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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3456–3462
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Physical basis for wideband resonant reflectors

Robert Magnusson and Mehrdad Shokooh-Saremi  »View Author Affiliations


Optics Express, Vol. 16, Issue 5, pp. 3456-3462 (2008)
http://dx.doi.org/10.1364/OE.16.003456


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Abstract

In this paper, we address resonant leaky-mode reflectors made with a periodic silicon layer on an insulating substrate. Our objective is to explain the physical basis for their operation and to quantify the bandwidth provided by a single resonant layer by illustrative examples for both TE and TM polarized incident light. We find that the number of participating leaky modes and their excitation conditions affect the bandwidth. We show that recently reported experimental [1, 2] wideband reflectors operate under leaky-mode resonance. These compact reflectors are new elements with many potential applications in photonic systems. The results presented explaining their physical basis will aid in their continued development.

© 2008 Optical Society of America

1. Introduction

Subwavelength periodic layers exhibit strong resonance effects that originate in quasi-guided, or leaky, waveguide modes [3–9

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

]. These compact elements yield versatile photonic spectra and surface-localized energy states with controllable Q factors. Using powerful electromagnetic design methods, the spectral bands of these subwavelength resonant leaky-mode elements can be engineered to achieve photonic devices with practical attributes. For example, we have shown that a single periodic layer with one-dimensional periodicity enables narrow-line filters, polarizers, reflectors, and polarization-independent elements [10

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

]. Potential applications include bandpass and bandstop filters, laser mirrors, ultrasensitive biosensors, absorption enhancement in solar cells, security devices, tunable filters, nanoelectromechanical display pixels, and others.

Multilayer thin films are widely applied to implement filters, polarizers, and reflectors for incorporation in various common optical systems [11

11. H. A. Macleod, Thin-Film Optical Filters, (McGraw-Hill, New York, 1989).

]. These devices typically consist of stacks of homogeneous layers deposited with precise thicknesses and tight control of index of refraction and absorption. In many cases, a large number of layers, perhaps ~10–100, may be needed to create the spectral and angular attributes required for a particular application. These optical devices operate on the basis of multiple reflections between the interfaces incorporated in a layer stack. In particular, periodic quarter-wave layer systems provide classical high reflectors for bulk laser cavities as well as integrated distributed Bragg reflectors for vertical cavity lasers. Bragg reflectors yield efficient reflection across wide spectral bands [12

12. A. E. Willner, “All mirrors are not created equal,” Nature Photonics 1, 87–88 (2007). [CrossRef]

, 13

13. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, New York, 2007).

]. The spectral expressions generated by resonant leaky-mode layers, in some ways, resemble spectral expressions associated with thin-film systems. In other ways, the resonance response is unique and not realizable with homogeneous thin films. Therefore, the functionality and applicability of thin films in optics and photonics technology can be complemented and enhanced by imbuing them with appropriate periodic modulation to achieve leaky-mode resonance.

Fig. 1. (a) A schematic view of a subwavelength guided-mode resonance element under normal incidence. A single silicon layer with thickness d, fill factor F, and a two-part period Λ is treated. 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. For the symmetric grating profile, a resonance appears at one edge. This picture applies to both TE (electric field vector normal to the plane of incidence) and TM (magnetic field vector normal to the plane of incidence) polarization states. K=2π/Λ, k0=2π/λ, and β is the propagation constant of a leaky mode.

Figure 1(a) shows the model device to be treated. It is a subwavelength reflector consisting of a modulated silicon layer (nH=nSi=3.48) on a SiO2 substrate (nS=nSiO2=1.48). It is necessary that the structure form a waveguide grating such that the periodic layer (Si) possesses higher refractive index than the adjacent regions (air, silica). The reflector works under a guided-mode resonance (GMR), which arises when the incident wave couples to a leaky waveguide mode by phase matching with the second-order grating [14

14. A. Hardy, D. F. Welch, and W. Streifer, “Analysis of second-order gratings,” IEEE J. Quantum Electron. 25, 2096–2105 (1989). [CrossRef]

