## Abstract

An analytical formula is derived for calculating the flux of radiation from a Gaussian source irradiating a spherical object. The formula was derived for the radiant intensity function represented by a paraxial approximate solution of the Halmholtz scalar wave equation. All calculations are presented in the Cartesian 0xyz coordinate system, where the coordinates, x, y and z, determine the center of the spherical object. The center of the source was located at the point P(0,0,0) in the plane z = 0. Some computer simulation results were illustrated graphically and analyzed.

© 2005 Optical Society of America

## 1. Introduction

The concept of Gaussian beams is often used in optics as a good approximation for the transversal intensity distribution of laser radiations. Other electromagnetic radiations from non-Gaussian sources can also be expanded as the sum of Gaussian beams [1–3]. Therefore, with increased application of lasers in the fields of laser communication, laser materials processing, optical storage and laser printing, the Gaussian source model has attracted much attention and been intensively studied. Examples include the works [4–7].

It is known that the power of radiation (or in other words the radiative flux) is a function of the radiation field and always depends on the position of the given object if the field is non-uniform, such as the Gaussian form of a laser beam. In this case, the analytical solution to the radiative power transfer equation requires quadruple-integral evaluation [19, 20]. Usually the exact analytical solution to this equation cannot be obtained in a closed form and calculations must be done numerically. However, the analytic solution to this problem is simplified considerably when the Gaussian source may be approximated by a point source model. Such approximation is frequently used in practice when the spatial extent of the source is less than about 1/20 of its distance from the irradiated object [20, 21]. Obviously, the extended Gaussian source may also be represented by a point source model with a Gaussian intensity profile when the far field approximation is applicable.

In this paper, our aim is to obtain an analytical formula for calculating the radiative flux, ΦG , from a point Gaussian source, G, incident on a spherical object located in far field. The formula is obtained for radiation propagating in a linear, homogeneous and nondispersive medium. To shorten the analytic part of the work we used the final formula derived in Ref. [22]. This formula was derived for calculating a radiative flux, ΦP , from a point source, P, when the radiation is incident on a spherical object lying at an arbitrary distance and any position with respect to the source. The formula with variables influencing the flux ΦP , is presented and characterized in Section 2. In Section 3 we present an analytical expression for the transversal intensity distribution of the Gaussian beam. Then, the single integral formula to calculate the radiative flux ΦG , incident on the spherical object from the point source with a Gaussian intensity profile, is given in Section 4. Next, in Section 5, the dependence of the flux ΦG on the distance from the beam waist center, radius of the object and location with respect to the beam axis is analyzed numerically. Finally, the conclusion is presented in Section 6.

## 2. The radiative flux from a point source illuminating a spherical object

The formula to calculate the flux ΦP from a point source, P, illuminating a spherical object was obtained in double-integral form. We provide a brief sketch of the theory, which is presently more extensively in Ref. [22].

An elementary radiative flux, dΦP (x, y, z), emitted by the source P lies at the point P(0, 0, 0) in the Cartesian coordinate system 0xyz (Fig. 1) is described by the expression [22]

$dΦP(x,y,z;x11,y11)=I(x,y,z;x11,y11)zdx11dy11[(x11−rxy)2+y112+z2]32,$

where x 11 and y 11 represent the coordinates of the Cartesian coordinate system 01x 11 y 11 z 11 oriented along the major and minor ellipse axes of the shadow made by the spherical object on the 0xy plane at z. The radial distances rxy and the distance r are given by

$rxy=x2+y2,$
$r=x2+y2+z2.$

Next the total radiative flux, ΦP (x, y, z), is calculated by integrating Eq. (1) with respect to x 11 and y 11.

The integral solution to Eq. (1), given in Ref. [22], may be rewritten as

$ΦP(x,y,z;ρef,rob)={∫−a2a1dx11∫−y11/y11/I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32,if0≤rxy≤ρef−a1,∫−a2ρef−rxydx11∫−y11/y11/I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32,if0<ρef−a1≤rxy,0,if0<ρef+a2

where

$y11/=ba2−[x11−(a1−a)]2a,$
$a=a1+a2=robzr2−rob2(z2−rob2),$
$a1=r2rob(zr2−rob2−robr2−z2),$
$a2=r2rob(zr2−rob2+robr2−z2),$
$b={rrob(rxy+a1−a)(rxyr2−rob2),ifrxy>0,robzz2−rob2,ifrxy=0,$

and ρef is the arbitrary chosen beam radius at the output plane 0xy limiting the area, within which the radiation is considered to be effective. The geometrical parameters and variables used in Eq. (3) are illustrated in Fig. 1.

Equation (3) is valid for the effective beam waist, ρef , clearly greater then the radius rob . Unfortunately, this equation is inaccurate when the surface made on the plane 0xy at distance z by the radiation incident on the spherical object lies outside the circle of the radius ρef as shows and explains Fig. 2(b).

Fig. 1. A spherical object with the center 01 at the distance r from the origin 0 of the main coordinate system 0xyz erected at the point P or at the center of the emitting Gaussian aperture G. Additional coordinate systems 01x1y1z1 and 01 x 11 y 11 z 11 are erected at the center 01 of the sphere. The parameter a 2, given by Eq. (4e), is always greater than the radius rob when the geometry is illustrated on the 0̀x 11z1 plane. Similar dependencies between geometrical variables used in Eq. (3) are shown in Ref. [22].

Fig. 2. Graphical illustrations of perpendicular projection of the radiative flux with the effective radius ρef at the distance z from the origin 0 of the main coordinate system 0xyz and the surface subtended by the spherical object when 0 ≤ rxyρef - a 1 (a) and 0 < ρef - a1rxy (b). The first function in Eqs. (3) and (5) was calculated for the surface S shown in the part (a). The second function in Eq. (3) was calculated for the surfaces S 1 + S 2 + S 3 shown in the part (b). The radiative flux related to the complete black surface S 3 illustrated in the part (b) was included erroneously into the second function of Eq. (3) in Ref. [22]. The first term of the second function in Eq. (5) was obtained at 0 < ρef - a1rxy for the surface S 1, while the second term of this function for the surface S 2.

The negative sign in the denominators of Eqs. (1) and (3) indicates that the coordinate system 01 xyz was linearly transformed on the distance rxy with respect to the center 01 of the system 01 x 11 y 11 z into positive direction of the coordinate x 11. Thus Eq. (3) is applicable when the radius rxy is treated as negative. The exact solution to the problem outlined above and obtained for r ≥ 0, rxy ≥ 0, and the variables defined in Figs. 1 and 2 is given by the formula

$ΦG(x,y,z;ρef,rob)={∫−a2a1dx11∫−y11/y11/I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32if0≤rxy≤ρef−a1,∫−a2x11/dx11∫−y11/y11/I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32+∫x11/ρef−rxy∫−y11//y11//I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32,if0<ρef−a1

which is valid for

$ρef≥b0=robzz2−rob2,$

and any distance and position of the spherical object with respect to the point 01, where b 0 represents the radius of the surface S made on the 01 x 1 y 1 plane at rxy = 0 by the radiation incident on the object from the point 0. The limits $x11/$ and $y11//$, in the two terms of the second expression in Eq. (5), are given by

$x11/=b2(a−a1)−a2rxy+aa2ρef2−2ab2(a1+rxy)+b2[(b2+ρef2)+(a1+rxy)2]a2−b2$
$y11//=ρef2−(x11+rxy)2,$

and were calculated from two equations describing the external contours on the plane 0xy at the distance z made by the surface S and by the radiation of effective radius ρef

$[x11−(a1−a)]2a2+y112b2=1,$
$(x11+rxy)2+y112=ρef2,$

respectively, see Figs. 1 and 2.

