Anisotropy of wood in the microwave region

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2010 Eur. J. Phys. 31 531

(http://iopscience.iop.org/0143-0807/31/3/010)

Download details:

IP Address: 132.248.199.250

The article was downloaded on 24/06/2011 at 14:16

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 31 (2010) 531–542 doi:10.1088/0143-0807/31/3/010

Anisotropy of wood in the microwaveregion

Sasa Ziherl1, Jurij Bajc1, Bernarda Urankar2 andMojca Cepic1,2

1 Faculty of Education, University of Ljubljana, Slovenia2 ‘Jozef Stefan’ Institut, Ljubljana, Slovenia

E-mail: sasa.ziherl@pef.uni-lj.si

Received 30 December 2009, in final form 9 February 2010Published 18 March 2010Online at stacks.iop.org/EJP/31/531

AbstractWood is transparent for microwaves and due to its anisotropic structure hasanisotropic dielectric properties. A laboratory experiment that allows for thequalitative demonstration and quantitative measurements of linear dichroismand birefringence in the microwave region is presented. As the proposedexperiments are based on the anisotropy (of wood), which is evident from theobservable anisotropic structure of wood, they may serve as a demonstrationfor explaining the anisotropic properties in crystals in the optical region.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

A simple look at a piece of wood reveals that its structure is anisotropic. One can easilysee annual rings forming sheets in certain directions and wood fibres. In an average pieceof wood with no knots, the wood fibres are oriented more or less in a single direction. Anumber of anisotropic properties of wood were reported, for example, thermal conductivity,thermal expansion, absorption of water, etc [1]. As wood is transparent for microwaves, onecan expect that also the optical properties of wood in the region of microwave frequencieswould reveal anisotropic properties [2, 3]. Indeed the anisotropic properties of wood in themicrowave region are used in technological processes for measurements of the water contentin some sorts of wood [1].

Usually, microwaves are discussed during undergraduate courses. The interference anddiffraction [4], Bragg reflection, evanescent absorption [5] and other phenomena [6, 7] arediscussed during lectures or studied in the laboratory. In spite of an intensive literature search,we were able to find only one study of the anisotropic properties of wood for educationalpurposes [2]. Perkalskis and Freeman presented a simple but effective means for demonstratingthe anisotropic properties of crystals by using microwaves and wood targets. They quoted

0143-0807/10/030531+12$30.00 c© 2010 IOP Publishing Ltd Printed in the UK & the USA 531

532 S Ziherl et al

a series of experiments that straightforwardly show the detection of target eigendirectionsusing crossed microwave receiver and transmitter, the existence of transmitted signal wheneigendirections are not parallel to the polarization of the incident microwave beam and theconstruction of microwave retarder (λ/4 plate). In a short comment a few years later, Chu andNoble pointed out the importance of linear dichroism, which makes the construction of theretarder more elaborate [3].

The aim of this contribution is to present a set of experiments that can easily be usedfor demonstration purposes or for laboratory work in undergraduate courses. Based on thefindings in [2, 3], our experiments allow for quantitative measurement of the anisotropy inthe absorption (linear dichroism) and the birefringence of wood. As the structure of woodin general is macroscopically anisotropic, wood can serve as a model for demonstration ofanisotropic optical properties in crystals in the visible region.

The paper is organized as follows. In section 2, we describe the theoretical backgroundrequired for understanding the influence of a material with anisotropic absorption andanisotropic refraction index on the propagation of microwaves. We discuss the circulardichroism and possible optical activity of wood and the circumstances in which these twoproperties may become important. In section 3, we present the experimental setup and theexperimental results of how to extract the polarization-dependent absorption coefficient andthe birefringence of various types of wood. We discuss possible obstacles when measuringthe anisotropic properties of wood. In section 3, we also discuss the application of presentedexperiments as laboratory or demonstration experiments during lectures and how wood canbe used as a model of anisotropic crystals or liquid crystals.

