Theory for Perfect Transmodal Fabry-Perot Interferometer

We establish the theory for perfect transmodal Fabry-Perot interferometers that can convert longitudinal modes solely to transverse modes and vice versa, reaching up to 100% efficiency. Two exact conditions are derived for plane mechanical waves: simultaneous constructive interferences of each of two coupled orthogonal modes, and intermodal interference at the entrance and exit sides of the interferometer with specific skew polarizations. Because the multimodal interferences and specific skew motions require unique anisotropic interferometers, they are realized by metamaterials. The observed peak patterns by the transmodal interferometers are similar to those found in the single-mode Fabry-Perot resonance, but multimodality complicates the involved mechanics. We provide their design principle and experimented with a fabricated interferometer. This theory expands the classical Fabry-Perot resonance to the realm of mode-coupled waves, having profound impact on general wave manipulation. The transmodal interferometer could sever as a device to transfer wave energy freely between dissimilar modes.


Fundamental equations
Consider a mechanical wave propagating in a two-dimensional x-y plane through an anisotropic elastic slab of width d that is sandwiched between layers of another base medium.
In the subsequent analysis, we will consider a longitudinal wave propagating along the x direction, which is normally incident from the base medium to the anisotropic slab. Before we derive the exact conditions for multimodal interference for Perfect Mode Conversion (PMC), the fundamental equations will be briefly explained.
Because waves propagating along the x axis will be considered, the stiffness coefficients needed for the analysis are C11, C66, and C16 for the slab medium. The stiffness terms with subscripts (11) and (66) denote the longitudinal stiffness along the x axis and the shear stiffness, respectively while the term with (16), the longitudinal-shear coupling term. If the base medium is isotropic, it will be characterized by Young's modulus (E0) and Poisson's ratio (0). If it is an anisotropic medium, it will be characterized by c11, c66, and c16, but c16=0 is assumed because purely longitudinal (transverse) wave incidence from the base medium to the sandwiched slab is considered. The symbols ρ0 and ρ will be used to denote the densities of the base material and slab material, respectively. Because the actual anisotropic slab will be made of a metamaterial, the stiffness Cij can be understood as effective stiffness.
For waves propagating along the x direction (see Figure. 1a of the main text), the x-and ydirectional displacements ux and uy can be assumed to vary as e j (kx-ωt) where k, ω and t are the x-directional wavenumber, angular frequency and time. The Christoffel equation [S1] can be written as . 2 Therefore, the general solution in the anisotropic slab can be expressed as 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.
The symbol Xk represents the amplitude ratio of ux and uy. Due to the orthogonality of the wave mode, the relation that PxQx+PyQy=0 always holds .
One can express the velocity (vx and vy) and stress (σxx and σxy) as 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, the x-directional wave numbers will be denoted α0 and β0.

Transmissions and reflections
To obtain the scattering (S) parameters, we first write the relationship between the velocity and stress fields on the left and the right boundaries of the anisotropic slab (see Figure.1a of the main text), By imposing the continuities between the field variables of the base medium and those of the anisotropic slab at x=0 and x=d, we can obtain the S matrix as where M0 is the M matrix for the base medium. The S matrix represents the relationship between the displacement amplitudes of the adjacent base medium at x=0and those at x=d + .
Once the components (Sij) of the S matrix are determined, the reflection and transmission coefficients for the case of longitudinal wave incidence can be expressed as: In Eq. (S10), RLL and RLT denote the reflection coefficients for the reflected L and T waves, respectively, and TLL and TLT, the transmission coefficients for transmitted L and T waves, respectively. The explicit expressions for the elements of the 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 the transmitted S-wave power intensity (P TLT ) to the incident L-wave power intensity (P I ).
Therefore, it can be expressed as TLT  TLT  TLT  2  T  I  I  I  0   real  conj  |  ,  real conj | Likewise, the same-mode L-to-L transmission power ratio (TL) representing the ratio of the transmitted T-wave power intensity (P TLL ) to the incident L-wave power intensity (P I ) can be obtained as Similarly, the reflections of the L wave and T wave can be written as S14)

