To study the applicability of the model developed in the previous chapter, Eq. (9.2) and Eq. (9.4) were used to determine the dispersion of a cast PDMS film (Sylgard®184) in the visible range. PDMS constitutes a well suited material for this purpose, since it allows for easy molding of holographically written SRGs. Unlike azobenzene-based materials, it is colorless in the visible region and imprinted gratings are stable if exposed to light of arbitrary wavelength. Using a molded grating also demonstrates that the presented technique is not limited to the determination of refractive indices of SRG-forming materials but also applies to any solid with a grating near its surface.
To measure the dispersion of Sylgard®184, a slightly modified version of the setup for the inscription of holographic gratings (cf. Chapter 7.1) was used. Instead of placing a photo-active medium into the sample holder on the rotational stage, a glass substrate covered with a PDMS layer was installed. The latter was cast from a molecular glass film with holographically generated SRGs according to the procedure described in Chap-ter 6.1.3. Thus, it featured several spatially separated sinusoidal SRGs with a periodicity ofΛ=1000.6±1.7nm on its surface. AFM measurements further confirmed the presence of defect-free, uncollapsed gratings. The substrate was mounted with one of the gratings located exactly at the position where usually the hologram would develop. Since the grating was already present on the PDMS film, illumination with a holographic interference pattern was not required and the writing beams were switched off during the whole experiment.
Determination ofn0as a function of the wavelength requires the use of diverse reading lasers. A multi-color He-Ne laser generated light at the wavelengths 593.932, 604.613, 611.802, and 632.816 nm. Further light sources were the laser at 489.20 nm, normally used for holographic grating inscription, and several diode lasers providing lines at 532.06, 660.3, 671.8, and 688.4 nm. In combination, a total number of nine different wavelengths were available. All laser beams were collimated and passed a polarizer to illuminate the PDMS grating withs-polarized light.
To measure the anglesθl, in90°, the sample was rotated. The pivot coincided with the center of the grating spot, so it was continuously hit by the reading laser. For normal incidence, the diffracted orders passed through the PDMS layer and the glass substrate behind it. With increasing rotation angle, the emerging orders became totally reflected at the air surfaces of glass substrate and PDMS film, acting as a waveguide. Further rotation of the sample caused a given diffracted order to propagate at an angle at which it did not reach the bound-ary between PDMS film and substrate anymore. The associated angle of incidence was θl, in90°, which could be determined for the 1st and 2nddiffraction orders (l=1 and 2). As a consequence of Eq. (9.2), higher orders were not observed.
The refractive indices determined with the above method were controlled by reference measurements performed with an Abbe refractometer (Zeiss Abbe-Refraktometer Modell A). To perfectly fit the size of the refracting prism, Sylgard®184 resin was cured in a cast-ing mold and prepared as a stripe with a size of 4×1cm2 and a thickness of 2.5 mm. Due to its low Young’s modulus, it easily adhered to the prism and, thus, no contact liquid was required. The PDMS stripe on the refracting prism was successively illuminated with the monochromatic light of the laser sources, while the compensator of the refractometer was adjusted to its neutral position. To obtain the refractive index, the internal scale was recal-culated for each wavelength as specified by the manufacturer.
The refractive indices of the PDMS elastomer determined by the different measurement procedures are plotted in Fig. 9.2 as a function of the wavelength. According to the data of the Abbe refractometer (solid squares),n0 decreases from 1.421 in the blue to 1.412 in the red spectral region. An uncertainty of 5×10−4arises from the reading accuracy of its
9.2 DISPERSION OF POLYDIMETHYLSILOXANE 79 internal scale. The data of the refractometer show normal dispersion, which is the expected behavior for a transparent, colorless material. Similar values for the refractive indices were calculated from Eq. (9.2) (solid circles) and Eq. (9.4) (solid triangles). The deviations from the refractometer data are small but show a systematic trend. Refractive indices determined from Eq. (9.4) have an almost constant offset of about −2×10−3, which is also present for wavelengths up to 650 nm if calculated from Eq. (9.2). For larger wavelengths,n0increases such that the last data point is located slightly above the refractometer data.
1.410 1.414 1.418 1.422
500 550 600 650 700
refractiveindexn0
wavelengthλr[nm]
Figure 9.2: Refractive index of the Sylgard®184 PDMS elastomer as a function of the wave-length. Its values are determined by Abbe refractometer measurements (solid squares) or calcu-lated either from Eq.(9.2)(solid circles) or Eq. (9.4)(solid triangles).
