Optics Communications 261 (2006) 13–18 www.elsevier.com/locate/optcom Total internal reflection diffraction grating in conical mounting L. Eisen *, M.A. Golub, A.A. Friesem Department of Physics of Complex Systems, Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel Received 17 August 2005; received in revised form 16 November 2005; accepted 28 November 2005 Abstract The main conditions and parameters for obtaining surface relief total internal reflection diffraction gratings in conical mounting are presented. Calculated and experimental investigations reveal that there are ranges of grating periods, incidence angles, diffraction angles and gratings depths for which such gratings could be obtained, both for TE and TM polarizations. With optimized grating parameters the diffraction efficiency of the total internal reflection diffraction gratings can be greater than 90%. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.40.Lx; 42.25.Fx; 42.40.Eq; 42.79.Dj Keywords: Diffraction grating; Holographic grating; Total internal reflection 1. Introduction ical mounting that have high diffraction efficiency and where the diffraction angle into the first order is equal During the last three decade surface relief diffraction to that of the incidence angle, so the diffracted light con- gratings became an integral part of many modern optical tinues to propagate by TIR. Such gratings in conical systems and devices including imaging systems [1,2], array mounting can be exploited in planar optics configurations illuminators [3], pulse shapers [4], and beam splitters and [9,10]. deflectors [5,6]. Some of the investigations were extended to include surface relief diffraction gratings that also 2. Basic conditions for TIR diffraction gratings in conical involve total internal reflection (TIR) in the substrate on mounting which the gratings are recorded. These included theoretical investigations of surface relief dielectric gratings in classical A typical three-dimensional conical diffraction geome- (Littrow) mounting [7] as well as subsequent experimental try for TIR diffraction gratings is schematically presented investigations [8]. In such classical mounting, the grating in Fig. 1. The surface relief TIR diffraction grating is vector and all diffraction orders lie in the plane of recorded on a transparent substrate of refractive index incidence. nsub that is surrounded with a material (typically air) In this paper, we investigate surface relief TIR diffrac- of refractive index nsup, where nsub > nsup. A linearly tion gratings in conical mounting, where the plane of polarized monochromatic light beam, propagating inside incidence does not contain the grating vector, and deter- the substrate, is obliquely incident onto the TIR diffrac- mine the relevant conditions and parameters for such tion grating at the polar angle hinc and azimuthal angle gratings. These are then used to design and experimen- /inc. The TE polarization is perpendicular to the plane tally record surface relief TIR diffraction gratings in con- of incidence and the TM polarization lies in the plane of incidence. * Corresponding author. Tel.: +972 8 9344433; fax: +972 8 9344109. The well known grating equations for the diffraction E-mail address: [email protected] (L. Eisen). gratings in conical mounting [11] can be written as 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.11.064 14 L. Eisen et al. / Optics Communications 261 (2006) 13–18 TIR grating y 3. Design and optimization procedures n sup Using Eqs. (4) and (5), we calculated the minimum and x θ θ diff ,1 maximum TIR diffraction grating periods Kmin and Kmax as inc a function of the polar incidence angle hinc for three TM TE selected azimuthal incidence angles /inc. For these calcula- n φ tions we assumed a sinusoidal surface relief diffraction sub inc φ diff ,1 grating formed on a glass substrate with refractive index nsub = 1.52, free-space wavelength of incident light as k = 0.6328 lm, and the selected three azimuthal incidence z angles were /inc =30°,45°,60°. The calculated results Fig. 1. Typical conical diffraction geometry for the TIR diffraction are presented in Fig. 2 as the boundaries of the allowable gratings. range of TIR diffraction grating periods. As shown, the range of TIR diffraction grating periods differs at each polar incidence angle hinc and each azimuthal incidence 2p angle /inc. This range broadens and the minimum period nsub sinðhdiff;mÞ cosð/diff;mÞ k Kmin is larger as the incidence angle /inc increases. 