2. Theoretical background

The visual anisotropy of wood as well as recognized anisotropies of other properties (bendingconstants, water absorption, etc) gives a hint about polarization of microwaves for which onecould expect both anisotropy of absorption and of refraction indices. In a typical piece of wood,the phloem vessels are elongated, and the annual rings form a layered structure. If a piece iscut from a big enough trunk of a tree and off the axis of the trunk, the layers are nearly paralleland one can easily identify three orthogonal directions with different mechanical properties.One is directed along the wood fibres, one is perpendicular to the layered annual sheets andthe third is perpendicular to these two directions. Although there is pronounced mechanicalanisotropy in these three directions, orientation of the fibres plays the most important role indielectrical properties [1], so uniaxial symmetry is expected in experiments with an electricfield. An electric field parallel to the fibres is expected to induce a different response from anelectric field perpendicular to the fibres. So refractive indices and absorption coefficients forpolarizations parallel and perpendicular to the wood fibres differ. From here on, the discussionis limited to simple well-defined samples like the one in figure 1, with the wood fibres alongthe annual ring lines (figure 1, right).

2.1. Anisotropy in absorption—linear dichroism

Microwaves are absorbed during propagation through wood. If the absorption depends onthe polarization, the amplitude and the direction of the electric field in the transmitted wavedepend on the initial direction of the polarization. The phenomenon is called linear dichroismas the different absorption is referred to linearly polarized light in contrast to circular dichroismwhere left- and right-handed circularly polarized light is absorbed differently.

Anisotropy of wood in the microwave region 533

Figure 1. Sample of wood appropriate for the measurements: side view (left), direct view (right).The lines indicate orientation parallel (‖) and perpendicular (⊥) to the wood fibres.

Let us assume that the typical structure of wood implies also the directions of polarizationsfor which the absorption is the highest and the lowest. If so, the amplitude of the electricfield for the incident polarized wave may be written in the coordinate system (figure 2) thatcorresponds to the wood sample as

�Ein = E0(cos β, sin β) = �E‖ + �E⊥ = (E‖, E⊥), (1)

where β is the angle between the transmission axis of the polarizer P and the x axis parallel tothe wood fibres (figure 2).

After passing through the wood, the component of the light polarized parallel to the woodfibres will be absorbed more (or less) than the component of the light polarized perpendicularto the wood fibres. Taking into account only absorbtion and no birefringence, the amplitudeof the electric field of the microwave after passing the wood is

�Eout = (E‖ e−λ‖d , E⊥ e−λ⊥d), (2)

where λ‖ and λ⊥ are the absorption coefficients for the parallel and perpendicular polarization,respectively, and d is the thickness of the wood. If λ‖ �= λ⊥, it follows from (2) that thetransmitted wave has a different direction of polarization and a different amplitude from theincident wave. However, it is hard to expect that a material with pronounced linear dichroismwould not exhibit significant anisotropy in the refraction indices, i.e. the birefringence, whichmakes the analysis more complicated.

534 S Ziherl et al

,x

,y

A PP A

z

d

Figure 2. Experimental geometry for the wood sample in figure 1. Side view (left): microwavespropagate along the z axis. The wood thickness is d; the letters P and A denote the polarizer andanalyser respectively. Direct view (right): the dark lines indicate wood fibres. The x axis is chosenparallel to the fibres and the y axis is perpendicular to the fibres. The arrows denoted by P and Ashow the transmission directions of the polarizer and analyser.

2.2. Anisotropy in refractive index—the birefringence

It is reasonable to assume that the orientations of the polarizations of largest and smallestabsorption coincide with the orientations of the polarizations having largest and smallestrefractive index. In other words, the eigenvectors for refractive index and absorption areassumed to be parallel. In the following, only the refractive indices of wood are consideredand the absorption is neglected. After transmission through the wood, the incident wave

�Ein(t, z) = (E0‖, E0⊥) cos(ωt − kz) (3)

becomes elliptically polarized. A polarized microwave is decomposed into two componentswith mutually perpendicular polarizations that are oriented along the eigenvectors of wood;one polarization is parallel to wood fibres and the other is perpendicular to the fibres. As therefractive indices of wood are different for each polarization, the beam with one polarizationis retarded with respect to the beam with the polarization perpendicular to it or vice versa. Thetransmitted microwave is

�Eout(t, z) = (E‖ cos(ωt − kz − ϕ‖), E⊥ cos(ωt − kz − ϕ⊥)), (4)

where ϕ‖ and ϕ⊥ are the phases gained by the microwaves of both polarizations while passingthrough the sample of wood. The phase difference between the microwaves polarized paralleland microwaves polarized perpendicular to the wood fibres is

δ = ϕ‖ − ϕ⊥ = 2π(n‖ − n⊥)

λ0d, (5)

where n‖ and n⊥ are refractive indices for microwaves polarized parallel and perpendicularto the wood fibres, respectively, and λ0 is the wavelength of the microwaves in the air.Renormalization of time

t ′ = t − kd

ω(n⊥ − 1) (6)

allows expression of the outcoming microwave at the position z = d as

�Eout(t′, d) = (E‖ cos(ωt ′ − δ), E⊥ cos(ωt ′)), (7)

which is a general expression for elliptically polarized light.

