Fano resonances in nanoscale structures

Andrey E. Miroshnichenko, Sergej Flach, and Yuri S. Kivshar

Rev. Mod. Phys. 82, 2257 – Published 11 August 2010

Abstract

Modern nanotechnology allows one to scale down various important devices (sensors, chips, fibers, etc.) and thus opens up new horizons for their applications. The efficiency of most of them is based on fundamental physical phenomena, such as transport of wave excitations and resonances. Short propagation distances make phase-coherent processes of waves important. Often the scattering of waves involves propagation along different paths and, as a consequence, results in interference phenomena, where constructive interference corresponds to resonant enhancement and destructive interference to resonant suppression of the transmission. Recently, a variety of experimental and theoretical work has revealed such patterns in different physical settings. The purpose of this review is to relate resonant scattering to Fano resonances, known from atomic physics. One of the main features of the Fano resonance is its asymmetric line profile. The asymmetry originates from a close coexistence of resonant transmission and resonant reflection and can be reduced to the interaction of a discrete (localized) state with a continuum of propagation modes. The basic concepts of Fano resonances are introduced, their geometrical and/or dynamical origin are explained, and theoretical and experimental studies of light propagation in photonic devices, charge transport through quantum dots, plasmon scattering in Josephson-junction networks, and matter-wave scattering in ultracold atom systems, among others are reviewed.

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DOI:https://doi.org/10.1103/RevModPhys.82.2257

Authors & Affiliations

Andrey E. Miroshnichenko^*

Nonlinear Physics Centre and Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS), Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia

Sergej Flach

Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany

Yuri S. Kivshar

Nonlinear Physics Centre and Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS), Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia

^*aem124@physics.anu.edu.au

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Vol. 82, Iss. 3 — July - September 2010

