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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 1.1 Research Goal The objective of this project was to obtain a realistic theoretical prediction of the transmission curve for linearly-polarized light incident normal on a [birefringent] retarder that behaves as a Fabry-Perot etalon. The birefringent retarder considered is a slab of birefringent crystal cut with the optic axis in the face of the slab, and with parallel faces, and the waveplate has no coating of any kind. Results of this project hoped to possibly explain the results of work done by a previous student, Nolan Woodley, for his Physics project course in the 2013/14 academic year. In his project, Woodley took many polarimeter scans which had a roughly sinusoidal shape [as they should], but adjacent peaks of different heights. These differing heights were completely unexplained, and not predicted by theory. 1.2 Methodology The projection of the normally-incident P-polarized (planepolarized) light’s E-vector onto the optic axis of the retarder is proportional to cos(β ), where β is the angle between the E-vector and the optic axis. The projection onto the other axis of the retarder (the axis perpendicular to the optic axis in the face of the waveplate) is proportional to sin(β ). The optic axis is called the fast axis if its refractive index is lower than the refractive index of the perpendicular axis, and it is called the slow axis if the refractive index is higher. Without loss of generality, it may be assumed that the optic axis is the fast axis. These axes each have associated characteristic refractive indices, and thus P-polarized light oscillating along one axis would be transmitted and reflected in different proportions, as sufficiently described by the Fresnel equations, assuming the air-retarder interfaces are completely lossless (no absorption). The reflectance of a dielectric at normal incidence is given as: R= nt − ni nt + ni 2 (1-1) where nt is the refractive index of the transmitting medium and ni is the refractive index of the incident medium. The transmitted intensity of along each axis must be considered separately, since equation (1-1) dictates that the reflectance of the fast- and slow-axis components of the incident light will be different. The reflectance along the fast axis (Ro ) and slow axis (Rs ) are may be expressed: n f − ni n f + ni 2 ns − ni ns + ni 2 Ro = Rs = (1-2) Assuming the sides of the etalon are essentially parallel, the retarder may now be treated as a low-finesse (lowreflectance, R << 1) Fabry-Perot etalon. Their coefficients of finesse (Fo and Fs , respectively) will also be different: Fo = Fs = 4Ro (1 − Ro )2 4Rs (1-3) (1 − Rs )2 The transmission of a Fabry-Perot etalon is typically given by the Airy Function: T= 1 1 + F · sin2 (1-4) δ 2 where F is the etalon’s coefficient of finesse and δ is the round trip phase shift of the wave; the accumulated phase associated with traversing the cavity back and forth once. The round trip phase shift of each component at normal incidence is expressed: δ = 2k · n · d (1-5) where k is the vacuum wavenumber of the incident light, n is the index of refraction of the material through which the light wave is propagating and d is the thickness of the etalon. Since the etalon being considered is a waveplate (made 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 specific desired phase shift upon propagating the length of the waveplate. We shall define the waveplate order q as: q≡ |δ o − δs | 1 · 2 2π (1-6) where δo and δs are the round trip phase shift for light waves whose electric fields are oscillating in the direction of the fast and slow axes, respectively. The thickness of a waveplate d may then be given by the following expression: d= q · λ0 . |ns − n f | Tnet To = Ts = 1 1 + Fo · sin2 2π·n f ·q |ns −n f | 2π·ns ·q |ns −n f | 1 1 + Fs · sin2 (1-8) Considering that the fast- and slow-axis projections of the P-polarized light’s electric field vector are proportional to cos[β ] and sin[β ], respectively, intensity of the transmitted fast- and slow-axes components scales with cos2 [β ] and sin2 [β ]. The net transmission of the waveplate may then be simply expressed in terms of equations (1-8a) and (1-8b): (1-7) where λ 0 is the vacuum wavelength of the incident light. Substituting equation (1.7) into (1.5) for each component, and then substituting that and equation (1.3) into (1.4), we get the 1 = 1 + Fs · sin2 2 transmission functions the fast- and slow-axis components: Tnet = cos2 (β ) · To + sin2 (β ) · Ts 1 1 = (Ts + To ) + (Ts − To )cos(2β ) 2 2 (1-9) Substituting equation (1-8) into (1-9) yields: !!−1 !!−1 2π · n · q 2π · ns · q f + + 1 + Fo · sin2 ns − n f ns − n f !!−1 !!−1 2π · n · q 1 2π · n · q f s cos (2β ) (1-10) 1 + Fs · sin2 − 1 + Fo · sin2 2 ns − n f ns − n f Equations (1-9) and (1-10) were implemented in Mathematica, and used to generate plots of net transmittance Tnet against waveplate rotation angle β . The parameters chosen 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 a fairly high birefringence (|n f − ns | > 0.1), unlike other commonly used materials such as quartz, so that any transmission properties related to the birefringence would be accentuated. 