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| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 9 — Sep. 26, 2007
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Widely tunable guided-mode resonance nanoelectromechanical RGB pixels

Robert Magnusson and Mehrdad Shokooh-Saremi  »View Author Affiliations


Optics Express, Vol. 15, Issue 17, pp. 10903-10910 (2007)
http://dx.doi.org/10.1364/OE.15.010903


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Abstract

Widely tunable display pixels are reported. The pixel consists of a subwavelength silicon-nitride/air membrane containing complementary fixed and mobile gratings. By altering the device refractive index profile and symmetry, using MEMS actuation methods, wavelength tuning across ~100 nm per pixel in the visible spectral region is shown to be possible. Initial results illustrating the influence of structural symmetry, pixel thickness, and polarization on the spectral response are provided. These pixels exhibit ~±4° angular acceptance aperture. Applications in compact display systems are envisioned.

© 2007 Optical Society of America

1. Introduction

Microelectromechanical systems (MEMS) methods [1

1. M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, 2nd ed. (CRC press, 2002).

] in conjunction with photonics provide interesting solutions in display technology. Well-known examples of MEMS-based systems for display include the Texas Instruments digital micromirror device (DMD) and the Silicon Light Machines grating light valve (GLV) technologies. The DMD pixel is a flat mirror that is tilted on an axis so that the illuminating light is directed towards the screen or away from it under electronic MEMS control [1

1. M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, 2nd ed. (CRC press, 2002).

, 2

2. Texas Instruments, DLP site: http://www.dlp.com/

]. Rapid tilting of the mirrors allows display of a large number of gray levels as well as a multitude of colors if the input white light is directed through a color wheel. This produces high contrast color images on display screens via suitable projector systems. The GLV system employs a MEMS reflection grating whose diffraction efficiency is controlled by applied voltage [3

3. D. M. Bloom, “The grating light valve: revolutionizing display technology,” Proc. SPIE 3013, 165–171 (1997). [CrossRef]

, 4

4. J. I. Trisnadi, C. B. Carlisle, and R. Monteverde, “Overview and applications of Grating Light ValveTM based optical write engines for high-speed digital imaging,” Proc. SPIE 5348, 1–13 (2004). [CrossRef]

]. A suspended ribbon structure in which alternate ribbons are deflected to control the level of diffracted intensity constitutes the GLV element. The GLV applies the first diffraction orders of the phase grating, possibly collecting the light in both the +1 and -1 orders for enhanced efficiency with additional optical components. Three different grating periods of the GLV are used to produce red, green, and blue colors for superposition on a display screen; thus 3 distinct pixels provide these 3 primary colors. The diffraction efficiency of each pixel is controlled rapidly by the MEMS feature thus achieving effective color mixing and balancing on the screen.

The interferometric modulator (IMOD) technology is a recent addition to the class of MEMS-based displays [5

5. Qualcomm, IMOD displays site: http://www.qualcomm.com/technology/imod

]. An IMOD pixel is basically a bistable optical element comprising a thin-film stack over a glass substrate and a movable reflective membrane. MEMS actuation varies the cavity dimension between the stack and the membrane to generate constructive interference of a desired color (red, blue, or green). Collapsing the cavity produces a dark state due to absorption of light. These reflective displays may be attractive in portable devices where low power consumption is required [5

5. Qualcomm, IMOD displays site: http://www.qualcomm.com/technology/imod

].

Tuning and switching of the resonance wavelength in guided-mode resonant (GMR) gratings can be realized by perturbing the physical parameters including refractive index of grating layers or surrounding media, layer thickness, period, or fill factor as described in [6

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

8

8. R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

]. Shu et al. reported analysis of a tunable structure consisting of two adjacent photonic crystal films, each composing a two-dimensional waveguide grating that could be displaced laterally or longitudinally by a mechanical force [9

9. W. Shu, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonances in photonic crystal slabs,” Appl. Phys. Lett 82, 1999–2001 (2003). [CrossRef]

]. Each periodic waveguide admitted guided-mode resonances whose coupling could be mechanically altered for spectral tuning. Carr et al. theoretically studied laterally deformable nanomechanical gratings under resonance conditions and, using a fixed input wavelength, found intensity modulation and polarization effects on application of grating lateral shifts [10

10. D. W. Carr, J. P. Sullivan, and T. A. Friedman, “Laterally deformable nanomechanical zeroth-order gratings: anomalous diffraction studied by rigorous coupled-wave theory,” Opt. Lett. 28, 1636–1638 (2003). [CrossRef] [PubMed]