Anisotropy of wood in the microwave region 535

The analyser (figure 2) is used for the analysis of the transmitted wave. The electricfield of the transmitted wave after passing through the analyser is linearly polarized along thetransmission direction of the analyzer �eA. The magnitude of the electric field after passing theanalyser is

Eout,A(t ′) = �Eout(t′) · �eA = cos β cos(ωt ′ − δ) cos α − sin β cos(ωt ′) sin α, (8)

where α and β are the angles shown in figure 2. The intensity of the transmitted electromagneticwave is

I = 12ε0cE

20A

= 12ε0cE

20

(cos2α cos2 β + sin2 α sin2 β − 1

2 sin 2α sin 2β cos δ), (9)

where E0A is the amplitude of the electric field of the outcoming microwave. Rotation ofthe transmitter and receiver allows for measurements of various properties of the ellipticallypolarized wave. The anisotropic properties of wood in the microwave region are discussed insection 3.

As there are in general two types of electromagnetic wave detectors, one has to be surewhich one is used. The output signal of one type of detector is proportional to the electric fieldamplitude of the incoming microwaves, and the output of the other type is proportional to theintensity of the incoming microwaves. The signal of the detector that we used is proportionalto the electric field amplitude3.

2.3. Simultaneous effects of linear dichroism and birefringence

Anisotropic materials are usually anisotropic in more than one property. For the microwavetransmission through the wood, one expects anisotropy in absorption as well as in refractiveindex and both have to be included in the analysis. With appropriate choice of polarizerand analyser orientation, the effect of absorption or birefringence can be studied separately.However, for a general orientation of the polarizer and analyser both effects have to beconsidered. In the further analysis, we assume that the wood is uniaxial in the microwaveregion with respect to the absorption coefficient and refractive index having the optical axisparallel to the wood fibres [1]. If so, (9) takes the following form:

I = 12ε0cE

20A

= 12ε0cE

20

[e−2λ‖d cos2 α cos2 β + e−2λ⊥d sin2 α sin2 β

− 12 e−(λ‖+λ⊥)d sin 2α sin 2β cos δ

]. (10)

It is evident from (10) that measurements with the appropriate orientation of the wood, analyserand polarizer can be used to determine both absorption coefficients and birefringence of wood.

The analysis presented above is valid if wood is not circularly dichroic and/or opticallyactive in the microwave region. Circular dichroism is the differential absorption of left-and right-handed circularly polarized electromagnetic waves in a material [9]. As a result,the linearly polarized incident microwave becomes elliptically polarized after transmissionthrough the material. On the other hand, optical activity of microwaves means that the speedof left- and right-handed circularly polarized microwaves is different. This results in rotationof the polarization plane of the transmitted wave. Both additional effects can make the analysisvery complicated. The question is: can one expect these phenomena in wood? The moleculesforming the fibres in wood are certainly chiral as they contain DNA. However, molecules

3 Equipment used in the experimental problem in the 37th International Physics Olympiad in Singapore in 2006 (seeipho.phy.ntnu.edu.tw/problems-and-solutions/2006/IPhO 2006 Expt Question.pdf).

536 S Ziherl et al

Figure 3. Experimental setup. Instrument ‘A’ is the microwave transmitter and instrument ‘B’ isthe microwave receiver. The detailed setup is shown in figure 2.

are much smaller than the wavelength of microwaves and observable effects of their chiralityare not expected. The effect could be large for macroscopic structures found in knots, butthis should only be considered as a warning not to use boards having knots. However, forboards used, it is still good to experimentally verify the absence of chiral dichroism and opticalactivity, as described in the continuation.

3. Experimental setup and experimental results

Polarizers and analysers for visible light are usually strongly linearly dichroic plastic sheets thatabsorb the light polarized in one direction much more than the light polarized perpendicularlyto this direction. The latter is called the transmission direction of the polarizer. However, in themicrowave region, polarizers and analysers are not needed as the transmitters emit polarizedmicrowaves and the receivers receive only the component of the light polarized in one specificdirection. Therefore, the polarizer and the analyser in figure 2 are considered symbolically asindicators for the direction of the incident microwave polarization and for the polarization ofthe receiver.