1.3 Notable Findings and Possible Next Steps When modelling a waveplate as a [lossless] birefringent FabryPerot etalon, the transmission curve (as a function of waveplate rotation angle, β ) is a sinusoid of period π (see Fig. 1). The β -dependence, and particularly the fact that the transmission curve is π-periodic in β , arises from reflection symmetry of the electric field vector’s projections on the fast and slow axes of the waveplate; the components along each axis are of equal magnitude whether it forms an angle β with the fast axis to the left or right of a particular axis. Though originally only quarter-wave plates were of interest, retarders of arbitrary thickness were shown to possess a similar transmission curve for [almost] all thicknesses. The amplitude of the sinusoid varies extremely with waveplate order q, and thus waveplate thickness. The maximum change in transmittance ∆T is twice the coefficient of the cos[2β ] term in equations (1-9) and (1-10). The trough-to-peak amplitude of the transmission curve is given as: 2 ∆T = 1 + Fo · sin 2π · q ·nf |ns − n f | −1 − 1 + Fs · sin2 2π · q · ns |ns − n f | −1 (1-11) where: • n f and ns are the characteristic indices of refraction of the waveplate’s slow and fast axes, respectively • ni is the index of refraction of the incident medium, Fo is the fast axis’ coefficient of finesse, Fo ≡ (n f –ni )2 /(n f + ni )2 • Fs is the slow axis’ coefficient of finesse, Fs ≡ (ns –ni )2 /(ns + ni )2 , • k is the vacuum wavenumber of the incident light, • 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, ... correspond to quarter-wave plates; 0.5, 1.5, 2.5, ... correspond to half-wave plates; 1, 2, 3, ... correspond to full-wave plates). the experimentally obtained data. 1.4 Relevant Literature • 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 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 minimum, such as the first plot in Fig. 1-1. ∆T can be seen to range between roughly -0.15 and 0.22, and it seems to have a periodic envelope. This theoretical groundwork is at the stage where it could be experimentally tested. A simple [but somewhat expensive] experimental setup could use an uncoated known-order (q) calcite waveplate aligned normal with a highly P-polarized collimated light-source of known power and a calibrated photodiode or power meter. The retarder should be mounted on a rotating-mount controlled by a step motor, and rotated through a full cycle of β while stopping to take power readings with the photodiode every 3.6o (or some other similarly small rotation angle). Beginning the rotation of the fast axis aligned with the fast axis of the calcite retarder, and scanning in small steps of β , one should be able to obtain a data set that, when normalized to the incident power and plotted, is comparable to those in Figure 1-1. Knowing the indices of refraction of the calcite waveplate for the wavelength of the incident light and the waveplate order q, a prediction of the intensity scan can be made using equation (1-10), and can be compared to 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 well have been, as the TM-mode is solely responsible for the Brewster’s angle phenomenon. It should be noted that the Eray and O-ray necessarily feel a different index of refraction, and thus should have different Brewster’s angles, but the same angle of reflection. 2.2 Methodology The focus of this project was definitely determining Brewster’s angle of the E-ray, θ pe , as Brewster’s angle of the O-ray, θ po , is easily determined using the conventional formula: no o θ p = arctan (2-1) ni where no is the characteristic ordinary refractive index of the birefringent material and ni is the refractive index of the incident medium. However, determining θ pe is not so simple, as the boundary conditions at the air-crystal interface give rise to a different set of Fresnel equations than those of the O-ray. This is due to the index of refraction of the E-ray, neo , actually being a function of the transmitted angle θeo , 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: ni sin(θi ) = neo (θeo ) sin(θeo ) 1 2 1 1 1 2 2 o o + − cos (φ ) sin (θe ) neo (θe ) = 2 2 2 (ne ) (no ) (ne ) (2-2) where no and ne are the characteristic ordinary and extraordinary refractive indices of the birefringent material and φ is the waveplate rotation angle; the angle formed between the plane of incidence and the optic axis of the waveplate (this angle is analogous to β from section 1). The symbol φ was used instead of β to simplify programming; Yang’s article used φ throughout, and thus it was easier to compare the equations in my program to those in the article. In his article, Yang describes the E-ray’s transmitted field, reflected field, and incident field. From these the TE(denoted ⊥) and TM-mode (denoted k) amplitude reflection coefficients (Re⊥ and Rek , respectively) and amplitude transmission coefficients (T⊥e and Tke , respectively) may be obtained. For each mode, each reflection and transmission coefficient is simply expressed as the ratio of reflected or transmitted field amplitude over the incident field amplitude. Finding θ pe at a given φ is now simply a matter of finding the θi -root of Rek . 