The conditions under which light interference in a transparent quarter-wave layer of refractive index n1 on a transparent substrate of refractive index n2 leads to 50% reflectance for incident unpolarized light at an angle φ are determined. Two distinct solution branches are obtained that correspond to light reflection above and below the polarizing angle, φp , of zero reflection for p polarization. The real p and s amplitude reflection coefficients have the same (negative) sign for the solution branch φ\u3eφp and have opposite signs for the solution branch φ\u3cφp . Operation at φ\u3cφp is the basis of a 50%–50% beam splitter that divides an incident totally polarized light beam (with p and s components of equal intensity) into reflected and refracted beams of orthogonal polarizations [ Opt. Lett. 31, 1525 (2006) ] and requires a film refractive indexn1⩾(2√+1)n2−−√ . A monochromatic design that uses a high-index TiO2 thin film on a low-index MgF2substrate at 488 nm wavelength is presented as an example

Azzam, R. M.A.

Sudradjat, F. F.

English

University of New Orleans

University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 2-14-2007 Quarter-wave layers with 50% reflectance for obliquely incident unpolarized light R. M.A. Azzam University of New Orleans, razzam@uno.edu F. F. Sudradjat Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs  Part of the Electrical and Electronics Commons, and the Optics Commons Recommended Citation R. M. A. Azzam and F. F. Sudradjat, "Quarter-wave layers with 50% reflectance for obliquely incident unpolarized light," J. Opt. Soc. Am. A 24, 850-855 (2007) 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. Quarter-wave layers with 50% reflectance forobliquely incident unpolarized lightR. M. A. Azzam and F. F. SudradjatDepartment of Electrical Engineering, University of New Orleans, New Orleans, Louisiana 70148Received June 15, 2006; accepted September 28, 2006;posted October 20, 2006 (Doc. ID 71932); published February 14, 2007The conditions under which light interference in a transparent quarter-wave layer of refractive index n1 on atransparent substrate of refractive index n2 leads to 50% reflectance for incident unpolarized light at an angle are determined. Two distinct solution branches are obtained that correspond to light reflection above andbelow the polarizing angle, p, of zero reflection for p polarization. The real p and s amplitude reflection coef-ficients have the same (negative) sign for the solution branch p and have opposite signs for the solutionbranch p. Operation at p is the basis of a 50%–50% beam splitter that divides an incident totallypolarized light beam (with p and s components of equal intensity) into reflected and refracted beams of or-thogonal polarizations [Opt. Lett. 31, 1525 (2006)] and requires a film refractive index n1 2+1n2. Amono-chromatic design that uses a high-index TiO2 thin film on a low-index MgF2 substrate at 488 nm wavelengthis presented as an example. © 2007 Optical Society of AmericaOCIS codes: 230.1360, 260.5430, 260.3160, 160.4330.1. INTRODUCTIONIn a recent letter1 it was shown that it is possible to splita monochromatic light beam into two beams of equalpower and orthogonal polarizations by reflection and re-fraction at the planar surface of a dielectric substrate atan angle of incidence below the Brewster angle. This re-quires a substrate of high refractive index, n23+22=5.8284, e.g., PbTe in the IR. It was noted that a quarter-wave layer of high refractive index, which is deposited ona substrate of low refractive index, can also be used to ac-complish the same novel beam splitting function. Thisthin-film coating design is now presented in this paper.For a succinct account of different types of beam