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Binary processes in holographic memories and application to the calculation of capacity
Binary processes in holographic memories and application to the calculation of capacity M. JEUDY
A theoretical model is used to calculate the capacity of both classical and Lippmann holograms, and experimental examples of reconstructed images are given. The greater capacity of the Lippmann hologram is thus shown, and the advantage of.using an alphabet composed of binary elements for storing text can be seen.
Previous studies have already been made to evaluate the amount of information which can be stored in a hologram* 1-3 In this paper the optimum use of the photographic emulsion as a function of the recording conditions is described for a Fourier hologram. Firstly the information recorded by a noiseless detector with an infinite passband is determined. The capacity of the emulsion is then obtained by considering the amount of information lost because of the granular nature of the detector and this is evaluated by means of the transfer function of the emulsion. We thus see the part played by the type of information stored and, in particular, the advantage of binary information.
For a photographic detector having a transfer function r(hr), the loss of information per degree of freedom in a frequency band (N1, N2) is required. According to Shannon and Weaver 4 and considering the emulsion as a linear filter, this is given by:
N2
/
WW’
(2)
NI
+Ar>
(3)
model
To distinguish the surface from the thickness of the emulsion, the latter is represented as a superposition of elementary layers having a binary response. If each of these contains only one sensitive grain in its thickness, the response of a layer to illumination E will, after development, give zero or unit transparency according to whether E has been greater than or less than the grain sensitivity. The absorption in the emulsion thickness, that is the number of layers, will serve to transform the binary response on the surface to a linear response, at least for a certain range of illumination. Calculation
log2
-N1
The capacity is then C, where: C=1,,(1
Photographic
N2
l
M=
Capacity for a hologram of the classical type If N is the modulation frequency in the hologram and IV the spectral width of the object, the limit W = N gives: N, =N-
W/2 =N/2
N2=N+Wf2=3N/2
(4)
To simplify the calculation of Al, the experimental transfer function of Kodak 649F emulsion 5 is resolved into three linear regions:
of the capacity of the hologram For 0
Method
For a spectral region (0, IV), the information recorded by an ideal detector having m levels is given by: For 900
I, = /
(1)
IT(
= 0.8 - 2
(N - 900)
U
For 1 400
de Physique GBnkale et BesanGon Cedex, France.
OPTICS AND LASER TECHNOLOGY.
Opt&e. Received
DECEMBER
1973
/r(N)1 = 0.4 -
0.4 =(N-
1400)
253
N [mm? Capacity for a hologram of the classical type. Capacity Fig.1 of Kodak 649F film as a function of modulation frequency
Fig.3 Reconstructed image from a Lippmann_$ype corresponding to a capacity of 4 x 10’ bits mm
hologram
The number m of levels is equal to the emulsion thickness (h = 16.5 pm) divided by the average distance between the grains (d = 0.18 pm) determined by electron microscopy. Finally: N2 C(N)=-log2m 2
log2
IdN)I fl1(5)
Values of the capacity C(N) as a function of modulation frequency N are given by the graph in Fig. 1. The maximum (1.4 x IO6 bits mmm2) is reached for N = 900 mm-l, that is for an interference angle of about 35O. Capacity
N [mm?
Capacity for a hologram of the Lippmann type. InformaFig.2 tion recorded in Kodak 649F film as a function of spectral width of the object
254
for a hologram
of the Lippmann
type
In this case one no longer has spectral modulation, but a system of stationary waves set up in the emulsion by the object and reference waves. For the same recorded information, the noise which appears in the photographic detector is different according to whether the recording is a classical or a Lippmann type.
OPTICS AND LASER TECHNOLOGY.
DECEMBER
1973
The capacity CL is defined as the maximum amount of information recorded as a function of the width W of the object spectrum:
The curve in Fig.2 shows that C, = 4.1 x lo6 bits mme2 for an object whose spectrum has a width of 1 550 mm-l.
Experiment Fig.3 shows the quality of a reconstructed image from a Lippmann type hologram corresponding to an information density of 4 x lo5 bits mm -2. The difference between the two types of hologram is apparent from the reconstructed texts shown in Fig.4. We are able to see that there is an advantage in using an alphabet composed of binary elements for storing text. If ri is the distribution function of the letter li in a given text, the spectrum of the text is written as:
TF[x(l&)] i
=>
($8~) i
(0 denotes a convolution product), of a binary distribution is:
whereas the spectrum
TF [(f 8 r)] = TF (f) . TF (r) One thus resolves shape and position for each letter, the latter becoming the only important factor because the shape f is already known.
Conclusion With the help of a theoretical model, we have shown the advantage of an arrangement of the Lippmann type for realizing a holographic store with photo-sensitive emulsion, and verified this ex erimentally. The corresponding capacity (4.1 x 10 6p.bits mmw2) is three times greater than that obtained with an arrangement of the classical type. Consideration of the spectral density of the information shows that there is an advantage in translating the information to a binary alphabet before putting it into store.
Acknowledgements I wish to thank Professor J. Ch. Vi&rot for the encouragement he has given me in this work, and C. Froehly for.his advice.
References Van Heerden,
Reconstructed text from holograms corresponding Fig.4 capacity of 4 x 10’ bits mm”. a - Classical hologram; b - Lippmann hologram
OPTICS AND LASER TECHNOLOGY.
to a
DECEMBER
1973
P. J. Appl Opt 2 (1963) 393 Aristov, V. V. Opt Comm3 (1971) 194 Lee, W. H. J Opt Sot Am 62 (1972) 797 Shannon, C. E., Weaver, W. The Mathematical Theory of Communication (University of Illinois Press 1949) Prenel, J. P. Nile Rev d’O@ Appl2 (1971) 263
255
A theoretical model is used to calculate the capacity of both classical and Lippmann holograms, and experimental examples of reconstructed images are given. The greater capacity of the Lippmann hologram is thus shown, and the advantage of.using an alphabet composed of binary elements for storing text can be seen.
