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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 2225–2241
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Bonding, antibonding and tunable optical forces in asymmetric membranes

Alejandro W. Rodriguez, Alexander P. McCauley, Pui-Chuen Hui, David Woolf, Eiji Iwase, Federico Capasso, Marko Loncar, and Steven G. Johnson  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 2225-2241 (2011)
http://dx.doi.org/10.1364/OE.19.002225


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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

1. Introduction

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).

Optical forces can arise due to radiation pressure or gradient forces [3

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]

, 36

36. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

]. Radiation pressure can be thought of as exchange of momentum between a photon (momentum h̄ω/c) and matter, and as a consequence it is easily seen that light with incident power P exerts a force F = P/c on a planar surface if the light is 100% absorbed or a force F = 2P/c if it is 100% reflected. Therefore, the ratio Fc/P is a useful dimensionless measure of the strength of an optical force. Gradient forces, as shown recently [5

5. 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). [CrossRef] [PubMed]

], can arise from the evanescent interaction between localized optical modes, and the resonant increase in the field intensity greatly enhances the force for a given input power [37

37. 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). [CrossRef] [PubMed]

], so that |F|c/P ≫ 1. Such large forces enable strong, tunable optomechanical interactions [13

13. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10(9), 095008 (2008). [CrossRef]

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15. 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). [CrossRef]

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22. 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). [CrossRef] [PubMed]

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28. 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]

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36. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

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38. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). [CrossRef] [PubMed]

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39. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef] [PubMed]

], which have applications such as optical cooling [1

1. A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980). [CrossRef] [PubMed]

,8

8. C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004). [CrossRef] [PubMed]

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12. 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). [CrossRef]

,40

40. T. Hansch and A. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13(1), 68–69 (1975). [CrossRef]

42

42. 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). [CrossRef] [PubMed]

], optical tweezers and traps [3

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]

,43

43. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

47

47. 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). [CrossRef] [PubMed]

], and optical switches [9

9. 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). [CrossRef]

,27

27. 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]

]. Successful demonstrations of other prominent resonant optomechanical effects include coherent mechanical oscillation (amplification) of mesoscopic objects with long vibrational lifetimes, and demonstrations of the photon–phonon strong-coupling regime via dressed optical states [28

28. 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]

, 36

36. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

, 48

48. M. Aspelmeyer and K. Schwab, “Focus on mechanical system at the quantum limit,” N. J. Phys. 10(9), 095001 (2008). [CrossRef]

, 49

49. 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]

].

We showed in Ref. [5

5. 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). [CrossRef] [PubMed]

] that if two identical waveguides or resonant cavities are bought together, the mutual interaction of these (degenerate) resonances or guided modes can induce a splitting of the modes into pairs characterized by attractive and repulsive mechanical forces, analogous to the well-known bonding/anti-bonding states formed by the level splitting (avoided crossings) of interacting degenerate states in quantum systems [50

50. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Butterworth-Heinemann, Oxford, 1977).

]. In submicron-scale photonic devices, these forces are strong enough to yield displacements and other mechanical effects that have been observed experimentally [4

4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009). [CrossRef] [PubMed]

, 19

19. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009). [CrossRef]

, 20

20. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

, 26

26. 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]

]. Furthermore, the frequency and/or phase of the optical excitation can be controlled to yield tunable optomechanical effects, even switching the sign of the force from attractive to repulsive. There are two ways to induce repulsive and attractive interactions, depending on whether the incident power comes in the form of a guided mode or external radiation (the focus of this paper). First, one can inject light parallel to the membranes/waveguides, exciting guided modes that propagate along the waveguides and interact evanescently. In this case, the sign of the interaction is controlled by the relative phase of the modes in the two waveguides [5

5. 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). [CrossRef] [PubMed]

, 14

14. H. Taniyama, M. Notomi, E. Kuramochi, T. Yamamoto, Y. Yoshikawa, Y. Torii, and T. Kuga, “Strong radiation force induced in two-dimensional photonic crystal slab cavities,” Phys. Rev. B 78(16), 165129 (2008). [CrossRef]

, 34

34. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystaloptomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009). [CrossRef] [PubMed]

