PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Mode conversion of elastic waves by using anisotropic
metamaterials
Xiongwei YANG; Yueming LI; Gang Chen
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong
University, No.28 Xianning West Road, 710049 Xi'an, China

ABSTRACTAs a fundamental phenomenon in elastic fields, mode conversion between longitudinal and transverse wave
modes can frequently occur when a wave is incident on an elastic discontinuity or passes through anisotropic
mediums. Due to the generation of a different wave mode, the phenomenon is widely used in industrial andmedical applications for nondestructive tests and ultrasonic inspections. Metamaterials are men-madesubwavelength composites and could be designed based on the local resonance mechanism. Due to local
resonance, effective physical properties of the metamaterials could be zero or negative in desired frequency
ranges, leading to novel wave phenomena, such as low frequency bandgap. Metamaterials could also be
designed with specific macroscopic anisotropy in specific direction. Such anisotropic metamaterials, if
designed to be nonresonant, barely exhibit material absorption losses, and could be more suitable for wave
manipulating, such as acoustic magnifying, invisibility cloaking. In this work, we are concerned whether the
resonance mechanism can be introduced into mode conversion. Because of the resonance mechanism, theeffective properties of the metamaterials can be highly frequency-dependent, which means vast possibilities
in finding new phenomena and obtaining new designs. Our final goal is to pave the way of using thefrequency-dependency concept for mode conversion.
Keywords: Elastic wave, Mode conversion, Anisotropic metamaterial

1. INTRODUCTION
The phenomenon of mode conversion between different wave modes can be frequently noticed inelastic fields, such as when an elastic wave is incident on an elastic discontinuity or passes throughanisotropic mediums. Because the wave mode defines the fundamental characteristics of how wave
energy propagates, changing of the wave mode, i.e., realization of mode conversion, could be of greatscientific importance and engineering significance [1-3]. For example, shear (transverse) waves, which
are difficult to generate, can be critically useful in structural non-destructive evaluation and medical
imaging, because of their sensitivity to elastic discontinuities or biomedical focuses. On the other hand,
longitudinal waves, are much easier in generation, yet less sensitive to elastic discontinuities orbiomedical focuses. Therefore, mode-conversion from longitudinal waves could be a potential way forthe efficient generation of transverse waves. In non-destructive evaluation, wedge-type mode
convertors, which work according to the Snell’s law, are commonly used [4]. However, due theinevitable generation of longitudinal wave mode, the mode-converting efficiency is rather low,normally about 25%. Researchers then noticed that if double- or triple- negative metamaterials areused to design the wedge-type mode convertors, complete mode-converting transmission could betheoretically possible, but requires that some additional conditions must be fulfilled between the
background test medium and the mode convertors [5-7]. Because the conditions are too restrictive, so far,no realization of complete mode-converting transmission has been attempted with double or triple
negative metamaterials. The mode-converting efficiency, is about 40%. Additionally, because of thehigh frequency-dependency of the effective material properties, the frequency range with a relatively
higher mode-converting efficiency could be rather narrow.
Recently, perfect transmodal Fabry-perot interference [8-9] allows a sole and maximal mode
conversion between longitudinal and transvers wave modes. In such a phenomenon, under incidenceof a longitudinal wave mode, there is only the transmission of the transverse wave. Because of themultimodal constructive interference, the displacement amplitudes before and after transmission are
the same. However, same displacement amplitudes indicate different energy, because the2582

longitudinal and transverse wave modes are determined by different stiffness coefficients. Even in
the same isotropic media, the transverse stiffness coefficient differs from the longitudinal stiffness
coefficient. Therefore, complete mode-converting transmission is only possible, when elastic waves
propagate in the same background media which has the same transverse and longitudinal stiffness
coefficients. Since the impedance mismatch is the reason that complete mode-convertingtransmission cannot be accomplished in the perfect transmodal Fabry -perot interference, a bimodal
quarter-wave impedance matching theory [10] was thereafter developed, which seeks complete
mode-converting transmission through matching impedances of the incident longitudinal wave modeand the transmitted transverse wave mode. Because the impedances can be matched regardless of th e
background media, complete mode-converting transmission between longitudinal and transverse
wave modes can be realized not only in the same background medium, but also between different
media.
Realization of the theories, either the perfect transmodal Fabry-perot interference theory or thebimodal quarter-wave impedance matching theory, requires that the interferometer or the matchingelement must have very specific anisotropy. Therefore, anisotropic metamaterials were designed forthe realization of the unique material properties. However, because the anisotropic metamaterials
were designed to be nonresonant, their dimensions must be much smaller than the wavelength. Thismeans that, if designed in the ultrasonic frequency range, unit cell of the metamaterials could be too
small to fabricate, while if designed in the low frequency range, unit cell could be too larger tofabricate. Therefore, in this work, we seek to introduce resonance into the design of anisotropic
metamaterials. This is possible, because the resonance mechanism can induce extraordinary effective
material properties for unusual wave phenomenon.

