Mie-Scattering Experiment for the Classroom … · Mie-Scattering Experiment for the Classroom...

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Mie-Scattering Experiment for the Classroom Manufactured by 3D Printing Christian Scholz, * Achim Sack, Michael Heckel, and Thorsten P¨ oschel Institute for Multiscale Simulations, Friedrich-Alexander-Universt¨at Erlangen-N¨ urnberg (Dated: January 19, 2016) Scattering experiments are fundamental for structure elucidation of matter on molecular, atomic and sub-atomic length scales. In contrast, it is not standard to demonstrate optical scattering exper- iments on the undergraduate level beyond simple diffraction gratings. We present an inexpensive Mie-scattering setup for the classroom manufactured by 3D printing. This experiment allows to determine the particle size in dilute monodisperse suspensions and is, thus, suitable to demonstrate relations between scattering measurements and microscopic properties of particles within under- graduate lab course projects. I. INTRODUCTION Many students at entry level lack a good understand- ing of the wave-nature of light 1 , therefore, there is a need for instructive experiments for the classroom and student lab courses. While diffraction and interference, e.g. from gratings, is commonly shown in the classroom 2 , scatter- ing experiments are less common. However, scattering experiments are fundamental for the understanding of experimental techniques used in modern physics, chem- istry, and biology, since they are used to investigate the microstructure of matter on many length scales 3,4 . An important representative of scatter experiments is Mie scattering, that is, the scattering of visible light by particles whose extension is similar to or larger than the wavelength of the incident light. Mie scattering is not only the physical effect underlying widely-used measur- ing instruments, but it is also the origin of common phe- nomena such as the white color of clouds 5 or the opti- cal properties of suspensions and emulsions such as milk or latex paint, e.g. 6 . Therefore, a propedeutic experi- ment introducing Mie scattering for classroom use would be desirable. Unfortunately, commercially available lab equipment for quantitative measurement based on Mie scattering is rather expensive. In this article we present an inexpensive Mie scatter- ing experiment manufactured by rapid prototyping. The experiment can be used for demonstration of Mie scat- tering in the classroom as well as for student lab courses. We illustrate its practical usage by determining the size of particles in a dilute suspension. As supplementary material 7 we provide all information to build and operate the experiment, including the files for 3D printing, a list of components with suppliers, the necessary software and additional information which may be helpful for calibration and sample preparation. II. MIE SCATTERING The basic setup of the experiment is sketched in Fig. 1: a laser illuminates a thin sample filled with a dilute monodisperse suspension of spherical particles such that the light is scattered by only a few particles in the plane α λ=635 nm sample detector FIG. 1. Sketch of the experiment. Monochromatic light (λ = 635 nm) is scattered from a dilute suspension of particles. A detector is placed at a fixed distance from the sample with adjustable angle α to measure the intensity of the scattered light. perpendicular to the optical axis. A detector with small sensitive area is then used to measure the intensity of the scattered light as a function of the angle α between incident and scattered light. The theory by Mie 8 describes the scattering of light by a single spherical object. For the analytical treatment of the problem, first the Maxwell equations are solved for a plane wave in the presence of a spherical dielectric to obtain the far-field solution of the scattering cross-section in dependence of the wavelength and particle diameter. One obtains the scattered intensity as a function of the scattering angle α, S tot (α) S 2 1 (cos(α)) + S 2 2 (cos(α)) (1) where S 1 and S 2 are functions of the refractive indexes of the particles and the ambient medium, n sph and n med , the magnetic permeability μ and the ratio (2πn med d)of particle diameter, d, and wavelength, λ. The latter dependences allows to determine the particle size, d, by means of Mie scattering experiments 9 . As an example Figure 2 shows S tot (α) for d = 3 μm and λ = 635 m. Clearly the full Mie-theory is beyond the typical un- dergraduate level 10,11 . For practical applications, how- ever, there are numerical programs available, e.g. 12,13 , to compute the functions S tot (α) and S k (α), S (α), i.e. the scattering intensity of the parallel and perpendicu- lar polarized component of the scattered light, for given

Transcript of Mie-Scattering Experiment for the Classroom … · Mie-Scattering Experiment for the Classroom...