, 35

35. D. Woolf, M. Loncar, and F. Capasso, “The forces from coupled surface plasmon polaritons in planar waveguides,” Opt. Express 17(22), 19996–20011 (2009). [CrossRef] [PubMed]

]. A similar effect occurs by controlling the relative phase of two coupled cavities (e.g. microspheres or microdisks), but in this case the bonding/anti-bonding resonances also have different frequencies that can be used to control the sign of the force [20

20. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

, 37

37. 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). [CrossRef] [PubMed]

, 51

51. 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). [CrossRef]

]. It is also possible to design asymmetric waveguide/cavity structures (e.g. a dielectric waveguide and a microdisk resonator) with repulsive and attractive interactions as long as both structures support propagating modes, again with light incident along the waveguide direction [4

4. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009). [CrossRef] [PubMed]

, 30

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

]. (Coupled propagating modes in asymmetric geometries were also recently shown to lead to non-monotonic forces [52

52. 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]

].) Second, one can shine light perpendicular to the membranes; if the membranes are perforated by periodic holes (or any other periodic modulation), normally incident radiation can couple via diffraction to guided-mode resonances within the membranes, which again couple evanescently. In this case, because the bonding/anti-bonding resonances have different frequencies, the sign of the force can be controlled by the frequency of the incident light. (By considering lateral shifts, one can also obtain lateral forces and other effects [24

24. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef] [PubMed]

].) As the periodic modulation (e.g. the hole radius) is made smaller, the lifetime (or quality factor Q) of the guided-wave resonances increases [24

24. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef] [PubMed]

], the resonant fields become stronger (intensity ∼ Q), and thus the resonant forces become stronger ∼ Q (albeit narrower in bandwidth ∼ 1/Q). This force enhancement is ultimately limited only by losses (absorption or scattering from finite size or disorder). Another limitation is that the narrow bandwidth translates into a sensitive dependence of the force on the separation of the two membranes (since the resonant frequency shifts with separation).

2. Computational method

If a broad-band force spectrum is desired, an attractive alternative is to compute the stress tensor via the Fourier transform of a short pulse in the time domain (e.g. finite-difference time-domain, FDTD [65

65. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Norwood, MA, 2000).

]), yielding the entire frequency spectrum at once. Here, one simply evolves Maxwell’s equations in response to a pulse source [e.g. ∼ J(x)δ(t)] in time, accumulating the discrete-time Fourier-transform [(ω) ∼ Σn f(nΔt)eiωtΔt] of both the electric E and magnetic H fields over the stress-tensor surface S, and at all desired frequencies ω [66

66. 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). [CrossRef] [PubMed]

,67

67. 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]

]. These Fourier-transformed fields then yield the stress tensor and hence the force. Of course, the force must be normalized in some way, and here the dimensionless Fc/P normalization is very convenient. One simply does a separate calculation, with no structure (vacuum), to compute the Fourier-transformed incident fields and hence the incident power P(ω). Dividing F(ω)c/P(ω) yields the dimensionless force spectrum, where all arbitrary normalization factors (e.g. the incident pulse spectrum or the normalization of the Fourier transform) have canceled. (Matters are more complicated in a nonlinear system, of course.)

In what follows, we exploit our FDTD approach to compute forces on the geometry of Fig. 1. All of the subsequent calculations were performed using MEEP, a free FDTD simulation software package develped at MIT [67

67. 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]

]. We find that discretization errors coming from our finite resolution of 40 pixels/a affect the computed force spectra by no more than a few percent.

3. Membrane forces

In this section, we explore attractive and repulsive resonances in the asymmetric membrane structure of Fig. 1, along with possible applications and the underlying theory.

