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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
ATA Time transforms Anisotropy Elasticity 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
ATA Time transforms Anisotropy Elasticity Conclusions
• 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
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
LABORATORY SPACE
1MEDIUM 1
/314
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
ABSTRACT RELATIVISTIC SPACETIME
2
LABORATORY SPACE
1MEDIUM 1
/315
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
ABSTRACT RELATIVISTIC SPACETIME
2
LABORATORY SPACE
1MEDIUM 1
/316
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
Rename:
ABSTRACT RELATIVISTIC SPACETIME
2 4
LABORATORY SPACE
1MEDIUM 1
/317
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
Rename:
ABSTRACT RELATIVISTIC SPACETIME
2 4
LABORATORY SPACE
Relation between MEDIUMS 1 and 2
1 5MEDIUM 1 MEDIUM 2
/318
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
Rename:
ABSTRACT RELATIVISTIC SPACETIME
2 4Related by coordinate
transformation
LABORATORY SPACE
Same solutionSame solution
1 5MEDIUM 1 MEDIUM 2
/319
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
Rename:
ABSTRACT RELATIVISTIC SPACETIME
2 4Related by coordinate
transformation
LABORATORY SPACE
Same solutionSame solution
1 5MEDIUM 1 MEDIUM 2Related by coordinate
transformation
/3110
ATA Time transforms Anisotropy Elasticity Conclusions
SOLUTION: AUXILIARY ANALOGUE SPACETIME
3Space‐time transformation:
Rename:
ABSTRACT RELATIVISTIC SPACETIME
2 4
No correspondence for this term
LABORATORY SPACE
No correspondence for this termSpace‐time
transformationE.g.:
1 5MEDIUM 1 MEDIUM 2
/3111
ATA Time transforms Anisotropy Elasticity Conclusions
• New problem: there is not a complete analogy between a general space‐time metric and the acoustic medium
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 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
ATA Time transforms Anisotropy Elasticity Conclusions
SPACETIME TRANSFORMATION ACOUSTICS
3Space‐time transformation:
ABSTRACT RELATIVISTIC SPACETIME
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).
ATA Time transforms Anisotropy Elasticity Conclusions
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).
ATA Time transforms Anisotropy Elasticity Conclusions
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).
ATA Time transforms Anisotropy Elasticity Conclusions
• 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).
ATA Time transforms Anisotropy Elasticity Conclusions
• COMSOL transient simulation:
EXAMPLE 1: DYNAMICALLY RECONFIGURABLE ABSORBER
Compressor
Ray 1Ray 1
Static omnidirectional
/3117
absorber (index gradient)
Ray 2
ATA Time transforms Anisotropy Elasticity Conclusions
• 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
/3118
−2 −1 0 1 23
x−2 −1 0 1 2
3x
1 M. W. McCall et al., J. Opt. 13, 024003 (2011).
ATA Time transforms Anisotropy Elasticity Conclusions
• 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).
ATA Time transforms Anisotropy Elasticity Conclusions
• 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).
ATA Time transforms Anisotropy Elasticity Conclusions
• Visualizing the effect:
EXAMPLE 4: SPACETIME COMPRESSOR
g
/3121
ATA Time transforms Anisotropy Elasticity Conclusions
• 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
/31
that depend on the constitutive materials
• A wide range of acoustic parameter values, even anisotropic, can be achieved. 22
ATA Time transforms Anisotropy Elasticity Conclusions
•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
ATA Time transforms Anisotropy Elasticity Conclusions
• 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).
ATA Time transforms Anisotropy Elasticity Conclusions
• 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).
ATA Time transforms Anisotropy Elasticity Conclusions
•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).
ATA Time transforms Anisotropy Elasticity Conclusions
• 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
ATA Time transforms Anisotropy Elasticity Conclusions
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)
ATA Time transforms Anisotropy Elasticity Conclusions
b il
ELASTICITY
• See presentation by Gil Jannes
/31‐‐
ATA Time transforms Anisotropy Elasticity Conclusions
• 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
ATA Time transforms Anisotropy Elasticity Conclusions
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
/3130
ATA Time transforms Anisotropy Elasticity Conclusions
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
/3131
‐ 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