RU2288785C2 - Method for converting ultrasound wave and apparatus for performing the same - Google Patents

Abstract

FIELD: acoustics-electronics, ultrasonic technique.

SUBSTANCE: in controlled sound conduit 1 having length Ls and made of non-polarized ferroelectric ceramics with abnormally high dielectric permeability, ultrasonic wave with frequency ω₀ is passed. Said wave is formed by converter 4 and it interacts with electric field rotating around propagation direction of wave and generated as result of action of AC voltage supplied by means of generator 6 and phase splitter 7 upon pairs of electrodes 2, 3 arranged in sound conduit 1. Rotation frequency of electric field coincides with frequency of ultrasonic wave Ω = ω₀. According to invention it is realized resonance interaction of circular component of ultrasonic wave (whose frequency and rotation sign coincide with frequency and rotation sign of electric field having intensity more than threshold value) with such electric field. In the result ultrasound absorption is prevented and in outlet of sound conduit elliptically polarized ultrasonic wave is created with intensity more than that of said wave.

EFFECT: possibility for increasing intensity of ultrasonic wave passing through sound conduit.

2 cl, 3 dwg

Description

The invention relates to acoustoelectronics and ultrasound equipment and can be used to create controlled devices in acoustoelectronics, namely to amplify an ultrasonic wave.

A known method and device for amplifying ultrasound in semiconductors by drift of charge carriers, made in the form of a shear wave transducer located on the input end face of a controlled sound duct connected to an active crystal having good piezoelectric properties and photoconductivity, on the end faces of which are applied metal electrodes, and subjected to the action of the illuminator [1]. An ultrasonic wave passing through a crystal is amplified if the drift velocity of the charge carriers in the direction of wave propagation exceeds its phase velocity.

The dynamic range of the amplifier is limited by noise and non-linear effects. To create the optimum conductivity of the crystal, it is necessary to select the intensity and spectral composition of the light from the illuminator. To prevent the destruction of the crystal due to overheating by direct current, the pulse mode of the amplifier is used. However, the pulse mode of operation, the need for the selection of lighting limit the possible applications of amplifiers.

The closest invention in technical essence to the claimed is a method of converting an ultrasonic wave, which consists in the action of a rotating electric field transverse to the direction of propagation of the ultrasonic wave on unpolarized ferroceramics with an anomalously high dielectric constant when an acoustic wave propagates in it [2].

The frequency of the ultrasonic wave and the frequency of rotation of the electric field should not be equal.

The technical essence of the known method lies in the fact that if an ultrasonic wave propagates in unpolarized ferroceramics with an anomalously high dielectric constant and is exposed to an electric field that rotates transversely to the direction of propagation of the ultrasonic wave, then depending on the ratio of the natural frequency of the ultrasonic wave, parameters of the ferroceramics, speed and the electric field, the wave changes its parameters. Taking into account these dependences, it is possible to control the parameters of the ultrasonic wave.

In the known method, subject to the condition that the frequency of the linearly polarized ultrasonic wave and the rotation frequency of the electric field should not be equal, the transformation of the ultrasonic wave consists in changing the angle of rotation of the plane of its polarization.

However, the known method for converting an ultrasonic wave does not fully take into account the relationship of the parameters and their influence on other characteristics of the ultrasonic wave.

Closest to the claimed is a device for converting an ultrasonic wave, containing a controlled sound duct made of non-polarized ferroceramics with an anomalously high dielectric constant, on which electrically isolated pairs of electrodes and a shear wave transducer, a phase shifter, an ultrasonic frequency alternating voltage generator are placed opposite, output which is connected to the input of the said phase shifter and one pair of electrodes, and you the phase shifter stroke is connected to another pair of electrodes [2].

Moreover, the known device necessarily contains two pairs of flat electrodes and a phase shifter with phase shift [2].

The known device provides rotation of the plane of polarization of the ultrasonic wave. The magnitude of the rotation of the plane of polarization is proportional to the length of the sound duct, which is chosen arbitrarily.

The known device does not take into account the influence of the length of the sound duct on ensuring the best conditions for interference of the eigenmodes of the acoustic field.

The technical problem solved by the claimed invention is to expand the technological capabilities of the method and device and to provide amplification of the ultrasonic wave.

The technical result achieved in this case is expressed in the possibility of increasing the intensity of the ultrasonic wave transmitted through the sound duct relative to the intensity of the incident wave.

