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Electromagnetic field enhancement in small liquid droplets using geometric optics Maurice A. Jarzembski and Vandana Srivastava

Maurice Jarzembski is with University of Alabama— Huntsville, Johnson Research Center, Huntsville, Ala­bama 35889, and Vandana Srivastava is with Universities Space Research Association, 4950 Corporate Drive, Huntsville, Alabama 35806. Received 24 August 1989. Sponsored by George W. Kattawar, Texas A&M University. 0003-6935/89/234962-04$02.00/0. © 1989 Optical Society of America.

The EM field enhancement in the forward direction of a dielectric sphere was derived leading to high-energy density in a critical ring region where, experimentally, the nonlin­ear processes are observed for small liquid droplets.

The generation of nonlinear processes like the stimulated Raman scattering (SRS) and laser-induced breakdown in small liquid droplets occurs in regions where the electromag­netic energy density is sufficiently high.1-6 Mie theory has been used to predict regions of high field enhancements, where the emission of the nonlinear processes is expected to take place. Since Mie theory calculations can be lengthy, a geometrical optics approach is presented, showing the loca­tion of the enhancement of the fields due to the droplet. For droplets of radius a and incident radiation of wavelength λ, geometrical optics is valid, provided that the size parameter x ≡ 2πa/λ » 1. Recent investigations of the interaction of high energy laser pulses with droplets have typically x > 100 in the visible region.

A nonabsorbing dielectric sphere, index of refraction m = 1.332, illuminated by a monochromatic light source of con­stant flux F0 is depicted by rays incident on the sphere [Fig. 1(a)]. The geometry shows an example of seven rays of radiation incident at angles φi and refracted to the forward direction of the sphere defined by corresponding angles θi (equivalent to the conventional scattering angle definition). The angle θi can be derived from the path of a ray inside the droplet as

the sphere [heavy line in Fig. 1(a)]. The illuminated side of the sphere [Fig. 1(b)] is subdivided into elemental areas (concentric rings) δA given by

with cross-sectional area cosφδA. For total energy E0 uni­formly incident on the sphere, E0δƒ (with δƒ = cosφδA/πa2) represents the fractional energy incident on each elemental area δA. The reflectance R and transmittance T in terms of Fresnel coefficients are polarization dependent and can be

which corresponds to the Descartes ray defining the rainbow. Substituting φc into Eq. (1), the critical refracted angle θc can be determined, defining the maximum angle from the axis to which the rays can be refracted in the forward direction of

Fig. 1. Schematic of a sphere illuminated by an uniform electro­magnetic field (a) creating a high energy density in the forward direction in the region at critical angle θc due to the refraction of rays around the critical incident angle φc ≡φi=1 [Eq. (2)]. (b) Part of the energy incident on the shaded ring region on the illuminated side of the sphere is reflected away. The transmitted energy is mapped into a smaller shaded ring region in the forward direction giving an enhancement of the fields S [Eqs. (8) and (12)]. (δφ,δθ correspond to

infinitesimal divisions; φi,θi correspond to discrete divisions.)

4962 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

The extremum of θi occurs at critical incident angle φc given by

evaluated for each ring region δA.7 Thus part of the frac­tional energy E0δƒ incident on each elemental area δA is reflected by RE0δƒ, and the transmitted energy is given by TE0δƒ. The energy transmitted TE0δƒ is mapped into a smaller ring region of area òA' in the forward direction [Fig. 1 (b)]. Since δA' is smaller than its corresponding area δA on the illuminated side, the energy density becomes greater giving rise to field enhancement. The elemental area δA' given in terms of corresponding angle θ is

A constant flux F0 of radiation incident on each ring region δA on the illuminated side of the sphere is given by

In the transmitted forward direction of the sphere, the flux F′ in each subsequent mapped ring region δA' is

The enhancement S due to the focusing effect is given by the ratio of F′ and F0 as

Eliminating δ(cos0) using Eq. (1), the enhancement S evalu­ated along the sphere's surface in the forward direction of the sphere is

where μ = cosφ and the transmittance T is evaluated at incident angle φ.

Since the sphere has been subdivided into infinitesimal areas δA, the field enhancement S at the critical angle θc becomes infinite by substitution of φc into Eq. (8). To apply geometrical optics theory to nonlinear processes which occur over finite dimensions of the material depending on material density and the availability of free electrons in the liquid droplet, it is important to also consider subdividing the illuminated side of the sphere into discrete areal divisions. Unlike the infinitesimal area case [Eq. (8)], discrete areal divisions of the sphere into concentric rings give rise to finite enhancements over a discrete area around θc, which is ob­served experimentally and discussed here later.

