Abstract

Microholographic data storage system model is presented that includes non-linear and non-local behavior of the storage material for accurate simulation of the system and optimization of the writing process. For the description of the photopolymer material a diffusion based nonlocal material model is used. The diffusion equation is solved numerically and the modulation of the dielectric constant is calculated. Diffraction efficiency of simulated microholograms and measurements were compared, and they show good agreement.

© 2007 Optical Society of America

1. Introduction

The microholographic storage principle [1,2] is one of the most promising candidates for the future high capacity optical data storage systems. In microholographic storage the write laser beam is focused into a photosensitive recording material and then focused back to the same point from a reflector unit on the opposite side of the disc. Consequently, microholograms are created by two counterpropagating, highly focused beams [Fig. 1]. Microholograms are recorded in a photopolymer as permittivity change of the material due to photopolymerization. The read beam is reflected from the microhologram to reconstruct the signal.

We have previously developed a computer model of microholographic data storage recording and readout [3] to investigate interhologram and interlayer crosstalk, multilayer recording possibilities and raw bit error rate of the system. In that system model we used a simple linear material model.

The goal of the present study is to include material properties in our microholographic system model. Typical photopolymer materials exhibit saturation behavior and non-local response to irradiance due to the diffusion of monomers.

 

Fig. 1. (a) Geometry of microholographic recording, and (b) calculated refractive index modulation of a microhologram. Grayscale shows the increasing of the refractive index due to exposure.

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Diffusion models of polymerization and hologram formation processes in photopolymer materials were studied in the literature [3–8]. They are using a nonlocal-response diffusion model and solve the diffusion equation in two and four-harmonic expansions. But the analytical results are provided only for infinite gratings formed by planar waves, conditions that are far from being applicable to microholographic storage.

There exists a semi-empirical saturation model widely used in holographic storage research that describes the saturation of the cumulated grating strength with a merit number (M#). [9] This model is well applicable to multiplexed volume gratings having sizes exceeding the typical range of polymer diffusion. However, the direct application of this saturation model does not describe the effect of a written microhologram to its proximity. According to experimental observations if one writes a series of highly localized and not overlapping microholograms next to each other the first written hologram will exhibit the highest diffraction efficiency further holograms show a monotonous reduction of the efficiency. This non-local interaction between neighboring holograms is a consequence of monomer diffusion from unexposed areas of the photopolymer towards the exposed regions.

Therefore the above results cannot be used directly in our microholographic system model. The diffusion of monomers has to be taken into account during and after recording for the given hologram geometry. In the next section the diffusion equation is solved for real microholographic system conditions. We verify the model by calculating the well known saturation curve for the case of large plane wave holograms, and then we apply it to a typical microholographic pattern containing a few written spot. A comparison of the simulation with experimental results is also provided.

2. Description of the model

This model describes the hologram formation and monomer diffusion process in dry photopolymers. Our microholographic system model is 3 dimensional and the material model discussed below is also applied in 3D form. The used coordinate system is rectangular, the typical calculation volume is (10μm)3, the sampling distance is 50nm in x,y and z directions. In the paper the model is presented in one dimensional form for easier understanding.

A simple description of the recording process is given below. Light exposure induces polymerization of the material increasing polymer concentration and decreasing monomer concentration in bright areas. Monomers will diffuse from high concentration dark areas to low concentration bright areas driven by the concentration gradient. Polymer density is rising on the exposed areas therefore the refractive index increasing. Thus the written hologram is considered in the model as a permanent index modulation proportional to the local exposure dose on one hand and the monomer concentration at the time of exposure on the other hand. The non-local effect of exposure is due to the monomer diffusion and the finite size of the polymers.

According to the above process the change rate of the monomer concentration is described by the following nonlocal diffusive transport equation [6]:

uxtt=x[Dxtuxtx]0tR(x,x;t,t)Fxtuxtdtdx,

where u(x,t) is the monomer concentration, D(x,t) is the diffusion constant, F(x′,t′) is the rate of the polymerization, which is proportional to the exposure. The nonlocal response function R represents the effect of monomer concentration at x′,t′ on the polymerization at x,t due to the finite size of polymers.

