Design of optimized photorefractive polymers: a novel class of chromophores

Article abstract of DOI:10.1063/1.472950

A fully tensorial formulation is presented of the photorefractive (PR) Kerr effect, providing a convenient formalism for the calculation of diffraction efficiencies for various geometries. A detailed microscopic interpretation of the PR Kerr effect is also presented in terms of the relevant molecular linear and nonlinear optical properties. This approach allows the identification of the dominant molecular contributions and a better establishment of design criteria for the optimization of functional dyes for organic PR materials. These generalized principles are illustrated with a prototype chromophore, 2,6-di-n-propyl-4H-pyran-4-ylidenemalonitrile (DPDCP). The linear and nonlinear optical properties of this compound are obtained from a complete set of refractive index, permittivity, depolarized Rayleigh scattering, electric field-induced second harmonic generation (EFISH), and UV-VIS spectroscopic measurements, and quantum chemical calculations.

Full text of DOI:10.1063/1.472950

Design of optimized photorefractive polymers: A novel class of chromophores
Rüdiger Wortmann, Constantina Poga, Robert J. Twieg, Christian Geletneky, Christopher R. Moylan, Paul M.
Lundquist, Ralph G. DeVoe, Patricia M. Cotts, Hans Horn, Julia E. Rice, and Donald M. Burland
Citation: The Journal of Chemical Physics 105, 10637 (1996); doi: 10.1063/1.472950
View online: http://dx.doi.org/10.1063/1.472950
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/105/23?ver=pdfcov
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Design of optimized photorefractive polymers: A novel class
of chromophores
R u¨ diger Wortmann, Constantina Poga, Robert J. Twieg, Christian Geletneky,
Christopher R. Moylan, Paul M. Lundquist, Ralph G. DeVoe, Patricia M. Cotts,
Hans Horn, Julia E. Rice, and Donald M. Burland
IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099
͑
Received 8 July 1996; accepted 10 September 1996͒
It is demonstrated that the microscopic mechanism of the photorefractive ͑PR͒ effect in organic
composites with low glass transition temperatures involves the formation of refractive index
gratings through a space-charge field-modulated Kerr effect. A tensorial formulation of the
macroscopic aspects of the PR Kerr effect and its microscopic interpretation is presented. The
second-order dipole orientation term containing the anisotropy of the first-order optical
polarizability ␣͑Ϫ␻;␻͒ is shown to yield the dominant contribution to the Kerr susceptibility
͑
3͒
␹
͑Ϫ␻;␻,0,0͒. A class of special chromophores having negligible second-order polarizabilities
͑Ϫ␻;␻,0͒ and large dipole moments ␮ has been identified in order to optimize this term. These
␤
chromophores are not subject to the efficiency-transparency tradeoff typically encountered with
second-order nonlinear optical ͑NLO͒ chromophores, providing highly transparent materials with
large PR Kerr response. Contrary to previous approaches in this field, the best-performing
PR polymers are then expected to employ chromophores that would be useless for second-
order
applications
͑negligible
␤͒.
We
report
PR
of
the
material
30%
2,6-di-n-propyl-4H-pyran-4-ylidenemalononitrile ͑DPDCP͒: 15% N,NЈ-bis͑3-methylphenyl͒-
N,NЈ-bis͑phenyl͒benzidine ͑TPD͒:55% poly͑methyl methacrylate͒ ͑PMMA͒:0.3% C60 as an
illustration of this principle. A 100 ␮m thick film of this material exhibits a steady-state diffraction
efficiency of ␩ϭ25% and net two-beam coupling of ⌫ϭ50 cmϪ1 at a bias field of 100 V/␮m and a
wavelength of 676 nm. The macroscopic Kerr susceptibility of the material is related to molecular
electronic properties of the chromophore DPDCP which were independently determined by
experiments in solution and by quantum chemical calculations. © 1996 American Institute of
Physics. ͓S0021-9606͑96͒50547-7͔
I. INTRODUCTION
ganic crystals, the origin of the photorefractivity of organic
composites or polymers with low glass transition tempera-
The photorefractive ͑PR͒ effect has long been known in
tures T can be quite different. These low-T materials are
g
g
1
noncentrosymmetric inorganic crystals such as LiNbO . In
3
isotropic in the absence of an externally applied bias field
PR crystals the interference pattern produced by two inter-
secting coherent beams generates a nonuniform space-charge
field Esc by photogeneration of free carriers and subsequent
carrier drift and diffusion processes. As a consequence, the
refractive index of the material is modulated by the interac-
E . Therefore, the first-order electro-optical ͑Pockels͒ sus-
ceptibility ␹ ͑Ϫ␻;␻,0͒ vanishes for reasons of symmetry
b
͑2͒
and the second-order electro-optical ͑Kerr͒ susceptibility
2,3
͑3͒
␹
͑Ϫ␻;␻,0,0͒ governs the nonlinear response. These mate-
rials are poled in situ and the diffraction grating is created by
tion of the field E with the Pockels susceptibility
11
sc
the combined effect of the fields E and E . In contrast to
͑2͒
b
sc
␹
͑Ϫ␻;␻,0͒ of the crystal. The refractive index grating is
spatially displaced from the intensity grating, by as much as
/2 rad ͑nonlocality of the PR effect͒. The PR effect gives
crystalline systems, reorientation of the dipolar chro-
mophores is an important, sometimes even dominant, mecha-
␲
11
nism in these materials. In order to distinguish clearly the
rise to a variety of two- and four-wave mixing phenomena,
͑
2͒
PR effect through ␹ ͑Ϫ␻;␻,0͒ in intrinsically noncen-
trosymmetric materials ͑such as permanently poled poly-
and has potential technological importance for a variety of
optical signal processing schemes.2,3 Recent developments in
mers͒ from that in isotropic low-T materials through the
enabling technologies for high-density holographic optical
g
͑
3͒
4
Kerr susceptibility ␹ ͑Ϫ␻;␻,0,0͒, we will use the nomen-
clature photorefractive Pockels effect for the former and pho-
torefractive Kerr effect for the latter. The PR Kerr effect
incorporates both the ‘‘conventional’’ PR mechanism and
the so-called ‘‘orientational enhancement’’ mechanism.11
In this paper we present a fully tensorial formulation of
the PR Kerr effect, providing a convenient formalism for the
calculation of diffraction efficiencies for various geometries.
We also present a detailed microscopic interpretation of the
data storage have stimulated research on PR materials, with
the goal of identifying inexpensive holographic materials.
Photorefractivity in organic materials was demonstrated
5,6
only recently, first in molecular crystals and later in poly-
7
mer systems. In particular, PR polymers have attracted
8
growing interest because of their ease of sample fabrication
and large diffraction efficiencies.9 While the PR effect in
molecular crystals and in permanently electric field-poled
polymers can be described in a similar fashion as in inor-
,10
J. Chem. Phys. 105 (23), 15 December 1996
0021-9606/96/105(23)/10637/11/$10.00
© 1996 American Institute of Physics
10637
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10638
Wortmann et al.: Photorefractive polymers
͑
1͒
͑2͒
͑0͒
⌬
␹
␣͑Ϫ␻;␻͒ϭ2␹␮␣␤͑Ϫ␻;␻,0͒E␤
␮
͑
3͒
͑0͒ ͑0͒
ϩ3␹ ␣␤␥͑Ϫ␻;␻,0,0͒E␤
E
ϩ•••
␥
͑5͒
␮
is the change of the first-order susceptibility induced by the
static field E , the Greek indices denoting Cartesian tensor
0
͑1͒
components in the laboratory frame. The susceptibilities ␹ ,
, and ␹ in the preceding equations use the convention
recommended by Butcher and Cotter.
͑
2͒
͑3͒
␹
FIG. 1. Molecule-fixed coordinate system and resonance structures for
DMDCP ͑R-methyl͒, DEDCP ͑Rϭethyl͒, and DPDCP ͑Rϭn-propyl͒.
1
2
͑
2͒
The second-order susceptibility ␹ gives rise to the PR
Pockels effect in noncentrosymmetric materials. In isotropic
͑
2͒
PR Kerr effect in terms of the relevant molecular linear and
nonlinear optical properties. This approach allows the iden-
tification of the dominant molecular contributions and a bet-
ter establishment of design criteria for the optimization of
functional dyes for organic PR materials. These general prin-
ciples are illustrated with a prototype chromophore, 2,6-di-
n-propyl-4H-pyran-4-ylidenemalononitrile ͑DPDCP, Fig. 1͒.
