Polarization characterization in surface second harmonic generation by nonlinear optical null ellipsometry

Analytica Chimica Acta 496 (2003) 133–142

Polarization characterization in surface second harmonic generation by nonlinear optical null ellipsometry Ryan M. Plocinik, Garth J. Simpson∗ Department of Chemistry, School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USA Received 14 October 2002; accepted 14 October 2002

Abstract Nonlinear optical null ellipsometry (NONE) is developed as a novel approach to evaluate second-order nonlinearities on and off resonance in transparent and absorbing thin surface films. Despite the growing use of second harmonic and sum frequency spectroscopy in characterizing interfaces, common approaches designed to extract the χ(2) nonlinear susceptibility tensor elements from relative intensity measurements are generally only reliable for ultra-thin transparent films at the interface between transparent media. The simple expressions routinely used to treat intensity-based polarization measurements far from resonance become significantly more complicated on or near resonance, for which the χ(2) tensor elements and the thin film refractive indices are complex-valued parameters. In contrast, the mathematical formalism developed for NONE measurements is well suited for spectroscopic studies. The combined results of NONE and polarization-dependent intensity (PDI) measurements at fused silica/dye solution interfaces demonstrate that NONE retains complex phase information between the χ(2) nonlinear susceptibility tensor elements that is lost using comparable intensity-based approaches. © 2003 Published by Elsevier B.V. Keywords: Ellipsometry; Nonlinear optics; Second harmonic generation; Sum frequency generation; Hyperpolarizability

1. Introduction Second harmonic generation (SHG) and sum frequency generation (SFG), which are the respective frequency doubling and frequency mixing of light, have emerged as powerful probes for characterizing oriented surface systems [1–21]. By nature of the unique symmetry of SHG and SFG measurements, the detected signals are often dominated by the oriented chromophores at interfaces and are largely insensitive to greater numbers of randomly oriented species in the bulk. This symmetry condition has been exploited with great success in surface-specific spectro∗ Corresponding author. E-mail address: [email protected] (G.J. Simpson).

0003-2670/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/S0003-2670(03)00994-2

scopic measurements by SHG and SFG at solid/liquid [4,6,16,19,21], liquid/liquid [4–6,13–15,19], liquid/air [4–7,11,13–16,18,20,21], solid/air [6,8,11], semiconductor [4,7,9,12,15,20], polymer [17], and biological [7,11] interfaces. The intensity of the second-order nonlinear beam generated from an oriented uniaxial organic film between two isotropic media is dependent on the nonlinear optical susceptibility (described by the rank three χ(2) tensor), the incident intensity or intensities, the polarization states of the incident and exigent light, and the linear optical properties of the interfacial system. A routine approach for isolating relative values of the χ(2) tensor elements is to compare the intensities of the s- or p-polarized nonlinear beams measured as functions of the polarization state(s) of the incident

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beam(s). For example, the expressions in Eqs. (1) and (2) (or variations thereof) are commonly used in reflection SHG measurements of achiral uniaxial interfacial films performed using linearly polarized incident light with the plane of polarization rotated away from purely p-polarized by the angle γ [4,22–26]. In standard reflection and transmission measurements of achiral films far from resonance, relative values of the tensor elements are easily obtained from the ratios of the second harmonic intensities measured under different polarization conditions [26–30] or from nonlinear curve-fitting [4,22–25,30,31]: Is2ω = C 41 sin2 (2γ)|s1 χXXZ |2 (I ω )2 (1) Ip2ω = C|s6 χZXX + cos2 γ(−s3 χXXZ + s5 χZXX − s6 χZXX + s7 χZZZ )|2 (I ω )2

(2)

