Measuring Mie Scattering from Ionized, Trapped …€¦ · measuring Mie scattering from a single...

WJP, PHY381 (2011) Wabash Journal of Physics v1.3, p.1

Measuring Mie Scattering from Ionized, Trapped Dielectric

Particles

Y. Tang, C.H. Gorman, and M.J. Madsen

Department of Physics, Wabash College, Crawfordsville, IN 47933

(Dated: October 20, 2011)

Light scattered by particles with sizes on the order of the wavelength is described

using Mie theory. Mie scattering is very sensitive to the scatterer’s size, as well as

the scattering angle. As a result, Mie scattering can be used to measure the size of

microspheres accurately. Previous undergraduate-level experiments have measured

Mie scattering from microspheres suspended in liquid. Alternatively, we propose

measuring Mie scattering from a single particle trapped in a double-needle Paul ion

trap. We present preliminary results and the ion radius is estimated to be about

10 µm.


Light scatters whenever it encounters a physical object. Its scattering behavior depends

on its wavelength, as well as the size of the object. For particles much larger than the light

wavelength, geometric optics is a good approximation of the scattered field; for particles

much smaller than the wavelength, one can use the Rayleigh scattering formula. However,

for particles whose sizes are on the order of the wavelength, there is no simple approximation.

Light scattering from such particles is described by the Mie solutions to Maxwell equations.

The remarkable feature of Mie theory is that it predicts the angular dependence of scattered

intensity, as illustrated in Fig. 1. Since this angular dependence is very sensitive to the

particle size, Mie scattering has been used as an accurate sizing method of microspheres

[1, 2]. In previous undergraduate-level experiments, particles suspended in liquid were used

to perform Mie-scattering measurements, assuming that the light scattering behavior of the

suspension is the same as that of a single particle [1, 3, 4]. Here we propose an experiment

that allows Mie-scattering measurement from a single particle levitated in an electrodynamic

trap. This is an improvement upon previous experiments in that we can measure the ac-

tual light scattering off an individual ion. Ion trapping is based on the principle that an

alternating electric field can trap an ion. Previous work has been devoted to developing ion

traps [5–7].

and is well known. The solution is obtained by solving theproblem of a plane electromagnetic wave incident on a uni-form uncharged sphere of arbitrary radius and refractive in-dex, as shown schematically in Fig. 1. The basic idea is toexpand the incident plane wave in a Fourier series usingappropriate basis functions that satisfy Maxwell’s equationsin spherical coordinates; spherical coordinates are used dueto the geometry of the scatterers. We then apply suitableboundary conditions at the surface of the sphere in order todeduce the Fourier expansions of the scattered fields insideand outside the sphere. Because the derivation of the scatter-ing coefficients is quite long, we defer it to Appendix A.!This derivation is mathematically advanced and will be achallenge for all but the most advanced undergraduates—however, it is our experience that undergraduates can masterthis material. We include the derivation, in its entirety, sothat the precocious student may find the relevant theoreticalframework and laboratory discussion self-contained in asingle document." As shown there, the final solution for thefield outside the sphere is given by

Es! #n!1

$

En! ianNe1n!3 " "bnMo1n!3 " ",

!1"Hs!

k%& #

n!1

$

En! ibnNo1n!3 " #anMe1n!3 " ",

where the coefficients are given by

an!&m2 jn!mx "'x jn!x "(!"&1 jn!x "'mx jn!mx "(!

&m2 jn!mx "'xhn!1 "!x "(!"&1hn

!1 "!x "'mx jn!mx "(!,

!2"bn!

&1 jn!mx "'x jn!x "(!"& jn!x "'mx jn!mx "(!

&1 jn!mx "'xhn!1 "!x "(!"&hn

!1 "!x "'mx jn!mx "(!.

