The group of optical radiation measurements (GIMRO), in the Institute of Optics of the Spanish National Research Council (IO-CSIC), has developed a method based on Zernike polynomials to analyze photometric properties of Light-Emitting Diodes (LEDs) from measurements of the angular luminous intensity distribution. The procedure basically consists in fitting the Zernike polynomials to absolute measurements of angular distribution of luminous intensity and relative measurements of the angular distribution of luminance. Since the Zernike polynomials describe the wavefront, the parameters of the fit polynomials can be easily related to the total luminous flux, the degree of Lambertianity, the divergence of the emission by the full width at half maximum (FWHM), its anisotropy and the direction of the optical axis. Zernike polynomials are a complete set of polynomials that are orthogonal over the interior of the unit circle. They allow aberration function to be expanded in a power series. They are usually expressed in polar coordinates (¿,¿), but we expressed them in spherical coordinates (¿,¿) by the transformation [¿=asin(¿), ¿=¿], in order to expand the luminous intensity and luminance distribution in power series. The advantage of this procedure is that it is possible to better understand the variation of the photometric magnitudes on the unit hemisphere from the evaluation of the weights corresponding to every Zernike polynomial. From goniophotometric measurements on 18 high power LED made with our near-field goniophotometer (consisting of two stages of rotation to vary the polar angle and azimuth angle and four stages of linear displacement for centering the LED), it was determined that of the Zernike polynomials: constant term (with weight CCons), defocus function (CDef), primary spherical (CSph), vertical coma (CComaV), horizontal coma (CComaH), vertical tilt (CTiltV), horizontal tilt (CTiltH), vertical astigmatism (CAstigV) and oblique astigmatism (CAstigO), describe well enough the angular distribution of the emission, provided that very high anisotropies aren¿t present. By fitting the luminous intensity to Zernike polynomials (letting weights as free parameters), some intrinsic properties of the emission are obtained, as the divergence, the optical axis, the total luminous flux and the anisotropy: LED's divergence (FWHM) is calculated from the weights of the ¿-independent polynomials (CCons, CDef and CSph). The direction of optical axis (in spherical coordinates [¿_0,¿_0]), solved from the two tilt polynomials and their weights, as the first derivative with respect to x and y. The anisotropy is defined as the standard deviation of the ratio of the ¿-dependent polynomials to the ¿-independent polynomials. Thus, there are an anisotropy value for every polar angle ¿. Total luminous flux (¿) is calculated by integration of the luminous intensity on the sphere. By integration of the Zernike expansion, this magnitude is simply expressed as: ¿=2*CCons*¿+2*CDef*¿/¿3+2*CSph*¿/¿5. A much easier expression for ¿ is obtained if, instead of the luminous intensity¿s, the angular relative luminance distribution is expressed as a Zernike expansion. In this case, the expression is ¿ = CCons * ¿. On the other hand, using this luminance-based approach, Lambertianity of the source can be accounted just by the relative value of the weight of the constant term respect to the remaining weights. The results for the angular luminous intensity distribution of the LEDs previously mentioned are shown in this work.

Kuala Lumpur, Malaysia, 23-26 April, 2014

Peer Reviewed