A Study of Reflection and Transmission of Birefringent Retarders

Home , Fresnel equations, Plane of incidence

JUST, Vol. IV, No. 1, 2016 Trent University

A study of reﬂection and transmission of birefringent retarders

James Godfrey

Keywords Optics — Computational Physics — Theoretical Physics Champlain College

1. Interference in Rotating Waveplates The transmitted intensity of along each axis must be considered separately, since equation (1-1) dictates that the re- 1.1 Research Goal flectance of the fast- and slow-axis components of the incident The objective of this project was to obtain a realistic theoreti- light will be different. The reflectance along the fast axis (Ro) cal prediction of the transmission curve for linearly-polarized and slow axis (Rs) are may be expressed: light incident normal on a [birefringent] retarder that behaves 2 as a Fabry-Perot etalon. The birefringent retarder considered n f − ni is a slab of birefringent crystal cut with the optic axis in the Ro = n f + ni face of the slab, and with parallel faces, and the waveplate has 2 no coating of any kind. ns − ni Rs = (1-2) Results of this project hoped to possibly explain the results ns + ni of work done by a previous student, Nolan Woodley, for his Physics project course in the 2013/14 academic year. In his Assuming the sides of the etalon are essentially paral- project, Woodley took many polarimeter scans which had a lel, the retarder may now be treated as a low-finesse (low- roughly sinusoidal shape [as they should], but adjacent peaks reflectance, R << 1) Fabry-Perot etalon. Their coefficients of of different heights. These differing heights were completely finesse (Fo and Fs, respectively) will also be different: unexplained, and not predicted by theory. 4Ro Fo = 2 1.2 Methodology (1 − Ro) The projection of the normally-incident P-polarized (plane- 4Rs polarized) light’s E-vector onto the optic axis of the retarder Fs = (1-3) (1 − R )2 is proportional to cos(β), where β is the angle between the s E-vector and the optic axis. The projection onto the other axis The transmission of a Fabry-Perot etalon is typically given of the retarder (the axis perpendicular to the optic axis in the by the Airy Function: face of the waveplate) is proportional to sin(β). The optic axis is called the fast axis if its refractive index is lower than 1 the refractive index of the perpendicular axis, and it is called T = (1-4) 1 + F · sin2 δ the slow axis if the refractive index is higher. Without loss 2 of generality, it may be assumed that the optic axis is the fast axis. where F is the etalon’s coefficient of finesse and δ is the round These axes each have associated characteristic refractive trip phase shift of the wave; the accumulated phase associated indices, and thus P-polarized light oscillating along one axis with traversing the cavity back and forth once. would be transmitted and reflected in different proportions, as The round trip phase shift of each component at normal sufficiently described by the Fresnel equations, assuming the incidence is expressed: air-retarder interfaces are completely lossless (no absorption). The reflectance of a dielectric at normal incidence is given as: δ = 2k · n · d (1-5)

where k is the vacuum wavenumber of the incident light, n is n − n 2 R = t i (1-1) the index of refraction of the material through which the light nt + ni wave is propagating and d is the thickness of the etalon. where nt is the refractive index of the transmitting medium Since the etalon being considered is a waveplate (made and ni is the refractive index of the incident medium. of birefringent material), it has two indices of refraction, and A study of reflection and transmission of birefringent retarders — 2/5 is manufactured to have a specific length as to impart a spe- transmission functions the fast- and slow-axis components: cific desired phase shift upon propagating the length of the 1 waveplate. We shall define the waveplate order q as: To = 2 2π·n f ·q 1 + Fo · sin |ns−n f | |δ − δ | 1 q ≡ o s · (1-6) 1 2 2π Ts = (1-8) 2 2π·ns·q 1 + Fs · sin |ns−n f | where δo and δs are the round trip phase shift for light waves whose electric fields are oscillating in the direction of the fast Considering that the fast- and slow-axis projections of and slow axes, respectively. the P-polarized light’s electric field vector are proportional The thickness of a waveplate d may then be given by the to cos[β] and sin[β], respectively, intensity of the transmit- 2 following expression: ted fast- and slow-axes components scales with cos [β] and sin2[β]. The net transmission of the waveplate may then be q · λ simply expressed in terms of equations (1-8a) and (1-8b): d = 0 . (1-7) |ns − n f | 2 2 Tnet = cos (β) · To + sin (β) · Ts 1 1 where λ is the vacuum wavelength of the incident light. = (Ts + To) + (Ts − To)cos(2β) (1-9) 0 2 2 Substituting equation (1.7) into (1.5) for each component, and then substituting that and equation (1.3) into (1.4), we get the Substituting equation (1-8) into (1-9) yields:

