Kevin Cahill
email: cahill@unm.edu

Simplicity and symmetry in physics

The uncertainty principle

In a lecture at Brookhaven National Laboratory in the summer of 1963, Robert Oppenheimer explained the simple ideas that underlie the uncertainty principle:

• To use an instrument to measure the position of something, the instrument must be attached to a known position.

• To use the recoil of an instrument to measure the momentum of something, the instrument must be free to recoil.

Since an instrument can not simultaneously be both attached and free, one can't simultaneously measure position and momentum to arbitrary accuracy. The limit is

∆x ∆p ≥ ℏ 2 . ( 1
)
One can use the uncertainty principle to estimate microscopic quantities. For example, setting the product of the average values of r and p equal to ℏ, one has 1/r = p/ℏ which gives the energy of the ground state of hydrogen as

E 0 = p 2 2m - e 2 r = p 2 2m - e 2 p ℏ . ( 2
)
Its minimum is at p = me 2 /ℏ. So

E 0 = (me 2 ) 2 2mℏ 2 - (me 2 ) 2 mℏ 2 = - mc 2 2 e 4 ℏ 2 c 2 = - mc 2 2 α 2 (3)
which is the nonrelativistic result. One may apply the uncertainty principle as in Fig. 1 and so derive an uncertainty relation for angle and angular-momentum:

ℏ 2 ≤ ∆x ∆p = ∆x r (r ∆p) = ∆ϕ ∆L ≈ ∆L. (4)
This says that particles must have spins greater than ℏ/2. Apart from the spinless Higgs particle, the known elementary particles obey this inequality having as they do spins ℏ/2, ℏ and 2ℏ. But why are they spinning? Why isn't the Higgs? For a system of several degrees of freedom q n , p n , the uncertainty principle (1) is generalized to

[q n , p m ] = iℏ δ nm . (5)
In quantum field theory, these relations become [ϕ n (t, x), φ † (t, y)] = iℏ δ nm δ(x -y) for bosons and

{ψ n (t, x), ψ † m (t, y)} = ℏ δ nm δ(x -y) for fermions.
(6)

Simplicity ⇐⇒ symmetry

The complexity of a thing is the length of its shortest description. The complexity of a circle of radius r is at most the length of its equation x 2 + y 2 = r 2 . The complexity of an arbitrary squiggle is at least twice the number of its pixels. The O(2) symmetry of the circle makes it simpler than a squiggle. The complexity of a sphere of radius r is less than the length of its equation

x 2 + y 2 + z 2 = r 2 . (7)
A rambutan has less symmetry and more complexity. The O(3) symmetry of the sphere makes it simpler.

The smallest things are presumably the simplest and the most symmetrical. Elementary particles are the smallest things we know of. They must be simple. They must be the simplest because they have the least room for complexity. They must be symmetrical.

Symmetry and conservation

Symmetry implies conservation. In mechanics, symmetry under rotations and under translations in time and space implies conservation of angular momentum, energy and momentum.

In field theory, if an action density L(ϕ α , ∂ i ϕ α ) depends upon the spacetime coordinates x i only through the fields ϕ α (x) and their derivatives ∂ i ϕ α (x), then the energy-momentum tensor

T i k = i ∂L ∂∂ i ϕ α ∂ϕ α ∂x k -δ i k L is conserved ∂ i T i k = 0. ( 8
)

Global symmetry

A global symmetry transformation changes fields in the same way at every spacetime point x. In the global U (1) symmetry transformation

ψ(x) → ψ ′ (x) = e iqθ ψ(x) (9) or ψ ′ 1 (x) ψ ′ 2 (x)
= cos(qθ) sin(qθ) -sin(qθ) cos(qθ)

ψ 1 (x) ψ 2 (x) , (10)
the charge q and the angle θ are independent of x. This symmetry conserves charge.

Local U (1) symmetry

The electric and magnetic fields of Maxwell and Faraday

E i = c ∂A 0 ∂x i - ∂A i ∂x 0 and B i = 1 2 ϵ ijk ∂A k ∂x j - ∂A j ∂x k = (∇ × A) i (11)
have a special symmetry. They are the 4-curl of the vector potential A k

F ik = ∂A k ∂x i - ∂A i ∂x k = -F ki (12)
and so are unchanged by the gauge transformation

A k (x) → A ′ k (x) = A k (x) + ∂ k θ(x) (13)
in which the angle θ(x) is an arbitrary function of x.

The action density ψ † γ k ∂ k ψ, in which {γ i , γ k } = η ik and η = diag(-1, 1, 1, 1), is not invariant under the local, i.e., spacetime dependent, transformation

ψ ′ (x) = e -iqθ(x) ψ(x) A ′ i (x) = A i (x) + ∂ i θ(x). (14)
But the Dirac action density x) is canceled by the change in the vector potential

ψ † γ k (∂ k + iqA k ) ψ is invariant because the derivative of the exponential e -iqθ(
A i (x) -iq(∂ k θ) e -iqθ ψ + iq(∂ k θ) e -iqθ ψ = 0. ( 15
)
This cancellation happens because the ordinary derivative ∂ k ψ in the action density ψ † γ k ∂ k ψ has been replaced by the "covariant" derivative (∂ k + iqA k ) ψ. Thus the local symmetry ( 14) requires ψ(x) to interact with A i (x). This local U (1) symmetry of classical electrodynamics allows the angle θ(x) to vary arbitrarily with the spacetime point x and that in turn requires the field ψ(x) to interact with the gauge field A k (x). This symmetry foreshadowed other local symmetries which require other interactions. In fact, all the known interactions are exactly those required by the known local symmetries.

Local nonabelian symmetry

Yang Chen-Ning and Robert Mills generalized local symmetry from the U (1) of electrodynamics (14) to SU (2). That was a big step forward.

Steven Weinberg, Sheldon Lee Glashow and Abdus Salam explained the electroweak interactions in terms of the group SU L (2) ⊗ U Y (1). They replaced the simple derivative ∂ k Ψ of a complex doublet Ψ such as

Ψ = ν e or Ψ = u d (16)
with the "covariant" derivative

D k Ψ(x) = ∂ k + ig σ • W (x) 2 + ig ′ Y B k (x) 2 Ψ(x) (17)
in which σ is the triplet of Pauli matriecs. The subscript L on SU L (2) means that the gauge bosons W (x) interact with the left-handed quark and lepton fields and not with the right-handed quark and lepton fields. Thus the electroweak interactions are lopsided and are not invariant under reflections, x → -x, i.e., they do not conserve parity.

Quantum chromodynamics

Murray Gell-Mann and George Zweig explained the strong interactions in terms of the group SU (3) which acts on "color" triplets of quarks. For example, the local SU c (3) transformation of the u quark triplet (u r , u g , u b ) is

Ψ ′ (x) =    u r (x) u g (x) u b (x)    ′ = U (x)    u r (x) u g (x) u b (x)    = U (x) Ψ(x) (18)
in which U (x) is a 3 × 3 unitary matrix of unit determinant, i.e., U † (x)U (x) = 1 and det U (x) = 1. This local symmetry requires that the ordinary derivative ∂ k in the action density

Ψ † γ k ∂ k Ψ be replaced by the covariant derivative D k Ψ = (∂ i + iA i (x)) Ψ(x) in which A i (x) is the linear combination A i (x) = 8 a=1 t a A a i (x). (19)
The resulting action density 18), and that invariance requires that the quarks Ψ(x) interact with the gluons A a i (x). The interactions of the standard model are exactly those required by local gauge invariance.

Ψ † (x)γ k (∂ i + iA i (x)) Ψ(x) is invariant under the local SU c (3) transformation (

General relativity

Einstein realized that we have no way of knowing which system of coordinates is the "true" or "best" system of coordinates. He concluded that we should write the equations of physics in such a way that they are the same in all coordinate systems. His implementation of these simple ideas was general relativity.

He required the equations of physics to remain the same under arbitrary transformations of the four coordinates of spacetime

x i → x ′i (t, x, y, z). ( 20
)
To achieve this symmetry, he replaced the simple derivative ∂ ℓ V k of a vector V k with the covariant derivative

∂ ℓ V k → D ℓ V k = ∂ ℓ V k + Γ k iℓ V i (21)
in which the Christoffel symbol is

Γ k iℓ = 1 2 g kn (∂ ℓ g ni + ∂ i g nℓ -∂ n g iℓ ) (22)
and g ik (x) is the metric of spacetime. The replacement (21) requires the vector V k (and eventually every other field) to interact with the gravitational field g ik (x).

In empty space, particles fall along geodesics independently of their masses

0 = δS = -m δ -g ik dx i dx k =⇒ 0 = d 2 x k dτ 2 + Γ k iℓ dx i dτ dx ℓ dτ . ( 23
)
General relativity was verified by measurements of the precession of Mercury and of the bending of light by the Sun as observed by Eddington during the eclipse of 1919.

It is said that when a journalist asked Eddington if it were true that only three people understood general relativity, Eddington replied, "Who's the other one?" It can be helpful to visualize the points p(x) = p(x 0 , x 1 , x 2 , x 3 ) of our curved spacetime as embedded in a flat euclidian spacetime E 1,n of one dimension of time and n ≤ 19 dimensions of space. The (n + 1)-dimensional vectors e i (x) = ∂ i p(x) are tangent to our curved spacetime. The displacement of a point p(x) due to changes dx i in the four coordinates is dp = e i dx i , and the invariant squared displacement ds 2 is the Minkowski dot-product in which the metric of spacetime g ik (x) appears as the dot-product of the tangent vectors e i (x) and e k (x)

ds 2 = dp • dp = (e i dx i ) • (e k dx k ) = e i • e k dx i dx k ≡ g ik dx i dx k (24)
g ik (x) = e i (x) • e k (x) = -e 0 i (x) e 0 k (x) + n a=1 e a i (x) e a k (x). (25)
For instance, one may visualize the closed Friedmann-Lemaître-Robinson-Walker universe with metric

g ij =      -c 2 0 0 0 0 a 2 /(1 -r 2 /L 2 ) 0 0 0 0 a 2 r 2 0 0 0 0 a 2 r 2 sin 2 θ      (26)
as the points

p(t, r, θ, ϕ) = t 0 c 1 + L 2 ȧ2 c 2 dt ′ , ar sin θ cos ϕ, ar sin θ sin ϕ, ar cos θ, a √ L 2 -r 2
(27) in the flat semi-euclidian space E (1,4) with metric (-1, 1, 1, 1, 1).

