Hologram dynamics: write and erasure
In the write phase, diffraction efficiency η as a function of write fluence can be modelled as5 (Fig. 3a)
$${{{{{{\rm{\eta }}}}}}}^{1/2}={{{\mbox{A}}}}_{{{\mbox{s}}}}[1-\exp (-{{\mbox{F}}}/{{{{{{\rm{\tau }}}}}}}_{{{\mbox{r}}}})]$$ (1)
where A s is the saturation diffraction efficiency, F is the write fluence with a unit of cm2 J−1, τ r is the write constant in unit of fluence. We have adopted this unit because the diffraction efficiency is contingent solely upon the light energy (in a defined area) for hologram write, rather than exposure duration, specifically in the low light intensity regime (Supplementary Note 1). We can also define the write efficiency A s /τ r that quantifies the rise in diffraction efficiency during the write phase because at F < <τ r for multiple page writing, \(\sqrt{\eta }\propto\) A s /τ r .
Fig. 3: Write and erasure. a Evolution of diffraction efficiency (log scale) of the first hologram in its write phase. The solid curve shows Eq. (1) fitted to the experimental measurements (dots). b Measured (dots) and fitted (solid curve) decay using Eq. (2) of the first hologram (log scale) because of write of later holograms in the erasure phase. Dashed line shows best fit using traditional exponential decal model. Full size image
As more holograms are written in the same area, the diffraction efficiency of previously written pages degrades with each subsequent write or read (Fig. 3b). In contrast to the widely adopted exponential decay6, we find the decay is best described by a stretched-exponential model
$${{{{{{\rm{\eta }}}}}}}^{1/2}={{{{{{\rm{\eta }}}}}}}_{0}^{1/2}\exp [-{(F/{{\tau }}_{e})}^{{{{{{\rm{\beta }}}}}}}]$$ (2)
where η 0 is the initial diffraction efficiency, F is the erasure fluence of subsequent holograms, τ e is the erasure constant, and β describes the degree of stretch, with β = 1 corresponding to conventional exponential decay. This stretched-exponential format was previously adopted to describe dark decay16 but suits us well here because of its similar reciprocity with respect to weakly doped crystals. Source data for Fig. 1 are submitted in Supplementary Data 1.
Optimization of energy profiles
Data durability is a challenge due to write and read erasure. To manage erasure, we optimize write and read profiles specifying the energy utilized for each IO, which improves both data durability and energy efficiency. As an example since the IO J−1 depends on the number of writes and reads done in a zone, we set the number of multiplexed pages in a zone at 400, and sweep the number of reads. Write profile optimization has been studied before10 that aims for all holograms to have the same diffraction efficiency when all holograms have been written. We adapt this approach by considering the more precise stretched-exponential erasure from our experiments. Our write profile can be obtained via backward-recursion with \(\sqrt{{{{\upeta }}}_{N}}=\sqrt{{{{\upeta }}}_{M}}\exp [{({\sum }_{i=N+1}^{M-1}{F}_{i}/{\tau }_{e})}^{{{\upbeta }}}]\), where M is the number of holograms, η N is the diffraction efficiency of the Nth hologram before further holograms being written. F i is the fluence used to write the ith hologram and can be deduced from Eq. (1). Having swept across various possible diffraction efficiency, best achievable energy efficiency is shown by the solid curve (adapted conventional write optimization) in Fig. 4a.