Mie-Scattering Experiment for the Classroom Manufactured by 3D Printing

Christian Scholz,∗ Achim Sack, Michael Heckel, and Thorsten PoschelInstitute for Multiscale Simulations, Friedrich-Alexander-Universtat Erlangen-Nurnberg

(Dated: January 19, 2016)

Scattering experiments are fundamental for structure elucidation of matter on molecular, atomicand sub-atomic length scales. In contrast, it is not standard to demonstrate optical scattering exper-iments on the undergraduate level beyond simple diffraction gratings. We present an inexpensiveMie-scattering setup for the classroom manufactured by 3D printing. This experiment allows todetermine the particle size in dilute monodisperse suspensions and is, thus, suitable to demonstraterelations between scattering measurements and microscopic properties of particles within under-graduate lab course projects.

I. INTRODUCTION

Many students at entry level lack a good understand-ing of the wave-nature of light1, therefore, there is a needfor instructive experiments for the classroom and studentlab courses. While diffraction and interference, e.g. fromgratings, is commonly shown in the classroom2, scatter-ing experiments are less common. However, scatteringexperiments are fundamental for the understanding ofexperimental techniques used in modern physics, chem-istry, and biology, since they are used to investigate themicrostructure of matter on many length scales3,4.

An important representative of scatter experiments isMie scattering, that is, the scattering of visible light byparticles whose extension is similar to or larger than thewavelength of the incident light. Mie scattering is notonly the physical effect underlying widely-used measur-ing instruments, but it is also the origin of common phe-nomena such as the white color of clouds5 or the opti-cal properties of suspensions and emulsions such as milkor latex paint, e.g.6. Therefore, a propedeutic experi-ment introducing Mie scattering for classroom use wouldbe desirable. Unfortunately, commercially available labequipment for quantitative measurement based on Miescattering is rather expensive.

In this article we present an inexpensive Mie scatter-ing experiment manufactured by rapid prototyping. Theexperiment can be used for demonstration of Mie scat-tering in the classroom as well as for student lab courses.We illustrate its practical usage by determining the sizeof particles in a dilute suspension.

As supplementary material7 we provide all informationto build and operate the experiment, including the filesfor 3D printing, a list of components with suppliers, thenecessary software and additional information which maybe helpful for calibration and sample preparation.

II. MIE SCATTERING

The basic setup of the experiment is sketched in Fig. 1:a laser illuminates a thin sample filled with a dilutemonodisperse suspension of spherical particles such thatthe light is scattered by only a few particles in the plane

α

λ=635 nm

sample

detector

FIG. 1. Sketch of the experiment. Monochromatic light (λ =635 nm) is scattered from a dilute suspension of particles. Adetector is placed at a fixed distance from the sample withadjustable angle α to measure the intensity of the scatteredlight.

perpendicular to the optical axis. A detector with smallsensitive area is then used to measure the intensity ofthe scattered light as a function of the angle α betweenincident and scattered light.

The theory by Mie8 describes the scattering of light bya single spherical object. For the analytical treatment ofthe problem, first the Maxwell equations are solved fora plane wave in the presence of a spherical dielectric toobtain the far-field solution of the scattering cross-sectionin dependence of the wavelength and particle diameter.One obtains the scattered intensity as a function of thescattering angle α,

Stot(α) ∝ S21 (cos(α)) + S2

2 (cos(α)) (1)

where S1 and S2 are functions of the refractive indexesof the particles and the ambient medium, nsph and nmed,the magnetic permeability µ and the ratio (2πnmedd)/λof particle diameter, d, and wavelength, λ. The latterdependences allows to determine the particle size, d, bymeans of Mie scattering experiments9. As an exampleFigure 2 shows Stot(α) for d = 3µm and λ = 635 m.