]. Later, they fabricated a prototype device in amorphous diamond with 600 nm period and nanoscale features and suggested uses for inertial sensing and modulation [11

11. B. E. N. Keeler, D. W. Carr, J. P. Sullivan, T. A. Friedman, and J. R. Wendt, “Experimental demonstration of a laterally deformable optical nanoelectromechanical system grating transducer,” Opt. Lett. 29, 1182–1184 (2004). [CrossRef] [PubMed]

]. More recently, Kanamori et al. fabricated two-dimensional, polarization independent GMR filters that could be moved, via MEMS actuators, within an air gap to add or drop selected wavelengths in a bit stream. The silicon-on-insulator device had ~65% efficiency at 1545 nm [12

12. Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007). [CrossRef]

].

In this paper, new display pixels based on tunable resonant leaky-mode nanoelectromechanical (NEMS) elements are presented. These elements are spectrally selective subwavelength devices with no higher propagating diffraction orders thus operating in the zero order regime. N/MEMS actuation methods may potentially be applied to tune their spectral response through mechanical micro/nano alteration of the structural parameters. Previously, we presented the characteristics of MEMS-tunable, guided-mode resonance (GMR) structures and explained their operational principles [8

8. R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

]. It was shown that such systems are highly tunable with only nanoscale displacements needed for wide-range tuning. Working with a silicon-on-insulator (SOI) device and fixed parameters, we quantified the level of tunability per unit movement for an example resonant structure. It was found that effective MEMS-based tuning can be accomplished by variation of grating profile symmetry, by changing the waveguide thickness, or both. Numerical examples of the particular MEMS-tunable, leaky-mode structure treated showed that the resonance wavelength could be shifted spectrally across ~300 nm with a horizontal movement of ~120 nm and the reflectance tuned in excess of 30 dB with a vertical movement of ~200 nm in the 1.3–1.7 µm wavelength band. The analysis showed that these structurally dynamic elements can function effectively as tunable filters, variable reflectors, and modulators in the optical telecommunication band [8

8. R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

].

It is clear that analogous tunable devices can be designed in numerous other material systems for operation in arbitrary spectral regions. Of course, as the operational wavelength diminishes the associated finer-feature patterning demands stricter tolerances in fabrication. In the present contribution, we investigate theoretically the operation of N/MEMS-tunable GMR membranes in the visible spectral region using a low-loss example medium. A chief motivation for the study is potential implementation of these compact tunable pixels in future display systems. In this application, a tunable pixelized display can be realized by employing a broadband illumination source and suitably N/MEMS-activated, reflection/transmission GMR filters. Advances in nanometric fabrication processes may allow utilization of such methods to spectrally tune the resonance position in the visible region.

In summary, widely tunable GMR pixels for display systems are introduced. Example results using a silicon nitride (Si3N4) periodic membrane are provided. We show that by altering the refractive index profile of the membrane, conceptually with a N/MEMS actuation approach, a high degree of wavelength tunability within the visible spectrum is achieved. We examine the effect of structural symmetry, device thickness, and input light polarization on the spectral response. These pixels provide tunable spectra with constant reflection efficiency.

2. Influence of structural symmetry in GMR devices

When an incident light beam becomes phase matched to a waveguide’s leaky mode through a grating, a guided-mode resonance (GMR) takes place. Under normal illumination, two counter-propagating leaky modes are created. These modes are excited by evanescent diffraction orders and the structure works in the second stop band [13

13. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express 15, 680–694(2007). [CrossRef] [PubMed]

]. The radiated fields generated by these leaky modes in a symmetric structure can be in phase or out of phase at the edges of the band [14

14. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985). [CrossRef]

]. At one edge, there is a zero phase difference and hence the radiation is enhanced while at the other edge, there is a π phase difference inhibiting the radiation. In this case, if β=βR+jβI is the complex propagation constant of the leaky mode, we have βI=0 at one edge, implying that no leakage is possible at that edge. Resonating the incident light in a layer with an asymmetric grating profile yields a nonzero βI at each edge [15

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

17

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

].

