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Nonlinear Optics and Quantum Optics, Vol. 38, pp. 99–140
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Second Harmonic Generation Technique
and its Applications
Ileana Rau1 and Francois Kajzar2
1 POLITEHNICA University of Bucharest,
Faculty of Applied Chemistry and Materials Science, 1 Polizu Str., Bucharest, Romania
E mail: [email protected]
2 Université d’Angers, UFR Sciences, Laboratoire POMA CNRS UMR 6136,
2, Bd Lavoisier, 49045 Angers, France
Received: July 12, 2007.
The principle of optical second harmonic generation in a noncentrosymmetric medium is described. Materials exhibiting high second order
NLO susceptibilities are reviewed. Practical applications, particularly for
material characterization, frequency conversion, optical microscopy and
surface studies of second harmonic generation technique are reviewed and
discussed.
Keywords: Nonlinear optics, SHG, electrooptic polymers, phase matching, polymer
liquid crystals, poling, relaxation of polar order, SHG microscope, surface SHG.
1 INTRODUCTION
Second harmonic generation (SHG) is a nonlinear optical process, in which
two input photons at ω frequency generate, in a nonlinear medium, one photon
with frequency 2ω, as it is shown schematically in Fig. 1. It is a typical three
photon process. The two, coherent, photons may come from the same or from
two different sources, provided they are collinear. In the last case it is a two
wave mixing process, although in general, the SHG is such a process.
In the quantum mechanical description this process is viewed as going
through virtual states as it is shown in Fig. 2. If one of the virtual state is close
to the excited state of unperturbed system a resonance will occur in SHG and
the NLO susceptibility, describing it, is complex.
Although the frequency doubling was observed earlier in electrical circuits
with low frequency fields, the first observation of the optical frequency conversion was done in 1961 by Franken et al. [1] in α-quartz single crystal and
99
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100
I. Rau and F. Kajzar
NL medium
2
FIGURE 1
Schematic presentation of SHG process.
2
1
2
0
FIGURE 2
Schematic representation of the second harmonic generation process. Solid lines show fundamental (0) and excited (1,2) levels of unperturbed system whereas dashed lines show virtual
states.
using a ruby laser (λω = 694.3 nm). Since that time the SHG technique has
known a significant development, driven not only by potential practical applications for e.g. laser frequency conversion, but also as a powerful tool for
second order NLO properties characterization, the study of metal surfaces,
polar order, poling kinetics and of the electronic structure of molecules. In
particular, it was very much used for the surface studies [2, 3]. It led also to
an important material development from the point of view of its conception
as well as in material engineering, particularly for application in electro-optic
modulators [4, 5].
In this paper we describe principles of optical second harmonic generation
process and overview some aspects of its practical applications.
Under the forcing external DC or AC electric field the dipole moments of
the material medium vibrate with frequencies being multiples and/or any linear
combination of the frequencies of forcing fields. Consequently the medium
polarization, at ωσ frequency (ωσ = ω1 +ω2 +ω3 +· · · ) varies. This variation
can be expanded into the power series of the external forcing field E. The ith
component of the polarization vector is given by
(1)
PI (ωσ ) = P0I (ωσ ) + K1 χIJ (−ωσ ; ωσ )EJωσ
(2)
ω2
+ K2 χIJK (−ωσ ; ω1 , ω2 )EJω1 EK
(3)
ω2 ω2
+ K3 χIJKL (−ωσ ; ω1 , ω2 , ω3 )EJω1 EK
EL + · · ·
(1)
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Second Harmonic Generation Technique and its Applications
101
where the expansion coefficients χ (n) are three dimensional (n + 1) rank
tensors, with 3n+1 components describing linear (n = 0) and nonlinear (n > 1)
properties of bulk material and P0 is the medium permanent polarization. The
coefficients K’s in Eq. (1) depend on conventions used for the definition of
electric field and the way how the frequency degeneracy is taken into account.
In Eq. (1) the Einstein’s convention is adopted.
Hereafter we use the convention in which the Fourrier transforms of the
electric field and the polarization are defined as follows (see also Refs. [6, 7])
1
[E(r)eiωt + c.c.]
2
1
P(r, t) = [P(r)eiωt + c.c.]
2
E(r, t) =
(2)
(3)
The K’s coefficients in Eq. (1) for different second order NLO processes are
products of two factors: W arising from the definition of Fourrier transform
of electric and polarization fields (cf. Eqs. (2)–(3)) and D which takes into
account the frequency degeneracy. They are listed in Table 1 for different
second-order NLO processes.
Second-order
NLO process
NLO susceptibility
term
W
D
K
Applications
Second
harmonic
generation
χ (2) (−2ω; ω, ω)
1
2
1
1
2
Two wave
mixing
χ (2) (−ω3 ; ω1 , ω2 )
1
2
2 (if no SHG)
1
Frequency tuning Nonlinear
spectroscopy Surface study
Vibronic excitations study
Linear
electrooptic
(Pockels)
effect
χ (2) (−ω; ω, 0)
1
2
2
Electro-optic modulation
Parametric
process
χ (2) (ω1 , ω2 ; ω3 )
ω1 + ω2 = ω3
1
2
2
1
Parametric amplification
Frequency tuning
Optical
rectification
χ (2) (0; ω, −ω)
1
2
2
Ultrashort electric pulses
generation
(THz generation)
Optical
parametric
generation
χ (2) (ω3 ; ω1 , ω2 )
ω3 = ω1 + ω2
6
2
1
Frequency tuning
Frequency doubling
Nonlinear spectroscopy
Surface study
TABLE 1
Second-order NLO processes, related NLO susceptibilities, multiplicative factors K
arising from electric field definition (W ) and degeneration (D) and their practical
applications
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102
I. Rau and F. Kajzar
For the second harmonic generation process, considered here, the nonlinear
polarization at 2ω frequency and within the assumed convention is given by
P (2ω) =
1 (2)
χ (−2ω; ω, ω)Eω2
2
(4)
As it follows from Eq. (4) χ (2) ≡ 0 in centrosymmetric media and in dipolar
approximation. Indeed, if we change the direction of the electric field in such
media, the polarization field will change too, while the product (−E)(−E)
will not. So the only possibility is χ (2) = 0.
As already mentioned the χ (2) tensor (n = 2) has 27 components. Due to
the material symmetry some of them vanish (tables of χ (2) tensor components
for different crystal symmetries can be found in a number of textbooks [8]).
The number of non-zero χ (2) tensor components is still reduced by applying
(2)
the Kleinman’s conditions [9]. According to them, the χijk components are
symmetrical with respect to any permutation of the indices i, j , and k
(2)
(2)
χijk (ω3 ; ω1 , ω2 ) = χikj (ω3 ; ω1 , ω2 )
(5)
The Kleinman’s conditions are valid far from the absorption band, in non
dispersive NL polarization media. Because of energy conservation conditions
the frequencies ω1 , ω2 , ω3 fulfill ω1 + ω2 + ω3 = 0.
Very often in literature, for historical reasons, another convention for electric and polarization fields and another tensor dijk are used to describe the
second order nonlinear optical properties of materials
dijk (−2ω; ω, ω) =
1 (2)
χ (−2ω; ω, ω)
2 ijk
(6)
(2)
As already mentioned in centrosymmetric media all χijk (dijk ) components vanish. In noncentrosymmetric crystals, comprising 21 crystallographic
(2)
classes from 32 total the number of χijk (dijk ) tensor components is reduced due
to the different symmetry elements. It gives, in the best case, 18 independent
tensor components. For this reason a reduced tensor notation dij is usually
used, defined as follows
dijk = din
for i =
1,
2,
3
and
n=1
2
3
4
5
6
for
(7)
jk = 11
jk =
22
33
23
31
12
32
13
21
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103
Second Harmonic Generation Technique and its Applications
Within this notation the polarization components at Pi (i = 1, 2, 3) at 2ω
frequency (Eq. (4)) are given by


P1


d11
P  
 2  = d21
P3
d31
d12
d22
d13
d23
d14
d24
d15
d25
d32
d33
d34
d35
E12

2 

 E2 
d16 

2
  E3 

d26  
2E E 
 2 3
d36 

2E1 E3 
(8)
2E1 E2
where Ei (i = 1, 2, 3) are components of the acting external field.
