Generalized model of holographic recording in photopolymer materials
The generalized diffusion model of holographic recording in photopolymer materials has been offered. The theoretical description of hologram formation process is based on the concept of free volume redistribution and using the generalized diffusion equation. Free volume is produced in photopolymer m...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Cite this: | Generalized model of holographic recording in photopolymer materials / H.M. Karpov, V.V. Obukhovsky, T.N. Smirnova // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 66-70. — Бібліогр.: 17 назв. — англ. |
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Karpov, H.M. Obukhovsky, V.V. Smirnova, T.N. 2017-06-10T08:03:39Z 2017-06-10T08:03:39Z 1999 Generalized model of holographic recording in photopolymer materials / H.M. Karpov, V.V. Obukhovsky, T.N. Smirnova // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 66-70. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS: 42.70.L, 42.40.Eq, 66.10.C. https://nasplib.isofts.kiev.ua/handle/123456789/119868 The generalized diffusion model of holographic recording in photopolymer materials has been offered. The theoretical description of hologram formation process is based on the concept of free volume redistribution and using the generalized diffusion equation. Free volume is produced in photopolymer medium due to the polymer shrinkage effect. The developed theory allows to take into account influence of inhomogeneous monomer distribution during recording process and shrinkage rate on a kinetics of a hologram formation. The principal influence of the diffusion/polymerization rates ratio on the hologram properties has been shown. The offered model allows also to describe relief formation process on the surface of a recording layer. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Generalized model of holographic recording in photopolymer materials Article published earlier |
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The generalized diffusion model of holographic recording in photopolymer materials has been offered. The theoretical description of hologram formation process is based on the concept of free volume redistribution and using the generalized diffusion equation. Free volume is produced in photopolymer medium due to the polymer shrinkage effect. The developed theory allows to take into account influence of inhomogeneous monomer distribution during recording process and shrinkage rate on a kinetics of a hologram formation. The principal influence of the diffusion/polymerization rates ratio on the hologram properties has been shown. The offered model allows also to describe relief formation process on the surface of a recording layer.
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Generalized model of holographic recording in photopolymer materials / H.M. Karpov, V.V. Obukhovsky, T.N. Smirnova // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 66-70. — Бібліогр.: 17 назв. — англ. |
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66 © 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 3. P. 66-70.
PACS: 42.70.L, 42.40.Eq, 66.10.C.
Generalized model of holographic recording
in photopolymer materials
H.M. Karpov1, V.V. Obukhovsky1, T.N. Smirnova2
1 Radiophysical Department, Kiev University, 60, Vladimirskaya St., 252033 Kiev,Ukraine
2 Institute of Physics of the NASU, 46, Prospekt Nauky, Kiev-22, Ukraine
(044)266-05-40; karpov@mail.univ.kiev.ua (H.M. Karpov)
(044)266-05-40; vvo@rpd.univ.kiev.ua (V.V. Obukhovsky)
(044)265-50-72; smirnova@iop.kiev.ua (T.N. Smirnova)
Abstract. The generalized diffusion model of holographic recording in photopolymer materials has
been offered. The theoretical description of hologram formation process is based on the concept of free
volume redistribution and using the generalized diffusion equation. Free volume is produced in
photopolymer medium due to the polymer shrinkage effect. The developed theory allows to take into
account influence of inhomogeneous monomer distribution during recording process and shrinkage rate
on a kinetics of a hologram formation. The principal influence of the diffusion/polymerization rates ratio
on the hologram properties has been shown. The offered model allows also to describe relief formation
process on the surface of a recording layer.
Keywords: photopolymer, holography, diffusion, shrinkage.
Paper received 26.07.99; revised manuscript received 30.09.99; accepted for publication 04.10.99.
1. Introduction
The quantitative theory of the holograms formation in
photopolymer materials has been developed only during last
10 years [1-7]. The theoretical model under the considera-
tion is based on the contemplation of two mutually connected
processes: photopolymerization and diffusion. Photopo-
lymerization occurs under the action of a inhomogeneous
light field and is accompanied with the photoinduced diffu-
sion redistribution of the photopolymer components. As a
result of these processes the stable phase hologram is formed.
In the papers [2,4] the consideration of matter transfer
processes is based on standard diffusion equation. This
method assumes the ability of free redistribution of the mono-
mer to a homogeneous equilibrium state. However, the given
assumption was not proved in the mentioned papers and its
insufficient correctness is obvious by closer consideration.
