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 con- sidered 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(b), where b 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(b). 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 d 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 d 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: d = 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 2p·n f ·q 1 + Fo · sin jns−n f j jd − d j 1 q ≡ o s · (1-6) 1 2 2p Ts = (1-8) 2 2p·ns·q 1 + Fs · sin jns−n f j where do and ds 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[b] and sin[b], respectively, intensity of the transmit- 2 following expression: ted fast- and slow-axes components scales with cos [b] and sin2[b]. The net transmission of the waveplate may then be q · l simply expressed in terms of equations (1-8a) and (1-8b): d = 0 : (1-7) jns − n f j 2 2 Tnet = cos (b) · To + sin (b) · Ts 1 1 where l is the vacuum wavelength of the incident light. = (Ts + To) + (Ts − To)cos(2b) (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: 0 !!−1 !!−11 1 2 2p · ns · q 2 2p · n f · q Tnet = @ 1 + Fs · sin + 1 + Fo · sin A+ 2 ns − n f ns − n f 0 !!−1 !!−11 1 2 2p · ns · q 2 2p · n f · q @ 1 + Fs · sin − 1 + Fo · sin Acos(2b) (1-10) 2 ns − n f ns − n f Equations (1-9) and (1-10) were implemented in Mathe- transmittance DT is twice the coefficient of the cos[2b] 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 b. The parameters chosen of the transmission curve is given as: were those of calcite for yellow incident light (l0 = 589 nm); n f = 1.4864 and ns = 1.6584. Calcite was chosen since it has −1 2 q a fairly high birefringence (jn −n j > 0:1), unlike other com- DT = 1 + Fo · sin 2p · · n f f s jn − n j 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 2p · · ns (1-11) jns − n f j 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, b) is a sinusoid of period p (see Fig. 1). the waveplate’s slow and fast axes, respectively The b-dependence, and particularly the fact that the transmis- sion curve is p-periodic in b, 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 b 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 or- der q, and thus waveplate thickness. The maximum change in • l 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 b. 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 p=2(l=4). Similarly, the second plot corresponds to a calcite plate of appropriate thickness such that the imparted phase difference is 9p=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, ... cor- respond 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 DT 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. DT 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.