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Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media
1 October 2002
Optics Communications 211 (2002) 15–30 www.elsevier.com/locate/optcom
Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media Stanislaw Tryka * Laboratory of Physics, Institute of Agricultural Sciences, University of Agriculture in Lublin, Szczebrzeska 102, PL-22-400 Zamosc, Poland Received 18 January 2002; received in revised form 23 July 2002; accepted 8 August 2002
Abstract A general formula is derived for calculating the optical radiation flux that illuminates a circular disk from an off-axis point source if the disk and the source are embedded in two different homogeneous media and if the media form a refractive planar interface parallel to the disk. This formula is represented by a set of four different expressions, each valid for a different range of distances determining the position between the source and the disk symmetry axis. These expressions are given as double-definite integrals with the boundary conditions described by some transcendental equations. All expressions represent functions depending on the disk radius, on the distances between the refractive plane from the disk and from the source, on the distance between the source and the disk symmetry axis, on the refractive indices and on the absorption coefficients of both media embedding the source and the disk. In many practical situations the formula simplifies to a set of five single integrals that can be easily calculated numerically. For illustrative purposes some examples of the formula used for calculating optical radiative fluxes are presented and illustrated graphically. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 07.60.Dq; 42.15.Eq; 42.25.Bs; 42.25.Gy Keywords: Radiometry; Radiative flux; Radiative transfer; Off-axis point source; Circular disk
1. Introduction The flux of radiation (the radiative flux) or the power of radiation (the radiative power) incident on an object of a given shape from a given source is necessary for calculations in various problems in physics, chemistry and biology. Such calculations are required in optical engineering for assembling, optimizing and *
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0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 6 3 - 1
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S. Tryka / Optics Communications 211 (2002) 15–30
calibrating radiometric devices that measure optical radiation from a given source illuminating a given detector. Therefore accurate estimation of the radiative flux incident on a given object from a given source is very important in radiometry, a discipline of optical science and engineering. Usually the radiative fluxes illuminating the objects can be estimated analytically only for some simple models of radiating sources and object shapes. However, for most models of radiating sources and objects of more complicated shapes these problems have often gone unsolved because of very obvious mathematical difficulties. These difficulties will increase as we consider various object–source geometries, separation of the source from the object by different media and radiation which is not distributed in space in a simple way [1,2]. Therefore, in practice most of the optical systems must be calibrated and so-called camera equations must be used for accurate detection of the radiative flux [3,4]. Of the many optical system geometries, rotational symmetry is considered most suitable for exact fabrication and reduction of assembly errors. Thus, optical systems of rotational symmetry have been intensively studied for over a century. Today various analytical and approximate solutions are known for the models of point [5–12], spherical [11] and circular [5,6,10,11,13] sources illuminating a circular disk if the source and the disk are both embedded in one homogeneous medium. However, in more complicated cases if the radiation propagates in two or more different homogeneous media, some solutions obtained so far give only approximate results [9,11,14] or are very complicated and are usually unsuitable for direct application. In many investigations the optical radiative flux incident on a circular detector from an off-axis point source embedded in a different medium from that of the detector is estimated. In practice the optical radiation may propagate through liquids or solids on a detector isolated by air. Centers of chemiluminescent reactions [4,11], luminescent points in scintillation cameras [15] and many small elements immersed in a different medium from that of the detector and emitting radiation [16] may be considered typical examples of similar situations. Obviously, mathematical formulas are necessary to determine such fluxes of radiation. By summing the fluxes from the respective single-point emitters or by integrating mathematical formulas with respect to the line, surface or volume elements, when the functions describing the distributions of single-point emitters are known, it may also be possible to estimate radiative fluxes from some types of multiple-point sources. In this paper, we derived a formula for optical radiative flux calculations from an off-axis point source illuminating the planar circular disk. This formula was derived for the disk embedded in a different medium than the source and if it is in rotational symmetry with the refractive planar interface between the two media. The study presented in the paper was limited only to classical radiometry in the domain of geometrical optics. The formula was derived from some fundamental laws of geometrical optics and then applied for two homogeneous and isotropic media that can attenuate the radiation. It was shown that the formula obtained could easily be programmed and evaluated numerically. Some graphics presenting examples of selected numerical evaluations are enclosed in the final part of the paper.
2. Fundamental assumptions The scheme of the optical system analyzed in the paper is shown in Fig. 1. S2 denotes the surface illuminated by the radiation from the point source P. In practical situations this surface may represent circular apertures in some optical systems or a planar circular surface of some collectors, detectors or other planar circular objects illuminated by the optical radiation. S1 is the planar refractive interface between two different homogeneous and isotropic media embedding P and S2 . H1 represents the distance between P and S1 and H2 is the distance between S1 and S2 , while n1 and n2 denote the refractive indices of the media embedding P and S2 , respectively. The point P1 is a virtual image of the source P and lies at a distance H0 from P. Both points P and P1 lie on the z1 -axis at a distance q from the z-axis. The z-axis is the symmetry axis of the optical system. Throughout the paper we will not consider propagation of the radiation through the
S. Tryka / Optics Communications 211 (2002) 15–30
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Fig. 1. Schematic diagram of an optical system showing the point source P contributing an infinitesimal radiative flux to an infinitesimal surface dS2 (a) and perpendicular projection of this diagram on 01 x1 z1 plane (b). The point P1 represents a virtual image of the source P. This image is seen by an observer stationed in the second medium. The ray trajectory are shown for the case of n2 > n1 .
lateral surfaces of the media. Therefore we shall assume that q, R1 and R2 fulfill the relations q < R1 , q < R2 and R2 < R1 . To facilitate physical interpretations of the mathematical formulas derived in this report all geometric and radiometric quantities and symbols used herein were adopted from generally accepted radiometric terminologies [2,17]. A list of these quantities, symbols and their units is presented in Table 1 in Appendix A. The study presented in this report will concern only the point source model. However such a source of radiation is only a mathematical concept and does not exist in material world because it must represent infinite radiant volume energy density in a geometrical point. The point source model is very useful for analyzing many optical problems where the spatial dimension of a real extended source is not resolved by the resolution of an optical system or by the human eye [2]. Obviously such a point source may represent any real source at an infinite distance. In practice, the finite extent of a source is sometimes ignored if its diameter or dimension is less than about 1/20 of the distance to the irradiated surface [17].
3. Basic equations Let us consider first the radiative flux in a hypothetical spherical coordinate system erected at a point P1 (see Fig. 1). An infinitesimal radiative flux, dUP 1;dS2 , incident on an infinitesimal surface, dS2 , at an azimuth angle, h2 , with respect to the x1 -axis, and forming a virtual image at the point P1 , will be given by [2,17]
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S. Tryka / Optics Communications 211 (2002) 15–30
dUP 1;dS2 ¼ Iðh2 ; u2 Þ dxP 1;dS2 ;
ð1Þ
where Iðh2 ; u2 Þ is the intensity of the radiation passing through dS2 and dxP1 ;dS2 is an infinitesimal solid angle, which dS2 subtends at P1 . After calculation of dxP 1;dS2 ¼ sin h2 dh2 du2 (see Eq. (B.2) in Appendix B), and substitution into Eq. (1) we get dUP 1;dS2 ¼ Iðh2 ; u2 Þ sin h2 dh2 du2 :
ð2Þ
In practice it may sometimes be inconvenient to express the radiant intensity as a function of h2 and u2 because it is usually defined in spherical coordinate systems originating at the point source P. Therefore it is desirable to express the radiant intensity as a function of the azimuth angle h1 and of the horizontal angle u1 . From Fig. 1(a) it is readily seen that u1 ¼ u2 and the differentiation of u2 yields du2 ¼ du1 . Thus by substituting Snell’s law [18,19] sin h2 ¼ n1 sin h1 =n2 ;
ð3Þ
with the derivative of h2 , obtained from Eq. (3), dh2 ¼ ½n1 cos h1 =ðn22 n21 sin2 h1 Þ
1=2
dh1
into Eq. (2), for the infinitesimal radiative flux, dUP ;dS2 , in spherical coordinate systems erected at the point P, we get the following expression: dUP ;dS2 ¼ AS2 ðn1 ; n2 ; h1 ÞIðh1 ; u1 Þ dh1 du1 ; where AS2 ðn1 ; n2 ; h1 Þ ¼ n21 sin h1 cos h1
ð4Þ
.h 1=2 i n2 n22 n21 sin2 h1
when 0 6 h1 < p=2 ^ 0 6 h1 < arcsinðn2 =n1 Þ:
ð5Þ
The differential radiative flux, dUP ;S2 , enclosed between h1 and h1 þ dh1 is equal to the definite integral of dUP ;dS2 with respect to the angle u1 . Calculating dUP ;S2 we will assume that the angle h1 is a parameter in Eq. (4). Thus this integration may be done in the cylindrical coordinate system 01 r2 u2 z1 (or 01 r2 u1 z1 because as was noted u1 ¼ u2 ). In this coordinate system, the functional dependence of dUP ;dS2 on the angle h1 may be disregarded. This manner we get: Z u00 Z u00 1 1 dUP ;S2 ¼ dUP ;dS2 ¼ AS2 ðn1 ; n2 ; h1 Þ dh1 Iðh1 ; u1 Þ du1 ; ð6Þ u01
u01
where the limiting angles u01 and u001 depend on whether the radiation confined between two conical cones of the apex angles h1 and h1 þ dh1 forms complete or incomplete rings on the surface S2 or in other words whether q 6 R2 r2 or q > R2 r2 . In Fig. 1(a) it can be seen that 0; q 6 R2 r2;h1 ; u01 ¼ ð7Þ p c2;h1 ; q > R2 r2;h1 ; 2p; q 6 R2 r2;h1 ; 00 u1 ¼ ð8Þ p þ c2;h1 ; q > R2 r2;h1 ; where c2;h1 and r2;h1 represent the angle c2 and the radius r2 , respectively, given as functions of the angle h1 . The angle, c2;h1 , calculated from the relation between R2 , q and r2 in Fig. 1(a), fulfills the following condition: 2 c2;h1 ¼ c2 ðR2 ; q; r2;h1 Þ ¼ arccos½ðq2 þ r2;h1 R22 Þ=2qr2;h1 ;
