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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 9322–9327
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High angular tolerance and reflectivity with narrow bandwidth cavity-resonator-integrated guided-mode resonance filter

X. Buet, E. Daran, D. Belharet, F. Lozes-Dupuy, A. Monmayrant, and O. Gauthier-Lafaye  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 9322-9327 (2012)
http://dx.doi.org/10.1364/OE.20.009322


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Abstract

Guided mode resonance filters (GMRFs) are a promising new generation of reflective narrow band filters, that combine structural simplicity with high efficiency. However their intrinsic poor angular tolerance and huge area limit their use in real life applications. Cavity-resonator-integrated guided-mode resonance filters (CRIGFs) are a new class of reflective narrow band filters. They offer in theory narrow-band high-reflectivity with a much smaller footprint than GMRF. Here we demonstrate that for tightly focused incident beams adapted to the CRIGF size, we can obtain simultaneously high spectral selecitivity, high reflectivity, high angular acceptance with large alignment tolerances. We demonstrate experimentally reflectivity above 74%, angular acceptance greater than ±4.2° for a narrow-band (1.4 nm wide at 847 nm) CRIGF.

© 2012 OSA

1. Introduction

Guided-mode resonance filters (GMRFs) are a promising new generation of reflective narrow-band filters, that combine structural simplicity with high efficiency. They consist of a simple sub-wavelength grating etched in the top layer of an anti-reflection coating. They have already been used to demonstrate extremely high-Q filters [1

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

, 2

2. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997). [CrossRef]

], and under certain conditions, maximum reflectivity of 100% combined with a high rejection rate [3

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

]. Their narrow-bandwidth, high-reflectivity and sensitivity to optical index changes have already been harnessed to demonstrate high-resolution optical index sensing [4

4. J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005). [CrossRef]

] or frequency stabilization of a laser diode [5

5. S. Block, E. Gamet, and F. Pigeon, “Semiconductor laser with external resonant grating mirror,” IEEE J. Quantum Electron. 41, 1049–1053 (2005). [CrossRef]

]. Moreover, several theoretical [6

6. E. Bonnet, A. Cachard, A. Tishchenko, and O. Parriaux, “Scaling rules for the design of a narrow-band grating filter at the focus of a free-space beam,” Proc. SPIE 5450, 217–222 (2004). [CrossRef]

8

8. F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]

] or experimental [9

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

] works have focused on improving the angular tolerance of GMRFs. However, their poor angular tolerance remains a key limitation to their widespread adoption. Indeed, high performances require both large area filters and an ideal case of perfectly-collimated large incident beam. This imposes stringent requirements on the fabrication of GMRFs [10

10. A.-L. Fehrembach, O. Gauthier-Lafaye, K. Chan, Shin Yu, A. Monmayrant, S. Bonnefont, E. Daran, P. Arguel, F. Lozes-Dupuy, and A. Sentenac, “Measurement and modeling of 2D hexagonal resonant-grating filter performance,” J. Opt. Soc. Am. A 27, 1535–1540 (2010). [CrossRef]

]. Ura & al. have really recently introduced the concept of cavity-resonator-integrated guided-mode resonance filter (CRIGF) [11

11. K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20, 1444–1449 (2012). [CrossRef] [PubMed]

,12

12. J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of Guided-Mode Resonance Filter by cavity resonator integration,” Appl. Phys. Express 5, 022201 (2012). [CrossRef]

] to allow for smaller GMRFs. In this approach, a small guided-mode resonance filter is inserted between two high reflectors made of distributed Bragg reflectors (DBR), providing a localized mode in the GMRF section that is coupled resonantly to an incident beam. This concept is quite similar to the high-Q resonators demonstrated in [13

13. Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008). [CrossRef] [PubMed]

], but relies on low index contrast structures, thus with larger optical modes that are more efficiently coupled to free space optics. Due to their small size, these filters are more easily fabricated using nanolithography tools (like e-beam) and less sensitive to stitching defects or non-homogeneity than GMRFs. This approach thus provides users with small-area, narrow-band, reflective filters with high-rejection rate, that will be useful in many application areas such as spectrometry, hyperspectral imaging, or as selective mirrors for diode laser in an external cavity. In this paper, we report on the design and performances of small filters under tightly focused incidence. We investigate both numerically and experimentally the sensitivity of the performances to the matching of incident beam size and filter size. We demonstrate experimentally high reflectivity (above 74%) together with a large angular tolerance of ±4.2° for a filter with extremely small footprint of 117×80 μm2 and narrow-band spectral reflectivity (1.4 nm wide at 847 nm).