The linear and nonlinear optical properties of this compound
are obtained from a complete set of refractive index, permit-
tivity, depolarized Rayleigh scattering, electric field-induced
second harmonic generation ͑EFISH͒, and UV-VIS spectro-
scopic measurements, and quantum chemical calculations.
centrosymmetric materials, ␹ vanishes and the nonlinear
response of the medium is due to the Kerr susceptibility
␹ ͑Ϫ␻;␻,0,0͒ only. The third-order susceptibility ␹ of an
isotropic medium has 21 nonvanishing tensor components of
which only three are independent. The general symmetry
relations12 between these can be written using Kronecker ␦
symbols in the compact form13
͑
3͒
͑3͒
12
͑3͒
␮
͑3͒
͑3͒
͑3͒
␹
␣␤␥ϭ␹ZZXX␦␮␣␦␤␥ϩ␹ZXZX␦␮␤␦␣␥ϩ␹ZXXZ␦␮␥␦␣␤
͑
6͒
with the additional linear relationship
͑3͒
͑3͒
͑3͒
͑3͒
ZXXZ
␹
ZZZZϭ␹ZZXXϩ␹ZXZXϩ␹
.
͑7͒
The Cartesian indices in Eqs. ͑6͒ and ͑7͒ refer to an arbi-
II. MACROSCOPIC THEORY OF THE
PHOTOREFRACTIVE KERR EFFECT
trarily chosen laboratory frame. In the particular case of
͑
3͒
␹
͑Ϫ␻;␻,0,0͒ two components are identical because of in-
trinsic permutation symmetry in the last two indices;
The PR Kerr effect is the local response of the medium
to the space-charge modulated static electric field. In this
section we present a tensorial formulation of the PR Kerr
effect and employ it in the calculation of diffraction efficien-
cies for different geometries and polarization conditions.
The principal indices of refraction of the medium at fre-
(
3)
(3)
␹
ϭ␹
. The most general Kerr susceptibility of an
ZXZX
ZXXZ
isotropic medium therefore has only two independent com-
ponents ␹(
3)
and ␹
(3)
ZZZZ
, and is of the form
13
ZZXX
͑
3͒
͑3͒
1
2
͑3͒
͑3͒
ZZXX
␹
␣␤␥͑
Ϫ␻;␻,0,0͒ϭ␹ZZXX␦␮␣␦␤␥
ϩ
͑
␹
ZZZZϪ␹
͒
␮
͑␻͒
ϫ͑␦␮␤␦␣␥ϩ␦␮␥␦␣␤͒.
͑8͒
quency ␻, n , are related to its first-order susceptibility
␣
͑
1͒
␹
␣͑Ϫ␻;␻͒ by the equation
␣
Using Eqs. ͑3͒, ͑5͒, and ͑8͒ it is straightforward to calculate
the diffraction efficiency ␩ of photorefractive gratings ob-
served in four-wave mixing ͑FWM͒ experiments for various
geometries and polarization conditions.
͑
␻͒
͑1͒
n
ϭ
ͱ
1ϩRe͓␹␣␣͑Ϫ␻;␻͔͒,
͑1͒
␣
where ␣ indicates the polarization direction of the light. In a
PR experiment, the total electric field E is a superposition of
The geometry of the FWM experiment of this work has
been described previously.11 The internal angles of incidence
of writing beams 1 and 2 are denoted ␪ and ␪ , and the
a static or slowly varying field E and an optical field with
0
amplitude E and frequency ␻. The time dependence of the
␻
1
2
total field is given by
angle of the grating vector relative to the surface normal is
␪G . The laboratory frame was chosen with the z axis parallel
to the surface normal, the xz plane as the plane of incidence,
and y normal to this plane. The polarization unit vectors of
the reading beam 3 and the diffracted beam 4 are then
E͑t͒ϭE ϩ 2 ͓E␻eϪi␻tϩ͑E ͒*ei␻t͔,
1
͑2͒
0
␻
where * indicates complex conjugation. The static field itself
is a superposition of the externally applied bias field E and
b
(
s)
(s)
T
the spatially periodic space-charge field with amplitude Esc
and grating vector KG
e3 ϭe4 ϭ(0,1,0) ͑T indicates tensor transposition͒ for
(
p)
T
s-polarized reading, and e3 ϭ͑cos ␪2,0,sin ␪2͒
and
(
4
p)
T
e
ϭ͑cos ␪ ,0,sin ␪ ͒ for p-polarized reading. The grating
1
1
E ϭE ϩE cos͑K •r͒.
͑3͒
T
0
b
sc
G
vector is K ϭK
͑sin ␪ ,0,cos ␪ ͒ , the bias field
G
G
G G
T
This spatially varying space-charge field is generated by in-
E ϭE (0,0,1) , and the space-charge field E ϭE
b b sc sc
2
T
ternal charge separation produced by a spatially varying op-
ϫ͑sin ␪ ,0,cos ␪ ͒ cos͑K •r͒.
G
G
G
tical field E . The Fourier component at frequency ␻ of the
The diffraction efficiency ␩ for Bragg-matched readout
␻
polarization induced by the field E(t) in the medium is given
by
of a slanted thin transmission grating of thickness d has been
shown1
1,14
to be the following:
͑
␻͒
͑1͒
͑1͒
͑␻͒
2
͑1͒
P
ϭ͓␹␮␣͑Ϫ␻;␻͒ϩ⌬␹␮␣͑Ϫ␻;␻͔͒E␣
,
͑4͒
␩ϭsin ͓H͑e •e ͒͑e •⌬␹ •e ͔͒
͑9͒
␮
3
4
3
4
where ͑sum convention used for repeated indices͒
with
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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Wortmann et al.: Photorefractive polymers
10639
tional distribution function w given in Eq. ͑28͒ below. The
␲
d
J
Fourier components p( of the induced dipole of the con-
␻)
Hϭ
.
/2
͑10͒
1
J␮
2
n␭ ͑cos ␪ cos ␪ ͒
0 1 2
stituent J are
For the geometry specified above, the results for
s-polarized and p-polarized reading are
͑
J␮
␻͒
͑␻͒
¯
͑␻͒ ͑0͒
p
¯
ϭ␣J␮␣
͑Ϫ␻;␻͒E␣ ϩ␤J␮␣␤͑Ϫ␻;␻,0͒E
E
␣ ␤
2
͑1͒
1
͑␻͒ ͑0͒ ͑0͒
␩sϭsin ͓H⌬␹ ͔,
͑11͒
ϩ 2
¯
␥
J␮␣␤␥͑Ϫ␻;␻,0,0͒E␣
E
E ϩ•••
␥
͑19͒
YY
␤
2
͑1͒
XX
¯
¯
J J
␩pϭsin ͓H cos͑␪ Ϫ␪ ͒͑cos ␪ cos ␪ ⌬␹
with ␣ , ␤ , and ␥J being effective first-, second-, and third-
¯
1
2
1
2
order polarizabilities of the molecules of type J in the me-
dium in the Taylor series convention. Applying Lorentz
͑
XZ
1͒
͑1͒
ZZ
ϩsin͑␪ ϩ␪ ͒⌬␹ ϩsin ␪ sin ␪ ⌬␹ ͔͒. ͑12͒
16
1
2
1
2
local field corrections, the effective polarizabilities are re-
lated to those of the free molecules by
The relevant tensor components are easily obtained from
Eqs. ͑3͒, ͑5͒, and ͑8͒ and contain spatial Fourier components
at 0 K , 1 K , and 2 K . The results for the 1 K compo-
G
G
G
G
¯
␣ ͑Ϫ␻;␻͒ϭL ␣ ͑Ϫ␻;␻͒,
͑20͒
͑21͒
͑22͒
J
␻
J
nent are:
͑
XX
1͒
͑1͒
͑3͒
¯
␤
͑Ϫ␻;␻,0͒ϭL L ␤ ͑Ϫ␻;␻,0͒,
␻ 0 J
⌬
⌬
⌬
␹
␹
␹
ϭ⌬␹YY ϭ6␹
E E cos ␪G ,
͑13͒
͑14͒
͑15͒
J
ZZXX
b
sc
͑
1͒
XZ
͑1͒
͑3͒
͑3͒
2
¯
␥J
͑Ϫ␻;␻,0,0͒ϭL L ␥ ͑Ϫ␻;␻,0,0͒,
0
␻ J
ϭ⌬␹ZX ϭ3͑␹ZZZZϪ␹
͒E E sin ␪G ,
ZZXX
b
sc
͑
1͒
ZZ
͑3͒
ZZZZ
with Lorentz factors L at frequencies 0 and ␻
ϭ6␹
E E cos ␪G .
␻
b
sc
The preceding equations can be used to calculate the
diffraction efficiency ␩ for various polarization conditions.
In particular, one obtains for the polarization anisotropy of
⑀␻ϩ2
L␻ϭ
.
͑23͒
3
More refined local field corrections and effects of the static
reaction field on effective molecular properties in condensed
media have been described elsewhere.
The energy of a molecule with permanent dipole ␮ and
static polarizability ␣ ͑0;0͒ in a static external field is orien-
the 1 K grating in the low diffraction limit:
G
͑
3͒
␩p
␩s
␹
␹
13,17
ZZZZ
2
ϭcos ͑␪ Ϫ␪ ͒ cos ␪ cos ␪ ϩ
sin ␪ sin ␪2
1
2
ͫ
1
2
͑3͒
ZZXX
1
J
J
͑
3͒
2
1
2
␹
␹
ZZZZ
tation dependent:
ϩ
Ϫ1 tan ␪ sin͑␪ ϩ␪ ͒
.
