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In holographic data storage, it is difficult to retrieve data if the temperature difference between recording and reading exceeds 2 K. To widen this tolerance, a compensation method—adjusting the wavelengths and incident directions of the recording and reading beams—has been proposed. In this paper, for the first time, a method for calculating the recording and reading temperature tolerance using this compensation is introduced. To widen the narrow tolerance, typically ± 10 K, it is effective to increase the coefficient of thermal expansion (CTE) of the substrate or decrease the CTE of the photopolymer. Although reducing the Numerical aperture of the objective lens is also effective, it degrades the recording density.
©2009 Optical Society of America
1. Introduction
Dhar et al. pointed out that thermal expansion is anisotropic in the photopolymer medium developed for holographic data storage (HDS); the coefficient of thermal expansion (CTE) in the thickness direction is identical to that of the photopolymer and the CTEs in the in-plane directions are identical to those of the substrate. These properties influence both the recorded grating period and its direction; therefore, it is difficult to retrieve data if the reading and recording temperatures do not match [1]. Toishi et al. confirmed experimentally that data retrieval is difficult if the temperature difference is more than 2 K. However, they showed that retrieval is possible, even for a difference of 20 K, by adjusting the wavelength and incident direction of the reading beam [2]. We proved this property with a 2-dimensional model and pointed out that the optimum wavelength and direction of the reading beam are linear functions of the temperature difference. We also pointed out that the optimum reading wavelength is independent of the reference beam direction during recording [3,4]. Tunable lasers have been developed to a practical level as the light source for the compensation method [3,5–9].
However, even when the compensation method is applied, a kind of bend remains (Fig. 3), and grows larger as the temperature difference increases. Although I ignored this bend in the calculations of previous papers, it narrows the temperature tolerance range within which the method can compensate for the deformation of a recorded grating. To take the bend into account, the temperature tolerances were calculated for glass and amorphous-polyolefin substrates; we assumed 0 and -1.0 × 10-3 polymerization expansion rates and 1.0 and 1.5 mm thicknesses of photopolymer. Two temperature tolerances are introduced in this analysis: reading temperature tolerance and the tolerance in recording and reading temperatures. Although tolerance in recording and reading temperatures is more useful than that in reading temperature, the tolerance in recording and reading is only half of that in reading.
2. Calculation method
Typically, the term reference beam refers to both one of the recording beams and the irradiating beam used for retrieval. However, these have different wavelengths and directions. To avoid confusion, here, reference beam refers only to the recording beam and reading beam refers to the irradiating beam used for retrieval. I assume that the thickness of the substrate is negligible. Beam directions are defined in the photopolymer and measured with respect to the z-axis.
2.1 Optimum wavelength and direction of a reading beam
The optimum wavelength and direction of the reading beam (the detail is shown in subsection 2A of [4]) are explained briefly. Figure 1 shows a two-beam optical setup for angle multiplexing. On recording, the reference beam direction is defined as Γ0, the signal beam direction is defined as Ψ0, and the temperature is T 0. As the reference beam direction is presumed to be from π/6 to π/3 in the air, 300 times multiplexing is possible with a 1.7 mrad ( = 0.1°) step of angle multiplexing. Using Snell’s law, the reference beam direction in a photopolymer, Γ0, is from 0.34 to 0.62 rad. Therefore, in this paper, 0.34 rad is referred to as the inner direction, 0.62 rad is the outer direction, and 0.49 rad is the center direction. On reading, the reading beam direction is defined as Ξ1, the diffracted beam direction is defined as Φ1, and the temperature is T 1.
∆Ξ is defined as Ξ1 - Γ0, which gives the difference between the directions of the reading and reference beams. ∆λ is defined as λ 1 - λ 0, where λ 1 and λ 0 are the wavelengths of the reading and recording beams, respectively.
A photopolymer thermally expands according to the CTE of the hard substrate in the x direction, and according to its own CTE in the z direction. Therefore, the photopolymer expansions, i.e., Gx in the x direction and Gz in the z direction, are written as
Fig. 1. Two-beam optical setup for (a) recording and (b) reading. The capital Greek letters stand for the beam directions in the photopolymer.
where αx is the CTE of a substrate, T 1 is the reading temperature, T 0 is the recording temperature, αz is the CTE of a photopolymer, and s is the expansion rate by polymerization (a photopolymer shrinks by polymerization on recording, therefore, s is negative).
