Design and Analysis of Resonant Leaky-mode Broadband Reflectors

Transcription

1 846 PIERS Proceedings, Cambridge, USA, July 6, 8 Design and Analysis of Resonant Leaky-mode Broadband Reflectors M. Shokooh-Saremi and R. Magnusson Department of Electrical and Computer Engineering, University of Connecticut 37 Fairfield Road, U-57, Storrs, CT , USA Abstract Devices based on the guided-mode resonance (GMR) effect are very promising elements in the areas of optics and electromagnetics. They can provide variety of spectral responses only founded on periodically patterning of a single optical layer on a substrate or as a free-standing membrane. Although the main manifestation of the GMR effect is sharp resonances in the reflection spectra, by proper selection of device parameters, attainment of variety of optical spectral responses such as narrow bandpass/bandstop filters, broadband reflectors, polarizers and so on is possible. Since in this kind of elements, light is coupled into waveguide leaky modes through a subwavelength grating structure, the device works in the second (leaky) stopband. Response of such devices is highly dependent on the modal characteristics of the guiding layer. Broadband high reflectors, based on periodically patterned single layers (with substrate or as a membrane), are promising and attractive elements in the area of optical devices and have recently found practical application as top mirror in tunable vertical cavity surface emitting lasers (VCSELs). In this paper, a single layer, strongly-modulated GMR-based broadband high reflector is designed for.45. µm band for TE and TM polarizations using particle swarm optimization (PSO) technique; a robust, easy to implement evolutionary technique inspired from the behavior of particles in a swarm searching for their requirements resources. A silicon-oninsulator (SOI) structure has been chosen in which the binary patterned silicon layer act as both grating and waveguide. The spectral and modal characteristics of these elements are analyzed utilizing rigorous electromagnetic techniques like rigorous coupled-wave analysis (RCWA) and modal techniques. The designed reflector for TM and TE polarizations provide 5 nm and 5 nm bandwidth at > 99% reflectance, respectively. Also, showing the reflection and transmission maps as well as the mode profiles, we investigate the effect of refractive index modulation and the thickness of the grating on device s spectral response.. INTRODUCTION Devices based on the guided-mode resonance (GMR) effect are promising in areas of optics and optical engineering [, ]. They can provide variety of spectral responses based on periodically patterning of a single optical layer on a substrate or as a free-standing membrane [3, 4]. Although the main manifestation of the GMR effect is a sharp, Lorentzian-shape resonance in the reflection spectra, by proper selection of device parameters, attainment of variety of optical spectral responses is possible [3 5]. Recently, there have been reports on the potential application of singlelayer, one-dimensional guided-mode resonance (GMR) gratings as optical elements like narrow bandpass/bandstop filters, broadband reflectors, and polarizers [3]. These devices are based on the resonant coupling of an incident light beam into a light-guiding layer through a surface relief diffraction grating. The period of the grating, its thickness, and the refractive indices of comprising materials and their distribution along the period, as well as the refractive indices of surrounding media, are basic parameters of a GMR element. These parameters can greatly influence the behavior of the device, since GMR devices are proven to be highly sensitive to their structural parameters. In a single layer device, the subwavelength grating acts as the incident light coupler as well as the light guiding medium. The subwavelength grating couples the incident light into leaky waveguide mode(s), provided that the phase matching condition is satisfied. Since the light is coupled into leaky modes, the device works in the second (leaky) stopband [6]. The spectral response of such devices is highly dependent on the modal characteristics of the guiding layer. Broadband high reflectors, based on periodically-patterned single layers (on a substrate or as membranes), are useful in many optics applications and have recently found practical application as mirrors in vertical cavity surface emitting lasers (VCSELs) [7]. The objective of this paper is to provide insights into the electromagnetic properties of these kinds of optical elements. Accordingly, in this paper, broadband high reflectors are designed for the.45. µm band using particle swarm optimization (PSO) technique and their spectral and modal behavior for both TE and TM polarizations is studied and discussed.

