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A. Model predictions
As a demonstration, we evaluate the local electric fields for an experimental setup where the beams are incident from an IR-grade fused silica prism, encounter a polydimethylsiloxane (PDMS) film at the PDMS–glass interface (z=0), and then reach the PDMS–air interface. In order to construct this model, we need the refractive indices of the fused silica (n1), PDMS film (N2), and the environmental side (N3), which happens to be transparent in our case, so N3≈n3≈1. Since our experimental configuration uses a hemicylindrical prism, the beam angle approaching the fused silica–PDMS interface is the angle θ1.6767. In cases where dove prisms or flat windows are used, one must calculate the incident angles by first considering refraction into the material that we are here calling phase 1. In this case, one should also take care if phase 1 is not sufficiently transparent. For example, the commonly used IR-grade fused silica Corning 7979 benefits from some transmission correction on account of absorption for IR wavenumbers in the range 2800–3000 cm −1. The model results shown below were obtained for ωvis=18797 cm−1 (λvis=532 nm), ωIR=2910 cm−1, and ωSFG=ωvis+ωIR. The values of the refractive indices used for all beams in the various phases are shown in Table II.
TABLE II. Refractive index values for IR-grade fused silica6868. I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965). https://doi.org/10.1364/JOSA.55.001205 and PDMS69–7169. F. Schneider, J. Draheim, R. Kamberger, and U. Wallrabe, Sens. Actuators A 151, 95 (2009). https://doi.org/10.1016/j.sna.2009.01.02670. M. R. Querry, “Optical constants of minerals and other materials from the millimeter to the ultraviolet,” Technical Report 88009, 1987.71. Dow Corning Corporation, “Sylgard 184 silicone elastomer product information,” Technical Report, 1999. used for the model demonstration. The third medium, air, was assumed to be transparent at all wavelengths. For all phases with negligible imaginary parts of the refractive index, see the text for the note about evaluation of the refracted angle above the critical angle.Vacuum wavelengthSilicaPDMSAirn1N2N3461 nm (SFG)1.4651.441 + 0i1 + 0i532 nm (vis)1.4611.435 + 0i1 + 0i3.436 μm (IR)1.4071.386 + 0.06203i1 + 0iWe first consider the case where θvis=43°, θIR=θvis+9°, and θSFG is calculated from the usual momentum conservation expression,11. Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, New York, 1984).
nvisωvissinθvis+nIRωIRsinθIR=nSFGωSFGsinθSFG,(41)where the refractive indices and angles correspond to those in the incident medium (fused silica). The first column in Fig. 6 plots |Lxx|2 (blue), |Lyy|2 (red), and |Lzz|2 (green) as a function of the PDMS film thickness in the case of the SFG (top row), visible (second row), and IR beams (third row) for the silica–PDMS (z=0) interface. In the bottom row, we have assembled the case of the SSP local field intensities defined by |Lyy,SFGLyy,visLzz,IR|2. In each case, the points indicate the model prediction with N′=N″=N2, i.e., the refractive index of the interfacial region is the same as the bulk PDMS film. The results plotted with solid lines correspond to the output obtained with N′=12(n1+N2) and N″=12(N2+N3). In the middle column, we have repeated this, but for the PDMS–air (z=d) interface. As a reality check, it makes sense that the period of the oscillations increases with increasing wavelength. The SFG wavelength is slightly shorter than the visible beam wavelength and so there are fewer maxima and minima for the visible beam with the same 1000 nm thickness range. Likewise, the IR wavelength is so much longer that there is not even a full period in this range. Overall, we also see that the two different choices for N′ and N″ result in the same values for |Lxx|2 and |Lyy|2. Since the interface has infinitesimal thickness, this is also expected as we require the tangential components of the field to be continuous. However, the effect of N′ and N″ is marked on |Lzz|2, especially at the PDMS–air interface. In the right column, we plot the ratio of these results, choosing z=d as the numerator so that larger values of the ratio highlight film thicknesses that are more selective for the PDMS–air interface. Here, we can observe several thickness ranges with significant field enhancement at the