Unique determination of the hyperpolarizability tensor ratio through a depolarized hyper-Rayleigh scattering under an external electric field

Unique determination of the hyperpolarizability

tensor ratio through depolarized

hyper-Rayleigh scattering under an external electric field

Jisoo Hwang, Byoungchoo Park, H. J. Chang, and J. W. Wu

Department of Physics, Ewha Womans University, Seoul 120-750, Korea Received July 30, 2001; revised manuscript received November 26, 2001

We have analyzed the depolarized hyper-Rayleigh scattering from a molecular system with partial macroscopic polar ordering. It is shown that the hyperpolarizability tensor ratio and the molecular dipole moment can be determined simultaneously by the measurement of depolarization as a function of external field strength. As an experimental example, we performed a quantitative analysis of the electric-field-dependent depolarized hyper-Rayleigh signal from a poly-␥-benzyl-L-glutamate solution, obtaining the dipole moment and the ratio of hyperpolarizability components as 4.0 D and ␤₃₁₁/␤_{333⫽ ⫺0.81, respectively. © 2002 Optical Society of} America

OCIS codes: 190.4160, 190.4710, 190.2620, 290.5870.

1. INTRODUCTION

In organic molecular systems, a macroscopic nonlinear optical response originates from molecular nonlinearities, and hyper-Rayleigh scattering (HRS) has been developed into a routine for measurement of microscopic molecular hyperpolarizabilities ␤.1–3 Incoherent double-quantum light scattering from molecules whose orientation fluctu-ates gives rise to a HRS signal.4,5 In determining␤ ex-perimentally, HRS is in strong contrast to electric-field-induced second-harmonic generation6–8(EFISHG) in that for HRS no external electric field is applied to the sample cell to break the macroscopic centrosymmetry. Hence it is possible to measure values of ␤ of nonpolar and ionic species9–11as well as of traditional polar nonlinear optical molecules. Another advantage of HRS compared with EFISHG is the absence of contributions from the second hyperpolarizability ␥ to 2␻ harmonic signals. In EFISHG, both the first and the second hyperpolarizabili-ties, ␤(⫺2␻; ␻, ␻) and ␥ (⫺2␻; ␻, ␻, 0), contribute to the second-harmonic signal at 2␻ as a sum of ␥ ⫹ ␤␮/5kT, and the precise determination of ␤ requires a temperature-dependent measurement of harmonics sig-nals to discern each contribution as well as an indepen-dent measurement of the molecular dipole moment.12 Note that, in comparing the reported experimental values of ␤, in EFISHG the projection of tensor components along the direction of the dipole moment具␤典 is experimen-tally measured, whereas in HRS the harmonic signal is proportional to the orientational average of the squares of a combination of all the tensor components present,具␤2_典. An important feature of HRS in this respect is that a strongly depolarized second harmonic occurs, owing to the incoherent nature of double-quantum scattering and the third-rank tensorial property of ␤; from the

depolariza-tion measurement each tensor component can be deter-mined individually.13

In double-quantum scattering from molecules in an iso-tropic distribution there are five independent observables among six quadratics of␤IJK(where I, J, and K are labo-ratory axes), namely, 具␤ZZZ2 典, 具␤XZZ2 典, 具␤XZZ␤XYY典,

具␤ZZZ␤ZYY典, 具(␤ZYZ⫹ ␤ZZY)2典, and具(␤XYZ⫹␤XZY)2典; in the experiment more than one depolarization ratio mea-surement is necessary for unique determination of each tensor component.4 For example, the intensities of 90° scattered harmonics with two orthogonal polarizations are measured for incident fundamentals with linear and circular polarization,3 or the incident polarization direc-tion is continually varied and the total scattered harmon-ics are measured.2 But the number of independent depo-larization ratios is determined by the molecular point-group symmetry, and sometimes it happens that the number of independent tensor components exceeds the maximum number of independent depolarization mea-surements, which is five. This is easily the case in the energy regime in which Kleinman symmetry is not valid. In other words, the depolarized HRS has limited value as a measurement technique in the determination of all the independent hyperpolarizability tensor components.