͑16͒
ͩ
͑3͒
ZZXX
ͪ
G
1
2
ͬ
͑0͒
1
2
͑0͒ ͑0͒
⌬
W ͑E ͒ϭϪ ␮ញ J␥E␥ Ϫ ␣ញ J␤␥͑0;0͒E␤
E Ϫ••• ͑24͒
␥
J
0
This expression is equivalent to Eq. 25 of Ref. 11.
with the effective quantities
⑀0
͑⑀ ϩ2͒
ϱ
III. MICROSCOPIC INTERPRETATION OF THE
PHOTOREFRACTIVE KERR EFFECT
␮ញ ϭ
␮J ,
͑25͒
͑26͒
J
2
⑀0ϩ⑀ϱ
2
3
⑀ ͑⑀ ϩ2͒
In the model of additive molecular contributions, the
0
ϱ
␣
ញ J͑0;0͒ϭ
␣ ͑0;0͒,
J
͑3͒
macroscopic susceptibility ␹ ͑Ϫ␻;␻,0,0͒ may be repre-
͑2⑀ ϩ1͒͑2⑀ ϩ⑀ ͒
0 0 ϱ
sented as a sum of terms that are proportional to the concen-
in Onsager’s local field approximation.13,18–20 The canonical
distribution function describing the molecular orientation is,
to second order in the static field13
trations c of the different constituents J of the medium
J
Jϭ1
1
͑
3͒
͑3͒
␹
␣␤␥͑Ϫ␻;␻,0,0͒ϭ
␨J␮␣␤␥͑Ϫ␻;␻,0,0͒cJ ,
␮
͚
⑀0
N
exp͑Ϫ⌬W /kT͒
J
͑17͒
w ͑␶͒ϭ
J
͑27͒
͗
exp͑Ϫ⌬W /kT͒
͘
J
(
3)
where ␨
are molar third-order polarizabilities of the
J␮␣␤␥
component J and ⑀ is the permittivity of free space. The
1
1
0
͑0͒
␥
(
3)
ϭ1ϩ
␮
¯
E
ϩ
͓3 ␮ញ J␤ ␮ញ J␥
J␥
6k2T2
quantities ␨
can be calculated on the basis of a specific
J␮␣␤␥
kT
molecular model and are closely related to partial molar
quantities that are well known in the formalism of chemical
1
2
͑0͒ ͑0͒
Ϫ ␮ញ ␦ ␦ ͔E
E ϩ
␥
͓3 ␣ញ J␤␥
1
3,15
J
␤
␥
␤
thermodynamics.
isfy the Euler theorem of homogeneous functions. The mo-
They do not, however, necessarily sat-
6kT
15
͑
0͒ ͑0͒
Ϫtr͑␣J
ញ ͒␦ ␦ ͔E
E
ϩ•••
␥
͑28͒
lar polarizabilities can be derived from13
␤ ␥
␤
ץ3
and consists of four terms corresponding to an isotropic con-
tribution, a first-order and a second-order dipole orientation,
and an orientation contribution arising from the anisotropy of
the static polarizability. Using standard methods of orienta-
tional averaging21 one obtains from Eqs. ͑18͒, ͑19͒, and ͑28͒
the two independent third-order molar polarizabilities
NA
͑
3͒
͑␻͒
␨
͑Ϫ␻;␻,0,0͒ϭ
lim
͗ ͘
wJpJ␮ ,
J␮␣␤␥
͑␻͒
͑0͒
0
␥
3
ץE␣ ץE␤ ␦E
͉
E
͉
→0
͑18͒
where NA is Avogadro’s constant and ͗ ͘ denotes a
statistical-mechanical average calculated with the orienta-
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10640
Wortmann et al.: Photorefractive polymers
TABLE I. Linear and nonlinear optical properties of DPDCP and pNA.
DPDCP
352a
pNA
j
␭ag
␮ag
␮
͑nm͒
354
16
Ϫ30
a
j
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
Cm͒
Cm͒
21
24.6
Ϫ30
Ϫ40
Ϫ40
Ϫ40
Ϫ40
Ϫ40
Ϫ50
Ϫ50
Ϫ60
Ϫ60
b
j
20.8
Ϫ1
2
c
j,k
tr͓␣͑Ϫ␻;␻͔͒
CV m ͒
110
57
Ϫ1
2
d
10f
13, 19, 25
10f
␦␣R
CV m ͒
26
Ϫ1
2
e
j,k
␣xx(Ϫ␻;␻),␣yy(Ϫ␻;␻),␣ (Ϫ␻;␻)
CV m ͒
21, 38, 51
22f
zz
Ϫ1
2
␦
␣
CV m ͒
K
͑
TLM͒
Ϫ1
2
g
g
␣
␤
␤
͑Ϫ␻;␻͒
CV m ͒
19
11
27
Ϫ2
3
h
j,l
͑Ϫ2␻;␻,␻͒
͑Ϫ␻;␻,0͒
CV m ͒
Ϫ2͑2͒
ϳ0
Ϫ2
3
i
i
CV m ͒
23
2
2
2
Ϫ3
4
4␮ ␦␣ /(3k T )
CV m ͒
105 000
ϳ0
34 000
7 000
k
Ϫ3
4
6␮␤͑Ϫ␻;␻,0͒/(kT)
CV m ͒
16
Second-order polarizabilities ␤ are given in the Taylor series definition and SI units. SI to cgs unit conversions
Ϫ30
Ϫ18
Ϫ40
2
Ϫ1
Ϫ23
Ϫ50
are: ␮, 10
Cmϭ0.2998ϫ10
esu; ␣, 10
Cm V ϭ0.089 88ϫ10
esu; ␤, 10
3
Ϫ2
Ϫ30
Cm V ϭ2.694ϫ10
From UV-VIS absorption in CHCl3 .
From permittivity measurements in CHCl3 .
From refractive index measurements in CCl at ␭ϭ633 nm.
From depolarized Rayleigh scattering in CCl at ␭ϭ633 nm.
From tr͓␣͔ and ␦␣R with an estimated value of ␣xx.
Calculated from ␣xx , ␣yy , and ␣zz
Calculated for ␭ϭ676 nm from ␮ag and ␭ag with Eq. ͑37͒.
From EFISH in CHCl at ␭ϭ1907 nm.
Calculated for ␭ϭ676 nm from ␤͑Ϫ2␻;␻,␻͒ by using Eqs. ͑32͒–͑34͒.
From Ref. 23.
At ␭ϭ589 nm in dioxane.
esu.
a
b
c
4
d
e
4
f
.
g
h
i
3
j
k
l
From EFISH in dioxane at ␭ϭ1064 nm. ␤ values are given in T convention, Ref. 16.
͑
3͒
Kerr response of organic PR materials with low T , in the
␨
␨
͑Ϫ␻;␻,0,0͒
g
JZZXX
identification of the dominant contributions, and in establish-
ing design criteria for its optimization, as will be discussed
below.
NA
2
1
¯
ϭ
¯
␥
ϩ
␮ញ Jr
␤
Ϫ
͓3 ␮ញ Jr ␮ញ Jt¯
␣
Jrt
ͭ
Jrrtt
Jrtt
3k2T2
90
kT
1
Ϫ␮
¯
␮ ␣
¯ ¯
͔Ϫ
͓3 ␣ញ Jrt
¯JrtϪ ␣ញ Jrr
␣
¯
␣
͔ ,
͑29͒
Jr Jr Jtt
Jtt
ͮ
3
kT
͑
3͒
JZZZZ
͑Ϫ␻;␻,0,0͒
IV. OPTIMIZATION OF FUNCTIONAL DYES FOR
ORGANIC PR MATERIALS
NA
6
1
¯
ϭ
¯
␥
ϩ
␮ញ Jr
␤
ϩ
͓6 ␮ញ Jr ␮ញ Jt¯
␣
Jrt
ͭ
Jrrtt
Jrtt
3k2T2
90
kT
It is illustrative to estimate the different contributions to
the molar Kerr susceptibility for a typical chromophore such
1
Ϫ2␮
¯
␮ ␣
¯ ¯
͔ϩ
͓6 ␣ញ Jrt
¯
␣
JrtϪ2 ␣ញ Jrr
¯
␣
͔ ,
Jr Jr Jtt
Jtt
ͮ
as 4-nitroaniline ͑pNA͒. The linear and nonlinear electric
3
kT
properties2
2,23
of this prototype NLO-phore were taken from
͑30͒
the literature and are collected in Table I. Neglecting local
field effects and correcting for the different dispersion behav-
ior of ␤͑Ϫ2␻,␻,␻͒ and ␤͑Ϫ␻;␻,0͒ ͓see Eqs. ͑33͒ and ͑34͒
below͔, one obtains for the molar susceptibility using Eq.
where the Latin indices refer to Cartesian tensor components
in the molecule-fixed frame and index permutation symmetry
was assumed for the tensors ␤ and ␥. Four molecular terms
therefore contribute to the PR Kerr effect. They are of type
, ␮␤, ␮ ␣, and ␣␣, and arise from the four different orien-
tation contributions in Eq. ͑28͒. The preceding equations rep-
resent an accurate description of the Kerr effect in dilute
solutions. For highly viscous or rigid polymer systems with
high concentrations of dopants, orientation correlations be-
come important and the assumption of independently and
freely reorienting molecules can be regarded as an approxi-
mation only. In addition, effects of electrostriction which are
neglected in Eqs. ͑29͒ and ͑30͒ can sometimes be important.
