1. Introduction

The rapid development of applications for photonic switching has led to aggressive efforts to find materials with large optical nonlinearities [1–3]. To this end, many different types of materials have been extensively studied [4–8], with organic materials having been identified as being particularly promising due to their ease of processing and large nonlinear figures of merit [9–13]. Here we are primarily interested in Kerr-like nonlinear refraction, where the irradiance-dependent refractive index is given by n(I) = n₀ + n₂I where n₀ and n₂ are the linear and nonlinear refractive indices, respectively, and I is the irradiance. One of the quickest and most efficient means of characterizing organics has been to study their optical properties in solutions by using such methods as the Z-scan technique which can simultaneously measure nonlinear absorption (NLA) and nonlinear refraction (NLR) utilizing a single Gaussian beam [14]. In this technique, as shown in Fig. 1(a) , the sample is scanned along the axis of a focusing beam, while measuring the transmittance T(Z) through an aperture in the far field. As described in [14], NLR results in self-lensing, causing changes in the far field beam radius, so that the aperture transmittance is sensitive to NLR. Removal of the aperture causes the Z-scan to be only sensitive to NLA, in which case, we refer to the Z-scan as an “open-aperture” (OA) Z-scan, and the transmittance will be a symmetric function of Z, where Z = 0 is defined at the beam waist (shown in Fig. 1(b)) and q₀ = α₂I₀L describes the transmission loss due to two-photon absorption (2PA) where α₂ is the 2PA coefficient, I₀ is the peak irradiance, and L is the sample thickness as detailed in [14]. Usually, we normalize T(Z) to unity for the sample position far from the beam waist, where the irradiance is low and the nonlinear effects are negligible. In the absence of NLA, T(Z) for the closed-aperture (CA) scan will be an antisymmetric function of Z showing a peak in transmittance prior to focus and a valley after focus for a negative n₂, with the order of the peak and valley being reversed for a positive n₂. In the case where NLA coexists with NLR, the CA Z-scan exhibits features due to both NLA and NLR, as can be seen from the closed aperture scan in Fig. 1(b). However, it is found that dividing the OA scan by the CA scan (CA/OA) yields a result, that is similar to that which would have been obtained for pure NLR (see Fig. 1(b)). In addition to the sample arm for CA Z-scans, a reference arm (Fig. 1(a)) in the same geometry as the sample arm with a matching closed aperture is used to reduce the noise [15].

 

Fig. 1 (a) The Z-scan experimental apparatus and (b) Z-scan open aperture (OA, red dotted line) and closed aperture (CA, black solid line) signals along with division (CA/OA, blue dot-dash line) for Δϕ₀ = −0.5, S = 0.33, and q₀ = 0.3.

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In the case of pure NLR (or for the divided result when small NLA is present), the difference between the peak and valley of T(Z), ΔT_p-v is directly proportional to the induced nonlinear phase shift Δϕ₀ (for small Δϕ₀ [14]) and hence to n₂, i.e. from [14] ΔT_p-v ≈0.406(1-S)^0.25Δϕ₀, in the absence of linear absorption where S is the linear transmittance of the aperture and Δϕ₀ = k₀n₂I₀L, k₀ = 2π/λ is the free space wavenumber where λ is the wavelength. Typically, a ΔT_p-v ~1% corresponds to a nonlinearly induced phase shift in the center of the Gaussian beam of ~λ/250. This interferometric sensitivity can be understood by noting the input beam induces a phase mask in the sample near focus, which has the greatest nonlinear phase shift (Δϕ₀) at the center of the beam, while the wings of the beam propagate linearly through the sample (i.e. Δϕ₀ → 0). This induced phase mask in the sample near focus alters the propagation of the beam to the far field, causing the redistribution of the irradiance at the closed aperture, i.e. diffraction, which can be thought of as interference between the wings of the beam and the center of the beam.

When measuring the nonlinear properties of molecules in solution, the NLA of the solvent is usually small and determination of α₂ for the solute is not problematic. However this is not the case for NLR. Typically, the NLR per molecule of the solvent is much less than that of the solute, but the large density of solvent molecules yields a large net NLR that may dominate the signal due to the solute. Additionally, there is a contribution to the measured n₂ due to the cells used to hold the samples. In cases where the n₂ of the solute is small, large discrepancies can arise when reporting the nonlinearity of the solute since the NLR of the solvent and cells must be subtracted from that of the solution. Thus, the determination of solute nonlinearities in regions where the NLR is similar to or much smaller than the solvent or cells has been difficult.