In this paper we introduce depolarized hyper-Rayleigh scattering under an external electric field (e-HRS) to in-crease the number of independent depolarization mea-surements, which enables us to uniquely determine the ratio of hyperpolarizability tensor components. e-HRS amounts to removing the degeneracy in具␤2_{典by imparting} a partial polar ordering to the molecules whose orienta-tion is fluctuating through an external dc electric field. It is shown that this novel e-HRS process permits deter-mination of the sign and the magnitude of the

larizability tensor ratios as well as the molecular dipole moment, through measurement of the depolarization ra-tio as a funcra-tion of external electric-field strength. As an experimental example, the ratio of two dominant hyper-polarizability tensor components and the permanent di-pole moment of poly-␥-benzyl-L-glutamate (PBLG), one of the synthetic polypeptides,8 are determined through e-HRS, subsequently yielding the sign and the magnitude of each tensor component of␤.

2. THEORETICAL ANALYSIS

Let us look at how the intensities of harmonic scattered light will change on application of an external static elec-tric field. In a coarse-graining picture the HRS signal as

a second-order nonlinear optical process comes from local instantaneous noncentrosymmetric orientation of mol-ecules in a centrosymmetric macroscopic distribution. The local instantaneous deviation from centrosymmetry, in fact, results from a thermal fluctuation of the molecu-lar orientations. When an external static field is applied to the orientationally fluctuating molecules, a certain amount of polar ordering will take place; the order pa-rameter is determined from minimization of the Helm-holtz free energy of the system at a temperature T. Even in the presence of the aligning static electric field, sto-chastic thermal fluctuations still exist, giving rise to a lo-cal noncentrosymmetry to generate harmonic double-quantum scatterings, i.e., the e-HRS signal. At a first glance, e-HRS seems merely another extension of coher-Table 1. Six Quadratics of␤IJK (I, J, and K; Laboratory Axes) Expressed in Terms of Coefficients

of Quadratic Functions of␤ in Molecular Axes, p, q, r, All Differenta

␣⫽

兺

p ␤ppp2 ␤1⫽

兺

p,q ␤ppp␤pqq ␤2⫽

兺

p,q ␤ppp共␤qpq⫹␤qqp兲 ␤3⫽

兺

p,q ␤pqq2 A1 具_⌽I p 6_典 f(␪) 2具⌽I4_p⌽I2_q典f(␪) 2具⌽4I_p⌽I2_q典f(␪) 具⌽I4_p⌽I2_q典f(␪) A2 具⌽_I p 2_⌽ Jp 4_典 f(␪) 2具⌽Ip 2_⌽ Jp 2_⌽ Jq 2_典 f(␪) 2具⌽I_p⌽I_q⌽Jp 3_⌽ J_q典f(␪) 具⌽Ip 2_⌽ Jq 4_典 f(␪) A3 _具⌽I p 2_⌽ J_p 2_⌽ K_p 2 _典 f(␪) 2具⌽I2_p⌽J2_p⌽2K_q典f(␪) 2具⌽I2_p⌽J_p⌽J_q⌽K_p⌽K_q典f(␪) 具⌽2I_p⌽J2_q⌽J2_r典f(␪) A4 8具_⌽I p 2_⌽ J_p 4_典 f(␪) 具⌽I2_p⌽J2_p⌽2J_q典f(␪) 具⌽Ip⌽Iq⌽Jp 3_⌽ Jq典f(␪) 具⌽Ip⌽Iq⌽Jp 3_⌽ Jq典f(␪) ⫹具⌽I2_p⌽J4_q典f(␪) ⫹具⌽I2_p⌽J2_p⌽J2_q典f(␪) A5 4具⌽_I p 2_⌽ Jp 4_典 f(␪) 8具⌽I_p⌽I_q⌽Jp 3 _⌽ J_q典f(␪) 4具⌽I_p⌽I_q⌽Jp 3_⌽ J_q典f(␪) 4具⌽Ip 2_⌽ Jp 2_⌽ Jq 2 _典 f(␪) ⫹4具⌽I2_p⌽J2_p⌽J2_q典f(␪) A6 4具_⌽I p 2_⌽ J_p 2_⌽ K_p 2 _典 f(␪) 8具⌽I2_p⌽Jp⌽Jq⌽Kp⌽Kq典f(␪) 8具⌽Ip 2_⌽ Jp⌽Jq⌽Kp⌽Kq典f(␪) 4具⌽Ip 2_⌽ J_p 2_⌽ K_q 2 _典 f(␪) ␤4⫽

