## Abstract

Conventionally, reading and writing of data holograms utilizes a fraction of the light power because of a trade off in write and read efficiencies. This system constraint can be mitigated by applying a resonator cavity. Cavities enable more efficient use of the available light leading to enhanced read and write data rates with no additional energy cost. This enhancement is inversely related to diffraction efficiency, so these techniques work well for large capacity holographic data storage having low diffraction efficiency. The enhancement in write data transfer rate is evaluated by writing plane wave holograms and image bearing holograms in Fe:LiNbO_{3} with a 532 nm wavelength laser. We confirmed 1.2 times enhancement in write data rate, out of a 1.4 theoretical maximum for materials absorption of 16%.

© 2016 Optical Society of America

## 1. Introduction

Data storage has seen a recent market shift with the advent of cloud storage, and media streaming. Before internet speeds made remote data storage a viable option, personal and local network storage devices were tending to higher and higher capacities, but cloud storage and media streaming services have caused consumers and businesses to offload their data to remote centers. Thus, the focus in data storage development is now a challenge of designing data centers that meet the rapidly growing consumer and industrial needs remotely.

The needs of data centers come from storing two kinds of data, “hot” and “cold”. The definition of hot versus cold data is application dependent. A familiar application would be personal computers, where the memory being actively used is read out from long term storage and kept on fast volatile memory called Random Access Memory (RAM). In personal computers, the data kept in RAM is clearly hot, while data remaining in long term storage is cold. However, data centers store much larger quantities of data with a wider range of access needs than personal computers, so it becomes meaningful to break long term storage into hot and cold archives. In archival storage hot data must be kept in fast media to prevent speed reductions during frequent access, but cold data can be kept on slower media to reduce the total cost of ownership of data centers. Hot data’s requirement of speed has led to the use of Solid State Drives (SSD) as the fastest medium available. On the other hand, cold data is infrequently accessed or modified, but needs to be stored for long periods of time. Infrequent access coupled with the relatively high cost of SSD establishes the need to find a cheaper alternative for cold data [1].

So far, this need has been met by magnetic tape and Hard Disc Drives (HDD). There have even been efforts made to use large Blu-ray optical disc arrays [2]. Blu-ray arrays have maintained the lowest cost of ownership mainly because they run cooler and more efficiently than other options while needing minimal backups and having an order of magnitude better longevity [3]; however, the scalability in capacity is limited for such bit based, multi-layer optical storage. Recording densities are limited by Numerical Aperture (NA), and wavelength restrictions, as well as the problem of absorption control to maintain write and read beam transmission throughout the increasing number of recording layers [4,5]. An additional drawback of optical discs is that readout is limited by serial access. These capacity and data rate limitations in current optical disc based cold storage are potentially overcome in Holographic Data Storage Systems (HDSS), which will likely outperform magnetic storage as well as Blu-ray [6,7]. HDSS can accomplish such high data densities through various multiplexing techniques by which multiple data pages are superimposed at the same location. The known multiplexing techniques of angular [6,8–10], shift [11], speckle shift [12–14], and orthogonal phase code [15] multiplexing provide increased capacity through multi-dimensional storage and increased speed through the parallel process of page based storage of bits [16]. Yet another available dimension of storage is utilized in phase shift keying in the coding domain [6,17], where the phase sensitive nature of holography is used to encode data rather than gray scale power modulation. These capacity and speed advantages as well as an anticipated cost of ownership similar to Blu-ray arrays combine to make HDSS a very attractive solution for cold data storage.

The appeal of HDSS is mainly attributed to multiplexing and parallel holographic data transfer. However, HDSS has an inherent efficiency problem due to this high degree of multiplexing. In HDSS data is stored by interfering two coherent beams inside of a photo-sensitive material, and holograms are formed from the interference pattern. This recording process is inherently inefficient in terms of using available light source power because only a portion of the light provided is used in the writing process and the rest of it is thrown away. Similar inefficiencies occur in readout as well. In addition, diffraction efficiencies of multiplexed holograms scale as ${(M\#/N)}^{2}$, where M# represents the material dynamic range, and N is number of holograms [18]. Since several hundred holograms are typically multiplexed in HDSS the diffraction efficiency of each hologram is very low [19]. Typical diffraction efficiencies are less than 0.3% for readout, so most of the readout power is wasted [19,20].

