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Nonlinear Optics and Quantum Optics, Vol. 38, pp. 99–140 Reprints available directly from the publisher Photocopying permitted by license only ©2008 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 99 — #1 100 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) “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 100 — #2 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 101 — #3 102 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 102 — #4 103 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 103 — #5 104 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) “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 104 — #6 105 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 105 — #7 106 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, “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 106 — #8 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. “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 107 — #9 108 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 108 — #10 109 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 109 — #11 110 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 110 — #12 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 111 — #13 112 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. “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 112 — #14 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 113 — #15 114 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 114 — #16 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. “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 115 — #17 116 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 116 — #18 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, “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 117 — #19 118 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 118 — #20 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) “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 119 — #21 120 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. “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 120 — #22 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 121 — #23 122 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]). “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 122 — #24 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) “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 123 — #25 124 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 124 — #26 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 125 — #27 126 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) “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 126 — #28 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]). “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 127 — #29 128 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 “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 128 — #30 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. “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 129 — #31 130 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]). “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 130 — #32 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]). “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 131 — #33 132 I. Rau and F. Kajzar 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]). “NLOQO” — “nlo-tok-2” — 2008/8/22 — 17:07 — page 132 — #34 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. 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