As new samples, unique substrates and diverse interfacial systems are increasingly investigated by SHG and SFG, many of the simplifying assumptions routinely used to treat polarization measurements may not hold. For example, the Fresnel factors that describe the electric fields at an interface will be complex-valued in studies conducted in total internal reflection [26,31–33] or in studies of multilayer films [28,34–36]. The Fresnel factors will also generally be complex in systems in which the nonlinear film, the substrate, or the ambient medium absorbs either the incident or exigent light (e.g. in spectroscopic studies) [28,33,35]. Finally, if the nonlinear surface layer absorbs one or more of the incident or exigent optical frequencies, the χ(2) tensor elements themselves will also be complex-valued [13,25,27,37–41]. In any or all of these instances, even the simple expressions in Eqs. (1) and (2) (or variations thereof) can become complicated. Expansion of the expressions in Eqs. (1) and (2) in the most general case of SHG measurements with complex fitting coefficients and complex tensor elements yields the expressions in Eqs. (3) and (4): Is = C 41 sin2 (2γ)[Re(s1 )2 Re(χXXZ )2 + Im(s1 )2 Im(χXXZ )2 + Re(s1 )2 Im (χXXZ )2 + Im(s1 )2 Re(χXXZ )2 ]

(3)

Ip = C{Re(s6 )Re(χZXX ) − Im(s6 )Im(χZXX ) + cos2 γ[−Re(s3 )Re(χXXZ ) + Re(s5 )Re(χZXX ) − Re(s6 )Re(χZXX ) + Re(s7 )Re(χZZZ )

+ Im(s3 )Im(χXXZ ) − Im(s5 )Im(χZXX ) + Im(s6 )Im(χZXX ) − Im(s7 )Im(χZZZ )]}2 + C{Re(s6 )Im(χZXX ) + Im(s6 )Re(χZXX ) + cos2 γ[−Re(s3 )Im(χXXZ ) + Re(s5 )Im(χZXX ) − Re(s6 )Im(χZXX ) + Re(s7 )Im(χZZZ ) − Im(s3 )Re(χXXZ ) + Im(s5 )Re(χZXX ) − Im(s6 )Re(χZXX ) + Im(s7 )Re(χZZZ )]}2

(4)

It is immediately apparent that extracting the real and imaginary components of the three independent tensor elements present in SHG from the second harmonic intensity measured as a function of γ is nontrivial. Even assuming the χ(2) tensor elements are purely real, Eqs. (3) and (4) yield a minimum of two possible solutions for the relative tensor elements (for χZXX and χXXZ of like and opposite sign) [28,42,43]. An alternative detection approach was pioneered during the emergence of nonlinear optical surface measurements of uniaxial films in which a rotating polarizer is placed between the sample and the detector [14,44,45]. Rather than compare normalized intensity ratios, the rotation angle of the polarizer that resulted in extinction of the nonlinear beam was used to determine relative values of the tensor elements and subsequently molecular orientation. Using this type of approach, the ambiguity regarding the relative sign between χZXX and χXXZ can in principle be removed. However, this rotating polarizer approach will only produce a true null provided the nonlinear beam is linearly polarized, which generally limits its reliable application to standard reflection and transmission measurements of vanishingly thin transparent films far from resonance (in which case both the χ(2) tensor elements and the Fresnel factors are purely real). Given the sensitive dependence of SHG and SFG on thin film optical properties [28,44,46–53] and the long tradition of ellipsometry as a successful tool for interpreting surface optics [34,54], it is tempting to suggest a new approach for acquiring and interpreting nonlinear optical surface measurements combining these linear and nonlinear techniques. Ellipsometry is a non-destructive technique for probing optical properties of films and surfaces [34,55]. In a typical ellipsometry measurement, an incident beam of known polarization is directed at the sample. The polarization state of the reflected light can be related back

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to the ellipsometric angles ψ and ∆, from which the optical properties of the surface can be determined. In linear ellipsometry, ρ is traditionally defined to be the ratio of the reflection coefficients of p- to s-polarized light and is related to ψ and ∆ through the following relation [34,56]: Rp ρ≡ = tan ψ × ei∆ (5) Rs Inherent characteristics of nonlinear optics prevent simple definitions of the nonlinear signals in terms of ellipsometric reflection or transmission coefficients. A more general notation will be adopted, in which the nonlinear parameter ρ2ω is defined as the complex ratio of the p- to s-polarized second harmonic electric field amplitudes: ρ2ω ≡

Ep2ω Es2ω

(6)

In this work, nonlinear optical null ellipsometry (NONE) is developed theoretically and tested experimentally as a novel method to directly determine the fully complex-valued χ(2) tensor element ratios from nonlinear polarization measurements. For simplicity, the discussion presented here focuses exclusively on SHG, although the approach is general enough to also be applicable in SFG investigations.