Nevertheless, these equations are useful in estimating the
(
3)
Ϫ36
4
3
ϩ
͑30͒ ␨
ϭ 240 ϫ 10
C m /V mol at the Kr laser
ZZZZ
2
2
␥
wavelength of 676 nm. The contributions of the ␥, ␮␤, ␮ ␣,
and ␣␣ terms to ␨
(
3)
are found to be 0.4%, 14%, 84%, and
ZZZZ
1%, respectively. This example demonstrates that the domi-
nant contribution in low-Tg organic materials is, in many
2
cases, due to the ␮ ␣ term ͑‘‘birefringence orientational con-
tribution’’͒, with the ␮␤ term ͑‘‘electrooptical contribu-
tion’’͒ being of minor importance, contrary to what was as-
sumed in early investigations of PR polymers. Contributions
of the ␥ and ␣␣ terms are usually negligible. It is, therefore,
appropriate to introduce the approximations
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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Wortmann et al.: Photorefractive polymers
10641
͑35͒
2
NA
2
2␮
ag
͑
3͒
¯
͑TLM͒
0
␨
␨
͑Ϫ␻;␻,0,0͒Ϸ
͑Ϫ␻;␻,0,0͒Ϸ
␮ញ ␤͑Ϫ␻;␻,0͒
ͭ
␣
ϭ
,
ZZXX
90
kT
ប␻ag
2
2
␻
2
␮ញ 2␦
ag
Ϫ
¯
␣ ͑Ϫ␻;␻͒ , ͑29a͒
␣͑TLM͒͑Ϫ␻;␻͒ϭ␣
͑TLM͒
k2T2
K
ͮ
.
2
͑36͒
3
0
͑
␻ Ϫ␻ ͒
ag
͑
TLM͒
NA
6
In the case of PNA ␣
contributes about 40% to ␣zz ͑cf.
Table I͒. For the moment, we assume that the off-axis com-
͑
3͒
ZZZZ
¯
␮ញ ␤͑Ϫ␻;␻,0͒
ͭ
90
kT
ponents of the polarizability, ␣xx and ␣ , and those of the
yy
4
second-order polarizability, ␤zxx and ␤zyy , are negligible.
Equations ͑35͒ and ͑36͒ can, therefore, be used as a guiding
principle to design functional dyes with optimized anisotropy
of ␣.
␮ញ 2␦
¯
ϩ
␣ ͑Ϫ␻;␻͒ , ͑30a͒
k2T2
K
ͮ
3
where the index for the constituent J and the Cartesian in-
dexes of the molecular tensors have been omitted for reasons
of simplicity. It is implied in Eqs. ͑29a͒ and ͑30a͒ that the
dipole is oriented along the z axis, which is also assumed to
be the main axis of the polarizability ellipsoid, and that the
vector part of the second-order polarizability refers to the
One-dimensional dipolar NLO-phores are usually
␲-conjugated donor–acceptor systems with resonance struc-
tures similar to those displayed in Fig. 1 for DPDCP. To a
first approximation, the ground and excited states can each
be represented by a linear combination of a neutral
͉
n
͘
and a
¯
component along the dipole axis [␤ ϭ␤zzzϩ␤zxxϩ␤zyy].
z
zwitterionic wave function
͉
z
͘
͘
͘
2
The ␮ ␣ term contains the anisotropy of the optical polariz-
͉
͉
g
a
͘
͘
ϭc
ϭc
͉
͉
z
͘
Ϫ͑1Ϫc2͒1/2
ϩ͑1Ϫc2͒1/2
͉
n
,
͑37͒
͑38͒
ability ␣͑Ϫ␻;␻͒ in the form
1
2
n
͘
͉
z
,
␦
␣ ϭ␣ Ϫ ͑␣ ϩ␣ ͒.
͑31͒
K
zz
xx
yy
where c is the mixing coefficient and zero differential over-
lap, ϭ0, is assumed. This description is reasonable for
As will become apparent below, it is, in fact, quite difficult
to design one-dimensional ͑1D͒ NLO chromophores for
which the ␮␤ term dominates. It thus seems appropriate in-
stead to focus design strategies on optimization of the ␮ ␦␣
term, as will now be discussed.
͗ ͘
n͉z
weakly interacting molecular complexes or spiro-conjugated
24
charge-transfer compounds. For ␲-conjugated systems, the
2
approximation will be crude in general, but nevertheless
some qualitative trends can be derived from it. The transition
dipole and the change of the dipole upon electronic excita-
tion are
It is well known that the second-order polarizability ␤ of
dipolar ‘‘charge-transfer’’ 1D NLO-phores with extended
␲-conjugated systems can be reasonably approximated by
ϭϪc͑1Ϫc2͒1/2⌬,
͑39͒
͑40͒
␮ag
ϭ
͗
͗
a
a
͉
␮
␮
͉
g
a
͘
͘
the two-level model ͑TLM, see, e.g., Ref. 22 and references
therein͒ taking into account only the ground and the first
optically allowed excited singlet state. The following expres-
sions can be readily derived using the methods of time-
dependent perturbation theory13
2
⌬
␮ϭ
͉
͉
͗ ͘
Ϫ g͉␮͉g ϭ͑1Ϫ2c ͒⌬,
where ⌬ is the difference between the dipole moments of the
neutral and the zwitterionic states, equal to the maximum
change of the dipole upon excitation in the given system.
Equations ͑39͒ and ͑40͒ allow for a straightforward estima-
tion of the TLM polarizabilities in Eqs. ͑32͒–͑35͒. The re-
sults are displayed in Fig. 2 as a function of the mixing
coefficient. The transition energy was assumed to be con-
stant. The predictions of this simple model are that ␣ is
maximized when ␤ is zero. Similar trends were recognized
͒2
ប␻ag͒2
⌬␮͑␮ag
͑
0
␤
TLM͒ϭ6
,
͑32͒
͑33͒
,
͑
2
2
2
␻
͑3␻ Ϫ␻ ͒
ag
ag
␤͑
TLM͒͑Ϫ␻;␻,0͒ϭ␤
͑TLM͒
0
,
2
ag
2
2
3
͑␻ Ϫ␻ ͒
4
ag
25
␻
in a two-state four-orbital model by Marder et al. The
␤͑
TLM͒͑Ϫ2␻;␻,␻͒ϭ␤
͑TLM͒
0
2
2
2
2
maximum for ␣ occurs when the coefficient c is equal to
͑
␻ Ϫ␻ ͒͑␻ Ϫ4␻ ͒
ag ag
Ϫ1/2
2
and the neutral and the zwitterionic wave functions
͑34͒
contribute equally to the ground and excited state. Since this
situation is naturally present in symmetric cyanine dyes, it is
sometimes called the ‘‘cyanine limit.’’ At the same time, the
change of the dipole upon excitation is zero and the magni-
tude of the transition dipole is maximized. The nature of the
electronic excitation changes from a ‘‘charge transfer’’ ͑CT͒
to a ‘‘charge resonance’’ ͑CR͒ transition. Geometry changes
upon excitation tend to be small in such CR systems. As a
consequence, Franck–Condon progressions are short and the
optical absorption spectra are unusually sharp.
͑
TLM͒
with ␤
⌬
being the TLM approximation to the static ␤,
0
␮, and ␮ag the change of the dipole and the transition di-
pole of the electronic excitation from the ground state to
the excited state , and ប␻ag the transition energy.
For first-order polarizabilities, it is, in general, necessary
to take contributions of higher excited ␲ states and of ␴
electrons explicitly into account. Nevertheless, the TLM con-
tribution to ␣ is often substantial and is always a lower
bound since all contributions to ␣ , ␣ , and ␣ are posi-
͉g
͘
͉a
͘
xx
yy
zz
tive. The TLM contribution can be estimated by applying the
following equations:
Let us briefly examine some of the approximations of
this model. It is assumed that ␦␣ is modeled by ␣ . The
K
zz
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10642
Wortmann et al.: Photorefractive polymers
26
since bathochromic shifts ͑solvatochromism ͒ and polar
environment-induced broadening of the absorption band are
minimized. The functional dye DPDCP ͑Fig. 1͒ is an ex-
ample of a system designed with these considerations in
mind.
V. SYNTHESIS
DPDCP was prepared by a four-step synthesis following
the methods of Deshapande,27 Yates et al., and Woods as
28
29
shown in Scheme I:
͑
TLM͒
, ␤ 0͑ TLM͒ , and ␥ 0͑ TLM͒ as a func-
FIG. 2. Relative TLM polarizabilities ␣
tion of the mixing coefficient c.
0
off-axis contributions to the polarizability, ␣xx and ␣ , are
yy
generally non-negligible. For molecules that maximize the
transition dipole ͑along the permanent dipole axis, z͒ and
that are predominantly one-dimensional in nature, however,
1
the value of (␣ ϩ␣ ) should be substantially smaller than
2
xx
yy
␣zz . For pNA, the difference is almost a factor of 2 ͑see
Table I͒. It should also be emphasized that molecules that
͑TLM͒
maximize ␣
will minimize ␤, within the two-level
model. Thus, ␤ may not in actual fact be zero, due to con-
tributions to all three components ͑␤zxx , ␤zyy , and ␤zzz͒ from
higher-order states.
Functional dyes that are close to the limiting case de-
scribed above meet the requirements for optimizing the PR
Kerr effect. They combine a high polarizability anisotropy
1
was obtained in 40%–50% yield by heating a threefold
excess of 1,3-acetonedicarboxylic acid with n-propionic an-
hydride and a few drops of concentrated sulfuric acid to
100 °C for 20–30 min. After this time, the dark brown mix-
ture was allowed to cool down to room temperature and the
crude product was precipitated by addition of a small amount
of water. 1 was isolated as a light brown solid after filtration,
washing with cold water, and air drying.
The decarboxylation step was carried out by heating 1 in
a 10% aqueous solution of sodium carbonate ͑pH must be
kept basic͒ for 30 min at 100 °C and an additional period of
1.5 h at 80–90 °C. After cooling to room temperature, the
yellow solution was acidified with 30% acetic acid. Since 2
did not crystallize after cooling to room temperature, it was
isolated by extraction of the aqueous reaction mixture with
ethyl acetate. The organic layer was dried over magnesium
sulfate and the solvent was evaporated to provide 2 as a light
brown oil in 70% yield.
For the second decarboxylation, 2 was boiled in 12 N
hydrochloric acid for at least 12 h. Reaction progress was
readily monitored by thin layer chromatography. After cool-
ing to room temperature, the brown solution was neutralized
with solid sodium carbonate and extracted with chloroform.
The 2,6-n-dipropyl-␥-pyrone 3 was obtained as a brown oil
in 80% yield after evaporation of the solvent from the dried
organic layer.
␦
␣K with a large resonantly enhanced ground state dipole
2
moment ␮ and maximize the ␮ ␣ term in Eqs. ͑29͒ and ͑30͒.
This optimization is achieved while maintaining a short con-
jugation length, which allows a small molecular size result-
ing in fast and efficient reorientation of the chromophore in
the polymer environment.
A molecular figure of merit for photorefractive chro-
mophores may be defined based on Eq. ͑30a͒. Substituting
w␳N /M for N, where w is the weight fraction of chro-
A
mophore in the material, ␳ is the density of the material, and
M is the molecular weight of the chromophore, and retaining
only terms involving molecular parameters, the figure of
merit F is determined ͓Eq. ͑41͔͒,
kT ␮ញ ␤͑Ϫ␻;␻,0͒ϩ2 ␮ញ 2␦
¯
␣ ͑Ϫ␻;␻͒
k
9
Fϭ
.
͑41͒
kTM
If one extracts explicit definitions of the coefficients CEO and
CBR from Ref. 11 and adds them together to get the coeffi-
cient C, and makes the same substitution and similar simpli-
fications as above, one arrives at an identical expression for
the molecular figure of merit F.
Because of the sharpness of the optical spectra, it is easy
to find chromophores that are nonabsorbing at the applied
laser wavelength. The vanishing dipole change upon excita-
tion also results in reduced absorption losses in the material,
The final condensation of the ␥-pyrone with malononi-
trile was carried out by heating 3 with a slight excess of
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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Wortmann et al.: Photorefractive polymers
10643
malononitrile in acetic anhydride to 140 °C for at least 2 h.
Reaction progress was readily monitored by thin layer chro-
matography. Compound 4 was isolated by adding water and
extraction of the aqueous reaction mixture with ethyl acetate.
The organic layer was dried over magnesium sulfate and
concentrated by rotary evaporation. Characterization of 4:
1
mp. ͑DSC, 20 °C/min͒: 68 °C. H-NMR ͑250 MHz, CDCl ͒
3
3
␦
ϭ0.91 ͑t, 6H, J ϭ7.3 Hz, —CH —CH —CH ͒, 1.65
H,H
2
2
3
3
͑
six, 4H, J ϭ7.4 Hz, —CH —CH —CH ͒, 2.48 ͑t, 4H,
H,H 2 2 3
3
JH,Hϭ7.3 Hz1, —CH —CH —CH ͒, 6.47 ͑s, 2H,
2
2
3
3
—
—
—
—
CHvC—͒. C-NMR ͑62.9 MHz, CDCl ͒ ␦ϭ13.4 ͑s,
3
CH —CH —CH ͒, 20.3 ͑s, —CH —CH —CH ͒, 35.5 ͑s,
2
2
3
2
2
3
CH —CH —CH ͒, 58.1 ͑s,͑NC͒ —CvC—͒, 106.2 ͑s,
2
2
3
2
CHvC͑Pr͒—O—͒, 115.0 ͑s, —CN͒, 156.9 ͑s,
͑
NC͒ —CvC—͒, 166.6 ͑s, —CHvC͑Pr͒—O—͒.
2
FIG. 3. Optical absorption spectrum of DPDCP in CHCl at room tempera-
3
ture.
VI. CHARACTERIZATION OF THE FUNCTIONAL DYE
DPDCP
was determined by depolarized Rayleigh scattering in the
same solvent following the method of Vuks.34 This experi-
ment was carried out with a BI-DS light scattering apparatus
͑Brookhaven Instruments Corporation͒ operating at 632.8
The molecular linear and nonlinear optical properties of
DPDCP were thoroughly investigated by a combination of
experimental techniques. The results are summarized in
Table I. The dipole moment was determined from the con-
Ϫ40
Ϫ1 2 34
nm. Diphenylacetylene ͑␦␣ ϭ22.5ϫ10
CV m ͒ was
R
used as an external calibration standard. Remarkably, the ob-
centration dependence of the relative permittivity15,30 ͑⑀ ͒ of
Ϫ40
Ϫ1
2
r
served anisotropy of DPDCP, 26ϫ10
CV m , is larger
solutions of DPDCP in chloroform using the
than that of the rod-shaped molecule diphenylacetylene.
The refractive index and the depolarized Rayleigh scat-
tering experiments yield only two equations for the three
components of the tensor ␣, so the component perpendicular
3
1,32
Guggenheim
method. The dipole moment was found to
Ϫ30
be ␮ ϭ24ϫ10
C m ͑7.3 D͒, which is significantly larger
z
Ϫ30
than the value of 12ϫ10
C m ͑3.6 D͒ expected from an
estimation based on group dipole moments. This indicates a
strong enhancement of the dipole moment due to zwitteri-
onic resonance structures as mentioned previously. The
second-order polarizability ␤͑Ϫ2␻;␻,␻͒ was measured in the
to the molecular plane, ␣ , was estimated from the group
xx
35
and bond polarizabilities given by Stuart. The reliability of
this method has been verified.1
5,30,34
The components ␣yy
and ␣zz were then calculated from tr͑␣͒ and ␦␣ ; results are
R
3
1
same solvent by EFISH as previously described. The ob-
served ␤ is small, as expected for a chromophore in the
charge resonance limit. The UV-VIS absorption spectrum
was recorded in chloroform solution on an HP8452A
UV-VIS spectrophotometer. DPDCP exhibits a strong ab-
sorption in the UV with a maximum at 352 nm, and is es-
sentially transparent above 400 nm ͑Fig. 3͒. The transition
listed in Table I. Estimated values for ␣ and ␣zz were 38
yy
Ϫ40
Ϫ1
2
and 33ϫ10
CV m , respectively. The discrepancy be-
tween the estimated and experimental ␣ can be attributed to
zz
the TLM contribution of the CR transition which is
͑
TLM͒
Ϫ40
Ϫ1
2
␣
͑Ϫ␻; ␻͒ϭ19ϫ10
CV m , Eq. ͑37͒. This contri-
bution is not included in the group and bond additivity
scheme. The Kerr anisotropy ␦␣ can now be calculated to
K
dipole ␮( is 21ϫ10 2 C m ͑6.3 D͒ as determined from the
z)
Ϫ30
Ϫ40
Ϫ1
2
ag
be 22ϫ10
CV m , which is larger than the Rayleigh
2,26
integrated absorption.
This value is large for a chro-
anisotropy ␦␣ . Only for rod-shaped molecules with
R
mophore of such a small size, again indicating resonant con-
tributions as implied by Eq. ͑39͒. Neither the position nor the
intensity of the absorption band changes significantly in sol-
vents of higher permittivity such as acetone or acetonitrile,
␣ ϭ␣ ϭ␣ and ␣ ϭ␣ are the anisotropies identical,
Ќ
xx
yy
ʈ
zz
␦␣ ϭ␦␣ ϭ␣ Ϫ␣ , Eqs. ͑31͒ and ͑41͒.
