Holographic data storage (HDS), in which both the amplitude and the phase of a signal beam are modulated, has been extensively studied with the goal of increasing its storage capacity. To detect such modulation during data retrieval, it is necessary to acquire the complex amplitude of the signal beam. In this study, we focus on the transport of intensity equation (TIE) method, which allows us to detect the phase distribution of the light wave quantitatively without using interferometry, contributing to miniaturization of the optical system and improvement of the vibration tolerance of HDS. We discuss the conditions of the modulation phase distribution of the signal beam required for accurate phase detection and propose a method to estimate and eliminate the noise that frequently appears in the phase distribution detected by the TIE method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Holographic data storage (HDS) technologies are expected to see continued development to provide storage for data archiving while accommodating the rapid increase in the amount of information being handled by recent information communication technologies [1–4]. In a conventional HDS system, the intensity distribution of a signal beam is two-dimensionally modulated into bright and dark binary values, and the interference fringe patterns of the signal beam and a reference beam are recorded as a hologram in a photosensitive recording material. Holograms are multiplexed in the same volume of the holographic material to achieve large storage densities and capacities. Furthermore, high-speed data retrieval is realized by simultaneously acquiring the two-dimensionally arrayed data symbols in the signal beam using an image sensor. Today, HDS is required to have larger recording capacities, faster retrieval speeds, and higher transfer rates. A technique called spatial quadrature amplitude modulation (SQAM), in which not only the intensity distribution but also the phase distribution of the signal beam are spatially modulated, has been actively studied to satisfy these requirements [5, 6]. SQAM is intended to improve the storage capacity further by increasing the amount of information per symbol by using multilevel spatial intensity modulation and multilevel spatial phase modulation of the signal beam in combination. SQAM has been investigated from various viewpoints, including generation and detection methods of the SQAM signal beam, and the optical system for hologram recording and retrieval [7–10].
To detect the SQAM signal beam modulation, it is necessary to acquire both the spatial intensity distribution and the spatial phase distribution of the beam, i.e., the complex amplitude of the signal beam. Interferometric methods, such as phase-shifting or Fourier fringe analysis, have often been used [11, 12]. While interferometric measurement techniques can realize accurate detection of the complex amplitude of the signal beam by using the interference of the signal beam and a reference beam, they may result in increased size and complication of the HDS optical system, causing deterioration of vibration tolerance. To avoid these problems, this study proposes the use of the transport of intensity equation (TIE) method for non-interferometric detection of SQAM signal beams. Using the TIE method, the spatial phase distribution of the light wave can be measured merely from the change in spatial intensity distribution due to its propagation [13, 14]. In addition, the TIE method can be applied not only to measurement of coherent light but also to that of partially coherent light. Because of these features, the TIE method has been studied for a wide range of fields such as X-ray imaging and microscopy [15, 16]. A unique optical system for HDS, in which a computer-generated hologram is used to generate a phase-modulated signal beam and the beam is detected by the TIE method, was also proposed . The TIE method allows accurate and vibration-tolerant detection of the SQAM signal beam because it uses a simple and compact optical design that excludes the need for reference beam used in interferometry. In this paper, we discuss various problems that may occur in detection of the complex amplitude of the SQAM signal beam in the HDS system. We propose their solutions and prove their effectiveness by numerical simulations and optical experiments.
In the next section, the TIE method for measuring the phase distribution of light waves and its application to complex amplitude detection of the SQAM signal beam in HDS are described in detail. In Section 3, a numerical simulation reveals the conditions of the modulation phase distribution of the SQAM signal beam required for accurately detecting the complex amplitude of the beam. A method to estimate and eliminate the spatial noise that typically appears in the phase distribution detected by the TIE method is also suggested and investigated. In Section 4, the detection of the complex amplitude of the SQAM signal beam holographically recorded and reconstructed from a photopolymer material is experimentally demonstrated, showing that accurate non-interferometric detection of the SQAM signal beam can be realized by the TIE method. In the final section, the conclusions of this study are presented.