3.1. Symmetric and asymmetric systems

In the symmetric case, it is well known that each resonance of the individual membranes splits into two resonances of the coupled two-membrane system: “bonding” and “anti-bonding” modes, in which the individual resonances are excited in phase and out of phase, respectively [5

5. 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). [CrossRef] [PubMed]

]. The frequencies of these two resonances as a function of membrane separation are shown as dashed lines in the left plot of Fig. 2, and as expected the frequency splitting vanishes as the separation increases (and hence the membrane coupling decreases) [5

5. 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). [CrossRef] [PubMed]

]. The corresponding Ex field patterns are shown as the right two insets of the left plot, and display the expected phases. Each resonant mode corresponds to a resonant peak in the optical force, and this peak force (at the resonant frequency) is plotted as a function of separation as dashed lines in the right part of Fig. 2. As expected, the bonding and anti-bonding modes correspond to opposite-sign attractive and repulsive forces between the membranes, respectively, and the force becomes stronger as the separation decreases (increasing the membrane interactions). The attractive force in the bonding case has slightly larger magnitude than the anti-bonding repulsion, which can be explained by the larger field overlap in the former case due to the lack of a node in Ex between the membranes [33

33. 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). [CrossRef] [PubMed]

, 37

37. 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). [CrossRef] [PubMed]

].

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.

In the asymmetric case, the mode of the isolated membrane is not the same frequency as the corresponding guided mode of the isolated layered-substrate structure (although the parameters can be adjusted to force a degeneracy if desired). The mode of the silicon on silica (oxide) system is actually a lossless waveguide mode (lifetime ∼ Q = ∞), not a resonance; it is only when the membrane is brought into proximity with the oxide that the membrane’s periodicity a allows guided modes at wavevector 2π/a (and multiples thereof) to couple to normal-incident radiation [61

61. 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, 2008). URL http://ab-initio.mit.edu/book.

]. In this case, the layered-substrate (waveguide) mode that is nearest in frequency to the isolated-membrane resonance frequency is the lowest-order waveguide mode of wavevector 2π/a. Because the two mode frequencies are no longer degenerate, when the resonant frequencies of the asymmetric case are plotted versus separation as solid lines in the left part of Fig. 2, the frequency splitting no longer vanishes as the separation increases. Nevertheless, there is a frequency splitting or “level repulsion,” explained below in terms of second-order perturbation theory, which becomes significant for small separations, and the corresponding field patterns display the qualitative phase characteristics of bonding/anti-bonding modes (insets). As a consequence, as considered theoretically below, the force spectrum in the asymmetric case (right, inset) indeed displays the characteristic attractive and repulsive resonant peaks of bonding/anti-bonding modes. The peak force (at resonance) versus separation is plotted as solid lines in the right part of Fig. 2, and has similar sign as in the symmetric case. Of course, the system is now more complicated than the symmetric case in a variety of ways (e.g. the field patterns are no longer symmetrical/anti-symmetrical and the lifetimes as well as the frequencies depend strongly on separation), so the peak force versus separation dependence is significantly different: First, the peak bonding (attractive) force decreases as s decreases and reaches a constant value as s → ∞. Second, the ratio of the antibonding (repulsive) to bonding force becomes increasingly larger at smaller separations (e.g. it is more than a factor of 2 larger at s = 0.1a), in contrast to what is normally observed [33

33. 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). [CrossRef] [PubMed]

, 37

37. 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). [CrossRef] [PubMed]

].

To understand these features of the force, we use the fact (reviewed in Sec. 3.3) that the force is proportional to both the separation (s) dependence /ds of the frequency and also the lifetime Q. As discussed below, perturbation theory indicates that /ds of a nondegenerate mode in one object decreases proportional to the square of its field overlap with the other object [68

68. 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). [CrossRef]

]. For a mode which as s → ∞ approaches a leaky mode of the perforated membrane—in our case, the antibonding mode—the lifetime Q asymptotes to a nonzero constant and hence the product Qdω/ds → 0; correspondingly, the force tends exponentially to zero with s. Similarly, the force must tend to zero for both the bonding and antibonding modes of a symmetric membrane (where all modes are leaky as s → ∞). On the other hand, for a mode that asymptotes as s → ∞ to a lossless guided mode of the unpatterned substrate—in this case, the bonding mode—the lifetime Q diverges as s → ∞. In fact, perturbative scattering theory [68

68. 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). [CrossRef]

] indicates that the scattered power, and hence 1/Q [61

61. 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, 2008). URL http://ab-initio.mit.edu/book.

