Relative illumination calculationsMatthew P.RimmerOptical Research Associates550 North Rosemead Boulevard,Pasadena,California 91107AbstractRelative illumination in an optical system is a function of many variables including distortion,vignetting and pupilaberration.It can be obtained by determining the size of the exit pupil in direction cosine space,using the appropriatecoordinate system.In general,the calculation is done by tracing bundles of rays through the system.In many cases ofinterest,it is possible to calculate relative illumination accurately by tracing one on-axis ray and three off-axis rays.Coordinate systemThe key to calculating relative illumination is to define the limiting rays in exit space in terms of direction cosines.1,2,3The limiting rays are those which just pass through the edge of the limiting aperture of the system.For a system with novignetting,the aperture stop is normally the limiting surface.In general,however,different rays will be limited by differentsurfaces.For systems with obstructions,the rays limited by the obstructions also have to be considered.The definition of the direction cosines is described in Figure 1.The origin of the coordinate system is at O'with the z-axis along the optical axis.Point Q'is the intersection of the chief ray with the image surface.Point E'is the position ofthe exit pupil of the system along the optical axis.A reference sphere is constructed whose center is at Q'and which passesthrough E'.An arbitrary ray intersects the reference sphere at P.The coordinates of interest are the direction cosines,(I,m,n),of the line segment between P'and Q'.Note that this segment does not coincide with the ray unless there is no aberration.Ray(I,m,n)Q'Chief RayOpticalSystemEOptical AxisReferenceSphereImageSurfaceFigure 1.Pupil coordinatesSPIE Vol.655 Optical System Design,Analysis,and Production for Advanced Technology Systems (1986)/99Downloaded From:http://proceedings.spiedigitallibrary.org/on 02/11/2016 Terms of Use:http://spiedigitallibrary.org/ss/TermsOfUse.aspxThe direction cosines must be defined in a coordinate system whose z-axis is parallel to the surface normal at Q'.For aflat image surface,this is the same as the coordinate system in which the surface is defined,the z-axis being along the opticalaxis.If the image surface is not flat,(1,m,n)must be rotated into the coordinate system of the surface normal.This is donewith the rotation matrix in equation (1),where (L,M,N)is the unit vector along the normal to the surface at point Q'.-LM/V1-L2-LN/V1-L20.0N/V1-12M/V1-L2LMNRelative illuminationWe assume a uniform light intensity over the object and that it behaves like a plane lambertian source.If we plot theexit pupil shape,as defined by the limiting rays,as a function of (l,m)we get a figure whose area is proportional to theillumination on the image surface.This is illustrated in Figure 2.Thus,the relative illumination is obtained by dividing theoff-axis pupil area by the on-axis pupil area.It should be noted that this approach automatically takes into account anyvariation of pupil size as a function of field,including the effects of vignetting and image distortion.It is valid for any generaloptical system and is not limited to rotationally symmetric systems.mm2Exit PupilDefined byLimiting Raysm1Figure 2.Exit pupil in direction cosine spaceThe effective F-number can also be obtained from Figure 2.If we denote the points where the exit pupil periphery crosses them-axis in the figure by ml and m2,the effective F-number in the m-direction is 1/(m2-m1).A corresponding number can beobtained in the 1-direction.100 SPIE Vol.655 Optical System Design,Analysis,and Production for Advanced Technology Systems(1986)Downloaded From:http://proceedings.spiedigitallibrary.org/on 02/11/2016 Terms of Use:http://spiedigitallibrary .org/ss/TermsOfUse.aspxThe cosine4 lawFigure 3 depicts the exit space of a thin lens with stop in contact and shows the limiting rays for the on-axis beam andfor an off-axis beam where the chief ray makes an angle,U,with the optical axis.R is the radius of the pupil,D is thedistance from the pupil to the image and H is the image height.We can algebraically define the direction cosines of thelimiting rays in the meridional and sagittal directions and,assuming the pupil can be represented by an ellipse,generate anexpression for the relative illumination.If we now take this expression and let R go to zero(the paraxial limit),we will findthat the relative illumination,RI,isRI cos4 UThis is the well known cosine4 law and agrees with the simpler notion that one cosine arises from the obliquity of the beamwith the image,one from the obliquity with the pupil,and two from the square of the reciprocal distance.This law moregenerally applies to a flat-field system whose exit pupil is constant in size and position with respect to field.The angle,U,is measured in image space.If,in addition,the pupils of the system are at the nodal points,the angle in object space is alsoU.UDStopImageFigure 3.Thin lens with stop in contactPractical calculationsThe straightforward way to calculate the pupil area on a computer is to turn the lens around and trace a grid of rays fromthe image point equally spaced in the direction cosines (1,m).The area is then proportional to the number of rays which getthrough the system.It is usually inconvenient to turn the lens around,so an alternate approach is to trace the rays from theobject point instead of the image point.This approach,with the following modifications,has proven to be sufficientlyaccurate for practical purposes.If the image is aberrated,then the rays will not coincide with the line segment between the image point and the referencesphere.In this case,the raytraced direction cosines in image space can be modified using equation(3),where(1',m')are theraytraced direction cosines,(1,m)are the modified direction cosines,(X,Y)are the transverse aberrations(deviations from theSPIE Vol.655 Optical System Design.Analysis,and Production for Advanced Technology Systems (1986)101Downloaded From:http:/proceedings.spiedigitallibrary.org/on 02/11/2016 Terms of Use:http://spiedigitallibrary.org/ss/TermsOfUse.aspx