Subwavelength gratings are gratings with periods smaller than the incident wavelength. They possess form birefringence, which depends on the grating parameters. This paper presents the results of an experimental method designed to measure the birefringent properties of diamond subwavelength gratings in the mid-infrared. The method consists of monitoring the intensity transmitted through one polarizer, a subwavelength grating, and a second polarizer for various orientations of the first polarizer. By fitting the intensity variation as a function of the orientation of the first polarizer, one can compute the phase shift induced by the grating, its local fast axis orientation, and the ratio of the transverse electric and transverse magnetic transmission efficiencies. The paper describes the method principle and its mathematical model. Then, several numerical simulations of different subwavelength gratings are presented and their results are discussed. Finally, the optical setup is described and the measurements of one subwavelength grating are displayed and compared with the values expected from the manufacturing process.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Subwavelength gratings are, as their name suggests, gratings with periods smaller than the incident wavelength . These gratings only allow the transmission of the zeroth order. The zero-order condition [1–3] is
These gratings act as birefringent plates and they can be described by four optical parameters, namely, the phase shift induced between the TE and TM components of the incident beam , the local fast axis orientation , and the transmission efficiency of the TE and TM components ( and ). These parameters depend on the grating parameters, namely, grating width on top , depth (), side wall angles (), and grating orientation (see Fig. 1). Using the effective medium theory and the rigorous coupled-wave analysis (RCWA), one can design the grating parameters to achieve a specific fast axis orientation and a desired phase shift over a specified bandwidth [2–6]. Another advantage of these gratings is the possibility to create space-variant elements, i.e., elements with a variation of their fast axis orientation: .
These elements have several applications such as in coronagraphy [1,2,7], polarization analysis [3,8], and in the generation of radially and azimuthally polarized beams . Finally, diamond has advantages of high transmission in the infrared region, robustness to external conditions, chemical resistance, low thermal expansion, and high thermal conductivity.
The efficiency of these components for the applications mentioned previously are affected by a deviation from the ideal fast axis orientation and by a variation of the phase shift with the wavelength; although this variation is mitigated by design , it still impacts performance [2,3,10]. For a proper spectral analysis of the optical parameters, the infrared source is needed to fulfill two requirements. It must possess a high intensity over a broadband domain to improve the signal-to-noise ratio and a fine spectral resolution to only acquire data over a small spectral range. Those requirements were achieved with a tunable quantum cascade laser (QCL). Using the measurements of the four optical parameters, one can compute the performance of the element. The paper will present the validation of an experimental method to measure , , and the efficiency ratio for one kind of subwavelength grating, namely, the annular groove phase mask (AGPM). AGPMs are constituted by concentric circular subwavelength gratings and they are used for coronagraphy [2,11–13]; they can be seen as half-wave plates with a radial orientation of their fast axis.
In this paper, first, the experimental method, the optical components, and the mathematical model will be described. Then, numerical simulations of the measurement process will be performed for three AGPM cases of increased complexity. Those numerical simulations aim to validate the principle of the measurement and to verify that the birefringent parameters can be extracted by fitting the intensity ratio curves. Therefore, the parameters obtained through the fitting will be compared with the parameters introduced to produce the intensity curves. Afterward, the components of the optical setup will be presented and the measurement results of one component will be shown and compared with RCWA simulations. Finally, the principle of the method and its results will be summarized.
2. DESCRIPTION OF THE EXPERIMENTAL METHOD
The experimental method is inspired by the one used by Yang and Yeh  to measure the phase shift and local fast axis orientation of birefringent elements. Our experimental setup is presented in Fig. 2. The QCL emits a collimated beam, which is transmitted through a linear polarizer and a quarter-wave plate to achieve a circular polarization. Then, the beam is transmitted through a computer-controlled linear polarizer , the sample (AGPM), and a second computer-controlled linear polarizer . Finally, the beam intensity is recorded with an imaging detector. The method principle is as follows: the transmission axis of will perform a full rotation in increments of 10°. For each orientation of with a transmission axis orientation of , two intensities, and , will be recorded. First, rotates to achieve parallel orientation of both transmission axes and is recorded with the detector. Then is rotated to achieve perpendicular orientation of transmission axes and is recorded. After a complete revolution of , the intensity ratio will be computed by dividing the intensities by each other for each orientation of and each pixel: (see Fig. 3 for the curve for one pixel). Afterward, will be fitted as a function of to extract , , and for each pixel.2) : 3) and (4): 5):
3. NUMERICAL SIMULATIONS
The numerical simulations will focus on the extraction of , , and for various AGPM cases. In these simulations, the intensity at the detector will be computed using the matrix multiplication presented in Section 2. Then the parameters , , and will be computed for each pixel by fitting the curves using Eq. (5) with the Matlab curve fitting tool for each pixel. For each AGPM case, three initial values and acceptable variation domains will be given to the fitting process of , , , and . For and , the initial values and computation domains are derived from the corresponding RCWA design simulations, whereas for , the initial values come from the visual interpretation of the transmitted intensities.
