Abstract

A general formula and some special integral formulas were presented for calculating radiative fluxes incident on a circular plane from a planar multiple point source within a coaxial cylindrical enclosure perpendicular to the source. These formula were obtained for radiation propagating in a homogeneous isotropic medium assuming that the lateral surface of the enclosure completely absorbs the incident radiation. Exemplary results were computed numerically and illustrated with three-dimensional surface plots. The formulas presented are suitable for determining fluxes of radiation reaching planar circular detectors, collectors or other planar circular elements from systems of laser diodes, light emitting diodes and fiber lamps within cylindrical enclosures, as well as small biological emitters (bacteria, fungi, yeast, etc.) distributed on planar bases of open nontransparent cylindrical containers.

© 2007 Optical Society of America

1. Introduction

With increased development of solid state lighting there is currently a great deal of interest in the use of laser diodes (LDs) and light emitting diodes (LEDs) to make LD and LED arrays and multi LED lamps. LD and LED arrays have already found numerous laboratorial [1-6], industrial [3, 7], medical [1, 3, 8, 9] and domestic applications [1, 3, 7], while multi LED lamps are used in agriculture [2, 5] medicine [8, 9-11] and as sources of white or colored lights for illumination. In addition, LED lamps are used at present for general illumination, landscape lighting, automotive lighting, and as indicators and portable flashlights [13, 14]. By combining high-power and high-brightness blue, green and red LEDs, LED full-color displays and lamps with high reliability, high durability and low energy consumption are now possible [7, 13-15]. The LED-based displays and lamps effectively replace the traditional bulbs and fluorescent lamps and will probably be the most popular and frequently used sources of light in the near future [7, 12-15]. However, in contrast to traditional extended light sources, LEDs and LDs are in fact multiple point sources even when are positioned at small distance from the irradiated objects.

In many applications of electromagnetic and corpuscular radiation it is necessary to calculate the radiative flux (the flux of radiation), or in other words the radiative power (the power of radiation) emitted by a given source and incident on an object of a given geometrical shape. Such calculations are necessary for measuring the fluxes of radiation incident on detectors and/or collectors [16-18] and for designing light sources for medical [8, 9] and agricultural [2, 5] experiments and uses. By using LEDs it is very simple to obtain multiple point sources of various geometrical shapes. However, when the high brightness multi LED sources are used the high power narrow-band radiations from individual or all LEDs may be hazardous to human and animal eyes. To protect humans and animals against ocular hazards the radiative fluxes incident on their eyes must be known [19, 20]. Therefore in a wide variety of practical applications of LD and LED arrays as well as LED lamps mathematical formulas related to various shapes of multiple point sources are indispensable.

This paper presents one general and some special mathematical formulas for calculating fluxes of radiation reaching planar circular elements from planar multiple point sources within a coaxial circular enclosure perpendicular to the source plane. These formulas were obtained for radiation propagated in a homogeneous attenuating medium when the lateral surface of the enclosure completely absorbs the incident radiation. The analyzed model of the multiple point source may represent LED and LD arrays, fiber sources within cylindrical enclosures or small biological objects emitting photons enclosed within open cylindrical containers. All presented formulas are directly applicable to planar coaxial circular multiple point source-detector and source-collector systems, for example.

2. Definition of variables and some fundamental expressions

The cylindrical coordinate system, 0ρφz, was used to obtain the mathematical formulas presented in the paper. The geometry of the analyzed optical system is shown schematically in Fig. 1. Throughout the paper we assume that the multiple point source of the surface, σ, and illuminated circular planar surface, S, are separated by a homogeneous isotropic medium and that the lateral surface of the enclosure completely absorbs the incident radiation so that only the radiation leaving the enclosure through its open end reaches the surface S.

The radiative flux, Φ σ→S, enclosed within the spectral region between wavelengths λ min and λ max and incident on the surface S from the planar circular coaxial multiple point source of surface σ can be calculated as [17]

ΦσS=λminλmaxΦλ,σSdλ,

where Φ λ,σ→S = dΦ σ→S/ dλ is the spectral radiative flux of wavelength λ from all point emitters given by

Φλ,σS=i=1Nλ,ρj=1Nλ,ϕΦλ,PS(ρi,ϕj).

The function Φ λ,PS(ρi, ϕj) in the above equation represents the spectral radiative flux at wavelength λ reaching the plane S from the single point emitter P(ρi, ϕj) while the symbols N λ,ρ and N λ,ϕ denote the number of point emitters radiating at wavelength λ and positioned at radial distances ρi and horizontal angles ϕj, respectively.

 

Fig. 1. Definition of some geometrical variables and the perspective view of the planar circular multiple point source of surface σ within the cylindrical enclosure contributing optical radiation to the planar circular coaxial surface S when aR and H > h (a) and the scheme of the σ-S system when a > R and H > h (b). The symbol r denotes the radial distance between the points P(ρi, ϕj) and P′(ρi, ϕj, r, φ).

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The radiative flux Φ, PS (ρi, ϕj) incident on the surface S from the single point emitter P(ρi, ϕj) is described by

Φλ,PS(ρi,ϕj)=SIλ,P(ρi,ϕj,θ,φ,H,αat)dωPdS(θ,φ),

where Iλ, P(ρi, ϕj, θ, φ, H, αλ, at) denotes the radiant intensity of wavelength λ at the point where P′(ρi, ϕj, r, φ) and dω P-dS(θ, φ) is an infinitesimal solid angle subtended at P by the surface element dS′ and is given as follows

dωPdS(θ,φ)=sinθdθdφ.

S′ represents the surface S or its part limited by the rays passing through the upper edge of the cylindrical enclosure. The symbol αλ, at denotes the coefficient describing the attenuation of the radiation along the ray between P(ρi, ϕj) and P′(ρi, ϕj, r, φ) usually expressed as [16, 17]

αλ,at=αλ,ab+αλ,sc,

where αλ, ab and αλ, sc are the absorption and scattering coefficients, respectively. In a homogeneous medium the intensity Iλ, P(ρi, ϕj, θ, φ, H, αλ, at) obeys the Bouguer–Lambert exponential law of intensity attenuation [16]

Iλ,P(ρi,ϕj,θ,φ,H,αλ,at)=Iλ,P(ρi,ϕj,θ,φ)τP(θ,H,αλ,at),

where Iλ, P(ρi, ϕj, θ, φ) is the intensity of radiation emitted by the emitter P(ρi, ϕj) and

τP(θ,H,αλ,at)=exp(Hαλ,atcosθ),0θ<π2,

is the internal transmittance.

