Nonlinear Optical Stokes Ellipsometry. 1. Theoretical Framework - The

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Nonlinear Optical Stokes Ellipsometry. 1. Theoretical Framework Nathan J. Begue, Andrew J. Moad, and Garth J. Simpson* Department of Chemistry, Purdue UniVersity, 560 OVal DriVe, West Lafayette, Indiana 47907 ReceiVed: December 3, 2008; ReVised Manuscript ReceiVed: March 31, 2009

The theoretical framework for nonlinear optical Stokes ellipsometry (NOSE) is developed as a faster alternative to previous nonlinear optical ellipsometry (NOE) techniques. NOE is the determination of the complexvalued elements within the Jones χ(n) tensor, from which all experimental observables in a given configuration can be predicted. By replacing the moving optics with a photoelastic modulator (PEM) operating at 50 kHz and employing a high repetition rate laser, full ellipsometric detection of the χ(2) tensor can in theory be performed in as little as 20 µs. Two complementary models were developed to analyze the proposed instrumental setup for NOSE. The first more general approach is based on a regression analysis method, which was compared to a second explicit analytical model valid for a particular experimental configuration. Additionally, the rigor of the regression analysis method against noise was investigated. I. Introduction Second harmonic generation (SHG) and sum frequency generation (SFG) (i.e., frequency doubling and frequency mixing, respectively) have developed into powerful tools for characterizing oriented thin films, surfaces, and interfaces.1-17 Because SHG and SFG disappear in centrosymmetric media, they provide exquisite surface selectivity. Furthermore, the nonlinear optical (NLO) nature of the wave-mixing processes typically results in the generation of coherent signal beams 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 theoretically and experimentally 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 as well as related approaches for polarization analysis have all relied on the physical movement of optical elements to perform the analysis, either by rotation or by translation. The time required to physically rotate 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. The primary focus of this work is to lay the theoretical foundations for a rapid NOE polarization analysis approach, with the goal to enable NOE microscopy measurements and NOE measurements of photolabile systems. NOE is defined as the experimental determination of the χ(n) Jones tensor describing the n + 1 wave-mixing process. NOE is a generalization of traditional, linear ellipsometry in which changes in polarization state upon reflection can routinely be used to detect changes in average film thickness of less than a single atomic layer. This sensitivity arises primarily from the detection of phase changes arising between the complex reflection coefficients for p-polarized and s-polarized light. In linear ellipsometry, the changes in amplitude and phase can be expressed concisely by a phenomenological (2 × 2) Jones matrix χ(1), which is effectively a transfer matrix describing the output polarization generated for any given input polarization.18 All * Corresponding author. E-mail: [email protected].

experimental observables in a given ellipsometry experiment can be predicted from precise knowledge of the Jones matrix. Once determined experimentally, ratios of elements within the Jones matrix can be subsequently related back to surface properties (e.g., film thickness, refractive index, etc.) by application of an appropriate thin film model in a separate step. Separation of the model-independent measurement of the Jones matrix with the model-dependent interpretation in terms of thin film properties allows the same data set to be fit by multiple different models. The formal mathematical framework for describing linear ellipsometry within the context of Jones matrices can be extended to NLO measurements of Jones tensors. By analogy with traditional linear ellipsometry, which yields the complexvalued elements of the χ(1) 2 × 2 Jones matrix describing the change in polarization upon reflection or transmission,19 NOE directly probes the 2 × 2 × 2 Jones tensor χ(2) (typically expressed with respect to p- and s-polarizations). Although providing a substantial improvement in information content when compared to conventional methods for NLO polarization analysis, previous methods for NOE have suffered from relatively long acquisition times. In nonlinear optical null ellipsometry (NONE), analysis times exceeding an hour were required to determine the complex-valued χ(2) Jones tensor elements of achiral surface monolayers to 2-3 significant figures, with the analysis time nearly doubling for chiral films.20 This time can be reduced to 20-30 min by performing rotating quarter waveplate ellipsometry (RQ-NOE),21 rotating analyzer ellipsometry (RA-NOE),21 or rotating half-waveplate ellipsometry (RH-NOE).21 More recently, discrete-retardance nonlinear optical ellipsometry (DR-NOE)22 has further reduced the acquisition time down to a few minutes for monolayer dye films. However, these previous approaches have generally exhibited a trade-off between precision in the measurements and the acquisition times, such that χ(2) tensor element ratios of dye films could be obtained to ( ∼20% in measurements performed in 16 min.22 In related methods, Kauranen and co-workers have developed several SHG techniques (both single beam and dual beam) for polarization analysis employing a total least-squares method to simultaneously fit multiple, independent sets of experimental results.23-26 Additional theoretical studies on