R
K
ʈ
Ќ
͑
TLM͒
It is interesting to note that ␣
contributes about 40%
to the long axis polarizability ␣zz and that the polarizability
2
6,33
͑TLM͒
as expected from the theory of solvatochromism
for an
anisotropy ␦␣K is almost entirely due to ␣
in both
excitation with a negligible change of the dipole moment
DPDCP and pNA. This suggests that the determination of
͑TLM͒
͑TLM͒
upon excitation. This is consistent with ␤
to Eq. ͑32͒.
ϳ0, according
␣
from UV-VIS spectroscopic measurements can be
used as a screening method in the search for chromophores
The tensor trace of the first-order polarizability ␣͑Ϫ␻;␻͒
with high ␦␣ . Table I provides a comparison of the differ-
ent molecular contributions to the molar susceptibilities ␨
K
was determined by refractive index measurements1
6
5,30
at
͑3͒
2
32.8 nm in tetrachloromethane solution with an Abbe-3L
refractometer ͑Bausch & Lomb͒. Information on the anisot-
in Eqs. ͑29͒ and ͑30͒. The ␮ ␦␣ term of DPDCP is about
K
36
three times larger than that of pNA due to its 18% larger
dipole moment and its two times larger polarizability anisot-
ropy. The ␮␤ term of DPDCP is nearly zero because of its
nearly vanishing second-order polarizability. For pNA, the
ropy ␦␣ ͑Ϫ␻;␻͒ of the first-order optical polarizability de-
R
fined by
2
R
1
2
2
2
2
͑3͒
␦
␣ ϭ ͓͑␣ Ϫ␣ ͒ ϩ͑␣ Ϫ␣ ͒ ϩ͑␣ Ϫ␣ ͒ ͔
͑42͒
xx
yy
xx
zz
yy
zz
␮␤ term is significant, but contributes only about 17% to ␨ .
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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10644
Wortmann et al.: Photorefractive polymers
TABLE II. Results of quantum chemical calculations ͑MP2͒ of dipole moments and static first- and second-
order polarizabilities of DMDCP and pNA.
DMDCP
pNAa
Ϫ30
Ϫ40
Ϫ40
Ϫ40
Ϫ40
Ϫ50
Ϫ50
Ϫ60
Ϫ60
␮
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
͑10
Cm͒
26.7
13.1, 24.2, 35.6
72.9
16.9
19.5
0.00, 1.58, 2.04
3.61
22.9
9.5, 17.3, 26.1
52.9
Ϫ1
2
␣xx ,␣yy ,␣zz
trace͑␣͒
CV m ͒
Ϫ1
2
CV m ͒
Ϫ1
2
␦
␦
␣K
␣R
CV m ͒
12.7
14.3
Ϫ1
2
CV m ͒
␤zxx ,␤zyy ,␤zzzb
CV m ͒
Ϫ2
3
Ϫ0.15, Ϫ0.25, 6.60
6.20
b
Ϫ2
3
␤ϭ␤zxxϩ␤ ϩ␤
CV m ͒
zyy
2
zzz
2
2
Ϫ3
4
4
6
␮ ␦␣ /(3k T )
␮␤/(kT)
CV m ͒
95 000
1400
52,500
2070
K
Ϫ3
4
CV m ͒
a
From Ref. 42.
Given in T convention, Ref. 16.
b
VII. QUANTUM CALCULATIONS
Kerr effect, ␤ would have to be at least ten times larger.
Thus, these calculations support the assumption that concen-
The geometry of DMDCP, in which the n-propyl groups
of DPDCP have been replaced by methyl groups, was opti-
2
trating on maximization of the ‘‘␮ ␦␣ ’’ contribution is a
K
more profitable approach to optimizing the chromophore for
mized in C
functional
symmetry using the MB3LYP density
in conjunction with the 6-31G* basis set.
2␯
3
7,38
39,40
PR applications. If the ‘‘␮␤’’ contribution has the same sign
2
as ‘‘␮ ␦␣ ,’’ this will increase the total value, but probably
K
The dipole and ͑hyper͒polarizability calculations were per-
formed at this optimized geometry using second-order per-
turbation theory ͑MP2͒ together with a double-zeta quality
basis set [3s2p/2s] augmented with diffuse polarization
by only 10%. Examination of the individual moments shows
that DMDCP has a larger dipole moment and a larger polar-
izability anisotropy ␦␣ than pNA. It is interesting to note
K
that the sum of ␣xx and ␣yy is close to ␣zz for both mol-
functions ͑␣ for C, N, and Oϭ0.2 and 0.05͒. This basis was
d
ecules, so that the anisotropy ␦␣K is about 1/2 ␣ . Thus,
zz
shown to give results in agreement with larger basis sets for
the polarizabilities of pNA.4
1,42
DMDCP has a larger polarizability anisotropy since ␣zz is
We have chosen to use
4
0% larger than for pNA. We note that in order to maximize
second-order perturbation theory since it is important to cal-
culate the hyperpolarizabilities of conjugated organic mol-
ecules with theoretical methods that include the effects of
electron correlation. For example, for pNA the electron cor-
relation contribution is responsible for 50% of the value of ␤.
The ͑hyper͒polarizabilities were calculated from finite differ-
ence calculations of the MP2 energy and MP2 dipole mo-
ment ͑where the dipole moment is defined as the first deriva-
tive of the energy with respect to an external electric field͒.
The field strengths used were 0.0, 0.000 667, 0.002 a.u.
Analysis of the values based on finite difference of energies
versus finite difference of dipole moment indicates that the
error based on finite field differentiation is not significant
the anisotropy, we must aim to increase ␣zz without signifi-
cantly increasing ␣xx or ␣ , since these latter components
yy
are not negligible for these molecules. Finally, if we sum the
major contributions to the photorefractive Kerr effect from
these theoretical calculations, we see that DMDCP is almost
two times more effective than pNA.
Let us turn now to comparison of the calculated values
for DMDCP with the experimental results for DPDCP ͑as
reported in Table I͒. The comparison between the calculated
and experimental para-nitroaniline values has been discussed
at length ͑see, for example, Ref. 41͒ and only comments
pertinent to the DMDCP discussion will be raised. First, we
chose to evaluate the properties of DMDCP rather than
DPDCP to make the calculations more computationally trac-
table. The difference between the DMDCP values and those
of DPDCP is expected to be small, since the major contribu-
tion to the polarizabilities arises from the mobility of the ␲
electrons in the conjugated system. We note that the dipole
moment overestimates the experimental value by 9%, not
unexpected since this is also seen for pNA. If we first exam-
ine ␦␣R , which may be directly compared with experiment,
͑less than 1% for ␤͒.
The values of the electrical properties ͑in the static field
limit͒ of DMDCP are reported in Table II together with those
4
1
for pNA. There are two important points to examine in the
context of this work. First, how do the theoretical values
compare and contrast for DMDCP and pNA, and second,
how do the calculations compare with experiment? We can
address the first point predominantly from examination of
the calculated gas phase properties in the static field limit. Of
Ϫ50
Ϫ1
2
main interest is the magnitude of the contribution to the pho-
we note good agreement ͑19.5 vs 18.1ϫ10
CV m ͒.
2
torefractive Kerr effect from the ‘‘␮ ␦␣ ’’ term and the
Comparison of trace ␣ with trace ␣͑Ϫ␻;␻͒ shows that the
theoretical static value is about 34% smaller than the experi-
mental value at 633 nm. It is unlikely that the frequency
dependence is responsible for more than 10%–20% of this
difference. It is possible that the remaining difference could
be attributed to the replacement of the methyl groups
͑DMDCP͒ by n-propyl groups ͑DPDCP͒. There is reason-
K
‘‘␮␤’’ term, as summarized in the last two lines of Table II.
It is clear that for both molecules the larger contribution
2
comes from the ‘‘␮ ␦␣ ’’ term and that the ‘‘␮␤’’ term is
K
considerably smaller even for pNA where ␤ is considered
significant ͑in the context of nonlinear processes͒. For this
latter term to play an equivalent role in the photorefractive
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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10645
able agreement for the percentage contributions of the indi-
vidual components of ␣ to the trace ␣. Since trace ␣ is
tin oxide ͑ITO͒. After evaporation of the solvent ͑nitroben-
zene͒, the plates were assembled face to face at an elevated
temperature of ϳ135 °C, at which the material is a viscous
liquid, using 100 ␮m spacers. The resulting samples were of
good optical quality and almost transparent throughout the
underestimated theoretically, however, so also is ␦␣ . We
K
note though that, most importantly, experiment and theory
2
are in clear agreement on the increasing trend of ‘‘␮ ␦␣ ’’
K
between DPDCP and pNA.
visible region. The absorption coefficient ␣ at 676 nm was 2
Ϫ1
There is a considerable difference between theory and
experiment in the prediction of ␤. We will discuss this dif-
ference, but it should be realized that since the ‘‘␮␤’’ con-
cm
.
