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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 9 — Apr. 26, 2010
  • pp: 9486–9495
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Spatial filtering technique to image and measure two-dimensional near-forward scattering from single particles

Matthew J. Berg, Steven C. Hill, Gorden Videen, and Kristan P. Gurton  »View Author Affiliations


Optics Express, Vol. 18, Issue 9, pp. 9486-9495 (2010)
http://dx.doi.org/10.1364/OE.18.009486


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Abstract

This work describes the design and use of an optical apparatus to measure the far-field elastic light-scattering pattern for a single particle over two angular-dimensions. A spatial filter composed of a mirror with a small through-hole is used to enable collection of the pattern uncommonly close to the forward direction; to within tenths of a degree. Minor modifications of the design allow for the simultaneous measurement of a particle’s image along with its two-dimensional scattering pattern. Example measurements are presented involving single micrometer-sized glass spherical particles confined in an electrodynamic trap and a dilute suspension of polystyrene latex particles in water. A small forward-angle technique, called Guinier analysis, is used to determine a particle-size estimate directly from the measured pattern without a priori knowledge of the particle refractive index. Comparison of these size estimates to those obtained by fitting the measurements to Mie theory reveals relative errors as low as 2%.

© 2010 Optical Society of America

1. Introduction

Electromagnetic scattering can be useful to study the physical characteristics of single and multiple particle systems in situ. As an example, the fractal-like morphology of carbon aggregates that form in hydrocarbon combustion can be estimated from the aggregates’ far-field scattering pattern [1

1. C. M. Sorensen, “Light scattering by fractal aggregates: A review,” Aerosol Sci. Technol. 35, 648–687 (2001).

, 2

2. R. Dhaubhadel, An Experimental Study of Dense Aerosol Aggregations, Doctoral Thesis, (Kansas State University, 2008), freely available at http://hdl.handle.net/2097/663.

]. This morphology can be examined as the aggregates form and evolve in the flame; such observations would not be possible using optical or electron microscopy since collection of the particles would likely disturb their structure. Additional applications of electromagnetic scattering can be found in atmospheric science and terrestrial environmental monitoring [3

3. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]

, 4

4. M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: A tutorial review,” J. Quant. Spectros. Radiat. Transfer 110, 808–832 (2009). [CrossRef]

].

The purpose of this paper is to describe a new technique to measure the scattering pattern from a single particle over a two-dimensional angular range extending close to the forward direction. This is achieved by exploiting the Fourier-transform-like properties of a lens in combination with a simple spatial filter, and is based on work by Ferri [9

9. F. Ferri, “Use of a charged coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2273 (1997). [CrossRef]

]. A brief discussion of the basic principles of the technique is presented along with measurements implementing the technique. Guinier analysis is applied to the scattering measurements for single microparticles to estimate their size without a priori knowledge of the refractive index. Comparisons to calculated scattering patterns are presented. Simple modifications are described that enable the measurement of a particle’s two-dimensional scattering pattern simultaneously with an image of the particle’s profile. This dual capability is illustrated with a real-time video of a dilute solution of polystyrene latex microspheres in water.

2. Spatial filtering technique

Scattering measurements on single microparticles are achieved using a spherical void electro-dynamic levitator (SVEL). The design and operation of this trap is similar to that described in [10

10. S. Arnold and L. M. Folan, “Spherical void electrodynamic levitator,” Rev. Sci. Instrum. 58, 1732–1735 (1987). [CrossRef]

]. In short, the trap consists of three slab-like copper conductors separated by thin rubber insulators. A spherical void (~ 25 mm dia.) is milled from these conductors to yield the configuration shown in Fig. 1(a). Small holes (~ 6.3 mm dia.) allow for the introduction of particles and optical access to the trapping region. The electrodynamic trapping potential is supplied by the driving circuit shown in Fig. 1(b). This supply provides an adjustable direct-current (DC) high-voltage to the top and bottom conductors of the SVEL. An adjustable alternating-current (AC) voltage is applied to the center conductor and referenced to the DC supply.

Particles must first be charged before they are trapped. This is done by rubbing them against the surface of a plastic Petri dish with a plastic pipette tip. The charged particles stuck to the tip are then transferred to the trap by tapping the tip over the top entrance hole in the SVEL, see Fig. 1(a). The majority of these particles fall through the trap and are lost; however, several are caught near the electrodynamic equilibrium point. The AC voltage is then decreased, weakening the trapping potential, until all but one of the trapped particles falls out of the SVEL.

