Anal. Chem. 2010, 82, 559–566
Chemically Selective Analysis of Molecular Monolayers by Nonlinear Optical Stokes Ellipsometry Nathan J. Begue and Garth J. Simpson* Department of Chemistry, Purdue University, 560 Oval Drive, West Lafayette Indiana 47907 The application of nonlinear optical Stokes ellipsometry (NOSE) coupled with principal component analysis (PCA) is demonstrated for the chemically selective analysis of molecular monolayers. NOSE allows for rapid polarization measurements of nonlinear optical materials and thin surface films, which in turn benefits from comparably fast data analysis approaches. PCA combined with linear curve fitting techniques greatly reduce the analysis time relative to nonlinear curve fitting. NOSE-PCA is first validated with studies of z-cut quartz, followed by analysis of four thin dye films with similar nonlinear optical properties. The high precision of NOSE measurements combined with the rapid analysis time enabled chemical discrimination between different dyes and the practical realization of NOSE microscopy. The recent development of nonlinear optical Stokes ellipsometry (NOSE)1,2 as a precise yet fast nonlinear optical ellipsometry (NOE) method has enabled NOE to expand into formerly unfeasible applications. In previous studies, 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.1-7 However, those previous methods, as well as related approaches for polarization analysis, have all relied on the physical movement of optical elements in order to perform the analysis. The time required to physically rotate the appropriate optical elements ultimately dictated the fastest analysis time possible in a given technique. Such long acquisition times have limited NOE analyses to systems exhibiting excellent photostability and made NOE imaging unfeasible. Nonlinear optical imaging has become a major tool in the biological sciences and other fields. For example, second harmonic generation (SHG) provides structural information based on its polarization depen* Corresponding author. E-mail:
[email protected]. (1) Begue, N. J.; Everly, R. M.; Hall, V. J.; Haupert, L.; Simpson, G. J. J. Phys. Chem. C 2009, 113, 10166–10175. (2) Begue, N. J.; Moad, A. J.; Simpson, G. J. J. Phys. Chem. C 2009, 113, 10158– 10165. (3) Dehen, C. J.; Everly, R. M.; Plocinik, R.; Simpson, G. J. Rev. Sci. Instrum. 2007, 78, 013106. (4) Moad, A. J.; Moad, C. W.; Perry, J. M.; Wampler, R. D.; Begue, N. J.; Shen, T.; Goeken, G. S.; Heiland, R.; Simpson, G. J. J. Comput. Chem. 2007, 28, 1996–2002. (5) Plocinik, R. M.; Everly, R. M.; Moad, A. J.; Simpson, G. J. Phys. Rev. B 2005, 72, 125409. (6) Plocinik, R. M.; Simpson, G. J. Anal. Chim. Acta 2003, 496, 133–142. (7) Polizzi, M. A.; Plocinik, R. M.; Simpson, G. J. J. Am. Chem. Soc. 2004, 126, 5001–5007. 10.1021/ac901832u 2010 American Chemical Society Published on Web 12/15/2009
dence. The primary contrast mechanism has been simple changes in the intensity of the signal (generally through increased number density or orientation of the chromophore).8-13 Adding NOSE analysis to this or other nonlinear optical imaging techniques may add additional information content. With the development of NOSE, meaningful polarizationdependent data can be acquired in as little as 20 µs, such that the bottleneck is shifted from data acquisition to data analysis. Reliable determination of the χ(2) tensor elements describing the nonlinear optical properties of the sample is currently performed by nonlinear curve fitting, which can take up to several minutes per measurement when using multiple guess values to screen against false minima (i.e., 3-5 orders of magnitude longer than typical data acquisition times). Depending on the application, it may not be necessary to or even desirable to calculate the complex-valued χ(2) tensor elements for every data point. For example, in applications such as imaging, mapping the changes in χ(2) as a function of position in each pixel may be excessive if image contrast based on differences in nonlinear