With Kleinman’s conditions the actual 18 components of d (or χ (2) ) tensor
reduce further to 10, with
d21 = d16
d31 = d15
d12 = d26
d14 = d25
d24 = d32
d13 = d35
d14 = d36
d36 = d25
It is worthy to note that the Kleinman’s relations are not always verified [10–13].
In the harmonic generation process, as it follows from the Maxwell equations with nonlinear polarization as source term (cf. e.g. Ref. [14]), two
harmonic waves propagates in the medium (cf. Fig. 3): one, called free
wave with velocity determined by the refractive index of harmonic wave and
the second one called the bound wave, propagating with the velocity given
by the refractive index of fundamental wave. This can be understood from the
following reasoning: the fundamental wave arriving at a given point will excite
dipoles which will radiate electric field at the harmonic frequency. The generated in this way harmonic wave will propagate in the medium (free wave). The
fundamental wave will propagate at the same time with its own velocity and
excite dipoles at other points, and consequently generate harmonic photons.
This may be seen as a fictive (or bound to the fundamental wave) harmonic wave propagating in the medium. As the nonlinear medium is usually
a dispersive one both waves propagate with different velocities and interfere
giving rise to the interference picture, called Maker fringes. An example of
Maker fringes for an y-cut single crystal slab of α-quartz is shown in Fig. 5.
The SH field generated in a nonlinear slab with thickness 1 is given by [15]
Et2ω =
4πP (2ω) iϕ2ω
T (θ )A(θ )[eiϕ − 1]
e
ε
(9)
where T (θ ) and A(θ ) are factors arising from transmission and boundary
conditions, respectively (for details see Ref. [14]), θ is the incidence angle
(cf. Fig. 6), ε = ε2ω − εω is the dielectric constant dispersion between
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104
I. Rau and F. Kajzar
n=0.4
OPTICAL ELECTRIC FIELD (arb. units)
1
2
-1
0
-2
lc
0
0.5
1.0
1.5
SECOND HARMONIC FIELD (arb. units)
resultan harmonic field at the output
bound wave
free wave
2.0
PROPAGATION LENGTH ( m)
FIGURE 3
Variation of bound, free (LHS) and resultant harmonic field (RHS) at the output of nonlinear
medium in function of the propagation length for the assumed refractive index dispersion n =
0.4 and at the fundamental wavelength λω = 1000 nm.
SECOND HARMONIC INTENSITY (arb. units)
1.5
lc
1.0
0.5
0
0
1
2
3
4
PROPAGATION LENGTH (l/lc)
FIGURE 4
Variation of the harmonic intensity as the function of the propagation length expressed in coherence
length (l/ lc ). The distance between the closest minimum and maximum is equal to the coherence
length.
fundamental and harmonic waves. The phase mismatch between harmonic
and fundamental beams is given by
ϕ = ϕω − ϕ2ω =
4π l
(nω cos θω − n2ω cos θ2ω )
λ
(10)
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105
Second Harmonic Generation Technique and its Applications
QUARTZ
SHG INTENSITY (arb. units)
400
300
200
100
0
-100
0
0.2
0.4
0.6
0.8
angle d'incidence (radians)
INCIDENCE ANGLE (radians)
FIGURE 5
Incident angle dependence of SHG intensity for an y-cut α-quartz single crystal. Points show the
measured values whereas solid line the fitted ones using Eq. (11).
where θω and θ2ω are propagation angles in nonlinear medium (cf. Fig. 6)
at fundamental and harmonic frequencies, respectively and nω(2ω) the
corresponding refractive indices.
From Eq. (9) we can calculate the harmonic field intensity at the output of
a nonlinear medium, which is given by


(2) 2
3
χ
ϕ
128π  eff 
I2ω =
|P (θ )A(θ )T (θ )|2 Iω2 sin2
(11)
ε
2
c2
(2)
where χeff is an effective second order NLO susceptibility for a given propagation direction defined by the propagation angle θω , Iω is the fundamental beam
intensity and P (θ ), A(θ ), T (θ ) are, respectively, projection and factors arising
from boundary and transmission conditions. The harmonic intensity reaches
a maximum for the phase mismatch ϕ = (2n + 1) π2 , n = ±1, ±2, ±3 . . . .
It defines the coherence length which is given by:
lc =
λω
4(n2ω − nω )
(12)
(see also Fig. 4). Here λω is the wavelength of fundamental beam.
It follows from Eq. (11) and Fig. 4 that the maximum conversion will
be achieved when the refractive indices at ω and 2ω frequencies are equal
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106
I. Rau and F. Kajzar
1
1
22
k
k 22
2
2
2
k2
f
2
b
2
FIGURE 6
SHG and light propagation in a nonlinear medium, kept in vacuum. Only incident fondamental
beam is shown and the harmonic free and bound waves generated in medium 2 are shown.
Superscripts b and f refer to the bound and free waves, respectively and the numbers to the
medium. No reflected beams are shown.
n = n2ω −nω = 0 (phase matching). In that case both bound and free waves
propagate with the same velocity. In Section 4.1 we will describe how such
situation can be realized. Although at ε ⇒ 0 we could expect a singularity
in Eq. (11), this is not the case, as at the same time ϕ ⇒ 0 and, consequently,
the last term on RHS of Eq. (11) tends to zero too.
The calculated variation of the harmonic intensity, as function of the propagation length (or the nonlinear medium thickness) is given in Fig. 4. The
distance between the closest minimum and maximum is equal to the already
mentioned coherence length lc .
Thus in SHG measurements it is important to vary the thickness of the
measured nonlinear medium to have a sufficient variation of the propagation
length. Usually it is done either by rotating the sample around a direction
perpendicular to the propagation direction, thus varying the incidence and
the propagation angles, or by using the wedge shaped samples and translating them perpendicular to the beam propagation direction. In that case the
incidence and the propagation angles are constant. An example of the variation of SHG intensity as function of the propagation length (Maker fringes) is
shown in Fig. 5 for α-quartz single crystal plate. For the propagation length
equal to the even multiples of the coherence length the harmonic intensity is
zero while it reaches a maximum for the propagation length equal to the odd
number of coherence lengths. The decreasing amplitude variation of the SHG
intensity is due to the decreasing amount of light coupled into the material,
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Second Harmonic Generation Technique and its Applications
107
because of reflections. The envelope function is determined here by the factors
P (θ ), A(θ ) and T (θ ) in Eq. (11). In the case of translated wedge the amplitude
of Maker fringes is constant. Equation (9) shows also that very important is
orientation of the studied material with respect to its symmetry directions, as
all quantities intervening in Eq. (11) depends on it, particularly the effective
(2)
susceptibility χeff .
The fit of experimental data gives, with a high precision and for a given
direction, the coherence length, defined by Eq. (12) and the nonlinear susceptibility if compared to a standard, measured at the same conditions, with
a known nonlinearity or by measuring the absolute intensities. However the
absolute measurements are usually difficult and a use of a standard is preferable. It is usually a y-cut α-quartz single crystal plate, with the recently
recommended value d11 (1064 nm) = 0.30±0.02 pm/V16 (see also discussion
by Roberts [17]). A lot of previous determinations were done using for d11 the
value reported by Choy and Byer (d11 (1064 nm) = 0.50 pm/V [18]).