10.1021/jp810643n CCC: $40.75  2009 American Chemical Society Published on Web 05/18/2009

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Figure 1. Schematic of the optical path of the NOSE instrument (the solid beam is the fundamental beam, and the dashed beam is the SH beam). VB, visible absorbing filter; QWP, quarter-wave plate; HWP, half-wave plate; GLP, Glan laser polarizer; PEM, photoelastic modulator; L, planoconvex lens; S, sample; SFS, sample filter stack; BS, partially polarizing beamsplitter; DFS, detector filter stack; PMT, photomultiplier tube.

polarization effects in NLO have been performed by Persoons and others.27-35 In all of these previous collective measurements, the long acquisition times have restricted analysis largely to stable monolayer or multilayer films, which generate sufficient signals and photostability to be amenable to such relatively long exposures to the incident beam. However, the broader utility of NOE would be significantly expanded by reduction of the analysis time down to less than a second, which would facilitate studies of the photolabile species and NOE microscopy measurements with full ellipsometric characterization. In this work, the theoretical foundation for a new NOE approach is described, in which rapid polarization modulation of the incident beam is combined with Stokes ellipsometric detection of the NLO polarization. Nonlinear optical Stokes ellipsometry (NOSE) has the potential to significantly reduce the NOE analysis time and/or improve the precision with which NOE measurements are performed. The mathematical framework for interpreting NOSE measurements is described in this work. Experimental validation of NOSE in measurements of dye monolayer films is described in a companion paper (DOI 10.1021/jp9013543). Although the primary focus of the NOSE technique is on SHG, it is also directly compatible with other NLO processes (e.g., coherent anti-Stokes Raman scattering or vibrational SFG). II. Theory A framework for NOE performed by the physical rotation of optical elements has been described previously and can serve as a basis for descriptions of NOSE.20,21,27,36-45 The key instrumental departure of the NOSE technique from previous NOE techniques is the introduction of a photoelastic modulator (PEM) for rapid modulation of the incident polarization state and the use of a Stokes ellipsometric detection configuration with no moving parts.46,47 Consistent with the experimental conditions described in a companion paper, the fundamental beam was assumed in all calculations to be produced by a Ti:Sapphire laser operating at ∼800 nm. The four-detector beam path used in the calculations consisted of a horizontally polarized light passing through a PEM; the polarization modified beam is then introduced to the sample, then to a partially polarizing beam-splitting cube, and finally through either a quarter-wave plate (QWP) or half-wave plate (HWP) as indicated in Figure 1. In an alternative simplified two-detector beam path, the second harmonic beam generated by the sample is assumed to pass through a QWP rotated at an angle γ and a polarizer.

Figure 2. Effect of the PEM on the polarization of light as a function of time for one complete cycle assuming horizontally polarized light incident upon the PEM at +45°. Adapted from the Hinds Instrument PEM-90 manual.

The intensity of the SH beam detected by the PMTs is a function of the PEM state. The PEM can be described mathematically as a wave plate at a fixed angle with a timevarying retardance (in this case, positioned at an angle of 45° and modulated sinusoidally between (λ/2), as illustrated in Figure 2. Consistent with the measurements described in the companion paper, the raw data consist of intensities measured by two or four photodetectors as functions of the incident polarization states acquired during the PEM cycle. The primary objective of NOE is to combine the experimental data to uniquely determine the complex-valued elements of the χ(2) tensor. In NOSE, this inversion is performed with no moving parts, with the PEM serving as the only time-varying element in the measurement. In this configuration, the PEM period (ca. 20 µs) ultimately dictates the lower limit on the acquisition time for complete polarization analysis. Two approaches were taken for describing NOSE measurements, a more general approach based on global least-squared regression analysis and complementary analytical expressions for NOSE performed in a twophotodetector configuration. Predictions of representative NOSE measurements for ubiquitous molecular symmetries in the weak orientation limit were considered to assess the strengths and limitations of the approach under conditions likely to be encountered experimentally. II.A. Regression Analysis for Determination of χ(2). By analogy with linear optics,18 the exiting electric field in secondorder NLO measurements is related to the incident electric field and the NLO χ(2) Jones tensor χJ(2) through eq 1.

