Abstract

We demonstrate that tunable attractive (bonding) and repulsive (anti-bonding) forces can arise in highly asymmetric structures coupled to external radiation, a consequence of the bonding/anti-bonding level repulsion of guided-wave resonances that was first predicted in symmetric systems. Our focus is a geometry consisting of a photonic-crystal (holey) membrane suspended above an unpatterned layered substrate, supporting planar waveguide modes that can couple via the periodic modulation of the holey membrane. Asymmetric geometries have a clear advantage in ease of fabrication and experimental characterization compared to symmetric double-membrane structures. We show that the asymmetry can also lead to unusual behavior in the force magnitudes of a bonding/antibonding pair as the membrane separation changes, including nonmonotonic dependences on the separation. We propose a computational method that obtains the entire force spectrum via a single time-domain simulation, by Fourier-transforming the response to a short pulse and thereby obtaining the frequency-dependent stress tensor. We point out that by operating with two, instead of a single frequency, these evanescent forces can be exploited to tune the spring constant of the membrane without changing its equilibrium separation.

© 2011 Optical Society of America

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2010

Y.-G. Roh, T. Tanabe, A. Shinya, H. Taniyama, E. Kuramochi, S. Matsuo, T. Sato, and M. Notomi, "Strong optomechanical interaction in a bilayer photonic crystal," Phys. Rev. B 81(12), 121101 (2010).
[CrossRef]

T. Stomeo, M. Grande, G. Rainò, A. Passaseo, A. D’Orazio, R. Cingolani, A. Locatelli, D. Modotto, C. D. Angelis, and M. D. Vittorio, "Optical filter based on two coupled PhC GaAs-membranes," Opt. Lett. 35(3), 411-413 (2010).
[CrossRef] [PubMed]

M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, "Quantum optomechanics—throwing a glance," J. Opt. Soc. Am. B 27(6), A189-A197 (2010).
[CrossRef]

S. Lin, E. Schonbrun, and K. Crozier, "Optical manipulation with planar silicon microring resonators," Nano Lett. 10(7), 2408-2411 (2010).
[CrossRef] [PubMed]

D. V. Thourhout, and J. Roels, "Optomechanical device actuation through the optical gradient force," Nat. Photonics 4, 211-217 (2010).
[CrossRef]

U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, "Single-photon opto-mechanics in the strong coupling regime," N. J. Phys. 12(8), 083030 (2010).
[CrossRef]

J. Ma, and M. L. Povinelli, "Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate," Appl. Phys. Lett. 97, 151102 (2010).
[CrossRef]

I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, "Programmable photonic crystal nanobeam cavities," Opt. Express 18(8), 8705-8712 (2010).
[CrossRef] [PubMed]

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," Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

T. P. M. Alegre, R. Perahia, and O. Painter, "Optomechanical zipper cavity lasers: theoretical analysis of tuning range and stability," Opt. Express 18(8), 7872-7885 (2010).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, "Analysis of short range forces in opto-mechanical devices with a nanogap," Opt. Express 18(12), 12615-12621 (2010).
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A. Mizrahi, and Y. Fainman, "Negative radiation pressure on gain medium structures," Opt. Lett. 35(20), 3405-3407 (2010).
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2009

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Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, "Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force," Phys. Rev. Lett. 103(10), 103601 (2009).
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G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, "Near-field cavity optomechanics with nanomechanical oscillators," Nat. Phys. 5(12), 909-914 (2009).
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S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, "Whispering gallery mode carousel: a photonic mechanism for enhanced nanoparticle detection in biosensing," Opt. Express 17(8), 6230-6238 (2009).
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S. Lin, J. Hu, L. Kimerling, and K. Crozier, "Design of nanoslotted photonic crystal waveguide cavities for single nanoparticle trapping and detection," Opt. Lett. 34(21), 3451-3453 (2009).
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S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, "Observation of strong coupling between a micromechanical resonator and an optical cavity field," Nature 460(7256), 724-727 (2009).
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W. H. P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, "Modeling of the optical force between propagating lightwaves in parallel 3D waveguides," Opt. Express 17(18), 16032-16037 (2009).
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V. Liu, M. Povinelli, and S. Fan, "Resonance-enhanced optical forces between coupled photonic crystal slabs," Opt. Express 17(24), 21897-21909 (2009).
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J. Roels, I. D. Vlaminck, L. Lagae, B. Maes, D. V. Throurout, and R. Baets, "Tunable optical forces between nanophotonic waveguides," Nat. Nanotechnol. 4, 510-513 (2009).
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2008

