Design of a binary grating with subwavelength features that acts as a polarizing beam splitter Lara Pajewski, Riccardo Borghi, Giuseppe Schettini, Fabrizio Frezza, and Massimo Santarsiero
A binary diffractive optical element, acting as a polarizing beam splitter, is proposed and analyzed. It behaves like a transmissive blazed grating, working on the first or the second diffraction order, depending on the polarization state of the incident radiation. The grating-phase profile required for both polarization states is obtained by means of suitably sized subwavelength groups etched in an isotropic dielectric medium. A rigorous electromagnetic analysis of the grating is presented, and numerical results concerning its performances in terms of diffraction efficiency as well as frequency and angular bandwidths are provided. © 2001 Optical Society of America OCIS codes: 050.1970, 050.2770, 260.5430.
1. Introduction
Polarizing beam splitters 共PBSs兲 are important devices for several optical information-processing applications, such as free-space optical switching networks,1–3 read–write magneto-optic data-storage systems,4 and polarization-based imaging systems.5 A PBS should present high efficiencies, high extinction ratios, wide angular bandwidth, broad wavelength range, and compact size. Conventional PBSs are based on the use of the natural birefringence of some crystals6 or on the polarization properties of multilayer dielectric coatings.7 Other solutions, which are more compact and easier to fabricate, have also been proposed.8 –10 Diffractive PBSs are based on the polarization-selective features of diffractive optical elements. Their cost is low, and they are compact and suitable for mass reproduction. The diffractive PBSs proposed to date present narrow angular bandwidths and have limited wavelength ranges because they work in the resonance domain of diffractive optics, where, by definition, the characterL. Pajewski, R. Borghi 共
[email protected]兲, and G. Schettini are with the Dipartimento di Ingegneria Elettronica and M. Santarsiero is with the Dipartimento di Fisica e Istituto Nazionale per la Fisica della Materia, Universita` degli Studi Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy. F. Frezza is with the Dipartimento di Ingegneria Elettronica, Universita` La Sapienza di Roma, Via Eudossiana 18, I-00184 Rome, Italy. Received 11 August 2000; revised manuscript received 26 March 2001. 0003-6935兾01兾325898-08$15.00兾0 © 2001 Optical Society of America 5898
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istic features are comparable with the incident wavelength.11–13 In the present paper a binary diffractive PBS, which belongs to a new class of devices working in the paraxial domain and exploiting the polarization properties of the incident field,14 is proposed. Moreover, the device provides subwavelength modulation of the refractive index. A rigorous electromagnetic diffraction theory must be used to design such an element to understand its limitations and to reach the best compromise between fabrication feasibility and quality of the device. In Section 2, we briefly review the formulation of the rigorous electromagnetic diffraction theory that we used. We follow the conventional approach described in Refs. 15 and 16 for TE polarization 共see also the recent review paper by Turunen et al.17兲. Moreover, for TM polarization the formulation presented in Refs. 18 and 19 was used to obtain a high convergence rate. Finally, to overcome numerical problems that are due to ill-conditioned matrices obtained when boundary conditions are imposed and to improve numerical stability and efficiency, we applied the technique presented in Ref. 20 to both polarizations. In Section 3, we propose a new type of diffractive PBS composed of a grating in which an isotropic material is suitably distributed within each period. Finally, in Section 4, we give some numerical results. Concluding remarks are given in Section 5. 2. Theoretical Analysis
Let us consider the reflection and the transmission of a monochromatic plane wave with a free-space wave-
Fig. 11. Transmission efficiencies of TE2 共filled circles兲 and TM1 共open circles兲 orders plotted as functions of N for the grating of Fig. 10 when d ⫽ 10.
efficiencies of the TE2 共filled circles兲 and the TM1 共open circles兲 orders plotted as functions of N for the grating of Fig. 10 when d ⫽ 10. Finally, we analyzed the tolerance to fabrication errors of our diffractive PBS. Let us denote with bf the maximum amplitude of the random fabrication error, that is, the difference between the ideal and the actual coordinates of each transition within the grating. Figure 12 shows the transmission efficiencies of the TE2 共filled circles兲 and the TM1 共open circles兲 orders for a grating with ds ⫽ 0.05d 共M ⫽ 20兲 as bf varies from 0% to 100% of the minimum detail of the structure. Because form birefringence is an average effect, the error for each transition is random, and the number of transitions is quite high; thus the device presents a very good tolerance to fabrication errors.