b e ω3 ) χJ(2):e bω1b e ω2

(1)

The vectors b eω1 and b eω2 represent the Jones vectors describing eω2 for SHG), the polarization of the incident beams (e bω1 ) b

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and b eω3 represents the Jones vector of the nonlinear beam (ω3 ) ω1 + ω2). Equation 1 holds for any second-order NLO process exhibiting weak scattering (such as IR-visible SFG and SHG). In partially scattering media, this Jones tensor approach can be extended through the use of Mueller matrices and Stokes vectors by an analogy with linear optics.18 The experimental second-order NOE measurements directly probe the generalized 2 × 2 × 2 χJ(2) tensor elements, expressed with respect to p- and s-polarizations in eq 1.21 In eq 2, the complex-valued χJ(2) tensor elements are written in vector form to facilitate subsequent manipulations, where T indicates the transpose of the vector.

F χJ(2) ) [χppp χpps χpsp χpss χspp χsps χssp χsss ]T

(2) Sample symmetry can reduce the number of independent and nonzero tensor elements in the Jones tensor. For example, SHG of an uniaxially oriented, achiral thin film results in the following relationships between the χJ(2) tensor elements: χsps ) χssp and χpps ) χpsp ) χspp ) χsss ) 0.21,36 In this case, only three independent and nonzero χJ(2) tensor elements remain, χppp, χpss, and χsps.21,36 In this vectorized format, the polarization states of the exiting beam (e bω3) from the sample can be related back to the four eω2 polarization state of the possible permutations of b eω1, the b incident beam and the χJ(2) tensor through eq 3 where ep and es represent electric field of the p- and s-polarized components of the fundamental beam.

b e ω3 )

[

]

epep epes esep eses 0 0 0 0 F · χJ e e e e e e e 0 0 0 0 p p p s s p ses (3)

Although this reformulation of eq 1 seems initially to be unnecessarily complicated, expressing the exiting polarization state in this form has advantages when combining data from measurements performed with many different incident polarization states as in NOSE. In this case, the 2 × 1 vector describing Eexiting, which describes b eω3 can be replaced by a 2n × 1 vector b the exiting polarization state for a combination of n different incident polarizations probed.

b Eexiting ) Eincident ·F χJ(2)

(4)

The b Eexiting vector contains 2n elements and describes the polarization of the SH beam directly after the sample with the top n elements describing the p-polarized component of the electric field and the bottom n elements describing the spolarized components of the electric field. In this representation the first element of b Eexiting and the n + 1 element completely describe the complex polarization of the first polarization state probed. The Eincident matrix consists of 8 columns and 2n rows (eq 5) and describes the polarization state of the fundamental beam directly before the sample.

Eincident

[

epep epes esep eses 0 0 0 0 l l l l l l l l ) 0 0 0 0 epep epes esep eses l l l l l l l l

]

(5)

The Jones χJ(2) tensor elements are related to the surface Cartesian χC(2) tensor elements expressed with respect to the Cartesian coordinate system with x, y, and z, through eq 6. The S matrix (eq 7) is composed of s coefficients comprised of

Fresnel factors and geometric projection terms used to describe the sample configuration. Explicit expressions for the Fresnel factors and s coefficients can be found in several previous publications.20,21,36,40

[

F χJ(2) ) S ·F χC(2)

szzz szxx sxzx sxxz 0 0 0 -sxzy szxy 0 0 0 0 0 sxyz -szyx 0 0 0 0 0 0 szyy 0 0 0 0 0 S) -s s 0 0 0 0 0 yxz yzx 0 0 0 0 0 0 syyz 0 0 syzy 0 0 0 0 0 0 0 0 0 0 0

]

(6)

(7)

As with the Jones χJ(2) tensor elements, symmetry properties in a uniaxial thin film reduce the number of independent and nonzero Cartesian χC(2) tensor elements from 27 possible elements down to only eight (eq 8). If the film is achiral, the last three elements in eq 8 are also zero-valued.

F χC(2) ) [χzzz χzxx χxzx χxxz χxyz χxyz χyzx χzxy ]T

(8) The detected photon counts (i.e., intensity of the SH beam) can be related to the polarization state of the electric field immediately following the sample by taking into account the effect of the polarizing optics between the sample and the detectors using traditional Jones matrices.19 The Jones matrix for a rotated optical element is given by R(-θ) · M(0°) · R(θ), where R(θ) is the rotation matrix for a rotation of θ degrees and M(0°) is the Jones matrix for an optical element at an angle of 0° (where all angles are relative to the laser table surface normal). In order for these matrices to be compatible with the large data set, the standard 2 × 2 Jones matrix representation for wave plates and other polarizing optics must be expanded. For example, by taking the direct product of the Jones matrices for a QWP at +45° with an n × n identity matrix, ID, the expanded Jones matrix is obtained (eq 9).