2. THEORY
Consider an elastic wave which is propagating in a two-dimensional x-y plane through ananisotropic elastic slab (with width d), which is sandwiched between layers of another base medium.
Here, we mainly consider a propagating longitudinal wave, which is normally incident from the basemedium to the anisotropic slab.
2.1 Theoretical background
For the analysis considerer here, the stiffness coefficients needed are C11 , C 66 , and C16 for the slab
medium. The stiffness terms with subscripts (11) and (66) denote the longitudinal stiffness along thex axis and the shear stiffness, respectively, while the term with (16) refers to the longitudinal-shear
coupling term. For common cases, we consider that the base medium is isotropic, which will becharacterized by Young’s modulus (E0 ) and Poisson’s ratio ( 0 ). The symbols ρ0 and ρ will be used todenote the densities of the base material and slab material, respectively.
For waves propagating along the x direction, the x- and y-directional displacements u x and u y can beassumed to vary as e j(kx- ωt) where k, ω and t are the x-directional wavenumber, angular frequency andtime. The Christoffel equation [1] can be written as
C11k 2   2
 u x 
C16 k 2(1)
  0,
2
2
2
C66 k    u y 
 C16 kNon-trivial solutions for Eq. Erreur ! Source du renvoi introuvable. can be found if k=±α or k=±β
where


 2  C11  C66    2

 C11  C66   4C162

  C11  C66   

 C11  C66   4C

2  C11C66  C162 

2



2

2

2  C11C66  C162 

,

(2)
2

2
16

.

Therefore, the general solution in the anisotropic slab can be expressed asu x   APx e j x  BPx e  j x  CQx e j x  DQx e  j x  e-jt ,u y   APy e j x  BPy e  j x  CQy e j x  DQy e  j x  e-jt .

(3)

The symbols A and B denote unknown amplitudes corresponding to k=+α and k=-α, respectively.
Likewise, C and D are unknown amplitudes corresponding to k=+β and k=-β, respectively.

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From Eq. Erreur ! Source du renvoi introuvable., the displacement components corresponding
to the α and β waves can be expressed asux   Ae j x  Be  j x  Px e-jt , uy   Ae j x  Be  j x  Py e-jt ,(4)u x   Ce j x  De  j x  Qx e-jt , u y   Ce j x  De  j x  Qy e-jt ,where Px, Py, Qx, and Qy represent the polarization vector components. They are found to bePx X
1  X

2, Py 

1
1  X

2, Qx 

X
1 X 

2, Qy 

1
1 X 

2

,

(5)

where
Xk  

C16 k 2C k 2   2
  66
 k   ,    .
2
2
C11k x  
C16 k x2

(6)

The symbol Xk represents the amplitude ratio of ux and u y. Due to the orthogonality of the wave mode,
the relation that PxQx+PyQy=0 always holds .
One can express the velocity (v x and v y) and stress (σxx and σxy) as vx 
 A
v 
 
 y 
B
   MN   .
 xx C 
 xy  D  (7)Please see Appendix for the elements of the M and N matrices. If Cij and ρ are replaced by cij and ρ0 in
the equations above, the corresponding results are applicable for the base medium. In this case, thex-directional wave numbers will be denoted α0 and β0 .
2.2 Transmissions and reflections
To obtain the scattering (S) parameters, we first write the relationship between the velocity and stressfields on the left and the right boundaries of the anisotropic slab (see Fig ure.1a of the main text),
 vx 
 vx 
 vx 
v 
v 
v 
 y 
 y 
1  y 

MN
M
=
T
.
 
 

xd
xd 


xx
xx
 
 
 xx  xy   xy   xy  
 x d x 0 x 0(8)

By imposing the continuities between the field variables of the base medium and those of theanisotropic slab at x=0 and x=d, we can obtain the S matrix as
S  M -10Tx  d M 0 ,(9)where M0 is the M matrix for the base medium. The S matrix represents the relationship between thedisplacement amplitudes of the adjacent base medium at x=0 - and those at x=d+.
Once the components (S ij ) of the S matrix are determined, the reflection and transmission
coefficients for the case of longitudinal wave incidence can be expressed as:RLL 

S 24 S 41  S 44 S 21,
S22 S 44  S42 S24

RLT 

S 42 S 21  S 22 S 41,
S22 S 44  S42 S24(10)

TLL  S11  S12 RLL  S14 RLT ,TLT  S31  S32 RLL  S34 RLT .

