CN114078353B - Fano resonance experimental instrument realization method based on coupling resonance circuit - Google Patents
Fano resonance experimental instrument realization method based on coupling resonance circuit Download PDFInfo
- Publication number
- CN114078353B CN114078353B CN202111412414.1A CN202111412414A CN114078353B CN 114078353 B CN114078353 B CN 114078353B CN 202111412414 A CN202111412414 A CN 202111412414A CN 114078353 B CN114078353 B CN 114078353B
- Authority
- CN
- China
- Prior art keywords
- oscillator
- resonance
- vibrator
- frequency
- fano
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- G—PHYSICS
- G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
- G09B—EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS
- G09B23/00—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes
- G09B23/06—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes for physics
- G09B23/18—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes for physics for electricity or magnetism
- G09B23/181—Models for scientific, medical, or mathematical purposes, e.g. full-sized devices for demonstration purposes for physics for electricity or magnetism for electric and magnetic fields; for voltages; for currents
Landscapes
- Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Physics & Mathematics (AREA)
- Business, Economics & Management (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Algebra (AREA)
- Pure & Applied Mathematics (AREA)
- Mathematical Physics (AREA)
- Computational Mathematics (AREA)
- Educational Administration (AREA)
- Educational Technology (AREA)
- Theoretical Computer Science (AREA)
- Oscillators With Electromechanical Resonators (AREA)
- Measurement Of Mechanical Vibrations Or Ultrasonic Waves (AREA)
Abstract
The invention discloses a method for realizing a Fano resonance experimental instrument based on a coupling resonance circuit. The invention can well show the Fano resonance phenomenon experimentally by constructing the coupling resonance circuit, and can also show the electromagnetic induction-like transparent EIT phenomenon as a special condition, and the simple classical circuit system can give out complete theoretical description on the aspect of common physics, thereby giving explanation to the experimental phenomenon by means of theoretical formula and calculation result and understanding the back physical mechanism; the experimental device and the measuring method required by the experiment are very basic, the displayed phenomenon has rich physical connotation, and the Fano resonance phenomenon and the similar EIT phenomenon can be displayed, and are linked with the forward research to stimulate the learning interest of students; meanwhile, the method is very helpful for students to understand the general physical law of the resonance phenomenon, particularly to deeply understand the physical significance of the phase in the resonance, and is suitable for being developed as high-order content in college physical experiments.
Description
Technical Field
The invention relates to a common physical experiment instrument, in particular to a realization method of a Fano resonance experiment instrument based on a coupling resonance circuit.
Background
Resonance is one of the basic phenomena commonly existing in nature, and is a typical resonance phenomenon from the resonance of a spring vibrator and the resonance of a string in a mechanical system, the resonance of an LCR circuit in an electromagnetic system (L represents inductance, C represents capacitance, and R represents resistance), various types of optical resonant cavities in an optical system, even atomic energy level transition in a quantum system, and the like. Generally, the simplest and most common resonance is a single resonance phenomenon, and an isolated damped oscillation system with a single resonance frequency is driven by an external environment with different frequencies, and the system shows a typical single resonance phenomenon, wherein the amplitude of the system shows a response spectrum of a Lorentzian line type along with the change of the external driving frequency, and the line type shows an approximately symmetrical distribution near the resonance frequency of the system, as shown in FIG. 1 (a). Another line type different from this is the Fano (Fano) resonant line type, whose response spectrum shows a clearly asymmetric distribution around the system resonance frequency, as shown in fig. 1 (b). The Fano resonance line was discovered by Fano in 1935, and was first observed in the absorption line of inert gas, and the microscopic mechanism thereof is the interference effect between different transition paths during the transition of atoms from the initial state to the final state: when one transition path exhibits a narrow spectral resonance (corresponding to a discrete state to discrete state transition) and the other transition path exhibits a broad spectral resonance (corresponding to a discrete state to continuous state transition), interference between the two transition paths causes the total absorption line of the system to exhibit an asymmetric fano-resonant line shape. Later, people observed the Fano resonance successively in various quantum systems such as semiconductor quantum wells, quantum dots and the like and various classical systems such as photonic crystals, whispering gallery micro-cavities and the like, and discovered a more special Electromagnetic Induction Transparency (EIT) phenomenon, which can be regarded as a special case of the Fano resonance. Due to the asymmetric characteristic of the Fano resonance line type, the response spectrum of the Fano resonance line type is steeper and sharper than that of the common Lorentz line type, and the Fano resonance line type has important application prospects in the fields of nonlinear optics, optical switches, sensing and the like, so that the Fano resonance line type also becomes one of the research hotspots in a plurality of leading-edge fields such as nanophotonics and the like in recent years.