2.1. Tensor elements from intensity measurements Provided that the electric-dipole allowed χ(2) tensor elements for surface second harmonic generation are purely real (or more generally of identical phase), their relative values can be interrogated from polarization-dependent intensity (PDI) measurements acquired under a minimum of three different polarization conditions, such as those described in Eqs. (7)–(9) [26,29]: 2ω Ipp = C| − s3 χXXZ + s5 χZXX + s7 χZZZ |2

(7)

2ω Ips = C|s6 χZXX |2

(8)

2ω Is45

(9)

=

beam and the second subscript the polarization state of the linearly polarized incident beam. For example, 2ω refers to the s-polarized intensity recorded for Is45 an incident beam linearly polarized with the plane of polarization rotated 45◦ with respect to the p and s coordinates. In all cases, it is assumed the second harmonic intensities are normalized by the square of the fundamental intensity. The relative values of χZXX and χXXZ are simple to determine:
2ω χZXX |s1 | Ips = ±2 (10) 2ω χXXZ |s6 | Is45 Since the relative sign of χZXX and χXXZ cannot be determined from the intensity alone, two solutions result (as indicated by the “±” symbol) [29]. In contrast to χZXX /χXXZ , evaluation of χZZZ /χXXZ 2ω is complicated by possible complex contrifrom Ipp butions in the fitting coefficients. Expansion of Eq. (7) yields the expression in Eq. (11): 2ω 2 2 2 Ipp = C{χXXZ |s3 |2 + χZXX |s5 |2 + χZZZ |s7 |2

− 2χXXZ χZXX [Re(s3 )Re(s5 ) + Im(s3 )Im(s5 )] − 2χXXZ χZZZ [Re(s3 )Re(s7 ) + Im(s3 )Im(s7 )] + 2χZXX χZZZ [Re(s3 )Re(s7 ) + Im(s5 )Im(s7 )]} (11) Combination and rearrangement of Eqs. (9) and (11) yields the following relationship between the ratio of 2ω /I 2ω and χ Ipp ZZZ assuming purely real tensor eles45 ments:

2. Theory

2 1 4 C|s1 χXXZ |

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The first subscript on I2ω in Eqs. (7)–(9) indicates the polarization component of the second harmonic

2 0 = aχZZZ + bχZZZ + c

(12a)

a = |s7 |2

(12b)

  b = χZXX |s5 + s7 |2 − |s5 |2 − |s7 |2   − χXXZ |s3 + s7 |2 − |s3 |2 − |s7 |2   2ω I 1 pp c = | − s3 χXXZ + s5 χZXX |2 − |s1 |2 2ω 4 Is45

(12c) (12d)

The quadratic formula can be used to determine two solutions for χZZZ /χXXZ using Eq. (12) for each solution in Eq. (10), for a total of four possible sets of relative tensor elements that are consistent with a given set of intensity measurements.

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VF

GLP

QWP

VF

QWP

IR

PBC

PMT

YAG (a)

IF

HWP

HWP

L

HWP

L

IR

AP DS

OR

(b)

Fig. 1. (a) Schematic of the optical path for the instrument used in NONE: VF, visible absorbing filter; HWP, half-wave plate; GLP, Glan laser polarizer; QWP, quarter-wave plate; L, plano-convex lens; IR, infrared absorbing filter; PBC, polarizing beam splitting cube; IF, 532 nm interference filter. (b) Schematic of the total internal reflection cell: DS, dye solution; AP, aluminum plate; OR, O-ring.