2 Progress In Electromagnetics Research Symposium, Cambridge, USA, July 6, DEVICE STRUCTURE AND DESIGN Figure illustrates the structure of the device. It is basically a silicon-on-insulator (SOI) element. The grating layer, which also acts as the waveguide, is defined by its period (Λ), thickness (d), and filling factor (F ). For designing the broadband reflector, normal incidence and both TE and TM polarizations are considered. The grating period has two sections consisting two materials with different refractive indices; one of them has higher refractive index (n H > n L ). y x I Λ Si R d z F -F T Silica Figure : Schematic of a typical single-layer, silicon-on-insulator GMR device. Λ, d and F are period, thickness, and filling factor, respectively. The incidence medium is air (n =.) and the substrate is silica (n =.48). The case of normal incidence is treated here. The particle swarm optimization (PSO) technique is utilized to design the broadband reflector. The design target is set to be % reflectance over the.45 µm to. µm band. The parameters to be optimized by the PSO to achieve the target reflectance are taken to be Λ, d, and F. The basics of the PSO technique for designing diffraction gratings have been reported elsewhere [8]. 3. FEATURES OF THE BROADBAND HIGH REFLECTORS 3.. TM Polarization Figures and show the reflectance and transmittance spectra (linear and logarithmic scales) of the designed two-part, SOI, broadband high reflector for TM polarization, in which Λ =.766 µm, d =.49 µm, and F =.76. The width of the high reflectance band (R >.99) is 5 nm over.45 to. µm range. As seen in Fig., there are three transmittance dips inside the reflection band, each of which corresponds to a guided-mode resonance. It should be noted that under guidedmode resonance conditions, in which the incident light is coupled into a leaky waveguide mode, the transmittance theoretically approaches zero. Therefore, Fig. shows co-existence, or a blend, of three modes. Figure 3 (c) illustrates the amplitude of the magnetic (modal) field (H y (z)) inside the grating structure and also in the surrounding media for the three resonances at.495,.6, and.839 µm and for the zeroth, first and second diffraction orders. As displayed, the modal Figure : Reflectance and transmittance spectra linear and logarithmic scale of the broadband TM reflector.

3 848 PIERS Proceedings, Cambridge, USA, July 6, 8 profiles for the first diffraction order (S ) at three resonant wavelengths show mixed mode states. Field amplitude 3.5 S S λ µ 3 S 4 λ µ S 3 λ µ S S =.495 m =.6 m =.839 m Field amplitude Field amplitude S S S Position along propagation (µ m) Position along propagation (µ m) Position along propagation (µ m) Figure 3: Amplitude of the magnetic (modal) field inside the grating structure and the surrounding media for the three resonances. Figure 4: Color-coded reflectance map R (λ, d) drawn versus wavelength and grating thickness. Transmittance map T (λ, d) in db. Figure 4 displays a color-coded reflectance map R (λ, d) drawn versus wavelength and grating thickness. This map shows the qualitative modal behavior of the grating structure, which is highly dependent on the thickness. As evident from Fig 4, S shaped high-reflection areas Position along propagation (µ m) Position along propagation (µ m) Figure 5: Modal field profiles for two resonance points far from the broadband reflection region, d =. µm and λ =.345m, d =.87 µm and λ =.99 µm.

4 Progress In Electromagnetics Research Symposium, Cambridge, USA, July 6, show the evolution of the reflection band from a nearly single narrow resonance to a broad one. Figure 4 illustrates the associated transmittance versus wavelength and thickness in db. As seen, the transmittance dip also has an S shape. The calculations for the reflection spectra are done using rigorous coupled-wave analysis (RCWA) [9] and modal analysis techniques []. Both methods yield results that are in complete agreement. Figure 5 shows the modal field profiles for two points on the reflection band in Fig. 4 but far from the broadband reflection region. As seen, for thicknesses well below the design thickness, TM is the main excited leaky mode, while above the design thickness TM is excited. TM,3.5 TM, d (µm) TM,.5 TM, λ (µm) Figure 6: R (λ, d) map for low refractive index modulation; n H =., n L =.347. Calculated modal curves for the first four modes excited by the first diffraction order considering a thin film waveguide with n f =.9. The R (λ, d) map in Fig. 4 can be considered as a qualitative counterpart for modal curves of an equivalent homogeneous slab waveguide. Modal curves can be obtained by solving the eigenvalue equation of the equivalent homogenous slab waveguide system [4]. To investigate the modal behavior of this system and provide a clearer view of the physics, we reduce the index modulation of the structure while keeping the average refractive index of the waveguide layer fixed at.734 (zeroorder effective index) []. This reduces the linewidth of the resonances and enhances the visibility of the modal curves. By considering n H =. and n ave =.734, n L is calculated to be.347. Figure 6 shows the R(λ, d) map for this low-index modulation structure as well as the calculated R & T - -4 Amplitude.5.5 S S S λ ( µ m) - - Position along propagation ( µ m) Figure 7: Reflectance (solid line) and transmittance (dashed line) spectra of the designed reflector for TE polarization (in logarithmic scale). Amplitude of electric field in the grating layer and incident and substrate media (λ =.559 µm).