2.3 Notable Findings and Suggested Next Steps Using the conventional TM-mode amplitude reflection coefficient for the O-ray, and the modified TM-mode reflection coefficient derived from Yang’s equations, it is now possible to determine the two Brewster angles θ po and θ pe . Figure 2-1 demonstrates that the theory in Yang’s article agrees well with previously-established theory; the plot shown below agrees with Figure 23-3 in Introduction to Optics, 3rd Ed. by Pedrotti. The program used to generate Figure 2-1 could be modified to generate a plot of Rok and Rek at a constant θi while the φ ranges from 0o to 90o , and analyzed to determine the estimated total intensity of the reflected ray at every φ . It may be beneficial to use an uncoated waveplate made of a material with a high birefringence (such as calcite) so that θ po and θ pe are noticeably different. A fairly simple experimental setup could be constructed to obtain data to compare to the theoretical prediction mentioned above. A suggested setup would include a waveplate on a rotating mount (preferably one controlled by a step motor), a source of P-polarized light (of known power/intensity) such as a diode laser, and a calibrated photodiode or power meter. The P-polarized light should be incident on the waveplate at an oblique angle, and polarized in the plane of incidence. If necessary, use a linear polarizer, although this may facilitate some measurement of the intensity of the incident light after it passes through the polarizer. The photodiode or power meter 1 yes, ‘Snel’ is actually the correct spelling Figure 3. A plot of the amplitude coefficients against θi for a quartz waveplate (no = 1.543, ne = 1.552). Dotted lines represent the amplitude reflection coefficients and solid thick lines represent amplitude transmission coefficients. The blue line is the height on the graph at which an interface coefficient has the value of 0. The black, dotted line (TM-mode reflection coefficient) can be seen vary such that [1 ≥ (Re|| + T||e ) ≥ −1] as θi ranges for 0o to 90o , and TE-mode coefficients behave such that Re⊥ + T⊥e = 1 for all θi . The point at which Rek crosses through 0 on the vertical axis corresponds to θ pe ≈ 57.5o for φ = 20o . should also be set up in the plane of incidence, normal to and in the path of the reflected beam. Rotating the waveplate should change the intensity of the reflected beam without changing its path. Since the incident light is polarized in the plane of incidence, the intensity of the reflected beam is a sum of the reflected intensities of the E- and O-ray TMmode components. By rotating the waveplate in small φ increments, taking power measurements of the reflected beam, and normalizing the power measurement to the incident power, an experimental data set comparable to the plot mentioned in the above paragraph could be taken. 2.4 Relevant Literature • Introduction to Optics, 3rd Ed. by F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti • W.Q. Zhang (2000): New phase shift formulas and stability of waveplate in oblique incident beam • T. Yang (2006): An improved description of Jones vectors of the electric fields of incident and refracted rays in a birefringent plate Please note that Yang’s article contained several errors: • in equation (14), the third component in the numerator should be positive (although it did not affect any calculated results) e should be squared under the square • in equation (21), Dtk root • in equations (24c) and (24d), remove/scratch out the first closing round bracket [)] in both, or add an opening e is not round bracket [(] between each cos and θe o; At⊥ 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 photodiodes 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 operating at near room temperature can be very sensitive to changes in the temperature of the gain medium; a 1o C 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 17.0o C, as opposed to about 18.6o C, 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 this summer: Wolfram Mathematica is a powerful, versatile, and simply fantastic program, and everybody should use it for everything. Perhaps the previous statement is an exaggeration, but it really is a fantastic program. From rendering beautiful high-resolution videos of time-evolving electric fields from arrays of dipole oscillators to preparing simple plots, Mathematica is a simple and effective choice of programming language. It’s got all of the numerical tools of Matlab, all of the analytical solving techniques of Maple, and even has builtin connectivity with Wolfram Alpha’s massive database. (And 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.) (a) (b) Figure 4. (a) A plot of transmitted intensity (arbitrary units) against waveplate rotation angle β ; intensity measurements taken with OPT101A photodiode. The difference between the maximum and minimum reading taken is 1.10%. The 3 coloured sets are 3 individual scans, and the black set is the mean of the 3 individual scans. The scans seems to be non-sinusoidal in shape. (b) A plot of transmitted intensity (arbitrary units) against waveplate rotation angle β ; intensity measurements taken with TSL251 photodiode. The difference between the maximum and minimum reading taken is 5.39%, much larger than the OPT101A scan. Scans seem to roughly follow a sinusoidal shape of period π/2. The shapes of the two plots are apparently fundamentally different.