If the tensor elements themselves are also allowed to be complex as would be expected in resonanceenhanced or spectroscopic investigations, determining the many unique solutions for the real and imaginary components of the tensor elements from intensity measurements alone is significantly more complicated. 2.2. Generalized null ellipsometric detection The nonlinear nature of second harmonic generation requires the use of a generalized ellipsometric approach. Using the instrumental configuration shown in Fig. 1a, a null at the detector must satisfy the expression in Eq. (13), written using Jones matrix notation:    π  0 2ω 2ω 2ω = MP2ω [MH2ω (α2ω H )][MQ (αQ )]e 2 0 (13) where e2ω describes the far-field complex second harmonic electric field polarization emanating from the interface, and the superscripts of 2ω indicate angles and matrices for the second harmonic beam. M indicate the Jones matrices for the rotated optical elements, with MP2ω representing the polarizing beam splitting cube, MH2ω the half-wave plate, and MQ2ω the quarter-wave plate. The Jones matrices for the rotated elements are determined from combinations of the Jones matrix evaluated with the fast axis of the optical element parallel with the p-polarized component of the field (i.e. M(0)) and rotation matrices describing the change in reference frame as

the optical element is rotated [34,57]. Mathematically, M(α) = R(−α)M(0)R(α), where R indicate the appropriate rotation matrices for rotation of the reference frame by the angles in parentheses [34]. In 2ω practice, the values for α2ω Q and αH yielding a null are determined experimentally for a given incident fundamental polarization state. Evaluation and simplification of the rotated Jones matrices in Eq. (13) yields the expression in Eq. (14):      0 0 0 cos(2α2ω sin(2α2ω H ) H ) = 2ω 0 0 1 sin(2α2ω H ) −cos(2αH )   sin(2α2ω cos(2α2ω Q )−i Q ) × −cos(2α2ω sin(2α2ω Q ) Q )−i   2ω ep × (14) e2ω s Further simplification of Eq. (14) using Euler’s formula leads to the following expression for the relationship between the complex polarization of the 2ω nonlinear beam and the angles of α2ω H and αQ that produce a null in the detected second harmonic intensity, for the instrument shown in Fig. 1a: ρ2ω ≡

e2ω p e2ω s

=

2ω 2ω cos(2α2ω H − 2αQ ) + i cos(2αH ) 2ω 2ω −sin(2α2ω H − 2αQ ) + i sin(2αH )

(15) A generalized notation has been adopted in which ρ is defined to be the complex ratio of the p- to s-polarized electric fields, rather than the ratio of reflection (or transmission) coefficients as in linear ellipsometry.

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2.3. Jones matrix description of surface second harmonic generation The polarization state of the second harmonic light generated at an interface is a function of the surface second-order nonlinear optical polarizability tensor χ(2) , the polarization state of the incident field, and the Fresnel factors relating the field amplitudes at the interface with the incident and exigent fields. For a uniaxial surface film with C∞ symmetry, only four nonzero independent tensor contributions to χ(2) remain in SHG, χZZZ , χZXX = χZYY , χXXZ = χXZX = χYYZ = χYZY , and χXYZ = χXZY = −χYXZ = −χYZX [26,58]. The last set of elements disappears for surfaces with mirror-plane C∞v symmetry (i.e. achiral). The nonlinear interfacial polarization for SHG measurements of an achiral film is given in Eq. (16) [26]:   P 2ω = 



2χXXZ eωL,X eωL,Z 2χXXZ eωL,Y eωL,Z

 

χZXX (eωL,X )2 + χZXX (eωL,Y )2 + χZZZ (eωL,Z )2 (16)

137

Evaluation of the appropriate Jones matrices for the polarizer/half-wave plate/quarter-wave plate arrangement with the polarizer set to pass s-polarized ◦ light and α2ω Q = −45 (Fig. 1a) yields the following expression for the polarization state of the incident field:    i 1 cos(2α2ω sin(2α2ω H ) H ) ω e ∝ 1 i sin(2α2ω ) −cos(2α2ω H )    ω H 0 0 ep × eωs in 0 1   exp(−2iαωH ) ∝ (18) i exp(−2iαωH ) 2ω Substitution of e2ω p and es into Eq. (17) yields the following expression describing the second harmonic polarization generated from an achiral uniaxial system for the configuration shown in Fig. 2:

ρ2ω (αωH ) =

e2ω p e2ω s [−s3 χXXZ + s5 χZXX + s7 χZZZ ]