Fig. 1. Cross-sectional view of the spherical void electrodynamic levitator (SVEL), diagram (a), and its driving circuit, diagram (b). Single charged particle can be trapped in the SVEL as described in the text.

Once a particle is confined in the SVEL, its scattering pattern is measured using the arrangement shown in Fig. 2. The particle is illuminated by a diode-pumped Q-switched Nd:YLF laser frequency-doubled to 532 nm (CrystaLaser, model QUV-002-266) and operating at a repetition rate of 9 kHz. This incident beam is linearly polarized and attenuated by an absorption neutral-density (ND) filter before reaching the particle. The forward region of the scattering pattern and the unscattered portion of the incident beam is intercepted by lens L 1 (50 mm dia., focal length f 1=75 mm, achromat), which is placed at a distance of f 1 from the particle. In this configuration, the lens collects the far-field pattern over an angular range of 0° – 18° and 0° – 360° in the polar θ and azimuthal ϕ angles, respectively, where θ is measured from the propagation direction of the incident beam. The pattern is collimated by this lens, while the unscattered portion of the incident beam is focused to a waist in the back focal plane of L 1. A 75 mm diameter silver mirror with a centered, 1 mm diameter through-hole at 45° to normal is placed at this focal plane such that the focused, unscattered beam passes through the hole, while the collimated scattering pattern is reflected. This spatial-filter mirror is denoted SFM. The unscattered light is then removed by a beam dump (BD). The scattering pattern, now separated from the incident beam, is imaged onto a charged coupled device (CCD) camera (Princeton Instruments Inc., model LN/CCD-1100-PB/UVAR/1) by lens L 2 (50 mm dia., f 2 = 100 mm, achromat). The distance z in Fig. 2 is adjusted such that the pattern fills as much of the CCD chip as possible. The reason that a 1 mm diameter through-hole is chosen for the SFM is because it is the smallest diameter available from the manufacturer (Lexon Laser, Inc.) for a 75 mm diameter mirror.

In the language of Fourier optics, lens L 1 is a Fourier-transform (FT) lens and the SFM acts as a spatial filter. This filter removes the small spatial-frequency portion of the light across the lens’ FT plane [12

12. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), pp. 103–108.

]. The size of the through-hole in the mirror determines the range of spatial frequencies rejected. Given that the diameter of this hole is 1 mm and the focal length of L 1 is f 1 = 75 mm, light propagating from θ=0° to θ = 0.3° will be rejected by the mirror. In practice, however, scattering of stray light from the through-hole’s edge widens this range to θ ≃ 0.8°. The diameter of lens L 1 establishes the maximum spatial frequency collected, which in terms of the corresponding maximum angle is θ = 18°. Limitations imposed by the lens mount and the noise floor of the CCD reduce this angle to θ ≃ 12°.

In order to analyze the measured patterns, a relationship is needed between each pixel (x,y) on the CCD chip and the corresponding scattering angles (θ,ϕ). This pixel-angle mapping is achieved by removing the SVEL and placing a 12 μm diameter pinhole where the particle would otherwise reside. The pattern collected by the CCD is then compared to the the calculated pattern following pinhole diffraction theory [13

13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999) pp. 490–492.

]. This comparison yields a scale factor accounting for the transverse demagnification introduced by lens L 2 and establishes the appropriate pixel-angle mapping.

Fig. 2. Optical arrangement used to obtain two-dimensional scattering patterns for single particles. The bottom-left inset diagram shows detail of the spatial-filter mirror (SFM). In short, the near-forward angular range of the measurements is achieved by the SFM and lens L 1; unscattered incident light is focused into the through-hole in the SFM while the scattering pattern is reflected. See text for further explanation.

Experiments are conducted with the optical design of Fig. 2 using glass-microsphere particles. Figure 3 shows an example of the measured patterns and their comparison to Mie theory for a single trapped particle. The azimuthal average of the measured pattern along with a calculated pattern is shown in Fig. 3(a) as a function of the polar scattering angle θ, while plots 3(b) and 3(c) show the corresponding two-dimensional patterns. The particle originates from a sample of NIST-traceable standard microspheres (Duke Scientific Corp., cat. no. 9008) with a diameter of D = 8.2 ± 0.8μm. The particle material is borosilicate glass with a manufacturer-provided refractive index of m = 1.56 + 0i at 589 nm. Note that there is no blocking of the two-dimensional pattern due to optical-hardware posts or access holes like in [3

3. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]

], or the presence of the “small mirror” in [9

9. F. Ferri, “Use of a charged coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2273 (1997). [CrossRef]

].