According to the invention, a method and apparatus for converting an ultrasonic wave, namely for its amplification, are provided.

The achievement of the specified technical result is ensured by the fact that in the method of converting an ultrasonic wave, which consists in the action of a rotating electric field transverse to the direction of propagation of the ultrasonic wave, on non-polarized ferroceramics with an anomalously high dielectric constant when an acoustic wave propagates in it, the frequency of rotation of the electric field is set in accordance with equality

and the electric field must exceed a threshold value

the threshold value of the electric field is set in accordance with the expression

Where

E _pore is the threshold value of the intensity of a rotating electric field;

 Ω is the frequency of rotation of the electric field;

ω ₀ is the frequency of the ultrasonic wave;

η ₄₄ is a component of the viscosity tensor;

α ₁₄₄ , α ₁₅₅ , are the tensor components that take into account the electrostrictive effect of the electric field on the elastic constant of the medium;

β ₁₄₄ , β ₁₅₅ are tensor components that take into account the electrostrictive effect of the electric field on the viscosity of the medium.

The inventive method is implemented using a device for converting an ultrasonic wave.

The achievement of the specified technical result is achieved by the fact that in the device for converting an ultrasonic wave containing a controlled sound duct made of non-polarized ferroceramics with an anomalously high dielectric constant, on which are placed electrically isolated pairs of electrodes and a shear wave transducer, phase shifter, alternating electric generator ultrasonic frequency voltage, the output of which is connected to the input of the phase shifter and one pair electrodes, and the phase shifter output is connected to other pairs of electrodes, the controlled sound duct has a length corresponding to the calculation formula

Where

L _s - the length of the controlled sound pipe;

ζ ₁ and ζ ₂ are arguments of complex ellipticities;

the parameter s takes values from the set of integers;

k ₁ and k ₂ are the wave numbers of the eigenmodes of the acoustic field;

ω ₀ is the frequency of the ultrasonic wave;

 Ω is the frequency of rotation of the electric field.

The basis of the proposed method is the phenomenon of suppressing the absorption of an ultrasonic wave. The essence of the phenomenon is that as a result of the interference of the incident acoustic wave and the reversed acoustic wave generated in ferroceramics by a rotating electric field, a standing ultrasonic wave is formed, with respect to which the viscous properties of the medium are very weakly manifested. As a result of this suppression of ultrasound absorption, there is a gigantic amplification of transmitted and reversed acoustic waves as a result of the highly efficient transfer of energy from a rotating electric field to ultrasound.

Consider the propagation of an ultrasonic wave along the Z axis in an electric field with amplitude E and components

rotating with a frequency Q around the Z axis. Such a field can be created by applying an electric potential with a phase shift to a system of parallel metal electrodes located on the surface of ferroceramics. In this case, the phase shift is determined by the number of electrodes and for the case of two pairs of electrodes is π / 2. The acoustic properties of ferroceramics can be described using the generalized Hooke's law [3], taking into account the viscosity of the medium

Here σ _ik , γ _lm and c _iklm are tensors of stresses, strains and elastic constants; η _iklm is the viscosity tensor. The action of a rotating electric field (5) can lead to a significant change in the acoustic properties of the crystal, as a result of which the propagation of an elastic wave with a displacement vector u will be described by the equation of motion

Here ρ is the density of the medium, tensors of elastic constants Λ (t) and viscosity B (t) take into account the unsteady effect of an external electric field (5) and have the following form:

In the expressions (8), the notation [4, 5] is used:

α and β are tensors that take into account the electrostrictive effect of the field E on the elastic constants and viscosity of the medium:

is the rotation matrix around the z axis by the angle φ = Ωt [6], the tilde (~) means transposition, and c ^x is the antisymmetric tensor dual to the vector c.

Using the technique proposed in [7, 8], solutions of the equation of motion (5) will be sought in the form of interconnected plane monochromatic waves

, где а и b - орты лабораторной декартовой системы координат.having the same wave numbers k (ω), different frequencies ω ± Ω and opposite circular polarizations specified by the vectors

, where a and b are the unit vectors of the laboratory Cartesian coordinate system.