3 MATERIALS
As we already mentioned, the principal requirement for the material is the lack
of the center of inversion, except in some isolated cases where the SHG was
also observed in centrosymmetric materials, such as phthalocyanines [19]
and fullerene C60 thin films [20–25] as due to the quadrupolar, magnetic
dipolar or combination of both contributions. There exists a large class of
phase matchable organic and inorganic single crystals, with large second order
NLO susceptibilities. Some examples of currently used and commercialized
non-centrosymmetric single crystals for the frequency up conversion, OPO or
electro-optic modulation are listed in Table 2.
However lot of applications are targeted in traveling wave configuration
where non-centrosymmetric thin films are needed. Moreover, the growth of the
required excellent optical quality single crystals is costly and time consuming.
For this reason a lot of effort has been paid in recent years towards the fabrication of artificial structures, such as poled polymers, Langmuir-Blodgett thin
films [26] , epitaxied single crystalline films [27] and self-assemblies [28].
Here the non-centrosymmetric molecules are assembled (LB films or self
assemblies) or oriented by the applied DC electric field or, alternatively,
by all optical poling into macroscopically non-centrosymmetric structures.
The molecules exhibiting high second order NLO response are the organic
charge transfer molecules, with electron donating (donor), electron accepting (acceptor) groups, linked by a conjugated π electron bond (cf. Fig. 7).
Such a molecule may be considered as a molecular diode [29]. It is polarized
strongly when the applied optical electric field is directed in the charge transfer
direction and weakly in the opposite case.
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108
Compound
I. Rau and F. Kajzar
Point
group
β -BaB2 O4
(BBO)
α -SiO2
(α -Quartz)
3m
KH2 PO4 (KDP)
4̄2m
NH4 H2 PO4
(ADP)
NH4 H2 PO4
(RDP)
CsH2AsO4
(CDA)
KH2AsO4
(KDA)
RbH2 PO4
(KDA)
4̄2m
32
Refractive
index at
1064 nm
no = 1.6551
ne = 1.5426
no = 1.5350
ne = 1.5438
at 1000 nm
no = 1.4938
ne = 1.4599
Transparency
range (nm)
198–2600
150–4500
dij (pm/V)
at 1064 nm
d22
d31
d11
d14
= ±(1.78 ± 0.09) [120]
= ±(0.12 ± 0.06)
= 0.50 [21]
= −0.014
176–1700
d36 = 0.435 ± 0.017 [122]
d36 = 0.39 ± 0.01 [123]
= 1.5065
= 1.4681
= 1.4920
= 1.4695
= 1.5514
= 1.5356
= 1.5476
= 1.5059
= 1.5405
= 1.5105
184–1500
d36 = 0.76 ± 0.01 [123]
220–1500
d36 = 0.402 ± 0.046 [122]
260–1430
d36 = 0.402 ± 0.046 [124]
216–1700
d36 = 0.52 ± 0.03 [125]
260–1460
d36 = 0.394 ± 0.040 [125]
6
no = 1.8571
ne = 1.7165
300–6000
LiNbO3
3m
no = 2.2340
ne = 2.1554
330–5500
CdSe
6̄2m
no = 2.5375
ne = 2.5572
750–20000
d31
d33
d31
d31
d31
d33
d15
GaSe
6̄2m
no = 2.9082
ne = 2.5676
650–18000
d22 = 24.4±0.1 [126] [127]
Co(NH2 )2 urea
4̄2m
d36 = 1.3 [128]
mm2
= 1.4811
= 1.5830
= 1.7386
= 1.7458
= 1.8287
200–1800
KTiOPO4 KTP
no
ne
nx
ny
nz
350–4500
KTiOAsO4 KTA
mm2
mm2
LiB3 O5 LBO
mm2
Ba2 NaNb5 O15
mm2
LiCOOH·H2 O
(LFM)
mm2
= 1.7820
= 1.7900
= 1.8680
= 2.1189
= 2.2199
= 2.2572
= 1.5656
= 1.5905
= 1.6055
= 2.2573
= 2.2571
= 2.1694
= 1.3593
= 1.4681
= 1.5035
350–4000
KNbO3 KBO
nx
ny
nz
nx
ny
nz
nx
ny
nz
nx
ny
nz
nx
ny
nz
d31 = ±6.5 [129]
d32 = ±5
d15 = ±6.1 d24 = ±7.6
d33 = 13.7
deff (KTA) = (1.6 ± 0.2)
deff (KTP) [130]
LiO3
4̄2m
4̄2m
4̄2m
4̄2m
no
ne
no
ne
no
ne
no
ne
no
ne
= −7.11 [18]
= −7.02
= −(5.53 ± 0.3) [122]
= d15 = −5.44 [122]
= −5.95 [18]
= −34.4
= 18 ± 1.8 [126]
400–4500
d31 = 11.5 [131–135]
d32 = −20.5 d33 = −20.1
370–5000
d31
d32
d33
d31
d32
d31
d31
d32
370–5000
230–1200
Remarks
d measured at
694 nm
d measured at
10600 nm
d measured at
10600 nm
= ±(1.09 ± 0.09) [135]
= ±(1.17 ± 0.14)
= ±(6.5 ± 0.6)
= −13.2 [136]
= −13.2 d33 = −18.2
= −14.6 [137]
= 0.1 [138]
= −1.16 d33 = 1.68
TABLE 2
Refractive indices (at given wavelength in nm) and second order NLO susceptibilities
of selected single crystals
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109
Second Harmonic Generation Technique and its Applications
Point
group
Compound
MAP
C10 H11 N3 O6
methy1-(2,4dinitrophenyl)amino-2propanoate-1oxide
POM 3-methy14-nitripyridine1-oxide
mNA
NO2 C6 H4 NH2
metanitroaniline
2
222
222
Refractive
index at
1064 nm
Transparency
range (nm)
dij (pm/V)
at 1064 nm
nx = 1.5079
ny = 1.5991
nz = 1.8439
500–2000
d16
d23
d14
d22
= 1.6242
= 1.6633
= 1.8287
= 1.6310
= 1.6780
= 1.7000
400–3000
d36 = 10 ± 1.5 [140]
nx
ny
nz
nx
ny
nz
Remarks
= d21 = 16.7 [139]
= d34 = 3.68
= d25 = d36 = −0.544
= 18.4
d31 = 20 [141] d32 = 1.6
d33 = 21
TABLE 2
(Continued)
+
DONOR
TRANSMITTER
_
ACCEPTOR
FIGURE 7
Schematic representation of a CT molecule designed for second order NLO applications.
Similarly, as in the macroscopic system, the molecule dipole moment is
varying under the applied electric field (this time the local field) and this can
be developed in the power series of the forcing field, which is the local field E,
experienced by the molecule.
µi (ωσ ) = µ0i (ωσ ) + K1 αij (−ωσ ; ωσ )Ejωσ + K2 βijk (−ωσ ; ω1 , ω2 )Ejω1 Ekω2
+ K3 γijkl (−ωσ ; ω1 , ω2 )Ejω1 Ekω2 Elω3 + · · ·
(13)
where the tensors α, β, γ , . . . are molecular linear, first and second hyperpolarizabilities, obeing the same selection rules and symmetry requirements as
the macroscopic susceptibilities, but this time restricted to the molecule itself.
In the charge transfer molecules the βzzz component of the first hyperpolarizability β tensor, responsible for the second order NLO response is enhanced
in the charge transfer direction z, and the others can be neglected with respect
to this one. These molecules are usually characterized by a large ground state
dipolar moment. Some examples of dipolar CT molecules are shown in Fig. 8.