splitters,see, e.g., the review by Dobrowolski.2In Section 2 an analytical design procedure is pre-sented for achieving 50% reflectance of unpolarized (orrandomly polarized) light that is obliquely incident on atransparent quarter-wave coating on a transparent sub-strate. The algorithm developed in Section 2 is applied inSubsection 3.A to the special case of a vanishing substraten2=1, i.e., an unsupported quarter-wave pellicle. It isalso applied to quarter-wave coatings of varying refrac-tive index n1 on a (glass or plastic) substrate of refractiveindex n2=1.5 in Subsection 3.B.In Section 4 a beam splitter design that uses a TiO2high-index quarter-wave coating on a low-index MgF2substrate for 488 nm (Ar-ion-laser) light is considered asa specific example. An error analysis shows the effect of±5° shifts in the angle of incidence and of ±5% shifts inwavelength or film thickness on the performance of thisbeam splitter. Section 5 gives a brief summary of the pa-per.2. QUARTER-WAVE LAYERS THATREFLECT 50% OF UNPOLARIZED LIGHTAT OBLIQUE INCIDENCE: ANALYTICALTREATMENTFor light reflection in air n0=1 by a transparent layer ofquarter-wave optical thickness (equal to half the film-thickness period) and refractive index n1 on a transparentsubstrate of refractive index n2, the amplitude reflectioncoefficients of the p and s polarizations at oblique inci-dence at an angle  are real and are given by3,4Rs =S0S2 − S12S0S2 + S12 , 1Rp =n14S0S2 − n22S12n14S0S2 + n22S12 , 2whereSi = ni2 − u1/2, i = 0,1,2, 3u = sin2 . 4All materials are assumed to be optically isotropic and areseparated by parallel plane boundaries.In terms of the quantityP = S0S2/S12, 5Eqs. (1) and (2) are rewritten as850 J. Opt. Soc. Am. A/Vol. 24, No. 3 /March 2007 R. M. A. Azzam and F. F. Sudradjat1084-7529/07/030850-6/$15.00 © 2007 Optical Society of AmericaRs =P − 1P + 1, Rp =n14P − n22n14P + n22 . 6To achieve 50% intensity reflectance for incident unpolar-ized light, the amplitude reflection coefficients of the pand s polarizations must satisfy the following condition:Rp2 +Rs2 = 1. 7If Eqs. (6) are substituted into Eq. (7), a quartic equationin P is obtained,a4P4 + a3P3 + a2P2 + a1P + a0 = 0, 8with coefficients given bya4 = n18,a3 = − 2n14n14 + n22,a2 = n14n14 − 12n22 + n24,a1 = − 2n22n14 + n22,a0 = n24. 9For a high-index transparent film on a low-index trans-parent substrate it is apparent from Eqs. (3)–(5) that P isreal and positive and in the range 0P1, hence, onlyroots of Eq. (8) that satisfy this condition are accepted.From Eqs. (3)–(5) we also obtainP2 = 1 − un22 − u/n12 − u2. 10Equation (10) can be rewritten as a quadratic equation inu:b2u2 + b1u + b0 = 0, 11with coefficients given byb2 = P2 − 1,b1 = n22 + 1 − 2n12P2,b0 = n14P2 − n22. 12The only acceptable roots of Eq. (11) are those for which0u1. Finally, the angle of incidence  at which the50% reflectance for incident unpolarized light is achieved,for given n1 and n2, is given by = arcsinu1/2. 133. QUARTER-WAVE LAYERS THATREFLECT 50% OF UNPOLARIZED LIGHTAT OBLIQUE INCIDENCE: NUMERICALRESULTSA. Case of a Vanishing Substrate, n2=1The algorithm of Section 2 is applied to determine theconstraint between n1 and  such that Eq. (7) is satisfiedwhen n2=1 (i.e., for an unsupported quarter-wave layeror pellicle), by assigning values to n1 between 1 and 6 insteps of 0.01 and solving for the corresponding values of.Figure 1 shows the two solution branches that we ob-tain: a high-angle branch (HAB), or solution 1, which isrepresented by the continuous line, and a low-anglebranch (LAB), or solution 2, which is represented by thedashed line. The dash-dot curve in the middle representsthe polarizing (Brewster) angle, p=B=tan−1n1.Figures 2 and 3 show the real amplitude reflection co-efficients Rp and Rs that are