optical properties is the primary objective. Even with the neglect of the time-constraints on analysis, subtle differences in nonlinear optical properties might not be easily identified by inspection of the resulting set of χ(2) tensor elements if those differences collectively span multiple tensor elements, which will often generally be the case. Complementary methods for rapid analysis of polarizationdependent nonlinear optical data could extend the range of applications. In the past 2 decades, chemometrics has been established as a set of statistical methods for mining useful information from highly dimensional data and for identifying relationships between subsets of data and physical properties.14-23 Principal component analysis (PCA) is particularly attractive for (8) Zoumi, A.; Lu, X.; Kassab, G. S.; Tromberg, B. J. Biophys. J. 2004, 87, 2778–2786. (9) Zipfel, W. R.; Williams, R. M.; Webb, W. W. Nat. Biotechnol. 2003, 21, 1369–1377. (10) Zipfel, W. R.; Williams, R. M.; Christie, R.; Nikitin, A. Y.; Hyman, B. T.; Webb, W. W. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 7075–7080. (11) Plotnikov, S.; Juneja, V.; Isaacson, A. B.; Mohler, W. A.; Campagnola, P. J. Biophys. J. 2006, 90, 328–339. (12) Zoumi, A.; Yeh, A.; Tromberg, B. J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 11014–11019. (13) Schenke-Layland, K.; Riemann, I.; Damour, O.; Stock, U. A.; Koenig, K. Adv. Drug Delivery Rev. 2006, 58, 878–896. (14) Cramer, J. A.; Morris, R. E.; Giordano, B.; Rose-Pehrsson, S. L. Energy Fuels 2009, 23, 894–902. (15) Johnson, K. J.; Parkinson, J. A.; Young, D. C.; Synovec, R. E. J. Sep. Sci. 2004, 27, 410–416.
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the analysis of NOSE data, as it provides unbiased analysis without requiring training sets. In PCA, high-dimensional data sets are transformed into a new set of uncorrelated, orthogonal coordinates constructed from a linear combination of the previously correlated variables. By maximization of the variance in each subsequent coordinate, PCA can effectively reduce the dimensionality of a data set. This is accomplished by a relatively simple procedure of determining the eigenvectors and eigenvalues of the covariance matrix of the mean-centered data. The high information content intrinsic in the NOSE measurements in combination with straightforward, fast linear curve-fitting allows for PCA without requiring detailed optical modeling or iterative nonlinear curve fitting. Additionally, since the eigenvectors illustrate the weighting of each original coordinate on the new coordinates, the two systems are easily relatable maintaining the detailed structural information inherent in the raw NOSE data. In the present study, analysis of NOSE data by PCA is described theoretically, validated experimentally with quartz, and used to discriminate chemical identity in dye films. The basic framework is sufficiently general to enable applications in studies of numerous thin films and materials of both high and low symmetry. THEORY The theoretical foundations for NOSE for uniaxial thin films have been discussed in detail in a previous publication.2 As an alternative to rigorous nonlinear curve fitting to the full tensor, analytical expressions for the polarization dependent intensity were derived for a uniaxial, achiral sample. In such a system only three unique χ(2) tensor elements are nonzero in the Jones tensor representation;5 χppp, χpss, and χsps (where χsps ) χssp due to the interchangeability of the two incident photons). The small number of nonzero unique tensor elements present in these initial studies provides a good test of the approach. In chiral uniaxial films, for example, two more, complex-valued, Jones tensor elements are nonzero resulting in an additional four parameters capable of contributing toward discrimination between chemical species.24-27 In principle, expansion of the analytical expressions to use all the independent Jones tensor elements allows complete generalization to systems of