1. Introduction

Volume holographic memory [1] has attracted much attention because of its large storage capacity and fast data transfer rate. The image information is stored as a volume hologram that is formed by an interference pattern between the signal and reference beams. The information is retrieved by reconstructing the image from the stored hologram, which is usually performed with a probe beam whose wavelength, incident angle, and wavefront are identical to those of the reference beam used in the recording process. This is because diffraction from the volume hologram is restricted by Bragg’s law. This restriction allows us to overwrite another hologram in the same volume (multiplexing) and is responsible for the large storage density of volume holographic memories. However, it also presents some obstacles for implementing practical memory systems. For example, in rewritable recording media like photorefractive materials [2], illumination with the probe beam affects the recorded hologram and erases the stored information [3]. Even in a photopolymer, which is currently the most promising write-once recording material, if some monomers still exist within the volume to be read, the probe beam will cause unintentional polymerization, which wastes the storage capacity. These issues are obviously caused by the destructive probe beam having the ability to expose the recording medium in a similar manner to the recording beam.

To avoid such a problem, several nondestructive readout methods [4–7] have been proposed so far, where the stored image is reconstructed at a longer wavelength, outside the sensitive spectral region of the recording material. These methods successfully retrieve the stored information from the volume hologram while avoiding the wavelength selectivity of Bragg’s law, but most of the methods may be not practical for a holographic memory system because the multiplexing capability is considerably lowered. For example, Petrov [5] successfully increased a reconstructed image field by utilizing anisotropic diffraction, but this approach requires a specific recording configuration and limits the number of multiplexed pages. Külich [6] employed a spherical probe beam having some bandwidth of the angular spectrum, but this technique tends to produce severe crosstalk noise from other multiplexed pages, which demands a large angular separation between two adjacent multiplexed holograms, and thus results in a small storage density.

Recently, we proposed another way to reconstruct an image at a wavelength different from the recording one [8]. Our method, which we call polychromatic reconstruction (PCR), utilizes a probe beam having some spectral bandwidth instead of monochromatic light. Each angular spectral component of the recorded gratings can be Bragg-matched with one particular wavelength within the broadband spectrum of the probe beam. Thus, the whole image can be reconstructed from the volume hologram even though the probe wavelength is very different from the recording one. However, the large spectral width of the polychromatic probe beam also leads to reduced angular selectivity. Thus, the resultant storage density decreases substantially, similar to the case of the spherical probe beam method.

In this paper, we propose a method of improving the multiplexing capability in PCR. The method is based on the selective detection of a target signal image that is submerged in noise waves. Inserting a suitable wavelength separator into the reconstructed image plane, we can retrieve the stored information without crosstalk even if the angular separation is not large enough to suppress the noise diffraction. Here, we first investigate the properties of the crosstalk noise theoretically, then we explain how to distinguish the signal from the noise using the wavelength separator, and finally we present some experimental results of the image reconstruction to confirm the effectiveness of this selective detection method.

2. Multiplexing capability in PCR and its improvement with selective detection

Figure 1 illustrates the recording and readout configurations studied here. Two recording beams with a wavelength of λ_w are incident on the recording medium at an internal crossing angle of 2θ_w. The reconstruction is performed with a polychromatic probe beam with center wavelength λ_p0, full spectral bandwidth Δλ_p, and internal incident angle θ_p. We assumed here that the refractive index of the recording medium is unity, for simplicity. Note that θ_p should be properly chosen so as to satisfy the Bragg condition λ_wsinθ_p = λ_p0sinθ_w.

 

Fig. 1 Schematic diagram of (a) the recording and (b) the readout schemes in the PCR method. λ_w is the recording wavelength; λ_p0 and Δλ_p are the center wavelength and the spectral bandwidth of the probe beam, respectively; θ_w is the internal half crossing angle of the recording beams; θ_p is the internal incident angle of the probe beam; and y_c is the crystal rotation axis for multiplexing. We assumed that the signal, reference, and probe beams lie in the same plane.

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Investigating the multiplexing capability, we consider here the crystal rotation with respect to the y_c-axis as a multiplexing method (hereafter, it will be called crystal angle multiplexing). Note that other multiplexing methods are also possible in principle, but most of them are probably not suitable for the PCR method because they require a complicated system to read out a target page. For example, the ordinary angle multiplexing method, where the incident angle of the reference beam is changed for recording another page, requires movement of the imaging system in addition to changing the incident angle of the probe beam, because the direction of the diffracted wave is different at each multiplexed page. The situation is similar in the case of the wavelength multiplexing method. Crystal angle multiplexing, in contrast, does not require any additional movement other than the crystal rotation. In addition, the imaging properties are almost unchanged with each multiplexed page because θ_w will be constant at every crystal angle. Thus, crystal angle multiplexing is most suitable for holographic memory systems employing PCR.

In crystal angle multiplexing, the recording of another page will be performed after crystal rotation by the proper angle. In order to retrieve the stored information without crosstalk, the rotation angle should be sufficiently large so that the other multiplexed holograms cannot produce noise diffracted waves that will disturb the detection of the target signal image. On the other hand, the theoretical limit of the storage density is inversely proportional to the angular separation between the adjacent multiplexed holograms. Therefore, the angular selectivity (i.e., how small the angular separation can be made) is an important figure of merit determining the total storage density of the system.

To investigate the angular selectivity in PCR method, we simulated the reconstructed images after a crystal rotation δθ_c, neglecting the off-Bragg diffraction. The detailed calculation method was described previously [8]. In our simulation, we assumed that an input image, which was an outline character “A” with dimensions 1 cm × 1 cm, shown in Fig. 2(a) , was recorded at λ_w = 532 nm and was reconstructed with the polychromatic probe beam with λ_p0 = 815 nm and Δλ_p = 40 nm. The images obtained at crystal rotation angles of 0°, −1°, and +1° are presented in Figs. 2(b), 2(c) and 2(d), respectively. Note that the PCR method essentially produces an image with a spectral dispersion of the Bragg-matched wavelength, and image magnification occurs in the y_d-direction [8]. Even after the crystal rotations of ±1°, the image was partially reconstructed by the polychromatic probe beam, and part of the image overlapped with an area where the target signal image will be reconstructed. Therefore, the crystal rotation of 1° is not a sufficient rotation angle for multiplexing. Such deterioration of the angular selectivity is due to the fact that the rotated gratings still satisfy the Bragg condition with respect to other spectral components within the broadband spectrum of the polychromatic light.

 

Fig. 2 The simulated results of image reconstruction by PCR. (a) Input image and the reconstructed images at crystal rotation angles of (b) δθ_c = 0°, (c) δθ_c = −1.0°, and (d) δθ_c = +1.0°. The colors in (b)-(d) represent the Bragg-matched wavelength of each diffracted wave. The detailed calculation parameters are as follows: θ_w = 30°; θ_p = 50°; and the focal lengths of the Fourier transform lenses in the recording and reconstruction processes are 100 mm.

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In Fig. 3 , the Bragg-matched wavelength at y_d = 0 is plotted as a function of x_d at each rotation angle. From the graph, we can find that the crystal rotation leads to translation of the Bragg-matched line. Theoretically, such a behavior of the Bragg-matched line is approximately expressed as

 

Fig. 3 The imaging location and the Bragg-matched wavelength at each rotation angle. The reconstructed image field is limited by (a) the probe spectral band and (b) the band-pass LVF. The dotted part of the Bragg-matched line corresponds to the portion of the image that will not be detected by the imager.

