Spherical object in radiation field from Gaussian source

Stanislaw Tryka

doi:10.1364/OPEX.13.005925

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].

Among a variety of natural object shapes, the spherical form is one of the most frequently observed in nature. This form is usually considered characteristic of astrophysical bodies (stars, planets, moons, asteroids, etc.), many fruits and seeds, numerous industrial products, various microorganisms, as well as different molecules, atoms and elementary particles. With the extensive development of lasers, many spherical objects are irradiated by Gaussian beams. Today, spherical objects are irradiated for cooling and trapping of micro-objects [8–10], during optical scattering for characterizing spherical particles [11, 12], in techniques for atmosphere monitoring [13], and so on. Moreover, in the last few decades a noticeable interest has developed in the possibility of using non-ionizing and ionizing radiation to destroy harmful pathogens and in extending the shelf life of some food products without any detectable physical and chemical changes [14–16]. The radiation emitted by lasers was used for irradiation of some agricultural products that may be considered spherical [17, 18]. In addition, some unsafe bacteria and fungi adhering to external surfaces of agricultural products and food may also be spherical. As shown [17, 18], the biological effect of laser radiation depends on many factors, among which the surface energy density of the radiation plays an important role. Therefore, to study the effects of laser radiation on such objects quantitatively, it is necessary to calculate the power of radiation incident on spherical bodies.

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, Φ_G , 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, Φ_P , 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 Φ_P , 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 Φ_G , 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 Φ_G 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 Φ_P 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, dΦ_P (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]

d Φ_{P} (x, y, z; x_{11}, y_{11}) = I (x, y, z; x_{11}, y_{11}) \frac{z d x_{11} d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}},

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 r_xy and the distance r are given by

r_{xy} = \sqrt{x^{2} + y^{2}},

r = \sqrt{x^{2} + y^{2} + z^{2}} .

Next the total radiative flux, Φ_P (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

Φ_{P} (x, y, z; ρ_{ef}, r_{ob}) = {\begin{matrix} \int_{- a_{2}}^{a_{1}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}}, if 0 \leq r_{xy} \leq ρ_{ef} - a_{1}, \\ \int_{- a_{2}}^{ρ_{ef} - r_{xy}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}}, if 0 < ρ_{ef} - a_{1} \leq r_{xy}, \\ 0, if 0 < ρ_{ef} + a_{2} < r_{xy}, \end{matrix}

where

y_{11}^{/} = b \sqrt{\frac{a^{2} - {[x_{11} - (a_{1} - a)]}^{2}}{a}},

a = a_{1} + a_{2} = r_{ob} z \frac{\sqrt{r^{2} - r_{ob}^{2}}}{(z^{2} - r_{ob}^{2})},

a_{1} = \frac{r^{2} r_{ob}}{(z \sqrt{r^{2} - r_{ob}^{2}} - r_{ob} \sqrt{r^{2} - z^{2}})},

a_{2} = \frac{r^{2} r_{ob}}{(z \sqrt{r^{2} - r_{ob}^{2}} + r_{ob} \sqrt{r^{2} - z^{2}})},

b = {\begin{matrix} \frac{r r_{ob} (r_{xy} + a_{1} - a)}{(r_{xy} \sqrt{r^{2} - r_{ob}^{2}})}, if r_{xy} > 0, \\ \frac{r_{ob} z}{\sqrt{z^{2} - r_{ob}^{2}}}, if r_{xy} = 0, \end{matrix}

and ρ_ef 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, ρ_ef , clearly greater then the radius r_ob . 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 ρ_ef 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 r_ob when the geometry is illustrated on the 0̀x ₁₁z₁ plane. Similar dependencies between geometrical variables used in Eq. (3) are shown in Ref. [22].

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Fig. 2. Graphical illustrations of perpendicular projection of the radiative flux with the effective radius ρ_ef at the distance z from the origin 0 of the main coordinate system 0xyz and the surface subtended by the spherical object when 0 ≤ r_xy ≤ ρ_ef - a ₁ (a) and 0 < ρ_ef - a₁ ≤ r_xy (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 < ρ_ef - a₁ ≤ r_xy for the surface S ₁, while the second term of this function for the surface S ₂.