These drawbacks in multiplexed holographic recording can be overcome by recycling the wasted light. Conveniently, that light which is normally thrown away can be recycled through the use of optical resonator cavities [21]. This recycling can be viewed in one of three ways: first, as a means reducing the total energy cost of the system; second, as a means of increasing write and read data transfer rates through a reduction in exposure time; and third, as an easing of material sensitivity and dynamic range requirements. While it is possible to increase data transfer rates by engineering more sensitive, higher dynamic range materials or increasing the power of the beam source, both of those options will eventually yield diminishing returns. For obvious reasons of energy costs, the light source power cannot be increased indefinitely, and cost effective, high sensitivity, high dynamic range materials tend to come with trade-offs related to increased shrinkage [22,23]. With these design limitations in mind, cavity techniques to enhance the performance of HDSS become an attractive improvement over conventional HDSS. The appeal of cavity enhancement techniques is further improved by its compatibility with the compact monocular system design [24] and the improved bit error rates of pseudo-phase conjugate readout and Gaussian apodization [16,25].

It should also be noted that cavity techniques have been demonstrated in hologram readout [21], and the lack of its effect on Bragg selectivity in readout has been observed [26]. However, quantitative formalization of cavity-based enhancement techniques in write data transfer rate has not been addressed in detail nor experimentally demonstrated.

In this paper, we focus on the process of cavity enhanced holographic recording. There are five unique enhancement geometries for cavity-based, plain wave, recording which are derived from the choice of one or two cavities and the choice of standing wave or traveling wave cavities. In Sec. 2 we will formulate theories for three of these possible geometries. Section 2.1 will address the general theory of cavity based irradiance enhancement, and Sec. 2.2 will cover the grating strengths of the four recording geometries considered. Within Sec. 2.2, Sec. 2.2.1 will cover the non-cavity writing case, Sec. 2.2.2 will cover the single standing wave cavity case, Sec. 2.2.3 will cover the single traveling wave cavity case, and Sec. 2.2.4 will cover the double traveling wave cavity case. The single traveling wave case is particularly interesting for angular multiplexing, but provides less improvement than the double traveling wave geometry, which provides the greatest enhancement while maintaining efficient use of dynamic range. The dual standing wave and standing/traveling wave hybrid theories will not be discussed due to their inefficient use of dynamic range and degree of complexity. Section 2.3 will then relate the grating strengths of Sec. 2.2 to enhancements in write data transfer rate. In Sec. 3 we experimentally verify the performance of single traveling wave cavity enhancement by applying a cavity to the reference arm while recording plane wave and image bearing holograms. Finally, in Sec. 4 we will discuss the impact of these techniques on HDSS system design.

## 2. Theory

#### 2.1 Cavity enhancement of irradiance

The fields inside the cavity can be represented by an infinite geometric summation of scalar plane waves [27]. Relevant quantities for the summation include the incident field amplitude ${U}_{in}$, entrance coupler transmission coefficient ${t}_{1}$, field amplitude transmission inside the cavity ${t}_{cav}$, and the round trip phase difference of the cavity $\delta $. Thus, in the coordinate system of for a wave vector and frequency we can write the forward propagating field inside the cavity:

#### 2.2 Cavity enhancement of writing irradiances

We can now apply the cavity enhancement described by Eq. (2) to the possible recording geometries depicted in Fig. 1. As mentioned before, the double standing wave and standing/traveling wave hybrid geometries will not be considered due to their inefficient use of dynamic range and higher degree of complexity. We will cover the enhancement in grating strength of the single standing wave, single traveling wave, and double traveling wave geometries. To aid in visualizing the geometries considered, the beam wave-vectors and grating vectors are summarized in Fig. 1, where $\overrightarrow{\rho}$ and $\overrightarrow{\sigma}$ are the reference and signal beam wave vectors, $-\overrightarrow{\rho}$ is the reverse propagating reference beam, ${\overrightarrow{K}}_{trans}$ is the desired transmission gating vector, ${\overrightarrow{K}}_{refl}$ is an extraneous reflection grating, and ${\overrightarrow{K}}_{s\mathrm{tan}d}$ is an extraneous distributed reflection hologram.