where the radius r2;h1 is given by
ð9Þ
S. Tryka / Optics Communications 211 (2002) 15–30
r2;h1 ¼ r2 ðH1 ; H2 ; n1 ; n2 ; h1 Þ ¼ H1 tan h1 þ H2 n1 sin h1 =ðn22 n21 sin2 h1 Þ when 0 6 h1 < p=2 ^ 0 6 h1 < arcsinðn2 =n1 Þ:
19
1=2
ð10Þ
Eqs. (9) and (10) clearly indicate that the angle c2;h1 must be considered as a function of several variables and c2;h1 ¼ c2 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; h1 Þ:
ð11Þ
4. Angular distribution of the radiative flux Eq. (6) can also be written as dUP ;S2 ¼ fP ;S2 dh1 ;
ð12Þ
where fP ;S2 ¼
dUP ;S2 ¼ AS2 ðn1 ; n2 ; h1 Þ dh1
Z
u001
u01
Iðh1 ; u1 Þ du1
ð13Þ
is a function representing the angular distribution of UP ;S2 with respect to the angle h1 . To obtain the formula for fP ;S2 , Eq. (13) must be integrated in two separate regions of the angle u1 given by Eqs. (7) and (8). These regions depend on whether the radiation between two conical cones of the apex angles h1 and h1 þ dh1 makes complete or incomplete rings on the surface S2 . Fig. 1(a) presents the situation were the radiation makes incomplete rings at q < R2 . Because at q ¼ R2 no complete ring will be formed, the calculation must be done in the four regions of q, similar to the calculation of integrals in Eq. (2.6) described in [12]. Here we obtain 8 R 2p 0 6 q < R2 ; 0 6 r2;h1 6 R2 q; AS2 ðn1 ; n2 ; h1 Þ 0 Iðh1 ; u1 Þ du1 ; > > > > A ðn ; n ; h Þ R pþc2;h1 Iðh ; u Þ du ; < 0 < q < R2 ; R2 q 6 r2;h1 6 R2 þ q; S2 1 2 1 1 1 1 2;h1 Rpc ð14Þ fP ;S2 ¼ pþc2R2;h1 > AS2 ðn1 ; n2 ; h1 Þ pc2R2;h1 Iðh1 ; u1 Þ du1 ; 0 < q ¼ R2 ; 0 6 r2;h1 6 2R2 ; > > > : A ðn ; n ; h Þ R pþc2;h1 Iðh ; u Þ du ; 0 < R2 < q; q R2 6 r2;h1 6 R2 þ q; S2 1 2 1 1 1 1 pc2;h1 where the boundary angle c2R2;h1 is equal to c2;h1 from Eq. (9) if q ¼ R. Thus, c2R2;h1 ¼ c2 ðR2 ; r2;h1 Þ ¼ arccosðr2;h1 =2R2 Þ
ð15Þ
and by Eq. (10) it is seen that c2R2;h1 ¼ c2 ðR2 ; H1 ; H2 ; n1 ; n2 Þ:
ð16Þ
From Eq. (12), it is clear that the radiative flux, UP ;S2 , will be described by the expression Z h1;R2þq UP ;S2 ¼ fP ;S2 dh1 ;
ð17Þ
0
where h1;R2þq denotes the maximal angle under which the radiation may be incident on the surface S2 . The angle h1;R2þq can be calculated from Eq. (10) and from the second boundary condition for r2;h1 of the last expression in formula (14) and is given by the following transcendental equation R2 þ q H1 tan h1;R2þq
H2 n1 sin H1;R2þq ðn22 n21 sin2 H1;R2þq Þ1=2
¼0
when 0 6 h1;R2þq < p=2 ^ 0 6 h1;R2þq < arcsinðn2 =n1 Þ:
ð18Þ
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5. Generalized formula for radiative flux calculations The respective expressions from formula (14) can be substituted into Eq. (17) and integrated in five domains of h1 and then written for the four separate regions of q, in a similar manner as in the approach presented during calculation of formula (4.25) in [12]. This way we obtain 8 R h1;R2 R 2p > AS2 ðn1 ; n2 ; h1 Þ dh1 0 ðh1 ; u1 Þ du1 ; 0 ¼ q < R2 ; > 0 > R 2p > R h1;R2q > > AS2 ðn1 ; n2 ; h1 Þ dh1 0 Iðh1 ; u1 Þ du1 > < 0 R h1;R2þq R pþc2;h1 þ h1;R2q AS2 ðn1 ; n2 ; h1 Þ dh1 pc2;h1 Iðh1 ; u1 Þ du1 ; 0 < q < R2 ; ð19Þ UP ;S2 ¼ > > R h1;2R2 A ðn ; n ; h Þ dh R pþc2R2;h1 Iðh ; u Þ du ; > 0 < q ¼ R2 ; > S2 1 2 1 1 pc 1 1 1 0 > 2R2;h1 > > : R h1;R2þq A ðn ; n ; h Þ dh R pþc2;h1 Iðh ; u Þ du ; 0 < R2 < q; S2 1 2 1 1 pc 1 1 1 h1;qR2 2;h1 where the limiting angles h1;R2 , h1;R2q , h1;2R2 and h1;qR2 , described by Eq. (10) and the second boundary conditions for r2;h1 in formula (14), similarly as h1;R2þq in Eq. (18), are given transcendentally by R2 H1 tan h1;R2
H2 n1 sin h1;R2 ðn22 n21 sin2 h1;R2 Þ
1=2
¼0
when 0 6 h1;R2 < p=2 ^ 0 6 h1;R2 < arcsinðn2 =n1 Þ; R2 q H1 tan h1;R2q
H2 n1 sin h1;R2q ðn22
n21 sin2 h1;R2q Þ
1=2
¼0
when 0 6 h1;R2q < p=2 ^ 0 6 h1;R2q < arcsinðn2 =n1 Þ; 2R2 H1 tan h1;2R2
H2 n1 sin h1;2R2 ðn22
n21 sin2 h1;2R2 Þ
1=2
H2 n1 sin h1;qR2 ðn22
ð21Þ
¼0
when 0 6 h1;2R2 < p=2 ^ 0 6 h1;2R2 < arcsinðn2 =n1 Þ; q R2 H1 tan h1;qR2
ð20Þ
1=2
n21 sin2 h1;qR2 Þ
ð22Þ
¼0
when 0 6 h1;qR2 < p=2 ^ 0 6 h1;qR2 < arcsinðn2 =n1 Þ:
ð23Þ
It is important to note that formula (19) was derived without any restriction made with respect to the angular dependencies of the radiant intensity function. We assumed only that the optical radiation propagates through two homogeneous isotropic media and across a refractive boundary, and that the first medium embeds the off-axis point source while the second medium embeds the circular disk. Therefore formula (19) is suitable for calculating various angularly distributed optical radiant fluxes in the optical system considered above. Additionally by substituting n1 instead of n2 in Eqs. (5), (10), (18), (20)–(23) formula (19) is simplified and is directly applicable for calculating radiative fluxes of optical radiations propagating in one homogeneous isotropic medium. From formula (19) it is easily seen that the flux UP ;S2 can be simply estimated if Iðh1 ; u1 Þ can be integrated analytically with respect to the angle u1 . Under such conditions the flux UP ;S2 will be given by a set of five single definite integrals that may sometimes be integrated analytically with respect to the angle h1 . Otherwise, these single integrals are easy to calculate numerically. If the radiant intensity is distributed in a complicated way around the z1 -axis, then Iðh1 ; u1 Þ may not be a simple function of u1 . In this situation the inner integrals in formula (19) may not be integrated analytically into the closed form expressions. When the inner integrals in formula (19) cannot be integrated analytically, the flux UP ;S2 must be estimated by using a numerical procedure for double integral evaluation such as the Monte Carlo method [11].
S. Tryka / Optics Communications 211 (2002) 15–30
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The directorial characteristics of the emitted radiation always depend on the physical properties of the source surface and on its geometry. Additionally these characteristics clearly depend on the source holder. Therefore the radiant intensity function Iðh1 ; u1 Þ is usually unknown. In practice this function is often approximated by the finite sum of the Fourier harmonics [2,20,21]. The respective coefficients in the Fourier series are usually calculated by using the last square method of approximation. This method leads us to a very accurate prediction of Iðh1 ; u1 Þ in most practical applications by calculating a given number of linear combinations of the integer powers of cosine and sine functions of the angles h1 and u1 . The elementary theory of the integral calculation says that such functions are always integrable analytically with respect to the angle u1 . Thus formula (19) may be simplified to a set of some single definite integrals that can be easily calculated numerically.
6. Physical conditions influencing the radiative flux Formula (19) shows that the radiative flux UP ;S2 may be estimated if the radiant intensity function Iðh1 ; u1 Þ is known. This function can be given a priori in an arbitrary mathematical form. However in this case formula (19) will have only strictly mathematical significance because it will not describe any physical properties influencing transmission of the optical radiation through the matter. Thus in this section we will consider some physical conditions influencing UP ;S2 most significantly. It is well known that when optical radiation passes from one medium to another across a refractive boundary between these two media a portion of the radiative energy is lost. In homogeneous and isotropic media, this lost energy may be connected with the absorption phenomenon and the Fresnel loss at the boundary [2]. When the first medium absorbs optical radiation, the radiant intensity, IS1 ðh1 ; u1 Þ, of the radiation leaving the surface S1 will always be less than the radiant intensity, IP ðh1 ; u1 Þ, from the point source P. In a wide range of practical applications, the relation between IS1 ðh1 ; u1 Þ and IP ðh1 ; u1 Þ describes Bouguer– Lambert’s law of absorption [18] IS1 ðh1 ; u1 Þ ¼ IP ðh1 ; u1 Þ expða1 H1 = cos h1 Þ;
ð24Þ
where a1 is the absorption coefficient of the first medium. However if the radiation is incident on the interface between two media of different refractive indices, a certain fraction of the radiant energy passing through this interface into the second medium undergoes refraction. The intensity I12 ðh1 ; u1 Þ of the radiation transmitted through S1 is described by the following expression [19]: I12 ðh1 ; u1 Þ ¼ IS1 ðh1 ; u1 Þs12 ðn1 ; n2 ; h1 Þ;
ð25Þ
where s12 ðn1 ; n2 ; h1 Þ is the single-surface transmission factor for the optical radiation passing through S1 . The factor s12 ðn1 ; n2 ; h1 Þ in Eq. (25) depends on the polarization of the incident radiation. Therefore it must be defined separately as s12jj ðn1 ; n2 ; h1 Þ for the electric field vectors parallel to the plane of the incidence angle, and as s12? ðn1 ; n2 ; h1 Þ for the electric field vector perpendicular to it. For the convenience of potential users of the above equation s12jj ðn1 ; n2 ; h1 Þ and s12? ðn1 ; n2 ; h1 Þ are given by [19] s12jj ðn1 ; n2 ; h1 Þ ¼
s12? ðn1 ; n2 ; h1 Þ ¼
4n1 n22 cos h1 ðn22 n21 sin2 h1 Þ1=2 ½ðn22 cos h1 þ n1 ðn22 n21 sin2 h1 Þ 4n1 cos h1 ðn22 n21 sin2 h1 Þ
1=2 2
1=2
½ðn1 cos h1 þ ðn22 n21 sin2 h1 Þ1=2 2
:
;
22
S. Tryka / Optics Communications 211 (2002) 15–30
In a simple case of natural unpolarized radiation we can assume that a portion of the radiation transmitted through S1 is proportional to the average of the factors s12jj ðn1 ; n2 ; h1 Þ and s12? ðn1 ; n2 ; h1 Þ. In this case, we can write s12 ðn1 ; n2 ; h1 Þ ¼ ½s12jj ðn1 ; n2 ; h1 Þ þ s12? ðn1 ; n2 ; h1 Þ=2: Assuming again that the second medium is homogeneous and isotropic and that it absorbs the radiation we have Iðh1 ; u1 Þ ¼ IS2 ðh1 ; u1 Þ ¼ I12 ðh1 ; u1 Þ expða2 H2 = cos h2 Þ;
ð26Þ
where Iðh1 ; u1 Þ is the intensity at the surface S2 and a2 is the absorption coefficient of the second medium. Combining Eqs. (3) and (24)–(26) we get Iðh1 ; u1 Þ ¼ IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ; where
ð27Þ
"
a1 H1 a2 H2 n2 sðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ ¼ s12 ðn1 ; n2 ; h1 Þ exp cos h1 ðn22 n21 sin2 h1 Þ1=2 ^
when 0 6 h1 < p=2
#
0 6 h1 < arcsinðn2 =n1 Þ:
If we now substitute Eq. (27) into formula (19) we readily obtain the following formula for UP ;S2 8 R h1;R2 R 2p > AS2 ðn1 ; n2 ; h1 Þ dh1 ; 0 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 ¼ q < R2 ; > 0 > R h1;R2q R 2p > > > AS2 ðn1 ; n2 ; h1 Þ dh1 0 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 > < 0 R h1;R2þq R pþc2;h1 þ AS2 ðn1 ; n2 ; h1 Þ dh1 pc2;h1 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 < q < R2 ; UP ;S2 ¼ h1;R2q R h1;2R2 R pþc2R2;h1 > > > AS2 ðn1 ; n2 ; h1 Þ dh1 pc2R2;h1 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 < q ¼ R2 ; > > R0 > > : h1;R2þq A ðn ; n ; h Þ dh R pþc2;h1 I ðh ; u ÞsðH ; H ; n ; n ; a ; a ; h Þ du ; 0 < R < q: h1;qR2
S2
1
2
1
1
pc2;h1
P
1
1
1
2
1
2
1
2
1
1
2