Fig. 4: Profile optimization and optimized energy efficiency. a Energy efficiency and page read gains that can be achieved by jointly optimizing the write and read energy profiles (orange dashed line) for a given multiplexed pages compared to the conventional write optimization (blue solid line). IO stands for input/output. b Optimized write/read energy profiles for workloads with 50% reads with realistic lab constraints. c Optimized write/read profiles for workloads with 100% reads. d Optimized energy efficiency for workloads with 50% reads achieved from calculations (solid curve) and measurements (cross and square markers). e Optimized energy efficiency in log scale for workloads with 100% reads achieved from calculations (solid curve) and measurements (cross markers). Full size image
We optimize the read profile by setting the minimum energy incident on camera to a minimum of e c , thereby mitigating the erasure of read holograms. Supplementary Note 2 provides the method of this energy determination process. The read fluence for the ith read can be determined from F i R=e c /(Dη i R) with η i R the diffraction of the holograms after the Nth read \(\sqrt{{{\eta }}_{N}^{R}}=\sqrt{{{\eta }}_{M}}{{{{{\mathrm{exp}}}}}}\,[-{({\sum }_{{{{{\rm{i=1}}}}}}^{{{{{\rm{N}}}}}}\,{F}_{i}^{R}/{\tau }_{e})}^{\beta }]\), D is the beam size, and η M the diffraction efficiency of the holograms after the writes. As shown by the dashed curve (joint write and read optimization) in Fig. 4a, the main impact of using variable reads is the possibility to do 1.9× better IO J−1 and 3.4× more reads at their best IO J−1.
Joint write and read optimization entails tuning the use of varying percentages of the material’s dynamic range to optimize for both the targeted number of writes and reads, which depend on the read/write operation ratio inherent in the workload the storage device can handle17,18. Consequently, profiles must be optimized according to the read/write ratio. In experiments we implement the above optimization protocol under constraints on the maximum read energy at the media of 10mJ, which is limited by the 2.78 W maximum read power and the maximum write time of 3.6 ms. Figure 4b and c shows selected examples of optimized write and read profiles for 400 multiplexed pages for workloads (assuming a device utilization of 90%) with two extreme read proportions (50% and 100% reads), at e c = 30nJ. As a consequence of the hologram erasure, the energy required to achieve the minimum required diffracted light incident on the camera for successful readout increases with each readout operation. For workloads with 50% reads the number of read operations in a zone before refresh equals the number of writes. Workloads with 100% reads require more write energy to start readout with higher diffraction efficiency and compensate for greater read erasure in order to maximize the number of reads before refresh is required.
After the plotted number of reads is completed, the energy required to readout exceeds 10mJ. At this point we can no longer successfully read out the stored data with the required access rate. Just prior to this point it is necessary to carry out a garbage collection and refresh operation, discarding the deleted pages and rewriting the live pages to a new location before they become unreadable to avoid data losses. To better understand useful Ios, we introduce net IO J−1 that refers to the number of user level IO J−1 (See Supplementary Note 3 for calculation of net IO J−1). This value includes the additional cost of the required system level operations such as garbage collection and refresh. Solid curves in Fig. 4d and e shows the trade-off between energy efficiency (in net IO J−1) and number of pages calculated from optimized write and read profiles.
Source data for Fig. 4 are submitted in Supplementary Data 1.
Demonstrations of energy efficiency, number of reads and density
As an example we experimentally demonstrated the achievable net IO J−1 in the crystal Fe0.015:r0.04 via writing, reading and decoding of data pages (and marked the results with cross markers in Fig. 4d, e). For workloads with 50% reads, the calculated optimized net IO J−1 was exactly achieved after writing, reading and decoding the data pages (see Supplementary Fig. 3a in Supplementary Note 4 for bit error rate (BER) which relates to user data recovery rate). For workloads with 100% reads, 9.3 K out of 12 K reads had lower BER (Supplementary Fig. 3b in Supplementary Note 4), which equates a smaller net IO J−1 as predicted because any uncertainty in the material parameters or experimental energies has big impact on the number of reads and BER that we can achieve. In both experiments, we write 400 multiplexed pages (limited by numerical aperture on the reference beam) with 47KB page size in a volume of 15.3 mm3.
