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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 22 — Oct. 27, 2008
  • pp: 18249–18263
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Wideband leaky-mode resonance reflectors: Influence of grating profile and sublayers

Mehrdad Shokooh-Saremi and Robert Magnusson  »View Author Affiliations


Optics Express, Vol. 16, Issue 22, pp. 18249-18263 (2008)
http://dx.doi.org/10.1364/OE.16.018249


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Abstract

We apply inverse numerical methods to design compact wideband reflectors in which a periodic silicon layer supports resonant leaky modes. Using particle swarm optimization to determine appropriate device thickness, period, and fill factors, we arrive at example reflector designs for both TE and TM polarized input light. As a properly configured grating profile provides added design freedom, we design reflectors with two and four subparts in the period. In TM polarization, a particular single-layer two-part reflector has 520 nm bandwidth whereas the four-part device reaches 600 nm bandwidth. In TE polarization, the corresponding numbers are 125 nm and 495 nm, respectively. We provide a qualitative explanation for the smaller TE-reflector bandwidth. We quantify the effects of deviation from the design parameters and compute the angular response of the elements. As the angle of incidence deviates from normal incidence, narrow transmission channels emerge in the response yielding a bandpass filter with low sidebands. The effects of adding a silica sublayer between a silicon substrate and the periodic silicon layer is investigated. It is found that a properly designed sublayer can extend the reflection bandwidth significantly.

© 2008 Optical Society of America

1. Introduction

Efficient reflection of light across wide spectral bands is essential in a plethora of common photonic systems. Classic mirrors are made with evaporated metal films and dielectric multilayer stacks. These ordinary devices have been widely studied for a long time and are well understood. A new method to achieve effective wideband reflection response has recently emerged. This approach is based on guided-mode resonance (GMR) effects [1

1. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979). [CrossRef]

19

19. V. A. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. C. Skolnick, T. F. Krauss, and R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2057 (1999). [CrossRef]

] that are native to one-dimensional (1D) and two-dimensional (2D) waveguide gratings, also called photonic crystal slabs. Indeed, it is increasingly being recognized by researchers around the world that even a single resonant layer can supply an extraordinary variety of spectra that are controlled by the layer’s patterning and refractive-index contrast as shown in [8

8. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004). [CrossRef] [PubMed]

].

In this paper, we employ resonant leaky modes to spectrally engineer reflective elements with emphasis on realizing wide flat bands. Briefly reviewing the relevant literature, the pursuit of resonant wideband response can be traced to Gale et al. [20

20. M. T. Gale, K. Knop, and R. Morf, “Zero-order diffractive microstructures for security applications,” Proc. SPIE 1210, 83–89 (1990). [CrossRef]

] and to Brundrett et al. [21

21. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstrations,” Opt. Lett. 23, 700–702 (1998). [CrossRef]

] who achieved experimental full-width half-maximum (FWHM) linewidths near 100 nm albeit not for flat spectra. Applying cascaded resonance structures, Jacob et al. designed narrow-band flattop filters which exhibited also lowered sidebands and steepened stopbands [22

22. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001). [CrossRef]

]. Alternatively, by coupling several diffraction orders into corresponding leaky modes in a two-waveguide system, Liu et al. found a widened spectral response and steep filter sidewalls generated by merged resonance peaks [23

23. Z. S. Liu and R. Magnusson, “Concept of multiorder multimode resonant optical filters,” IEEE Photon. Technol. Lett. 14, 1091–1093 (2002). [CrossRef]

]. Suh et al. designed a flattop reflection filter using a 2D-patterned photonic crystal slab [10

10. W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004). [CrossRef]

]. Emphasizing new modalities introduced by asymmetric profiles, Ding et al. presented single-layer elements exhibiting both narrow and wide flat-band spectra [24

24. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12, 1885–1891 (2004). [CrossRef] [PubMed]

]. Using a subwavelength grating with a low-index sublayer on a silicon substrate, Mateus et al. designed flattop reflectors with linewidths of several hundred nanometers operating in TM polarization [25

25. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladding subwavelength grating,” IEEE Photon. Technol. Lett. 16, 518–520 (2004). [CrossRef]

]. Subsequently, they fabricated a reflector with reflectance exceeding 98.5% over a 500 nm range and compared the response with numerical simulations [26

26. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12-1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004). [CrossRef]

]. To emphasize the numerous new device possibilities afforded by properly designed resonant leaky-mode elements, we extended ref. [24

24. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12, 1885–1891 (2004). [CrossRef] [PubMed]

] and showed single-layer elements with ~600 nm flattop reflectance in both TE and TM polarization [8

8. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004). [CrossRef] [PubMed]

]. Most recently, we further addressed the detailed physical basis for such reflectors by treating the simplest possible case which is a single-layer waveguide grating patterned in 1D with a two-part period [27

27. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008). [CrossRef] [PubMed]

]. We quantified the bandwidth provided by a single resonant layer by illustrative examples for both TE and TM polarized incident light and showed that reported experimental [26

26. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12-1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004). [CrossRef]

] wideband reflectors operate under leaky-mode resonance. This work defines the minimal structure capable of extensive reflectance bands [27

27. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008). [CrossRef] [PubMed]

].

2. Device structure and design

Figure 1 shows the most general structure treated in this article. It is basically a silicon-oninsulator (SOI) element. For most of the designs, a silica substrate is used such that dL→∞. However, in some of our examples a silica film deposited on the silicon wafer defines a sublayer with low refractive index. The grating layer, which also acts as the waveguide, is defined by its period (Λ), thickness (d), and fill factors (Fi; ∑i Fi=1) that are the fractions of the period filled with each constituent material, as shown in Fig. 1. We emphasize the case of normal incidence for both TE and TM polarizations while also examining the angular sensitivity of the reflection spectra. The grating period has two or four sections consisting of two materials with different refractive indices nH and nL where nH > nL.

Fig. 1. Basic structure and parameters of the GMR reflector. Λ, d, and F1 to F4 denote the grating period, thickness, and fill factors, respectively. dL is the silica sublayer thickness. The incidence medium is air, substrate is silicon, nH=nSi=3.48, nL=nair=1.0, and nsilica=1.48.

3. Resonant broadband reflectors with two-part period and no sublayer

3.1. TM polarization

Fig. 2. Reflectance and transmittance spectra (a) linear and (b) logarithmic of a PSO-designed broadband reflector for TM polarization. The resonance wavelengths are (i) 1.495 µm, (ii) 1.620 µm, and (iii) 1.839 µm.
Fig. 3. Amplitude of the magnetic (modal) field (Hy(z)) inside the grating structure and in the surrounding media for the three resonances in Fig. 2 at (a) 1.495 µm, (b) 1.620 µm, and (c) 1.839 µm.

Fig. 4. (a) Reflectance map R0(λ,d) drawn versus wavelength and grating thickness. (b) Corresponding transmittance map T0(λ,d) in dB.

Fig. 5. (a) Magnified transmission map (in dB, Fig. 4(b)) of the reflector for thicknesses between 0.2 µm and 0.9 µm around the optimum design thickness (dotted line). (b) Reflection spectra for thicknesses at two points near the optimal thickness (0.455 and 0.520 µm) in comparison to that for the optimal thickness (0.490 µm); note vertical axis scale change. Amplitude of the leaky-mode magnetic field for two points on the resonance locus in Fig. 5(a) but far from the optimal condition: (c) d=0.217 µm, λ=1.363 µm, {point (i)}, and (d) d=0.843 µm, λ=1.883 µm {point (ii)}.

Figure 7(a) estimates the angular sensitivity of the reflection spectra associated with this device. The response is highly sensitive to the angle of incidence and ±1° deviation from normal incidence splits the band into two shorter bands. Figure 7(b) provides sampled spectra under normal (θ=0°) and oblique (θ=+5°) incidence. A narrow transmission channel emerges within the reflection spectra under off-normal incidence, yielding a resonance bandpass filter response [28

28. S. Tibuleac and R. Magnusson, “Narrow-linewidth bandpass filters with diffractive thin-film layers,” Opt. Lett. 26, 584–586 (2001). [CrossRef]

].

