The first analysis to our knowledge of the optical data storage density of photon-echo storage is presented. Mainly considering signal-to-noise ratio performance, we calculate the obtainable storage density for data storage and processing using photon echoes to be approximately 100 times the theoretical limit for conventional optical data storage. This limit is similar to that theoretically calculated for data storage by use of persistent spectral hole burning. For storage times longer than the upper-state lifetime the highest densities can, however, be obtained only if all the excited atoms decay, or are transferred, to a different state than that from which they were originally excited. The analysis is restricted to samples with low optical density, and it also assumes that for every data sequence, writing is performed only once. It is therefore not directly applicable to accumulated photon echoes. A significant feature of photon-echo storage and processing is its speed; e.g., addressing 1 kbyte/(spatial point) permits terahertz read and write speeds for transitions with transition probabilities as low as 1000 s^{−1}.

Boon K. Lee, Robert Chih-Jen Chi, David Liang-Chiun Chao, Ji Cheng, Irving Yeong-Ning Chry, Fred R. Beyette, and Andrew J. Steckl Appl. Opt. 40(21) 3552-3558 (2001)

Byoung S. Ham J. Opt. Soc. Am. B 29(3) 463-474 (2012)

References

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Accumulated photon echoes3–6 and hybrid techniques.12,13

These methods would have higher storage capacities at the expense of slower data rates.

Stored information is intensity coded.

Phase-coded information storage14,15 and methods that use external fields.^{a}

Photon-echo storage is unique in the way it pre-serves the phase information of the stored information. This is still a largely unexplored area.

See the Note added in proof at the end of this paper.

Table 2

Definition of Variables

Symbol

Quantity

Comment

N

Number of bits stored at each spatial location.

A

Einstein coefficient of the transition.

τ

Duration of one data bit.

η

Probability that an atom excited by a data pulse will store information.

The analysis shows that the material parameters predominantly determining the obtainable storage capacity are η and the product NAτ.

λ

Wavelength of the excitation light.

In the calculations, λ = 600 nm.

k

A_{x} = (kλ)^{2}.

A_{x} is the area in which the N data bits are stored.

N/k^{2}

Fractional increase in storage density in comparison with conventional optical storage.

It is assumed that conventional optical storage is capable of storing 1 bit/λ^{2}.

L

Length of storage volume (L = 2A_{x}/λ).

For consistency with previous investigations10,11 the length is chosen in terms of the Rayleigh length of the focused laser beam.

S

Number of photons in the detected signal (S = 400)

This value certifies that the required SNR is the same (20 dB) as in Refs. 10 and 11.

n

Density of centers that may store information.

n_{max}

Maximum density of centers for storing information (n_{max} = 10^{26}/m^{3})

Higher concentrations are assumed to lead to too large an interaction between the active centers. The present value is already a bit high for larger molecules, but when the active center is an atomic ion or a small molecule, this concentration is feasible.

NσL

The sample transmission is exp(−nσL). The maximum value of nσL in this paper is (nσL)_{max} = 0.3, which corresponds to 25% absorption.

The reason for this upper limit is that the analysis is made assuming an optically thin sample. Higher absorption is expected to yield somewhat higher storage density. But the limit used here is expected to give results of the correct order of magnitude.

T_{2}

Homogeneous dephasing time.

T

Inhomogeneous dephasing time.

I_{th}

Maximum light intensity that can be applied without increasing the sample temperature.

Chosen to be 10^{4} W/cm^{2} for crystalline materials submerged in superfluid liquid helium.10

η_{0}

Probability the reading process will destroy the stored information

Assumed to be zero when the equations are displayed in the figures.

m

Number of times a stored bit of data is read.

ξ

Detector quantum efficiency.

In the calculations, ξ = 0.75.

∊

Index of refraction squared.

In the calculations, ∊ = 2.25.

Table 3

Maximum Storage Density (N/k^{2})^{a} Versus Writing Efficiency (η)

η

N/k^{2}

0.008

1

0.08

10

0.8

100

Fractional improvement in storage density in comparison with conventional optical storage.