2. Detection of the complex amplitude of the SQAM signal beam using the TIE method
Figure 1 shows the conceptual diagrams of (a) recording of the SQAM signal beam in HDS and (b) its holographic reconstruction and detection using the TIE method. The SQAM signal beam can be generated by various methods, such as a cascaded configuration of a spatial light intensity modulator and a spatial light phase modulator, a parallel arrangement of two spatial light phase modulators, or a computer-generated hologram [18–22]. Multilevel modulation of both the intensity and the phase of the signal beam at each symbol position, which is two-dimensionally arranged in the signal beam, increases the amount of information transmitted per symbol and contributes to the improvement of recording capacity compared to conventional binary intensity modulation. The complex amplitude of the SQAM signal beam on the SLM plane is relayed to the camera plane by using a 4-f imaging system [23, 24]. The interference pattern between the SQAM signal beam and the reference beam is recorded as a hologram in the holographic material, and can be multiplexed in the same volume within the material to increase the storage density of the material. During data reproduction, the SQAM signal beam is reconstructed by irradiating the recorded hologram with only the reference beam. Both the intensity and the phase distributions of the reproduced signal beam then have to be detected simultaneously, and we propose applying the TIE method to the detection. Equation (1) is the TIE in the coordinate system described in Fig. 1(b), as follows.13]. This equation can be solved as Eq. (2) by using the Fourier transform at z = 0 : Eq. (2), the phase distribution of the signal beam can be obtained without dependence on interferometry. Here, in our numerical simulation and the optical experiment below, the signal beam reconstructed from the hologram is assumed to be in focus at the z = 0 plane.
There are several reports that use only two intensity distributions for the TIE method to implement the measurement of the phase distribution quickly, although there is a possibility that the phase measurement accuracy may be slightly reduced [26–28]. TIE methods in which the signal beam intensity distributions are captured at more axial positions are also reported to improve the accuracy of the approximated , consequently improving the accuracy of the phase measurement by the TIE method [29–32].
3. Numerical simulation of SQAM signal beam detection by the TIE method
In this section, the detection of the complex amplitude of the SQAM signal beam using the TIE method is examined by numerical simulation. Subsection 3.1 clarifies the modulation phase distribution suitable for accurate detection of the SQAM signal beam with the TIE method. Subsection 3.2 proposes and examines a technique for estimating and eliminating low spatial frequency noise, which frequently appears on the phase distribution detected by the TIE method. In the simulations below, is set at 1mm. The propagation of the signal beam is calculated by the angular spectrum method .
3.1. Modulation phase distribution of a SQAM signal beam suitable for accurate phase detection by the TIE method
Figures 2(a) and 2(b) show the spatial intensity and the phase distributions of a signal beam typically used in conventional SQAM, where the number of symbols has been reduced to 10 × 10 so that the phase distribution can be easily understood. In Fig. 2(b), α is the maximum phase modulation depth, and fixed values are specified in each of the following calculations. The intensity and phase of the signal beam at each symbol position arranged two-dimensionally have values of multiple levels, and the phase between adjacent symbols is changed by discrete amounts owing to the step-type phase modulation. Figures 2(c) and 2(d) show the spatial phase distributions when the SQAM signal beam is detected by the TIE method. Here, the number of the intensity modulation levels is two, the number of phase modulation levels is four (8SQAM), and the maximum phase modulation depth (α) of the modulated SQAM signal beam is set to 90 for Fig. 2(c) and 360 for Fig. 2(d). Figures 2(e) and 2(f) show the phase distributions of the signal beam on the positions denoted by the red lines in Figs. 2(c) and 2(d), respectively. From these simulation results, it is evident that the difference between the modulated phase and the detected phase increases and the detection accuracy deteriorates as α increases when the discretely modulated phase of the signal beam is detected by the TIE method. Figure 2(g) shows the constellation diagram of the 10 × 10 = 100 symbols in Fig. 2(d), where each symbol is composed of 50 × 50 pixels. Here, the average values of the 2 × 2 pixels at the center of the symbol are used as the values of the amplitude and the phase of each symbol in the constellation diagram obtained by the numerical simulations in this study. In this figure, the horizontal and vertical axes represent the real and imaginary parts of the complex value of each symbol. It is impossible to separate and recognize the eight symbol groups of 8SQAM and, in this case, it is difficult to correctly detect the SQAM signal without error.