Disclosure of Invention
In order to show the Fano resonance phenomenon at the level of a common physical experiment and establish the connection between a basic experiment and a forward research, the invention provides a realization method of a Fano resonance experiment instrument based on a coupling resonance circuit, the Fano resonance phenomenon is shown experimentally, the similar EIT phenomenon is shown as a special case, and the experiment result can be explained by the calculation result of a theoretical formula.
The invention discloses a realizing method of a Fano resonance experimental instrument based on a coupling resonance circuit, which comprises the following steps:
1) setting up an experimental instrument:
i. the first inductor, the first capacitor, the first resistor and a primary coil of the mutual inductor are sequentially connected to form a first oscillator, and then connected with an alternating current signal source to form a first loop;
a second inductor, a second capacitor, a second resistor and a secondary coil of the mutual inductor are sequentially connected to form a second oscillator and form a second loop;
the quality factor of the first oscillator is controlled by changing the resistance value of the first resistor of the first oscillator, and the spectrum width of the first oscillator is further changed, wherein the spectrum width is the full width at half maximum of a resonance peak of a response spectrum, so that the first oscillator is a oscillator with Lorentz-type wide-spectrum resonance; the quality factor of the second oscillator is controlled by changing the resistance value of a second resistor of the second oscillator, and the spectrum width of the second oscillator is further changed, so that the second oscillator is a oscillator with Lorentz line type narrow spectrum resonance;
the first oscillator and the second oscillator are coupled through a mutual inductor to form a coupled resonant circuit;
2) the alternating current signal source takes a sine wave as an excitation signal to excite the first vibrator to vibrate;
3) the first vibrator is coupled with the second vibrator through the mutual inductor so as to excite the second vibrator to vibrate, the vibration of the second vibrator reacts on the first vibrator through the coupling of the mutual inductor to form feedback on the vibration of the first vibrator, and the total equivalent impedance of the first vibrator in the first loop in the coupled resonant circuit is changed;
4) in the process that the frequency of the excitation signal scans the resonant frequency of the second vibrator, the second vibrator resonates, and the feedback effect of the second vibrator on the vibration of the first vibrator is obvious; the second oscillator is narrow spectrum resonance, the vibration phase of the second oscillator is rapidly and obviously changed along with the increase of the frequency of the excitation signal, the vibration phase is obviously larger than pi/2, and the phase tends to be in reverse phase; the first oscillator is in broad-spectrum resonance, the vibration phase of the first oscillator is regarded as being kept unchanged, the feedback effect of the second oscillator on the first oscillator and the coherent superposition state of the self vibration of the first oscillator correspondingly change along with the change of the vibration phase of the second oscillator, namely the coherent superposition state is changed from constructive to destructive or is changed from destructive to constructive, so that the reciprocal of the total equivalent impedance of the first oscillator in the first loop in the coupled resonant circuit generates an asymmetric Fano resonance linear response spectrum; the coupling strength of the first oscillator and the second oscillator is determined by the mutual inductance value of the mutual inductor, the larger the mutual inductance value of the mutual inductor is, the stronger the coupling strength is, the coupling strength of the first oscillator and the second oscillator is adjusted by adjusting the mutual inductance value of the mutual inductor, so that the coupling strength is between loss factors of the first oscillator and the second oscillator, the coupling resonance circuit generates obvious Fano resonance, the obvious Fano resonance is expressed that the first oscillator has an asymmetric response spectrum of a Fano resonance line type, and the second oscillator still has a response spectrum of a narrow-spectrum Lorentz line type;
5) the coupled resonant circuit has the following two conditions by setting parameters:
a) if the resonance frequency of the second vibrator is far away from the resonance frequency of the first vibrator and the deviation amount is larger than the spectrum width of the first vibrator, the first vibrator in the coupling resonance circuit generates Fano resonance near the resonance frequency of the second vibrator; when the resonance frequency of the second vibrator is respectively greater than and less than the resonance frequency of the first vibrator, the asymmetry of the asymmetric Fano resonance line type of the Fano resonance is opposite;
b) the resonance frequency of the second oscillator is equal to that of the first oscillator, the EIT-like phenomenon occurs on the first oscillator in the coupled resonance circuit, the EIT-like phenomenon is a special case of Fano resonance, and at the moment, a narrow-band valley occurs in the middle of a symmetrical broad peak of a response spectrum.