, 15

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

]. Under normal incidence, counter-propagating leaky modes form a standing wave in the grating as indicated in Fig. 1(a). As the modes interact with the waveguide grating, they reradiate reflectively [16

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

]. A schematic dispersion diagram is shown in Fig. 1(b). The device works in the second stop band corresponding to the second-order grating [15

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

]. A given evanescent diffraction order can excite not just one but several leaky modes. Thus, in Fig. 1(b), we show the stop bands for the first two TE modes to emphasize this point. At each stop band, a resonance is generated as denoted in Fig. 1(b) also. The fields radiated by these leaky modes in a grating with a symmetric profile can be in phase or out of phase at the edges of the band [17

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

, 3

3. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979). [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, βI=0 at one edge, implying that no leakage is possible at that edge. In this paper, for clarity, we treat resonance elements with two-part periods which can only have symmetric profiles.

2. Numerical methods and assumptions

3. Results

Fig. 2. (a) Zero-order reflectance and transmittance spectra of an example broadband reflector operating in TM polarization. The R0>0.99 bandwidth is ~520 nm. For clarity, the spectra are plotted on both linear and logarithmic scales. (b) Amplitudes of the magnetic modal fields inside the grating layer and in the surrounding media at the wavelength of the center resonance.
Fig. 3. (a) Reflectance map R(λ,d) pertaining to the broadband reflector and the results in Fig. 2. (b) Transmittance map T(λ,d) in dB. (c) Reflectance map R(λ,d) for a resonance layer with reduced contrast (nH=2.0 and nL=1.3417). (d) Modal curves for the first four modes excited by the first evanescent diffraction order setting nfilm=1.92.

Figure 3(a) displays a color-coded reflectance map R(λ,d) drawn versus wavelength and grating thickness for the same example. Numerous S-shaped high reflection regions appear showing the evolution of the reflectance from narrow to broad resonance bands. Figure 3(b) illustrates the associated transmittance versus wavelength and thickness in dB, clearly revealing the sharp resonance loci. Note the flat-band locus near d=0.5 µm which corresponds to the results in Fig. 2.

Fig. 4. (a) TE polarized spectra for a simple single-layer reflector. The bandwidth is ~125 nm for reflectance R0>0.99. (b) Electric field distribution pattern at the resonance wavelength of 1.559 µm.

Wideband resonance reflectors are not limited to the TM polarization state. Figure 4(a) provides computed reflectance and transmittance spectra for TE polarization. For this design, Λ=0.986 µm, d=0.228 µm, and F=0.329. The spectral width is ~125 nm for a flat band with R0>0.99. Figure 4(b) displays the electric field distribution pattern at the resonance wavelength. A dominant TE0 leaky mode is generated by the first evanescent diffraction order. Moreover, by dividing the period into four parts ({Si, air, Si, air} with corresponding fill factors {F1, F2, F3, F4; F1+F2+F3+F4=1.0}), very broad reflection bandwidths with R0>0.99 are realizable; these may exceed ~600 nm in both TE and TM polarization for a single Si layer without substrate enhancement [10

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

]. In essence, the four-part period imbues the periodic layer with a rich set of Fourier harmonics with concomitant emergence of additional spectral features not available for the two-part case in Fig. 1(a).

Mateus et al. reported an experimental wideband reflector operating around the 1.55 µm wavelength [1

1. 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]

]. The element consists of a silicon grating over a silica sublayer (with thickness dL) on a silicon substrate. In a prior paper focusing on reflector design [23

23. 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]

], they provided a set of optimized parameters as Λ=0.7 µm, d=0.46 µm, dL=0.83 µm, and F=0.75. We use this set in Fig. 5(a) which shows the spectra of this reflector including a sharp leaky-mode transmission minimum consistent with the results presented above. The computed reflection bandwidth is ~467 nm for R0>0.99 with TM polarization. Figure 5(b) shows the effect of the sublayer/substrate combination on the reflectance spectrum. Without the sublayer, the bandwidth for this design is ~385 nm. Therefore, in this case, the sublayer extends the flat band by ~80 nm or ~20%.