On the other hand, the accuracy of the detected phase differs greatly from the results mentioned above when the phase of the signal beam is modulated so that the phase value between adjacent symbols changes continuously. Figure 3(a) shows an example of the continuously modulated phase distribution of the SQAM signal beam. Here, a pyramid type phase distribution is adopted as one of the most basic continuous phase modulations. The phase distribution of the signal beam detected by the TIE method is shown in Fig. 3(b), where the modulation intensity is the same as that in Fig. 2(a). Figure 3(c) shows the phase distributions of the signal beam on the positions denoted by the red lines in Fig. 3(b). From these results, it can be seen that the detected phase substantially coincides with the modulation phase even when the maximum phase modulation depth (α) is 360. Figures 3(d)–3(f) also show the simulation results when the sinusoidal modulation of the signal beam phase is adopted. Here, for instance, the phase of a symbol whose modulation phase value is equal to the maximum phase modulation depth is modulated to the distribution of in the sinusoidal modulation, where are coordinates in each symbol whose origin is at the symbol center, and W is the symbol width. For the symbol whose modulation phase value is smaller than α, the value of α in the above expression is replaced by the modulation phase value of the symbol. From these results, it can also be seen that the phase of the signal beam can be accurately detected by the TIE method when the phase of the signal beam within the symbol is modulated in a sinusoidal waveform such that the phase value becomes zero at the boundary of the symbols. Figures 3(g) and 3(h) show the constellation diagrams of the symbols in Figs. 3(b) and 3(e), respectively. It is possible to separate and recognize the eight symbol groups of 8SQAM completely, and it is understood that the phase of the signal beam can be detected with high accuracy.
Figure 4 shows the root mean square error (RMSE) of the detected phase as a function of the maximum phase modulation depth (α) when the discrete, pyramidal, and sinusoidal modulations of the signal beam phase are used. Here, the number of symbols has been increased to 32 × 32 in order to obtain calculation results with parameters closer to practical conditions. The RMSE is defined as
Here Nx and Ny are the numbers of pixels along the x and y directions within the region where the SQAM symbols are two-dimensionally arrayed in the signal beam, and and indicate the modulation and the detection phases at the th pixel within the region. The smaller the value of RMSE, the closer the detected phase is to the modulation phase. Because the phase value detected by the TIE method may contain an offset phase, the spatially constant offset phase (which minimizes RMSE) is subtracted to eliminate this influence. From Fig. 4, the RMSE for the discrete modulation phase rapidly deteriorates as the maximum phase modulation depth (α) increases, whereas the increase in RMSE is small even if α increases when pyramidal and sinusoidal phase modulations are adopted. Therefore, the continuous phase modulation is effective for improving the accuracy of the SQAM signal detection by the TIE method.
As explained in the references [28, 34, 35], the phase distribution obtained by the TIE method needs to be a continuous and smooth (twice differentiable) function. The discrete phase distribution shown in Fig. 2(b) obviously deviates from this condition, and the degree of the deviation is considered to become worse as the maximum phase modulation depth (α) increases. We believe that this deviation causes the deterioration of the phase detection accuracy. On the other hand, it seems that the two continuous phase modulation methods introduced in this paper alleviate this deviation and effectively improve the phase detection accuracy. Here, it should also be noted that the TIE is derived under the paraxial approximation. In the simulation in this section, the condition for the approximation is satisfied because where and are the maximum spatial frequencies along the x and y directions treated in the simulation, respectively. The validity of the paraxial approximation in the experiment was confirmed in a similar manner. The TIE is also derived under Teague’s assumption . We confirmed that the combinations of the intensity and the phase distributions of the SQAM signal beam used in this paper satisfy Teague’s assumption in most cases. Although has non-zero values at the corners of the symbols only in the case of the combination of the step-type intensity and the step-type phase distributions, we numerically confirmed that its effect on the RMSE is small.
3.2. Technique to eliminate the low spatial frequency noise on the detected phase distribution
The simulation of Subsection 3.1 does not take into consideration any noise that exists in the three diffracted intensity distribution images of the SQAM signal beam used for the phase calculation of the TIE method. In the actual experimental circumstances, however, electrical noise of the camera and optical noise originating from scattered light exist. It is known that these noise sources frequently cause additional cloud-like phase pattern with low spatial frequency on the phase distribution obtained by the TIE method [36–38]. In the simulation in this subsection, Gaussian noise is assumed to be superimposed on the intensity distribution images of the SQAM signal beam, and a method to estimate and eliminate the resultant cloud-like noise on the detected phase is proposed and investigated. In the following, pseudo-random Gaussian noise with zero mean and standard deviation equal to 2.5%
of the noise-free intensity at each pixel is independently added to the three diffraction intensity distribution images. It should also be noted that the phase within each symbol is modulated in a pyramidal distribution in the following simulation.
The left image in Fig. 5(a) shows an example of the phase distribution of the SQAM signal beam detected by the TIE method when the Gaussian noise is independently added to the three diffraction intensity distribution images of the signal beam. The right graph in Fig. 5(a) represents the phase value on the red line in the phase map image. Here, the phase of the SQAM signal beam is modulated in four levels. It can be seen that the cloud-like noise is superimposed on the phase distribution. In the constellation diagram shown in Fig. 5(b), it is impossible to identify the symbol groups separately. This noise is considered to originate from the low-frequency component of the Gaussian noise added to the diffracted intensity distribution images used in the TIE calculation. This low-frequency component is amplified by multiplication by in Eq. (2) in the spatial frequency domain, leading, therefore, to the cloud-like noise that has a spatially slow phase change compared with the phase variation in the SQAM modulation in the space domain [36, 39]. We focus on this feature of the noise and propose a method to estimate and eliminate the cloud-like noise.
