Long-lived scattering resonances and bragg structures

SIAM Journal on Applied Mathematics

Volumn 73, Issue 2, 2013, Pages 827-852

  Osting, Braxton ^a   Weinstein, Michael I ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Columbia University ^*  (United States)

Author keywords

Bragg condition; Fabry P rot cavity; Helmholtz equation; Quality factor; Quarter wave stack; Resonance; Spectral optimization; Spherical resonator

Indexed keywords

BRAGG CONDITIONS; QUALITY FACTORS; QUARTER WAVE STACKS; SPECTRAL OPTIMIZATION; SPHERICAL RESONATORS;

BRAGG CELLS; CONSTRAINED OPTIMIZATION; EIGENVALUES AND EIGENFUNCTIONS; HELMHOLTZ EQUATION; REFRACTIVE INDEX; RESONANCE; STRUCTURAL DESIGN; STRUCTURAL OPTIMIZATION; WAVE EQUATIONS;

SCATTERING;

EID: 84878029541     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/110856228     Document Type: Article

References (47)

1. V. Akcelik, G. Biros, O. Ghattas, D. Keyes, K. Ko, L.-Q. Lee, and E. G. Ng, Adjoint methods for electromagnetic shape optimization of the low-loss cavity for the international linear collider, J. Phys. Conf. Ser., 16 (2005), pp. 435-445.
2. D. Brill and G. Gaunaurd, Resonance theory of elastic waves ultrasonically scattered from an elastic sphere, J. Acoust. Soc. Amer., 81 (1987), pp. 1-21.
3. M. Burger, S. Osher, and E. Yablonovitch, Inverse problem techniques for the design of photonic crystals, IEICE Trans. Electron., 87 (2004), pp. 258-265.
4. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, Periodic nanostructures for photonics, Phys. Rep., 444 (2007), pp. 101-202.
5. C. A. Bauer, G. R. Werner, and J. R. Cary, Truncated photonic crystal cavities with optimized mode confinement, J. Appl. Phys., 104 (2008), 053107.
6. S. J. Cox and D. C. Dobson, Maximizing band gaps in two-dimensional photonic crystals, SIAM J. Appl. Math., 59 (1999), pp. 2108-2120.
7. S. J. Cox and D. C. Dobson, Band structure optimization of two-dimensional photonic crystals in h-polarization, J. Comput. Phys., 158 (2000), pp. 214-224.
8. S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys., 214 (2000), pp. 315-337.
9. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer, Berlin, 1998.
10. S. J. Cox and J. R. McLaughlin, Extremal eigenvalue problems for composite membranes, I, Appl. Math. Optim., 22 (1990), pp. 153-167.
11. S. J. Cox and J. R. McLaughlin, Extremal eigenvalue problems for composite membranes, II, Appl. Math. Optim., 22 (1990), pp. 169-187.
12. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, Circular dielectric cavity and its deformations, Phys. Rev. A (3), 77 (2008), 013804.
13. C. L. Dolph and S. K. Cho, On the relationship between the singularity expansion and the mathematical theory of scattering, IEEE Trans. Antennas and Propagation, 28 (1980), pp. 888-897.
14. D. C. Dobson and F. Santosa, Optimal localization of eigenfunctions in an inhomo-geneous medium, SIAM J. Appl. Math., 64 (2004), pp. 762-774.
15. D. Englund, I. Fushman, and J. Vuckovic, General recipe for designing photonic crystal cavities, Opt. Express, 13 (2005), pp. 5961-5975.
16. M. Felici, K. A. Atlasov, A. Surrente, and E. Kapon, Semianalytical approach to the design of photonic crystal cavities, Phys. Rev. B, 82 (2010), 115118.
17. A. Gondarenko and M. Lipson, Low modal volume dipole-like dielectric slab, Opt. Express, 16 (2008), 17689.
18. A. Gondarenko, S. Preble, J. Robinson, L. Chen, H. Lipson, and M. Lipson, Spontaneous emergence of periodic patterns in a biologically inspired simulation of photonic structures, Phys. Rev. Lett., 96 (2006), 143904.
19. J. M. Geremia, J. Williams, and H. Mabuchi, Inverse-problem approach to designing photonic crystals for cavity qed experiments, Phys. Rev. E (3), 66 (2002), 066606.