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Figure 1
(Color online) Ugo Fano (1912–2001). “Outstanding interpreter of how radiation interacts with atoms and cells” (64) and much more (this review).Reuse & Permissions
Figure 2
(Color online) Resonances of parametrically driven coupled oscillators. (a) Schematic view of two coupled damped oscillators with a driving force applied to one of them; (b) the resonant dependence of the amplitude of the forced oscillator $| c_{1} |$ , and (c) the coupled one $| c_{2} |$ . There are two resonances in the system. The forced oscillator exhibits resonances with symmetric and asymmetric profiles near the eigenfrequencies $ω_{1} = 1$ and $ω_{2} = 1.2$ (b), respectively. The second coupled oscillator responds only with symmetric resonant profiles (c). Adapted from 167.Reuse & Permissions
Figure 3
(Color online) Fano resonance as a quantum interference of two processes: direct ionization of a deep inner-shell electron and autoionization of two excited electrons followed by the Auger effect. This process can be represented as a transition from the ground state of an atom $| g ⟩$ either to a discrete excited autoionizing state $| d ⟩$ or to a continuum $| c ⟩$ . Dashed lines indicate double excitations and ionization potentials.Reuse & Permissions
Figure 4
(Color online) Illustration of the Fano formula (1) as a superposition of the Lorentzian line shape of the discrete level with a flat continuous background.Reuse & Permissions
Figure 5
(Color online) Normalized Fano profiles (1) with the prefactor $1 ∕ (1 + q^{2})$ (2) for various values of the asymmetry parameter $q$ .Reuse & Permissions
Figure 6
Tuning of the asymmetry parameter of the Fano resonance. Measured (upper row) and calculated (bottom row) absorption spectra of a single quantum dot for various laser powers. The absorption profile varies from a symmetrical to an asymmetrical one with increase of the laser power, indicating the enhancement of the continuum transition. From 195.Reuse & Permissions
Figure 7
(Color online) Two-channel model for a Feshbach resonance. Atoms that are prepared in the open channel undergo a collision at low incident energy. In the course of the collision, the open channel is coupled to the closed channel. When a bound state of the closed channel has energy close to zero, a scattering resonance occurs. From 42.Reuse & Permissions
Figure 8
(Color online) Variations in the Fano-Anderson model. (a) Schematic view of the Fano-Anderson model with an additional defect in the chain. (b) Transmission coefficient for different distances between the Fano site and the additional defect for parameters $C = 1$ , $V_{F} = 0.5$ , $E_{F} = 0$ , and $E_{L} = 1$ . (c) Schematic view of the Fano-Anderson model with a locally coupled $N$ -defect chainlet. (d) Transmission coefficient of the $N$ -site chainlet model. All sites in the chainlet are identical and with zero eigenfrequencies $E_{m} = 0$ , and the couplings are $C = V_{m} = 1$ . Adapted from 253.Reuse & Permissions
Figure 9
Resonant reflection of a soliton in topological networks. (a) The reflection coefficient vs wave number $k$ for two Cayley trees of length $M = 5$ (line) and $M = 6$ (line) attached to the discrete array. Empty circles and stars correspond to direct numerical simulations of the soliton propagation. (b) Example of the soliton reflection by a Fano-like defect. From 52.Reuse & Permissions
Figure 10
(Color online) Fano-Anderson model as a discrete one-dimensional system with a single side-coupled Fano state defect. (a) The array of circles corresponds to a linear chains, and the isolated circle is the Fano state. Arrows indicate the coupling between different states. (b) Transmission coefficient [Eq. (10)] for various values of the coupling coefficient $V_{F}$ . Other parameters are $C = 1$ , and $E_{F} = 0$ . (c) Areas of bistability of the nonlinear Fano resonance (dashed line) in the parameter space $(α_{k}, γ_{k})$ . (d) Nonlinear transmission coefficient versus input intensity for various frequencies $ω_{k}$ for $C = 1$ , $V_{F} = 0.8$ , $E_{F} = 0$ , and $λ = 1$ . Regions of bistability are indicated by dashed lines, corresponding to unstable solutions. Adapted from 257.Reuse & Permissions
Figure 11
(Color online) Light scattering in an array of channel waveguides with quadratic nonlinearity. (a) Schematic view of a one-dimensional array of channel waveguides with nonlinear defects, created by periodic poling. Arrows indicate the scattering process. (b) Comparison of the transmission coefficients of plane waves (solid line) and a Gaussian beam (crosses). Bottom: Example of the Gaussian beam scattering by a single nonlinear defect showing (c) the resonant reflection part of the beam at the fundamental frequency and (d) resonant excitation of the second harmonic. Adapted from 256.Reuse & Permissions
Figure 12
(Color online) Time-periodic scattering potentials. (a) Schematic view of the open channel $X$ and closed channel $Y$ from Eqs. (24, 25, 26). The dashed line indicates the localized state of the closed channel $Y$ inside the open channel $X$ . (b) Schematic view of the virtual states, generated by a possibly infinite number of harmonics of the time-periodic scattering potential.Reuse & Permissions
Figure 13