Clearly the full Mie-theory is beyond the typical un-dergraduate level10,11. For practical applications, how-ever, there are numerical programs available, e.g.12,13,to compute the functions Stot(α) and S‖(α), S⊥(α), i.e.the scattering intensity of the parallel and perpendicu-lar polarized component of the scattered light, for given

2

(a)

(b)

0◦

20◦160◦

180◦

200◦ 340◦

S⊥S‖Stot

−40 −20 0 20 4010−3

10−2

10−1

100

α(◦)

S/S

(α=

0)

S⊥S‖Stot

FIG. 2. (Color online) (a) Polar plot of the scattered lightintensity (logarithmic) around the target (d = 3µm, λ =635 nm, nmed = 1.33, nsph = 1.58, concentration c = 0.01spheres/µm3). (b) Intensity of the scattered light as a func-tion of the scatter angle α.

parameters of the experiment. In particular, we can com-pute the values of α where S‖(α) and S⊥(α) assume theirlocal extrema. The positions of these extrema allow todetermine the diameter of the particles in the suspension.When we refer to S(α) without lower index, the corre-sponding statement refers equally to Stot(α), S‖(α) andS⊥(α).

III. EXPERIMENT

A. Idea of the Measurement

As explained in the preceding Section, the intensity ofthe scattered light, S(α), as a function of the scatter an-gle (Fig. 2) depends on the particle size. Therefore, inorder to determine the particle size, we have to find theoptimal value of the particle size for the theoretical curvesuch that S(α) obtained from an experiment agrees withthe corresponding theoretical function. For the measure-ment, we are faced with some problems:

a) For a fine angular resolution of S(α), we have tomeasure the light intensity in small steps of α,where each measurement should be repeated sev-eral times to reduce statistical scatter. Therefore,to determine the particle size for a single probe, alarge total number of measurements of the light in-

tensity with a precise value of α is needed. Thissuggests an automated experiment whose setup isdescribed in Sec. III B.

b) The experimental function S(α) will deviate fromthe ideal theoretical function for several reasonswhich will be discussed in Sec. III E. While the ab-solute deviations between theory and experimentmay be considerable (see, e.g., Fig. 8b below), thevalues of α corresponding to local extrema of S(α)agree very well and are, thus, characteristic for theparticle size. Therefore, we determine the parti-cle diameter from the values of α corresponding tolocal extrema of S(α). Figure 2 shows that theseextrema correspond to slightly different values of αfor S‖ and S⊥. Therefore, a revolvable polarizationfilter is used in the experiment to discriminate S‖and S⊥, see Sec. III D.

c) The intensities of the higher order extrema are or-ders of magnitude smaller than the zeroth (Fig. 2)such that inhomogeneous background illuminationas typically found under classroom conditions pre-cludes an accurate measurement of S(α). There-fore, we have to correct for inhomogeneous back-ground illumination, see Sec. III D.

A set of python scripts7 was developed for the controlof the experiment, including the calibration and the dataacquisition. These programs control both servo motorsand the laser module and read the light intensity signalfrom the photodiode.

B. 3D Printed Experimental Setup

The experimental setup shown in Fig. 3 consists of asolid mount that holds the laser light source, the sam-ple, the servo motor for the angular adjustment of thedetector and an Arduino Uno which is used for servomovement, laser and photodiode control and data read-out. The servo motor carries a second mount that holdsthe photodiode detector and a small servo motor whichturns a polarizer in front of the detector.

The mechanical mount of the experiment consists offour parts, manufactured by 3D printing (all parts ap-pearing in white color in Fig. 3), whose full technical de-scription is given as supplementary material7, includingthe files needed for 3D printing.

In order to measure the angular dependent intensity ofthe scattered light the larger servo motor turns the de-tector. To select the desired polarization for S‖(α) andS⊥(α), the smaller servo motor turns a round polariza-tion filter foil. A small segment of the circle of about40◦ is cut out to allow for recording the incoming lightunfiltered in order to obtain the total intensity, Stot(α).