× exp(−4iαωH ) + s6 χZXX exp(4iαωH ) The polarization vector of the incident beam within the = ω interfacial layer eL is given by matrix multiplication is1 χXXZ of the far-field incident polarization e␻ with a 3 × 2 (19) transformation matrix TL projecting the s and p coorThe relationship in Eq. (19) provides a link between dinate system to the surface coordinate system and a the complex value of ρ2ω measured from the null ␻ diagonal 3 × 3 matrix L containing the appropriate angles αωH and αωQ given in Eq. (15) and the χ(2) Fresnel factors describing the local field components nonlinear optical tensor elements. Complex ratios of (eωL = Lω TLω eω ) [59]. Similar matrix algebra can be the tensor elements can be determined by combining used to relate the nonlinear polarization within the inmeasurements of ρ2ω acquired for multiple rotation terfacial film to the second harmonic intensities far angles of αωH . Using the relations e0 = 1, eiπ/2 = i, from the interface (e2ω = TL2ω L2ω e2ω L ) [59]. Performand e−iπ/2 = −i the following equalities can be ing the appropriate matrix algebra yields the relation in Eq. (17) for SHG measurements acquired in reflection:   ω )2 + s χ ω )2 + s χ ω )2 + s χ ω )2 −s χ (e (e (e (e 3 XXZ ZXX ZXX 7 ZZZ 5 6 p p s p e2ω = (17) s1 χXXZ eωp eωs The sn fitting coefficients include all the angular terms and the appropriate Fresnel factors [59]. As defined, these coefficients are valid both for chiral and achiral interfaces for an arbitrary polarization of the incident beam. Explicit expressions for the fitting coefficients used for these measurements are presented in other work [59].

easily derived: π

s6 χZXX (20) ρ2ω + iρ2ω (0) = −2 8 s1 χXXZ   π

s5 χZXX s7 χZZZ s3 − iρ2ω (0) = −2 + − ρ2ω 8 s1 χXXZ s1 χXXZ s1 (21)

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ω in

2 ωout 1 = Ambient 3 = Nonlinear Film 2 = Substrate

Fig. 2. Second harmonic generation in an ultra-thin film. Solid lines indicate the incident beam, dotted lines the second harmonic source initially propagating upward, and dashed lines the initially downward propagating source.

The ratios of the tensor elements are straightforward to determine from the measured values of ρ2ω :  χZXX s1  2ω π

=− ρ (22) + iρ2ω (0) χXXZ 2s6 8  χZZZ s3 s5 χZXX s1  2ω π

= − − ρ − iρ2ω (0) χXXZ s7 s7 χXXZ 2s7 8 (23) Since both the fitting coefficients and the ρ2ω values in Eqs. (22) and (23) are allowed to be complex, the real and imaginary components of the relative tensor elements can be uniquely determined from the ellipsometric nulling angles without the need for complicated nonlinear curve-fitting algorithms and with no a priori knowledge of the surface hyperpolarizability.

3. Experimental All experiments were performed using an instrument as described in Fig. 1a. A New Wave Research Polaris Nd:YAG laser generated 5–7 ns pulses of 1064 nm light at 20 Hz. The infrared light was passed through a visible-blocking RG695 filter and the beam power was attenuated through the combination of a half-wave plate and Glan laser polarizer set to pass s-polarized light. After attenuation, the beam passed through a half-wave plate in a rotatable mount, a quarter-wave plate fixed at −45◦ , a plano-convex lens, and a final visible-blocking filter. Approximately 1 mJ per pulse of incident light was then focused onto