Incidentally, in terms of the tangent vectors e i and e k = g kℓ e ℓ , the Christoffel symbol

(22) is Γ k iℓ = e k • ∂ ℓ e i .

Einstein the humanist

Einstein had a way with words:

• I never think of the future. It comes soon enough.

• Once we accept our limits, we go beyond them.

• Any fool can know. The point is to understand.

• Imagination is more important than knowledge.

• Two things are infinite. The universe and human stupidity -I'm not so sure about the universe.

• I never teach my pupils; I only provide the conditions in which they can learn.

• Everybody is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.

• I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.

• The most beautiful thing we can experience is the Mysterious.

Path integrals

Feynman's path integrals are based on two simple ideas. One is that for an infinitesimal interval dt, the time-evolution operator exp(-iHt/ℏ) for a non-relativistic particle of mass m is approximately e -iHdt/ℏ = e -i(p 2 /2m+V (q))dt/ℏ ≈ e -ip 2 dt/2mℏ e -iV (q)dt/ℏ . (

) 28
The other is that the amplitude for the particle to go from position q a to position q 1 in time dt is

⟨q 1 |e -ip 2 dt/2mℏ e -iV (q)dt/ℏ |q a ⟩ = ⟨q 1 | e -ip 2 dt/2mℏ ∞ -∞ |p ′ ⟩⟨p ′ | dp ′ e -i V (q) dt/ℏ |q a ⟩ = e -i V (qa) dt/ℏ ∞ -∞ e -ip ′2 dt/2mℏ e i(q 1 -qa) p ′ /ℏ dp ′ 2πℏ = m 2πiℏdt 1/2 exp i m 2 q 1 -q a dt 2 -V (q a ) dt ℏ . ( 29
)
To link two of these matrix elements together, one integrates over the complete set of outer products

|q 1 ⟩⟨q 1 | ⟨q 2 | e -ip 2 dt/2mℏ e -iV (q)dt/ℏ 2 |q a ⟩ = m 2πiℏdt ∞ -∞ exp i m q2 a 2 -V (q 1 ) + m q2 a 2 -V (q a ) dt ℏ dq 1
where now qa = (q 1 -q a )/dt and q1 = (q 2 -q 1 )/dt. By stitching together n = (t b -t a )/dt time intervals each of length dt and letting n → ∞, we get the path integral for a particle of mass m to go non-relativistically from q a to q b in time t

⟨q b |e -iHt/ℏ |q a ⟩ = m 2πiℏdt n/2 exp i ℏ m 2 q2 (t) -V (q(t)) dt ≡ e iS[q]/ℏ Dq (30)
in which the action is the integral of the lagrangian S[q] = L(q, q) dt. The amplitude is the integral over all paths from q a to q b each weighted by the phase exp(iS[q]/ℏ).

The amplitude for any state |Ψ⟩ to become |Φ⟩ in time t is the integral over the fields A that represent all processes that go from |Ψ⟩ to |Φ⟩ weighted by the phase exp(iS/ℏ) in which S[A] is the action of the process ⟨Φ|e -itH/ℏ |Ψ⟩ = e iS[A]/ℏ DA.

(31)

Supersymmetry

All the predictions of the standard model and of general relativity have been verified experimentally.

In contrast, supersymmetry predicts that every particle of spin s has a superpartner of spin s±ℏ/2. None have been observed, although they could be lurking at much higher energies.

Mysteries

The most beautiful thing we can experience is the Mysterious. We now have two huge mysteries:

1. Dark matter, i.e., invisible matter, is 84% of all matter.

2. Dark energy is 68% of the energy of the universe.

What are they?

And we have other mysteries. Why the Higgs? Why does it couple so weirdly? Why are the electroweak interactions so lopsided?

Interactions with right-handed fields

One way to balance the gauge group SU L (2) is to add the gauge group SU R (2) with a triplet of gauge bosons (W + R , W 0 R , W - R ) that interact only with right-handed quarks and leptons. One also would add a new Higgs field H R that transforms under SU R (2)⊗U Y ′ (1) in which the new weak hypercharge Y ′ has the eigenvalues of Y but with left and right interchanged.

There would be a total of seven new fields -three W R 's and the four components of the SU R (2) doublet H R . They would have to be very heavy or couple very weakly or both.

Summary

The equations of the standard model are invariant under the local symmetries SU c (3) ⊗ SU L (2) ⊗ U Y (1) and also under the general coordinate transformations of general relativity. All the known interactions-the strong, weak, electromagnetic, and gravitational interactions-are required by these local symmetries.

Figure 1 :

 1 Figure 1: How ∆x ∆p ≥ ℏ 2 suggests ∆ϕ ∆L ≥ ℏ 2 .

Figure 2 :

 2 Figure 2: Eclipse of 1919 (ESO/Landessternwarte Heidelberg-Königstuhl/F. W. Dyson, A. S. Eddington, C. Davidson).

Figure 3 :

 3 Figure 3: Albert Einstein and Arthur Eddington in 1930 (Royal Astronomical Society/Science Photo Library).


Rahman Sharaf

Sara Darbari
email: s.darbari@modares.ac.ir

Abdelkrim Khelif

A PnC-based SAW splitter, as a building block for double-stage surface acoustic wave Mach-Zehnder Vertical Surface Phononic Mach-Zehnder interfeometer

Keywords: Surface acoustic wave, double-stage phononic crystal, elastic Mach-Zehnder, UV and hydrogen sensor

In this work, we propose and evaluate a novel non-planar design of a doublestage phononic crystal (PnC) consisting of a Si substrate, a bottom layer of ZnO, Si pillars and another ZnO layer on top of the pillars. The proposed double-stage PnC exhibits interesting properties such as routing of incoming elastic wave based on source polarization and frequency. In addition, the elastic energy profile of the surface coupled modes are distributed between surface layers including two ZnO layers and pillars dividing the incoming elastic wave between top and bottom ZnO layers. Therefore, the proposed doublestage PnC can be considered as a building block in the design of a super-sensitive structure. The top ZnO layer's conductivity is modulated based on acoustoelectric effect when exposed to an external excitation like UV light illumination and environmental gases such as hydrogen. The proposed super-sensitive device, benefits from higher coupling between local resonators and therefore, the higher phase change between two branches of the elastic Mach-Zehnder which leads to a sensitivity and transmission as high as 23 and -8 dB at 8.6 GHz. Moreover, the proposed elastic Mach-Zehnder has the advantage of avoiding the non-sensing parts such as bottom ZnO layer as the reference branch, pillars and substrate to be affected by the external excitation due to the vertical configuration.

I. INTRODUCTION

Surface acoustic wave (SAW) devices were introduced in 1965 to replace the bulky electromagnetic elements (mm-m) in radar application with their micron-sized elastic counterparts through the idea of using slow elastic phonons instead of high speed photons [1]. Since then, SAWs have been an interesting research field for a variety of devices and applications, including disparate types of sensors [2][3][4], on-chip RF filters [START_REF] Group | IEEE 5G and Beyond Technology Roadmap White Paper[END_REF], microfluidics [START_REF] Winkler | [END_REF], quantum technology [7], acoustoelectric and acousto-optic phenomena [8][9][10], and nonreciprocity [11][12][13]. Phononic crystals (PnC) allow the controlling the propagation characteristics of acoustic and elastic waves by manipulating dispersion in applications such as filters [14], demultiplexers [9,15,16], sensors [17,18], heat transfer [19,20], energy harvesting [20,21], and resonators [22]. The PnCs for controlling surface elastic waves can be fabricated by holes or pillars on the surface of a substrate. The latter has the advantage of opening up low frequency locally surface resonance (LSR) gaps [23,24].

In piezoelectric semiconductor materials such as ZnO, elastic wave propagation has been shown that give rise to a traveling electric field, which in turn produces electric current [25]. This phenomenon, called acoustoelectric effect (AE), disturb both the velocity and attenuation of elastic waves. The use of such materials in SAW delay lines in the form of thin films or pillars opens up the possibility of modulating the elastic wave velocity and phase by controlling the number of free carriers and changing the conductance (σ) with external excitation [26]. This effect can be exploited to design sensitive SAW devices for applications such as hydrogen [27], ultraviolet (UV) light [28], and biological [29] sensors, as well as devices such as tunable elastic waveguides and demultiplexers [9,14]. In the Si substrate/ZnO pillar SAW configuration, it is shown that the elastic wave transmission peak, shifts by about 100 MHz due to the phase change caused by ZnO conductivity modulation when illuminated by a 365 nm UV light source [14]. A convenient way to effectively collect phase change data is to convert this variation into a change in elastic strength through interference in a Mach-Zehnder design and measure it at the output of the interdigital transducers IDTs. However, such Mach-Zehnder that uses light or hydrogen gas to control the interference of elastic wave not yet been reported. In a such design, two ZnO waveguides form Mach-Zehnder branches, and external excitation is applied to one of them, while the other branch is unaffected and serves as a reference for interference. However, in planar design of such structure, the reference branch should cover with mask to avoid the modulation of its properties by external excitation. Another drawback is that the elastic properties of the substrate are also affected by the external excitation, and addressing this problem was difficult with conventional Mach-Zehnder designs made up of planar thin films or single-stage PnCs. In addition, the surface area of the detection region in the planar structure is limited to the surface area of the involved waveguide.

In this research, we proposed for the first time an ultra-sensitive elastic Mach-Zehnder that detects external excitation such as UV and environmental gases such as hydrogen with high sensitivity and output transmission. To address the potential shortcomings of traditional planar Mach-Zehnder configurations, we introduce a double-stage design consisting of two ZnO layers that vertically connected to each other by Si pillars. The top ZnO layer acts as the sensing branch that its conductivity is modulated by the external excitation and the bottom ZnO layer serves as the reference one. This double-stage PnC has the advantage of a higher coupling coefficient between the local resonators for a particular propagation length compared with traditional single-stage PnCs without the top layer. This higher coupling of the proposed device's local resonators provides high phase resolution at elastic frequencies in the GHz range with a small dimensional structure. The double-stage PnC, serves as an efficient splitter to split the elastic wave excited at the surface of the bottom ZnO layer to both the top and bottom thin films. We examine the characteristics of the proposed two-stage PnC in detail and describe potential applications such as splitting and routing incoming waves on different ports. This study shows that the proposed PnC produces elastic wave propagation with negative group velocity at some specific dispersion points. Another ZnO layer between top and bottom waveguides is used to serve as the interference region in the proposed elastic Mach-Zehnder. Due to the vertical design of the proposed double-stage super-sensitive Mach-Zehnder, the external excitation only modulates the top ZnO layer elastic properties, which solve the challenging of the masking and influence of the external excitation on the substrate. Moreover, the area of the top layer is as high as the surface area of the substrate and therefore, the sensing area can be high with respect to conventional planar Mach-Zehnder. The present report, is a super-sensitive double-stage PnC elastic Mach-Zehnder which is a high sensitive high transmission, CMOScompatible device for several GHz range frequencies that is a promising candidate for real-time detection of the external excitations such as UV light and hydrogen gas.