The circuit diagram for connecting the components tothe Arduino is shown in Fig. 4 and the complete list of all

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FIG. 3. Mechanical setup of the experiment manufactured by3D printing.

components including the detailed technical descriptionis provided online7.

FIG. 4. Circuit diagram for the Arduino Uno microcontroller,as used for controlling the servo motors, laser, photodiodedetector and data recording.

C. Preparation of the Sample

Mie theory is based on the assumption that the incom-ing light interacts with a single scatterer. Therefore, weprepare a flat cell that contains only a thin section ofparticle suspension, in direction of the incoming laserbeam to reduce multiple scattering from the particles.We put a small drop of the sample solution on a mi-croscope slide where two stripes of cellotape are used asspacers for a cover glass of thickness < 0.17 mm. Finally,the cell is sealed by means of epoxy glue. The width ofthe cell is approximately 0.14 mm due to the thickness of

the cellotype spacers. An image of the sample is shownin Fig. 5.

FIG. 5. Image of the sample of a 3µm suspension.

The experiments presented here use suspensions ofpolystyrene particles of size d = (3, 4.5, 6) µm in water14

at bulk concentration 26 g/l.

D. Calibration of the Orientation of the Polarizer

We position the detector manually such that α ≈ 0,that is, the detector measures the intensity of the incom-ing laser beam. To calibrate the angular control of thepolarizer, the servo motor turns the polarizer in smallsteps of about 1◦ from the minimum to the maximumposition. At each position we measure the light inten-sity. To account for electronic noise, we average over 200such measurements for each orientation of the polarizer.From this calibration procedure we find the orientationϕ‖ of the polarizer for which the measured intensity ismaximal. This is the orientation needed in order to mea-sure S‖. This procedure delivers also ϕ⊥ = ϕ‖ ± π/2corresponding to S‖. The range of the servo motor is ap-proximately 180◦ such that independently of the initialorientation of the polarizer we find always ϕ‖ and ϕ⊥.

In order to measure the total intensity Stot, a smallsegment [ϕ1, ϕ2] is cut from the polarizer (see Fig. 3 suchthat the laser does not interact with the polarizer forϕ ∈ [ϕ1, ϕ2]. When cutting out the segment, one has totake care that ϕ‖ and ϕ⊥ are both outside the interval[ϕ1, ϕ2].

E. Calibration of the Detector Position

For the calibration of the angular position of the detec-tor, we manufacture a diffraction grating from a compactdisc (CD), see Fig. 6. We remove the all non-transparentlayers, including the reflecting layer from the CD usingcellotape. A small piece from the outer rim of a CD iscut out and attached to a microscope slide. When thissample is placed in the sample holder (Fig. 3), the in-terference pattern is visible on the horizontal plane, suchthat the interference patterns are in line with the accessi-ble positions of the sensor. Knowing the line spacing of aCD (1.5µm for 80 min CD-R) and the laser wavelength,

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FIG. 6. Preparation of the sample for calibration (right) of thedetector position from a compact disc (left). The orientationof the cut-out is equal in both images.

λ = 635 nm, the angular scale can be calibrated by meansof the diffraction peaks of zeroth and first order at anglesα = 0◦ and α = ±25.045◦. To obtain a good contrast,the polarizer is rotated such that the laser hits the de-tector directly, ϕ ∈ [ϕ1, ϕ2], to maximize the intensity ofthe incoming light.

F. Correction due to the water-glas-air interface ofthe sample

Mie theory describes the scatter of light in a suspen-sion. In our experiment we have, however, a slightly morecomplex situation. The optical path of the light emittedby the laser crosses several interfaces sketched in Fig. 7,namely (the markers refer to the sketch)

(A) the incoming beam crosses an air-glass interface atthe outer side of the microscope slide

(B) the incoming beam crosses a glass-liquid interfaceat the inner side of the microscope slide

(C) the scattered light crosses a liquid-glass interface atthe inner side of the cover glass

(D) the scattered light crosses a glass-air interface atthe outer side of the cover glass

For quantitative comparison of the measurement and Mietheory the diffraction of the light at these interfaces shallbe considered, according to Snell’s law. The interfacesdue to the microscope slide, A and B, are irrelevant sincethey are perpendicular to the laser beam.