The proposed tunable GMR pixel, in its general state, is an asymmetric grating structure exhibiting a resonance at each edge of the band. However, as the element approaches symmetry points under dynamic mechanical tuning, one band-edge resonance disappears. Figure 1(a) shows a schematic view of a subwavelength GMR device under normal incidence indicating leaky-mode excitation and their reflective reradiation. Figure 1(b) depicts schematically the dispersion diagram of the device at the second stop band as well as the associated resonances. GMR structures with two different media per period may be categorized as symmetric whereas elements with multi-part periods can be generally taken as having asymmetric profiles.

3. Device structure

Figure 2 shows the structure of the tunable GMR pixels under study. It is a membrane consisting of air (nL=1.0) and silicon nitride (Si3N4, nH=2.1) parts. One of the high-index (nH) parts of the structure is fixed and the other part is mobile using MEMS actuation as indicated. For each pixel, the period (Λ), thickness (d), and fill factors (width of each grating subdivision divided by the period) F1 and F3 are assumed to be fixed. F2 and F4 are taken to be the variable fill factors, although the operation of the device can be characterized by F2 alone. Here, F2 is considered to be the variable fill factor and thus the tuning parameter. By changing F2, the refractive index distribution and symmetry of the structure change. While the average refractive index (in an effective-medium sense) remains unchanged, other refractive index Fourier components change accordingly. The second Fourier component of the refractive index profile has the main influence on the width of the second (leaky) stop band [8

8. R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

].

Fig. 1. (a) A general schematic view of a subwavelength GMR device under normal incidence. When phase matching occurs between evanescent diffraction orders and a waveguide mode, a reflection resonance takes place. I, R, and T denote the incident wave, reflectance, and transmittance, respectively. (b) Schematic dispersion diagram of a GMR device at the second stop band (blue). For a grating with an asymmetric profile, a resonance occurs at each edge of the stop band as shown. For a symmetric element, a resonance appears only at one edge. Here, K=2π/Λ, k0=2π/λ, and β is the propagation constant.
Fig. 2. Proposed idealized silicon nitride membrane structure for the tunable GMR pixels addressed in the paper with F1=0.15 and F3=0.1. For the blue-green pixel, Λ=0.385 µm and d=0.2 µm. For the red pixel, Λ=0.5 µm and d=0.25 µm.

Two sets of parameters are selected for operation of the device in the blue-green and red portions of the visible spectrum. In both sets, F1=0.15, F3=0.1, with operation at normal incidence illumination assumed. For the blue-green pixel we set Λ=0.385 µm and d=0.2 µm. For the red pixel, Λ=0.5 µm and d=0.25 µm. The computed results provided are based on idealized conditions (infinite number of periods and plane wave illumination) as in the rigorous coupled-wave analysis [18

18. M. G. Moharam, D. A. Pommet, 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 (1995). [CrossRef]

] code applied. In practice, of course, the incident light beam may be ~Gaussian with finite diameter and the pixels will have finite extent.

4. Results

Figure 3 illustrates the reflection spectra of the two pixels for TE-polarized illumination. For the wavelength ranges shown, the two gratings work in the subwavelength (Λ<λ) regime and hence only zero-order reflection appears without higher diffraction orders being present. The reflectance map R(λ,F2) has similar appearance for both color regions as the pixel parameters are not very different. In each case, the longer-wavelength region shows a clear, well-shaped reflectance spectrum that responds favorably to tuning by variation of F2. This branch corresponds to the lower stop band edge [8

8. R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

, 13

13. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express 15, 680–694(2007). [CrossRef] [PubMed]

] with the resonant leaky mode being the fundamental TE0 mode. The blue-green pixel of Fig. 3(a) has a tuning capability of ~100 nm over the 0.45–0.55 µm range whereas the red pixel in Fig. 3(b) shows a tuning range of ~100 nm across 0.60–0.70 µm, corresponding to a rate of ~0.10 µm wavelength tuning per ~0.125 µm lateral shift. As shown in Fig. 3, the R(λ,F2) maps are symmetric around the F2=0.375 profile symmetry point.

Fig. 3. Reflection spectra of the tunable pixels versus the air-gap fill factor F2 for TE polarization (electric field vector perpendicular to the incidence plane). (a) 2D map of reflectance R(λ,F2) for the blue-green pixel with Λ=0.385 µm and d=0.2 µm. (b) R(λ,F2) for the red pixel where Λ=0.5 µm and d=0.25 µm.

