TIME TRANSFORMATIONS, ANISOTROPY AND ANALOGUE TRANSFORMATION ELASTICITY

Transcription

1 TIME TRANSFORMATIONS, ANISOTROPY AND ANALOGUE TRANSFORMATION ELASTICITY ACT ARIADNA PROJECT C. García Meca,, S. Carloni, C. Barceló, G. Jannes, J. Sánchez Dehesa, and A. Martínez TECHNICAL REPORT T R

2 OUTLINE. FUNDAMENTALS OF ANALOGUE TRANSFORMATION ACOUSTICS 2. SPACE TIME TRANSFORMATIONS 3. ANISOTROPIC TRANSFORMATIONS 4. ANALOGUE TRANSFORMATION ELASTICITY 5. CONCLUSIONS

3 TRANSFORMATION ACOUSTICS We started t dby analyzing the pressure wave equation typically used in transformation ti acoustics: = Bulk modulus Space time transformation E.g.: No correspondence for this term = isotropic density = Inverse inhomogeneous anisotropic density VIRTUAL SPACE (Cartesian coordinates and homogeneous isotropic medium) PHYSICAL SPACE (Cartesian coordinates and general medium) Acoustic equations are not invariant under general transformations that mix space and time: We can design static devices such as standard cloaks We cannot design dynamic devices such as time cloaks or frequency converters as in optics 2 /3

4 SOLUTION: AUXILIARY ANALOGUE SPACETIME Analogue gravity: searches laboratory analogues of relativistic phenomena with formally identical equations Wave equation for a relativistic massless scalar field in a curved spacetime (form invariant) = Spacetime metric ABSTRACT RELATIVISTIC SPACETIME Formally identical in some coordinate systems en.wikipedia.org 2 Acoustic equation (form variant) LABORATORY SPACE C. Barceló et al., Living Rev. Relativity 4, 3 (20) 2 By Alain r (CC BY SA 2.5), via Wikimedia commons 3 /3

5 SOLUTION: AUXILIARY ANALOGUE SPACETIME MEDIUM LABORATORY SPACE 4 /3

6 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME 2 MEDIUM LABORATORY SPACE 5 /3

7 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 2 MEDIUM LABORATORY SPACE 6 /3

8 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Rename: 2 4 MEDIUM LABORATORY SPACE 7 /3

9 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Rename: 2 4 Relation between MEDIUMS and 2 LABORATORY SPACE 5 MEDIUM MEDIUM 2 8 /3

10 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Related by coordinate transformation 2 4 Rename: Same solution Same solution LABORATORY SPACE 5 MEDIUM MEDIUM 2 9 /3

11 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Related by coordinate transformation 2 4 Rename: Same solution Same solution LABORATORY SPACE 5 MEDIUM Related by coordinate MEDIUM 2 transformation 0 /3

12 SOLUTION: AUXILIARY ANALOGUE SPACETIME ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Rename: 2 4 Space time transformation E.g.: No correspondenceforthis term LABORATORY SPACE 5 MEDIUM MEDIUM 2 /3

13 SOLUTION: AUXILIARY ANALOGUE SPACETIME New problem: there is not a complete analogy between a general space time metric and the acoustic medium, which has not as many degrees of freedom as the metric Spacetime transformations not yet possible We need a more general system: allow the background fluid to move VELOCITY POTENTIAL WAVE EQUATION c = Speed of sound (directly related to B and ) v = background velocity This equation is not form invariant under general spacetime transformations but has more degrees of freedom able to mimic many spacetime transformations using analogue transformations 2 /3

14 SPACETIME TRANSFORMATION ACOUSTICS ABSTRACT RELATIVISTIC SPACETIME Space time transformation: 3 Rename: Form invariant 2 equation 4 LABORATORY SPACE 5 Form variant MEDIUM velocity potential wave equation MEDIUM 2 C. García Meca et al., Sci. Rep. 3, 2009 (203). 3 /3

15 MOVING BACKGROUND Additionally, the use of this equation allows us to work with moving media Example: cloaking a bump in a moving aircraft a c Flat wall Bump Bump y x Cloak (no background velocity correction) Cloak (corrected background velocity) C. García Meca et al., Sci. Rep. 3, 2009 (203). 4 /3

16 WHEN IS ATA INDISPENSABLE? Find under which conditions the velocity potential wave equation preserves its shape (no new terms appear) : There is almost no possibility of performing a spacetime transformation without the appearance of new terms We explored the application of spacetime transformations by designing gseveral devices that do not fulfill any of these conditions C. García Meca et al., Wave Motion 5, 785 (204). 5 /3

17 EXAMPLE : DYNAMICALLY RECONFIGURABLE ABSORBER Selective absorption of acoustic rays: Transformation Compressor Ray.0 Space compression f 0(t) Ray enters the box Ray 2 enters the box Ray enters the absorber t (ms) Implementation Static omnidirectional absorber (index gradient ) A. Climente, D. Torrent, and J. Sánchez Dehesa, Appl. Phys. Lett. 00, 4403 (202). Ray 2 6 /3

18 EXAMPLE : DYNAMICALLY RECONFIGURABLE ABSORBER COMSOL transient simulation: Compressor Ray Static omnidirectional absorber (index gradient) Ray 2 7 /3