By assuming an equivalent instantaneous response and assuming, that the D(x,t) diffusion constant is really a constant and independent from the location, then equation 1 can be simplified to the shape of the classical diffusion equation.

uxtt=D(t)2uxtx2RxxFxtuxtdx

The solution of this differential equation can be calculated numerically for simple hologram configurations. In case the hologram writing procedure is much faster than the diffusion process (this is the usual case for microholographic storage), we can consider that the monomer inhomogeneity induced by the exposure is instantaneous. Thus, the calculation can be split into two steps as follows.

The first step is to calculate the integral, which describes the hologram formation: the decrease of u(x,t) monomer concentration which is equal to the increase of the p(x,t) polymer concentration. The second step is to solve the diffusion equation with the obtained monomer distribution as initial condition.

An initial monomer concentration u(x,t0) is used to evaluate the integral at t0 with an exposure and polymerization occuring during a short time ∆t. Given that the time-scale of the diffusion is much longer than ∆t, equation 2 can be written in finite difference form as

uxt+Δtuxt=Δuxt=ΔtRxxFxtuxtdx

while change of the polymer concentration is ∆p(x,t)=-∆u(x,t).

Now we must calculate the integral in Equation 3. According to the literature [6] the nonlocal material response function R(x,x′) has a Gaussian form:

Rxx=exp[(xx)22σ]2πσ,

where the square root of the variance √σ is the nonlocal response length, typically given by the length of the polymer chain.

The function F(x′,t) is the rate of the polymerization. Cationic ring opening polymerization type photopolymer was used for measurements. The polymerization in this type of photopolymer depends linearly on exposure [10], this yields

Fxt=κIxt,

where κ is a fixed constant, and I(x,t) is the recording intensity distribution. Substituting equation 4 and 5 into equation 3 we got:

Δuxt=Δtexp[(xx)22σ]2πσκIxtuxtdx

We solved this convolution integral numerically.

The second step is solving the diffusion equation in Equation 2. For this we can use Green’s theorem [11]. The Green function of the classical diffusion equation is again a Gaussian spreading in time: Gdiff=14πDτexp(x2/4Dτ). The monomer concentration diffuses from neighboring locations to the point x. The monomer concentration at time t+τ can be written as

uxt+τ=uxt14πDτexp(x24).

The result is the convolution of the monomer concentration at time t and the Green function, which describes the spreading of the monomers. This convolution is easily computable numerically.

The solving process repeats these two steps (Equation 3 and 7) until all holograms are written into the material.

It is assumed, that the modulation of the dielectric constant is linearly related to the polymer concentration [6–8]. Therefore the change of the dielectric constant can be written as

Δεxt=Δεmaxpxtpmax

where pmax is the maximal achievable polymer concentration and ∆εmax is the maximal achievable dielectric constant value. Equation 8 provides the modulation of the dielectric constant, which describes holograms written into the material and can be directly used in our microholographic system model [3] to calculate diffracted electric fields from microholograms.

We assumed in the model, that the diffusion constant D(x,t) is independent from location. The diffusion equation can be solved without this assumption by applying finite-difference time domain (FDTD) method [12,13], but this slows down the solving process.

3. Verification of the model on plane wave gratings

We tested the model on a simple configuration of two counterpropagating plane waves creating a reflection holographic grating.

The animation in Fig. 2. shows the evolution of the plane wave grating during the recording process. Monomer concentration, polymer concentration and index modulation were calculated as the function of position and time. The diffraction efficiency and the cumulative diffraction efficiency as the function of exposure can also be calculated. The literature discusses this configuration, the results is a saturating curve [14]. Our model returns the same result [See right plot in Fig. 2.]. This saturation is mainly affected by the integral part of Equation 2, however, the situation is completely different in the case of microholographic recording. Microholograms have small effective volume with high exposure and the monomer concentration is strongly decreased. This generates significant diffusion from the unlit areas to the exposed areas.

 

Fig. 2. 2.2 Mbyte animation of plane wave hologram recording in photopolymer material [Media 1]

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4. Application to microholographic recording and comparison to measurements

We have carried out a series of measurements to investigate the influence of the material non-linearity and the diffusion on the hologram writing process. Animation in Fig. 3 shows the writing and readout process of three microholograms spaced 0.8μm. This animation shows the change of monomer and polymer concentration in the photopolymer material during recording, then shows the data readback with diffraction efficiency of microhologram calculated by our microholographic system model [3]. Figures 4 and 5 were measured and calculated in the same way as you can see in the animation.

