FDTD solutions for the distribution of radiation from dipoles embedded in dielectric particles

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1 Journal of Quantitative Sectroscoy & Radiative Transfer 106 (007) FDTD solutions for the distribution of radiation from dioles embedded in dielectric articles Changhui Li a,, George W. Kattawar a, Yu You a, Pengwang Zhai a, Ping Yang b a Deartment of Physics, Texas A&M University, College Station, TX, USA b Deartment of Atmosheric Sciences, Texas A&M University, College Station, TX, USA Abstract The finite-difference time-domain (FDTD) method is used to simulate the electromagnetic radiation emitted by an infinitesimal electric diole embedded in a small article with an arbitrary shae and internal comosition. The far-field attern of the radiation ertaining to dioles embedded in a host article is highly sensitive to the article shae. Thus, it is ossible to discriminate host articles according to their radiation atterns. The method reorted here is also alicable to the study of induced Raman scattering and fluorescence henomena and the detection of biological agents. r 007 Published by Elsevier Ltd. Keywords: Finite-difference time-domain method; Electric diole radiation; Non-sherical article 1. Introduction Raman scattering and fluorescence effects are of great interest to many discilines including remote sensing [1], biological agent detection [], and cell structure identification. Both Raman scattering and fluorescence rocesses are quantum-state transitions at molecular level, which, however, are widely assumed as electric diole transitions. The sies of articles that contain the molecules triggering Raman scattering or fluorescence radiation are usually comarable with the wavelength of the radiated field. Thus, the alicability of geometric otics breaks down in this case. The analytical solutions for the radiation emitted by an electric diole embedded within a small host article with sherical or ellisoidal geometry have been reorted in the literatures [3 5]. It is shown that both the total radiation ower and the satial distribution of radiated energy deend on the osition and orientation of the diole and the refractive index of the host article. Usually, the shaes of host articles exhibiting fluorescence and Raman scattering are irregular. Thus, it is imortant to understand the radiation field induced by infinitesimal electric dioles embedded in small articles with arbitrary shaes and comositions. To our best knowledge, only a handful of studies have been carried out to numerically simulate the radiation by infinitesimal electric dioles in small articles or cavities [6]. Corresonding author. address: cli@biomed.wustl.edu (C. Li) /$ - see front matter r 007 Published by Elsevier Ltd. doi: /j.jqsrt

2 Δx 58 ARTICLE IN PRESS C. Li et al. / Journal of Quantitative Sectroscoy & Radiative Transfer 106 (007) Simulation method In this study, we use the finite-difference time-domain (FDTD) [7,8] method to simulate the radiation ertaining to an infinitesimal diole with an arbitrary orientation and osition. A time function is needed to describe the diole source before the simulation. For examle, a Gaussian function is used in the time domain in the form of ðtþ ¼ 0 e ðt T 0Þ =s, (1) where 0 is a constant vector reresenting the strength and orientation of the diole, s and T 0 are two arameters secifying, resectively, the width and center of the Gaussian ulse in the time domain. The diole is assumed to be a hard source, that is, the strength of the diole is indeendent of the ambient electric field. After several simle mathematical derivations [9], the electric field can be calculated in the form of Eðr; tþ ¼ 1 r=cþ r rðt, () 4 r where is the medium ermittivity. Fig. 1 shows two tyical FDTD grid cells with a grid resolution secified by the arameters Dx; Dy; and D. A satial location in the discretied sace is denoted by the indices ði; J; KÞ ¼ðIDx; JDy; KDÞ. As the same as in the conventional FDTD algorithm, we define the electric and magnetic fields at the centers of the cell edges and the center of the cell faces, resectively. Note that electric fields E x and E y shown in Fig. 1 also reresent the ath-averaged (along the grid edges) values. Similarly, the magnetic fields reresent the values averaged over the cell faces. Because the field values along the grid edges close to the diole are quite sensitive to the distance and direction of the diole, field values at the centers of these edges are not good aroximations to reresent the corresonding ath-averaged values. Therefore, to simulate the radiation from infinitesimal electric diole, as shown in Fig. 1, we modified the conventional FDTD by calculating the exact ath-average field values from Eq. () on edges of a grid that contains the diole. As shown in Fig. 1, we aly the Faraday s law to the shaded area in Fig. 1 so that we have Z E dl ¼ q Z B ds, (3) l qt s where the ath of the receding integral is along the direction shown in Fig. 1. According to our modification, the ath integral of the electric field values is relaced by the roduct of the field value at the center of the edge and the grid sie excet for the edge nearest to the diole. The surface integral for the magnetic field is also relaced aroximately by the roduct of the area of the cell face and the field value at the center of the cell B (I,J,K+1/) E y Δy E x 3 1 Δ E y Fig. 1. Modification of FDTD in simulation of the diole radiation.