the solid/liquid interface of a total internal reflection sample cell (Fig. 1b). After exiting the cell, 532 nm light generated at the interface was passed through an infrared absorbing KG3 filter and recollimated with a second lens. The light then passed through a quarter-wave plate, a half-wave plate, a polarizing beam splitting cube oriented to pass s-polarized light, a second KG3 filter, and a 532 nm interference filter before introduction to a photomultiplier tube (Burle 8850). The intensity of the nonlinear response was read directly off an oscilloscope (Tektronix TDS 540) following current-to-voltage conversion and preamplification (EG&G Ortec 9305). In a given NONE measurement, the intensity of the nonlinear response was manually nulled by iteratively rotating the 532 nm quarter-wave plate and half-wave plate until a minimum in the peak-to-peak potential from the PMT was observed on the oscilloscope (∼30 averages). Background signal from dark noise was ∼10 mV, a prism with no sample would generate a ∼20 mV background, and a sample with a nulled signal produced ∼30 mV. The signal detected under nulled conditions was typically more than two orders of magnitude smaller than the maximized signal (obtained by rotating the 532 nm half-wave plate 45◦ from the null angle). Solutions of 1.0 × 10−3 M rhodamine 6G (R-6G, Aldrich, ∼99% pure) and 5.0 × 10−4 M disperse red 19 (DR-19, Aldrich, ∼95% pure) in 2-propanol (Mallinckrodt, HPLC grade) were introduced to the prism/solution interface using a home-built sample cell. Prior to use, the prism was cleaned by immersion in chromic acid (LabChem Inc., 10%) for 15 min followed by a thorough rinsing with ultrapure water (>17.5 M cm). The prism was then rinsed with 2-propanol before being placed into the sample cell. To contain the dye solutions; an O-ring was sandwiched between the total internal reflection face of the prism and a thin piece of aluminum (2.5 cm × 2.5 cm) machined with a ∼3 mm hole for solution introduction/extraction.

4. Results and discussion UV-Vis spectra for both dye solutions are shown in Fig. 3. The vertical bar in the figure indicates the second harmonic wavelength (532 nm). In the case

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139

1.4

532 nm DR-19

1.2

1.0

R-6G

Absorbance

0.8

0.6

0.4

0.2

0.0

300

400

500

600

700

800

Wavelength (nm) Fig. 3. UV-Vis spectra of disperse red 19 (solid line, 5 × 10−5 M in 2-propanol) and rhodamine 6G (dashed line, 1 × 10−5 M in 2-propanol), with a vertical line indicating 532 nm.

of R-6G, the second harmonic wavelength is resonant with a strong transition within the chromophore (λmax = 529.5 nm). In contrast, the second harmonic for DR-19 is within the absorption envelope but not perfectly on resonance (λmax = 475.2 nm). The null rotation angles for the half-wave plate and quarter-wave plate are compiled in Table 1. Consistent with Eqs. (20) and (21), null measurements were made for both circular and the linear incident polarizations of light (αωH = 0 and 22.5◦ , respectively). The complex-valued χ(2) tensor elements were obtained from sums and differences of ρ2ω values as described in Eqs. (22) and (23) (Tables 2 and 3). In contrast to the NONE measurements, the PDI approach required the assumption of purely real χ(2) tensor elements and yielded a set of four possible solutions. In solving for the normalized value of the

χZXX tensor element, the square root of the intensity ratio was taken Eq. (10) for which χZXX and χXXZ could be either of like or opposite sign. Subsequent use of the quadratic formula to calculate the χZZZ tensor element produced two solutions for each of the two sign possibilities for χZXX . From the intensity Table 1 Null angles (◦ ) detected using NONE for disperse red 19 and rhodamine 6G DR-19

αωH = 0◦ αωH = 22.5◦

R-6G

α2ω Q

α2ω H

α2ω Q

α2ω H

132 (±2) −119 (±2)

−185 (±2) −17 (±2)

27 (±5) 36 (±1)

210 (±3) 153 (±1)

The averages and standard deviations were determined from five replicates.

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Table 2 Values of the χ(2) tensor elements for R-6G in 2-propanol, normalized for χXXZ = 1 NONE

PDI (+)

χXXZ χZXX χZZZ χZZZ /χZXX

1 −0.5 (±0.2) − 0.52i (±0.09i) 1.7 (±0.1) + 0.7i (±0.1i) −2.3 (±0.3) + 1.2i (±0.4i)

(−)

(+)

1 0.90 (±0.02) 1.54 (±0.08) −1.78 (±0.08) 1.7 (±0.1) −2.0 (±0.1)

(−)

1 −0.90 (±0.02) 1.50 (±0.08) −1.83 (±0.08) −1.7 (±0.1) 2.0 (±0.1)

