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J. Phys. Chem. C 2009, 113, 10166–10175
Nonlinear Optical Stokes Ellipsometry. 2. Experimental Demonstration Nathan J. Begue, R. Michael Everly, Victoria J. Hall, Levi Haupert, and Garth J. Simpson* Department of Chemistry, Purdue UniVersity, 560 OVal DriVe, West Lafayette, Indiana 47907 ReceiVed: February 13, 2009; ReVised Manuscript ReceiVed: March 31, 2009
Second harmonic generation (SHG) has developed into a powerful tool for characterizing oriented thin films, surfaces, and interfaces. Furthermore, the nonlinear optical nature of wave-mixing processes typically results in the generation of a coherent signal beam with a well-defined polarization state. This coherence offers unique opportunities for the extraction of detailed molecular and surface properties from polarization analysis. In previous studies, nonlinear optical ellipsometry (NOE) has been developed as a means to retain sign and phase information between the different nonzero χ(2) tensor elements present in a given sample. However, those previous methods and related approaches for polarization analysis have all relied on the physical movement of optical elements to perform the analysis. The time required to physically move the appropriate optical elements ultimately dictates the fastest analysis time possible in a given technique. Such long acquisition times have limited NOE analyses to systems exhibiting excellent photostability and have precluded the feasibility of NOE imaging. Development of nonlinear optical Stokes ellipsometry (NOSE) was shown to address many of these problems. By increasing the repetition rate of the laser system and replacing previously slow rotating polarization optics with a rapid photoelastic modulator, the acquisition time with full polarization analysis was reduced from several hours to less than a second. This technique was validated against established NOE techniques using z-cut quartz as a reference sample and then demonstrated on a dye thin film. Additionally, an orientation analysis of the thin film was performed. These studies resulted in an order of magnitude improvement in precision relative to previous NOE techniques, while simultaneously accompanied by a reduction in acquisition time of more than four orders of magnitude. I. Introduction The orientation selectivity of even ordered nonlinear optical (NLO) processes has been demonstrated as an effective tool for investigating interfacial systems. Specifically, second harmonic generation (SHG) and sum frequency generation (SFG) have been used to probe solid/liquid,1-5 liquid/liquid,1,3,5-10 liquid/air,1,2,4-8,10-12 solid/air,5,12,13 and other interfaces,1,10-12,14 just to cite a few applications. Previous intensity based techniques for polarization analysis, while widespread, have limitations related to determination of the χ(2) tensor elements, which are generally complex-valued numbers. In studies of thin surface films, the phase difference between tensor elements can arise from the presence of molecular resonances, from complex-valued Fresnel factors, from interference between multiple contributions (e.g., nonresonant background), or from any combination thereof. To address this limitation, nonlinear optical ellipsometry (NOE) was established as an alternative method, by which detailed polarization analysis allows the determination of the relative phase/sign of the each nonzero χ(2) tensor element.15-17 These previous techniques employed mechanically manipulated optics, which ultimately limit the fastest possible analysis times. Investigation of dynamic systems or microscopy with NOE analysis using such approaches would not be practical. This paper demonstrates a new instrument, nonlinear optical Stokes ellipsometry (NOSE), capable of precise and rapid polarization analysis. In the NOSE beam path, the waveplates on the incident beam path were replaced by a photoelastic modulator (PEM) to rapidly query many polarization states, while the detection arm optics were replaced by a bank of * Corresponding author. E-mail:
[email protected].
two or four photomultiplier tubes (PMTs), modeled loosely after a static Stokes ellipsometry configuration.18,19 The PEM enables a set of polarization states to be queried in as little as 20 µs, while in principle the static four PMT Stokes detection scheme allows the complete, complex polarization state of the signal beam to be determined for each pulse of the laser. Whereas the primary focus of this work is on SHG, the NOSE technique is also compatible with other coherent multiphoton processes. The theoretical foundation for NOSE is presented in a companion paper (DOI 10.1021/jp810643n). In short, NOSE analysis is performed by recording signal intensity as a function of the PEM state on multiple detectors. Determination of χ(2) (the second order susceptibility) is performed by minimizing χ2 (the sum of the squared residuals of the fit, not to be confused with χ(2)), as shown in eq 1.