arbitrary symmetry. (16) Pedroso, M. P.; de Godoy, L. A. F.; Ferreira, E. C.; Poppi, R. J.; Augusto, F. J. Chromatogr., A 2008, 1201, 176–182. (17) Arey, J. S.; Nelson, R. K.; Xu, L.; Reddy, C. M. Anal. Chem. 2005, 77, 7172–7182. (18) Prazen, B. J.; Johnson, K. J.; Weber, A.; Synovec, R. E. Anal. Chem. 2009, 73, 5677–5682. (19) Morris, R. E.; Hammond, M. H.; Cramer, J. A.; Johnson, K. J.; Giordano, B. C.; Kramer, K. E.; Rose-Pehrsson, S. L. Energy Fuels 2009, 23, 1610– 1618. (20) Morris, R. E.; Hammond, M. H.; Shaffer, R. E.; Gardner, W. P.; RosePehrsson, S. L. Energy Fuels 2004, 18, 485–489. (21) Kramer, K. E.; Morris, R. E.; Rose-Pehrsson, S. L.; Cramer, J.; Johnson, K. J. Energy Fuels 2008, 22, 523–534. (22) Johnson, K. J.; Morris, R. E.; Rose-Pehrsson, S. L. Energy Fuels 2006, 20, 727–733. (23) Cramer, J. A.; Morris, R. E.; Hammond, M. H.; Rose-Pehrsson, S. L. Energy Fuels 2009, 23, 1132–1133. (24) Haupert, L. M.; Simpson, G. J. Annu. Rev. Phys. Chem. 2009, 60, 345–365. (25) Kauranen, M.; Verbiest, T.; Maki, J. J.; Persoons, A. J. Chem. Phys. 1994, 101, 8193–8199. (26) Ostroverkhov, V.; Ostroverkhov, O.; Petschek, R. G.; Singer, K. D.; Sukhomlinova, L.; Tweig, R. J. IEEE. J. Sel. Top. Quantum Electron. 2001, 7, 781–792. (27) Petralli-Mallow, T.; Wong, T. M.; Byers, J. D.; Yee, H. I.; Hicks, J. M. J. Phys. Chem. 1993, 97, 1383–1388.
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Generalization of the previous expressions for uniaxial and achiral thin films allows straightforward derivation of the Jones 2ω vector for the electric field measured at the detector Edet (2) generated with all six possible nonzero independent, Jones χ tensor elements present in a given SHG measurement (eq 1).
[
2ω Edet ) ei∆
[
]
M11 M12 M21 M22
( ∆2 ) - 2iχ ∆ sin ( ) - 2iχ 2
-χppp sin2 -χspp
psp
2
ssp
( ∆2 )cos( ∆2 ) + χ ∆ ∆ sin( )cos( ) + χ 2 2 sin
pss
( ∆2 ) ∆ cos ( ) 2 cos2 2
sss
]
(1)
A general detection optics matrix, M, is included in eq 1 to allow for the collective effects of polarization-dependent optical elements placed between the sample and the detector (e.g., wave-plates, dichroic mirror, angled windows, beam splitters, etc.). Simplification of eq 1 results in the following general expression for the polarization dependent intensity (eq 2).
( ∆2 ) + B cos ( ∆2 )sin( ∆2 ) + C cos ( ∆2 )sin ( ∆2 ) + ∆ ∆ ∆ D cos( )sin ( ) + E sin ( ) (2) 2 2 2
2ω Idet ∝ A cos4
3
2
2
3
4
where ∆ is the time dependent retardance of the PEM, ∆ ) 2πλ sin(ωt), λ is the retardance setting of the PEM (typically 0.5 for half-wave operation) and ω is the angular frequency of the PEM. Relationships between the five coefficients and the Jones tensor elements can be obtained by combining the relationships in eqs 2ω 2ω 2 1 and 2, together with the relationship Idet ∝ |Edet |.
|∑ 2
An )
Mnmχmpp
[( ∑
|
2
)( ∑ )] | ∑ | [( ∑ )( ∑ [( ∑ )( ∑ )]
m)1
2
Bn ) -4 Im
2
Mnmχmpp
m)1
2
Cn ) 4
Mnmχmsp
m)1
2
Mnmχmsp
2
- 2 Re
m)1
m)1
2
Dn ) 4 Im En )
|
2
Mnmχmss
m)1
2
∑M
m)1
2
Mnmχmpp
nmχmss
|
m)1
)] *
Mnmχmss
Mnmχmsp
m)1
2
(3) In eq 3, the summation is performed with indices 1 and 2 corresponding to p and s-polarized light, respectively. For example, for the A coefficient on the s-polarized detector would result in the eq 4 for As. As ) |M21χppp + M22χspp|2
(4)
In turn, the Jones tensor elements are given by linear combinations of Cartesian χ(2) tensor elements, with the coefficients related to the thin film optical constants and experimental geometry.1,2,5 It is important to note that the intensity measured at a given detector is linearly dependent on the A-E coefficients in eq 2 but nonlinearly dependent on both the Jones and Cartesian tensor
Figure 1. Schematic of the optical path of the simplified 2PMT NOSE instrument (the solid beam is the fundamental beam and the dashed beam is the SH beam). VB, visible absorbing filter; HWP, half-wave plate; GLP, glan laser polarizer; PEM, photoelastic modulator; L, planoconvex lens; S, sample; SFS, sample filter stack; DFS, detector filter stack; PMT, photomultiplier tube.