Fig. 6. (a) R0(λ,d) map for a low-refractive-index contrast structure (nH=2.0 and nL=1.3417). (b) Calculated modal curves for the first four leaky modes excited by the first diffraction order in an equivalent homogenous film with refractive index of 1.92. (c,d) Magnetic modal field amplitudes corresponding to curves I (at λ=1.1849 µm and d=0.3293 µm (TM0)) and IV (at λ =1.1466 µm and d=1.913 µm (TM3)) in Fig. 6(a), respectively.
Fig. 7. (a) Angular sensitivity of the reflection spectra of the broadband reflector. ~ ±1° deviation from normal incidence induces a transmission channel in the reflection band. (b) Samples of reflection spectra under normal and off-normal incidence.

It is known that the spectral response of GMR-based devices is highly dependent on the structural parameters. Thus, we evaluate the response of our reflector under variation in the period, thickness, and fill factor. Figures 8(a-c) show the sensitivity of the spectra when each parameter deviates from its optimal value by up to ±10%. Such results are used to estimate fabrication tolerances. A statistical sensitivity analysis can also be performed by randomly deviating all of the parameters together in a predetermined parameter space.

Fig. 8. Sensitivity of the reflectance R0 to the structural parameters. In each part only one parameter is variable and the other two parameters are kept at their optimal values (a) Reflectivity map with period deviation, (b) Reflectivity map with thickness deviation, and (c) Reflectivity map with fill factor deviation. In each case, the optimum value is shown by a dotted line.

3.2. TE polarization

Broadband high reflectors can be also designed to operate under TE-polarized incident light. Applying the PSO algorithm, a two-part-period reflector is designed for the 1.45-2.0 µm band using a SOI structure. The resulting parameters are found to be Λ=0.986 µm, d=0.228 µm, F1=0.3294, F2=1-F1, F3=F4=0, and dL →∞. Figure 9(a) illustrates the spectra of the device on linear and logarithmic scales. The only transmission dip, which shows the resonance wavelength, falls at 1.558 µm. The reflection bandwidth of the filter for R0 > 0.99 is ~125 nm. In addition, Fig. 9(b) displays the amplitude distribution of the total electric field (Ey(z)) in the grating and surrounding media. This figure clearly shows that the leaky-mode has ~TE0 characteristic and how the total field concentrates in parts of the grating period.

Fig. 9. (a) Reflectance and transmittance spectra of a broadband reflector for TE polarization on linear and logarithmic scales. (b) Map of the amplitude of the total electric field (Ey(z)) in the grating and surrounding media at the resonance wavelength (λ=1.558 µm).
Fig. 10. (a) Color-coded R0(λ,d) map for the TE reflector. (b) Reflectance map for reduced refractive contrast with nH=3.0 and nL=1.5898. (c) Reflectance map for reduced contrast with nH=2.3 and nL=2.0856, and (d) Modal characteristic curves for the equivalent homogeneous slab waveguide corresponding to (c) with the layer’s refractive index set to 2.167. The optimal thickness is shown by the dotted line in part (a).

Figure 12(a) shows the angular sensitivity of this reflector. The reflection response is highly dependent on the angle of incidence and ~±0.7° deviation from normal incidence generates a transmission channel and thus a bandpass filter response. Figure 12(b) displays sample spectra for normal and off-normal incidence. Figures 13(a–c) illustrate reflectance maps under variations in period, thickness, and fill factor. These maps show the sensitivity of the reflection spectra when each parameter deviates from its optimal value (dotted lines) up to ~±10%.

