arXiv:physics/0702225v2 [physics.optics] 28 Feb 2007 ∗ from the are of data square the The waveplate. with the linearly size. of grows vs size Price waveplates catalog. of CASIX price a The 1: FIG. aertre o ih at quarter- constructed light a be of for can construction retarder Recently, fa- wave plates his wave material. in this and vis- Feynman from [4], for Richard thickness lectures by half-wave mentioned mous a as to light, close pack- ible very commercial For be in can interest. used aging practical cellophane great inex- ordinary of from example, be plates can wave materials of which than pensive in construction important plates the more wave quality, are of availability high applications rapid In and price 3]. low 2, [1, mica or nti ae eepoetepsiiiyo sn most using of possibility the explore laboratory. we a in paper found film this commonly this not In While is it investigated. inexpensive, were is film this of properties PP2500 lcrncades [email protected] address: Electronic ovninly eadr r aeo act,quartz, calcite, of made are retarders Conventionally, Price, $ 100 120 140 160 180 200 220 240 260 280 300 M T 10 030 050 0700 600 500 400 300 200 100 0 oyse l a rpsd[,6 n some and 6] [5, proposed was film polyester ASnmes 22.a4.0-,27.e 81.05.-t,81. 42.25.Ja,42.70.-a,42.79.-e, numbers: PACS combinations of out waveplates rotatio Constructing on retardation temperature. of dependence Various the investigated: transparencies. are available commercially ordinary edsrb rcdrsfrcntutn nxesv wavep inexpensive constructing for procedures describe We .INTRODUCTION I. ula cec iiin arneBree ainlLabo National Berkeley Lawrence Division, Science Nuclear Diameter squared,mm eateto hsc,Uiest fClfri,Berkeley California, of University Physics, of Department aelt eadr ae noeha transparencies overhead on based retarders Waveplate λ 632 = . mfo h 3M the from nm 8 grSvkvadDir Budker Dmitry and Savukov Igor 2 05.Lg I.2: FIG. ln.Te“oain sdfie stertto rudthe around rotation this of the to angle plane as through perpendicular the defined axis is in is z axis “rotation” lie of z The right, y the the plane. and while on x film, shown axes transparency principal ellipsoid, The index waveplate. the transparency a of rudtetl xsoine tsm angle some at oriented axis tilt the around ytewvpae h itaon tecs when case (the y around tilt The waveplate. the by lisi n t nescinwt h ln perpendicula plane normal the wave with a intersection plane. its transparency and the ellipsoid in x to axes l rmwihsm vredtasaece r made are transparencies overhead some which (boPET) as from such terephthalate film materials, stretch- polyethylene properties. biaxial film desirable produce biaxially-oriented achieving Additional may for stretching extruded employed side. Such where is be one roll can film from turning ing the polished quenched chilled is which a it onto in slit process a through a by produced Fig.1. in wave-plate illustrated the as with often linearly area grows are price the waveplates while mag- needed, large-size In applications, [7]. measurements of netometer polarization detection light and using light optical spins polarized on circularly based with magnetometers pumping exper- need atomic and the commercial optical of by building imental for motivated class plates also broad wave appli- cheap a is for retarder in project in interest this quality general applications, from their Apart test companies) and cations. different plates from wave color, as white, and (black different types of transparencies overhead inexpensive common illustrated. is vredtasaece r aeo oye film polymer of made are transparencies Overhead Tilt axis ftefim it aeegh oiin and position, wavelength, tilt, film, the of n n eeatpoete ftetransparencies the of properties relevant a y ftasaec hesi loexplored. also is sheets transparency of and nteleft the On ae fdsrdrtrainotof out retardation desired of lates n b aoy eklyC 94720 CA Berkeley ratory, x a k r epnil o h eadto produced retardation the for responsible are ∗ A97070 and 94720-7300 CA , h nescini nelpewoehalf- whose ellipse an is intersection The . q θ ento ftecodnt system coordinate the of Definition : h iti enda h rotation the as defined is tilt The . t/n z y nteright the On n n a y D n a b x D b n x α t/n ihrespect with h index The : k α n z 90 = z to r ◦ ) 2