2. CRIGFs design and simulation

CRIGFs are composed of five sections (see Fig. 1(a)), namely a GMRF section (length LG, grating of period a) at center flanked on each side with a phase adjusting section (length δ, no grating) and a DBR mirror section (length LDBR, grating period a/2). These five sections form a cavity with a mode that is localized inside the GMRF section by the two DBRs. This mode is resonantly excited by an incident beam that is coupled into the waveguide by the GMRF section. The vertical stack is similar than in [10

10. A.-L. Fehrembach, O. Gauthier-Lafaye, K. Chan, Shin Yu, A. Monmayrant, S. Bonnefont, E. Daran, P. Arguel, F. Lozes-Dupuy, and A. Sentenac, “Measurement and modeling of 2D hexagonal resonant-grating filter performance,” J. Opt. Soc. Am. A 27, 1535–1540 (2010). [CrossRef]

], with a silicon nitride waveguide and a silica grating layer. The layers thicknesses and optical indexes are e = 111 nm and n1 = 1.46 for silica, t = 165 nm and n2 = 1.97 for silicon nitride (the optical index corresponds to those obtained with our Plasma Enhanced Chemical Vapor Deposition (PECVD) deposition system). These two layers deposited on a glass substrate provide an anti-reflection coating at λ = 850 nm. The GMRF and DBR sections have identical 100 nm wide and 120 nm deep grooves. For CRIGF modeling we use 2D FDTD [14

14. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Com. 181, 687–702 (2010). [CrossRef]

]. A 3D simulation would be more accurate but requires a much too heavy computation. The GMRF period is a = 532 nm and all calculations are done using a resolution of a/50, with PMLs on the four sides of the computational cell. The input source has gaussian amplitude, flat phase and is placed just above the grating. The electric field is aligned along the grooves direction (TE polarization). 2D FDTD calculations were first used to evaluate the length of phase adjusting section δ (see Fig. 1(a)) that optimizes the reflectivity of the CRIGF. From our calculations, the optimal length is δ = 1.05a in our geometry. This value is close to one period and reflects the fact that since the grating duty cycle is low, the effective index of the in-plane guided mode is almost the same in all sections of the filter, thus leading to almost phase matched sections. We also used 2D FDTD to adjust the DBR length to achieve high reflectivity whilst maintaining a reasonable length for this section [11

11. K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20, 1444–1449 (2012). [CrossRef] [PubMed]

]. A length of 200 periods for the DBR sections was found to be a good compromise between efficiency and size and is used in all the following. Figure 2(a) shows a typical reflection spectrum calculated for a small CRIGF (GMRF length LG = 21a) illuminated with a gaussian source of waist w0 = 10 a. As seen, calculated reflectivity is high (≃90%) and spectrally narrow (Full Width at Half maximum FWHM ≃ 0.6 nm). We numerically investigated the reflectivity dependence of the filters on both GMRF section size LG, and input beam waist w0. Figure 3(a) shows the calculated evolution of the maximum reflectivity with the input beam waist for three grating lengths, 21a, 41a and 101a. For more clarity, the beam waist was normalized to the grating section size. The first result is that the input mode size and grating section size have to be matched for optimum performances. Maximum reflectivity is achieved when the waist in amplitude is ≃ 0.5 times the GMRF length. For larger input beam size, the beam extends outside of the GMRF section and part of the energy is not coupled, inducing a decrease in reflectivity. For smaller input beams, the reflectivity also decreases, as part of the angular components of the small beam exceed the angular tolerance of the CRIGF. The second result from this study (not shown here) is that the resonant wavelength and FWHM of the reflection peak increase with the GMRF section length, but are independent of the input beam size. We believe that the independence to the input beam parameters proves that the reflectivity of the device steams from the excitation of a localized mode. We understand the evolution of the resonant wavelength as a result of the hybridization between the GMRF mode and a mode confined in-between the two DBR sections. As for the evolution of the FWHM, it is better understood in terms of Q-factor. The computed Q factors are around 1380 (FWHM 0.62 nm), 1070 (FWHM 0.8 nm) and 875 (FWHM 0.98 nm) for respectively the 21 periods, 41 periods and 101 periods GMRF sections. These results agree with a localized excited mode analysis : as the GMRF section resonantly couples energy out of the cavity, mode losses scale with its length, thus decreasing the Q factor.