], is proportional to the square of the field overlap with the periodic membrane (the source of the scattering loss). Hence Q diverges at the same rate at which /ds vanishes, and thus the force should asymptote to a nonzero constant as s → ∞. These behaviors are precisely what is observed in Fig. 2 (right): the peak forces of the symmetric system and the asymmetric antibonding mode decrease monotonically to zero with increasing s, while the peak force of the asymmetric bonding mode increases monotonically to a constant with increasing s. The corresponding variation in Q is shown in Fig. 3 (left). Note that for s ≲ 0.2a, the Q of the antibonding mode increases rapidly with decreasing s, leading to an increasing antibonding force that is many times larger than the corresponding bonding force. In a practical system, the behavior of the bonding mode will be further modified by the presence of loss in the isolated-substrate guided mode (from finite-size effects, roughness, absorption, etcetera)—this will cause its Q to saturate to a finite value. In this case, the force will behave nonmonotonically: it will initially increase, but will then decrease to zero as s goes beyond the saturation point of Q (while /ds continues to decrease). Thus, the lifetimes of both the membrane and substrate could be exploited to tailor the s dependence of the force in this and other similar systems.

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.

3.2. Tunable mechanical properties

Because the frequency and magnitude of the resonant forces change as s is varied, it is interesting to study also the separation dependence of the force for light incident at a single frequency, which alters the mechanical dynamics of the membrane. Here, we consider the effect of light incident at a single frequency, and then extend our analysis to the case of two frequencies. (More generally, it may prove interesting to study the dynamics of the membrane for modulated pulses.) Figure 3 plots the optical force Fc/P as a function of separation s for incident light at various frequencies ω. As expected, the lifetime ∼ Q of the attractive peak, with Ex concentrated in the layered substrate (shown on the inset), becomes infinite (Q → ∞) as s → ∞ due to the reduced coupling between the infinite-Q substrate mode and the finite-Q PhC resonance. At a fixed frequency, changing s can move the system into or out of resonance, leading to a dramatic s-dependence of the force. (Indeed, the s dependence can be much sharper than shown here, e.g. if the hole diameter is shrunk to increase the Q of the resonances.) One can obtain transitions in the sign of the force, from attractive to repulsive and vice versa, as s is varied, leading to stable and unstable equilibria, not yet including the mechanical restoring force from the membrane supports, and even multiple equilibria.

When mechanical forces are included, two things can happen. If the optical force is nonzero, the mechanical equilibrium point of the membrane will shift and the slope of the force curve (the spring constant κo = dF/ds) will be altered. If one operates at a point where the optical force is zero, then the equilibrium position is unaltered but the spring constant is changed. The total spring constant, including linear mechanical restoring forces on the membrane, will be given by κ = κo + κm, where κm denotes the mechanical spring constant. Whereas κm is frequency- and power-independent, κo exhibits a very sensitive dependence on both, and therefore by choosing ω and the incident power it is possible to tune the total spring constant of the system [69

69. 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). [CrossRef]

72

72. 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]

]. On the one hand, if one chooses ω so that κo/κm > 0, then optical forces act to increase κ and therefore stiffen the stable equilibrium. In systems driven by undesirable thermal fluctuations, this effect has been exploited for “cooling” the resulting vibrations [20

20. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009). [CrossRef]

, 30

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

, 73

73. R. M. Camacho, J. Chan, M. Eichenfield, and O. Painter, “Characterization of radiation pressure and thermal effects in a nanoscale optomechanical cavity,” Opt. Express 17(18), 15726–15735 (2009). [CrossRef] [PubMed]

]. On the other hand, if one chooses ω so that κo/κm < 0, then κ can be decreased and even flip sign as the optical power increases, leading to an unstable equilibrium and bistable behavior [69

69. 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). [CrossRef]

,71

71. A. Mizrahi and L. Schächter, “Two-slab all-optical spring,” Opt. Lett. 32(6), 692–694 (2007). [CrossRef] [PubMed]

]. Near an exact cancellation κo ≈ −κm, the linear term in the s-dependence of the force is decreased relative to the higher-order nonlinear terms (which include both optical and mechanical terms), allowing arbitrarily strong nonlinear mechanical effects, and even a strictly nonlinear regime of operation (κo = −κm) where effects like bistability, hysteresis, and frequency conversion should be readily observable [36

36. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

, 74

74. 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). [CrossRef] [PubMed]

].