Fig.11. a) Modal curves for equivalent homogeneous slab waveguides corresponding to the high-contrast reflectors in Figs. 2 (TM) and 9 (TE) using second-order effective indices. (a) TE reflector (nf=3.365), and (b) TM reflector (nf=3.1577).
Fig.12. a) Angular sensitivity of the spectra of the TE reflector. (b) Samples of the reflection spectra for normal (θ=0°) and off-normal (θ=+3°) incidence.
Fig. 13. Maps showing the sensitivity of the reflection spectra to (a) period, (b) thickness, and (c) fill factor for TE polarization. The optimal parameters are denoted by the dotted line in each case.

4. Resonant broadband reflectors with multi-part periods and sublayers

4.1. TM and TE reflectors with two-part periods and sublayers

Similarly, a reflector with a silica sublayer is designed for TE polarization using PSO. An optimal parameter set is found as Λ=0.965 µm, d=0.221 µm, F=0.326, and dL=3.354 µm. Figure 15(a) shows the spectra of this device. Figure 15(b) depicts the spectral response of the reflector with and without a silica sublayer. The sublayer with optimal thickness enhances the reflection bandwidth from 118 nm to 128 nm in this example.

Fig. 14. (a) Reflectance and transmittance of a resonant reflector with a silica sublayer for TM polarization (logarithmic scale). (b) Reflection spectra with and without the sublayer.
Fig. 15. (a) Reflectance and transmittance spectra of the TE reflector with silica sublayer on a logarithmic scale. (b) Reflection spectra with and without the sublayer.

4.2. TM and TE reflectors with four-part periods and without sublayers

Dividing the period into more than two parts increases the number of design parameters and provides design freedom and control over the Fourier series component distribution of the structure resulting in enhanced spectra for broadband reflectors. Two-part and four-part period gratings can have the same zero-order effective index albeit with completely different spectral response. Also, multi-part period gratings provide both symmetric and asymmetric leaky-mode structures, which can result in excitation of degenerate and non-degenerate leaky modes [24

24. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12, 1885–1891 (2004). [CrossRef] [PubMed]

]. In this section, GMR elements whose period is divided into four parts (with fill factors F1, F2, F3, and F4; F1+F2+F3+F4=1.0) are designed to provide maximum reflection bandwidth coverage. The first design is for TM polarization with parameters Λ=1.0 µm, d=0.81 µm, and [F1, F2, F3, F4]=[0.5, 0.125, 0.25, 0.125] established by our PSO algorithm; it is identical to the corresponding device reported in [8

8. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004). [CrossRef] [PubMed]

]. The device layer is again comprised of silicon and air parts and the substrate is silica. This reflector has a symmetric structure. Figure 16(a) displays the spectra of this filter on linear and logarithmic scales. A bandwidth of ~600 nm for R0 > 0.99 over the target wavelength range is found. In comparison to the TM reflector with two-part period with spectrum in Fig. 2, this design exhibits ~80 nm enhancement in the bandwidth.

Fig. 16. (a) Reflectance (solid line) and transmittance (dashed line) spectra of the four-part broadband reflector for TM polarization. (b) R0(λ,d) map for this device. (c) Transmittance map T0(λ,d) in dB. (d-f) Magnetic field amplitude distribution in the device and surrounding media for three leaky-mode resonances at 1.627 µm, 1.744 µm, and 2.015 µm, respectively.

Fig. 17. (a) Angular sensitivity of the four-part broadband reflector for TM polarization. (b) Samples of reflection spectra for normal and off-normal incidence.
Fig. 18. (a) Reflectance and transmittance spectra of the four-part broadband reflector for TE polarization. (b) Color-coded R0(λ,d) map. (c) T0(λ,d) map in dB. (d–f) Amplitudes of electric field modal profiles for the three resonance leaky-modes at 1.489 µm, 1.642 µm, and 1.872 µm, respectively.

