of the results from the 2D-FDTD calculation.
Fig. 9. Results of 2D-FDTD calculation using graded-index distribution. (a) Schematic diagram of calculation region. (b) Refractive index distribution of calculation region. (c)–(n) Twelve examples of intensity distributions of $z$ component of electric field (see Visualization 3). (o)–(z) Enlarged pictures of the part surrounded by the red frame in (c)–(n), respectively. The dashed line in (d) and (p) indicates the boundary between the graded-index medium and the free space.
Fig. 10. Reconstructed moving pictures of pulsed-light propagation in the graded-index medium (see Visualization 4). The time in each figure indicates the relative value with that of the image reconstructed from the left-end sub-hologram as 0 s.
The diffuser plate was used as the scattering medium to obtain the object light using Fig. 1. In this case, the holographic images to be reconstructed as the moving pictures have only 2D information because the object light generated from the cross-section between the ultrashort pulsed light and the diffuser plate was recorded [29]. As described earlier, LIF holography can record and reconstruct 3D moving pictures. We can obtain 3D moving pictures of light propagation both in 3D volume using a volumetric scattering medium [27]. By adjusting the optical setup, we can also obtain the moving pictures of light propagation in various situations such as refraction, reflection, and diffraction [25]. Moreover, LIF holography can realize the observation of light propagation in the microscopic area [28] and multiple polarization states [34].
5. Conclusions
We proposed a hologram calculation method for LIF holography. The proposed method applied the 2D-FDTD method to simulate the behavior of pulsed light on the diffuser plate. We assumed the total internal reflection as the target situation to be recorded by LIF holography and calculated its hologram using the results from the 2D-FDTD method. The pulsed-light behavior in the moving pictures reconstructed from the calculated hologram agreed well with both the results from the 2D-FDTD calculation and those obtained from the optical experiment. We also calculated a hologram of pulsed-light propagation in the graded-index medium as a more complex situation and successfully reconstructed the behavior of the pulsed-light propagation as the moving pictures. In future research, the pulsed-light propagation will be simulated in a volume medium [27] with multiple polarization states [34], thereby enabling a more detailed analysis of the present experimental results. Based on the results of our proposed method, we must also compensate the spatiotemporal distortions of moving pictures reconstructed from an optically recorded hologram.
Funding
Kenjiro Takayanagi Foundation; Shimadzu Science Foundation.
Disclosures
The authors declare no conflicts of interest.
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