{ ( )[ ] ( )}

Mexiting ) R -

π π 1 0 R 4 0 -i 4

X ID

(9)

If multiple polarizing elements are in the beam path, an effective Jones matrix can be determined and then expanded by taking the direct product as shown in eq 10 for the case of transmission through a partially polarizing beam splitter followed by a QWP at +45° where tp and ts are the transmission coefficients for pand s-polarized light, respectively.

{ ( )[ ] ( )[ ]}

Mexiting ) R -

π tp 0 π 1 0 R 4 0 -i 4 0 ts

X ID

(10)

The real and imaginary components of the χC(2) tensor elements are determined from the experimental measurements by minimizing χ2 (the sum of the squared residuals, not to be confused with χ(2)) (eq 11), where the χC(2) tensor elements are treated as the free parameters in the minimization.

b - A|Mexiting · Eincident · S ·F χ2 ) |C χC(2) | 2 | 2

(11)

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. The elements in A are determined prior to performing an experiment and are fixed throughout the fitting. If the χJ(2) tensor elements are

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desired, then the equation given in eq 12 can be used to fit the experimental data.

b - A|Mexiting · Eincident ·F χ2 ) |C χJ(2) | 2 | 2

(12)

In a two-photodetector configuration, the partially polarizing beam splitter is removed from the beam path, which is equivalent to setting the Jones matrix for the beam splitter equal to the identity matrix in transmission and zero-valued in reflection. The data analysis and minimization are otherwise identical. II.B. Analytical Expressions for χJ(2) for Two-Detector NOSE. Insights into trends in the results obtained by the global regression analysis described by eqs 11 and 12 can be obtained by explicitly deriving expressions for the χ(2) tensor elements in terms of the phase retardance of the PEM and the rotation angle of the QWP in two-detector NOSE. Although analogous expressions for four-detector NOSE can also be derived, they are significantly less concise (and are presented in Appendix II). For the specific case of a 2PMT setup with a detection QWP at γ, horizontally polarized (s-polarized) light into the PEM at +45° and for a uniaxial, achiral thin film, the following analytical expressions can be derived for the intensity detected on the vertically polarized detector, eq 13.

( ∆2 ) + B cos ( ∆2 ) sin( ∆2 ) + ∆ ∆ ∆ ∆ ∆ C cos ( ) sin ( ) + D cos( ) sin ( ) + E sin ( ) 2 2 2 2 2

2ω Ip-det ∝ A cos4 2

3

2

3

4

1 (2) 2 |χ | [1 + cos2(2γ)] 2 pss (2) (2) (2) sin(2γ)[Im(χpss ) cos(2γ) - Re(χpss )] B ) 2χsps

A )

C ) D ) E )

systems were chosen largely because of their ubiquity in both surface and materials applications of second-order nonlinear optics. The ensemble-averaged χC(2) tensor of a thin surface layer depends both on the molecular tensor and on the orientation distribution of the molecules at the surface. Rather than assume a particular form for the orientation distribution in the simulations, calculations were performed for a uniaxial system in the weak order limit. In this limit, which is routinely observed experimentally both in poled polymer films and surface monolayers, the first Legendre moment of the orientation distribution is by definition much larger than the higher moments. Under these conditions, χzzz = χzxx + χxzx + χxxz from which χzxx/χxxz ) 1 for rod-like chromophores (βzzz-dominated) and χzxx/χxxz ) -2/3 for Λ-like chromophores (βxxz-dominated). In addition to these resonant contributions, there was also assumed to be a nonresonant background, phase-shifted 90° (i.e., purely real and positive) with a magnitude equal to 0.1 × |χxxz|, chosen arbitrarily. Generation of the 2 × 2 × 2 Jones tensor χJ(2) from the 3 × 3 × 3 Cartesian χC(2) was achieved using established expressions for the Fresnel factors36 assuming measurements performed in reflection at the air/glass interface for an incident angle of +45° for an average film thickness of 2 nm and an incident wavelength of 800 nm. The nonzero elements of the χJ(2) tensor are reported in Table 1 where all values have been normalized to χsps. To facilitate direct comparisons with the anticipated experimental data, plots of simulated data are presented as a function of incident polarization (or equivalently as a function of PEM cycle time) and not a function of PEM retardance (∆). III. Results and Discussion