Perfect Mode Conversion with Full Transmission
In this section, we will derive the condition for perfect mode conversion with full transmission (i.e., with TT=100%). As mentioned earlier, we will mainly consider the L-wave incidence but the obtained result is equally valid for the S-wave incidence. For perfect mode conversion, one mode can be perfectly converted only to the other mode, i.e. TL=0 at PMC (Perfect Mode Conversion) frequencies. TT can reach 100% if the mechanical impedances of the longitudinal and transverse waves in the base medium are the same. We begin with this special case before discussing more general cases of perfect mode conversion.
As we are interested in the incidence of a pure L wave from a base medium, no coupling between longitudinal and transverse stiffness is assumed, i.e., c16=0. In this section, the base medium is assumed to be anisotropic.
Because TT given in Eq. (S11) is very complicated, to find the exact condition for the perfect mode conversion, we use the following equation for sake of simplification based on the As the components Sij in Eq. (S16) still appear to be complicated, we consider the case of a special anisotropic base material satisfying 11 66 . c c  (S18) This condition is equivalent to the condition that the mechanical impedance of the longitudinal wave is equal to that of the transverse wave. With Eq. (S18) (and c16=0), one can have ξ=1.
In this case, Eq. (S16) can be simplified to where the symbol <> denotes the complex conjugate of the term inside. Clearly Note that we used Eq. (S10) to obtain (S20). Because all Vi (i=1,2,3,…,6) are real, Eq. (S20) requires that 1 2 3 4 5 6 =0; If the following conditions are met, Eq. (S21) can be identically satisfied: To extract more useful information from (S22), the explicit expressions for  and  are Because α≤β, NSS must be larger than NFS. If Eq. (S23) is expressed in terms of the wavelengths λFS and λSS, we have Eq. (S24) states that perfect and full (100%) conversion from the L wave mode to the T wave mode is possible if the slab width d is a multiple of half the wavelength λFS and a multiple of half the wavelength of λSS simultaneously.
If the coprime integers of NFS and NSS are denoted by nFS and nSS with the corresponding fundamental PMC frequency fMC Eq. (S23) can be written as , To obtain Eq. (S27), α and β in Eq. (S25) were expressed as (with m=1) Solving Eq. (S2) with Eq. (S29), one can obtain Eq. (S27).
The analysis above show that the conditions for perfect and full (100%) mode-conversion transmission (TT=1 and TL=0) are obtained as Eqs. (S15) and (S26) that apply to the interferometer slab and Eq. (S18) that applies to the base medium. FIG. S1 shows the transmission curves exhibiting interference behavior when perfect and full mode conversion occurs. At the PMC frequencies f=fMC, 3fMC, ···, TT=100% and TL = 0.

Theory of perfect mode conversion
The conditions for full and perfect mode conversion derived in the previous section require a special anisotropic base medium satisfying c11=c66 and c16=0. This base material requirement is very restrictive. Now, we consider general isotropic base media and look for the conditions for PMC in which 100% transmodal transmission is relaxed. Therefore, we will remove the condition given by Eq. (S18) imposed on the base medium and consider the two remaining conditions given by Eqs. (S26) and (S15) imposed on the TFPI modeled as an anisotropic slab.
First, let us consider the condition (S26), equivalently, (S22). Without the loss of generality, only the following equations can be considered The frequencies that satisfy Eq. (S30) will be denoted by fMC, 3fMC, …, as before. When Eq.
(S30) is substituted into the S matrix given in Appendix, we have If Eq.(S31) is used, TLT and TLL at the PMC frequencies f=fMC, 3fMC, … become Let us now consider the consequence of Eq. (S15 For an isotropic base medium (E0, ν0), the parameter ξ appearing in Eq. (S37) becomes To obtain Eq. (S39), we used the relationship,

Remarks on displacement field inside the interferometer at frequency 2fMC
It has been shown in Figure. 1 of the main text that at the frequencies, 2fMC, 4fMC,… RL=RT=TT=0 and TL=1. This means that at these frequencies, the incident wave mode is preserved after it passes through the TFPI. To illustrate the mode-preserving mechanism through the TFPI, we calculate the distributions of ux and uy displacement components inside the TFPI in FIG. S3. It shows that the ux components of the FS and SS modes can interfere constructively while their uy components interfere destructively. This interference behavior at f=2fMC, 4fMC, … differs from that observed at f=fMC, 3fMC,…, in which the uy components interfere constructively while the ux components interfere destructively (see Figure 1e in the main text).

FIG. S3
The normalized displacement components of the FS and SS modes at f=2fMC for the TFPI used in Figure 1 of the main text.

Effect of the unit cell size
The Therefore, we used a=d/100 to obtain the numerical results in Figure 1c of the main text.

Conversion from T wave to L wave
As mentioned earlier, PMC is also valid when a T wave is normally incident. Therefore, a perfect T-to-L mode-conversion phenomenon can also be realized with the same interferometer shown in Figure 1 of the main text. The transmission and reflection power ratios for normally incident T waves are shown in FIG S5a. Note that the T-to-L mode conversion ratio at the PMC frequencies is 92.88%, which is the same as the L-to-T mode conversion ratio.

FIG S5b shows the snapshots of the transient displacement when a harmonic T wave
propagates through the TFPI. When the harmonic wave is excited at the PMC frequencies fMC and 3fMC, the incident T wave is perfectly converted to an L wave. When the harmonic wave is excited at the mode-persevering frequency (2fMC), the incident T wave is transmitted without any mode conversion.