30

20

10

0

-10 Ellipticity, degrees -20 FIG. 5: Large-scale retardation uniformity of a transparency sheet in two dimensions. The transparency is placed between -30 a 17” thin film transistor - liquid crystal display (TFT-LCD) computer monitor, source of polarized light with polarization ◦ 0 50 100 150 200 250 300 350 at 45 with respect to vertical, and a square polaroid film Rotation, degrees of similar size oriented to extinguish light in the absence of transparency. Transparency has its edges aligned with the screen. The color segments are generated in the computer by drawing filled rectangles in a drawing program. Three “pure” FIG. 3: Ellipticity as a function of transparency rotation. color quadrants provide information on uniformity at different The data shown with squares agree well with a fit using Eq. wavelengths. In general it is expected that birefringence vari- (3) (the solid line), in which δ = 0.91 rad and an arbitrary off- ation caused by non-uniformity of stretching should be similar set in the rotation angle is allowed. Some aperiodicity can be for the three different colors, although somewhat smaller for observed due to the fact that under rotation slightly different red. regions of the transparency are probed. The rotation of the transparency is the most convenient way to adjust ellipticity to a desirable level between maximal and minimal values.

-50

-40

-30

-20 FIG. 6: The uniformity of the zero-ellipticity across large area of a transparency film, meaning that the orientation of Ellipticity, degrees -10 the index ellipsoid does not depend on position.

0 for copiers or laser printers. Refractive indices of boPET 0 10 20 30 40 50 60 70 were studied in Ref. [8]. The extrusion and stretch- Displacement, mm ing processes in general lead to biaxial birefringence; however, it is possible that one preferential direction of molecular orientation is defined and one refractive index FIG. 4: Ellipticity as a function of transparency translation. is more pronounced than the others and the medium can The size of probing laser beam is approximately 2 mm. The be approximated as uniaxial. For example, in Ref.[9], the data exhibit smooth behavior with typical continuous varia- structural properties of HDPE (High Density Polyethy- tion 5 degrees per 10 mm and ellipticity changing slowly in lene) film have been studied and it was found that draw a large range. From applications point of view, this means ratio affected crystalline orientation and refractive in- that waveplates of given retardation and of the size 10-30 dices approximately in the same way in two directions mm depending retardation tolerance can be made by selecting but quite differently in the third direction. Birefringence an appropriate region. This behavior is similar for different is a very general property of transparent polymers, in- transparencies. cluding overhead transparencies. The theory of birefrin- 3

azimuth depend only on two variables, 60 1 χ = arcsin[sin 2θ sin δ], (3) 2 40

1 20 ψ = arctan[tan 2θ cos δ]. (4) 2

0 Equation 3 will be used throughout the paper to relate the polarimeter’s output to a more fundamental property of optical material such as retardation and to fit experi- -20 mental data in which θ or δ are varied. Ellipticity, degrees First, to verify that transparency can operate as a wave -40 plate, we study the dependence of ellipticity on the trans- parency rotation angle. The experimental arrangement

