## Spherical object in radiation field from Gaussian source

Optics Express, Vol. 13, Issue 16, pp. 5925-5938 (2005)

http://dx.doi.org/10.1364/OPEX.13.005925

Acrobat PDF (3534 KB)

### 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 0*xyz* 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

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]

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

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

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]

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]

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

*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

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]

*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

*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

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]

*Φ*

_{P}(

*x*,

*y*,

*z*), emitted by the source

*P*lies at the point

*P*(0, 0, 0) in the Cartesian coordinate system 0

*xyz*(Fig. 1) is described by the expression [22

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]

*x*

_{11}and

*y*

_{11}represent the coordinates of the Cartesian coordinate system 01

*x*

_{11}

*y*

_{11}

*z*

_{11}oriented along the major and minor ellipse axes of the shadow made by the spherical object on the 0

*xy*plane at

*z*. The radial distances

*r*

_{xy}and the distance

*r*are given by

*Φ*

_{P}(

*x*,

*y*,

*z*), is calculated by integrating Eq. (1) with respect to

*x*

_{11}and

*y*

_{11}.

**12**, 512 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-512. [CrossRef] [PubMed]

*ρ*

_{ef}is the arbitrary chosen beam radius at the output plane 0

*xy*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.

*ρ*

_{ef}, clearly greater then the radius

*r*

_{ob}. Unfortunately, this equation is inaccurate when the surface made on the plane 0

*xy*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).

_{1}

*xyz*was linearly transformed on the distance

*r*

_{xy}with respect to the center 0

_{1}of the system 0

_{1}

*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

_{1}, where

*b*

_{0}represents the radius of the surface

*S*made on the 0

_{1}

*x*

_{1}

*y*

_{1}plane at

*r*

_{xy}= 0 by the radiation incident on the object from the point 0. The limits

*xy*at the distance

*z*made by the surface

*S*and by the radiation of effective radius

*ρ*

_{ef}

*xy*at

*z*and for

*r*

_{xy}= 0 is greater than the radius

*ρ*

_{ef}. This solution obtains the form

*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

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

*E*(

*x*,

*y*,

*z*;

*A*, ρ

_{0}, 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

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

*A*is the electric field amplitude of the beam waist center, ρ

_{0}is the minimal radius of the beam waist at

*z*= 0,

*k*= 2π/λ is the wave number, and

*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.

*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

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

*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}= ρ

_{0}/[1+(

*z*

_{R}/

*f*)

^{2}]

^{1/2}and

*f*is the focal length of the lens [24

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]

*z*component of the time-average Pointing vector, S

*(*

_{av}*x*,

*y*,

*z*;

*A*, ρ

_{0},

*λ*,

*ϵ*

_{r},

*μ*

_{r}), defined as [25]

**(**

*H**x*,

*y*,

*z*;

*A*, ρ

_{0},

*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*, ρ

_{0},

*λ*,

*ϵ*

_{r},

*μ*

_{r}) can be expressed by the electric field intensity

*E*(

*x*,

*y*,

*z*;

*A*, ρ

_{0},

*λ*) as follows [25]

*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 (

*μ*

_{0}= 4 π×∙10

^{-7}H m

^{-1}). From Eqs. (11), (14

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]

*L*(

*x*,

*y*,

*z*;

*A*, ρ

_{0},

*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

*c*= 1/(ϵ

_{0}μ

_{0})

^{1/2}= 2.998×10

^{8}m s

^{-1}denotes the speed of light in a vacuum. Substituting Eq. (11) into (16) leads to

*n*= (

*ϵ*

_{r}

*μ*

_{r})

^{1/2}is the refractive index of the medium surrounding the source.

*I*

_{0}(

*A*, ρ

_{0},

*ϵ*

_{r},

*n*) denotes on-axis intensity at the distance

*z*

_{0}given by

*L*(

*x*,

*y*,

*z*;

*A*, ρ

_{0}, λ,

*ϵ*

_{r},

*n*) in Eqs. (18) is expressed in W m

^{-2}sr

^{-1}, while the intensity

*I*(

*x*,

*y*,

*z*;

*A*, ρ

_{0}, λ,

*ϵ*

_{r},

*n*) in Eq. (17) is expressed in W sr

^{-1}.

