Analytic expressions for in-field scattered light distributions

Transcription

1 Analytic expressions for in-field scattered light distributions Gary L. Peterson Breault Research Organization, 6400 East Grant Road, Suite 350, Tucson, AZ ABSTRACT Light that is scattered from lenses and mirrors in an optical system produces a halo of stray light around bright objects within the field of view. The angular distribution of scattered light from any one component is usually described by the Harvey model. This paper presents analytic expressions for the scattered irradiance at a focal plane from optical components that scatter light in accordance with the Harvey model. It is found that the irradiance is independent of the location of an optical element within the system, provided the element is not located at or near an intermediate image plane. It is also found that the irradiance has little or no dependence on the size ofthe element. Keywords: stray light, scatter, in-field I. INTRODUCTION When we speak of in-field stray light, we are referring to sources of stray light that are within the field of view of an optical system. For example, if we are using a telescope to see a dim star in the neighborhood of a much brighter companion, a low-level halo of stray light around the bright companion is formed by scatter from the telescope's mirrors and lenses and diffraction from its apertures and obscurations. Ifthe halo of stray light is brighter than the image ofthe dim star, then the dim star is lost. Analytic expressions and numerical methods for calculating the irradiance from diffraction are well known. Not so with scattered light. This paper derives and discusses analytic expressions for the magnitude and distribution of scattered light around bright images in the focal plane of optical systems. II. IN-FIELD SCATTERED LIGHT IRRADIANCE In-field scattered light is troublesome because it is difficult to use baffles to block the bright source. Often the only solution is to reduce the scattered light by using well-polished components and keeping them clean. To set specifications on the quality and cleanliness ofthe optical components in the telescope we must find a way to calculate the magnitude and distribution of in-field stray light on the focal plane, given the bidirectional scatter distribution function (BSDF) ofthe optical components. Imaging systems are often complex. Most have several separated lenses and mirrors combined with various stops and apertures. What is the magnitude and distribution ofthe stray light irradiance in a complex imaging system? It may seem that the answer to this question depends upon the detailed geometry and optics of each system. In fact, this is not the case. We shall see that the functional form ofthe stray light halo follows the BSDF of each lens or mirror, and that the size and irradiance of the halo have a very simple dependence upon the BSDF of each lens and the beam size at each lens. The in-field stray light irradiance is the sum ofthe contributions from each optical element in a system. To calculate the contribution from an arbitrary element to the total stray distribution we consider an axially symmetric system and make use ofthe optical invariant, which we shall denote by H.' Suppose a small bright source on-axis. We wish to calculate the stray light irradiance at a point in the focal plane that is a distance r away from the bright source image. To defme an optical invariant for the given point, first trace a marginal ray backward from the bright on-axis source image. This ray intersects the edge of the aperture stop. We denote the distance of this ray from the optical axis at any given axial plane in the system by y and the slope of the ray at this same plane by u. Now trace achief ray from a point that is r away from the bright source image. This ray passes through the center of the aperture stop. Denote its 184 Optical Modeling and Performance Predictions, edited by Mark A. Kahan, Proceedings of SPIE Vol (SPIE, Bellingham, WA, 004) X/04/$15 doi: /

2 height by y and its slope by U, both measured at the same plane as the marginal ray. Figure (1) illustrates the defmitions of the above quantities for an arbitrary plane (perpendicular to the optical axis) in a portion of an optical system. The optical invariant for these two rays is defmed as H=yu yu. (1) This quantity has the same value at any plane within the system, including the source and image plane. In particular, at the image plane, where y is zero, the value of H is easily evaluated as H= ru =r(na), () where na is the numerical aperture of the system. Let us now turn our attention to an arbitrary optical element, denoted by a labelj, within an imaging system. Let the height ofthe marginal ray at this element be y. The absolute value ofy, denoted by au,, is the semi-diameter of the beam of light from the source that passes through the element. We assume only that y is not zero. That is, the element is not a field lens, reticle, or window that sits directly at an intermediate image plane. Now the light that passes through the element converges toward a real image of the focal plane, as illustrated in Figure (), or diverges from a virtual image. The image of a point that is a distance r away from the bright-source image on the focal plane is represented in the figure by a vertical arrow with height r. A marginal ray is shown which has heighty at the jth element. This marginal ray intersects the axis at a detector image, which is a distance z, from thejth element. A postulated element labeledj+1 is shown which intersects and diverts the marginal ray from thejth element. The focal-plane image is found by projecting the rays from the jth element through the next element, so the presence of Elementj+l has no effect on the size or location of the focal-plane image. We now derive the brightness of the scattered light from the lens as seen from a location on the intermediate image plane that is conjugate to the point that is r away from the source image on the fmal image plane. As shown Stop Axial Plane Focal Plane Figure (1): Illustration of marginal and chief rays traced back from a bright on-axis image and a point r away from this image. in Figure (), let the distance (possibly negative) to the intermediate image plane be z,, and let the evaluation point in the intermediate image plane be a distance i (possibly negative) from the axis. Now inspection of Figure () shows that the scatter angle, relative to the specular direction, is given by I Figure (): Illustration of intermediate image height and location, and scatter angle (. zi 0. = r Jzf (3) Proc. of SPIE Vol