3. The radiative flux reaching the circular plane from the single point emitter within a coaxial cylindrical enclosure

Substituting Eqs. (4) and (5) into (3) leads to the integral

Φλ,PS(ρi,ϕj)=SIλ,P(ρi,ϕj,θ,φ)τP(θ,H,αλ,at)sinθdθdφ,

which can be calculated similarly to the integral in Eqs. (2.7) from Ref. [21]. This way, for the boundary conditions made by the external contour of the circular surface S with the radius R and by the top circular edge of the cylindrical enclosure with the radius a, as shown in Fig. 1, we obtain a set of seven double-definite integral expressions:

Φλ,PS(ρi,ϕj)=0π2sinθdθ02πIλ,P(θ,φ)dφ,ρi=0,

if h = H = 0 and a = 0,

Φλ,PS(ρi,ϕj)=0,ρi=0,

if 0 < hH and a = 0,

Φλ,PS,(ρi,ϕj)=0π2sinθdθ02πIλ,P(θ,φ)dφ,ρi=0,

if h = H = 0 and a > 0,

Φλ,PS(ρi,ϕj)={0θaτP(θ,H,αλ,at)sinθdθ02πIλ,P(θ,φ)dφ,0=ρi<a,0θaρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θaρiθa+ρiτP(θ,H,αλ,at)sinθdθγa,ρiγa,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<ρi<a,0θ2aτP(θ,H,αλ,at)sinθdθγ2aγ2aIλ,P(ρi,ϕj,θ,φ)dφ,0<ρi=a,

if 0 < hH and 0<ahR2HhR,

Φλ,PS(ρi,ϕj)={0θaτP(θ;H;αλ,at)sinθdθ02πIλ,P(θ,φ)dφ,0=ρi<a,0θaρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θaρiθa+ρiτP(θ,H,αλ,at)sinθdθγa,ρiγa,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<ρiYa,0θaρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θaρiθa+YτP(θ,H,αλ,at)sinθdθγa,ρiγa,ρiIλ,P(ρi,ϕj,θ,φ)dφ+θa+YθR+ρiτP(θ,H,αλ,at)sinθdθγR,ρiγR,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<Y<ρi<a,

if 0 < h < H and 0<hR2Hh<ahRH<R,

Φλ,PS(ρi,ϕj)={0θRτP(θ,H,αλ,at)sinθdθ02πIλ,P(θ,φ)dφ,0=ρi<a,0θRρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θRρiθR+ρiτP(θ,H,αλ,at)sinθdθγR,ρiγR,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<Yρia,0θaρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θaρiθa+YτP(θ,H,αλ,at)sinθdθγa,ρiγa,ρiIλ,P(ρi,ϕj,θ,φ)dφ+θa+YθR+ρiτP(θ,H,αλ,at)sinθdθγR,ρiγR,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<ρiYa,

if 0 < h < H and 0<hRH<a<R,

Φλ,PS(ρi,ϕj)={0θRτP(θ;H;αλ,at)sinθdθ02πIλ,P(θ,φ)dφ,0=ρi<R,0θRρiτP(θ,H,αλ,at)sinθdθ02πIλ,P(ρi,ϕj,θ,φ)dφ+θRρiθR+ρiτP(θ,H,αλ,at)sinθdθγR,ρiγR,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<ρi<R,0θ2RτP(θ,H,αλ,at)sinθdθγ2Rγ2RIλ,P(ρi,ϕj,θ,φ)dφ,0<ρi=R,θρiRθR+ρiτP(θ,H,αλ,at)sinθdθγR,ρiγR,ρiIλ,P(ρi,ϕj,θ,φ)dφ,0<R<ρi,

if 0 < hH and R < a, where the limiting horizontal angles

γa,ρi=arccos[(ρi2+h2tan2θa2)(2ρihtanθ)],
γR,ρi=arccos[(ρi2+H2tan2θR2)(2ρiHtanθ)],
γ2a=arccos[htanθ(2a)],
γ2R=arccos[Htanθ(2R)],

the limiting polar angles

θa=arctan(ah),
θaρi=arctan[(aρi)h],
θa+ρi=arctan[(a+ρi)h],
θ2a=arctan(2ah),
θa+Y=arctan[(a+Y)h],
θR+ρi=arctan[(R+ρi)H],
θR=arctan(RH),
θRρi=arctan[(Rρi)H],
θ2R=arctan(2RH),
θρiR=arctan[(ρiR)H],

and

Y=(hRHa)(Hh).

It can be noted here that the limiting angles γa, ρi and γR, ρi in Eqs. (9a) and (9b) obey the cosines law of trigonometry as shown in Figs. 2(a) and (b). The angles γ 2a and γ 2R in Eqs. (9c) and (9d) are defined similarly to γa, ρi and γR, ρi at ρi = a and ρi = R, respectively.

 

Fig. 2. Geometrical dependencies between a, ρi, r and γa, ρi at aR (a) and between R, ρi, r and γR, ρi at a > R (b). The radial distance r is defined as r = h tan θ (a) and as r = H tan θ (b).

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Equations (8a)-(8g) represent the general double-integral solution to the spectral radiative flux Φ λ, PS(ρi, ϕj) reaching the planar circular surface S from a given point emitter P(ρi, ϕj) within the coaxial cylindrical enclosure perpendicular to the source σ provided that the lateral surface of the enclosure completely absorbs the incident radiation. Putting R = ∞ in Eqs. (8a)-(8g) and assuming that the radius a, height h and distance H are not infinite, we obtain Eqs. (8a)-(8d) for calculating the total radiative fluxes, Φ λ, σ(ρi, ϕj), at wavelength λ emitted by any point emitter P(ρi, ϕj) into the space surrounding the cylindrical enclosure.