The experimental setup is shown in figure 3. One is allowed to rotate separately thetransmitter, the sample, as well as the receiver. The transmitter used was IPHO Microwavetransmitter (A) (see footnote 3). The signal was measured by IPHO Microwave receiver (B)(see footnote 3). Both transmitter and receiver are equipped with a goniometer with an accuracyof 5◦, which is common for undergraduate laboratory work equipment. The wavelength ofmicrowaves was 2.8 cm. The receiver signal was measured with a digital multimeter. Asmentioned above, the measured electric current is proportional to the amplitude of the electricfield and not to the energy flux of the microwaves absorbed by the receiver. The samples ofwood were boards of different wood types and thicknesses (table 1). Different thicknesses ofspruce boards were obtained by appropriately clamping together two triangular-shaped piecesof spruce wood as suggested in the paper by Perkalskis and Freeman [2].

3.1. Circular anisotropy

To control the absence of both phenomena originating from chiral wood structures, thepreliminary control of these properties is necessary. The main consequence of both

Anisotropy of wood in the microwave region 537

0.0

0.4

0.8

0 50 100 150[°]

detectorsignal [a.u.]

Figure 4. Dependence of measured intensities on the angle between the analyser and wood fibres(α) for three thicknesses of spruce wood (squares—5.3 cm, diamonds—4.6 cm, and triangles—3.4 cm). The angle between the polarizer and analyser is fixed to 90◦.

Table 1. Samples of wood used in the study.

Wood Thickness (mm)

Spruce 34.0, 46.0, 53.0Beech 15.0Elm 14.5Pine 15.0Linden 14.5Oak 15.0Chestnut 15.0Splinter board 15.0

phenomena, circular dichroism and optical activity, is that the incident electromagnetic waveof general polarization exits the material as an elliptically polarized wave instead of a rotatedlinearly polarized wave. Therefore, some transmission through crossed polarizer and analyseris present regardless of the orientation relative to wood fibres. It may happen that in a veryunusual case sample thickness, linear and circular absorption coefficients, and circular andlinear refraction indices combine to give a linearly polarized outcoming wave, but this wouldnot be true for a different thickness.

The control of circular anisotropy is therefore simple: if the polarizer and analyser areperpendicular and the wood in between is gradually rotated, than if the transmitted intensityfalls to zero twice during the rotation for 180◦, circular anisotropy is negligible.

The results of such a measurement for three different thicknesses of spruce wood arepresented in figure 4. It is clearly seen that the received signal falls to zero twice within therotation. A closer look shows that these two directions correspond to the directions parallel tothe wood fibres and perpendicular to the wood fibres. We have measured other wood samplesusing this setup as well. The intensity of detected microwaves in their minima was negligiblefor all studied woods, a clear sign of the absence of circular anisotropy. Another control is

538 S Ziherl et al

the position of the maximal detected intensities. If the material is not circularly anisotropic,the maxima should be detected for parallel orientation of the transmitter and receiver, whichis also true for the measured specimens of wood.

The ‘V’ shaped dependence of the detector signal around the minimum confirms that thesignal of the detector is proportional to the electric field and not to the intensity of microwaves.In the latter case, the signal close to the minimum should depend quadratically on the angledisplaying well-known ‘U’ shape around the minimum.

3.2. Measurements of absorption anisotropy

To measure the absorption coefficients for the polarizations parallel and perpendicular to thewood fibres, the polarizer and analyser are oriented parallel (α = −β) (see (10)). To measurethe absorption coefficient for the polarization parallel to the wood fibres, β is set to 0◦. Tomeasure the absorption coefficient for the polarization perpendicular to the wood fibres, β isset to 90◦. In the first case (β = 0◦), (10) takes the form

I‖ = 12ε0cE

20 e−2λ‖d , |E0‖| = E0 e−λ‖d , (11)

while in the second case (β = 90◦), it becomes

I⊥ = 12ε0cE

20 e−2λ⊥d , |E0⊥| = E0 e−λ⊥d . (12)

For precise measurement of the absorption coefficient, one needs boards of studied wood withdifferent thicknesses or one can use the method of variable thickness described above. In ourcase, three different thicknesses of the spruce wood were used. For other types of wood, ouraim was only to obtain an estimate of the absorption coefficient and only one board of eachwood was used. It is crucial that the wood is as homogeneous as possible and does not haveknots (figure 1). To extract both absorption coefficients, (11) and (12) are written in the form

logE0‖E0

= −λ‖d logE0⊥E0

= −λ⊥d . (13)

For the analysis, one has to bear in mind that the output signals of some receivers areproportional to the magnitude of the intensity for the received microwave, but some, like ours,are proportional to the magnitude of the microwave electric field.