II. THEORY AND IMPLEMENTATION OF SIMULATION

It is well known that surface local resonators, such as pillars, can be coupled to surface acoustic waves at certain resonance frequencies. Accordingly, PnCs consisting of an array of local resonators leads to low frequency surface coupled guiding modes with dispersion behavior depending on the elastic coupling between the resonators. The weaker elastic coupling leads to the less dispersive surface coupled bands, and a probable local resonance band gap is opened at low frequencies. Conversely, stronger elastic coupling between the individual resonators results in highly dispersive bands, and closing the local resonance band gaps. Elastic coupling can generally be tuned by the interspacing distance between the resonators, which is the basic principle of designing coupled resonator surface acoustic waveguides (CRSAWs) with tunable guiding mode dispersion and pass band FWHM, consequently [14]. Here, for the first time, we introduce a top layer on top of the pillar array to add a second degree of freedom to tune the elastic coupling between the resonators with a fixed interspacing distance. Thus, we expect stronger elastic coupling and more dispersive surface coupled modes, while increasing the thickness of the top layer. This double-stage PnC-based configuration can allow emerging more surface coupled modes, with different contribution portions of the bottom and top surfaces. In other words, we can achieve different low frequency surface coupled modes, in which the elastic wave is dominantly propagating in the bottom or top surface, or both of them simultaneously. This behavior can be applied as a vertical SAW splitting behavior, in which pillars couple the elastic wave to the bottom or top surface with a frequency-depended coupling strength. Here, we elaborate the full behavior of the proposed double-stage PnC as the vertical splitter in our SAW Mach-Zender design. In this case, splitting portions are equal, so that the top and bottom surfaces play as the two waveguides for SAW propagation. In such a Mach-Zender design the bottom surface is protected from the external stimulations such as light illumination or gas absorption, by the top layer. Hence, the output behavior can be intensively affected by slight variations in the elastic properties of the top layer, in response to external stimulations. Here, we propose acoustoelectric effect of piezoelectric layers such as ZnO, as the transduction mechanism in the top branch. Hence, the ZnO conductivity (σ) is assumed to be modulated in the top layer, when exposed to external UV illumination or gases like hydrogen [25]. To calculate the band structure of the proposed double-stage PnC, we have assumed a unit cell with lattice constant of a, which consists of Si cylindrical pillar with radius of r and height of h p , connecting the top and bottom ZnO thin films (Fig. 1(a)). Moreover, the thickness of the top and bottom ZnO thin films are h u and h d , and the substrate is made up of Si with thickness of h s . Periodic boundary conditions (PBC) are assumed along ±x and ±y directions, as shown in Fig. 1(a), while boundary condition at the substrate bottom (-z direction) is assumed to be fixed. To investigate the SAW transmission behavior of the proposed double-stage PnC structure, a supercell with 11 rows of pillars along the SAW propagation (+x direction) is considered, as shown in Fig. 1(b). Sidewalls of the supercell along ±y directions, are assumed to be PBC, while sidewalls along ±x and -z axes are assumed perfectly matched layer (PML) to effectively suppress the boundary reflections. Here, we considered a SAW line source on the surface of the bottom ZnO thin film, which is labeled as In in Fig. 1(b). Moreover, we consider the double-stage PnC in Fig. 1(b), as a 3-output terminal component, and define three different output cut-lines as FD, FU, and RU, to calculate the corresponding transmissions. To achieve the transmission (T), we calculate the integral of the elastic energy along the output cut-lines by Eq. ( 1), normalized to the input elastic energy. PBC boundary condition obeys the Bloch-Floquet theorem for the elastic waves as following [30]:

(2) 𝑢 𝑖 (𝑥 + 𝑎, 𝑦 + 𝑎) = 𝑢 𝑖 (𝑥, 𝑦)𝑒 -𝑖(𝑘 𝑥 𝑎+𝑘 𝑦 𝑎)

Wherein k x and k y are the Bloch wave vectors.

The values of all the structural parameters are summarized in Table I. The lattice constant of the PnC is set at 200 nm to achieve the surface coupled splitting behavior in the frequency range of GHz. We have presented a comprehensive investigation on the structural parameters of a single-stage PnC (r, h p ) in our previous work, which has led us to choose similar dimensions here (Table I), to achieve reasonable transmission values [11]. Moreover, it should be noted that the thicknesses of both top and bottom ZnO layers (h u , h d ) are found through optimization of the behavior of the final proposed SAW Mach-Zehnder device, which will be discussed in the next section. However, taking into account that the surface elastic waves are penetrated about one wavelength into the substrate, we assumed h s +h d =5a to avoid the SAW confinement effects in the substrate along z-direction.

I. Parameters of the unit cell and supercell structures shown in Fig. Fig1 It is well established that for linear non-piezoelectric materials such as Si, two coupled equations ( 3) and ( 4), govern the elastic wave propagation [14]:

(3) 𝑆 𝑖𝑗 = 1 2 (𝜕 𝑗 𝑢 𝑖 + 𝜕 𝑖 𝑢 𝑗 ) (4
) 𝑇 𝑖𝑗 = 𝐶 𝑖𝑗 𝑆 𝑖𝑗 Wherein S ij , T ij , and C ij show the strain, stress, and the elastic stiffness components along i,j=x,y,z directions. Si is a symmetric crystalline semiconductor material, and has three independent anisotropic elastic constants including C 11 =165.7 GPa, C 12 =63.9 GPa, and C 44 =79.9, and the mass density of ρ=2331 kg/m 3 [30]. On the other hand, in piezoelectric semiconductors such as ZnO, propagation of the elastic waves coincides with a strain-induced electric field. Hence, elastic wave propagation in ZnO can be described by the coupled equations of ( 5) and ( 6) :

(5) 𝑇 = 𝐶𝑆 -𝑒 𝑇 𝐸 (6) 𝐷 = 𝑒𝑆 -𝜀𝐸
Wherein e, ɛ, D, E are the piezoelectric coupling and dielectric matrices, and the electric displacement and electric field, respectively.

According to acoustoelectric effect in piezoelectric semiconductor materials like ZnO, variation of carrier concentration and the consequent conductivity variation affect the elastic wave propagation. This effect is attributed to the screened electric field, associated with the elastic wave propagation, by the free carriers. Acoustoelectric effect leads to a modulation of elastic wave velocity and attenuation by the carrier concentration or the correlated conductivity value. Assuming plane wave propagation for D and E in a conductive volume, Eq. ( 7) describes the relation between the current density (J) and the electric displacement and electric field []:

(7) 𝐷 = - 𝑗 𝜔 𝐽 = - 𝑗 𝜔 𝜎𝐸
Wherein σ is the conductivity of the piezoelectric semiconductor, and ω is the angular frequency.

Defining an effective acoustoelectric elastic matric (C՛) for the piezoelectric semiconductor, equations ( 5), [START_REF] Winkler | [END_REF], and ( 7) can be combined to describe the coupling effect between strain and stress as [31]:

(8) 𝑇 = 𝐶′𝑆
For ZnO, as an anisotropic crystal, elastic matrix elements are C 11 =C 22 =209.71 GPa, C 12 =121.14 GPa, C 13 =105.39 GPa, C 33 =211.19 GPa, C 44 =C 55 =42.37 GPa, and C 66 =44.24 GPa, and the mass density is ρ=5680 kg/m 3 . Accordingly, the effective acoustoelectric matrix is established as [14]: Wherein Y is a 3×3 matrix with elements defined by Eq. ( 10):

(9) [𝐶 ′ ] = [ 𝑐 11 +
(10) 𝑌 = [ 1 𝑗 𝜔 𝜎 + 𝜀 11 0 0 0 1 𝑗 𝜔 𝜎 + 𝜀 22 0 0 0 1 𝑗 𝜔 𝜎 + 𝜀 33]
All simulations to achieve the band structures and transmissions are carried out by using finite element method (FEM). Figures 2(d)-2(f) display the band structures for the double-stage PnC with the top ZnO layer thickness (h u ) of 50, 100, and 150 nm, respectively. It is notable here that the top and bottom ZnO layers conductivities are assumed zero (σ u =σ d =0), as the extreme limit of the intrinsic conductivity in such a wide-gap semiconductor. This intrinsic condition corresponds to absence of any external excitation that can modulate the conductivity of the top ZnO layer, such as exposure to UV-illumination or hydrogen environment. Moreover, all the bands are depicted in colors, determining their sagittal to shear polarization ratio. To achieve this ratio at each point of the bands, we have calculated the maximum displacement components in sagittal plane (xz plane) to the maximum shear displacement component i.e., sagittal to shear ratio=max(u x ,u z )/max(u y ). The dark gray area in the band structures covers the bulk modes with higher velocities, and the rest indicates the surface elastic modes with lower group velocities. The interface between the bulk and surface modes is defined by the sonic cone, corresponding to the bulk modes with lowest velocities. We have shown the constant frequencies of 5, 8, and 10.5 GHz by three dashed lines in all the band structures.