_{1}

*xy*at

*z*the intensity

*I*(

*x*,

*y*,

*z*;

*A*, ρ

_{0}, λ,

*ϵ*

_{r},

*n*) becomes

_{1}

*x*

_{11}

*y*

_{11}

*z*

_{11}illustrated in Figs. 1 and 2 for

*r*

_{xy}≥ 0 takes the form

*I*

_{0}=

*I*

_{0}(

*A*, ρ

_{0},

*ϵ*

_{r},

*n*).

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

*r*

_{xy}

*ρ*

_{ef}-

*a*

_{1}and

*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}, ρ

_{0},

*r*

_{ob},

*λ*) is calculated for a Gaussian source.

*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

*ρ*

_{ef}≥

*b*

_{0}=

*r*

_{ob}

*z*/[(

*z*

^{2}-

^{1/2}]. The functions

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef},ρ

_{0},

*r*

_{ob},λ),

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef},ρ

_{0},

*r*

_{ob},λ) and

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef},ρ

_{0},

*r*

_{ob},λ),are given by

*ρ*

_{ef}<

*b*

_{0}=

*r*

_{ob}

*z*/[(

*z*

^{2}-

^{1/2}] is

*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}, ρ

_{0},

*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

*Φ*

_{G}(

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*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}< ρ

_{0}, that is when the radius

*r*

_{ob}of the spherical object was less then the beam waist radius ρ

_{0}at

*z*= 0. However the series were converged quickly with increased

*i*and slowly with increased

*k*when

*r*

_{ob}> ρ

_{0}. 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.

*Φ*

_{G}(

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*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}, ρ

_{0},

*r*

_{ob},

*λ*) in Fig. 4(a), at

*r*

_{ob}= 0.001 [m], clearly less then the beam waist radius ρ

_{0}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}, ρ

_{0},

*r*

_{ob}, λ) reaches maximal value, which does not depend on the further increase of the radius

*r*

_{ob}as shows Fig. 4(d).

*Φ*

_{G}(

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*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}, ρ

_{0},

*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}, ρ

_{0},

*r*

_{ob},

*λ*) at the same values of

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*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}, ρ

_{0},

*r*

_{ob},

*λ*) is lager for the lager objects when the radius

*r*

_{ob}is less than ρ

_{0}. When the radius

*r*

_{ob}is greater than ρ

_{0}the total flux

*Φ*

_{G}(

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*r*

_{ob},

*λ*) obtains its maximal axial value independent of the subsequent growth of

*r*

_{ob}.

*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}, ρ

_{0},

*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

*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

**12**, 512 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-512. [CrossRef] [PubMed]

*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.

*r*

_{ob}obeying the relation

*r*

_{ob}≤

*ρ*

_{ef}(

*z*

^{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}, ρ

_{0},

*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}, ρ

_{0},

*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}-

^{1/2}/

*z*then the total flux

*Φ*

_{G}(

*x*,

*y*,

*z*;

*I*

_{0},

*ρ*

_{ef}, ρ

_{0},

*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}, ρ

_{0},

*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}, ρ

_{0},

*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.

*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}, ρ

_{0},

*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}, ρ

_{0},

*r*

_{ob},

*λ*) from the point Gaussian source shown in Figs. 5(a) and (b).

## Appendix A

*x*

_{11}+

*r*

_{xy})

^{2}+

*z*

^{2}]

^{3/2}we have

*x*

_{11}+

*r*

_{xy})

^{2}+

*z*

^{2}]

^{3/2}} can be represented by the following power series of

*x*

_{11}+

*r*

_{xy})

^{2}+

*z*

^{2}]:

*r*

_{xy}and

*z*because no real solutions for

*r*

_{xy}can be obtained from the inequality

*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).

*r*

_{xy}≤

*ρ*

_{ef}-

*a*

_{1}, and

*ρ*

_{ef}-

*a*

_{2}≤

*r*

_{xy}<

*ρ*

_{ef}+

*a*

_{2}, where

*f*

^{/}(

*x*,

*y*,

*z*; ρ

_{0}, λ) and

*f*

^{//}(

*x*,

*y*,

*z*; ρ

_{0}, λ) differ only in the variables

*y*

_{11}in these functions may then be expanded in the power series of

## References and Links

1. | X. Deng, Y. Li, D. Fan, and Y Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. |