ð28Þ The formula given above determines the amount of radiation emitted by the point source at the angles h1 and u1 and transmitted through two different homogeneous isotropic media absorbing this radiation and forming the boundary obeying Snell’s law of refraction. In fact, the radiation may also be scattered in both transparent mediums. The scattering phenomenon is usually described by an exponential function, similarly as the absorption process [2]. Thus the radiative flux scattered in two different homogeneous and transparent media will be given by formula (28) if we substitute for the absorption coefficients a1 and a2 in Eqs. (24) and (26) the scattering coefficients kS1 and kS2 , respectively. In optical systems the absorbed and scattered radiation do not transfer any information from the source to the image plane. Therefore the absorption and scattering phenomena decrease the signal-to-noise ratio [2]. These two phenomena originate the stray radiation problems influencing on the quality of many optical instruments, such as infrared telescopes for example [1]. 7. Some practical examples of optical radiative flux calculations It was assumed in Section 6 that formula (19) is valid for the optical radiation of any state of its angular distribution. Unfortunately the angular distribution of radiation from most types of temporally emitting natural radiators is unknown and we cannot use formulas (19) or (28) directly to estimate UP ;S2 . In this case, we may solve an inverse problem to obtain the radiant intensity waiting function IP ðh1 ; u1 Þ. Here we will illustrate the usefulness and simplicity of the solutions obtained using formula (28) for analyzing some
S. Tryka / Optics Communications 211 (2002) 15–30
23
practical examples from illumination engineering. In these examples, we will assume that the radiant intensity from the point source model is distributed in some different ways with respect to the angles h1 and u1 and that the radiation is unpolarized. 7.1. Rectilinear source Let us consider the optical radiation emitted by a small rectilinear source and distributed analogously to the radiation from a dipole antenna. Obviously such radiation is angularly distributed non-uniformly with respect to the symmetry axis perpendicular to this source and is distributed uniformly with respect to the axis lying along this source. In the examples presented below we will consider these two cases separately. 7.1.1. Rotationally unsymmetrical radiation When the small rectilinear source lies on the x-axis the radiant intensity will be described by [19] IP ðh1 ; u1 Þ ¼ I0 sin2 u1 ;
ð29Þ
where I0 is the maximal radiant intensity at u1 ¼ p=2. The angular distribution of IP ðh1 ; u1 Þ from Eq. (29) is plotted in Fig. 2(a). The center of the rectilinear source is placed at the point P ð0; 0; H0 Þ of the spherical coordinate system with the coordinates obtained by setting: x1 ¼ cos h1 , y1 ¼ sin h1 sin u1 and z1 ¼ sin h1 cos u1 . Substituting Eq. (29) into (28) and calculating the inner integrals in formula (28) we get 8 Rh > pIh0 0 1;R2 BS2 dh1 ; > > i 0 ¼ q < R2 ; > R h1;R2q R h1;R2þq > < BS2 dh1 þ h1;R2q BS2 C2;h1 dh1 ; 0 < q < R2 ; I0 p 0 ð30Þ UP ;S2 ¼ R h1;2R2 > I0 BS2 C2R2;h1 dh1 ; 0 < q ¼ R2 ; > > 0 > R > : I0 h1;R2þq BS2 C2;h1 dh1 ; 0 < R2 < q; h1;qR2
where BS2 ¼ AS2 ðn1 ; n2 ; h1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 ½r2;h1 ðR2 qÞ ðq2 þ r2;h1 R22 Þ=ð2r2;h1 qÞ ; C2;h1 ¼ c2;h1 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; h1 Þ ½ðR2 þ qÞ r2;h1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C2R2;h1 ¼ c2R2;h1 ðR1 ; H1 ; H2 ; n1 ; n2 ; h1 Þ 4R22 r2;h1 ð31Þ =ð2R2 Þ ; and by Eq. (10) it is seen that r2;h1 ¼ r2 ðH1 ; H2 ; n1 ; n2 ; h1 Þ. After integration of the respective expressions in formula (30) the flux UP ;S2 will depend on several variables and UP ;S2 ¼ UP ;S2 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; a1 ; a2 Þ.
Fig. 2. The angular distribution of the radiant intensity from a small rectilinear source emitting radiation identically as a dipole antenna lying on the x1 -axis (a) and on the z1 -axis (b). The center of the rectilinear source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
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Fig. 3. The radiative flux UP ;S2 from a small rectilinear source as a function of R2 and q for the source lying along the x1 -axis (a) and along the z1 -axis (b). The results were calculated numerically using Mathematica 2.2.3 [22] from formulas (30) and (33) for the cases (a) and (b), respectively. These calculations were done for the relative values of R2 , q, H1 and H2 when R2 and q both vary from 0 to 10 and at constant values of H1 ¼ H2 ¼ 0:5, n1 ¼ 1:0, n2 ¼ 1:5, and a1 ¼ a2 ¼ 0.
Fig. 3(a) presents the results of numerical calculations of UP ;S2 from formula (30) obtained at relative values of H1 ¼ H2 ¼ 0:5 for R2 and q varying from 0 to 10. The surface plotted in this figure shows that in the regions where q and R2 are comparable the radiative flux UP ;S2 strongly increases with an increase of R2 or with a decrease of the distance q between the point source and the z1 -axis. For q sufficiently smaller than R2 the flux UP ;S2 is slowly increased with an increase of R2 or with a decrease of q. However in the region of q greater than R2 the value of UP ;S2 tends to zero as q is increased or R2 is decreased. Generally it can be stated here that a small displacement of the rectilinear point source from the symmetry axis of the optical system considered here leads only to a small variation in the UP ;S2 values if the radius R2 is sufficiently greater than q. 7.1.2. Rotationally symmetrical radiation For a source lying on the z1 -axis the radiant intensity will be distributed symmetrically with respect to it and is described by IP ðh1 ; u1 Þ ¼ I0 sin2 h1 ;
ð32Þ
where I0 is the maximal intensity at h1 ¼ 0. Fig. 2(b) presents the angular dependence of IP ðh1 ; u1 Þ from Eq. (32) plotted in the spherical coordinate system. The center of the source lies at the point P ð0; 0; H0 Þ of this system and the respective coordinates are obtained by setting x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 sin u1 and z1 ¼ cos h1 . Here by substituting Eq. (32) into (28) we obtain formula 8 Rh > 2pIh0 0 1;R2 BS2 sin2 h1 dh1 ; > > i 0 ¼ q < R2 ; > > < 2I0 R h1;R2q BS2 sin2 h1 dh1 þ R h1;R2þq BS2 c sin2 h1 dh1 ; 0 < q < R2 ; 2;h1 0 h1;R2q ð33Þ UP ;S2 ¼ R h1;2R2 2 > 2I0 BS2 c2R2;h1 sin h1 dh1 ; 0 < q ¼ R2 ; > > 0 > R > : 2I0 h1;R2þq BS2 c sin2 h1 dh1 ; 0 < R2 < q; h1;qR2
2;h1
which is simpler than formula (30) for the rotationally unsymmetrical radiation. Fig. 3(b) illustrates the dependence of UP ;S2 on R2 and q calculated from formula (33). The results plotted here were obtained at the same values of the variables as in Fig. 3(a). Although the values of UP ;S2 and
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surface profiles in these both figures are similar it can be seen that UP ;S2 in Fig. 3(b) changes more slowly with the variation of R2 and q. However, this change occurs in the wider regions of R2 and q variations. Therefore any displacement of the source from the axial position in the optical system changes the values of the UP ;S2 more quickly than in the example presented above. 7.2. Lambertian source The Lambertian source is an extended surface with azimuthal symmetry of the radiance Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ independent on the direction of observation. This radiance is described by [19] Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ ¼ Lðx1 ¼ 0; y1 ¼ 0Þ ¼ L0 ;
ð34Þ
where the radiance L0 does not depend to the angle h1 . Because L0 does not depend on h1 this radiation is sometimes regarded as isotropic with respect to h1 (for h1 ranging from 0 to p=2). When the Lambertian source is considered as the point source the intensity IP ðh1 ; u1 Þ obeys Lambert’s law of radiation [19] IP ðh1 ; u1 Þ ¼ L0 S cos h1 ¼ I0 cos h1 ;
ð35Þ
where I0 ¼ L0 S is the maximal intensity at h1 ¼ 0. Fig. 4 illustrates the angular dependence of IP ðh1 ; u1 Þ from Eq. (35) plotted in spherical coordinates determined by x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 sin u1 and z1 ¼ cos h1 . The center of the Lambertian source represented in this figure by the planar circular disk lies at the point P ð0; 0; H0 Þ. Putting Eq. (35) into formula (28) and then calculating the inner integrals we have 8 Rh > 2pIh0 0 1;R2 BS2 cos h1 dh1 ; > > i 0 ¼ q < R2 ; > R h1;R2þq R h1;R2q > < BS2 cos h1 dh1 þ h1;R2q BS2 c2;h1 cos h1 dh1 ; 0 < q < R2 ; 2I0 0 ð36Þ UP ;S2 ¼ R h1;2R2 > BS2 c2R2;h1 cos h1 dh1 ; 0 < q ¼ R2 ; > 2I0 0 > > R > : 2I0 h1;R2þq BS2 c cos h1 dh1 : 0 < R2 < q: h1;qR2
2;h1
Typical results of numerical simulations of UP ;S2 from formula (36) are presented in Fig. 5(a) and (b). These simulations were done for the same values of the variables as in Figs. 3(a) and (b) but for two different cases when n1 < n2 and when n1 > n2 . Here the dependencies of UP ;S2 on R2 and q are also similar as in Fig. 3(a) but UP ;S2 is clearly greater than in Fig. 3(a). It is also seen that UP ;S2 attains greater values for the radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)), than from the medium of n1 < n2 to the medium of n2 > n1 (Fig. 5(a)). Also greater UP ;S2 variation can be observed in Fig. 5(b) than in Fig. 5(a) with the variation of the distance q of the source from the z-axis.
Fig. 4. The angular distribution of the radiant intensity from a Lambertian source lying in Px1 y1 plane. The center of the planar circular disk representing the Lambertian source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
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Fig. 5. The radiative flux UP ;S2 from a Lambertian source as a function of R2 and q for the radiation emitted from a medium of n1 ¼ 1:0 to a medium of n2 ¼ 1:5 (a) and for the radiation emitted from a medium of n1 ¼ 1:5 to a medium of n2 ¼ 1:0 (b). The results were calculated numerically from formula (36) for the relative values of R2 , q, H1 and H2 when R2 and q are both varied from 0 to 10 and at constant values of H1 ¼ H2 ¼ 0:5, and a1 ¼ a2 ¼ 0.
7.3. Spherical source Let us consider the optical radiation emitted by a small spherical source. If each element of the surface of the source has identical physical properties, the radiation may be considered as isotropic and IP ðh1 ; u1 Þ ¼ I0 ;
ð37Þ
where I0 is constant. Fig. 6 presents angular dependence of IP ðh1 , u1 ) plotted in the spherical coordinate system with its origin in the point 01 . The spherical coordinates are defined here by the transformation x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 cos u1 and z1 ¼ cos h1 . The center of the source is fixed at the point P ð0; 0; H0 Þ. Inserting Eq. (37) into formula (28) and calculating the inner integrals in this formula the radiative flux will be given by 8 Rh > 2pIh0 0 1;R2 BS2 dh1 ; > > i 0 ¼ q < R2 ; > > < 2I0 R h1;R2q BS2 dh1 þ R h1;R2þq BS2 c dh1 ; 0 < q < R2 ; 2;h1 0 h1;R2q ð38Þ UP ;S2 ¼ R h1;2R2 > BS2 c2R2;h1 dh1 ; 0 < q ¼ R2 ; 2I0 0 > > > R > : 2I0 h1;2Rþq BS2 c dh1 ; 0 < R2 < q: h1;qR2
2;h1
Fig. 6. The angular distribution of the radiant intensity from a spherical source. The center of the source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
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Fig. 7. The radiative flux UP ;S2 from the spherical source as a function of R2 and q for the radiation emitted through two different homogeneous isotropic media across the refractive boundary (a) and for the radiation emitted through one homogeneous isotropic medium (b). The results were calculated numerically from formula (38) for H1 ¼ H2 ¼ 0:5, n1 ¼ 1:0 and n2 ¼ 1:5 in the case (a) and for H1 þ H2 ¼ 1:0 and n1 ¼ n2 in the case (b). These calculations were done for the relative values of R2 , q, H1 and H2 when R2 and q both vary from 0 to 10 and at constant values of a1 ¼ a2 ¼ 0.