It can be useful to give an explicit example showing the physical dimensions of the subwavelength feature sizes and amplitudes of our diffractive PBS. It might be preferable to use a high-refractive-index material to fabricate the grating to keep the aspect ratio 共which is defined as the ratio between the grating depth and the minimum linewidth兲 as low as possible. A good choice should be GaAs, which is frequently used in the fabrication of subwavelength transmission structures. The refractive index of GaAs at ⫽ 10 m is n1 ⫽ 3.13, and its transmittance is approximately 60%.27 The required grating depth is h ⫽ 10.77 m. If each period of the grating is divided into M ⫽ 20 subperiods 共as in the case of the grating of Fig. 10兲 the fill factor b decreases linearly within the period, starting from 0.81 and decreasing to 0.04. This decrease results in a minimum linewidth of 200 nm, and the maximum aspect ratio turns out to be approximately 50, which is an undoubtedly high value. Nevertheless, the rapid growth of nanotechnology manufacturing should make such structures achievable in the relatively near future. In any case it is always possible to find nonoptimal solutions 共for instance, by the avoidance of etching the smallest details兲 to verify current manufacturing constraints with a compromise in terms of diffraction efficiency. 5. Conclusions
A diffractive PBS that consists of a binary grating in which an isotropic dielectric material is suitably distributed in a nonuniform way within each period has been proposed. When the characteristic sizes of the structure are small enough with respect to the wavelength form-birefringence theory predicts that this device will reproduce the behavior of a blazed grating that is made up of a uniaxial birefringent material, thus separating the two polarization components of an incident field. A full-wave electromagnetic approach has been employed to characterize the grating performances. In particular, diffraction efficiencies of the desired orders have been obtained for different values of the subperiod of the subwavelength microstructures as well as for different incidence angles and wavelengths. The results obtained show that the proposed device is rather insensitive to variations of such parameters and seems to behave better than other similar structures presently described in the literature. Finally, the tolerance to random fabrication errors has been investigated, showing that the feasibility of the device should not be critical. References
Fig. 12. Average diffraction efficiencies of the transmitted orders TE2 共filled circles兲 and TM1 共open circles兲 plotted as functions of the maximum amplitude bf of the random fabrication error for ds ⫽ 0.05d 共M ⫽ 20兲. 5904
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1. G. A. De Biase, “Optical multistage interconnection networks for large-scale multiprocessor systems,” Appl. Opt. 27, 2017– 2021 共1988兲. 2. S. H. Song, E. H. Lee, C. D. Carey, D. R. Selviah, and J. E. Midwinter, “Planar optical implementation of crossover interconnects,” Opt. Lett. 17, 1253–1255 共1992兲. 3. D. M. Marom and D. Mendlovic, “Compact all-optical bypass– exchange switch,” Appl. Opt. 35, 248 –253 共1996兲. 4. M. Ojima, A. Saito, T. Kaku, M. Ito, Y. Tsunoda, S. Takayama,
5.
6. 7.
8.
9.
10.
11.
12.
13.
14.
15.
and Y. Sugita, “Compact magneto-optical disk for coded data storage,” Appl. Opt. 25, 483– 489 共1986兲. P. Kunstmann and H. J. Spitschan, “General complex amplitude addition in a polarization interferometer in the detection of pattern differences,” Opt. Commun. 4, 166 –168 共1971兲. M. Born and E. Wolf, Principles of Optics, 7th ed. 共extended兲 共Cambridge U. Press, Cambridge, UK, 1999兲. J. L. Pezzaniti and R. A. Chipman, “Angular dependence of polarizing beam-splitter cubes,” Appl. Opt. 33, 1916 –1929 共1994兲. R. M. A. Azzam, “Polarizing beam splitters for infrared and millimeter waves using single-layer-coated dielectric slab or unbacked films,” Appl. Opt. 25, 4225– 4227 共1986兲. K. Shiraishi and S. Kawakami, “Spatial walk-off polarizer utilizing artificial anisotropic dielectrics,” Opt. Lett. 15, 516 –518 共1990兲. H. Tamada, T. Doumuki, T. Yamaguchi, and S. Matsumoto, “A wire-grid polarizer using the s-polarization resonance effect at the 0.8-m wavelength band,” Opt. Lett. 22, 419 – 421 共1997兲. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, and M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206 –1213 共1992兲. M. Schmitz, R. Bra¨ uer, and O. Bryngdahl, “Gratings in the resonance domain as polarizing beam splitters,” Opt. Lett. 20, 1830 –1831 共1995兲. P. Lalanne, J. Hazart, P. Chavel, E. Cambril, and H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 共1999兲. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 共2000兲. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206 –1210 共1978兲.
16. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. 共North Holland, Amsterdam, 1984兲, Vol. XXI, pp. 1– 68. 17. J. Turunen, M. Kuittinen, and F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics E. Wolf, ed. 共North Holland, Amsterdam, 2000兲, Vol. XL, pp. 327– 341. 18. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779 –784 共1996兲. 19. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019 –1023 共1996兲. 20. D. A. Pommet, M. G. Moharam, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 共1995兲. 21. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870 –1876 共1996兲. 22. C. Brosseau, Fundamentals of Polarized Light 共Wiley, New York, 1998兲. 23. F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “On anisotropic gratings,” Atti Fond. Giorgio Ronchi LIV, 59 – 67 共1999兲. 24. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 共1996兲. 25. J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 共1996兲. 26. M. L. Lee, T. Lalanne, and P. Chavel, “Blazed-binary diffractive elements with period much larger than the wavelength,” J. Opt. Soc. Am. A 17, 1250 –1255 共2000兲. 27. M. Bass, ed., Handbook of Optics, Devices, Measurements, and Properties, 2nd ed. 共McGraw-Hill, New York, 1995兲, Vol. 2.
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