A. Ideal Case
As expressed before, the ideal element exhibits a constant phase shift of 180°, a perfect radial orientation of its fast axis, and a perfect transmission of both TE and TM components: , , , and . The parameters introduced in the fitting routine are , , and . Figure 4 presents and for several values of . Table 1 presents the results of the fitting process with , the introduced phase shift; , the spatially averaged computed phase shift; , the spatial standard deviation of the computed phase shift; , the introduced efficiency ratio; , the spatially averaged computed ; , the spatial standard deviation of the computed ; and , the spatially averaged error between the introduced fast axis orientation and the computed one. Figure 5 presents the fast axis orientation computed for this component.
By observing the results, it can be seen that the fitting of the curve leads to accurate computation of the element parameters. This good agreement allows validating the fitting principle and to proceed with a more complex case.
B. Chromatic Phase Shift Case
For the second case, the phase shift depends on the incident wavelength, . For this case, the phase shift values were obtained through RCWA. Those values correspond to grating parameters designed to achieve a phase shift close to 180° in the L-band region (3.5–4.1 μm) and were calculated in the 3.5–11.5 μm region for our analysis. The grating parameters are , , and . They correspond to the expected grating parameters of AGPM L13r presented by Vargas et al.  along with the manufacturing and tuning processes. Afterward, the intensities and curves were computed for each pixel and each wavelength. Afterward, the extraction process for , , and was performed as before. The parameters introduced in the fitting routine are , , and . The phase domain extremes are derived from RCWA simulations. Those simulations were performed with a deviation of 20% from the ideal set of gratings parameters: , , and . Figure 6 presents and at 5.7 μm for the same values of as in Fig. 4. Table 2 presents some results for different wavelengths and Fig. 7 shows the comparison between over the range of 3.5–11.5 μm and .
From these results, it appears that the fitting process is also reliable for the chromatic phase shift case. The next case, which combines the wavelength-dependent phase shift and transmission ratio, can therefore be studied.
C. Chromatic Phase Shift and Chromatic Efficiency Ratio Case
Finally, a more realistic component combines the wavelength-dependent phase shift and efficiency ratio, both obtained through RCWA using the same grating parameters as in case B. The fitting principle is the same as in the previous case, only the values are changed: . As in case B, the domain extremes are also derived from RCWA in the 20% error combination of the grating parameters. Figure 8 presents and at 5.7 μm. Table 3 presents the fitting results for the same wavelength as before and Fig. 9 presents the results for the computation of in the 3.5–11.5 μm domain.
From these results, it appears that the fitting routine is able to compute both the chromatic phase shift and efficiency ratio. This corresponds to the expected component. The experimental measurement of a real component is presented in the next section.
4. EXPERIMENTAL RESULTS
Our components are:
- • A tunable QCL (Daylight Solutions, MIRcat-QT) from 5.2 to 11 μm, with a pulse width of 400 ns and a duty cycle of 10 μs.
- • Three linear polarizers with transmission higher than 80% and an extinction ratio of 150 each for our bandwidth (, , and ).
- • Two computer-controlled rotators each with an accuracy of 5 arcmin and a repeatability of 10 arcsec.
- • A quarter-wave plate () at 6.55 μm with a phase shift between 65° and 110° in our measurement domain.
- • A pyrocam (; Ophir Photonics, Pyrocam IIIHR) sensitive from 1.1 to 30 μm, with a pixel size of and a sensor size of .
- • An AGPM optimized for the 3.5–4.1 μm wavelength domain (component L13r from Vargas et al. ).
The measurements are acquired automatically using Labview scripts (National Instruments V 2015). For image processing reasons, a measurement cycle includes the acquisition of two darks when the laser is not lasing ( and ), one after the acquisition of , and one after the acquisition of . The measurement cycle is divided into the following six steps:
- 1. rotates to achieve position and rotates accordingly to achieve parallel orientation in relation to .
- 2. frames are acquired with the pyrocam while the laser is active .
- 3. frames are acquired with the pyrocam while the laser is passive .
- 4. rotates to achieve a crossed configuration where the transmission axis of is perpendicular to axis.
- 5. frames are acquired with the pyrocam while the laser is active .
- 6. frames are acquired with the pyrocam while the laser is passive .
From these images, it can be observed that the transmitted intensity at the center of the component is very low. This is due to the singularity at the center. Therefore, pixels with low intensity, which cannot be separated from the background noise, will be omitted by the fitting algorithm. Consequently, an intensity criterion, which is based on the sum of the intensity of all positions, will be designed:12 and 13 present the results at 5.7 μm as function of the distance to the component singularity in the binned configuration, .