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The negative sign in the denominators of Eqs. (1) and (3) indicates that the coordinate system 0₁ xyz was linearly transformed on the distance r_xy 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 r_xy is treated as negative. The exact solution to the problem outlined above and obtained for r ≥ 0, r_xy ≥ 0, and the variables defined in Figs. 1 and 2 is given by the formula

Φ_{G} (x, y, z; ρ_{ef}, r_{ob}) = {\begin{matrix} \int_{- a_{2}}^{a_{1}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}} if 0 \leq r_{xy} \leq ρ_{ef} - a_{1}, \\ \int_{- a_{2}}^{x_{11}^{/}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}} \\ + \int_{x_{11}^{/}}^{ρ_{ef} - r_{xy}} \int_{- y_{11}^{//}}^{y_{11}^{//}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}}, if 0 < ρ_{ef} - a_{1} < r_{xy}, \\ 0, if 0 < ρ_{ef} + a_{2} < r_{xy}, \end{matrix}

which is valid for

ρ_{ef} \geq b_{0} = \frac{r_{ob} z}{\sqrt{z^{2} - r_{ob}^{2}}},

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 r_xy = 0 by the radiation incident on the object from the point 0. The limits $x_{11}^{/}$ and $y_{11}^{/ /}$ , in the two terms of the second expression in Eq. (5), are given by

x_{11}^{/} = \frac{b^{2} (a - a_{1}) - a^{2} r_{xy} + a \sqrt{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}^{//} = \sqrt{ρ_{ef}^{2} - {(x_{11} + r_{xy})}^{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 ρ_ef

\frac{{[x_{11} - (a_{1} - a)]}^{2}}{a^{2}} + \frac{y_{11}^{2}}{b^{2}} = 1,

respectively, see Figs. 1 and 2.

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

ρ_{ef} < b_{0} = \frac{r_{ob} z}{\sqrt{z^{2} - r_{ob}^{2}}},

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

Φ_{P} (x, y, z; ρ_{ef}, r_{ob}) = {\begin{matrix} \int_{- ρ_{ef} - r_{xy}}^{ρ_{ef} - r_{xy}} d x_{11} \int_{- y_{11}^{//}}^{y_{11}^{//}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}} if 0 \leq r_{xy} \leq a_{2} - ρ_{ef}, \\ \int_{- a_{2}}^{x_{11}^{/}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}} \\ + \int_{x_{11}^{/}}^{ρ_{ef} - r_{xy}} d x_{11} \int_{- y_{11}^{//}}^{y_{11}^{//}} \frac{I (x, y, z; x_{11}, y_{11}) z d y_{11}}{{[{(x_{11} - r_{xy})}^{2} + y_{11}^{2} + z^{2}]}^{\frac{3}{2}}}, if 0 \leq {a_{2} - ρ}_{ef} < r_{xy} \leq a_{2} + ρ_{ef}, \\ 0, if 0 < ρ_{ef} + a_{2} < r_{xy} . \end{matrix}

where $y_{11}^{/}$ , $x_{11}^{/}$ , and $y_{11}^{/ /}$ 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, ρ_{0}, λ) = A \frac{ρ_{0}}{ρ} \exp (- \frac{r_{xy}^{2}}{ρ^{2}}) \exp {- i [\frac{k r_{xy}^{2}}{2 R} + k z - \arctan (\frac{z}{z_{R}})]},

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 = ρ_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}},

R = z [1 + {(\frac{z_{R}}{z})}^{2}] .

The parameter

z_{R} = \frac{π ρ_{0}^{2}}{λ},

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 r_xy = 0. The beam divergence in the far field (for z » z_R ) is illustrated by the dashed straight lines and is equal to the half-angle θ = λ/(π ρ₀) [24]. Here z_R = 31.03 [m] and θ = 8.06×10^-5 [rad]. For the distance z » z_R the extended Gaussian source may be considered as a point Gaussian source. In these plots the radius r_xy 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 » z_R . When the Gaussian beam is focused by a thin lens of focal length, f, then the focused spot size, ρ_0f = ρ₀/[1+(z_R /f)²]^1/2, and the far field approximation may be used at z » z_Rf = π (ρ_0f)²/λ [24].