Previous studies have already been made to evaluate the amount of information which can be stored in a hologram* 1-3 In this paper the optimum use of the photographic emulsion as a function of the recording conditions is described for a Fourier hologram. Firstly the information recorded by a noiseless detector with an infinite passband is determined. The capacity of the emulsion is then obtained by considering the amount of information lost because of the granular nature of the detector and this is evaluated by means of the transfer function of the emulsion. We thus see the part played by the type of information stored and, in particular, the advantage of binary information.
For a photographic detector having a transfer function r(hr), the loss of information per degree of freedom in a frequency band (N1, N2) is required. According to Shannon and Weaver 4 and considering the emulsion as a linear filter, this is given by:
N2
/
WW’
(2)
NI
+Ar>
(3)
model
To distinguish the surface from the thickness of the emulsion, the latter is represented as a superposition of elementary layers having a binary response. If each of these contains only one sensitive grain in its thickness, the response of a layer to illumination E will, after development, give zero or unit transparency according to whether E has been greater than or less than the grain sensitivity. The absorption in the emulsion thickness, that is the number of layers, will serve to transform the binary response on the surface to a linear response, at least for a certain range of illumination. Calculation
log2
-N1
The capacity is then C, where: C=1,,(1
Photographic
N2
l
M=
Capacity for a hologram of the classical type If N is the modulation frequency in the hologram and IV the spectral width of the object, the limit W = N gives: N, =N-
W/2 =N/2
N2=N+Wf2=3N/2
(4)
To simplify the calculation of Al, the experimental transfer function of Kodak 649F emulsion 5 is resolved into three linear regions:
of the capacity of the hologram For 0
Method
For a spectral region (0, IV), the information recorded by an ideal detector having m levels is given by: For 900
I, = /
(1)
IT(
= 0.8 - 2
(N - 900)
U
For 1 400
de Physique GBnkale et BesanGon Cedex, France.
OPTICS AND LASER TECHNOLOGY.
Opt&e. Received
DECEMBER
1973
/r(N)1 = 0.4 -
0.4 =(N-
1400)
253
N [mm? Capacity for a hologram of the classical type. Capacity Fig.1 of Kodak 649F film as a function of modulation frequency
Fig.3 Reconstructed image from a Lippmann_$ype corresponding to a capacity of 4 x 10’ bits mm
hologram
The number m of levels is equal to the emulsion thickness (h = 16.5 pm) divided by the average distance between the grains (d = 0.18 pm) determined by electron microscopy. Finally: N2 C(N)=-log2m 2
log2
IdN)I fl1(5)
Values of the capacity C(N) as a function of modulation frequency N are given by the graph in Fig. 1. The maximum (1.4 x IO6 bits mmm2) is reached for N = 900 mm-l, that is for an interference angle of about 35O. Capacity
N [mm?
Capacity for a hologram of the Lippmann type. InformaFig.2 tion recorded in Kodak 649F film as a function of spectral width of the object
254
for a hologram
of the Lippmann
type
In this case one no longer has spectral modulation, but a system of stationary waves set up in the emulsion by the object and reference waves. For the same recorded information, the noise which appears in the photographic detector is different according to whether the recording is a classical or a Lippmann type.
OPTICS AND LASER TECHNOLOGY.
DECEMBER
1973
The capacity CL is defined as the maximum amount of information recorded as a function of the width W of the object spectrum:
The curve in Fig.2 shows that C, = 4.1 x lo6 bits mme2 for an object whose spectrum has a width of 1 550 mm-l.
Experiment Fig.3 shows the quality of a reconstructed image from a Lippmann type hologram corresponding to an information density of 4 x lo5 bits mm -2. The difference between the two types of hologram is apparent from the reconstructed texts shown in Fig.4. We are able to see that there is an advantage in using an alphabet composed of binary elements for storing text. If ri is the distribution function of the letter li in a given text, the spectrum of the text is written as:
TF[x(l&)] i
=>
($8~) i
(0 denotes a convolution product), of a binary distribution is:
whereas the spectrum
TF [(f 8 r)] = TF (f) . TF (r) One thus resolves shape and position for each letter, the latter becoming the only important factor because the shape f is already known.
Conclusion With the help of a theoretical model, we have shown the advantage of an arrangement of the Lippmann type for realizing a holographic store with photo-sensitive emulsion, and verified this ex erimentally. The corresponding capacity (4.1 x 10 6p.bits mmw2) is three times greater than that obtained with an arrangement of the classical type. Consideration of the spectral density of the information shows that there is an advantage in translating the information to a binary alphabet before putting it into store.
Acknowledgements I wish to thank Professor J. Ch. Vi&rot for the encouragement he has given me in this work, and C. Froehly for.his advice.
References Van Heerden,
Reconstructed text from holograms corresponding Fig.4 capacity of 4 x 10’ bits mm”. a - Classical hologram; b - Lippmann hologram
OPTICS AND LASER TECHNOLOGY.
to a
DECEMBER
1973
P. J. Appl Opt 2 (1963) 393 Aristov, V. V. Opt Comm3 (1971) 194 Lee, W. H. J Opt Sot Am 62 (1972) 797 Shannon, C. E., Weaver, W. The Mathematical Theory of Communication (University of Illinois Press 1949) Prenel, J. P. Nile Rev d’O@ Appl2 (1971) 263
255