Certainly, the presence of monomer concentration gra-
dient, which occurs during the exposure of the medium in a
inhomogeneous light field, is not a sufficient condition for
the occurrence of the diffusion matter transfer. Under con-
dition of total volume conservation, the monomer flow from
dark to bright regions must be accompanied with the flow
of substance in the opposite direction. This circumstance
was not considered in the papers mentioned above.
In real photopolymer compositions (which we named
«photoformers» [8]) the appearance of diffusion flows be-
comes possible due to polymer shrinkage effect [9-13] or
due to the presence of an additional mobile component in
the photosensitive medium [14-18].
The generalization of the diffusion equation for the case
of diffusion redistribution of two mobile components (mono-
mer and neutral component) with presence of the third mo-
tionless component (polymer) was carried out in [6].
In this paper the approach to the description of diffusion
processes in photopolymer materials in presence of the
shrinkage effect is considered on the basis of generalized
diffusion equation obtained in [6].
2. Polymer shrinkage effect and diffusion
transfer of matter
In this paper the shrinkage effect is considered as the forma-
tion of «free volume», which allows the monomer redistri-
bution within medium bulk.
The main idea of our approach in the description of the
diffusion process is the considering of the free volume as
additional mobile component, which we designate as X. Thus,
we have a model of the medium consisted of three compo-
nents and two of them (monomer and X component) are
mobile. In this case the total volume of the system (with X
component) can be considered as the constant one.
H.M. Karpov et al.: Generalized model of holographic recording ...
67SQO, 2(3), 1999
It is known that under the condition of total volume con-
servation there are n − 1 independent gradients and n − 1
independent flows in a n-component mixture. Taking into
account an immovability of the polymer there are only one
independent flow and two independent gradients in our case.
The generalized diffusion equation for a similar media (bi-
nary photopolymer compositions) was obtained in our paper
[6]. In this case the diffusion equation can be written down as
follows
[ ]),(),(),(),( tXtMtMtXDXM rrrrjj ∇−∇−=−= , (1)
where M, jM, X, jX are volume fractions (normalized con-
centration) and flows of monomer and X component accord-
ingly; D is diffusion coefficient. Note, that in a general case
D = D(M, P, ...) is a function of system state depending on
coordinates and time.
In the phase of partial polymerization diffusion proc-
esses aspire to redistribute of monomer to the state, which is
intermediate between homogeneous monomer distribution
and homogeneous medium density. Thus, distributions
),( tM e r and ),( tX e r , corresponding to an equilibrium
state for fixed time t, satisfy the condition
)(),(/),( tftMtX ee =rr , (2)
where f(t) does not depend on spatial coordinates. This con-
dition corresponds to homogeneous monomer distribution
throughout free volume formed as polymer shrinkage re-
sult. Note, that in general case equilibrium distribution of
monomer ),( tM e r is inhomogeneous due to the presence
of the polymer.
It should be emphasized that the Eq. (1), in contrast to
the Fick�s diffusion equation, is nonlinear even in the case
when diffusion coefficient D is constant.
3. General system of equations for
inhomogeneous photopolymerization
Basing on the model considered above, it is possible to write
down the general system of equations for inhomogeneous
polymerization of the photoformer.
We write down a conservation law of an arbitrary physi-
cal quantity A in the standard form
),(),(
),(
tSt
t
tA
AA rrj
r =∇+
∂
∂
, (3)
where ),( tSA r is sources density of quantity A.
In general case the sources density function of monomer is
),(),(),( tMtQtSM rrr −= , (4)
where ),( tQ r is local polymerization rate. Note that sources
density of monomer is negative, that reflects its consump-
tion in polymerization reaction.
Generally, a local polymerization rate is connected to
distribution of irradiation in the interference pattern by
nonlinear fashion. Besides, this connection is also nonlocal
(at least, for time). This is proved, for example, by presence
of postexposure self-amplification of the holograms in the
photopolymer media [19].
Obtaining an explicit relationship between ),( tQ r and
),( tI r is a complicated problem which exceeds the bounds
of this paper. Note only that the approaches of the solution
of this problem can be based on the consideration of chemi-
cal reactions kinetics or on the analysis of physical kinetics
of phase transformation processes. In the first case ),( tQ r ,
to within coefficient, corresponds to distribution of poly-
mer radicals concentration which can be connected to ),( tI r
by the system of differential equations [6].