As it is shown in Fig. 2(a), the Si substrate only supports two pure sagittal surface bands in Γ-X direction, as we presented in our previous report [11]. Adding a ZnO thin film on top of the Si substrate (Fig. 2(h)), leads to emerging more surface modes with both sagittal and shear polarizations, as shown in Fig. 2(b). It can be observed that the first and second surface bands are pure sagittal and pure shear polarized in Γ-X direction, respectively. Moreover, ZnO/Si substrate has led to negative slope or negative group velocity along Γ-X direction, in some surface bands such as the third and fourth ones in Fig. 2(b). These negative group velocities force the phase of elastic wave to propagate in opposite directions, and can be used for designing lens-like behavior to focus the elastic wave in specific directions [32]. It can be observed in Fig. 2(c) that adding pillars on top of the bottom ZnO layer, the negative group velocity of the surface bands is vanished out and the low frequency surface coupled bands are nearly isolated apart, as compared with part (b). In this PnC, surface coupled modes are constructed between resonating modes of pillars and the surface modes of the substrate. The localized resonating behavior of the emerged surface coupled modes leads to lower dispersive modes and flatter bands, and leading to the observed partial isolation of the lower bands in part (c). Finally, adding the top ZnO layer on the pillars in Fig 2(j), the negative slope is reappeared in some of the intermediate surface bands (Figs. 2(d)-2(f)). For instance, negative slope has been emerged in fifth to eighth bands along Γ-X direction in Fig. 2(e). It is shown in Figs. 2(d)-2(f) that increasing h u leads to low frequency surface coupled bands with higher dispersive behavior, which can be attributed to stronger elastic coupling between the local resonating pillars through the top ZnO layer. Moreover, the polarization purity of the low frequency surface coupled bands are increased for thicker top ZnO layers, which is again related to the stronger coupling between the resonating pillars that can break the degeneracy in the bands, splitting them into individual highly dispersive single mode surface coupled bands, which are purely sagittal or shear. The other observation is emergence of higher number of surface modes for thicker top ZnO layers, which is related to emergence of new elastic modes across the top ZnO layer that can take part in new coupled elastic modes in the whole structure. Another phenomenon that should be noted is the polarization exchange between two modes that are approaching to each other. For example, the first two bands in Fig. 2(e), have shear and sagittal polarizations in Γ point, while after approaching at around 2 GHz, they exchange their polarizations.

To evaluate the effect of conductivity variations in the investigated double-stage PnC on the eigenmodes, we consider the upper and lower extreme limits of conductivity value exclusively, in which the frequency-dependent term in Eq. ( 8) vanishes and allows calculation of the relating band structure. On the other hand, considering the UV penetration depth of about 100 nm in ZnO, the top ZnO layers thicker than about 100 nm in the double-stage PnC can totally absorb the UVillumination, and prevent the bottom ZnO layer to be affected by the external excitation. Moreover, we have approximated the conductivity modulation through the top ZnO layer thickness to be uniform.

Figures 3(a) and 3(b) show the band structures of the double-stage PnC (Fig. 1(a)) with h u =150 nm, for the lower (σ u =0) and upper (σ u =∞) extreme limits of the top ZnO layer conductivity. It is notable that we have assumed conductivity of the bottom bare ZnO layer fixed at σ d =0, regardless of external stimulation presence.

As it is shown in Fig. 3(a) and 3(b), when σ u is increased from 0 to ∞, nearly all the surface modes are down shifted for about 100 MHz, without any other significant variation, which is in agreement with our previous results on acoustoelectric-induced maximum frequency shift in the line defect guiding modes in a single-stage PnC [14]. Figure 3(c) shows the total displacement for the points A, B, C, D, E, and F in Fig. 3(a), wherein A, C, D, E correspond to different surface modes at the X point, and the corresponding displacements are mostly confined around the two ZnO layers and pillar. F corresponds to a bulk mode at X point, in which the elastic energy is mostly spread through the substrate. Points B 1 -B 4 show the intersection of the surface coupled bands with the horizontal dashed line in Fig. 3(a), revealing simultaneous excitation of the corresponding eigenmodes (Fig. 3(c)) for the incident excitation frequency of 5 GHz. It can be also observed in Fig. 3(c), the elastic energy can be localized in the top ZnO layer (point A), the bottom ZnO layer (point D), the top and bottom ZnO layers (point C), and the both ZnO layers and pillar (point E), depending on the incident frequency. Figures 3(d) and 3(e) show the displacement components for modes B 1 and B 2 , wherein x and z displacement components are dominant for sagittal polarized modes like B 1 and y component is dominant for a shear polarized mode like B 2 . These data are in agreement with the presented sagittal to shear ratio depicted by band colors in the band structures. Moreover, Figs. 3(f) and 3(g), show the total displacement profiles relating to constant frequencies of 8 and 10.5 GHz in Fig. 3(a), revealing three and two excited surface modes, respectively.

Based on the results in Fig. 3, we can select out desirable behavior of the proposed double-stage PnC among the different possible applications. To design a vertical SAW splitter with the capability of dividing the elastic energy into the top and bottom ZnO layers, eigenmodes of point C in Fig. 3(a) should be excited, so that SAW waves can propagate in both the top and bottom ZnO layers along x direction. Eigenmodes like point A and D, support SAW propagation in one of the ZnO layers dominantly, so that their excitation can be used to efficient and locally couple the elastic wave from one layer to another. This application can lead to a new generation of vertical acoustic via in active acousto-optic components in integrated photonic chips, such as electro-optic modulators. To elaborate the presented double-stage PnC behavior, we have plotted the isofrequency contours at σ u =0, for 5 GHz and 8 GHz in Fig. 4(a) and 4(d). The isofrequency plots are obtained through calculation of eigen frequencies for all possible combinations in the k x -k y plane, finding their intersections with constant frequencies. The color in these iso-frequency contours depicts the sagittal to shear polarization ratio at that point, and the filled green regions correspond to the bulk modes. Γ, X, and M show the high symmetric points of the first Brillouin zone in Fig. 4(a), and the displayed data in this figure are in complete agreement with the discussed band structure in Fig. 3(a). Figures 4(b) and 4(e) show the elastic wave propagation from a central sagittal point source on the bottom ZnO layer in the presented double-stage PnC, wherein we have assumed 30 rows of pillars in both x and y directions. These figures display the in-plane (x-y plane) displacement profile on the surface of the bottom ZnO layer, which are similar to the displacement distributions on the top ZnO layer. It can be obviously observed in part (c) that sagittaly excited elastic waves propagate nearly isotropically for 5 GHz, which is consistent with part (a). At 8 GHz, Fig. 4(d) reveals a band with negative group velocity around X point along Γ-X direction, which can lead to a suppressed elastic wave propagation along x direction, as shown in Fig. 4(e) in which the elastic wave is completely focused in Γ-M direction. Figures 4(c) and 4(f) demonstrate the cross section (x-z plane) of the total displacement of the corresponding supercells with sagittal line sources along x direction at 5 and 8 GHz, respectively. Here, the input line source, the forward (+x direction) output on the top and bottom layers are labeled as In, FU, and FD, respectively. Figure 4(c) demonstrates the excited elastic waves propagate along +x direction both in the top and bottom ZnO layers, at 5 GHz. However, Fig. 4(f) reveals the dominant elastic wave propagation in the opposite direction (-x) through the top ZnO layer, at 8 GHz. Here, the reverse output on the top ZnO layer is labeled as RU. Therefore, the proposed double-stage PnC has the capability to guide different surface coupled modes anisotropically through excitation of different eigenmodes, depending on polarization and frequency. The sinusoidal profile of the coupling between the top and bottom ZnO waveguides is obviously observed in further propagation lengths of coupled elastic waveguides, such as Figs. 5(c), 5(e), and 5(g). Figures 5(b) (5(d)) and 5(c) (5(e)) show the coupled elastic wave propagation at 5 GHz (8 GHz). Moreover, there are some specific lengths that the elastic waves can propagate both in top and bottom waveguides simultaneously, as shown in Fig. 5(e) for 30 pillars at 8 GHz. This configuration case can serve as a vertical elastic splitter, which will be used in the next section of this work to design a super-sensitive elastic Mach-Zehnder device. Hence, the dominant output port in elastic energy transmission depends on the number of pillars (coupled elastic propagation length) and the excitation frequency. In Figs. 5(f) and 5(g), it can be observed that elastic waves are mostly reflected from the top ZnO layer at 10.5 GHz, so that the dominant output transmission at RU is independent of the pillars number.

The assumed shear horizontal line source in the transmission calculations is allowed to couple to the top or bottom ZnO waveguides without imposing significant polarization change, owing to the assumed PBCs along the y direction in the investigated supercell (Fig. 1(a)). In other words, we expect that the assumed supercell fully supports the polarization of the shear excitation, because the purely y-directed source displacements are not exposed to any y-direction confinements in the supercell. In Figure 6 we have explored this polarization behavior by superimposing the calculated total output transmission spectra (u x 2 +u y 2 +u z 2 ) at FD, FU, and RU onto the separated sagittal (u x 2 +u z 2 ) and shear (u y 2 ) transmission components, for a shear polarized source. According to Fig. 6, sagittal components are negligible and all outputs kept the shear polarization in all frequencies, without any polarization exchange in the structure.

IV. THE PROPOSED MACH-ZEHNDER DEVICE

The proposed super-sensitive elastic Mach-Zehnder device, based on the discussed double-stage PnC as the vertical SAW splitter, is shown in Fig. 7(a). It can be observed that the proposed device includes a Si substrate, a bottom ZnO layer on the substrate, a double-stage PnC consisting of 5×11 Si pillars in the x-y plane, and a top ZnO layer. Shear polarized SAW is assumed to be excited on the surface of the bottom ZnO layer, while the output detection port is assumed on the top ZnO layer, as shown in Fig. 7(a). Because of the vertical non-planar design of the structure, only the top ZnO layer is affected by the external excitation, leading to a consequent conductivity modulation. The proposed non-planar SAW Mach-Zehnder design with the top ZnO layer as a thin sensing layer, benefits from a small chip area and a supersensitive sensing behavior, without the need for focusing the external excitation (e. g. UV-illumination) on the sensitive branch, as compare with the conventional planar Mach-Zehnder configurations. Moreover, here we don't need an extra dummy SAW device as the reference in conventional differential measurements, and a highly sensitive real-time sensing with minimal cross sensitivity can be achieved in a single double-stage Mach-Zehnder configuration.