兺

p,q ␤pqq共␤qpq⫹␤qqp兲 ␤5⫽

兺

p,q 共␤qpq⫹␤qqp兲2 ␥1⫽

兺

p,q,r ␤pqq␤prr ␥2⫽

兺

p,q,r 共␤qpq⫹␤qqp兲共␤rpr⫹␤rrp兲 A1 2具⌽_I p 4_⌽ Iq 2_典 f(␪) 具⌽Ip 4_⌽ Iq 2_典 f(␪) 具⌽Ip 2_⌽ Iq 2_⌽ Ir 2_典 f(␪) 具⌽Ip 2_⌽ Iq 2_⌽ Ir 2_典 f(␪) A2 2具⌽I_p⌽I_q⌽J3_p⌽J_q典f(␪) 具⌽I2_p⌽J_p⌽J_q⌽K_p⌽K_q典f(␪) 具⌽I2_p⌽2J_q⌽J2_r典f(␪) 具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) A3 2具_⌽I p 2_⌽ Jp⌽Jq⌽Kp⌽Kq典f(␪) 具⌽Ip 2_⌽ Jp⌽Jq⌽Kp⌽Kq典f(␪) 具⌽Ip 2_⌽ J_q 2_⌽ K_r 2_典 f(␪) 具⌽I_p⌽I_q⌽J_p⌽J_r⌽K_q⌽K_r典f(␪) A4 具_⌽I_p⌽I_q⌽J3_p⌽J_q典f(␪) 具⌽I_p⌽I_q⌽J3_p⌽J_q典f(␪) 具⌽I2_p⌽J2_q⌽J2_r典f(␪) 具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) A5 4具⌽I_p⌽I_q⌽Jp 3 _⌽ J_q典f(␪) 2具⌽I_p⌽I_q⌽Jp 3 _⌽ J_q典f(␪) 4具⌽I_p⌽I_q⌽J_p⌽J_q⌽Jr 2_典 f(␪) 具⌽Ip 2_⌽ Jq 2_⌽ Jr 2_典 f(␪) ⫹4具⌽I2_p⌽J2_p⌽2J_q典f(␪) ⫹具⌽I2_p⌽4J_q典f(␪) ⫹3具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) ⫹具⌽I2_p⌽J2_p⌽J2_q典f(␪) A6 8具⌽_I p 2_⌽ J_p⌽J_q⌽K_p⌽K_q典f(␪) 2具⌽I_p 2_⌽ J_p 2_⌽ K_q 2 _典 f(␪) 4具⌽I_p 2_⌽ J_q⌽J_r⌽K_q⌽K_r典f(␪) 2具⌽I_p 2_⌽ J_q⌽J_r⌽K_q⌽K_r典f(␪) ⫹2具⌽I2_p⌽J_p⌽J_q⌽K_p⌽K_q典f(␪) ⫹2具⌽I_p⌽I_q⌽J_p⌽J_r⌽K_q⌽K_r典f(␪) ␥3⫽

兺

p,q,r ␤pqq共␤rpr⫹␤rrp兲 ␦1⫽

兺

p,q,r 共␤pqr⫹␤prq兲2 ␦2⫽

兺

p,q,r 共␤qpq⫹␤prp兲共␤qpr⫹␤qrp兲 A1 2具_⌽I p 2_⌽ I_q 2_⌽ I_r 2_典 f(␪) 12具⌽I2_p⌽I2_q⌽I2_r典f(␪) 具⌽I2_p⌽I2_q⌽I2_r典f(␪) A2 2具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) 12具⌽2I_p⌽J2_q⌽J2_r典f(␪) 具⌽Ip⌽Iq⌽Jp⌽Jq⌽Jr 2_典 f(␪) A3 2具⌽_I p 2_⌽ J_q⌽J_r⌽K_q⌽K_r典f(␪) 12具⌽Ip 2_⌽ J_q⌽J_r⌽K_q⌽K_r典f(␪) 具⌽Ip 2_⌽ J_q⌽J_r⌽K_q⌽K_r典f(␪) A4 具_⌽I p 2_⌽ J_q 2_⌽ J_r 2_典 f(␪) 12具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) 具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) ⫹具⌽Ip⌽Iq⌽Jp⌽Jq⌽Jr 2_典 f(␪) A5 8具⌽I_p⌽I_q⌽J_p⌽J_q⌽J_r 2_典 f(␪) 具⌽I_p 2_⌽ J_q 2_⌽ J_r 2_典 f(␪) 具⌽I_p 2_⌽ J_q 2_⌽ J_r 2_典 f(␪) ⫹具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) ⫹3具⌽I_p⌽I_q⌽J_p⌽J_q⌽J2_r典f(␪) A6 8具_⌽I_p⌽I_q⌽J_p⌽J_r⌽K_q⌽K_r典f(␪) 具⌽I2_p⌽2J_q⌽K2_r典f(␪) 2具⌽I2_p⌽J_q⌽J_r⌽K_q⌽K_r典f(␪) ⫹2具⌽I_p⌽I_q⌽J_p⌽J_r⌽K_q⌽K_r典f(␪) a_A1⫽具␤ ZZZ 2 _典

