Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit

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Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit

Peter T. Rakich, Charles Reinke, Ryan Camacho, Paul Davids, and Zheng Wang

Phys. Rev. X 2, 011008 – Published 30 January 2012

Abstract

Stimulated Brillouin scattering (SBS) is traditionally viewed as a process whose strength is dictated by intrinsic material nonlinearities with little dependence on waveguide geometry. We show that this paradigm breaks down at the nanoscale, as tremendous radiation pressures produce new forms of SBS nonlinearities. A coherent combination of radiation pressure and electrostrictive forces is seen to enhance both forward and backward SBS processes by orders of magnitude, creating new geometric degrees of freedom through which photon-phonon coupling becomes highly tailorable. At nanoscales, the backward-SBS gain is seen to be $10^{4}$ times greater than in conventional silica fibers with 100 times greater values than predicted by conventional SBS treatments. Furthermore, radically enhanced forward-SBS processes are $10^{5}$ times larger than any known waveguide system. In addition, when nanoscale silicon waveguides are cooled to low temperatures, a further $10 - 100$ times increase in SBS gain is seen as phonon losses are reduced. As a result, a $100 − μ m$ segment of the waveguide has equivalent nonlinearity to a kilometer of fiber. Couplings of this magnitude would enable efficient chip-scale stimulated Brillouin scattering in silicon waveguides for the first time. More generally, we develop a new full-vectorial theoretical formulation of stimulated Brillouin scattering that accurately incorporates the effects of boundary-induced nonlinearities and radiation pressure, both of which are seen to have tremendous impact on photon-phonon coupling at subwavelength scales. This formalism, which treats both intermode and intramode coupling within periodic and translationally invariant waveguide systems, reveals a rich landscape of new stimulated Brillouin processes when applied to nanoscale systems.

Received 22 August 2011

DOI:https://doi.org/10.1103/PhysRevX.2.011008

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Peter T. Rakich^1,*, Charles Reinke¹, Ryan Camacho¹, Paul Davids¹, and Zheng Wang^2,3

¹Sandia National Laboratories, P.O. Box 5800 Albuquerque, New Mexico 87185-1082, USA
²Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
³Department of Electrical and Computer Engineering, Microelectronics Research Center, The University of Texas at Austin, Austin, Texas 78758 USA

^*rakich@alum.mit.edu.

Popular Summary

Stimulated Brillouin scattering (SBS) in an optical medium is a nonlinear process through which intense laser light interacts with a material to produce coherent phonons through photon-phonon coupling. Since its discovery in 1964, stimulated Brillouin scattering in bulk macroscopic) media and (micron-scaled) optical fibers has been extensively studied and exploited for manipulating both phonons and photons. Realizations of coherent phonon generation, slow light, and a host of new light sources are just a few examples of its importance. Through stimulated Brillouin scattering, the photon-phonon coupling is generally mediated by electrostriction—a nonlinear contraction of the material under the influence of the light—and depends very little on the geometry or size of the material. In this theoretical paper, we open up a new direction in the study of stimulated Brillouin scattering by demonstrating that a new, powerful, and geometry-dependent form of stimulated Brillouin scattering emerges in the absence of intrinsic material electrostriction as light is confined to nanoscales.

The systems we have investigated are nanoscale waveguides. The new form of stimulated Brillouin scattering has its origin in the enormous radiation pressure generated by light confined to the nanoscale and in a boundary-induced nonlinearity. We show that the radiation pressure and the boundary-induced nonlinearity can already generate powerful SBS nonlinearity on their own. When coherently combined with electrostriction, they are seen to produce giant SBS effects that are several orders of magnitude stronger than what has been seen to date in any other waveguide system. Such enhanced couplings could enable efficient stimulated Brillouin scattering in silicon waveguides for the first time.

More generally, we have developed a new theoretical formulation of stimulated Brillouin scattering that accurately incorporates the effects of boundary-induced nonlinearities and radiation pressure, both of which are shown to have tremendous impact on photon-phonon coupling at subwavelength scales. Our application of this formalism already reveals a rich landscape of new stimulated Brillouin processes within nanoscale systems. More applications of this formalism in the rapidly developing field of optomechanics are certainly to be expected.

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Vol. 2, Iss. 1 — January - March 2012

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Figure 1
(a) Sketch showing one experimental embodiment of a suspended silicon waveguide. The black wire-frame represents the portion of the waveguide examined here. (b) The cross-sectional sketch of the waveguide within the wire-framed region with the $E_{x}$ field of fundamental TE-like mode superposed. (c) The energy-level diagram of a Brillouin process. Here the optical pump and Stokes waves are shown in blue and red, respectively, while the phonon is shown in black. Phase-matching diagrams for forward (d) and backward (e) SBS. (f) The waveguide cross section (f) along with mode profiles for the (g) $E_{x}$ , (h) $E_{y}$ , and (i) $i E_{z}$ components of the fundamental TE-like mode for $λ = 1550 nms$ , $a = 300 nm$ , and $b = 280 nm$ .Reuse & Permissions
Figure 2
Computed SBS gain shown in arbitrary units (a.u.) for (a) FSBS and (c) BSBS processes, for comparison with the computed phonon dispersion of the lowest-order symmetric phonon modes (labeled E1–E5) seen in (b). The FSBS and BSBS resonances match well with computed dispersion of (b). In (b), the wave vector is normalized by $K = 2 k_{p}$ for wavelength $λ = 1550 nms$ . Colored insets of (a) show the displacement amplitudes and shape deformation of the dominant E2 and E5 phonons about $K = 0$ excited through FSBS, while those of (c) show the E1 and E2 modes at $K = 2 k_{p}$ in the BSBS case. High (low) displacement regions are shaded red (blue). Calculations assume silicon waveguide dimensions of $a = 300 nm$ and $b = 280 nm$ and wavelength $λ = 1550 nms$ . Note, elastic modes which do not contribute to SBS have been omitted from the dispersion plot in (b). For clarity, the computed elastic modes seen in (b) were restricted to those with reflection symmetries about the $x − z$ and $y − z$ mirror planes centered on the waveguide core. However, the frequency dependent elastic-wave models used to compute SBS gain in (a) and (c) do not limit the number of modes by use of symmetry. Hence, SBS gain shown in (a) and (c) includes the contribution to photon-phonon coupling arising from all acoustic modes in the system.Reuse & Permissions
Figure 3
Detailed analysis of FSBS: (a) The computed FSBS gain vs the phonon frequency near the E2 and E5 resonances centered at 13.0 and 18.8 GHz. The individual contributions of electrostriction (red) and radiation pressure (green) to the SBS gain are shown along with their combined effect (blue). Simulations assume a phononic loss $Q$ of $1000$ and optical wavelength $λ = 1.55 μ ms$ . (b) A schematic of a waveguide segment ( $L = 300 nm$ ). (c) and (d) are the characteristic displacements of the E2 and E5 elastic modes. High-displacement regions are shown in red, and low-displacement regions are shown in blue. Schematics showing the amplitudes and directions of the dominant components of ${\tilde{f}}_{Ω} (x, y)$ are seen in (e), (f), and (g). Radiation pressure contributes only to the transverse force components (e), while electrostriction contributes to both the transverse components (f) and the longitudinal components (g). Note that the longitudinal forces are vanishingly small for FSBS.Reuse & Permissions
Figure 4
Detailed analysis of BSBS: (a) The computed BSBS gain vs the phonon frequency near the E1 and E2 resonances centered at 13.8 and 15.0 GHz. The individual contributions of electrostriction (red) and radiation pressure (green) to the SBS gain are shown along with their combined effect (blue). Simulations assume a phononic loss $Q$ of $1000$ and optical wavelength $λ = 1.55 μ m$ . (b) A schematic of a waveguide segment ( $L_{1} = 300 nm$ ), and characteristic mode displacements (c) and (d) of the E1 and E2 elastic modes. High-displacement regions are shown in red, and low-displacement in blue. Schematics (e), (f), and (g) show the amplitudes and directions of the dominant components of ${\tilde{f}}_{Ω} (x, y)$ within a waveguide segment of length $L_{2} = (π / Δ k) ≅ 150 nm$ . Radiation pressure contributes only to the transverse force components (e), while electrostriction contributes to both the transverse components (f) and the longitudinal components (g).Reuse & Permissions
Figure 5
Computed SBS contributions from electrostriction (ES) and radiation pressure (RP) as the waveguide dimension, $a$ , is varied from 0.2–10 microns (for $b = 0.93 a$ ). (a) BSBS gain resulting from ES (red), RP (green), and the coherent combination of both ES and RP (blue). (b) An identical set of curves for FSBS. In comparison, the conventional microscale SBS theory (gray curves) underestimates the BSBS gain by up to 2 orders of magnitude for length scales between 250 nm and a few microns. For further details on the microscale SBS trend (gray curves), see Appendix . The dimensions where radiation pressure is dominant are shaded gray.Reuse & Permissions
Figure 6
Computed BSBS gain versus the waveguide dimension, $a$ , for $b = 0.93 a$ . Comparison of a model based on MS-SBS theory (shown in red circles) is made with the integral form of the SBS gain (shown in triangles) and the simplified SBS trend (shown in dashes).Reuse & Permissions

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