 !!−1 !!−1 1 2 2π · ns · q 2 2π · n f · q Tnet =  1 + Fs · sin + 1 + Fo · sin + 2 ns − n f ns − n f  !!−1 !!−1 1 2 2π · ns · q 2 2π · n f · q  1 + Fs · sin − 1 + Fo · sin cos(2β) (1-10) 2 ns − n f ns − n f

Equations (1-9) and (1-10) were implemented in Mathe- transmittance ∆T is twice the coefficient of the cos[2β] term matica, and used to generate plots of net transmittance Tnet in equations (1-9) and (1-10). The trough-to-peak amplitude against waveplate rotation angle β. The parameters chosen of the transmission curve is given as: were those of calcite for yellow incident light (λ0 = 589 nm); n f = 1.4864 and ns = 1.6584. Calcite was chosen since it has −1 2 q a fairly high birefringence (|n −n | > 0.1), unlike other com- ∆T = 1 + Fo · sin 2π · · n f f s |n − n | monly used materials such as quartz, so that any transmission s f −1 properties related to the birefringence would be accentuated. 2 q − 1 + Fs · sin 2π · · ns (1-11) |ns − n f | 1.3 Notable Findings and Possible Next Steps where: When modelling a waveplate as a [lossless] birefringent Fabry- Perot etalon, the transmission curve (as a function of wave- • n f and ns are the characteristic indices of refraction of plate rotation angle, β) is a sinusoid of period π (see Fig. 1). the waveplate’s slow and fast axes, respectively The β-dependence, and particularly the fact that the transmission curve is π-periodic in β, arises from reflection symmetry • ni is the index of refraction of the incident medium, Fo is of the electric field vector’s projections on the fast and slow 2 the fast axis’ coefficient of finesse, Fo ≡ (n f –ni) /(n f + axes of the waveplate; the components along each axis are 2 ni) of equal magnitude whether it forms an angle β with the fast 2 axis to the left or right of a particular axis. • Fs is the slow axis’ coefficient of finesse, Fs ≡ (ns–ni) /(ns + 2 Though originally only quarter-wave plates were of inter- ni) , est, retarders of arbitrary thickness were shown to possess a similar transmission curve for [almost] all thicknesses. The • k is the vacuum wavenumber of the incident light, amplitude of the sinusoid varies extremely with waveplate order q, and thus waveplate thickness. The maximum change in • λ is the vacuum wavelength of the incident light A study of reflection and transmission of birefringent retarders — 3/5

Figure 1. Two plots of simulated transmission data for two different thicknesses of calcite waveplate rotated through a full cycle of β. The ﬁgure on the left is a waveplate with q = 0.25; the two orthogonal components of the incident P-polarized light entering in-phase emerge with a phase difference of π/2(λ/4). Similarly, the second plot corresponds to a calcite plate of appropriate thickness such that the imparted phase difference is 9π/2.

• q is the waveplate mode number (0.25, 1.25, 2.25, ... the experimentally obtained data. correspond to quarter-wave plates; 0.5, 1.5, 2.5, ... correspond to half-wave plates; 1, 2, 3, ... correspond to 1.4 Relevant Literature full-wave plates). • Optics, 4th Ed. by E. Hecht • Introduction to Optics, 3rd Ed. by F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti

2. Transmission and Reflection from Birefringent Crystals 2.1 Research Goal The goal of this project was to determine how Brewster’s angle of a birefringent waveplate (in a waveplate the optic axis is necessarily in the face of the optic) changes with the orientation of the waveplate. The light considered was not necessarily polarized in the plane of incidence, but might as Figure 2. A plot of equation (1-11) over the range 0 ≥ q ≥ 1. Negative values of ∆T correspond two transmission curves that start at a well have been, as the TM-mode is solely responsible for the minimum, such as the first plot in Fig. 1-1. ∆T can be seen to range Brewster’s angle phenomenon. It should be noted that the E- between roughly -0.15 and 0.22, and it seems to have a periodic ray and O-ray necessarily feel a different index of refraction, envelope. and thus should have different Brewster’s angles, but the same angle of reflection. This theoretical groundwork is at the stage where it could be experimentally tested. A simple [but somewhat expensive] 2.2 Methodology experimental setup could use an uncoated known-order (q) The focus of this project was definitely determining Brewster’s calcite waveplate aligned normal with a highly P-polarized e o angle of the E-ray, θp, as Brewster’s angle of the O-ray, θp , is collimated light-source of known power and a calibrated pho- easily determined using the conventional formula: todiode or power meter. The retarder should be mounted on a rotating-mount controlled by a step motor, and rotated through o no a full cycle of β while stopping to take power readings with θp = arctan (2-1) ni the photodiode every 3.6o (or some other similarly small rotation angle). Beginning the rotation of the fast axis aligned where no is the characteristic ordinary refractive index of with the fast axis of the calcite retarder, and scanning in small the birefringent material and ni is the refractive index of the steps of β, one should be able to obtain a data set that, when incident medium. e normalized to the incident power and plotted, is comparable However, determining θp is not so simple, as the boundary to those in Figure 1-1. Knowing the indices of refraction of conditions at the air-crystal interface give rise to a different the calcite waveplate for the wavelength of the incident light set of Fresnel equations than those of the O-ray. This is due and the waveplate order q, a prediction of the intensity scan to the index of refraction of the E-ray, neo, actually being a o can be made using equation (1-10), and can be compared to function of the transmitted angle θe , unlike no. The version of A study of reflection and transmission of birefringent retarders — 4/5

Snel’s Law of Refraction1 that must be satisﬁed by the E-ray is given by:

o o ni sin(θi) = neo(θe )sin(θe ) 1 2 o 1 1 1 2 2 o neo(θe ) = 2 + 2 − 2 cos (φ)sin (θe ) (ne) (no) (ne) (2-2) where no and ne are the characteristic ordinary and extraordi- nary refractive indices of the birefringent material and φ is the waveplate rotation angle; the angle formed between the Figure 3. A plot of the amplitude coefficients against θi for a quartz plane of incidence and the optic axis of the waveplate (this waveplate (no = 1.543, ne = 1.552). Dotted lines represent the angle is analogous to from section 1). amplitude reflection coefficients and solid thick lines represent β amplitude transmission coefficients. The blue line is the height on the The symbol φ was used instead of β to simplify program- graph at which an interface coefficient has the value of 0. The black, ming; Yang’s article used φ throughout, and thus it was easier dotted line (TM-mode reflection coefficient) can be seen vary such e e o o to compare the equations in my program to those in the ar- that [1 ≥ (R|| + T|| ) ≥ −1] as θi ranges for 0 to 90 , and TE-mode coefficients behave such that Re + T e = 1 for all . The point at which ticle. In his article, Yang describes the E-ray’s transmitted ⊥ ⊥ θi Re crosses through 0 on the vertical axis corresponds to θ e ≈ 57.5o field, reflected field, and incident field. From these the TE- k p for φ = 20o. (denoted ⊥) and TM-mode (denoted k) amplitude reflection e e coefficients (R⊥ and Rk, respectively) and amplitude transmis- e e sion coefficients (T⊥ and Tk , respectively) may be obtained. should also be set up in the plane of incidence, normal to For each mode, each reflection and transmission coefficient is and in the path of the reflected beam. Rotating the waveplate simply expressed as the ratio of reflected or transmitted field should change the intensity of the reflected beam without e amplitude over the incident field amplitude. Finding θp at a changing its path. Since the incident light is polarized in e given φ is now simply a matter of finding the θi-root of Rk. the plane of incidence, the intensity of the reflected beam is a sum of the reflected intensities of the E- and O-ray TM- 2.3 Notable Findings and Suggested Next Steps mode components. By rotating the waveplate in small φ- Using the conventional TM-mode amplitude reflection coef- increments, taking power measurements of the reflected beam, ficient for the O-ray, and the modified TM-mode reflection and normalizing the power measurement to the incident power, coefficient derived from Yang’s equations, it is now possible an experimental data set comparable to the plot mentioned in o e to determine the two Brewster angles θp and θp. Figure 2-1 the above paragraph could be taken. demonstrates that the theory in Yang’s article agrees well with previously-established theory; the plot shown below agrees 2.4 Relevant Literature with Figure 23-3 in Introduction to Optics, 3rd Ed. by Pe- • Introduction to Optics, 3rd Ed. by F. L. Pedrotti, L. S. drotti. Pedrotti, and L. M. Pedrotti The program used to generate Figure 2-1 could be mod- o e • W.Q. Zhang (2000): New phase shift formulas and ified to generate a plot of R and R at a constant θi while k k stability of waveplate in oblique incident beam the φ ranges from 0o to 90o, and analyzed to determine the estimated total intensity of the reflected ray at every φ. It may • T. Yang (2006): An improved description of Jones vec- be beneficial to use an uncoated waveplate made of a material tors of the electric fields of incident and refracted rays o e with a high birefringence (such as calcite) so that θp and θp in a birefringent plate are noticeably different. A fairly simple experimental setup could be constructed to Please note that Yang’s article contained several errors: obtain data to compare to the theoretical prediction mentioned above. A suggested setup would include a waveplate on a • in equation (14), the third component in the numera- rotating mount (preferably one controlled by a step motor), a tor should be positive (although it did not affect any source of P-polarized light (of known power/intensity) such calculated results) as a diode laser, and a calibrated photodiode or power meter. e The P-polarized light should be incident on the waveplate at • in equation (21), Dtkshould be squared under the square an oblique angle, and polarized in the plane of incidence. If root necessary, use a linear polarizer, although this may facilitate • in equations (24c) and (24d), remove/scratch out the some measurement of the intensity of the incident light after it first closing round bracket [)] in both, or add an opening passes through the polarizer. The photodiode or power meter e round bracket [(] between each cos and θeo; At⊥ is not 1 yes, ‘Snel’ is actually the correct spelling divided by neo cos(θi) A study of reflection and transmission of birefringent retarders — 5/5

