January 15, 2013 / Vol. 38, No. 2 / OPTICS LETTERS
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Controlling the spectral width in compound waveguide grating structures Wenxing Liu,1 Yunhui Li,1,3 Haitao Jiang,1 Zhenquan Lai,2 and Hong Chen1,* 1
Key Laboratory of Advanced Micro-structure Materials, Ministry of Education, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2
Department of Physics, Nanchang University, Nanchang 330031, China 3 e-mail:
[email protected] *Corresponding author:
[email protected]
Received November 19, 2012; revised December 7, 2012; accepted December 11, 2012; posted December 12, 2012 (Doc. ID 180189); published January 9, 2013 Spectral responses in compound waveguide grating structures composed of two ridges with identical widths in each period are presented. For the proposed structures, we show that the spectral width of the guided-mode resonance (GMR) can be tailored in an independent way without modifying the spectral lineshapes and sideband levels. The method described in this Letter offers a very simple and efficient way to control spectral responses in GMR structures. © 2013 Optical Society of America OCIS codes: 050.0050, 050.6624, 310.0310, 310.2790.
The guided-mode resonance (GMR) refers to rapid variations in the intensity of propagating waves that are diffracted by a waveguide grating structure. This resonance effect occurs within small parameter ranges such as the wavelength, the angle of incidence, or the refractive index. Physically, resonant grating structures utilize a periodic modulation of the refractive index to couple the diffracted fields from an incident wave into a leaky guided mode supported by the structure [1,2]. GMR structures have a multilayer configuration, the most basic of which is composed of a substrate, a dielectric waveguide layer, and an additional layer in which a grating is etched, as shown in Fig. 1(a). When such a structure is illuminated with an external incident light, the resonances occur, while evanescent diffraction orders of the grating are phased matched to the guided modes localized in the waveguide layer. Generally, the spectral width of the GMR depends mainly on the mode confinement and the coupling process in the resonant structure. Several geometrical parameters have been exploited to tailor the spectral width, such as refractive index [1,3], fill factor [4], and layer thickness [5–9]. In prior publications, the most effective method of tailoring the spectral width of the GMR was proposed by introducing a low-index separation layer between the grating and the waveguide layers to control the coupling between the grating and the leaky guided mode [7]. Although this is true, the decreased spectral width comes at the price of increased layer thickness. Specifically, the increased layer thickness would lead to the mechanical and thermal instability for the device illuminated with the high-intensity pulsed laser. In this Letter, we resort to a different GMR configuration, in which each period is composed of two grating ridges with identical width, as shown in Fig. 1(b). In essence, the four-part period grating enables a rich set of Fourier harmonics with concomitant emergence of additional spectral features not available for the classic two-part period grating. Based on the configurations, we demonstrate that an ultra-narrow linewidth resonant reflection/transmission can be obtained easily by adjusting grating modulation profile. 0146-9592/13/020163-03$15.00/0
For the proposed structure in Fig. 1(b), the spatially modulated permittivity in the grating region can be expanded into Fourier series as X 2nπx ; (1) εn exp i εx Λ n where the grating Fourier harmonics εn are given by ε0 1 − f b − f c n2H f b f c n2L ;
(2)
sinnπ1 − f c − sinnπf b ; nπ n 1; 2; … N…:
(3)
εn n2H − n2L
According to the rigorous coupled-wave theory [10], the grating Fourier harmonics εn control the amplitude of the evanescent diffraction fields and are responsible for the mutual interaction of the evanescent diffraction
Fig. 1. Scheme of the two types of waveguide gratings. (a) two-part period and (b) four-part period. The first layer is a grating; the second layer is a waveguide layer. nH and nL are the high and low refractive indices of the grating; n2 , nC , and nS are the refractive indices of the waveguide layer, cover layer and substrate; d1 and d2 are the layer thicknesses; Λ is the grating period; f a , f b , and f c are the fill factors. © 2013 Optical Society of America