From those results, we decided that only pixels at a distance larger than 20 camera pixels (1.5 mm) from the component center would be processed by the fitting algorithm. Afterward, the fitting process was performed for pixels fulfilling those two criteria. As in case C, the parameters introduced into the fitting algorithm are , , and . The results of the measurements are presented in Table 4. Figures 14 and 15 compare the measured parameters with the parameters obtained through RCWA for the ideal grating parameters and a realistic error on them of 2.5% expected from previously cracked similar components . Finally, Fig. 16 shows a comparison of and .
From the table and graphics, it can be observed that the measurement parameters are in the same range as the expected ones. However, the spatial deviations are important and . To determine the origins of these variations, and were presented as functions of the distance to the singularity (Figs. 17 and 18, respectively).
By observing those graphs for every wavelength, emerging trends were noticed: and show radial variations and their spatial deviation at constant radius is smaller than the global one. To explain the radial variations and dispersions at constant radius, we propose two origins: an imperfect wavefront and a radial variation of the grating geometry. For a non-planar wavefront at non-normal incidence, the incident angle and the optical path in the component are not uniform; radial variations could be observed for a diverging beam and dispersion at constant radius could originate from astigmatism. Moreover, our RCWA simulations considered an ideal anti-reflective grating (ARG) [11,16] located on the back-side of the component. Those gratings are 2D symmetrical subwavelength gratings, whose purpose is to reduce the internal reflection at the back-side of the component, to increase the intensity at the exit of the component and its performance as a coronagraph . However, the ARG present on the component was designed for wavelengths smaller than the measured range, and so its maximum efficiency is not reached. Therefore, a fraction of the beam exiting the component experiences internal reflections and impacts the measured and . Also, small radial variations of the grating geometry have been observed on previously cracked components ; variations such as alterations of the grating shape or fluctuations of the grating parameters can also induce a change in the measured and .
To reduce the spatial deviation, statistical smoothing was implemented. Our method is based on the computation of the lower and upper quartiles, which excludes values that are separated by more than 1.5 times the interquartile range from the lower or upper quartile. First, the and values belonging to the pixels located inside our area of interest are grouped into vectors corresponding to rings with a radius of one pixel. Then, the tests are performed on those sub-vectors and only the elements passing the tests for their and are preserved. Afterward, the sub-vectors are merged together to reform a global vector and the tests are performed once again to exclude the remaining outliers. Finally, the averaged values and spatial deviations are computed; they are presented in Table 5 and compared with unfiltered data.
As a result of the statistical smoothing, the spatial dispersions were reduced. However, the radial variations of and remained, confirming our track on the impact of an aberrated incident wavefront and of the radial variation of the grating parameters. Moreover, the first results for an AGPM designed to be used in the 11–13.2 μm region and measured at 11 μm (, , and ) presented smaller radial variations, confirming our hypothesis on the role of the ARG in reducing the radial variations. To achieve a better understanding of those radial variations and to improve the measurement accuracy, our future plan is to change our measurement protocol and to improve our performance prediction tool to investigate the effect of aberrated wavefronts. Our new procedure will include the measurement of the wavefront before the beam hits the component, to assess its characteristics. The wavefront will be measured between and the component for each wavelength before running the rotation of . Besides, finite-difference time domain (FDTD) simulations will be used to investigate the origin of the radial variations of and . Using the Comsol Multiphysics (COMSOL Inc., V5.2) electric field propagation module, several cases of incident beam and component geometry will be studied to determine their effects on and . In those simulations, realistic aberrated wavefronts will be coupled to components with ARGs and more advanced simulations will tackle the radial variations of the grating parameters.
5. CONCLUSIONS AND PERSPECTIVES
In this paper, an experimental method to measure the birefringent parameters of space-variant subwavelength gratings was described. The method principle and its mathematical model using the Jones formalism were presented. Then, the results of numerical simulations representing increasing component complexity were displayed and discussed. The simulations of the measurement process showed that the experimental method was suited to optically evaluate space-variant subwavelength gratings. Afterward, the optical setup and measurement process were detailed. Finally, the measurement results for one AGPM were summarized and compared with the parameters expected from its design. It was observed that the measured parameters were in the same range as the expected ones. However, both and exhibited large spatial dispersions when they were supposed to be constant over the whole component. Statistical smoothing reduced those dispersions but radial variations remained. The effect of an aberrated wavefront combined with the impact of an ARG and the radial variations of the gratings were proposed to explain those behaviors.
In the future, the experimental procedure will include a measurement of the incident wavefront and FDTD simulations will be performed to correlate its effect on the measured parameters. Moreover, due to the improved accuracy on and , the identification of the grating parameters within a significant domain would be possible and performance prediction would be a reality, making this method an excellent complement to the already existing measurement methods.
Vetenskapsrådet (VR) (621-2014-5959); Carl Tryggers Stiftelse för Vetenskaplig Forskning (CTS 15:259); FP7 Ideas: European Research Council (IDEAS-ERC) (337569).
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