A study of reﬂection and transmission of birefringent

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

A study of reflection and transmission ofbirefringent retardersJames Godfrey

KeywordsOptics — Computational Physics — Theoretical Physics

Champlain College

1. Interference in Rotating Waveplates

1.1 Research GoalThe objective of this project was to obtain a realistic theoreti-cal prediction of the transmission curve for linearly-polarizedlight incident normal on a [birefringent] retarder that behavesas a Fabry-Perot etalon. The birefringent retarder consideredis a slab of birefringent crystal cut with the optic axis in theface of the slab, and with parallel faces, and the waveplate hasno coating of any kind.

Results of this project hoped to possibly explain the resultsof work done by a previous student, Nolan Woodley, for hisPhysics project course in the 2013/14 academic year. In hisproject, Woodley took many polarimeter scans which had aroughly sinusoidal shape [as they should], but adjacent peaksof different heights. These differing heights were completelyunexplained, and not predicted by theory.

1.2 MethodologyThe projection of the normally-incident P-polarized (plane-polarized) light’s E-vector onto the optic axis of the retarderis proportional to cos(β ), where β is the angle between theE-vector and the optic axis. The projection onto the other axisof the retarder (the axis perpendicular to the optic axis in theface of the waveplate) is proportional to sin(β ). The opticaxis is called the fast axis if its refractive index is lower thanthe refractive index of the perpendicular axis, and it is calledthe slow axis if the refractive index is higher. Without lossof generality, it may be assumed that the optic axis is the fastaxis.

These axes each have associated characteristic refractiveindices, and thus P-polarized light oscillating along one axiswould be transmitted and reflected in different proportions, assufficiently described by the Fresnel equations, assuming theair-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 mediumand ni is the refractive index of the incident medium.

The transmitted intensity of along each axis must be con-sidered separately, since equation (1-1) dictates that the re-flectance of the fast- and slow-axis components of the incidentlight will be different. The reflectance along the fast axis (Ro)and slow axis (Rs) are may be expressed:

Ro =

(n f −ni

n f +ni

)2

Rs =

(ns−ni

ns +ni

)2

(1-2)

Assuming the sides of the etalon are essentially paral-lel, the retarder may now be treated as a low-finesse (low-reflectance, R << 1) Fabry-Perot etalon. Their coefficients offinesse (Fo and Fs, respectively) will also be different:

Fo =4Ro

(1−Ro)2

Fs =4Rs

(1−Rs)2 (1-3)

The transmission of a Fabry-Perot etalon is typically givenby the Airy Function:

T =1

1+F · sin2(

δ

2

) (1-4)

where F is the etalon’s coefficient of finesse and δ is the roundtrip phase shift of the wave; the accumulated phase associatedwith traversing the cavity back and forth once.

The round trip phase shift of each component at normalincidence is expressed:

δ = 2k ·n ·d (1-5)

where k is the vacuum wavenumber of the incident light, n isthe index of refraction of the material through which the lightwave is propagating and d is the thickness of the etalon.

Since the etalon being considered is a waveplate (madeof 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-cific desired phase shift upon propagating the length of thewaveplate. We shall define the waveplate order q as:

q≡|δ o−δs|

2· 1

2π(1-6)

where δo and δs are the round trip phase shift for light waveswhose electric fields are oscillating in the direction of the fastand slow axes, respectively.

The thickness of a waveplate d may then be given by thefollowing expression:

d =q ·λ0

|ns−n f |. (1-7)

where λ 0 is the vacuum wavelength of the incident light.Substituting equation (1.7) into (1.5) for each component, andthen substituting that and equation (1.3) into (1.4), we get the

transmission functions the fast- and slow-axis components:

To =1

1+Fo · sin2(

2π·n f ·q|ns−n f |

)Ts =

1

1+Fs · sin2(

2π·ns·q|ns−n f |

) (1-8)

Considering that the fast- and slow-axis projections ofthe P-polarized light’s electric field vector are proportionalto cos[β ] and sin[β ], respectively, intensity of the transmit-ted fast- and slow-axes components scales with cos2[β ] andsin2[β ]. The net transmission of the waveplate may then besimply expressed in terms of equations (1-8a) and (1-8b):