-60 for this measurement is simple: a helium-neon laser beam 780 800 820 840 860 880 900 920 940 (632.8 nm wavelength), linearly polarized with a Polaroid Wavelength, nm film, is sent to the detection head of the polarimeter through a fragment of transparency attached to a rota- tion mount with angular resolution of 1 degree. The ro- tation is defined in Fig.2, left panel. From this definition FIG. 7: Ellipticity as a function of light wavelength for 3M if transparency behaves as a waveplate the ellipticity of black and white transparency. The data follow Eq. (3) with ◦ the transmitted light as a function of the transparency θ = 45 and δ[rad] = 46120/λ[nm]. At 840 nm, the waveplate rotation must satisfy Eq. (3) in which θ is the trans- retardation is 17.5π. parency rotation angle. The ellipticity of the light as a function of this angle is shown in Fig.3. The behavior is gence induced by mechanical deformation for polymers periodic as expected for a wave plate, in accordance with and liquid crystals is given in Ref.[10]. Optical proper- Eq. (3), but some deviation from periodicity is observed ties are of great practical interest because they can be which can be explained by the shift of the point where used to analyze the material structure and to control the the laser beam intersects the transparency when it is not extrusion process. exactly in the center of rotation combined with spatial variations in the retardation that will be discussed later. The retardation δ is about 0.91 rad and because in the II. POLARIZATION PROPERTIES OF Taylor expansion of arcsine the first term is much larger OVERHEAD TRANSPARENCIES than the others, we obtain that the dependence on the transparency rotation angle is dominantly a sine wave as We investigate polarization properties of overhead is seen in Fig.3. The maximum value of ellipticity in this transparencies with a commercial polarimeter, Thorlabs particular measurement is less than maximum possible PAX5710IR1-T, which outputs the state of polarization 45 degrees of a λ/4 plate. For constructing a λ/4 plate in the form of a polarization ellipse with numerical val- a more appropriate part of the transparency has to be ues of the ellipticity χ and the azimuth ψ as well as chosen or the plane of the transparency has to be tilted. the Poincar´esphere with coordinates that are the three The next graph, Fig.4 demonstrates how the position Stokes parameters s1, s2, and s3. The Stokes parameters of the transparency for a fixed orientation affects the can be found from the ellipticity and azimuth, or from induced ellipticity and hence retardation. Apart from the set of Jones vector parameters such as field ampli- demonstrating how the retardation can be optimized by spatial selection, this graph also gives information about tudes ax and ay and phase shift δ, or vice versa [11]. For example, the ellipticity and azimuth are related to the typical retardation spatial non-uniformity in one dimen- Jones vector parameters through the following equations sion which is important for the determination of the use- ful size of the transparency wave plate. From the data, 1 2axay sin δ we see that the size is of order 1 cm. By using a computer χ = arcsin 2 2 , (1) screen and a crossed polaroid film it is possible to find 2 ax + ay regions with larger area of uniformity as demonstrated in Fig.5. 1 s2 1 2axay cos δ To understand the order of the transparency waveplate ψ = arctan = arctan 2 − 2 . (2) 2 s1 2 ax ay and to estimate the useful wavelength range where the transparency can function as a waveplate of a given type, It is convenient to parameterize normalized field ampli- we measured the wavelength dependence of the elliptic- tudes as ax = cos θ and ay = sin θ, so ellipticity and ity of a transparency using a tunable Ti:Sapphire laser as 4 light source (Fig.7). For a particular angle θ = 45◦, cho- sen in the experiment to give maximum ellipticity, Eq. 0.15

(3) predicts the dependence of ellipticity on retardation 0.14 as a triangle wave with an amplitude 45◦. Because the 0.13 retardation changes approximately linearly with wave- length in some range 0.12 0.11 2π∆n(λ)l 2π∆n(λ0)l 2π∆n(λ0)l(λ − λ0) ≈ − 0.10 δ = 2 , (5) λ λ0 λ0 y 0.09 -n x the ellipticity follows a quasi-periodic triangle wave as a n 0.08 function of wavelength as observed experimentally. The 0.07 distance between zeros can be used to obtain total retar- 0.06 dation. It is necessary in general to have the dependence of re- 0.05 fractive index on wavelength to obtain total retardation. 0.04 Such dependence is often approximated with the Cauchy 0.03 2 4 dispersion formula n(λ)= n0 + n1/λ + n2/λ + ... which 400 500 600 700 800 is accurate when wavelength of the light is far from res- Wavelength, nm onances. For wavelengths shown in the Fig.7 the con- stancy of refractive index is a good approximation. In a larger range, from 300 nm to 900 nm, this assumption FIG. 8: Refractive index difference nx − ny of 3M black is not satisfied since a strong absorption band is located and white transparency (circles and dotted line) and color near 300 nm. We used a Varian CARY219 spectrom- transparency (solid squares and line) measured with a spec- eter to investigate the refractive index variation in the trometer and two crossed polarizers with a transparency in range 400-800 nm. A transparency film was inserted be- between. The dotted line is the fit with nx − ny = 0.0806 − ◦ tween two crossed polaroid films with its x axis at 45 , 13100/λ2 + 3.5 × 109/λ4, and the solid line is the fit with 2 9 4 so the transmission depended on the retardation. The nx − ny = 0.05628 − 5500/λ + 2.2 × 10 /λ . wavelength of minima (or maxima) of the transmitted intensity could be used to calculate the refractive index difference nx − ny as a function of wavelength. The re- sults are plotted in Fig.8. Below 600 nm the wavelength dependence of refractive indices becomes significant, and a good fit requires retaining 1/λ2 and 1/λ4 terms as it is x shown in Fig.8. 3 It is interesting to note that a transparency between 2 two polarizers can serve as a birefringent filter. Inves- 2 tigation of such filters based on low-cost polymers has -1

