Second Harmonic Generation Ppt

Recently, the second harmonic generation in an one-dimensional, nonlinear acoustic metamaterial with negative mass density, composed of an air-ﬁlled waveguide periodically loaded by clamped elastic plates have been analytically and numerically reported 28. The specific slot waveguide structure is exploited to significantly enhance the pump-harmonic modal overlap by greatly reducing the counteractive electric field portion of the second mode at TH. According to our simulation, THG conversion efficiency reaches 1.4732 × 10 −5 with 3 W pump power at a waveguide length of 12.3 μ m.

Second Harmonic Generation Slot Waveguide

Second Harmonic Generation Ppt

ACTA ACUSTICA UNITED WITH Vol. 104 (2018) 235 – 242
ACUSTICA DOI 10.3813/AAA.919165
Second-Harmonic Generation in Acoustic Waveguides Loaded with an Array of Side Holes Jiangyi Zhang1) , Vicente Romero-García1) , Georgios Theocharis1) , Olivier Richoux1) , Vassos Achilleos1) , Dimitrios J. Frantzeskakis2)
1)
2)
Laboratoire d’Acoustique de l’Université du Maine – CNRS UMR 6613, Le Mans, France. [email protected] Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
Summary We study analytically and numerically second-harmonic generation in a one-dimensional weakly lossy nonlinear acoustic metamaterial composed of an air-ﬁlled waveguide periodically loaded by side holes. Based on the transmission line approach, we derive a lossy nonlinear dispersive lattice model which, in the continuum limit, leads to a nonlinear, dispersive and dissipative wave equation. The latter is studied by means of a perturbation method, which leads to analytical expressions for the ﬁrst and second harmonics, in very good agreement with numerical results. PACS no. 43.25.-x, 43.25.Jh
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1. Introduction The study of metamaterials in diﬀerent ﬁelds of wave physics has seen an explosion of interest during the last years, leading to signiﬁcant developments both from basic research and applications point of view. Metamaterials are artiﬁcially engineered structures, exhibiting physical properties not found in nature [1, 2, 3]. In the case of acoustics, Liu et al. [4] proposed the ﬁrst acoustic metamaterial, originally called locally resonant sonic material, based on an array of coated spheres presenting negative mass density. Later, Fang et al. [5] investigated an acoustic metamaterial composed by an array of Helmholtz resonators embedded in a waveguide, presenting negative bulk modulus. Although such systems have been previously analyzed in the context of acoustics by Sugimoto [6] and Bradley [7], the novelty in this case is to characterize the system by using the eﬀective properties. The linear properties of such acoustic metamaterials including the eﬀect of viscothermal losses [8] have been exploited over the last decade, showing several possibilities to control wave propagation in acoustics [9, 10, 11, 12, 13, 14, 15, 16]. In addition to the more standard case of linear metamaterials, the study of nonlinear ones has been receiving increased attention during the last years [17, 18, 19, 20]. However, in the case of acoustic metamaterials, the presence of nonlinearity is less studied, and only a few works have exploited the combined role nonlinear eﬀects and
Received 25 August 2017, accepted 27 November 2017.
© S. Hirzel Verlag · EAA
other fundamental features of the system, such as dispersion or losses. The combination of nonlinearity and dispersion has revealed diﬀerent eﬀects, such as the formation of solitons [21, 22, 23], the self-demodulation eﬀect [24, 25] and the generation of higher harmonics [26, 27, 28]. More recently, in a similar setup with negative mass density, the formation of envelope (bright and gap) solitons has been analytically and numerically studied in the presence of viscothermal losses [29]. In this work, we analytically and numerically study the second-harmonic generation in an one-dimensional (1D) weakly lossy nonlinear acoustic metamaterial with negative bulk modulus composed by an air-ﬁlled waveguide periodically loaded by side holes. The main motivation is to study the combined eﬀects of dispersion, nonlinearity and dissipation in an acoustic metamaterial with negative bulk modulus (our setup). The model used in this