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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 z_R 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 » z_R for non-focused beam or at a distance z » z_Rf = π(ρ_0f)²/λ for a beam in a vacuum focused by a thin lens, where ρ_0f = ρ₀/[1+(z_R /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, S_av(x, y, z; A, ρ₀, λ, ϵ_r , μ_r ), defined as [25]

S_{av} (x, y, z; A, ρ_{0}, λ, ε_{r}, μ_{r}) = \frac{1}{2} Re [S_{c} (x, y, z; A, ρ_{0}, λ)],

where

S_{c} (x, y, z; A, ρ_{0}, λ, ε_{r}, μ_{r}) = E (x, y, z; A, ρ_{0}, λ) \times H^{*} (x, y, z; A, ρ_{0}, λ, ε_{r}, μ_{r}),

is the complex Pointing vector, H(x, y, z; A, ρ₀, λ, ϵ_r , μ_r ) is the magnetic field strength, ϵ_r and μ_r 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, ϵ_r , μ_r ) can be expressed by the electric field intensity E(x, y, z; A, ρ₀, λ) as follows [25]

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),

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

S_{av} (x, y, z; A, ρ_{0}, λ, ε_{r}, μ_{r}) = \frac{1}{2} \sqrt{\frac{ε_{r} ε_{0}}{μ_{r} μ_{0}}} E (x, y, z; A, ρ_{0}, λ) E^{*} (x, y, z; A, ρ_{0}, λ) .

The radiance, L(x, y, z; A, ρ₀, λ, ϵ_r , μ_r ), 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, ρ_{0}, λ, ε_{r}, μ_{r}) = \frac{1}{2 π} S_{av} (x, y, z; A, ρ_{0}, λ, ε_{r}, μ_{r})

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, ρ_{0}, λ, ε_{r}, n) = \frac{A^{2} c ε_{r} ρ_{0}^{2}}{4 π n ρ^{2}} \exp (- \frac{2 r_{xy}^{2}}{ρ^{2}}),

where n = (ϵ_r μ_r )^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, ρ_{0}, λ, ε_{r}, n) = I_{0} (A, ρ_{0}, ε_{r}, n) \exp (- 2 \frac{r_{xy}^{2}}{ρ^{2}}),

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

I_{0} (A, ρ_{0}, ε_{r}, n) = \int_{- \infty}^{\infty} [\int_{- \infty}^{\infty} L (x, y, z; A, ρ_{0}, λ, ε_{r}, n) dy] d x = \frac{A^{2} c ε_{r} ε_{0} ρ_{0}^{2}}{(8 n ρ^{2})} .

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

I (x, y, z; A, ρ_{0}, λ, ε_{r}, n) = I_{0} (A, ρ_{0}, ε_{r}, n) \exp {- 2 \frac{[{(x + x_{1})}^{2} + {(y + y_{1})}^{2}]}{ρ^{2}}},

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

I (x, y, z; I_{0}, ρ_{0}, λ) = I_{0} \exp {- 2 \frac{[{(x_{11} + r_{xy})}^{2} + y_{11}^{2}}{ρ^{2}}},

where I ₀ = I ₀(A, ρ₀, ϵ_r , 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

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{- a_{2}}^{a_{1}} e^{- 2 \frac{{(x_{11} + r_{xy})}^{2}}{ρ^{2}}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{e^{\frac{- 2 y_{11}^{2}}{ρ^{2}}} d y_{11}}{{[{(x_{11} + r_{xy})}^{2} + z^{2} + y_{11}^{2}]}^{\frac{3}{2}}},

if 0 < r_xy ρ_ef - a ₁ and

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z {\int_{- a_{2}}^{x_{11}^{/}} e^{- 2 \frac{{(x_{11} + r_{xy})}^{2}}{ρ^{2}}} d x_{11} \int_{- y_{11}^{/}}^{y_{11}^{/}} \frac{e^{\frac{- 2 y_{11}^{2}}{ρ^{2}}} d y_{11}}{{[{(x_{11} + r_{xy})}^{2} + z^{2} + y_{11}^{2}]}^{\frac{3}{2}}}

if 0 < ρ_ef - a ₁ ≤ r_xy < ρ_ef + a ₂. The subscript G in Eqs. (22a) and (22b) indicates that the flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = {\begin{matrix} F_{1}^{/} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ), if 0 \leq r_{xy} \leq 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} \leq r_{xy} < ρ_{ef} + a_{2}, \\ 0, if 0 < ρ_{ef} + a_{2} \leq r_{xy}, \end{matrix}

valid for ρ_ef ≥ b ₀ = r_obz/[(z ² - $r_{ob}^{2}$ )^1/2]. The functions $F_{1}^{/}$ (x,y,z;I ₀,ρ_ef ,ρ₀, r_ob ,λ), $F_{2}^{/}$ (x,y,z;I ₀,ρ_ef ,ρ₀, r_ob ,λ) and $F_{3}^{/ /}$ (x,y,z;I ₀,ρ_ef ,ρ₀, r_ob ,λ),are given by

