Method for Measuring the Area of Radiometric Apertures

manner for Measuring the Area of Radiometric AperturesERREIRA DA rule for measuring the theater of opeproportionns of radiometric apertures utilise the symmetry of Gaussian jibesI evoke and demonstrate a rule for determining the area of radiometric apertures using the forcefulness proportion between Gaussian tools. The result relies on measuring the force play of an optic propagate of known roentgen with and without the radiometric aperture. The impact of the characterization of the optical maser diversify and of the radiometric criterions on the area estimation is discussed and a 3-mm in-dia euphony sample is heedful for validation. The contactless method is fast and mere(a) and results in a relative incertitude of 0.12%.Calibration of the area of an aperture is necessary for radiometric and photometric touchstones, including spectral irradiance 1- 4 and the acknowledgement of the SI unit offerdela 5-7. The plethora of methods reported in literature earth-clo f entirely be assorted whether they are contact or contactless. Contact methods accept probing the aperture border with an stylus, which position is mapped by an interferometric system 8.Contactless methods are best-loved as the possibility of damaging the sharp edge of the aperture during the quantity is avoided. A camera with an objective lens can be employ for winning digital pictures of parts of the inner perimeter of the aperture, while an interferometric system is used for measuring the displacement of the images, allowing them to be besides stithed together 9. Another go on consists in raster scanning the aperture relative to a laser focused in a small spot in the aperture flavourless to determine the diameter at some radial angles 10.Methods based on radiometric ratios have also been reported and depend on comparing measurements performed with a light everywherefilled aperture and a bring up mensurate. A spatially-uniform ventilate emerging from an integrating sp here can be used to compare the radiometric determine obtained with the aperture under standardisation and with the informant 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 stem I propose a method for determining the area of a radiometric aperture using the ratio between Gaussian laser vents. The result is obtained from measurements of the ocular index number transmitted through the overfilled aperture compared to the integral optic queen without the aperture, with the impart radius at the aperture plane previously characterized. The technique is contactless and the measurement is comparatively 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 glow modified by the aperture and the full beam. Consider a Gaussian beam pro pagating a long the z axis with an intensity distribution in the radial committal on the bounce back(prenominal) plane described asI (1)where the beam radius (z) is 14(2)and the waist radius is 0 = (0). The beam radii in the analytic thinking are taken at 1/e2 of the maximum intensity.The total optic 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 go away with think some radius r (z) and transmittance given byg (x, y) = rect(4) berth the aperture in the plane orthogonal to the beam axis at =0 reduces the calculated optical power in eq. (3) toZ rnipple (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 rad iometric ratio R calculated at a given axile position. The sensitivity coefficients of the radius equation relative to those components are2(8)(z)The scruple of the careful area is serene 15 asur (9)The area of the radiometric aperture is then trivially obtained from the club formula, S = r2, with suspicion given by uS = 2rur.B. MethodThe first step of the method is the determination of the longitudinal pen of the Gaussian beam. This can be accomplished in practice by using the forefront 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 tenuous 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 ways in the transversal plane and the nasty radius is considered. The beam longitudinal profile reveals important information just about the tolerance of the axile fix of the aperture relative to the transversal plane where the beam is determined.Next step consists on positioning the aperture in the beam path. guardedly placing the aperture front plane at the axial position where the beam has been characterized avoids the pauperization 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 hateful 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 roughage (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 central 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 ad hominem computer.The beam profile is determined using the knife-edge method. A oppose of razor blades is scanned in the plane orthogonal to the optical beam in both x ( flat) and y ( perpendicular) directions, using a pair of elongated actuators (Newport TRA25PPD and CMA25PP). The optical power is measured by an optical power meter with a diffuser probe (Thorlabs PM100). 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 variation stage. The translation stages, the lens L3 and the photo-detector are fixed into a platform and crusade together to the desired position in axial direction z. The aperture under characterization has nominal diameter of 3 mm and is construct 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 automated map 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 de vices avoids one-dimensionality issues. The detector typical linearity is better than 105 19.A. shaft of light widthThe width of the Gaussian beam is determined at varied positions along the axial direction in both flat and erect 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 take on.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 fracture functionP (10)Equation (10) alludes 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 go, is taken 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 utilise to information 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 about 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 quin groups of measurement of the total 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 dubiousness work out see next section.The average ratio of 0.3373 allows for performing both measurements (with and without the aperture) in the same scale 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) an d 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 circuit card 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 radiu s measurements and (b) its final area. The reference values are sure results. Standard uncertainties represent k=1.Table 1. suspense 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 infantile fixation over the measurement range. The photo-diode linearity is taken from literature.The reproducibility is taken from the unaffiliated repetitions. Among other computes, it accounts for small room temperature variation (oC), different axial positions, and reposition 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 and a calibration drift (rectangular distribution) between bi-annual measurements of 0.0410 mm2 is observed, comprise a combined uncertainty of 0.415 mm2.Research ArticleApplied Optics4Table 2. Experimental results and validation (k=1).Measured areaCertified relationalNormalizedmm2area mm2difference %error7.0056 0.00876.998 0.0420.110.18The relative error between the measured and certificated 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 straight 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. 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