The solutions represented by Eqs. (8a)-(8g) are appropriate for calculating the flux Φ λ, σS from Eq. (2) for arbitrary distributed point emitters P(ρi, ϕj) composing the source σ within the cylindrical enclosure if the attenuation coefficient αλ, at of the homogeneous medium is defined and the radiant intensities Iλ, P(ρi, ϕj, θ, φ) from each point emitter are known. However the radiative fluxes Φ λ, PS(ρi, ϕj) in Eq. (2) must be calculated at given wavelengths λ from Eqs. (8a)-(8g). Obviously by substituting Eqs. (8a)-(8d) into (2) at R = ∞ and finite a, h and H, we obtain a formula for calculating the radiative flux, Φ λ, σ, emitted by the source σ into the space surrounding this source within the enclosure.

For many functions Iλ, P(ρi, ϕj, θ, φ) Eqs. (8a)-(8g) can be integrated with respect to the angle φ and expressed by single definite integrals. Even though these equations can be expressed by single definite integrals, further computation of the radiative fluxes Φ λ, σ→S for large numbers of emitters P(ρi, ϕj) composing the source σ from general Eq. (2) may be time-consuming and impractical. Therefore, it is desirable to look for a simpler analytical solution to the problem outlined above. We obtain such a solution when the emitters P(ρi, ϕj) are spread uniformly and each of them emits identically angularly distributed radiation.

4. Identically radiating emitters of a uniformly spread multiple point source

Equation (2) can also be rewritten as

Φλ,σS=Nλ<Φλ,σS>,

where Nλ represents the total number of point emitters radiating at wavelength λ and

<Φλ,σS>=1Nλi=1Nλ,ρj=1Nλ,ϕΦλ,PS(ρi,ϕj),

is the average radiative flux from all point emitters P(ρi, ϕj) radiating at wavelength λ.

For identically radiating emitters P(ρi, ϕj) from a uniformly distributed multiple point source we can divide the surface σ into the surface elements π(ρ 2 i - ρ 2 i-1) containing Nλ(ρi) emitters so that

i=1mNλ(ρi)=Nλ,i=1m(ρiρi1)=i=1mΔρi=a,

where m is the number of elements Δρi and a is the radius of the source σ or radius of the cylindrical enclosure. Then due to the uniform distribution we have

Nλ(ρi)(2πρiΔρi)=Nλ(πa2)

and for R > 0 Eq. (12) obtains the following form

<Φλ,σS>={0π2sinθdθ02πIλ,P(θ,φ)dφ,h=H=0,0,0<hH,

if a = 0, and

<Φλ,σS>=1Ni=1mN(ρi)Φλ,PS(ρi)=2a2i=1mΦλ,PS(ρi)ρiΔρi,0<hH,

if a > 0, where Φ λ, PS(ρi) is the radiative flux from the point emitter P(ρi) lying at a distance ρi. For extremely small elements Δρi→0 we obtain m→∞ so that Eq. (14) becomes

<Φλ,σS>=2a20aΦλ,PS(ρ)ρdρ,0<hH.

To integrate Eq. (15) we can use the method for calculating the average solid angle subtended by a circular disk from uniformly spread multiple points on the coaxial circular plane described in Ref. [22] during integration of Eq. (22). Applying to Eq. (15) the boundary conditions made by the geometrical variables defined in Figs. 1(a) and (b) we get

<Φλ,σS>=2a2[0arctan(ah)τP(θ,H,αλ,at)sinθdθ0ahtanθρdρ02πIλ,P(θ,φ)dφ+0arctan(ah)τP(θ,H,αλ,at)sinθdθahtanθ0ρdργa,ργa,ρIλ,P(θ,φ)dφ+arctan(ah)arctan(2ah)τP(θ,H,αλ,at)sinθdθhtanθaaρdργa,ργa,ρIλ,P(θ,φ)dφ],

if 0 < ah R/(2H - h) ≤ R,

<Φλ,σS>=2a2[0arctan(ah)τP(θ,H,αλ,at)sinθdθ0ahtanθρdρ02πIP(θ,φ)dφ+0arctan(ah)τP(θ,H,αλ,at)sinθdθahtanθaρdργa,ργa,ρIλ,P(θ,φ)dφ+arctan(ah)arctan[(a+Y)h]τP(θ,H,αλ,at)sinθdθhtanθaaρdργa,ργa,ρIλ,P(θ,φ)dφ+arctan[(a+Y)h]arctan[(R+a)H]τP(θ,H,αλ,at)sinθdθHtanθRaρdργR,ργR,ρIλ,P(θ,φ)dφ],

if 0 < Rh R/(2H - h)≤ ah R/H,

<Φλ,σS>=2a2[0arctan[(R+Y)H]τP(θ,H,αλ,at)sinθdθ0ahtanθρdρ02πIλ,P(θ,φ)dφ+arctan[(R+Y)H]arctan(RH)τP(θ,H,αλ,at)sinθdθ0RHtanθρdρ02πIλ,P(θ,φ)dφ+arctan[(R+Y)H]arctan(RH)τP(θ,H,αλ,at)sinθdθRHtanθYρdργR,ργ,R,ρIλ,P(θ,φ)dφ+arctan[(R+Y)H]arctan(RH)τP(θ,H,αλ,at)sinθdθYahtanθρdργa,ργa,ρIλ,P(θ,φ)dφ+0arctan(RH)τP(θ,H,αλ,at)sinθdθahtanθaρdργa,ργa,ρIλ,P(θ,φ)dφ+arctan(RH)arctan[(R+a)H]τP(θ,H,αλ,at)sinθdθHtanθRaρdργR,ργR,ρIλ,P(θ,φ)dφ],

if 0 < h R/H < a < R,

<Φλ,σS>=2a2[0arctan(RH)τP(θ,H,αλ,at)sinθdθ0RHtanθρdρ02πIλ,P(θ,φ)dφ+0arctan(RH)τP(θ,H,αλ,at)sinθdθRHtanθR+HtanθρdργR,ργR,ρIλ,P(θ,φ)dφ+arctan(RH)arctan[(aR)H]τP(θ,H,αλ,at)sinθdθHtanθRR+HtanθρdργR,ργR,ρIλ,P(θ,φ)dφ+arctan[(aR)H]arctan[(R+a)H]τP(θ,H,αλ,at)sinθdθHtanθRaρdργR,ργR,ρIλ,P(θ,φ)dφ],

if 0 < Ra/2, and

<Φλ,σS>=2a2[0arctan(RH)τP(θ,H,αλ,at)sinθdθ0RHtanθρdρ02πIλ,P(θ,φ)dφ+0arctan[(aR)H]τP(θ,H,αλ,at)sinθdθRHtanθR+HtanθρdργR,ργR,ρIλ,P(θ,φ)dφ+arctan[(aR)H]arctan(RH)τP(θ,H,αλ,at)sinθdθRHtanθaρdργR,ργR,ρIλ,P(θ,φ)dφ+arctan(RH)arctan[(R+a)H]τP(θ,H,αλ,at)sinθdθHtanθRaρdργR,ργR,ρIλ,P(θ,φ)dφ],

if 0 < a/2 < R < a. The limiting angles γa, ρ and γR, ρ are described by Eqs. (9a)-(9b) without the subscript i.