Elastic waves in a crystal with a rotating structure (10) substantially depend on the value of ω determined by the frequency and polarization of the incident acoustic wave. We consider the case when a circularly polarized ultrasonic wave is excited at the crystal boundary at z = 0

the elastic displacement vector of which has the same direction of rotation over time as the external electric field. Due to the continuity of the elastic displacement vector at the crystal boundary, the incident wave excites in the crystal, first of all, a wave with the same polarization and frequency described by the second term in (10). Hence,

ω ₀ = ω + Ω,

and the intrinsic acoustic mode of the crystal (10) can be written as

Consider the case

ω ₀ = Ω,

when the incident wave coincides with the external electric field not only in the direction of rotation, but also in frequency. Then

those. the eigenmode of a crystal consists of two circularly polarized waves interconnected, propagating in opposite directions and having the same frequencies. Since the polarization is determined depending on the direction of wave propagation, the circular components of the crystal’s own wave have virtually the same polarizations.

Substituting expression (10) into equation (7) and taking into account the explicit form of tensors (8), we obtain the following system of linear homogeneous equations:

Equating the determinant of system (11) to zero, we find the wave numbers

k _3.4 (ω) = - k _2.1 (ω)

and amplitude ratios of circular components

eigenmodes (10) of the acoustic field.

The following notation is used in expressions (15), (16):

The quantities ξ _k characterize the coupling coefficient between circular waves that form the eigenmodes of the acoustic field. For brevity, we will call ξ _{k the} ellipticities of the eigen acoustic modes for brevity. The reason for using this term is the fact that the quantities ξ _{k are} really equal to the ellipticities of the natural waves of the acoustic field, if we consider these waves in a rotating coordinate system.

We turn to the rotating coordinate system accompanying the rotation of the external electric field (5) and substitute the expression (10) transformed for this system into equation (7), taking into account that in the rotating coordinate system, unlike the laboratory coordinate system, tensors of elastic constants and viscosity of the medium ( 8) do not depend on time, which makes it possible to search for solutions of the equation of motion in the form of plane monochromatic waves

with frequency ω 'and wavenumber k (ω'). In the laboratory coordinate system, the wave displacement vector (17) has a form similar to (10);

Let us consider the case when a circularly polarized acoustic wave (11) is excited at the crystal boundary at z = 0, the elastic displacement vector of which has the same direction of rotation in time as the external electric field. The continuity condition for the vector u at the boundary implies ω ₀ = ω '+ Ω, i.e. ω '= ω ₀ - Ω. If the ultrasound frequency ω ₀ coincides with the frequency of the electric field Ω (the case of resonance interaction), then the relation ω '= 0 is satisfied, and the eigenmodes (17) of the acoustic field have the form of standing waves.

Equating the determinant of the resulting system to zero, we find the expressions for the wave numbers of the eigenmodes of the acoustic field:

A rotating electric field can lead to a significant change in the acoustic properties of a crystal with an abnormally high dielectric constant. As follows from expression (18), with an increase in the anisotropy of the elastic constant tensor δ and the anisotropy of the viscosity tensor χ induced by the electric field, the imaginary parts of the wave numbers monotonically decrease to zero. Starting from the threshold values of the parameters δ and χ determined by the condition

the wave numbers (18) become real, and the eigenmodes of the acoustic field cease to attenuate in the crystal. In accordance with condition (19), the threshold value of the intensity of a rotating electric field is expressed in terms of the crystal parameters as follows:

If the ultrasound frequency ω differs from the frequency of the electric field Ω, then the wave numbers (18) are complex regardless of the electric field strength. As numerical estimates show, with the following values of the parameters characteristic of ceramics based on barium titanate with an anomalously high dielectric constant [9–11]:

=10¹² дин/см², δ=-4,465·10¹⁰ дин/см²,

=1000 ед. СГС,

= 10 ¹² dyne / cm ² , δ = -4,465 · 10 ¹⁰ dyne / cm ² ,

= 1000 units GHS

χ = 50 units GHS, ρ = 5.7 g / cm ³ , Ω = 10 ⁷ rad / s,

the fulfillment of threshold condition (17) is achieved at an electric field strength of the order of several kV / cm.

The solution of the boundary problem allows us to determine the value of the normalized intensity of the transmitted wave

Where

- amplitude of the transmitted wave,

u ₀ is the amplitude of the incident wave (11)

The expression for the normalized intensity of the transmitted ultrasonic wave (21) allows us to evaluate the possibility of its amplification.