(2) 2 (2)* 2 2(χsps ) [1 - cos2(2γ)] - Re[χ(2) ppp χpss ][1 + cos (2γ)] (2) (2) sin(2γ)[-Im(χ(2) 2χsps ppp) cos(2γ) + Re(χppp)]

1 (2) 2 |χ | [1 + cos2(2γ)] 2 ppp

(13) where ∆ is the time-dependent retardance of the PEM, ∆ ) 2πλ 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. Additional information on this derivation and the derivation of the s-polarized intensity can be found in Appendix I. Analogous expressions can be derived for the s-polarized detected intensity, where the general expression remains the same, but in eq 13, B and D must be multiplied by -1; additionally, in A, C, and D, each 1 ( cos2(2γ) is replaced by 1 - cos2(2γ). Consequently, only the C coefficient on the second detector supplies additional unique information, although the quality of the fit may be improved by using the combined data set. II.C. Predictions for Model Systems. The Jones tensors used in the simulations were generated for two different assumed molecular tensors (β(2)): one in which the β(2) tensor was assumed to be dominated by the βzzz tensor element (characteristic of “rod-like” charge-transfer chromophores) and another in which the βxxz ) βxzx elements dominate the Cartesian tensor (characteristic of “Λ-like” chromophores). These two model

Figure 3 contains the results of 250 independent fits using the regression method for both Λ-like and rod-like chromophores assuming no detection arm waveplate (or equivalently, a QWP at γ ) 0°). Each guess value resulted in a unique solution, all equidistant from the origin. From inspection of the figure, the regression method appears to only recover the magnitude of the χJ(2) tensor elements. The results of Figure 3 can be understood by setting γ ) 0° in eq 13. A, C, and E are the only surviving fitting coefficients, with A and E generating the squared magnitudes of χppp and χpss, respectively, and insufficient complementary information in C to allow definitive determination of the sign/phase relationships among the three nonzero elements. The incorporation of a rotated QWP prior to the final polarizer significantly changes the predicted results of the fits, as shown in Figure 4. A small number of solutions produced fits of comparable quality. In the case of the 2PMT configuration with the QWP rotated +45°, four solutions emerge with approximately equal probability in which one solution yields the true values for the tensor elements and a second solution predicts the incorrect sign of the imaginary component of χppp and χpss. Rotation of the QWP to smaller or larger angles produces a general preference for solutions close to the actual results used to generate the data sets, but false minima still yield incorrect tensor elements in ∼50% of the trials. As in the previous case, setting γ equal to 45° provides

TABLE 1: Non-zero Elements of the χ(2) Tensor Used To Generate the Simulation Dataa Λ-like rod-like a

χppp

|χppp|

χpss

|χpss|

χsps

-1.170 - 0.649i 1.438 + 0.112i

1.338 1.442

-0.678 - 0.503i 1.219 + 0.623i

0.844 0.1369

1.00 1.00

Each tensor element is normalized to χsps, which is assumed to be purely real.

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Figure 3. Plots of the fits to theoretical NOSE data with no detection arm waveplate. Left: data predicted assuming a Λ-like χ(2) tensor. Right: data assuming a rod-like χ(2) tensor. Top: example fits from one of the 250 trials. Bottom: results of 250 trial fits using randomly generated initial guess values.

minimum for all guess values used. While the general intensity expression (eq 13) is the same for 2- and 4PMT NOSE, the complexity introduced by the complex-valued transmission and reflection coefficients results in significantly more complex analytical expressions for the coefficients (which are presented in Appendix II), but the excellent stability in the algorithm is not entirely unexpected. Treating the beamsplitter Jones matrix as an identity matrix and assuming equal sensitivities for all four PMTs results in the following expressions for χpss:

Figure 4. Summary of results for 250 fits for several instrumental setups assuming a Λ-like chromophore. Black indicates the occurrences of the correct solution, light gray for correct magnitudes but with the incorrect sign for the imaginary component of the χ(2) tensor elements, and dark gray for incorrect solutions.