To determine the NLR of a solute in solution, typically two sequential Z-scans are performed, one for the solution and one for the solvent. Each Z-scan curve is individually fit to determine the nonlinear refractive index for the solution, n_2,S and the solvent, n_2,V, separately. These values are then subtracted to yield the nonlinear refractive index of the solute, n_2,U. Literature values for solvents cannot be used since there is a wide range of values reported using different techniques. For example, Gong et al. [16] reported a value of 2.2 × 10⁻¹⁵ cm²/W at 820 nm for toluene using the optical Kerr effect, while Couris et al. [17] reported a value of 0.88 × 10⁻¹⁵ cm²/W and 1.3 × 10⁻¹⁵ cm²/W at 800 nm for toluene using spectral shearing interferometry and Z-scan, respectively. Subtracting a literature value for n_2,V from n_2,S could lead to varying values of n_2,U; therefore, it is essential to measure the solvent at each time a nonlinear measurement is made to avoid such large discrepancies.

Cases where n_2,U is much smaller than n_2,S are often of interest. For example, a molecule designed for use in thin films may have low solubility, but may need to be characterized in solution. Also, in characterizing the frequency dependence of the nonlinear response, there may be regions where the NLR changes sign, so that n_2,U is inevitably small. To resolve these problems, we note that a significant source of the noise in a closed aperture Z-scan is due to laser beam fluctuations. In subtracting two separate Z-scans, this noise cannot be eliminated because the noise in the two sequential scans is uncorrelated. However, if the two Z-scans (for solvent and solution, respectively) are taken simultaneously using identical Z-scan arms, as shown in Fig. 2 , much of the noise (energy fluctuations, pointing instabilities, etc…) is correlated between the two arms and hence this noise will be eliminated upon subtraction, leaving only the uncorrelated noise (detector amplifier noise, air currents, etc…). This assumes that the measured solution normalized transmission T_S(Z) is the sum of the solvent normalized transmission T_V(Z) and the solute normalized transmission T_U(Z) (see appendix). This expands upon the work of Ma et al. [15], where a reference arm with a focusing lens and aperture matched to the signal arm was used to increase the signal-to-noise ratio. While using a reference arm with no sample goes part way toward correlating the noise, including a solution reference in the second arm and doing Z-scan simultaneously (i.e. dual-arm Z-scan) does a considerably better job to further remove the correlated noise as confirmed in the experiments reported here.

 

Fig. 2 Schematic of dual-arm Z-scan. The items labeled CA and OA represent the closed aperture and open aperture detectors for each arm, respectively. The reference beam used for energy monitoring is not shown.

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In this work, we carefully prepare the dual Z-scan arms to be as similar to each other as possible by using matched optics and equal arm path lengths and pulse energies to give equal irradiances. This correlates the noise in both Z-scans due to laser pulse energy variation, beam pointing instability, etc., thus enhancing the signal-to-noise ratio (SNR) for determining n_2,U By performing simultaneous Z-scans on the two arms: one arm with the solution and the second arm with only the solvent, we find that we can accurately extract n_2,U from typically large solvent NLR signals. This allows us to determine solute nonlinearities up to an order of magnitude smaller than that without using this procedure.

2. Experimental setup

The experimental configuration is shown in Fig. 2. A Ti:Sapphire laser system (Clark-MXR, CPA 2110), with a pulse energy output of approximately 1 mJ and a pulse width of ~150 fs (FWHM), operating at a 1 kHz repetition rate, is used to pump an optical parametric generator/amplifier (Light Conversion, Ltd., TOPAS-C) which generates a wide range of wavelengths for nonlinear optical characterization. The output beam is passed through a half-wave plate and polarizer (not shown) to provide attenuation, then spatially filtered and collimated to obtain a Gaussian beam. Approximately 10% of the energy is split off by a beam sampler (not shown) used to monitor the input energy. The remaining 90% is evenly split between two arms (Arm A and Arm B) using a 50/50 dielectric beam splitter. On one of the arms (here arm B in Fig. 2), a continuously variable neutral density (ND) filter is introduced for energy equalization, which will be discussed later.

In order to ensure that the signal from the solvent can be properly removed, it is essential that the two arms be equalized. This requires that the pulse width, beam waist, pulse energy, relative Z-position of the two samples, aperture linear transmittance, and distance from each sample to its respective aperture are matched. In addition to matching the irradiance parameters, the equipment in each arm is also matched, i.e. photodetectors, sample cuvettes matched to within tight tolerances (path length of 1 mm, matched to within 1 micron) [18], and corresponding matched optics.

