final presentation - technical report v pdf · ata time transforms anisotropy elasticity...

TIME TRANSFORMATIONS, ANISOTROPY AND ANALOGUE TRANSFORMATION ELASTICITYANALOGUE TRANSFORMATION ELASTICITY

C. García Meca, S. Carloni, C. Barceló,T R

ACT ARIADNA PROJECT, , ,

G. Jannes, J. Sánchez‐Dehesa, and A. Martínez TECHNICAL REPORT

ATA Time transforms Anisotropy Elasticity Conclusions

OUTLINE

1. FUNDAMENTALS OF ANALOGUE TRANSFORMATION ACOUSTICS

2. SPACE‐TIME TRANSFORMATIONS

3. ANISOTROPIC TRANSFORMATIONS

4. ANALOGUE TRANSFORMATION ELASTICITY

5. CONCLUSIONS


W t t d b l i th ti t i ll d i t f ti ti

TRANSFORMATION ACOUSTICS

• We started by analyzing the pressure wave equation typically used in transformation acoustics:

No correspondence for this termSpace‐time

= isotropic density

= Bulk modulus

Inverse inhomogeneoustransformationE.g.:

= 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

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‐ We cannot design dynamic devices such as time cloaks or frequency converters as in optics


• Analogue gravity: searches laboratory analogues of relativistic phenomena with formally identical equations1

SOLUTION: AUXILIARY ANALOGUE SPACETIME

ABSTRACT RELATIVISTIC SPACETIMEWave equation for a

g g y y g p y q

Wave equation for a relativistic massless scalar field in a curved spacetime

(form invariant)

Formally identical = Spacetime metricen.wikipedia.org2

LABORATORY SPACEA ti ti

in some coordinate systems

Acoustic equation(form variant)

/3131 C. Barceló et al., Living Rev. Relativity 14, 3 (2011)2 By Alain r (CC‐BY‐SA‐2.5), via Wikimedia commons



LABORATORY SPACE

1MEDIUM 1

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ABSTRACT RELATIVISTIC SPACETIME

2

LABORATORY SPACE

1MEDIUM 1

/315



3Space‐time transformation:


2

LABORATORY SPACE

1MEDIUM 1

/316




Rename:


2 4

LABORATORY SPACE

1MEDIUM 1

/317




Rename:


2 4

LABORATORY SPACE

Relation between MEDIUMS 1 and 2

1 5MEDIUM 1 MEDIUM 2

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Rename:


2 4Related by coordinate

transformation

LABORATORY SPACE

Same solutionSame solution


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Rename:


2 4Related by coordinate

transformation

LABORATORY SPACE

Same solutionSame solution

1 5MEDIUM 1 MEDIUM 2Related by coordinate

transformation

/3110




Rename:


2 4

No correspondence for this term

LABORATORY SPACE

No correspondence for this termSpace‐time

transformationE.g.:


/3111


• New problem: there is not a complete analogy between a general space‐time metric and the acoustic medium


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 c = Speed of sound (directly related to B and )b k d l it

VELOCITY POTENTIAL WAVE EQUATION

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

/3112


SPACETIME TRANSFORMATION ACOUSTICS



Rename:

2 4Form‐invariant

equation

LABORATORY SPACE

1 51 5MEDIUM 1 MEDIUM 2

Form‐variant velocity potential wave equation

/3113C. García‐Meca et al., Sci. Rep. 3, 2009 (2013).


Additi ll th f thi ti ll t k ith i di

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

yCloak Cloak

x(no backgroundvelocity correction)

(corrected backgroundvelocity)

/3114C. García‐Meca et al., Sci. Rep. 3, 2009 (2013).


1

WHEN IS ATA INDISPENSABLE?

• Find under which conditions the velocity potential wave equation preserves its shape (no new terms appear)1 :

• 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 several devices that do not fulfill any

/3115

p pp p y g g yof these conditions

1 C. García‐Meca et al., Wave Motion 51, 785 (2014).


• Selective absorption of acoustic rays: Transformation

EXAMPLE 1: DYNAMICALLY RECONFIGURABLE ABSORBER

p y

1 0Space compression

Compressor

Ray 1

f (t)

0 6

0.7

0.8

0.9

1.0

Ray 2 entersthe box

Ray 1 entersthe absorber

Ray 1entersthe box0

Ray 1

t (ms)

0.5

0.6

0 1 2 3 4 5 6

the absorber

Implementation

Static omnidirectional

/3116

absorber (index gradient 1)

Ray 2

1 A. Climente, D. Torrent, and J. Sánchez‐Dehesa, Appl. Phys. Lett. 100, 144103 (2012).


• COMSOL transient simulation:

EXAMPLE 1: DYNAMICALLY RECONFIGURABLE ABSORBER

Compressor

Ray 1Ray 1

Static omnidirectional

/3117

absorber (index gradient)

Ray 2


• Any transformation mixing time and one space variable can be implemented:

EXAMPLE 2: SPACETIME CLOAK

y g p p

Implementation

Transformation

• Transformation for a spacetime cloak1:

Curtain map 1

2

3

kedn

SimulationTheory

1

2

3

−3

−2

−1

0ct Cloake

region

−3

−2

−1

0ct

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−2 −1 0 1 23

x−2 −1 0 1 2

3x

1 M. W. McCall et al., J. Opt. 13, 024003 (2011).


• A simple transformation of the time variable changes the frequency of the input acoustic wave1:

EXAMPLE 3: FREQUENCY CONVERTER

p g q y p

Transformation

• Verified with full‐wave COMSOL transient simulations

• Useful to prevent oscillations of undesired frequencies from entering a given region or to accommodate the

/31

wave frequency to the spectral range of our detector

191 S. A. Cummer and R. T. Thompson, J. Opt. 13, 024007 (2011).


• Increases the density of events within a spacetime region by simultaneously compressing space and time1.

EXAMPLE 4: SPACETIME COMPRESSOR

• 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 ctTransformation

• COMSOL transient simulation:

x x

SimulationTheory

ct

/3120x

1 C. García‐Meca et al. Photon. Nanostruct. Fudam. Appl. 12, 312 (2014).


• Visualizing the effect:

EXAMPLE 4: SPACETIME COMPRESSOR

g

/3121


• Prescribed acoustic parameters are smooth functions of the coordinates and show an anisotropic character.

ANISOTROPY

•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

Homogenization

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that depend on the constitutive materials

• A wide range of acoustic parameter values, even anisotropic, can be achieved. 22


•We initially focused on the static background case:

HOMOGENIZATION

y gVelocity potential Pressure

MICROSCOPIC ACOUSTIC EQUATION

• Two‐scale homogenization procedure (medium properties change much faster than the acoustic wave):

1. Periodic acoustic parameters 2. Ellipticity condition

•Under these assumptions:Effective properties

HOMOGENIZED ACOUSTIC EQUATION

/3123

Cell problem


• Based on a multilayer structure 1 Acoustic properties of each layer

CLOAKING THE VELOCITY POTENTIAL

Based on a multilayer structure

• Potential transformation is physically different from a pressure transformation

p p y

Scatterer surrounded by 50‐layer cloakScatterer

/31241 C. García‐Meca et al., Phys. Rev. B 90, 024310 (2014).


• Typical configuration: wood inclusions in air

HOMOGENIZATIONSupersonic speeds achievable!yp ca co gu at o : ood c us o s a

/3125• Different parameters for the same microstructure depending on which equation we homogenize?

1 C. García‐Meca et al., Phys. Rev. B 90, 024310 (2014).


•We investigated the origin of this unexpected result from basic fluid mechanics

HOMOGENIZATION

POTENTIAL EQ.EQ. OF STATEFLUID MECHANICS

PRESSURE EQ.

Same medium everywhere S h

/3126

‐ Same medium everywhere‐Acoustic parameters may vary due to a background pressure gradient

‐ Same pressure everywhere‐ Acoustic parameters may vary if there are different media at each point

1 C. García‐Meca et al., Phys. Rev. B 90, 024310 (2014).


• New way of implementing acoustic metamaterials unveiled!

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 time‐varying pressure distribution:

/3127


d ff f d ff •Microparticle suspension: ferromagnetic flakes

PRACTICAL IMPLEMENTATION

• PIPES: different pressures for different cross‐sections (Bernoulli's theorem)

High

•Microparticle suspension: ferromagnetic flakes orientation dynamically reconfigurable through an external magnetic field1

High pressure Low

pressure

/31281 M.J. Seitel et al., Appl. Phys. Lett. 101, 061916 (2012)


b il

ELASTICITY

• See presentation by Gil Jannes

/31‐‐


• ANALOGUE TRANSFORMATIONS

CONCLUSIONS

• ANALOGUE TRANSFORMATIONS

1. 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

1. 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

/3129


PUBLICATIONS

Nature’s Scientific Reports 3 2009 (2013) Wave Motion 51 785 (2014)Nature’s Scientific Reports 3, 2009 (2013) Wave Motion 51, 785 (2014)

Physical Review B 90, 024310 (2014) Photonics and Nanostructures 12, 312 (2014) INVITED

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FUTURE WORK

ANALOGUE TRANSFORMATIONS OPEN A NEWWORLD OF POSSIBILITIESANALOGUE TRANSFORMATIONS OPEN A NEWWORLD OF POSSIBILITIES

1. 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

ELECTROMAGNETISM

way many possible different transformational theories for thesame field!

‐ Interpret the transformation of an equation as the equation of adifferent physical phenomena to connect their solutions

EXPERIMENT PHASE

Th l t ti l f b i i ATA t lit th t ld lik t t t

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‐ There are several potential ways for bringing ATA to reality that we would like to test

‐ Experimental capabilities in our team: Sanchez‐Dehesa’s LAB + Nanophotonics Technology Center

final presentation - technical report v pdf · ata time transforms anisotropy elasticity...

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