insights into these trends. In the analytical expressions (eq 13), the imaginary component of χppp and χpss is multiplied by a factor of sin(2γ) which is zero for 45°. Thus the real component and the absolute magnitude of the tensor elements are still constrained resulting in sign ambiguity on the imaginary component. QWP angles that maximize the value of the C coefficient, such as 15° or 75°, should minimize this risk. In contrast to the 2PMT configurations, calculations performed for the 4PMT instrumental designs converged to the correct

[

]

1 Bp1 cos(2γ2) sin(2γ2) - Bp2 sin(2γ1) cos(2γ1) × 2 sin(2γ1) sin(2γ2) cos(2γ1) cos(2γ2) -1 1cos(2γ1) B sin(2γ ) B cos(2γ2) -1 s1 2 s2 sin(2γ1) 1 (2) 1Im[χpss ]) 2 sin(2γ1) sin(2γ2) cos(2γ2) cos(2γ1)

(2) Re[χpss ])

[

(

)

](

)

(14)

where the subscripts 1 and 2 indicate the two separate detection arms of the 4PMT instrumental setup. Analogous expressions for χppp are found by simple substitution of D for the B coefficients. For the case of a perfect, nonpolarizing, 50:50 beamsplitter with both QWPs at the same angle, eq 14 goes to zero, which is expected as no new information is gained from the second detection arm. More realistic beamsplitter behavior would result in a weighting of fitting coefficients between the two detection arms as they are sensitive to the intensity of the signal. The complexity of 4PMT expressions which treat the transmission and reflection coefficients as complex values preclude the derivation of concise expressions for the individual tensor elements, but the use of four photodetectors in the

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Figure 5. Demonstration of the resilience of the 4-PMT NOSE regression method against noise. (a) Example of a fit to a Λ-like chromophore with an SNR of 10 and its corresponding fit. (b) Relative standard deviation of 25 fits to theoretical data with unique noise for several SNRs.

configuration shown allows for Stokes ellipsometry. This provides for the unique determination of the complete polarization state of the NLO beam immediately after the sample by the collective intensities measured at the detectors.18,46-48 Influence of Noise. In all fits previously described, only the guess values were altered using noise-free simulated data. For the model systems described, the signal is expected to be sufficiently weak as to benefit from the detection of the SH signal by photon counting. To characterize the NOSE regression algorithm in the presence of counting noise, hypothetical data for a Λ-like chromophore were generated by transforming the noise-free data into a probability distribution function normalized to give a photon count rate of one signal photon for every 100 laser pulses (or 16 photon counts per sweep assuming 20 µs sweeps and 80 MHz laser repletion rate). The data set was built up on a per laser pulse basis thus ensuring Poisson distributed noise. An example of a noisy 4PMT data set for a signal-to-noise ratio (SNR) of 10 and its corresponding fit are presented in Figure 5a. Errors were calculated by fitting 25 uniquely generated data sets with increasing numbers of sweeps (from 25 to 100 000 sweeps) to yield SNRs of between 3 and 200, and the results are summarized in Figure 5b. As shown, the error rapidly approaches zero, with even a modest SNR of 10, resulting in uncertainty in the tensor elements of less than (0.03 (at the 95% confidence interval). Even in the low SNR limit the NOSE algorithm attains the correct tensor elements. For example, five sweeps (less than seven counts for any PEM state) resulted in a SNR of 1.4, χppp ) -1.21 ( 0.16 - 0.56 ( 0.24i, and χpss ) -0.67 ( 0.13 - 0.45 ( 0.18i. Precise tensor elements to three decimal places are predicted for a SNR of 20 (about 1000 sweeps in this example). In higher signal-to-noise regimes, Poisson noise incurred in photon counting will likely be subsumed by systematic errors in instrumentation, such as the finite bandwidth of the waveplates or the nonideal behavior of the PEM. While the number of sweeps to attain a given SNR will change on the basis of the signal count rate, the rapid convergence of the error with increasing SNR should be maintained.