associated with the HAB andLAB, respectively, as functions of the film refractive indexn1. Rp and Rs are both negative for the HAB, Fig. 2, andhave opposite signs (Rp0, Rs0) for the LAB, Fig. 3.Figure 4 is a plot of Rp versus Rs and shows two sepa-rate arcs of the unit circle, Eq. (7), in the third and fourthquadrants, respectively. Points A and B correspond to thesimilarly marked points in Fig. 1.Fig. 1. Angle of incidence  versus film refractive index n1 suchthat 50% of incident unpolarized light is reflected by a quarter-wave pellicle. The continuous and dashed curves represent twoindependent solution branches. The middle curve gives theBrewster angle as a function of n1, B=tan−1n1.Fig. 2. Amplitude reflection coefficients Rp and Rs that are as-sociated with the HAB of Fig. 1 plotted as functions of the filmrefractive index n1.R. M. A. Azzam and F. F. Sudradjat Vol. 24, No. 3 /March 2007/J. Opt. Soc. Am. A 851Figures 5 and 6 show the intensity reflectances Rp2 andRs2 for the HAB and LAB, respectively, as functions of n1.The average value of Rp2 and Rs2 is constant =0.5, inde-pendent of n1 as is required by Eq. (7).An interesting feature of Fig. 1 is that  of the HABreaches a minimum, min=71.8225°, at point A where n1=1.78. Therefore, a pellicle of 1.78 refractive index andquarter-wave optical thickness (or an odd integral mul-tiple thereof) functions as a 50%–50% BS for incident un-polarized light at 72° angle of incidence. The startingpoint B of the LAB in Fig. 1 is at =0 and corresponds ton1=2+1=2.414. Film materials that have this refractiveindex include diamond in the visible and ZnSe in theIR.5,6B. General Case of n21For an arbitrary substrate of refractive index n2, thestarting value of n1 for the LAB is obtained by noting thatat =0, Rp2=Rs2=1/2. It follows from Eqs. (6) thatRs = P − 1/P + 1 = − 1/2, 14P = 2 − 12. 15If u=0 is substituted in Eq. (10), we getn1 = n2/P1/2. 16When Eqs. (15) and (16) are combined, we obtainn1 = 2 + 1n2. 17As a specific example we take quarter-wave layers on a(glass or plastic) substrate with n2=1.5. Figure 7 showsthe two solution branches that we obtain using the algo-rithm of Sec. 2. When compared with Fig. 1 (for n2=1),the HAB in Fig. 7 is displaced upward to higher angles,Fig. 3. Amplitude reflection coefficients Rp and Rs that are as-sociated with the LAB of Fig. 1 plotted as functions of the filmrefractive index n1.Fig. 4. Plot of Rp versus Rs associated with the high- and low-angle branches (Figs. 2 and 3) yields two arcs of the unit circle inthe third and fourth quadrants, respectively. The arrows indicatethe direction of increasing n1, and points A and B correspond tothe similarly marked points in Fig. 1.Fig. 5. Intensity reflectances Rp2 and Rs2 that are associatedwith the HAB of Fig. 1 plotted as functions of the film refractiveindex n1. The average reflectance, Rp2+Rs2 /2, is constant at 0.5.Fig. 6. Intensity reflectances Rp2 and Rs2 that are associatedwith the LAB of Fig. 1 plotted as functions of the film refractiveindex n1. The average reflectance, Rp2+Rs2 /2, is constant at 0.5.852 J. Opt. Soc. Am. A/Vol. 24, No. 3 /March 2007 R. M. A. Azzam and F. F. Sudradjatand its minimum at point A now occurs at min=81.6208°, n1=1.98. The starting value n1=2.9568 of theLAB at point B is predicted by Eq. (17).In Fig. 7 the middle (dash-dot) curve gives the polariz-ing angle p of the film-substrate system at which Rp=0as a function of the film refractive index n1 with n2=1.5.To calculate the polarizing angle p we note from Eqs. (6)that Rp=0 ifP = n22/n14. 