work could pave the way to study the linear and nonlinear propagation in double negative metamaterials considering the diﬀerent physical eﬀects playing role as dispersion, nonlinearity and dissipation. The nonlinearity is activated here by using high-amplitude incident waves. Based on the electroacoustic analogy and the transmission line approach, we derive a weakly lossy lattice model describing the system, which, in the continuum limit, leads to a nonlinear, dispersive and dissipative wave equation. By using a perturbative scheme we derive analytical expressions for the ﬁrst and second harmonics, in the presence of losses. The analytical results are found to be in very good agreement with direct numerical simulations in the framework of the lattice model.
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ACTA ACUSTICA UNITED WITH Vol. 104 (2018)
ACUSTICA
Zhang et al.: Second-harmonic generation in waveguides
The article is structured as follows. In Section 2 we introduce the electro-acoustic analogue modeling based on the transmission line (TL) approach, which has been shown to be a powerful tool for studying electromagnetic and acoustic metamaterials [29, 30, 31, 32, 33, 34]. In Section 2.1, we describe the setup and introduce the 1D nonlinear dissipative lattice model. Then, in Section 2.2, we obtain a nonlinear dispersive and dissipative wave equation, stemming from the continuum limit of the lattice model. Section 2.3 describes the linear properties of the proposed metamaterial. In Section 3, we present the analytical and numerical results regarding the second harmonic generation. Finally, in Section 4, we summarize and present our conclusions.
2. Electro-Acoustic Analogue Modeling
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2.1. Setup and model We consider low-frequency nonlinear wave propagation in the acoustic metamaterial shown in Figure 1a. The structure is composed by a waveguide of radius r periodically loaded with an array of side holes of radius rH and length lH . The distance between two consecutive side holes is d. The frequency range considered is well below the ﬁrst cut-oﬀ frequency, therefore the problem is considered as one-dimensional. We adopt the electro-acoustic analogy, where the voltage corresponds to the acoustic pressure and the current to the volume velocity ﬂowing through the waveguide’s cross-sectional area [35, 36]. Our aim is to derive a nonlinear discrete wave equation, describing wave propagation in an equivalent electrical transmission line (TL). To do this, we consider the unit-cell circuit of the equivalent TL model of our setting, shown in Figure 1b. The unit-cell circuit is composed by two parts. The ﬁrst one, corresponding to the waveguide, is modeled by the acoustic masses Mω , the resistance Rω and shunt acoustic compliances Cω . In the linear regime, the acoustic masses and acoustic compliances are given by Mω0 = ρ0 d/S and Cω0 = Sd/(ρ0 c02 ), where ρ0 , c0 and S are respectively the density, the speed of sound and the cross-section area of the waveguide. The resistance Rω = Im(kZc )d describes the viscothermal losses, where the wavenumber and acoustic characteristic impedance of the waveguide are given by [8]: 0005 ω 1−j k= 1+ (1) (1 + (γ − 1)/ P r) , c0 s ρ0 c0 Zc = S
0005
1−j 1+ (1 − (γ − 1)/ P r) . s
(2)
Here, γ is the speciﬁc heat ratio, P r is the Prandtl number, and s = ωρ0 r2 /η, with η being the shear viscosity. The second part of the unit-cell circuit, corresponding to the side hole, is modeled by a shunt RM circuit composed by the series combination of an acoustic masses MH
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2rH
lH
r
(a)
d un
(b)
M L !0001
pn-1
un +1 uH
R0001 C0001
L MHH
RH d
waveguide
pn
side holes
Figure 1. (Colour online) (a) Acoustic metamaterials composed by a waveguide loaded with an array of side holes. (b) Corresponding unit-cell circuit.
and a resistance RH . In the regime where the geometric characteristics of the side holes are much smaller than the wavelength λ of the sound wave, i.e., for krH 1, the corresponding acoustic masses and resistance are given by MH = ρ0 lH /SH and RH = ρ0 ω2 /(2πc0 ), where SH is the area of the side holes [37]. Here we should mention that we approximate the frequency-dependent

Second Harmonic Generation Slot Waveguide