2. | F. Gori, “Flattened Gaussian beams,” Opt. Commun. |

3. | L. Zeni, S. Campopiano, A. Cutolo, and G. D“Angelo, “Power semiconductor laser diode characterization,” Opt. Lasers Eng. |

4. | L.D. Dickson, “Characteristics of propagating Gaussian beam,” Appl. Opt. |

5. | C. Hu and T.N. Baker, “Prediction of laser transformation hardening depth using a line source model.” Acta Metal. Mater. |

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. |

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. |

8. | G. Grynberg and C Robilliard, “Cold atom in dissipative optical lattices,” Phys. Rep. |

9. | R.C. Gauthier, “Optical trapping: a tool to assist optical machining,” Opt. Laser Technol. |

10. | C.S. Adams and E. Riis, “Laser cooling and trapping of neutral atoms,” Prog. Quantum Electron. |

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. |

12. | A.R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. , |

13. | C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quantum Spectrosc. Radiat. Transfer |

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. |

15. | J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. |

16. | G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. |

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 |

18. | W.H.A. Wilde, W.H. Parr, and D.W. McPeak, “Seeds bask in laser light,” Laser Focus |

19. | F Grum and R.J. Becherer, |

20. | M. Strojnik and G Paez, “Radiometry” in |

21. | J.R. Meyer-Arendt, “Radiomatry and photometry: units and conversion factors,” Appl. Opt. |

22. | S. Tryka, “Spherical object in radiation field from a point source,” Opt. Express |

23. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

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 , |

25. | E.J. Rethwell and M.J. Cloud, |

26. | S. Wolfram, |

27. | J.H. Habbell, R.L. Bach, and R.J. Herbold, “Radiation field from a circular disk source,” J. Res. Natl. Bur. Stand. |

28. | R.P. Gardner and K. Verghese, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods , |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(120.5240) Instrumentation, measurement, and metrology : Photometry

(120.5630) Instrumentation, measurement, and metrology : Radiometry

(140.3570) Lasers and laser optics : Lasers, single-mode

(230.6080) Optical devices : Sources

(350.5500) Other areas of optics : Propagation

(350.5610) Other areas of optics : Radiation

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 23, 2005

Revised Manuscript: July 19, 2005

Published: August 8, 2005

**Citation**

Stanislaw Tryka, "Spherical object in radiation field from Gaussian source," Opt. Express **13**, 5925-5938 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-5925

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### References

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- F. Gori, Flattened Gaussian beams,�?? Opt. Commun. 107, 335-341 (1994). [CrossRef]
- L. Zeni, S. Campopiano, A. Cutolo and G. D�??Angelo, �??Power semiconductor laser diode characterization,�?? Opt. Lasers Eng. 39, 203-217 (2003). [CrossRef]
- L.D. Dickson, �??Characteristics of propagating Gaussian beam,�?? Appl. Opt. 9, 1854 (1970). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- G. Grynberg and C Robilliard, �??Cold atom in dissipative optical lattices,�?? Phys. Rep. 355, 335-451 (2001). [CrossRef]
- R.C. Gauthier, �??Optical trapping: a tool to assist optical machining,�?? Opt. Laser Technol. 29, 389-399 (1997). [CrossRef]
- C.S. Adams and E. Riis, �??Laser cooling and trapping of neutral atoms,�?? Prog. Quantum Electron. 21, 1-79 (1997). [CrossRef]
- 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]
- A.R. Jones, �??Light scattering for particle characterization,�?? Prog. Energy Combust. Sci., 25, 1-53 (1999). [CrossRef]
- C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quantum Spectrosc. Radiat. Transfer 55, 673-681 (1996). [CrossRef]
- 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.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]
- 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]
- W.H.A. Wilde, W.H. Parr and D.W. McPeak, �??Seeds bask in laser light,�?? Laser Focus 5, 41-42 (1969).
- F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) p. 37.
- 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.R. Meyer-Arendt, �??Radiomatry and photometry: units and conversion factors,�?? Appl. Opt. 7, 2081-2084 (1968). [CrossRef] [PubMed]
- 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]
- H. Kogelnik and T. Li, �??Laser beams and resonators,�?? Appl. Opt. 5, 1550-1567 (1966). [CrossRef] [PubMed]
- 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]
- 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.
- R.P. Gardner and K. Verghese, �??On the solid angle subtended by a circular disc,�?? Nucl. Instrum. Methods, 93, 163-167 (1971). [CrossRef]
- 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).

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