T. J. Kippenberg, and K. J. Vahala, "Cavity optomechanics: Back-action at the mesoscale," Science 321, 1172-1176 (2008).
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A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, "Resolved-sideband cooling of a micromechanical oscillator," Nat. Phys. 4, 415-419 (2008).
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M. Aspelmeyer, and K. Schwab, "Focus on mechanical system at the quantum limit," N. J. Phys. 10(9), 095001 (2008).
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D. C. Benito, S. H. Simpson, and S. Hanna, "FDTD simulations of forces on particles during holographic assembly," Opt. Express 16(5), 2942-2957 (2008).
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M. Lo, W. H. P. Pernice, T. Baehrs-Jones, M. Hochberg, and H. X. Tang, "Harnessing optical forces in integrated photonics circuits," Nano Lett. 456, 480-484 (2008).

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2007

A. Mizrahi, and L. Schächter, "Two-slab all-optical spring," Opt. Lett. 32(6), 692-694 (2007).
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F. Capasso, J. N. Munday, D. Iannuzzi, and H. B. Chan, "Casimir forces and quantum electrodynamical torques: Physics and nanomechanics," IEEE J. Sel. Top. Quantum Electron. 13(2), 400-415 (2007).
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P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, "Trapping, corralling and spectral bonding of optical resonances through optically induced potentials," Nat. Photonics 1, 658-665 (2007).
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T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, "An all-optical trap for a gram-scale mirror," Phys. Rev. Lett. 98, 150802 (2007).
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B. S. Schmidt, A. H. Yang, D. Erickson, and M. Lipson, "Optofluidic trapping and transport on solid core waveguides within a microfluidic device," Opt. Express 15(22), 14322-14334 (2007).
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I. Favero, C. Metzger, S. Camerer, D. Konig, H. Lorenz, J. P. Kotthaus, and K. Karrai, "Optical cooling of a micromirror of wavelength size," Appl. Phys. Lett. 90, 104101 (2007).
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T. J. Kippenberg, and K. J. Vahala, "Cavity Opto-Mechanics," Opt. Express 15(25), 17172-17205 (2007).
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2006

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
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A. Mizrahi, and L. Schächter, "Electromagnetic forces on the dielectric layers of the planar optical Bragg acceleration structure," Phys. Rev. E 74(3), 036504 (2006).
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A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, "Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Back action," Phys. Rev. Lett. 97(24), 243905 (2006).
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S. Gigan, H. R. Bohme, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, "Self-cooling of a micromirror by radiation pressure," Nature 444, 67-70 (2006).
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D. Kleckner, and D. Bouwmeester, "Sub-Kelvin optical cooling of a micromechanical resonator," Nature 444, 75-78 (2006).
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O. Arcizet, P. F. Cohadon, T. Briant, and A. Heidmann, "Radiation-pressure cooling and optomechanical instability of a micomirror," Nature 444, 71-74 (2006).
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2005

R. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13(10), 3707-3718 (2005).
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F. W. DelRio, M. P. de Boer, J. A. Knaap, E. D. J. Reedy, P. J. Clews, and M. L. Dunn, "The role of van der Waals forces in adhesion of micromachined surfaces," Nat. Mater. 4, 629-634 (2005).
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K. L. Ekinci, and M. L. Roukes, "Nanoelectromechanical systems," Rev. Sci. Instrum. 76, 061101 (2005).
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T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity," Phys. Rev. Lett. 95(3), 033901 (2005).
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K. Dholakia, "Micromanipulation: optoelectronic tweezers," Nat. Mater. 4, 579-580 (2005).
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J. Ng, C. T. Chan, P. Sheng, and Z. Lin, "Strong optical force induced by morphology-dependent resonances," Opt. Lett. 30(15), 1956-1958 (2005).
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M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso, and J. Joannopoulos, "High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators," Opt. Express 13(20), 8286-8295 (2005).
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M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, "Evanescent-wave bonding between optical waveguides," Opt. Lett. 30, 3042 (2005).
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W. Suh, O. Solgaard, and S. Fan, "Displacement sensing using evanescent tunneling between guided resonances in photonic crystal slabs," J. Appl. Phys. 98(3), 033102 (2005).
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K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72(7), 075429 (2005).
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2004

C. H. Metzger, and K. Karrai, "Cavity cooling of a microlever," Nature 432, 1002-1005 (2004).
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D. Zhang, X. Yuan, S. Tjin, and S. Krishnan, "Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping," Opt. Express 12(10), 2220-2230 (2004).
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B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004).
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M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, "Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide," Appl. Phys. Lett. 85, 1466-1468 (2004).
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2003