elements. Fits of the intensity-dependent data using the either the Jones and Cartesian χ(2) tensor elements as fitting parameters generally requires iterative searches on a high-dimensional surface to find the global minimum of χ2-space, corresponding the most probable values for the coefficients. This approach can be susceptible to false minima or “troughs” in χ2-space, for which multiple solutions are of comparable probability, even in such simple systems as achiral thin films.1,2 However, linear fits, such as in eq 2, are deterministic, such that a single, unique result is obtained from simple matrix inversion, requiring no guess values or iterative searching. As such, linear fits are highly desirable when short data-analysis times are required. Once the coefficients are obtained, the Jones tensor elements can still be determined from numerical combinations of the coefficients.2 METHODS AND MATERIALS The NOSE instrument has been described previously.1,2 For this work, a simplified, 2PMT configuration of the instrument was used (see Figure 1). In short, the polarization state of a Ti:sapphire laser (Spectra-Physics, Mai Tai HP, 800 nm, 80 MHz, >2.5 W for dye films and 300 mW for quartz) was modulated by a photoelastic modulator (Hinds, PEM-90, 50 kHz). The incident beam was softly focused on the sample using a 50 mm focal length planoconvex lens, and the signal beam was recollimated using a matching lens. The simplified 2PMT NOSE setup was used for all experiments, and no waveplate was placed in the detection side of the instrument to maximize the linear independence of the analytical fitting coefficients in eq 2. Imaging was performed using a pair of automated translation stages (Newport, VP-25XA). The intensity of the signal beam was detected on a pair of PMTs (Burle 83112502), amplified using fast preamplifiers (Ortec, 9305) and collected using time-gated photon counting cards (Becker and Hickl GmbH, PMS-400A). The intensity of the signal was monitored as a function of PEM state using a custom Labview application. The total counts obtained for all 50 000 cycles of the PEM were summed to generate a complete data set. Before data processing, each detector’s signal was corrected for dark counts. Thin film dye samples were prepared by dip coating from 0.220 mM methylene blue (MB, Matheson Colman and Bell), 0.103 mM bromocresol green (BG, Sigma-Aldrich), 0.136 mM solution of crystal violet (CV, Sigma-Aldrich), and 0.0954 mM allura red (AR, Aldrich) all in methanol (Mallinckrodt, ChromAR). Data were collected at a 45° of incidence with the imaging being performed
by sample scanning, with 0.5 mm steps between pixels, 9 steps per line and 9 lines to generate a data set containing 81 NOSE traces for each dye. An additional CV sample was prepared for NOSE imaging. A 120 × 120 pixel CV image was acquired with a 4PMT NOSE configuration.1 All chemicals were used as received, and all glassware was cleaned with piranha (3:1 sulfuric acid/ 20% hydrogen peroxide) prior to use. The z-cut quartz sample was obtained from Boston Piezo-Optics and used as received. Data analysis was performed in several steps to compare the performance of total intensity, polarization dependent intensity, analytical coefficient, and principal component images. The total intensity image was developed by summing over all incident polarizations and detectors to obtain the total counts on each detector for each pixel, while the polarization dependent images were obtained by extracting a single 250 ns data bin from the NOSE trace. To generate images based on the analytical coefficients, the polarization dependent intensity traces for each PMT at each pixel were fit to the analytical expression, eq 2, resulting in two five element vectors containing the fitting coefficients. Like the polarization dependent traces, the single coefficient images were obtained by generating images using only the value of one fitting coefficient. Mean-centered PCA was performed using the intensity normalized coefficients. PCA for the dye films was performed on the combined data set from all four dye films, normalized by intensity. RESULTS Z-Cut Quartz. Prior to studies of more complex samples, comparisons of different analysis approaches were performed for z-cut quartz, for which the nonlinear optical properties are well established and strictly dictated by symmetry.28 Two approaches to determine the rotation angle of a single crystal of z-cut quartz interrogated along the z-axis through NOSE analysis were performed and the results compared. The first was a nonlinear curve fitting approach, where the χ(2) tensor elements were held fixed to the symmetry defined values expected from z-cut quartz, and the orientation parameters were used as the fitting coefficients. The second approach generated a calibration curve in principal component space for the crystal rotation angle. For z-cut quartz interrogated along the z-axis, three nonzero Jones tensor elements exist and are related as -χppp ) χpss ) χsps1,29 when positioned with the crystallographic x-axis coparallel with the p-polarization plane. The rotation angle of the z-cut quartz was determined by extending the previous NOSE nonlinear curve fitting methodologies. The crystal tensor can be rotated into the laboratory frame by eq 5. f (2) χlab (θ, φ, ψ)