According to the proposed method, an ultrasonic wave in the form of a beam is introduced into non-polarized ferroceramics with an abnormally high dielectric constant. In this case, ferroceramics is subjected to a rotating electric field transverse to the direction of propagation of the ultrasonic wave, in other words, the field strength vector is directed normally to the wave vector of the ultrasonic wave. The intensity E _{0 of the} rotating electric field is set in accordance with the calculation formula (2).

The rotation frequency Ω of the rotating electric field is set equal to the frequency of the ultrasonic wave ω ₀ in accordance with the calculation formula (1).

In this case, the wave numbers (18) become real, the eigenmodes of the acoustic field cease to attenuate in the crystal, and the intensity of the ultrasonic wave transmitted through the sound duct increases relative to the intensity of the incident wave as a result of highly efficient transfer of the energy of the rotating electric field to ultrasound.

Examples of the method are shown in table 1.

Table 1 Frequency of a rotating electric field Ω Wave frequency ω ₀ Electric field strength E The value of the normalized intensity of the transmitted wave T Ω = 10 ⁷ rad / s ω ₀ = 10 ⁷ rad / s E ₀ = 4 kV / cm> E _then 2 Ω = 10 ⁷ rad / s ω ₀ = 10 ⁷ rad / s E ₀ = 1 kV / cm <E _then 0.62 Ω = 10 ⁷ rad / s ω ₀ = 1.03 · 10 ⁷ rad / s E ₀ = 4 kV / cm> E _then 0.9 Ω = 10 ⁷ rad / s ω ₀ = 1.04 · 10 ⁷ rad / s E ₀ = 4 kV / cm> E _then 0.8

As follows from table 1, the amplification of the ultrasonic wave is possible if the electric field strength E ₀ exceeds the threshold value and the frequency of the ultrasonic wave ω ₀ coincides with the rotation frequency of the electric field Ω.

Thus, the claimed method potentially has a wider dynamic range and greater control over the parameters of the ultrasonic wave.

A device for implementing the method provides the possibility of performing the operations of the method and selecting the necessary parameters. Performing a controlled sound duct with a length L _s corresponding to the calculation formula (4) provides the best interference conditions for the eigenmodes of the acoustic field and maximizes amplification of the transmitted wave relative to the incident one.

Figure 1 shows the acoustoelectric assembly of the device (side view), figure 2 - acoustoelectric assembly (front view), figure 3 is an electrical diagram of the device.

A device for amplifying an ultrasonic wave contains (see FIGS. 1 and 2) a guided sound duct 1 having a length L _s and made of non-polarized ferroceramics with an anomalously high dielectric constant, which can be made in the form of a rectangular parallelepiped, on opposite sides of which are parallel the axis of the sound duct, pairs of electrodes 2, 3 are installed opposite each other, forming respectively the first and second pairs. The number of pairs of electrodes determines the phase shift and for the case of two pairs of electrodes is π / 2. On the end face perpendicular to the axis of the sound duct, a shear wave transducer 4 is placed. The controlled sound duct 1 with the pairs of electrodes 2, 3 and the transducer 4 placed on it makes up the acoustoelectric assembly 5. The first pair of electrodes 2 is connected through the first input of the acoustoelectric assembly 5 to the output of the ultrasonic frequency alternating voltage generator 6, and the second pair of electrodes 3 is connected through the second input acoustoelectric node 5 to the output of the phase shifter 7.

The connection of the pairs of electrodes 2, 3 with the ultrasonic frequency alternating voltage generator 6 and the phase shifter 7 is carried out by means of coaxial cables, the central conductor of each cable being connected to one pair electrode, and the screening to the second. The indicated mutual arrangement of the pairs of electrodes 2, 3 on the sound duct 1 and their connection to the phase shifter 7 and the generator 6 of the alternating electric voltage of the ultrasonic frequency provide the device with constant amplitudes and a constant phase shift of the voltage applied to the pairs of electrodes 2, 3. The output of the alternator 6 ultrasonic frequency voltage is connected to the input of the phase shifter 7 by the first input of the acoustoelectric unit 5, and the output of the phase shifter 7 is connected to the second input of the acoustoel electric node 5 (see figure 3).

To prepare the device for operation, a sound duct 1 is preliminarily made, and depending on the frequency of the ultrasonic wave ω _0, its length L _s must be equal to the value calculated by the formula (4).