In the following paragraphs, we describe a method for estimating the cloud-like noise by spatial linear interpolation and eliminating the noise by subtracting the noise estimates from the detected phase. As shown in Fig. 5(a), the phase distribution obtained by the TIE behaves as if it is the sum of the SQAM modulation phase and the cloud-like noise. Furthermore, in the case of the continuous phase modulation, the modulation phase at the boundary of each symbol is designed to be 0 as shown in Figs. 3(c) and 3(f). Therefore, the detected phase at the boundary of each symbol consists of only the noise contribution, and such phase values are denoted by the blue circles in the right graph of Fig. 5(a). It is considered that the cloud-like noise within a symbol area can be sufficiently estimated by connecting the phase values on the opposite sides of a square symbol boundary with linear interpolation in either the vertical direction or the horizontal direction. Figure 5(c) shows the cloud-like noise estimated by the proposed method, where the interpolation is implemented in the vertical direction over the entire region where the SQAM symbols exist. The phase value on the red line in the phase map image is also shown on the right graph in Fig. 5(c).
By subtracting this estimated cloud-like phase from the phase distribution obtained by the TIE method, a noise-free phase distribution of the SQAM signal beam can be recovered. Figure 5(d) shows the phase distribution of the SQAM signal beam detected by the TIE method for the case of uniform amplitude and four-level phase modulation (4SQAM), Gaussian noise of 2.5%, and a two-dimensional arrangement of 32 × 32 SQAM symbols, where each symbol consists of 16 × 16 pixels. Obvious cloud-like noise exists in the phase distribution. Figure 5(e) shows the phase distribution with the noise removed by the method described above. The phase is recovered by the proposed noise elimination technique, and each symbol group can be separated and identified in the constellation diagram of Fig. 5(f), where the RMSE of the noise-eliminated phase distribution is 5.3. The constellation diagram of Fig. 5(g) is also obtained for the case of binary amplitude and four-level phase modulation (8SQAM). The RMSE of the noise-eliminated phase distribution is 6.5, and it is understood that the SQAM signal can be detected even when the intensity modulation is applied. Practically, sync marks commonly used in the conventional HDS system to locate symbol positions in a data page  can be used to locate the symbol boundary in our noise reduction technique.
4. Experiment to confirm detection of SQAM signal beam by the TIE method
We experimentally confirm that the amplitude and the phase distributions of an SQAM signal beam can be detected by the TIE method by using the noise estimation and elimination technique described in Section 3. Figure 6 shows the experimental setup. A diode-pumped solid-state laser with a wavelength of 532 nm was used as the light source, and a photopolymer with a thickness of 400 μm was used for the holographic material. Liquid crystal spatial light modulators (SLMs) were used for intensity modulation and phase modulation of the signal beam (Holoeye LC-2002 and Hamamatsu Photonics LCOS-SLM X10468-01, respectively). Although this optical system can apply SQAM using both intensity modulation and phase modulation, uniform amplitude and four-level phase modulations (4SQAM) were applied to the signal beam in this experiment.
The SQAM signal beam contained 15 × 15 symbols, as shown in Fig. 7(a). Each symbol was composed of 16 × 16 SLM pixels at the phase SLM plane and 49 × 49 CMOS pixels at the CMOS sensor plane. The actual dimensions of each symbol were 320 μm × 320 μm for both the SLM and the image sensor planes, and the difference in the number of pixels was caused by the difference between the pixel pitch of the phase SLM (20 μm) and that of the CMOS sensor (6.5 μm). The maximum phase modulation depth was set to 360 in the experiment.