The parallel shift of the beam at the cover glass can beneglected given that the ratio of the thickness of the coverglass and the distance between the cover glass and thedetector is small (0.17 mm/40 mm ≈ 4 ·10−3). Therefore,the interfaces C and D can be considered as one water/airinterface. The relation between entrance angle α and exit

α

β

nmed nair

A B C D

FIG. 7. Sketch of the optical path between the laser and thedetector. The light crosses several interfaces, A-D, leadingto refraction and potentially total reflection. The measuredangle, β, must be corrected correspondingly to obtain the trueMie scatter angle, α.

angle β is then easily computed from Snell’s law

α = arcsin

(nairnmed

sinβ

), (2)

where nmed = 1.33 is the index of refraction of water andnair = 1. Experimentally we only have access to S(β),therefore, to obtain S(α) we have to employ Eq. (2). Animportant consequence is, that total reflection limits ourmeasurement to |α| < αcrit = arcsin(1/1.33) ≈ 48.75◦.Only extrema of the intensity for which −αcrit < α <αcrit can be exploited for the measurement of the particlediameter since higher order extrema are not accessible.

G. Comparison of Experimental and TheoreticalResults

Following the descriptions in Secs. III A-III F we mea-sure the scatter intensity for a system with the param-eters given in Tab. I. The function S(β) is obtained

λ = 635 nm wavelength of the lasernmed = 1.33 optical index of the fluid (water)nsph = 1.58 optical index of the particles

(polysterene)nair = 1 optical index of airc = 26 g/l concentration of the suspension

(mass/volume)

TABLE I. Experimental parameters for the sample measure-ments.

by sweeping the detector over the entire angular rangelimited by mechanical constraints of the servo motor,−90◦ ≤ β ≤ 90◦, in steps of ∆β = 1◦. At each posi-tion we measure the difference of light intensity for thelaser switched on and off, in order to account for inho-mogeneous background illumination. As in Sec. III D, toaccount for electronic noise, we average over 200 suchmeasurements for each orientation. For the measure-ment, the polarizer is set fixed to the position of maxi-mal intensity, which corresponds to the parallel polarized

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scattering signal. The intensity of Mie scattering, S(α),is then obtained via Eq. (2). Figure 8 shows S(α) for twodifferent particle sizes, d = 6µm and d = 3µm.

(a)

(b)

10−3

10−2

10−1d = 6µm

S‖/S‖(α=

0)

100

101

102

103

current(nA)

exptheory

−40 −20 0 20 40

10−2

10−1

d = 3µm

α(◦)

S‖/S‖(α=

0)

101

102

current(nA)

FIG. 8. S(α) obtained from the experiment due to the pa-rameters given in Tab. I in comparison with Mie theory. a)particle diameter d = 6µm; b) d = 3µm.

Regarding the values of the scattered light intensity,S‖(α), Fig. 8 shows qualitative agreement of the experi-mental results and Mie theory for both cases, d = 6µmand d = 3µm. Considerable deviations between theoryand experiment appear since Mie scattering is superim-posed by diffuse scattering, that is, scattering by morethan one particle in the suspension. This effect becomesparticularly pronounced for small particles. Despite thedifferences in absolute intensity, we find excellent agree-ment between Mie theory and experiment regarding thepositions of local extrema of S‖(α) up to a precision . 1◦,see Fig. 9. This good agreement allows for a reliable mea-surement of the particle sizes based on the positions ofthe local extrema, as explained in Sec. IV.