Tnet = cos2(β ) ·To + sin2(β ) ·Ts

=12(Ts +To)+

12(Ts−To)cos(2β ) (1-9)

Substituting equation (1-8) into (1-9) yields:

Tnet =12

(1+Fs · sin2

(2π ·ns ·q∣∣ns−n f

∣∣))−1

+

(1+Fo · sin2

(2π ·n f ·q∣∣ns−n f

∣∣))−1

+

12

(1+Fs · sin2

(2π ·ns ·q∣∣ns−n f

∣∣))−1

−

(1+Fo · sin2

(2π ·n f ·q∣∣ns−n f

∣∣))−1

cos(2β ) (1-10)

Equations (1-9) and (1-10) were implemented in Mathe-matica, and used to generate plots of net transmittance Tnetagainst waveplate rotation angle β . The parameters chosenwere those of calcite for yellow incident light (λ0 = 589 nm);n f = 1.4864 and ns = 1.6584. Calcite was chosen since it hasa fairly high birefringence (|n f −ns|> 0.1), unlike other com-monly used materials such as quartz, so that any transmissionproperties related to the birefringence would be accentuated.

1.3 Notable Findings and Possible Next StepsWhen modelling a waveplate as a [lossless] birefringent Fabry-Perot etalon, the transmission curve (as a function of wave-plate rotation angle, β ) is a sinusoid of period π (see Fig. 1).The β -dependence, and particularly the fact that the transmis-sion curve is π-periodic in β , arises from reflection symmetryof the electric field vector’s projections on the fast and slowaxes of the waveplate; the components along each axis areof equal magnitude whether it forms an angle β with the fastaxis to the left or right of a particular axis.

Though originally only quarter-wave plates were of inter-est, retarders of arbitrary thickness were shown to possess asimilar transmission curve for [almost] all thicknesses. Theamplitude of the sinusoid varies extremely with waveplate or-der q, and thus waveplate thickness. The maximum change in

transmittance ∆T is twice the coefficient of the cos[2β ] termin equations (1-9) and (1-10). The trough-to-peak amplitudeof the transmission curve is given as:

∆T =

(1+Fo · sin2

(2π · q|ns−n f |

·n f

))−1

−(

1+Fs · sin2(

2π · q|ns−n f |

·ns

) )−1

(1-11)

where:

• n f and ns are the characteristic indices of refraction ofthe waveplate’s slow and fast axes, respectively

• ni is the index of refraction of the incident medium, Fo isthe 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


Figure 1. Two plots of simulated transmission data for two different thicknesses of calcite waveplate rotated through a full cycle of β . The figureon 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 phasedifference of π/2(λ/4). Similarly, the second plot corresponds to a calcite plate of appropriate thickness such that the imparted phase differenceis 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, ... cor-respond to half-wave plates; 1, 2, 3, ... correspond tofull-wave plates).

Figure 2. A plot of equation (1-11) over the range 0≥ q≥ 1. Negativevalues of ∆T correspond two transmission curves that start at aminimum, such as the first plot in Fig. 1-1. ∆T can be seen to rangebetween roughly -0.15 and 0.22, and it seems to have a periodicenvelope.

This theoretical groundwork is at the stage where it couldbe experimentally tested. A simple [but somewhat expensive]experimental setup could use an uncoated known-order (q)calcite waveplate aligned normal with a highly P-polarizedcollimated light-source of known power and a calibrated pho-todiode or power meter. The retarder should be mounted on arotating-mount controlled by a step motor, and rotated througha full cycle of β while stopping to take power readings withthe photodiode every 3.6o (or some other similarly small ro-tation angle). Beginning the rotation of the fast axis alignedwith the fast axis of the calcite retarder, and scanning in smallsteps of β , one should be able to obtain a data set that, whennormalized to the incident power and plotted, is comparableto those in Figure 1-1. Knowing the indices of refraction ofthe calcite waveplate for the wavelength of the incident lightand the waveplate order q, a prediction of the intensity scancan be made using equation (1-10), and can be compared to

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 fromBirefringent Crystals

2.1 Research GoalThe goal of this project was to determine how Brewster’sangle of a birefringent waveplate (in a waveplate the opticaxis is necessarily in the face of the optic) changes with theorientation of the waveplate. The light considered was notnecessarily polarized in the plane of incidence, but might aswell have been, as the TM-mode is solely responsible for theBrewster’s angle phenomenon. It should be noted that the E-ray and O-ray necessarily feel a different index of refraction,and thus should have different Brewster’s angles, but the sameangle of reflection.