Here jn’s are the spherical Bessel functions of the first kind,hn’s are the spherical Hankel functions, &1 and & are themagnetic permeability of the sphere and surrounding me-dium, respectively, N and M are the vector spherical har-monics whose expressions in terms of commonly known ba-sis functions are given in Appendix A, and the primeindicates a derivative. If we take &1!& , both drop out of the

expressions. The coefficients of Eq. !2" are dependent uponthe quantity

x!ka!2)Na

*, !3"

where a is the radius of the sphere, * is the wavelength ofincident light in vacuum, and N is the refractive index of thesurrounding medium. The quantity x, called the ‘‘size param-eter,’’ is what determines the scattering curve, rather than theradius or wavelength independently. It is this parameter towhich we fit in our routines. We display sample plots of thetheoretical angular scattering distribution for a range of sizeparameters in Fig. 2. The size parameters x!5, 10, and 20correspond, with our setup (*!633 nm), to sphere diametersof 0.76, 1.51, and 3.03 &m, respectively. Although the inten-sities are expressed in arbitrary units, the scaling of intensi-ties is the same for the three plots. Thus, we can note that asthe scatterer’s size parameter increases, scatter in the forwarddirection increases dramatically. Also, we begin to see oscil-latory behavior in the angular scattering distribution only atsize parameters around ten or higher; thus spheres of sizeparameter smaller than this are more difficult to fitaccurately.

III. EXPERIMENTAL METHODS

Figure 3 shows a diagram of the experimental setup. Anunpolarized 1-mW helium–neon !He–Ne" laser !EdmundScientific, #F61337", with a wavelength of 632.8 nm, is di-rected through the center of a glass cuvette with square crosssection !a square of side length 1 cm", containing the spheresuspension !one drop of spheres in 100 ml of water; +1010spheres/ml". The suspension is created using filtered water toeliminate any scatterers other than the latex spheres. We di-rected the beam using two mirrors, which aided in laser

Fig. 1. Illustration of Mie scattering. N and N1 refer to the indices of re-fraction of the medium surrounding the scatterer and the particle material,respectively; * is the wavelength of the incident light.

Fig. 2. The angular scattering distribution for incident light which is unpo-larized, parallel, and perpendicularly polarized. Scattered light is shown forparticles of size parameter x!!a" 5, !b" 10, and !c" 20.

130 130Am. J. Phys., Vol. 69, No. 2, February 2001 Weiner, Rust, and Donnelly

FIG. 1. Illustration of wavefronts in Mie scattering. N and N1 are the indices of refraction of

ambient medium and the scatterer, respectively. λ is the wavelength of incident light. Scattering

intensity depends on the angle θ [1].

The derivation of Mie theory is mathematically advanced and can be found in the liter-


ature [8, 9]. Essentially, Mie scattering depends on the size parameter x, which is defined

as

x =2πNa

λ, (1)

where N is the index of refraction of the surrounding medium, a is the radius of the sphere

and λ is the wavelength of incident light in vacuum. It is interesting to note that Mie

scattering depends on the ratio of a/λ rather than solely on radius or wavelength. Fig. (2)

shows calculated angular scattering distribution for x = 5, 10, and 20 [2]. As x increases,

oscillations in the angular distribution curve becomes more prominent. For our experiment,

we used λ = 532 nm laser, an average particle radius of a = 15 µm, and, assuming a reflective

index of refraction of air N = 1, we have x = 177.

and is well known. The solution is obtained by solving theproblem of a plane electromagnetic wave incident on a uni-form uncharged sphere of arbitrary radius and refractive in-dex, as shown schematically in Fig. 1. The basic idea is toexpand the incident plane wave in a Fourier series usingappropriate basis functions that satisfy Maxwell’s equationsin spherical coordinates; spherical coordinates are used dueto the geometry of the scatterers. We then apply suitableboundary conditions at the surface of the sphere in order todeduce the Fourier expansions of the scattered fields insideand outside the sphere. Because the derivation of the scatter-ing coefficients is quite long, we defer it to Appendix A.!This derivation is mathematically advanced and will be achallenge for all but the most advanced undergraduates—however, it is our experience that undergraduates can masterthis material. We include the derivation, in its entirety, sothat the precocious student may find the relevant theoreticalframework and laboratory discussion self-contained in asingle document." As shown there, the final solution for thefield outside the sphere is given by

Es! #n!1

$

En! ianNe1n!3 " "bnMo1n!3 " ",

!1"Hs!

k%& #

n!1

$

En! ibnNo1n!3 " #anMe1n!3 " ",

where the coefficients are given by

an!&m2 jn!mx "'x jn!x "(!"&1 jn!x "'mx jn!mx "(!