To generate the calculated patterns, a Mathematica (Wolfram Research, Inc.) program is created to implement the Mie solution to the Maxwell equations following the analytical framework of Bohren et al. [14

14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983) Ch. 4.

]. The manufacturer-provided D and m are used as initial values to calculate a scattering curve in the horizontal scattering plane; the plane perpendicular to and containing the polarization and propagation directions of the incident light. This curve is then compared to the measured curve and D and m are adjusted until the two curves agree. The polarization dependence of the patterns is neglected here since only the near-forward portions of the patterns are of interest, which is where such dependence is weakest. Figure 3(a) shows the result of this Mie-theory fitting procedure, where the values for D and m yielding the best fit are 8.36μm and m = 1.56+0i, respectively. Comparison between the measured and calculated two-dimensional patterns, Figs. 3(b) and 3(c), also shows good qualitative agreement. One can see that the scattering pattern is measured very close to the forward direction, down to an angle of θ ~ 0.74° and extending to θ ~ 12.3°. The portion of the pattern lost or distorted due to the hole in the SFM is shaded in Fig. 3(c).

Fig. 3. Comparison between scattering-pattern measurement and calculation for a single trapped glass-microsphere. Plot (a) shows the azimuthally averaged scattering curve while plots (b) and (c) show the two-dimensional measured and calculated patterns, respectively. The horizontal streak seen in plot (b) is due to blooming on the CCD chip and is not a feature of the scattering pattern.

3. Size estimation

One technique used to obtain size estimates from the scattering curve for a spherical particle is to fit the curve to that predicted by Mie theory by varying both D and m in the calculations. Indeed, this is how the particle size is determined in Fig. 3, where a good estimate for the refractive index is known a priori from the manufacturer. However, estimation of the particle size is possible without any knowledge of the refractive index using the so-called Guinier analysis [8

8. A. Guinier, G. Fournet Small-Angle Scattering of X-Rays (Wiely, 1955).

]. This is useful when size estimates for spherical particles of unknown composition is desired. The following will demonstrate this sizing analysis on single-particle patterns collected with the optical arrangement shown in Fig. 2.

DestπqowhereI(qo)0.6I(0).
(1)

In Eq. (1), q o represents the value for q at which the scattering curve has decreased to 0.6I(0) and identifies the Guinier crossover. This crossover denotes the angular location where the onset of destructive interference begins between the largest extents of the particle along the q direction, i.e., D. Equivalently, Eq. (1) relates to the decay of the particle’s primary diffraction peak, i.e., the peak in the forward direction.

Fig. 4. Application of Guinier analysis to estimate particle size D est without knowledge of the refractive index. The particles here are single trapped glass-microspheres with four nominal sizes ranging from 8–20μm in diameter and labeled (a)–(d). The Guinier-analysis determined size-estimates are shown along with those resulting from Mie-theory fits. The polynomial fit to the forward most portion of the measured data, used to determine I(0), is shown by the fine-dashed curves.

To determine the value of the scattering curve in the exact forward direction I(0), a polynomial fit is made to the smallest-q portion of the curve surrounding the crossover, excluding the portion distorted by the hole in the SFM. This procedure is essentially a fit of the data to the Guinier equation, i.e., Eq. (4) of Ref. [5

5. C. M. Sorensen and D. Shi, “Guinier analysis for homogeneous dielectric spheres of arbitrary size,” Opt. Commun. 178, 31–36 (2000). [CrossRef]

], except the measured pattern is not normalized since the value of I(0) is absent due to the hole in the SFM. Consequently, the Guinier equation cannot be used directly, and hence the polynomial fit is used in its place. The portion of the measured pattern disturbed by the SFM hole is estimated from the scattering curve around the forward direction: The hole causes a sharp decrease in the forward scattering peak and hence its influence can be easily discerned from the angular structure of the scattering pattern. This can be seen in the gray shaded portion of Fig. 3. In passing, the reader should note that Guinier analysis can be presented in different forms, e.g., see [5

5. C. M. Sorensen and D. Shi, “Guinier analysis for homogeneous dielectric spheres of arbitrary size,” Opt. Commun. 178, 31–36 (2000). [CrossRef]

].

