An all-transparent symmetric trilayer structure, which consists of a high-index center layer coated on both sides by a low-index film and embedded in a high-index prism, can function as an efficient polarizer or polarizing beam splitter under conditions of frustrated total internal reflection over a wide range of incidence angles. For a given set of refractive indices, all possible solutions for the thicknesses of the layers that suppress the reflection of either the p or s polarization at a specified angle, as well as the reflectance of the system for the orthogonal polarization, are determined. A 633 nm design that uses a MgF2-ZnS-MgF2 trilayer embedded in a ZnS prism achieves an extinction ratio (ER)\u3e40 dB from 50° to 80° in reflection and an ER\u3e20 dB from 58° to 80° in transmission. IR polarizers that use CaF2-Ge-CaF2 trilayers embedded in a ZnS prism are also considered

Azzam, Rasheed M.A.

Perla, Siva R

English

University of New Orleans

University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 3-1-2006 Polarizing properties of embedded symmetric trilayer stacks under conditions of frustrated total internal reflection Rasheed M.A. Azzam University of New Orleans, razzam@uno.edu Siva R. Perla Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs  Part of the Engineering Commons Recommended Citation Rasheed M. A. Azzam and Siva R. Perla, "Polarizing properties of embedded symmetric trilayer stacks under conditions of frustrated total internal reflection," Appl. Opt. 45, 1650-1656 (2006) This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact scholarworks@uno.edu. Polarizing properties of embedded symmetrictrilayer stacks under conditions of frustratedtotal internal reflectionRasheed M. A. Azzam and Siva R. PerlaAn all-transparent symmetric trilayer structure, which consists of a high-index center layer coated onboth sides by a low-index film and embedded in a high-index prism, can function as an efficient polarizeror polarizing beam splitter under conditions of frustrated total internal reflection over a wide range ofincidence angles. For a given set of refractive indices, all possible solutions for the thicknesses of thelayers that suppress the reflection of either the p or s polarization at a specified angle, as well as thereflectance of the system for the orthogonal polarization, are determined. A 633 nm design that uses aMgF2–ZnS–MgF2 trilayer embedded in a ZnS prism achieves an extinction ratio ER  40 dB from50° to 80° in reflection and an ER  20 dB from 58° to 80° in transmission. IR polarizers that useCaF2–Ge–CaF2 trilayers embedded in a ZnS prism are also considered. © 2006 Optical Society ofAmericaOCIS codes: 230.5440, 240.0310, 260.5430, 310.0310.1. IntroductionThin-film polarizers and polarizing beam splitters(PBS) are widely known1–4 and are based on the de-structive interference of light for one linear polariza-tion (p or s) and the nearly full constructiveinterference for the orthogonal polarization.In this paper we consider the polarizing propertiesof a symmetric trilayer stack of refractive indicesn1, n2, n1, which is embedded in a high-index medium(prism) of refractive index n0; see Fig. 1. All media areconsidered to be transparent, optically isotropic, andseparated by parallel-plane boundaries. We also as-sume that n0  n1, n2  n0 and that light is incidentfrom medium 0 at an angle 0, which is greater thanthe critical angle c01  arcsinn1n0 of the 01 in-terface, so that frustrated total internal reflection(FTIR) takes place.In a recent paper5 Li and Dobrowolski reviewed theearlier literature about polarizers that employ FTIRand proposed a new high-performance PBS that re-flects the p polarization and transmits the s polariza-tion. (In the conventional MacNeille design6,7 thatoperates at Brewster’s angle, the p polarization istransmitted and the s polarization is reflected.) TheLi and Dobrowolski design uses a symmetric trilayer(3) of thin films as a basic unit (period), which isrepeated many times (9–27). Therefore the full mul-tilayer consists of as many as 81 thin films.In