Tuesday, June 4, 2019
Method for Measuring the Area of Radiometric Apertures
Method for Measuring the Area of Radiometric AperturesERREIRA DAMethod for measuring the ara of radiometric apertures apply the dimension of Gaussian institutionalizesI propose and demonstrate a method for determining the area of radiometric apertures utilize the effect ratio between Gaussian beams. The allow for relies on measuring the power of an ocular beam of known wheel spoke with and without the radiometric aperture. The impact of the characterization of the laser beam and of the radiometric measuring rods on the area estimation is discussed and a 3-mm in-diameter sample is measured for validation. The contactless method is fast and simple and results in a copulation doubt of 0.12%.Calibration of the area of an aperture is necessary for radiometric and photometric measurements, including spectral irradiance 1- 4 and the actualisation of the SI unit candela 5-7. The plethora of methods reported in literature can be assorted whether they are contact or contactless. C ontact methods implicate probing the aperture border with an stylus, which position is mapped by an interferometric system 8.Contactless methods are preferable as the possibility of damaging the sharp edge of the aperture during the measurement is avoided. A camera with an objective lens system can be used for taking digital pictures of parts of the inner perimeter of the aperture, while an interferometric system is used for measuring the teddy of the images, allowing them to be further stithed together 9. Another approach consists in raster scanning the aperture coitus to a laser focused in a small get along in the aperture savourless to determine the diameter at some radial angles 10.Methods based on radiometric ratios have also been reported and depend on comparability measurements performed with a light overfilled aperture and a reference value. A spatially-uniform beam emerging from an integrating sphere can be used to compare the radiometric set obtained with the apert ure to a lower place calibration and with the reference one 11. Similarly a matrix of small-spot laser sources can be used 12, 13, with the reference provided by the known uniform irradiance distribution.In this paper I propose a method for determining the area of a radiometric aperture using the ratio between Gaussian laser beams. The result is obtained from measurements of the ocular power transmitted through the overfilled aperture compared to the total optical power without the aperture, with the beam radius at the aperture plane previously characterized. The technique is contactless and the measurement is relatively fast, providing an alternative way for measuring radiometric apertures.A. ModelThe method proposed for determining the area of the aperture is based on measuring the radiometric ratio between the beam limited by the aperture and the lavish beam. Consider a Gaussian beam propagating along the z axis with an intensity distribution in the radial direction on the t ransversal plane described asI (1)where the beam radius (z) is 14(2)and the waist radius is 0 = (0). The beam radii in the analysis are interpreted at 1/e2 of the uttermost intensity.The total optical power of the beam is obtained by integrating its intensity over the transversal area as Ptotal /2(3)The circular radiometric aperture is modelled as a Boxcar function with mean radius r (z) and transmittance abandoned byg (x, y) = rect(4)Positioning the aperture in the plane orthogonal to the beam axis at =0 reduces the measured optical power in eq. (3) toZ rPap (z) =I (, z) 2d(5)0The ratio between the optical power limited by the aperture at position z and the total optical power of the beam is thus 14(z)2r2R(6)The radius of the aperture is obtained by inverting eq. (6), resulting inr (7)Equation (7) reveals the dependence of the aperture radius on the beam radius and radiometric ratio R measured at a given axial position. The sensitivity coefficients of the radius equation rela tive to those components are2(8)(z)The incertitude of the measured area is make up 15 asur (9)The area of the radiometric aperture is therefore trivially obtained from the circle formula, S = r2, with precariousness given by uS = 2rur.B. MethodThe first step of the method is the finish of the longitudinal visibility of the Gaussian beam. This can be accomplished in practice by using the poke-edge scanning method 16 or using a spatially-resolving photodetector (for example, a CMOS or CCD camera). While the later can be troubling for beams wider than the sensitive area of the camera, the primer requires caution relative to radial asymmetries in the beam profile. The astigmatism of the beam must be verified by knife edge scanning along orthogonal directions in the transversal plane and the mean