3 To calculate a numerical value for this angle we need to know the distance to the intermediate image plane and the height ofthe evaluation point. To fmd the distance to the image plane, start with an equation for the slope of the marginal ray (which is taken to be negative in Figure ()), uj=-, (4) wherej' is the height ofthe marginal ray at thejth element. We also use the following equation for the optical invariant at the intermediate image plane: H=-r,u3. (5) Eliminate i, between these two equations to get 3). z=-_l=yj ry. 3 U] (-H/ri) H Substitute this equation for z3 into Equation (3) for the scatter angle to get :, r H o.= = =. 7 izi ry3/hy The dependence ofthe scatter angle on the intermediate image height and location has dropped entirely out of this equation! The scatter angle depends only upon the optical invariant, which is a constant everywhere within the system, and the marginal ray height y, whose absolute value is simply the beam size a, atthis particular element. We are now able to calculate the stray light irradiance from thejth element at the focal plane. To do this, first consider the radiance of the element as seen from the intermediate image plane. The BSDF is p(o) where 9 is the scatter angle given by Equation (7). If the irradiance on the jth element is L,,then the brightness of the scattered light from the lens is ' = P(0 )E = ( ). (8) Now the optical power that propagates through the system is conserved (except for transmission losses, which we are neglecting). If the irradiance from the source at the first element of the system is Eeng,andthe beam semi-diameter at the first element is aen,, then the power passing through the system is 186 Proc. of SPIE Vol. 5178

4 P=Eenttat, (9) and this is equal to the power passing through thejth element: P = Etltaent jj. (10) Solving for E gives aent 3 ent E.= E Substituting for E in Equation (8) gives aent H L= -p Eent (1) a3 a Now comes an essential point. Because of conservation of brightness, the brightness ofthe lens as seen from the intermediate image plane is equal to the brightness ofthe lens as seen from the system focal plane (except for transmission losses). Furthermore, the half-angle subtended by the illuminated area ofthe element, as seen from the final focal plane, is approximately equal to the system numerical aperture, so the stray light irradiance at the fmal focal plane from the jth element is given by = ir(na) =(na) ( ) E. (13) Equation () can be used to replace H in the argument of p, and we multiply by the system transmission Tto include the losses that we neglected to obtain E51(r) =(na) TL =(na)t (na) Eent (14) c1 We now have an expression for the stray light irradiance from thejth element as a function of distance r from the source image. Equation (14) for the in-field stray light irradiance has several features of interest. First, the stray light Proc. of SPIE Vol