Figure 7(b) shows the cross-section scheme of the proposed structure, along the dashed line (AA') in part (a). Fig. 7(b) evidently illustrates the typical parts of a Mach-Zehnder device, consisting of the splitting PnC region to divide the incoming SAW into the top and bottom ZnO-based waveguides, the phase region wherein the elastic properties of the top branch is modulated individually by the external excitation, and the interference region wherein the top and bottom elastic waves interfere, producing the output signal at the end. It is notable that the elastic properties of the top ZnO layer are supposed to be modulated based on the acoustoelectric effect and the resulting modulation of effective stiffness matrix elements versus the conductivity modulation, which in turn is a consequence of external excitation. As shown in part (b), elastic waves propagate along a length of L p in the phase region, affecting the total phase different between the top and bottom branches, and the final output signal consequently. Moreover, replacing the isolating air gap between the top and bottom ZnO layers by an intermediate ZnO film, we make the elastic branches to merge in the interference region with length of L i , and define the output probe at the end of this region. It is notable that L i affects the acoustic impedance of the interference region, and the elastic reflections from the phase/interference interface in the top and bottom branches. The geometrical parameters of the proposed Mach-Zehnder device, excluding L p and L i , are defined according to our previous discussions and data about the double-stage PnC in Table I. As observed in Fig. 7, we have assumed the thickness of the intermediate ZnO layer in the interference region equal to the height of Si pillars to achieve a feasible and well-defined device structure. It is notable for each set of geometrical parameters in Table I, we can achieve different optimized L p and L i values for the proposed device. To optimize L p and L i values in the proposed Mach-Zehnder device, first we investigate the transmission values of the corresponding supercell at two extreme limits of the top ZnO layer conductivity: (i) σ u =0 for the absence of external excitation, and (ii) σ u =ꝏ for the intensive presence of external excitation, corresponding to transmission values of T 0 and T ∞ for the Mach-Zehnder device, respectively. To achieve the maximum sensitivity of the designed Mach-Zehnder device, L p and L i are optimized so that the resulting extinction ratio (|10×log10(T ∞ /T 0 )|) is maximized, wherein the output signal changes significantly by exposure to external excitation. Moreover, we pick up designs in which both extinction ratio and maximum transmission (max (T 0, T ∞ )) value are maximized simultaneously, so that we can achieve a highly sensitive device in addition to an acceptable and measurable output signal. Figures 8(a)-8(c) show the calculated extinction ratios of the proposed SAW Mach-Zehnder supercell (Fig. 7(b)), versus varying the phase region length (L p ) from 50 nm to 6000 nm and incident frequency, for interference region lengths of L i =2, 5, and 10 μm, respectively. As it is shown in this figure, higher L i values lead to extension of high extinction ratios and the relating operation frequency range of the device into lower frequencies. This observation is attributed to elastic impedance modulation of the interference region by varying L i , so that impedance matching can occur at L i =10 μm, which minimizes the elastic reflections at the phase/interference interface and allows emerging of elastic interference in the interference region. Moreover, for lower operation frequencies in Fig. 8(c), a periodic dependence of the extinction ratio to L p can be obviously observed at each frequency. For more clarification, Fig. 8(d) depicts the extinction ratios versus L p (L i =10 μm) at three different frequencies of 3 (bottom), 8 (middle), and 14 GHz (up), shown by dashed lines in part (c). As expected from the periodic nature of the produced phase delay in the top ZnO layer, the observed periods in this part grow shorter for higher incident frequencies. This figure confirms that interference effect is dominantly responsible for the observed extinction ratio variations for L i >5 μm at frequencies lower than about 10 GHz. Considering the latter study, we assume L i =10 μm in our design to achieve a well-behaved Mach-Zehnder behavior, based on successful elastic interference in the interference region. Considering extinction ratio hot spots in Fig. 8(c), we determined L i =10 μm, L p =2.8 μm at 8.3 GHz for our highly sensitive Mach-Zehnder design, to achieve the highest possible output signal (max(T 0, T ∞ )). Fig. 9(a) indicates the resulting T 0 (solid-blue) and T ∞ (dashed-red) transmission spectra of the supercell, revealing a high transmission contrast of about 21 dB at 8.3 GHz. Figures 9(b) and 9(c) show the total displacement distributions in the supercell of the determined design at 8.3 GHz, corresponding to 0 and T ∞ respectively. As shown in parts (b) and (c), incident elastic waves are successfully coupled to both top and bottom waveguides in the splitter regions, propagate through the same length but with different phase delays along the phase regions, then merge and interfere in the interference regions with different interference profile patterns, leading to an extinction ratio of about 21 dB at a fixed output probe (shown by Out). Vertical dashed lines at some specific x-positions in Fig. 9(b) and 9(c) clarify the observed elastic wave lag between the un-excited and excited conditions after the phase regions, which is originated from the phase delay difference induced by external excitation. Thus, our optimized design has led to the maximum output sensitivity of about 21 dB for the investigated supercell in response to extreme limits of external excitations (σ u =0, ꝏ), while a high output signal of T 0= -1.5 dB has been achieved. To elaborate the exact output sensitivity, we will investigate the acoustoelectric-induced conductivity modulation trend in the final device with finite structure, from σ u =0 to ꝏ. According to the supercell results of Figs. 8(c) and 10, we assume L i =10 μm in the final Mach-Zehnder device with finite dimensions, and re-optimize L i value. Here, we use the configuration shown in Fig. 7(a) to carry out three-dimensional simulations. Geometrical parameters are set according to Table I, while the double-stage PnC consists of five pillar rows along x-axis, and 11 pillar rows along y-axis. Figure 10 depicts the calculated extinction ratios versus varying the incident frequency from 1 to 10 GHz, and L p from 50 nm to 5000 nm with a step of 50 nm. Among the observed hot spots in this plot, L p =1650 nm leads to both maximum extinction ratio of 23.5 dB and output signal of -8 dB for the complete finite design, at 8.6 GHz. Moreover, the periodic variation of extinction ratios versus varying L p can be evidently observed in Fig. 10. To elaborate the sensitivity of the designed Mach-Zehnder device, we have varied σ u from 10 -3 S/m to 10 4 S/m, and calculated the corresponding output transmission spectra (T σu ). Figure 12(a) shows 10log 10 (T σu ) spectra for different σ u values, which reveals the larger σ u , the smaller transmission value at 8.6 GHz. Figure 12(b) shows the variation trend of transmission values versus varying σ u , at 8.6 GHz. It is observed that increasing σ u low levels up to 1 S/m (violet zone), T σu value doesn't change significantly and is approximately equal to T 0 . Similarly, increasing σ u from about 100 S/m toward high levels (pink zone), T σu value does not change considerably and remains about equal to T ∞ . However, for 2.5 S/m<σ u <50 S/m (yellow zone), T σu value reveals a nearly linear correlation to σ u with the output sensitivity of about 2.3 dB per 0.1 S/m (23 dB/(S/m)). The observed variation trend in Fig. 13(b) is in good agreement with the conventional trend of SAW-velocity modulation by conductivity in the literature [14,33]. Moreover, this trend is consistent with the conductivity-induced modulation of the defect mode guiding frequencies in a single stage PnC, in our previous work [14]. Here, we have proposed and designed of a novel non-planar SAW Mach-Zehnder device, based on double-stage PnCs, and discussed the relating physical operation behavior. Moreover, we proposed this configuration as a highly sensitive sensor in response to external excitations, and evaluated the relating sensing performance.

V. CONCLUSION

In this paper, we proposed a new double stage surface acoustic wave structure which shows interesting behaviours. This new structure benefited from two layer of ZnO which are connected to each other through Si pillars. This structure, can route the elastic wave to different ports based on frequency or polarization, as well as guiding the elastic energy with 180 degrees rotation at the top layer if the source is excited at the surface of the bottom ZnO layer. Moreover, it was shown that for different surface eigenmodes, the confinement of the displacement is distributed between bottom thin film, PnCs, and top thin film layers. Therefore, in specific condition, this structure can be used as an SAW splitter to divide elastic wave between bottom and top ZnO waveguides.

According to these results, we design a super-sensitive non-planar structure for detecting any external excitation such as UV-light or gases like hydrogen that affect on the top ZnO layer conductivity. The variation of the conductivity, can leads to different propagation properties of the elastic wave through acoustoelectric phenomenon. We used this effect to design an elastic Mach-Zehnder to increase the sensitivity in conjunction to real-time detection of the external excitation. The proposed device benefits from two major advantages: (1) the large area of the sensing that is the entire of the top ZnO layer, while the properties of the bottom layers do not affect and remains constant, and (2) realtime detecting which is based on constructive and destructive interference in an elastic Mach-Zehnder.

The simulation results, show that by optimizing the structure parameters, 23.5 dB contrast is achieved with the maximum transmission of -8 dB. Also the sensitivity of the proposed device in the linear region is equal to 2.3 dB decrement in transmission per 0.1 S/m increment of the top ZnO layer conductivity. Based on these results, the proposed super-sensitive double stage device is a promising structure for applications such as UV-light and hydrogen detections.

2

 2 Wherein u x,y,x are the displacement components. To evaluate the shear horizontal (sagittal) transmission, we consider shear (sagittal) polarized source with displacement component(s) in y (x, z) direction(s).

Fig1. ( a )

 a Fig1. (a) The unit cell of the PnC, used for calculating the relating band structure. (b) The supercell of the PnC with 11 rows of pillars along the wave propagation direction (+x), used for calculating the transmission spectrum.

Figures 2 (

 2 Figures 2(a)-2(c) show the calculated band structures for the first 35 bands of the unit cells shown in Figs. 2(g)-2(i), which correspond to the bare Si substrate, ZnO-coated Si substrate, and the single-stage PnC consisting of Si pillars array on the ZnO/Si substrate, respectively. The inset of Fig. 2(a) shows the irreducible path in the first Brillouin zone of the investigated PnCs with square lattice.Figures2(d)-2(f) display the band structures for the double-stage PnC with the top ZnO layer thickness (h u ) of 50, 100, and 150 nm, respectively. It is notable here that the top and bottom ZnO layers conductivities are assumed zero (σ u =σ d =0), as the extreme limit of the intrinsic conductivity in such a wide-gap semiconductor. This intrinsic condition corresponds to absence of any external excitation that can modulate the conductivity of the top ZnO layer, such as exposure to UV-illumination or hydrogen environment. Moreover, all the bands are depicted in colors, determining their sagittal to shear polarization ratio. To achieve this ratio at each point of the bands, we have calculated the maximum displacement components in sagittal plane (xz plane) to the maximum shear

Fig2. (

  Fig2. (a), (b), and (c) The band structures of the unit cell shown in parts (g), (h), and (i) respectively. (d), (e), and (f) The band structure of the unit cell shown in part (j) with top ZnO layer thickness (hu) equal to 50, 150, and 150 nm respectively. (g) The unit cell consists of Si substrate. (h) The unit cell with Si substrate and a ZnO thin film layer at the top. (i) The unit cell with Si substrate, ZnO thin film layer and Si Pillars. (j) The unit cell containing Si substrate, bottom ZnO thin film layer, Si pillars, and top ZnO thin film layer. Inset in part (a), shows the irreducible path for the first Brillouin zone of the square lattice.

Fig3.