For light incident at a single frequency ω, sign transitions in the force occur as the structure moves toward or past a force resonance, as illustrated in Fig. 3. Away from these resonances, the “background” dimensionless force Fc/P is attractive and bounded from above by 2 (see Sec. 4), leading to negligible optical spring constants (small κo) at the corresponding equilibria separations [see Fig. 3 (inset)]. This generally does not preclude a strong modification of the mechanical properties of the membrane (achieving large κoκm) since it is also possible to operate at separations where Fc/P is large and has linear slope (dF /dss), although this inevitably causes a change in the initial mechanical equilibrium separation of the membrane [39

39. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef] [PubMed]

]. For applications in which achieving a large κo without modifying the initial mechanical separation (i.e. achieving a large dF/ds at a position where Fc/P = 0) is important, then a different scheme is required. For example, rather than operating with incident light at a single frequency ω, one can instead consider the combined effect of light incident at two different frequencies ω+ and ω, near the attractive (bonding) and repulsive (antibonding) resonances, respectively. This idea is illustrated in Fig. 4 (left), which shows the optical force Fc/P as a function of separation s for incident light of power P = P+ + P consisting of two frequencies, ω+ [chosen in the region ω+ ∈ [0.41, 0.424] (2πc/a)] and ω = 0.495 (2πc/a), of corresponding power P+ and P, respectively. From Fig. 3, it is clear that incident light at ω leads to a repulsive peak at s ≈ 0.2a whereas incident light in the ω+ range leads to attractive resonances in the range s ∈ [0.1, 0.3]a. As the attractive and repulsive peaks excited by ω and ω+ come close to one another, the transitions in the sign of the force become more pronounced, leading to larger κo. To quantify the enhancement in the spring constant, Fig. 4 (right) plots the absolute value of the optical spring constant |κo| (units of P/ca) as a function of ω+, for different values of the ratio η = P/P+ of power in ω versus ω+, where dashed/solid lines correspond to negative (unstable) and positive (stable) κo, demonstrating orders-of-magnitude enhancement in κo. For example, the peak spring constant |κo| in the case where η = 1 is ≈ 104 (P/ca), whereas it is ≈ 1 in the case of incident light only at ω, corresponding to the limit η → ∞. An alternative scheme that allows tailoring of the optical spring constant near equilibrium was explored in Ref. [51

51. 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). [CrossRef]

], although in that case the effect is achieved by the presence of multiple bonding/antibonding pairs in which opposite-sign force resonances were designed to occur at closely spaced frequencies, whereas here there is no need for the resonances to be closely spaced.

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.

Additional forces on the membrane arise at small separations due to residual static charges and also due to quantum/thermal fluctuations (Casimir and van der Waals forces), which are typically attractive [75

75. K. A. Milton, “The Casimir effect: recent controversies and progress,” J. Phys. A 37, R209–R277 (2004). [CrossRef]

78

78. G. L. Klimchitskaya, U. Mohideen, and V. M. Mostapanenko, “The Casimir force between real materials: experiment and theory,” Rev. Mod. Phys. 81(4), 1827–1885 (2009). [CrossRef]

] and may lead to “stiction” problems in micromechanical (MEMS) systems where moving parts are forced into contact [55

55. F. M. Serry, D. Walliser, and M. G. Jordan, “The role of the Casimir effect in the static deflection of and stiction of membrane strips in microelectromechanical systems MEMS,” J. Appl. Phys. 84, 2501 (1998). [CrossRef]

,56

56. 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). [CrossRef] [PubMed]

,79

79. 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). [CrossRef] [PubMed]

]. Here, the separation dependence of the classical optical force can potentially be used to combat such stiction effects [80

80. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061101 (2005). [CrossRef]

, 81

81. 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). [CrossRef] [PubMed]

]. Not only can one exploit a repulsive resonance to oppose stiction, but the separation dependence means that such a repulsion can be designed to take effect only if s inadvertently falls below some threshold. That is, a repulsive resonance for small s can be used as a feedback effect to reduce the chance of stiction without significantly altering the mechanical dynamics at larger s where the incident light is out of resonance. In an upcoming manuscript, we will demonstrate how these effects can also be exploited to design integrated, all-optical [82

82. The interaction of normal-incident light with the membrane in this system can be exploited to simultaneously control and measure the membrane’s equilibrium separation.