ent second-harmonic generation (SHG), similar to EFISHG. But e-HRS is fundamentally different from EFISHG in that it is still an incoherent process measured at the 90° scattering angle, with no necessity for phase matching between free and bound 2␻ waves. The e-HRS intensity, then, will be related to the rotational invariants of the orientational average of products of the directional cosines for the given macroscopic polar ordering intro-duced into the molecular system. This implies that on application of a static electric field there will be a change in the intensities of double-quantum scattered light with polarization parallel and perpendicular to the incident po-larization, IZ2␻and IX2␻, respectively, with a constant sum

IZ2␻⫹ IX2␻ that is independent of the electric-field strength. For a molecule with the single nonvanishing component␤333it is easy to see that a partial alignment of polar molecules along the Z direction will at once in-crease IZ2␻and decrease IX2␻, resulting in an overall reduc-tion of the depolarizareduc-tion ratio I_X2␻/I_Z2␻to a value less than the isotropic value of 1/5. However, when off-diagonal

tensor components in addition to ␤333are present in the molecule, we expect that the hyperpolarizability tensor components that are parallel and perpendicular to the permanent dipole moment will contribute in a different way to the e-HRS signal. The resultant depolarization ratio, in general, will be a complicated function of the electric-field strength, containing information on the structure of hyperpolarizability tensor components.

The orientational correlation of the molecular dipoles that are responsible for the HRS signal is described through the product of molecular hyperpolarizabilities in laboratory coordinates:

具␤IJK␤LMN典⫽具␾Ip␾Jq␾Kr␾Ls␾Mt␾Nu典␤pqr␤stu, (1) where␾Ipare the direction cosines that relate laboratory axes (IJK, LMN) to molecular axes ( pqr, stu) and the angle brackets denote the orientational average. The ro-tational invariants of the product of the directional cosine are well tabulated in the literature for the case of a ran-dom orientational distribution.4,5 In the presence of Table 2. Averaged Products of Six Direction Cosinesain the Presence of Polar Ordering, xÆ(␮EÕkT)

具␾I6_p典f(␪) 1 7i0共x兲 ⫹ 10 21i2共x兲 ⫹ 24 77i4共x兲 ⫹ 16 231i6共x兲 具␾Ip 4_␾ Iq 2_典 f(␪) 1 35i0共x兲 ⫹ 1 21i2共x兲 ⫺ 16 385i4共x兲 ⫺ 8 231i6共x兲 具␾I2_p␾I2_q␾I2_r典f(␪) 1 105i0共x兲 ⫺ 1 55i4共x兲 ⫹ 2 231i6共x兲 具␾I2_p␾J4_p典f(␪) 1 35i0共x兲 ⫺ 3 55i4共x兲 ⫹ 2 77i6共x兲 具␾I2_p␾J4_q典f(␪) 3 35i0共x兲 ⫹ 3 14i2共x兲 ⫹ 201 3080i4共x兲 ⫹ 3 308i6共x兲 具␾I2_p␾J2_p␾J2_q典f(␪) 2 105i0共x兲 ⫹ 1 42i2共x兲 ⫺ 23 770i4共x兲 ⫺ 1 77i6共x兲 具␾I2_p␾J2_q␾J2_r典f(␪) 1 35i0共x兲 ⫹ 1 14i2共x兲 ⫹ 67 3080i4共x兲 ⫹ 1 308i6共x兲 具␾I_p␾I_q␾3J_p␾J_q典f(␪) ⫺ 1 70i0共x兲 ⫹ 3 770i4共x兲 ⫺ 1 77i6共x兲 具␾I_p␾I_q␾J_p␾J_q␾J2_r典f(␪) ⫺ 1 210i0共x兲 ⫺ 1 168i2共x兲 ⫹ 23 3080i4共x兲 ⫹ 1 77i6共x兲 具␾I2_p␾J2_p␾K2_p典f(␪) 1 105i0共x兲 ⫺ 1 385i4共x兲 ⫹ 2 231i6共x兲 具␾I2_p␾J2_p␾K2_q典f(␪) 1 35i0共x兲 ⫹ 1 42i2共x兲 ⫺ 37 770i4共x兲 ⫺ 1 231i6共x兲 具␾I_p 2_␾ J_q 2_␾ K_r 2_典 f(␪) 8 105i0共x兲 ⫹ 9 42i2共x兲 ⫹ 529 6160i4共x兲 ⫹ 1 231i6共x兲 具␾I2_p␾Jp␾Jq␾Kp␾Kq典f(␪) ⫺ 1 210i0共x兲 ⫹ 1 55i4共x兲 ⫺ 1 231i6共x兲 具␾I2_p␾J_q␾J_r␾K_q␾K_r典f(␪) ⫺ 1 42i0共x兲 ⫺ 1 14i2共x兲 ⫺ 19 616i4共x兲 ⫹ 4 231i6共x兲 具␾I_p␾I_q␾J_p␾J_r␾K_q␾K_r典f(␪) 1 105i0共x兲 ⫹ 1 168i2共x兲 ⫺ 51 3080i4共x兲 ⫹ 1 231i6共x兲 a