The first hyperpolarizability β in CT molecules depends strongly on the
conjugation length of the transmitter and on the strength of electron donating
and electron accepting end groups. The scaling law dependence on number of
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110
I. Rau and F. Kajzar
(a)
CH2
CH2
( C CH2 ) ( C CH2 )
1-n
n
C
C
O
O
OMe
O
(b)
NH2
NH2
N
N
NO2
+
CH2
O
O
CH2 C H
2 5
O
O
N
N
O
(c)
NH2
C2H5
N
N
C2H5
NO2
N
N
NO2
+
( CH2
CH3
)
OCO
CH
CH
FIGURE 8
Chemical structure of side chain polymer PMMA-DR1 (a), epoxy matrix and active disperse
orange chromophore which cross link under heating (b) or UV irradiation with a polyvinyl (c).
double bond was checked experimentally for polyphenyl [30] and for polyene
oligomers [31] as dependence of β on number of double bonds N . A strong
dependence was observed, of the type β ∝ N δ , with δ ≈ 2.5. Concerning
the strength of donors and acceptors their strength can be described by the
following ranking [32] (a larger list of organic donors and acceptors is given
in Ref. [33]):
Acceptors: N(CH3 )2 > NH2 > OCH3 > OH
Donors:
NO > NO2 > CHO > CN
Octupolar molecules form another class of molecules with enhanced first hypzerpolarizability β tensor components [34,35]. However, these potentially very
interesting for practical application molecules, are difficult to process into
oriented, noncentrosymmetric structures. Only all optical poling technique
has been shown to create noncentrosymmetry in thin films made of these
molecules [36].
3.1 Poled polymers
Because of strong dipole-dipole interaction, leading usually to their antiparallel alignment, minimizing the ground state energy, it is very difficult to growth
noncentrosymmetric materials, either in the form of bulk single crystals, or
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Second Harmonic Generation Technique and its Applications
a
b
c
d
111
FIGURE 9
Schematic representation of different ways of making functionalized polymers for second order
NLO applications: (a) guest-host systems, (b) side chain polymers, (c) main chain polymers,
(d) photo-or thermally crosslinking polymers Arrows represent chromophore dipole moments
orientation.
single crystalline thin films. Therefore these molecules are usually dissolved
in polymer matrix. Here one exploits excellent optical wave propagation properties of optically inert polymer matrix and highly responsive NLO dipolar
molecules[37]. Four types of solid solutions are usually made, as shown in
Fig. 9 (for a review see Refs. [3, 4]):
(i) guest-host systems
(ii) side chain polymers
(iii) main chain polymers
(iv) thermally- or photo crosslinking systems
In the first case the active chromophores are simply dissolved in the polymer
matrix (Fig. 9(a)). In the second case (ii) the active chromophores are covalently bond to the polymer chains (side chain polymers, Fig. 9(b)). The main
chain polymers (iii) consists on the introduction of the active chromophores
into the polymer chain (cf. Fig. 9(c)) whereas in the case of the thermally [38]
or photo crosslinking [39] polymers the active chromophores are binding
elements between the polymer chains (Fig. 9(d)). These different approaches
exhibit some advantages and some drawbacks. The guest-host systems are easy
to prepare and don’t require an advanced chemistry. However, because of the
already mentioned strong dipole-dipole interaction, at the higher solute concentrations, formation of centrosymmetric aggregates takes place. As its result
one observes a decrease of the NLO response and at the same time undesirable
increase of the propagation losses [40]. Finally it reduces the concentration
of active molecules to a low value (e.g. 5% in the case od PMMA-DR1
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I. Rau and F. Kajzar
guest – host system). Although the concentration of active molecules can
be increased in side chain polymers, it is still subject of aggregation [41].
Chromophores in main chain of polymers are difficult to orient because
of limited mobility of polymers and steric constraints. Therefore the most
promising are the photo- and thermally crosslinking systems. However the
orientation of chromophores has to be done very carefully as the glass transition temperature increases during the crosslinking process with a possibility of
cracks formation in thin film because of strain. It leads to a significant increase
of the propagation losses. Obviously the chromophore orientation procedure
has to be done under the applied poling field. Examples of thermally and
photocrosslinking polymers are shown in Fig. 8.
To orient chromophores with dipolar moments pointing preferably in one
direction, one uses optical and static electric fields. Two techniques were set
up, such as
(i) static field poling
Electrode (Fig. 10) or corona (Fig. 11)
(ii) all optical poling
In static field poling [42] procedure thin films are heated close to the glass
transition temperature, in order to increase the chromophore mobility, and an
ELECTRODES
Polymer thin film
HV
SUBSTRATE
HEATING BLOCK
FIGURE 10
Schematic representation of an electrode poling set-up.
mA
HV
NEEDLE
ELECTRODE
mA
HV
++++++++++++++++++++++++++
THIN FILM
METALLIC
GRID
ELECTRODE
SUBSTRATE
HEATING BLOCK
FIGURE 11
Schematic representation of a corona poling set-up.
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Second Harmonic Generation Technique and its Applications
113
FIGURE 12
Optical absorption spectrum of a thin film of the side chain liquid crystalline polymer functionalized with a cyanobiphenyle mesogen before (solid line) and after (dashed line) poling ([42]).
electric field either through electrodes or by corona discharge is applied. After
poling thin films are cooled down to room temperature under the applied electric field. In all optical poling the poling field originates from the interference
of two beams at ω and 2ω frequency. In that case the poling can be done at
room temperature.
A first prove of the polar chromophore orientation is the decrease of the
chromophore absorption spectrum [43] and its red or blue shift [44], as it is
seen in Fig. 12 for a side chain liquid crystalline polymer (SCLCP), functionalized with a cyanobiphenyle chromophore. Important decrease and shift of
the optical absorption spectrum is due to a large amount of polar orientation.
Indeed, the poling field tends to orient the molecules with dipolar moments
perpendicular to the thin film substrate. In CT molecules ground state and
dipolar transition moments are collinear. So with observation field parallel to
the thin film surface less molecules are excited. As the linear optical absorption
spectrum does not distinguish between polar and axial order only the order
parameter P2 can be extracted from such measurements. This is given by:
P2 = 1 −
Ap
A0
(14)
where A0 and Ap are optical absorbances of thin film measured before and
after poling, respectively. Figure 12 shows the optical absorption spectrum of
a side chain liquide crystalline polymer functionalized with cyanobiphenyle
chromophore. An important variation of absorbance is observed due to poling,
corresponding to the order parameter P2 ≈ 0.6.
The only way to distinguish between the polar and the axial order is
the use of NLO techniques, such as, eg SHG discussed here. The poled
polymers exhibit an ∞ mm point symmetry. For this crystal class and tak(2)
ing into account the Kleinman’s relations there are two nonzero χijk tensor
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I. Rau and F. Kajzar
(2)
(2)
components: the diagonal χZZZ , and the off diagonal χXXZ components, where
Z is the poling direction (perpendicular to the thin film surface). These NLO
susceptibilities depend on the microscopic βzzz hyperpolarizability (we assume
CT molecules with CT direction along the molecular z axis), on the number
density of active chromophores N and on the degree of orientation, expressed
by the configurational averages of cos3 and sin2 cos , where is
the angle between the molecular axis (CT direction) and Z axis. These two
components are given by
(2)
χZZZ (−2ω; ω, ω) = NFβzzz (−2ω; ω, ω)cos3 (15)
and
(2)
χXXZ (−2ω; ω, ω) =
1
NFβzzz (−2ω; ω, ω)sin2 cos 2
(16)
respectively.
In Eqs. (15)–(16) F is the local field factor, which for molecules with
symmetry close to the spherical one is given by
2
n + 2 2 n22ω + 2
F = ω
(17)
3
3
where nω(2ω) are refractive indices at ω(2ω) frequencies, respectively.
The charge transfer molecules are characterized not only by the enhanced
component of first hyperpolarizability βzzz but also by a large fundamental state
dipole moment. The first is important for the efficiency of the NLO response
while the second is important for the degree of orientation when poling with a
DC field. In that case the nonzero components of χ (2) susceptibility are given
by (see e.g. Ref. [45])
NFβzzz 3
2
(2)
χZZZ (−2ω; ω, ω) =
(18)
P1 + P3 kT
5
5
and
(2)
χXXZ (−2ω; ω, ω) =
NFβzzz 1
1
P1 − P3 kT
5
5
(19)
where Pl (l = 1, 3) are the polar order parameters (for details see e.g.