M. Vogel, C. Mooser, K. Karrai, and R. J. Warburton, "Optically tunable mechanics of microlevers," Appl. Phys. Lett. 83, 1337 (2003).
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K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
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F. Shanhui, W. Suh, and J. D. Joannopoulos, "Temporal coupled-mode theory for the Fano resonance in optical resonators," J. Opt. Soc. Am. A 20(3), 569-572 (2003).
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2002

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
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2001

H. B. Chan, V. A. Aksyuk, R. N. Kleinman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectromechanical systems by the Casimir force," Science 291, 1941-1944 (2001).
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1999

M. I. Antonoyiannakis, and J. B. Pendry, "Electromagnetic forces in photonic crystals," Phys. Rev. B 60(4), 2363-2374 (1999).
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1998

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1996

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1992

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1988

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1986

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1983

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A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, "LIGO: The Laser Interferometer Gravitational-Wave Observatory," Science 256(5055), 325-333 (1992).
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U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, "Single-photon opto-mechanics in the strong coupling regime," N. J. Phys. 12(8), 083030 (2010).
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Aksyuk, V. A.

H. B. Chan, V. A. Aksyuk, R. N. Kleinman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectromechanical systems by the Casimir force," Science 291, 1941-1944 (2001).
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Alegre, T. P. M.

T. P. M. Alegre, R. Perahia, and O. Painter, "Optomechanical zipper cavity lasers: theoretical analysis of tuning range and stability," Opt. Express 18(8), 7872-7885 (2010).
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A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, "LIGO: The Laser Interferometer Gravitational-Wave Observatory," Science 256(5055), 325-333 (1992).
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G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, "Near-field cavity optomechanics with nanomechanical oscillators," Nat. Phys. 5(12), 909-914 (2009).
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A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, "Resolved-sideband cooling of a micromechanical oscillator," Nat. Phys. 4, 415-419 (2008).
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Angelis, C. D.

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Arcizet, O.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, "Near-field cavity optomechanics with nanomechanical oscillators," Nat. Phys. 5(12), 909-914 (2009).
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A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, "Resolved-sideband cooling of a micromechanical oscillator," Nat. Phys. 4, 415-419 (2008).
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O. Arcizet, P. F. Cohadon, T. Briant, and A. Heidmann, "Radiation-pressure cooling and optomechanical instability of a micomirror," Nature 444, 71-74 (2006).
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Arnold, S.

S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, "Whispering gallery mode carousel: a photonic mechanism for enhanced nanoparticle detection in biosensing," Opt. Express 17(8), 6230-6238 (2009).
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Ashkin, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, "Experimental Observation of Optically Trapped Atoms," Phys. Rev. Lett. 57(3), 314-317 (1986).
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U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, "Single-photon opto-mechanics in the strong coupling regime," N. J. Phys. 12(8), 083030 (2010).
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S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, "Observation of strong coupling between a micromechanical resonator and an optical cavity field," Nature 460(7256), 724-727 (2009).
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M. Aspelmeyer, and K. Schwab, "Focus on mechanical system at the quantum limit," N. J. Phys. 10(9), 095001 (2008).
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S. Gigan, H. R. Bohme, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, "Self-cooling of a micromirror by radiation pressure," Nature 444, 67-70 (2006).
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Baehrs-Jones, T.

M. Lo, W. H. P. Pernice, T. Baehrs-Jones, M. Hochberg, and H. X. Tang, "Harnessing optical forces in integrated photonics circuits," Nano Lett. 456, 480-484 (2008).

Baets, R.

J. Roels, I. D. Vlaminck, L. Lagae, B. Maes, D. V. Throurout, and R. Baets, "Tunable optical forces between nanophotonic waveguides," Nat. Nanotechnol. 4, 510-513 (2009).
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Bäuerle, D.

S. Gigan, H. R. Bohme, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, "Self-cooling of a micromirror by radiation pressure," Nature 444, 67-70 (2006).
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Benito, D. C.

D. C. Benito, S. H. Simpson, and S. Hanna, "FDTD simulations of forces on particles during holographic assembly," Opt. Express 16(5), 2942-2957 (2008).
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Bermel, P.

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," Comput. Phys. Commun. 181, 687-702 (2010).
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A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Burr, J. D. Joannopoulos, and S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
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Bishop, D. J.