The change of the refractive index ∆n is written as
where n 0 is the refractive index of the photopolymer at T 0 and v is the temperature coefficient of the refractive index.
Using these equations, the combination of ∆Ξ and ∆λ satisfying the Bragg condition is written as,
It is significant that ∆Ξ depends strongly, not only on ∆λ, but also on the signal beam direction,Ψ0.
Figure 2 shows ∆Ξ versus Ψ0 calculated with Eq. (4). The parameter is the wavelength of the reading beam. The recording temperature is 25 °C, the recording wavelength is 405 nm, the reading temperature is 35 °C, and other conditions are listed in Tables 1, 2, and 3. If the reading wavelength is 402.7 nm, ∆Ξ is the same on both sides of Ψ0, i.e., at -0.66 and 0.08 rad.
When ∆Ξ satisfies the condition—it is the same on both sides—the wavelength is defined as the optimum wavelength, ∆Ξ is defined as the optimum ∆Ξ, and the curve of ∆Ξ versus Ψ0 is defined as the optimum ∆Ξ curve. (If a longitudinal axis is Ξ1 instead of ∆Ξ, it is called an optimum Ξ1 curve.) Under these conditions, ∆Ξ is almost constant from end to end. At that time, a diffracted image is clear from end to end, because the Bragg condition is satisfied at all Ψ0. This is the principle of the compensation method that adjusts the wavelength and incident direction. In [3,4], I reported the characteristics of the method by assuming that an optimum ∆Ξ curve is a straight line.
Fig. 2. The correction of the reading beam versus signal beam direction. The parameter is the wavelength of the reading beam. The recording temperature is 25 °C, the recording wavelength is 405 nm, the reading temperature is 35 °C, and the reference beam direction is 0.34 rad in the medium. The other conditions for the calculation are listed in Tables 1, 2, and 3.
Table 1. The directions of reference and signal beams at 25 °
Table 2. Medium parameters
Table 3. Recording conditions
2.2 Distribution of the optimum reading beam direction
Here, it is confirmed that the optimum Ξ1 curve is not a straight line. Figure 3 shows the reading beam versus the signal beam direction, where the recording temperature is 25 °C and the reading temperature is 45 °C. The two figures are the same except for the scale of the longitudinal axes. The scale in the left Fig. (1 mrad) is adopted in my previous paper [4], and the graph seems to be a horizontal straight line. However, the right figure, which has a longitudinal scale 20 thousand times that of the left figure, shows that it bends. The extent of bend is important; therefore, Ω. is defined as the absolute difference between the maximum and minimum in the right figure. The properties of the curve at the optimum wavelength are as follows: Ξ1 is the same at both ends, this being the definition of the optimum wavelength; the curve has an extremum near -0.4 rad of Ψ0, regardless of the reference beam direction and temperature; Ω. is smallest under various wavelengths. (It is apparent when a wavelength is much different from the optimium, as shown in Fig. 2. When a wavelength is slightly different from the optimum, The difference between Ξ1 at an end and at -0.4 rad of Ψ0 is larger, and the difference between Ξ1 at another end and at -0.4 rad of Ψ0 is smaller, compared to Ω. at optimum wavelength. Therefore, Ω. is smallest at optimum wavelength.)
Fig. 3. Reading beam versus signal beam direction, and the definition of Ω. The recording temperature is 25 °C, the reading temperature is 45 °C, and the other conditions for the calculation are listed in Tables 1, 2, and 3.
Fig. 4. The extent of the bend, Ω, versus the temperature difference between reading and recording, T 1 - T 0. The parameter is the reference beam direction, the recording temperature is 25 °C, the expansion rate by polymerization is 0, and the other conditions for the calculation are listed in Tables 1, 2, and 3.