The pulse widths in both arms will be identical so long as the dispersion in the optical components is the same in each arm. This is typically not a problem for pulse widths greater than 100 fs (FWHM), as in our experiments. However, for shorter pulses path differences due to the beam splitter and the ND filter used to match the energies on both arms may cause pulse widths to differ. In such cases a compensator plate in one arm should be used to account for this.

Matching the beam waists is done by collimating the beam prior to the first beam splitter, using matched optics in each arm, and equalizing the arm path lengths. To ensure that signal fluctuations due to pointing instability of the laser are correlated between the arms, the number of mirror reflections beyond the 50/50 beam splitter shown in Fig. 2 is matched so that any asymmetries in the beam profile are clipped in the same manner by the apertures placed after the samples.

Once the beam waists and pulse widths along each arm have been equalized, the pulse energy can then be equalized. Attempts to do so using energy meters failed as the accuracy is not sufficient for our purposes. Instead this was accomplished by scanning two cells filled with the same solvent, preferably with large NLR, e.g. carbon disulfide (CS₂), and attenuating one of the beams using the continuously variable ND filter until ΔT_p-v on each arm is equal. To ensure that the two samples experience the same irradiance and noise simultaneously at all points along the Z-axis, the relative Z-position of the samples is adjusted until the CA signals from both arms lie directly on top of each other. When subtracting the two curves from each arm, the resultant signal should yield a curve whose nonlinear effects are minimized to below the noise floor as discussed in the conclusion. This can be checked by increasing the pulse energy and verifying that the resultant signal does not change. More information on the equalization methodology is given in the appendix.

Once the difference between CA signals of both arms has been minimized, the cells used for alignment are replaced with cells containing the solution in one arm and the solvent in the other arm. Following Fig. 3 , a scan is performed at a low irradiance (typically less than 2 GW/cm² in our experiments using 1 mm cells), where the nonlinear signal from either arm is less than the noise level, to correct for any remaining systematic differences in the arms due to linear effects such as cell mismatch (Fig. 3(a)). This is similar to the procedure of background subtraction described in [14]. The subtraction of the solvent CA signal T_V(Z) from the solution CA signal T_S(Z) at this low energy yields a signal that we refer to as the Low Energy Background LEB(Z) (Fig. 3(b)), which is the residual signal due to linear differences between the two arms. This is primarily due to variations in linear transmittance and/or inhomogeneities of the two cells. The LEB(Z) can be minimized by using a set of high quality matched cells whose manufacturing tolerances have been tightly controlled as mentioned previously.

 

Fig. 3 Procedure of processing dual-arm Z-scan data using the low-energy background (LEB(Z)) and the corresponding Z-scan curves of the solution and solvent at 695nm at (a) low energy (< 1 nJ) scan of a squaraine solution (SD-O 2405 [19]) in toluene at a concentration (C) of 47 μM and solvent and their (b) subtraction, (c) high energy (11 nJ, I₀ = 18 GW/cm²) scan of solution and solvent and their (d) subtraction, (e) direct comparison of (b) and (d), and (f) corrected solute signal and fit with Δϕ₀ = −0.06, q₀ = 0.03, using S = 0.33.

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To quantify the relative merits of the dual-arm vs. single-arm Z-scan techniques, we compare the fitting error associated with each. Considering only the error due to the fitting, the range of n₂ that can be fit for a given ΔT_p-v is dependent on the noise in the signal. Typically, our CA signal will have some amount of noise δ which can obscure the determination of ΔT_p-v. Thus the determination of n₂ is bound by some fitting error, Δn₂. For small signals where $n_2\propto \Delta T_{p-v}$ this amounts to

3. Results and discussion

To experimentally investigate this technique, we compare the use of single-arm and dual-arm Z-scan techniques to measure the nonlinear refraction of a squaraine molecule, SD-O 2405 [19], in toluene solution in a 1 mm quartz cuvette. We perform both single and dual-arm Z-scans on this solution at wavelengths from 695 nm to 920 nm, using pulses of duration 115 fs (FWHM) generated from a TOPAS optical parametric generator/amplifier (OPG/OPA), as described in Section 2. Typically, there are two different approaches to analyzing the data using sequential Z-scans of the solution and solvent. First we use the single-arm method of subtracting the CA Z-scan of the solvent, T_V(Z), from the CA Z-scan of the solution, T_S(V), independently and fitting the respective n_2,U value which we refer to as the subtract-fit method. Since we typically restrain our fits to within one standard deviation σ from the peak and the valley, we can write the error in ΔT_p-v as