Refs. [43, 44, 46]).
The ratio
(2)
a=
χZZZ
(2)
(20)
χXXZ
which varies between 1 and ∞, gives information about the amount of polar
order. The last value is reached for perfectly ordered structures (all dipole
moments pointing in the same direction). The parameter a = 3 for a free
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Second Harmonic Generation Technique and its Applications
115
electron gas (isotropic model). For side chain liquid crystalline polymers a
values as high as (17 ± 3) were obtained [47, 48].
3.2 Intermolecular charge transfer systems
Another interesting approach in fabrication of thin films for second-order
NL optics present the intermolecular charge transfer complexes [49, 50], as
shown in Fig. 13. In contrast to the intramolecular charge transfer molecules,
discussed previously, the charge transfer, and as consequence the macroscopic noncentrosymmetry is realized by the alternating electron donating
and electron accepting molecules in successive, alternate thin layers.
Three types of electron donating molecules were used: 5,10,15,20tetraphenyl-21H ,23H -porphine (TPP); 5,6,11,12-tetraphenylnaphthacene
(rubrene) (TPN) and N, N -bis-(3-methylphenyl)-N, N -diphenyl-N, N -bis
(3-methylphenyl)-1,1 -biphenyl-4,4 -diamine (TPD) were used. As electron
accepting the fullerene C60 was chosen. The molecules were deposited by
sublimation in an ultrahigh vacuum chamber. Two types of structures were
fabricated (cf. Fig. 13) of ABABAB . . . and ABCABC . . . staking, where layer
A was made from electron accepting molecules, layer B from electron donating
and layer C from neutral molecules (MgF2 ). The typical thickness of individual
layers was of about 2 nm. Almost no SHG signal was observed inABABAB . . .
structure. In contrary, a significant SHG activity was observed in ABCABC. . .
type structures, where an effective charge transfer is expected with creation of a noncentrosymmetry in the perpendicular direction to the layers.
FIGURE 13
Two types of intermolecular charge transfer structures tested: ABABAB. . . (a) and ABCABC. . .
(b) with A – electron donating, B – electron accepting and C – neutral layers. Only the first type
ensures a full acentricity in direction perpendicular to the layers.
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I. Rau and F. Kajzar
The largest χ (2) (−2ω; ω, ω) susceptibility was obtained with multilayers
(2)
containing rubrene (χpp = (9.2 ± 1.2) pm/V). The SHG generation experiments performed on thin films with different thicknesses have shown that the
nonlinear optical response is coming from the bulk material and not from the
interfaces. A quadratic dependence of SHG intensity on input power was also
observed [51], showing that the charge transfer takes place at the ground state.
4 APPLICATIONS
Here we will discuss briefly the most pertinent applications developed by using
the second harmonic generation technique. This field is still in movement and
new applications are under development.
4.1 Frequency conversion
One of the important applications of SHG process is the frequency up
conversion. Its efficiency, calculated from Eq. (9) is given by


(2) 2
π l
512π 3  deff 
I2ω
=
|P (θ )A(θ )T (θ )|2 Iω sin2
(21)
η=
2
Iω
ε
2 lc
c
where c is the light velocity, deff is the effective SHG susceptibility for a given
propagation direction, l is the propagation length and lc is the coherence length,
respectively.
There are two important limits:
1. very thin films (l lc ). In that case
η=
32π 5
d 2 |P (θ )A(θ )T (θ )|2 Iω l 2
+ n22ω ) eff
c2 (n2ω
(22)
2. very large coherence length (lc → ∞)
In that case Eq. (20) holds too, but the propagation length can be large and the
conversion efficiency high.
The second case is the case of phase matching, realized when the refractive
indices at ω and 2ω frequencies are the same. For the wave vector it reads :
k = k2ω − 2kω = 0
(23)
It can be realized in birefringent materials only, where the refractive index
depends not only on the wavelength but also on the propagation angle and the
optical field direction. There are two types of birefringent materials:
(a) uniaxial. In that case the refractive index (which is a tensor) described
by an ellipsoid, with so called ordinary index of refraction no (constant
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Second Harmonic Generation Technique and its Applications
117
in the plane perpendicular to the evolution axis of the ellipsoid) and
extraordinary ne . There are two kind of uniaxial materials:
(i) positive with ne > no
(ii) negative with ne < no
(b) biaxial. The index of refraction depends on the propagation direction
in a more complex way, as there are 3 refractive indices describing the
material: nx , ny , nz in the Cartesian reference frame.
Therefore in birefringent materials it is usually possible to find a propagation direction and fundamental beam polarization direction to realize a phase
matching.
In uniaxial crystals one distinguishes two types of phase matching:
1. Phase matching I
In that case the refractive indices describing the SHG process are:
o+o→e
or
e+e →o
The first type of phase matching is obviously possible in negative crystals
while the second one in positive ones, respectively
2. Phase matching II
e+o→e
in negative crystals
e+o→o
in positive ones.
An example of the realization of phase matching I with o+o → e is shown
in Fig. 14 for a negative uniaxial single crystal.
Knowing the dispersion of both ne and no refractive indices as well as
dependence of ne on the propagation direction it is possible to calculate the
phase matching angles (see e.g. Nikogosyan et al. [52] and consequently cut
NLO single crystal to get phase matching at normal incidence. One decreases
in this way the losses by reflection on the input face.
In biaxial crystals situation is more complicated. However knowing the
dispersion and the three refractive indices nx , ny , nz it is possible to calculate
the phase matching angles for a given single crystal [53–55].
In birefringent materials, as refractive indices depend on the propagation direction and on the wavelength, the free and bound waves propagate
in slightly different directions and their overlap depends on the propagation
length. This phenomenon is known as walkoff and the conversion efficiency,
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118
I. Rau and F. Kajzar
3
REFRACTIVE INDEX
no
ne
2
1
0
0.5
2
1.0
1.5
2.0
WAVELENGTH ( m)
FIGURE 14
Schematic representation of phase matching I in a negative uniaxial single crystal.
depends, in fact, on the effective length which is smaller than the propagation
length in material: leff < l.
Presently a number of SHG crystals are commercially available. Usually
they are made from adequately cut single crystals of KDP, KTP, LBO or BBO
(cf. Table 2) to assure phase matching at a given wavelength. Three types of
SHG conversion configurations are realized with:
(i) extra – cavity SHG generation. In that case the NLO crystal is placed
in the beam outside the laser cavity as it is shown schematically in
Fig. 15(a)
(ii) intracavity SHG conversion. Two configurations are present:
1. the NLO crystal is in a cavity adapted for SHG beam (Fig. 15(b))
2. both the NLO and the laser crystal are in the cavity (Fig. 15(c))
The second configuration is more favorable for the conversion efficiency as
the fundamental beam intensity is larger. SHG conversion efficiencies of up to
100% were reported with the following single crystals (cf. Table 2), LiIO3 [56],
LiNbO3 [57] and Ba2 NaNb5 O15 [58, 59].
Another example of phase matching is the modal phase matching, which
is realized in optical waveguides. The principle is similar to the discussed one
here, except that condition is realized between different modes propagating
in the film. An example of such a phase matching in electro-optic polymers is
described in Ref. [60]. Another approach used is the phase matching through
Cerenkov radiation in optical waveguides, in which the fundamental wave
propagates whereas the harmonic wave is decoupled from the waveguide, by
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Second Harmonic Generation Technique and its Applications
119
LASER
F
G
(a)
LASER
M1
G
M2
F
(b)
LASER
(c)
M1
G
M2
F
FIGURE 15
Off laser cavity (a), inside (b) and (c) SHG igurations. In (c) both laser and SHG media are
insiede the cavity. M are mirrors, F are cavity filters and G are SHG crystals, usually mounted on
a goniometer.
an adequate choice of refractive indices of substrate and of waveguide [61].