In a similar way, we may obtain the integral solution to Eq. (1) for the case

$ρef

corresponding to the situations where the shadow made on the plane 0xy at z and for rxy = 0 is greater than the radius ρef . This solution obtains the form

$ΦP(x,y,z;ρef,rob)={∫−ρef−rxyρef−rxydx11∫−y11//y11//I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32if0≤rxy≤a2−ρef,∫−a2x11/dx11∫−y11/y11/I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32+∫x11/ρef−rxydx11∫−y11//y11//I(x,y,z;x11,y11)zdy11[(x11−rxy)2+y112+z2]32,if0≤a2−ρef

where $y11/$, $x11/$, and $y11//$ are given by Eqs. (4a), (7), and (8), respectively.

Equations (5) and (10) may be integrated analytically with respect to y 11 and x 11 for some radiant intensity functions I(x, y, z; x 11, y 11). However, for a number of these functions the integration with respect to y 11 and x 11 does not lead us to analytical closed form expressions. In this paper, we will present the solution to Eqs. (5) and (10) for the Gaussian profile (in transversal direction) of the intensity I(x, y, z; x 11, y 11).

## 3. Some fundamental expressions for the Gaussian beam

The propagation of Gaussian beams in free space and through some optical elements has been discussed in several papers. An excellent description of this problem may be found in Ref. [23].

The expression for the spatial distribution of the electric field intensity, E(x, y, z; A, ρ0, λ), of the Gaussian wave propagating along the z-axis in an isotropic, homogeneous and non-dispersive medium can be derived from the scalar Helmholtz wave equation after applying the paraxial approximation [23], and is given by

$E(x,y,z;A,ρ0,λ)=Aρ0ρexp(−rxy2ρ2)exp{−i[krxy22R+kz−arctan(zzR)]},$

where A is the electric field amplitude of the beam waist center, ρ0 is the minimal radius of the beam waist at z = 0, k = 2π/λ is the wave number, and

$ρ=ρ01+(zzR)2,$
$R=z[1+(zRz)2].$

The parameter

$zR=πρ02λ,$

is known as the Rayleigh range at the wavelength λ. R is the curvature of the wave front and ρ denotes the spot size which is equal to the distance in the transverse direction at which the field amplitude decays to 1/e of its maximum value.

Fig. 3. The transverse distribution of the radiant intensity I(x, y, z; I 0, ρ 0, λ) and spot sizes ρ0, ρ1 and ρ2 given by Eq. (12a) at distances z 0 = 0 [m], z 1 = 100 [m] and z 2 = 200 [m] from the emitting Gaussian aperture. For the dependence of I(x, y, z; I 0, ρ0, λ) on the variable I 0, ρ0 and λ see Eqs. (12a), (13) and (19). The data were obtained for ρ0 = 2.5×10-3 [m] and λ = 632.8×10-9 [m] corresponding to the wavelength of He-Ne laser emission. Solid lines represent the influence of the distance z on the radiant intensity I(x, y, z; I 0, ρ0, λ) equal to 2I 0/e, I 0/e, and I 0/(16 e), where I 0 denotes the maximal intensity at rxy = 0. The beam divergence in the far field (for z » zR ) is illustrated by the dashed straight lines and is equal to the half-angle θ = λ/(π ρ0) [24]. Here zR = 31.03 [m] and θ = 8.06×10-5 [rad]. For the distance z » zR the extended Gaussian source may be considered as a point Gaussian source. In these plots the radius rxy is expressed in millimeters and z in meters, so it is clearly seen that the extended Gaussian source will be well approximated by the point Gaussian source at z » zR . When the Gaussian beam is focused by a thin lens of focal length, f, then the focused spot size, ρ0f = ρ0/[1+(zR /f)2]1/2, and the far field approximation may be used at z » zRf = π0f)2/λ [24].

As the Gaussian beam propagates along the z-axis it expands in the manner presented in Fig. 3. The dashed straight lines clearly demonstrate that at a distance z sufficiently greater than zR the Gaussian beam propagates similarly to the radiation emitted by a point source, represented here and in the rest of the paper, by the symbol G. As noted previously, in practical situations many extended sources are often represented by point sources if their spatial extents are less than about 1/20 of the distances to the irradiated surfaces [20, 21]. These representations may also be used for shorter distances to the irradiated surface in the case of radiation converged or diverged by optical lenses. Clearly a point source model may represent a Gaussian beam in the case of far field approximation, that is when z » zR for non-focused beam or at a distance z » zRf = π(ρ0f)2/λ for a beam in a vacuum focused by a thin lens, where ρ0f = ρ0/[1+(zR /f)2]1/2 and f is the focal length of the lens [24]. Therefore, for the point Gaussian source, we may calculate the radiative flux emitted into the space subtended by a spherical object using Eqs. (5) and (10).

The time-average power transferred across the boundary surface of the source is equal to the value of the z component of the time-average Pointing vector, Sav(x, y, z; A, ρ0, λ, ϵr , μr ), defined as [25]

$Sav(x,y,z;A,ρ0,λ,εr,μr)=12Re[Sc(x,y,z;A,ρ0,λ)],$

where

$Sc(x,y,z;A,ρ0,λ,εr,μr)=E(x,y,z;A,ρ0,λ)×H*(x,y,z;A,ρ0,λ,εr,μr),$

is the complex Pointing vector, H(x, y, z; A, ρ0, λ, ϵr , μr ) is the magnetic field strength, ϵr and μr are called, respectively, the relative permittivity and relative permeability of the medium and the asterisk * denotes the complex conjugate function. The magnetic field H(x, y, z; A, ρ0, λ, ϵr , μr ) can be expressed by the electric field intensity E(x, y, z; A, ρ0, λ) as follows [25]

$μrμ0H2(x,y,z;A,ρ0,λ,εr,μr)=εrε0E2(x,y,z;A,ρ0,λ),$

where, ϵ 0 and μ 0 are, respectively, the free-space permittivity (ϵ 0 = 8.854×10-12 F m-1), and the free-space permeability (μ0 = 4 π×∙10-7 H m-1). From Eqs. (11), (14) and (15) we obtain

$Sav(x,y,z;A,ρ0,λ,εr,μr)=12εrε0μrμ0E(x,y,z;A,ρ0,λ)E*(x,y,z;A,ρ0,λ).$

The radiance, L(x, y, z; A, ρ0, λ, ϵr , μr ), of the radiation (or the power of radiation emitted by a surface unit in the direction z > 0 or within a solid angle equal to 2 π) for the electromagnetic wave can be calculated as

$L(x,y,z;A,ρ0,λ,εr,μr)=12πSav(x,y,z;A,ρ0,λ,εr,μr)$
$=cε04πεrμrE(x,y,z;A,ρ0,λ)E*(x,y,z;A,ρ0,λ),$

where c = 1/(ϵ0 μ0)1/2 = 2.998×108 m s-1 denotes the speed of light in a vacuum. Substituting Eq. (11) into (16) leads to

$L(x,y,z;A,ρ0,λ,εr,n)=A2cεrρ024πnρ2exp(−2rxy2ρ2),$

where n = (ϵr μr )1/2 is the refractive index of the medium surrounding the source.

The intensity of the radiation from the point source representing an extended Gaussian source can be calculated as

$I(x,y,z;A,ρ0,λ,εr,n)=I0(A,ρ0,εr,n)exp(−2rxy2ρ2),$

where I 0(A, ρ0, ϵr , n) denotes on-axis intensity at the distance z 0 given by

$I0(A,ρ0,εr,n)=∫−∞∞[∫−∞∞L(x,y,z;A,ρ0,λ,εr,n)dy]dx=A2cεrε0ρ02(8nρ2).$

It can easily be seen that the radiance L(x, y, z; A, ρ0, λ, ϵr , n) in Eqs. (18) is expressed in W m-2 sr-1, while the intensity I(x, y, z; A, ρ0, λ, ϵr , n) in Eq. (17) is expressed in W sr-1.