3. Serendipitous Learning A beautiful part of research is that, in attempting to answer a single question, you end up answering an incredible number of questions on the way to answering the first one. This section is dedicated to some of the wonderful unexpected [practical] learning that during the summer. When doing polarimetry, it is important to consider the type of photodiode being used; between different silicon pho- todiodes the results of a rotating waveplate scan were vastly different (i.e. different percentage change, and completely different characteristic shape of intensity profile). Perhaps some PDs have different sensitivities to certain polarizations of light; the TSL251 had large percentage changes (≈ 5 − 12%) in its scans, and had [roughly] sinusoidal intensity profile of period π/2, whereas the OPT101A had a non-periodic shape and small percentage changes (≈ 0.5 − 2%). The TSL251 also seemed to hit its local minima when circularly-polarized light was incident (at β =45o, 135o, 225o, 315o), and maxima when linearly-polarized light was incident (at β = 0o, 90o, 180o, 270o, 360o). Another fun fact, the output power of a diode laser operat- ing at near room temperature can be very sensitive to changes (a) in the temperature of the gain medium; a 1oC increase in the temperature of the gain medium increased the laser’s output power as much 2 percent. Although somewhat puzzling, it seemed to be easier to keep the diode laser at a constant temperature at a temperature of about 17.0oC, as opposed to about 18.6oC, which was the initial set point of the laser’s cooling device. Intuitively, one might think it would be easier to maintain a temperature closer to room temperature. When taking any measurement with a photodiode, devise an experiment where you never have to move the photodiode; it will make your life significantly less difficult 100% of the time. In an experiment where the photodiode is moved manually, it is an extremely onerous task to keep all of the components of the apparatus aligned for every intensity measurement. Finally, arguably the most important thing that was learned (b) this summer: Wolfram Mathematica is a powerful, versatile, Figure 4. (a) A plot of transmitted intensity (arbitrary units) against and simply fantastic program, and everybody should use it for waveplate rotation angle β; intensity measurements taken with everything. Perhaps the previous statement is an exaggeration, OPT101A photodiode. The difference between the maximum and minimum reading taken is 1.10%. The 3 coloured sets are 3 individual but it really is a fantastic program. From rendering beau- scans, and the black set is the mean of the 3 individual scans. The tiful high-resolution videos of time-evolving electric fields scans seems to be non-sinusoidal in shape. (b) A plot of transmitted from arrays of dipole oscillators to preparing simple plots, intensity (arbitrary units) against waveplate rotation angle β; intensity Mathematica is a simple and effective choice of programming measurements taken with TSL251 photodiode. The difference between the maximum and minimum reading taken is 5.39%, much language. It’s got all of the numerical tools of Matlab, all of larger than the OPT101A scan. Scans seem to roughly follow a the analytical solving techniques of Maple, and even has built- sinusoidal shape of period π/2. The shapes of the two plots are in connectivity with Wolfram Alpha’s massive database. (And apparently fundamentally different. no, the research in this report was not sponsored by Wolfram in any way, although Mathematica was an indispensable tool in all research mentioned in this report.)