  Fig3. (a) and (b) The band structure of the unit cell with σu=0 and σu=∞ respectively. (c) The total displacement profiles for the points shown in the band structure in part (a). (d) and (e) The displacement components of the eigenmode of the point B1 and B2

Fig4

  Fig4. (a), (b), and (c) The iso-frequency contours of the unit cell shown in Fig. 2(e), sagittal polarized point source propagation at the surface of the bottom ZnO layer, and the total displacement profiles of the supercell along Γ-X direction for sagittal source respectively at 5 GHz. (d), (e), and (f) the same results of part (a), (b) and (c) respectively at 8 GHz.

Figure 5 (

 5 Figure 5(a) depicts the reduced band structure of the double-stage PnC (unit cell in Fig.1(a)) in the Gama-X direction, beside the calculated shear transmissions through the output ports of the corresponding supercell (Fig.1(b)). Both band structure and transmission calculations are carried out for σ u =σ d =0, when there is no external excitation. Here we have assumed a shear input line source at the In cutline, and calculated the transmission spectra corresponding to the output forward transmissions on the bottom (FD) and top (FU) ZnO layers in the +x direction, and the reverse transmission on the top ZnO layer in -x direction (RU). Individual transmission spectra in Fig.5(a) reveal that input elastic wave can be coupled to either the top or bottom ZnO layers, at each excitation frequency. The top and bottom ZnO layers behave like elastic waveguides that are coupled through the pillars, so that they periodically exchange the elastic energy during wave propagation along x direction. This elastic energy exchange can be observed in Figs. 5(b)-5(g), depicting the total displacement profile of the supercell in the cross section view (x-z plane). Here, we have assumed 11 pillars in parts (b), (d), and (f), and 30 pillars in parts (c), (e), and (g) along the x axis, proving that increasing the coupled elastic propagation length can switch the outward elastic energy from FD to FU, or vice versa.

Fig5.

  Fig5. (a) The band structure of the unit cell in dark mode in Γ-X direction and the transmissions of the supercell. (b), (d), and (f) the cross sectional of the total displacement profiles of the supercell in x-z plane at 5, 8, and 10.5 GHz respectively for 11 rows of pillars along x. (c), (e), and (g) the cross sectional of the total displacement profiles of the supercell in x-z plane at 5, 8, and 10.5 GHz respectively for 30 rows of pillars along x.

Fig6.

  Fig6. The total transmissions (ux 2 +uy 2 +uz 2 ) of FD, FU, and RU and their sagittal (ux 2 +uz 2 ) and shear (uy 2 ) components of the supercell excited with shear source.

Fig7

  Fig7. (a) Three-dimensional perspective of the proposed surface elastic Mach-Zehnder. (b) Schematic of a typical Mach-Zehnder and Supercell of the proposed device in part (a).

Fig8.

  Fig8. Profile of the transmission ratio of the proposed elastic Mach-Zehnder supercell, in terms of SAW frequency and phase length (Lp) for interference length of (a) Li=2 μm, (b) Li=5 μm, and (c) Li=10 μm. (d) transmission ratio of the supercell versus Lp at 14 GHz (up), 8 GHz (middle), and 3 GHz (bottom).

Fig 9 .

 9 Fig 9. (a) Spectrum of the supercell transmissions with Li=10 μm and Lp=2.8 μm. (b) and (c) Total displacement profiles at 8.3 GHz for T0 and T∞ respectively with shear polarized source.

Fig10.

  Fig10. Profile of the transmission ratio in the proposed super-sensitive elastic device.

Figure 11 (

 11 Figure 11(a) shows the T 0 (solid-blue) and T ∞ (dashed-red) transmission spectra of the designed complete device for L p =1650 nm, confirming output signal of T 0= -8 dB, and extinction ratio or output sensitivity of 23.5 dB to extreme variation limits of σ u , at 8.6 GHz. Figures 11(b) and 11(c) exhibit the total displacement cross-sectional (xz plane) profiles corresponding to T 0 and T ∞ , respectively. Again, the lag between the elastic waves in part (b) and (c) is observable, which origins form the different phase of elastic wave at the top ZnO layer. This acoustoelectric-induced phase difference has led to different interference profile patterns in parts (b) and (c), and the observed high extinction ratio consequently. On the other hand, displacement profiles reveal that the splitting behavior of the double-stage PnC and the coupling strength to the top and bottom ZnO layers are also affected by external excitation and the relating conductivity modulation. These elastic wave profiles show that external excitation has led to decrease the elastic coupling strength of the incident wave to the top ZnO layer for T ∞ . This splitting modulation behavior is attributed to the aforementioned down shift of surface bands of the double-stage PnC as in Figs. 3(a) and 3(b), versus increasing σ u from 0 to ꝏ. In other words, T 0 is high here, due to constructive interference of the top and bottom elastic waves, and the superficial confinement of elastic energy in the vicinity of output surface probe. Similarly, T ∞ is low because of the destructive interference of the elastic waves propagated at top and bottom waveguides, and the weak elastic coupling to the top ZnO layer for σ u =ꝏ. Considering 12(a) and 10(a), an enhanced extinction ratio of about 2.5 dB has been observed for the optimized design of the complete finite device, as compared

Fig11.

  Fig11. (a) Spectrum of the proposed device transmissions with Li=10 μm and Lp=1.65 μm. (b) and (c) Total displacement profiles at 8.6 GHz for T0 and T∞ respectively with shear polarized source.

Fig12.

  Fig12. (a) The spectrum of the T∞ for different σu. (b) The sensitivity of the proposed device at 8.6 GHz.

  .

	Parameter	Symbol	Value
	Pillar radius	r	85 nm
	Pillar height	hp	10 nm
	Lattice constant	a	200 nm
	Top ZnO layer	hu	150 nm
	Bottom ZnO layer	hd	200 nm
	Si substrate thickness	hs	800 nm

  𝑒 31 𝑒 33 𝑌 33 𝑐 13 + 𝑒 31 𝑒 33 𝑌 33 𝑐 23 + 𝑒 31 𝑒 33 𝑌 33

	𝑒 31 2 𝑌 33 𝑐 12 + 𝑒 31 2 𝑌 33	𝑐 12 + 𝑒 31 2 𝑌 33 𝑐 11 + 𝑒 31 2 𝑌 33	𝑐 13 + 𝑒 31 2 𝑌 33 𝑐 23 + 𝑐 33 + 𝑒 33 2 𝑌 33	0 0 0	0 0 0	0 0 0
	0	0	0	𝑐 44 𝑒 15 2 𝑌 22	0	0
	0 0	0 0	0 0	0 0	𝑐 44 + 𝑒 15 2 𝑌 11 0 0 𝑐 66 ]

AKNOWLEDGEMENT

The authors acknowledge the financial support from Tarbiat Modares University through Grant #IG-39703.

Disclosures

The authors declare no conflicts of interest.


Alexandros Eskenazis

Haonan Zhang

APPROXIMATION OF POLYNOMIALS FROM WALSH TAIL SPACES

Keywords: 2020 Mathematics Subject Classification. Primary: 42C10; Secondary: 41A17, 60E15 Discrete hypercube, polynomial inequalities, Rademacher sum, tail space

published or not. The documents may come

Introduction

Given n ∈ N, every function f : {-1, 1} n → R admits a unique Fourier-Walsh expansion

∀ x ∈ {-1, 1} n , f (x) = S⊆{1,...,n} f (S)w S (x), (1)
where the Walsh function w S is given by w S (x) = i∈S x i for x = (x 1 , . . . , x n ) ∈ {-1, 1} n . We shall say that f is of degree at most k ∈ {1, . . . , n} if f (S) = 0 for every subset S of {1, . . . , n} with |S| > k.

Similarly, we say that f belongs on the k-th tail space, where k ∈ {1, . . . , n}, if f (S) = 0 for every subset S with |S| ≤ k. More generally, given a nonempty subset I ⊆ {0, 1, . . . , n}, we denote by

P n I def = f : {-1, 1} n → R : f (S) = 0 for every S with |S| I . ( 2
)
We shall also adopt the natural notations P n >k = P n {k+1,...,n} , P n ≤k = P n {0,1,...,k} , P n =k = P n {k} and so on. Many modern developments in discrete analysis (see [START_REF] Ryan | Analysis of Boolean functions[END_REF]) are centered around quantitative properties of functions whose spectrum is bounded above or below, in analogy with estimates established for polynomials in classical approximation theory on the torus T n or on R n . One of the first results of this nature, going back at least to [START_REF] Bonami | Étude des coefficients de Fourier des fonctions de L p (G)[END_REF][START_REF] Bourgain | Walsh subspaces of L p -product spaces[END_REF], is the important fact that all finite moments of low-degree Walsh polynomials are equivalent to each other up to dimension-free factors. Namely, given any 1 ≤ p ≤ q < ∞ and k ∈ N, there exists a (sharp) constant M p,q (k) such that for any n ≥ k, every polynomial f : {-1, 1} n → R of degree at most k satisfies

f q ≤ M p,q (k) f p , (3)
where • r always denotes the L r norm on {-1, 1} n with respect to the uniform probability measure. Note that the reverse of (3) holds trivially with constant 1 by Hölder's inequality. We refer to [START_REF] Eskenazis | Polynomial inequalities on the Hamming cube[END_REF][START_REF] Ivanisvili | Comparison of moments of Rademacher chaoses[END_REF][START_REF] Levhari | Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Rényi entropy[END_REF] for the best known bounds on the implicit constant M p,q (k). In the special case k = 1, (3) is the celebrated Khintchine inequality [START_REF] Khintchine | Über dyadische Brüche[END_REF] for Rademacher sums. Our starting point is the simple observation that the moment comparison estimates (3) have the following (equivalent) dual formulation in terms of distances from tail spaces. Proposition 1. For every 1 ≤ p ≤ q < ∞ and d ∈ N, the constant M p,q (k) in inequality (3) is also the least constant for which every function

f : {-1, 1} n → R, where n ≥ k, satisfies inf g∈P n >k f -g p * ≤ M p,q (k) inf g∈P n >k f -g q * , ( 4
)
where the conjugate exponent

r * of r ∈ [1, ∞] satisfies 1 r * + 1 r = 1.
This material is based upon work supported by the NSF grant DMS-1929284 while the authors were in residence at ICERM for the Harmonic Analysis and Convexity program.

Again, the reverse of (4) holds with constant 1. In the special case k = 0, inequality (4) becomes trivial with M p,q (0) = 1 as both sides are equal to |Ef |. When k = 1, which corresponds to the dual of the classical Khintchine inequality, one can derive the following more precise formula for the distance from the tail space P n >1 .