FIG S5 PMC with a normally incident T wave. TL (a) (T-to-L transmission, in this case) and TT (T-to-T transmission, in this case) and the reflection ratios RL (T-to-L reflection, in this case) and RT (T-to-T reflection, in this case) through the TFPI, calculated by the theoretical analysis (lines) and full-wave numerical simulation (dots). (b) Snapshots of transient displacement
for a normally incident harmonic T wave. The prescribed vertical displacement (uy) at x=0 is u0sin(2πft). All the snapshots are captured at t=29.25/f. All parameters are the same as those in Figure 1 of the main text.

Perfect mode-converting interference with a different set of nFS and nSS
We showed that perfect mode-converting interference occurs if the two sets of conditions stated in Eq. (S42) and Eq. (S43) (i.e. Eq. (S26) and Eq. (S15)) are exactly satisfied. In the main text, the effective properties of the designed TFPI in Figure 1 satisfy Eq. (S15) with nFS=2 and nSS=3; For Figure. 2a of the main text, we also used nFS=2 and nSS=3.
Here, we will investigate PMC corresponding to a different set of nFS=1 and nSS=2.
The TFPI is assumed to be sandwiched by aluminum.

A design procedure
We showed that perfect mode-converting interference occurs when the two conditions stated by Eqs. (S42) and (S43) are satisfied. Because it is difficult or nearly impossible to find a natural material the stiffness of which satisfies (S42) and (S43) exactly, we realize the interferometer by a metamaterial.
To facilitate the design of the perfect TFPI exhibiting PMC, the condition (S42) (equivalently, (S27) For the design of the TFPI, therefore, we can check the satisfaction degree of the conditions that κ=1 and γ=1. Since neither κ nor γ involves the density ρ of the anisotropic metamaterial, we only need to determine the values of Cij that can make κ=1 and γ=1 as precisely as possible.
After a specific metamaterial satisfying (S47)  Although fMCd is not known until an actual perfect TFPI is obtained, one can choose d arbitrarily once fMC is given.

Practical design
We will explain how the unit cell shown in Figure 1c of the main text was designed. After choosing nFS=2 and nSS=3, void slits are inserted in the unit cell. The unit cell is made of aluminum which is the same material as the base material. After trials and errors, l2=0.0800a, l3=0.1200a, l4=0.1850a, r=0.1000a, and θ=45 o were determined. We used the remaining parameter l1 as a tuning parameter to ensure that κ=1 and γ=1. The unit cell is also shown in

Experimental details
This section explains the details of the fabrication of the sample and experimental procedure. The corresponding fMCd is found to be, which is virtually the same as that of the original one. Although the stiffness values in Eq. (S53) are not exactly the same as those in Eq. (S51), they satisfy the two conditions given by (S15) and (

Sound transmission suppression
A promising application would be to minimize acoustic transmission through a panel. We can also find that fMCd =1.03 kHz·m. The fundamental PMC frequency fMC can be set to be fMC≈10 kHz by choosing d=0.1m.
As illustrated in FIG S11a, a speaker generating acoustic pressure of unit magnitude is placed just in front of the left sides of the panel. We calculate the wave field in air located on the right side of the panel and then compute the transmission loss (TL). FIG S11b compares the TL around f= fMC. Although the TFPI is effective over a narrow frequency range, the increase in the TL becomes as much as 112 dB. FIG S11c shows the displacement field inside the TFPI at f= fMC =10kHz. It shows that the incident longitudinal wave is converted to a transverse wave involving shearing deformation. Therefore, the incident acoustic wave can hardly propagate into the adjacent air located on the right side of the interferometer.

Efficient wave generation of transverse (or shear) waves
A method using Snell's critical angle is a traditional method to convert L waves to T waves.
Because the direct generation of high-power T waves is not easy, T waves are typically generated by mode conversion from L waves that are much easier to generate by available (and widely used) piezoelectric transducers. As T waves have received much attention in medical applications where ultrasound waves are to be delivered to the human brain through its skull, efficient T-wave generation by PMC could be critically useful. Also, the T-wave mode is used in ultrasound-based flow meters. In either case, a wedge based on Snell's critical angle, typically made of acrylic resin, is needed, but large impedance mismatch between the wedge and a test object hinders efficient power transmission Typically, the mode-conversion transmission efficiency by the wedge approach is 20~30% 20 . Note that the TFPI, which can exactly satisfy the PMC conditions derived in this work, is shown to convert an incident L wave to a T wave with as high as 92.88% mode-conversion transmission for an aluminum base material. (The related results are presented in Figure 1b of the main text.)