Inside the shadow made by the spherical object on the plane 01 xy at z the intensity I(x, y, z; A, ρ0, λ, ϵr , n) becomes

$I(x,y,z;A,ρ0,λ,εr,n)=I0(A,ρ0,εr,n)exp{−2[(x+x1)2+(y+y1)2]ρ2},$

and in the coordinate system 01 x 11 y 11 z 11 illustrated in Figs. 1 and 2 for rxy ≥ 0 takes the form

$I(x,y,z;I0,ρ0,λ)=I0exp{−2[(x11+rxy)2+y112ρ2},$

where I 0 = I 0(A, ρ0, ϵr , n).

## 4. The single integral formula for the radiative flux from a point Gaussian source

Substituting Eq. (21) into the first two expressions in Eq. (5) we get

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫−a2a1e−2(x11+rxy)2ρ2dx11∫−y11/y11/e−2y112ρ2dy11[(x11+rxy)2+z2+y112]32,$

if 0 < rxy ρef - a 1 and

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)=I0z{∫−a2x11/e−2(x11+rxy)2ρ2dx11∫−y11/y11/e−2y112ρ2dy11[(x11+rxy)2+z2+y112]32$
$+∫x11/ρef−rxye−2(x11+rxy)2ρ2dx11∫−y11//y11//e−2y112ρ2dy11[(x11+rxy)2+z2+y112]32},$

if 0 < ρef - a 1rxy < ρef + a 2. The subscript G in Eqs. (22a) and (22b) indicates that the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) is calculated for a Gaussian source.

The integrals in Eqs. (22a) and (22b) may be evaluated into superpositions of some elementary and hypergeometric functions after expansion of exp[-2(y 11/ρ)2] into the power series of 2 (y 11/ρ)2 and integration [26]. In Appendix A we present a simpler and more practical alternative solution. This solution leads us to the formula

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)={F1/(x,y,z;I0,ρef,ρ0,rob,λ),if0≤rxy≤ρef−a1,F2/(x,y,z;I0,ρef,ρ0,rob,λ)+F3//(x,y,z;I0,ρef,ρ0,rob,λ),if0<ρef−a1≤rxy<ρef+a2,0,if0<ρef+a2≤rxy,$

valid for ρefb 0 = robz/[(z 2 - $rob2$)1/2]. The functions $F1/$(x,y,z;I 0,ρef0, rob ,λ), $F2/$(x,y,z;I 0,ρef0, rob ,λ) and $F3//$(x,y,z;I 0,ρef0, rob ,λ),are given by

$F1/(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫−a2a1f/(x,y,z;ρ0,λ)dx11$
$=I0z{∑i=1∞∑k=1∞(−1)i+k(2i−1)!!(2i−3)!!ρ2(i−k)(2i−2)!!(2k−1)(k−1)!22i−k−2$
$×∫−a2a1(y11/)2k−1e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2dx11$
$+∑i=1∞∑j=1i−l(−1)i2(2i−1)!!(2i−3)!!(2i−2)!![2(i−j)−1]!!(ρ2)2j$
$×∫−a2a1(y11/)2(i−j)−1e[−2(x11+rxy)2+(y11/)2]ρ2[(x11+rxy)2+z2](2i+1)2dx11},$
$F2/(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫−a2x11/f/(x,y,z;ρ0,λ)dx11$
$=I0z{∑i=1∞∑k=1∞(−1)i+k(2i−1)!!(2i−3)!!ρ2(i−k)(2i−2)!!(2k−1)(k−1)!22i−k−2$
$×∫−a2x11/(y11/)2k−1e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2dx11$
$+∑i=1∞∑j=1i−l(−1)i2(2i−1)!!(2i−3)!!(2i−2)!![2(i−j)−1]!!(ρ2)2j$
$×∫−a2x11/(y11/)2(i−j)−1e−2[(x11+rxy)2+(y11/)2]ρ2[(x11+rxy)2+z2](2i+1)2dx11,}$
$F3//(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫x11/ρef−rxyf//(x,y,z;ρ0,λ)dx11$
$=I0z{∑i=1∞∑k=1∞(−1)i+k(2i−1)!!(2i−3)!!ρ2(i−k)(2i−2)!!(2k−1)(k−1)!22i−k−2$
$×∫x11/ρef−rxy(y11//)2k−1e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2dx11$
$+∑i=1∞∑j=1i−l(−1)i2(2i−1)!!(2i−3)!!(2i−2)!![2(i−j)−1]!!(ρ2)2j$
$×∫x11/ρef−rxy(y11//)2(i−j)−1e−2[(x11+rxy)2+(y11//)2]ρ2[(x11+rxy)2+z2](2i+1)2dx11}.$

Similarly, the single integral solution to Eq. (10) for ρef < b 0 = robz/[(z 2 - $rob2$)1/2] is

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)={F4//(x,y,z;I0,ρef,ρ0,rob,λ)if0≤rxy≤a−ρef,F2/(x,y,z;I0,ρef,ρ0,rob,λ)+F3//(x,y,z;I0,ρef,ρ0,rob,λ)if0≤a2−ρef≤rxy

where

$F4//(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫−ρef−rxyρef−rxyf//(x,y,z;ρ0,λ)dx11$
$I0z{∑i=1∞∑k=1∞(−1)i+k(2i−1)!!(2i−3)!!ρ2(i−k)(2i−2)!!(2k−1)(k−1)!22i−k−2$
$×∫−ρef−rxyρef−rxy(y11//)2k−1e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2dx11$
$+∑i=1∞∑j=1i−l(−1)i2(2i−1)!!(2i−3)!!(2i−2)!![2(i−j)−1]!!(ρ2)2j$
$×∫−ρef−rxyρef−rxy(y11//)2(i−j)−1e−2[(x11+rxy)2+(y11//)2]ρ2[(x11+rxy)2+z2](2i+1)2dx11}.$

It is important to note that $y11/$ and $y11//$ in Eqs. (24a)-(24c) and (26) depend on x 11 as shown by Eqs. (4a) and (8). Therefore, the integrals with respect to x 11 may be calculated only into impractical extensive combinations of various elementary and special functions. This calculation is not presented here because it is usually easier to compute the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) from the formulas (23) and (25) applying one of well known simple numerical procedures for single integral evaluation than from extensive analytical expressions.

## 5. Numerical calculation results

Numerical results were calculated using our subroutine program written in Mathematica 2.2.3 [26]. The flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) was computed for various numbers of terms in the series given by Eqs. (23) and (25) and was then tested for convergence to obtain the precision desired. The convergence was tested at the distance z > 40 ρ 0, usually sufficient to approximate an extended circular source by a point source model. The successive terms in these series decreases very quickly with increased i and k when rob < ρ0, that is when the radius rob of the spherical object was less then the beam waist radius ρ0 at z = 0. However the series were converged quickly with increased i and slowly with increased k when rob > ρ0. All results in the paper were calculated with accuracy to a twelve decimal places, and precision markedly better than 0.1 %, which is acceptable in problems of radiative flux calculation [27–28], was always obtained for i = 20 and k = 30.