2.2 MethodologyThe focus of this project was definitely determining Brewster’sangle of the E-ray, θ e

p, as Brewster’s angle of the O-ray, θ op , is

easily determined using the conventional formula:

θop = arctan

(no

ni

)(2-1)

where no is the characteristic ordinary refractive index ofthe birefringent material and ni is the refractive index of theincident medium.

However, determining θ ep is not so simple, as the boundary

conditions at the air-crystal interface give rise to a differentset of Fresnel equations than those of the O-ray. This is dueto the index of refraction of the E-ray, neo, actually being afunction of the transmitted angle θ o

e , unlike no. The version of


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

ni sin(θi) = neo(θoe )sin(θ o

e )

neo(θoe ) =

(1

(ne)2 +

(1

(no)2 −1

(ne)2 cos2(φ)sin2(θ oe )

)) 12

(2-2)

where no and ne are the characteristic ordinary and extraordi-nary refractive indices of the birefringent material and φ isthe waveplate rotation angle; the angle formed between theplane of incidence and the optic axis of the waveplate (thisangle is analogous to β from section 1).

The symbol φ was used instead of β to simplify program-ming; Yang’s article used φ throughout, and thus it was easierto compare the equations in my program to those in the ar-ticle. In his article, Yang describes the E-ray’s transmittedfield, reflected field, and incident field. From these the TE-(denoted ⊥) and TM-mode (denoted ‖) amplitude reflectioncoefficients (Re

⊥ and Re‖, respectively) and amplitude transmis-

sion coefficients (T e⊥ and T e

‖ , respectively) may be obtained.For each mode, each reflection and transmission coefficient issimply expressed as the ratio of reflected or transmitted fieldamplitude over the incident field amplitude. Finding θ e

p at agiven φ is now simply a matter of finding the θi-root of Re

‖.

2.3 Notable Findings and Suggested Next StepsUsing the conventional TM-mode amplitude reflection coef-ficient for the O-ray, and the modified TM-mode reflectioncoefficient derived from Yang’s equations, it is now possibleto determine the two Brewster angles θ o

p and θ ep. Figure 2-1

demonstrates that the theory in Yang’s article agrees well withpreviously-established theory; the plot shown below agreeswith Figure 23-3 in Introduction to Optics, 3rd Ed. by Pe-drotti.

The program used to generate Figure 2-1 could be mod-ified to generate a plot of Ro

‖ and Re‖ at a constant θi while

the φ ranges from 0o to 90o, and analyzed to determine theestimated total intensity of the reflected ray at every φ . It maybe beneficial to use an uncoated waveplate made of a materialwith a high birefringence (such as calcite) so that θ o

p and θ ep

are noticeably different.A fairly simple experimental setup could be constructed to

obtain data to compare to the theoretical prediction mentionedabove. A suggested setup would include a waveplate on arotating mount (preferably one controlled by a step motor), asource of P-polarized light (of known power/intensity) suchas a diode laser, and a calibrated photodiode or power meter.The P-polarized light should be incident on the waveplate atan oblique angle, and polarized in the plane of incidence. Ifnecessary, use a linear polarizer, although this may facilitatesome measurement of the intensity of the incident light after itpasses through the polarizer. The photodiode or power meter

1yes, ‘Snel’ is actually the correct spelling

Figure 3. A plot of the amplitude coefficients against θi for a quartzwaveplate (no = 1.543, ne = 1.552). Dotted lines represent theamplitude reflection coefficients and solid thick lines representamplitude transmission coefficients. The blue line is the height on thegraph at which an interface coefficient has the value of 0. The black,dotted line (TM-mode reflection coefficient) can be seen vary suchthat [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 whichRe‖ crosses through 0 on the vertical axis corresponds to θ e

p ≈ 57.5o

for φ = 20o.