Fig. 5. (a) Spectra of a broadband reflector with Λ=0.7 µm, d=0.46 µm, dL=0.83 µm, and F=0.75. (b) Computed results showing the effect of the sublayer/substrate combination on the reflectance spectra.

4. Conclusions

Leaky-mode resonance devices belong to the class of periodic nanophotonic structures, photonic crystals, and diffractive elements that are of growing importance for applications in active and passive devices such as filters, lasers, displays, and sensors. The results presented explain the physics of their operation as broadband reflectors and may contribute to their further development and utility. Excitation of leaky modes is a necessary, but not sufficient, condition for broadband reflectance. Additionally, the leaky-mode spectra must be shaped by proper choice of the device parameters. As in this work, such optimized parameters can be effectively established using inverse mathematical design methods like PSO. Future research will explore bandwidth enhancements achievable by more complex architectures including sublayer-substrate enhancements, multi-component periods, and additional layers.

Acknowledgements

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

1.

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]

2.

M. C. Y Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-indexcontrast subwavelength grating,” Nature Photonics 1, 119–122 (2007). [CrossRef]

3.

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

4.

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

5.

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

6.

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]

7.

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]

8.

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

9.

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

10.

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

11.

H. A. Macleod, Thin-Film Optical Filters, (McGraw-Hill, New York, 1989).

12.

A. E. Willner, “All mirrors are not created equal,” Nature Photonics 1, 87–88 (2007). [CrossRef]

13.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, New York, 2007).

14.

A. Hardy, D. F. Welch, and W. Streifer, “Analysis of second-order gratings,” IEEE J. Quantum Electron. 25, 2096–2105 (1989). [CrossRef]

15.

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

16.

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

17.

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

18.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985). [CrossRef]

19.

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]

20.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975). [CrossRef]

21.

R. Eberhart and J. Kennedy, “Particle swarm optimization,” in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995) 1942–1948.

22.

M. Shokooh-Saremi and R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. 32, 894–896 (2007). [CrossRef] [PubMed]

23.

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]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(130.2790) Integrated optics : Guided waves
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: December 10, 2007
Revised Manuscript: February 23, 2008
Manuscript Accepted: February 24, 2008
Published: February 29, 2008

Citation
Robert Magnusson and Mehrdad Shokooh-Saremi, "Physical basis for wideband resonant reflectors," Opt. Express 16, 3456-3462 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3456


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References

  1. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 ?m) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004). [CrossRef]
  2. M. C. Y Huang, Y. Zhou, and C. J. Chang-Hasnain, "A surface-emitting laser incorporating a high-index- contrast subwavelength grating," Nature Photonics 1, 119-122 (2007). [CrossRef]
  3. P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979). [CrossRef]
  4. L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Comm. 55, 377-380 (1985). [CrossRef]
  5. E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986). [CrossRef]
  6. 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]
  7. 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]
  8. R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61,1022-1024 (1992). [CrossRef]
  9. S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32,2606-2613 (1993). [CrossRef] [PubMed]
  10. Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004). [CrossRef] [PubMed]
  11. H. A. Macleod, Thin-Film Optical Filters, (McGraw-Hill, New York, 1989).
  12. Q4. A. E. Willner, "All mirrors are not created equal," Nature Photonics 1, 87-88 (2007). [CrossRef]
  13. A. Yariv and P. Yeh., Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, New York, 2007).
  14. A. Hardy, D. F. Welch, and W. Streifer, "Analysis of second-order gratings," IEEE J. Quantum Electron. 25, 2096-2105 (1989). [CrossRef]
  15. Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express 15, 680-694 (2007). [CrossRef] [PubMed]
  16. D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997). [CrossRef]
  17. R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. 21, 144-150 (1985). [CrossRef]
  18. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985). [CrossRef]
  19. 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]
  20. S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975). [CrossRef]
  21. R. Eberhart and J. Kennedy, "Particle swarm optimization," in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995) 1942-1948.
  22. M. Shokooh-Saremi and R. Magnusson, "Particle swarm optimization and its application to the design of diffraction grating filters," Opt. Lett. 32, 894-896 (2007). [CrossRef] [PubMed]
  23. 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]

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