Fig. 4. The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of x and y at rob = 0.0001 [m] (a), rob = 0.0025 [m] (b), rob = 0.0050 [m] (c), and rob = 0.0500 [m] (d). The data were calculated for I 0 = 1 [W sr-1], z = 0.2 [m], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Figures 4(a)–(d) show four plots of the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), in absolute units, as functions of x and y at some radii rob and at given values of the remaining variables. The transversal distribution of ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) in Fig. 4(a), at rob = 0.001 [m], clearly less then the beam waist radius ρ0 at z = 0, is almost identical to the Gaussian distribution. However as rob increases, at the constant distance z, the transversal distribution of the total flux diverges from the Gaussian in the manner illustrated by the subsequent Figs. 4(b), (c) and (d). For a given radius rob the axial flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) reaches maximal value, which does not depend on the further increase of the radius rob as shows Fig. 4(d).

Fig. 5. The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated for I 0 = 1 [W sr-1], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Fig. 6. The absolute values of the total radiative flux Φp (x, y, z; I 0, ρef , rob ), denoted by ΦP , from the isotropic point source P as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated from Eq. (5) at I 0 = 1 [W sr-1] and ρef = 0.005 [m] for the radiant intensity function I(x, y, z; x 11, y 11) = I 0 z 2/[(x 11 + rxy )2 + z 2 + $y112$], where I 0 = I(x=0, y=0, z).

The dependence of the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) on the radial distance rxy with respect to the beam axis and on the axial distance z from the source G at given values of I 0, ρef , ρ0, rob and λ is plotted in Fig. 5(a). The data illustrate that the total flux decreases as rxy and z increase but depends more sharply on the radial distance rxy than on the axial distance z. In Fig. 5(b) we observe the influence of the object radius rob and the distance rxy on the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) at the same values of I 0, ρef , ρ0, rob and λ as in Fig. 5(a). It is not difficult to observe that the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) is lager for the lager objects when the radius rob is less than ρ0. When the radius rob is greater than ρ0 the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) obtains its maximal axial value independent of the subsequent growth of rob .

Figures 6(a) and (b) show the total flux ΦG (x, y, z; I 0, ρef , rob ) emitted by the isotropic point source P into the direction of spherical object of the radius rob as dependencies on these same variables as in Figs. 5(a) and (b). Comparing the data from Figs. 5 and 6 it is seen that the flux Φp (x, y, z; I 0, ρef , rob ) is significantly greater than the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) at radial distance rxy = 0. Then, at given distance z the flux Φp (x, y, z; I 0, ρef , rob ) continuously decayed with increased rxy and is smaller at smaller rob until Eq. (6) is fulfilled.

## 6. Conclusions

In this paper, we have obtained a single integral formula to calculate the flux of radiation from a Gaussian source (in transversal direction) propagating in one linear, homogeneous and nondispersive medium in the directions subtended by a spherical object. The formula was solved in the Cartesian coordinate system erected at the center of the object. The solution was given for an arbitrary value of the object radius rob and for axial distances z allowing the approximation of an extended Gaussian source by a point Gaussian source model. In particular, the formula satisfies the conditions of far field approximation theory. The analytical part of this work begins with a concise presentation of the fundamental radiometric equations leading to the formula derived elsewhere [22] and used for calculating the radiative flux emitted by a point source into the space subtended by a spherical object. In this paper, we showed that the total radiative flux incident on spherical objects from a point Gaussian source may be evaluated using Eqs. (23) and (25) determined by the double sum of single integral expressions. However, the accuracy of these calculations depends on the number of terms summed with respect to the indices i and k, but is even more dependent on the number of terms summed with respect to the index k. Some selected numerical calculation results, given in absolute units, were illustrated graphically and analyzed.

It was shown that when a Gaussian beam is incident on a spherical object with the small radius rob obeying the relation robρef (z 2-$rob2$)1/2/z (see Eq. (6) and condition under Eq. (23)), the transversal distribution of the radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) has a nearly Gaussian shape. When the radiation is incident on an object with the bigger radius rob then its axial flux is larger then the value for the smaller radius rob , but the transversal distribution of the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) diverges more from Gaussian. Such relationships are illustrated in Figs. 4(a) and 4(b) When the radius rob > ρef (z 2-$rob2$)1/2/z then the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) must be calculated from Eq. (25) and represents flattened shapes similar to those in Figs. 4(c) and 4(d). The larger the radius rob , the more flattened the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ). As the spherical objects move across the radiation field from the point Gaussian source, the radial distance rxy changes and the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) varies considerably as shown in Figs. 5(a) and (b). This variation may be important in processes applying Gaussian beams to irradiation of spherical objects.

Finally the total flux Φp (x, y, z; I 0, ρef , rob ) emitted by the isotropic point source P, was calculated from Eq. (5) for these same values of the variables as the total flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) shown in Figs 5(a) and (b). The data obtained were illustrated in Figs. 6(a) and (b) for comparison. The results presented in Figs 6(a) and (b) indicate that the flux Φp (x, y, z; I 0, ρef , rob ) reaches the greater value at rxy = 0 and is varied more slowly with increased rxy at given z than the flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ) from the point Gaussian source shown in Figs. 5(a) and (b).

## Appendix A

Dividing the denominators in the second integrals of Eqs. (24a) and (24b) by [(x 11 + r xy)2 + z 2]3/2 we have

$1[(x11+rxy)2+z2+y112]32=1[(x11+rxy)2+z2]32{1+y112[(x11+rxy)2+z2]}32,$

and if

$y112[(x11+rxy)2+z2]<1,$

then the function 1/{1 + $y112$/[(x 11 + rxy )2 + z 2]3/2} can be represented by the following power series of $y112$/[(x 11 + rxy )2 + z 2]:

$1{1+y112[(x11+rxy)2+z2]}32=∑i=1∞(−1)i−1(2i−1)!!(2i−2)!!(y11)2(i−1)[(x11+rxy)2+z2]i−1.$

valid for any value of $y112$, rxy and z because no real solutions for rxy can be obtained from the inequality $y112$/[(x 11 + rxy )2 + z 2] ≥ when the major ellipse axis a and the minor ellipse axis b are expressed by r, z, and rob using Eqs. (4b)–(4e).

Substituting Eq. (A.3) into (A.1) the inner integrals in Eqs. (24a) and (24b) may be given by

$∫y112(i−1)e−2(y11)2ρ2dy11=(2i−3)!!22i2πρ(2i−1)Erf(2y11ρ)$
$−∑j=1i−1(2i−3)!!y11(2i−2j−1)(2i−2j−1)!!(ρ2)2je−2(y11ρ)2+C,$

where i = 1, i = 2, i = 3, …, and C is a constant. Thus Eqs. (24a) and (24b) take the form

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)=I0z∫−a2a1f/(x,y,z;ρ0,λ)dx11,$

if 0 < rxyρef - a 1, and

$ΦG(x,y,z;I0,ρef,ρ0,rob,λ)=I0z[∫−a2x11/f/(x,y,z;ρ0,λ)dx11$
$+∫x11/ρef−rxyf//(x,y,z;ρ0,λ)dx11],$

if ρef - a 2rxy < ρef + a 2, where

$f/(x,y,z;ρ0,λ)=∑i=1∞(−1)i−1(2i−1)!!(2i−3)!!(2i−2)!!〈2(ρ2)2i−1{∫−2y11/ρ2y11/ρe−(y11)2dy11$
$−∑j=1i−1(ρ2)2j−12(y11/)2i−2j−1e−2(y11/ρ)2[2(i−j)−1]!!}e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2〉,$

and

$f//(x,y,z;ρ0,λ)=∑i=1∞(−1)i−1(2i−1)!!(2i−3)!!(2i−2)!!〈2(ρ2)2i−1{∫−2y11//ρ2y11//ρe−(y11)2dy11$
$−∑j=1i−1(ρ2)2j−12(y11//)2i−2j−1e−2(y11//ρ)2[2(i−j)−1]!!}e−2(x11+rxy)2ρ2[(x11+rxy)2+z2](2i+1)2〉.$

Note that the functions f /(x, y, z; ρ0, λ) and f //(x, y, z; ρ0, λ) differ only in the variables $y11/$ and $y11//$. The integrals with respect to y 11 in these functions may then be expanded in the power series of $y112$ as

$∫e−(y11)2dy11=∑k=1∞(−1)k+112k−1(y11)2k−1(k−1)!,$

and after some simple algebra Eqs. (A-5a) and (A-5b) obtain the form of Eq. (23).