Ω versus the temperature difference between reading and recording, T 1 - T 0, is shown in Fig. 4, where the parameter is the reference beam direction on recording, Γ0, the recording temperature is 25 °C, the expansion rate by polymerization is 0, and the other conditions for
the calculation are listed in Tables 1, 2, and 3. This figure shows that Ω increases with an increasing temperature difference because the photopolymer expands anisotropically. This suggests that the compensation method—adjusting the wavelength and direction of the reading beam—does not work if the temperature difference exceeds a certain value.
2.3 Angle tolerance of a reading beam
To retrieve recorded data, it is necessary to get a good diffracted image. This can be achieved if a reading beam is optimum in both its wavelength and direction. However, the optimum direction is distributed, as shown in Fig. 3. Therefore, it is impossible to irradiate with an optimum direction from end to end, because the reading beam is a parallel beam. As a result, the intensity of a diffracted light is not uniform.
To consider the intensity, I introduce two definitions and one assumption: the relative diffracted efficiency is defined as the diffracted efficiency normalized by the one in the Bragg condition. Θ, the angle tolerance of a reading beam, is defined as the absolute difference between the angles corresponding to 1 and 0.7 of the relative diffracted efficiency. The assumption is that data retrieval is possible if the relative diffracted efficiency exceeds 0.7.
The relative diffracted efficiency is calculated according to [10]. Figure 5 shows the angle tolerance of the reading beam, Θ, versus the signal beam direction, Ψ0, where the parameter is the direction of the reference beam, Γ0. The photopolymer is 1.0 mm thick, the refractive index of the photopolymer is 1.5 + 5.7 × 10-5 sin(Kw), K is the wave number of the grating, and w is the coordinate whose axis is parallel to the grating normal. Θ increases with decreasing Γ0 and increasing Ψ0 because the grating period becomes large. This graph is used in the next subsection.
Fig. 5. Angle tolerance of the reading beam, Θ, versus the signal beam direction, Ψ0. The parameter is the reference beam direction, Γ0. The photopolymer is 1.0 mm thick, the refractive index of the photopolymer is 1.5 + 5.7 × 10-5sin(Kw), K is the wave number of the grating, and w is the coordinate whose axis is parallel to the grating normal.
2.4 Comparison between the extent of bend and the angle tolerance of the reading beam
The reading direction, Ξ1, is minimum (maximum) at -0.66 rad of Ψ0, and maximum (minimum) near -0.4 rad of Ψ0 when T 1 - T 0 is positive (negative). Considering this property, the relationship between the extent of bend, Ω., and the angle tolerance of the reading beam, Θ, at the limit of a reading temperature is shown in Fig. 6, where Θ at -0.4 rad of Ψ0 is written as Θ-0.4, and the relative diffracted efficiencies are 0.7 at the minimum and maximum of Ξ1. This is written as
The reading temperature tolerance, calculated from Figs. 4 and 5 , is shown in Table 4, where the recording temperature is 25 °C, the substrate is made from glass, the photopolymer is 1.0 mm thick, and the expansion rate by polymerization, s, is 0. (To avoid confusion, the unit of temperature is °C and the unit of temperature difference is K.) The tolerance depends on the incident direction of the reference beam; however, only the narrowest tolerance is important because all data must be retrieved. Therefore, the tolerance is decided from -19.0 to + 20.0 K (from 6.0 to 45.0 °C). I call this the reference, although 0 expansion by polymerization is not real.
Fig. 6. The relationship between the extent of the bend, Ω, and the reading beam angle tolerance, Θ, at the limit of a reading temperature. The solid line represents Ξ1, and the two double dotted lines represent a 0.7 relative diffracted efficiency.
Table 4. The reading temperature tolerance of the grating recorded at 25 °C. The substrate is glass, the photopolymer is 1.0 mm thick, and the expansion rate by polymerization is 0.
3. Some reading temperature tolerances
Here, a few reading temperature tolerances are compared with the reference. (The calculated tolerances are listed in Table 5 of section 4.)