It worth noting that the obtained formula is directly applicable for calculating radiative fluxes of the isotropic radiations propagating in one homogeneous isotropic medium after substitution of n1 for n2 in Eqs. (5), (10), (18), (20)–(23) and (27). The resulting formula will be of the form UP ;S1 ¼ I0 xP ;S1 ;
ð39Þ
where S1 represents here the surface S2 of the disk and xP ;S1 is the solid angle that S1 subtends at P. Various solutions for xP ;S1 may be found elsewhere [5,7–10,12]. However the results evaluated numerically from formula (38) for the radiation propagating in one homogenous isotropic medium and the results obtained from Eq. (39) are nearly identical within the range of declared accuracy of numerical calculations. Fig. 7(a) shows results of numerical evaluations of UP ;S2 from formula (38) as a function of R2 and q at the same values of the variables as in Fig. 3(a). Here the radiative flux UP ;S2 depends on the position of the off-axis source with respect to the z-axis in a similar way as in the example for the Lambertian source from Fig. 5(a). The radiative flux UP ;S2 from the spherical source in Fig. 7(a) is clearly greater than UP ;S2 from the Lambertian source in Fig. 5(a). The surface plotted in Fig. 7(b) represents the results of UP ;S1 calculations for the radiation propagating in one homogeneous isotropic medium. This plot was simulated from formula (38) and was identical to the plot obtained using formula (39). The results in Fig. 7(b) suggest that the radiative flux UP ;S1 of the radiation propagating in one homogeneous isotropic medium attains values comparable to those obtained for the radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 , see Fig. 5(b). However it is also seen that UP ;S1 in Fig. 7(b) is more sensitive to the source position variation with respect to the z-axis than in Fig. 7(a).
8. Conclusion We derived the general formula (19) for calculating the fluxes UP ;S2 of the optical radiation illuminating a circular disk in optical systems where the off-axis point source and the disk are embedded in two different homogeneous isotropic media that can absorb the radiation and form a refractive surface parallel to the disk.
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This general formula was described by a set of four expressions, valid in different ranges of distance determining the point source position with respect to the disk symmetry axis. Each of these expressions has a different form of the double-definite integral of the same function represented by the product AS2 ðn1 ; n2 ; h1 Þ Iðh1 ; u1 Þ, where AS2 ðn1 ; n2 ; h1 Þ is described by Eq. (5) and Iðh1 ; u1 Þ is the radiant intensity function. It has been shown that it is quite simple to introduce the transmission coefficient into formula (19) for calculating UP ;S2 transmitted through the refractive surface between two different homogeneous and isotropic media. In addition this general formula can be easy modified into a form suitable for calculating UP ;S2 of the radiation propagated through the media absorbing the radiation according to Bouguer–Lambert’s law. Such considerations lead us to Eq. (27) representing the dependence of Iðh1 ; u1 Þ on the single-surface transmission factor s12 ðn1 ; n2 ; h1 Þ and on the absorption coefficients a1 and a2 of the first and second medium, respectively. Eq. (27) was then introduced into formula (19) to obtain a new formula directly applicable for calculating various fluxes of unpolarized optical radiations. In this manner we obtain Eq. (28) valid for the calculation of UP ;S2 if the optical radiation propagates in two homogeneous isotropic media and across the refractive boundary. Next we presented some examples of UP ;S2 calculations using formula (28). In these examples the radiation emitted by the rectilinear, Lambertian and spherical sources considered as the point source models was analyzed. It was shown that the smallest radiative fluxes UP ;S2 occur for the rectilinear, higher for the Lambertian and the highest for the spherical point source models. Comparing the results of UP ;S2 calculation for the Lambertian source it was also shown that the radiative fluxes depend on the relation between n1 and n2 . Greater values of UP ;S2 were obtained for radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)) than from the medium of n1 < n2 to the medium of n2 > n1 (Fig. 5(a)). However the highest radiative fluxes UP ;S1 were calculated for the isotropic radiation from a spherical source propagating in one homogeneous isotropic medium (see the case for n1 ¼ n2 illustrated in Fig. 7(b)). Analyzing the influence of the distance q of these sources from the symmetry axis of the optical system, it can be stated that the radiative flux UP ;S2 always depends on the variation of q. In the examples presented in the paper the clearest dependence of /P ;S2 on q was observed in the regions where q attains similar values as R2 . However this variation was most clear for the radiation from a spherical point source model propagating in one homogeneous isotropic medium (Fig. 7(b)) and then for the radiation from the Lambertian point source model propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)). According to intuitive expectation, increases of H1 and H2 as well as a1 and a2 always lead to the decrease of UP ;S2 . To analyze such dependence there is a need to show some new graphical illustrations. In this report, we do not present such relations because their plots are less interesting than those described above and look similar to the plot in Fig. 12 of [12]. The presented general formula (19) is suitable for numerical evaluations of many types of angularly distributed optical radiant fluxes. After integration with respect to a given line or surface elements this formula can also be suitable for calculating the radiative fluxes from linear and extended surface sources. In many practical situations the inner integrals can be calculated into closed form expressions and the obtained formula can be expressed by a set of five single definite integrals that sometimes may be calculated analytically or are easy to estimate numerically in a number of ways. Therefore the method described in this paper is accurate, fast and not difficult and appears to have some advantages over the series method based on Monte Carlo estimations of UP ;S2 from double or multiple integrals evaluations. Finally it can be stated that the mathematical dependencies presented in this paper were expressed by the total quantities of classical radiometry. However, all formulas derived herein are directly applicable to the total quantities of visual photometry and to the spectral radiometric and photometric quantities. Complete characteristics of these total and spectral radiometric and photometric quantities can be found in [2,17].
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Acknowledgements The author would like to express his thanks to the anonymous referees for their careful review of the manuscript and constructive suggestions that helped to improve the paper significantly.
Appendix A. Quantities and symbols In this appendix, we summarize in Table 1 the quantities and symbols used in this paper and their units. Table 1 A list of the quantities and symbols used, their units and nomenclature Quantity or symbol
Units
Nomenclature
a1 ; a2 AS2 ðn1 ; n2 ; h1 Þ dUP ;dS2 ; dUP 1;dS2 dUP ;S2 dxP 1;dS2 fP ;S2 ¼ dUP ;S2 =dh1 H0 ; H1 ; H2 Iðh2 ; u2 Þ; Iðh1 ; u1 Þ,IS1 ðh1 ; u1 Þ; I12 ðh1 ; u1 Þ; IP ðh1 ; u1 Þ; I0 ks1 ; ks2 Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ, Lðx1 ¼ 0; y1 ¼ 0Þ; L0 n1 ; n2 P ; P1 r1 ; u1 ; z1 and r2 ; u2 ; z1
1/m – Wa , lmb Wa , lmb sr (W/rad)a , (lm/rad)b m ðW=srÞa , cdb
Absorption coefficients Superposition of sine and cosine functions Infintesimal radianta or luminousb fluxes Differential radianta or luminousb flux Infinitesimal solid angle Angular distribution of UP ;S2 with respect to h1 Distances Radiant or luminous intensities as functions of various variables
1/m (W/m2 )a (1m/m2 )b – – –
Scattering coefficients Radianta or luminousb radiances as functions of various varibles
r1 ; r2 ; r1;h1 ; r2;h1 R1 ; R2 S1 ; S2 x,y,z and x1 ; y1 ; z1 z z1
m m m2 – – –
u1 ; u2 UP ;S1 ; UP ;S2 c1;h1 , c2R1;h1 , c2;h1 , c2R2;h1 q h1 ; h2 h1;R2 ; h1;R2q ; h1;2R2 , h1;qR2 ; h1;R2þq , h1;R1 ; h1;R1q , h1;2R1 , h1;qR1 ; h1;R1þq , s12 ðn1 ; n2 ; h1 Þ, s12jj ðn1 ; n2 ; h1 Þ, s12? ðn1 ; n2 ; h1 Þ, xP ;S1
rad Wa , lmb rad m rad rad
Refractive indices Point source and its virtual image, respectively axes of the cylindrical 0r1 u1 z1 and 01 r2 u2 z1 coordinate systems, respectively Radii as functions of various variables Radii Surfaces Axes of the Cartesian 0xyz and 01 x1 y1 z1 coordinate systems, respectively Symmetry axis of the optical system Symmetry axis of the conical surfaces made by optical ray trajectories at given h1 Horizontal angles Radiativea or luminousb fluxes Boundary angles Distance from the source to the z-axis Incidence and refraction angles, respectively Boundary angles
–
Single-surface transmission factors
–
Solid angle at P subtended by S1
a b
Radiometric units. Photometric units.
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Appendix B. Elementary solid range An infinitesimal solid angle, dxP 1;dS2 , is defined by the following equation [5,8] dxP 1;dS2 ¼ dS2 cos h2 =l2 ;
ðB:1Þ
where l represents the linear distance between P1 and an infinitesimal surface dS2 . The infinitesimal surface dS2 in Fig. 1(a) can be calculated as dS2 ¼ r2 dr2 du2 and the distance l ¼ ðH0 þ H1 þ H2 Þ= cos h2 , when 0 6 h2 < p=2. Then from simple trigonometric dependencies we have: r2 ¼ ðH0 þ H1 þ H2 Þ= tan h2 and taking the derivative of r2 we get: dr2 ¼ ðH0 þ H1 þ H2 Þdh2 = cos2 h2 . Therefore if (H0 þ H1 þ H2 Þ > 0 and h2 is limited to the region 0 6 h2 < p=2 Eq. (B.1) can be rewritten in the equivalent form dxP 1;dS2 ¼ sin h2 dh2 du2 :
ðB:2Þ
References [1] M.S. Scholl, G. Paez Padilla, Infrared Phys. Technol. 38 (1997) 25. [2] M. Strojnik, G. Paez, in: D. Malacara, B.J. Thompson (Eds.), Handbook of Optical Engineering, Marcel Dekker, New York, 2001. [3] P. Apian-Bennewitz, J. von der Hardt, Sol. Energy Mater. Sol. Cells 54 (1998) 309. [4] J.W. Hastings, G. Weber, J. Opt. Soc. Am. 53 (1963) 1410. [5] A.H. Jaffey, Rev. Sci. Instrum. 25 (1954) 349. [6] J.H. Smith, M.L. Storm, J. Appl. Phys. 25 (1954) 519. [7] A.V. Masket, Rev. Sci. Instrum. 28 (1957) 191. [8] F. Paxton, Rev. Sci. Instrum. 30 (1959) 254. [9] J.H. Hubbell, R.L. Bach, R.J. Herbold, J. Res. Natl. Bur. Stand. 65C (1961) 249. [10] R.P. Gardner, K. Verghese, Nucl. Instrum. Meth. 93 (1971) 163. [11] S.L. Jacques, Photochem. Photobiol. 67 (1988) 23. [12] S. Tryka, Opt. Commun. 137 (1997) 317. [13] S. Tryka, Rev. Sci. Instrum. 70 (1999) 3915. [14] P. Olivier, S. Rioux, D. Gagnon, Opt. Eng. 32 (1993) 2266. [15] W.J. Price, Nuclear Radiation Detection, McGraw-Hill, New York, 1958. [16] W.B. Joyce, R.Z. Bachrach, R.W. Dixon, D.A. Sealer, J. Appl. Phys. 45 (1974) 2229. [17] J.R. Mayer-Arendt, Appl. Opt. 7 (1968) 2081. [18] J.R. Meyer-Arendt, Introduction to Classical and Modern Optics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1972. [19] M. Born, E. Wolf, Principles of Optics, Pergamon, London, 1959. [20] T.E. Gureyev, K.A. Nuget, Opt. Commun. 133 (1997) 339. [21] R.J.D. Spurr, T.P. Kurosu, K.V. Chance, J. Quant. Spec. Rad. Trans. 68 (2001) 689. [22] S. Wollfram, Mathematica – A System for Doing Mathematics by Computer, Addison-Wesley, Reading, MA, 1993.