Figure 4(a) shows sequential single-arm Z-scans of the solution at 695nm using a peak irradiance of 51 GW/cm². At this pulse energy (E) of 31 nJ and corresponding irradiance and concentration (C) of 47 μM, the differences between the Z-scan signals of the solvent and solution are just barely distinguishable, given the noise amplitude and thus, $\Delta T_{p-v,S}-\Delta T_{p-v,V}\approx \sqrt{{\delta }_S{}^2+{\delta }_V{}^2}$. Also, notice the level and uncorrelated nature of the noise between the two scans. Applying the high-pass filter to the individual single-arm Z-scans in Fig. 4(a) yields δ_S = 1.2% and δ_V = 1.3%. This translates to Δn_2,S = 0.10 × 10⁻¹⁵ cm²/W and Δn_2,V = 0.11 × 10⁻¹⁵ cm²/W, which yields Δn_2,U = 0.15 × 10⁻¹⁵ cm²/W using the irradiance parameters listed in Fig. 4(a). Subtracting T_V(Z) from T_S(Z) yields the signal shown in Fig. 4(b). Taking a high-pass filter of the curve in Fig. 4(b) yields the same result, δ = 1.8% corresponding to Δn_2,U = 0.16 × 10⁻¹⁵ cm²/W. Because of the uncorrelated nature and amplitude of the noise between the two Z-scans, the total signal of the solute is nearly the same as the noise so that n_2,U cannot be accurately determined. With the respective signal and noise values for this case, the single-arm Z-scan technique cannot accurately determine n_2,U, regardless of which of the two methods is used.

 

Fig. 4 (a) Sequential CA single-arm Z-scans of the solvent toluene (open red triangles) and the solution SD-O 2405 in toluene (closed black squares) at 695 nm where the concentration C = 47 μm, the pulse energy E = 31 nJ (I₀ = 51 GW/cm²) and (b) the subtraction of the solvent CA signal from the solution CA signal (open green squares); (c) Simultaneous CA dual-arm Z-scans of the solvent toluene (open red triangles) and the solution SD-O 2405 in toluene (closed black squares) at 695 nm using the same pulse and (d) the subtraction of the solvent CA signal from the solution CA signal after LEB(Z) subtraction (open green squares) and corresponding fit of both 2PA and NLR (solid blue line) with Δϕ₀ = −0.16, q₀ = 0.077, using S = 0.33.

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Next, we perform a dual-arm Z-scan with the same experimental parameters as in Fig. 4(a). Figure 4(c) shows T_V(Z) and T_S(Z) measured simultaneously using the dual-arm technique. Figure 4(d) shows the corrected CA signal of SD-O 2405 after solvent subtraction when using the dual-arm Z-scan technique. To fit both nonlinear parameters concurrently (also shown in Fig. 4(d)), we free-space propagate the electric field after the sample using a Huygens-Fresnel integral to an aperture placed in the far field [14]. Because most of the noise is correlated between the arms, the correlated noise is cancelled out and the SNR is increased, thus reducing the fitting error. In this case δ = 0.19% which corresponds to a Δn_2,U = 0.016 × 10⁻¹⁵ cm²/W, more than a 9 times reduction in Δn_2,U compared to the single-arm Z-scan and hence a 9.3 × enhancement in SNR when using the dual-arm technique. Again at this wavelength there is some NLA in addition to the NLR and the same comments as for Fig. 3 apply. Here α_2,U = 0.015 cm/GW.

Even in cases where the solute CA signal is readily apparent from the single-arm Z-scan, the dual-arm Z-scan technique is still advantageous compared to the single-arm technique. Figure 5(a) shows sequential single-arm Z-scans of the solution and solvent at a wavelength of 780 nm, irradiance of 88 GW / cm², and concentration of 0.60 mM, where the difference between the solvent and solution signals is large. Note that in our laser system the noise at different wavelengths can vary significantly as seen between Fig. 4 and Fig. 5. In this case $\Delta T_{p-v,S}-\Delta T_{p-v,V}>>\sqrt{{\delta }_S{}^2+{\delta }_V{}^2}$, thus n_2,U can be easily extracted from the noise. Taking a high-pass filter of T_S(Z) and T_V(Z) independently yields δ_S = 0.31% and δ_V = 0.32%, respectively. This translates into Δn_2,S = 0.017 × 10⁻¹⁵ cm²/W and into Δn_2,V = 0.018 × 10⁻¹⁵ cm²/W, yielding Δn_2,U = 0.025 × 10⁻¹⁵ cm²/W.