IV. COMPUTATION OF THE PARTICLE SIZEFROM S(α)

A. Method

If all system parameters are fixed, the function S(α)is specific for the particle diameter. In particular, thevalues αi, i = 0, 1, 2, . . . , where S assumes local extremaare indicative for the particle diameter.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

20

40

d = 6µm

d = 3µm

order of extremum

αextr

exptheory

FIG. 9. Values of α corresponding to local extrema of S(α)for d = (3, 4.5, 6)µm, see Figs. 8 and 10. For all cases wherelocal extrema can be determined from the measurement, thecorresponding values of α agree with Mie theory up to a highprecision.

Following this idea we compute the vectors

AMiek ≡

(αMie0 (dk) , αMie

1 (dk) , . . . , αMieI (dk)

)dk =k∆d

k ∈ [bdmin/∆dc , bdmax/∆dc+ 1]

(3)

according to Mie theory, Eq. (1). Here the bracketsbxc denote the integer part of x. The argument (dk)of the angles means that the corresponding S(α) wascomputed with the assumption that the particle diam-eter is dk and I denotes a maximum order of the lo-cal extrema, e.g., I = 8. Thus, for each value ofd ∈ (dmin −∆d/2, dmax + ∆d/2) the set

{AMie

k

}contains

a vector whose elements are the angles of local extremaaccording to Mie theory due to a certain particle diame-ter D with |D − d| ≤ ∆d/2.

In reverse, the element of{AMie

k

}which minimizes the

distance ‖AMiek − Aex‖ where Aex ≡ (αex

i , i = 1, 2, . . . ),corresponds to a certain value of dk which can be consid-ered as an approximation of the particle diameter d usedin the experiment.

For the definition of ‖AMiek − Aex‖ we have to take

into account that not all extrema which exist accordingto Mie theory can be seen in the experimental data, seeFig. 8. Therefore, we define

‖AMiek (α1, .., αI)−Aex (αex

1 , . . . , αexI′ )‖

≡ 1

I ′

I′∑i=1

min (|αj − αexi | , j ∈ [1, I]) , (4)

where I ′ denotes the number of experimental extrema.The value of k which minimizes ‖AMie

k −Aex‖ correspondsto the experimentally determined particle diameter.

The set of vectors{AMie

k

}can be conveniently com-

puted by means of available programs, e.g.15. The choiceof the parameters dmin, dmax follows from the minimumand maximum of the expected particle size used in theexperiment and ∆d determines the precision of the mea-sured value of d.

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B. Example

As an example measurement, we investigate a di-lute suspension of polystyrene particles of size d =4.5µm in water14 (parameters given in Tab. I) and ob-tain the intensity S‖(α) shown in Fig. 10. We per-

−40 −20 0 20 40

101

102

103

α(◦)

current(nA)

−40 −20 0 20 40

101

102

103

α(◦)

current(nA)

FIG. 10. S‖(α) (black) for a suspension of polystyrene par-ticles, d = 4.5µm, in water. The red line is a spline fit toestimate the position of the extrema (blue marks).

form a spline fit to smoothen the data (see inset inFig. 10) and obtain local extrema of S‖(α) for αex

i =(23.0, 26.1, 34.0, 36.4, 42.4, 45.2)◦ and for α = 0◦, corre-sponding to the incoming beam.

Using Mie theory15 we generate the vectors AMiek for

dmin = 1µm, dmax = 8µm, ∆d = 0.01µm, that is k =100, 101, . . . , 800 and compute the distances ‖AMie

k −Aex‖according to Eq. (4); see Fig. 11 (red line). The dis-tance is minimal for k = 434 corresponding to the par-ticle diameter d = 4.34µm. This result agrees with thedata provided by the manufacturer of the suspension14,d = 4.5µm.

When we perform the same analysis also for suspen-sions with d = 3.0µm and d = 6.0µm, using the datashown in Fig. 8 we obtain global optima for k = 299 andk = 601 corresponding to d = 2.99µm and d = 6.01µm,respectively, see Fig. 11b. Thus we obtain even betteragreement of our measured diameters with the specifica-tions by the manufacturer.

