How To Design Birefringent PolarizersWhat is Birefringence?Normal glasses are homogeneous and isotropic,that is,they have the same refractiveindex no matter what direction light travels through them.Uniaxial materials,such asCalcite,have a crystal axis which defines an axis of symmetry.These materials refractrays differently,depending on the polarization state of the ray and the angle the ray makeswith respect to the crystal axis.Therefore there are two possible refraction angles for anyray,representing two orthogoanl polarization states.This phenomenon is known asdouble refraction or birefringence.Birefringent materials always bend rays according to Snell's law,but the effective index ofrefraction in the media depends on the input polarization state of the ray and the angle theray makes to the crystal axis."Ordinary"rays are refracted by:nsin nosinwhere no is the ordinary refractive index.which is just Snell's law."Extraordinary"rays arerefracted by.nsin =n(w)sinwhich is also Snell's law,but note that the refractive index is now a function of theanglew.which is the angle between the crystal axis vector a and the refractedwavevector k.Now here is the hard part.The ray vector S,which points in the direction of energy flow,does not follow the wave vector k but instead makes a small angle with respect to it.Innormal glasses k and S are the same vector and we just keep track of k.In birefringentmedia we must consider the ray and wave vectors as being different.The vectors k and Sboth lie in the same plane as the crystal axis vector a,andThe effective refractive index seen by the extraordinary ray is defined by:(1/n(w))2=(cos w/no)2+(sin w/ne)where ne is the extraordinary index of refraction.The Birefringent-In SurfaceClearly tracing birefingent rays is more complex than ordinary ray-tracing:we mustconsider two refractive indices and the orientation of the ray and wave vectors relative to acrystal axis vector.So,birefringent ray tracing is performed only when called for.It startswhen a ray hits a Birefringent In surface,and ends when the ray hits a Birefringent Outsurface.Only Coordinate Break surfaces are allowed between a Birefringent IN andBirefringent Out surface.In normal ray-tracing the ray vector S and the wave vector k point in the same direction:the direction of energy flow.Therefore in normal ray-tracing the k vector componentsdefine the ray's direction cosines.In birefringent ray-tracing,k and S are not identical but are coplanar with the crystal axisvector a.The components of S now define the ray's direction cosines.Here is an example of a ray entering a block of calcite.The dotted line represents thecrystal axis:3:3D Layout 3Update SettingsPrint Window Text ZoomThe ray enters the calcite and appears to be split(more on this later)into two rays.Theordinary ray refracts normally,and as the surface is flat,it is not bent.The extraordinaryray undergoes double refraction,once at the surface and again at the crystal axis,and sois deviated,even though the surface is flat and the ray is at normal incidence.This is how this is entered in ZEMAX:Lens Data Editor:Config 1/2Radiusgtandard20.00050.000CALCITEInfinitz25.000The ray is traced surface-by-surface nommally until it hits the Birefringent In surface.Thishas the same surface shape as the Standard surface (i.e.,a conic asphere)Note theglass used is CALCITE:this is used as the ordinary index.ZEMAX will look in the samecatalog for a glass named CALCITE-E.This contains the extraordinary index of thematerial.By using two glasses,every piece of data about the glass(transmission,dispersion,thermal properties etc)can be incorporated for both indices.The crystal axis orientation relative to the surface normal is defined by the parameter dataof the Birefringent-In surface:Lens Data Editor:Config 1/2Edit Solves Options HelpDraw Axis?ModeX-cosineY-cosineZ-cosineOBJStandardStandard15,00000.0000.7070.707IMAStandardThe direction cosines of the crystal axis are entered directly,in the local coordinates of theBirefringent-In surface.The "Draw Axis"parameter specifies the length of the dotted lineshown in layout plots to represent the axis,in lens units.This may be set to zero to avoiddrawing the axis,if required.Now the layout plot above is misleading:it appears to show a ray being split into ordinaryand extraordinary components.In sequential ray-tracing,rays can never split:one ray inmust give only one ray out.The Mode flag in the parameters of the Birefringent In surfacetells ZEMAX which ray to trace:if mode 0,then ZEMAX traces the ordinary ray;if mode 1,then ZEMAX traces the extraordinary rayThis layout plot is computed by using one configuration of the design with mode=0,andanother with mode =1,and overlaying the plots for the two configurations: