Supplementary Information: Manipulating Acoustic Wavefront by Inhomogeneous Impedance and Steerable Extraordinary Reflection

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1 Supplementar Information: Manipulating Acoustic Wavefront b Inhomogeneous Impedance and Steerable Extraordinar Reflection Jiajun Zhao 1,, Baowen Li,3, Zhining Chen 1, and Cheng-Wei Qiu 1, 1 Department of Electrical and Computer Engineering, National Universit of Singapore, Singapore , Republic of Singapore Department of Phsics and Centre for Computational Science and Engineering, National Universit of Singapore, Singapore , Republic of Singapore and 3 Center for Phononics and Thermal Energ Science, School of Phsical Science and Engineering, Tongji Universit, Shanghai 0009, People s Republic of China (Dated: Ma 8, 013) I. DERIVATION OF IGSL IN ACOUSTICS We mathematicall derive the connection between the interface specific acoustic impedance (SAI) and the manipulation of wavefronts, which gives birth to the proposed impedance-governed generalized Snell s law of reflection (IGSL) as the design rule of SAI. In addition, we mathematicall predict the double reflections and the situation when the ordinar reflection can be switched off. We assume the time-harmonic factor in this appendix is e iωt, where ω is the circular frequenc, and the coordinate sstem is that in Fig. 1(a). The incident acoustic pressure can be expressed as: p i (, z, ω) = p i0 (ω) exp[ik 0 ( sin θ i z cos θ i )], (1) where k 0 = ω/c 0 is the wave number in free space, θ i the incident angle and p i0 (ω) the amplitude. Z n (, ω) = p(, 0, ω)/[n v(, 0, ω)] as the specific acoustic impedance (SAI) [1] of a locall reacting boundar is laid at the interface, where n is the unit vector opposite to z direction and v is the acoustic velocit. The boundar condition of this problem can be paraphrased as []: z p(, 0, ω) + ik 0β(, ω)p(, 0, ω) = 0, () where β(, ω) = ρ 0 c 0 /Z n (, ω) (ρ 0 and c 0 being the given densit and sound speed respectivel in the upper space) is the normalized acoustical admittance of the locall reacting surface. We expand β to be β(, ω) = β(, ω) + β 0 (ω), where β 0 is a real constant. The ordinar reflection is expressed as: p ro (, z, ω) = p i0 (ω)r(θ i, β 0 ) exp[ik 0 ( sin θ ro + z cos θ ro )], (3) where R is the reflection coefficient and θ ro the angle of p ro. Because p ro observes the usual Snell s law, θ ro = θ i. In order to find the expression of R, we introduce the constant SAI: Z 0 (ω) = ρ 0c 0 β 0 (ω) = p i (, 0, ω) + p ro (, 0, ω) n v i (, 0, ω) + n v ro (, 0, ω), (4)

2 (a) p i S θi θro ρ 0c0 D SAI Zn n o z p ro θre dl p re (b) r0 point source image source r0 z θ * o θ * n 1 ρ 0c 0 r observing point Ζ0=1/β0(ω) FIG. 1: (a) Illustration for some notations. The orange line indicates the contour of the Green s integral. S is the semicircular contour; D is the flat one along the surface. p i, p ro and p re denote the incidence, the ordinar reflection, and the extraordinar reflection, respectivel. n is the unit vector opposite to z direction. Z n is set for the flat interface (z = 0). (b) Schematic diagram for the effective paths of acoustic radiation. The introduced θ can be interpreted as the effective incident angle. r, r 0, and r 0 are the location vectors for the point source, the image source and the observer, respectivel. where n is the normal vector indicated in Fig. 1(a), v i and v ro the acoustic velocities of p i and p ro. Substituting Eq.(1) and Eq.(3) into Eq.(4) and appling Euler equation ρ 0 tv = p, we obtain: R(θ i, β 0 ) = cos θ i β 0 (ω) cos θ i + β 0 (ω). (5) In Fig. 1(a), the total acoustic field can be written in the integral form: p(, z, ω) = [G(, z, ω; 0, z 0 ) p( 0, z 0, ω) p( 0, z 0, ω) G(, z, ω; 0, z 0 )]dl, (6) n 0 n 0 S+D where dl( 0, z 0 ) is the infinitesimal segment along the integral contour, n 0 = n( 0, z 0 ) and G(, z, ω; 0, z 0 ) is the Green s function corresponding to the following partial differential problem: G + k 0 G = δ( 0 )δ(z z 0 ), z > 0; [ G + ik 0 β 0 (ω)g] z 0 = 0. (7) z0=0 When the radius of the semicircular contour S approaches, we can regard the contour integral along S is mainl contributed b p i and p ro. Therefore Eq.(6) changes into p(, z, ω) = p i (, z, ω) + p ro (, z, ω) [G z 0 p( 0, z 0, ω) p( 0, z 0, ω) z 0 G]d 0. (8) We can simplif Eq.(8) b substituting Eq.(7) and Eq.() into it. B defining the last part in Eq.(9) as the extraordinar reflection p re (, z, ω), which is the unique extra component beond p ro, we obtain p re (, z, ω) = ik 0 The explicit solution of G(, z, ω; 0, z 0 ) in Eq.(7) is G = i 4 H(1) 0 (k 0 r r 0 ) + i 4π β( 0, ω)p( 0, 0, ω)g(, z, ω; 0, 0)d 0. (9) 1 k z k z ωβ 0 /c 0 k z + ωβ 0 /c 0 exp[ik z (z + z 0 ) + ik ( 0 )]dk, (10) where r = (, z), r 0 = (, z), and k 0 = k + k z. When r is awa from the surface D, k z k 0 cos θ holds, where θ is introduced as a constant. Via this approximation and another definition r 0 = ( 0, z 0 ), it turns out that [3] cos θ z ( z 0) r r constant. (11) 0