PDMS–air interface. There are occasionally thicknesses (albeit within a narrow range) where the enhancement over the PDMS–substrate interface exceeds 106. Achieving such a large selectivity is important as, in general, we do not know the relative magnitudes of χ(2) at the two surfaces. Finally, it is worth pointing out that although the ratio of |LLL|2 factors for SSP can get large, we do not observe thicknesses for which the ratio is small enough to indicate selectivity for the PDMS–substrate interface. That requires a different set of beam angles.We now repeat the analysis with all of the same parameters, except now with θvis=75°. To model our experimental geometry, we still have θIR=θvis+9°, so the IR beam is approaching at a grazing angle. The model results in Fig. 7 show all of the same basic behavior that we illustrated for the lower set of beam angles with two notable exceptions. First, the period of the oscillations has increased. More importantly, the SSP field intensity ratio (in the bottom right subplot) no longer has such large enhancement factors with the two maxima around 700 and 900 nm being factors of approximately 1000. However, we now have factors approaching values as small as 10−4 for a thickness in the range of 350 nm. Based on the way we defined this ratio, we can take the reciprocal to say that the signals at the PDMS–prism interface are now enhanced by factor of 104 over those at the PDMS– air interface.We can visualize such scenarios for all combinations of beam angles and thicknesses by plotting maps of the ratio of the electric field amplitudes for the beam polarization combination of interest. This is less obvious in the case of PPP7272. C. Cai, M. S. Azam, and D. K. Hore, J. Phys. Chem. C 125, 12382 (2021). https://doi.org/10.1021/acs.jpcc.1c02584 but, for polarization combinations (SSP, SPS, PSS) where a single element of χ(2) is probed, this is readily calculated as we have demonstrated for SSP. Assembling all such slices into a map results in a depiction that is challenging to interpret unless a logarithmic scale is used to readily identify the local maxima (selectivity for PDMS–air) and minima (targeting PDMS–glass). An alternative depiction is to plot select contours on this map, as shown in Fig. 8. In this way, we plot ratios greater than 10 in yellow, greater than 100 in orange, and above 10 000 in red. Similarly, regions with ratios less than 0.1 are shaded light blue, below 0.01 medium blue, and regions below 10−4 dark blue. The vertical dashed lines correspond to the θvis=43° case described in detail in Fig. 6 and the θvis=75° case from Fig. 7. The horizontal line illustrates that, for a film thickness of 350 nm, switching between the two sets of angles enables either interface to be probed in turn. Of particular importance are the white regions, where there is no selectivity. This means that χ(2) from each interface contributes with equal weighting since the ratio of the field strengths is close to unity.A final point concerns the magnitude of the local electric fields at the desired interface. The map presented in Fig. 8 was colored based on the value of the ratios alone. Having a large (or small) ratio provides the sought selectivity but does not guarantee that an observable SFG signal (large enough |χeff(2)|2) can be measured, even if |χ(2)|2 itself is large enough. For example, if we wish to study the PDMS–glass interface in SSP polarization, a large ratio of |LLL|2 at z=0 over z=d (dark region in Fig. 8) is not helpful in itself if both |LLL|2 values are vanishingly small. This is in fact the case for angles greater than 75°. Furthermore, all of the analysis here has been at a single IR wavelength. The recommended approach here is to look at such ratios together with the predicted field magnitudes at each interface (before taking the ratio) and for all IR wavelengths of interest.7272. C. Cai, M. S. Azam, and D. K. Hore, J. Phys. Chem. C 125, 12382 (2021). https://doi.org/10.1021/acs.jpcc.1c02584C. Extension to other systems