In that specific case the NLO medium may exhibit a limited absorption at
harmonic wavelength.
4.1.1 Artificial structures
The phase matching condition (cf. Eq. (22)), leading to the maximum energy
conversion at a given fundamental beam intensity, is not always easy to satisfy in real systems. Especially it is true for poled polymers, where always
no < ne . Therefore another, artificial structures were proposed, allowing an
optimized frequency conversion rate. Here we will describe shortly and discuss only the case of quasi phase-matching (QPM). According to Fig. 4 for
the propagation length l equal to the coherence length lc one obtains the maximum conversion to harmonic intensity. For lc < l < 2lc , depending on the
refractive index dispersion (normal or anomalous), the free and bound wave
cancel and the harmonic intensity is decreasing. It is possible to prevent this
effect in two ways:
(i) by introducing a linear, dispersive, medium which will allow to put
both waves at the same phase (as at the entry of nonlinear medium).
In that case, as shown in Fig. 16(a) one obtains an efficient frequency
conversion on the whole nonlinear medium thickness
(ii) by reversing, alternatively, dipolar moments in layers with the thickness
equal to the coherence length of the material (cf. Fig. 16(b)). In that
case a larger second harmonic conversion efficiency is obtained for the
same medium thickness as in case (i) (cf. Fig. 17)
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120
I. Rau and F. Kajzar
(a)
(b)
FIGURE 16
Structures used for SHG in thin layers: (a) inversed domains, (b) periodically poled polymers.
SHG intensity
SHG intensity
2
4
(a)
l/lc
1
2
l/lc
(b)
FIGURE 17
Variation of SHG in thin layers: of (a) inversed domains, (b) periodically poled polymers.
Important problem occurring with this kind of structures is the light scattering
due to the refractive index variations. Phase matched SHG was realized in
periodically poled polymer films (Fig. 16(b)) was demonstrated by several
research groups (see e.g. Khanarian et al. [62], Norwood and Khanarian [63]).
4.2 Characterization of second order NLO properties of materials
SHG is a relatively simple and viable technique used very frequently to
characterize the second-order NLO properties of bulk single crystals and of
non-centrosymmetric thin films. A powder SHG method [64] allows also
the synthetic chemists to screen rapidly the newly synthesized compounds
and compare their SHG efficiency with that of the already known material. It measures the fast, electronic part of second-order NLO susceptibility
χ (2) (−2ω; ω, ω). Using accordable laser sources it is also possible to determine relatively easy the dispersion of this susceptibility. An example is shown
in Fig. 18 for a side chain liquid crystalline polymer functionalized with
cyanobiphenyle. The dispersion is fitted with a two level model.
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Second Harmonic Generation Technique and its Applications
121
FIGURE 18
Dispersion of second-order NLO susceptibility for a side chain liquid crystalline polymer functionalized with cyanobiphenyle. Solid line shows a fitted with a two level model, as described
in text.
4.2.1 Poling and relaxation kinetic studies
Although the poled polymers were developed mainly for electro-optic modulation applications, the SHG technique appears to be very convenient to
characterize them, and particularly to study the kinetics of the growth of polar
order and its relaxation, two important phenomena in processing these materials for their practical application. Here of particular importance are in situ
SHG measurements as a useful tool for the study of the poling and relaxation
of polar order in electro-optic polymers [65–70]. A schematic presentation of
in situ experimental set up is shown in Fig. 19. The poling system is located
in the laser beam path. The bottom electrode (usually made from copper) has
a hole, with diameter of about 1 cm. The size of the hole is important to pass
both fundamental and harmonic beams The poling field as well as the poling
temperature are controlled and the kinetics of both the temporal and the temperature dependence of poling efficiency can be monitored by the observed
variation of SHG intensity.
It appears that at normal incidence the SHG intensity is equal to zero
because of the isotropic distribution of dipole moments and their projections
within the thin film surface (cf. Fig. 20). Thus it is necessary to orient the
studied film in such a way that the normal to its surface make a sufficiently
large angle with the polarization of the incident optical beam in order to get
a comfortable SHG signal (usually 45 degrees is fine). For comparison a
SHG dependence on incident angle is shown in Fig. 21 for an LB monolayer,
deposited on both sides of a glass substrate [71]. The interference fringes
originate from the interference of harmonic fields generated in both layers.
The phase mismatch is due to the refractive index dispersion in substrate. The
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I. Rau and F. Kajzar
FIGURE 19
Schematic representation of the experimental set-up for in situ SHG measurements of poling
kinetics and relaxation in electro-optic polymer:. F – filters, P – polarizer, L – lense, NE – needle
electrode, HV – high DC voltage.
5000
SHG INTENSITY (arb. units)
S-P CONFIG FIT
S-P CONFIG EXPER
P-P CONFIG FIT
P-P CONFIG EXPER
3000
1000
-1000
0
0.5
1.0
1.5
INCIDENCE ANGLE (RADIANS)
FIGURE 20
Incidence angle dependence of SHG intensity for a poled functionalized PVK thin film for s-p and
p-p fundamental-harmonic beam polarization configurations. Squares and closed circles show
fitted values using Eqs. (24)–(26) whereas open circles and triangles the measured ones (after
Bermudez et al. [73]).
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Second Harmonic Generation Technique and its Applications
123
FIGURE 21
Interference fringe pattern of SHG from a both side single coated glass substrate with an LB
monolayer at 853 nm fundamental wavelength. Points show measured values whereas solid line
the fitted values as described in text (after Kajzar and Ledoux [71]).
phase difference between the fundamental and the harmonic wave is varied
when rotating the film with substrate and is due to the refractive index dispersion in substrate. It is seen that it is very well described by theory; otherwise
the variation would be similar to that shown in Fig. 20.
For a single layer deposited on the glass substrate the SHG intensity as
function of the incidence angle θ is given by the following equation
ffh
I2ω (θ ) =
(2)
32π 3 χffh (−2ω; ω, ω) 2
2 iϕ
− 1|2 Iω2
|Tfh (θ )Afh (θ )| |e
ε
c2 (24)
where the indices f and h refer to the polarizations of fundamental and harmonic waves, respectively. Tfh (θ ) and Afh (θ ) are, respectively, incidence
angle dependent factors arising from transmission and boundary conditions
(cf. Eq. (9)), Iω is the incident light intensity and ϕ is the phase mismatch
between the fundamental and harmonic beam in the studied film (Eq. (10)).
(2)
The effective χIJK susceptibilities for a given fundamental-harmonic beam
polarization configurations contain the projection factors and are given by [72]
(2)
(2)
s
χssp
(−2ω; ω, ω) = χsp
sin 2θωs cos θ2ω
(25)
for s-p fundamental harmonic beam polarization and
p
(2)
(2)
χppp
(−2ω; ω, ω) = χpp
sin2 θωp sin θ2ω
p
(2)
+ χsp
cos θωp (cos θωp + 2 sin θωp cos θ2ω )
(26)
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I. Rau and F. Kajzar
SUBSTRATE
POLARIZER
2
ANALYZER
FILTER
THIN FILM
Rotation axis
FIGURE 22
Schematic representation of experimental arrangement for the study of the symmetry of induced
χ (2) susceptibilty.
Equations (25) and (26) show that determination of both tensor components requires two independent measurements in s-p and p-p fundamentalharmonic beams polarization configurations. First measurements gives the off
(2)
diagonal value χsp . The second (p-p configuration) measurements will yield
(2)
the diagonal χpp susceptibility, injecting the previously determined value of
(2)
χsp into Eq. (26).