3 3 Through Eq.(11), it can be obtained that k z ωβ 0 /c 0 cos θ β 0 (ω) k z + ωβ 0 /c 0 cos θ + β 0 (ω) constant R(θ, β 0 ). (1) Appling Eq.(1) into Eq.(10) and using the formula of the clindrical wave expansion in terms of plane waves, we approach a neat form of the Green s function: G(, z, ω; 0, z 0 ) i 4 H(1) 0 (k 0 r r 0 ) + R(θ, β 0 ) i 4 H(1) where H (1) 0 ( ) the Hankel function of the first kind[4]. 0 (k 0 r r 0 ), (13) From the phsical insight into Eq.(13), the first part of G is the direct contribution of the point source to the observer through path in Fig. 1(b). The second part is the product of the Green s function excited b the image source and the reflection coefficient R, denoting p ro. According to our interpretation, Fig. 1(b) illustrates path 1 and path, visualized as p ro and p re respectivel[5]. Due to the expression of R, we figure out that θ is the effective incident angle regarding to Fig. 1(b). Furthermore, it is reasonable to sa that the major contribution of the integral in Eq.(10) is attributed to the vicinit of θ, in which wa R can be regarded as a constant and put outside the integral. B far-field approximation, we are able to get these expansions: r r 0 = r( 0 sin θ + z 0 cos θ); r r 0 = r( 0 sin θ z 0 cos θ); H (1) 0 (x) x πx ei(x π 4 ), (14) where r is the length of r; sin θ = /r; cos θ = z/r. Substituting Eq.(14) and Eq.(5) into Eq.(13), we obtain: 1 G(, z, ω; 0, 0) i πk 0 r ei(k 0r π 4 ) e ik 0 0 sin θ cos θ cos θ + β 0 (ω). (15) After substituting Eq.(15) into Eq.(9), the extraordinar reflection becomes: p re k0 πr ei(k 0r π 4 ) cos θ cos θ + β 0 (ω) β( 0, ω)p( 0, 0, ω)e ik 0 0 sin θ re d 0. (16) Further, after appling Born approximation to Eq.(16) and expanding it b Eq.(1) and Eq.(3), p re becomes: k0 p i0 (ω) exp[i(k 0 r π 4 p re )] cos θ cos θ i β( πr [cos θ 0, ω)e ik 0 0 (sin θ i sin θ re ) d 0. (17) + β 0 (ω)][cos θ i + β 0 (ω)] Now we consider our proposed SAI: [ Z n (, ω) = A 1 i tan ψ() ] ; β 0 (ω) = ρ 0c 0 A. (18) After substituting Eq.(18) into Eq.(3) and Eq.(17), we obtain the ordinar reflection and the extra reflection: extraordinar reflection: p ro A cos θ i ρ 0 c 0 A cos θ i + ρ 0 c 0 exp[ik 0 ( sin θ ro + z cos θ ro )], (19) p re e iψ() e ik0(sin θi sin θre) d. (0)