3.1 Amorphous-polyolefin substrate
Because of the nature of the substrate material, this case differs from the reference. The graph of the extent of bend versus T 1 - T 0 is similar to that of the reference. As the CTE of an amorphous polyolefin (APO) substrate is 8 × 10-5, it is nearer to that of photopolymer (5 × 10-4) than that of the glass substrate (7 × 10-6). Therefore, Ω. is relatively small, compared to the reference, for all temperature differences. In contrast, Θ is the same as the reference, because the photopolymer thicknesses are the same. Therefore, the reading temperature tolerance is ± 3.5 K wider than that of the reference—from -22.5 to + 23.5 K. An APO substrate is superior to a glass substrate in terms of this property.
3.2 Expansion rate of -1.0 × 10-3
Only the expansion rate by polymerization is different from 0 of the reference. -1.0 × 10-3 is close to the smallest for the present photopolymers. Ω. versus the temperature differences between reading and recording, T 1 - T 0, is shown in Fig. 7, where the parameter is the reference beam direction when recording. The lines of the graph shift + 2 K compared to the reference (Fig. 4); therefore, the reading temperature tolerance also shifts + 2 K, from -17.0 to + 22.0 K. The amount of shift depends on the interaction of shrinkage by polymerization and the thermal expansion of + 2 K. There seems to be no problem with this property—it is narrow when T 1 - T 0 is negative and wide when T 1 - T 0 is positive. However, it influences another temperature tolerance, which is introduced in section 4.
Fig. 7. The extent of the bend, Ω, versus the temperature difference between reading and recording, T 1 - T 0. The parameter is the direction of the reference beam. The expansion rate by polymerization is -1.0 × 10-3. The other conditions for the calculation are listed in Tables 1, 2, and 3.
3.3 Amorphous-polyolefin substrate and an expansion rate of -1.0 × 10-3
This combination of substrate and expansion rate is realistic, and the temperature tolerance is between -20.0 and + 26.0 K. It has the properties of both subsections, 3.1 and 3.2, i.e., a widening of temperature tolerance and a shift. The shift does not coincide with the + 2 K of subsection 3.2; it is + 2.5 K, because both the photopolymer polymerization and the APO thermal expansion need to be compensated for.
3.4 The photopolymer thickness of 1.5 mm
Here, only the photopolymer thickness differs from the 1.0 mm of the reference. The reading temperature tolerance is from -12.5 to + 13.0 K and is 2/3 of the reference value. This is because Θ is inversely proportional to the thickness, as shown in Fig. 8, where -0.34 rad of the signal beam direction is selected as an example.
Recording density increases with an increase in the photopolymer thickness because the M/# of the medium is proportional to the thickness and the angle tolerance of the reading beam is inversely proportional to the thickness. For example, the multiplexing number is 300 times for a 1.0 mm thickness and 450 times for a 1.5 mm thickness. The bit capacity at a position is calculated by multiplying the pixel number of a spatial light modulator and the multiplexing number.
Therefore, a tradeoff exists between the recording density and the tolerance of temperature difference. It seems the reading temperature tolerance is sufficient for a 1.0 mm thickness, because the tolerance is near to that of Blu-ray, ± 25 K. However, this is not expected in section 4.
Fig. 8. Angle tolerance of the reading beam, Θ, versus the photopolymer thickness. -0.34 rad of the signal beam direction is selected as an example. The other conditions for the calculation are listed in Tables 1,2, and 3.
3.5 Comparing calculations and experiment
The experimental upper limit of the reading temperature is reported in [11], where the photopolymer thickness is 1.5 mm, the expansion rate by polymerization is about 1 × 10-3, and the numerical aperture (N.A.) of the objective lens is 0.6. The direction of the reference beam is not reported. It shows that SNR abruptly worsen at + 17 K.
The calculated upper limit distributes between + 15 and + 17 K, depending on the reference beam direction. Because the experimental value is within this range, the experiment and the calculation agree well.
4. Recording and reading temperature tolerance
The recording and reading temperature tolerance is different from the tolerance of reading temperature, recording at 25 °C. First I explain the difference, then define the recording and reading temperature tolerance, whose standard recording temperature is 25 °C.
Fig. 9. The readable temperature region when a medium is recorded at various temperatures. The bold lines are the tolerances for reading temperature recorded at temperatures of T w1, T w2, and T w3, and the arrow shows the common temperature for each tolerance.