To obtain the equation for the density of polymer and X
component sources , we define shrinkage coefficient as fol-
lows:
M
PM
V
VV
s
−= , (5)
where VM, VP are volumes of monomer and polymer, re-
spectively.
Taking into account (5), the equations for the density of X
and P components sources can be written down as follows:
MX sSS −= ,
MP SsS )1( −−= , (6)
where SM is the monomer sources density determined by
expression (4).
As the result, using the equations (1,4,6), we can obtain
the following system of equations which describes polym-
erization of the photoformer:
( )[ ]),(),(),(),(),(),(
),(
tMtXtXtMDtMtsQ
t
tX
rrrrrr
r ∇−∇∇+=
∂
∂
,
(7a)
( )[ ]),(),(),(),(),(),(
),(
tXtMtMtXDtMtQ
t
tM
rrrrrr
r ∇−∇∇+−=
∂
∂
,
(7b)
),(),()1(
),(
tMtQs
t
tP
rr
r −=
∂
∂
. (7c)
The system (7) has to be complemented with the initial con-
ditions:
1)0,( ==tM r , 0)0,( ==tX r , 0)0,( ==tP r . (8)
If the volume fractions of the initiator and intermediate
products of polymerization reaction are small (that is a typi-
cal situation), the law of total volume conservation is
1),(),(),( =++ tXtPtM rrr . (9)
In turn, the expression (9) allows excluding one equation
from the system (7).
4. Model analysis and discussion of results
The analysis of the developed model of holographic record-
ing was carried out by the numerical solution of the system
(7) for the case of recording of the transmitting hologram
grating in the interference pattern of two laser beams. In the
simplest case the spatial distribution of irradiance in the re-
cording layer is
)]/2cos(1[)( 0 Λ+= xmIxI π , (10)
H.M. Karpov et al.: Generalized model of holographic recording ...
68 SQO, 2(3), 1999
where 210 III += is the total intensity of recording beams;
)/(2 2121 IIIIm += is the fringe visibility; Λ is the fringe
spacing (the x axis is directed along recording layer surface).
Comparability with results of papers [13,14] was
achieved by taking local polymerization rate function as
follows
)(),( 2/1 xIktxQ p= . (11)
As it follows from [7] this form of Q(x, t) can be considered
as zero approach for compositions with the radical polym-
erization mechanism. The representation Q(x, t) as (11) does
not allow describing some important effects (for example,
postexposure self-amplification of the holograms), however,
it allows us to be concentrated on the most essential aspects
of holograms formation in the photopolymers.
For the convenience of the further analysis we rewrite
system (7) using dimensionless coordinates with excluding
one of the equations accordingly to (9):
,
),(
),(
),(
),(
),()f(
),(
2
2
2
2
′
′′′′−
′
′′′′+
+′′′=
′
′′
x
txM
txX
x
txX
txMD
txMxs
t
txX
ef
∂
∂
∂
∂
∂
∂
(12a)
),()f()1(
),(
txMxs
t
txP ′′′−=
′
′′
∂
∂
. (12b)
Here x,= x/Λ, t,= t /τp are dimensionless coordinate and
time; 12/1
0 )( −= Ik ppτ is characteristic time of polymeriza-
tion process; f ( )2cos(1)( xmx ′+=′ π ; 2
0 / Λ= DD pef τ is ef-
fective diffusion coefficient. In the system (12) we also as-
sume that the diffusion coefficient during the recording proc-
ess is constant and equal to D0.
Diffraction properties of the holographic grating are de-
termined by the amplitudes of spatial harmonics of refrac-
tive index modulation and can be computed, for example,
by using coupled wave theory which includes these ampli-
tudes as parameters. Therefore, for the description of dif-
fraction properties of the grating, we will use amplitudes of
spatial harmonics of component volume fraction modula-
tion which are determined by expression
∫
Λ
Λ−
Λ
Λ
=
2/
2/
)/2cos(),(
2
)( dxixtxYtYi π , (13)
where i = 1, 2, ... and Y can be M, P and X. Further we will
designate amplitude of the i-th harmonic by index «i» in
corresponding symbols.
The numerical experiments have shown the principal
influence of effective diffusion coefficient value on holo-
gram formation process. This quantity, effecctively, is ratio
of the characteristic polymerization time to the monomer
diffusion time at a distance of the Λ order.