radius is considered. The beam longitudinal profile reveals important information intimately the tolerance of the axial positioning of the aperture relative to the transversal plane where the beam is impelled.Next step consists on positioning the aperture in the beam path. Carefully placing the aperture comportment plane at the axial position where the beam has been characterized avoids the need for a correction on the beam radius value. The aperture under measurement must then be centralized relative to the beam axis. A recursive gradient search can be performed along the plane axes until convergence at the maximum optical power, where 0.The value of the optical power measured with the aperture is compared to the total optical power measured without it. This ratio and the mean beam radius are substituted in eq. (7) and the aperture radius is determined.Research ArticleApplied Optics2A laser diode with continuous-wave emission at 633 nm is collected with an objective lens into a meter-long single-mode optical fiber (Thorlabs SM600 17), which acts as a spatial filter by selecting the LP01 transversal mode. The beam is launched into free-space through the tip of an FC-PC connector and collimated using an 1-large AR-coated plano-convex lens (L2) with a focal length of 38.2 mm, as illustrated in Fig. 1. A similar lens (L3) with 150-mm long focal length focuses the beam into the photo-detector.Fig. 1. Experimental setup. LD laser diode L plano-convex lens C fiber connector PD photo-detector PC individual(prenominal) computer.The beam profile is determined using the knife-edge method. A pair of razor blades is scanned in the plane orthogonal to the optical beam in both x (horizontal) and y ( vertical) directions, using a pair of linear actuators (Newport TRA25PPD and CMA25PP). The optical power is measured by an optical power meter with a diffuser probe (Thorlabs PM one hundred). Data acquisition and transversal positioning of the knives and aperture are performed with a personal computer.Flip mounts allow for selecting either the knives or the aperture, which are placed in the same x y translation stage. The translation stages, the lens L3 and the photo-detector are inflexible into a platform and move together to the desired position in axial direction z. The aperture under characterization has nominal diameter of 3 mm and is built in anodized aluminium with sharp edges.The offset distance between the planes of the knives and the aperture is set within 0.05 mm using a multi-probe optical reflectometer 18. An automatise routine is used to position the aperture in the transversal plane relative to the optical beam by scanning it along x and y directions until it is centralized.The radiometric ratio is obtained by removing and reinserting the aperture using the flip mount while the power is measured using a silicon photodiode (Hamamatsu S1227-1010BQ) in photovoltaic mode. Calibrated trans-impedance amplifier (LabKinetics Vinculum) and digital voltmeter (Agilent 34401A) are used. Conditioning the signals for using a single range of these devices avoids linearity issues. The detector typical linearity is better than 105 19. A. Beam largenessThe width of the Gaussian beam is determined at different positions along the axial direction in both horizontal and vertical axes. Figure 2 shows a sample of the transversal beam profile Fig. 2. Sample of the transversal intensity profile of the beam. The slices in the details cross the center and are Gaussian fit.The longitudinal profile of the beam is evaluated by applying the knife-edge analysis at different axial positions. The optical power measured as a function of the knife position in x direction is modelled as the integral of the Gaussian intensity, resulting in the error functionP (10)Equation (10) indicates that the measured power profile reveals the horizontal beam radius x (z). The procedure performed along the y direction returns a similar result as a function of the vertical beam radius y (z).Figures 3a and 3b show the power measured with the knifeedge method along both x and y directions, respectively. A group of 10 scans, with 0.25-mm steps, is int erpreted at a given axial position. Data is interpolated to steps of 0.1 mm using piecewise cubic Hermite interpolating polynomials 20. Non-linear curve fit (Levemberg-Marquadt method) is globally applied to entropy with the beam radius parameter shared by all curves in the group. The beam radius values as a function of the axial distance to the collimating lens are shown in Fig. 3c.Observe that the beam profile behaves linearly at the sampled axial positions. Fitting data with eq. (2) reveals the horizontal and vertical waists localized at or so 3.3 m and 3.7 m, respectively. The slope of 104 indicates that a positioning error between the knives and the aperture