an external electric field, the dipolar interaction

U⫽ ⫺␮E cos ␪ changes the orientational distribution

from the isotropic f0⫽ 1/4␲ to the Maxwell– Boltzmannian f(␪) ⬀ exp(⫺U/kT) ⫽ exp(x cos ␪), where ␪ is the angle between the molecular dipole moment and the electric field and x ⬅␮E/kT.14 With the anisotropic orientational distribution f(␪), the double-quantum scat-tered light intensities are related to 具␤ZZZ2 典f(␪) and

具␤XZZ 2 _典

f(␪). For example,具␤ZZZ 2 _典

f(␪)is expressed in terms of three kinds of rotational invariant of the products of the directional cosines:

具␤ZZZ 2 _典 f共␪兲⫽

兺

p Ap具␾Zp 6 _典 f共␪兲⫹

兺

p,q Bpq具␾Zp 4 _␾ Zq 2 _典 f共␪兲 ⫹

兺

p,q,r Cpqr具␾Z2_p␾2Z_q␾Z2_r典f共␪兲, (2) where Ap, Bpq, and Cpqrare sums of quadratic functions of␤.4 Similar expressions can be obtained for具␤_XZZ2 典f(␪),

具␥ZZZZ2 典f(␪), and具␥XZZZ2 典f(␪).

In the presence of static electric field E0along the Z di-rection, 2␻ harmonic signals are generated through the processes␥ (⫺2␻; ␻, ␻, 0) as well as ␤(⫺2␻; ␻, ␻), and the double-quantum scattered light intensity with polar-ization along the X and Z directions in e-HRS will be

Ii2␻⫽具␤iZZ2 典f(␪)⫹具␥iZZZ2 典f(␪)E02, with i ⫽ X, Z for a

Z-polarized incident fundamental. Let us make an order-of-magnitude estimate of relevant quantities to en-able us to compare e-HRS with EFISHG. For typical val-ues ␤ ⬇ 10⫺30 and ␥ ⬇ 10⫺36esu of nonlinear organic molecules we get ␤␮/5kT兩T⫽300 K⬇ 2.4 ⫻ 10⫺35 and ␥E0兩E0⫽500 V/cm⬇ 1.7 ⫻ 10

⫺36_{esu. It is important to} note that ␥ is only 1 order of magnitude smaller than ␤␮/5kT and hence is an approximation required for ig-noring the contribution of␥ in EFISHG, whereas ␥E0is 6 orders of magnitude smaller than ␤ for e-HRS. These considerations lead to the conclusion that there is no es-sential contribution from␥ (⫺2␻; ␻, ␻, 0) to the e-HRS signal as a second-order nonlinear ␹(2) process. Hence the depolarization ratio DZZZX(e-HRS) can be introduced in the same way as defined for HRS3:

DZZZX共e-HRS兲 ⫽ I_X2␻ I_Z2␻ ⫽ 具␤XZZ 2 _典 f共␪兲 具␤ZZZ 2 _典 f共␪兲 . (3)

For a given molecular system the expressions具␤_XZZ2 典f(␪) and 具␤ZZZ2 典f(␪) can be obtained from the polar-order-imparted orientational average of the products of the mo-lecular hyperpolarizabilities; the nonvanishing compo-nents are determined from molecular point-group symmetry. Details of the relationship between具␤2_典

f(␪)in laboratory coordinates and the molecular hyperpolariz-abilities are listed in Tables 1 and 2.