20. E. M. Harrell, General lower bounds for resonances in one dimension, Comm. Math. Phys., 86 (1982), pp. 221-225.
21. P. Heider, D. Berebichez, R. V. Kohn, and M. I. Weinstein, Optimization of scattering resonances, Struct. Multidiscip. Optim., 36 (2008), pp. 443-456.
22. E. M. Harrell and R. Svirsky, Potentials producing maximally sharp resonances, Trans. Amer. Math. Soc., 293 (1986), pp. 723-736.
23. P. Heider and M. I. Weinstein, Unpublished notes, 2008.
24. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed., Princeton University Press, Princeton, NJ, 2008.
25. I. M. Karabash, Optimization of Quasi-normal Eigenvalues for 1-d Wave Equations in Inhomogeneous Media: Description of Optimal Structures, preprint, arXiv:1103.4117[math.SP], 2011.
26. C.-Y. Kao, S. Osher, and E. Yablonovitch, Maximizing band gaps in two-dimensional photonic crystals using level set methods, Appl. Phys. B, 81 (2005), pp. 235-244.
27. M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Transl. (2), 1 (1955), pp. 163-187.
28. C.-Y. Kao and F. Santosa, Maximization of the quality factor of an optical resonator, Wave Motion, 45 (2008), pp. 412-427.
29. H. Koch and D. Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001), pp. 339-360.
30. H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Comm. Math. Phys., 267 (2006), pp. 419-449.
31. P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York, 1989.
32. R. P. Lipton, S. P. Shipman, and S. Venakides, Optimization of resonances in photonic crystal slabs, in Physics, Theory, and Applications of Periodic Structures in Optics II, P. Lalanne, ed., Proc. SPIE 5184, SPIE Publications, Bellingham, WA, 2003, pp. 168-177.
33. M. Lenoir, M. Vullierme-Ledard, and C. Hazard, Variational formulations for the determination of resonant states in scattering problems, SIAM J. Math. Anal., 23 (1992), pp. 579-608.
34. R. B. Melrose, Geometric Scattering Theory, Cambridge University Press, Cambridge, UK, 1995.
35. J. U. Nockel, Resonances in Nonintegrable Open Systems, Ph.D. thesis, Yale University, New Haven, CT, 1997.
36. B. Osting, Spectral Optimization Problems Controlling Wave Phenomena, Ph.D. thesis, Columbia University, New York, 2011.
37. B. Osting, Bragg structure and the first spectral gap, Appl. Math. Lett., 25 (2012), pp. 1926-1930.
38. B. Osting and M. I. Weinstein, Emergence of periodic structure from maximizing the lifetime of a bound state coupled to radiation, Multiscale Model. Simul., 9 (2011), pp. 654-685.
39. M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1979.
40. J. Scheuer, W. M. J. Green, and A. Yariv, Annular Bragg resonators (ABR): The ideal tool for biochemical sensing, nonlinear optics, and cavity QED, in Integrated Optics: Devices, Materials, and Technologies X, Y. Sidorin and C. A. Waechter, eds., Proc. SPIE 6123, SPIE Publications, Bellingham, WA, 2006, pp. 234-245.
41. O. Sigmund and K. Hougaard, Geometric properties of optimal photonic crystals, Phys. Rev. Lett., 100 (2008), 153904.
42. O. Sigmund and J. S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 361 (2003), pp. 1001-1019.
43. R. Svirsky, Maximally resonant potentials subject to p-norm constraints, Pacific J. Math., 129 (1987), pp. 357-374.
44. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235-286.
45. S. H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure Appl. Math., 53 (2000), pp. 1305-1334.
46. P. Yeh, Optical Waves in Layered Media, John Wiley & Sons, New York, 1988.
47. P. Yeh, A. Yariv, and E. Marom, Theory of Bragg fiber, J. Opt. Soc. Amer., 68 (1978), pp. 1196-1201.