f
(2) ) [RM(θ) X RM(φ) X RM(ψ)] · χxtal
(5)
For the specific case of z-cut quartz interrogated along the crystallographic z-axis, θ ) 0 and φ and ψ describe equivalent rotation operations. Consequently, determination of the crystal rotation angle can be accomplished by nonlinear curve fitting to just two parameters, the rotation angle, φ, and an instrument sensitivity factor, a, to account for the overall intensity and the (28) Shen, Y. R. The Principles of Nonlinear Optics; John Wiley & Sons: New York, 1984. (29) Boyd, R. W. Nonlinear Optics; Academic Press: San Diego, CA, 2003.
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instrument response function, as shown in eq 6. The value of χ2 defined in eq 6 is minimized in the nonlinear curve-fitting approach used to determine the two parameters φ and a. (2) χ2 ≡ |F C - A|MexitingEincidentaχFlab (φ)|2|2
(6)
b is a vector of signal counts, A is a matrix defining the where C calibrated relative PMT sensitivities, Mexiting is the Jones matrix describing the polarizing optics after the sample, and Eincident is a matrix composed of the different products of the Jones vector elements describing the incident polarization states. Further details on this expression can be found in a pair of previous publications.1,2 Least-squares fits for each rotation angle of z-cut quartz were performed 250 times using initial guess values generated randomly from a uniform probability distribution function (PDF) with limits of 0° e φ e 30° and 0 e a e 1. Because of the C3 symmetry of quartz, one can only determine the nominal rotation angle away from the crystallographic x-axis, which repeats every 30° in the measured intensity. Multiple fits were performed to test the convergence performance of the nonlinear curve fitting. The experimentally determined angle was taken as the mean of the resulting fit values. Further simplification of the data set was achieved by averaging the symmetrically equivalent rotation angles as shown in Figure 5a. The error bars represent 1 standard deviation from 12 mean fit values, with the exception of 0° and 30°, which use 6 equivalent angles. Linear fits to the analytical expressions (eq 2) resulted in a single solution for each experimental rotation angle. However, determination of crystal orientation based on individual coefficients from such fits was still problematic. An example of the analytical fit results is shown in Figure 2b for the Dp coefficient. Because of the D3 symmetry, the intensity should be periodic, repeating every 60°. A simplified equation for the angle dependent intensity is given in eq 7. I ) A cos(υφ + δ)
(7)
where the three fitting parameters are A, the amplitude, υ, the repeat period, and δ, a phase term, and the variable φ, the rotation angle. While the periodic nature of the rotation is evident in the data, fitting to the Dp coefficients predicts a repeat period of 67.52(1)° (error is estimated by the inverse Hessian of the fit, representing a lower limit on the error), and the quality of the fit is rather poor. Similarly, fits to the other nine coefficients give comparably inaccurate results for the repeat period (see Table 1). For comparison, the orientation of the crystal was determined by PCA of the analytical coefficients reported above, the results of which are shown in Figure 2c. To minimize the effect of laser power fluctuations, the analytical fit coefficients were normalized by the total photon count for each rotation angle. Application of mean centered PCA to the 10-dimensional 2PMT NOSE data set resulted in a first principal component (PC1) that accounted for greater than 98% of the variance in the data. This analysis approach show remarkable agreement with the theoretical response given by eq 7. Fits to PC1 resulted in a repeat period of 59.98(8)° with an amplitude of 0.035(5) and δ ) -1(1)°, in good agreement with expectations based on symmetry. 562
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Figure 2. Experimental results for determination of the rotation angle of z-cut quartz: (a) calculated angles from nonlinear curve fitting, (b) calculated angles for the Dp analytical coefficient, and (c) first principal component (PC1) of the analytical fit.