Optics Communications 211 (2002) 15–30 www.elsevier.com/locate/optcom
Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media Stanislaw Tryka * Laboratory of Physics, Institute of Agricultural Sciences, University of Agriculture in Lublin, Szczebrzeska 102, PL-22-400 Zamosc, Poland Received 18 January 2002; received in revised form 23 July 2002; accepted 8 August 2002
Abstract A general formula is derived for calculating the optical radiation flux that illuminates a circular disk from an off-axis point source if the disk and the source are embedded in two different homogeneous media and if the media form a refractive planar interface parallel to the disk. This formula is represented by a set of four different expressions, each valid for a different range of distances determining the position between the source and the disk symmetry axis. These expressions are given as double-definite integrals with the boundary conditions described by some transcendental equations. All expressions represent functions depending on the disk radius, on the distances between the refractive plane from the disk and from the source, on the distance between the source and the disk symmetry axis, on the refractive indices and on the absorption coefficients of both media embedding the source and the disk. In many practical situations the formula simplifies to a set of five single integrals that can be easily calculated numerically. For illustrative purposes some examples of the formula used for calculating optical radiative fluxes are presented and illustrated graphically. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 07.60.Dq; 42.15.Eq; 42.25.Bs; 42.25.Gy Keywords: Radiometry; Radiative flux; Radiative transfer; Off-axis point source; Circular disk
1. Introduction The flux of radiation (the radiative flux) or the power of radiation (the radiative power) incident on an object of a given shape from a given source is necessary for calculations in various problems in physics, chemistry and biology. Such calculations are required in optical engineering for assembling, optimizing and *
Tel.: +48-84-63-96-034x460; fax: +48-84-63-99-511. E-mail address: [email protected] (S. Tryka).
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 8 6 3 - 1
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calibrating radiometric devices that measure optical radiation from a given source illuminating a given detector. Therefore accurate estimation of the radiative flux incident on a given object from a given source is very important in radiometry, a discipline of optical science and engineering. Usually the radiative fluxes illuminating the objects can be estimated analytically only for some simple models of radiating sources and object shapes. However, for most models of radiating sources and objects of more complicated shapes these problems have often gone unsolved because of very obvious mathematical difficulties. These difficulties will increase as we consider various object–source geometries, separation of the source from the object by different media and radiation which is not distributed in space in a simple way [1,2]. Therefore, in practice most of the optical systems must be calibrated and so-called camera equations must be used for accurate detection of the radiative flux [3,4]. Of the many optical system geometries, rotational symmetry is considered most suitable for exact fabrication and reduction of assembly errors. Thus, optical systems of rotational symmetry have been intensively studied for over a century. Today various analytical and approximate solutions are known for the models of point [5–12], spherical [11] and circular [5,6,10,11,13] sources illuminating a circular disk if the source and the disk are both embedded in one homogeneous medium. However, in more complicated cases if the radiation propagates in two or more different homogeneous media, some solutions obtained so far give only approximate results [9,11,14] or are very complicated and are usually unsuitable for direct application. In many investigations the optical radiative flux incident on a circular detector from an off-axis point source embedded in a different medium from that of the detector is estimated. In practice the optical radiation may propagate through liquids or solids on a detector isolated by air. Centers of chemiluminescent reactions [4,11], luminescent points in scintillation cameras [15] and many small elements immersed in a different medium from that of the detector and emitting radiation [16] may be considered typical examples of similar situations. Obviously, mathematical formulas are necessary to determine such fluxes of radiation. By summing the fluxes from the respective single-point emitters or by integrating mathematical formulas with respect to the line, surface or volume elements, when the functions describing the distributions of single-point emitters are known, it may also be possible to estimate radiative fluxes from some types of multiple-point sources. In this paper, we derived a formula for optical radiative flux calculations from an off-axis point source illuminating the planar circular disk. This formula was derived for the disk embedded in a different medium than the source and if it is in rotational symmetry with the refractive planar interface between the two media. The study presented in the paper was limited only to classical radiometry in the domain of geometrical optics. The formula was derived from some fundamental laws of geometrical optics and then applied for two homogeneous and isotropic media that can attenuate the radiation. It was shown that the formula obtained could easily be programmed and evaluated numerically. Some graphics presenting examples of selected numerical evaluations are enclosed in the final part of the paper.
2. Fundamental assumptions The scheme of the optical system analyzed in the paper is shown in Fig. 1. S2 denotes the surface illuminated by the radiation from the point source P. In practical situations this surface may represent circular apertures in some optical systems or a planar circular surface of some collectors, detectors or other planar circular objects illuminated by the optical radiation. S1 is the planar refractive interface between two different homogeneous and isotropic media embedding P and S2 . H1 represents the distance between P and S1 and H2 is the distance between S1 and S2 , while n1 and n2 denote the refractive indices of the media embedding P and S2 , respectively. The point P1 is a virtual image of the source P and lies at a distance H0 from P. Both points P and P1 lie on the z1 -axis at a distance q from the z-axis. The z-axis is the symmetry axis of the optical system. Throughout the paper we will not consider propagation of the radiation through the
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17
Fig. 1. Schematic diagram of an optical system showing the point source P contributing an infinitesimal radiative flux to an infinitesimal surface dS2 (a) and perpendicular projection of this diagram on 01 x1 z1 plane (b). The point P1 represents a virtual image of the source P. This image is seen by an observer stationed in the second medium. The ray trajectory are shown for the case of n2 > n1 .
lateral surfaces of the media. Therefore we shall assume that q, R1 and R2 fulfill the relations q < R1 , q < R2 and R2 < R1 . To facilitate physical interpretations of the mathematical formulas derived in this report all geometric and radiometric quantities and symbols used herein were adopted from generally accepted radiometric terminologies [2,17]. A list of these quantities, symbols and their units is presented in Table 1 in Appendix A. The study presented in this report will concern only the point source model. However such a source of radiation is only a mathematical concept and does not exist in material world because it must represent infinite radiant volume energy density in a geometrical point. The point source model is very useful for analyzing many optical problems where the spatial dimension of a real extended source is not resolved by the resolution of an optical system or by the human eye [2]. Obviously such a point source may represent any real source at an infinite distance. In practice, the finite extent of a source is sometimes ignored if its diameter or dimension is less than about 1/20 of the distance to the irradiated surface [17].
3. Basic equations Let us consider first the radiative flux in a hypothetical spherical coordinate system erected at a point P1 (see Fig. 1). An infinitesimal radiative flux, dUP 1;dS2 , incident on an infinitesimal surface, dS2 , at an azimuth angle, h2 , with respect to the x1 -axis, and forming a virtual image at the point P1 , will be given by [2,17]
18
S. Tryka / Optics Communications 211 (2002) 15–30
dUP 1;dS2 ¼ Iðh2 ; u2 Þ dxP 1;dS2 ;
ð1Þ
where Iðh2 ; u2 Þ is the intensity of the radiation passing through dS2 and dxP1 ;dS2 is an infinitesimal solid angle, which dS2 subtends at P1 . After calculation of dxP 1;dS2 ¼ sin h2 dh2 du2 (see Eq. (B.2) in Appendix B), and substitution into Eq. (1) we get dUP 1;dS2 ¼ Iðh2 ; u2 Þ sin h2 dh2 du2 :
ð2Þ
In practice it may sometimes be inconvenient to express the radiant intensity as a function of h2 and u2 because it is usually defined in spherical coordinate systems originating at the point source P. Therefore it is desirable to express the radiant intensity as a function of the azimuth angle h1 and of the horizontal angle u1 . From Fig. 1(a) it is readily seen that u1 ¼ u2 and the differentiation of u2 yields du2 ¼ du1 . Thus by substituting Snell’s law [18,19] sin h2 ¼ n1 sin h1 =n2 ;
ð3Þ
with the derivative of h2 , obtained from Eq. (3), dh2 ¼ ½n1 cos h1 =ðn22 n21 sin2 h1 Þ
1=2
dh1
into Eq. (2), for the infinitesimal radiative flux, dUP ;dS2 , in spherical coordinate systems erected at the point P, we get the following expression: dUP ;dS2 ¼ AS2 ðn1 ; n2 ; h1 ÞIðh1 ; u1 Þ dh1 du1 ; where AS2 ðn1 ; n2 ; h1 Þ ¼ n21 sin h1 cos h1
ð4Þ
.h 1=2 i n2 n22 n21 sin2 h1
when 0 6 h1 < p=2 ^ 0 6 h1 < arcsinðn2 =n1 Þ:
ð5Þ
The differential radiative flux, dUP ;S2 , enclosed between h1 and h1 þ dh1 is equal to the definite integral of dUP ;dS2 with respect to the angle u1 . Calculating dUP ;S2 we will assume that the angle h1 is a parameter in Eq. (4). Thus this integration may be done in the cylindrical coordinate system 01 r2 u2 z1 (or 01 r2 u1 z1 because as was noted u1 ¼ u2 ). In this coordinate system, the functional dependence of dUP ;dS2 on the angle h1 may be disregarded. This manner we get: Z u00 Z u00 1 1 dUP ;S2 ¼ dUP ;dS2 ¼ AS2 ðn1 ; n2 ; h1 Þ dh1 Iðh1 ; u1 Þ du1 ; ð6Þ u01
u01