&m2 jn!mx "'xhn!1 "!x "(!"&1hn

!1 "!x "'mx jn!mx "(!,

!2"bn!

&1 jn!mx "'x jn!x "(!"& jn!x "'mx jn!mx "(!

&1 jn!mx "'xhn!1 "!x "(!"&hn

!1 "!x "'mx jn!mx "(!.

Here jn’s are the spherical Bessel functions of the first kind,hn’s are the spherical Hankel functions, &1 and & are themagnetic permeability of the sphere and surrounding me-dium, respectively, N and M are the vector spherical har-monics whose expressions in terms of commonly known ba-sis functions are given in Appendix A, and the primeindicates a derivative. If we take &1!& , both drop out of the

expressions. The coefficients of Eq. !2" are dependent uponthe quantity

x!ka!2)Na

*, !3"

where a is the radius of the sphere, * is the wavelength ofincident light in vacuum, and N is the refractive index of thesurrounding medium. The quantity x, called the ‘‘size param-eter,’’ is what determines the scattering curve, rather than theradius or wavelength independently. It is this parameter towhich we fit in our routines. We display sample plots of thetheoretical angular scattering distribution for a range of sizeparameters in Fig. 2. The size parameters x!5, 10, and 20correspond, with our setup (*!633 nm), to sphere diametersof 0.76, 1.51, and 3.03 &m, respectively. Although the inten-sities are expressed in arbitrary units, the scaling of intensi-ties is the same for the three plots. Thus, we can note that asthe scatterer’s size parameter increases, scatter in the forwarddirection increases dramatically. Also, we begin to see oscil-latory behavior in the angular scattering distribution only atsize parameters around ten or higher; thus spheres of sizeparameter smaller than this are more difficult to fitaccurately.

III. EXPERIMENTAL METHODS

Figure 3 shows a diagram of the experimental setup. Anunpolarized 1-mW helium–neon !He–Ne" laser !EdmundScientific, #F61337", with a wavelength of 632.8 nm, is di-rected through the center of a glass cuvette with square crosssection !a square of side length 1 cm", containing the spheresuspension !one drop of spheres in 100 ml of water; +1010spheres/ml". The suspension is created using filtered water toeliminate any scatterers other than the latex spheres. We di-rected the beam using two mirrors, which aided in laser

Fig. 1. Illustration of Mie scattering. N and N1 refer to the indices of re-fraction of the medium surrounding the scatterer and the particle material,respectively; * is the wavelength of the incident light.

Fig. 2. The angular scattering distribution for incident light which is unpo-larized, parallel, and perpendicularly polarized. Scattered light is shown forparticles of size parameter x!!a" 5, !b" 10, and !c" 20.

130 130Am. J. Phys., Vol. 69, No. 2, February 2001 Weiner, Rust, and Donnelly

FIG. 2. Calculated angular scattering distributions with (a) x =5, (b) x =10, and (c) x =20 for

differently polarized light [1].

During the course of developing this project, we realized that a double-needle electrody-

namic trap is ideal for this measurement due to its simplicity. We constructed a double-

needle trap.The distance between the two Tungsten needles is 1.56 ± 0.12 mm, while the

distance between the two stainless steel cylinders is 4.60 ± 0.03 mm. The needles have a


0.62±0.01 mm diameter; the cylinders have a inner diameter of 1.73±0.02 mm and an outer

diameter of 2.43±0.01 mm. The needles and the cylinders are separated by a ceramic tube.

The schematic is shown in Fig. 3. The radiofrequency (RF) voltage on the trap electrodes

are provided by a microwave transformer; the input voltage into the microwave transformer

is 50 V at 60 Hz. The voltage on the needles are Vtrap = 845.0 ± 0.5 Vrms (95% CI). The

trap is placed in an acrylic box which provides shielding against air currents. The lower

part of the trap is fixed onto the optical table via a post. The upper trap is connected to a

three-dimensional translational stage, which allows fine adjustments of the relative position

between the two electrodes.

