IMAGE ILLUMINATION ( OR +?) BACKGROUND Publications abound wit two differing expressions for calculating image illumination, te amount of radiation tat transfers from an object troug an optical system to an image,. Te discrepancy frequently appears in equations defining te response of an optical or infrared detector to an emitting object in a viewed scene, te two competing forms being R and R, () Were R is te response and F is te optical f/number. Te f/number is defined by f F, () D were f is te effective focal lengt of te optical system and D is te diameter of te entrance pupil. Te popular understanding is tat te first form is an approximation of te second. Te purpose of tis paper is to demonstrate tat te first is rigorously correct for a well designed optical system, and te second results from a paraxial approximation. BASIC OPTICS Figure sows an optical system imaging an object located in object space between te first focal plane F and infinity, te image occurring in image space between te second focal plane F and infinity. Tere are several points and surfaces of interest. Te image space focal plane F is te location of te image of an object located at infinity in object space. Tat is, all parallel rays incident on te optical system in object space will converge to a single point in image space, and tat point will be in te plane F. Similarly, all rays emanating from a point in te plane F will emerge as a parallel bundle in image; i.e., tey will create image of te point at infinity in image space. F P N N P A A F R M R FIGURE KEY FEATURES OF AN OPTICAL SYSTEM. Page of 6
Te planes N and N in Figure are, respectively, te first and second nodal planes, wic are te planes normal to te optical axis and passing troug te nodal points. Te nodal points are te on axis points for wic a ray passing troug one of tem at a small angle emerges from te oter at te same angle. Te effective focal lengt (efl), or simply focal lengt, of an optical system is te distance between te focal plane and te corresponding nodal plane. Te tird set of surfaces of interest are te principal surfaces, often call te principal planes because tey can be approximated by planes in te paraxial approximation. Te principal surface P in image space is defined by te intersection of rays parallel to te optical axis in object space wit teir corresponding converging rays in image space. Te principal surface P in object space is similarly defined. A pair of planes defined suc tat an object in one of te planes creates an image in te oter is known a set of conjugate planes. Tus, te plane conjugate to F is at infinity in image space, and te plane conjugate to F is at infinity in object space. In Figure, te planes A and A are conjugate. A well known teorem of optics, te Abbe sine condition 3, states tat for rays in conjugate planes nysin nysin, (3) were n is te index of refraction, y is te ray eigt in te object or image plane, is te angle between te ray and te optical axis, and te subscripts and designate object space and image space, respectively. Tis condition must be valid if te optical system is well corrected for coma. Distances are defined as positive above te optical axis and positive to te rigt of te nodal point in eac space. Note tat for a converging optic y and y will generally ave opposite signs, as will and. If we assume te object and te image bot exist in te same medium (usually air), ten te condition is y sin y sin. (4) ANALYSIS GENERAL CASE A S N N S A F 3 F 4 Z FIGURE MAPPING OF RAYS FROM CONJUGATE PLANES OF AN OPTICAL SYSTEM. Plane A in Figure is located a distance (< 0) to te left of te first nodal plane N, and plane A is located a distance (>0) to te rigt of te second nodal plane N. Te surface S is a spere of radius Page of 6
centered at point, and surface S is a spere of radius centered at point 4, te axial point in te image plane. Consider an object space ray tat emanates from a source at point, and tat strikes surface S at a eigt above te axis. We ave and terefore, from Eq. (3), Now te magnification of te instrument is sin sin, (5) y y. (6) M y y, (7) even toug y may be infinitesimally small, so sin. (8) Tis means tat a ray emitted from point emerges from sperical surface S at te same eigt tat it strikes sperical surface S. Now, if we allow, ten 0, S becomes a plane coincident wit te first nodal plane, f (te effective focal lengt), and S becomes a spere centered at te intersection of F and te optical axis. If we ten consider a marginal ray (i.e., a ray tat intersects S at te maximum aperture), ten sin D f F, (9) were F is te f/number. In words, tis says tat a ray parallel to te optical axis in object space produces a ray tat passes troug te image space focal point, and te two rays intersect on a spere of radius f centered at te focal point in image space. Tat is, te principal surface in image space is a spere 4. A similar argument wit te spaces reversed would sow tat te same olds true in object space. Tese sperical surfaces are represented in Figure by P and P. Te power received from infinity by a small, Lambertian detector at te focal plane of a lossless optical system is max image cos sin 0 sin max P B A d B A, (0) were B 0 is te brigtness of te source (W/cm sr), te cos in te integral reflects te fact tat te effective area of te detector for te off axis contributions is te projected area, and max is te cone alfangle of te ray bundle as viewed from te detector. P B A image, () Page 3 of 6
PARAXIAL APPROXIMATION Note tat if we worked in te paraxial approximation were te principal surface in image space is a plane, ten we would ave and so sin In tis case we would ave tan, () F. (3) B0 Adet Pimage. (4) Te form Eq. (4) is terefore only valid in te paraxial approximation (i.e., large f/numbers), Eq. () being te generally valid form. A SIMPLE ALTERNATIVE DERIVATION OF THE CORRECT FORM Anoter way to look at te problem is to consider te source side; i.e., to consider te portion of radiation emitted by te source, collected by te optics, and incident on te detector. Consider an onaxis detector pixel of small x det y det sie, and consider te rays emitted from te peripery of te pixel and passing troug te image space nodal point, as illustrated in Figure 3. Tese rays will emerge from te object space nodal point and form a rectangle of sie x tgt y tgt in object space. Tis defines te area of object space tat is exactly imaged onto te axial pixel; i.e., all te radiation tat is emitted from tat target area and collected by te lens will impinge on te pixel, but no oter scene radiation will. Te area of te target is simply R R R A x y x y A f f f tgt det det det det det Lens. (5) x tgt y tgt x det y det R f FIGURE 3 ENERGY COLLECTED BY A SINGLE PIXEL. Te radiation collected by te optics from tat area of te target is given by P B0 A sin, (6) image tgt opt were opt is te alf angle subtended by te optical aperture as viewed from te target. Now, Page 4 of 6
so, from Eqs. (5), (6) and (7), sin opt R D D, (7) R D 4 B A R B A Pimage B A f R D 4 R D 4 R 0 det. (8) OTHER CONSIDERATIONS CAVEATS It sould be noted tat image illumination is not necessarily te same as te radiation sensed by te detector, and for more reasons tat just absorption efficiency. If te only issue were te absorption efficiency we could simply introduce a multiplier, wic could be wavelengt dependent, and tis is te usual approximation. Tat approac assumes te detector to be Lambertian; i.e., it assumes tat te absorption efficiency is independent of te angle of incidence. A real detector does not generally beave tat way, for a couple of reasons. First, wen an electromagnetic wave is incident upon a surface, te reflection and transmission depend on bot te angle of incidence and te polariation, according to te Fresnel equations. In addition to tis, most igly efficient absorbers make use of anti reflection coatings or oter suc structures wose beavior depends in some resonant manner on te wavelengt of te radiation. Canging te angle of incidence canges te effective optical tickness of eac layer of tese structures, tereby canging overall efficiency. Canging te f/number canges te cone angle of te radiation incident on te detector. A small cange in f/number will cange te cone by a tin cone saped sell, and te absorption of tat energy will, in general, differ on average from te bulk of te cone. Depending on te design of te absorption structure and on te spectral composition of te incident energy, tis can eiter improving or degrading te average absorption efficiency, tereby distorting te apparent dependence on f/number of Eq. (). ILLUMINATION VS. RANGE Note tat Eq. (8) (before te limit R ) also predicts a cange in image illumination vs. distance to te target. For many sensors tis is irrelevant because R>>D, but tere are instances were tis is not valid. An important example is te indoor use of a radiometric termal imaging sensor, were te objects viewed can be very close. Pysically, two tings appen as te viewed object moves closer to te optical aperture. Assuming te position of te focal is adjusted to keep te object in focus, te magnification increases as te object come closer. Tat means a smaller portion of te image cover on a single pixel, tereby reducing te illumination. At te same time te solid angle subtended by te optical aperture (relative to te object) increases. Tis as te effect of increasing te illumination. Te first effect wins by a sligt margin, resulting in a small net loss of illumination. It is also important to remember tat te distance to te viewed object sould be measured from te first nodal point, not from te first surface of te lens or te lens ousing. Tis effect can be important, but it can also be overwelmed by a cange in te effective optical aperture as te object moves closer. If te entrance pupil is pysical stop in front of te lens, a close object can emit rays tat pass troug te stop, but teir angle is sufficiently steep tat tey strike anoter stop beind te entrance pupil. If te stop is internal to te lens te effect will be different still. Page 5 of 6
Niclaus, F., Decarat, A., Jansson, C. and Stemme, G., Performance model for uncooled infrared bolometer arrays and performance predictions of bolometers operating at atmosperic pressure, Infrared Pysics & Tecnology, Vol. 5, No. 3, Jan 008, p.68. Ricwine, R., Balcerak, R., Rapac, C., Freyvogel, K. and Sood, A., A Compreensive Model for Bolometer Element and Uncooled Array Design and Imaging Sensor Performance Prediction, Proc. SPIE, Vol. 694, 008. 3 Born, M. and Wolf, E., Principles of Optics, 4 t Ed., Pergamon Press, 970, p. 68. 4 Smit, Warren J., Modern Optical Engineering, 4 t Ed., McGraw Hill, 008, p. 3. Page 6 of 6