 

Fig. 3. 1.1 Mbyte animation of microholographic recording and readout. [Media 2]

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Figure 4(a) shows the measured diffraction efficiency of a 3 by 3 bit pattern, and Fig. 4(b) shows the calculated diffraction efficiency of the same bit pattern. Bits are 0.8 μm far from each other and are written into the material in one layer with 0.532μm wavelength and 0.6 numerical aperture in a series starting from left to right and lines are written from bottom to top. The exposure power was 50nW, the writing time of each hologram were 100ms and the positioning time of the movement stage to the place of the next hologram required about 1 second. The material is an Aprilis CROP photopolymer. The first hologram has the highest diffraction efficiency, because it was written into empty material with maximal monomer concentration. Monomer concentration decreases at the position of the first hologram, and then diffusion averages the monomer concentration. This lowers the monomer concentration at the position of the other holograms, and results in a weaker index modulation after writing them into the material and finally smaller diffraction efficiency during readout. Figure 5 is the cross sectional view of the right three holograms from Fig. 4. Figure 5 and the comparison of Fig. 4(a) and 4(b) show, that modeling results fit well to the experiments.Continuous black line is the calculated and dashed red line is the measured diffraction efficiency of the microholograms. The scale is normalized to the strongest (bottom left) holograms. The middle hologram is shifted with about 200nm due to inaccuracy of the movement stage in the lab setup. The left three holograms are much thinner in x directions than in y direction. This effect can be caused by various reasons from destructive interference of the neighboring holograms to simple beam positioning problem during readout.

Diffraction efficiency of this bit pattern was also calculated with a saturating but nondiffusive material model. With nondiffusive material model the diffraction efficiency of the microholograms does not depend on the order of exposure, it is only a function of the total flux received by the given area of the material. These results show that both saturation and diffusion must be taken into account to properly describe the recording process and to find an optimal writing strategy.

 

Fig. 4. Measured (a) and calculated (b) diffraction efficiency from a 3x3 bit pattern. Holograms are written in order from left to right and lines are written from bottom to top. Holograms are equally exposed. See cross sectional view on Fig. 5.

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Fig. 5. Cross sectional view of Fig. 4 at the right three holograms. Continuous black line is the calculated and dashed red line is the measured diffraction efficiency of the microholograms. The scale is normalized to the strongest (bottom left) holograms.

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5. Summary

We presented a new model of microholographic data storage system including the nonlinear and non-local behavior of the photopolymer storage material. It is based on nonlocal-response diffusion model and the diffusion equation is solved numerically for real microholographic gratings. We have verified the model by applying it on well known plane wave gratings, and observed a very good match with saturation curves described in the literature. Monomer concentration, polymer concentration and modulation of dielectric constant can be calculated. The modulation of the dielectric constant describes the hologram, and diffraction efficiency of microholograms was calculated. We have carried out a series of measurements to investigate the influence of the material non-linearity and the diffusion on the hologram writing process. Calculated diffraction efficiency fit well to the experiments. Results show that diffusion must be taken into account to properly describe the recording process.

Acknowledgments

This study was supported by the European Union under the project Microholas, the Hungarian Academy of Sciences (BO/00097/03) and the Hungarian Scientific Research Fund (OTKA) grant No. T 046667.

References and links

1. H.J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, “High-density disk storage by multiplexed microholograms,” IEEE J. Sel. Top. Quantum Electron 4,840–848 (1998) [CrossRef]  

2. S. Orlic, S. Ulm, and H. J. Eichler, “3D bit-oriented optical storage in photopolymers,” J. Opt. A 3,72–81 (2001) [CrossRef]  

3. Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lörincz, “Modeling of multilayer microholographic data storage,” Appl. Opt (to be published, 2007) [CrossRef]   [PubMed]  

4. W. S. Colburn and K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt 10,1636–1641 (1971) [CrossRef]   [PubMed]  

5. G. Zhao and P. Mouroulis, “Diffusion model of hologram formation in dry photopolymer materials,” J. Mod. Opt 41,1929–1939 (1994) [CrossRef]  