The tensor components determined from the intensity-based measurements were evaluated assuming purely real χ(2) elements and yielded four possible solutions consistent with the set of experimental results. Table 3 Values of the χ(2) tensor elements for DR-19 in 2-propanol, normalized for χXXZ = 1 NONE

PDI (+)

χXXZ χZXX χZZZ

1 0.9 (±0.1) − 0.14i (±0.09i) 2.5 (±0.3) + 1.0i (±0.2i)

(−)

(+)

1 0.97 (±0.04) 2.5 (±0.1) −2.6 (±0.1)

data, the sets of possible solutions for the three independent nonzero χ(2) tensor elements, normalized with respect to χXXZ , are summarized in Tables 2 and 3. The fitting coefficients used to calculate values for the relative tensor elements for both the NONE and PDI measurements have been reported elsewhere [59].The dielectric constant of the thin film was assumed to be the average of the corresponding prism and solution dielectric constants, such that n2film = (n2prism + n2substrate )/2. Previous investigations indicate an intermediate value for the refractive index of the nonlinear layer yields physically reasonable results in nonlinear optical polarization measurements of submonolayer films [28,30,42,51]. Since NONE measurements retain phase information in the fully complex tensor ratios while the PDI measurements do not, the NONE results should correctly predict the observed relative intensity data (although the reverse is not generally true). A comparison of the measured relative intensities and those predicted from the NONE measurements is summarized in Table 4 for both DR-19 and R-6G. With DR-19, the predicted intensity ratios were within experimental error of the measured values for both Ips /Is45 and Ipp /Is45 . Although qualitatively similar results were observed comparing the measured intensity ratios for R-6G with those predicted from NONE, this system

(−)

1 −0.97 (±0.04) 2.5 (±0.1) −2.1 (±0.1)

yielded values outside the range of experimental error for Ips /Is45 and Ipp /Is45 . A possible explanation for this discrepancy could be the relatively simple treatment used to model the interfacial optics. Previous studies with disperse red dyes and R-6G suggest that DR-19 should be dominated by the βZ Z Z element [43] and that R-6G is dominated by the βX X Z element [25,43,60] at the wavelengths used in these investigations. Consistent with these expectations, the ratio of χZXX /χXXZ should be ∼1 for DR-19 and the ratio of χZZZ /χZXX should be ∼−2 for R-6G [1]. NONE measurements of DR-19 yielded χZXX /χXXZ = 0.9 (±0.1) − 0.14i (±0.09i), a value within error of unity and almost purely real. The most closely-matched set of tensor elements from PDI analysis (with χXXZ and χXXZ ) of like sign yielded Table 4 Comparison of the experimental intensity ratios and those predicted from the NONE data Observed

Predicted from NONE

Disperse red 19 Ips /Is45 Ipp /Is45

0.88 (±0.08) 7.2 (±0.5)

0.8 (±0.2) 8 (±1)

Rhodamine 6G Ips /Is45 Ipp /Is45

0.76 (±0.03) 2.8 (±0.3)

0.5 (±0.2) 4.1 (±0.6)

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a ratio of 0.97 (±0.04) in good agreement with both expectations and the NONE measurements. In the case of R-6G, NONE yielded χZZZ /χZXX = −2.3 (±0.3) + 1.0i (±0.2i). Of the four possible solutions from PDI measurements assuming purely real tensor elements, the tensor ratio that most closely matches the NONE results (i.e. χZXX = −0.90 ± 0.02 and χZZZ = 1.50 ± 0.08) yielded a ratio of χZZZ /χZXX = −1.7 (±0.1). The poor correspondence between the NONE results and the PDI results presumably stems from the improper application of the PDI approach in systems on resonance, in which the assumption of purely real tensor components is no longer justified. The fully-resonant results for R-6G summarized in Table 2 demonstrate the importance of retaining relative phase information in surface second harmonic generation investigations. The imaginary component of the χZZZ /χZXX ratio is half as great at the real component in the R-6G film and is large enough to be significant. Consequently, care must be taken when interpreting χ(2) tensor elements from intensity-based measurements in resonance-enhanced and/or spectroscopic investigations. Even near resonance as in the case of DR-19, imaginary contributions can be non-negligible (Table 3). Using expressions that are only simple to apply far from resonance such as those in Eqs. (10) and (12) can lead to significant errors for systems with complex-valued tensor elements on or near resonance. In contrast to PDI methods, NONE measurements can be reliably and easily employed both on and off resonance for systems with real or complex Fresnel coefficients (e.g. for dyes absorbing incident or exigent light, for multilayer films, and/or for measurements acquired in total internal reflection).