b - A|Mexiting · Eincident ·F χ2 ) |C χ J(2) | 2 | 2
(1)
b represents the adjusted number of photon counts (dark where C counts subtracted) and A is a calibration matrix correcting for the difference in sensitivities of the detectors. Mexiting is an expanded Jones matrix for the detection arm polarizing optics, Eincident is a matrix containing the set of possible two photon polarization permutations for each PEM state, and χFJ(2) is a vectorized representation of Jones χJ(2) tensor. For the purpose of this work, all NLO susceptibilities are determined in a modelindependent Jones tensor representation.16 The Jones tensor formalism has the advantage of allowing separation of the model-independent experimental observables with the modeldependent determination of the Cartesian χC(2) tensor elements, calculated for a given set of Fresnel factors and geometric
10.1021/jp9013543 CCC: $40.75 2009 American Chemical Society Published on Web 05/18/2009
NOSE II: Experimental Demonstration projection terms used to describe the sample configuration. Expressions for the Fresnel factors and s coefficients can be found in several previous publications16,17,20,21 and were addressed explicitly in the companion theory paper (DOI 10.1021/ jp810643n).
J. Phys. Chem. C, Vol. 113, No. 23, 2009 10167
( ∆2 ) + B cos ( ∆2 ) sin( ∆2 ) + ∆ ∆ ∆ ∆ ∆ C cos ( ) sin ( ) + D cos( ) sin ( ) + E sin ( ) 2 2 2 2 2
2ω Idet ∝ A cos4 2
z-Cut quartz provides an excellent standard, as it generates a large NLO response, has a high damage threshold, and has been previously studied by other methodologies.22 The D3 symmetry of quartz results in up to four unique χC(2) tensor elements.22 If the sample is interrogated along the crystallographic z-axis, the number of independent sets of tensor elements is reduced to one, leaving only the relationship: χxxx ) -χxyy ) -χyyx ) -χyxy. If quartz is oriented such that the crystallographic x-axis is coparallel with p-polarized light for normal illumination (thus defining the y-axis to be analogous to s-polarization), transformation of the tensor to a Jones representation yields the following symmetry relationships: χppp ) -χpss ) -χssp ) -χsps. Accounting for the degeneracy of the second and third index in SHG makes χssp and χsps equivalent. Applying the symmetry operations of quartz using NLOPredict23 generates a six-lobed hyperellipsoid when probing a sample along its C3 axis (Figure 1a). Importantly, these are precisely the same nonzero tensor elements expected for a uniaxial, achiral thin film, such that algorithms validated for z-cut quartz will be directly applicable for studies of thin films. In the special case of quartz, the relative signs and magnitudes of the χ(2) tensor elements are already known a priori. On the basis of these expected outcomes, settings for the maximum retardance of the PEM and the rotation angles of the waveplates in the Stokes ellipsometry configuration can be optimized, as not all angles will be comparably sensitive. Analytical expressions for signal intensity were derived previously in the companion paper (DOI 10.1021/jp810643n) and are reproduced in eq 2 for a 2PMT NOSE configuration with a quarter-wave plate (QWP) in the beam path rotated at an angle γ.
2
3
4
1 (2) 2 |χ | [1 ( cos2(2γ)] 2 pss (2) (2) (2) ) (2χsps sin(2γ)[Im(χpss ) cos(2γ) - Re(χpss )]
Ap/s ) Bp/s
II. Theoretical Predictions of NOSE for a Model System: z-Cut Quartz
3
(2)
(2) 2 (2)* 2 ) [1 - cos2(2γ)] - Re[χ(2) Cp/s ) 2(χsps pppχpss ][1 ( cos (2γ)] (2) (2) Dp/s ) (2χsps sin(2γ)[-Im(χ(2) ppp) cos(2γ) + Re(χppp)] 1 Ep/s ) |χ(2) | 2[1 ( cos2(2γ)] 2 ppp
where ∆ is the time-dependent retardance of the PEM, ∆ ) 2λπ sin(ωt), λ is the retardance setting of the PEM (typically 0.5 for a half-wave operation), and ω is the angular frequency of the PEM. The Jones tensor elements are ultimately determined from numerical combinations of the coefficients, suggesting that the sign/phase information will be generally retained for configurations that yield non-negligible values for all five coefficients. Using the symmetry relationships required for the χ(2) tensor of quartz, the fitting coefficients can be predicted as a function of γ. Taking the product of these coefficients, eq 3, and plotting them against γ provide a simple measure of the likelihood of generating a collective set of nonzero coefficients (A-E). Figure 1b shows this coefficient-product is symmetric around 45° and repeats every 90°, with maxima for P(γ) at 30° and 60° and a maximum for S(γ) at 45°. Inspection of eq 2 reveals a sign ambiguity in the imaginary component of the χJ(2) tensor elements for a QWP at (45°. Therefore, QWP angles around 35° and 55° can be expected to provide unambiguous fits.