Fig. 1 (a) Schematic view of the filter. The dark section corresponds to the waveguide. (b) Experimental setup used to study filters under tightly focused excitation.
Fig. 2 (a) Theoretical response of a filter with a GMRF section length LG = 21a and DBR length LDBR = 200a/2. (b) Experimental spectra of the small grating (blue, with 25mm focal length lens), medium grating (red, with 40 mm focal length lens) and large grating (black, with 75 mm focal length lens), with δ = 1.0a
Fig. 3 Evolution of maximum reflectivity with the input mode size 2w0 normalized to the GMRF section length LG : (a) theoretical evolution, (b) experimental evolution respectively for gratings with LG = 21a (full blue circles), 41a (open red squares) and 101a (black crosses).

3. Samples fabrication and characterization

Samples corresponding to the three designed geometries were fabricated using a generic MOS-type process described in [15

15. S. Hernandez, O. Bouchard, E. Scheid, E. Daran, L. Jalabert, P. Arguel, S. Bonnefont, O. Gauthier-Lafaye, and F. Lozes-Dupuy, “850 nm wavelength range nanoscale resonant optical filter fabrication using standard microelectronics techniques,” Microelectron. Eng. 84, 673–677 (2007). [CrossRef]

]. The process relies on the combination of PECVD deposition, e-beam lithography and dry etching. Layers thickness and optical index are measured by ellipsometry and identical to those used in the simulations. Gratings with respectively LG = 21a (small), 41a (medium) and 101a (large) were fabricated, with DBR section length LDBR = 200a/2 on each side. The grating period was fixed at 528 nm to adjust the reflection wavelength towards 850 nm and the DBR period is 264 nm, whilst the phase adjustment factor δ was varied from 0.8a to 1.3a in step of 0.1a. The gratings width was set to 80 μm. After fabrication, the samples were checked using both scanning electron and atomic force microscopy. The grooves depth was found to be 120 ±5 nm. For optical characterization, we use the set-up shown on Fig. 1(b). A polarized collimated beam from a super-luminescent diode is focused on the sample surface using a plano-convex lens L. Different lenses were used to achieve different spot sizes. The polarization is set so that the E field is along the grooves direction (TE polarization). A 10x microscope objective mounted on the back side of the sample is used to image the sample on a CCD for monitoring of the beam size and position on the sample. The reflected light is self-collimated by the focusing lens L and a 50/50 beam-splitter BS sends it to a 1m focal length spectrometer. For reference purposes, a gold mirror of 97% reflectivity was mounted just above the sample.

We investigated the reflection dependence on the phase section length and found that best reflectivity are achieved for δ = 1.0a, in relatively good agreement with our simulations. Figure 2(b) shows the reflection spectra for the three GMRF section length, with δ = 1.0a. The maximum measured reflectivities are 78%, 72% and 74% for respectively the large, medium and small grating filters. The Q factors of the reflection peaks are equal to 426 (FWHM 2 nm), 567 (FWHM 1.5 nm) and 605 (FWHM 1.4 nm) for respectively the large, medium and small grating filters. To achieve the best performances, we used different focusing lenses, resulting in different spot sizes on the sample. Figure 3(b) shows the reflection value versus the beam size. As in Fig. 3(a), the beam size is normalized to the grating section size. The maximum reflection value occurs for a beam size of ≃ 0.5 times the GMRF section length. However the decay of reflectivity for large spot sizes seems less marked than expected from our calculation. The Q factor of the reflection peak is found to be independent of the spot size within our experimental accuracy, in good agreement with the localized mode analysis. For large gratings with a small spot size, additional reflection peaks could be observed, we attribute them to the excitation of higher order localized modes within the CRIGF. To summarize, we found a good agreement between simulations and experiments but for the Q factor and the decay of reflectivity with large waist size. We believe that this deviation steams from the difference between the 3D geometry of the real sample as opposed to the 2D geometry used for simulations. Divergence of light along the grooves axis, – not taken into account in our simulations –, broadens the k selection rule of the grating coupling into the waveguide. Indeed tilted plane waves get the correct k vector along the periodicity axis for resonance to occur at longer wavelength. This will broaden the resonance towards long wavelength, transforming the theoretical Fano lineshape into a lorentzian peak, as experimentally observed. As first order calculations, we computed an averaged response of the filter in the same way as in [10