Figure 22 shows the experimental configuration to measure the two nonzero
χ (2) susceptibility components for poled polymers. Both polarizations of
incident fundamental and harmonic output are controlled by the polarizer
(fundamental beam) and the analyzer (harmonic beam). In order to decrease
the effect of light scattering on thin film surface this one is located on the side
of detection system. An example of a fit of Eqs. (24)–(26) to experimental
data is shown in Fig. 20 for a poled functionalized PVK thin film for both sp and p-p fundamental-harmonic beam polarization configurations. Squares
and closed circles show fitted values using equations whereas open circles
and triangles the measured ones (after Bermudez et al. [73]). An excellent
agreement is observed between the fitted and the measured values.
Figure 23 compares kinetics of poling in a side chain liquid crystalline
polymer (polyacrylate, TgN = 51.5◦ C), (a) and in an isotropic polymer (polymethacrylate, Tg = 97◦ C, virtual isotropic/nematic phase between 70 and
90◦ C), both functionalized with cyanobiphenyle [74] and measured by in situ
SHG technique. It shows that the kinetics is not the same in both compounds.
When increasing the temperature chromophores orient smoothly in isotropic
polymer while a percolation type behavior is observed in SCLCP (see also
Fig. 24). It is due to the specific domain structure of liquid crystal and their
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Second Harmonic Generation Technique and its Applications
125
FIGURE 23
Kinetics of electric field poling in a side chain liquid crystalline polymer (polyacrylate, TgN =
51.5◦ C, (a)) and in an isotropic polymer (polymethacrylate, Tg = 97◦ C, virtual isotropic/nematic
phase between 70 and 90◦ C), both functionalized with cyanobiphenyle (after Kajzar and
Noel [69]).
collective motion when approaching the nematic/isotropic phase transition
temperature.
4.2.2 Relaxation
One of the important parameters determining practical applicability of poled
polymers is the stability of the induced polar orientation. This is studied usually
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126
I. Rau and F. Kajzar
(a)
(b)
FIGURE 24
Temperature variation of the order parameter P2 for an isotropic polymer (a) and for SCPLC
(b) (after Kajzar and Noel [69]).
through the temporal behavior of the χ (2) susceptibility or of the electro-optic
coefficient r at elevated temperatures. Relaxation studies were almost done
for static field poled polymers. The temporal decay with time t of the χ (2)
susceptibility is usually described by the Kohlrausch-Williams-Watt (KWW)
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Second Harmonic Generation Technique and its Applications
127
stretched exponential function:
χ (2) (t) = χ (2) (t = 0)e−( τ )
t β
(27)
where τ is the relaxation time constant, depending on temperature and
β(0 < β < 1) describes the width of the relaxation (departure from a
monoexponential behavior) [75]. Although the stretched exponential function describes well the relaxation process, there is no physical meaning behind.
Therefore it is preferable to describe it by using a sum of exponential functions,
which takes account of different relaxation processes in polymeric materials.
Usually the relaxation of polar order in poled polymers (isotropic or liquid
crystalline) may be well described by a biexponential function:
1
l
p
I2ω (t)
r (t)
I2ω
1
2
= R1 e−t/τ1 + R2 e−t/τ2 + C
(28)
p(r)
where l is the thin film thickness, I2ω is the second harmonic intensity of
the studied thin film (p) and of reference r, respectively. The constant C in
Eq. (28) characterizes the residual orientation at the experimental time scale
and is important for practical applications. R’s in Eq. (28) are the relaxation
rates and τ1 and τ2 are relaxation times of different processes contributing
to the molecular disorientation, respectively. All these parameters depend
on the measurement temperature. Closer is the glass transition temperature
to the measurement temperature, smaller are time constants τ and larger are
relaxation rates. This behavior is true for both isotropic and liquid crystalline
polymers, as it was observed by Dantas de Morais et al. [65]. Figure 25 shows
FIGURE 25
Measured by in situ SHG technique temporal variation of SHG intensity, at elevated temperature,
fitted by a monoexponential (dashed line) and bi-exponential (solid line) function (after Ref. [65]).
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I. Rau and F. Kajzar
an example of such mono- and biexponential fits of χ (2) relaxation curve.
A better description of the observed polar order relaxation is obtained with the
biexponential function.
4.3 Nonlinear spectroscopy
SHG is a three photon process. As it is seen from Fig. 1 this process is going
through virtual states. If one of the virtual states is close to the excited state of
unperturbed system, a resonance enhancement in the quadratic susceptibility
will occur. Moreover in that case the χ (2) susceptibility is complex.
The wavelength dispersion of χ (2) susceptibility may be well described by
a two level model, introduced primarily by Oudar and Chemla [76]. For the
resonant case this susceptibility is given by the following expression [77]
χ (2) (−2ω; ω, ω) =
1
NF|µ01 |2 G(ω, ω0 , )
22
(29)
where µ01 is the dipolar transition moment between fundamental and excited
states and µ = µ11 − µ00 is the difference of dipole moments at excited
and fundamental states, respectively. The function G(ω, ω0 , ) describes the
dispersion of χ (2) susceptibility
G(ω, ω0 , ) =
1
1
+ ∗
( − 2ω)( − ω) ( + 2ω)( − ω)
1
+ ∗
( + ω)( − ω)
(30)
where = ω0 − i and ∗ = ω0 + i. Here is the damping term and ω0
is the transition energy (in units) between states 0 and 1 (cf. Fig. 1). Figure 26 shows the measured and the fitted incident photon energy dependence
(2)
χpp quadratic susceptibility measured in a Langmuir Blodgett monolayer of
an azo dye (4-[4-(N-m-dodecyl-N-methylamino)phenylazo]-3-nitrobenzoic)
(2)
acid (DPNA) . The observed increase of χpp when approaching the one photon transition energy is due to two photon resonant enhancement. The splitting
in the resonance curve was interpreted in terms of resonance with vibronic
levels.
4.4 Surface study
Shen and coworkers [2,3] observed large SHG signal from the silver surface. It
originates from the breaking of symmetry at the metal surface. The importance
of SHG signal depends obviously on the surface itself and the largest effect is
observed on already mentioned Ag. The effect is basically due to the fact that
there is a number of unbalanced charges on the surface creating a large electric
field. The phenomenon is largely used to study different metal surfaces and the
orientation of molecules deposited on them. The break of centrosymmetry at
the interface leads to the second order nolinear polarization at the interface, or
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Second Harmonic Generation Technique and its Applications
129
FIGURE 26
(2)
Dispersion of χpp susceptibility in a Langmuir Blodgett monolayer of an azo dye
(4-[4-(N-m-dodecyl-N-methylamino)phenylazo]-3-nitrobenzoic) acid (DPNA) (after Kajzar and
Ledoux [71]).
in other words directly on the metallic surface. It allows to observe SHG even
from one monolayer [78]. It was successfully used to determine the orientation
of molecular absorbates on a fused silica substrate [79], check the structural
symmetry of the crystal and of the surface as it was done in case of Si(111) [80]
or study adsorption of molecules on metal surfaces [81].
In a recent study Arfaoui et al. [82] reported observation of a large SHG
signal from silver surface with deposited very thin layers of rotaxanes, even
from a single layer. In order to explain the observed large SHG intensity,
and larger on silver than on gold surface Arfaoui et al. performed theoretical
calculations of the electric field experienced by the molecules on both surfaces.
It is well known that the silver surface breaks isotropy in presumably isotropic
thin films deposited on it because of the large electric field created by silver
atoms on the surface. The calculations were carried out for a single molecule
of fumaramide [2]rotaxane deposited on Au(111) and Ag(111). The substrates
were chosen for the sake of comparison. The rotaxane structure was minimized
using TINKER molecular mechanics/dynamics software package [83–85].