The absorption coefficients are extracted from the slope of log ErecE0

(d), where Erec standsfor either E0‖ or E0⊥, and E0 is the magnitude of the electric field of the incoming wave(figure 5).

The absorption coefficients (λ‖, λ⊥) of various woods are given in table 2. For easierinterpretation, half lengths (z‖, z⊥) for absorption are included in table 2:

z‖ = ln 2

λ‖, z⊥ = ln 2

λ⊥. (14)

One has to note a few interesting details. The absorption coefficient for polarizationparallel to the wood fibres is consistently larger than the absorption coefficient for thepolarization perpendicular to the wood fibres. The ratio between the two absorption coefficientsis significantly greater than 1, as already suggested in the comment [3], and it varies from 1.5to 2.5 for the wood studied (table 2—last column).

3.3. Measurements of birefringence

To measure the birefringence of wood in the microwave region, the incident microwaves arepolarized in the direction 45◦ with respect to the wood fibres (β = 45◦). The analyser, i.e. thereceiver, is slowly rotated and α occupies values between 0 and π in steps of 5◦. The signal

Anisotropy of wood in the microwave region 539

y = 0.063x - 0.0019

y = 0.167x - 0.0012

-1

-0.6

-0.2

0 1 2 3 4 5 6

d[cm]log(Erec/E0)

Figure 5. Dependence of the ratio of the transmitted and incident electric field amplitudes on thethickness of the spruce samples. The lines represent the best linear approximation of the measuredpoints polarized parallel (upper line, squares) and perpendicular (lower line, diamonds) to woodfibres. Both equations are added beside the lines.

Table 2. Absorption coefficients and corresponding half lengths for different types of wood. Thespruce wood was measured at three different thicknesses, the other woods at a single thickness.The letters z‖ and z⊥ denote half lengths for parallel and perpendicular polarization with respectto wood fibres.

Type of wood λ‖ (cm−1) λ⊥ (cm−1) z‖ (cm) z⊥ (cm)λ‖λ⊥

Spruce 0.167 0.063 6.0 15.9 2.65Beech 0.13 0.052 7.69 19.2 2.50Chestnut 0.213 0.097 4.7 10.3 2.19Linden 0.145 0.068 6.9 14.7 2.13Pine 0.251 0.126 3.98 7.94 1.99Elm 0.192 0.133 5.21 7.52 1.44Oak 0.211 0.148 4.74 6.76 1.43Splinter board 0.056 0.057 17.9 17.5 0.98

is measured at each orientation of the receiver. Equation (10) becomes

I = 12ε0cE

20A

= 14ε0cE

20[e−2λ‖d cos2 α + e−2λ⊥d sin2 α − e−(λ‖+λ⊥)d sin 2α cos δ]. (15)

The measured dependence of the received signal on the angle between the polarization directionof the receiver and wood fibres (α) for three different thicknesses of spruce wood is plotted infigure 6.

Since the absorption coefficients are known from the previous measurements, the onlyundetermined parameter is the birefringence. It is determined by the minimization of thedifference between the observed and theoretical intensities for all three wood thicknesses.Theoretical predictions are calculated from (15) using different trial birefringence values.

To get the value of the phase difference between the two polarized microwave beams, onehas to vary cos δ to obtain the best agreement between measurements and theoretical values

540 S Ziherl et al

0.4

0.8

1.2

0 50 100 150[°]

detectorsignal [a.u.]

Figure 6. Measured dependence of intensities on the angle between the analyser and wood fibres(α) for three thicknesses of spruce wood (squares—5.3 cm, diamonds—4.6 cm, and triangles—3.4 cm). The polarizer is rotated 45◦ (β = 45◦) with respect to fibres. The fit of this measurementaccording to (15) is given by corresponding solid curves.

Table 3. Birefringence values for different types of measured wood.