Formulas (13) and (16a)-(16e) enable calculation of the average radiative fluxes <Φ λ, σ→S> from any uniformly distributed multiple point source, provided that each point emitter of this source identically emits angularly distributed radiation with respect to the z-axis. When the number Nλ of point emitters is known it is simple to obtain the total spectral radiative flux Φ λ, σ→S using Eq. (11).

The average radiative flux, <Φ λ, σ>, emitted into the space surrounding the source σ within a cylindrical enclosure is described by Eqs. (13) and (16a) because the remaining Eqs. (16b)-(16e) disappear for R = ∞ and finite values of a, h, and H.

For the intensity Iλ, P (θ, φ) not dependent on the horizontal angle φ, as in the case of radiation rotationally symmetrical around z-axis, the inner integrals with respect to the angle φ and distance ρ in Eqs. (16a)-(16e) can be expressed by simple elementary functions and we obtain single integral solutions to Eq. (12). In the following two subsections, we present such solutions for radiation not dependent on the angle φ and then for isotropic radiation.

4.1 Rotationally symmetrical radiation with respect to the z-axis

For the radiant intensity Iλ, P (θ, φ) not dependent on the horizontal angle φ we have Iλ, P (θ, φ) = Iλ, P (θ) and Eqs. (13) and (16a)-(16e) simplify to

<Φλ,σS>={2π0π2Iλ,P(θ)sinθdθ,h=H=0,0,0<hH,

if a = 0,

<Φλ,σS>=2a20arctan(2ah)Iλ,P(θ)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ,

if 0 < ah R/(2H - h) ≤ R,

<Φλ,σS>=2a2[0arctan[(a+Y)h]Iλ,P(θ)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ+arctan[(a+Y)h]arctan[(R+a)H]Iλ,P(θ)τP(θ,H,αλ,at)gR,a(θ,R,a,h)sinθdθ],

if 0 < Rh R/(2H - h) ≤ ah R/H,

<Φλ,σS>=2a2[0arctan(RH)Iλ,P(θ)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ+arctan[(R+Y)H]arctan(RH)Iλ,P(θ)τP(θ,H,αλ,at)gR,Y(θ,R,a,H)sinθdθarctan[(R+Y)H]arctan(RH)Iλ,P(θ)τP(θ,H,αλ,at)ga,Y(θ,a,h)sinθdθ+arctan(RH)arctan[(R+a)H]Iλ,P(θ)τP(θ,H,αλ,at)gR,a(θ,R,a,H)sinθdθ],

if 0 < h R/H < a < R,

<Φλ,σS>=2a2[πR20arctan[(aR)H]Iλ,P(θ)τP(θ,H,αλ,at)sinθdθ+arctan[(aR)H]arctan[(R+a)H]Iλ,P(θ)τP(θ,H,αλ,at)gR,a(θ,R,a,H)sinθdθ],

if 0 < R < a, where

ga,a(θ,a,h)=2a2arccos[htanθ(2a)]12htanθ4a2h2tan2θ,
ga,Y(θ,R,a,H)=Y2arccos[a2h2tan2θY22Yhtanθ]+a2arccos[a2+h2tan2θY22ahtanθ]12[(aY)2h2tan2θ][h2tan2θ(a+Y)2],
gR,Y(θ,R,a,H)=Y2arccos[R2H2tan2θY22YHtanθ]+R2arccos[R2+H2tan2θY22RHtanθ]12[(RY)2H2tan2θ][H2tan2θ(R+Y)2],
gR,a(θ,R,a,H)=a2arccos(a2+H2tan2θR22aHtanθ)+R2arccos(R2+H2tan2θa22RHtanθ)12[(R+a)2H2tan2θ][H2tan2θ(Ra)2].

Equations (17a)-(17e) can be easily turned into a computer code for calculating the fluxes <Φ λ, σ→S> if the radiant intensity Iλ, P (θ) is defined. Figures 3(a)–3(d) demonstrate four examples of such calculations obtained for the point emitters of the source σ emitting rotationally symmetrical radiation around z-axis and when the radiant intensity is given by the expression Iλ, P (θ) = Iλ, Pcos2 θ. The data plotted in Figs. 3(a) and 3(c) were computed for nonattenuated radiation while the data shown in Figs. 3(b) and 3(d) were calculated for radiation attenuated within a homogeneous isotropic medium.

 

Fig. 3. The average spectral radiative flux <Φ λ, σ→S> within wavelength interval Δλ = 1 nm obtained for IP (θ 1) = I 0 cos2 θ 1 as a function of R and H at a = 5 and αat = 0 (a), as a function of R and H at a = 5 and αat = 0.1 (b), as a function of a and H at R = 5 and αat = 0 (c), and as a function of a and H at R = 5 and αat = 0.1 (d). The data were calculated for the relative units of R, a, H, and αat i.e. if R, a, and H are given in m then αat is expressed in m-1. The radiant intensity was taken as I 0 = 1 W∙sr-1, so the flux <Φ λ, σ→S> is given in W∙nm-1.

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Figures 3(a) and (3c) show that the flux <Φ λ, σ→S> in a non-attenuating medium is always greater at the lower distance H and clearly depends on both radii R and a. The flux <Φ λ, σ→S> shown in Fig. 3(a) increases with increased R and obtains maximal value at a small distance H in the region limited by the relation R > a. In Fig. 3(c) we observe that at a given distance H the flux <Φ λ, σ→S> monotonically increases with increasing a until aR. Then the flux <Φ λ, σ→S> monotonically decreases.