5. Conclusions Nonlinear optical null ellipsometry was developed as a novel approach for extracting the complex χ(2) tensor components from nonlinear optical polarization measurements. Comparisons with a routine intensity-based methodology demonstrated the greater information content afforded by NONE measurements. The assumption of purely real χ(2) tensor elements commonly employed when treating PDI measurements is not generally valid on or near resonance. In contrast, the retention of complete phase

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information in NONE measurements allows straightforward application both on and off resonance in reflection, transmission, or total internal reflection for ultra-thin, multilayer, and/or absorbing films. Additionally, the experimental and numerical approach developed to extract complex ratios of the surface tensor elements is significantly simpler than comparable intensity-based algorithms. Continuing studies in our laboratory are designed to extend the capabilities of NONE through the development of more detailed models for treating the interfacial optics. Acknowledgements The authors would like to acknowledge funding from a Research Corporation Research Innovation Award and from a Camille and Henry Dreyfus Foundation New Faculty Award. We would also like to acknowledge Robert M. Everly and Robert E. Santini for their helpful insights. References [1] T.F. Heinz, Nonlinear Surface Electromagnetic Phenomena, North-Holland, New York, 1991, 354 pp. [2] M. Buck, M. Himmelhaus, J. Vac. Sci. Technol. 19 (2001) 2717. [3] C.D. Bain, Curr. Opin. Colloid Interface Sci. 3 (1998) 287. [4] R.M. Corn, D.A. Higgins, Chem. Rev. 94 (1994) 107. [5] K.B. Eisenthal, Annu. Rev. Phys. Chem. 43 (1992) 627. [6] Y.R. Shen, Annu. Rev. Phys. Chem. 40 (1989) 327. [7] Y.R. Shen, Nature 337 (1989) 519. [8] G.A. Somorjai, G. Rupprechter, J. Phys. Chem. B 103 (1999) 1623. [9] Y.-M. Chang, L. Xu, H.W.K. Tom, Chem. Phys. 251 (2000) 283. [10] F.M. Geiger, A.C. Tridico, J.M. Hicks, J. Phys. Chem. B 103 (1999) 8205. [11] V. Vogel, Curr. Opin. Colloid Interface Sci. 1 (1996) 257. [12] T. Rasing, Appl. Phys. A 59 (1994) 531. [13] G.L. Richmond, Annu. Rev. Phys. Chem. 52 (2001) 357. [14] K.B. Eisenthal, Acc. Chem. Res. 26 (1993) 636. [15] K.B. Eisenthal, J. Phys. Chem. 100 (1996) 12997. [16] P.B. Miranda, Y.R. Shen, J. Phys. Chem. B 103 (1999) 3292. [17] D.H. Gracias, Z. Chen, Y.R. Shen, G.A. Somorjai, Acc. Chem. Res. 32 (1999) 930. [18] H.C. Allen, E.A. Raymond, G.L. Richmond, Curr. Opin. Colloid Interface Sci. 5 (2000) 74. [19] C.D. Bain, P.R. Greene, Curr. Opin. Colloid Interface Sci. 6 (2001) 313. [20] Y.R. Shen, Solid State Commun. 102 (1997) 221.

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Garth J. Simpson has been on the faculty at Purdue University since 2001 as an assistant professor. He received his PhD degree from the University of Colorado at Boulder in 2000 and was a Life Sciences Research Foundation postdoctoral fellow working with Richard N. Zare at Stanford from 2000 to 2001. He has recently received the ACS Victor K. LaMer Award in colloid and interface science, a New Faculty Award from the Camille and Henry Dreyfus Foundation, and a Research Innovation Award from the Research Corporation.