As a concrete example of the above relationship to the experimental measurement reported here, we consider a cylindrically shaped molecule (C_⬁v symmetry) that pos-sesses seven nonvanishing hyperpolarizability tensors, namely,␤333, ␤311⫽␤322, ␤131⫽␤232, and␤113⫽ ␤223, when the Kleinman symmetry is not assumed. From Tables 1 and 2, the relevant terms for a C_⬁v symmetry molecule are expressed explicitly in terms of the spherical modified Bessel functions in(x):

具␤XZZ2 典f共␪兲⫽ ␤3332

再

冋

1 35i0共x兲 ⫺ 3 55i4共x兲 ⫹ 2 77i6共x兲

册

⫹ 4

冋

2 105i0共x兲 ⫹ 1 42i2共x兲 ⫺ 23 770i4共x兲 ⫺ 1 77i6共x兲

册冉

␤311 ␤333

冊

⫹ 4

冋

⫺1 70i0共x兲 ⫹ 3 770i4共x兲 ⫺ 1 77i6共x兲

册冉

␤131⫹ ␤113 ␤333

冊

⫹ 4

冋

1 70i0共x兲 ⫹ 1 56i2共x兲 ⫺ 69 3080i4共x兲

册

⫻

冉

␤131␤113 ␤3332

冊

⫹ 4

冋

⫺ 2 105i0共x兲 ⫺ 1 168i2共x兲 ⫹ 1 88i4共x兲

册

冉

␤311␤131⫹␤311␤113 ␤333 2

冊

⫹ 2

冋

4 35i0共x兲 ⫹ 2 7i2共x兲 ⫹ 67 770i4共x兲 ⫹ 1 77i6共x兲

册冉

␤311 ␤333

冊

2 ⫹ 2

冋

1 70i0共x兲 ⫹ 1 56i2共x兲 ⫺ 69 3080i4共x兲

册

⫻

冋冉

␤131 ␤333

冊

2 ⫹

冉

␤113 ␤333

冊

2

册

冎

, 具␤ZZZ2 典f共␪兲⫽ ␤3332

再冋

1 7i0共x兲 ⫹ 10 21i2共x兲 ⫹ 24 77i4共x兲 ⫹ 16 231i6共x兲

册

⫹ 4

冋

1 35i0共x兲 ⫹ 1 21i2共x兲 ⫺ 16 385i4共x兲 ⫺ 8 231i6共x兲

册

⫻

冉

␤311⫹ ␤131⫹␤113 ␤333

冊

⫹ 2

冋

4 105i0共x兲 ⫹ 1 21i2共x兲 ⫺ 23 385i4共x兲 ⫺ 2 77i6共x兲

册

⫻

冉

␤311⫹ ␤131⫹␤113 ␤333

冊

2

冎

. (4)

From Eqs. (3) and (4) we find that permanent dipole moment ␮ plus three sets of hyperpolarizability ratios, i.e., ␤113/␤333, ␤131/␤333, and ␤311/␤333, can be deter-mined experimentally from the four macroscopic depolar-ization measurements of D_ZZZX(e-HRS) with different ex-ternal electric-field strengths applied. It is easy to see that for a given molecular symmetry group all the nonva-nishing hyperpolarizability tensor ratios can be obtained in principle because one can increase the number of

inde-pendent macroscopic quadratics of␤IJK by changing the magnitude of the external field.

3. EXPERIMENT AND RESULT

Now we turn to experimental measurement. We employ PBLG (molecular weight⬃300,000; degree of polymeriza-tion, Dp⬃ 1300; from Sigma Chemicals) as the nonlinear optical molecule with which to study e-HRS, as it is known to possess ferroelectric-phase characteristics with

C_⬁vmolecular point-group symmetry.15 PBLG is a well-known second-order nonlinear optical molecular system that shows virtually complete field-induced polar orienta-tion in ␣-helix-forming solvents at modest electric-field strengths.8,16 Also, the hyperpolarizabilities of PBLG measured by EFISHG are already available from the literature,8,15and we can make a direct comparison of val-ues measured by e-HRS and EFISHG techniqval-ues.

We prepared a solution with a concentration of 4 ⫻ 10⫺2 g/cm3_{in ethylene dichloride; after mixing for 24} h, the solution was contained in a long glass cell (7.5 cm ⫻ 2.5 cm ⫻ 1.0 cm), with indium tin oxide electroded glasses at the top and bottom to apply an external static electric field. The distance between the indium tin oxide electrodes was 1 cm, and the dc voltage applied between two indium tin oxide electrodes inside the sample cell pro-duced an electric field E0 of as much as 500 V/cm (Z di-rection). The experimental optical setup is the same as that for the usual HRS measurement. A Q-switched Nd:YAG laser operating at 1064 nm with a pulse width of 10 ns and a repetition rate of 10 Hz was employed. A combination of a half-wave plate and a linear polarizer varied the fundamental light intensity, and, after an IR pass filter, a lens with f⫽ 17.5 cm focused the laser beam propagating in the X direction into the sample cell (Fig. 1). At a 90° scattering angle (Y direction), 2␻ harmonics scattered from the sample solution were collected with an