Theorem 2. For every 1 < r ≤ ∞ and n ∈ N, every f :

{-1, 1} n → R satisfies 1 inf g∈P n >1 f -g r |Ef | + max i∈{1,...,n} f ({i}) + r -1 r n i=1 f ({i}) 2 1/2 . ( 5
)
This is the dual to a well-known result of Hitczenko [START_REF] Hitczenko | Domination inequality for martingale transforms of a Rademacher sequence[END_REF] (see also [START_REF] Stephen | The distribution of Rademacher sums[END_REF][START_REF] Hitczenko | On the Rademacher series[END_REF]), obtaining pindependent upper and lower bounds for the L p -norms of Rademacher sums, where p ∈ [1, ∞).

At this point, we should point out that in both Proposition 1 and Theorem 2, the exponents of the norms are always strictly greater than 1. For instance, choosing

f 1 (x) = n i=1 x i , (4) gives ∀ r ∈ (1, ∞],
inf

g∈P n >k f 1 -g r r,k inf g∈P n >k f 1 -g 2 = √ n. ( 6
)
On the other hand, it follows from a result of Oleszkiewicz [START_REF] Oleszkiewicz | On mimicking Rademacher sums in tail spaces[END_REF], which is the main precursor to this work, that the L 1 -distance of f 1 from the k-th tail space satisfies inf

g∈P n >k f 1 -g 1 min{k, √ n}, ( 7
)
and thus exhibits a starkly different behavior as n → ∞ from the L r norms with r > 1.

More generally, it is shown in [START_REF] Oleszkiewicz | On mimicking Rademacher sums in tail spaces[END_REF] that for every

a 1 ≥ • • • ≥ a n ≥ 0, we have inf g∈P n >k f a a a -g 1 min r∈{0,1,...,n} r i=1 a 2 i 1/2 + ka r+1 , (8)
where for a a a = (a 1 , . . . , a n ) we denote f a a a (x) = n i=1 a i x i and we make the convention that a n+1 = 0. The quantity appearing on the right hand side of (8) can be rephrased in terms of the Kfunctional of real interpolation (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Chapter 3]). Recall that if (A 0 , A 1 ) is an interpolation pair, then the Lions-Peetre K-functional is defined for every t ≥ 0 and a ∈ A 0 + A 1 as

K(a, t; A 0 , A 1 ) def = inf a 0 A 0 + t a 1 A 1 : a = a 0 + a 1 . ( 9
)
It is elementary to check (see [START_REF] Holmstedt | Interpolation of quasi-normed spaces[END_REF]), that if

a 1 ≥ • • • ≥ a n ≥ 0 and k ∈ N, then min r∈{0,1,...,n} r i=1 a 2 i 1/2 + ka r+1 K(a a a, k; n 2 , n ∞ ). ( 10
)
Note that the right-hand side is invariant under permutations of the entries of a a a. The main result of this work is an appropriate extension of the upper bound in Oleszkiewicz's result [START_REF] Eskenazis | Polynomial inequalities on the Hamming cube[END_REF] to polynomials of arbitrary degree on the discrete hypercube.

Theorem 3. For every d ∈ N, there exists

C d ∈ (0, ∞) such that for any n ≥ k ≥ d, every polynomial f : {-1, 1} n → R of degree at most d satisfies inf g∈P n >k f -g 1 ≤ K f , C d k d ; m 2 , m 2d d-1 , ( 11
)
where f is the vector of Fourier coefficients of f , viewed as an element of

R m with m = n 0 + • • • + n d .
As was already pointed out by Oleszkiewicz, the method of [START_REF] Oleszkiewicz | On mimicking Rademacher sums in tail spaces[END_REF] does not appear to extend beyond Rademacher sums. Instead, in our proof we shall employ the discrete Bohnenblust-Hille inequality from approximation theory (see [START_REF] Blei | Analysis in integer and fractional dimensions[END_REF][START_REF] Defant | The Bohnenblust-Hille cycle of ideas from a modern point of view[END_REF][START_REF] Defant | On the Fourier spectrum of functions on Boolean cubes[END_REF][START_REF] Defant | Dirichlet series and holomorphic functions in high dimensions[END_REF]) along with a classical bound of 1 Throughout the paper we shall use standard asymptotic notation. For instance, ξ η (or η ξ) means that there exists a universal constant c > 0 such that ξ ≤ cη and ξ η stands for (ξ η)∧(η ξ). We shall use subscripts of the form t , t , t when the implicit constant c depends on some prespecified parameter t.

Figiel on the Rademacher projection of polynomials. A discussion concerning the size of the implicit constant C d appearing in [START_REF] Hitczenko | On the Rademacher series[END_REF] is postponed to Section 2 (see Remark 8 there).

Unlike the two-sided inequality (8), our bound ( 11) is only one-sided and as a matter of fact there are examples in which it is far from optimal. In particular, for functions which are permutationally symmetric, we obtain a more accurate estimate. In what follows, we shall denote by T k (x) = k =0 c(k, )x the k-th Chebyshev polynomial of the first kind characterized by the property T k (cos θ) = cos(kθ), where θ ∈ R. Moreover, we shall use the ad hoc notation

c(k, ) def =      c(k, ), if k -is even c(k -1, ), if k -is odd . ( 12
)
For ∈ {1, . . . , n}, let f be the -th elementary symmetric multilinear polynomial

∀ x ∈ {-1, 1} n , f (x) def = S⊆{1,...,n}: |S|= w S (x). ( 13
)
We have the following bound on the distance of symmetric polynomials from tail spaces.

Theorem 4. Let n, k, d ∈ N with n ≥ k ≥ d. Then, every symmetric polynomial f = d =0 α f (14) of degree at most d on {-1, 1} n satisfies inf g∈P n >k f -g 1 ≤ d =0 |α || c(k, )|. ( 15
)
This bound can sometimes be reversed and, in particular, it gives a sharp estimate as n → ∞ for the L 1 -distance of the elementary symmetric polynomial f d from the k-th tail space.

Corollary 5. For every n, k, d ∈ N with n ≥ k ≥ d, there exists ε n (k, d) > 0 such that | c(k, d)| -ε n (k, d) ≤ inf g∈P n >k f d -g 1 ≤ | c(k, d)| ( 16
)
and

lim n→∞ ε n (k, d) = 0.
The main motivation behind the work [START_REF] Oleszkiewicz | On mimicking Rademacher sums in tail spaces[END_REF] was a question of Bogucki, Nayar and Wojciechowski, asking to estimate the L 1 -distance of the Rademacher sum f 1 from the k-th tail space. Corollary 5 extends (at least asymptotically in n) the answer given by Oleszkiewicz to all symmetric homogeneous polynomials. We point out though that for k = 1, ( 16) is sharper than Oleszkiewicz's bound [START_REF] Defant | The Bohnenblust-Hille cycle of ideas from a modern point of view[END_REF] as n → ∞, as ( 7) is tight only up to a multiplicative constant.

Acknowledgements. We are grateful to Krzysztof Oleszkiewicz for valuable discussions. H. Z. is grateful to Institut de Mathématiques de Jussieu for the hospitality during a visit in 2023.

Proofs

We proceed to the proofs of our results. We start with the simple duality argument leading to Proposition 1, variants of which will be used throughout the paper.

Proof of Proposition 1. Consider the identity operator acting as id(h) = h on a function of the form h : {-1, 1} n → R. Then, the optimal constant M p,q (k) can be expressed as

M p,q (k) = id : (P n ≤k , • p ) → (P n ≤k , • q ) = id * : (P n ≤k , • q ) * → (P n ≤k , • p ) * (17)
by duality. Moreover, observe that since (P n ≤k , • r ) is a subspace of L r , its dual is isometric to

(P n ≤k , • r ) * = L r * (P n ≤k ) ⊥ = L r * P n >k , ( 18
)
where A ⊥ is the annihilator of A. Since it is also clear that id * = id, (17) concludes the proof.

Using a theorem of Hitczenko [START_REF] Hitczenko | Domination inequality for martingale transforms of a Rademacher sequence[END_REF] as input and the same duality, we deduce Theorem 2.

Proof of Theorem 2. The result of [START_REF] Hitczenko | Domination inequality for martingale transforms of a Rademacher sequence[END_REF] asserts that if a a a = (a 0 , a 1 , . . . , a n ) and f a a a (x) = a 0 + n i=1 a i x i ,

f a a a r * = E n i=0 a i x i r * 1/r * K(a a a, √ r * ; n+1 1 , n+1 2 ), (19)
where x 0 , x 1 , . . . , x n are independent Bernoulli random variables, and the first equality holds due to symmetry. In other words, the linear operator

T : R n+1 , K(•, √ r * ; n+1 1 , n+1 2 ) → (P n ≤1 , • r * ) (20)
given by T a a a = f a a a is an isomorphism, and thus the same holds for its adjoint. Recalling that

K(a a a, √ r * ; n+1 1 , n+1 2 ) = inf b b b n+1 1 + √ r * c c c n+1 2 : a a a = b b b + c c c (21)
and the duality between sums and intersections of normed spaces [1, Theorem 2.7.1], we see that the dual space of (R n+1 , K(•,

√ r * ; n+1 1 , n+1 2 )
) can be identified with

∀ y ∈ R n+1 , y (R n+1 ,K(•, √ r * ; n+1 1 , n+1 2 )) * = max y n+1 ∞ , y n+1 2 √ r * . ( 22
)
By Parseval's identity, the action of the adjoint

T * : L r P n >1 → (R n+1 , K(•, √ r * ; n+1 1 , n+1 2 )) * (23)
is given by

T * (f + P n >1 ) = Ef , f ({1}), . . . , f ({n}) (24)
and thus the conclusion is equivalent to fact that T * is an isomorphism.

We now proceed to the proof of the general upper bound for polynomials given in Theorem 3. The first ingredient for the proof is a discrete version of the classical Bohnenblust-Hille inequality from approximation theory (see the survey [START_REF] Defant | The Bohnenblust-Hille cycle of ideas from a modern point of view[END_REF]) proven in [START_REF] Blei | Analysis in integer and fractional dimensions[END_REF][START_REF] Defant | On the Fourier spectrum of functions on Boolean cubes[END_REF]. This asserts that for every d ∈ N, there exists a (sharp) constant B d ∈ (0, ∞) such that for any n ≥ d, every polynomial f : {-1, 1} n → R of degree at most d satisfies

S⊆{1,...,n} | f (S)| 2d d+1 d+1 2d ≤ B d f ∞ . ( 25
)
Moreover, 2d d+1 is the least exponent for which the implicit constant becomes independent of the ambient dimension n. The best known upper bound B d ≤ exp(C d log d) for the constant B d is due to the work of Defant, Mastyło and Pérez [START_REF] Defant | On the Fourier spectrum of functions on Boolean cubes[END_REF].

