Schematic diagram of NOE instrument

Development of Nonlinear Optical Stokes Ellipsometry (NOSE)

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.^1-13 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.^{14, 15} The schematic of a SHG instrument with the incorporated NOSE detection technique is presented in Figure 1 and detailed descriptions can be found in a pair of publications.^{16, 17}

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; 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.

The development of NOSE marked a radical departure in data analysis methodologies employed for the interpretation of NOE data.¹⁶  Previous NOE methods used analytical expressions to fit the polarization dependant signal and calculate the complex polarization state of the signal beam (ρ, Eq. 1).

                                                                         (1)

 The χ⁽²⁾ tensor elements were then determined by mathematically combining ρ values for several incident polarization states. For example:

                                                                                                       (2)

In contrast, 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. 3.

                                                                               (3)

Where  represents the adjusted number of photon counts (dark counts subtracted) and A is a calibration matrix correcting for the difference in sensitivities of the detectors. M_exiting is an expanded Jones matrix for the detection arm polarizing optics and E_incident is a matrix containing the set of possible two photon polarization permutations for each PEM state and  is a vectorized representation of Jones χ_J⁽²⁾ tensor. While the Jones representation of light polarization has been around since the 40’s, the expanded Jones matrix formulation is a novel adaptation which enables the simultaneous analysis of large data sets. This analysis method uses the multiplex advantage to attain highly precise measurements from noisy data. Previous fast NOE methods achieved a precision of ~20% relative error after 15 minutes of data acquisition. In contrast, data for 5 sweeps of the PEM (100 ms of data acquisition) with a signal-to-noise ratio of 1.4 resulted in equivalently precise measurements. Even moderate data acquisition times (on the order of one second) reduce the error in the NOSE measurements to less than one percent.

Experimental Results

            Validation of the NOSE technique was performed on a sample of z-cut quartz and a thin film of disperse yellow 7 (DY7) dye. While both systems are SHG active, they represent vastly different samples in sample parameter space. As a non-centrosymetric crystal, quartz generates a large response from the three dimensional crystal and also has a high damage threshold. Additionally quartz has been previously studied and with minimal resonance with the SHG transition the χ⁽²⁾ tensor elements should be purely real valued. In contrast, DY7 provides a two dimensional monolayer and have complex valued χ⁽²⁾ tensor elements due to its electronic resonance at the double excitation frequency. To aid in interpretation of the data, the second detection arm of the NOSE instrument was removed resulting in a 2 PMT instrument as detailed in the NOSE theory paper.¹⁶

The results of the NOSE analysis showed remarkable agreement with the results acquired from the RQ-NOE method, previously developed in our lab,² Figure 2. For quartz, the χ⁽²⁾ tensor elements were determined with up to four significant figures of precision and agreed with the expected values required by symmetry. Additionally, up to four significant figures of precision were attained for the thin films of DY7.

Figure 2. Compairison to RQ-NOE.  The complex polarization of the signal beam r is plotted as a function of the incident polarization, where the Re[ρ] and the Im[ρ] are represented by closed and open shapes, respectively. a) Results for z-cut quartz, b) results for DY7 and c) blow up of the elliptically polarized region for DY7.  The large uncertainties for the s- and p-polarized incident polarization are a mathematical artifact as r approaches ±∞ for these polarizations.

In linear ellipsometry, a common method for reporting ρ is not as a complex number but as a magnitude (tanψ) and a phase angle (Δ) as shown in Eq. 4.

                                                                                                               (4)

Notably, for the DY7 data the phase angle of the χ⁽²⁾ tensor elements were determined with six significant figures of precision, while the magnitude was only to three significant digits. This balance in precision is sensitive to the rotation angle of the detection waveplates such that the analyst can effectively tune the instrument to be more sensitive for magnitude or phase as demanded by individual experimental requirements. The average errors of NOSE provide a means of drawing comparisons with previously established NOE techniques. Table 1 summarizes the relative error for eight NOE methods. Comparison by the normalized error shows that NOSE exhibits a more than a seven order of magnitude improvement in performance over previous NOE techniques.

Table 1. Comparison of NOE methodologies.

Normalized Error in ρ weighted for differences in acquisition times and laser repetition rates. *Indicate methods that give direct access to χ⁽²⁾ without first measuring ρ.

(2)          Plocinik, R. M.; Everly, R. M.; Moad, A. J.; Simpson, G. J. Phys. Rev. B 2005, 72, 125409.