19 EXAMPLE 2: SPACETIME CLOAK Any transformation mixing time and one space variable can be implemented: Implementation Transformation Transformation for a spacetime cloak : 3 3 ct 2 0 Curtain map ct 2 0 Simulation Theory Cloaked region x x M. W. McCall et al., J. Opt. 3, (20). 8 /3

20 EXAMPLE 3: FREQUENCY CONVERTER A simple transformation of the time variable changes the frequency of the input acoustic wave : Transformation Verified with full wave COMSOL transient simulations Useful to prevent oscillations of undesired frequencies from entering a given region or to accommodate the wave frequency to the spectral range of our detector S. A. Cummer and R. T. Thompson, J. Opt. 3, (20). 9 /3

21 EXAMPLE 4: SPACETIME COMPRESSOR Increases the density of events within a spacetime region by simultaneously compressing space and time. Changes the frequency and wavelength within the compressed region The medium of the region where we have the compressed wave needs not be changed ct Transformation ct x x COMSOL transient simulation: Simulation Theory ct C. García Meca et al. Photon. Nanostruct. Fudam. Appl. 2, 32 (204). x 20 /3

22 EXAMPLE 4: SPACETIME COMPRESSOR Visualizing the effect: 2 /3

23 ANISOTROPY Prescribed acoustic parameters are smooth functions of the coordinates and show an anisotropic character. We only have a discrete set of isotropic acoustic properties available. How to connect the theoretical results of ATA and the technological realization of the required media? MICROSCOPIC WAVE EQUATION MACROSCOPIC WAVE EQUATION Homogenization For low frequency oscillations, a composite behaves as a homogeneous medium with different properties that depend on the constitutive materials A wide range of acoustic parameter values, even anisotropic, can be achieved. 22 /3

24 HOMOGENIZATION We initially focused on the static background case: MICROSCOPIC ACOUSTIC EQUATION Velocity potential Pressure Two scale homogenization procedure (medium properties change much faster than the acoustic wave):. Periodic acoustic parameters 2. Ellipticity condition Under these assumptions: Effective properties HOMOGENIZED ACOUSTIC EQUATION Cell problem 23 /3

25 CLOAKING THE VELOCITY POTENTIAL Based on a multilayer structure Acoustic properties p of each layer Potential transformation is physically different from a pressure transformation Scatterer Scatterer surrounded by 50 layer cloak C. García Meca et al., Phys. Rev. B 90, (204). 24 /3

26 HOMOGENIZATION Typical configuration: o wood inclusions cuso sin air Supersonic speeds achievable! Different parameters for the same microstructure depending on which equation we homogenize? C. García Meca et al., Phys. Rev. B 90, (204). 25 /3

27 HOMOGENIZATION We investigated the origin of this unexpected result from basic fluid mechanics FLUID MECHANICS EQ. OF STATE POTENTIAL EQ. PRESSURE EQ. Same medium everywhere Same pressure everywhere Acoustic parameters may vary due to a Acoustic parameters may vary if there are background pressure gradient different media at each point C. García Meca et al., Phys. Rev. B 90, (204). 26 /3

28 PRACTICAL IMPLEMENTATION New way of implementing acoustic metamaterials unveiled! A continuous material variation can be achieved without the need for homogenization EXAMPLE: The interference of several high amplitude background waves can produce the desired timevarying pressure distribution: 27 /3

29 PRACTICAL IMPLEMENTATION PIPES: dff different pressures for different dff cross Microparticle suspension: ferromagnetic flakes sections (Bernoulli's theorem) orientation dynamically reconfigurable through an external magnetic field High pressure Low pressure M.J. Seitel et al., Appl. Phys. Lett. 0, 0696 (202) 28 /3

30 ELASTICITY See presentation by Gil Jannes /3

31 CONCLUSIONS ANALOGUE TRANSFORMATIONS. Allows us to generalize transformational techniques to non form invariant equations 2. The main requirement is to find an relativistic equation which is analogue to the form variant one 3. We applied the method to acoustics extension to spacetime transformations open the door to dynamically tunable devices based on transformational techniques 4. Application to elasticity: we have obtained important preliminary results but more work needs to be done WE HAVE UNVEILED A NEW METHOD FOR CONSTRUCTING ACOUSTIC METAMATERIALS PRESSURE GRADIENTS. No need to combine different materials to change the acoustic properties of a medium 2. Extension of ATA to the anisotropic case 3. Possibility of achieving acoustic media with time varying properties 29 /3

32 PUBLICATIONS Nature s Scientific Reports 3, 2009 (203) Wave Motion 5, 785 (204) Physical Review B 90, (204) Photonics and Nanostructures 2, 32 (204) INVITED 30 /3

33 FUTURE WORK ANALOGUE TRANSFORMATIONS OPEN A NEW WORLD OF POSSIBILITIES. Application to more fields of physics with an incomplete transformational technique: Thermodynamics Electronics 2. Reinterpretation of the transformational properties of an equation potentially has far richer applications: Interpret the elements of a form invariant equation in a different way many possible different transformational theories for the same field! ELECTROMAGNETISM Interpret the transformation of an equation as the equation of a different physical phenomena to connect their solutions EXPERIMENT PHASE There are several potential ti ways for bringing i ATA to reality that t we would like to test t Experimental capabilities in our team: Sanchez Dehesa s LAB + Nanophotonics Technology Center 3 /3