Thermal limit prevents higher transition probability A when NAτ = 10^{−6} (see Fig. 4). We assume that storage density is 10 times the conventional limit, that there are 1000 bits/point, and that writing efficiency is 0.1 (i.e., N/k^{2} = 10, N = 1000, and η = 0.1).

We assume that storage density is 100 times the conventional limit, that there is unity writing efficiency, and that the transition probability is 1000/s (i.e., N/k^{2} = 100, η = 1, and A = 1000 s^{−1}).

Tables (5)

Table 1

Limitations of the Analysis

Assumptions

Methods Not Covered

Comments

Single-shot read and write.

Accumulated photon echoes3–6 and hybrid techniques.12,13

These methods would have higher storage capacities at the expense of slower data rates.

Stored information is intensity coded.

Phase-coded information storage14,15 and methods that use external fields.^{a}

Photon-echo storage is unique in the way it pre-serves the phase information of the stored information. This is still a largely unexplored area.

See the Note added in proof at the end of this paper.

Table 2

Definition of Variables

Symbol

Quantity

Comment

N

Number of bits stored at each spatial location.

A

Einstein coefficient of the transition.

τ

Duration of one data bit.

η

Probability that an atom excited by a data pulse will store information.

The analysis shows that the material parameters predominantly determining the obtainable storage capacity are η and the product NAτ.

λ

Wavelength of the excitation light.

In the calculations, λ = 600 nm.

k

A_{x} = (kλ)^{2}.

A_{x} is the area in which the N data bits are stored.

N/k^{2}

Fractional increase in storage density in comparison with conventional optical storage.

It is assumed that conventional optical storage is capable of storing 1 bit/λ^{2}.

L

Length of storage volume (L = 2A_{x}/λ).

For consistency with previous investigations10,11 the length is chosen in terms of the Rayleigh length of the focused laser beam.

S

Number of photons in the detected signal (S = 400)

This value certifies that the required SNR is the same (20 dB) as in Refs. 10 and 11.

n

Density of centers that may store information.

n_{max}

Maximum density of centers for storing information (n_{max} = 10^{26}/m^{3})

Higher concentrations are assumed to lead to too large an interaction between the active centers. The present value is already a bit high for larger molecules, but when the active center is an atomic ion or a small molecule, this concentration is feasible.

NσL

The sample transmission is exp(−nσL). The maximum value of nσL in this paper is (nσL)_{max} = 0.3, which corresponds to 25% absorption.

The reason for this upper limit is that the analysis is made assuming an optically thin sample. Higher absorption is expected to yield somewhat higher storage density. But the limit used here is expected to give results of the correct order of magnitude.

T_{2}

Homogeneous dephasing time.

T

Inhomogeneous dephasing time.

I_{th}

Maximum light intensity that can be applied without increasing the sample temperature.

Chosen to be 10^{4} W/cm^{2} for crystalline materials submerged in superfluid liquid helium.10

η_{0}

Probability the reading process will destroy the stored information

Assumed to be zero when the equations are displayed in the figures.

m

Number of times a stored bit of data is read.

ξ

Detector quantum efficiency.

In the calculations, ξ = 0.75.

∊

Index of refraction squared.

In the calculations, ∊ = 2.25.

Table 3

Maximum Storage Density (N/k^{2})^{a} Versus Writing Efficiency (η)

η

N/k^{2}

0.008

1

0.08

10

0.8

100

Fractional improvement in storage density in comparison with conventional optical storage.

Thermal limit prevents higher transition probability A when NAτ = 10^{−6} (see Fig. 4). We assume that storage density is 10 times the conventional limit, that there are 1000 bits/point, and that writing efficiency is 0.1 (i.e., N/k^{2} = 10, N = 1000, and η = 0.1).

We assume that storage density is 100 times the conventional limit, that there is unity writing efficiency, and that the transition probability is 1000/s (i.e., N/k^{2} = 100, η = 1, and A = 1000 s^{−1}).