Monolayer Films. A total of 81 separate NOSE measurements were performed for each of four dye films, the results of which were analyzed by PCA. Representative NOSE traces are shown in Figure 3. The four dyes studied were achiral, “Λ-like”, charge transfer dyes, which likely adopt broad orientation distributions, such that the polarization dependent responses are reasonably expected to be quite similar. Consistent with these expectations, only subtle differences were observed between the data sets, corroborated by the large correlation coefficients (greater than 0.9946 for any dye pair using intensity-normalized data). From visual inspection alone, the differences in the polarization-dependent data are quite subtle, including small changes in the crossover for the detected p- and s-polarized intensities for s-polarized incident polarization. The χ(2) tensors determined for these four films are also quite similar as shown in Table 2. The calculated values of |χppp/χsps| agree within error for the allura red and
Table 1. Results of Fitting Analytical Coefficients of Z-Cut Quartz Data to Equation 7
a
coefficient
A
υ (deg)
δ (deg)
χ2/105
PCA coefficient
Ap Bp Cp Dp Ep As Bs Cs Ds Es PC1a
442.33(5) -22.39(5) 827.51(5) -25.26(5) 442.49(5) 246.34(5) -14.57(5) 502.63(5) -8.57(5) 246.22(5) 34.91(5)
60.062(7) 67.56(1) 59.621(3) 67.52(1) 60.0905(7) 59.899(1) 66.48(2) 60.2624(6) 68.60(3) 59.900(1) 59.989(8)
-0.49(1) 37.5(3) 3.023(7) 43.9(2) -2.44(1) 0.49(2) 16.7(3) -4.43(1) 9.7(7) -0.70(2) -1.1(2)
21.1 13.8 514 13.9 20.0 5.36 4.99 179 3.78 4.73 0.002 01
-0.086 7 -0.032 86 0.689 554 -0.037 45 -0.0869 0.088 551 0.014 392 -0.700 68 0.013 262 0.088 214
Values of PC1 were scaled before fitting by a factor of 103 to bring the amplitude factor, A, into the range of the other coefficients.
Figure 3. Example NOSE traces for several thin films of achiral, Λ-like, charge transfer dyes. Table 2. Calculated χ(2) Tensor Elements for Four Thin Film Dyes
a
film
|χppp|a
|χpss|a
allura red bromocresol green crystal violet methylene blue
2.34(6) 2.68(9) 2.29(1) 2.77(5)
0.58(4) 0.7(1) 0.48(1) 0.93(3)
Tensor elements normalized by χsps.
crystal violet films while the bromocresol green and methylene blue films also agree within error. Both pairs also show similar values for |χpss/χsps|. Enabled by the reduced analysis time from PCA, microscopy measurements of monolayer films were performed with full NOSE characterization at each pixel. Figure 4a shows the total intensity image generated by integrating the intensities over all incident
polarizations for a glass slide dip coated in a solution of CV. The intensity image is dominated by a bright vertical line that denotes the initial air/methanol interface of the dipping solution. By comparison, the corresponding image of PC1 (Figure 4b) exhibits contrast across the entire field of view, attributed to structural differences in the dye film. DISCUSSION Z-Cut Quartz. Although the nonlinear least-squares fitting approach for quartz was performed with just two parameters (quartz rotation angle and instrument sensitivity), the fits still produced systematic errors, overestimating the nominal crystal rotation angles for small angles and underestimating them for large angles. This trend can be explained by inspection of χ2space. Figure 5b shows χ2-space for quartz rotated at φ ) 30° (relative to one of the three degenerate principal crystallographic axes), where the nonlinear curve fitting was Analytical Chemistry, Vol. 82, No. 2, January 15, 2010
563
Figure 4. Images of dip coated thin film of crystal violet. False color images of the (a) total integrated intensity image and (b) an image of the first principal component (PC1).