Similarly, a four-part TE reflector has design parameters Λ=0.979 µm, d=0.465 µm, and [F1, F2, F3, F4]=[0.071, 0.265, 0.399, 0.265], which is a symmetric profile. Figure 18(a) shows the associated spectra. This reflector has a bandwidth of ~495 nm for R0 > 0.99. Considerable enhancement (~370 nm) is achieved relative to the two-part structure in Fig. 9. The effect of thickness on the spectra is displayed in Figs. 18(b) and (c). The electric-field profiles of the modes contributing to the bandwidth of the reflector are displayed in Figs. 18(d–f); these show ~TE0 and ~TE1 mode features. Figure 19(a) displays the angular sensitivity of this structure, which is ~±0.6° whereas Fig. 19(b) shows samples of reflection spectra for normal and off-normal incidence. It is interesting to note that the spectrum splits at its extremes rather than in the center.

Fig. 19. (a) Angular sensitivity of the four-part broadband reflector. (b) Samples of reflection spectra for normal (θ=0°) and oblique (θ=+3°) incidence.

4.3. TM and TE reflectors with four-part periods and sublayers

Additional layers directly affect the resonance response. Thus, to study the influence of sublayers on the performance of GMR reflectors, we directly simulate insertion of a silica layer between the periodic device layer and a silicon substrate for the four-part reflectors designed in Sec 4.2. Thus, in this case, the sublayer thickness is not included in the PSO optimization. The results are shown as R0(λ,dL) maps in Figs. 20(a,b). Proper sublayer thickness enhances the bandwidth to some extent. The periodic undulation seen as function of sublayer thickness is principally a thin-film effect.

Fig. 20. (a) Color-coded R0(λ,dL) map showing the effect of adding a silica sublayer to the fourpart reflector designed in section 4.2 for TM polarization. (b) Same for TE polarization.

5. Conclusions

Acknowledgments

The authors thank Y. Ding for his contributions in developing parts of the analysis codes used. This material is based, in part, upon work supported by the National Science Foundation under Grant No. ECS-0524383.

References and links

1.

P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979). [CrossRef]

2.

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985). [CrossRef]

3.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]

4.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985). [CrossRef]

5.

I. A. Avrutsky and V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989). [CrossRef]

6.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992). [CrossRef]

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S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993). [CrossRef] [PubMed]

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Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004). [CrossRef] [PubMed]

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S. Peng and M. Morris, “Resonant scattering from two-dimensional gratings ,” J. Opt. Soc. Am. A 13, 993–1005 (1996).

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W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84, 4905–4907 (2004). [CrossRef]

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S. Boonruang, A. Greenwell, and M. G. Moharam, “Multiline two-dimensional guided-mode resonant filters,” Appl. Opt. 45, 5740–5747 (2006). [CrossRef] [PubMed]

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A.-L. Fehrembach, A. Talneau, O. Boyko, F. Lemarchand, and A. Sentenac, “Experimental demonstration of a narrowband, angular tolerant, polarization independent, doubly periodic resonant grating filter,” Opt. Lett. 32, 2269–2271 (2007). [CrossRef] [PubMed]

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M. T. Gale, K. Knop, and R. Morf, “Zero-order diffractive microstructures for security applications,” Proc. SPIE 1210, 83–89 (1990). [CrossRef]

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D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstrations,” Opt. Lett. 23, 700–702 (1998). [CrossRef]

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D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001). [CrossRef]

23.

Z. S. Liu and R. Magnusson, “Concept of multiorder multimode resonant optical filters,” IEEE Photon. Technol. Lett. 14, 1091–1093 (2002). [CrossRef]

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Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12, 1885–1891 (2004). [CrossRef] [PubMed]

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26.

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27.

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OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(120.2440) Instrumentation, measurement, and metrology : Filters
(130.2790) Integrated optics : Guided waves
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 3, 2008
Revised Manuscript: October 14, 2008
Manuscript Accepted: October 21, 2008
Published: October 23, 2008

Citation
Mehrdad Shokooh-Saremi and Robert Magnusson, "Wideband leaky-mode resonance reflectors: Influence of grating profile and sublayers," Opt. Express 16, 18249-18263 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18249


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References

  1. P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979). [CrossRef]
  2. L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Commun. 55, 377-380 (1985). [CrossRef]
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