4 4 (a) p i p ro θi θro GSL 0 0 p ro p ra ρ c Z GSL inhomogeneous phase change (b) p i ρ c 0 0 θi Z θi on or off θ re p ro inhomogeneous acoustic impedance IGSL p re FIG. : (a) For a flat interface with an inhomogeneous phase change, the angle of p ro, i.e., θ ro, is tweaked in a fashion of GSL. The manipulated ordinar reflection is called to be the anomalous reflection p ra in terms of GSL. [6] (b) For a flat interface with an inhomogeneous SAI, θ ro = θ i without influence, while p re occurs simultaneousl and θ re is controlled b IGSL, impling double reflections. If SAI is properl controlled, p ro can be switched off. Here note that in our case we are able to create double reflections b means of SAI inhomogeneit. Eq.(0) is a Dirac Delta if we consider ψ() to be a linear term as the first order approximation. Or else, we know that the integral in Eq.(0) will reach the maximum b imposing the stationar phase approximation, i.e., sin θ re sin θ i = 1 k 0 dψ() d. (1) Although Eq.(1) corresponds to the form of the generalized Snell s law of reflection (GSL) [6], the variables in the two situations are different. Starting from Eq.(18) and ending up with Eq.(1), we provide the insight between our designed SAI and the direction of p re, without considering the phase in terms of wave propagation. We name Eq.(1) as IGSL in acoustics, as the design principle of the SAI Eq.(18). According to Eq.(19), if A = (ρ 0 c 0 )/( cos θ i ), we can switch off p ro. Therefore Eq.(18) becomes Z n (, ω) = ρ [ 0c 0 1 i tan ψ() ]. () cos θ i II. DIFFERENCES BETWEEN GSL AND IGSL Although GSL is not our topic in this paper, the same appearance of IGSL and GSL ma cause the false impression that our IGSL is the same as GSL. Actuall their mechanisms are totall different. In terms of phase inhomogeneit, the anomalous reflection p ra actuall corresponds to the situation when the ordinar reflection p ro is steered toward a wrong direction governed b GSL [6], illustrated in Fig. (a). There is onl one single direction of reflection all the while. On the contrar in terms of SAI inhomogeneit, it is found that IGSL cannot alter p ro b an SAI interface, but can turn off p ro so as to provide insight into the engineering of special wavefronts b SAI interface, illustrated in Fig. (b). Moreover, the extraordinar reflection p re governed b IGSL is an additionall unique component in acoustics, which can be geared along arbitrar directions, simultaneousl with vanishing p ro. Therefore, our proposed IGSL opens up rich effects and unprecedented applications in the communit of acoustics. Additionall, GSL can even be phenomenologicall considered as one subset of IGSL, when p ro is turned off. In order to stress the irrelevance between IGSL and GSL again, we list the differences: 1. GSL is initiated in electromagnetism with electric properties; IGSL is initiated in acoustics with mechanical properties.. GSL is derived

5 z(m) Reflected Pressure Field (Pa) p rb 30 Degree (m) -60 Degree incidence FIG. 3: The SAI Eq.() with ψ() = ( ) is set along the flat surface z = 0. In the upper space, the medium is water (ρ 0 = 1kg/m 3 ; c 0 = 1500m/s). An audible plane wave with unit amplitude and ω = 30Krad/s is 60 obliquel incident. Onl reflected acoustic pressure is plotted. The propagating path of p re is noted as an arrow with purple crossbars. from Fermat Principle, i.e., the conservation of the wave number along an interface; IGSL is derived from Green s function. The fundamental phsics is distinguished. 3. The variable of GSL is phase inhomogeneit; the variable of IGSL is impedance inhomogeneit. The methods are independent. 4. GSL will onl generate single reflection; IGSL not onl can generate single reflection, but also can generate double reflections. 5. GSL acts upon p ro ; IGSL acts upon p re. 6. In GSL, the anomalous reflection corresponds to the situation where p ro is tweaked toward a different direction governed b GSL; in acoustics, IGSL cannot alter p ro b SAI interfaces, but is capable of turning on or turning off p ro. III. SIMULATION FOR OBLIQUE INCIDENCE In Fig. 3, we assume water (ρ 0 = 1kg/m 3 ; c 0 = 1500m/s [1]) as the background medium in the upper space. The SAI Eq.() with the linear parameter ψ() = ( ) is set along the flat surface, and an audible (ω = 30Krad/s) plane wave with a unit amplitude is obliquel incident with the incident angle 60. These parameters theoreticall lead to the angle of p re 30 according to our proposed IGSL Eq.(1). Furthermore, p ro vanishes thanks to the specific A chosen in Eq.(). In Fig. 3, we find out the simulation confirms the prediction via IGSL accuratel, and p ro disappears as expected. Moreover, the incident audible plane wave and p re are at the same side of the normal line, confirming the possibilit of the negative extraordinar reflection. The singularit due to tan[ψ()/] in the imaginar part of Eq.() does not pla a significant role because the mathematical singularit ±i just occurs at singular positions, phsicall meaning the total reflection (reflection coefficient equals +1). [1] D. T. Blackstock, Fundamentals of phsical acoustics (Wile, 000). [] M. A. Nobile, and S. I. Haek, Acoustic propagation over an impedance plane. J. Acoust. Soc. Am. 78, 135(1985).

6 6 [3] G. Taraldsen, A note on reflection of spherical waves. J. Acoust. Soc. Am. 117, 3389 (005). [4] C. F. Chien, and W. W. Soroka, Sound propagation along an impedance plane. J. Sound Vib. 43, 9 (1975). [5] G. Taraldsen, The complex image method. Wave Motion 43, 91 (005). [6] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso and Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333 (011).