the least accurate. In the crystal rotation dimension, χ2-space is virtually flat with fits of comparable quality based on the values of χ2 obtained for rotation angles of φ ranging over ±10°. The underestimation of φ near 0° and 30° is an artifact of the PDF used to generate the guess values. While values of φ ) 31° and φ ) 29° would yield identical fits, as they are symmetrically equivalent angles (nomillay the same), limiting the guess values to φ e 30° effectively removes larger angles from the gradient search routine. At 15° there is an equal likelihood of a guess value either greater than or less than the true value resulting in a mean fit value unbiased by the PDF used to generate the initial guess values. All other angles show similar behavior, with the effect intensified for the values of φ farther from the center of the PDF. By comparison, the calibration of the quartz rotation angle by PCA of the analytical coefficients produced unbiased fits in excellent agreement with eq 7. Interestingly, PC1, used in the calibration, is constructed primarily from contributions from Cp and Cs, which independently gave the poorest fits (see Table 564
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Figure 5. Comparison of the nominal rotation angle of a quartz single crystal (as determined by independent calibration) and the values obtained by least-squares fitting (a) and by PCA (c). (a) Calculated NOSE angles generated by comparatively time-consuming iterative least-squares minimization, averaged over symmetrically equivalent angles. (b) Graphical representation of χ2 space for a rotation angle of φ ) 30° that defines the least-squares minimization surface, indicating a virtually flat parameter space that in turn complicates accurate determination of the fitting parameters (i.e., the quartz rotation angle and the photodetector sensitivity). (c) Calibration curve for quartz rotation angle by PCA, averaged over symmetrically equivalent angles. The error bars in both parts a and c represent 1 standard deviation of the replicate data (N ) 6).
2) in terms of the value of χ2. Yet a new ordinate based largely on these two analytical coefficients results in a curve that yields a value of χ2 nearly 5 orders of magnitude smaller. Monolayer Films. While the raw polarization traces of the dye films shown in Figure 3 and the corresponding χ(2) tensor elements are qualitatively similar for the four dye films, fitting the NOSE data to the analytical expressions (eq 2) reveals subtle differences. As shown in Figure 6a, separation arises in coefficient space for the A and B coefficients for the p-polarized detector. For this particular pair of coefficients, good separation is observed
Figure 6. Separation of dye populations in data component space. (a) Separation based on Ap and Bp coefficients, one of 45 possible combinations of coefficient pairs. (b) Separation of dyes in principal component space, plotting PC2 vs PC1.