Figure 4 shows the measured scattering curves for single, trapped glass-microspheres of several sizes. The Guinier-crossover size estimates D est are shown along with the values for D and m resulting from the Mie-theory fits described in Sec. 2. One can see that the estimated sizes D est are close to those obtained by the Mie-theory fits. The relative errors between the results using Guinier analysis and Mie theory are approximately 6% in 4(a), 2% in 4(b), 7% in 4(c), and 19% in 4(d). One may notice that the Giunier estimated sizes are all less than the Mie-theory estimates. Given the limited scope of this work, it is not clear if this underestimation is coincidental, or is indicative of a general property of the Guinier law like that found by Sorensen et al. [5

5. C. M. Sorensen and D. Shi, “Guinier analysis for homogeneous dielectric spheres of arbitrary size,” Opt. Commun. 178, 31–36 (2000). [CrossRef]

]. Further work is needed to clarify this issue.

4. Particle-profile imaging

The optical arrangement in Fig. 2 can be modified to obtain images of a particle’s profile simultaneously with its two-dimensional scattering pattern. This is done by splitting the scattered light with a beam splitter (BS) following the SFM (see Fig. 5). The same lens L 2 as in Fig. 2 is used along with a relay lens L 3 (50 mm dia, f = 60 mm) to project the scattering pattern onto an 8.5“x11” sheet of copy paper, which acts as an observation screen. The other portion of the scattered light is used to image the particles onto the paper screen. This is done with an additional mirror (M) and a short focal length lens L 4 (50 mm dia, f = 40 mm). The lens is positioned such that, in combination with L 1, the object plane resides within approximately 1 mm of the front focal plane of L 1; this images any particle contained in that plane on the screen. The numerical aperture of the particle-imaging arrangement is N.A. ~ 0.3.

A quartz cuvette containing a dilute solution of polystyrene latex microspheres in water is placed at the front focal plane of L 1. The microspheres (Duke Scientific Corp. cat. no. 4220A, NIST-traceable) are 20±0.1μm in diameter. A commercial digital camera (Canon Inc., model PowerShot G9) is used to capture a video of the images appearing on the paper screen (recall Fig. 5). Figure 6 shows this video, where particle images can be seen on the left and the simultaneous two-dimensional scattering patterns on the right. The particles move due to convection in the cuvette. As they move, they pass in and out of the object plane mentioned above and hence fade in and out of focus. Unlike Figs. 3 and 4, the scattering pattern here is not a single-particle pattern, but rather is the accumulated pattern due to all particles passing through the illumination beam during the course of the video. The pattern shows well-defined fringes indicative of a single-particle because the particle size-distribution is highly monodisperse and because the solution is dilute, thus minimizing multiple scattering effects.

A unique property of this design is its substantial insensitivity to particle position. For example, the elliptical mirror in [3

3. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]

] requires a particle to reside within 10 μm of the mirror’s focus before off-axis aberration coma causes angular distortions in the pattern exceeding 1°. In contrast, the particle position in the designs of this work, shown in Figs. 2 and 5, can vary by as much as the width of the illuminating beam, which is approximately 2 mm. Figure 6 illustrates this; the scattering pattern’s structure is invariant while the particles move distances many times their diameter. This position insensitivity can be understood from the translational invariance of the FT-like operation performed by lens L 1. In Fourier optics, the particle is modeled by an amplitude transmittance profile [12

12. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), pp. 103–108.

]. The intensity distribution in the lens’ back focal plane is proportional to the absolute square of the profile’s FT. Any translation of the particle introduces a complex-valued phase factor in the transform via the FT shift theorem. This phase factor becomes irrelevant once the absolute square is taken, and hence does not affect the angular structure of the light intensity in the back focal plane.

Fig. 5. Optical arrangement used to obtain simultaneous particle images and two-dimensional scattering patterns. The design is similar to Fig. 2 with the modifications described it the text. Figure 6 presents a movie showing an example of the images and patterns observed on the copy paper screen.

5. Comments

Although the particles in this work are spherical, there is nothing inherent to the optical designs that restrict their application to more complex-shaped particles. The primary reason that spheres are used here is due to the need for high-quality reference particles with known size and shape for validation purposes.

Fig. 6. Single-frame excerpt from a video recording of the images seen on the copy-paper screen in Fig. 5. The particles are 20μm diameter polystyrene latex microspheres in solution. Images of these particles along with their two-dimensional far-field scattering pattern can be seen on the left and right side of the screen, respectively. Notice that the motion of the particles does not disturb the pattern, thus illustrating the translational invariance of the optical design. A real-time video is linked to this figure (Media 1).