Section 2 we consider polarizers that consist ofonly one period (trilayer), and we do not resort to thethin-film approximation of Ref. 5. Also, instead ofusing Herpin’s equivalent-layer theory,8,9 we intro-duce an explicit technique that is based on the fullexpressions of the complex-amplitude reflection coef-ficients of the trilayer structure. We also allow foreither the p or s polarization to be reflected or trans-mitted.In Section 3 we assume a set of refractive indices,and we determine all possible embedded symme-tric trilayers that suppress the reflection of p- ors-polarized light at each one of several angles of in-cidence 0, while boosting the reflection of the orthog-onal polarization toward 100%. We find a continuumof solutions at each angle and for each polarization.Results are presented for visible-laser 633 nm po-larizers that use MgF2–ZnS–MgF2 trilayers embed-ded in a Cleartran10 (ZnS) prism.In Section 4 a specific trilayer stack is describedthat functions as a dual polarizer (or PBS) at twowidely separated angles of choice (e.g., 55° and 75°).The authors are with the Department of Electrical Engineering,University of New Orleans, New Orleans, Louisiana 70148.R. M. A. Azzam’s e-mail address is razzam@uno.edu.Received 7 March 2005; accepted 13 August 2005.0003-6935/06/071650-07$15.00/0© 2006 Optical Society of America1650 APPLIED OPTICS  Vol. 45, No. 7  1 March 2006One such MgF2–ZnS–MgF2 trilayer embedded in aCleartran (ZnS) prism achieves an extinction ratio inreflection ERr  40 dB over the 50°–80° angularrange and an extinction ratio in transmission ERt 20 dB over the 58°–80° range. The angular rangemeasured in air outside the prism is even larger if oneaccounts for refraction at the entrance face of theprism by using Snell’s law. The response of this wide-angle design to a 600–700 nm spectral scan is alsoconsidered.In Section 5 we present results for infrared polar-izers that use the CaF2–Ge–CaF2 trilayer embeddedin a ZnS prism. Finally, Section 6 gives a brief sum-mary of the paper.2. Design ProcedureConsider a monochromatic light beam traveling in anambient medium (solid prism) of refractive index n0and incident on an embedded symmetric trilayerstructure at an angle of incidence 0 with respect tothe normal to the interfaces, as shown in Fig. 1. Theoverall complex-amplitude reflection coefficient for-polarized light   p or s can be expressed as11rlmX1nX2mX1 X2nX12 lX2 X121 bX1 cX2dX12 bX1 X2 eX2 X12 , (1)l r01,m r12 1 r012 ,nr01 r122 , (2)b 2r01 r12,cr122 ,d r012 r122 ,er012 . (3)In Eq. (1) X1 and X2 are complex exponential func-tions of film thickness given byXi expjZi cos i, (4)where Zi is the thickness of the ith film i  1, 2normalized to the quarter-wave thickness at normalincidence, i.e.,Zi4dini	. (5)In Eq. (4) i is the angle of refraction in the ith layer,and in Eq. (5) ni, di are the refractive index and met-ric thickness of the ith layer, respectively, and  is thevacuum wavelength of light.The Fresnel complex-amplitude reflection coeffi-cients of the ij interface for the p and s polarizationsare given by11rijpnj cos ini cos jnj cos ini cos j, rijsni cos inj cos jni cos inj cos j.(6)To suppress the  polarization on reflection, we setr 0 (7)in Eq. (1). This gives us the following constraint be-tween X1 and X2:X2lmX1nX12nmX1 lX12. (8)When the refractive indices and angle of incidenceare such that n1n0  sin 0, FTIR takes place at the01 interface at 0 and the light field becomes evanes-cent in medium 1 (the low-index film). In this casecos 1 is pure imaginary, and X1 is real in the range0 X1 1. Because we also choose n2 n0, the angleof refraction 2 in the middle layer is real, whichmakes X2 a pure phase factor, so that |X2|  1. Foreach real value of X1 in the range 0  X1  1, we findthat |X2| 1 by using Eq. (8). (An analytical proof ofthis statement is given in Appendix A.) Thereforethere is an infinite number of solutions X1, X2 thatsatisfy Eq. (8) so that r  0. The corresponding so-lution sets of normalized film thicknesses Z1, Z2 aredetermined subsequently by using Eq. (4).An acceptable design must