5 irradiance has the same functional form as the BSDF ofthe optical element. Second, other than the BSDF, the only variable that affects the contribution of one element relative to another is the beam radius on the element, a,. The appearance ofthe beam radius in the argument ofthe BSDF function p causes the width ofthe distribution on the focal plane to expand and contract with a,. There is also a factor of 1/a that scales the magnitude ofthe stray light irradiance. It is tempting to state the magnitude ofthe stray light irradiance varies inversely as the square ofthe beam radius, but we shall now see that this is often not correct. III. HARVEY SCATTER MODELS To make further progress we must know the BSDF ofthe optical elements. Empirically, it is observed that the BSDF of clean, polished, optical surfaces follows a two-parameter equation that is known as the Harvey model: p = b (1 00 Isin(O) sin(00) I). (15) The angles 9 and O are the scatter and specular angles, measured relative to the surface normal. The parameter b is the value ofthe BSDF at 0.01 radians (0.57 degrees) from the specular direction, and s is the slope ofthe BSDF when it is plotted as a function of sin(o)-sin(00) on a log-log scale. In general, numerical values for b and s must be obtained from a scatter measurement on the actual part, or a sample that is polished under the same conditions as the part. They are a property ofthe surface fmish, not of the material out of which the lens or mirror is made. Good quality, clean, optical components have values for b and s that lie typically in the ranges O.O1<b<1, (16) and 3<s< 1. (17) For rough stray light estimates we might choose intermediate values of 0. 1 and -1.8 for b and s. We pick a slope thatis shallower than - so that the total integrated scatter is fmite. Scatter from particulate contamination exhibits a roll-off in the BSDF at small scatter angles.3 This roll-off is accommodated by a three-parameter Harvey model: p = b 1 + sin(o) sin(90) 0 s/ (18) The shoulder parameter / is the value of sin(o)-sin(00) at which the BSDF changes from a constant value to a function that decays with the sth power of sin(o)-sin(00). The parameter b0 is the value of the BSDF in the specular direction. The parameter s is the slope of the BSDF, plotted on a log-log scale, for values of sin(o)-sin(00)above the should parameter Proc. of SPIE Vol. 5178

6 Iv. SCATTERED LIGHT IRRADIANCE WITH HARVEY MODEL BSDF Recognizing sin(o)-sin(00)as an approximation for the scatter angle relative to the specular direction, we substitute the two-parameter Harvey model, Equation (15), into Equation (14) to get S ent r Eent E5(r) =ir(na) T b 1 00 (na) a = it: 1OOTb (na) aentr Eent s+ (19) Two important points should be noted. First, the functional form ofthe stray light halo around a small bright source goes as the sth power of distance from the point source, where s is the slope ofthe Harvey BSDF. Second, for a given distance r, the contribution ofthejth element scales inversely with the s+ power ofthe beam size on the element. We are now in a position to answer the following question. Given an optical system in which the beam size varies from one element to the next, but in which all elements have the same BSDF, which elements make the largest contribution to the in-field stray light irradiance; the large elements or the small elements? Let us assume that the slope ofthe BSDF is -. The dependence ofequation (19) on beam size, a3, drops out, because the exponent s+ is zero. Therefore, all elements make an equal contribution to the in-field stray light irradiance, regardless of position (as long as a is not zero) within the optical system. While this result is exact only for a BSDF slope of -, it is approximately true for slopes that are close to -. For example, ifthe slope is -1.8, then a factor of 10 reduction in the beam size results in only a factor of 1.6 increase in the contribution to the stray light irradiance. Finally,we note that ifthe slope is shallower than -, then elements for which the beam is small make the largest stray light contribution. Conversely, if the slope is steeper than -, then elements with large beam sizes contribute more. Of course, in general all elements do not have the same BSDF, so it is necessary to evaluate Equation (19) explicitly to obtain both the total BSDF and a relative ranking of the contributions from each element. We now look at the effect of the three-parameter Harvey model on the in-field stray light irradiance. Substituting Equation (18) in for the value of p that appears in Equation (14), we get E5 (r) = ir T(na) b0 1 + a J s/ (na) r la J (0) Here the story is a bit more complex than it was for the two-parameter Harvey model. We can distinguish two regions of interest. The first region is for small radii, where (na)r<i (1) In this region Proc. of SPIE Vol

7 aent E(r) - ir T(na) b0 Eent () a3 and the stray light irradiance varies inversely as the square ofthe beam size on each element that satisfies Inequality (1). Here small elements, provided they satisfy Inequality (1), are the dominant contributors to the in-field stray light background. The second region is for large radii, where (na)r > la (3) In this region E.(r)iiTb s (na)sf aentr E Si 0 s+ ent This equation is the same as Equation (19) for the two-parameter Harvey model if we set b0=b(100l)5, (5) and the discussion on the relative contributions of small and large elements that follows Equation (19) applies. V. COMPARISON WITH DIFFRACTION IRRADIANCE We now compare the irradiance produced by scatter with the irradiance produced by diffraction. The discussion will be limited to the two-parameter Harvey model and diffraction from circular apertures in diffractionlimited systems. Such diffraction produces the so-called Airy disk. Attention is often confmed the central region and first few rings of the Airy disk. But, we want to look outside this area and obtain expressions for the behavior of the irradiance at large distances from the center. The irradiance distribution on the focal plane that is produced by a diffraction limited system with a circular aperture has the following functional form E(r) = E0somb(!) =E0 (6) where J1(r) is a first-order Bessel function of the first kind, and somb(r) is the sombrero function defmed by Gaskill.4 The width of the function is scaled by the parameter d, which has the value 190 Proc. of SPIE Vol. 5178