-3 been carried out in Ref.[12] where filters with bandwidth 1

50 nm were demonstrated. -6 It might be convenient to adjust the retardation to a 0 -8 desirable level by tilting a transparency waveplate. The dependence on the tilt is determined by two factors: one -11 y is the increase in the optical path, and the other is the -1 -13 change in the retardation per unit length. In the gen- Tilt 2, degrees eral case of a biaxial medium the dependence of retarda- -16 -2 tion and ellipticity on the tilt angle is quite complicated -18 and for the analysis it is important to find the principal axes of the index ellipsoid, see Fig.2. To first approxi- -3 mation one in-plane principal axis, x or y, with the as- -3 -2 -1 0 1 2 3 sumption to be justified in a moment that z is normal Tilt 1, degrees to the transparency, can be found easily by determining tilt direction for which the change in retardation for a ◦ small tilt angle (±3 ) is the largest. From the depen- FIG. 9: The dependence of ellipticity in degrees on the tilt in dence of retardation on the tilt angle (Eq.6) for this tilt two directions (first around the vertical axis [Tilt 1] and then direction we estimated that nz − nx was about 0.2, much around the horizontal axis [Tilt 2]) for 3M black and white larger than nx − ny ≈ 0.06 obtained from the wavelength transparency. The ellipticity was maximized by rotating the measurements at normal incidence (Fig.8) for the trans- transparency around its normal. parency of this type. Such small retardation for normal 5

is known, then from the fitting for tilt is sufficient to ob- tain nz − ny or nz − nx. If we neglect small effect due to -5 path elongation, then nz − nx ≃ 0.23, close to our initial -10 estimate. Different transparencies have slightly different proper- -15 ties, although qualitatively they are similar, including the orientation of the index ellipsoid. In Fig.9 we show tilt -20 measurements for a 3M black and white transparency. To -25 give an idea of variations, in Fig.10 we also present tilt measurements for the color transparency, with direction -30 of the tilt around the x axis. This graph also illustrates

Ellipticity, degrees the behavior of ellipticity in a larger tilt range and shows -35 that 45◦ ellipticity can be achieved by tilting at a rela- ◦ -40 tively small angle of 7 . The dependence in Fig.10 is quite symmetric in accordance to the previously drawn conclu- -45 sion that the z axis is orthogonal to the plane as depicted -10 -5 0 5 10 in Fig.2 on the left panel. We found using crossed polar- ◦ Tilt angle, degrees izers that the x axis makes a constant angle about 20 with respect to the edges of the transparency sheet across a large transparency area, although it varies for different samples. Using parabolic fits, we obtain nz − ny ≃ 0.18, FIG. 10: Dependence of ellipticity on the tilt angle t of a color − transparency sheet for the tilt around a principal axis of the again much larger than nx ny =0.06 in accordance with refractive index ellipsoid. Solid line is the fit based on Eqs. the z axis perpendicular alignment to the transparency (3) and (6). plane. We have qualitatively investigated the dependence of ellipticity of this transparency on the tilt angle for dif- ferent tilt direction, around x, y, and the xy bisector. incidence means that the z axis of the index ellipsoid has The results are consistent with theoretical predictions: a relatively small projection in the transparency plane for the tilt around x - the dependence is a downward and hence our zero-approximation assignment of refrac- parabola, around y - upward, and for the tilt around xy tive index subscripts is justified. After finding the z axis the dependence is very slow (see Fig.9). approximately, we can determine the principal refractive To complete the analysis it was also necessary to know indices more accurately by investigating the retardation the average refractive indices. We have measured them for small deviations of the transparency film normal from using Snell’s law by sending a 633-nm light beam at the the z direction. If na and nb are defined as the semi- edges of a stack of transparencies and by tracing inci- axes of the ellipse obtained by the intersection of the dent and refracted rays. Several measurements for dif- plane normal to the k vector with the index ellipsoid ferent incidence angles resulted in the average value n = (Refs.[11, 13]), the retardation is proportional to na −nb. 1.54 ± 0.05. For comparison similar values at the same For tilt of the ellipsoid around the x axis (Ref.[11], section wavelength are obtained in Ref.[8] for biaxially-stretched 14.3.3), poly-ethyleneterephthalate (PET) films, n = 1.59, and in Ref.[14] for polyester biaxial stretched films, n =1.66, − − − − 2 na nb = nx ny (nz ny) sin(tx/n) (6) while measurements in Ref.[6] for 3M PP2500 films gave different result, n ≈ 1.83. Such a high refractive index is (t is the tilt angle, n is the average refractive index), and uncommon for transparent organic materials which con- the retardation decreases with tilt if nx