have a reflectanceRr2 (9)for the orthogonal polarization = that is as close aspossible to 1. In general this reflectance increases asthe normalized thickness Z1 (of the low-index layersthat support the evanescent field) and the angle ofincidence 0 are increased.Fig. 1. Embedded trilayer structure as a polarizing beam splitter(PBS). A PBS in which the p polarization is transmitted and the spolarization is reflected is also considered. See text for discussion.1 March 2006  Vol. 45, No. 7  APPLIED OPTICS 16513. MgF2–ZnS–MgF2 Trilayer Polarizers forthe Visible SpectrumFigure 2 shows Z2 as a function of Z1 such that rp 0 at angles of incidence 0 from 45° to 85° in stepsof 5° for MgF2–ZnS–MgF2 trilayers embedded in aZnS substrate with refractive indices n0  2.35ZnS, n1  1.38 MgF2, and n2  2.35 ZnS in thevisible spectrum.In Fig. 3 the associated reflectance Rs  |rs|2 forthe s polarization is plotted versus Z1 at the sameangles of incidence as in Fig. 2. It is apparent that Rsincreases monontonically with Z1 at a given 0 andwith 0 at a given Z1. Figure 4 shows a magnifiedview of the region of high reflectance in Fig. 3. Effi-cient polarizers Rs  99% are readily obtained byusing many combinations of Z1 and 0.An interesting feature in Fig. 2 is that all thecurves appear to pass through one common point A. Ifthis were true, it would signify that one multilayercan force rp  0 at many angles of incidence simul-taneously. However, as Fig. 5 shows, there is notstrictly one common point of intersection for all thecurves in Fig. 2. Furthermore, because Z1 at A issmall 0.2, the reflectance Rs of such a wide-anglepolarizer is low except at high incidence angles, as isevident from Fig. 3.We now consider the other class of rs 0 polarizersby using the same MgF2–ZnS–MgF2 trilayer embed-ded in a ZnS substrate.Figure 6 shows Z2 as a function of Z1 such that rs 0 at angles of incidence 0 from 45° to 85° in stepsof 5°. In Fig. 7 the corresponding reflectance Rp |rp|2 for the p polarization is plotted versus Z1 at theFig. 2. Plot of Z2 as a function of Z1 such that rp  0 at angles ofincidence 0 from 45° to 85° in steps of 5°. We assume MgF2–ZnS–MgF2 trilayers embedded in a ZnS substrate with refractive indi-ces n0  2.35 (ZnS), n1  1.38 (MgF2), and n2  2.35 (ZnS) in thevisible spectrum. Notice that all curves appear to have a commonpoint of intersection at A.Fig. 3. Reflectance Rs  |rs|2 for the s polarization is plottedversus Z1 at the same angles of incidence as in Fig. 2 and under therp  0 condition.Fig. 4. Magnified view of the region of high reflectance inFig. 3.Fig. 5. High-resolution view of the family of curves of Fig. 2 nearpoint A shows that there is not strictly one common point of inter-section for all the curves.1652 APPLIED OPTICS  Vol. 45, No. 7  1 March 2006same angles of incidence. Again we find that Rp in-creasesmonontonically with Z1 at a given 0 and with0 at a given Z1, as one may intuitively expect.Efficient polarizers with high ER in transmissionRp  99% are realizable by using many combina-tions of Z1 and 0, as is shown more clearly in Fig. 8.In Fig. 6 all the curves appear to pass through acommon point B. Although this is not strictly true, asis shown in Fig. 9, it still suggests that one and thesame multilayer can satisfy the condition rs  0accurately (although not exactly) at many angles ofincidence simultaneously. The value of Z1 at B islarge enough 0.7 to make the reflectance Rp 90% at all angles 50°, as one can see from Fig. 7.This important wide-angle polarizer is consideredfurther in Section 4.4. Wide-Angle Polarizer DesignThis specific design is based on operation at thepoint of intersection Z1 0.6831, Z2 1.7472 of thetwo curves in Fig. 6 that correspond to 0  55°and 75°. The corresponding metric thicknesses of theMgF2 and ZnS films, calculated at a wavelength 	 633 nm, are d1  78.305 nm and d2  117.618nm, respectively.Figure 10 shows the reflectancesRp and Rs as func-tions of the angle of incidence 0 from 40° to 85° whenthe metric film thicknesses and wavelength are keptconstant. Note that Rs  0.002 over this entire rangeof angles and that Rp  90% for 0  52°.In Fig. 11 the extinction ratios in reflection andtransmission ERr and ERt in decibels are plottedversus the angle of incidence 0 from 50° to 80°. Goodpolarization properties ERr  40 dB and ERt  20dB are achieved over an extended range of angles. Ifone accounts for the refraction of light from air to theFig. 