 

Fig. 5 (a) Sequential Z-scans of the solution SD-O 2405 in toluene (closed black squares) and solvent toluene (open red triangles) at 780 nm, E = 50 nJ (I₀ = 88 GW/cm²), C = 0.60 mM and (b) subtraction of solution and solvent CA Z-scan signals (open green squares) along with the OA Z-scan of the solution (closed black circles) and corresponding 2PA and CA fit (solid red and blue line, respectively) with α_2,U-Fit = 0.013 cm/GW and n_2,U-Fit = −0.35 × 10⁻¹⁵ cm²/W; (c) Dual arm Z-scans of solution (closed black squares) and solvent (open red triangles) taken simultaneoulsy; (d) Simultaneous subtraction of solution and solvent yielding solute signal (open green squares) and fit incorporating both 2PA and NLR (solid blue line) with Δϕ₀ = −0.25, q₀ = 0.11 using S = 0.33.

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Figure 5(b) shows the subtraction of T_V(Z) from T_S(Z) for the sequential single-arm Z-scans. Taking a high pass digital high pass filter of this subtraction yields δ = 0.42% which corresponds to Δn_2,U = 0.023 × 10⁻¹⁵ cm²/W. The fit gives n_2,U = −0.35 × 10⁻¹⁵ cm²/W.

Figure 5(c) shows dual-arm Z-scans of the solution and solvent using the same experimental parameters as in Fig. 5(a). Figure 5(d) shows T_U(Z) after subtraction of the curves in Fig. 5(c) and its associated LEB(Z). From this dual-arm subtraction, we find δ = 0.23% corresponding to Δn_2,U = 0.013 × 10⁻¹⁵ cm²/W. This gives a reduction in Δn_2,U by a factor of 1.9 compared to fitting the two single-arm Z-scans individually (Fig. 5(a)) and then subtracting their n₂ values, and a factor of 1.8 compared to fitting the subtraction of the two sequential scans as shown in Fig. 5(b). These reductions are approximately equal as expected, the only difference coming from the high-pass filtering of the data sets. At this wavelength the solution shows considerable 2PA. The OA Z-scan is shown in Fig. 5(b) along with the 2PA fit with q₀ = 0.11, or α_2,U = 0.013 cm/GW. Dividing this OA signal from the CA signal of the solution in Fig. 5(a) yields n_2,S and n_2,V of 1.05 × 10⁻¹⁵ cm²/W and 1.4 × 10⁻¹⁵ cm²/W for the solution and toluene, respectively. The difference in the values agrees with the n_2,U = −0.35 × 10⁻¹⁵ cm²/W (Δϕ₀ = −0.25) of the subtracted curve of Fig. 5(d), thus showing the agreement between measurements with the single-arm and dual-arm techniques when the signal is far enough above the noise. Correspondingly, the enhancement of SNR is 1.9 × for the dual-arm technique compared to the single-arm technique. Again, a two parameter fit to the data of Fig. 5(d) gives the same α_2,U as the OA Z-scan. Note that the noise level of the OA Z-scan is approximately the same as that of the dual-arm as shown in Figs. 5(b) and 5(d). We discuss this more in the conclusions.

Figures 6(a) and 6(b) show dual-arm Z-scans for the solvent and solution, respectively, at 880 nm with the same concentration of 0.6 mM. At this wavelength and irradiance, ΔT_p-v from the solvent and solution are similar. To illustrate how important it is to determine n_2,U by the subtract-fit method as opposed to the fit-subtract method, especially for small signals, we analyze this data as if these were single-arm Z-scans even though they were taken using the dual-arm technique. The best fit for n_2,V and n_2,S in Figs. 6(a) and 6(b) is 1.63 × 10⁻¹⁵ cm²/W and 1.47 × 10⁻¹⁵ cm²/W, respectively. Applying a high-pass filter to the data in Figs. 6(a) and 6(b) yields δ_V = 0.16% and δ_S = 0.16%, which corresponds to a Δn_2,V = 0.040 × 10⁻¹⁵ cm²/W and Δn_2,S = 0.040 × 10⁻¹⁵ cm²/W. Upper and lower bound fits for n₂ ± Δn₂ are also shown in Figs. 6(a) and 6(b). Given the noise values, it could be reasonable to report a value of n_2,V ± Δn_2,V or n_2,S ± Δn_2,S which would yield a range of values for n_2,U from −0.080 to −0.24 × 10⁻¹⁵ cm²/W when subtracting the extrema values, i.e. up to 50% error. For the case of simultaneous subtraction and then fitting the resultant signal (Fig. 6(c)), the best fit for n_2,U is −0.15 × 10⁻¹⁵ cm²/W. Taking a high-pass filter yields δ = 0.11% for T_U(Z) which corresponds to Δn_2,U = 0.026 × 10⁻¹⁵ cm²/W. The uncertainty is 17%, an improvement of nearly 3 times compared to fitting the Z-scan signals independently. Therefore, it is important to reduce the fitting error by fitting the subtraction rather than the solution and solvent independently specifically in cases where ΔT_p-v,S – ΔT_p-v,V is close to the amplitude of the noise levels. It is worth noting that the measured n_2,U is less than the n₂ (~0.25 × 10⁻¹⁵ cm²/W [22]) from the quartz cell walls (typically ~1mm thick on each side). Since these cells are tightly matched for both solvent and solution, one of the advantages of the dual-arm Z-scan is that the subtraction automatically eliminates the nonlinear refraction signal from each cell wall, leaving the pure signal from the solute. Also note that the α_2,U reported in Fig. 6(c) is too small to determine at this irradiance with the OA Z-scan, thus the value is reported from two-photon fluorescence measurements given in [19] and verified by OA Z-scans performed at higher irradiances.