The device operates as follows.

The transducer 4 excites shear linearly polarized with a certain orientation ultrasonic waves with a frequency of ω ₀ and directs them in the form of a beam into a controlled sound duct 1 parallel to its axis. Using an alternating voltage generator 6 and a phase shifter 7, alternating voltage is applied to the pairs of electrodes 2, 3, and the voltage supplied to the first pair of electrodes 2 differs in phase by π / 2 from the voltage supplied to the second pair of electrodes 3 from the phase shifter 7. In this case, as a result of a superposition of homogeneous mutually perpendicular electric fields that change in time according to a harmonic law with a relative phase shift π / 2, inside the sound duct 1 there is a rotating around the axis of the sound duct 4 is an electric field, the intensity vector of which rotates over time around the axis of the sound duct 1 with a frequency Ω. The frequency Ω is pre-set on the alternating voltage generator 6 in the same way as the amplitude of the alternating voltage. Due to the effect of electrostriction, a rotating electric field influences the acoustic properties of unpolarized ferroceramics.

A linearly polarized wave can be represented as the sum of two circularly polarized waves having the same displacement amplitude and opposite directions of rotation. The circular component of the ultrasonic wave, which coincides with the external rotating electric field in the direction of rotation and frequency, is resonantly affected by the rotating electric field. The resonant interaction of ultrasound with a rotating electric field for a wave with opposite circular polarization does not occur at any frequency. As a result, the ultrasonic wave at the exit from the zone of action of the rotating electric field, that is, at the output face of the controlled sound duct 1, is elliptically polarized. As a result of the resonant interaction of the circular component of the ultrasonic wave, the frequency ω ₀ and the direction of rotation of which coincide with the frequency Ω and the direction of rotation of the electric field E ₀ , the intensity of which exceeds the threshold value E _pore , the effect of suppressing the absorption of ultrasound arises, due to which it forms elliptically at the output from the sound duct polarized ultrasonic wave, the intensity of which exceeds the intensity of the incident wave.

The influence of the value of the length L _{s of the} sound duct on the intensity of the transmitted wave T is presented in table 2.

table 2 The frequency of the wave ω ₀ , the frequency of the rotating electric field Ω (Ω = ω ₀ ) Sound pipe length L _s The value of the normalized intensity of the transmitted wave T 10 ⁷ rad / s 3.5 cm 0.5165610 ⁵ 10 ⁷ rad / s 9.52 cm 0.8117710 ⁶ 10 ⁷ rad / s 15.47 cm 0.17272 · 10 ⁷

Such a giant amplification of ultrasound is a consequence of the suppression of the absorption of acoustic waves in a rotating electric field and the result of interference of the eigenmodes of the acoustic field.

The gain can also be controlled by deviating the frequency of the rotating electric field Ω from the frequency of the wave ω ₀ (Ω ≠ ω ₀ ).

Examples are presented in table 3.

Table 3 Wave frequency ω ₀ · 10 ⁷ rad / s The frequency of the rotating electric field Ω, · 10 ⁷ rad / s Sound duct length L _s , cm The value of the normalized intensity of the transmitted wave T 1,000 1,000 3,5 0.5165610 ⁵ 1.001 1,000 3,5 0.3498310 ³ 1.002 1,000 3,5 0.1125110 ³ 1.003 1,000 3,5 0.53171 · 10 ² 1.004 1,000 3,5 0,3071210 ² 1.005 1,000 3,5 0.1999510 ² 1.006 1,000 3,5 0.1408810 ² 1.007 1,000 3,5 0.1049910 ² 1.008 1,000 3,5 0.8159710 ¹ 1.009 1,000 3,5 0.6553010 ¹ 1.010 1,000 3,5 0.5403210 ¹ 1.011 1,000 3,5 0.4553110 ¹ 1.012 1,000 3,5 0.3907810 ¹ 1.013 1,000 3,5 0.3407010 ¹ 1.014 1,000 3,5 0.3011210 ¹ 1.015 1,000 3,5 0.2693610 ¹ 1.016 1,000 3,5 0.2435410 ¹ 1.017 1,000 3,5 0.2223210 ¹ 1.018 1,000 3,5 0.2047210 ¹ 1.019 1,000 3,5 0.1900110 ¹

Thus, the inventive method and device for converting an ultrasonic wave have three degrees of control: the gain control conditions can be changed due to a joint change in the frequency of the incident wave ω ₀ and the frequency of the rotating electric field Ω (Ω = ω ₀ ), due to the deviation of the frequency of the rotating electric field Ω from the wave frequency ω ₀ (Ω ≠ ω ₀ ), due to a change in the length of the sound duct L.