We use the same optimization protocol for a different setup and crystal Fe0.03:r0.08. In this experiment, we increased the page size to 128KB by improving the numerical aperture on the signal beam and moving from 1 bit/symbol to 2 bits/symbol. We increased to 705 multiplexed pages by increasing the reference beam numerical aperture. With data recovery improved by machine learning, we achieved a record net density of 9.6 GB cm−3 in a reduced volume of 5.4 mm3 (see Supplementary Fig. 3c for BER). It’s worth noting that with our conventional decoding pipeline, the calculation of net density was not feasible due to excessively high BER. Average write and read times were 13 ms and 2 ms, respectively, resulting in an energy efficiency of 108 raw IO J−1 (Supplementary Fig. 3d in Supplementary Note 4).
Optimization across various Fe:LiNbO3 crystals: measured results and projected trend
We have used framework described above to quantify the best energy efficiency and number of pages across a set of crystals with various doping and annealing conditions to understand the impact of the Fe concentrations on the system performance. We then used the end-to-end optimizer to predict the best achievable performance with an optimally doped and annealed Fe:LiNbO 3 Crystal.
Figure 5a shows write efficiency of the first hologram in a fresh crystal as a function of the Fe2+ concentration (c Fe2+ ) across all the crystal crystals measured in this work. We find that the write efficiency primarily depends on the Fe2+ concentration, with the role of Fe3+ concentration being comparatively less impactful. In the lower light regime ( <8.3 W cm-2), it increases with increased Fe2+ concentrations up to 2.5 × 1017cm−3 but diminishes at higher Fe2+ concentration. Near optimal Fe2+ concentrations can be achieved via reducing more Fe3+ to Fe2+ through post growth annealing. In lowly-doped crystals (e.g. Fe0.001:r6.01), growing crystals with ideal as-grown Fe2+ concentration (Fe0.02:r0.09), or oxidising more Fe2+ to Fe3+ in higher doped crystals (not done with the crystals in this work). The finding that the diffraction efficiency is at the maximum at specific Fe2+ concentration is consistent with previous studies19, but the point at which \({A}_{s}/{\tau }_{r}\) starts to decrease varies with the recording geometry, crystal size and beam profiles. We expect write efficiency of higher doped crystals with optimal Fe2+ concentration will be worse than that of lowly doped crystal with optimal Fe2+ because of absorption introduced by more Fe3+.
Fig. 5: Optimized write, erasure and energy efficiency across various crystals. a Write efficiency of the first hologram in a fresh crystal as a function of Fe2+ concentration. The crystals that will be discussed in the main text are labelled. Dashed line is a trend line. b Erasure constant (log scale) as a function of ratio of Fe2+ to Fe3+ concentrations (log scale). Dashed curve describes the trend. c Net IO J−1 (log scale) at various number of pages calculated from characterized write and erasure parameters for a workload with 50% reads. IO stands for input/output. d Net IO J−1 (log scale) at various number of pages calculated from characterized write and erasure parameters for a workload with 100% reads. e Number of pages (log scale) inferred from characterization experiments (triangles) and estimated number of pages approximated for 50% and 100% reads (solid and dashed curves) from one-centre charge transport model in combination with optimal energy search. The number of pages is threshold at 100 net IO J−1 (which corresponds to 27nJ per bit when writing 47KB data pages). f Analysis of optimized energy efficiency (log scale) for workload with 50% reads in co-doped crystals inferred from literature at e c = 30nJ. The angles indicated in the legend represent the angles at which the write and erasure measurements for the materials were conducted. Full size image
The erasure constant τ e is determined primarily by the ratio of Fe2+ to Fe3+ concentration, as the dielectric relaxation time increases inversely proportional to photoconductivity3, which was experimentally shown to increase linearly with the concentration ratios20. We validated this when writing multiple data pages (Fig. 5b). In crystals with similar Fe2+ concentration (e.g. Fe0.001:r6.01 v.s. Fe0.02:r0.09), erasure is slower and more ideal in the higher doped crystal (Fe0.02:r0.09). This is beneficial as less energy is required to compensate for both the write and read erasure, thereby optimizing energy usage.