Discrete concentric rings of cross-sectional area A; for the ith region given by

The magnitude of S(m,φi) at a particular region between θi and θi+1 corresponds to the field enhancement in that region evaluated along the sphere's surface in the forward direction. The location of the maximum of S indicates the region of maximum field enhancement where nonlinear processes originate. This theory predicts the enhancement near the droplet surface, neglecting internal reflections within the droplet which would not reduce the enhancement.

In the visible spectral region with the refractive index for water m = 1.332, the critical angles calculated using Eqs. (2) and (1) are φc = 59.469° and θc = 21.112°. Figure 2 shows calculated enhancement S as a histogram for various regions (θi to θi+1) in the forward direction for three different discrete areal subdivisions corresponding to fractions ƒi = 1.0, 0.5, 0.2 [Eq. (12)] and as a continuous curve for infinitesimal divi­sions δƒ [Eq. (8)] for unpolarized light. Increasing the num­ber of areal subdivisions decreases the area of the concentric ring Ai on the illuminated side giving higher resolution for studying local enhancement effects. The trivial case for ƒi = 1.0 [Fig. 2(a)] with no resolution shows a uniform enhance­ment in the forward direction of the sphere with no local enhancement. Decreasing the fraction ƒi [Figs. 2(b), (c), (d)] shows the emergence of the critical ring region located at θC where a high electromagnetic energy density is created due to the refraction of rays around the critical angle φc.

Geometrical optics predicts a dramatic enhancement of the fields near the surface at the critical ring located at angle θc and low field enhancement at the axis and area around the axis 0 = 0. Therefore, nonlinear processes initiated in re-

can be generated beginning at critical incident angle φc as shown in Fig. 1(a) so that each ring region generated has the same cross-sectional area; thus the same fractional energy E0ƒi (with ƒi = |sin2φi+1 - sin2φi|) is incident on each ring region. The reflectance Ri and transmittance T i can be evaluated at angle φc

i{φci = sin -1[(sin2φ i+1 + sin2φi)/2]1/2, mid­

dle angle Ai} representative for each ring region. The ring region A′i, in the forward mapped region is written like Eq. (9) with angle φi replaced by angle θi. The incident flux F0 and transmitted flux Fi are given, respectively, as

where A′i = {[1 - (sinφci/m)2]1/2/cosθc

i}A′i and θci is calculated

using φci from Eq. (1). The enhancement S(m,φi) using

discrete areas is given by

Fig. 2. Electromagnetic field enhancement S(m,φ) shown as a func­tion of θ [Eq. (1)] for increasing the number of areal subdivisions of the sphere from (a) to (c) for a discrete case [Eq. (12)] and (d) for infinitesimal divisions [Eq. (8)]. Decreasing the fraction ƒi from 1.0 to infinitesimal of increases the resolution showing the dramatic increase of field enhancement for the critical ring region located at

the critical refracted angle θc.

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Fig. 3. Experimental setup (a) used to observe visually the critical ring located at the critical refracted angle θc; (b) an ~120-μm diam water droplet showing SRS on the inner critical ring at θc, which has not been detected before, we believe. The SRS observed at the periphery of the droplet corresponds to the structural reso-

nances.1,2,5

At the threshold of the SRS [Fig. 4(a)] both the ring of the SRS located at the periphery of the droplet and the critical ring located at θc appear almost simultaneously. With in­creasing laser power [Figs. 4(b), (c), (d)], still no SRS is visible inside the critical ring, only the critical ring becomes more luminous. The critical ring, however, is not of uniform intensity but has regions of localized high electromagnetic energy density. The laser-induced breakdown also origi­nates from the critical ring; however, it is initiated at regions where there is highest electromagnetic energy density local­ized on the critical ring. Figure 4(e) depicts the breakdown threshold signified by the appearance of a plasma and plume (very luminous regions in the photographs). Very frequent­ly, the breakdown occurs at more than one region on the critical ring. Figure 4(f) shows the laser-induced breakdown above its threshold showing two regions at which it is initiat­ed from the critical ring.