18Substitution of P from Eq. (18) into Eq. (10) gives a qua-dratic equation,c2u2 + c1u + c0 = 0, 19in u=sin2 p with coefficients given byc2 = n18 − n28,Fig. 7. Angle of incidence  versus the refractive index n1 of atransparent quarter-wave coating on a transparent substraten2=1.5 that reflects 50% of incident unpolarized light. The con-tinuous and dashed curves represent two distinct solutionbranches. The middle dash-dot curve gives the polarizing anglep as a function of n1 as obtained from Eqs. (19) and (20).Fig. 8. Amplitude reflection coefficients Rp and Rs that are as-sociated with the HAB of Fig. 7 plotted as functions of the filmrefractive index n1. The significance of the point of intersection Cis discussed in the text.Fig. 9. Amplitude reflection coefficients Rp and Rs that are as-sociated with the LAB of Fig. 7 plotted as functions of the filmrefractive index n1.Fig. 10. Intensity reflectances Rp2 and Rs2 that are associatedwith the HAB of Fig. 7 plotted as functions of the film refractiveindex n1. The average reflectance, Rp2+Rs2 /2, is constant at 0.5.Fig. 11. Intensity reflectances Rp2 and Rs2 that are associatedwith the LAB of Fig. 7 plotted as functions of the film refractiveindex n1. The average reflectance, Rp2+Rs2 /2, is constant at 0.5.R. M. A. Azzam and F. F. Sudradjat Vol. 24, No. 3 /March 2007/J. Opt. Soc. Am. A 853c1 = 2n12n24 − n18n22 + 1,c0 = n14n22n14 − n22. 20Figures 8 and 9 show the amplitude refection coeffi-cients Rp and Rs as functions of n1 for the HAB and LABof Fig. 7, respectively. Again, Rp and Rs are both negativefor the HAB, Fig. 8, and have opposite signs (Rp0, Rs0) for the LAB, Fig. 9.The point of intersection C of the curves of Rp and Rs inFig. 8 is located atRp =Rs = − 1/2 = − 0.7071, n1 = n2 = 1.2247. 21This corresponds to a polarization-preserving (orpolarization-independent) 50%–50% beam splitter in bothreflection and transmission.3The associated intensity reflections Rp2 and Rs2 asfunctions of n1 are shown in Figs. 10 and 11 for the HABand LAB, respectively. Again, the average of Rp2 and Rs2is constant =0.5 independent of n1 as required by Eq.(7).4. BEAM SPLITTER FOR PRODUCINGREFLECTED AND TRANSMITTED BEAMS OFEQUAL POWER AND ORTHOGONALPOLARIZATIONS IN THE VISIBLEConsider a high-index n1=2.895 thin film of TiO2 on alow-index n2=1.386 MgF2 substrate at wavelength =488 nm. (Because we assume isotropic phases, the re-fractive indices are taken as averages of the ordinary andextraordinary indices no and ne published in Ref. 6.) Byuse of the algorithm of Section 2, the operating angle ofincidence =46.037° is obtained. The film-thickness pe-riod D1 and quarter-wave metric film thickness d1 are de-termined byD1 = /2n12 − sin2 −1/2,d1 =D1/2, 22which give D1=87.02 nm and d1=43.51 nm. Under theideal operating conditions, the average reflectance for in-cident unpolarized light is 1/2 and the reflection andtransmission ellipsometric parameters4 satisfy the follow-ing relations:1,7r = ,t = 0,	r + 	t = /2. 23To couple the refracted light out of the substrate into air,a prismatic substrate with prism angle equal to the angleof refraction (31.287°) can be used, so that light leaves theprism normal to its exit face which is antireflectioncoated.8When the wavelength of light and metric film thicknessare kept constant, and the angle of incidence is varied byup to ±5°, the average reflectance and ellipsometric pa-Fig. 12. Average reflectance Rp2+Rs2 /2 and ellipsometric pa-rameters 	r ,	t ,r ,t as functions of the angle of incidence  for abeam splitter that consists of a quarter-wave 43.51 nm thinfilm of TiO2 on a MgF2 substrate at 488 nm wavelength. Theangle of incidence is varied by ±5° around the design angle =46.04°.Fig. 13. Average reflectance Rp2+Rs2 /2 and ellipsometric pa-rameters 	r ,	t ,r ,t of a beam splitter as functions of the thick-ness d1 of a TiO2 film on a MgF2 substrate at =488 nm and =46.04°. The film thickness is varied by ±5% around the designvalue of 43.5 nm.854 J. Opt. Soc. Am. A/Vol. 24, No. 3 /March 2007 R. M. A. Azzam and F. F. Sudradjatrameters change in the manner shown in Fig. 12. FromFig. 