Before we discuss the particular geometries, let us begin with the most general field distributions we will encounter. As seen in Fig. 1, the single and double traveling wave geometries only have two waves involved in grating formation, while the single standing wave cavity introduces a third wave to the single cavity geometry due to the collinear nature of the standing wave cavity. Thus, the most general field distribution that we will encounter is a three wave addition as follows:

Taking the magnitude squared of this plane wave summation, we arrive at the general irradiance distribution of a recording geometry:

Assuming, we are trying to record transmission type holograms the only grating of Eq. (5) that we want is the second cosine term. Similar to the work of Liangcai Cao the use of standing wave cavities is seen to consume extra dynamic range in the form of additional offset terms and unwanted gratings [28].

### 2.2.1 Non-cavity grating strength

In order to theoretically demonstrate cavity enhancement in write data transfer rate, we must first establish the base-line for comparison. This base-line will be simple plane wave, non-cavity recording. We begin by assuming a plain wave input of irradiance ${E}_{e,in}$ to be split between the two recording beams, so that we have a field amplitude of

If we then write the irradiance splitting ratio S aswhere ${E}_{e,\rho}$ is the irradiance of the reference arm, ${E}_{e,\sigma}$ is the irradiance of the signal arm, and we constrain ${E}_{e,\rho}+{E}_{e,\sigma}={E}_{e,in}$, we can then write the reference and signal arm irradiances as Looking back at Eqs. (4) and (5), with ${A}_{R}=\sqrt{{E}_{e,\rho}}$, ${A}_{B}=0$, and ${A}_{S}=\sqrt{{E}_{e,\sigma}}$ we can now write the coherent scalar field addition in the recording medium as### 2.2.2 Enhancement of grating strength by single standing wave cavity

To find the grating strength enhancement of a single standing wave cavity applied to the reference arm we need to formulate the ${G}_{F}$ of Eq. (3) and the field amplitudes of Eq. (4). Beginning with the cavity enhancement factor${G}_{F}$, we find that circulating fields in a two mirror, standing wave resonator encounter losses from the mirror reflection coefficient magnitudes ${r}_{1}$ and ${r}_{2}$, the hologram diffraction efficiency $\eta $, and the recoding material absorption loss b, as shown in Fig. 2.

Thus, we can write the cavity transmission, ${t}_{cav}$ of Eq. (1), as

Here the term in parentheses is technically an irradiance transmission term, but in a standing wave resonator the round trip of the beam takes it through the crystal twice, so that taking its square root is negated. Thus, for a perfectly tuned cavity length we may write the ${G}_{F}$ for the standing wave cavity [21]:Applying Eq. (14) to the coefficients of Eq. (4) we can write the field amplitudes asIdeally, we would like to have unit fringe visibility where the offset terms are equal to the amplitude of this cosine term, but there is no real value of the splitting ratio that will accomplish this. Thus, it is more advantageous to maximize the grating strength of this particular term. The splitting ratio for maximum grating strength is then found by setting the derivative of the grating strength to zero and solving for the splitting ratio S:

### 2.2.3 Enhancement of grating strength by single traveling wave cavity

Again, starting from the definition of the cavity transmission for a traveling wave, bow-tie cavity we can derive the form of ${G}_{F}$. Figure 3 shows a schematic of such a cavity where ${r}_{1}$ to ${r}_{4}$ are the magnitudes of the mirror reflection coefficients, ${\rho}_{1}$ to ${\rho}_{4}$ are the reference beam paths in the cavity, B is the roundtrip amplitude transmission for the recording material, b is the material power loss, and $\eta $ is the hologram base diffraction efficiency. Here, B is the square root of the power transmission because the reference beam only passed through the material once per round trip. This gives us a round trip transmission coefficient of

Thus, for a perfectly tuned cavity length we may write the ${G}_{F}$ for the traveling wave cavity:As in Sec. 2.2.1, we only have one cosine term which is our desired grating. Comparing the grating strengths of Eqs. (25) and (12), we can see that the general enhancement in grating strength is still given by Eq. (17). The only change is that ${G}_{F}$ is now given by Eq. (23). Once again, we would like to have unit fringe visibility, and this time it is possible. If we set the splitting ratio to the cavity gain, $S={G}_{F}$, we get unit fringe visibility, and we can write the grating strength enhancement as

It is also important to note that since the general form of the grating strength enhancement is still given by Eq. (17), we can also choose to disregard the fringe visibility and maximize the grating strength according to Eq. (18). Maximizing grating strength at the cost of fringe visibility would be an attractive option if the constant irradiance terms of Eq. (25) do not consume dynamic range, but losses in fringe visibility are generally considered to be undesirable if the constant terms consume dynamic range.

### 2.2.4 Enhancement of grating strength by double traveling wave cavity

The final writing geometry we will consider is the double traveling wave cavity. The cavity enhancement in irradiance, ${G}_{F}$, still follows Eq. (23) assuming we use identical cavities, so we only need to define the coefficients of Eq. (4) as

#### 2.3 Cavity enhancement of write data rates in photorefractive materials

Now that we know how the grating strength is enhanced via cavities, we need to consider how these enhancements in grating strength transfer to the write rate with photorefractive materials such as Fe:LiNbO_{3}. We begin by defining a functional form for the time evolution of a hologram based on the band transport equations in photorefractive materials, and the coupled wave equations [29,30]. Under the band transport model the refractive index modulation depth, ${n}_{1}$, evolves in time according to a simple differential equation:

During our single cavity recording experiments, the available power is split equally between the cavity and non-cavity beam, and the cavity beam is enhanced by a factor of 2. This will result in a time dependent ${n}_{ss}$ which depends on the energy exchange of the unbalanced beams. Heaton et. all analyzed this transient behavior by solving the coupled equations. An iterative, numerical method was used, where each time step used a version Eq. (31) to solve for the instantaneous refractive index modulation. The intermediate solution was used to solve for the new beam irradiances [30]. This indicates that for exposure times $t\ll \tau $ the time dependence of ${n}_{ss}$ may be ignored. Thus, for $t\ll \tau $ we may write the time dependence of the refractive index modulation depth as

The assumption of $t\ll \tau $ is justified in Sec. 3.1, so that we now have a functional form for the time dependence of the refractive index modulation depth. We can apply this form to Kogelnik’s Eq. (45) to arrive at an equation for the time evolution of diffraction efficiency [31]:

where $A$ is a constant dependent upon ${n}_{ss}$, the grating thickness, recording wavelength, and Bragg angle. $A$ is independent of recording irradiance for continuous wave recording [32] and $\tau $ is linearly dependent on irradiance [29,32], so the enhancement in write rate is applied to $\tau $. Thus, the enhancement of write-rate can be computed by taking the ratio of the non-cavity recording time constant to the cavity recording time constant. These ratios will theoretically follow the grating enhancements derived in Sec. 2.2.1 to 2.2.4, and are summarized in Table 1.## 3 Experimental results of single standing wave cavity enhanced writing

#### 3.1 Plane wave recording with a cavity enhanced reference

### 3.1.1 Plane wave setup and procedure

As mentioned in Sec. 2.3, $t\ll \tau $. To prove this we compare our experimental setup to that of Maxein et. all [32]. For our experiments we used a combined recording irradiance of approximately 2 kW/m^{2} and an Fe doping concentration in LiNbO_{3} of 0.015 mole%. For a recording irradiance of 2 kW/m^{2} and a doping concentration on the order of 0.1 mole% Maxein et. all demonstrated a time constant of ~300 sec. Since our doping concertation is an order of magnitude lower we would expect our time constant to similarly increase to the order of at least $1\times {10}^{3}$ sec. Thus, Eq. (32) is applicable to our experimental results for exposure times on the order of 10 sec.

Next, to prove that single cavity enhancement of write-rates is possible we compared the photo-refractive time constants of normal and single cavity recording. To do this we wrote single holograms in an anti-reflection coated, 0.015 mole % Fe:LiNbO_{3} crystal from Deltronic Crystal Industries using a diode pumped Nd:YAG laser operating with a single longitudinal mode at a 532 nm wavelength (Model: Compass 315M, Manufacturer: Coherent, Germany). The time evolution of the diffraction efficiency was monitored by a 633 nm wavelength HeNe laser which was optically chopped at 80 Hz to allow for lock-in amplification of the diffracted beam using a Princeton Applied Research lock-in amplifier model 5210. A single trial is composed of two holograms recorded at separate but adjacent locations in the crystal, so that the *A* term of Eq. (33) is not changed by writing successive holograms at the same location. One hologram in the set is recorded using normal methods, while the other is recorded with the reference beam enhanced by a 100 mm long standing wave cavity with a planar entrance coupler and a 100 mm radius of curvature, concave, mirror. Switching from cavity recording to normal recording is accomplished by flipping the entrance coupler out of the beam path and blocking the spherical mirror. Each hologram was started with a 30 sec pre-exposure to allow for alignment of the read beam to the hologram. No time evolution data was collected during this exposure, so data collection starts from 30 sec. in the data displayed in Sec. 3.2. To compensate for fluctuations in readout power and errors in Bragg tuning of the read beam, each data set is normalized by the read beam power and scaled by the diffracted power measured after recording the hologram and retuning the read beam. Writing beams of 532 nm wavelength and ~36.5 μW power record a hologram inside an AR coated, 5x10x20 mm, 0.015 mole % Fe:LiNbO3 crystal. The writing beams have an estimated diameter of ~206 μm inside the crystal, and are separated by a 28.1° angle outside of the crystal. The 633 nm read beam has a power of ~2.3 μW, is chopped at 80 Hz, and has an estimated beam diameter of ~291 μm inside the crystal. The standing wave cavity is formed by a 1 inch diameter, 41.8% transmission, planar, entrance coupler, and a 1 inch diameter, 100 mm radius of curvature, 99% reflectance, dielectric mirror. The circulating power of the cavity is monitored via a 1 inch diameter beam sampler with 94% transmission. The light reflected off of that beam splitter is monitored via a Thorlabs DET110 photodiode connected to a Tektronix TDS220 oscilloscope. By monitoring the oscilloscope signal, while scanning the cavity length with the Piezo-Electric Transducer (PZT) mirror mount, alignment of the cavity mirrors was tuned to maximize the cavity finesse. Prior to writing a cavity enhanced hologram, the cavity length is tuned via a constant voltage applied to the PZT. To aid in Bragg matching the readout beam, before each data set is recorded, a weak hologram is written with a 30 second exposure, and the diffraction efficiency of the read beam is maximized by adjusting its angle. Diffracted readout power is monitored via a lock-in amplified photo diode. For non-cavity recording trials the entrance coupler is flipped out of the beam path and the spherical mirror is blocked. This procedure’s apparatus is shown in Fig. 4.