3 C. Li et al. / Journal of Quantitative Sectroscoy & Radiative Transfer 106 (007) face. Thus, the ath integral oeration is only alied to the edges of the grid that contains the diole. After some mathematical maniulations, the numerical scheme for udating can be written as follows: B nþ1= I; J; K þ 1 B n 1= I; J; K þ 1 ¼ 1 Dt Dx En y I 1 ; J; K þ 1 E n y I þ 1 ; J; K þ 1 þ 1 Z Iþ1 E n x l; J þ 1 Dy I ; K þ 1 dl E n x I; J 1 ; K þ 1, ð4þ where the suerscrits reresent the discretied temoral indices. Although the simulations are in the time domain, we can aly a Fourier transformation to both the diole source and the simulated values in the time domain to obtain the diole radiation in the frequency domain. 3. Simulation results Fig. shows the simulated angular distribution of the radiated field from a diole within a shere in comarison with the analytical counterart. The shere has a radius of 8:0 mm and the radiated field wavelength is 6:8 mm. The refractive index of the shere is The diole is located at 1 of the radius along the radial direction. The results are normalied by the maximum value. Clearly, the resent results are quite consistent with the analytical solutions in most of the angular region. For the radiation from dioles embedded in non-sherical shaes, we comuted the far-field radiation attern for a diole inside a cube or a cylinder as shown in Fig. 3, where angular distribution of the radiation in aimuthal angle f ¼ 0 lane are shown. Same wavelength and refractive index values are used as in Fig. for both cases. The width of the cube is 1:90 mm. The radius of the cylinder is 6:99 mm and the height of the cylinder is twice the radius. Both the cube and cylinder have the same volume as the shere in Fig.. The infinitesimal dioles are embedded at the centers of the articles with the directions along the -axis. The value of the intensity is normalied by its maximum value as did in Fig.. We also comare our results with the discrete diole aroximation (DDA) [10] solutions. As shown in the figure, those atterns are quite different from those for shere in that, for the latter, the maximum of the radiated field by the centered diole is observed for a olar angle of y ¼ 90. The far-field radiation attern is highly sensitive to the article shae. Thus, it is ossible to discriminate host articles according to their radiation atterns. Normalied angular radiation intensity Analytical Result FDTD result y Angle (degrees) x Fig.. Comarison of the angular distribution of the radiation field from a diole in a sherical article with radium 8:0 mm, wavelength is 6:8 mm, and the refractive index m ¼ 1:33. The diole is located at 1 of the radius and oriented in the radial direction.

4 60 C. Li et al. / Journal of Quantitative Sectroscoy & Radiative Transfer 106 (007) Normalied angular radiation intensity a b FDTD DDA FDTD DDA Angle (degrees) x Fig. 3. Radiation from the diole inside cube and cylinder which has a same volume and refractive index as the shere in Fig., in both cases, the direction of the diole is along the -axis: (a) normalied angular radiation intensity vs. olar angle for a centered diole inside the homogenous cube; (b) normalied angular radiation intensity vs. olar angle for a centered diole inside the homogenous cylinder. 4. Discussions Although we only simulated the radiation field ertaining to a single diole, it is straightforward to aly the resent method to the case for multi-dioles. Comared with the method used in [6], our method is based on a real infinitesimal electric diole instead of a uniformly distributed diole source in a grid. Comared with the DDA method, FDTD has some advantages such as the latter can be used for larger articles and it can simulate the radiation rocess in the time domain. Moreover, the FDTD method is more convenient for simulating surface enhanced cases where the article is close to a metal surface. It is also ossible to aly this method to study induced Raman scattering and fluorescence henomena if the molecules embedded in host articles are known. Acknowledgments George Kattawar s effort is suorted by the Office of Naval Research under contracts N and N Ping Yang s effort is suorted by the National Science Foundation Physical Meteorology Program (ATM ) managed by Dr. Andrew Detwiler and a research grant (NNG04GL4G) from NASA Radiation Sciences Program managed by Dr. Hal Maring (reviously by Dr. Donald Anderson). References [1] Immler F, Engelbart D, Schrems O. Fluorescence from atmosheric aerosol detected by a lidar indicates biogenic articles in the lowermost stratoshere. Atmos Chem Phys 005;5: [] Pinnick RG, Hill SC, Nachman P, Pendleton JD, Fernande GL, Mayo MW, et al. Fluorescence article counter for detecting airborne bacteria and other biological articles. Aerosol Sci Technol 1995;3: [3] Chew H, McNulty PJ, Kerker M. Model for Raman and fluorescent scattering by molecules embedded in small articles. Phys Rev A 1976;13: [4] Chew H. Transition rates of atoms near sherical surfaces. J Chem Phys 1987;87: [5] Chew H. Radiation lifetime of atomos inside dielectric articles. Phys Rev A 1988;38: [6] Xu Y, Vuc ković JS, Lee RK, Painter OJ, Scherer A, Yariv A. Finite-difference time-domain calculation of sontaneous emission lifetime in a microcavity. J Ot Soc Am B 1998;36:

5 C. Li et al. / Journal of Quantitative Sectroscoy & Radiative Transfer 106 (007) [7] Yee KS. Numerical solution of initial boundary value roblems involving Maxwell s equations in isotroic media. IEEE Trans Antennas Proag 1966;14:30 7. [8] Taflove A, Hagness SC. Comutational electromagnetics, nd ed. Boston, MA: Artech House; 000. [9] Jackson JD. Classical electrodynamics. New York: Wiley; [10] Draine BT. The discrete-diole aroximation and its alication to interstellar grahite grains. Astrohys J 1988;333:848 7.