] and accurate techniques for measuring the Casimir effect that rely on measuring static displacements rather than forces or force gradients.

Figures 3 and 4 only show a small sample of the kinds of optical effects that can be observed in evanescently-coupled systems. In particular, there are many degrees of freedom and possibilities to explore, especially if one is not restricted to symmetric structures or operating at a single frequency. For example, the magnitude of the optical forces shown in Fig. 3 are by no means the largest possible, since larger forces can be obtained merely by increasing Q at the expense of bandwidth (and s insensitivity). Even more complicated behaviors can be obtained by increasing the number of resonances, and the choice of resonance offers a corresponding choice of lengthscales or operating frequencies.

3.3. Level repulsion in asymmetric membranes

For well-defined (long lifetime) resonant modes, perturbation theory has been used to analyze the relationship between the force and the resonant frequency/lifetime [34

34. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystaloptomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009). [CrossRef] [PubMed]

, 37

37. 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). [CrossRef] [PubMed]

, 72

72. 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]

], and this relationship can also be used to illuminate the relationship between the sign of the force and the field distribution, as well as the physical origin of resonant repulsion.

The dependence of ω on s can be predicted by perturbation theory. In particular, the first-order change δω(1) to the frequency ω coming from a small change Δɛ in the permittivity of a system with original permittivity ɛ is readily expressed as:
δω(1)ω=12Eω|Δɛ|EωEω|ɛ|Eω
(3)
In this case, however, Δɛ is not small: at a given point near the interface, ɛ is changing discontinuously as that interface moves past the point. In this case, perturbation theory must be derived more carefully [68

68. 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). [CrossRef]

]. For an interface from ɛ1 to ɛ2 that is moving by Δs (towards ɛ2), assuming isotropic materials, the numerator of Eq. (3) changes to:
E|Δɛ|EE|||Δs(ɛ1ɛ2)|E||D|Δs(ɛ11ɛ21)|D
(4)
Without loss of generality, we can hold the upper membrane fixed and move the substrate (or lower membrane) away by Δs. From Eq. (4), the way to obtain attractive (/ds > 0) and repulsive (/ds < 0) resonant effects is clear. If we hold one membrane fixed and move the substrate (or the other membrane), /ds will be positive (attractive) if |E|2 is peaked where ɛ1 < ɛ2, i.e. on the air/silicon interface (adjacent to the upper membrane). Conversely, /ds will be negative (repulsive) if |E|2 is peaked where ɛ1 > ɛ2, i.e. on the silicon/oxide interface (away from the upper membrane). Precisely such field patterns can be observed in the insets of Fig. 2 and Fig. 3: the repulsive and attractive modes have Ex peaked at the expected interfaces. Note that a homogeneous substrate, e.g. semi-infinite silicon or oxide, has no interface except for the air interface adjacent to the upper membrane, so in this case a repulsive force cannot arise by this mechanism, as demonstrated for the h2 = 0 case on the inset of Fig. 5 below. An exception to this rule is discussed in Sec. 4, in which repulsive forces arising from radiative modes are analyzed, a situation where a lack of normalizability causes the perturbation theory to break down.

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).