P(γ) ) |Ap(γ) · Bp(γ) · Cp(γ) · Dp(γ) · Ep(γ)| S(γ) ) |As(γ) · Bs(γ) · Cs(γ) · Ds(γ) · Es(γ)|
(3)
III. Methods and Materials The schematic of a SHG instrument with the incorporated NOSE detection technique is presented in Figure 2. The fundamental beam was produced by a Ti:Sapphire laser (Spectra-Physics Mai Tai HP, 800 nm, ∼100 fs, 80 MHz, 16.5 MΩ/ cm) and methanol to remove any residual piranha solution. Thin films of the organic dye were prepared by dip coating and then air-dried. The z-cut quartz sample was obtained from Boston Piezo-Optics and used as received. 2PMT NOSE data were acquired for z-cut quartz rotation angles from 0°, 60°, and 120° and QWP rotation angles from 0° to 360° in 5° steps with the PEM operating at a nominal 0.5λ retardance. 4PMT NOSE data were similarly acquired but
with additional sample rotation angles of 180°, 240°, and 300°. For all experimental data presented here for quartz, the beam was incident upon the sample normal to the surface and attenuated such that a photon counting event occurred less than once in 100 laser pulses to minimize bias from undercounting of two-photon events. Thin film, DY7 data was acquired with a 45° angle of incidence with QWP rotation angles from 0° to 360° in 5° steps with the PEM operating at a nominal 0.5λ retardance. Analysis of the data was performed by nonlinear curve fitting to theoretical models detailed in the companion paper. Typically, NOSE data sets were fit using the nonzero, χJ(2) tensor elements and the PMT sensitivities as fitting parameters. Instrumental timing was defined by the PEM trigger signal, while the timing of the laser was left unrestrained. Quantum chemical calculations were performed with Gaussian 03.24 The structure of DY7 was calculated using the Hartree-Fock method with the 3-21+G* basis set, and the hyperpolarizability was derived from semiempirical ZINDO. A program to calculate the permanent excited state dipoles from the Gaussian 03 ZINDO output was adapted from the work of Mitchell and Moffatt et al.25,26 IV. Results and Discussion To assess the reliability of NOSE, the technique was used to analyze z-cut quartz, and the results of this were compared to those expected by symmetry and to the results obtained by the previously established RQ-NOE method. The results of NOSE measurements from thin films are also presented, and the figures of merits of NOSE relative to alternative methods are discussed. IV.A. Instrument Validation via Quartz Analysis. Experimental data were acquired as a two-dimensional (2D) data set by recording signal intensity as a function of PEM state ∆ along one dimension and the rotation angle of the detection arm QWP γ along the other dimension, Figure 3. Taking a “slice” of data along one dimension or the other results in either a NOSE experiment or a RQ-NOE experiment, allowing validation for the NOSE technique using a single data set, thus reducing 1/f noise. SHG measurements of the z-cut quartz as a function of rotation angle (acquired for coparallel incident and exiting polarization) are shown in Figure 3. 2PMT NOSE. In fitting the NOSE data, the ratios of χppp/ χsps and χpss/χsps were first constrained to be purely real and then expanded to be complex-valued. Additionally the relative sensitivity of the two PMTs was treated as a fitting parameter, but little variation was observed in this value (relative standard deviation of less than 4% in the ratio of sensitivities of PMT
NOSE II: Experimental Demonstration
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Figure 3. Example of z-cut quartz data. (a) Composite 2D data with the signals from the v-polarized detector on the left and the h-polarized detector on the right. (b) Example of 2-PMT NOSE trace taken for a QWP at 25°. (c) Example of RQ-NOE trace taken for a right circularly polarized incident beam.
1h/1v from six measurements). Therefore, four independent fitting parameters were used to perform the fits to purely real χJ(2) and six fitting parameters for complex-valued χJ(2). Reliable recovery of the phase terms in the NOSE measurements was found to demand high accuracy in the calibration of the sample rotation angle, the waveplate rotation angles, and the maximum PEM retardance. Each of the waveplates was calibrated to within