100

III. APPLICATIONS 90

80 The simplest method to construct a desired wave plate for a given wavelength is to use a light source of this 70 wavelength and to choose an appropriate area in a trans- 60 parency sheet, monitoring retardation with either a com- 50 mercial polarimeter or some other optical polarimetric ar- Retardation, degrees rangement. On the other hand, tuning can be achieved 40 by tilting. Wavelengths for which transparency retarders 30 can be constructed are in the wide range from visible to infrared. In particular, we tested operation at 633 0 20 40 60 80 100 nm with a helium-neon laser, at 790-930 nm with a Angle b/w optical axes, degrees Ti:Sapphire laser, and in the range of 300-900 nm with a spectrometer, as well as using a computer monitor with pure color settings. FIG. 11: Dependence of retardation of a combination of two In addition, we have also investigated the possibility wave plates on the angle between their symmetry axes. Points of superposition of two transparency sheets for achieving are experimental data for black&white 3M transparencies. desirable retardation. Since the eigenvectors of the Jones They allow accurate fit with analytical equation Eq. (8), a matrix of a combination of two waveplates are in gen- solid line, with φ2 = 72.3 degrees and φ1 = 38.7 degrees. Measurements for individual wave plates gave φ2 = 74.6 de- eral complex, the system has elliptical-polarization eigen- grees and φ1 = 37 degrees, in agreement. Small deviation is states and is not equivalent to a single waveplate which due to spatial variations of retardation when transparency is has eigenstates of linear polarization, except in some de- rotated. Analytical equation for retardations φ1 = 23 degrees generate cases. In other words, the two-transparency and φ2 = 57 degrees, another solid line, is also tested by com- combination can be viewed as a waveplate combined with parison with numerical simulations based on the Jones-matrix a gyrotropic plate, whose effect is just a rotation of the formalism, dotted line. polarization ellipse. If the purpose of the waveplate is, for example, to produce a polarization state with a given ellipticity starting with linearly polarized light, then a the optical axis; so a single transparency behaves as a combination waveplate can generally be used for this. true birefringent wave plate. From the Jones-matrix formalism one finds that a two- It is interesting to note that when the angle between transparency combination produces the same output el- the fast axes of two transparency sheets is 90◦, the com- lipticity as a single waveplate, which has retardation φ bined retardation will be lower, meaning that the order between φ1 − φ2 and φ1 + φ2 depending on the angle is lower, and the sensitivity to variations in parameters, between the fast axes of the two wave plates θ12. For such as temperature and wavelength, will be also lowered practical purposes this dependence can be interpolated due to cancelation. This is a common technique for build- with sufficient accuracy with ing low-order wave plates and is described for example in 2 2 Ref.[3]. φ = (φ1 + φ2)cos θ12 + (φ1 − φ2) sin θ12. (8) Although the fast wavelength dependence of the retar- A comparison of numerical simulations with this inter- dation for a single sheet would limit the range of opera- polation equation for a typical case is shown in Fig.11. tion of a transparency wave plate to ∼10 nm (Fig.7), a To confirm this theory, we investigated experimentally combined plate can work in a much larger range. the behavior of two-sheet waveplates by rotating them One additional possibility is to combine a high-quality and by varying the angle between the axes of the two wave plate, which has some retardation offset due to constituent transparencies. First we tested that the el- operation at a non-nominal wavelength, with a two- lipticity of two transparencies under the common rota- transparency wave plate to compensate the offset. If we tion follows Eq. (3). Second, from the fits by Eq. (3) we choose a 90-degree relative orientation in the combined found retardation for different relative orientations of the transparency wave plate, the temperature stability of the transparencies. Figure 11 summarizes the analysis and resulting wave plate will not be significantly reduced. shows that the interpolation equation (Eq. (8)) agrees Similarly, although the temperature dependence of the with the experimental data. In principle, even a single retardation of a transparency is substantial, as shown in transparency sheet may have gyrotropy. Experimentally, Fig.12, the combined wave plate has a much weaker tem- we found that a single sheet of transparency does not perature dependence. The same is also true for the more change polarization if input polarization is aligned with conventional composite low-order optical, for example, 7