of 0.05 mm has negligible impact on estimated radius.The beam is slightly astigmatic (horizontal radius about 1% greater than the vertical one), so the average radius is computed from both horizontal and vertical radii as/2(11)B. Radiometric ratioThe radiometric ratio is determined from five groups of measurement of the t otal beam power, alternated with four measurements of the power limited by the aperture. Interleaved measurements allows for data interpolation and avoids slow drift effects. Each measurement is composed by a group of 30 data points, corrected by the dark measurement. Three measurement were performed at each axial position. The calibration data of the trans-impedance amplifier and voltmeter are used for correction and considered in the uncertainty budget see next section.The average ratio of 0.3373 allows for performing both measurements (with and without the aperture) in the same get over of the amplifier and voltmeter. Keeping the measurement range of the equipment fixed avoids linearity issues, which must otherwise be corrected and could burden on the uncertainty budged.C. Aperture radius/area and uncertainty budgetThe aperture radius is computed from the measured values of (z) and R (z) using eq. (7). The result obtained at three different axial distances from the collimating lens are presented in Fig. 4a.The uncertainty budget for the radius measurement is presented in Table 1. The uncertainty of the beam width and power ratio are combined with the reproducibility of the measurement. The radius measurement is obtained from the global fit of the knife-edge scan measurements. The impact of the beam divergence is obtained by multiplying this value by the maximum axial offset between the knife-edge and the aperture plane. The beam width uncertainty is dominant over all other components. Improvements over this estimation would greatly benefit the final uncertainty.The repeatability comes from the statistics of the ratio measurements. Stability of the laser source is the major component and could be iproved using a further power stabilization closedloop. The amplifier and voltmeter uncertainties are obtainedFig. 4. Experimental results (a) aperture radius measurements and (b) its final area. The reference values are certified results. Standard uncertainties represent k=1.Table 1. Uncertainty budget for the measurement of the aperture radius (relative values).ComponentTypeUncertainty (k=1)Radius measurementsB5.3 - 104Beam divergence mmB2.3 - 105Trans-impedance amplifierB6.3 - 105VoltmeterB5.5 - 105Photodiode linearityB6.2 - 106Power ratio0.00017Reproducibility mmA0.00027Aperture radius mm0.00062from their calibration uncertainty and from the linear regression over the measurement range. The photo-diode linearity is taken from literature.The reproducibility is taken from the independent repetitions. Among other factors, it accounts for small room temperature variation (oC), different axial positions, and repositioning of the aperture center relative to the beam axis.The final relative uncertainty obtained for the measurement of area is 0.12%. The validation of the method is assessed by comparing the results to a certified value, as shown in Table 2. The certificates present a relative uncertainty (k=1) of 0.0065 mm2 for the area value an d a calibration drift (rectangular distribution) between bi-annual measurements of 0.0410 mm2 is observed, composing a combined uncertainty of 0.415 mm2.Research ArticleApplied Optics4Table 2. Experimental results and validation (k=1).Measured areaCertifiedRelativeNormalizedmm2area mm2difference %error7.0056 0.00876.998 0.0420.110.18The relative error between the measured and certificate values is 0.11%, while the normalized error 15 is below unit, indicating the compatibility of the results. The coverage factor of the measurements, calculated for a confidence interval of 95.45%, is k=2.The area of an aperture impacts directly on the determination of some radiometric and photometric quantities. This paper presents a simple and fast contactless method for characterizing an aperture area through the measurement of radiometric ratio of characterized Gaussian beams. The model is presented and the measurement uncertainty budget is discussed. The results are validated and indicate the m ethod as suitable for metrology applications.References M. White, N. P. Fox, V. E. Ralph, and N. J. Harrison, The characterization of a high-temperature black body as the basis for the NPL spectralirradiance carapace, Metrologia 32, 431-434 (1995/96).P. Sperfeld, K.-H. Raatz, B. Nawo, W. Mller, and J. Metzdorf, Spectralirradiance scale based on radiometric black-body temperature measurements, Metrologia 32, 435-439 (1995/96).P. Krh, P. Toivanen, F. Manoochehri, and E. 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