12, it is apparent that the ideal conditions of Eqs. (23)are nearly maintained; hence, this novel beam splitter istolerant to small angle-of-incidence errors.If the wavelength of light and angle of incidence arefixed, but the film thickness is varied by ±5% around itsdesign value d1=43.51 nm, the average reflectance andellipsometric parameters change in the manner shown inFig. 13. Again, the ideal conditions of Eqs. (23) are ap-proximately satisfied, to within a small error, and thecoating design is not overly sensitive to small film-thickness errors around the quarter-wave thickness.Finally, when the angle of incidence and metric thick-ness of the coating are kept constant, and the wavelengthis shifted by ±5% around =488 nm, the average reflec-tance and ellipsometric parameters change in the mannershown in Fig. 14. In Fig. 14, the deviations from the idealconditions of Eqs. (23) are more pronounced; hence, thebeam splitter is not considered achromatic.5. SUMMARYA transparent layer of quarter-wave optical thickness, apellicle or a coating deposited on a transparent substrate,reflects 50% of incident unpolarized light under certainconditions that are completely determined. Two distinctsolution branches are obtained that correspond to inci-dence below and above the p-suppressing polarizing angleof the film-substrate system. Operation below the polariz-ing angle makes possible a novel thin-film beam splitter1that divides incident totally polarized light into reflectedand refracted beams of equal power (50% each) and or-thogonal polarizations. The performance of one such de-vice that uses a high-index TiO2 quarter-wave coating ona low-index MgF2 substrate at 488 nm wavelength is pre-sented.R.M.A. Azzam’s e-mail address is razzam@uno.edu.REFERENCES1. R. M. A. Azzam, “Dividing a light beam into two beams oforthogonal polarizations by reflection and refraction at adielectric surface,” Opt. Lett. 31, 1525–1527 (2006).2. J. A. Dobrowolski, “Optical properties of films andcoatings,” in Handbook of Optics, M. Bass, E. W. VanStryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 42.3. R. M. A. Azzam, “Simultaneous reflection and refraction oflight without change of polarization by a single-layer-coated dielectric surface,” Opt. Lett. 10, 107–109 (1985).4. R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolarized Light (North-Holland, 1987).5. Handbook of Optical Constants of Solids, E. D. Palik, ed.(Academic, 1985).6. W. J. Tropf, M. E. Thomas, and T. J. Harris, “Opticalproperties of crystals and glasses,” in Handbook of Optics,M. Bass, E. W. Van Stryland, D. R. Williams, and W. L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 33.7. R. M. A. Azzam and A. De, “Optimal beam splitters for thedivision-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A20, 955–958 (2003).8. R. M. A. Azzam, “Variable-reflectance thin-filmpolarization-independent beam splitters for 0.6328 and10.6 m laser light,” Opt. Lett. 10, 110–112 (1985).Fig. 14. Average reflectance Rp2+Rs2 /2 and ellipsometric pa-rameters 	r ,	t ,r ,t as functions of the wavelength  (in nanom-eters) for a beam splitter that consists of a 43.51 nm TiO2 thinfilm on a MgF2 substrate at an angle of incidence of =46.04°.The wavelength  is changed by ±5% around 488 nm.R. M. A. Azzam and F. F. Sudradjat Vol. 24, No. 3 /March 2007/J. Opt. Soc. Am. A 855

Quarter-wave layers with 50% reflectance for obliquely incident unpolarized light

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Quarter-wave layers with 50% reflectance for obliquely incident unpolarized light

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