In step 1), the relationship between the resistance values of the first resistor of the first oscillator and the second resistor of the second oscillator and the quality factors and the spectrum widths of the first oscillator and the second oscillator satisfies the following conditions: the larger the resistance values of the first resistor of the first oscillator and the second resistor of the second oscillator are, the smaller the quality factors of the first oscillator and the second oscillator are, and the larger the spectrum widths of the first oscillator and the second oscillator are.
In step 4), the frequency of the excitation signal sweeping the resonance frequency of the second transducer means that the frequency f of the excitation signal is from f 2 -3Δf 2 Change to f 2 +3Δf 2 I.e. the frequency f of the excitation signal satisfies: f. of 2 -3Δf 2 ≤f≤f 2 +3Δf 2 Wherein f is 2 Is the resonance frequency of the second vibrator, Δ f 2 The spectral width of the second oscillator. The vibration phase of the second vibrator changes rapidly and remarkably along with the increase of the frequency of the excitation signal, the phase tends to be in opposite phase, namely the change of the phase approaching pi refers to that: when the frequency of the excitation signal is less than the resonance frequency of the second vibrator and the deviation amount is far greater than the spectrum width of the second vibrator, the phase of the second vibrator tends to pi/2, and the phase of the second vibrator is equal to pi/2 when the frequency of the excitation signal is zero; when the frequency of the excitation signal is greater than the resonance frequency of the second oscillator and the deviation amount is much greater than the spectrum width of the second oscillator, the second oscillator tends to have a phaseAt-pi/2, when the frequency of the excitation signal is infinite, the phase of the second oscillator is equal to-pi/2; in the process that the frequency of the excitation signal is gradually increased from being far less than the resonance frequency of the second vibrator to being far more than the resonance frequency of the second vibrator, the phase of the second vibrator is continuously changed from tending to pi/2 to tending to-pi/2, the change quantity is approximately equal to pi, the phase change of the second vibrator is equal to pi when the frequency of the excitation signal is changed from zero to infinity, and the frequency is zero and infinity, which are two extreme states which cannot be achieved. Variation of vibration phase of second vibrator
The relation with the sweep range of the frequency of the excitation signal satisfies:
n is the ratio of half of the scanning range of the frequency of the excitation signal to the spectral width of the second oscillator; further, the frequency of the excitation signal is swept over a range from f 2 -3Δf 2 To f 2 +3Δf 2 In the process of (1), the phase of the second oscillator changes from atan (6) to-atan 6 to-0.45 pi, and the phase changes to 3
In the step 5), in the case a), when the resonance frequency of the second oscillator is greater than that of the first oscillator, the inverse number of the total equivalent impedance of the first oscillator in the coupled resonant circuit in the first loop generates an asymmetric Fano resonant line type from a valley to a peak in the process of scanning the frequency of the excitation signal through the resonance frequency of the second oscillator; when the resonant frequency of the second oscillator is smaller than that of the first oscillator, in the process that the frequency of the excitation signal scans the resonant frequency of the second oscillator, the inverse number of the total equivalent impedance of the first oscillator in the coupling resonant circuit in the first loop generates an asymmetric Fano resonant line type from peak to valley. F in the vicinity of the resonance frequency of the second vibrator 2 -3Δf 2 ～f 2 +3Δf 2 Wherein f is 2 Is the resonance frequency of the second vibrator, Δ f 2 The spectral width of the second oscillator. The resonant frequency is related to the capacitance of the capacitor and the inductance of the inductor.
