Original article J. Optics (Paris), 1995, vol. 26, no 4, pp. 161-174 (1) LABORATONO DE OFTICA, DEPARTAMENTO INTERUNIVERSITARIO DE OFTICA, UNIVERSIDAD DE ALICANTE Apdo. no 99. Alicanre E 03080, Spain (2) DEPARTAMENTO DE INGENIE& DE SISTEMAS Y COMUNICACIONES, UNIVERSIDAD DE ALICANTE Apdo. n ' 99, Alicanre E 03080, Spain THE COMPUTATION AND STATISTICAL ANALYSIS OF ABERRATIONAL DIFFRACTION PATTERNS IN HOLOGRAPHIC OPTICAL ELEMENTS L. CARFSTERO (I), A. FIMI.4 (I), A. BELhJIEZ (2) SUMMARY : Three numerical methods for calculating the point spread function for holographic optical elements in the presence of aberrations are analyzed. The stdng point for the analytical formulation of the diffmtion theory of aberrations for holo- graphic optical elements is the Fresnel-Kirchhoff diffraction inte- gral. To calculate the diffraction integral, the exit pupil of the el- ement is broken up into subdomains each with the same area, using both Cartesian and polar coordinates. A statistical analysis of the point spread function is introduced by defining a set of ir- radiance distribution moments. These moments can be used to characterize the imaging properties and aberrations of the holo- graphic optical element. The minimum number of area subdo- mains needed an the exit pupil of the element to obtain the cor- rect point spread function is obtained by optimizing the statistical moments of the irndiance distribution. KEY WORDS : MOTS c&s : Holographic Optical Elements El.5ments optiques Aberrations holographiques DitfractiO" Aberrations Diffraction Calcul et analyse statistique de la diffraction en pr6- sence d'aherrations dans des elements optiques holo- graphiques F&SUMI? : Trois methodes numeriques pour le calcul de la re- ponse impulsionnelle des 616ments optiques holographiques en presence d'abermtions on1 61.5 analys6es. La formulation analyti- que de la theorie de diffraction des aberrations dans des dl6ments optiques holographiques repose sur l'integde de diffraction de Fresnel-Kirchhoff. Pour le calcul de l'integrale de diffraction, la pupille de sortie de r616ment se divise. selon les caardonndes car- tdsiennes et polaires, en sous-domains ayant la mCme surface. L'analyse statistique de la reponse impulsionnelle est intcoduite par la definition des moments de distribution de l'imdiance. Ces moments sont utilises pour caractdriser lea propri6t6s d'imagerie et les aberrations de l'bldment optique holographique. Les mo- ments de la distribution de l'kadiance son1 optimises afin d'avoir un nombre minimal de sous-domaines n6cessaires donnant "ne 16- ponse impulsionnelle c o m t e pour la pupille de sortie. INTRODUCTION ing quality is possible by using the diffraction theory of aberrations. The diffraction image given bv an L I optical system from a point is known as the point spread function (psF) and this optical response This is still of great use to study the of Research done on the imaging quality of classical ally includes both the estimation of the values of the coefficients characterizing the particular aberrations an system with aberrations. and an evaluation of the wave aberration or calcula- Optical systems or holoWPhic Optical systems usu- func60n be calculated from the design data. tion of the aberration spot using the 'Yay tracing" method. These methods, however, do not allow one to directly calculate the light intensity distribution on the image plane. An evaluation of the lens imag- Generally, in order to obtain the PSF it is neces- sary to solve the diffraction integral numerically. This can be done with the fast Fourier transform al- gorithms [l] although it has been remarked that the