The above discussion indicates which field patterns might be expected to lead to repulsion and attraction, but does not explain how such field patterns can arise. For the case of a symmetric membrane, symmetry considerations predict that the degenerate modes of two isolated membranes will split into even/odd bonding/anti-bonding pairs by degenerate first-order perturbation theory [84

84. A. Messiah, Quantum Mechanics: Vol. II (Wiley, New York, 1976). Ch. 17.

]. The presence of a nodal plane bisecting the anti-bonding mode (assuming its field is dominated by Exy and not Ez) means that the field pattern will be stronger on the far sides of the membranes, leading to a repulsive interaction as observed in Fig. 2 and Fig. 3, and as predicted above. For asymmetric membranes, however, there is typically no degeneracy, and the corresponding two-mode interaction must instead be analyzed by second-order perturbation theory, which plays a role for sufficiently small separations (large interactions). When two isolated waveguides each have a mode with nearby frequencies, and one brings the waveguides together so that the mode fields overlap, second-order perturbation theory predicts a contribution to Δω that tends to split the two frequencies:
δω(2)ω=14|Eω|Δɛ|Eω|2|Eω|ɛ|Eω|212ωω(ω3ω2ω2)|Eω|Δɛ|Eω|2Eω|ɛ|EωEω|ɛ|Eω
(As above, the overlap integrals are modified into E|| and D components for motion of discontinuous interfaces [68

68. 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). [CrossRef]

].) Note that the ω2ω2 term pushes ω away from ω′, and becomes stronger as the frequencies become closer (with the most dramatic case being degenerate modes, where the derivation is modified). Because of the competition between the first- and second-order terms, which may be comparable in magnitude for small separations (large overlaps), it is possible for the force near a resonance to switch signs with separation, a possibility that is demonstrated in Sec. 3.4 below.

3.4. Multi-modal interactions

Previously, we considered resonances at a relatively low frequency (compared to 2πc/a), where the only relevant interactions were between two modes (one for each isolated membrane or substrate). At higher frequencies, the density of states generally increases, and thus many more resonant modes are typically present. Correspondingly, the inter-modal interactions become more complicated, and it is not always possible to identify individual pairs of bonding/anti-bonding modes. However, the qualitative features of repulsive and attractive resonances are still present, although the additional complexity provides more degrees of freedom leading to more complicated force phenomena.

For example, the force and reflection spectra for the asymmetric membrane system of Fig. 1 are shown in Fig. 5 in a higher frequency window (about double the frequencies in Fig. 3). As before, the force spectrum (left) shows both repulsive and attractive resonances. In this case, however, we actually observe a force resonance changing sign as a function of separation, which physically can be interpreted as different terms dominating in the perturbation theory [Eq. (5)] at different separations. In the reflection spectrum (right), these resonances correspond to Fano shapes (adjacent peaks and dips), a well-known consequence of the coherent combination of a resonant process with direct transmission through the slabs [9

9. 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). [CrossRef]

, 24

24. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef] [PubMed]

, 85

85. 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). [CrossRef]

]. Because the reflection spectrum depends sensitively on the separation, the peak locations from a broad-bandwidth low-intensity source could be used to accurately determine the separation in an experiment. Note also that the resonant peaks in this frequency window approach one another as the separation decreases, which means that the largest contribution to level repulsion in this case is coming from interactions with other modes (outside this frequency window, not shown); this is verified by examining the field patterns (upper insets), which clearly correspond to completely different modes in the membrane and not just a relative-phase change. As in Sec. 3.2, fixing the frequency and plotting the force versus separation reveals a force that changes both magnitude and sign as a function of separation.

4. Fabry–Perot forces

Resonant radiation pressure within Fabry–Perot cavities has a long history, dating back to work in the 1960s on interferometer sensitivity [96

96. V. B. Braginsky and A. B. Manukin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 25, 653–655 (1967).

], and has since been considered both theoretically and experimentally for many applications [36

36. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

, 89

89. 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). [CrossRef]

], such as nonreciprocal phenomena [97

97. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef] [PubMed]

], optical cooling [93

93. S. Gigan, H. R. Böhm, 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). [CrossRef] [PubMed]

95

95. O. Arcizet, P. F. Cohadon, T. Briant, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micomirror,” Nature 444, 71–74 (2006). [CrossRef] [PubMed]

], and tunable optical springs [47

47. 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). [CrossRef] [PubMed]

]. If the space between two partially reflecting mirrors (e.g. Bragg mirrors) is viewed as a waveguide, then the resonance frequency for normal-incident light corresponds to a slow-light (zero group-velocity) band edge where radiation pressure is enhanced [70

70. 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). [CrossRef]

, 89

89. 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). [CrossRef]

]. In all of these cases, the pressure is repulsive, as one might expect for light bouncing between the two objects (and was argued in general for two semi-infinite objects [64

64. M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60(4), 2363–2374 (1999). [CrossRef]

]). We show this in general below, for radiating modes (not guided modes) and any unpatterned multi-layered structure (in the absence of gain).