The invention has the advantages that:
the invention well shows the Fano resonance phenomenon experimentally by constructing the coupling resonance circuit, and also shows the similar electromagnetic induction transparent phenomenon as a special condition, and the simple classical circuit system can give out complete theoretical description on the aspect of common physics, thereby giving explanation to the experimental phenomenon by means of theoretical formula and calculation result and understanding the back physical mechanism; the experimental device and the measuring method required by the experiment are very basic, the displayed phenomenon has rich physical connotation, and the Fano resonance phenomenon and the similar EIT phenomenon can be displayed, and are linked with the forward research to stimulate the learning interest of students; meanwhile, the method is very helpful for students to understand the general physical law of the resonance phenomenon, particularly to deeply understand the physical significance of the phase in the resonance, and is suitable for being developed as high-order content in college physical experiments.
Drawings
FIG. 1 is a schematic diagram of two linear types according to the present invention, wherein (a) is a Lorentzian linear type and (b) is a Fano resonance linear type;
FIG. 2 is a circuit diagram of a Fano resonance experimental instrument based on a coupled resonance circuit according to the present invention;
fig. 3 is a graph of experimental measurement results of a mode (amplitude-frequency characteristic curve) and a phase (phase-frequency characteristic curve) of an inverse complex impedance of an isolated and coupled oscillator in an embodiment of a coupled resonant circuit according to the present invention, wherein (a) is the amplitude-frequency characteristic curve of the isolated first oscillator (solid line) and second oscillator (dotted line), and for convenience of display, the amplitude of the second oscillator is compressed to 1/25 of an original value, (b) is the phase-frequency characteristic curve of the isolated first oscillator (solid line) and second oscillator (dotted line), (c) is the amplitude-frequency characteristic curve of the first oscillator in the coupled resonant circuit composed of the first oscillator and the second oscillator, and (d) is the phase-frequency characteristic curve of the first oscillator in the coupled resonant circuit composed of the first oscillator and the second oscillator;
FIG. 4 is a graph showing the result of theoretical formula calculation in an embodiment of the Fano resonance experiment apparatus based on coupled resonant circuit of the present invention, wherein (a) is the complex impedance Z of the first oscillator 1 (solid line) and complex impedance Z of the second oscillator for feedback action of the second oscillator on the first oscillator under coupling action 21 A frequency characteristic diagram of a mode (dotted line), and (b) a total equivalent complex impedance Z ═ Z of the coupled resonant circuit in the first loop 1 +Z 21 The frequency characteristic diagram of the mode (c) is the complex impedance Z of the first oscillator 1 (solid line) and complex impedance Z of the second oscillator for feedback action of the second oscillator on the first oscillator under coupling action 21 A frequency characteristic diagram of the phase (dotted line), and (d) the inverse number Z of the total equivalent complex impedance of the coupled resonant circuit in the first loop -1 A frequency characteristic diagram of the mode of (a);
FIG. 5 is a graph showing the response of the coupled resonant circuit under the condition of changing parameters in an embodiment of the Fano resonance experiment apparatus based on the coupled resonant circuit of the present invention, wherein (a) is when C 2 The amplitude-frequency characteristic curve of the first oscillator in the coupling resonance circuit is measured in an experiment (b) is the complex impedance Z of the corresponding first oscillator calculated by using a theoretical formula 1 (solid line) and complex impedance Z of the second oscillator with respect to the feedback of the first oscillator under coupling 21 A frequency characteristic diagram of the phase (dotted line) and (C) when C 2 The amplitude-frequency characteristic curve of the first oscillator in the coupling resonance circuit is measured in an experiment (d) is the complex impedance Z of the corresponding first oscillator calculated by using a theoretical formula 1 (solid line) and complex impedance Z of the second oscillator for feedback action of the second oscillator on the first oscillator under coupling action 21 Frequency characteristic diagram of phase (dotted line).