5 85 PIERS Proceedings, Cambridge, USA, July 6, 8 modal curves for the first four modes excited by the first diffraction order considering n f =.9. It is seen that there is a good qualitative agreement between the two graphs. Also, this shows that in single layer broadband high reflector, a modal mixture takes place when a high refractive index contrast is present and will make different modes contribute to the broadband reflection. The value of nf is also qualitatively in reasonable agreement to the second-order effective index of the equivalent homogenous layer which is.8 []. 3.. TE Polarization Broadband high reflector can be also designed to work with TE polarized incident light. Using PSO as the design tool, a two-part period reflector is designed for the.45.7 µm band based on a SOI structure. The designed parameters are: Λ =.986 µm, d =.8 µm, and F =.39. Figure 7 illustrates the reflectance and transmittance spectra of the device in logarithmic scale. The transmission dip, which shows the resonance wavelength, falls at.559 µm. The reflection band of the filter for R >.99 is 5 nm. Figure 7 shows the amplitude of electric field in the grating layer as well as in incident and substrate media. As evident from this figure, the leaky mode has TE characteristics. Figure 8 displays a color-coded map R (λ, d), which qualitatively represents the modal behavior of the device. As seen in this figure, at the design thickness (d =.8 µm), the reflection spectrum exhibits maximum width. Figure 8 shows the T (λ, d) map in db, in which the blue traces represent the position of the resonance. In this structure, in part due to higher modal cut-off wavelength than for TM, a kind of folding back regions appear. The zero-order effective medium refractive index (average refractive index) of the structure is.59 and the second-order effective d (µm) (c) λ (µm) (d) Figure 8: R (λ, d) map for the reflector designed for TE polarization. T (λ, d) map in db. (c) Reflection map for the reduced refractive index modulation structure (n H =.3 and n L =.86) and (d) Modal characteristic curves for the equivalent homogeneous layer with refractive index of n f =.696.

6 Progress In Electromagnetics Research Symposium, Cambridge, USA, July 6, 8 85 medium refractive index is Keeping the average refractive index fixed, by reducing the refractive index contrast, the reflection map can qualitatively represent the leaky-mode curves of the equivalent homogenous layer. Figure 8(c) shows the reflection map for the reduced modulation structure (n H =.3 and n L =.86) and Fig. 8(d) illustrates the modal characteristic curves for the equivalent homogeneous layer with refractive index assumed to be n f =.696, which is the second order effective medium refractive index of the structure at.7 µm. There is a clear resemblance between the two figures, showing the modal origin of the broad reflection spectra, as in the TM case. 4. CONCLUSIONS In this paper, we have presented the basics of the broadband high reflector rooted in guided-mode resonance effect. The modeling and simulation results of this device show that the high refractive index contrast of the grating can be responsible for exciting a blend of leaky modes and providing the maximum reflection linewidth for the optimal design parameters. It is important to use effective inverse methods to determine the parameters providing the proper spectra. Broadband high reflectors with two-part periods have been designed for both TE and TM polarizations. The design and analysis results show that a maximum reflection linewidth can be obtained by incorporating a mixture of TE or TM modes. ACKNOWLEDGMENT This material is based, in part, upon work supported by NSF under Grant No. ECS REFERENCES. Magnusson, R. and D. Shin, Diffractive optical components, Encyclopedia of Physical Science and Technology, 3rd ed., Vol. 4, 4 44,.. Kikuta, H., H. Toyota, and W. Yu, Optical elements with subwavelength structured surfaces, Optical Review, Vol., 63 73, Ding, Y. and R. Magnusson, Resonant leaky-mode spectral-band engineering and device applications, Opt. Express, Vol., , Wang, S. S. and R. Magnusson, Theory and applications of guided-mode resonance filters, Appl. Opt., Vol. 3, 66 63, Tibuleac, S. and R. Magnusson, Reflection and transmission guided-mode resonance filters, J. Opt. Soc. Am A, Vol. 4, 67 66, Ding, Y. and R. Magnusson, Band gaps and leaky-mode effects in resonant photonic-crystal waveguides, Opt. Express, Vol. 5, , Huang, M. C. Y., Y. Zhou and C. J. Chang-Hasnain, A surface-emitting laser incorporating a high-index-contrast subwavelength grating, Nature Photonics, Vol., 9, Shokooh-Saremi, M. and R. Magnusson, Particle swarm optimization and its application to the design of diffraction grating filters, Opt. Lett., Vol. 3, , Gaylord, T. K. and M. G. Moharam, Analysis and applications of optical diffraction by gratings, Proc. IEEE, Vol. 73, , Peng, S. T., T. Tamir, and H. L. Bertoni, Theory of periodic dielectric waveguides, IEEE Trans. Microwave Theory Tech., Vol. 3, 3 33, 975.