Information sources

1. Ultrasound / Ch. ed. I.P. Golyamin. Moscow: Soviet Encyclopedia, 1979. S.355-360.

2. RF patent №2123895, publ. 1998 (prototype).

3. Gurevich V.L. Kinetics of phonon systems. Moscow, 1980.400 p.

4. Bely V.N., Sevruk B.B. Control of the polarization of elastic waves by an electric field creating a spiral anisotropy // Acoust. journal 1983. V.29, No. 2. S.157-161.

5. Bely V.N., Sevruk B.B. Parametric interaction of circularly polarized electromagnetic and acoustic waves in crystals with electrostrictive nonlinearity // Sat .: Covariant methods in theoretical physics. Minsk, 1986. P.132-141.

6. Fedorov F.I. The theory of gyrotropy. Minsk: Science and Technology, 1976. 456 p.

7. Semchenko I.V., Serdyukov A.N. The propagation of light in a medium with a rotating cholesteric anisotropy structure // Journal of Applied Spectroscopy, 1984. T.41, No. 5. S.827-830.

8. Belyi V.N., Sevruk B.B. Parametric electro-acoustic effects in crystals with an induced external electric field, rotating acoustic anisotropy // Journal of Technical Physics, 1987. V. 57, No. 2. S.336-340.

9. Pekar S.I., Demidenko A.A., Zdebsky A.P. et al. Study of electrostriction constants of the first and second order in substances with high dielectric constant // Dokl. USSR Academy of Sciences. Ser. Fiz. 1976.V.230, No. 5. S.1089-1091.

10. Zhabitenko N.K., Kucherov I.Ya. Investigation of the influence of an electric field on the propagation velocity of elastic waves in isotropic solids // Ukr. physical Zh., 1978. V.23, No. 2. S.263-266.

11. The influence of a constant electric field on the propagation of surface acoustic waves in the piezoceramics of the PZT system / Ed. A.N. Rybyanets, A.V. Turik, N.V. Dorokhova, E.S. Miroshnichenko // Journal of Technical Physics. 1986.V.56, No. 12. S.2371-2375.

Claims (2)

1. The method of converting an ultrasonic wave, which consists in the action of a rotating electric field transverse to the direction of propagation of the ultrasonic wave, on non-polarized ferroceramics with an anomalously high dielectric constant when an acoustic wave propagates in it, characterized in that the rotation frequency and intensity of the rotating electric field are set in accordance with equality

and the electric field must exceed a threshold value E ₀ > E _then wherein

Where  Ω is the frequency of rotation of the electric field; ω ₀ is the frequency of the ultrasonic wave; E _pore is the threshold value of the intensity of a rotating electric field; η ₄₄ is a component of the viscosity tensor; α ₁₄₄ , α ₁₅₅ , are the tensor components that take into account the electrostrictive effect of the field E on the elastic constant of the medium; β ₁₄₄ , β ₁₅₅ are tensor components that take into account the electrostrictive effect of field E on the viscosity of the medium. 2. Device for converting an ultrasonic wave, containing a controlled sound guide made of non-polarized ferroceramics with an abnormally high dielectric constant, on which are placed electrically isolated pairs of electrodes and a shear wave transducer, phase shifter, ultrasonic frequency alternating voltage generator, the output of which is connected with the input of the phase shifter and one pair of electrodes, and the output of the phase shifter is connected to other pairs of elec trodes, characterized in that the controlled sound duct has a length corresponding to the calculation formula L _s = (ζ ₁ -ζ ₂ + 2πs) / [k ₁ (ω ₀ - Ω) -k ₂ (ω ₀ - Ω)], where L _s is the length of the controlled sound pipe; ζ ₁ and ζ ₂ are arguments of complex ellipticities; the parameter s takes values from the set of integers; k ₁ and k ₂ are the wave numbers of the eigenmodes of the acoustic field; ω ₀ is the frequency of the ultrasonic wave;  Ω is the frequency of rotation of the electric field.

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