The flux <Φ λ, σ→S> of the radiation propagated in an attenuated homogeneous medium is dependent on the radii R, a, and distance H, similarly to the flux <Φ λ, σ→S> propagated in an non-attenuating medium. However, it is clearly seen that the fluxes <Φ λ, σ→S> in Figs. 3(b) and 3(d) are lower in comparison to those presented in Figs. 3(a) and 3(c) and this effect is due to the attenuation phenomena described by Eq. (6).

The computer simulated surface-plots for the radiant intensity function Iλ, P (θ) = I λ, 0 cosθ were similar to the plots in Figs. 3(a)-3(d) although the fluxes <Φ λ, σ→S> were varied within the wider ranges of the spectral power obtaining maximal values of 1.200 W nm-1 and 0.657 W nm-1 at I λ, 0 = 1 W sr-1 within wavelength interval Δλ = 1 nm in a non-attenuating and attenuating medium respectively. Therefore from these data, it is easy to deduce that the fluxes <Φ λ, σ→S> reaching the surface S clearly depend on the angular distribution of the emitted radiation.

Some representative data of the absolute radiative fluxes <Φ λ, σ→S> computed for Iλ, P (θ) = I λ, 0 cos2 θ and Iλ, P (θ) = I λ, 0 cosθ with accuracy to twelve decimal places when the radii R and a, and height H were expressed in m and when the attenuation coefficient αat was expressed in m-1 are given in the fifth and sixth column of Table 1 in Appendix A. All these data were computed from Eqs. (17a)-(17e) for the radiant intensity I λ, 0 = 1 W∙sr-1. Therefore one can use the data from Table 1 for checking one’s own calculations from the formulas presented above.

4.2 Isotropic radiation

Sometime the angular distribution of radiation emitted by point emitters is isotropic or is unknown and considered isotropic. In this case the intensity Iλ, P (θ) does not depend on the angle θ so that Iλ, P (θ) = I λ, 0 and Eqs. (17a)-(17e) become

<Φλ,σS>={2πIλ,0,h=H=0,0,0<hH,

if a = 0,

<Φλ,σS>=2Iλ,0a20arctan(2ah)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ,

if 0 < ah R/(2H - h) ≤ R,

<Φλ,σS>=2Iλ,0a2[0arctan[(a+Y)h]τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ+arctan[(a+Y)h]arctan[(R+a)H]τP(θ,H,αλ,at)gR,a(θ,R,a,H)sinθdθ],

if 0 < Rh R/(2H - h) ≤ ah R/H,

<Φλ,σS>=2Iλ,0a2[0arctan(RH)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ+arctan[(R+Y)H]arctan(RH)τP(θ,H,αλ,at)gR,Y(θ,R,a,H,h)sinθdθarctan[(R+Y)H]arctan(RH)τP(θ,H,αλ,at)ga,Y(θ,R,a,H,h)sinθdθ+arctan(RH)arctan[(R+a)H]τP(θ,H,αλ,at)gR,a(θ,R,a,H)sinθdθ],

if 0 < h R/H < a < R, and

<Φλ,σS>=2Iλ,0a2[πR20arctan[(aR)H]τP(θ,H,αλ,at)sinθdθ+arctan[(aR)H]arctan[(R+a)H]τP(θ,H,αλ,at)gR,a(θ,R,a,H)sinθdθ],

if 0 < R < a.

The data computed from Eqs. (18a)-(18e) for isotropic radiation, i.e. when Iλ, P (θ) = I λ, 0 were illustrated identically as the surface-plots in Figs. 3(a)-3(d) for comparison. From these plots we observe that the fluxes <Φ λ, σ→S> of isotropic radiation at I λ, 0 = 1 W sr-1 and within wavelength interval Δλ = 1 nm attain significantly higher values than in Figs. 3(a)-(d) and in case of Iλ, P (θ) = I λ, 0 cosθ although the plotted dependencies look similar.

The absolute values of the fluxes <Φ λ, σ→S> of isotropic radiation at some radii R, a, height H, and coefficient αat are presented in the seventh column of Table 1 in Appendix A. Here it can indeed seen that these data are higher than those obtained for the radiant intensities Iλ, P (θ) = I λ, 0 cosθ and Iλ, P (θ) = I λ, 0 cos2 θ.

4.2.1 Isotropic radiation in non-attenuating medium

When the radiation is not attenuated then τ P (θ, H, αλ, at) = 1 and the flux <Φ λ, σ→S> at a = 0 is given by Eq. (18a), rewritten here for convenience,

<Φλ,σS>={2πIλ,0,h=H=0,0,0<hH,

while at a > 0 from Eqs. (18b)-(18e) we obtain

<Φλ,σS>=2Iλ,0a20arctan(2ah)ga,a(θ,a,h)sinθdθ,

if 0 < ah R/(2H - h) ≤ R,

<Φλ,σS>=2Iλ,0a2[0arctan[(a+Y)h]ga,a(θ,R,a,H)sinθdθ+arctan[(a+Y)h]arctan[(R+a)H]gR,a(θ,R,a,H)sinθdθ],

if 0 < Rh R/(2H - h) ≤ ah R/H,

<Φλ,σS>=2Iλ,0a2[0arctan(RH)ga,a(θ,a,h)sinθdθ+arctan[(R+Y)H]arctan(RH)gR,Y(θ,R,a,H)sinθdθarctan[(R+Y)H]arctan(RH)ga,Y(θ,R,a,h)sinθdθ+arctan(RH)arctan[(R+a)H]gR,a(θ,R,a,H)sinθdθ],

if 0 < h R/H < a < R, and

<Φλ,σS>=2Iλ,0a2{πR2[1HH2+(Ra)2]+arctan[(aR)H]arctan[(R+a)H]gR,a(θ,R,a,H)sinθdθ},

if 0 < R < a.

The integrals in Eqs. (19b)-(19e) can be expressed by complete Legendre-Jacobi elliptic integrals of all three kinds, like the integrals in Eqs. (39) and (40) from Ref. [22]. However these elliptic integrals are not presented in the paper because so many of them must be used represent Eqs. (19b)-(19e) and the obtained single integral formulas are easy to use in numerical evaluation.