2p 2p We now consider a specific example of TIR diffraction ¼ nsub sinðhincÞþ m cosð/ Þ; ð1Þ k K inc grating in conical mounting where the polar diffraction 2p 2p angle h equals that of the incidence angle h , and n sinðh Þ sinð/ Þ¼ m sinð/ Þ; ð2Þ diff,1 inc k sub diff;m diff;m K inc the azimuthal diffraction angle differs from that of the inci- where k is the wavelength of the incident light, K the grat- dence angle. With hinc = hdiff,1, Eqs. (1) and (2) lead to an expression for the TIR grating period K(h = h ), as ing period, hdiff,m the polar diffraction angle of m diffraction inc diff,1 order, and /diff,m the azimuthal diffraction angle of the m k cosð/incÞ diffraction orders. The condition for TIR is that no trans- Kðhinc ¼ hdiff;1Þ¼ n sinðh Þð1 þ cosð/ ÞÞ mitted diffraction orders exist while reflected orders do [7], sub inc diff;1 k sinð/ Þ so that ¼ inc . ð7Þ n sin h sin / nsup sub ð incÞ ð diff;1Þ 6 jjsinðhincÞ 6 1. ð3Þ nsub According to Eq. (7) the azimuthal diffraction angle is the Using Eqs. (1) and (2) together with Eq. (3), we determined twice that of the incident angle that at certain wavelength, incidence angles and refractive / ¼ 2/ . ð8Þ indices, there will be an upper and lower bounds for the diff;1 inc range of grating periods beyond which the TIR condition Alternatively, Eq. (7) could be readily obtained from will not hold. Specifically, the maximum and minimum direct observation of the TIR diffraction grating geometry allowable periods Kmax and Kmin are of Fig. 1. The calculated results for K(hinc = hdiff,1)asa function of hinc are shown by the solid curve in Fig. 2. krffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kmax ¼ As evident this period decreases to K at the higher inci- 2 min nsup 2 nsub sinðhincÞ cosð/incÞþ 2 2 À sin ð/incÞ dence angles hinc for all three azimuthal diffraction angles. n sin ðhincÞ sub Using rigorous coupled wave analysis algorithms [13],we ð4Þ initially numerically calculated first order diffraction effi- and ciency for a TIR diffraction grating with K(hinc = hdiff,1)as a function of grating depth d for a number of selected inci- hik Kmin ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . dence angles and /diff,1 = 120°. The results for TE and TM n sinðh Þ cosð/ Þþ 1 À sin2ð/ Þ polarizations are presented in Fig. 3. As evident, the diffrac- sub inc inc sin2ðh Þ inc inc tion efficiency has alternating maxima and minima, with the ð5Þ highest occurring at grating depths that range from 1 lmto In this range of grating periods only the 0 and 1st reflected 2 lm for TE polarization. For example, with an incidence diffraction orders exist ( i.e. m = 0,1). Note that in classical angle of and a grating period of K(hinc = hdiff,1) = 0.6224 lm, mounting, where /inc = 0, Eqs. (4) and (5) readily reduce to the diffraction efficiency is about 0.9831 for a grating depth the usual simple relation of Kmax,L and Kmin,L for Littrow of d = 1.3900 lm with a corresponding aspect ratio of mounting [12],as K/d = 2.23. For TM polarization, high diffraction efficien- cies occur at grating depths ranging between 0.2 and k K ¼ ; 0.5 lm, indicating a significantly lower aspect ratio which max;L n 2 sup is more practical than with TE polarization. k We then calculated the optimal TIR diffraction grating Kmin;L ¼ . ð6Þ 2nsub depth d that will provide the maximum first order diffrac- L. Eisen et al. / Optics Communications 261 (2006) 13–18 15 0.38 0.36 0.34 m) μ 0.32 0.30 0.28 Grating period ( 0.26 a 0.24 0.22 40 45 50 55 60 65 70 75 80 85 90 a Polar incidence angle (deg) 0.60 0.56 0.52 m) μ 0.48 0.44 b 0.40 Fig. 3. First order diffraction efficiencies for the TIR gratings with four different periods K(hinc = hdiff,1) and four different incidence angles hinc as a Grating period ( 0.36 function of grating depth d for /diff,1 = 120° and TE and TM light polarizations: (a) TE polarization; (b) TM polarization. 0.32 0.28 40 45 50 55 60 65 70 75 80 85 90 hinc =45° and a corresponding grating period K(hinc = hdiff,1) = 0.3400 lm, the diffraction efficiency is about the b Polar incidence angle (deg) same for both TE and TM polarizations, at a fixed opti- mized grating depth of d = 0.4200 lm. The procedure for 0.90 optimizing the grating depth for sinusoidal profiles can 0.84 be extended for optimizing the depth of gratings with more 0.78 general surface profiles. Finally, we considered the spectral dispersion of a TIR m) 0.72 (μ diffraction grating in conical mounting. We started with 0.66 the grating dispersion relation for the first diffraction order, 0.60 which was derived from Eq. (1),as 0.54 dhdiff;1 cosð/incÞ Grating period ¼ 0.48 dk K cosðh Þ cosð/ Þ diff;1 diff;1 0.42 cosð/ Þ 0.36 inc .