### 3.1.2 Plane wave results

Taking the scaled data from Sec. 3.1.1, a conversion factor is applied to convert it from a 633 nm wavelength diffraction efficiency to a 532 nm wavelength diffraction efficiency. This conversion factor was derived from averaging the ratio of the two wavelength diffraction efficiencies for twelve holograms recorded after the original data set. With the data now in units of 532 nm diffraction efficiency, we then fit the data to Eq. (33) with a least squares non-linear regression to find an average value for *A* of ~7. Using this fixed value of *A* we then fit the data again while only varying $\tau $. The write-rate enhancement of each trial pair was then computed by taking the ratio of the time constants. Taking all eleven pairs resulted in an average enhancement of 1.07 with a standard deviation of 0.1, a maximum of 1.2, and a minimum of 0.9. However, this mean is brought down by trials which showed no enhancement or a loss in write-rate, so it becomes meaningful to look at the average of those pairs which display an enhancement greater than 1.1. There are five such trials, which give us an average of 1.16 with a standard deviation of 0.05. Figure 5 shows the diffraction efficiency data and fitting curves for the best trial with a 1.22 enhancement in write rate, and includes a histogram of the write rate enhancements for the 11 trials. The histogram shows that recording speeds were generally enhanced, but some trials with de-enhancement were present.

#### 3.2 Image recording with a cavity enhanced reference arm

### 3.2.1 Imaging setup and procedure

Since an image bearing hologram cannot be fully Bragg matched with a reconstruction wavelength that differs from the recording, we changed to recording with a cavity in the reference arm and readout by a pseudo phase conjugate method similar to Cao et. all [28]. The reverse propagating beam in the standing wave cavity is used to read out the hologram while recording it.

Figure 6 shows the experimental setup. In lieu of a data encoded bitmap pattern, we used a Newport USAF-1951 RES-1 resolution test target as the object. The object was placed at the front focal plane of a Rolyn Optics 80.3020, 53 mm focal length, microscope objective, and the rear focus of the lens overlapped with the reference beam inside of the crystal. The object was illuminated by a ~5 mm diameter Gaussian beam. Diffraction efficiency was monitored via the phase conjugate diffraction from the hologram being recorded which was sampled with a bare microscope cover glass inserted between the object and Fourier Transform (FT) lens. The sampled diffraction was then optically chopped and lockin amplified by a photodiode. At the end of each recording the normal diffraction efficiency was measured without the cavity, and these measurements were used to convert the data from voltages to actual diffraction efficiencies. The reconstructed image was obtained by placing a Thorlabs DCC1445M CMOS camera at the focal plane of the FT lens during reconstruction, and the object was recorded by taking a picture of the signal beam at a large distance from the focal plane of the FT lens. The reference cavity was also stabilized by a proportional gain feedback loop tied to the PZT mirror and intra-cavity power.

### 3.2.2 Imaging results

Taking the scaled data from Sec. 3.2.1, the same fitting procedure of Sec. 3.1.2 was applied to find the write rate enhancement. Three trial pairs were carried out as described in Sec. 3.2.1 with the reference arm irradiance enhanced by a factor ~1.48. The mean write rate enhancement of the three trials is 1.19 with a standard deviation of 0.1, which is ~98% of the expected 1.22. Figure 7 displays the diffraction efficiency data for the best trial pair and a histogram showing the distribution of write rate enhancements.

A representative pair of object and reconstruction images are shown in Fig. 8. The object is seen to be clearly reconstructed from the recording created with a 1.54 irradiance enhanced reference beam. We would expect a write rate enhancement of 1.24.

## 4. Discussion

Returning to the theoretical discussion of Sec. 2, it is clear from Eqs. (14) and (23) that the cavity enhancement of irradiance is limited by the losses of the cavity. In particular ${G}_{F}$ is inversely related to the hologram diffraction efficiency. This relationship is advantageous for HDSS where several hundred holograms are typically multiplexed to constrain individual hologram diffraction efficiencies to less than 0.3% [19,20]. Comparing the diffraction efficiency loss to the recoding medium absorption (typically on the order of tens of percent), the diffraction efficiency makes a negligible change to the enhancement in write data rate. This makes cavity enhancement of holographic processes particularly well suited to HDSS, but recording medium absorption must be balanced against the cavity enhancement for maximum performance.