It is worth noting that the data shown in Figs. 5(b) and 5(d) and in the seventh column of Table 1 in Appendix A were identical when Eqs. (16a)-(16f), (18a)-(18e), and (19a)-(19e) were used and the computation was performed with accuracy to twelve decimal places by a simple subroutine integration procedure written in Mathematica 2.2.3 [23].

5. Conclusions

In many radiometric problems it is very important to know the radiative fluxes reaching a given body from a given source of radiation [15-17]. Because of many possible source and body shapes, source-body geometrical configurations, physical properties of the separating media and radiation distributions, various mathematical formula for calculating the radiative fluxes are necessary.

This paper presents one general and some special formulas for calculating the radiative fluxes reaching a circular plane from planar multiple point sources enclosed within a coaxial cylindrical enclosure. Formulas were obtained for radiation propagated in a homogeneous isotropic attenuating medium for a planar source perpendicular to the symmetry axis of the enclosure provided that the lateral surface of the enclosure completely absorbs the incident radiation. The analyzed model of the source may represent LD and LED arrays, multi LED and fiber lamps within cylindrical enclosures and small biological emitters (bacteria, yeast, fungi, and etc.) distributed on the base of an open cylindrical container. The mathematical formulas obtained are appropriate for calculating radiative fluxes from single point and multiple point sources reaching coaxial planar circular elements such as collectors or detectors. The formulas presented are directly applicable for calculating the radiative fluxes passing through a circular aperture on a coaxial circular plane from a single point and planar circular multiple point source if the source and the aperture radii are equal. The formulas may also be easily simplified for calculating the radiative fluxes emitted into the space surrounding the cylindrical enclosure. The above-mentioned equations are applicable for any values of the variables characterizing the system shown in Fig. 1 and for media reducing radiation according to the exponential law of attenuation.

Equations (8a)-(8g), (17a)-(17e), (18a)-(18e) and (19a)-(19e) were turned into a computer code for computing the average fluxes <Φ λ, σ→S>. Examples of numerical computations were presented in Figs. 3(a)-3(d) for the radiant intensity function I(θ) = I 0cos2 θ 1. From these plots, one can see that the flux <Φ λ, σ→S> decreases generally with increased distance H and decreased radii a and R. A similar dependencies were also observed for the fluxes <Φ λ, σ→S> calculated for the radiant intensities I(θ) = I 0cosθ 1 and I(θ) = I 0. All these dependencies clearly indicate geometrical conditions at which the grater fluxes <Φ λ, σ→S> reach the plane S. Such conditions are particularly important when extremely weak intensity radiation is measured. Therefore Eqs. (8a)-(8g), (17a)-(17e), (18a)-(18e) and (19a)-(19e) are particularly valuable for optimizing the geometry of the optical system shown schematically in Fig. 1. Additionally, some computed representative data are presented in Appendix A.

Appendix A

In the fifth, sixth and seventh columns of Table 1 we present some values of the average fluxes <Φ λ, σ→S> computed with accuracy to twelve decimal places by Mathematica 2.2.3 [23] from programmed Eqs. (16a)-(16e), (17a)-(17e), and (18a)-(18e), respectively.

Tables Icon

Table 1. The average spectral radiative fluxes <Φ λ, σ→S> calculated for I λ, 0 = 1 W∙sr-1 and Δλ = 1 nm , at h = 1 and selected values of R, a, H, and αλ, at given in the relative units.

References and Links

1. V.P. Gribkovskii, “Injection lasers,” Prog. Quant. Electr. 23,41–88 (1995). [CrossRef]  

2. A.C. Schuerger, C.S. Brown, and E.C. Stryjewski, “Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light,” Ann. Botany 79,273–282 (1997). [CrossRef]  

3. R. Szweda, “Lasers at the cutting edge,” III-Vs Rev. 12,28–31 (1999). [CrossRef]  

4. L. Botter-Jensen, E. Bulur, G.A.T. Duller, and A.S. Murray, “Advances in luminescence instrument systems,” Radiat. Meas. 32,523–528 (2000). [CrossRef]  

5. O. Monje, G.W. Stutte, G.D. Goins, D.M. Porterfield, and G.E. Bingham, “Farming in space: environmental and biophysical concerns,” Adv. Space. Res. 31,151–167 (2003). [CrossRef]   [PubMed]  

6. K.T. Lau, W.S. Yerazunis, R.L. Shepherd, and D. Diamond, “Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array,” Sens. Actuators B-Chem. 114,819–825 (2006). [CrossRef]  

7. S. Nakamura, S. Pearton, and G. Fasol, The blue laser diode: The complete story (Springer-Verlag, Berlin, 2000).

8. C. Curachi, A.M. Toboy, D.V. Magalhaes, and V.S. Bagnato, “Hardness evaluation of a dental composite polymerized with experimental LED-based devices,” Dent. Mat. 17,309–315 (2001). [CrossRef]  

9. G. Glickman, B. Byrne, C. Pineda, W.W. Hauck, and G.C. Brainard, “Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs),” Biol. Psych. 59,502–507 (2006). [CrossRef]  

10. A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani, and J. Moan, “Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy,” Lasers Med. Sci. 19,139–149 (2004). [CrossRef]   [PubMed]  

11. B. Link, S. Ruhl, A. Peters, A. Junemann, and F.K. Horn, “Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system,” Doc. Opthalmol. 12,1–11 (2006). [CrossRef]  

12. J.M. Gaines, “Modeling of multichip LED packages for illumination,” Lighting Res. Technol. 38,152–165 (2006). [CrossRef]  

13. A. Mills, “Trends in HB-LED markets,” III-Vs Rev. 14,38–42 (2001). [CrossRef]  

14. C. Gardner, “The use of misuse of coloured light in the urban environment,” Opt. Lasers Tech. 38,366–376 (2006). [CrossRef]  

15. A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska, and M.S. Shur, “Optimization of white polychromic semiconductors lamps,” Appl. Phys. Lett. 80,234–236 (2002). [CrossRef]  

16. F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) pp.30–52,81–83.

17. M. Strojnik and G Paez, “Radiometry” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) pp.649–699.

18. E. Sparrow and R. Cess, Radiation Heat Transfer, (McGraw-Hill, New York , 1978) pp.77–136.

19. D.H. Sliney, “Laser effect on vision and ocular exposure limit,” App. Occup. Environ. Hyg. 11,313–319 (1996). [CrossRef]  

20. T.R. Fry, “Laser safety,” Vet. Clin. Small Anim. 32,535–547 (2002). [CrossRef]  

21. S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Optics Comm. 137,317–333 (1997). [CrossRef]  

22. S. Tryka, “Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc,” J. Mod. Opt. 47,769–791 (2000).

23. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp.44–186.