(Color online) Light scattering by optical solitons. (a) Sketch of the scattering setup by an optical soliton in a one-dimensional waveguide array. The soliton beam is sent along the $z$ axis, while the probe beam propagates in the $x − z$ -plane at some angle to the soliton. (b) Top view of the scattering process (c) Transmission coefficient vs $k_{x}$ for plane waves under oblique incidence. There is total suppression of the transmission near $k_{x} \approx 0.181$ . (d) Fourier spectrum of the incident (dashed line) and transmitted (solid line) beams. The suppression of the resonant frequency [see (c)] in the spectrum is observed. Adapted from 113.Reuse & Permissions
Figure 14
(Color online) Plasmon scattering by discrete breathers in Josephson-junction ladders. (a) Schematic setup for the measurement of plasmon scattering with the use of controlled bias currents $γ_{i}$ ; (b) oscillating power $P_{ac, R}$ at the right end with (solid line) and without (dashed line) the DB; (c) transmission coefficient $T$ , derived from (b) by using Eq. (34). Adapted from 258.Reuse & Permissions
Figure 15
(Color online) Scattering scheme in an optical lattice. The incoming, reflected, and transmitted beams of atoms are represented as plane waves. The atoms interact only around $n = n_{c}$ , where the BEC is centered. From 345.Reuse & Permissions
Figure 16
(Color online) Transmission $T$ vs momentum $k$ . Lines, analytic solution; symbols, real time numerical simulations of Eq. (35) using wave packets for $g = 0.36$ (line and boxes), $g = 0.6$ (line and diamonds), and $g = 0.9$ (line and triangles). From 345.Reuse & Permissions
Figure 17
Typical resonant structures. Schematic setup for (a) a waveguide directly coupled to a cavity and (b) a waveguide side-coupled to a cavity.Reuse & Permissions
Figure 18
(Color online) Modeling Fano resonances in photonic crystals. Schematic view of (a) photonic crystal waveguide with an isolated side-coupled cavity and (b) effective discrete system. (c) Typical profile of the Fano resonance.Reuse & Permissions
Figure 19
Light propagation in a photonic crystal waveguide with a side-coupled cavity. (a) Photonic crystal waveguide formed by removing a single row of rods. Within the line defect there are two smaller rods. A point defect, created by reducing the radius of a single rod, is placed away from the waveguide. (b) Transmission spectra through the structure (a) with (solid) and without (dashed) the two defects in the waveguide. From 89.Reuse & Permissions
Figure 20
(Color online) Add-drop filter. (a) Schematic diagram of two waveguides coupled through an element which supports a localized resonant state. (b) Electric field pattern of the photonic crystal at the resonant frequency. The white circles indicate the position of the rods. From 92.Reuse & Permissions
Figure 21
(Color online) Electric field distributions in a photonic crystal for (a) high and (b) low transmission states. The same color scale is used for both panels. The black circles indicate the positions of the dielectric rods. From 366.Reuse & Permissions
Figure 22
(Color online) Ultralow all-optical switching in the slow-light regime. (a) Quality factor $Q$ vs group velocity $v_{g}$ at resonance for the waveguide-cavity structure. (b) Nonlinear bistable transmission at the frequencies with 80% of linear light transmission vs the incoming light power for different values of the rod radius. (c) Switch-off bistability threshold vs the group velocity at resonance. From 243.Reuse & Permissions
Figure 23
(Color online) Dynamical instability of the nonlinear Fano resonance. (a) Nonlinear transmission coefficient and (b) imaginary part of eigenvalues of the stability problem vs input power. In the vicinity of the nonlinear Fano resonance the plane-wave excitation becomes dynamically unstable. Temporal evolution of (c) and (e) the field inside the side-coupled cavity and (d) and (f) the transmission coefficient for two different values of the input power values, indicated in (a). Near the resonance the dynamics of the field inside the nonlinear cavity yields a buildup of a modulation instability in time. Adapted from 252.Reuse & Permissions
Figure 24
(Color online) Two types of the geometries of a photonic crystal waveguide side-coupled to two nonlinear optical resonators. Light transmission and bistability are qualitatively different for (a) on-site and (b) intersite locations of the resonator along the waveguide. Adapted from 244.Reuse & Permissions
Figure 25
(Color online) Overlapping Fano resonances. Typical transmission curves for four different cases. (a) Two identical side-coupled defects $ω_{α} = ω_{β}$ (solid). Transmission for a single side-coupled cavity is shown by a dashed line. (b) Two side-coupled cavities with strongly detuned eigenfrequencies $| ω_{α} - ω_{β} | ⪢ 1$ . (c) and (d) Two side-coupled cavities with slightly detuned eigenfrequencies $| ω_{α} - ω_{β} | ⪡ 1$ for (c) on-site coupling and (d) intersite coupling. From 244.Reuse & Permissions
Figure 26
Light scattering by photonic crystal slabs. (a) Geometry of the photonic crystal film. (b) Transmission and (c) reflection spectra. The solid lines are for the photonic crystal structure, and the dashed lines are for a uniform dielectric slab with a frequency-dependent dielectric constant. Adapted from 90.Reuse & Permissions
Figure 27
(Color online) Measured scattering spectra (dots) and fitting by the Fano formula (solid lines) of a photonic crystal nanocavity for two different excitation conditions: (a) a tightly focused and (b) a slightly defocused laser beam of diameters $d_{1}$ and $d_{2}$ , respectively, indicated by circles. Note that the actual profiles are inverted ones because of the use of cross-polarized detection. From 124.Reuse & Permissions