 

Fig. 6 CA dual-arm Z-scans at 880 nm, E = 13 nJ, C = 0.60 mM, and I₀ = 22 GW/cm² for (a) toluene (open black circles) and (b) solution (open black circles) along with independent fits for n₂ – Δn₂ (solid blue line) and n₂ + Δn₂ (solid green line); (c) T_U(Z) (open green squares) of SD-O 2405 and fit (solid blue line) with Δϕ₀ = −0.023, q₀ = 0.0020 (see note in text), using S = 0.33.

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Figure 7(a) shows the dispersion of NLR and spectrum of 2PA in terms of cross sections defined as δ_NLR = 10⁵⁸(hν)k₀n_2,U /N (refractive Göppert-Mayer, RGM) [25] and δ_2PA = 10⁵⁸(hν)α_2,U /N (Göppert-Mayer, GM), respectively, where h is Planck’s constant, ν is the optical frequency in Hz, and N is the concentration in molecules/m³ of SD-O 2405 using a 0.6 mM concentration measured by dual-arm Z-scan as well as the structure and linear absorption. The units of GM and RGM are 10⁻⁵⁰ cm ^4·s·molecule⁻¹·photon⁻¹. To convert between units of RGM and GM to real and imaginary parts of the hyperpolarizability γ in MKS units we use the conversion relations Re(γ) = (2c²n₀²ε₀²δ_NLR × 10⁻⁵⁸)/(3πf ⁴hν²) and Im(γ) = (c²n₀²ε₀²δ_2PA × 10⁻⁵⁸)/(3πf ⁴hν²), where c is the speed of light, n₀ is the refractive index of the solvent, ε₀ is the vacuum permittivity, and f is the local field enhancement factor defined as f = (n₀² + 2)/3 [26,27]. γ in cgs units can be converted from γ in MKS units via γ(cgs) = γ(MKS) × 81 × 10²³ [27]. Note that the figure of merit (FOM) can be written as FOM = |Re(γ)/Im(γ)| = 2|δ_NLR/δ_2PA|. The experimental results agree with the theoretical calculation results based on a three-level sum-over-states model. The n_2,U fits of the solute at wavelengths from 800 nm to 920 nm range from −0.1 to −0.25 x 10⁻¹⁵ cm²/W (corresponding to δ_NLR from −500 to −1250 RGM).

 

Fig. 7 (a) 2PA cross section (open black squares) and NLR cross section (open red circles) versus wavelength of SD-O 2405 measured by the dual-arm Z-scan technique. GM and RGM represent the units of cross section for 2PA and NLR, respectively, as 10⁻⁵⁰ cm^4·s·molecule⁻¹·photon⁻¹. The solid black line and red line are from the three-level sum-over-states model [2,23,24] for the 2PA and NLR cross sections, respectively. The inset shows the structure and linear absorption of the molecule dissolved in toluene. (b) NLR cross sections along with the three-level sum-over-states model for the region of small NLR with the vertical axis expanded.

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From our dual-arm measurements, n_2,U remains negative across the wavelengths measured as shown in Fig. 7(b). Given these small magnitudes, in the single-arm Z-scan method, any small dispersion of the solvent over the wavelength range could be buried under the noise floor of the Z-scan signal, which, if not taken into account, could erroneously result in an incorrect or even positive n_2,U. Hence another advantage of the dual-arm technique is that neither the dispersion of n_2,V nor its absolute value need to be known. In the case that this information is required, it can be readily determined through fitting of T_V(Z). The dual-arm Z-scan technique automatically eliminates any such variations in the solvent nonlinearity, so that any error in determining n_2,V is not transferred to determining that of the solute.