where the limiting angles u01 and u001 depend on whether the radiation confined between two conical cones of the apex angles h1 and h1 þ dh1 forms complete or incomplete rings on the surface S2 or in other words whether q 6 R2 r2 or q > R2 r2 . In Fig. 1(a) it can be seen that 0; q 6 R2 r2;h1 ; u01 ¼ ð7Þ p c2;h1 ; q > R2 r2;h1 ; 2p; q 6 R2 r2;h1 ; 00 u1 ¼ ð8Þ p þ c2;h1 ; q > R2 r2;h1 ; where c2;h1 and r2;h1 represent the angle c2 and the radius r2 , respectively, given as functions of the angle h1 . The angle, c2;h1 , calculated from the relation between R2 , q and r2 in Fig. 1(a), fulfills the following condition: 2 c2;h1 ¼ c2 ðR2 ; q; r2;h1 Þ ¼ arccos½ðq2 þ r2;h1 R22 Þ=2qr2;h1 ;
where the radius r2;h1 is given by
ð9Þ
S. Tryka / Optics Communications 211 (2002) 15–30
r2;h1 ¼ r2 ðH1 ; H2 ; n1 ; n2 ; h1 Þ ¼ H1 tan h1 þ H2 n1 sin h1 =ðn22 n21 sin2 h1 Þ when 0 6 h1 < p=2 ^ 0 6 h1 < arcsinðn2 =n1 Þ:
19
1=2
ð10Þ
Eqs. (9) and (10) clearly indicate that the angle c2;h1 must be considered as a function of several variables and c2;h1 ¼ c2 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; h1 Þ:
ð11Þ
4. Angular distribution of the radiative flux Eq. (6) can also be written as dUP ;S2 ¼ fP ;S2 dh1 ;
ð12Þ
where fP ;S2 ¼
dUP ;S2 ¼ AS2 ðn1 ; n2 ; h1 Þ dh1
Z
u001
u01
Iðh1 ; u1 Þ du1
ð13Þ
is a function representing the angular distribution of UP ;S2 with respect to the angle h1 . To obtain the formula for fP ;S2 , Eq. (13) must be integrated in two separate regions of the angle u1 given by Eqs. (7) and (8). These regions depend on whether the radiation between two conical cones of the apex angles h1 and h1 þ dh1 makes complete or incomplete rings on the surface S2 . Fig. 1(a) presents the situation were the radiation makes incomplete rings at q < R2 . Because at q ¼ R2 no complete ring will be formed, the calculation must be done in the four regions of q, similar to the calculation of integrals in Eq. (2.6) described in [12]. Here we obtain 8 R 2p 0 6 q < R2 ; 0 6 r2;h1 6 R2 q; AS2 ðn1 ; n2 ; h1 Þ 0 Iðh1 ; u1 Þ du1 ; > > > > A ðn ; n ; h Þ R pþc2;h1 Iðh ; u Þ du ; < 0 < q < R2 ; R2 q 6 r2;h1 6 R2 þ q; S2 1 2 1 1 1 1 2;h1 Rpc ð14Þ fP ;S2 ¼ pþc2R2;h1 > AS2 ðn1 ; n2 ; h1 Þ pc2R2;h1 Iðh1 ; u1 Þ du1 ; 0 < q ¼ R2 ; 0 6 r2;h1 6 2R2 ; > > > : A ðn ; n ; h Þ R pþc2;h1 Iðh ; u Þ du ; 0 < R2 < q; q R2 6 r2;h1 6 R2 þ q; S2 1 2 1 1 1 1 pc2;h1 where the boundary angle c2R2;h1 is equal to c2;h1 from Eq. (9) if q ¼ R. Thus, c2R2;h1 ¼ c2 ðR2 ; r2;h1 Þ ¼ arccosðr2;h1 =2R2 Þ
ð15Þ
and by Eq. (10) it is seen that c2R2;h1 ¼ c2 ðR2 ; H1 ; H2 ; n1 ; n2 Þ:
ð16Þ
From Eq. (12), it is clear that the radiative flux, UP ;S2 , will be described by the expression Z h1;R2þq UP ;S2 ¼ fP ;S2 dh1 ;
ð17Þ
0
where h1;R2þq denotes the maximal angle under which the radiation may be incident on the surface S2 . The angle h1;R2þq can be calculated from Eq. (10) and from the second boundary condition for r2;h1 of the last expression in formula (14) and is given by the following transcendental equation R2 þ q H1 tan h1;R2þq
H2 n1 sin H1;R2þq ðn22 n21 sin2 H1;R2þq Þ1=2
¼0
when 0 6 h1;R2þq < p=2 ^ 0 6 h1;R2þq < arcsinðn2 =n1 Þ:
ð18Þ
20
S. Tryka / Optics Communications 211 (2002) 15–30
5. Generalized formula for radiative flux calculations The respective expressions from formula (14) can be substituted into Eq. (17) and integrated in five domains of h1 and then written for the four separate regions of q, in a similar manner as in the approach presented during calculation of formula (4.25) in [12]. This way we obtain 8 R h1;R2 R 2p > AS2 ðn1 ; n2 ; h1 Þ dh1 0 ðh1 ; u1 Þ du1 ; 0 ¼ q < R2 ; > 0 > R 2p > R h1;R2q > > AS2 ðn1 ; n2 ; h1 Þ dh1 0 Iðh1 ; u1 Þ du1 > < 0 R h1;R2þq R pþc2;h1 þ h1;R2q AS2 ðn1 ; n2 ; h1 Þ dh1 pc2;h1 Iðh1 ; u1 Þ du1 ; 0 < q < R2 ; ð19Þ UP ;S2 ¼ > > R h1;2R2 A ðn ; n ; h Þ dh R pþc2R2;h1 Iðh ; u Þ du ; > 0 < q ¼ R2 ; > S2 1 2 1 1 pc 1 1 1 0 > 2R2;h1 > > : R h1;R2þq A ðn ; n ; h Þ dh R pþc2;h1 Iðh ; u Þ du ; 0 < R2 < q; S2 1 2 1 1 pc 1 1 1 h1;qR2 2;h1 where the limiting angles h1;R2 , h1;R2q , h1;2R2 and h1;qR2 , described by Eq. (10) and the second boundary conditions for r2;h1 in formula (14), similarly as h1;R2þq in Eq. (18), are given transcendentally by R2 H1 tan h1;R2
H2 n1 sin h1;R2 ðn22 n21 sin2 h1;R2 Þ
1=2
¼0
when 0 6 h1;R2 < p=2 ^ 0 6 h1;R2 < arcsinðn2 =n1 Þ; R2 q H1 tan h1;R2q
H2 n1 sin h1;R2q ðn22
n21 sin2 h1;R2q Þ
1=2
¼0
when 0 6 h1;R2q < p=2 ^ 0 6 h1;R2q < arcsinðn2 =n1 Þ; 2R2 H1 tan h1;2R2
H2 n1 sin h1;2R2 ðn22
n21 sin2 h1;2R2 Þ
1=2
H2 n1 sin h1;qR2 ðn22
ð21Þ
¼0
when 0 6 h1;2R2 < p=2 ^ 0 6 h1;2R2 < arcsinðn2 =n1 Þ; q R2 H1 tan h1;qR2
ð20Þ
1=2
n21 sin2 h1;qR2 Þ
ð22Þ
¼0
when 0 6 h1;qR2 < p=2 ^ 0 6 h1;qR2 < arcsinðn2 =n1 Þ:
ð23Þ
It is important to note that formula (19) was derived without any restriction made with respect to the angular dependencies of the radiant intensity function. We assumed only that the optical radiation propagates through two homogeneous isotropic media and across a refractive boundary, and that the first medium embeds the off-axis point source while the second medium embeds the circular disk. Therefore formula (19) is suitable for calculating various angularly distributed optical radiant fluxes in the optical system considered above. Additionally by substituting n1 instead of n2 in Eqs. (5), (10), (18), (20)–(23) formula (19) is simplified and is directly applicable for calculating radiative fluxes of optical radiations propagating in one homogeneous isotropic medium. From formula (19) it is easily seen that the flux UP ;S2 can be simply estimated if Iðh1 ; u1 Þ can be integrated analytically with respect to the angle u1 . Under such conditions the flux UP ;S2 will be given by a set of five single definite integrals that may sometimes be integrated analytically with respect to the angle h1 . Otherwise, these single integrals are easy to calculate numerically. If the radiant intensity is distributed in a complicated way around the z1 -axis, then Iðh1 ; u1 Þ may not be a simple function of u1 . In this situation the inner integrals in formula (19) may not be integrated analytically into the closed form expressions. When the inner integrals in formula (19) cannot be integrated analytically, the flux UP ;S2 must be estimated by using a numerical procedure for double integral evaluation such as the Monte Carlo method [11].
S. Tryka / Optics Communications 211 (2002) 15–30
21
The directorial characteristics of the emitted radiation always depend on the physical properties of the source surface and on its geometry. Additionally these characteristics clearly depend on the source holder. Therefore the radiant intensity function Iðh1 ; u1 Þ is usually unknown. In practice this function is often approximated by the finite sum of the Fourier harmonics [2,20,21]. The respective coefficients in the Fourier series are usually calculated by using the last square method of approximation. This method leads us to a very accurate prediction of Iðh1 ; u1 Þ in most practical applications by calculating a given number of linear combinations of the integer powers of cosine and sine functions of the angles h1 and u1 . The elementary theory of the integral calculation says that such functions are always integrable analytically with respect to the angle u1 . Thus formula (19) may be simplified to a set of some single definite integrals that can be easily calculated numerically.
6. Physical conditions influencing the radiative flux Formula (19) shows that the radiative flux UP ;S2 may be estimated if the radiant intensity function Iðh1 ; u1 Þ is known. This function can be given a priori in an arbitrary mathematical form. However in this case formula (19) will have only strictly mathematical significance because it will not describe any physical properties influencing transmission of the optical radiation through the matter. Thus in this section we will consider some physical conditions influencing UP ;S2 most significantly. It is well known that when optical radiation passes from one medium to another across a refractive boundary between these two media a portion of the radiative energy is lost. In homogeneous and isotropic media, this lost energy may be connected with the absorption phenomenon and the Fresnel loss at the boundary [2]. When the first medium absorbs optical radiation, the radiant intensity, IS1 ðh1 ; u1 Þ, of the radiation leaving the surface S1 will always be less than the radiant intensity, IP ðh1 ; u1 Þ, from the point source P. In a wide range of practical applications, the relation between IS1 ðh1 ; u1 Þ and IP ðh1 ; u1 Þ describes Bouguer– Lambert’s law of absorption [18] IS1 ðh1 ; u1 Þ ¼ IP ðh1 ; u1 Þ expða1 H1 = cos h1 Þ;
ð24Þ
where a1 is the absorption coefficient of the first medium. However if the radiation is incident on the interface between two media of different refractive indices, a certain fraction of the radiant energy passing through this interface into the second medium undergoes refraction. The intensity I12 ðh1 ; u1 Þ of the radiation transmitted through S1 is described by the following expression [19]: I12 ðh1 ; u1 Þ ¼ IS1 ðh1 ; u1 Þs12 ðn1 ; n2 ; h1 Þ;
ð25Þ
where s12 ðn1 ; n2 ; h1 Þ is the single-surface transmission factor for the optical radiation passing through S1 . The factor s12 ðn1 ; n2 ; h1 Þ in Eq. (25) depends on the polarization of the incident radiation. Therefore it must be defined separately as s12jj ðn1 ; n2 ; h1 Þ for the electric field vectors parallel to the plane of the incidence angle, and as s12? ðn1 ; n2 ; h1 Þ for the electric field vector perpendicular to it. For the convenience of potential users of the above equation s12jj ðn1 ; n2 ; h1 Þ and s12? ðn1 ; n2 ; h1 Þ are given by [19] s12jj ðn1 ; n2 ; h1 Þ ¼
s12? ðn1 ; n2 ; h1 Þ ¼
4n1 n22 cos h1 ðn22 n21 sin2 h1 Þ1=2 ½ðn22 cos h1 þ n1 ðn22 n21 sin2 h1 Þ 4n1 cos h1 ðn22 n21 sin2 h1 Þ
1=2 2
1=2
½ðn1 cos h1 þ ðn22 n21 sin2 h1 Þ1=2 2
:
;
22
S. Tryka / Optics Communications 211 (2002) 15–30
In a simple case of natural unpolarized radiation we can assume that a portion of the radiation transmitted through S1 is proportional to the average of the factors s12jj ðn1 ; n2 ; h1 Þ and s12? ðn1 ; n2 ; h1 Þ. In this case, we can write s12 ðn1 ; n2 ; h1 Þ ¼ ½s12jj ðn1 ; n2 ; h1 Þ þ s12? ðn1 ; n2 ; h1 Þ=2: Assuming again that the second medium is homogeneous and isotropic and that it absorbs the radiation we have Iðh1 ; u1 Þ ¼ IS2 ðh1 ; u1 Þ ¼ I12 ðh1 ; u1 Þ expða2 H2 = cos h2 Þ;
ð26Þ
where Iðh1 ; u1 Þ is the intensity at the surface S2 and a2 is the absorption coefficient of the second medium. Combining Eqs. (3) and (24)–(26) we get Iðh1 ; u1 Þ ¼ IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ; where
ð27Þ
"
a1 H1 a2 H2 n2 sðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ ¼ s12 ðn1 ; n2 ; h1 Þ exp cos h1 ðn22 n21 sin2 h1 Þ1=2 ^
when 0 6 h1 < p=2
#
0 6 h1 < arcsinðn2 =n1 Þ:
If we now substitute Eq. (27) into formula (19) we readily obtain the following formula for UP ;S2 8 R h1;R2 R 2p > AS2 ðn1 ; n2 ; h1 Þ dh1 ; 0 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 ¼ q < R2 ; > 0 > R h1;R2q R 2p > > > AS2 ðn1 ; n2 ; h1 Þ dh1 0 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 > < 0 R h1;R2þq R pþc2;h1 þ AS2 ðn1 ; n2 ; h1 Þ dh1 pc2;h1 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 < q < R2 ; UP ;S2 ¼ h1;R2q R h1;2R2 R pþc2R2;h1 > > > AS2 ðn1 ; n2 ; h1 Þ dh1 pc2R2;h1 IP ðh1 ; u1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ du1 ; 0 < q ¼ R2 ; > > R0 > > : h1;R2þq A ðn ; n ; h Þ dh R pþc2;h1 I ðh ; u ÞsðH ; H ; n ; n ; a ; a ; h Þ du ; 0 < R < q: h1;qR2
S2
1
2
1
1
pc2;h1
P
1
1
1
2
1
2
1
2
1
1
2
ð28Þ The formula given above determines the amount of radiation emitted by the point source at the angles h1 and u1 and transmitted through two different homogeneous isotropic media absorbing this radiation and forming the boundary obeying Snell’s law of refraction. In fact, the radiation may also be scattered in both transparent mediums. The scattering phenomenon is usually described by an exponential function, similarly as the absorption process [2]. Thus the radiative flux scattered in two different homogeneous and transparent media will be given by formula (28) if we substitute for the absorption coefficients a1 and a2 in Eqs. (24) and (26) the scattering coefficients kS1 and kS2 , respectively. In optical systems the absorbed and scattered radiation do not transfer any information from the source to the image plane. Therefore the absorption and scattering phenomena decrease the signal-to-noise ratio [2]. These two phenomena originate the stray radiation problems influencing on the quality of many optical instruments, such as infrared telescopes for example [1]. 7. Some practical examples of optical radiative flux calculations It was assumed in Section 6 that formula (19) is valid for the optical radiation of any state of its angular distribution. Unfortunately the angular distribution of radiation from most types of temporally emitting natural radiators is unknown and we cannot use formulas (19) or (28) directly to estimate UP ;S2 . In this case, we may solve an inverse problem to obtain the radiant intensity waiting function IP ðh1 ; u1 Þ. Here we will illustrate the usefulness and simplicity of the solutions obtained using formula (28) for analyzing some
S. Tryka / Optics Communications 211 (2002) 15–30
23
practical examples from illumination engineering. In these examples, we will assume that the radiant intensity from the point source model is distributed in some different ways with respect to the angles h1 and u1 and that the radiation is unpolarized. 7.1. Rectilinear source Let us consider the optical radiation emitted by a small rectilinear source and distributed analogously to the radiation from a dipole antenna. Obviously such radiation is angularly distributed non-uniformly with respect to the symmetry axis perpendicular to this source and is distributed uniformly with respect to the axis lying along this source. In the examples presented below we will consider these two cases separately. 7.1.1. Rotationally unsymmetrical radiation When the small rectilinear source lies on the x-axis the radiant intensity will be described by [19] IP ðh1 ; u1 Þ ¼ I0 sin2 u1 ;
ð29Þ
where I0 is the maximal radiant intensity at u1 ¼ p=2. The angular distribution of IP ðh1 ; u1 Þ from Eq. (29) is plotted in Fig. 2(a). The center of the rectilinear source is placed at the point P ð0; 0; H0 Þ of the spherical coordinate system with the coordinates obtained by setting: x1 ¼ cos h1 , y1 ¼ sin h1 sin u1 and z1 ¼ sin h1 cos u1 . Substituting Eq. (29) into (28) and calculating the inner integrals in formula (28) we get 8 Rh > pIh0 0 1;R2 BS2 dh1 ; > > i 0 ¼ q < R2 ; > R h1;R2q R h1;R2þq > < BS2 dh1 þ h1;R2q BS2 C2;h1 dh1 ; 0 < q < R2 ; I0 p 0 ð30Þ UP ;S2 ¼ R h1;2R2 > I0 BS2 C2R2;h1 dh1 ; 0 < q ¼ R2 ; > > 0 > R > : I0 h1;R2þq BS2 C2;h1 dh1 ; 0 < R2 < q; h1;qR2
where BS2 ¼ AS2 ðn1 ; n2 ; h1 ÞsðH1 ; H2 ; n1 ; n2 ; a1 ; a2 ; h1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 ½r2;h1 ðR2 qÞ ðq2 þ r2;h1 R22 Þ=ð2r2;h1 qÞ ; C2;h1 ¼ c2;h1 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; h1 Þ ½ðR2 þ qÞ r2;h1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C2R2;h1 ¼ c2R2;h1 ðR1 ; H1 ; H2 ; n1 ; n2 ; h1 Þ 4R22 r2;h1 ð31Þ =ð2R2 Þ ; and by Eq. (10) it is seen that r2;h1 ¼ r2 ðH1 ; H2 ; n1 ; n2 ; h1 Þ. After integration of the respective expressions in formula (30) the flux UP ;S2 will depend on several variables and UP ;S2 ¼ UP ;S2 ðR2 ; q; H1 ; H2 ; n1 ; n2 ; a1 ; a2 Þ.
Fig. 2. The angular distribution of the radiant intensity from a small rectilinear source emitting radiation identically as a dipole antenna lying on the x1 -axis (a) and on the z1 -axis (b). The center of the rectilinear source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
24
S. Tryka / Optics Communications 211 (2002) 15–30
Fig. 3. The radiative flux UP ;S2 from a small rectilinear source as a function of R2 and q for the source lying along the x1 -axis (a) and along the z1 -axis (b). The results were calculated numerically using Mathematica 2.2.3 [22] from formulas (30) and (33) for the cases (a) and (b), respectively. These calculations were done for the relative values of R2 , q, H1 and H2 when R2 and q both vary from 0 to 10 and at constant values of H1 ¼ H2 ¼ 0:5, n1 ¼ 1:0, n2 ¼ 1:5, and a1 ¼ a2 ¼ 0.
Fig. 3(a) presents the results of numerical calculations of UP ;S2 from formula (30) obtained at relative values of H1 ¼ H2 ¼ 0:5 for R2 and q varying from 0 to 10. The surface plotted in this figure shows that in the regions where q and R2 are comparable the radiative flux UP ;S2 strongly increases with an increase of R2 or with a decrease of the distance q between the point source and the z1 -axis. For q sufficiently smaller than R2 the flux UP ;S2 is slowly increased with an increase of R2 or with a decrease of q. However in the region of q greater than R2 the value of UP ;S2 tends to zero as q is increased or R2 is decreased. Generally it can be stated here that a small displacement of the rectilinear point source from the symmetry axis of the optical system considered here leads only to a small variation in the UP ;S2 values if the radius R2 is sufficiently greater than q. 7.1.2. Rotationally symmetrical radiation For a source lying on the z1 -axis the radiant intensity will be distributed symmetrically with respect to it and is described by IP ðh1 ; u1 Þ ¼ I0 sin2 h1 ;
ð32Þ
where I0 is the maximal intensity at h1 ¼ 0. Fig. 2(b) presents the angular dependence of IP ðh1 ; u1 Þ from Eq. (32) plotted in the spherical coordinate system. The center of the source lies at the point P ð0; 0; H0 Þ of this system and the respective coordinates are obtained by setting x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 sin u1 and z1 ¼ cos h1 . Here by substituting Eq. (32) into (28) we obtain formula 8 Rh > 2pIh0 0 1;R2 BS2 sin2 h1 dh1 ; > > i 0 ¼ q < R2 ; > > < 2I0 R h1;R2q BS2 sin2 h1 dh1 þ R h1;R2þq BS2 c sin2 h1 dh1 ; 0 < q < R2 ; 2;h1 0 h1;R2q ð33Þ UP ;S2 ¼ R h1;2R2 2 > 2I0 BS2 c2R2;h1 sin h1 dh1 ; 0 < q ¼ R2 ; > > 0 > R > : 2I0 h1;R2þq BS2 c sin2 h1 dh1 ; 0 < R2 < q; h1;qR2
2;h1
which is simpler than formula (30) for the rotationally unsymmetrical radiation. Fig. 3(b) illustrates the dependence of UP ;S2 on R2 and q calculated from formula (33). The results plotted here were obtained at the same values of the variables as in Fig. 3(a). Although the values of UP ;S2 and
S. Tryka / Optics Communications 211 (2002) 15–30
25
surface profiles in these both figures are similar it can be seen that UP ;S2 in Fig. 3(b) changes more slowly with the variation of R2 and q. However, this change occurs in the wider regions of R2 and q variations. Therefore any displacement of the source from the axial position in the optical system changes the values of the UP ;S2 more quickly than in the example presented above. 7.2. Lambertian source The Lambertian source is an extended surface with azimuthal symmetry of the radiance Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ independent on the direction of observation. This radiance is described by [19] Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ ¼ Lðx1 ¼ 0; y1 ¼ 0Þ ¼ L0 ;
ð34Þ
where the radiance L0 does not depend to the angle h1 . Because L0 does not depend on h1 this radiation is sometimes regarded as isotropic with respect to h1 (for h1 ranging from 0 to p=2). When the Lambertian source is considered as the point source the intensity IP ðh1 ; u1 Þ obeys Lambert’s law of radiation [19] IP ðh1 ; u1 Þ ¼ L0 S cos h1 ¼ I0 cos h1 ;
ð35Þ
where I0 ¼ L0 S is the maximal intensity at h1 ¼ 0. Fig. 4 illustrates the angular dependence of IP ðh1 ; u1 Þ from Eq. (35) plotted in spherical coordinates determined by x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 sin u1 and z1 ¼ cos h1 . The center of the Lambertian source represented in this figure by the planar circular disk lies at the point P ð0; 0; H0 Þ. Putting Eq. (35) into formula (28) and then calculating the inner integrals we have 8 Rh > 2pIh0 0 1;R2 BS2 cos h1 dh1 ; > > i 0 ¼ q < R2 ; > R h1;R2þq R h1;R2q > < BS2 cos h1 dh1 þ h1;R2q BS2 c2;h1 cos h1 dh1 ; 0 < q < R2 ; 2I0 0 ð36Þ UP ;S2 ¼ R h1;2R2 > BS2 c2R2;h1 cos h1 dh1 ; 0 < q ¼ R2 ; > 2I0 0 > > R > : 2I0 h1;R2þq BS2 c cos h1 dh1 : 0 < R2 < q: h1;qR2
2;h1
Typical results of numerical simulations of UP ;S2 from formula (36) are presented in Fig. 5(a) and (b). These simulations were done for the same values of the variables as in Figs. 3(a) and (b) but for two different cases when n1 < n2 and when n1 > n2 . Here the dependencies of UP ;S2 on R2 and q are also similar as in Fig. 3(a) but UP ;S2 is clearly greater than in Fig. 3(a). It is also seen that UP ;S2 attains greater values for the radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)), than from the medium of n1 < n2 to the medium of n2 > n1 (Fig. 5(a)). Also greater UP ;S2 variation can be observed in Fig. 5(b) than in Fig. 5(a) with the variation of the distance q of the source from the z-axis.
Fig. 4. The angular distribution of the radiant intensity from a Lambertian source lying in Px1 y1 plane. The center of the planar circular disk representing the Lambertian source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
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Fig. 5. The radiative flux UP ;S2 from a Lambertian source as a function of R2 and q for the radiation emitted from a medium of n1 ¼ 1:0 to a medium of n2 ¼ 1:5 (a) and for the radiation emitted from a medium of n1 ¼ 1:5 to a medium of n2 ¼ 1:0 (b). The results were calculated numerically from formula (36) for the relative values of R2 , q, H1 and H2 when R2 and q are both varied from 0 to 10 and at constant values of H1 ¼ H2 ¼ 0:5, and a1 ¼ a2 ¼ 0.