8 d=(na) (7) The first zero occurs at a radius of approximately r=1.d=o.61. (na) (8) The total volume, or power, contained within this irradiance distribution is 4d P= E0. (9) If an optical system has an entrance pupil with a radius aeni, atransmission of T, and is illuminated by a radiation from an in-field source with irradiance Eeng, then the total power in the image is also given by P =E7ta T. (30) Equating Equations (9) and (30), solving for E0,and using Equation (7) for d gives the following expression for the irradiance at the center of the Airy disk: it(na) = E7c aent T (31) Let us now consider the diffracted irradiance for large values of r. An approximate expression for the Bessel function when r is large is5 1/ cos r (3) 4 Substituting this approximation for J1 into Equation (6), and using Equation (31) for E0 gives the following equation for the diffracted irradiance at large r: E(r) = ET?aent cos( ir(na)r -. (33) iz(na) r3 4 Proc. of SPIE Vol

9 The irradiance has the form of a sinusoidal variation modulated by a power-law envelope. If we drop the sinusoidal variation, and focus our attention on the envelope we get E T?a E(r) = ent ;nt. (34) iz(na) r Inspection shows that the irradiance falls of as the inverse cube ofthe distance from the image point. Plotted on a loglog scale, the irradiance takes the form of a straight line with a slope of -3. The irradiance from scattered light typically has a slope that is around -. From this we might guess that most ofthe stray light at small radii comes from diffraction, but the irradiance from scattered light takes over as the radius gets larger. We now compare the irradiance from scattered light with that from diffracted light, and derive an expression for the radius at which they make equal contributions. Confming our attention to the two-parameter Harvey model, we use Equation (19) for the irradiance from scattered light and Equation (34) for the irradiance from diffraction. Inspection shows that these equations have many factors in common, so we expect to obtain a relatively simple expression if we divide one by the other: 5 1 1OOTb (na)s+ aentr E. E (r) E(r) = EentT Xa IZ(na) r (35) 3+5 = b 1OO(na)3_r An interesting feature ofthis expression is that it has almost no dependence on the parameters ofthe imaging system. The system parameters enter only through the numerical aperture and the beam diameter, c, at thejth element. Furthermore, the dependence on the beam diameter is weak. For example, ifthe slope ofthe BSDF is -, then the exponent on the beam diameter is zero and the dependence on a drops out altogether. We conclude with an example in which we solve for the radius at which scatter and diffraction are equal. Consider a system with five elements in which the BSDF of every element is characterized by a Harvey model with an intercept of 0. 1 and slope of -. Let the system operate at a visible wavelength of 550 nm, and let the numerical aperture be 0.. Entering a slope of - into Equation (35), setting the left hand side to 1, and solving for r gives us an expression for the distance from the image point at which diffraction and scatter are equal: i' 1 1 O.55x103=1.8mm, 3 (36) Nb(na) 3 5(0. 1)(O.) where N is the number of elements in the system. For this system, diffraction is the dominant source of in-field stray light inside 1.8 mm, while scatter is largest outside 1.8 mm. 19 Proc. of SPIE Vol. 5178

10 REFERENCES 1. Robert R. Shannon, The Art and Science ofoptical Design, p. 41, Cambridge University Press, New York, James Elmer Harvey, Light-Scattering Characteristics of Optical Surfaces, Ph.D. Dissertation, University of Arizona, Michael G. Dittman, "Contamination Scatter Functions for Stray-Light Analysis", Proceedings of the SPIE, Optical System Contamination: Effects, Measurements, and Control, 4774, pp , Jack D. Gaskill, Linear Systems, Fourier Transforms, and Optics, p. 7, John Wiley and Sons, New York, Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, p. 364, National Bureau of Standards, Washington D.C., 197. Proc. of SPIE Vol