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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 5 — May. 17, 2007
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Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane

Stanislaw Tryka  »View Author Affiliations


Optics Express, Vol. 15, Issue 7, pp. 3777-3790 (2007)
http://dx.doi.org/10.1364/OE.15.003777


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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

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

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.

]

ΦσS=λminλmaxΦλ,σSdλ,
(1)

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).
(2)

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).

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),
(3)

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.
(4)

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

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

, 17

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.

]

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

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

]

Iλ,P(ρi,ϕj,θ,φ,H,αλ,at)=Iλ,P(ρi,ϕj,θ,φ)τP(θ,H,αλ,at),
(5)

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,
(6)

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,
(7)

which can be calculated similarly to the integral in Eqs. (2.7) from Ref. [21

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

]. 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,
(8a)

if h = H = 0 and a = 0,

Φλ,PS(ρi,ϕj)=0,ρi=0,
(8b)

if 0 < hH and a = 0,

Φλ,PS,(ρi,ϕj)=0π2sinθdθ02πIλ,P(θ,φ)dφ,ρi=0,
(8c)

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,
(8d)

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,
(8e)

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,
(8f)

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,
(8g)

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

γa,ρi=arccos[(ρi2+h2tan2θa2)(2ρihtanθ)],
(9a)
γR,ρi=arccos[(ρi2+H2tan2θR2)(2ρiHtanθ)],
(9b)
γ2a=arccos[htanθ(2a)],
(9c)
γ2R=arccos[Htanθ(2R)],
(9d)

the limiting polar angles

θa=arctan(ah),
(10a)
θaρi=arctan[(aρi)h],
(10b)
θa+ρi=arctan[(a+ρi)h],
(10c)
θ2a=arctan(2ah),
(10d)
θa+Y=arctan[(a+Y)h],
(10e)
θR+ρi=arctan[(R+ρi)H],
(10f)
θR=arctan(RH),
(10g)
θRρi=arctan[(Rρi)H],
(10h)
θ2R=arctan(2RH),
(10i)
θρiR=arctan[(ρiR)H],
(10j)

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).

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>,
(11)

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

<Φλ,σS>=1Nλi=1Nλ,ρj=1Nλ,ϕΦλ,PS(ρi,ϕj),
(12)

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,
(13)

if a = 0, and

<Φλ,σS>=1Ni=1mN(ρi)Φλ,PS(ρi)=2a2i=1mΦλ,PS(ρi)ρiΔρi,0<hH,
(14)

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.
(15)

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

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).

] 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],
(16a)

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],
(16b)

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],
(16c)

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],
(16d)

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],
(16e)

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,
(17a)

if a = 0,

<Φλ,σS>=2a20arctan(2ah)Iλ,P(θ)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ,
(17b)

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],
(17c)

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],
(17d)

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],
(17e)

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.

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,
(18a)

if a = 0,

<Φλ,σS>=2Iλ,0a20arctan(2ah)τP(θ,H,αλ,at)ga,a(θ,a,h)sinθdθ,
(18b)

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],
(18c)

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],
(18d)

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],
(18e)

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,
(19a)

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

<Φλ,σS>=2Iλ,0a20arctan(2ah)ga,a(θ,a,h)sinθdθ,
(19b)

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],
(19c)

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],
(19d)

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},
(19e)

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

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).

]. 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

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

].

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

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]

]. 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

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

] from programmed Eqs. (16a)-(16e), (17a)-(17e), and (18a)-(18e), respectively.

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.

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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.

OCIS Codes
(080.2720) Geometric optics : Mathematical methods (general)
(120.5240) Instrumentation, measurement, and metrology : Photometry
(120.5630) Instrumentation, measurement, and metrology : Radiometry
(230.6080) Optical devices : Sources
(350.5610) Other areas of optics : Radiation

History
Original Manuscript: September 29, 2006
Revised Manuscript: February 16, 2007
Manuscript Accepted: March 1, 2007
Published: April 2, 2007

Virtual Issues
Vol. 2, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Stanislaw Tryka, "Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane," Opt. Express 15, 3777-3790 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-7-3777


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References

  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.

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