for AR and MB while BG and CV overlap in this two-dimensional slice of the 10 dimensional coefficient space. Ap vs Bp is only one of 45 possible pairs of analytical coefficients that can be chosen, and it does not necessarily provide the best or worse separation of the chemical populations. In contrast, PCA maximizes the variance along each principal component, which allows the rapid screening of data by focusing only on the first few coefficients with the greatest information content. For this particular case, the first two principal components (PC1 and PC2) account for 94.5% of the data variance, which increases to 99.5% if the third principal component is included. Separation of the four dyes in principle component space is shown in Figure 6b, which demonstrates that the CV and BG are well separated in both principal component dimensions. Additionally, PC1 separates well the two pairs of dyes that were indistinguishable by their χ(2) tensor elements alone (i.e., χppp ≈ 2.3 for AR and CV and χppp ≈ 2.7 for BG and MB). Some outlying data points that do not group with their appropriate populations were observed for all dye films. In general,
the NOSE data for these points have either significantly larger photon fluxes than other locations (by at least an order of magnitude) and/or exhibit visually different behavior in the polarization dependent NOSE traces. These sample spots are tentatively attributed to inhomogeneities in the thin film samples (e.g., from aggregation, local structure in the glass slides, etc.). Consistent with the explanation for the outliers in the four monolayer films, subtle inhomogeneities in the polarizationdependent responses from the dip-coated films were clearly present in the mean-centered NOSE-PCA image of a CV film shown in Figure 4. Interestingly, the major differences appear to be dependent on the location along the dipping axis, potentially arising from changes in solvent concentration from slow evaporation during dipping. More importantly for the present purposes, the differences responsible for contrast in the PC1 image arise from relatively minor changes in the polarization-dependent NLO response across the slide. However, the high precision of NOSE analysis combined with the intrinsic preference of PCA for Analytical Chemistry, Vol. 82, No. 2, January 15, 2010
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selecting the significant contrasting features of information-rich data sets clearly allows discrimination of these subtle differences. CONCLUSIONS Principal component analysis of nonlinear optical Stokes ellipsometry measurements was used to discriminate between different chemical species with similar nonlinear optical properties. With the use of linear curve-fitting of the polarization-dependent data instead of nonlinear curve-fitting, the analysis time was substantially reduced. Fast data analysis combined with fast data acquisition allows for the incorporation of NOSE measurements in microscopy applications. These proof-of-concept studies provide a base for subsequent NOSE microscopy measurements of more complex systems. When compared to previous nonlinear optical ellipsometry approaches based on rotating optical elements,3,5-7,30-36 NOSE-PCA provides substantial improvement in three key areas: (i) data acquisition time, (ii) speed of data analysis, and (iii) discrimination between qualitatively similar data sets. As demonstrated in previous work,1,2 NOSE provides a ∼6 order of magnitude improvement in data acquisition time compared to previous NOE measurements using rotating optical elements (e.g., waveplates or polarizers), while simultaneously improving precision by an additional 1-2 orders of magnitude. Multiple NOSE (30) Canfield, B. K.; Laiho, K.; Kauranen, M. J. Opt. Soc. Am., B: Opt. Phys. 2007, 24, 1113–1121. (31) Cattaneo, S.; Kauranen, M. Phys. Rev. B 2005, 72, 033412. (32) Cattaneo, S.; Siltanen, M.; Kauranen, M. J. Nonlinear Opt. Phys. Mater. 2003, 12, 513–523. (33) Corn, R. M.; Higgins, D. A. Chem. Rev. 1994, 94, 107–125. (34) Eisert, F.; Dannenberger, O.; Buck, M. Phys. Rev. B 1998, 58, 10860. (35) Kauranen, M.; Elshocht, S. V.; Verbiest, T.; Persoons, A. J. Chem. Phys. 2000, 112, 1497. (36) Maki, J. J.; Kauranen, M.; Persoons, A. Phys. Rev. B: Condens. Matter 1995, 51, 1425–1434.
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measurements can be acquired in rapid succession, greatly increasing the information content of polarization-dependent SHG measurements. PCA of the NOSE data provides a simple means for data analysis on timeframes commensurate with the data acquisition rates. For the particular example of the CV image presented here, full nonlinear curve fitting of each of the 14 400 pixels/NOSE traces in the 120 × 120 image would have taken approximately 2 weeks of continuous processing on a standard desktop computer and can be prone to inaccuracies from local minima in χ2 space. Expanding to a 512 × 512 image would increase the data analysis time to greater than 8 months. With the use of the same computer, the presented images were generated in less than 5 minutes, consisting of linear fits of each data set to obtain the analytical coefficients and subsequently performing PCA. Finally, PCA has the distinct advantage of preferentially emphasizing the key differences between data sets to maximize their separation, which is particularly useful for interpretation of data with greater similarities than differences such as those included in this study. ACKNOWLEDGMENT The authors gratefully acknowledge financial support from the National Science Foundation (Grants NSF-CHE-0640549 and NSFMRI-ID-0722558). The authors additionally thank the Jonathan Amy Facility for Chemical Instrumentation at Purdue for their help in developing the LabView data acquisition software and consultation on instrument design and thank the Laser Facility at Purdue University’s Chemistry Department for laboratory space and support. Received for review August 14, 2009. Accepted November 24, 2009. AC901832U