The results shown in Figs. 6–8 were obtained for the silica–PDMS–air system, for λvis=532 nm and ωIR=2910 cm−1, and for a geometry where θIR=θvis+9.8°. For other materials, other wavelengths, or for experimental geometries that use a different method of varying the beam angles, the expressions we have provided, particularly the compact form summarized in Table I, can be used to arrive at the predicted thickness and angle combination that best isolates the interface of interest. In some cases, the modeling may reveal that no selectivity can be achieved—and this is valuable information too.However, it is interesting to generalize this for the case of common polymers, as there is not that large of a variation in the (real part of the) refractive index for a fairly wide range of polymers. Furthermore, Corning 7979 is a popular choice of material for a silica prism that offers a reasonable amount of IR transparency in the C–H stretching window. We first make three observations: (1) the dispersion of the IR refractive index of the polymer, although important for the actual magnitude and phase of the local fields, does not significantly affect the general shape of the ratios of the combined fields at the two interfaces. (2) The IR wavenumber itself has a greater influence on the interference. In other words, even if the IR refractive index was the same at 1000 and 3000 cm−1, this factor of three difference in wavenumber would significantly influence the thickness and angle planning. However, if we limit ourselves to the C–H stretching region, the results throughout the 2800–3200 cm−1 range are more or less the same as those presented at 2910 cm−1 in Fig. 8. (3) The difference in the visible and IR beam angles again affects the value of the fields at each interface but does not strongly affect the ratios. We have checked this up to 15° difference between θvis and θIR, and the results again appear as in Fig. 8.Finally, this puts us in a position where we can evaluate the more general effect of differences in the visible and IR refractive indices (taking average values in the 2800–3200 cm−1 region and neglecting dispersion) corresponding to different polymers. Many polymers have refractive indices in the visible region between 1.45 and 1.60.76,7776. N. Sultanova, S. Kasarova, and I. Kokolov, Acta Phys. Polonica A 116, 585 (2009). https://doi.org/10.12693/APhysPolA.116.58577. X. Zhang, J. Qiu, X. Li, J. Zhao, and L. Liu, Appl. Opt. 59, 2337 (2020). https://doi.org/10.1364/AO.383831 If we consider poly(methyl methacrylate) (PMMA), the visible refractive index is in the neighborhood of 1.495 and 1.472 in the C–H stretching region of the midinfrared, with the largest κ value around 0.03 in that region.7878. X. Zhang, J. Qiu, J. Zhao, X. Li, and L. Liu, J. Quant. Spectrosc. Radiat. Transfer 252, 107063 (2020). https://doi.org/10.1016/j.jqsrt.2020.107063 The resulting 2D map (not shown) bears striking similarity to that of Fig. 8, to the point where the same set of film thickness and angles can be used to target either side of a PMMA film. At the other extreme, we find materials such as polystyrene, with a refractive index close to 1.6 in the visible and 1.57 in this region of the IR, again with κ as large as 0.03.7878. X. Zhang, J. Qiu, J. Zhao, X. Li, and L. Liu, J. Quant. Spectrosc. Radiat. Transfer 252, 107063 (2020). https://doi.org/10.1016/j.jqsrt.2020.107063 Here, partially owing to the fact that the polymer refractive index is significantly larger than that of the fused silica substrate, the situation is different. The results in Fig. 12 plot the same vertical lines at θvis=43° and θIR=75°. Here, we can see that the 350 nm thickness considered for PDMS (point A in Fig. 12) now misses the region of highest polystyrene–air selectivity, although this could be compensated by a slight increase in the beam angles, if the experimental configuration allows. Of greater consequence, however, is that the high angle set for this thickness (point B) has barely any selectivity for the polystyrene–glass interface. Since there is no dark blue region for this combination of materials (polystyrene, glass, air), a better choice of the film thickness is then 110 nm where θvis=43° achieves a strong polystyrene–air selectivity (point C). It is interesting to note that if PDMS–glass is not of interest in a particular experiment, then a 110 nm film would also be desirable for the PDMS–air interface. It then seems that the combination of a visible beam angle in the range 43–47° with a film 100–120 nm thickness on an IR-grade fused silica prism should provide reasonable selectivity for the polymer–air interface for many common polymers.
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