The photographs of the SRS at threshold can be used as a visual aid to obtain an idea of the effective enhancement of the fields Seff due to the droplet by estimating the region of illumination by the SRS on the critical ring, ~3° around θc. In this way, the Seff is estimated to be ~20 corresponding to

gions of high electromagnetic energy densities would origi­nate at the critical ring region in the forward direction of the droplet. Thus at the threshold of SRS one expects the SRS to be localized at the critical ring around θc and not the axis (0 = 0). The laser-induced breakdown would also be initiated from the critical ring and not at the axis, as has been accepted so far.4"6

To observe visually the critical ring in the forward direc­tion, a very simple experiment was set up [Fig. 3(a)]. Water droplets with diameter ~120 μm (x ~ 700) were generated from a Berglund-Liu particle generator and synchronized with Nd:YAG laser pulses, λ = 0.532 μm. The laser beam was directed at an angle θ ~ 37° (as close to the forward direction of the droplet as possible without damaging the microscope objective). Photographs of the interaction of the high energy laser radiation with a water droplet were taken as observed through a microscope close to the forward direction of the droplet [Figs. 3(b) and 4]. The nonlinear processes could be seen through a red filter, removing the intense incident and elastically scattered green radiation. As geometrical optics predicts, a thin well defined inner ring of SRS is clearly visible located at the critical refracted angle θc, a region encircling the axis of the droplet in the forward direction, with no visible SRS in the center at θ = 0. The other ring at the periphery of the droplet corresponds to the SRS seeking out the structural resonances of the droplet.1,2,5

Figure 4 shows the interaction of the liquid water droplet and the laser beam as a function of increasing laser power.

Fig. 4. Interaction of a ~120-μm diam water droplet and laser beam [same setup as shown in Fig. 3(a)] as a function of increasing laser power. In the forward direction of the droplet, the SRS occurs at regions of high electromagnetic field intensity on the well defined inner critical ring (a)-(d). At the breakdown threshold (e) and above the threshold (f), the breakdown originates from this ring. The SRS and the laser-induced breakdown is not observed along the

axis of the droplet at the threshold levels.

4964 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

the geometrical optics prediction shown in the histogram of Fig. 2(c), using Eq. (12). This represents the effective en­hancement of the fields in going from bulk water to a 120-μm spherical water droplet.

The emission of nonlinear processes from the critical ring was even visually observed in water droplets as small as 50 μm in diameter. Thus nonlinear processes continue to be confined to the well defined ring for droplets with smaller size parameters. Further investigation of this critical ring for varying droplet sizes and droplet material would add to the understanding of the generation and emission of nonlin­ear processes inside small liquid droplets.

This research was supported in part by U.S. Army Re­search Office grant DAAL03-87-K-0093 and NASA under contract NAS8-37135. The authors gratefully thank A. Bis­was, R. G. Pinnick, and P. Shah for the use of the laboratory facilities at the Physics Department, New Mexico State Uni­versity and G. W. Kattawar for reviewing the manuscript and making suggestions.

References 1. J. B. Snow, S.-X. Qian, and R. K. Chang, "Stimulated Raman

Scattering from Individual Water and Ethanol Droplets at Mor­phology-Dependent Resonances," Opt. Lett. 10, 37-39 (1985).

2. S.-X. Qian and R. K. Chang, "Multiorder Stokes Emission from Micrometer-Size Droplets," Phys. Rev. Lett. 56, 926-929 (1986).

3. P. Chylek, M. A. Jarzembski, N. Y. Chou, and R. G. Pinnick, "Effect of Size and Material of Liquid Spherical Particles on Laser-Induced Breakdown," Appl. Phys. Lett. 49, 1475-1477 (1986).

4. P. Chylek, M. A. Jarzembski, V. Srivastava, R. G. Pinnick, J. D. Pendleton, and J. P. Cruncleton, "Effect of Spherical Particles on Laser-Induced Breakdown of Gases," Appl. Opt. 26, 760-762 (1987).

5. R. G. Pinnick et al., "Aerosol-Induced Laser Breakdown Thresh­olds: Wavelength Dependence," Appl. Opt. 27, 987-996 (1988).

6. R. K. Chang, J. H. Eickmans, W.-F. Hsieh, C. F. Wood, J.-Z. Zhang, and J. Zheng, "Laser-Induced Breakdown in Large Trans­parent Water Droplets," Appl. Opt. 27, 2377-2385 (1988).

7. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 204.

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