Type of wood d (cm) δ (rad) n

Pine 1.5 0.574 0.17Elm 1.45 0.403 0.13Oak 1.5 0.403 0.12Chestnut 1.5 0.723 0.22Linden 1.45 0.403 0.13Beech 1.5 0.555 0.17Spruce 3.4 1.115 0.15

(figure 6). We used the trial and error method to find the best single n, defined as

δ = 2πd

λ0n, n = n‖ − n⊥, (16)

for all thicknesses of spruce samples. According to the literature, the average refractive indexfor dry wood is around 1.5 [1]. The values of measured birefringences for wood samples arebetween 0.12 and 0.22 (table 3), which indicates a relatively large anisotropy. For example,liquid crystals, which are known to have very large birefringence in the optical region, can haven up to 0.3 [8]. As mentioned before, the anisotropy of wood depends on a number of factorssuch as the presence of knots, moisture, etc, and the method is not very accurate; therefore,the values should be considered more as an estimate used to demonstrate the usefulness of themethod rather than an accurate laboratory value.

3.4. Arbitrary setup

As an additional verification of the analysis presented in the theoretical section, one canmeasure the intensity, detected by the receiver for an arbitrary setup. As an example of an

Anisotropy of wood in the microwave region 541

0.4

0.8

0 100 200 300 [°]

detectorsignal [a.u.]

Figure 7. Dependence of the measured amplitudes of the transmitted electric field (Erec) on theangle between the analyser and wood fibres (α) for the spruce wood (thickness—4.6 cm). Theangle between the polarizer and analyser is fixed to 45◦. Theoretical dependance according to (10)is shown by a solid line.

arbitrary setup, we set the angle between the polarizer and analyser to 45◦. The board ofwood between them is rotated and the intensity of the transmitted wave is measured. Thedependence of the received signal on the orientation of the board is rather interesting, as can beseen in figure 7. Irrespective of the wood fibres’ direction, the microwaves are always partiallytransmitted. The maxima differ due to linear dichroism. The maximum for the incidentpolarization perpendicular to the wood fibres is lower than the maximum for the incidentpolarization parallel to them. Although the measured signal seems rather complicated, it canbe straightforwardly reproduced by expression (10) using measured values for absorption andbirefringence for the piece of wood in question. The solid line in figure 7, obtained by insertingthe previously measured birefringence and absorption coefficient data for the board specimenin question in (10), almost perfectly fits the measured values.

4. Conclusions

As wood is transparent in the microwave region and has anisotropic structure, woodand microwaves offer an ideal system in which the anisotropy is easily illustrated. Wehave presented a few simple experiments, in which the consequences of anisotropy aredemonstrated. The experiments show the anisotropic properties of wood in the microwaveregion that are generally found in crystals for visible light. These experiments can easilybe used for demonstration purposes or as laboratory work. The experimental setup, whichallows for measurements of the linear dichroism and the birefringence of wood in theundergraduate courses, is presented.

Acknowledgments

The authors would like to thank Joze Vreze, Goran Iskric and Gregor Tarman for their technicalsupport.

542 S Ziherl et al

References

[1] Bucur V 2003 Nondestructive Characterization and Imaging of Wood (Berlin: Springer)[2] Perkalskis B S and Freeman J R 1995 Demonstrating crystal optics using microwaves on wood targets Am. J.

Phys. 63 762–4[3] Chu K C and Noble J D 1997 Comment on ‘Demonstrating crystal optics using microwaves on wood targets’

Am. J. Phys. 65 797[4] Perkalskis B S and Gluck P 2007 Demonstrating edge diffraction with microwaves Eur. J. Phys. 28 1091–5[5] Planinsic G and Vollmer M 2008 The surface-to-volume ratio in thermal physics: from cheese cube physics to

animal metabolism Eur. J. Phys. 29 369–84[6] Lozano-Rogado J, Miranda Pantoja J M and Sebastian J L 2001 Measurement of velocities in noisy environments

with a microwave Doppler-effect radar Eur. J. Phys. 22 249–55[7] Allen B P and Whitehouse J E 1982 A Josephson effect apparatus for the undergraduate laboratory Eur. J.

Phys. 3 136–8[8] Chandrasekhar S 1992 Liquid Crystals 2nd edn (Cambridge: Cambridge University Press)[9] Rodger A and Norden B 1997 Circular Dichroism and Linear Dichroism (Oxford: Oxford University Press)