At shorter wavelengths, as shown in Fig. 3, the patterns exhibit secondary branches due to appearance of higher resonant modes. At F2~0 there is a wide-band reflectance region arising from a mixture of leaky modes. Centered on the profile symmetry point F2=0.375 there appears a second band of high reflectance due to a TE1 type resonant leaky mode. This elliptical region, just as the fundamental band, has a lower-frequency continuous edge and a gap at the upper edge around the symmetry point. Calculations show that the leaky modes (~TE0 and TE1) constituting these bands are excited mainly by the first (±1) diffraction orders. The leftmost branch, which vanishes at the symmetry point, supports mixed modes.

Figure 4 displays samples of the reflectance spectra for the blue-green and red pixels for selected values of the fill factor from the two-dimensional maps of Fig. 3. It is apparent that the resonances are well shaped with useful linewidths of several nanometers maintaining their shape well on tuning. The minimum useful pixel size is determined by the parameters used in each case. The key idea is to resonate the input light effectively within the lateral size of the pixel. As the leaky mode has a finite decay length, the lateral size has to exceed this characteristic length. For example, in case of the blue-green pixel, we can estimate the decay length (Ld) by the formula Ld~Λλ/4πΔλ [19

19. 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]

] where Δλ is the spectral linewidth in Fig. 4(a). This yields Ld ~3 µm with a transverse pixel size of 10 µm being feasible in this case.

For illumination with practical, imperfectly collimated sources, there needs to be an accommodating angular aperture. Figure 5 shows angular spectra for two example values of F2 for the red pixel. It is seen that this pixel will work properly with ~±4° tolerance in angle of incidence which is reasonable for display applications.

Fig. 4. Samples of the reflection spectra for (a) blue-green and (b) red pixels for different values of F2 from the 2D maps in Fig. 3.
Fig. 5. Angular spectra of the resonant peaks for two different values of F2 and wavelength for the red pixel.
Fig. 6. Reflection spectra map R(λ,F2) of the red pixel for TM polarization.

Figure 7 displays the mode field patterns and profiles inside the red pixel structure at the middle of the tuning diagram (symmetry point F2=0.375) for TE and TM polarizations. In Fig. 7(a), the mode on the main branch is a TE0 type excited mainly by the first diffraction order thus denoted TE0,1. Similarly, the mode in Fig. 7(b) can be considered to be a TE1,1 mode; this mode resides on the sub-branch in Fig. 3(b). In the TM case at the symmetry point, as shown in Fig. 7(c), the mode is TM0,1 with λ=0.5051 µm (upper band edge) and with no resonance appearing on the right branch. As seen in Figs. 6 and 7(c), this pixel exhibits high-Q behavior (Q=λ/Δλ) with the attendant narrow linewidth and high resonance field value in the leaky mode whose evanescent tail extends several wavelengths out of the structure in this case.

Fig. 7. Mode patterns (left) and cross-sectional mode profiles (right) for the red pixel. (a) λ=0.7125 µm (main branch TE), (b) λ=0.5365 µm (sub-branch TE) and (c) λ=0.5051 µm (TM). F2 is equal to 0.375. Electric field profiles due to the zero, first, and second diffraction orders are shown as blue, red, and purple, respectively.

Fig. 8. Reflectance spectra map R(λ,F2) of the red pixel for (a) d=0.15 µm and (b) d=0.35 µm. Note that λ=0.5 µm defines the border between the zero order (subwavelength) and multiorder diffraction regimes for this structure.

5. Conclusions

Acknowledgements

The authors thank Y. Ding for his help with the analysis code. This material is based, in part, upon work supported by the National Science Foundation under Grant No. ECCS-0702307.

References and links

1.

M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, 2nd ed. (CRC press, 2002).

2.

Texas Instruments, DLP site: http://www.dlp.com/

3.

D. M. Bloom, “The grating light valve: revolutionizing display technology,” Proc. SPIE 3013, 165–171 (1997). [CrossRef]

4.

J. I. Trisnadi, C. B. Carlisle, and R. Monteverde, “Overview and applications of Grating Light ValveTM based optical write engines for high-speed digital imaging,” Proc. SPIE 5348, 1–13 (2004). [CrossRef]

5.

Qualcomm, IMOD displays site: http://www.qualcomm.com/technology/imod

6.

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

7.

R. Magnusson and S. S. Wang, “Optical guided-mode resonance filter,” US patent number 5,216,680, June 1, 1993.

8.

R. Magnusson and Y. Ding, “MEMS tunable resonant leaky mode filters,” IEEE Photon. Technol. Lett. 18, 1479–1481 (2006). [CrossRef]

9.