of model chromophores illustrated that this simplified, 2PMT, NOSE system is prone to give two solutions for the real and imaginary components of the χ(2) tensor elements without some a priori knowledge of the NLO properties of the system being studied. With sufficient knowledge of the sample it may be possible to reject one of the solutions on the basis of symmetry or other arguments. Despite this trade-off, the simplicity of the instrumental setup and the high degree of accuracy predicted in the absolute magnitudes of the χJ(2) tensor elements make this an attractive NOE method. Lastly, the addition of a second detection arm, which includes two additional detectors, consistently provides unique solutions for the complex-valued χJ(2) tensor elements. The primary advantages of incorporating a PEM and a high repetition rate laser into a NLO instrument are the drastic reduction of the measurement time and the decrease in 1/f noise. For the case of a 50 kHz PEM and a 80 MHz repetition rate (typical of fs Ti:Sapphire lasers), a single 20 µs sweep of the PEM sufficiently samples the possible incident polarization states. In the case of RQ-NOE with the QWP spinning continuously at 5°/s, a single sweep would take over a minute. Even if several thousand sweeps of the PEM are necessary to achieve the desired SNR, the NOSE setup is faster than traditional techniques by several orders of magnitude. This drastic reduction in analysis time opens the door to several new avenues of exploration. Specifically, full polarization analysis of dynamic systems is now theoretically possible as is the incorporation of detailed polarization analysis into NLO microscopy. Additionally, the resulting values for the χJ(2) tensor elements acquired from theoretically noisy data are 1-2 orders of magnitude more precise than previously developed NOE techniques.20,21,40,45

IV. Conclusions

In deriving eq 13 for a particular experimental configuration, the polarization state of the beam incident upon the sample was first considered. The beam was assumed to be generated by passing horizontally polarized light through a PEM, which was assumed to be rotated +45° relative to the laser table surface normal. The signal polarization state was calculated mathematically by rotating the frame of reference for the PEM and then ω , as multiplying by the Jones vector for the incident beam, Ein shown in eq A.1.

The theoretical framework for NOSE has been presented and predicted to have several advantages over previous NOE techniques. This regression technique is sufficiently general to be easily adaptable to many different instrument configurations, allowing for a high degree of instrumental flexibility. To aid in the interpretation of the general regression algorithm, analytical expressions were developed for a specific 2PMT instrumental configuration. While potentially useful, the simulated studies

Acknowledgment. The authors gratefully acknowledge financial support from the National Science Foundation (NSFCHE-0640549, NSF-MRI-ID-0722558). Appendix I: Analytical Expression for 2PMT NOSE

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]()

[

( π4 )e

Eω ) R -

i∆/2

e-i∆/2 0 π ω R Ein i∆/2 4 0 e

[ ]

) ei∆/2

Begue et al.

Bp/s

(2) 2 ) [1 - cos2(2γ)] Cp/s ) 2(χsps

∆ 2 ∆ cos 2

() ()

-i sin

[

ω 2 (2) (2) ω ω (2) ω 2 χ(2) ppp(Ep ) + (χpps + χpsp)Ep Es + χpss(Es ) (2) (2) (2) (2) ω 2 χspp (Eωp )2 + (χsps + χpsp )Eωp Esω + χsss (Es )

[

) ei∆

( ∆2 ) + χ cos ( ∆2 ) ∆ ∆ -2iχ sin( ) cos( ) 2 2

2 -χ(2) ppp sin

(2) pss

2

(2) sps

]

]

[

∆ ∆ (2) + χpss cos2 2 2 ∆ ∆ (2) cos sin 2γ 2iχsps sin 2 2 (2) (2) 2 ∆ 2 ∆ sin 2γ -χppp sin + χpss cos + 2 2 ∆ ∆ (2) sin cos (cos 2γ - i) 2iχsps 2 2

(

(

()

()

( )) ( ( ) ( )) ( )) ( ( ) ( ))

Cp/s

( ∆2 ) + B cos ( ∆2 ) sin( ∆2 )+ ∆ ∆ ∆ ∆ cos ( ) sin ( ) + D cos( ) sin ( ) + E 2 2 2 2 2

]

3

p/s

tp 0 2ω 1 0 R(γ) E 0 -i 0 ts

(A.6)

Expanding eq A.6 results in an expression for the electric field with 110 terms for each polarization component. Collecting the terms by their trigonometric power series of ∆ results in the same general expression for the intensity on each PMT as shown in eq A.4 but with significantly less consise expressions than the B and D coefficients (eqs A.7a and A.7b). Expressions for the detection beam reflected from the beamsplitter are obtained by replacing tp and ts with rp and rs, respectively.