6. J. T. Sheridan and J. R. Lawrence, “Nonlocal-response diffusion model of holographic recording in photopolymer,” J. Opt. Soc. Am. A 17,1108–1114 (2000) [CrossRef]  

7. F. T. O’Neill, J. R. Lawrence, and J. T. Sheridan, “Comparison of holographic photopolymer materials by use of analytic nonlocal diffusion models,” Appl. Opt 41,845–852 (2002) [CrossRef]   [PubMed]  

8. J. R. Lawrence, F. T. O’Neill, and J. T. Sheridan, “Adjusted intensity nonlocal diffusion model of photopolymer grating formation,” J. Opt. Soc. Am. B 19,621–629 (2002) [CrossRef]  

9. F. Mok, G.W. Burr, and D. Psaltis, “A system metric for holographic memory systems,” Opt. Lett 21,896–898 (1996) [CrossRef]   [PubMed]  

10. L. Dhar, A. Hale, H.E. Katz, M.L. Schilling, M.G. Schnoes, and F.G. Schilling, “Recording media that exhibit high dynamic range for digital holographic data storage,” Opt. Lett 24,487–489 (1999) [CrossRef]  

11. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (Dover Publications, 2000), Chapter 10

12. S.-D. Wu and E. Glytsis, “Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis,” J. Opt. Soc. Am. B 20,1177–1188 (2003) [CrossRef]  

13. J. V. Kelly, M. R. Gleeson, C. E. Close, F. T. O’Neill, J. T. Sheridan, S. Gallego, and C. Neipp, “Temporal analysis of grating formation in photopolymer using the nonlocal polymerization driven diffusion model,” Opt. Express 13,6990–7004 (2005) [CrossRef]   [PubMed]  

14. H. J. Coufal, D. Psaltis, and G. T. Sincerbox, Holographic data storage, (Springer, 2000), Part II -Photopolymer systems

References

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  1. H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
    [CrossRef]
  2. S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
    [CrossRef]
  3. Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
    [CrossRef] [PubMed]
  4. W. S. Colburn and K. A. Haines, "Volume hologram formation in photopolymer materials," Appl. Opt. 10, 1636-1641 (1971).
    [CrossRef] [PubMed]
  5. G. Zhao and P. Mouroulis, "Diffusion model of hologram formation in dry photopolymer materials," J. Mod. Opt. 41, 1929-1939 (1994).
    [CrossRef]
  6. J. T. Sheridan and J. R. Lawrence, "Nonlocal-response diffusion model of holographic recording in photopolymer," J. Opt. Soc. Am. A 17, 1108-1114 (2000).
    [CrossRef]
  7. F. T. O'Neill, J. R. Lawrence, and J. T. Sheridan, "Comparison of holographic photopolymer materials by use of analytic nonlocal diffusion models," Appl. Opt. 41, 845-852 (2002).
    [CrossRef] [PubMed]
  8. J. R. Lawrence, F. T. O'Neill, and J. T. Sheridan, "Adjusted intensity nonlocal diffusion model of photopolymer grating formation," J. Opt. Soc. Am. B 19, 621-629 (2002).
    [CrossRef]
  9. F. Mok, G.W. Burr, and D. Psaltis, "A system metric for holographic memory systems," Opt. Lett. 21, 896-898 (1996).
    [CrossRef] [PubMed]
  10. L. Dhar, A. Hale, H. E. Katz, M. L. Schilling, M. G. Schnoes, and F. G. Schilling, "Recording media that exhibit high dynamic range for digital holographic data storage," Opt. Lett. 24, 487-489 (1999).
    [CrossRef]
  11. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (Dover Publications, 2000), Chapter 10.
  12. S.-D. Wu and E. Glytsis, "Holographic grating formation in photopolymers: analysis and experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis," J. Opt. Soc. Am. B 20, 1177-1188 (2003).
    [CrossRef]
  13. J. V. Kelly, M. R. Gleeson, C. E. Close, F. T. O'Neill, J. T. Sheridan, S. Gallego, and C. Neipp, "Temporal analysis of grating formation in photopolymer using the nonlocal polymerization driven diffusion model," Opt. Express 13, 6990-7004 (2005).
    [CrossRef] [PubMed]
  14. H. J. Coufal, D. Psaltis and G. T. Sincerbox, Holographic data storage, (Springer, 2000), Part II - Photopolymer systems

2007 (1)

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

2005 (1)

2003 (1)

2002 (2)

2001 (1)

S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

1996 (1)

1994 (1)

G. Zhao and P. Mouroulis, "Diffusion model of hologram formation in dry photopolymer materials," J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

1971 (1)

Burr, G.W.