Detailed Description
The invention will be further elucidated by means of specific embodiments in the following with reference to the drawing.
As shown in fig. 2, the franco resonance experiment instrument based on the coupled resonant circuit of the present embodiment includes: the circuit comprises a first inductor, a second inductor, a first capacitor, a second capacitor, a first resistor, a second resistor, an alternating current signal source and a mutual inductor; the mutual inductor comprises a primary coil and a secondary coil; the first inductor, the first capacitor, the first resistor and a primary coil of the mutual inductor are sequentially connected to form a first oscillator, and then connected with an alternating current signal source to form a first LOOP LOOP 1; the second inductor, the second capacitor, the second resistor and a secondary coil of the mutual inductor are sequentially connected to form a second oscillator, and a second LOOP LOOP2 is formed; the quality factor of the first oscillator is controlled by changing the resistance value of the first resistor of the first oscillator, and the spectrum width of the first oscillator is further changed, wherein the spectrum width is the full width at half maximum of a resonance peak of a response spectrum, so that the first oscillator is a oscillator with Lorentz-type wide-spectrum resonance; the quality factor of the second oscillator is controlled by changing the resistance value of a second resistor of the second oscillator, and the spectrum width of the second oscillator is further changed, so that the second oscillator is a oscillator with Lorentz line type narrow spectrum resonance; the first oscillator and the second oscillator are coupled through a mutual inductor to form a coupled resonant circuit; the alternating current signal source takes a sine wave as an excitation signal to excite the first vibrator to vibrate, the first vibrator is coupled with the second vibrator through the mutual inductor so as to excite the second vibrator to vibrate, the vibration of the second vibrator reacts on the first vibrator through the coupling of the mutual inductor to form feedback on the vibration of the first vibrator, and the total equivalent impedance of the first vibrator in the first loop in the coupled resonant circuit is changed; in the process that the frequency of the excitation signal scans the resonance frequency of the second vibrator, the second vibrator resonates, and the feedback effect of the second vibrator on the vibration of the first vibrator is obvious; the second oscillator is narrow spectrum resonance, the vibration phase of the second oscillator is rapidly and obviously changed along with the increase of the frequency of the excitation signal, the vibration phase is obviously larger than pi/2, and the phase tends to be in reverse phase; the first oscillator is in broad-spectrum resonance, the vibration phase of the first oscillator is regarded as being kept unchanged, the feedback effect of the second oscillator on the first oscillator and the coherent superposition state of the self vibration of the first oscillator correspondingly change along with the change of the vibration phase of the second oscillator, namely the coherent superposition state is changed from constructive to destructive or is changed from destructive to constructive, the phases are nearly the same and are coherent constructive, the phase difference is close to pi and is coherent destructive, and the reciprocal of the total equivalent impedance of the first oscillator in the coupled resonant circuit in the first loop generates an asymmetric Fano resonant line type response spectrum; the coupling strength of the first oscillator and the second oscillator is determined by the mutual inductance value of the mutual inductor, the coupling strength is stronger when the mutual inductance value of the mutual inductor is larger, the coupling strength of the first oscillator and the second oscillator is adjusted by adjusting the mutual inductance value of the mutual inductor, the coupling strength is enabled to be between loss factors of the first oscillator and the second oscillator, the coupling resonance circuit generates obvious Fano resonance, the obvious Fano resonance is expressed that the first oscillator has an asymmetric response spectrum of a Fano resonance line type, and the second oscillator still has a response spectrum of a narrow-spectrum Lorentz line type.