References

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  • |

  1. V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
    [CrossRef]
  2. A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
    [CrossRef]
  3. R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
    [CrossRef]
  4. L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
    [CrossRef]
  5. O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
    [CrossRef] [PubMed]
  6. K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
    [CrossRef]
  7. S. Nakamura, S. Pearton and G. Fasol, The blue laser diode: The complete story (Springer-Verlag, Berlin, 2000).
  8. C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
    [CrossRef]
  9. G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
    [CrossRef]
  10. A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
    [CrossRef] [PubMed]
  11. B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
    [CrossRef]
  12. J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
    [CrossRef]
  13. A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
    [CrossRef]
  14. C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
    [CrossRef]
  15. A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
    [CrossRef]
  16. F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) pp. 30-52, 81-83.
  17. M. Strojnik and G Paez, "Radiometry" in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) pp. 649-699.
  18. E. Sparrow and R. Cess, Radiation Heat Transfer, (McGraw-Hill, New York, 1978) pp. 77-136.
  19. D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
    [CrossRef]
  20. T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
    [CrossRef]
  21. S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
    [CrossRef]
  22. S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).
  23. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

2006

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

2004

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

2003

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

2002

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

2001

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

2000

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

1999

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

1997

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

1996

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

1995

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Bagnato, V. S.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Bingham, G. E.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Botter-Jensen, L.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Brainard, G. C.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Brown, C. S.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Bulur, E.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Byrne, B.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Curachi, C.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Diamond, D.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Duller, G. A. T.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Fry, T. R.

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

Gaines, J. M.

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

Gardner, C.

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

Gaska, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Glickman, G.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Goins, G.D.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Gribkovskii, V. P.

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Hauck, W. W.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Horn, F.K.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Iani, V.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Ivanauskas, F.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Junemann, A.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Juzenas, P.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Juzeniene, A.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Lau, K. T.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Link, B.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Ma, L.-W.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Magalhaes, D. V.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Mills, A.

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

Moan, J.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Monje, O.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Murray, A. S.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Peters, A.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Pineda, C.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Porterfield, D. M.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Ruhl, S.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Schuerger, A. C.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Shepherd, R. L.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Shur, M. S.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Sliney, D. H.

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

Stryjewski, E. C.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Stutte, G. W.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Szweda, R.

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

Toboy, A. M.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Tryka, S.

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

Vaicekauskas, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Yerazunis, W. S.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Zukauskas, A.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Adv. Space. Res.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Ann. Botany

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

App. Occup. Environ. Hyg.

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

Appl. Phys. Lett.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Biol. Psych.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Dent. Mat.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Doc. Opthalmol.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

III-Vs Rev.

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

J. Mod. Opt.

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

Lasers Med. Sci.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Lighting Res. Technol.

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

Opt. Lasers Tech.

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

Optics Comm.

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

Prog. Quant. Electr.

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Radiat. Meas.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Sens. Actuators B-Chem.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Vet. Clin. Small Anim.

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

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S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

S. Nakamura, S. Pearton and G. Fasol, The blue laser diode: The complete story (Springer-Verlag, Berlin, 2000).

F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) pp. 30-52, 81-83.

M. Strojnik and G Paez, "Radiometry" in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) pp. 649-699.

E. Sparrow and R. Cess, Radiation Heat Transfer, (McGraw-Hill, New York, 1978) pp. 77-136.

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Figures (3)

Fig. 1.
Fig. 1.

Definition of some geometrical variables and the perspective view of the planar circular multiple point source of surface σ within the cylindrical enclosure contributing optical radiation to the planar circular coaxial surface S when aR and H > h (a) and the scheme of the σ-S system when a > R and H > h (b). The symbol r denotes the radial distance between the points P(ρi , ϕj ) and P′(ρi , ϕj , r, φ).

Fig. 2.
Fig. 2.

Geometrical dependencies between a, ρi , r and γa, ρi at aR (a) and between R, ρi , r and γR, ρi at a > R (b). The radial distance r is defined as r = h tan θ (a) and as r = H tan θ (b).

Fig. 3.
Fig. 3.

The average spectral radiative flux <Φ λ, σ→S > within wavelength interval Δλ = 1 nm obtained for IP (θ 1) = I 0 cos2 θ 1 as a function of R and H at a = 5 and αat = 0 (a), as a function of R and H at a = 5 and αat = 0.1 (b), as a function of a and H at R = 5 and αat = 0 (c), and as a function of a and H at R = 5 and αat = 0.1 (d). The data were calculated for the relative units of R, a, H, and αat i.e. if R, a, and H are given in m then αat is expressed in m-1. The radiant intensity was taken as I 0 = 1 W∙sr-1, so the flux <Φ λ, σ→S > is given in W∙nm-1.

Tables (1)

Tables Icon

Table 1. The average spectral radiative fluxes <Φ λ, σ→S > calculated for I λ, 0 = 1 W∙sr-1 and Δλ = 1 nm , at h = 1 and selected values of R, a, H, and αλ, at given in the relative units.