Figure 28
(Color online) Exact Mie solution of the light scattering by a plasmonic nanoparticle. The radius of the nanoparticle is much smaller than the light wavelength $a ∕ λ = 0.083$ . (a) Frequency dependence of the scattering light intensity in the vicinity of the dipole and quadrupole resonances. In the latter case both forward- (solid lines) and backward- (dashed lines) scattering profiles exhibit asymmetric Fano resonances. (b) and (c) The angular dependence of the light scattering in the vicinity of the quadrupole resonance. The plasmonic frequency is normalized to $ω_{p} a ∕ c = 1$ . Adapted from 214.Reuse & Permissions
Figure 29
(Color online) A metallic nanostructure consisting of a disk inside a thin ring supports superradiant and very narrow subradiant modes. Symmetry breaking in this structure enables a coupling between plasmon modes of differing multipolar order, resulting in a tunable Fano resonance: (a) extinction spectra as a function of incident angle $θ$ ; (b) the impact of filling the cavity with a dielectric material on the extinction spectrum or permittivity $ϵ = 1$ (solid line), 1.5 (dashed line), and 3 (dotted line). Adapted from 145.Reuse & Permissions
Figure 30
Light scattering by metallic gratings. (a) Focused ion beam image of a two-dimensional hole array in a polycrystalline silver film. (b) Observed transmission intensity as a function of photon energy and $k_{x}$ with predicted energy dispersion of surface plasmon-polaritons (solid) and loci of Wood’s anomaly (dashed lines). From 128.Reuse & Permissions
Figure 31
Planar wave-vector diagram illustrating the phase-matching condition for RFWM. From 331.Reuse & Permissions
Figure 32
Two-color resonant four-wave mixing. (a) Nonparametric and (b) parametric TCRFWM process. The autoionizing level (Fano state) above the ionization potential is indicated by $| η ⟩$ . $| i ⟩$ and $| e ⟩$ are ground and intermediate states, respectively. (c) TCRFWM and Fano profiles. Inset: the slopes of two profiles. Note the separation between the slope zeros which correspond to profile minima. From 330.Reuse & Permissions
Figure 33
Electron micrograph of a single-electron transistor based on a $Ga As ∕ Al Ga As$ heterostructure. The split gates (I) define the tunnel barriers and the additional gate electrode (II) adjusts the potential energy on the quantum dot. From 136.Reuse & Permissions
Figure 34
Conductance vs gate voltage. Comparison of conductance measurements in the (a) Fano regime, (b) intermediate regime, and (c) Coulomb-blockade regime. From (c) to (a) the lead-dot coupling increases. Fits to the Fano formula (1) are shown for the center and right resonances in (a). The respective asymmetry parameters are $q = - 0.03$ and $- 0.99$ . From 136.Reuse & Permissions
Figure 35
Conductance vs gate voltage. (a) Temperature dependence of the conductance for two Fano resonances. (b) Conductance as a function of the gate voltage for various magnetic fields applied perpendicular to the two-dimensional electron gas. Adapted from 136.Reuse & Permissions
Figure 36
(Color online) Channel conductance data (squares) and fits (curves) vs gate voltage in the Fano regime. Bars show fitting ranges. Inset: scanning electron microscope image of a similar sample. From 168.Reuse & Permissions
Figure 37
(Color online) Tunable Fano interferometer. (a) Experimental data with 12 Fano resonances. (b) Fits of one resonance using different fitting parameters. (c)–(e) $g_{tot}$ , $g_{coh}$ , $Γ$ , and $q$ from (a). (f) Data exhibiting reversals of $q$ . (g) Extracted $q$ values. From 168.Reuse & Permissions
Figure 38
An Aharonov-Bohm ring with an embedded QD in one of its arms. (a) Schematic representation of the experimental setup. (b) Scanning electron micrograph of the fabricated device. From 182.Reuse & Permissions
Figure 39
Coulomb oscillation at $V_{c} = - 0.12 V$ with the arm pinched off and asymmetric Coulomb oscillation at $V_{c} = - 0.086 V$ with the arm transmissible. Here $T = 30 mK$ and $B = 0.91 T$ . Adapted from 182.Reuse & Permissions
Figure 40
(Color online) Control of the Fano resonance via magnetic field. Conductance of (a) two Fano peaks at $30 mK$ and selected magnetic fields, (b) one Fano peak vs $V_{g}$ and $B$ , (c) same as (b) but for larger windows of $V_{g}$ variations. The white line represents the AB phase as a function of $V_{g}$ . From 182.Reuse & Permissions
Figure 41
Typical resonant structures with quantum dots. Schematic representation of (a) a serial model of leads and a quantum dot [Eqs. (48, 49)], and (b) a T-shaped model of leads and a side-coupled quantum dot [Eq. (50)].Reuse & Permissions
Figure 42
Conductance as a function of $ϵ_{d}$ for different values of background transmission $T_{b}$ . The AB phase $ϕ = 0$ . From 156.Reuse & Permissions
Figure 43
Spin filter with Fano dot. Left plots: $⟨ n_{σ} ⟩$ vs $ϵ_{d}$ for a finite splitting $Δ = 1.6$ , $U = 0$ (top) and $U = 12$ (bottom). The black (white) dots represent the numerical results for the spin-up (spin-down) occupation number. Right plots: $g_{σ}$ vs $ϵ_{d}$ for finite splitting $Δ = 1.6$ , $U = 0$ (top), and $U = 12$ (bottom). The conductance is computed numerically for spin-up (black dots) and spin-down (white dots) electrons. The solid and dashed lines on the top figures represent the exact results of Eq. (56). Adapted from 335.Reuse & Permissions

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