The dark conductivity ␴dark of samples of DPDCP:
PMMA:TPD:C60 was measured to be typically 3.2
Ϫ12
Ϫ1
tribution to the photorefractive Kerr effect is so much
ϫ 10
͑⍀ cm͒ . The specific photoconductivity ␴ /I at
ph
2
Ϫ12
Ϫ1
2
smaller than the ‘‘␮ ␦␣ ’’ contribution, this discrepancy
676 nm was 1.2ϫ10
of 5 V/␮m and 3ϫ10
V/␮m.
͑⍀ cm͒ /͑W/cm ͒ at a bias field
K
Ϫ12
͑⍀ cm͒Ϫ1/͑W/cm ͒ at E ϭ30
2
does not have any impact on the conclusions in this paper.
First, we note that there is a difference of more than a factor
of 3 between ␤calc and ␤exp for pNA. This difference is
mainly due to the fact that the experiments are done in polar
solvents, whereas the calculations refer to gas-phase
b
The PR properties of DPDCP:PMMA:TPD:C60 were in-
vestigated by using the holographic optical techniques of
four-wave mixing ͑FWM͒47 and two-beam coupling ͑2BC͒.
The continuous-wave writing beams 1 and 2 ͑␭ϭ676 nm, 6
mW͒ were overlapped in the sample, the external angles of
incidence being 60° and 30° as in previous studies.11 This
49
1
3,43
species.
We note that the long axis component of ␤ ͑␤zzz
͒
for DMDCP is three times smaller than for pNA and that it is
the other in-plane component ␤zyy that is responsible for the
overall ␤ of DMDCP being less than a factor of 2 smaller
than for pNA. Given the fact that DPDCP is much less of a
geometry corresponds to internal angles of ␪ ϭ33.5° and
1
␪2ϭ18.6° in the polymer ͑index of refraction nϭ1.57͒ and a
‘‘charge-transfer’’ molecule than pNA, it is possible that the
grating period of ⌳ ϭ1.66 ␮m. The spot diameter ͑FWHM͒
G
solvent effect for DPDCP is smaller and, therefore, that there
is a much wider difference in hyperpolarizability between
these two molecules in solution. This rationale is borne out
by the fact that the charge resonance transition of DPDCP
does not solvatochromically shift in a polar environment.
of the weakly focused writing beams at the sample was de-
termined with a silicon photodiode array ͑Reticon RL0512G͒
to be 760 ␮m. The writing beams were s-polarized in the
FWM experiment and the grating was probed with a
p-polarized beam 3 ͑676 nm, 16 ␮W, spot diameter 370 ␮m͒
counterpropagating to beam 2. The smaller spot size of beam
3
3
was chosen in order to guarantee complete overlap of beam
with the grating. Diffraction efficiencies were corrected for
VIII. PHOTOREFRACTIVE PROPERTIES OF THE
DOPED POLYMER SYSTEM DPDCP:TPD:PMMA:C60
reflection losses at the outer surfaces of the sample. In 2BC
experiments, the writing beams 1 and 2 were p-polarized and
their intensities were monitored directly.
A doped polymer material consisting of 30% DPDCP,
15% TPD, 55% PMMA, and 0.3% C60 was chosen as a
model system to illustrate the potential of functional dyes in
the CR limit for PR organic materials. DPDCP was charac-
terized as a transparent, highly dipolar chromophore with
large anisotropy of the optical polarizability and negligible
second-order nonlinearity ͑Table I͒. Poly͑methylmeth-
Samples of DPDCP:PMMA:TPD:C60 exhibited steady-
state diffraction efficiencies ␩ of up to 25% at a bias field of
p
100 V/␮m. This corresponds to an index modulation in the
(
1)
PR grating of ⌬nϭ(e •⌬ •e )/(2n)ϭ0.0010 ͓calculated
3
␹
4
from Eqs. ͑1͒, ͑9͒, and ͑10͔͒. The index modulation showed
a quadratic dependence, ⌬nϳE2
.06Ϯ0.05
, on the applied bias
acrylate͒ ͑PMMA, purchased from Aldrich, M ϭ33 000,
W
b
secondary standard͒ was used as an inert polymer binder
with good optical quality and low scattering.
N,NЈ-bis͑3-methylphenyl͒-N,NЈ-bis͑phenyl͒benzidine͑TPD,
purchased from H. W. Sands͒ was employed as an efficient
charge transport agent ͑CTA͒.44,45 The fullerene C60 ͑Ald-
rich͒ was used as a sensitizer with low absorption at 676 nm
for the photogeneration of free carriers.4
field ͑Fig. 4͒, indicating proportionality of Esc and E . This
behavior is predicted by the standard PR model when the
trap density-limited space-charge field Eq is much larger
b
than both the bias field and the diffusion field E ϳ0.1
d
V/␮m.8
The initial growth of the PR grating was single exponen-
6,47
1/2
tial, ␩ ϰ1Ϫexp͑Ϫt/␶ ͒, with a characteristic rise time ␶ .
r
r
Hole mobilities in vapor-deposited glasses and doped
polymers containing TPD were investigated recently, and
charge transport in these materials was demonstrated to oc-
cur via a simple disorder-controlled hopping mechanism.44,45
C60 was observed to form a charge transfer complex with
This functional dependence is consistent with the conven-
tional single carrier model for the formation of the space-
charge field Esc .50 The rise time ␶ was 850 ms at E ϭ50
r
b
Ϫ1
V/␮m. The inverse rise time ␶ was a sublinear function of
r
Ϫ1
r
0.42
the writing intensity, ␶ ϳI , similar to observations in
48
other systems.8
TPD in organic nanocomposites and multilayer structures,
the maximum of the CT band lying near 675 nm. The func-
tion of C60 in the photogeneration of charges and in charge
transport processes have also been investigated in
The photorefractive nature of the observed gratings was
confirmed by two-beam coupling and grating-phase
measurements.5 In these experiments, gratings were writ-
ten with two writing beams of equal intensity. Figure 5
shows the transmitted intensities of beams 1 and 2 in the
steady state ͑at negative times͒, with beam 1 being amplified
,49
photoconducting4 and photorefractive polymers.
6
47
Samples for PR studies were prepared by casting the
polymer from solution onto glass plates coated with indium
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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Wortmann et al.: Photorefractive polymers
this work, ␪ ϭ33.5°, ␪ ϭ18.6°, ␪ ϭ64.0°, sample thickness
1
2
G
Ϫ3
dϭ100 ␮m, refractive index nϭ1.57, density ␳ϭ0.9 g cm
ϳn , ␮ញ ϳ2␮ ͑assumed͒, molecular weight of DPDCP
Mϭ228.3 g mol , one obtains
,
ϱ
2
⑀
Ϫ1
5
͑3͒
2
␩sϷ1.91ϫ10 ϫ͑␹
E E ͒ ,
͑43͒
͑44͒
ZZXX
b
sc
5
͑3͒
2
␩pϷ1.73ϫ10 ϫ͑␹
E E ͒ ,
b sc
ZZZZ
where a contribution of less than 2% of ␹(3) in Eq. ͑43͒
ZZXX
was neglected. Thus, with ␩ and ␩ , one probes in this
p
s
(
3)
(3)
ZZXX
geometry almost exclusively ␹
and ␹
, respectively.
ZZZZ
From Eqs. ͑43͒ and ͑44͒, one expects the depolarization ratio
to be ␩ /␩ ϭ3.6 in the limit of the purely ‘‘birefringent’’
p
s
␤
-less chromophore DPDCP, Eqs. ͑29͒ and ͑30͒. In good
agreement with this prediction, we observed experimentally
the ratio 4.0Ϯ1.6. A quantitative estimate of the diffraction
efficiency and the index modulation assuming a maximum
space-charge field of E ϭE cos ␪ based on Eqs. ͑17͒,
FIG. 4. Refractive index modulation ⌬n and two-beam coupling gain ⌫ as
a function of the bias field E . The curves are quadratic fits.
b
sc
b
G
͑
30a͒, and ͑44͒ yields ␩ ϭ7% and ⌬nϭ0.000 55 at E ϭ50
p
b
V/␮m. The observed index modulation was ⌬nϳ0.0003
Fig. 4͒. This is a reasonable agreement, in view of the ap-
proximate character of the outlined theory for highly doped
polymer systems and the small index changes involved.
by asymmetric energy transfer. The two-beam coupling co-
͑
efficient ͑energy transfer coefficient͒ showed a quadratic de-
2
b
.05Ϯ0.22
pendence, ⌫ϳE
, on the bias field. Because of the
low absorption coefficient ␣ of the material, net two beam
coupling, ⌫Ͼ␣, was achieved at very low fields, E Ӷ20
V/␮m.
b
IX. CONCLUSIONS
We have demonstrated photorefractivity in a low-Tg
polymer system where the dopant chromophore DPDCP has
Starting at time zero in Fig. 5, the sample was translated
for 2 s with rate 3.2 ␮m/s. The resulting periodic intensity
variations of beams 1 and 2 were clearly out of phase, con-
sistent with a dominant index grating. Fitting of these curves,
as described in Ref. 49, was impeded by grating decay and
intensity fluctuations. Typical phase shifts between index
and intensity gratings could thus be determined only to be
between 60° and 90° at fields greater than 30 V/␮m, corrobo-
rating the PR origin of the index gratings.
a negligible second-order polarizability ␤. The modulation of
the refractive index in this material was shown to be due to
the interaction of the square of the total electric field in the
͑
3͒
material with the Kerr susceptibility ␹ ͑Ϫ␻;␻,0,0͒, the total
electric field being a superposition of the bias field and the
spatially periodic space-charge field. A tensorial formulation
of the photorefractive Kerr effect and its microscopic inter-
pretation was presented. A molecular figure of merit for pho-
torefractive chromophores was derived ͓Eq. ͑41͔͒. Predic-
tions of the theory were in reasonable agreement with
experimental findings. The second-order dipole orientation
term containing the anisotropy of the first-order optical po-
larizability and the square of the dipole moment was shown
to provide the major molecular contribution to PR perfor-
mance. Functional dyes in the charge resonance limit,
DPDCP representing a prototype, were identified to optimize
this term. These dyes combine high dipole moments with a
large polarizability anisotropy which is produced by the in-
tense CR transition. The CR transition in DPDCP contributes
Some estimates of the PR properties of the material can
be made on the basis of Eqs. ͑11͒–͑15͒, ͑17͒, and ͑20͒–͑30͒.