0.62±0.01

1.73±0.02

2.43±0.01

4.60±0.03 1.56±0.12

Units: mm (95% CI)

RFGND tungsten

ceramics stainless steel

FIG. 3. Schematics of the doulbe-needle trap. The metal needle is placed inside a metal tube, with

a ceramic insulating tube in between (not shown in graph). A RF high voltage is applied to the

needles, while the tubes are grounded. Dimensions of the key parts are labeled as above.

Since the acrylic box needs to be enclosed to perform scattering measurements, we built a

robotic arm with LEGO components to perform the sweep automatically. The arm is driven

by a pair of gears with ratio 7 : 1. The motor is controlled by the NXT (1.0) controller, whose

internal sensor gives digital feedback on the angular location of the motor in arbitrary units.

We mounted a photomultiplier tube (PMT) at the end of the arm. The distance between

the ion and the PMT is 30.5 ± 0.4 cm (95% CI). To collect more light, a lens with focal

length f = 3.81 cm was mounted in front of the PMT via a lens tube, 22.6 ± 0.4 cm (95%

CI) away from the trap center, shown in Fig. 4. We inserted two neutral density filters with

total attenuation 7.0 at θ = 0 to prevent the laser from saturating the PMT. Input voltage

for the PMT was 460 V. A National Instruments USB-6009 computer data acquisition board


digitalized the analog signal from the PMT through a LM741 inverting unity amplifier with

R1 = R2 = 510 kΩ. Under these conditions, background signal level was about 0.2 V. We

used a single LabVIEW program to control the NXT and data acquisition from the PMT.

Laser 2

θ

PMT

DoubleNeedleTrap

Laser 1Neutral density lters

f = 3.81 cm

FIG. 4. A photomultiplier tube (PMT) is mounted at the end of the LEGO arm, whose motor is

placed below the double-needle trap. Two lasers were used to characterize the sweeping motion of

the arm. Laser 1 was turned off during actual scattering light measurements.

Two lasers were used to the characterize the motion of the LEGO arm. Both lasers were

adjusted so that they illuminated the ion in the trap while their beams were parallel with

the table. Angular separation of the two beams was ∆θ = 90.0 ± 1.1 (95% CI). With

both lasers turned on and no ion in the trap, light intensity was recorded as the PMT swept

from approximately θ = 110 to θ = −5 continuously. Data collected from 7 trials were

interpolated and summed to produce the points in Fig. 5 at an interval of 0.2 NXT ticks

(arbitrary units). The points were fitted to Gaussian functions, which are the continuous

curves in Fig. 5. By measuring the number of ticks between the two peaks, we found the

corresponding average of 0.143 ± 0.002 per tick (95% CI).

Microspheres were manually inserted into the trapping region using a ionizing plastic

rod. The spheres were trapped by the RF potential and were loaded probabilistically. We

found that trapped ions were extremely susceptible to air flow; we loaded with the acrylic

box completely closed except for a small hole used to inset the rod. We first performed


110 115 120 125 130 135 140

0

5

10

15

NXT Ticks a.u.

Ligh

tint

ensi

tya

.u.

740 745 750 755 760 765 770

0

5

10

15

20

25

30

NXT Ticks a.u.

Ligh

tint

ensi

tya

.u.

Peak 1: 124.27±0.04 (95% CI)

Peak 2: 754.62±0.04 (95% CI)

(a) (b)

FIG. 5. Data from seven sweeps were converted into seven interpolation functions in Mathematica.

The points were generated using the sum of the interpolation functions, at a step size of 0.2. The

points were fitted to a Gaussian function, which is shown as the curves.

ten measurements of the background, and then ten measurements with an ion in the trap,

shown in Fig. 6 and Fig. 7, respectively.

The background data in Fig. 6 show that the background level is mostly low and constant

when the PMT is not directly illuminated by the laser beam. In addition, aside from the

horizontal shifts due to the difficulty in starting the PMT from the same location for each

trial, the graphs are essentially the same. Looking at Fig. 7, we see that there are numerous

oscillations in the scattered light intensity in the trials in with a loaded ion that is not present

in the background intensity. We suspect that the oscillations are due to the constructive

and destructive interference patterns that resulted from Mie scattering. The averages of the

background and ion data were shown in Fig. 8. The scattered light intensity from the ion is

significantly higher than background. The difference between the two is not isotropic, which

qualitatively agrees with the Mie theory.