F_{1}^{/} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{- a_{2}}^{a_{1}} f^{/} (x, y, z; ρ_{0}, λ) d x_{11}

F_{2}^{/} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{- a_{2}}^{x_{11}^{/}} f^{/} (x, y, z; ρ_{0}, λ) d x_{11}

F_{3}^{/ /} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{x_{11}^{/}}^{ρ_{ef} - r_{x y}} f^{/ /} (x, y, z; ρ_{0}, λ) d x_{11}

Similarly, the single integral solution to Eq. (10) for ρ_ef < b ₀ = r_obz/[(z ² - $r_{ob}^{2}$ )^1/2] is

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = {\begin{matrix} F_{4}^{/ /} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) if 0 \leq r_{xy} \leq 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 \leq a_{2} - ρ_{ef} \leq r_{xy} < a_{2} + ρ_{ef}, \\ 0, if 0 < ρ_{ef} + a \leq r_{xy}, \end{matrix}

where

F_{4}^{/ /} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{- ρ_{e f} - r_{x y}}^{ρ_{ef} - r_{x y}} f^{/ /} (x, y, z; ρ_{0}, λ) d x_{11}

It is important to note that $y_{11}^{/}$ and $y_{11}^{/ /}$ 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 Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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 Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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 r_ob < ρ₀, that is when the radius r_ob 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 r_ob > 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 Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ), denoted by Φ_G , as a function of x and y at r_ob = 0.0001 [m] (a), r_ob = 0.0025 [m] (b), r_ob = 0.0050 [m] (c), and r_ob = 0.0500 [m] (d). The data were calculated for I ₀ = 1 [W sr^-1], z = 0.2 [m], ρ_ef = 0.005 [m], ρ₀ = 0.0025 [m] and λ = 632.8×10^-9 [m].

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Figures 4(a)–(d) show four plots of the total flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ), in absolute units, as functions of x and y at some radii r_ob and at given values of the remaining variables. The transversal distribution of Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ) in Fig. 4(a), at r_ob = 0.001 [m], clearly less then the beam waist radius ρ₀ at z = 0, is almost identical to the Gaussian distribution. However as r_ob 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 r_ob the axial flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ) reaches maximal value, which does not depend on the further increase of the radius r_ob as shows Fig. 4(d).

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

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Fig. 6. The absolute values of the total radiative flux Φ_p (x, y, z; I ₀, ρ_ef , r_ob ), denoted by Φ_P , from the isotropic point source P as a function of r_xy and z at r_ob = 0.0050 [m] (a), and as a function of r_xy and r_ob at z = 0.2 [m] (b). The data were calculated from Eq. (5) at I ₀ = 1 [W sr^-1] and ρ_ef = 0.005 [m] for the radiant intensity function I(x, y, z; x ₁₁, y ₁₁) = I ₀ z ²/[(x ₁₁ + r_xy )² + z ² + $y_{11}^{2}$ ], where I ₀ = I(x=0, y=0, z).

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

Figures 6(a) and (b) show the total flux Φ_G (x, y, z; I ₀, ρ_ef , r_ob ) emitted by the isotropic point source P into the direction of spherical object of the radius r_ob 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 Φ_p (x, y, z; I ₀, ρ_ef , r_ob ) is significantly greater than the flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ) at radial distance r_xy = 0. Then, at given distance z the flux Φ_p (x, y, z; I ₀, ρ_ef , r_ob ) continuously decayed with increased r_xy and is smaller at smaller r_ob 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 r_ob 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 r_ob obeying the relation r_ob ≤ ρ_ef (z ²- $r_{ob}^{2}$ )^1/2/z (see Eq. (6) and condition under Eq. (23)), the transversal distribution of the radiative flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ) has a nearly Gaussian shape. When the radiation is incident on an object with the bigger radius r_ob then its axial flux is larger then the value for the smaller radius r_ob , but the transversal distribution of the flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ) diverges more from Gaussian. Such relationships are illustrated in Figs. 4(a) and 4(b) When the radius r_ob > ρ_ef (z ²- $r_{ob}^{2}$ )^1/2/z then the total flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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 r_ob , the more flattened the flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , λ). As the spherical objects move across the radiation field from the point Gaussian source, the radial distance r_xy changes and the total flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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 Φ_p (x, y, z; I ₀, ρ_ef , r_ob ) emitted by the isotropic point source P, was calculated from Eq. (5) for these same values of the variables as the total flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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 Φ_p (x, y, z; I ₀, ρ_ef , r_ob ) reaches the greater value at r_xy = 0 and is varied more slowly with increased r_xy at given z than the flux Φ_G (x, y, z; I ₀, ρ_ef , ρ₀, r_ob , 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

