
University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 7-1-2011 Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack R. M.A. Azzam University of New Orleans, [email protected] 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, "Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack," Appl. Opt. 50, 3316-3320 (2011) 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 [email protected]. Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack R. M. A. Azzam Department of Electrical Engineering, University of New Orleans, New Orleans, Louisiana 70148, USA ([email protected]) Received 28 March 2011; accepted 6 May 2011; posted 17 May 2011 (Doc. ID 144834); published 1 July 2011 An analytically tractable design procedure is presented for a polarizing beam splitter (PBS) that uses frustrated total internal reflection and optical tunneling by a symmetric LHL trilayer thin-film stack em- bedded in a high-index prism. Considerable simplification arises when the refractive index of the high- index center layer H matches the refractive index of the prism and its thickness is quarter-wave. This leads to a cube design in which zero reflection for the p polarization is achieved at a 45° angle of incidence in- dependent of the thicknesses of the identical symmetric low-index tunnel layers L and L. Arbitrarily high reflectance for the s polarization is obtained at subwavelength thicknesses of the tunnel layers. This is illustrated by an IR Si-cube PBS that uses an embedded ZnS–Si–ZnS trilayer stack. © 2011 Optical Society of America OCIS codes: 230.1360, 230.5440, 240.0310, 260.5430, 310.6860. 1. Introduction the low-index tunnel layers and is propagating in Thin-film polarizing beam splitters (PBSs) are versa- the prism and center layer. In Sections 2 and 3 we tile optical elements that use destructive interfer- consider an important and analytically tractable spe- ence of light for one linear polarization (p or s) and cial case in which the refractive index of the center nearly full constructive interference for the ortho- layer is deliberately matched to that of the prism, gonal polarization in a multilayer stack at oblique i.e., n2 ¼ n0. This simple design employs only two incidence [1–10]. optical materials and depends on fewer design pa- This paper follows up on previous work by Azzam rameters, namely only one index ratio N ¼ n0=n1, and Perla [11,12] that dealt with the polarizing prop- angle of incidence ϕ0, and film thicknesses d1, d2. erties of a symmetric LHL trilayer stack (that con- A novel PBS cube (ϕ0 ¼ 45°)p isffiffiffi introduced in Section 4 1=2 sists of a high-index center-layer H of refractive that uses index ratio N ¼ð 2 þ 1Þ ¼ 1:55377 and index n2 and thickness d2 sandwiched between two center layer of quarter-wave optical thickness. This identical low-index tunnel layers L of refractive PBS, which uses tunnel layers of subwavelength index n1 and thickness d1), which is embedded in thicknesses, fully transmits the p polarization and a prism of high refractive index n0 > n1, Fig. 1. All nearly totally reflects the s polarization. Section 5 media are considered to be transparent and optically provides a brief summary of this paper. isotropic and are separated by parallel-plane bound- aries. Light is incident from medium 0 at an angle 2. Constraint on Film Thicknesses for Zero Reflection ϕ0 > the critical angle ϕc01 ¼ arcsinðn1=n0Þ of the 01 of the p or s Polarization interface, so that frustrated total internal reflection In [11] it was shown that the condition of zero reflec- (FTIR) and optical tunneling through the trilayer tion of the ν polarization (ν ¼ p, s) by an embedded stack take place. The optical field is evanescent in symmetric trilayer stack can be put in the form ℓ þ mX − nX2 X ¼ 1 1 : ð Þ 0003-6935/11/193316-05$15.00/0 2 2 1 © 2011 Optical Society of America −n þ mX1 þ ℓX1 3316 APPLIED OPTICS / Vol. 50, No. 19 / 1 July 2011 Fig. 1. Embedded symmetric trilayer stack as a PBS operating under conditions of FTIR. p and s are the linear polarizations parallel and perpendicular to the plane of incidence, respectively, and ϕ0 is the angle of incidence. When index matching (n2 ¼ n0) is satisfied, ℓ, m,and citly on the index ratio N and angle of incidence n in Eq. (1) are determined by only one Fresnel ϕ0 via the 01-interface reflection phase shift δ. reflection coefficient r01ν at the 01 interface: From the round-trip phase thickness θ2 of the high-index center layer, the corresponding metric 2 3 ℓ ¼ r01ν; m ¼ −r01νð1 þ r01νÞ; n ¼ −r01ν: ð2Þ thickness d2 is determined by −1 Upon substitution of ℓ, m, and n from Eqs. (2)in d2=λ ¼½ð4πn0Þ sec ϕ0θ2: ð7Þ Eq. (1), the condition of zero reflection of the ν polar- ization reduces to Likewise, the metric thickness of the low-index X 1 − r2 X tunnel layers is obtained from 1 by X ¼ 01ν 1 : ð Þ 2 2 3 r − X1 − X 01ν d =λ ¼ ln 1 : ð Þ 1 2 2 1=2 8 In Eqs. (1) and (3) X1 and X2 are exponential ð4πn1ÞðN − sin ϕ0Þ functions of film thicknesses given by Figure 2 shows a family of curves of the normalized Xi ¼ exp½−j4πðnidi=λÞ cos ϕi; i ¼ 1; 2: ð4Þ center-layer phase thickness θ2=π as a function of the tunnel-layer thickness parameter X1 in the range 0 ≤ In Eq. (4), ϕi is the angle of refraction in the ith layer, X1 ≤ 1 as calculated from Eq. (6) for discrete values of and λ is the vacuum wavelength of light. the interface reflection phase shift δ ¼ qπ=8, q ¼ Since FTIR takes place at the 01 interface at ϕ0, 0; 1; 2; …; 8 as a parameter. Limiting cases are iden- the light field is evanescent in the low-index tunnel tified from Eq. (6) and Fig. 2 as follows. layers, cos ϕ1 is pure imaginary, and X1 is pure real in the range 0 ≤ X1 ≤ 1. Also, because n2 ¼ n0, the angle 1. When δ ¼ 0, θ2 ¼ 0; and when δ ¼ π, θ2 ¼ 2π of refraction in the high-index center layer ϕ2 ¼ ϕ0 independent of X1. These limiting values of δ occur and X2 is a pure phase factor, i.e., jX2j¼1. at the critical angle and grazing incidence [13], re- To further clarify the thickness constraint for zero spectively, and are of little or no practical interest. reflection of the ν polarization, we substitute 2. In the limit as X1 → 0, i.e., for tunnel-layer thickness of the order of a wavelength or greater, r01ν ¼ expðjδÞ; X2 ¼ expð−jθ2Þ; ð5Þ θ2 → 2δ. Such a condition is desirable for achieving high reflectance of the unextinguished orthogonal in Eq. (3) and solve for θ2 in terms of δ, X1: polarization as discussed in Section 3. 3. When δ ¼ π=2 (01-interface reflection phase θ2 ¼ 2δ þ 2 arg½1 − X1 expð−j2δÞ: ð6Þ shift is quarter-wave), θ2 ¼ π (center layer is of quar- ter-wave optical thickness at oblique incidence) inde- Equation (6) is significant in that it is equally valid pendent of X1. This interesting special case provides for the p and s polarizations (for simplicity the sub- the basis of the unique PBS cube design described in script ν is dropped from δ) and depends only impli- Section 4. 1 July 2011 / Vol. 50, No. 19 / APPLIED OPTICS 3317 Fig. 2. (Color online) Normalized center-layer phase thickness θ2=π is plotted as a function of the tunnel-layer thickness parameter X1 using Eq. (6) at discrete values of the 01-interface reflection phase shift δ ¼ qπ=8, q ¼ 0; 1; 2; …; 8 such that zero reflection is achieved for the p or s polarization. 3. Embedded Symmetric Trilayer with Index-Matched layer stack with index-matched center layer n2 ¼ n0 Quarter-Wave-Thick Center Layer: Reflectance of the that is embedded in a cube prism of refractive index Orthogonal Polarization n0. The 01-interface reflection phase shift δp ¼ π=2 is For an embedded symmetric trilayer stack with achieved at ϕ0 ¼ 45° if the index ratio N is selected index-matched quarter-wave center layer (i.e., n2 ¼ such that [13] n0, θ2 ¼ π, X2 ¼ −1) the complex-amplitude reflection 2 2 4 coefficient for the orthogonal polarization (o) is given sin ϕ0 ¼ 1=2 ¼ðN þ 1Þ=ðN þ 1Þ: ð11Þ by Equation (11) yields ð1 − X Þ2 ðjδ Þ R ¼ 1 exp o : ð Þ o 2 9 pffiffiffi 1=2 1 − 2X1 sec δo expðjδoÞþX1 expðj2δoÞ N ¼ n0=n1 ¼ 2 þ 1 ¼ 1:55377: ð12Þ In Eq. (9) δo is the 01-interface reflection phase shift for the orthogonal polarization. The associated N ¼ 1:55377 intensity reflectance is obtained from Eq. (9)as It is interesting to note that causes maximum separation between the critical angle ð1 − X Þ4 δ − δ jR j2 ¼ 1 : ð Þ and the angle at which p s is maximum [13]. o 4 2 2 10 Table 1 lists some prism and film refractive indices ð1 − X1Þ þ 4ðsec δo − cos δoÞ X1 that satisfy Eq.
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