1. Experimental device
In this embodiment, as shown in fig. 2, the inductance of the first inductor is L 1 The capacitance value of the first capacitor is C 1 The resistance value of the first resistor is R 1 The self-inductance value of the primary coil of the mutual inductor is LM 1 Mutual inductance value of the mutual inductor is M, and inductance value of the second inductor is L 2 The capacitance value of the second capacitor is C 2 The resistance value of the second resistor is R 2 The self-inductance value of the secondary coil of the mutual inductor is LM 2 And the line end voltage output outwards by the alternating current signal source is U S . Reciprocal Z of total equivalent complex impedance Z of coupled resonance circuit formed by first oscillator and second oscillator in first loop -1 Characterizing the response of the coupled resonant circuit (corresponding to the current I in the first loop with a constant voltage at the output terminal of the fixed signal source 1 ) More specifically, the modulus | Z of the reciprocal of the complex impedance -1 The variation curve of | along with the frequency f of the excitation signal represents the amplitude-frequency characteristic of the coupled resonance circuit, and the phase arg (Z) of the reciprocal of the complex impedance is used -1 ) The curve of the frequency f of the excitation signal characterizes the phase-frequency behavior of the coupled resonant circuit. In the experiment, the line end voltage U output by the AC signal source is measured S And a voltage U across the first resistor R1 As a function of the frequency f of the excitation signal and using Z -1 ＝I 1 /U S ＝U R1 /(R 1 ·U S ) To obtain Z -1 As a function of the frequency f of the excitation signal.
In this embodiment, the capacitor used is an RX7-0A type capacitor box, the resistor is a ZX96 type resistor box, and both the inductor and the transformer are made by a manufacturer, and a coil using a soft ferrite ring as a magnetic core is used. Alternating current signal source and measuring instrumentThe device can adopt a common sine wave signal generator and an oscilloscope, an automatic measurement system is designed for improving the measurement speed, a signal source adopts a DG1022U type programmable signal generator of RIGOL company, the signal generator is controlled by LabVIEW software to output sine waves with certain frequency and amplitude, and then the LabVIEW software is used for controlling two channels of a USB-6343 type data acquisition card of NI company to respectively record the output line end voltage U of the alternating current signal source in the first loop S And a voltage U across the first resistor R1 The data are processed by LabVIEW programming, and the experimental measurement results of the amplitude-frequency and phase-frequency characteristic curves of the system are conveniently obtained at one time.
2. Experimental results of typical Fano resonance phenomena
In the experiment, the mutual inductance value of the mutual inductor is set to be M-8 mH, the coupling of the two oscillators is too weak under an excessively small mutual inductance value, and the Fano resonance phenomenon is not obvious; too strong coupling of the two oscillators under an excessively large mutual inductance value can cause a complex strong coupling phenomenon of a coupled resonant circuit, and a typical Fano resonance occurs in a weak coupling region. The self-inductance value LM of the primary coil of the mutual inductor 1 And self-inductance value LM of secondary coil 2 Also both 8 mH. Other parameters of the first oscillator are: l is 1 ＝32 mH、C 1 ＝0.08μF、R 1 The mode of the inverse complex impedance measured when the first transducer is in the isolated state, i.e., the amplitude-frequency characteristic curve, is shown as a solid line in fig. 3 (a). The first resistor of the first oscillator has a large resistance value, a small quality factor and a wide corresponding resonance spectrum, and corresponds to the wide spectrum resonance required in the Fano resonance. Other parameters of the second oscillator are: l is 2 ＝32 mH、C 2 ＝0.02μF、R 2 The modulus of the reciprocal of the complex impedance measured when the second oscillator is in an isolated state is shown by the dotted line in fig. 3(a), and the resistance value of the second resistor of the second oscillator is small, the quality factor is large, and the corresponding resonance spectrum is narrow, which corresponds to the narrow spectrum resonance required in the fanno resonance. The amplitude-frequency characteristic curves of the isolated first oscillator and the isolated second oscillator are approximately represented by Lorentz lines, and resonance peaks in the vicinity of respective resonance frequencies are approximately symmetrically distributed. Measured to obtain isolationThe first and second oscillators of (1) have resonance frequencies of about 2804Hz and 5551Hz, respectively, and the resonance frequencies theoretically calculated from the element parameter values are 2813Hz and 5627Hz, respectively, and the measured resonance frequencies substantially match the theoretically calculated values with deviations of 0.3% and 1.4%, respectively, the small deviations being mainly attributed to errors in the parameter values of the respective elements used in the experiment.