6. Plot of Z2 as a function of Z1 such that rs  0 at angles ofincidence 0 from 45° to 85° in steps of 5°. All the curves appear topass through a common point B.Fig. 7. Reflectance Rp  |rp|2 for the p polarization is plottedversus Z1 at the same angles of incidence as in Fig. 6 and under thers  0 condition.Fig. 8. Expanded view of Fig. 7 in the region of high reflec-tance.Fig. 9. Magnified view of the family of curves of Fig. 6 in theneighborhood of point B shows that there is not strictly one com-mon point of intersection for all the curves.1 March 2006  Vol. 45, No. 7  APPLIED OPTICS 1653high-index ZnS substrate, the angular bandwidth inair is even larger that that indicated by Fig. 11.Although this wide-angle polarizer is intended foruse with monochromatic 633 nm light, it is of inter-est to consider its performance over a range of wave-lengths. For this purpose, the metric film thicknessesd1  78.305 nm and d2  117.618 nm and the angleof incidence0 65° are kept constant, and the wave-length  is scanned over the 600–700 nm range. Inthis calculation the dispersion of MgF2 and ZnS overthis limited spectral range is ignored. Figure 12shows the extinction ratios ERr and ERt in decibelsas functions of . Note that ERt remains greater than20 dB over this 100 nm bandwidth, whereas ERr ishigh only in the immediate neighborhood of the de-sign wavelength 633 nm.5. CaF2–Ge–CaF2 Trilayer Infrared PolarizersFigure 13 shows the constraint between Z2 and Z1such that rp  0 at angles of incidence 0 from 45° to85° in 5° steps for CaF2–Ge–CaF2 trilayers embed-ded in a ZnS substrate with refractive indices n0 2.2 ZnS, n1  1.4 CaF2, and n2  4 Ge in theinfrared. The family of curves in Fig. 13 is substan-tially different from that of Fig. 2 for the MgF2–ZnS–MgF2 trilayer polarizers in the visible spectrumunder the same rp  0 condition, and a common (ornear-common) point of intersection among all thecurves does not exist. This significant change is duemostly to the substantially higher refractive index ofthe center layer Ge, n2  4.Figure 14 shows the associated reflectance Rs Fig. 10. Reflectances Rp and Rs as functions of the angle of inci-dence 0 from 40° to 85° for a polarizer using a MgF2–ZnS–MgF2trilayer embedded in a ZnS substrate with refractive indices n0 2.35 (ZnS), n1  1.38 (MgF2), and n2  2.35 (ZnS). The metric filmthicknesses (d1  78.31 nm, d2  117.62 nm) and wavelength(633 nm) are kept constant.Fig. 11. Extinction ratios in reflection and transmission (ERr andERt) in decibels are plotted versus the angle of incidence 0 from50° to 80° for the wide-angle polarizer of Fig. 10.Fig. 12. Extinction ratios ERr and ERt in decibels as functions ofwavelength  for the wide-angle polarizer of Fig. 10. The metricfilm thicknesses (d1  78.31 nm, d2  117.62 nm) and angle ofincidence 0 (65°) are kept constant.Fig. 13. Constraint between Z2 and Z1 such that rp  0 at anglesof incidence 0 from 45° to 85° in 5° steps for CaF2–Ge–CaF2trilayers embedded in a ZnS substrate with refractive indices n02.2 (ZnS), n1  1.4 (CaF2), and n2  4 (Ge) in the infrared.1654 APPLIED OPTICS  Vol. 45, No. 7  1 March 2006|rs|2 for the s polarization plotted versus Z1 at thesame angles of incidence as in Fig. 13. Again, Rsincreases monontonically with Z1 at a given 0and with 0 at a given Z1. Efficient polarizers Rs 99% are readily obtained for many combinationsof Z1 and 0.Figure 15 shows Z2 versus Z1 such that rs  0 atangles of incidence 0 from 45° to 85° in 5° steps forthe same CaF2–Ge–CaF2 trilayers embedded in ZnSsubstrate in the infrared. All the curves in Fig. 15appear to share a common point of intersection C.The associated reflectance Rp  |rp|2 for the p polar-ization is plotted versus Z1 at the same angles as inFig. 