1. X. Deng, Y. Li, D. Fan, and Y Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997). [CrossRef]

2. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). [CrossRef]

3. L. Zeni, S. Campopiano, A. Cutolo, and G. D“Angelo, “Power semiconductor laser diode characterization,” Opt. Lasers Eng. 39, 203–217 (2003). [CrossRef]

4. L.D. Dickson, “Characteristics of propagating Gaussian beam,” Appl. Opt. 9, 1854 (1970). [CrossRef]   [PubMed]

5. C. Hu and T.N. Baker, “Prediction of laser transformation hardening depth using a line source model.” Acta Metal. Mater. 43, 3563–3569 (1995). [CrossRef]

6. K. Shinozaki, C. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997). [CrossRef]

7. M.R. Taghizadeh, P. Blair, K. Ballüder, N.J. Waddie, P. Rudman, and N. Ross, “Design and fabrication of diffractive elements for laser material processing applications,” Opt. Lasers Eng. 34, 289–307 (2000). [CrossRef]

8. G. Grynberg and C Robilliard, “Cold atom in dissipative optical lattices,” Phys. Rep. 355, 335–451 (2001). [CrossRef]

9. R.C. Gauthier, “Optical trapping: a tool to assist optical machining,” Opt. Laser Technol. 29, 389–399 (1997). [CrossRef]

10. C.S. Adams and E. Riis, “Laser cooling and trapping of neutral atoms,” Prog. Quantum Electron. 21, 1–79 (1997). [CrossRef]

11. Y.P. Han, L. Méès, K.F. Ren, G. Gouesbet, S.Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002). [CrossRef]

12. A.R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. , 25,1–53 (1999). [CrossRef]

13. C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quantum Spectrosc. Radiat. Transfer 55, 673–681 (1996). [CrossRef]

14. G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Eng. 60, 31–39 (2003). [CrossRef]

15. J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002). [CrossRef]

16. G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996). [CrossRef]

17. S.R. Govil, D.C. Agrawal, K.P. Rai, and S.N. Thakur, “Growth responses of Vigna radiata seeds to laser irradiation in the UV-A region,” Phys. Plantarum 63, 133–134 (1985). [CrossRef]

18. W.H.A. Wilde, W.H. Parr, and D.W. McPeak, “Seeds bask in laser light,” Laser Focus 5, 41–42 (1969).

19. F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) p. 37.

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

21. J.R. Meyer-Arendt, “Radiomatry and photometry: units and conversion factors,” Appl. Opt. 7, 2081–2084 (1968). [CrossRef]   [PubMed]

22. S. Tryka, “Spherical object in radiation field from a point source,” Opt. Express 12, 512 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-512. [CrossRef]   [PubMed]

23. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef]   [PubMed]

24. S. Liu, H. Guo, M. Liu, and G. Wu, “A comparison of propagation characteristics of focused Gaussian beam and fundamental Gaussian beam in vacuum,” Phys. Lett. A , 327, 254–262 (2004). [CrossRef]

25. E.J. Rethwell and M.J. Cloud, Electromagnetics, (CRC, Boca Raton, 2001) Chap. IV, 6–51.

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

27. J.H. Habbell, R.L. Bach, and R.J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. 65C, 249–264 (1961).

28. R.P. Gardner and K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods , 93, 163–167 (1971). [CrossRef]

### References

• View by:
• |

1. X. Deng, Y. Li, D. Fan and Y Qiu, �??Propagation of paraxial circular symmetric beams in a general optical system,�?? Opt. Commun. 140, 226-230 (1997).
[CrossRef]
2. F. Gori, Flattened Gaussian beams,�?? Opt. Commun. 107, 335-341 (1994).
[CrossRef]
3. L. Zeni, S. Campopiano, A. Cutolo and G. D�??Angelo, �??Power semiconductor laser diode characterization,�?? Opt. Lasers Eng. 39, 203-217 (2003).
[CrossRef]
4. L.D. Dickson, �??Characteristics of propagating Gaussian beam,�?? Appl. Opt. 9, 1854 (1970).
[CrossRef] [PubMed]
5. C. Hu and T.N. Baker, �??Prediction of laser transformation hardening depth using a line source model.�?? Acta Metal. Mater. 43, 3563-3569 (1995).
[CrossRef]
6. K. Shinozaki, C. Xu, H. Sasaki and T. Kamijoh, �??A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,�?? Opt. Commun. 133, 300-304 (1997).
[CrossRef]
7. M.R. Taghizadeh, P. Blair, K. Ballüder, N.J. Waddie, P. Rudman and N. Ross, �??Design and fabrication of diffractive elements for laser material processing applications,�?? Opt. Lasers Eng. 34, 289-307 (2000).
[CrossRef]
8. G. Grynberg and C Robilliard, �??Cold atom in dissipative optical lattices,�?? Phys. Rep. 355, 335-451 (2001).
[CrossRef]
9. R.C. Gauthier, �??Optical trapping: a tool to assist optical machining,�?? Opt. Laser Technol. 29, 389-399 (1997).
[CrossRef]
10. C.S. Adams and E. Riis, �??Laser cooling and trapping of neutral atoms,�?? Prog. Quantum Electron. 21, 1-79 (1997).
[CrossRef]
11. Y.P. Han, L. Méès, K.F. Ren, G. Gouesbet, S.Z. Wu and G. Gréhan, �??Scattering of light by spheroids: the far field case,�?? Opt. Commun. 210, 1-9 (2002).
[CrossRef]
12. A.R. Jones, �??Light scattering for particle characterization,�?? Prog. Energy Combust. Sci., 25, 1-53 (1999).
[CrossRef]
13. C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quantum Spectrosc. Radiat. Transfer 55, 673-681 (1996).
[CrossRef]
14. G. Brescia, R. Moreira, L. Braby and E. Castell-Perez, �??Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,�?? J. Food Eng. 60, 31-39 (2003).
[CrossRef]
15. J.F. Diehl, �??Food irradiation-past, present and future,�?? Radiat. Phys. Chem. 63, 211-215 (2002).
[CrossRef]
16. G. W. Gould, �??Potential of irradiation as a component of mild combination preservation procedures,�??Radiat. Phys. Chem. 48, 366 (1996).
[CrossRef]
17. S.R. Govil, D.C. Agrawal, K.P. Rai and S.N. Thakur, �??Growth responses of Vigna radiata seeds to laser irradiation in the UV-A region,�?? Phys. Plantarum 63, 133-134 (1985).
[CrossRef]
18. W.H.A. Wilde, W.H. Parr and D.W. McPeak, �??Seeds bask in laser light,�?? Laser Focus 5, 41-42 (1969).
19. F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) p. 37.
20. M. Strojnik and G Paez, �??Radiometry�?? in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) 649-699.
21. J.R. Meyer-Arendt, �??Radiomatry and photometry: units and conversion factors,�?? Appl. Opt. 7, 2081-2084 (1968).
[CrossRef] [PubMed]
22. S. Tryka, �??Spherical object in radiation field from a point source,�?? Opt. Express 12, 512 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-512">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-512</a>
[CrossRef] [PubMed]
23. H. Kogelnik and T. Li, �??Laser beams and resonators,�?? Appl. Opt. 5, 1550-1567 (1966).
[CrossRef] [PubMed]
24. S. Liu, H. Guo, M. Liu and G. Wu, �??A comparison of propagation characteristics of focused Gaussian beam and fundamental Gaussian beam in vacuum,�?? Phys. Lett. A, 327, 254-262 (2004).
[CrossRef]
25. E.J. Rethwell and M.J. Cloud, Electromagnetics, (CRC, Boca Raton, 2001) Chap. IV, 6-51.
26. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), 44-186.
27. R.P. Gardner and K. Verghese, �??On the solid angle subtended by a circular disc,�?? Nucl. Instrum. Methods, 93, 163-167 (1971).
[CrossRef]
28. J.H. Habbell, R.L. Bach and R.J. Herbold, �??Radiation field from a circular disk source,�?? J. Res. Natl. Bur. Stand. 65C, 249-264 (1961).