4. Conclusion

We have introduced the dual-arm Z-scan technique as a method to extract small nonlinear refractive signals from solutes of low concentration in the presence of large solvent nonlinearities. A squaraine dye was used to test the sensitivity of the new technique which yielded results consistent with previous data [19]. Measurements across a wide wavelength spectrum were taken and compared to conventional single-arm Z-scans. The fitting error of the solute NLR, n_2,U, is reduced due to the correlation of noise in the two equalized arms from measuring both solution and solvent with the same pulse, thus increasing the signal-to-noise ratio (SNR). Utilizing this new method allows the determination of solute nonlinear refractive indices considerably smaller than when using the single-arm Z-scan.

From our experiments we calculated that for ΔT_p-v,U ~0.2% (which is the noise floor for several of the wavelengths used) that the minimum detectable nonlinear phase shift (using Eq. (13b) from [14]) is less than λ/1000. Using the associated irradiance and fit parameters this corresponds to a minimum detectable |n_2,U| ≈10⁻¹⁷ cm²/W at an irradiance of 100 GW/cm² in a 1 mm path length. In terms of the nonlinear refractive cross section at a given concentration of 10⁻³ M, the minimum |δ_NLR| ≈30 RGM (10⁻⁵⁰ cm^4·s·molecule⁻¹·photon⁻¹). The increased sensitivity for measuring nonlinear refractive indices is dependent on the noise properties of the optical source used. Thus our results are not strictly applicable to other laboratories using different sources. What is important for the improvement is the correlation of the noise between the two arms. For example, if the optical source has large beam pointing instabilities, it would be expected that the dual-arm Z-scan technique would give a large increase in sensitivity. The enhancement is most pronounced in those cases where the signal from the solute is small compared to the noise. Such cases may arise with solutes for which the nonlinear phase change is low due to concentration or path length limitations as in thin films, or solutions where low damage thresholds preclude their study using high irradiances. When the solute signal is much larger than the noise in a single-arm Z-scan, the dual-arm method is less advantageous. In general, this technique could be extended to measure not only solutes in solutions, but dopants embedded in solid hosts so long as the composite material is homogeneous.

The increase in SNR for NLR measurements does not transfer to NLA measurements in the solution studied since the solvent normally has immeasurably small NLA. Thus the SNR for e.g. 2PA measurements using the usual Z-scan is about the same as the SNR for n₂ measurements using the dual arm Z-scan, as shown in Fig. 5. However, for cases where the solvent exhibits NLA, it would be expected that a portion of the noise in the dual-arm Z-scan would then be correlated and thus the SNR would increase in a similar way to that of the CA Z-scan.

Appendix A: Arm Equalization Tolerance

One of the primary challenges in utilizing this technique is equalizing the arms sufficiently. Using an approximation to the Gaussian decomposition code referred to in [14] we can determine the sensitivity of the system to mismatches of parameters. This is done by calculating the equalization error from the normalized transmittances, i.e. $T_{err}(Z)=T_{armA}(Z)-T_{armB}(Z)$.

We can get a reasonable approximate analytical determination of these errors by using the first two terms in the Gaussian decomposition, which is valid for small phase shifts as outlined in [14] and assuming the far field condition d₀ >> z₀ is met where d₀ is the distance from the position of the beam waist to the closed aperture. For negligible absorption and reflection losses the normalized CA transmittance T(Z) can be approximated [14] as

(A1)$$T(Z)\approx 1+\frac{4\Delta \phi {}_0(Z/z_0)}{({(Z/z_0)}^2+9)({(Z/z_0)}^2+1)}$$

(A2)$$\Delta {\phi }_0=k_0{n}_2{I}_{0}L$$

(A3)$$I_0=\frac{2E}{{\pi }^{3/2}\tau w_0{}^2}$$

where τ is the half-width at 1/e of the maximum (HW 1/e M) pulse width. At the extrema of T(Z) (peak and valley, Z ≈ ± 0.85z₀), the peak-to-valley transmittance change reduces to ΔT_p-v ≈0.406Δϕ₀ for S ≈0, and the peak-to-valley equalization error ΔT_err due to mismatches in irradiance can be expressed as,

(A4)$$\Delta T_{err}=\Delta T_{p-v,armB}-\Delta T_{p-v,armA}\approx 0.406\Delta \Delta {\phi }_0=0.406k_0{n}_2\Delta I_{0}L$$

where ΔΔϕ₀ and ΔI₀ are the differences in the peak nonlinear phase shift and irradiance between arms due to parameter mismatch. From this, the maximum allowable mismatch in the peak irradiance can be calculated to give,

(A5)$$\frac{\Delta I_0}{I_0}\cong \frac{\Delta \Delta {\phi }_0}{\Delta {\phi }_0}\cong \frac{\Delta T_{err}}{\Delta T_{p-v}}\cong \frac{\Delta E}{E}\cong 2\frac{\Delta w_0}{w_0}$$

where the last equalities show the linear dependence of irradiance on energy and inverse quadratic dependence of irradiance on spot size. Typically, the irradiance is not directly equalized. Rather the beam waist and pulse energy (assuming the pulse width is the same in both arms) as well as cell Z-positioning are equalized.