7.3. Spherical source Let us consider the optical radiation emitted by a small spherical source. If each element of the surface of the source has identical physical properties, the radiation may be considered as isotropic and IP ðh1 ; u1 Þ ¼ I0 ;
ð37Þ
where I0 is constant. Fig. 6 presents angular dependence of IP ðh1 , u1 ) plotted in the spherical coordinate system with its origin in the point 01 . The spherical coordinates are defined here by the transformation x1 ¼ sin h1 cos u1 , y1 ¼ sin h1 cos u1 and z1 ¼ cos h1 . The center of the source is fixed at the point P ð0; 0; H0 Þ. Inserting Eq. (37) into formula (28) and calculating the inner integrals in this formula the radiative flux will be given by 8 Rh > 2pIh0 0 1;R2 BS2 dh1 ; > > i 0 ¼ q < R2 ; > > < 2I0 R h1;R2q BS2 dh1 þ R h1;R2þq BS2 c dh1 ; 0 < q < R2 ; 2;h1 0 h1;R2q ð38Þ UP ;S2 ¼ R h1;2R2 > BS2 c2R2;h1 dh1 ; 0 < q ¼ R2 ; 2I0 0 > > > R > : 2I0 h1;2Rþq BS2 c dh1 ; 0 < R2 < q: h1;qR2
2;h1
Fig. 6. The angular distribution of the radiant intensity from a spherical source. The center of the source lies at the point P ð0; 0; H0 Þ of the Cartesian coordinate system 01 x1 y1 z1 .
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Fig. 7. The radiative flux UP ;S2 from the spherical source as a function of R2 and q for the radiation emitted through two different homogeneous isotropic media across the refractive boundary (a) and for the radiation emitted through one homogeneous isotropic medium (b). The results were calculated numerically from formula (38) for H1 ¼ H2 ¼ 0:5, n1 ¼ 1:0 and n2 ¼ 1:5 in the case (a) and for H1 þ H2 ¼ 1:0 and n1 ¼ n2 in the case (b). These calculations were done for the relative values of R2 , q, H1 and H2 when R2 and q both vary from 0 to 10 and at constant values of a1 ¼ a2 ¼ 0.
It worth noting that the obtained formula is directly applicable for calculating radiative fluxes of the isotropic radiations propagating in one homogeneous isotropic medium after substitution of n1 for n2 in Eqs. (5), (10), (18), (20)–(23) and (27). The resulting formula will be of the form UP ;S1 ¼ I0 xP ;S1 ;
ð39Þ
where S1 represents here the surface S2 of the disk and xP ;S1 is the solid angle that S1 subtends at P. Various solutions for xP ;S1 may be found elsewhere [5,7–10,12]. However the results evaluated numerically from formula (38) for the radiation propagating in one homogenous isotropic medium and the results obtained from Eq. (39) are nearly identical within the range of declared accuracy of numerical calculations. Fig. 7(a) shows results of numerical evaluations of UP ;S2 from formula (38) as a function of R2 and q at the same values of the variables as in Fig. 3(a). Here the radiative flux UP ;S2 depends on the position of the off-axis source with respect to the z-axis in a similar way as in the example for the Lambertian source from Fig. 5(a). The radiative flux UP ;S2 from the spherical source in Fig. 7(a) is clearly greater than UP ;S2 from the Lambertian source in Fig. 5(a). The surface plotted in Fig. 7(b) represents the results of UP ;S1 calculations for the radiation propagating in one homogeneous isotropic medium. This plot was simulated from formula (38) and was identical to the plot obtained using formula (39). The results in Fig. 7(b) suggest that the radiative flux UP ;S1 of the radiation propagating in one homogeneous isotropic medium attains values comparable to those obtained for the radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 , see Fig. 5(b). However it is also seen that UP ;S1 in Fig. 7(b) is more sensitive to the source position variation with respect to the z-axis than in Fig. 7(a).
8. Conclusion We derived the general formula (19) for calculating the fluxes UP ;S2 of the optical radiation illuminating a circular disk in optical systems where the off-axis point source and the disk are embedded in two different homogeneous isotropic media that can absorb the radiation and form a refractive surface parallel to the disk.
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This general formula was described by a set of four expressions, valid in different ranges of distance determining the point source position with respect to the disk symmetry axis. Each of these expressions has a different form of the double-definite integral of the same function represented by the product AS2 ðn1 ; n2 ; h1 Þ Iðh1 ; u1 Þ, where AS2 ðn1 ; n2 ; h1 Þ is described by Eq. (5) and Iðh1 ; u1 Þ is the radiant intensity function. It has been shown that it is quite simple to introduce the transmission coefficient into formula (19) for calculating UP ;S2 transmitted through the refractive surface between two different homogeneous and isotropic media. In addition this general formula can be easy modified into a form suitable for calculating UP ;S2 of the radiation propagated through the media absorbing the radiation according to Bouguer–Lambert’s law. Such considerations lead us to Eq. (27) representing the dependence of Iðh1 ; u1 Þ on the single-surface transmission factor s12 ðn1 ; n2 ; h1 Þ and on the absorption coefficients a1 and a2 of the first and second medium, respectively. Eq. (27) was then introduced into formula (19) to obtain a new formula directly applicable for calculating various fluxes of unpolarized optical radiations. In this manner we obtain Eq. (28) valid for the calculation of UP ;S2 if the optical radiation propagates in two homogeneous isotropic media and across the refractive boundary. Next we presented some examples of UP ;S2 calculations using formula (28). In these examples the radiation emitted by the rectilinear, Lambertian and spherical sources considered as the point source models was analyzed. It was shown that the smallest radiative fluxes UP ;S2 occur for the rectilinear, higher for the Lambertian and the highest for the spherical point source models. Comparing the results of UP ;S2 calculation for the Lambertian source it was also shown that the radiative fluxes depend on the relation between n1 and n2 . Greater values of UP ;S2 were obtained for radiation propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)) than from the medium of n1 < n2 to the medium of n2 > n1 (Fig. 5(a)). However the highest radiative fluxes UP ;S1 were calculated for the isotropic radiation from a spherical source propagating in one homogeneous isotropic medium (see the case for n1 ¼ n2 illustrated in Fig. 7(b)). Analyzing the influence of the distance q of these sources from the symmetry axis of the optical system, it can be stated that the radiative flux UP ;S2 always depends on the variation of q. In the examples presented in the paper the clearest dependence of /P ;S2 on q was observed in the regions where q attains similar values as R2 . However this variation was most clear for the radiation from a spherical point source model propagating in one homogeneous isotropic medium (Fig. 7(b)) and then for the radiation from the Lambertian point source model propagating from the medium of n1 > n2 to the medium of n2 < n1 (Fig. 5(b)). According to intuitive expectation, increases of H1 and H2 as well as a1 and a2 always lead to the decrease of UP ;S2 . To analyze such dependence there is a need to show some new graphical illustrations. In this report, we do not present such relations because their plots are less interesting than those described above and look similar to the plot in Fig. 12 of [12]. The presented general formula (19) is suitable for numerical evaluations of many types of angularly distributed optical radiant fluxes. After integration with respect to a given line or surface elements this formula can also be suitable for calculating the radiative fluxes from linear and extended surface sources. In many practical situations the inner integrals can be calculated into closed form expressions and the obtained formula can be expressed by a set of five single definite integrals that sometimes may be calculated analytically or are easy to estimate numerically in a number of ways. Therefore the method described in this paper is accurate, fast and not difficult and appears to have some advantages over the series method based on Monte Carlo estimations of UP ;S2 from double or multiple integrals evaluations. Finally it can be stated that the mathematical dependencies presented in this paper were expressed by the total quantities of classical radiometry. However, all formulas derived herein are directly applicable to the total quantities of visual photometry and to the spectral radiometric and photometric quantities. Complete characteristics of these total and spectral radiometric and photometric quantities can be found in [2,17].
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Acknowledgements The author would like to express his thanks to the anonymous referees for their careful review of the manuscript and constructive suggestions that helped to improve the paper significantly.
Appendix A. Quantities and symbols In this appendix, we summarize in Table 1 the quantities and symbols used in this paper and their units. Table 1 A list of the quantities and symbols used, their units and nomenclature Quantity or symbol
Units
Nomenclature
a1 ; a2 AS2 ðn1 ; n2 ; h1 Þ dUP ;dS2 ; dUP 1;dS2 dUP ;S2 dxP 1;dS2 fP ;S2 ¼ dUP ;S2 =dh1 H0 ; H1 ; H2 Iðh2 ; u2 Þ; Iðh1 ; u1 Þ,IS1 ðh1 ; u1 Þ; I12 ðh1 ; u1 Þ; IP ðh1 ; u1 Þ; I0 ks1 ; ks2 Lðx1 ¼ 0; y1 ¼ 0; h1 ; u1 Þ, Lðx1 ¼ 0; y1 ¼ 0Þ; L0 n1 ; n2 P ; P1 r1 ; u1 ; z1 and r2 ; u2 ; z1
1/m – Wa , lmb Wa , lmb sr (W/rad)a , (lm/rad)b m ðW=srÞa , cdb
Absorption coefficients Superposition of sine and cosine functions Infintesimal radianta or luminousb fluxes Differential radianta or luminousb flux Infinitesimal solid angle Angular distribution of UP ;S2 with respect to h1 Distances Radiant or luminous intensities as functions of various variables
1/m (W/m2 )a (1m/m2 )b – – –
Scattering coefficients Radianta or luminousb radiances as functions of various varibles
r1 ; r2 ; r1;h1 ; r2;h1 R1 ; R2 S1 ; S2 x,y,z and x1 ; y1 ; z1 z z1
m m m2 – – –
u1 ; u2 UP ;S1 ; UP ;S2 c1;h1 , c2R1;h1 , c2;h1 , c2R2;h1 q h1 ; h2 h1;R2 ; h1;R2q ; h1;2R2 , h1;qR2 ; h1;R2þq , h1;R1 ; h1;R1q , h1;2R1 , h1;qR1 ; h1;R1þq , s12 ðn1 ; n2 ; h1 Þ, s12jj ðn1 ; n2 ; h1 Þ, s12? ðn1 ; n2 ; h1 Þ, xP ;S1
rad Wa , lmb rad m rad rad
Refractive indices Point source and its virtual image, respectively axes of the cylindrical 0r1 u1 z1 and 01 r2 u2 z1 coordinate systems, respectively Radii as functions of various variables Radii Surfaces Axes of the Cartesian 0xyz and 01 x1 y1 z1 coordinate systems, respectively Symmetry axis of the optical system Symmetry axis of the conical surfaces made by optical ray trajectories at given h1 Horizontal angles Radiativea or luminousb fluxes Boundary angles Distance from the source to the z-axis Incidence and refraction angles, respectively Boundary angles
–
Single-surface transmission factors
–
Solid angle at P subtended by S1
a b
Radiometric units. Photometric units.
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Appendix B. Elementary solid range An infinitesimal solid angle, dxP 1;dS2 , is defined by the following equation [5,8] dxP 1;dS2 ¼ dS2 cos h2 =l2 ;
ðB:1Þ
where l represents the linear distance between P1 and an infinitesimal surface dS2 . The infinitesimal surface dS2 in Fig. 1(a) can be calculated as dS2 ¼ r2 dr2 du2 and the distance l ¼ ðH0 þ H1 þ H2 Þ= cos h2 , when 0 6 h2 < p=2. Then from simple trigonometric dependencies we have: r2 ¼ ðH0 þ H1 þ H2 Þ= tan h2 and taking the derivative of r2 we get: dr2 ¼ ðH0 þ H1 þ H2 Þdh2 = cos2 h2 . Therefore if (H0 þ H1 þ H2 Þ > 0 and h2 is limited to the region 0 6 h2 < p=2 Eq. (B.1) can be rewritten in the equivalent form dxP 1;dS2 ¼ sin h2 dh2 du2 :
ðB:2Þ
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