#### Acta Metal. Mater. (1)

C. Hu and T.N. Baker, �??Prediction of laser transformation hardening depth using a line source model.�?? Acta Metal. Mater. 43, 3563-3569 (1995).
[CrossRef]

#### Handbook of Optical Engineering (1)

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

#### J. Food Eng. (1)

G. Brescia, R. Moreira, L. Braby and E. Castell-Perez, �??Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,�?? J. Food Eng. 60, 31-39 (2003).
[CrossRef]

#### J. Quantum Spectrosc. Radiat. Transfer (1)

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quantum Spectrosc. Radiat. Transfer 55, 673-681 (1996).
[CrossRef]

#### J. Res. Natl. Bur. Stand. (1)

J.H. Habbell, R.L. Bach and R.J. Herbold, �??Radiation field from a circular disk source,�?? J. Res. Natl. Bur. Stand. 65C, 249-264 (1961).

#### Laser Focus (1)

W.H.A. Wilde, W.H. Parr and D.W. McPeak, �??Seeds bask in laser light,�?? Laser Focus 5, 41-42 (1969).

#### Nucl. Instrum. Methods (1)

R.P. Gardner and K. Verghese, �??On the solid angle subtended by a circular disc,�?? Nucl. Instrum. Methods, 93, 163-167 (1971).
[CrossRef]

#### Opt. Commun. (4)

Y.P. Han, L. Méès, K.F. Ren, G. Gouesbet, S.Z. Wu and G. Gréhan, �??Scattering of light by spheroids: the far field case,�?? Opt. Commun. 210, 1-9 (2002).
[CrossRef]

K. Shinozaki, C. Xu, H. Sasaki and T. Kamijoh, �??A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,�?? Opt. Commun. 133, 300-304 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan and Y Qiu, �??Propagation of paraxial circular symmetric beams in a general optical system,�?? Opt. Commun. 140, 226-230 (1997).
[CrossRef]

F. Gori, Flattened Gaussian beams,�?? Opt. Commun. 107, 335-341 (1994).
[CrossRef]

#### Opt. Laser Technol. (1)

R.C. Gauthier, �??Optical trapping: a tool to assist optical machining,�?? Opt. Laser Technol. 29, 389-399 (1997).
[CrossRef]

#### Opt. Lasers. Eng. (2)

L. Zeni, S. Campopiano, A. Cutolo and G. D�??Angelo, �??Power semiconductor laser diode characterization,�?? Opt. Lasers Eng. 39, 203-217 (2003).
[CrossRef]

M.R. Taghizadeh, P. Blair, K. Ballüder, N.J. Waddie, P. Rudman and N. Ross, �??Design and fabrication of diffractive elements for laser material processing applications,�?? Opt. Lasers Eng. 34, 289-307 (2000).
[CrossRef]

#### Phys. Lett. (1)

S. Liu, H. Guo, M. Liu and G. Wu, �??A comparison of propagation characteristics of focused Gaussian beam and fundamental Gaussian beam in vacuum,�?? Phys. Lett. A, 327, 254-262 (2004).
[CrossRef]

#### Phys. Plantarum (1)

S.R. Govil, D.C. Agrawal, K.P. Rai and S.N. Thakur, �??Growth responses of Vigna radiata seeds to laser irradiation in the UV-A region,�?? Phys. Plantarum 63, 133-134 (1985).
[CrossRef]

#### Phys. Rep. (1)

G. Grynberg and C Robilliard, �??Cold atom in dissipative optical lattices,�?? Phys. Rep. 355, 335-451 (2001).
[CrossRef]

#### Prog. Energy Combust. Sci. (1)

A.R. Jones, �??Light scattering for particle characterization,�?? Prog. Energy Combust. Sci., 25, 1-53 (1999).
[CrossRef]

#### Prog. Quantum Electron. (1)

C.S. Adams and E. Riis, �??Laser cooling and trapping of neutral atoms,�?? Prog. Quantum Electron. 21, 1-79 (1997).
[CrossRef]

J.F. Diehl, �??Food irradiation-past, present and future,�?? Radiat. Phys. Chem. 63, 211-215 (2002).
[CrossRef]

G. W. Gould, �??Potential of irradiation as a component of mild combination preservation procedures,�??Radiat. Phys. Chem. 48, 366 (1996).
[CrossRef]

#### Other (3)

F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) p. 37.

E.J. Rethwell and M.J. Cloud, Electromagnetics, (CRC, Boca Raton, 2001) Chap. IV, 6-51.

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

### Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (6)

Fig. 1.

A spherical object with the center 01 at the distance r from the origin 0 of the main coordinate system 0xyz erected at the point P or at the center of the emitting Gaussian aperture G. Additional coordinate systems 01x1y1z1 and 01 x 11 y 11 z 11 are erected at the center 01 of the sphere. The parameter a 2, given by Eq. (4e), is always greater than the radius rob when the geometry is illustrated on the 0̀x 11z1 plane. Similar dependencies between geometrical variables used in Eq. (3) are shown in Ref. [22].

Fig. 2.

Graphical illustrations of perpendicular projection of the radiative flux with the effective radius ρef at the distance z from the origin 0 of the main coordinate system 0xyz and the surface subtended by the spherical object when 0 ≤ rxyρef - a 1 (a) and 0 < ρef - a1rxy (b). The first function in Eqs. (3) and (5) was calculated for the surface S shown in the part (a). The second function in Eq. (3) was calculated for the surfaces S 1 + S 2 + S 3 shown in the part (b). The radiative flux related to the complete black surface S 3 illustrated in the part (b) was included erroneously into the second function of Eq. (3) in Ref. [22]. The first term of the second function in Eq. (5) was obtained at 0 < ρef - a1rxy for the surface S 1, while the second term of this function for the surface S 2.

Fig. 3.

The transverse distribution of the radiant intensity I(x, y, z; I 0, ρ 0, λ) and spot sizes ρ0, ρ1 and ρ2 given by Eq. (12a) at distances z 0 = 0 [m], z 1 = 100 [m] and z 2 = 200 [m] from the emitting Gaussian aperture. For the dependence of I(x, y, z; I 0, ρ0, λ) on the variable I 0, ρ0 and λ see Eqs. (12a), (13) and (19). The data were obtained for ρ0 = 2.5×10-3 [m] and λ = 632.8×10-9 [m] corresponding to the wavelength of He-Ne laser emission. Solid lines represent the influence of the distance z on the radiant intensity I(x, y, z; I 0, ρ0, λ) equal to 2I 0/e, I 0/e, and I 0/(16 e), where I 0 denotes the maximal intensity at rxy = 0. The beam divergence in the far field (for z » zR ) is illustrated by the dashed straight lines and is equal to the half-angle θ = λ/(π ρ0) [24]. Here zR = 31.03 [m] and θ = 8.06×10-5 [rad]. For the distance z » zR the extended Gaussian source may be considered as a point Gaussian source. In these plots the radius rxy is expressed in millimeters and z in meters, so it is clearly seen that the extended Gaussian source will be well approximated by the point Gaussian source at z » zR . When the Gaussian beam is focused by a thin lens of focal length, f, then the focused spot size, ρ0f = ρ0/[1+(zR /f)2]1/2, and the far field approximation may be used at z » zRf = π0f )2/λ [24].