1 (2) 2 2 |χ | |t | (1 ( cos2 2γ) 2 pss p (2) ) [(Rs(1 + cos2 2γ) ( βs(1 - cos2 2γ)]χsps sin 2γ

(2) 2 ) |ts | 2(1 - cos2 2γ) + Cp/s ) 2(χsps

(A.3)

3

2

[ ] [ ]

Bp/s

2ω 2ω 2 Ip/s ∝ |Ep/s | p/s

The derivation of analytical expression for 4PMT NOSE is equivalent to that of 2PMT NOSE with the exception that the Jones matrix for the beamsplitter must be inserted between the signal source, E2ω, and the rotated Jones matrix for the waveplate as shown in eq A.6, where tp and ts are the complex-valued transmission coefficients for the beamsplitter. Expressions for the reflection detection arm are equivalent upon symbolic substitution of reflection coefficients for the transmission coefficients.

Ap/s )

Lastly, the beam is separated into its orthogonal polarization components by a polarizer, and the PMTs detect the intensity, which is proportional to the squared magnitude of the electric field as shown in eq A.4 for the two separated polarizations.

2ω Ip/s ) Ap/s cos4

Appendix II: Analytical Expression for 4PMT NOSE

(A.2)

1+i × 2

2 (cos 2γ - i) -χ(2) ppp sin

(2) (2) sin(2γ)[-Im(χ(2) Dp/s ) (2χsps ppp) cos(2γ) + Re(χppp)] 1 Ep/s ) |χ(2) | 2[1 ( cos2(2γ)] (A.5) 2 ppp

2ω Edet ) R(-γ)

Once the second harmonic signal has been predicted directly after the sample, it can be propagated though any polarizing optics. In a similar manner to eq A.1 but for the case of a QWP at γ0, the detected electric field is given by eq A.3.

2ω Edet ) ei∆

(2)* 2 Re[χ(2) ppp χpss ][1 ( cos (2γ)]

(A.1)

In eq A.1, R(γ) is the 2 × 2 rotation matrix, and ∆ is the timedependent retardance of the PEM, given by ∆ ) 2πλ sin(ωt). To predict the electric field of the second harmonic signal, one can apply the second-order NLO susceptibility. In the case of a uniaxial, achiral thin film only three independent, nonzero (2) is assumed to be purely real, in elements remain, of which χsps eq A.2.

E2ω )

1 (2) 2 |χ | [1 ( cos2(2γ)] 2 pss (2) (2) (2) ) (2χsps sin(2γ)[Im(χpss ) cos(2γ) - Re(χpss )]

Ap/s )

(A.4) 4 ∆ p/s sin 2

()

While the general equations for the p- and s-detectors are equivalent, slight differences are seen in the expressions for the coefficients as shown in eq A.5 (where the upper sign corresponds to the p-polarized detector and the lower sign corresponds to the s-polarized detector).

(2)* 2 2 Re[χ(2) ppp χpss ]|tp | (1 ( cos 2γ) (2) sin 2γ Dp/s ) [(Rp(1 + cos2 2γ) ( βp(1 - cos2 2γ)]χsps 1 (2) 2 2 2 (A.7a) Ep/s ) |χppp | |tp | (1 ( cos 2γ) 2

where R and β are linear combinations of the real and imaginary components of the χ(2) tensor elements and the transmission/ reflection coefficients of the beamsplitter as shown in eq A.7b.

Rx ) (∆p xx pR - ∑p xx pI)sR + (∑p xx pR + ∆p xx pI)sI βx ) (∑p xx pR + ∆p xx pI)sR - (∆p xx pR - ∑p xx pI)sI

(A.7b)

For simplicity, ∑pxx is the sum and ∆pxx is the difference between Re[χpxx] and Im[χpxx], and the R and I subscripts represent the real and imaginary component of the p- and s-component of the beamsplitter coefficients (either transmission or reflection). References and Notes (1) Allen, H. C.; Raymond, E. A.; Richmond, G. L. Curr. Opin. Colloid Interface Sci. 2000, 5, 74–80. (2) Bain, C. D. Curr. Opin. Colloid Interface Sci. 1998, 3, 287–292. (3) Buck, M.; Himmelhaus, M. J. Vac. Sci. Technol., A 2001, 19, 2717– 2736. (4) Chang, Y.-M.; Xu, L.; Tom, H. W. K. Chem. Phys. 2000, 251, 283–308. (5) Corn, R. M.; Higgins, D. A. Chem. ReV. 1994, 94, 107–125. (6) Eisenthal, K. B. Annu. ReV. Phys. Chem. 1992, 43, 627–626. (7) Eisenthal, K. B. Acc. Chem. Res. 1993, 26, 636–643. (8) Eisenthal, K. B. J. Phys. Chem. 1996, 100, 12997–13006.

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