In the experiment, the phases of the inverse complex impedances of the isolated first and second oscillators were also measured, and the obtained phase-frequency characteristic curves are shown as a solid line and a broken line in fig. 3(b), respectively. In accordance with a conventional LCR series resonant circuit, the phases of the first and second elements gradually transition from a tendency toward pi/2 to a tendency toward-pi/2 (in arg (Z) around the resonant frequency -1 ) The phase frequency characteristic is characterized so that the positive and negative of the phase are opposite to the rule adopted in the usual resonance circuit), the phase at the resonance frequency is equal to 0. The phase change of the first oscillator near the resonance frequency in the phase-frequency characteristic curve is more gentle, the phase change of the second oscillator is more rapid, and the wide spectrum and narrow spectrum resonance behaviors respectively shown by the first oscillator and the second oscillator in the amplitude-frequency characteristic curve are consistent, so that the general rule of a resonance system is reflected: i.e. the resonant behavior of amplitude with frequency and the phase variation with frequency are related to each other.
Next, the amplitude-frequency characteristic curve obtained by measuring the coupled resonant circuit formed by the first and second transducers, that is, the circuit shown in fig. 2 is shown in fig. 3 (c). It can be seen that the amplitude-frequency characteristic curve of the coupled resonant circuit rapidly changes from a valley to a peak in the vicinity of the resonant frequency of the second oscillator, showing a significantly asymmetric fano resonant line shape. The measurement results of the phase frequency characteristic curves of the corresponding coupled resonant circuits are shown in fig. 3(d), and the phase change behavior of the coupled resonant circuits is also correlated with the amplitude change behavior.
3. Theoretical analysis and explanation of Fano resonance phenomenon in coupled resonance circuit
Comparing fig. 3(a) and fig. 3(c), it is found that the asymmetric peak-valley of the fanuo resonance line type in the coupled resonance circuit is located close to the resonance frequency of the second vibrator, which suggests that the occurrence of such asymmetric peak-valley is related to the feedback effect of the resonance of the second vibrator on the first vibrator. The experimental results are explained below by means of theoretical formulas and associated calculation results.
For the coupled resonant circuit of the experiment, a theoretical expression of the total equivalent complex impedance Z of the coupled resonant circuit formed by the first oscillator and the second oscillator in the first loop is derived according to a circuit equation, and is as follows:
wherein the content of the first and second substances,
L′ 1 ＝L 1 +LM 1 ；
L′ 2 ＝L 2 +LM 2 。
here, Z 1 And Z 2 Respectively represents the complex impedance, L 'of the isolated first and second oscillators' 1 And L' 2 Each of the isolated first and second transducers has a total inductance value, ω is an angular frequency of the excitation signal, ω is 2 pi f, and j is an imaginary unit. As seen from equation (1), the total equivalent complex impedance Z of the coupled resonant circuit in the first loop consists of two parts: a part being the complex impedance Z of the first oscillator 1 I.e. the complex impedance of the isolated first oscillator itself; complex impedance Z of the other part and the second oscillator 2 In relation to, reflecting the feedback effect of the second oscillator on the first oscillator under the coupling effect, using Z 21 To represent this portion of the complex impedance, i.e. Z 21 Defining the complex impedance of the second oscillator to the feedback effect of the first oscillator under the coupling effect
It is easy to see the complex impedance Z of the second oscillator to the feedback of the first oscillator under the coupling effect 21 Complex impedance Z with isolated second oscillator 2 In an inverse relationship.