6. Conclusion

Acknowledgments

This work was supported by a National Research Council Postdoctoral Fellowship, funded by the United States Defense Threat Reduction Agency, contract no. DAAD17-03-0070. The authors are thankful for assistance provided by Melvin Felton, Chatt Williamson, and Drs. David Ligon, Leonid Beresnev, Chris Sorensen, and comments provided by two anonymous reviewers.

References and links

1.

C. M. Sorensen, “Light scattering by fractal aggregates: A review,” Aerosol Sci. Technol. 35, 648–687 (2001).

2.

R. Dhaubhadel, An Experimental Study of Dense Aerosol Aggregations, Doctoral Thesis, (Kansas State University, 2008), freely available at http://hdl.handle.net/2097/663.

3.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]

4.

M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: A tutorial review,” J. Quant. Spectros. Radiat. Transfer 110, 808–832 (2009). [CrossRef]

5.

C. M. Sorensen and D. Shi, “Guinier analysis for homogeneous dielectric spheres of arbitrary size,” Opt. Commun. 178, 31–36 (2000). [CrossRef]

6.

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008). [CrossRef]

7.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis (Eds.) Light Scattering by Nonspherical Particles (Academic Press, San Diego2000), pts. IV-V.

8.

A. Guinier, G. Fournet Small-Angle Scattering of X-Rays (Wiely, 1955).

9.

F. Ferri, “Use of a charged coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2273 (1997). [CrossRef]

10.

S. Arnold and L. M. Folan, “Spherical void electrodynamic levitator,” Rev. Sci. Instrum. 58, 1732–1735 (1987). [CrossRef]

11.

S. Arnold and L. M. Folan, “Fluorescence spectrometer for a single electrodynamically levitated microparticle,” Rev. Sci. instrum. 57, 2250–2253 (1986). [CrossRef]

12.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), pp. 103–108.

13.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999) pp. 490–492.

14.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983) Ch. 4.

15.

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Explanation of the patterns in Mie theory,” J. Quant. Spectros. Radiat. Transfer 111, 782–794 (2010). [CrossRef]

OCIS Codes
(010.1110) Atmospheric and oceanic optics : Aerosols
(070.6110) Fourier optics and signal processing : Spatial filtering
(290.4020) Scattering : Mie theory
(290.5820) Scattering : Scattering measurements
(290.5850) Scattering : Scattering, particles
(290.2558) Scattering : Forward scattering

ToC Category:
Scattering

History
Original Manuscript: February 22, 2010
Revised Manuscript: April 2, 2010
Manuscript Accepted: April 7, 2010
Published: April 21, 2010

Citation
Matthew J. Berg, Steven C. Hill, Gorden Videen, and Kristan P. Gurton, "Spatial filtering technique to image and measure two-dimensional near-forward scattering from single particles," Opt. Express 18, 9486-9495 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9486


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References

  1. C. M. Sorensen, “Light scattering by fractal aggregates: A review,” Aerosol Sci. Technol. 35, 648–687 (2001).
  2. R. Dhaubhadel, An Experimental Study of Dense Aerosol Aggregations, Doctoral Thesis, (Kansas State University, 2008), freely available at http://hdl.handle.net/2097/663.
  3. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]
  4. M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: A tutorial review,” J. Quant. Spectrosc. Radiat. Transf. 110, 808–832 (2009). [CrossRef]
  5. C. M. Sorensen, and D. Shi, “Guinier analysis for homogeneous dielectric spheres of arbitrary size,” Opt. Commun. 178, 31–36 (2000). [CrossRef]
  6. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Reflection symmetry of a sphere’s internal field and its consequences on scattering: a microphysical approach,” J. Opt. Soc. Am. A 25, 98–107 (2008). [CrossRef]
  7. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic Press, San Diego 2000), pts. IV–V.
  8. A. Guinier, and G. Fournet, Small-Angle Scattering of X-Rays (Wiely, 1955).
  9. F. Ferri, “Use of a charged coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2273 (1997). [CrossRef]
  10. S. Arnold, and L. M. Folan, “Spherical void electrodynamic levitator,” Rev. Sci. Instrum. 58, 1732–1735 (1987). [CrossRef]
  11. S. Arnold, and L. M. Folan, “Fluorescence spectrometer for a single electrodynamically levitated microparticle,” Rev. Sci. Instrum. 57, 2250–2253 (1986). [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), pp. 103–108.
  13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999) pp. 490–492.
  14. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983) Ch. 4.
  15. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Explanation of the patterns in Mie theory,” J. Quant. Spectrosc. Radiat. Transf. 111, 782–794 (2010). [CrossRef]

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