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

Φ σ S = λ min λ max Φ λ , σ S d λ ,
Φ λ , σ S = i = 1 N λ , ρ j = 1 N λ , ϕ Φ λ , P S ( ρ i , ϕ j ) .
Φ λ , P S ( ρ i , ϕ j ) = S I λ , P ( ρ i , ϕ j , θ , φ , H , α at ) d ω P d S ( θ , φ ) ,
d ω P d S ( θ , φ ) = sin θ d θ d φ .
α λ , at = α λ , ab + α λ , sc ,
I λ , P ( ρ i , ϕ j , θ , φ , H , α λ , at ) = I λ , P ( ρ i , ϕ j , θ , φ ) τ P ( θ , H , α λ , at ) ,
τ P ( θ , H , α λ , at ) = exp ( H α λ , at cos θ ) , 0 θ < π 2 ,
Φ λ , P S ( ρ i , ϕ j ) = S I λ , P ( ρ i , ϕ j , θ , φ ) τ P ( θ , H , α λ , at ) sin θ d θ d φ ,
Φ λ , P S ( ρ i , ϕ j ) = 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , ρ i = 0 ,
Φ λ , P S ( ρ i , ϕ j ) = 0 , ρ i = 0 ,
Φ λ , P S , ( ρ i , ϕ j ) = 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , ρ i = 0 ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ a τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i < a , 0 θ 2 a τ P ( θ , H , α λ , at ) sin θ d θ γ 2 a γ 2 a I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i = a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ a τ P ( θ ; H ; α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i Y a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + Y τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a + Y θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < Y < ρ i < a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ R τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ R ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ R ρ i θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < Y ρ i a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + Y τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a + Y θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i Y a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ R τ P ( θ ; H ; α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < R , 0 θ R ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ R ρ i θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i < R , 0 θ 2 R τ P ( θ , H , α λ , at ) sin θ d θ γ 2 R γ 2 R I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i = R , θ ρ i R θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < R < ρ i ,
γ a , ρ i = arccos [ ( ρ i 2 + h 2 tan 2 θ a 2 ) ( 2 ρ i h tan θ ) ] ,
γ R , ρ i = arccos [ ( ρ i 2 + H 2 tan 2 θ R 2 ) ( 2 ρ i H tan θ ) ] ,
γ 2 a = arccos [ h tan θ ( 2 a ) ] ,
γ 2 R = arccos [ H tan θ ( 2 R ) ] ,
θ a = arctan ( a h ) ,
θ a ρ i = arctan [ ( a ρ i ) h ] ,
θ a + ρ i = arctan [ ( a + ρ i ) h ] ,
θ 2 a = arctan ( 2 a h ) ,
θ a + Y = arctan [ ( a + Y ) h ] ,
θ R + ρ i = arctan [ ( R + ρ i ) H ] ,
θ R = arctan ( R H ) ,
θ R ρ i = arctan [ ( R ρ i ) H ] ,
θ 2 R = arctan ( 2 R H ) ,
θ ρ i R = arctan [ ( ρ i R ) H ] ,
Y = ( hR Ha ) ( H h ) .
Φ λ , σ S = N λ < Φ λ , σ S > ,
< Φ λ , σ S > = 1 N λ i = 1 N λ , ρ j = 1 N λ , ϕ Φ λ , P S ( ρ i , ϕ j ) ,
i = 1 m N λ ( ρ i ) = N λ , i = 1 m ( ρ i ρ i 1 ) = i = 1 m Δ ρ i = a ,
N λ ( ρ i ) ( 2 π ρ i Δ ρ i ) = N λ ( π a 2 )
< Φ λ , σ S > = { 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 1 N i = 1 m N ( ρ i ) Φ λ , P S ( ρ i ) = 2 a 2 i = 1 m Φ λ , P S ( ρ i ) ρ i Δ ρ i , 0 < h H ,
< Φ λ , σ S > = 2 a 2 0 a Φ λ , P S ( ρ ) ρ d ρ , 0 < h H .
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ 0 ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( a h ) arctan ( 2 a h ) τ P ( θ , H , α λ , at ) sin θ d θ h tan θ a a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I P ( θ , φ ) d φ + 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( a h ) arctan [ ( a + Y ) h ] τ P ( θ , H , α λ , at ) sin θ d θ h tan θ a a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan [ ( R + Y ) H ] τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ Y ρ d ρ γ R , ρ γ , R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ Y a h tan θ ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a R ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = { 2 π 0 π 2 I λ , P ( θ ) sin θ d θ , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 a 2 0 arctan ( 2 a h ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan [ ( a + Y ) h ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , h ) sin θ d θ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , Y ( θ , R , a , H ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , Y ( θ , a , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 a 2 [ π R 2 0 arctan [ ( a R ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) sin θ d θ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
g a , a ( θ , a , h ) = 2 a 2 arccos [ h tan θ ( 2 a ) ] 1 2 h tan θ 4 a 2 h 2 tan 2 θ ,
g a , Y ( θ , R , a , H ) = Y 2 arccos [ a 2 h 2 tan 2 θ Y 2 2 Yh tan θ ] + a 2 arccos [ a 2 + h 2 tan 2 θ Y 2 2 ah tan θ ] 1 2 [ ( a Y ) 2 h 2 tan 2 θ ] [ h 2 tan 2 θ ( a + Y ) 2 ] ,
g R , Y ( θ , R , a , H ) = Y 2 arccos [ R 2 H 2 tan 2 θ Y 2 2 YH tan θ ] + R 2 arccos [ R 2 + H 2 tan 2 θ Y 2 2 RH tan θ ] 1 2 [ ( R Y ) 2 H 2 tan 2 θ ] [ H 2 tan 2 θ ( R + Y ) 2 ] ,
g R , a ( θ , R , a , H ) = a 2 arccos ( a 2 + H 2 tan 2 θ R 2 2 aH tan θ ) + R 2 arccos ( R 2 + H 2 tan 2 θ a 2 2 RH tan θ ) 1 2 [ ( R + a ) 2 H 2 tan 2 θ ] [ H 2 tan 2 θ ( R a ) 2 ] .
< Φ λ , σ S > = { 2 π I λ , 0 , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 I λ , 0 a 2 0 arctan ( 2 a h ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan [ ( a + Y ) h ] τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) g R , Y ( θ , R , a , H , h ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) g a , Y ( θ , R , a , H , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ π R 2 0 arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = { 2 π I λ , 0 , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 I λ , 0 a 2 0 arctan ( 2 a h ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan [ ( a + Y ) h ] g a , a ( θ , R , a , H ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan ( R H ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) g R , Y ( θ , R , a , H ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) g a , Y ( θ , R , a , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 { π R 2 [ 1 H H 2 + ( R a ) 2 ] + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ } ,

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