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]

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

The function Φ _λ,P→S(ρ_i, ϕ_j) in the above equation represents the spectral radiative flux at wavelength λ reaching the plane S from the single point emitter P(ρ_i, ϕ_j) 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 ρ_i and horizontal angles ϕ_j, 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 a ≤ R 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(ρ_i, ϕ_j) and P′(ρ_i, ϕ_j, r, φ).

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The radiative flux Φ, P→S (ρ_i, ϕ_j) incident on the surface S from the single point emitter P(ρ_i, ϕ_j) is described by

where I_{λ, P}(ρ_i, ϕ_j, θ, φ, H, α_{λ, at}) denotes the radiant intensity of wavelength λ at the point where P′(ρ_i, ϕ_j, 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

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

where α_{λ, ab} and α_{λ, sc} are the absorption and scattering coefficients, respectively. In a homogeneous medium the intensity I_{λ, P}(ρ_i, ϕ_j, θ, φ, H, α_{λ, at}) obeys the Bouguer–Lambert exponential law of intensity attenuation [16]

where I_{λ, P}(ρ_i, ϕ_j, θ, φ) is the intensity of radiation emitted by the emitter P(ρ_i, ϕ_j) and

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

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:

if h = H = 0 and a = 0,

if 0 < h ≤ H and a = 0,

if h = H = 0 and a > 0,

if 0 < h ≤ H and $0<a\le \frac{\mathit{hR}}{2H-h}\le R$,

if 0 < h < H and $0<\frac{\mathit{hR}}{2H-h}<a\le \frac{\mathit{hR}}{H}<R$,

if 0 < h < H and $0<\frac{\mathit{hR}}{H}<a<R$,

if 0 < h ≤ H and R < a, where the limiting horizontal angles

the limiting polar angles

and

It can be noted here that the limiting angles γ_{a, ρi} and γ_{R, 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 γ_{a, ρi} and γ_{R, ρi} at ρ_i = a and ρ_i = R, respectively.

 

Fig. 2. Geometrical dependencies between a, ρ_i, r and γ_{a, ρi} at a ≤ R (a) and between R, ρ_i, r and γ_{R, 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, P→S}(ρ_i, ϕ_j) reaching the planar circular surface S from a given point emitter P(ρ_i, ϕ_j) 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}(ρ_i, ϕ_j), at wavelength λ emitted by any point emitter P(ρ_i, ϕ_j) into the space surrounding the cylindrical enclosure.

The solutions represented by Eqs. (8a)-(8g) are appropriate for calculating the flux Φ _{λ, σ→S} from Eq. (2) for arbitrary distributed point emitters P(ρ_i, ϕ_j) composing the source σ within the cylindrical enclosure if the attenuation coefficient α_{λ, at} of the homogeneous medium is defined and the radiant intensities I_{λ, P}(ρ_i, ϕ_j, θ, φ) from each point emitter are known. However the radiative fluxes Φ _{λ, P→S}(ρ_i, ϕ_j) 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 I_{λ, P}(ρ_i, ϕ_j, 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(ρ_i, ϕ_j) 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(ρ_i, ϕ_j) 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

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

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

For identically radiating emitters P(ρ_i, ϕ_j) 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 N_λ(ρ_i) emitters so that

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

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

if a = 0, and

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

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

if 0 < a ≤ h R/(2H - h) ≤ R,

if 0 < R ≤ h R/(2H - h)≤ a ≤ h R/H,

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

if 0 < R ≤ a/2, and

if 0 < a/2 < R < a. The limiting angles γ_{a, ρ} and γ_{R, 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 N_λ 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 I_{λ, 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 I_{λ, P} (θ, φ) not dependent on the horizontal angle φ we have I_{λ, P} (θ, φ) = I_{λ, P} (θ) and Eqs. (13) and (16a)-(16e) simplify to

if a = 0,

if 0 < a ≤ h R/(2H - h) ≤ R,

if 0 < R ≤ h R/(2H - h) ≤ a ≤ h R/H,

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

if 0 < R < a, where

Equations (17a)-(17e) can be easily turned into a computer code for calculating the fluxes <Φ _{λ, σ→S}> if the radiant intensity I_{λ, 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 I_{λ, P} (θ) = I_{λ, P}cos² 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 I_P (θ ₁) = I ₀ cos² θ ₁ as a function of R and H at a = 5 and α_at = 0 (a), as a function of R and H at a = 5 and α_at = 0.1 (b), as a function of a and H at R = 5 and α_at = 0 (c), and as a function of a and H at R = 5 and α_at = 0.1 (d). The data were calculated for the relative units of R, a, H, and α_at i.e. if R, a, and H are given in m then α_at 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 a ≤ R. 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 I_{λ, 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.