Initially, the pyramidal phase modulation explained in the previous section was used to continuously modulate the phase of the signal beam. The SQAM signal beam and the reference beam simultaneously illuminated the photopolymer to record the hologram. There corded SQAM signal beam in the photopolymer was reconstructed by illuminating the hologram with the reference beam, and the reconstructed signal beam was captured by the CMOS camera (Hamamatsu Photonics ORCA-Flash4.0 V3). Figures 7(b)–7(d) are the images captured by the CMOS camera when the position of the camera in the optical axis direction in Fig. 6 was set to z = −5 mm, 0 mm, and 5 mm, respectively, by using the stepping motor underneath the camera. It is well known that increasing is effective for reducing the cloud-like noise, while an increase in degrades the accuracy of the approximated in the TIE and may decrease the phase measurement accuracy. In our experiments, we determined by trial and error. Figure 7(e) shows the phase distribution of the SQAM signal beam obtained by the TIE method using the three captured images, and Fig. 7(f) shows the noise distribution estimated by the spatial linear interpolation. By subtracting the noise estimate from the phase distribution obtained by the TIE [Fig. 7(e)], the phase distribution of the signal beam was obtained as shown in Fig. 7(g). The complex values of the symbols in Fig. 7(g) were plotted in the constellation diagram shown in Fig. 7(h). In this figure, the amplitudes of 35 × 35 pixels and the phases of 2 × 2 pixels at the center of each symbol in Fig. 7(g) were averaged and plotted, and the color of each point in the diagram corresponds to the modulation value of each symbol. From Fig. 7(h), it is shown that each symbol group can be separated and identified.
Figures 8(a)–8(h) show the experimental results when sinusoidal phase modulation was used as the continuous phase modulation instead of the pyramidal phase modulation. The experimental procedure, other than the phase distribution of the signal beam, was the same as before. These figures show that, as in the case of the pyramidal phase modulation, the complex amplitude of the SQAM signal beam can be detected by the TIE method, even when the sinusoidal phase modulation is used as the continuous phase modulation. The experimental results given in this section reveal that the complex amplitude distribution of the SQAM signal beam, which is recorded in the photopolymer and reconstructed from it, can be detected by the TIE method with the noise estimation and elimination technique using spatial linear interpolation.
Although the SQAM symbol groups in the constellation diagrams in Fig. 7(h) and Fig. 8(h) can be identified separately, there are variations in the points in the plots. Scattered light noise from photopolymers and optical elements, imperfection of hologram recording / reproduction with photopolymers, non-uniformity and wavefront distortion of light waves irradiated to the SLMs, and low fill factor of transmission type SLM for the intensity modulation can cause this variation. We believe that more accurate optical system alignment, noise reduction, and the use of a reflective SLM for intensity modulation will be effective in improving this variation. In addition, it is considered that the accuracy of the cloud-like noise estimation by linear interpolation proposed in this paper can be improved by reducing the size of each symbol because the smaller the symbol is, the denser the sample points of the actual cloud-like noise become. Thus, proper reduction of symbol size may be effective for more accurate signal recovery, although other tradeoffs may be induced by the symbol size reduction.
The capability of the TIE method to measure the phase distribution of a signal beam without interferometry has been proven to be useful for detecting the complex amplitude of the SQAM signal beam for HDS. First, we have shown that continuous phase modulation is effective for accurate phase detection by the TIE method. Secondly, we have proposed the spatial linear interpolation method to estimate and eliminate the low spatial frequency noise that typically emerges in the phase distribution detected by the TIE method. The effectiveness of this method for noise estimation and elimination has been demonstrated by a numerical simulation and an optical experiment. By combining the continuous phase modulation and the noise elimination technique, it has been possible to detect the complex amplitude distribution of the SQAM signal beam with high accuracy.
The complex amplitude detection of the SQAM signal beam by the TIE method proposed in this paper numerically solves Eq. (2) using a total of seven times of Fast Fourier transforms, and the computational load is relatively high. At the present stage, the interferometric methods seem to be superior in terms of calculation speed owing to the ease of their numerical processing. On the other hand, as opposed to interferometry, the TIE method can be carried out by a simple and small optical system, and high vibration tolerance can be realized by the feature. The application of the TIE method to various fields has been widely studied, and the processing time and measurement system of the TIE method can still be improved. As the TIE method can also be applied to the measurement of partially coherent light, it might enable readout of holograms of the SQAM signal beam by partially coherent illumination in order to reduce speckle noise and alleviate degradation of the hologram readout due to shrinkage of the holographic material. Aside from the TIE method, a method that uses a non-interferometric iterative technique to detect phase modulated signal beams in HDS has also been proposed . Detection methods for phases and complex amplitudes that do not rely on interferometry bring about new possibilities for HDS.
We are planning to examine the detection performance of the TIE method for SQAM signal beams with various intensity and phase modulation levels and greater symbol densities. In practice, complex amplitude detection of the SQAM signal beam by single-shot image capture is essential, and its realization is also a future task.
Japan Society for the Promotion of Science (JSPS KAKENHI) (17K06406).
The authors would like to thank Dr. Kazutaka Kanno of the Graduate School of Science and Engineering, Saitama University, for helpful discussions.
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