The typical kinetics of the first harmonic of spatial com-
ponents distribution during recording process shown in Fig.1.
These curves are obtained by the numerical solution of the
system (12) with 2.0=s , m = 0.98 for different values of
Def. As follows from this figure, for any values of effective
diffusion coefficient there is a significant gradient of
monomer concentration during the recording process. This con-
clusion differs essentially from the results obtained in [4].
Certainly, if diffusion process proceeds much faster than
polymerization process (in our case this corresponds to
Def >> 1), in each specific time moment the system is in the
state close to equilibrium from the point of view of diffu-
sion matter transfer. Used in [4] the standard diffusion equa-
tion means uniformity of the monomer equilibrium state,
from that the conclusion about absence of an significant
monomer concentration gradient during recording follows.
As the optical properties of the monomer and polymer
are different, the monomer distribution influences on modu-
lation of optical properties of recording layer in a direct way.
Possibility of account of monomer distribution heterogene-
ity allows describing hologram kinetics more accurately as
compared with the approach used in [2,4].
A typical dry photopolymer system consists of a mono-
mer, polymeric binder and photoinitiator. In this case, the
main result of recording process is spatial modulation of
density of polymer and its optical properties. With absence
of a polymeric binder (or its insufficient rigidity), however,
there are conditions for formation of the relief on the layer
surface during recording process.
The discussed theoretical model, in contrast to earlier
offered models, allows describing formation of surface re-
lief in a natural way. This process can be considered as a
«displacement» of the X component (free volume) onto the
recording layer surface. There is possibility to analyze the
surface relief formation by considering the X component re-
distribution kinetics during holographic recording.
For example, as follows from Fig.1, for small values of
effective diffusion coefficient Def, the sign of the first har-
monic of X component distribution changes during record-
ing. In a context of surface relief consideration, it means
that at initial stage of hologram formation the hollows of the
relief correspond to bright regions, and after recording com-
pletion they correspond to dark regions.
The explicit accounting of shrinkage effect allows also
investigating dependence of holographic recording efficiency
Fig.1. Kinetics of the first harmonic of spatial components distribution
during recording process for different values of Def.
1 2 3 4 5 6 7 8 9 10 11
D = 1 0e f
D = 1e f
D = 0 .1e f
1
1
1
t/
M
X
P
P1
X1
M1
H.M. Karpov et al.: Generalized model of holographic recording ...
69SQO, 2(3), 1999
(steady state value of the first spatial harmonic of polymer
distribution stP1 ) on shrinkage rate. This dependence with
various values Def is represented on Fig.2. As follows from
this figure, the optimal value of shrinkage coefficient ex-
ists. It corresponds to the peak recording efficiency.
The theoretical dependencies of the first three harmon-
ics of steady state polymer volumetric part distribution
)(xPst ′ from Def are shown in Fig.3. As follows from this
figure, over the range 0< Def < 0.1 the hologram is charac-
terized by low efficiency and large relative amplitude of
higher ( 1>i ) harmonics, which is an exidence of a strong
nonlinearity of the recording process. Efficiency and lin-
earity of recording rise considerably with increasing Def and
reach saturation at Def 1≅ . The specified dependencies
qualitatively agree with the results obtained in papers [4,7].
Fig.4 shows the profiles of steady state polymer distri-
bution stP for different Def values. Under small Def val-
ues the profile of stP strongly differs from the irradiation
distribution and has characteristic two-peak shape.
Thus, Def can be considered as criterion of holographic
recording efficiency. The condition of obtaining of effec-
tive recording can be written as Def > 1. As follows from
the definition of effective diffusion coefficient Def its value
decreases with increase of grating period and, hence, the
diffusion process rate limits material performance over high
spacing range.
Conclusions
The generalized diffusion model of hologram formation in
photopolymer materials has been offered. The model is based
on using the nonlinear diffusion equation for the description
of the matter transfer processes in presence of the polymer
shrinkage effect.
The obtained results allow taking into account influence
of monomer distribution heterogeneity during recording on
hologram diffraction efficiency kinetics. The model also al-
lows analyzing the influence of shrinkage rate on holographic
recording efficiency.
The analysis of the theoretical model has shown the prin-
cipal influence of the diffusion/polymerization rates ratio on
hologram properties.
Using the offered model allows also tj describe the sur-
face relief formation process which can take place during
hologram recording.
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2
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