f _{⫽ 20 cm lens and fed into a photomultiplier tube after a}

532-nm spike filter and a linear analyzer. With the inci-dent polarization fixed along the Z direction, the scattered harmonic intensities with polarization parallel IZ2␻ and perpendicular IX2␻to the direction of incident polarization were measured. Both I_Z2␻and I_X2␻had a quadratic depen-dence on the fundamental intensity, ensuring that the harmonics were from a second-order nonlinear process. Figure 2 shows the measured depolarization ratio (filled circles) as a function of external field strength. At the

zero external field, the depolarization ratio was_{⬃1.17, a} rather large value compared to that of 0.2 for one-dimensional shaped molecules, which implies that in PBLG the off-diagonal tensor components are as impor-tant as diagonal component␤333. The data points DZZZX for E0⫽ 300, 400, 500 V/cm are also shown in Fig. 2.

Now, from Eqs. (3) and (4), we can obtain the values of the permanent dipole moment and the hyperpolarizabil-ity ratios. The least-squares fit provided the fit values of ␮ ⬃ 5200 D at T ⫽ 300 K with the literature value of lo-cal field factor L⬃ 4.7 of the solvent ethylene

dichloride,17 ␤311/␤333⬃ ⫺0.81, ␤131/␤333⬃ 0, and ␤113/␤333⬃ 0. The maximum allowed values ␤131/␤333 and ␤113/␤333within the error bars of the experimental data points are found to be less than 10% of the ratio ␤311/␤333, which means that␤311is dominant among the off-diagonal tensor components and that other compo-nents are negligible. From the fit value of the dipole mo-ment we can deduce the dipole momo-ment of a monomeric unit of PBLG as␮/Dp⬃ 4.0 D, in a good agreement with the reported literature value of⬃4 D.15

4. DISCUSSION

To assess the versatility of e-HRS measurement we per-formed several analyses to determine how the relation-ship between the macroscopic depolarization ratio mea-surements and the microscopic hyperpolizability ratios behaves in various conditions. First, we assumed that the Kleinman symmetry holds among the off-diagonal components that appear in Eqs. (4), which means that ␤311⫽ ␤131⫽␤113, and attempted a least-squares fit for the ratio of tensor components,␤311/␤333, which resulted in the complex numbers _{⫺0.36 ⫾ i0.31 from the} mea-sured D_ZZZX of 1.17 at zero external field. Furthermore, the macroscopic depolarization ratio was found to in-crease as the external electric-field strength, contrary to the experimental observation. Second, we looked for an-other value of the ratio␤311/␤333, in addition to the value ⬃⫺0.81 obtained above, that could be obtained from the traditional HRS depolarization measurement, meaning Fig. 1. Laboratory coordinates for incident fundamental and

scattered harmonics. An external electric field was applied to the sample cell along the Z direction.

Fig. 2. Measured depolarization ratio D_ZZZX_{as a function of} ex-ternal electric field strength. Solid and dashed curves,␤311/␤333 ⬃⫺0.81 and␤311/␤333⬃⫹1.22, respectively. Inset, example of the lift of degeneracy in D_ZZZX_.

that no external static electric field was applied. For simplicity, we assumed that there are only two free pa-rameters, ␮ and ␤311/␤333, with ␤131⫽␤113⫽ 0 in Eqs. (4). The resultant value was␤311/␤333⬃ ⫹1.22, a posi-tive value, in contrast to the value_{⬃⫺0.81. To see how} e-HRS measurement discriminates two tensor ratios, we attempted a least-squares fit of the experimental data points to dipole moment ␮ as a free parameter with ␤311/␤333⬃ ⫹1.22 fixed, and we could not get the appro-priate value for the dipole moment. When the literature value of the dipole moment, instead, was adopted, the change in the depolarization ratio with changes in the electric-field strength was not at all pronounced, as shown by the dashed curve in Fig. 2, which deviates sig-nificantly from the experimental data points. Third, as a simple example illustrating the ambiguity in determining tensor ratios through HRS, we considered a molecular system exhibiting DZZZX⫽ 0.5 when there was no external static field, to which correspond two values, 0.43 and ⫺0.53, of tensor ratio ␤311/␤333, for which both magni-tude and sign are different, when we again assumed that ␤131⫽ ␤113⫽ 0. As we pointed out above, the ambiguity in the tensor ratio comes from the fact that the intensity of harmonic scattered light is proportional to具␤2_{典, not to} 具␤典, in HRS. But the application of a static electric field lifts the degeneracy in D_ZZZX_{, as depicted in the inset of} Fig. 2 (filled squares, 0.43; filled circles, ⫺0.53), which shows the main advantage of using e-HRS measurement in determining the sign and relative value of ␤311/␤333. This situation is in contrast to that of EFISHG measure-ment, in which one measures the total sum of hyperpolar-izability tensor components projected in the direction of the dipole moment without being able to distinguish the diagonal from the off-diagonal components.