In Eq. Erreur ! Source du renvoi introuvable., RLL and RLT denote the reflection coefficients forthe reflected L and T waves, respectively, and TLL and TLT, the transmission coefficients fortransmitted L and T waves, respectively. The explicit expressions for the elements of t he T and S
matrices are given in Appendix.
The mode conversion ratio, i.e. the L-to-T transmission power ratio (TT) represents the ratio of thetransmitted S-wave power intensity (P TLT) to the incident L-wave power intensity (PI). Therefore, it
can be expressed as

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



TLT
TLT
P TLT real  xy  conj  v y  |x  d 
2TT  I   TLT ,I
I
Preal  xx  conj  vx  | 





where

=

(11)

x 0

 0 c66.
 0 c11(12)

Similarly, the L-to-L transmission power ratio (TL) representing the ratio of the transmitted T-wave
power intensity (PTLL) to the incident L-wave power intensity (PI) can be obtained as





TLL
TLL
P TLL real  xx  conj  vx  |x  d 
2TL  I  TLL .
I
I
P
real  xx  conj  vx  | 





(13)

x 0

The reflections of the L wave and T wave can be written as
2

2

(14)

RL  RLL , RT   RLT .

2.3 Theory for Full mode-converting transmission
Here, we mainly consider the bimodal quarter-wave impedance matching theory [10], which allows
full mode-converting transmission. In this theory, four necessary conditions are needed:
1) Bimodal quarter-wave phase matching condition:(15a)d  nFS FS ,
4
SS
(15b)
d  nSS,4nSS nFS(15c)

=odd ,
22
where n FS and n SS are integers. These conditions are equivalent to sin(αd)=±1 and sin(βd)=∓1.
2) Bimodal impedance matching condition:
Z  Z0 ,(16)

where Z = C11C66  C162 and Z 0 =0 c11c66 denote the bimodal impedances of the anisotropic
metamaterial and background isotropic medium, respectively.
3) Polarization condition:
C11  C66 ,(17)owhich implies that the fast skew and slow skew modes have ±45 polarizations, respectively.
4) Weak mode-coupling condition:
 =C16 / C11  0 .
(18)
With the above four conditions, one can find that at the bimodal quarter-wave phase matchingfrequency,
lim RL  f BQW   0,lim RT  f BQW   0,lim TL  f BQW   0,lim TT  f BQW   1.
 0

 0

 0

 0

The anisotropic slab must have specific anisotropy, which can be descried as follows,Z nn C11 =C66 = 0  SS  FS 
2   nFS nSS 
C16 =Z 0  nSS nFS ,2   nFS nSS 

(19)

(20a)
(20b)

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

Z 0 nFS nSS
4 f BQW d

.

(20c)

3. DESIGN WITH RESONANCE MECHANISM
In this work, we consider to introduce the resonance mechanism by designing anisotropic
metamaterials in the short wavelength limit. Because the target wavelength could be much larger
than the unit cell dimension, resonance could naturally occur. Based on this idea, we aim to design
the bimodal quarter-wave impedance matching element by cutting slits on an aluminum plate.
Figure 1 shows the designed metamaterial, which is obtained by cutting two columns of slits on
the aluminum matrix. As for its geometric information, h=2cm, w=3cm, l=0.18cm, r=1.8mm, θ=31o .

Figure 1 Designed anisotropic metamaterial
Its mode-converting performance is plotted in Figure 2, which shows that the mode -converting
efficiency reaches as high as 75% around f=150kHz.

Figure 2 Mode conversion efficiency
Although efforts are still needed to improve the mode-converting efficiency, one may still see thepotentials in designing anisotropic metamaterials by introducing resonance mechanism. Firstly, i f theanisotropic metamaterial were design in the long wavelength limit, on the other hand, it would
behave as a nonresonant composite. It can then be noticed that the bimodal impedance matchingcondition in Eq. (16) can never be satisfied by cutting silts on a matrix. However, the satisfaction
could be possible in the short wavelength limit, because the metamaterial can havefrequency-dependent effective material properties.
Secondly, if subwavelength unit cell means that the microstructure should be much smaller than

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the wavelength. This would make the actual fabrication very difficult, not to mention that much
more unit cells need to be fabricated. In this design case, the minimum feature of the unit cell is
r=1.8mm, which is very easy for fabrication, and only two layers need to be machined. This could
significantly broaden the application of elastic metamaterials, especially in ultrasonic ranges.
Because a dynamic homogenization method needs to be develop to reveal the instinct influence ofthe resonance mechanism through evaluating the effective properties of the metamaterial, the actual
evaluation of the effective properties will not be discussed here.

4. SUMMARY
In this work, we have a brief discussion on designing metamaterials in short wavelength range.
Because working in the short wavelength range means the resonance could occur, effective material
properties of the metamaterials could exert much more possibilities in satisfying strict material
requirements for extraordinary wave phenomena. Motivated by this observation, we tried to realize
an efficient mode-converting efficiency through cutting two columns of slits on an elastic matrix.
With this simple design, the mode-converting efficiency reached over 70%.

ACKNOWLEDGEMENTS
The authors sincerely acknowledge the financial support from the National Natural Science
Foundation of China (11802220, 11772251), the Natural Science Basic Research Plan in Shaanxi
Province of China (2019JQ-293), and the 111 Project (B18040).

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