A natural limitation of cavities is their need for extreme stability. Generally, high values of ${G}_{F}$ require that the cavity length be controlled with a sub 100th wave tolerance over the duration of a single hologram exposure. This requires the use of an environmental enclosure, vibration isolation table, and high-speed automated control of the cavity length for long exposure times. Fortunately, exposure times for HDSS are on the order of 3 ms and decreasing [20], so the cavity only needs to be stabilized over a millisecond time scale allowing the system to be un-stabilized during down time and reducing the overall stabilization need.

Stability aside, let us consider the theoretical limits on cavity enhancement for typical HDSS parameters. If we design our cavity to be critically coupled [21] the entrance coupler has a power reflectance equal to the product of the remaining loss terms in the cavity, so for a standing wave cavity the maximum irradiance enhancement of Eq. (14) becomes

From Table 2 we can see that cavity enhancements are very sensitive to the losses introduced by the recording medium. According to Gleeson et. all the primary means of adjusting the absorption of photopolymers is to vary its thickness and doping concentration [33]; however, adjusting the thickness will change Bragg selectivities while changing the absorption will directly affect the sensitivity. Thus, cavity enhancement is subject to some trade-offs while providing additional degrees of freedom for system design.

Experimentally, our cavity was under coupled, ${r}_{1}<{r}_{2}(1-b-\eta )$, and we had a system with ${r}_{2}^{2}={R}_{2}=0.99,$$b=0.16$, $\eta \cong 4\times {10}^{-4}$, and ${r}_{1}^{2}={R}_{2}=0.572,$. We also had an additional transmission loss, ${t}_{samp}^{2}={T}_{samp}^{}=0.94$, due to the beam sampler in the cavity so that we would expect a maximum ${G}_{F}$ of

We managed to tune the system to ${G}_{F}\cong 2$, which is 71% of the prediction. This discrepancy is likely due to the presence of other losses in the cavity such as poor coupling or surface imperfections in the optical elements. We would expect similar reductions in for implementation in photopolymer based HDSS systems where non-idealities such as scattering, shrinkage, and media vibrations [34,35] will introduce their own losses in the cavity. However our crystal recording medium is much thicker than the interaction length of the holograms recorded, so losses could be further reduced by using a thinner crystal.

Further deviations from theory were observed: with ${G}_{F}\cong 2$ we expected a write data rate enhancement of 1.41, but saw a maximum of 1.2 with multiple trials showing no enhancement or a decrease in data rate. This performance degradation over long exposures (tens of seconds) can be attributed to the instability of the cavity: these trials were done on a vibration isolation table with no enclosure around the cavity and no cavity length stabilization. To characterize the degree of instability, the sampled and fast Fourier transformed photo diode voltage, which monitors the circulating power inside the cavity, is plotted in Fig. 9.

Figure 6 clearly shows cavity length oscillations around 2 Hz and 6.4 Hz indicating the presence of slow cavity instabilities during recording. It should also be noted that the sampling interval for Fig. 9 was 20 ms, so we have no information about higher frequency instabilities; however, both high and low frequency instabilities may be eliminated by actively tuning the cavity length. If the instabilities are only at such low frequencies the sub 3 ms exposures of current generation HDSS [20] may not require stabilization. That being said, a proportional gain feedback loop and enclosure were used in image recording, and system performance was greatly improved.

A further challenge in implementing cavity enhanced HDSS is applying the cavity to the signal arm of FT or near FT image recording used in page based storage. The plane wave theory used here would need to be replaced with an expansion in the cavity eigenmodes. The arbitrary Guoy phase shift of a given cavity limits the modes which can be coupled into, but a π phase shift would allow for propagation of all modes. Thus any image could be coupled into a π Guoy phase shift cavity, such as a concentric standing wave cavity, as an expansion in Laguerre or Hermite Gaussian beams due to these polynomials forming complete set of orthogonal basis functions [27]. Thus, the only challenge in implementing an image carrying, cavity enhanced, signal beam would be integrating the resonator with the Fourier transform recording geometry. This may be accomplished by designing the FT lens and entrance coupler such that the FT of the object is at the beam waist of the cavity. The spatial frequencies would then be limited by the choice of cavity length and clear aperture.