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fields. Therefore, grating Fourier harmonics εn are related to the coupling between evanescent diffraction fields and the leaky guided modes. Figure 2 illustrates the amplitude of the grating Fourier harmonics jεn j as functions of fill factor f b . Here, the total width of the two grating grooves in each period is kept constant, that is, f b f c 0.4. As shown in Fig. 2, jεn j is highly dependent on the fill factor f b , which means that the coupling can be controlled by adjusting the fill factor f b . With appropriate design for waveguide gratings the leaky guided modes can be excited solely through the first evanescent diffracted order of the grating. In the case of the coupling it is mainly determined by the first Fourier harmonic ε1 . Generally, smaller jε1 j indicates a weaker coupling, which, in turn, yields decreased spectral width. In principle, the spectral lineshapes and sideband levels of the GMR are mainly determined by the equivalent homogeneous layers. For the proposed structure in Fig. 1(b), the refractive index of the grating layer could be represented using an effective medium theory approximation [11], where the value for the effective medium was mainly determined by the zero-order effective relative permittivity ε0 . From Eq. (2), it is not difficult to find that ε0 is fixed at some value (i.e., ε0 3.2815) for different fill factor f b , which indicates that these spectral characteristics will almost not change when the fill factor f b changes. To identify the theoretical analysis, we illustrate the reflection responses of the compound waveguide grating structure with various fill factor f b for the normal incidence TE-polarized light (electric-field parallel to y direction), as shown in Fig. 3. The spectral responses with symmetrical lineshape and low-reflection sideband are achieved by combining GMR effects in waveguide grating with antireflection effects of equivalent homogeneous layers [12]. The spectral analysis of the dielectric waveguide gratings can be exactly performed by the rigorous coupled wave analysis (RCWA) [13], the rigorous integral equation method [14], and the boundary element method [15]. In our results, numerical calculations are performed using RCWA. To ensure excellent numerical convergence of the results, 10 Fourier harmonics are retained in our calculations. The calculated spectral responses reveal a single resonance peak associated with the leaky guided mode TE0 excited by first evanescent diffraction order. As shown in Fig. 3, the decreased spectral widths are achieved perfectly by adjusting the fill factor f b and the spectral symmetry and sideband suppression are nearly preserved. At the fill factor of f b 0 and f b 0.19, the spectral widths are 1.6 × 10−2 and
Fig. 2. (Color online) Amplitude of the Fourier grating harmonics jεn j as functions of fill factor f b for the four-part period grating profile. The parameters are nH 1.95 and nL 1.
Fig. 3. (Color online) Reflection responses of the compound waveguide grating structure with various fill factor f b for the normal incidence TE-polarized light. The remaining parameters are nc nL 1 (air), nH n2 1.95 (HfO2 ), ns 1.46 (SiO2 ), Λ 0.4 um, f a 0.3, d1 0.26 um, and d2 0.12 um.
4 × 10−6 um, respectively. Here, the spectral width is nearly decreased by 4000 times. And the spectral width can be further decreased by increasing f b and reaches an exactly vanishing width at f b 0.2. It is easy to know that jε1 j equals to zero at f b 0.2 (see Fig. 2) and hence the guided mode will not couple with the incident light. Generally, the GMR is highly sensitive to the changes of the structural parameters. Though the changes are minor, the shift of the resonance peak may be large. In Fig. 3, it is interesting to note that the resonance wavelength almost does not change when the fill factor f b changes. The unusual phenomenon can be explained by applying the theory of periodic dielectric waveguides [16]. The equivalent homogeneous slab waveguides corresponding to the waveguide grating structures remain almost unchanged, as the zero-order effective relative permittivity of the grating is kept constant for different fill factors f b . This leads to the dispersion relations of the leaky guided modes supported by the structures, which are nearly identical. Therefore, the same resonance wavelength is nearly preserved for different structures. Figures 4(a) and 4(c) display the electric field distributions at the wavelength of the center resonance for fill
Fig. 4. (Color online) Effect of the fill factor f b . Electric field distributions (left) and amplitude of the electric modal fields (right) at the wavelength of the center resonance for fill factor f b 0 (a), (b) and f b 0.16 (c), (d). White (gray) lines indicate interfaces of the different layers.
January 15, 2013 / Vol. 38, No. 2 / OPTICS LETTERS
Fig. 5. (Color online) (a) Single-layer waveguide grating with four-part period. (b)–(e) Reflection responses of the structure with various fill factor f b for the normal incidence TE-polarized light. The total fill factor of the two grating grooves in each period is kept at 0.3. The remaining parameters are nc nL 1 (air), nH 3.48 (Si), ns 1.48 (SiO2 ), Λ 0.862 um, f a 0.35, and d 0.16 um.