Fig. 4.

The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of x and y at rob = 0.0001 [m] (a), rob = 0.0025 [m] (b), rob = 0.0050 [m] (c), and rob = 0.0500 [m] (d). The data were calculated for I 0 = 1 [W sr-1], z = 0.2 [m], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Fig. 5.

The absolute values of the total radiative flux ΦG (x, y, z; I 0, ρef , ρ0, rob , λ), denoted by ΦG , as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated for I 0 = 1 [W sr-1], ρef = 0.005 [m], ρ0 = 0.0025 [m] and λ = 632.8×10-9 [m].

Fig. 6.

The absolute values of the total radiative flux Φp (x, y, z; I 0, ρef , rob ), denoted by ΦP , from the isotropic point source P as a function of rxy and z at rob = 0.0050 [m] (a), and as a function of rxy and rob at z = 0.2 [m] (b). The data were calculated from Eq. (5) at I 0 = 1 [W sr-1] and ρef = 0.005 [m] for the radiant intensity function I(x, y, z; x 11, y 11) = I 0 z 2/[(x 11 + rxy )2 + z 2 + $y112$], where I 0 = I(x=0, y=0, z).

### Equations (70)

$d Φ P ( x , y , z ; x 11 , y 11 ) = I ( x , y , z ; x 11 , y 11 ) z d x 11 d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 ,$
$r xy = x 2 + y 2 ,$
$r = x 2 + y 2 + z 2 .$
$Φ P ( x , y , z ; ρ ef , r ob ) = { ∫ − a 2 a 1 d x 11 ∫ − y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 ≤ r xy ≤ ρ ef − a 1 , ∫ − a 2 ρ ef − r xy d x 11 ∫ − y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 < ρ ef − a 1 ≤ r xy , 0 , if 0 < ρ ef + a 2 < r xy ,$
$y 11 / = b a 2 − [ x 11 − ( a 1 − a ) ] 2 a ,$
$a = a 1 + a 2 = r ob z r 2 − r ob 2 ( z 2 − r ob 2 ) ,$
$a 1 = r 2 r ob ( z r 2 − r ob 2 − r ob r 2 − z 2 ) ,$
$a 2 = r 2 r ob ( z r 2 − r ob 2 + r ob r 2 − z 2 ) ,$
$b = { r r ob ( r xy + a 1 − a ) ( r xy r 2 − r ob 2 ) , if r xy > 0 , r ob z z 2 − r ob 2 , if r xy = 0 ,$
$Φ G ( x , y , z ; ρ ef , r ob ) = { ∫ − a 2 a 1 d x 11 ∫ − y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 if 0 ≤ r xy ≤ ρ ef − a 1 , ∫ − a 2 x 11 / d x 11 ∫ − y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 + ∫ x 11 / ρ ef − r xy ∫ − y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 < ρ ef − a 1 < r xy , 0 , if 0 < ρ ef + a 2 < r xy ,$
$ρ ef ≥ b 0 = r ob z z 2 − r ob 2 ,$
$x 11 / = b 2 ( a − a 1 ) − a 2 r xy + a a 2 ρ ef 2 − 2 a b 2 ( a 1 + r xy ) + b 2 [ ( b 2 + ρ ef 2 ) + ( a 1 + r xy ) 2 ] a 2 − b 2$
$y 11 // = ρ ef 2 − ( x 11 + r xy ) 2 ,$
$[ x 11 − ( a 1 − a ) ] 2 a 2 + y 11 2 b 2 = 1 ,$
$( x 11 + r xy ) 2 + y 11 2 = ρ ef 2 ,$
$ρ ef < b 0 = r ob z z 2 − r ob 2 ,$
$Φ P ( x , y , z ; ρ ef , r ob ) = { ∫ − ρ ef − r xy ρ ef − r xy d x 11 ∫ − y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 if 0 ≤ r xy ≤ a 2 − ρ ef , ∫ − a 2 x 11 / d x 11 ∫ − y 11 / y 11 / I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 + ∫ x 11 / ρ ef − r xy d x 11 ∫ − y 11 // y 11 // I ( x , y , z ; x 11 , y 11 ) z d y 11 [ ( x 11 − r xy ) 2 + y 11 2 + z 2 ] 3 2 , if 0 ≤ a 2 − ρ ef < r xy ≤ a 2 + ρ ef , 0 , if 0 < ρ ef + a 2 < r xy .$
$E ( x , y , z ; A , ρ 0 , λ ) = A ρ 0 ρ exp ( − r xy 2 ρ 2 ) exp { − i [ k r xy 2 2 R + k z − arctan ( z z R ) ] } ,$
$ρ = ρ 0 1 + ( z z R ) 2 ,$
$R = z [ 1 + ( z R z ) 2 ] .$
$z R = π ρ 0 2 λ ,$
$S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 Re [ S c ( x , y , z ; A , ρ 0 , λ ) ] ,$
$S c ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = E ( x , y , z ; A , ρ 0 , λ ) × H * ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) ,$
$μ r μ 0 H 2 ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = ε r ε 0 E 2 ( x , y , z ; A , ρ 0 , λ ) ,$
$S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 ε r ε 0 μ r μ 0 E ( x , y , z ; A , ρ 0 , λ ) E * ( x , y , z ; A , ρ 0 , λ ) .$
$L ( x , y , z ; A , ρ 0 , λ , ε r , μ r ) = 1 2 π S av ( x , y , z ; A , ρ 0 , λ , ε r , μ r )$
$= c ε 0 4 π ε r μ r E ( x , y , z ; A , ρ 0 , λ ) E * ( x , y , z ; A , ρ 0 , λ ) ,$
$L ( x , y , z ; A , ρ 0 , λ , ε r , n ) = A 2 c ε r ρ 0 2 4 π n ρ 2 exp ( − 2 r xy 2 ρ 2 ) ,$
$I ( x , y , z ; A , ρ 0 , λ , ε r , n ) = I 0 ( A , ρ 0 , ε r , n ) exp ( − 2 r xy 2 ρ 2 ) ,$
$I 0 ( A , ρ 0 , ε r , n ) = ∫ − ∞ ∞ [ ∫ − ∞ ∞ L ( x , y , z ; A , ρ 0 , λ , ε r , n ) dy ] d x = A 2 c ε r ε 0 ρ 0 2 ( 8 n ρ 2 ) .$
$I ( x , y , z ; A , ρ 0 , λ , ε r , n ) = I 0 ( A , ρ 0 , ε r , n ) exp { − 2 [ ( x + x 1 ) 2 + ( y + y 1 ) 2 ] ρ 2 } ,$
$I ( x , y , z ; I 0 , ρ 0 , λ ) = I 0 exp { − 2 [ ( x 11 + r xy ) 2 + y 11 2 ρ 2 } ,$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ − a 2 a 1 e − 2 ( x 11 + r xy ) 2 ρ 2 d x 11 ∫ − y 11 / y 11 / e − 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 ,$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z { ∫ − a 2 x 11 / e − 2 ( x 11 + r xy ) 2 ρ 2 d x 11 ∫ − y 11 / y 11 / e − 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2$