Close, C. E.

Colburn, W. S.

Dhar, L.

Dietz, E.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

Eichler, H. J.

S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
[CrossRef]

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

Frohmann, S.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

Gallego, S.

Gleeson, M. R.

Glytsis, E.

Haines, K. A.

Hale, A.

Katz, H. E.

Kelly, J. V.

Koppa, P.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

Kümmel, P.

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

Lawrence, J. R.

Lorincz, E.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

Mok, F.

Mouroulis, P.

G. Zhao and P. Mouroulis, "Diffusion model of hologram formation in dry photopolymer materials," J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

Nagy, Zs.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

Neipp, C.

O'Neill, F. T.

Orlic, S.

Zs. Nagy, P. Koppa, E. Dietz, S. Frohmann, S. Orlic, and E. Lőrincz, "Modeling of multilayer microholographic data storage," Appl. Opt.(to be published, 2007).
[CrossRef] [PubMed]

S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
[CrossRef]

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

Psaltis, D.

Schilling, F. G.

Schilling, M. L.

Schnoes, M. G.

Sheridan, J. T.

Ulm, S.

S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
[CrossRef]

Wappelt, A.

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

Wu, S.-D.

Zhao, G.

G. Zhao and P. Mouroulis, "Diffusion model of hologram formation in dry photopolymer materials," J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

Appl. Opt. (3)

IEEE J. Sel. Tops. Quantum Electron. (1)

H. J. Eichler, P. Kümmel, S. Orlic, and A. Wappelt, "High-density disk storage by multiplexed microholograms," IEEE J. Sel. Tops. Quantum Electron. 4, 840-848 (1998).
[CrossRef]

J. Mod. Opt. (1)

G. Zhao and P. Mouroulis, "Diffusion model of hologram formation in dry photopolymer materials," J. Mod. Opt. 41, 1929-1939 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

S. Orlic, S. Ulm, and H. J. Eichler, "3D bit-oriented optical storage in photopolymers," J. Opt. A, Pure Appl. Opt. 3, 72-81 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (2)

Other (2)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (Dover Publications, 2000), Chapter 10.

H. J. Coufal, D. Psaltis and G. T. Sincerbox, Holographic data storage, (Springer, 2000), Part II - Photopolymer systems

Supplementary Material (2)

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Figures (5)

Fig. 1.
Fig. 1.

(a) Geometry of microholographic recording, and (b) calculated refractive index modulation of a microhologram. Grayscale shows the increasing of the refractive index due to exposure.

Fig. 2.
Fig. 2.

2.2 Mbyte animation of plane wave hologram recording in photopolymer material [Media 1]

Fig. 3.
Fig. 3.

1.1 Mbyte animation of microholographic recording and readout. [Media 2]

Fig. 4.
Fig. 4.

Measured (a) and calculated (b) diffraction efficiency from a 3x3 bit pattern. Holograms are written in order from left to right and lines are written from bottom to top. Holograms are equally exposed. See cross sectional view on Fig. 5.

Fig. 5.
Fig. 5.

Cross sectional view of Fig. 4 at the right three holograms. Continuous black line is the calculated and dashed red line is the measured diffraction efficiency of the microholograms. The scale is normalized to the strongest (bottom left) holograms.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

u x t t = x [ D x t u x t x ] 0 t R ( x , x ; t , t ) F x t u x t dt dx ,
u x t t = D ( t ) 2 u x t x 2 R x x F x t u x t dx
u x t + Δ t u x t = Δ u x t = Δ t R x x F x t u x t dx
R x x = exp [ ( x x ) 2 2 σ ] 2 πσ ,
F x t = κ I x t ,
Δ u x t = Δ t exp [ ( x x ) 2 2 σ ] 2 πσ κ I x t u x t dx
u x t + τ = u x t 1 4 πDτ exp ( x 2 4 ) .
Δε x t = Δ ε max p x t p max

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