16. A wide-angle infrared polarizer that is basedon operation at point C of Fig. 15 is possible, and itsperformance is comparable with that of the visiblepolarizer discussed in Section 4.6. SummaryThe polarizing properties of a symmetric trilayerstack embedded in a high-index prism are thoroughlyexamined by using analytical and numerical calcula-tions. All possible solutions (infinite in number) thatsuppress the p or s polarization in reflection are de-termined, and the associated throughput for theorthogonal polarization is calculated. Conditions forobtaining wide-angle designs are clearly demonstra-ted.Presented are specific examples of polarizers forthe visible spectrum that use MgF2–ZnS–MgF2 tri-layers embedded in a ZnS prism to achieve extinctionratios greater than 40 dB in reflection and greaterthan 20 dB in transmission over a wide range of in-cidence angles. Infrared polarizers that use CaF2–Ge–CaF2 trilayers embedded in a ZnS prism are alsoconsidered.Appendix AIn this Appendix we prove that, for each real value ofX1 in the range 0  X1  1, X2 as given by Eq. (8) isa pure phase factor; hence |X2|  1. To do this wesubstitute l, m, and n from Eqs. (2) into Eq. (8) andcast the result in the following form:X2 r122r122 r121r011 r01X1X12r122 r12r011 r01X1X12. (A1)Under conditions of FTIR, all interface reflection co-efficients are pure phase factors; hencerij expjij,rij1 rij* expjij,rij1 rij 2 cosij, (A2)Fig. 14. Reflectance Rs  |rs|2 for the s polarization plottedversus Z1 under the rp  0 condition and at the same angles ofincidence as in Fig. 13.Fig. 15. Plot of Z2 versus Z1such that rs 0 at angles of incidence0 from 45° to 85° in 5° steps for CaF2–Ge–CaF2 trilayers embed-ded in ZnS substrate in the infrared. All the curves appear to sharea common point of intersection at C.Fig. 16. Reflectance Rp  |rp|2 for the p polarization plottedversus Z1 under the rs  0 condition and at the same angles ofincidence as in Fig. 15.1 March 2006  Vol. 45, No. 7  APPLIED OPTICS 1655where ij is the interface reflection phase shift.12Equations (A2), and the fact that X1 is real, allow Eq.(A1) to be written asX2 r122WW*. (A3)Because each of the bracketed terms in Eq. (A3) is apure phase factor, it follows thatX2 1. (A4)For simplicity the polarization subscript  wasdropped in Eqs. (A1)–(A3).We thank Amrit De for his participation in theinitial phase of this project.References1. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (McGraw-Hill, 1986).2. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988).3. J. A. Dobrowolski, “Optical properties of films and coatings,” inHandbook of Optics, Vol. I, M. Bass, E. W. van Stryland, D. R.Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995).4. J. M. Bennett, “Polarizers,” in Handbook of Optics, Vol. II,M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe,eds. (McGraw-Hill, 1995).5. L. Li and J. A. Dobrowolski, “High-performance thin-film po-larizing beam splitter operating at angles greater than thecritical angle,” Appl. Opt. 39, 2754–2771 (2000).6. S. M. MacNeille, “Beam splitter,” U.S. patent 2,403,731 (6 July1946).7. M. Banning, “Practical methods of making and using multi-layer filters,” J. Opt. Soc. Am. 37, 792–797 (1947).8. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42,806–810 (1952).9. M. C. Ohmer, “Design of three-layer equivalent films,” J. Opt.Soc. Am. 68, 137–139 (1978).10. CVD, Inc., 35 Industrial Parkway, Woburn, Mass. 01801.11. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polar-ized Light (North-Holland, 1987).12. R. M. A. Azzam, “Phase shifts that accompany total internalreflection at a dielectric–dielectric interface,” J. Opt. Soc. Am.A 21, 1559–1563 (2004).1656 APPLIED OPTICS  Vol. 45, No. 7  1 March 2006

Polarizing properties of embedded symmetric trilayer stacks under conditions of frustrated total internal reflection

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Polarizing properties of embedded symmetric trilayer stacks under conditions of frustrated total internal reflection

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