4.1 Difference between the two temperature tolerances
The reading temperature tolerance defined in section 3 is premised on the recordings at 25 °C. Here, I consider a medium recorded on at various temperatures. Because all parts of the medium must be read at a given reading temperature, the temperature must be in the range of common temperatures for each reading temperature tolerance, as shown in Fig. 9. Therefore, the recording and reading temperature tolerance is narrower in comparison to the reading temperature tolerance for recording at 25 °C.
4.2 Definition of the recording and reading temperature tolerance
The reading temperature tolerance is used to determine the tolerance of recording and reading temperature. Here, the reading temperature tolerance is written as -∆T low [K] and ∆T high [K]. Although these values are defined when the recording temperature is 25.0 °C, they are almost the same even when the recording temperature is different. The procedure for obtaining the recording and reading temperature tolerance is to select the smaller of ∆T low and ∆T high; half of the value is defined as ∆T rr (rr means recording and reading.); then, the recording and reading temperature tolerance is determined as ± ∆T rr [K].
As an example, the reference is checked. As its tolerance of reading temperature is -19.0 and + 20.0 K, half of the smaller is 9.5 K. Therefore, the recording and reading temperature tolerance is ± 9.5 K (from 15.5 to 34.5 °C, because the standard recording temperature is 25.0 °C). If the medium is recorded at 15.5 °C, the lower limit, it can be read at 34.5 °C, the upper limit, because the temperature difference is less than + 20 K of the upper limit in the tolerance of reading temperature. On the contrary, if the medium is recorded at 34.5 °C, it can be read at 15.5 °C, because the temperature difference is -19 K of the lower limit of the tolerance of the reading temperature.
4.3 Some tolerances of recording and reading temperature
The recording and reading temperature tolerances, which are also studied in section 3, are shown in Table 5. As the tolerances are about a half of the reading temperature, they are narrow. Furthermore, the tolerances become smaller if polymerization shrinkage occurs, because ∆T low becomes small. As HDS requires a tolerance of more than ± 10 K, the tolerances shown in Table 5 are insufficient. (An expansion rate of 0 is impractical.)
Table 5. Two types of temperature tolerances.
4.4 Discussion for widening the tolerance of recording and reading temperature
Reducing the N.A. of the objective lens in a holographic data storage equipment is an effective way to widen the range, because this decreases Ω—the extent of the bend. Figure 10 shows reading beam direction, Ξ1, versus signal beam direction, Ψ0, where the N.A. is 0.3, the recording temperature is 25 °C, the reading temperature is 45 °C, and the other conditions are listed in Tables 2 and 3. As the range of Ψ0 is narrow, due to the small N.A, Ω is 0.06 mrad, which is 0.3 times that shown in Fig. 3. Therefore, the reading temperature tolerance, calculated with Ω versus T 1 - T 0, is four times as large as that for a N.A. of 0.6. However, a small N.A. deteriorates the recording density because the beam spot size is large.
It is also effective to increase the CTE of the substrate or decrease the CTE of the photopolymer because the extent of the optimum ∆Ξ curve becomes small.
Fig. 10. Reading beam versus signal beam direction. The N.A. is 0.3, the recording temperature is 25 °C, and the reading temperature is 45 °C. The other conditions are listed in Tables 2 and 3.
5. Summary
As shown in previous papers, the data retrieval in holographic data storage is difficult if the temperature difference between recording and reading exceeds 2 K. However, it is possible if the wavelengths and incident directions of the recording and reading beams are adjusted. In this paper, the reading temperature tolerance of a photopolymer medium recorded at 25 °C, and the tolerance of the recording and reading temperature are calculated under the compensation method for the first time. Typically, the former tolerance is from -20.0 to + 26.0 K, and the latter tolerance is ± 10.0 K. The latter tolerance is about half that of the former and is narrow.
To widen the tolerance, it is effective to increase the CTE of the substrate or to decrease the CTE of the photopolymer, reducing the difference of the CTEs. Although lessening the N.A. of the objective lens is also effective for widening the tolerance, it deteriorates the recording density.
Acknowledgments
The author is grateful to Professor Tsutomu Shimura of the University of Tokyo for his advice and to Mitsuru Toishi of SONY for checking the manuscript.
References and links
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