We calculate the trade-off between energy efficiency and number of pages across various crystals from measured write efficiency and erasure constants (see Fig. 5c for workloads with 50% reads and Fig. 5d for 100% reads). The superior performance of better materials manifests in two dimensions: energy efficiency (vertical axis in Fig. c-d) and the potential for a greater number of pages (horizontal axis). Whist obvious improvement in number of pages is achievable due to the exponential improvement of data durability at lower Fe2+ to Fe3+ concentration ratios, the gains in energy efficiency are more modest due to the inherent limit on optimal write efficiency. Raising energy efficiency beyond 100 net IO J−1 is challenging with a sufficient number of pages. Among the crystals we have measured, Fe0.02:r0.09 exhibits the best performance because its Fe2+ concentration approaches the peak of the write efficiency (Fig. 5a), and its Fe2+ to Fe3+ concentration ratio is sufficiently low to ensure slow erasure (Fig. 5b). At 100 net IO J−1, we can potentially multiplex 1482 pages for a workload with 100% reads, and 1918 pages for a workload with 50% reads, corresponding to densities of 4.4 and 5.7 GB cm−3 (assuming a small page size of 47kB in a 15.3 mm3 volume).
Finally, we investigate the best performance achievable across untested Fe:LiNbO 3 . Using 100 net IO J−1 as a threshold, we estimate the maximum achievable number of pages at this energy efficiency in relation to the doping level and optimized Fe2+ concentration (solid curves in Fig. 5e). (The trend is consistent with the previously shown increase in dynamic range with doping level21). Here the media write efficiency and erasure constants are estimated from a one-centre charge transport model (Supplementary Note 5). For comparison, measured performance for the subset of crystals that had a Fe2+ concentration close to the optimum 2.5 × 1017cm−3 are shown as solid triangles. Our estimates present a conservative trend, as the simplified model does not account for the degree of stretch β in the erasure procedure that we have seen in the experiments. The lowly doped crystals are better in experiments than in estimation because the beneficial stretch exponential is not considered in the estimation. We also include an example where a higher doping level, but non-optimal concentrations, results in a lower number of pages (empty triangles). The experimental data point at 0.03% Fe drops to lower than estimated optimal because we do not have this crystal sample at the optimal Fe2+ concentration. This result highlights the goal of maximizing doping while keeping Fe2+ concentration near its optimum.
Our study suggests the potential of writing 2201 pages at 100 net IO J−1 for a workload with 100% reads, and 2910 pages for a workload with 50% reads in a crystal doped at 0.05% Fe and at its optimal oxidation rate. These number of pages may translate to raw densities of 6.6 and 8.7 GB cm−3, assuming a raw page size of 47KB in a 15.3 mm3 volume as we used in media characterization experiments. While the prospect of achieving high density is promising, it remains uncertain whether we can successfully attain the ideal oxidation state in a crystal doped with 0.05% Fe. Moreover, changes in media parameters (e.g. Glass constant) may affect the fidelity of the media’s write and erasure characteristics so that they deviate from the current trend.
Our framework also enables performance prediction based on adjustable parameters. For instance, Fig. 5f shows the calculated energy efficiency for workload with 50% reads, considering the write efficiency and erasure constants of co-doped crystals provided by literature11,14. In Sc:Ru:Fe:LiNbO 3 crystals, the write efficiency, which limits the maximum achievable energy efficiency (to approximately 100 net IO J−1) across all Fe:LiNbO 3 crystals, could potentially be enhanced. Whilst these results suggest that these materials could provide energy efficiency and density gains, in a holographic storage system we need to caution that the parameters taken from the literature studies were obtained with small angles in a colinear setup and further work is needed to see if these would translate into a 90° setup that is capable of realizing the predicted number of pages. Refer to Supplementary Note 6 for energy efficiency associated with workloads with 100% reads.
Source data for Fig. 5 are submitted in Supplementary Data 1.