We compare our result with other literature values of␤. Park et al.18 reported SHG measurement of poly( ␥-benzyl-L-glutamate-co-␥-methyl-L-glutamate) (PBMLG) with a Dp of 800; this is one of the lyotropic cholesteric liquid crystals of poly(L-glutamate). They investigated the ferroelectric liquid-crystalline phase of PBMLG by measuring the polarization behavior of SHG. With the s-and p-polarization fundamental inputs, only p-polarized second harmonics are observed in the Maker’s fringe mea-surement, which is characteristic of C_⬁vsymmetry. In a liquid-crystalline system with C_⬁vsymmetry, macroscopic second-order nonlinear optical susceptibility ␹(2) _is re-lated to the molecular hyperpolarizabilities through the order parameters具Pi(cos␪)典(i⫽ 1, 2, 3) (Ref. 19):

␹ZZZ共2兲 ⫽ 1 5N

冋冉

3⫺ 2 ␤311 ␤333

冊

具P1共cos␪兲典⫹ 2

冉

1 ⫺ ␤311 ␤333

冊

⫻具P3共cos␪兲典

册

␤333, ␹ZXX共2兲 ⫽ 1 5N

冋冉

1 ⫹ 4 ␤311 ␤333

冊

具P1共cos␪兲典⫺

冉

1 ⫺ ␤311 ␤333

冊

⫻具P3共cos␪兲典

册

␤333. (5)

They reported that ␹ZZZ(2) ⫽ 2.1⫻10⫺10 and ␹ZXX(2) ⫽ ⫺2.4 ⫻ 10⫺10_{esu. By introducing a generalized} mean-field potential we calculated the order parameters

具Pi(cos␪)典 (i⫽ 1, 2, 3) with ␤333⫽ ⫹3.52 ⫻ 10⫺28esu, which corresponds to ␤333(monomer)⫽ ⫹4.4 ⫻ 10⫺31_esu.19 _{It is worthwhile, based on this value and} on the hyperpolarizability tensor ratio that we obtained, to estimate the magnitude and the sign of ␤ that would have been measured by EFISHG for the PBLG solution. That is, ␤EFISHG_{(monomer)⫽␤}

333⫹ 2␤311⫽ 关1 ⫹ 2(␤311/␤333)兴␤333⫽ ⫺2.7 ⫻ 10⫺31esu. In Ref. 8, Le-vine and Bethea report an EFISHG value of ␤ for PBLG in solution. With Dp of 2500, they obtained ␤ ⫽ ⫹5 ⫻ 10⫺28 _{⫾ 50% esu, corresponding to the monomeric} value␤EFISHG_{(monomer)⫽ ⫹2 ⫻ 10}⫺31_{⫾ 50 % esu. The} two values are comparable in magnitude.

5. CONCLUSIONS

In conclusion, we have used depolarized hyper-Rayleigh scattering under an external electric field to determine the hyperpolarizability tensor ratio. We demonstrated that for polar nonlinear optical molecules the absolute values including signs of the hyperpolarizability tensor ratio can be determined through depolarized hyper-Rayleigh scattering under an external field, even in the resonance regime where Kleinman symmetry is not valid. It should be noted that the e-HRS measurement allows one to discriminate between two possible hyperpolariz-ability ratios, including signs, experimentally in a unique way. As a specific example of this kind of measurement technique, we measured the electric-field-dependent de-polarized hyper-Rayleigh signal from a poly-␥-benzyl- L-glutamate solution to obtain the dipole moment and the ratio of hyperpolarizability components as 4.0 D and ␤311/␤333⫽ ⫺0.81, respectively.

ACKNOWLEDGMENTS

This research is supported by the Brain Korea 21 project and by Korea Research Foundation grant KRF-2001-015-DP0175.

J. W. Wu’s e-mail address is [email protected].

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