W. Shu, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonances in photonic crystal slabs,” Appl. Phys. Lett 82, 1999–2001 (2003). [CrossRef]

10.

D. W. Carr, J. P. Sullivan, and T. A. Friedman, “Laterally deformable nanomechanical zeroth-order gratings: anomalous diffraction studied by rigorous coupled-wave theory,” Opt. Lett. 28, 1636–1638 (2003). [CrossRef] [PubMed]

11.

B. E. N. Keeler, D. W. Carr, J. P. Sullivan, T. A. Friedman, and J. R. Wendt, “Experimental demonstration of a laterally deformable optical nanoelectromechanical system grating transducer,” Opt. Lett. 29, 1182–1184 (2004). [CrossRef] [PubMed]

12.

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90, 031911 (2007). [CrossRef]

13.

Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express 15, 680–694(2007). [CrossRef] [PubMed]

14.

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985). [CrossRef]

15.

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

16.

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

17.

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

18.

M. G. Moharam, D. A. Pommet, 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 (1995). [CrossRef]

19.

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]

OCIS Codes
(120.2040) Instrumentation, measurement, and metrology : Displays
(130.2790) Integrated optics : Guided waves

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 24, 2007
Revised Manuscript: July 26, 2007
Manuscript Accepted: July 30, 2007
Published: August 15, 2007

Virtual Issues
Vol. 2, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Robert Magnusson and Mehrdad Shokooh-Saremi, "Widely tunable guided-mode resonance nanoelectromechanical RGB pixels," Opt. Express 15, 10903-10910 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-17-10903


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References

  1. M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, 2nd ed. (CRC press, 2002).
  2. Texas Instruments, DLP site: http://www.dlp.com/
  3. D. M. Bloom, "The grating light valve: revolutionizing display technology," Proc. SPIE 3013, 165-171 (1997). [CrossRef]
  4. J. I. Trisnadi, C. B. Carlisle, and R. Monteverde, "Overview and applications of Grating Light ValveTM based optical write engines for high-speed digital imaging," Proc. SPIE 5348, 1-13 (2004). [CrossRef]
  5. Qualcomm, IMOD displays site: http://www.qualcomm.com/technology/imod
  6. S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993). [CrossRef] [PubMed]
  7. R. Magnusson and S. S. Wang, "Optical guided-mode resonance filter," US patent number 5,216,680, June 1, 1993.
  8. R. Magnusson and Y. Ding, "MEMS tunable resonant leaky mode filters," IEEE Photon. Technol. Lett. 18, 1479-1481 (2006). [CrossRef]
  9. W. Shu, M. F. Yanik, O. Solgaard, and S. Fan, "Displacement-sensitive photonic crystal structures based on guided resonances in photonic crystal slabs," Appl. Phys. Lett 82, 1999-2001 (2003). [CrossRef]
  10. D. W. Carr, J. P. Sullivan, and T. A. Friedman, "Laterally deformable nanomechanical zeroth-order gratings: anomalous diffraction studied by rigorous coupled-wave theory," Opt. Lett. 28, 1636-1638 (2003). [CrossRef] [PubMed]
  11. B. E. N. Keeler, D. W. Carr, J. P. Sullivan, T. A. Friedman, and J. R. Wendt, "Experimental demonstration of a laterally deformable optical nanoelectromechanical system grating transducer," Opt. Lett. 29, 1182-1184 (2004). [CrossRef] [PubMed]
  12. Y. Kanamori, T. Kitani, and K. Hane, "Control of guided resonance in a photonic crystal slab using microelectromechanical actuators," Appl. Phys. Lett. 90, 031911 (2007). [CrossRef]
  13. Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express 15, 680-694 (2007). [CrossRef] [PubMed]
  14. R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation losses," IEEE J. Quantum Electron. 21, 144-150 (1985). [CrossRef]
  15. P. Vincent and M. Nevière, "Corrugated dielectric waveguides: a numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979). [CrossRef]
  16. Y. Ding and R. Magnusson, "Use of nondegenerate resonant leaky modes to fashion diverse optical spectra," Opt. Express 12, 1885-1891 (2004). [CrossRef] [PubMed]
  17. Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004). [CrossRef] [PubMed]
  18. M. G. Moharam, D. A. Pommet, 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 (1995). [CrossRef]
  19. 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]

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