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To avoid the crosstalk, the crystal should be rotated until the solid part of the Bragg-matched line does not overlap with the signal imaging area. Since the signal imaging area is represented by

From Eq. (3), the requirement for the rotation angle is written as

Obviously, an excessive Δλ_p increases the minimum angular separation and reduces the multiplexing capability. Therefore, Δλ_p should be identical to the spectral width required for the reconstruction of a single hologram. In this case, the first and second terms inside the parenthesis in Eq. (4) become equal [8]. Then, the minimum angular separation, δθ_cMin, is written as

Note that the minimum angular separation should, to be exact, include the term related to the hologram dimensions that represents the conventional angular selectivity in the monochromatic readout. In most cases, however, it is very small compared with the term presented in Eq. (5). Therefore it can be usually neglected in the PCR method. For example, if x_sMax = 5 mm, f_s = 100 mm, θ_w = 30°, λ_w = 532 nm, and λ_p0 = 815 nm, then θ_p = 50° and the resultant δθ_cMin calculated from Eq. (5) is 1.5°. On the other hand, in the case of the conventional monochromatic readout, the minimum angular separation is typically on the order of 0.01° when the hologram thickness is 1 mm. Therefore, the multiplexing capability in the PCR method is lowered by two orders of magnitude. This reduction of the multiplexing capability is comparable to that in the spherical probe beam method [9].

Due to the translation property, the Bragg-matched wavelength at some fixed position x_d will linearly shift with the rotational angle δθ_c. Furthermore, its wavelength shift is almost independent of the position x_d. Thus, taking advantage of the wavelength difference, we can distinguish the signal from the noise and selectively detect the signal image alone even if the diffracted images after rotation (i.e., crosstalk noise) are overlapped with the signal imaging area. Such a selective detection can be achieved by inserting a suitable optical component, like a wavelength filter or grating, into the reconstructed image plane. Let us consider utilizing a special wavelength filter whose transmittance T_LVF is represented by,

Note that the transmitting spectrum of this filter depends on the illuminated location on the filter; that is, the optical waves passing through different spatial positions will undergo different spectral filtering by this filter. Such a wavelength filter is known as a linear variable filter (LVF) since the spectral shift of the transmitting band is proportional to the spatial shift of the illuminated position.

Figure 3(b) illustrates the concept of this selective detection utilizing a band-pass LVF. Since the transmitting wavelength of the LVF coincides with the spectral dispersion of the target signal image, every diffracted wave that constitutes the signal image can go through the LVF and will be detected by the imager. On the contrary, the LVF will reject the noise diffracted wave whose wavelength lies outside the transmitting band of the LVF. In this case, the minimum angular separation δθ_cMin is determined by the transmitting bandwidth of the LVF, that is,

A smaller bandwidth leads to a smaller δθ_cMin, and thus results in a larger multiplexing capability. For example, when the transmitting bandwidth Δλ _LVF is 10 nm, the minimum angular separation δθ_cMin is 0.2° with our calculation conditions, and if Δλ _LVF is 1 nm, then δθ_cMin reduces to 0.02°. In principle, the degraded multiplexing capability normally encountered in PCR may be completely recovered if Δλ _LVF is sufficiently small. Note that, in such case, the minimum angular separation is limited by the hologram dimensions, which is similar to the conventional monochromatic readout. We should also note that the probe spectral width Δλ_p need not be equal to the spectral width required for the reconstruction of a single hologram because Eq. (8) no longer includes Δλ_p.

3. Experiment

To confirm the effectiveness of the selective detection method, we performed experiments employing a band-pass LVF produced by Barr Associates, Inc. The experimental setup is shown in Fig. 4(a) . The recording light source was a frequency-doubled Nd:YAG laser operating at a wavelength of 532 nm. Image information was applied to the signal wave by a digital micro-mirror device. The hologram was recorded in the 90°-geometry with an internal crossing angle 2θ_w of about 60°. The recording material was a 45°-cut Fe-doped LiNbO₃ crystal with dimensions 10 mm × 10 mm × 10 mm, which was mounted on the rotation stage to control the crystal angle. Readout was performed by using a superluminescent diode (SLD) as a polychromatic light source whose center wavelength was 840 nm and whose full width at half maximum (FWHM) spectral width was about 50 nm. The input image was typically about 7 mm in width (x_s-direction) and 3 mm in height (y_s-direction). The focal lengths of the Fourier transform lenses used in the recording and reconstruction processes were 80 mm and 15 mm, respectively. The band-pass LVF was placed at the first reconstructed image plane. The resultant reconstructed image passing through the LVF was detected by CCD2. The transmitting properties of the LVF used in this experiment are shown in Fig. 4(b). The wavelength dispersion of the LVF was 20 nm/mm and its full bandwidth, which was defined here as a width down to e ⁻⁴ of its peak value, was 12 nm. Note that the size of the reconstructed image on the LVF was adjusted with the focal length f_d so that the wavelength dispersion of the signal image coincided with that of the LVF.