Here, we consider a general class of geometries, depicted in Fig. 6, consisting of two unpatterned (translationally invariant in two directions) planar multilayer objects separated by distance s in vacuum, denoted as objects (1) and (2), characterized by complex reflection/transmission amplitudes r1/t1 and r2/t2 at a given frequency, respectively (satisfying |rk|2 + |tk|2 = 1 in the absence of absorption or gain). The dimensionless force Fc/P on object (1) due to light normally incident from above at frequency ω can be readily computed (using a simple transfer-matrix analysis to obtain the stress tensor [98

98. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

]) to be:
FcP=1+|r1+t1r2e2iδF+|2|F+|2,
(5)
where F+ = t1/[1 − r1r2 exp(2)] is the induced field at the lower interface of object (1), and δ ≡ 2πωs is the phase associated with the air gap.

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.

One can construct multilayer objects supporting exponentially localized resonances which couple to normally incident radiation. For example, this is the case if each object consists of a multilayer Bragg mirror with an embedded defect layer [61

61. 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, 2008). URL http://ab-initio.mit.edu/book.

]. In this case, degenerate perturbation theory implies that two symmetric objects [such as those in Fig. 6 (right)], each with an identical embedded defect/resonance, should couple to form bonding/anti-bonding states where the resonances are in/out-of phase. Naively, one might suppose that this will lead to repulsive and attractive resonances, as in the coupled guided-mode case, but Eq. (5) indicates that this is impossible. The explanation is straightforward: although such defect modes are exponentially localized within the Bragg mirrors composing each object, they are propagating in the region between the objects where there are no mirrors (because the input beam is propagating in free space and no diffraction occurs). This means that the frequency splitting does not depend exponentially on the separation between the two objects, and hence there is no resonant force enhancement via these modes, by the analysis of Sec. 3.

5. Conclusion

Optomechanical interactions are a rich subject of current research, and the use of evanescently coupled guided resonances enables an especially rich set of phenomena because of the presence of both attractive and repulsive resonances. The ability to tailor and exploit guided resonances coupled via periodic modulations offers an exciting opportunity to procure complicated force effects at small separations. In this paper, we showed that functionality similar to that of previously studied symmetric-membrane systems can be obtained in asymmetrical membrane-substrate structures. From an experimental point of view, such asymmetrical structures are attractive in that only a single membrane need be suspended and patterned. From a theoretical viewpoint, because the resonant modes of asymmetrical structures need not come in degenerate pairs (unless degeneracies are forced), more than one pair of modes can have strong interactions, leading to the possibility of richer force phenomena [51

51. 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). [CrossRef]

]. Correspondingly, the distance dependence of a force spectrum with multiple attractive and repulsive resonances can exhibit richer “optical spring” phenomena than are possible with repulsive resonances alone (as in Fabry–Perot resonances between mirrors); for example, one can operate at a zero of the optical force to tune the optical spring constant (in either direction) without altering the equilibrium position.

Acknowledgments

We are grateful to Aristeidis Karalis and Peter Bermel at MIT for useful discussions. This work was supported by the Army Research Office through the ISN under Contract No. W911NF-07-D-0004, and by the Defense Advanced Research Projects Agency (DARPA) under contract N66001-09-1-2070-DOD.

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OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(230.4320) Optical devices : Nonlinear optical devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 19, 2010
Revised Manuscript: January 14, 2011
Manuscript Accepted: January 16, 2011
Published: January 21, 2011

Citation
Alejandro W. Rodriguez, Alexander P. McCauley, Pui-Chuen Hui, David Woolf, Eiji Iwase, Federico Capasso, Marko Loncar, and Steven G. Johnson, "Bonding, antibonding and tunable optical forces in asymmetric membranes," Opt. Express 19, 2225-2241 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2225


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