factor f b 0 and f b 0.16, respectively. Obviously, the enhanced electric field localized in the waveguide layer is obtained by adjusting the grating modulation profile. To gain further physical insight into the origin of the different electric field intensity, we calculate the amplitude of the electric modal fields at the wavelength of the center resonance for the zeroth, first (evanescent), and second (evanescent) diffraction orders, as shown in Figs. 4(b) and 4(d). The zeroth field and the second evanescent field remain unchanged in both cases and do not contribute to intensity enhancement, while the first evanescent field is responsible for intensity enhancement in the structure. It is noteworthy that the first evanescent field shows a predominant TE0 shape, which directly demonstrates that the resonance reflection occurs by excitation of leaky guided mode TE0 by the first evanescent diffracted order. The same principles for controlling the spectral widths can also be applied to other GMR geometries. Figure 5(a) shows a single-layer compound waveguide grating structure. Note that the total fill factor of the two grating grooves in each period is kept at 0.3, in the case jε1 j equals to zero at f b 0.15, and thus an ultra-narrow linewidth resonant reflection/transmission can be obtained when the fill factor f b approaches 0.15. Formerly, Ding and Magnusson have shown that a similar single-layer four-part period GMR structure, in which each period is composed of two grating ridges with different widths, enables diverse spectral responses, including bandpass filters, polarizers, and polarization-independent elements [17]. Here, the spectral responses appear asymmetric lineshapes, as shown in Figs. 5(b)–5(e). The asymmetric resonance responses belong to a large family of characteristically asymmetric Fano resonances, which occur from interference between a discrete GMR in the waveguide grating structures and a continuum Fabry–Perot etalon resonance of the equivalent homogeneous structures [18,19]. Over the past few years, Fano resonances have attracted considerable attention in a number of physical systems by
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virtue of their various applications [20]. Intriguingly, the design presented here offers a much simpler geometry and more efficient way to realize the sharp Fano resonance. In conclusion, a method of tailoring the spectral width in compound waveguide grating structures is presented. The proposed structures, in which each period is composed of two grating ridges with identical widths, can result in significant reduction of the spectral width without modifying the spectral lineshapes and sideband levels. Although the structures proposed in this Letter were illuminated by the normal incidence TE-polarized light, it should be pointed out that the polarization and angle of the incident light are not mandatory for the purpose of achieving the decreased spectral width in our designs. Such compound waveguide grating structures may have a variety of potential applications. Here are a few. First, the structure with an ultra-narrow linewidth resonance reflection may serve as a perfect filter or laser reflector for a particular wavelength. Second, controlling the spectral width of the resonance or the coupling strength of the leaky guided modes offers a possibility of a strong enhancement of electromagnetic fields within the structures and, hence, enhancement of fluorescence or nonlinear phenomena. This work is supported by the National Basic Research Program (973) of China (No. 2011CB922001), National Natural Science Foundation of China (Nos. 11234010, 51007064, 61137003, and 11074187), Program of the Shanghai Science and Technology Committee (No. 11QA1406900), and Program of National Laboratory for Infrared Physics (No. 201005). References 1. R. Magnusson and S. S. Wang, Appl. Phys. Lett. 61, 1022 (1992). 2. A. Sharon, D. Rosenblatt, and A. A. Friesem, J. Opt. Soc. Am. A 14, 2985 (1997). 3. T. Sang, Z. Wang, J. Zhu, L. Wang, Y. Wu, and L. Chen, Opt. Express 15, 9659 (2007). 4. D. Shin, S. Tibuleac, T. A. Maldonado, and R. Magnusson, Opt. Eng. 37, 2634 (1998). 5. D. Rosenblatt, A. Sharon, and A. A. Friesem, IEEE J. Quantum Electron. 33, 2038 (1997). 6. S. M. Norton, T. Erdogan, and G. M. Morris, J. Opt. Soc. Am. A 14, 629 (1997). 7. S. T. Thurman and G. M. Morris, Appl. Opt. 42, 3225 (2003). 8. W. Liu, Z. Lai, H. Guo, and Y. Liu, Opt. Lett. 35, 865 (2010). 9. Y. L. Tsai, J. Y. Chang, M. L. Wu, Z. R. Tu, C. C. Lee, C. M. Wang, and C. L. Hsu, Opt. Lett. 35, 4199 (2010). 10. M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. 71, 811 (1981). 11. S. M. Rytov, Sov. Phys. JETP 2, 466 (1956). 12. S. S. Wang and R. Magnusson, Opt. Lett. 19, 919 (1994). 13. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, J. Opt. Soc. Am. A 12, 1077 (1995). 14. N. L. Tsitsas, N. K. Uzunoglu, and D. I. Kaklamani, Radio Sci. 42, RS6S22 (2007). 15. Y. Nakata and M. Koshiba, J. Opt. Soc. Am. A 7, 1494 (1990). 16. S. S. Wang and R. Magnusson, Appl. Opt. 34, 2414 (1995). 17. Y. Ding and R. Magnusson, Opt. Express 12, 1885 (2004). 18. S. H. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002). 19. G. D’Aguanno, D. de Ceglia, N. Mattiucci, and M. J. Bloemer, Opt. Lett. 36, 1984 (2011). 20. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