1.4 in agreement with the common values for such materials, Ref.[15]. Apart from being a parasitic effect, the temper-

1.3 ature dependence can be used in some applications. For example, it is possible to construct multi-channel ther- mometers based on monitoring retardation using broad 1.2 light beams. Another application is the conversion of UV or IR images into visible ones by measuring retardation across the transparency (the local retardation is affected 1.1 by heating with the absorbed UV or IR light). For small sizes, the transparency retardation is uniform and imag- Retardation, rad 1.0 ing can be implemented directly, but for large sizes non- uniformity can be taken into account with a computer. It is interesting to note that using sensitive polarimeters −9 0.9 with fundamental photon shot noise on the order of 10 rad/Hz1/2, it should be possible to measure temperature −7 20 25 30 35 40 45 changes as small as 10 Kelvin with a single sheet and Temperature, C even better using a stack of transparencies. Other pa- rameters such as ambient pressure or electric field (Kerr effect), can be measured with high sensitivity. Electric FIG. 12: Dependence of retardation of a color transparency field imaging can be also of interest. on temperature at 633 nm. Open circles show a set of data in The wavelength dependence of the retardation (Fig.7) one run, and the squares the second run after heating-cooling can be also used for sensitive wavelength measure- cycle to check reproducibility of the data. The dependence ments. For example, using again shot-noise limit of order fits well with a line, φ = 0.50(1) + 0.0187(4)T , where T is the 10−9 rad/Hz1/2 in angular resolution, we estimate that temperature. Large offset ∼ 20π rad which can be estimated −8 changes in wavelength on the order of 3 × 10 nm can from refractive index for the color transparency (Fig.8) is not ∼ included. be detected for 1 second of integration time. In conclusion, we have analyzed polarization proper- ties of overhead transparencies and found that they can crystalline quartz, plates [3]. This dependence is linear be used as inexpensive wave plates in a broad range of in a wide range, which can be explained by linear ther- wavelengths. mal expansion. Assuming that the expansion is isotropic and ∆l/l = α∆T , where α is the coefficient of linear expansion and ∆l/l is the relative change in the length IV. ACKNOWLEDGMENT for temperature increment ∆T , the density is reduced by 1/(1 + α∆T )3 and the total retardation is reduced by This work is supported by DOD MURI grant # N- 1/(1 + α∆T )2 ≈ 1 − 2α∆T , where we have taken into 00014-05-1-0406. The authors are grateful to A. Cingoz account the change in the plate’s length. From the total and D. English for helping with measurements of wave- retardation of the transparency of the order 20π rad (for length dependence, and to A. Sushkov, A. Park, J. Hig- nx − ny = 0.05 − 0.06, [Fig.8]), we can estimate the co- bie, T. Karaulanov, and M. Ledbetter for useful com- efficient of linear expansion to be α ≈ 1.5 × 10−4 K−1, ments on the manuscript.

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Chemical Constants (Longman, United Kingdom, 1995), 16th ed.