The level -Rademacher projection of a function f : {-1, 1} n → R is defined as

∀ x ∈ {-1, 1} n , Rad f (x) def = S⊆{1,...,n}: |S|= f (S)w S (x). ( 26
)
Moreover, we write Rad ≤d = ≤d Rad . Apart from the discrete Bohnenblust-Hille inequality (25), we will also use a standard bound on the norm of the -Rademacher projections which is usually attributed to Figiel (see also [START_REF] Eskenazis | Learning low-degree functions from a logarithmic number of random queries[END_REF]Section 3] for a short proof).

Proposition 6. Let n ≥ k ≥ d. Then, every function f : {-1, 1} n → R of degree at most k satisfies Rad ≤d f ∞ ≤ d =0 Rad f ∞ ≤ d =0 | c(k, )| f ∞ , ( 27
)
where c(k, ) is given by [START_REF] Holmstedt | Interpolation of quasi-normed spaces[END_REF]. It is moreover known that | c(k, )| ≤ k ! . Combining the above with Parseval's identity, we deduce the following bound.

Lemma 7. Let n ≥ k ≥ d. Then, every function f : {-1, 1} n → R of degree at most k satisfies max Rad ≤d (f ) m 2 , σ (k, d) -1 Rad ≤d (f ) m 2d d+1 ≤ inf g∈P n >d ∩P n ≤k f -g ∞ , ( 28
)
where m = n 0 + • • • + n d and σ (k, d) = B d d =0 | c(k, )|. Proof. Fix a function g ∈ P n >d ∩ P n ≤k . Then, Rad ≤d (f ) m 2 ≤ f -g M 2 = f -g 2 ≤ f -g ∞ , ( 29
)
where

M = n 0 + • • • + n k . Moreover, we have Rad ≤d (f ) m 2d d+1 (25) ≤ B d Rad ≤d (f ) ∞ = B d Rad ≤d (f -g) ∞ (27) ≤ B d d =0 | c(k, )| f -g ∞ .
Equipped with Lemma 7, we can complete the proof of Theorem 3.

Proof of Theorem 3. Consider the normed spaces

X = (P n ≤k , • ∞ ) and Y = (R m , • Y ) with ∀ y ∈ R m , y Y = max y m 2 , σ (k, d) -1 y m 2d d+1 (30)
and

m = n 0 + • • • + n d .
Moreover, let Z = P n >d ∩ P n ≤k ⊂ X, viewed as a normed subspace of X. Lemma 7 asserts that the linear operator A : X/Z → Y given by

∀ f ∈ X, A(f + Z) = f (S) |S|≤d (31)
has norm A ≤ 1. Therefore, the same holds for its adjoint A * : Y * → (X/Z) * . By the usual duality between sums and intersections of normed spaces [1, Theorem 2.7.1], we see that the space Y * is isometric to

∀ w ∈ R m , w Y * = K w, σ (k, d); m 2 , m 2d d-1 . ( 32
)
Moreover, as X/Z is a quotient of X, its dual is the subspace of X * = L 1 /P n >k which is identified with the annihilator of Z inside X * . In other words, it is the set (X/Z) * = f + P n >k : E[f g] = 0 for every g ∈ Z = f + P n >k : f ∈ P n ≤d = span(P n ≤d ∪ P n >k ) P n >k (33) equipped with the L 1 quotient norm. Finally, for a sequence a a a = (a S ) |S|≤d ∈ Y * and an equivalence class f + Z ∈ X/Z, we have

a a a, A(f + Z) = S⊆{1,...,n}: |S|≤d a S f (S) = S⊆{1,...,n}: |S|≤d a S w S + P n >k , f + Z = A * (a a a), f + Z , (34)
where the first brackets •, • denote the duality in Y and the following brackets denote the duality in X/Z. Therefore, we conclude that

∀ a a a ∈ Y * , A * (a a a) = S⊆{1,...,n}: |S|≤d a S w S + P n >k ( 35
)
and thus, the condition A * ≤ 1 means that for any f :

{-1, 1} n → R of degree at most d, inf g∈P n >k f -g 1 = A * f (X/Z) * ≤ f Y * = K f , σ (k, d); m 2 , m 2d d-1 . ( 36
)
Finally, since

σ (k, d) ≤ B d d =0 | c(k, )| ≤ B d d =0 k ! ≤ eB d k d , ( 37
k . ( 39
)
This can be proven using similar ideas as in the proof of Theorem 3.

A slight variant of the arguments above also yields Theorem 4 for symmetric functions.

Proof of Theorem 4. Let f be a symmetric function of the form f = d =0 α f where f is the -th elementary symmetric polynomial. Then, the Hahn-Banach theorem gives inf

g∈P n >k f -g 1 = sup 0 h∈P n ≤k E[f h] h ∞ . ( 40
)
Observe now that we can write

E[f h] = d =0 α E[f h] = d =0 α S⊆{1,...,n}: |S|= h(S) = d =0 α Rad h(1, . . . , 1). (41)
Thus, by Figiel's bound (27),

E[f h] ≤ d =0 |α | Rad h ∞ ≤ d =0 |α || c(k, )| h ∞ (42)
and the desired inequality follows from (40).

Equipped with Theorem 4, we present the proof of Corollary 5.

Proof of Corollary 5. The upper bound in [START_REF] Levhari | Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Rényi entropy[END_REF] follows immediately from Theorem 4. For the lower bound, consider the auxiliary symmetric function H k,n : {-1, 1} n → R given by

∀ x ∈ {-1, 1} n , H k,n (x) def = T k x 1 + • • • + x n n = k =0 β ,k,n f (x), ( 43
)
where f is the -th elementary symmetric polynomial, and notice that H k,n has degree at most k. As T k (x) is odd or even when k is odd or even respectively, it follows that β ,k,n = 0 if kis odd. We distinguish two cases depending on the parity of kd.

• Suppose that kd is even and consider the function

ϕ d,k,n : {-1, 1} n → R given by ∀ x ∈ {-1, 1} n , ϕ d,k,n (x) def = 0≤ ≤d: 2|d- sign(β ,k,n )f (x) (44)
that is also symmetric and of degree at most d. Then, on one hand we know that inf

g∈P n >k ϕ d,k,n -g 1 (15) ≤ 0≤ ≤d: 2|d- | c(k, )| = 0≤ ≤d: 2|d- |c(k, )|. ( 45
)
On the other hand, we have the following lower estimate, inf

g∈P n >k ϕ d,k,n -g 1 (40) = sup 0 h∈P n ≤k E[ϕ d,k,n h] h ∞ ≥ |E[ϕ d,k,n H k,n ]| H k,n ∞ . ( 46
thus concluding the proof of the lower bound in [START_REF] Levhari | Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Rényi entropy[END_REF].

• If kd is odd, we use the identity

f d = sign(β d,k-1,n )(ϕ d,k-1,n -ϕ d-2,k-1,n ). ( 52
)
The rest of the argument is identical.

Remark 11. In this paper, we studied dual versions of moment comparison estimates on the hypercube (3) and investigated the endpoint case of their duals (4) for polynomials. By formal reasoning similar to the proof of Proposition 1, one can derive dual versions of various other polynomial inequalities, including Bernstein-Markov inequalities and their reverses and bounds for the action of the heat semigroup. We refer to [START_REF] Eskenazis | Polynomial inequalities on the Hamming cube[END_REF] for a systematic treatment of such estimates.

≥

  bound from below the L 1 -distance of f d from the tail space, we writef d = sign(β d,k,n )(ϕ d,k,nϕ d-2,k,n ) 0≤ ≤d: 2|d-|c(k, )|ε n (k, d) -0≤ ≤d-2: 2|d-2-|c(k, )| = |c(k, d)|ε n (k, d),

  )we deduce the conclusion of the theorem with C d = eB d . Remark 8. To the best of our knowledge, there are no nonconstant lower bounds on the size of the discrete Bohnenblust-Hille constant B d , so it is even conceivable that the constant C d in[START_REF] Hitczenko | On the Rademacher series[END_REF] can be chosen to be independent of d. Remark 9. A duality argument similar to that employed for Theorem 3 shows that for every d ∈ N, the constant B d in inequality (25) is also the least constant for which every function f : {-1, 1} n → R, where n ≥ d, satisfies Remark 10. It was pointed out to us by Oleszkiewicz that the main result (8) of[START_REF] Oleszkiewicz | On mimicking Rademacher sums in tail spaces[END_REF] also admits a dual formulation. Namely, for every a a a = (a 1 , . . . , a n ) ∈ R n , we have inf f a a ag ∞ : g ∈ P n

							d-1
	inf >d g∈P n	f -g 1 ≤ B d	S⊆{1,...,n}:	| f (S)|	2d d-1	2d	.	(38)
			|S|≤d
		{0}∪{2,...,k}	max a a a n 2 ,	a a a n 1

)

  By definition, H k,n∞ ≤ sup x∈[-1,1] |T k (x)| ≤ 1 and H k,n (1, . . . , 1) = T k (1) = 1. Therefore,To further estimate this sum, we use [14, Lemma 27] which implies that there exists a positive constantε n (k, d) > 0 with ε n (k, d) = O k,d (1/n) as n → ∞, such that 0≤ ≤d: 2|d-Rad H k,n (1, . . . , 1) ≥

	inf g∈P n >k	ϕ d,k,n -g 1 ≥ |E[ϕ d,k,n H k,n ]| =	0≤ ≤d: 2|d-	sign(β ,k,n )Rad H k,n (1, . . . , 1)	(47)
				=		Rad H k,n (1, . . . , 1) .
					0≤ ≤d: 2|d-
						|c(k, )| -ε n (k, d).	(48)
					0≤ ≤d: 2|d-
	Hence, combining the above we conclude that
		inf g∈P n >k	ϕ d,k,n -g 1 ≥	0≤ ≤d: 2|d-

|c(k, )|ε n (k, d).

(A. E.) CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université, France and Trinity College, University of Cambridge, UK.

Email address: alexandros.eskenazis@imj-prg.fr, ae466@cam.ac.uk (H. Z.) Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA.

Email address: haonanzhangmath@gmail.com, haonanz@mailbox.sc.edu