$+ ∫ x 11 / ρ ef − r xy e − 2 ( x 11 + r xy ) 2 ρ 2 d x 11 ∫ − y 11 // y 11 // e − 2 y 11 2 ρ 2 d y 11 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 } ,$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = { F 1 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) , if 0 ≤ r xy ≤ ρ ef − a 1 , F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) + F 3 // ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) , if 0 < ρ ef − a 1 ≤ r xy < ρ ef + a 2 , 0 , if 0 < ρ ef + a 2 ≤ r xy ,$
$F 1 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ − a 2 a 1 f / ( x , y , z ; ρ 0 , λ ) d x 11$
$= I 0 z { ∑ i = 1 ∞ ∑ k = 1 ∞ ( − 1 ) i + k ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ρ 2 ( i − k ) ( 2 i − 2 ) ! ! ( 2 k − 1 ) ( k − 1 ) ! 2 2 i − k − 2$
$× ∫ − a 2 a 1 ( y 11 / ) 2 k − 1 e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11$
$+ ∑ i = 1 ∞ ∑ j = 1 i − l ( − 1 ) i 2 ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ( 2 i − 2 ) ! ! [ 2 ( i − j ) − 1 ] ! ! ( ρ 2 ) 2 j$
$× ∫ − a 2 a 1 ( y 11 / ) 2 ( i − j ) − 1 e [ − 2 ( x 11 + r xy ) 2 + ( y 11 / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } ,$
$F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ − a 2 x 11 / f / ( x , y , z ; ρ 0 , λ ) d x 11$
$= I 0 z { ∑ i = 1 ∞ ∑ k = 1 ∞ ( − 1 ) i + k ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ρ 2 ( i − k ) ( 2 i − 2 ) ! ! ( 2 k − 1 ) ( k − 1 ) ! 2 2 i − k − 2$
$× ∫ − a 2 x 11 / ( y 11 / ) 2 k − 1 e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11$
$+ ∑ i = 1 ∞ ∑ j = 1 i − l ( − 1 ) i 2 ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ( 2 i − 2 ) ! ! [ 2 ( i − j ) − 1 ] ! ! ( ρ 2 ) 2 j$
$× ∫ − a 2 x 11 / ( y 11 / ) 2 ( i − j ) − 1 e − 2 [ ( x 11 + r xy ) 2 + ( y 11 / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 , }$
$F 3 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ x 11 / ρ ef − r x y f / / ( x , y , z ; ρ 0 , λ ) d x 11$
$= I 0 z { ∑ i = 1 ∞ ∑ k = 1 ∞ ( − 1 ) i + k ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ρ 2 ( i − k ) ( 2 i − 2 ) ! ! ( 2 k − 1 ) ( k − 1 ) ! 2 2 i − k − 2$
$× ∫ x 11 / ρ e f − r x y ( y 11 / / ) 2 k − 1 e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11$
$+ ∑ i = 1 ∞ ∑ j = 1 i − l ( − 1 ) i 2 ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ( 2 i − 2 ) ! ! [ 2 ( i − j ) − 1 ] ! ! ( ρ 2 ) 2 j$
$× ∫ x 11 / ρ e f − r x y ( y 11 / / ) 2 ( i − j ) − 1 e − 2 [ ( x 11 + r xy ) 2 + ( y 11 / / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } .$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = { F 4 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) if 0 ≤ r xy ≤ a − ρ ef , F 2 / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) + F 3 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) if 0 ≤ a 2 − ρ ef ≤ r xy < a 2 + ρ ef , 0 , if 0 < ρ ef + a ≤ r xy ,$
$F 4 / / ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ − ρ e f − r x y ρ ef − r x y f / / ( x , y , z ; ρ 0 , λ ) d x 11$
$I 0 z { ∑ i = 1 ∞ ∑ k = 1 ∞ ( − 1 ) i + k ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ρ 2 ( i − k ) ( 2 i − 2 ) ! ! ( 2 k − 1 ) ( k − 1 ) ! 2 2 i − k − 2$
$× ∫ − ρ ef − r xy ρ e f − r x y ( y 11 / / ) 2 k − 1 e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11$
$+ ∑ i = 1 ∞ ∑ j = 1 i − l ( − 1 ) i 2 ( 2 i − 1 ) ! ! ( 2 i − 3 ) ! ! ( 2 i − 2 ) ! ! [ 2 ( i − j ) − 1 ] ! ! ( ρ 2 ) 2 j$
$× ∫ − ρ ef − r xy ρ e f − r x y ( y 11 / / ) 2 ( i − j ) − 1 e − 2 [ ( x 11 + r xy ) 2 + ( y 11 / / ) 2 ] ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 d x 11 } .$
$1 [ ( x 11 + r xy ) 2 + z 2 + y 11 2 ] 3 2 = 1 [ ( x 11 + r xy ) 2 + z 2 ] 3 2 { 1 + y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] } 3 2 ,$
$y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] < 1 ,$
$1 { 1 + y 11 2 [ ( x 11 + r xy ) 2 + z 2 ] } 3 2 = ∑ i = 1 ∞ ( − 1 ) i − 1 ( 2 i − 1 ) ! ! ( 2 i − 2 ) ! ! ( y 11 ) 2 ( i − 1 ) [ ( x 11 + r xy ) 2 + z 2 ] i − 1 .$
$∫ y 11 2 ( i − 1 ) e − 2 ( y 11 ) 2 ρ 2 d y 11 = ( 2 i − 3 ) ! ! 2 2 i 2 π ρ ( 2 i − 1 ) Erf ( 2 y 11 ρ )$
$− ∑ j = 1 i − 1 ( 2 i − 3 ) ! ! y 11 ( 2 i − 2 j − 1 ) ( 2 i − 2 j − 1 ) ! ! ( ρ 2 ) 2 j e − 2 ( y 11 ρ ) 2 + C ,$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z ∫ − a 2 a 1 f / ( x , y , z ; ρ 0 , λ ) d x 11 ,$
$Φ G ( x , y , z ; I 0 , ρ ef , ρ 0 , r ob , λ ) = I 0 z [ ∫ − a 2 x 11 / f / ( x , y , z ; ρ 0 , λ ) d x 11$
$+ ∫ x 11 / ρ ef − r xy f / / ( x , y , z ; ρ 0 , λ ) d x 11 ] ,$
$f / ( x , y , z ; ρ 0 , λ ) = ∑ i = 1 ∞ ( − 1 ) i − 1 ( 2 i − 1 ) ! ! ( 2 i − 3 ) !! ( 2 i − 2 ) ! ! 〈 2 ( ρ 2 ) 2 i − 1 { ∫ − 2 y 11 / ρ 2 y 11 / ρ e − ( y 11 ) 2 d y 11$
$− ∑ j = 1 i − 1 ( ρ 2 ) 2 j − 1 2 ( y 11 / ) 2 i − 2 j − 1 e − 2 ( y 11 / ρ ) 2 [ 2 ( i − j ) − 1 ] ! ! } e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 〉 ,$
$f / / ( x , y , z ; ρ 0 , λ ) = ∑ i = 1 ∞ ( − 1 ) i − 1 ( 2 i − 1 ) ! ! ( 2 i − 3 ) !! ( 2 i − 2 ) ! ! 〈 2 ( ρ 2 ) 2 i − 1 { ∫ − 2 y 11 // ρ 2 y 11 // ρ e − ( y 11 ) 2 d y 11$
$− ∑ j = 1 i − 1 ( ρ 2 ) 2 j − 1 2 ( y 11 // ) 2 i − 2 j − 1 e − 2 ( y 11 // ρ ) 2 [ 2 ( i − j ) − 1 ] ! ! } e − 2 ( x 11 + r xy ) 2 ρ 2 [ ( x 11 + r x y ) 2 + z 2 ] ( 2 i + 1 ) 2 〉 .$
$∫ e − ( y 11 ) 2 d y 11 = ∑ k = 1 ∞ ( − 1 ) k + 1 1 2 k − 1 ( y 11 ) 2 k − 1 ( k − 1 ) ! ,$