H. B. Chan, V. A. Aksyuk, R. N. Kleinman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectromechanical systems by the Casimir force," Science 291, 1941-1944 (2001).
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Figures (6)

Fig. 1
Fig. 1

Schematic of single-membrane (asymmetric) structure: a photonic-crystal (holey) membrane (thickness h1 = 0.2a) consisting of a square-lattice of air holes (radius R = 0.2a) on silicon is suspended (separation s) on top of an unpatterned (homogeneous) silicon slab (thickness h2 = 0.2a) sitting on top of a semi-infinite silica substrate. Light is incident on the membrane from the normal direction (top).

Fig. 2
Fig. 2

(Left:) Resonance frequency ω (units of 2πc/a) a function of separation s (units of membrane period a), for both the single-membrane (asymmetric) structure of Fig. 1 (solid lines) as well as the double-membrane (symmetric) structure of Ref. [24] (dashed lines). The insets show the electric field component Ex in the xz plane (y = 0) near ω at a particular s = 0.3a. In the symmetric case, the attractive and repulsive modes are in-phase and out of phase, respectively, as expected. (Right:) Resonant (peak) force Fc/P (units of incident power P/c), at the resonant frequencies ω plotted on the left figure, as a function of s. The bottom inset shows the broad-bandwidth force spectrum of the asymmetric structure at a particular s = 0.2a, showing both the bonding (F > 0) and antibonding (F < 0) resonances. The inset also denotes what is meant by resonance frequency ω and peak force F.

Fig. 3
Fig. 3

(Left:) Optical force Fc/P on the single-membrane structure of Fig. 1, as a function of the frequency ω of incident light of power P, for various separations s. The insets show typical Ex field patterns (in the xz plane, at y = 0) for both the attractive (left) and repulsive (right) resonances. (Right:) Optical force Fc/P as a function of separation s for incident light input at various frequencies ω ∈ [0.48, 0.5] (2πc/a). The bottom inset shows Fc/P for light input over a lower frequency range ω ∈ [0.41, 0.43] 2πc/a. The force versus s plot was obtained by fitting the force spectrum obtained via FDTD at a few s to a sum of Lorentzian resonances, and then interpolating the resulting Lorentzian parameters over a denser s range.

Fig. 4
Fig. 4

(Left:) Optical force Fc/P as a function of separation s, for light incident at two frequencies ω+ (varied) and ω = 0.495 (2πc/a), with corresponding power P+ and P, respectively. Dashed lines show the force for P = 0. (Right:) Absolute value of optical spring constant |κo| (units of P/ca) as a function of frequency ω+. Dashed and solid lines correspond to negative (unstable) and positive (stable) values of κo, plotted for different values of η = P/P+. Both Fc/P and κo are normalized against the total input power P = P+ + P.

Fig. 5
Fig. 5

(Left:) Optical force Fc/P as a function of frequency ω for light of power P incident on the single-membrane structure of Fig. 1, for various separations s. The bottom inset shows the force (solid lines) and reflection (dashed line) of the same geometry but for h2 = 0 and s = 0.3a. (Right:) Corresponding reflection spectrum R as a function of ω. The open circles indicate frequencies for which there exist force minima or maxima. The insets show the electric field component Ex in the xz plane (y = 0) at a particular s = 0.2a, and at the indicated frequency points ω = 0.64(2πc/a) (left) and ω = 0.681(2πc/a) (right).

Fig. 6
Fig. 6

Schematic of system consisting of two multilayer objects [labeled as (1) and (2)] separated by a distance s. A two-dimensional cross-section for the particular case of two quarter-wave stack mirrors with a defect (yellow) is shown on the right.

Equations (6)

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T i j = 1 2 Re [ ɛ 0 ( E i E j * 1 2 k E k E k * ) + μ 0 ( H i H j * 1 2 k H k H k * ) ]
F c P = c P d d s ( P Q ω ) = Q c ω 2 d ω d s ,
δ ω ( 1 ) ω = 1 2 E ω | Δ ɛ | E ω E ω | ɛ | E ω
E | Δ ɛ | E E | | | Δ s ( ɛ 1 ɛ 2 ) | E | | D | Δ s ( ɛ 1 1 ɛ 2 1 ) | D
δ ω ( 2 ) ω = 1 4 | E ω | Δ ɛ | E ω | 2 | E ω | ɛ | E ω | 2 1 2 ω ω ( ω 3 ω 2 ω 2 ) | E ω | Δ ɛ | E ω | 2 E ω | ɛ | E ω E ω | ɛ | E ω
F c P = 1 + | r 1 + t 1 r 2 e 2 i δ F + | 2 | F + | 2 ,

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