\frac{1}{{[{(x_{11} + r_{xy})}^{2} + z^{2} + y_{11}^{2}]}^{\frac{3}{2}}} = \frac{1}{{[{(x_{11} + r_{xy})}^{2} + z^{2}]}^{\frac{3}{2}} {1 + \frac{y_{11}^{2}}{[{(x_{11} + r_{xy})}^{2} + z^{2}]}}^{\frac{3}{2}}},

and if

\frac{y_{11}^{2}}{[{(x_{11} + r_{xy})}^{2} + z^{2}]} < 1,

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

\frac{1}{{1 + \frac{y_{11}^{2}}{[{(x_{11} + r_{xy})}^{2} + z^{2}]}}^{\frac{3}{2}}} = \sum_{i = 1}^{\infty} {(- 1)}^{i - 1} \frac{(2 i - 1)!!}{(2 i - 2)!!} \frac{{(y_{11})}^{2 (i - 1)}}{{[{(x_{11} + r_{xy})}^{2} + z^{2}]}^{i - 1}} .

valid for any value of $y_{11}^{2}$ , r_xy and z because no real solutions for r_xy can be obtained from the inequality $y_{11}^{2}$ /[(x ₁₁ + r_xy )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

\int y_{11}^{2 (i - 1)} e^{- \frac{2 {(y_{11})}^{2}}{ρ^{2}}} d y_{11} = \frac{(2 i - 3)!!}{2^{2 i}} \sqrt{2 π} ρ^{(2 i - 1) Erf} (\frac{\sqrt{2} y_{11}}{ρ})

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

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z \int_{- a_{2}}^{a_{1}} f^{/} (x, y, z; ρ_{0}, λ) d x_{11},

if 0 < r_xy ≤ ρ_ef - a ₁, and

Φ_{G} (x, y, z; I_{0}, ρ_{ef}, ρ_{0}, r_{ob}, λ) = I_{0} z [\int_{- a_{2}}^{x_{11}^{/}} f^{/} (x, y, z; ρ_{0}, λ) d x_{11}

if ρ_ef - a ₂ ≤ r_xy < ρ_ef + a ₂, where

f^{/} (x, y, z; ρ_{0}, λ) = \sum_{i = 1}^{\infty} {(- 1)}^{i - 1} \frac{(2 i - 1)!! (2 i - 3)!!}{(2 i - 2)!!} 〈 \sqrt{2} {(\frac{ρ}{2})}^{2 i - 1} {\int_{- \sqrt{2} \frac{y_{11}^{/}}{ρ}}^{\sqrt{2} \frac{y_{11}^{/}}{ρ}} e^{- {(y_{11})}^{2}} d y_{11}

and

f^{/ /} (x, y, z; ρ_{0}, λ) = \sum_{i = 1}^{\infty} {(- 1)}^{i - 1} \frac{(2 i - 1)!! (2 i - 3)!!}{(2 i - 2)!!} 〈 \sqrt{2} {(\frac{ρ}{2})}^{2 i - 1} {\int_{- \sqrt{2} \frac{y_{11}^{//}}{ρ}}^{\sqrt{2} \frac{y_{11}^{//}}{ρ}} e^{- {(y_{11})}^{2}} d y_{11}

Note that the functions f ^/(x, y, z; ρ₀, λ) and f ^//(x, y, z; ρ₀, λ) differ only in the variables $y_{11}^{/}$ and $y_{11}^{/ /}$ . The integrals with respect to y ₁₁ in these functions may then be expanded in the power series of $y_{11}^{2}$ as

\int e^{- {(y_{11})}^{2}} d y_{11} = \sum_{k = 1}^{\infty} {(- 1)}^{k + 1} \frac{1}{2 k - 1} \frac{{(y_{11})}^{2 k - 1}}{(k - 1)!},

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

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Spherical object in radiation field from Gaussian source

Abstract

1. Introduction

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

3. Some fundamental expressions for the Gaussian beam

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

5. Numerical calculation results

6. Conclusions

Appendix A

References and Links

Cited By

Figures (6)

Equations (70)

Optics Express