10. A.-L. Fehrembach, O. Gauthier-Lafaye, K. Chan, Shin Yu, A. Monmayrant, S. Bonnefont, E. Daran, P. Arguel, F. Lozes-Dupuy, and A. Sentenac, “Measurement and modeling of 2D hexagonal resonant-grating filter performance,” J. Opt. Soc. Am. A 27, 1535–1540 (2010). [CrossRef]

], assuming that the divergence along the direction perpendicular to the simulation plane induces an homothetic shift of the spectral response. That is, the light sees an effective grating with a period increased by the out of axis angle of propagation. As a result, we observed a reduced reflectivity (75% instead of 91%) and a broadening of the peak FWHM from 0.6 nm to 0.8 nm, in good agreement with our observations and hypothesis. In the same way, using a spot larger than the numerically optimal one, that is a spot with a lower numerical aperture, may reduce these effects and then compensate slightly the reflectivity loss, shifting slightly the optimum reflectivity towards larger beams and reducing the reflectivity decay rate.

We investigated both angular and positioning tolerances of the small filter when used with the most appropriate beam size. Figure 4 left shows the evolution of the reflection peak when the filter position is moved away from the beam waist along the optical axis and Fig. 4 right shows the measured evolution of the reflection spectra when the filter is rotated along the grooves axis. We see that on both figures, the reflection peak wavelength is fixed at the resonance, as expected from a localized optical state. The FWHM (and thus the Q factor) of the reflection peak is also constant. The filter positioning tolerance is quite large ≃ 400μm, which is 3 times larger than the Rayleigh range of our beam (calculated to be ≃ 130 μm). For the angular tolerance, we find a FWHM of ±4.2°. This large tolerance has never been reported before for this kind of small filter. Moreover, it is above one order of magnitude larger than what can be achieved with a large area GMRF offering similar spectral bandwidth. Indeed, using RCWA, we computed the angular tolerance of an infinite GMRF with the same geometry and found it to be ±0.2°. The CRIGF thus exhibits an angular tolerance more than 20 times larger than a standard resonant grating filter, with a similar Q factor (theoretical FWHM of 1.5 nm). Moreover, as can be seen on Fig. 4, both central wavelength and Q factor are constant when the filter is moved away from the designed incidence configuration, as opposed to the behaviour of a normal GMRF.

Fig. 4 bottom left: 2D map of reflection spectrum versus CRIGF displacement. top left: plot of the evolution of maximum reflection versus CRIGF displacement. bottom right: 2D map of reflection spectrum versus CRIGF angular displacement. top right: plot of the evolution of maximum reflection versus CRIGF angular displacement

4. Conclusion

We investigated reflection properties of CRIGFs under tightly focused excitation. These filters are almost 40 times smaller than CRIGFs found previously in litterature [11

11. K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20, 1444–1449 (2012). [CrossRef] [PubMed]

,12

12. J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of Guided-Mode Resonance Filter by cavity resonator integration,” Appl. Phys. Express 5, 022201 (2012). [CrossRef]

]. Experimental high reflectivity (above 70%) together with narrow bandwidth (FWHM≃ 1.5nm, Q > 600) and extremely large angular tolerance (±4.2°) have been demonstrated, showing promising properties for practical applications. Even narrower filters can certainly be achieved either using shallower gratings or using 2D gratings to avoid in-plane dispersion of light along the grooves axis. The large optical alignment tolerances of these filters are extremely interesting for future use. More specifically, these filters show an extremely large angular tolerance, more than one order of magnitude larger than GMRFs using more complicated patterns [8

8. F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]

, 9

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

]. This property will allow an easy use of CRIGFs with focused beams in applications such as imaging spectroscopic sensors or for laser diode external cavity setup.

Acknowledgments

Authors acknowledge funding by the Centre National d’Etudes Spatiales under grant number 104331/00. X. Buet acknowledges financial support from a DGA grant under P. Adam supervision.

References and links

1.

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

2.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997). [CrossRef]

3.

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

4.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005). [CrossRef]

5.

S. Block, E. Gamet, and F. Pigeon, “Semiconductor laser with external resonant grating mirror,” IEEE J. Quantum Electron. 41, 1049–1053 (2005). [CrossRef]

6.

E. Bonnet, A. Cachard, A. Tishchenko, and O. Parriaux, “Scaling rules for the design of a narrow-band grating filter at the focus of a free-space beam,” Proc. SPIE 5450, 217–222 (2004). [CrossRef]

7.