As confirmed by the Abbe refractometer data, the increasing values ofn0derived from Eq. (9.2) at longer wavelengths cannot be caused by anomalous dispersion in PDMS. To understand their origin, a closer inspection of the experimental error sources is required. The accuracy to which the refractive index can be determined from Eq. (9.4) depends only on two subsequent angular measurements. Equation (9.2), on the other hand, requires additional precise knowledge of the grating period and the wavelength of the reading beam. Thus, the error bars of the filled triangles in Fig. 9.2 are smaller than those of the filled circles. An error of the grating period would not only changen0at longer wavelengths but introduce an offset to the whole data set. Incorrect determination of the critical diffraction angles would influence the refractive indices determined from both Eq. (9.2) and Eq. (9.4). Thus, the wavelengthλrremains the only parameter which can affect Eq. (9.2) exclusively in part of the data points.
Depending on the angular resolution, the 1st-order critical diffraction angle can be mea-sured with almost arbitrary accuracy if the incident plane wave is monochromatic. For a light source of non-negligible spectral width, however, each spectral component is diffracted by the grating at its own angle, so the collimated beam splits up into its different colors. Equa-tion (9.1) implies that light with shorter wavelength emerges at a smaller angle inside the medium. Upon rotation, the red part of the probing beam becomes evanescent first, fol-lowed by the light at its wavelength peak. Hence, for quasi- or non-monochromatic light, the criterion that the 1st order must not propagate into the substrate is fulfilled by the blue part of its spectrum. Specification ofn0to the third decimal place requires the wavelength
of the light source to be known to an accuracy of better than 1nm. Otherwise, the spectral splitting of the probing beam reduces the precision. Spectrometer measurements showed that this was actually the case for the diode lasers at the long wavelengths 660.3, 671.8, and 688.4 nm. They had a spectral width of 2.8 nm (full width at half-maximum). For all other laser sources it was below 0.1 nm.
The systematic error caused by the uncertainty of the peak wavelength λr in Eq. (9.2) can be compensated by replacing it with an adequately blue-shifted wavelength λr0. For diode lasers with a Gaussian spectral profile, it is reasonable to defineλ0ras the wavelength at which their intensity decreases to 1/e2 of its maximum. This yields 658.8, 669.9, and 686.9 nm as the corrected wavelengthsλ0r for the red diode lasers. The refractive indices calculated from these wavelengths are plotted in Fig. 9.3 together with the (unchanged) values determined before. Refractive indices calculated from both Eq. (9.2) and Eq. (9.4) now show normal dispersion and match each other. As a consequence of the broader spectral width of the laser sources above 650 nm, the error bars become quite large in this range. The constant offset to the Abbe refractometer data remains present in both data sets.
1.410 1.414 1.418 1.422
500 550 600 650 700
refractiveindexn0
wavelengthλr[nm]
Figure 9.3: Data are identical to the one shown in Fig. 9.2 except for the refractive indices calculated from Eq.(9.2)for the red diode lasers. Instead of the peak wavelengths 660.3, 671.8, and 688.4 nm, the corrected wavelengths 658.8, 669.9, and 686.9 nm were inserted, respectively.
The grating periodΛenters only in Eq. (9.2) and cannot be responsible for both shifts.
An independent reference value is the average grating period ¯Λ, which is the mean value of the grating periods determined from Eq. (9.3) for each individual laser source. It is calculated as ¯Λ=999.7±1.0nm, which agrees with the value ofΛwithin the error margins.
Therefore, substitution of ¯Λinto Eq. (9.2) only gives rise to minor changes of the refractive indices. This proves thatΛis calculated correctly from Eq. (2.3).
Several other error sources can also be excluded to cause the offset. The experiment was repeated with different laser polarizations. For boths- andp-polarized light, the critical diffraction angles were identical. Thus, no anisotropy or birefringence was introduced into the PDMS film during preparation. Also, switching to a different grating spot did not have any significant effect on the refractive-index data. The critical diffraction angles of the−1st and−2nd diffracted orders, θ−1, in90° andθ−2, in90° , were determined by reversing the direction of rotation. Because negative and positive orders propagate at angles with opposite sign,