should also be set up in the plane of incidence, normal toand in the path of the reflected beam. Rotating the waveplateshould change the intensity of the reflected beam withoutchanging its path. Since the incident light is polarized inthe plane of incidence, the intensity of the reflected beam isa sum of the reflected intensities of the E- and O-ray TM-mode 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 inthe 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 andstability of waveplate in oblique incident beam

• T. Yang (2006): An improved description of Jones vec-tors of the electric fields of incident and refracted raysin a birefringent plate

Please note that Yang’s article contained several errors:

• in equation (14), the third component in the numera-tor should be positive (although it did not affect anycalculated results)

• in equation (21), Det‖should be squared under the square

root

• in equations (24c) and (24d), remove/scratch out thefirst closing round bracket [)] in both, or add an openinground bracket [(] between each cos and θeo; Ae

t⊥ is notdivided by neo cos(θi)


3. Serendipitous LearningA beautiful part of research is that, in attempting to answer asingle question, you end up answering an incredible number ofquestions on the way to answering the first one. This sectionis dedicated to some of the wonderful unexpected [practical]learning that during the summer.

When doing polarimetry, it is important to consider thetype of photodiode being used; between different silicon pho-todiodes the results of a rotating waveplate scan were vastlydifferent (i.e. different percentage change, and completely dif-ferent characteristic shape of intensity profile). Perhaps somePDs have different sensitivities to certain polarizations oflight; the TSL251 had large percentage changes (≈ 5−12%)in its scans, and had [roughly] sinusoidal intensity profile ofperiod π/2, whereas the OPT101A had a non-periodic shapeand small percentage changes (≈ 0.5− 2%). The TSL251also seemed to hit its local minima when circularly-polarizedlight was incident (at β =45o, 135o, 225o, 315o), and maximawhen 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 changesin the temperature of the gain medium; a 1oC increase inthe temperature of the gain medium increased the laser’s out-put power as much 2 percent. Although somewhat puzzling,it seemed to be easier to keep the diode laser at a constanttemperature at a temperature of about 17.0oC, as opposed toabout 18.6oC, which was the initial set point of the laser’scooling device. Intuitively, one might think it would be easierto maintain a temperature closer to room temperature.

When taking any measurement with a photodiode, devisean experiment where you never have to move the photodi-ode; it will make your life significantly less difficult 100% ofthe time. In an experiment where the photodiode is movedmanually, it is an extremely onerous task to keep all of thecomponents of the apparatus aligned for every intensity mea-surement.

Finally, arguably the most important thing that was learnedthis summer: Wolfram Mathematica is a powerful, versatile,and simply fantastic program, and everybody should use it foreverything. Perhaps the previous statement is an exaggeration,but it really is a fantastic program. From rendering beau-tiful high-resolution videos of time-evolving electric fieldsfrom arrays of dipole oscillators to preparing simple plots,Mathematica is a simple and effective choice of programminglanguage. It’s got all of the numerical tools of Matlab, all ofthe analytical solving techniques of Maple, and even has built-in connectivity with Wolfram Alpha’s massive database. (Andno, the research in this report was not sponsored by Wolframin any way, although Mathematica was an indispensable toolin all research mentioned in this report.)

(a)

(b)Figure 4. (a) A plot of transmitted intensity (arbitrary units) againstwaveplate rotation angle β ; intensity measurements taken withOPT101A photodiode. The difference between the maximum andminimum reading taken is 1.10%. The 3 coloured sets are 3 individualscans, and the black set is the mean of the 3 individual scans. Thescans seems to be non-sinusoidal in shape. (b) A plot of transmittedintensity (arbitrary units) against waveplate rotation angle β ; intensitymeasurements taken with TSL251 photodiode. The differencebetween the maximum and minimum reading taken is 5.39%, muchlarger than the OPT101A scan. Scans seem to roughly follow asinusoidal shape of period π/2. The shapes of the two plots areapparently fundamentally different.

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A study of reﬂection and transmission of birefringent