The small oscillations in our data have rather short period (compare with Fig. 2 in

Ref. [1]), indicating that the ion has a large x. The approximated Mie scattering formula

provided by Drake and Gordon was used to estimate our ion size [4]:

I(θ) = I0(3|J1(β)|/β + β−3/2)2(1 + cos2 θ)/2, (2)

where I is the scattered intensity, J1(β) = (sin β−β cos β)/β2, and β = ka(1+N2−2N cos θ).


0 20 40 60 800

2

4

6

8

10

Θ HdegreesL

Lig

htin

tens

ityHa

.u.L

FIG. 6. Ten sets of background data plotted with arbitrary vertical offsets. The main difference

between each of the trials is horizontal translation of the peak. This is due to the difficulty in

starting the detection arm at the same location each time. However, the background level is

mostly low and constant, meaning there was little light scattering off the trap electrodes.

0 20 40 60 800

10

20

30

40

Θ HdegreesL

Lig

htin

tens

ityHa

.u.L

FIG. 7. Ten sets of data with ion in the trap plotted with arbitrary vertical offsets. At the top in

red is the average of the ten trials. We can see that there are definite places of constructive and

destructive interference due to the Mie scattering.


0 20 40 60 800.0

0.5

1.0

1.5

2.0

2.5

3.0

Θ degrees

Ligh

tint

ensi

tya

.u.

Signal with the ion

Background

FIG. 8. Above are the average trials of the background light intensity and light reflecting off of

an ion in the trap. As we can see, it is obvious that there is definite Mie scattering and definite

places of constructive and destructive interference that is predicted by Mie theory that is not in

the background light.

Numerical calculation for a particle with radius a = 10 µm is shown in Fig. 9. Since the

computed curve demonstrates oscillations with similar period as in our data, we estimate

the ion radius to be about 10 µm.

From the data we see that our PMT is detecting light that is reflecting off the ion and

that it is nontrivial; we measured real Mie scattering. One improvement that will be made

on the present apparatus is to attach the motor onto the end of the rotating arm connected

to a wheel and have that turn the PMT. This should reduce the systematic errors in the

ability to restart the PMT sensor in the same location after every trial. Additionally, another

improvement that can be made is loading ions of known size into the double needle trap

via electro-ionization spray. This would allow one to have a value for x, the ratio of the

wavelength to particle radius, and compare the theory to the measured intensity of light.

A further enhancement to our current design would be to place the entire apparatus in a

vacuum chamber. This would help decrease the possibility of the ion being knocked out of


0 20 40 60 80107

106

105

104

0.001

0.01

0.1

Θ degrees

Ligh

tint

ensi

tya

.u.

Particle size = 10 μm

FIG. 9. Mie scattering curve computed for a particle with radius a = 10µm using an approximation

formula from Ref. [4]. It exhibits similar small oscillations as in our data.

the trap by air currents within the enclosure.

[1] I. Weiner, M. Rust, and T. D. Donnelly, “Particle size determination: an undergraduate lab in

Mie scattering,” Am. J. Phys. 53 (10), 955-962 (1985).

[2] C. Mund and R. Zellner, “Optical Levitation of Single Microdroplets at Temperatures Down

to 180 K,” ChemPhysChem. 4, 630-638 (2003).

[3] A. J. Cox, Alan J. DeWeerd, and Jennifer Linden, “An experiment to measure Mie and Rayleigh

total scattering cross sections,” Am. J. Phys. 70 (6), 620-625 (2002).

[4] R. M. Drake and J. E. Gordon, “Mie scattering,” Am. J. Phys. 53 (10), 955-962 (1985).

[5] C. E. Pearson, et. al. Phys. Rev. A, 73, 032307 (2006)

[6] M.J. Madsen and C.H. Gorman, “Compact toroidal ion trap design and optimization,” Phys.

Rev. A. 82, 043423 (2010).

[7] H.G. Dehmelt, Adv. Atom. Mol. Opt. Phys., 3, p. 53 (1967)

[8] H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

[9] Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles

(Wiley, New York, 1983).

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