The Embedded Atom Model [86] was used in describing the metal-metal
interactions, MM3 force field [87] for the organic-organic part and a modified
Morse potential in the description of metal-organic interactions. In order to get
the charges used in the metal-organic Coulomb interactions [88] the charge
equilibration scheme of Rappe and Goddard was applied throughout the whole
system. A metal(111) surface model consisting of five layers of 20 by 20
atoms was used and the lowest layer of metal atoms was kept fixed during the
calculations. The top four layers of the surface model were allowed to relax
or reconstruct to achieve the lowest energy.
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I. Rau and F. Kajzar
The charge distribution generated by the adsorption of the rotaxane
molecule on the metal(111) surface was investigated with Delphi4 program [89,90] to compute the electrostatic potential and the electric field on top
of the surface. The interaction of the deposited molecules with metal surface
was taken into account. It is well known that the rotaxane molecules orient on
the surface in a way that one benzene ring and one carbonyl group of the thread
as well as one benzene ring and two carbonyl groups of the macrocycle are
close to the surface, changing the metal charge distribution, as it was observed
from birefringence studies of vacuum deposited thin films [91]. The electric
field was calculated on top of the metal surface on ∼17000 evenly distributed
points which were selected so, that they would be inside the space occupied
by the rotaxane van der Waals volume. The calculated average field strength
inside the rotaxane volume of space is roughly 15 MV/cm and of a very short
range, disappearing ∼9 Å away from the surface.
The calculations was also done for the rotaxane/Au(111) system and the
obtained field strength was lower, about 7 MV/cm. The metal atoms in the top
layer of the Au(111) surface show a larger variety of metal partial charges, but
the field is weakened by the presence of negatively charged atoms, whereas
in the case of Ag(111) the atoms are, on average, more positively charged.
Figure 27 displays the fundamental beam polarization dependence of SHG
intensity from a monolayer and a multilayer of fumaramide [2]rotaxane on
Ag(111) surface. The harmonic beam polarization is filtered to be of p type
and fixed as such during the measurements which are done as function of the
angle between both polarizations. The film is tilted with respect to the normal
SHG intensity (Arb. units)
200
150
100
50
0
90
180
270
360
Incident light polarization (degrees)
FIGURE 27
SHG intensity from a monolayer (----) and a multilayer (--•--) of fumaramide [2]rotaxane on
Ag(111) as a function of the angle (in degrees) between the fundamental beam and the harmonic
(fixed to be p-type) polarization directions (after Arfaoui et al. [82]).
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Second Harmonic Generation Technique and its Applications
131
to the beam propagation direction. First of all it is seen that the SHG intensity
for a monolayer is slightly larger than for the multilayer. It confirms that the
electric field ofAg acts on a very short distance as gives the theory. The incident
beam polarization is rotated. First of all it is seen that the SHG intensity goes
through maxima and minima, the calculated value of electric field, its depth
and the value of cubic susceptibility χ (3) (−3ω; ω, ω, ω), measured by optical
third harmonic generation technique for the rotaxane. Arfaoui et al. [82])
observed a very good agreement between determined in this way effective
(2)
χEFISH defined as
(2)
χEFISH (−2ω; ω, ω) = NFγ (−2ω; ω, ω, 0)E
(31)
where E is the calculated electric field on Ag surface and γ (−2ω; ω, ω, 0) the
molecular second hyperpolarizability γ (−2ω; ω, ω, 0), assumed to be equal
to the THG second hyperpolarizability γ (−3ω; ω, ω, ω), and the measured
directly χ (2) (−2ω; ω, ω) susceptibility. Also the SHG measurements have
shown a better orientation of rotaxane molecules in monolayers than in
multilayers (for details see Arfaoui et al. [82]).
4.5 Checking the centro-noncentrosymmetry in materials
As already mentioned, the χ (2) susceptibility vanishes in centrosymmetric
media. This shows that the SHG technique may be a useful tool to check
the presence or the lack of center symmetry in single crystals. Examples are
the recent studies [92, 93], on reversible solid-state transitions in crystals of
organic disulfide-based iodoplumbate. Example of such a study, made on a
powder sample of H3 N(CH2 )2 S-S(CH2 )2 NH3 )PbI5 ·H2 O is shown in Fig. 28.
It is seen that when increasing the sample temperature the SHG intensity at
ca. 78◦ C drops abruptly to zero. The SHG signal is recovered when cooling
the sample, at ca. 58◦ C. It means that there is a reversible transition between
acentric and centrosymmetric phases of this compound, with a hysteresis of
FIGURE 28
Reversible transition from noncentrosymmetric to centrosymmetric crystal structure in
H3 N(CH2 )2 S-S(CH2 )2 NH3 )PbI5 · H2 O (after Ref. [93]).
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about 20◦ C (for details see Ref. [91]). Similar behavior was observed for other
compounds from the disulfide-based iodoplumbate family [91,92].
4.6 Second harmonic generation microscope
The second harmonic generation microscope [94–97] (SHGM) is a fine tool
allowing to observe very small polar domains or molecules using the SHG
phenomenon [98]. It is very similar to the two photon absorption (TPA) fluorescence microscope (FM) [99], with the difference that in TPA fluorescence
microscope the observation beam is the fluorescence beam with frequency
smaller than the double of that of the fundamental beam. Also the fluorescence
beam is broad. In SHGM the observation beam is exactly the double of the fundamental beam frequency and is significantly narrower than the fluorescence
beam, narrower even than the fundamental beam. Both techniques take advantage of the two photon process, creating the observation beam, thus giving the
possibility of 3D imaging. Also they use the fact that material is transparent
for fundamental beam, thus it is possible to visualize inside the studied body.
In SHGM one utilizes the interference effect between the second SH wave
from the studied object and that from an uniform SHG source (e.g. a plate),
as shown in Fig. 29. The highest contrast can be obtained by varying effective
optical path-lengths of the SHG plate and the phase plate. Similarly as the TPA
fluorescence microscope the studied material is moved in the X, Y, Z directions.
The SHGM was successfully used to observe ferroelectric domains and
periodically poled thin layers of the quasi phase matched single crystal structures. It found a particular interest and application in imaging biological
species and supramolecules [100–104], such as proteins in biological tissues [105], neurones [106], orientation of collagen fibers [107–112], imaging
P
F
F
SHG
plate
sample
stage
IR
P
Mirror
lens
FIGURE 29
Schematic presentation of a SHG microscope: P – polarizer, F – filter (after Ref. [98]).
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Second Harmonic Generation Technique and its Applications
133
of other biopolymers [113]. Using an excitation beam with controlled polarization and controlling the orientation of optical electric field polarization with
a spatial light modulator. Yoshiki et al. [114] have shown the ability of SHGM
to determine the 3D orientation of collagen fibers in an Achilles tendon. The
main advantage of both techniques: TPA fluorescence microscope and SHGM
is that they allow a deep material observation of the structure, as in both cases
the fundamental (excitation) beam is not absorbed [115].
5 CONCLUSIONS
In this overview we gave a brief description of the principles of second harmonic generation in nonlinear media and discussed its practical applications in
different fields: fundamental research, material research and characterisation,
in laser industry (frequency conversion which leads to several important applications in microelectronic industry, biology, medicine, data storage, etc.). We
described also and discussed material development on microscopic (molecular
engineering) and macroscopic (material engineering) levels. The noncentrosymmetric materials developed for SHG applications, except some specific
cases like artificial quasi phase matching structures, find other important applications such as: electro-optic modulators for optical signal transmission and
for sensing [3,4], generation of ultrashort electrical pulses through the optical
rectification technique [116, 117], otherwise unachievable through electrical
circuitry. Another demonstrated example is an integrated optical parametric
amplifier [118]. An electro-optic polymer was also used to make an all optical
switching device, although its operation doesn’t base on exploring its secondorder NLO properties [119]. SHG single crystals are also used presently in
commercial optical parametric oscillators, such as, particularly BBO and KTP
(cf. Table 2). However these applications are not in the scope of the present
review.
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