A.-L. Fehrembach and A. Sentenac, “Unpolarized narrow-band filtering with resonant gratings,” Appl. Phys. Lett. 86, 121105 (2005). [CrossRef]

8.

F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]

9.

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

10.

A.-L. Fehrembach, O. Gauthier-Lafaye, K. Chan, Shin Yu, A. Monmayrant, S. Bonnefont, E. Daran, P. Arguel, F. Lozes-Dupuy, and A. Sentenac, “Measurement and modeling of 2D hexagonal resonant-grating filter performance,” J. Opt. Soc. Am. A 27, 1535–1540 (2010). [CrossRef]

11.

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20, 1444–1449 (2012). [CrossRef] [PubMed]

12.

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of Guided-Mode Resonance Filter by cavity resonator integration,” Appl. Phys. Express 5, 022201 (2012). [CrossRef]

13.

Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008). [CrossRef] [PubMed]

14.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Com. 181, 687–702 (2010). [CrossRef]

15.

S. Hernandez, O. Bouchard, E. Scheid, E. Daran, L. Jalabert, P. Arguel, S. Bonnefont, O. Gauthier-Lafaye, and F. Lozes-Dupuy, “850 nm wavelength range nanoscale resonant optical filter fabrication using standard microelectronics techniques,” Microelectron. Eng. 84, 673–677 (2007). [CrossRef]

OCIS Codes
(230.1950) Optical devices : Diffraction gratings
(050.5298) Diffraction and gratings : Photonic crystals
(050.6624) Diffraction and gratings : Subwavelength structures
(230.7408) Optical devices : Wavelength filtering devices

ToC Category:
Optical Devices

History
Original Manuscript: February 24, 2012
Revised Manuscript: March 30, 2012
Manuscript Accepted: March 30, 2012
Published: April 6, 2012

Citation
X. Buet, E. Daran, D. Belharet, F. Lozes-Dupuy, A. Monmayrant, and O. Gauthier-Lafaye, "High angular tolerance and reflectivity with narrow bandwidth cavity-resonator-integrated guided-mode resonance filter," Opt. Express 20, 9322-9327 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9322


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References

  1. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993). [CrossRef] [PubMed]
  2. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997). [CrossRef]
  3. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]
  4. J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005). [CrossRef]
  5. S. Block, E. Gamet, and F. Pigeon, “Semiconductor laser with external resonant grating mirror,” IEEE J. Quantum Electron. 41, 1049–1053 (2005). [CrossRef]
  6. E. Bonnet, A. Cachard, A. Tishchenko, and O. Parriaux, “Scaling rules for the design of a narrow-band grating filter at the focus of a free-space beam,” Proc. SPIE 5450, 217–222 (2004). [CrossRef]
  7. A.-L. Fehrembach and A. Sentenac, “Unpolarized narrow-band filtering with resonant gratings,” Appl. Phys. Lett. 86, 121105 (2005). [CrossRef]
  8. F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998). [CrossRef]
  9. A.-L. Fehrembach, A. Talneau, O. Boyko, F. Lemarchand, and A. Sentenac, “Experimental demonstration of a narrow-band, angular tolerant, polarization independent, doubly periodic resonant grating filter,” Opt. Lett. 32, 2269–2271 (2007). [CrossRef] [PubMed]
  10. A.-L. Fehrembach, O. Gauthier-Lafaye, K. Chan, Shin Yu, A. Monmayrant, S. Bonnefont, E. Daran, P. Arguel, F. Lozes-Dupuy, and A. Sentenac, “Measurement and modeling of 2D hexagonal resonant-grating filter performance,” J. Opt. Soc. Am. A 27, 1535–1540 (2010). [CrossRef]
  11. K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20, 1444–1449 (2012). [CrossRef] [PubMed]
  12. J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of Guided-Mode Resonance Filter by cavity resonator integration,” Appl. Phys. Express 5, 022201 (2012). [CrossRef]
  13. Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008). [CrossRef] [PubMed]
  14. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Com. 181, 687–702 (2010). [CrossRef]
  15. S. Hernandez, O. Bouchard, E. Scheid, E. Daran, L. Jalabert, P. Arguel, S. Bonnefont, O. Gauthier-Lafaye, and F. Lozes-Dupuy, “850 nm wavelength range nanoscale resonant optical filter fabrication using standard microelectronics techniques,” Microelectron. Eng. 84, 673–677 (2007). [CrossRef]

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