Challenges aside, cavity enhancement in both the reference and signal beams can be integrated into the current monocular optical system design [24] by building the entrance coupler in to the optical system and building the rest of the cavity around the optical disc. The standing wave cavity is also compatible with pseudo-phase conjugate readout [16], but the presence of the extraneous holograms and non-unit fringe visibility, as mentioned in Sec. 2, will require some fine tuning to make it suitable for recording.

Regarding the extra irradiance terms of standing wave recording in Eq. (5), the constant terms may consume dynamic range via a bulk refractive index change depending on the recording medium physics. The first cosine term (${\overrightarrow{K}}_{s\mathrm{tan}d}$of Fig. 1) will be a non-localized, reflection type hologram with no data, which will also consume dynamic range. The second and third cosine terms (${\overrightarrow{K}}_{trans}$ and ${\overrightarrow{K}}_{refl}$ of Fig. 1) will both provide data storage because one is the transmission type data hologram while the other is the reflection type data hologram; however, the reflection type data hologram can only be accessed by pseudo phase conjugate readout. All three of these terms have recorded gratings of comparable strength according to Eq. (20), so we expect at least a $1/\sqrt{3}$ effective reduction in the dynamic range of the recording material. As such, standing wave cavities require further optimization for use in HDSS, and more suitably, traveling wave cavities will need to be used to avoid the extra grating formation and achieve the most efficient use of dynamic range of the recording material.

It would also be reasonable to expect the extra gratings of Sec. 2.2.2 to play a part in pseudo phase-conjugate readout of Sec. 3.2, but the orientation of the Fe:LiNbO_{3} electroptic tensor prevents the formation of gratings while still consuming dynamic range. The extra gratings would certainly play a role in media with isotropic responses, and would cause additional cavity losses as well as increased data bearing diffraction efficiencies. However, the extra holograms were experimentally observed to be absent in Fe:LiNbO_{3}.

One way of removing these extra holograms in isotropic media is a bow-tie cavity, as mentioned earlier, but the extraneous holograms could also be eliminated by bracketing the crystal with quarter wave plates [36] and circularly polarizing the input to the cavity. Circularly polarized light is converted to a linear polarization before entering the recording medium and is reconverted to circularly polarized light after exiting the crystal. The circular polarization would then change handedness upon reflection and would pass back through the medium while orthogonally polarized relative to the forward propagating beam. This prevents the reverse propagating beam from interfering with the other beams, but it only works for isotropic media since birefringence would cause the orthogonal polarizations to take different paths through the medium. This removes the extra gratings, but the constant terms of Eq. (20) could still consume dynamic range, and would need to be optimized for.

## 5. Conclusions

Cavity enhanced writing provides theoretical boosts to write data transfer rates, and double cavity, traveling wave, recording provides the greatest improvement. The double cavity, traveling wave, case provides a theoretical enhancement of ${G}_{F}\simeq 4.4$ for typical HDSS system parameters, and this enhancement can be increased further by reducing hologram diffraction efficiencies and recording medium losses.

In the case of single standing wave cavity enhanced reference arm recording, we have demonstrated a maximum write data rate enhancement of 1.2, which is 85% of the theoretical 1.414 enhancement for plane waves, and a mean of 1.19, which is 98% of the theoretical 1.22 enhancement for image bearing waves. Cavity instabilities are observed to reduce system performance, but their low frequencies will limit their impact on the sub 3 ms exposures used in current HDSS, and instabilities can be compensated by feedback control of the cavity length [37,38].

## Acknowledgments

We would like to thank Kenichi Shimada for instruction on holographic theory, as well as Toshiki Ishii and Yusuke Nakamura for their collaboration on the project.

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