 

Fig. 4 (a) The experimental setup for PCR with the selective detection method. L, lens; M, mirror; HWP, half wave plate; PBS, polarized beam splitter; SLM, spatial light modulator; A, aperture; F, fiber; RS, rotation stage; LVF, band-pass linear variable wavelength filter; SLD, superluminescent diode; and CCD, charge coupled device. The LVF was placed at the first reconstructed image plane. (b) Transmitting properties of the LVF at several illuminated positions. x _LVF is a coordinate defined on the LVF.

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The experimental results of the angular selectivity without and with the LVF are presented in Figs. 5(a) and 5(b), respectively. In this experiment, a single hologram (i.e., not multiplexed) was recorded within the crystal, and then the reconstruction was performed with the SLD at each rotation angle δθ_c. From Fig. 5(a), we can see that the spatial shift of the diffracted image increased with increasing crystal rotation, and part of the image faded away, as is consistent with the theoretical prediction. Even after the crystal rotation of δθ_c = 0.6°, the image was still reconstructed and overlapped with the signal imaging area. Eventually, the minimum angular separation was estimated to be about 1.5° in this case. On the other hand, in Fig. 5(b), the diffracted image at δθ_c = 0.6° almost disappeared, while that at δθ_c = 0° was still observed. This clearly shows that the LVF successfully selected the image to be detected. Consequently, the multiplexing capability increased by a factor of 2.5 with the LVF used in this experiment. If we take the refraction at the crystal surfaces into account, the minimum angular separation of δθ_cMin = 0.6° corresponds to Δλ _LVF = 13 nm, which is in very good agreement with the actual bandwidth of the LVF.

 

Fig. 5 The experimental results of the angular selectivity (a) without and (b) with the LVF.

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An experiment for reconstructing multiplexed holograms was also performed. The three different images shown in Fig. 6(a) were recorded with an angular separation of 0.6°. The reconstructed images obtained without and with the LVF are shown in Figs. 6(b) and 6(c), respectively. Since the angular separation of 0.6° was not a sufficient rotation angle, as was seen in Fig. 5(a), the severe crosstalk noise disturbed the detection of the target signal image in Fig. 6(b). Employing the LVF, however, although the polychromatic probe beam produced noise images from the other multiplexed holograms, only the target signal image was successfully detected at each angle.

 

Fig. 6 The reconstructed images of multiplexed holograms. (a) Input images at each rotation angle and the reconstructed images (b) without and (c) with the LVF. θ_c is the crystal angle.

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4. Conclusion

We have investigated the multiplexing capability in the polychromatic reconstruction (PCR) method. While a probe beam with broadband spectrum is essential for the PCR method, it also causes deterioration of the angular selectivity and results in considerable lowering of the multiplexing capability. However, such a drawback of PCR can be overcome by the selective detection method proposed here. An additional optical component acting as a wavelength separator enables us to detect the signal image alone, even if the noise images overlap with the target signal image. Such selective detection is responsible for a unique imaging property of PCR, namely that the diffracted wave has a different Bragg-matched wavelength depending on the crystal rotation angle. Therefore, PCR has a great advantage in terms of the multiplexing capability compared with other nondestructive readout methods proposed so far. In our experiment, the improvement factor was only 2.5 due to the relatively wide bandwidth of the LVF, but it should be possible to increase it further if we utilize another LVF with a narrower bandwidth or employ a specially designed volume holographic grating as a wavelength separator. Therefore, PCR with the selective detection method is a promising way to achieve nondestructive readout in volume holographic memories without sacrificing the multiplexing capability.