We estimate the needed precision for two 1 mm cells filled with the same solvent using a value of n₂ = 1 × 10⁻¹⁵ cm²/W with E = 15 nJ, w₀ =20 µm, and τ = 70 fs. For ΔT_err equal to the system noise of 0.2% (equivalent to a total nonlinear phase shift of less than λ/1000) and ΔT_p-v = 6.8% this evaluates to an irradiance difference of 2.9%. This allows a ΔE/E of 2.9% and Δw₀/w₀ of 1.5%. Having used matched optics for the focusing lenses, equalizing the beam waists is dependent on equalizing the path lengths of each arm from the beam splitter to the respective focusing lens. For a well collimated beam, this level of path length equalization is typically not difficult to achieve. For instance, a beam collimated by focusing over a long distance (a 2 mm beam focused to a 0.5 mm spot over a length of 7 m) the maximum allowable path length difference is an easily attainable 6 cm.

The cell Z-positioning mismatch is calculated by taking one copy of T(Z) and shifting it with respect to another copy by some small distance ΔZ and subtracting the two curves. In this way, the mismatch is similar to a derivative, and as such, it is minimum at the peak and valley of T(Z) and maximum at Z = 0. Unlike for the errors from pulse energy and beam waist mismatch, the cell Z-positioning mismatch does not affect ΔT_p-v. However, as in the case of pulse energy and beam waist, mismatches should be minimized by proper cell Z-positioning so as to not introduce unwanted artifacts into the solute CA signal that may complicate fitting. Performing the same analysis as above, the Z-positioning tolerance ΔZ/z₀ is found to be 2.7%.

Mismatches in the CA linear transmittance and position must be calculated numerically and are determined to have little effect, with mismatches of up to 25% still resulting in ΔT_err below the system noise.

Because the maximum allowable parameter mismatch is inversely proportional to ΔT_p-v it is best to perform the equalization at high energy using a calibration solvent with a large n₂, e.g. CS₂ which has n₂ ~ 4 × 10⁻¹⁵ cm²/W (for ~130 fs FWHM pulses) at 700 nm, similar to recently reported values [28–30]. If T_err(Z) can be reduced below the noise floor under these conditions, then T_err(Z) will remain under the noise floor when switching to a solvent with n₂ lower than the calibration solvent, as is usually the case.

To process the data we operate on the assumption that the signals from the solvent and solute are additive, and hence

(A6)$$\Delta T_{p-v,S}\approx \Delta T_{p-v,V}+\Delta T_{p-v,U}$$

for small ΔT_p-v,S. Figure 8 shows ΔT_p-v,U for a solute alone with n_2,U = 0.1 × 10⁻¹⁵ cm²/W, and ΔT_p-v,U derived via subtraction in the case where n_2,V = 0.9 × 10⁻¹⁵ cm²/W and n_2,S = 1.0 × 10⁻¹⁵ cm²/W. While the two curves diverge for large phase shifts, for the small signal limit (|Δϕ₀| < 0.5 radians [14]) the difference between ΔT_p-v,U of the pure solute and ΔT_p-v,U derived via subtraction is small. This error from subtraction is due to higher order terms of the Gaussian decomposition which become significant for large phase shifts. However, so long as we remain in the range where the small signal approximation holds, i.e. where Eq. (A1) is valid, we can apply the subtraction technique without incurring any significant error. Because the subtraction error is deterministic, even in the case where the small signal approximation does not apply, it is possible to account for this via simulation to extract the proper value of n_2,U.

 

Fig. 8 ΔT_p-v,U for a CA solute signal and CA solute signal extracted via subtraction technique.

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Acknowledgments

This work was performed under the AFOSR MURI grant FA9550-10-1-0558. We would also like to thank Dr. Joel Hales of the School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia, USA for his critical review and comments and Dr. Yurii L. Slominsky of the Institute of Organic Chemistry, National Academy of Sciences, Kiev, Ukraine and Dr. Olga V. Przhonska of the Institute of Physics, National Academy of Sciences, Kiev, Ukraine for supplying the molecule SD-O 2405 used in this study.

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