For the actual FWM geometry and the material properties of
3
7% to the long axis polarizability and 86% to the polariz-
ability anisotropy, assuming that the contribution from
higher-order states is equivalent for ␣ , ␣ , and ␣ . Ex-
xx
yy
zz
perimental values of the dipole moment and first-order po-
larizability of DPDCP were in good agreement with those
obtained from quantum chemical calculations. The ground
state of DPDCP is stabilized by contributions of aromatic
resonance structures and no significant change of the dipole
moment occurs upon excitation in the CR band. As a conse-
quence, the CR transition occurs at high energy in the UV
region and does not solvatochromically shift in a polar envi-
ronment. Highly transparent and efficient PR polymers can,
FIG. 5. Two-beam coupling and grating phase measurement. The transmit-
ted intensities of beams 1 and 2 are shown as a function of the translation
time. The signal for negative times is the steady state. The translation starts
at time zero, the rate being 3.2 ␮m/s.
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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Wortmann et al.: Photorefractive polymers
10647
2
2
2
3
4
5
therefore, be devised on the basis of functional dyes in the
CR limit. Since it is possible in this limit to achieve large
anisotropies of the polarizability for relatively short mol-
ecules, it can also be expected that efficient PR materials
with fast orientational response can be prepared.
L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, R. Rikken, and S.
R. Marder, J. Phys. Chem. 95, 10631 ͑1991͒.
P. Maslak, A. Chopra, C. R. Moylan, R. Wortmann, S. Lebus, A. L.
Rheingold, and G. P. A. Yap, J. Am. Chem. Soc. 118, 1471 ͑1996͒.
S. R. Marder, D. N. Beratan, and L.-T. Cheng, Science 252, 103 ͑1991͒.
W. Liptay, Angew. Chem. Intern. Ed. 8, 177 ͑1969͒.
26
2
7
S. S. Deshapande, J. Indian Chem. Soc. 9, 303 ͑1932͒.
P. Yates, E. S. Hand, P. Singh, S. K. Roy, and I. W. J. Still, J. Org. Chem.
1
28
F. S. Chen, J. Appl. Phys. 38, 3418 ͑1967͒.
Electrooptic and Photorefractive Materials, Springer Proceedings in
2
34, 4046 ͑1969͒.
Physics 18, edited by P. G u¨ nter ͑Springer, Berlin, 1987͒.
P. Yeh, Introduction to Photorefractive Nonlinear Optics ͑Wiley, New
29
30
L. L. Woods, J. Am. Chem. Soc. 80, 1440 ͑1958͒.
R. Wortmann, K. Elich, S. Lebus, W. Liptay, P. Borowicz, and A.
Grabowska, J. Phys. Chem. 96, 9724 ͑1992͒.
C. R. Moylan, R. D. Miller, R. J. Twieg, K. Y. Betterton, T. J. Matray, and
C. Nguyen, Chem. Mater. 5, 1499 ͑1993͒.
E. A. Guggenheim, Trans. Faraday Soc. 45, 203 ͑1949͒.
W. Z. Liptay, Naturforsch. 21a, 1605 ͑1966͒.
3
York, 1993͒.
G. T. Sincerbox, Opt. Mater. 4, 370 ͑1995͒.
K. Sutter and P. G u¨ nter, J. Opt. Soc. Am. B 7, 2274 ͑1990͒.
K. Sutter, J. Hulliger, and P. G u¨ nter, Solid State Commun. 74, 867 ͑1990͒.
4
3
1
5
6
3
3
2
3
7
S. Ducharme, J. C. Scott, R. J. Twieg, and W. E. Moerner, Phys. Rev.
Lett. 66, 1846 ͑1991͒.
W. E. Moerner and S. M. Silence, Chem. Rev. 94, 127 ͑1994͒.
K. Meerholz, B. L. Volodin, Sandalphon, B. Kippelen, and N. Peygham-
8
34
35
K. Eidner and M. F. Vuks, Opt. Spektroskop. 45, 697 ͑1978͒.
H. A. Stuart, in Landolt-B o¨ rnstein ͑Springer, Berlin, 1951͒, Vol. I/3, p.
9
barian, Nature 371, 497 ͑1994͒.
509.
1
1
1
0
1
2
O. Zobel, M. Eckl, P. Strohriegl, and D. Haarer, Adv. Mater. 7, 911
1995͒.
36
`
M. J. Aroney, K. E. Calderbank, R. J. W. LeFevre, and R. K. Pierens, J.
Chem. Soc. B 1968, 561.
͑
W. E. Moerner, S. M. Silence, F. Hache, and G. C. Bjorklund, J. Opt. Soc.
Am. B 11, 320 ͑1994͒.
P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics ͑Cam-
bridge University Press, Cambridge, 1990͒, Chap. 2.
R. Wortmann, Habilitation thesis, Mainz 1993.
J. Feinberg, in Optical Phase Conjugation, edited by R. A. Fisher ͑Aca-
demic, New York, 1983͒.
W. Liptay, J. Becker, D. Wehning, W. Lang, and O. Burkhard, Z. Natur-
forsch 37a, 1396 ͑1982͒.
37
38
A. D. Becke, J. Chem. Phys. 98, 5648 ͑1993͒.
V. J. Barone, Chem. Phys. 101, 6834 ͑1994͒.
R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys. 54, 724 ͑1971͒.
W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 56, 2257
͑1972͒.
F. Sim, S. Chin, M. Dupuis, and J. E. Rice, J. Phys. Chem. 97, 1158
͑1993͒.
C. Daniel and M. Dupuis, Chem. Phys. Lett. 171, 209 ͑1990͒.
M. St a¨ helin, C. R. Moylan, D. M. Burland, A. Willetts, J. E. Rice, D. P.
Shelton, and E. A. Donley, J. Chem. Phys. 98, 5595 ͑1993͒.
S. Heun and P. M. Borsenberger, Chem. Phys. 200, 245 ͑1995͒.
S. Heun and P. M. Borsenberger, Physica B 216, 43 ͑1995͒.
Y. Wang, Nature 356, 585 ͑1992͒.
S. M. Silence, C. A. Walsh, J. C. Scott, and W. E. Moerner, Appl. Phys.
Lett. 61, 2967 ͑1992͒.
H. W. Sarkas and Z. H. Kafafi, in Organic Thin Films for Photonic Ap-
plications, Vol. 21 of OSA Technical Digest Series ͑Optical Society of
America, Washington DC, 1995͒, pp. 321–324.
C. A. Walsh and W. E. Moerner, J. Opt. Soc. Am. B 9, 1642 ͑1992͒.
G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 ͑1983͒.
3
4
9
0
1
1
3
4
41
1
1
1
5
6
7
42
43
A. Willetts, J. E. Rice, D. M. Burland, and D. P. Shelton, J. Chem. Phys.
9
7, 7590 ͑1992͒.
R. Wortmann, C. Glania, P. Kr a¨ mer, R. Matschiner, J. J. Wolff, E. Barbu,
44
45
46
47
and D. L a¨ ngle ͑unpublished results͒.
L. Onsager, J. Am. Chem. Soc. 58, 1486 ͑1937͒.
C. J. F. B o¨ ttcher, Theory of Electric Polarization, Vol. I. Dielectrics in
Static Fields ͑Elsevier, Amsterdam, 1973͒.
W. F. Brown, in Encyclopedia of Physics, edited by S. Fluegge ͑Springer,
Berlin, 1956͒ Vol. XVII, p. 1.
D. L. Andrews and T. Thirunamachandran, J. Chem. Phys. 67, 5026
1
1
8
9
2
2
2
0
1
2
48
͑1977͒.
49
R. Wortmann, P. Kr a¨ mer, C. Glania, S. Lebus, and N. Detzer, Chem.
Phys. 173, 99 ͑1993͒.
50
J. Chem. Phys., Vol. 105, No. 23, 15 December 1996
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