V. SUMMARY

We present an inexpensive Mie-scattering experimentwhich allows for the measurement of particle sizes inmonodisperse colloidal suspensions to a surprising preci-sion. The main body of the experiment is manufacturedby means of 3D printing. Together with all other com-ponents such as laser module, servo motors, Arduino

(a)

(b)0

5

10

15

20

k = 443

‖AM

iek−A

ex‖

4.5µm

200 400 6000

5

10

15

20

k = 299 k = 601

k‖A

Mie

k−A

ex‖

3µm6µm

FIG. 11. (a) ‖AMiek −Aex‖ as a function of k for a suspension

of particles of size d = 4.5µm. The function assumes itsminimum (arrow) for k = 434, corresponding to d = 4.34µm.(b) Same for d = 3.0µm and d = 6.0µm, based on the datashown in Fig. 8. Here the global optima (arrows) at k = 299and k = 601 correspond to d = 2.99µm and d = 6.01µm,respectively.

microcontroller, power supply, polarizer sheet and smallcomponents such as wire, screws, etc. the total cost ofthe experiment sums up to about 50 US$.

In our institute, we use the experiment in the class-room as a demonstration experiment and also in one-daystudent lab courses. Constructing the setup requires ba-sic electronics, optics and manufacturing techniques, allof which are on the entry level. Moreover, each taskcan be distributed individually among a group of stu-dents. Therefore, we use the experiment also as a shortlab course project for a small group of 2-3 undergradu-ate students where the setup of the experiment (hardwareand software) is included as part of the lab project.

As supplementary material we provide the files for 3Dprinting, the full list of required components and mate-rials, detailed hands-on instructions, the Python scriptsfor the control of the experiment and data processing andlinks to the programs needed to compute S(α) due to Mietheory.

The experiment can be improved in several aspects:

• The local extrema of S‖ and S⊥ correspond to dif-ferent values of α. Therefore, it would be desirableto use also S⊥(α) to improve the data used for thecomputation of ‖AMie

k −Aex‖ according to Eq. (4)resulting in a higher reliability and precision of the

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measurement. The absolute intensity of S⊥ is, how-ever, much lower than S‖. Therefore, in order toexploit S⊥ we need either a more sensitive detector,that is, a phototransistor or an additional amplifier,requiring additional electronics or a more powerfullaser which might entail measures for safety ofwork.

• Shielding the experiment from the ambient lightwould improve the contrast such that further ex-trema of S(α) could be identified. Performing theexperiment in a closed box would, however, not al-low to observe the progress of the experiment andthe experiment might be considered as a “blackbox”.

• The range of particle diameter, d, which can be de-termined by this experiment is limited from bothsides: For small d, the local minima correspondto large values of α such that the intensity, S(α),drops to very small values already for the first or-ders, see Fig. 8b. Moreover, only a small number ofextrema can be observes because of total reflectionfor values α & 48.75◦, see Sec. III F. For large d,the local extrema come rather close, such that theprecision of the servo motors are not sufficient toprecisely measure the angles corresponding to local

extrema.

A simple solution to increase the interval for dwould be to use light from more than one laserof different color, possibly outside the visible inter-val. This improvement would cause extra costs andpossibly safety issues.

• The interval of d can be extended towards largervalues by replacing the servo motor by a steppermotor which operates at much better precision.This improvement is simple as well, but would needsome extra effort regarding hardware and softwareand extra costs due to the driver for the motor.

While all these extensions are rather basic in this paperit was our aim to present a low budget version of theexperiment which is simple enough to be accessible foraverage undergraduate students.

ACKNOWLEDGMENTS

We thank the German Research Foundation (DFG)for funding through the Cluster of Excellence “Engineer-ing of Advanced Materials” and the Collaborative Re-search Center SFB814 (Additive Manufacturing). Theschematic drawing in Fig. 4 was created using Fritzing16

software.

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