Substituting the element parameters used in the previous experiment into the theoretical formula to calculate to obtain Z 1 And Z 21 The results are shown as solid and dashed lines in fig. 4(a), respectively. It can be seen that | Z 1 The overall variation trend of | is exactly opposite to the amplitude-frequency characteristic curve of the isolated first oscillator measured experimentally in fig. 3(a), because the amplitude-frequency characteristic curve is characterized by the modulus of the reciprocal of the complex impedance, and | Z 1 I is in inverse proportion; i Z 1 The change of | is more gentle because the first oscillator is wide spectrum resonance; i Z 1 L at the resonance frequency (f) of the first vibrator 1 2813Hz) because the total impedance of the series resonant circuit is minimal at the resonant frequency. In contrast, | Z 21 The variation trend of | is the same as the amplitude-frequency characteristic curve of the isolated second oscillator experimentally measured in fig. 3(a), because
Therefore | Z 21 L is proportional to the modulus of the reciprocal of the complex impedance of the second oscillator; i Z 21 At the resonance frequency (f) of the second vibrator 2 5627Hz) shows a very narrow peak because the resonance of the second oscillator is a narrow spectrum resonance. Comparison of | Z in 4(a) 1 I and I Z 21 The numerical value of | Z can be found only in the vicinity of the resonance frequency of the second vibrator 21 Numerical value of and | Z 1 I can be compared, and the second oscillator has obvious feedback effect on the first oscillator; and at other frequencies, | Z 21 |＜＜|Z 1 I.e. Z is approximately equal to Z 1 And the feedback effect of the second oscillator on the first oscillator can be ignored, and the response of the whole coupled resonant circuit is almost the same as that of the isolated first oscillator. Fig. 4(b) shows | Z | ═ Z calculated by a theoretical formula 1 +Z 21 The result of | the variation curve with the frequency f of the excitation signal is consistent with the analysis and prediction: the curve is clearly different only in the vicinity of the resonance frequency of the second vibrator from the isolated first vibrator, which explains why the experimentally measured region of the coupled resonance circuit in fig. 3(c) outside the vicinity of the resonance frequency of the second vibrator is almost identical to the amplitude-frequency characteristic of the isolated first vibrator in fig. 3 (a).
For further analysis of the vicinity Z of the resonance frequency of the second vibrator 21 The contribution to the coupled resonant circuit is calculated by using a theoretical formula to obtain Z 1 And Z 21 The phase of (c) is varied with frequency, and the results are shown as a solid line and a broken line in fig. 4(c), respectively. There is a similar rule to the result of FIG. 4(a), arg (Z) 1 ) The variation trend of (a) is opposite to the phase-frequency characteristic curve of the isolated first oscillator experimentally measured in fig. 3(b), because the phase-frequency characteristic curve is characterized by the phase of the reciprocal of the complex impedance; and arg (Z) 21 ) Is the same as the phase-frequency characteristic curve of the isolated second oscillator experimentally measured in fig. 3(b), because Z is 21 And with
Is in direct proportion. Arg (Z) of the first oscillator that resonates in a broad spectrum in the vicinity of the resonance frequency of the second oscillator that resonates in a narrow spectrum 1 ) Can be regarded as a constant, and since the resonance frequency of the second vibrator is larger than that of the first vibrator, arg (Z) at this time 1 ) Close to pi/2; and arg (Z) of a second vibrator of narrow spectrum resonance 21 ) The total equivalent impedance of the coupled resonant circuit, IZI, will change rapidly from Zn/2 to- π/2 as the frequency of the excitation signal increases 1 And Z 21 The maximum state given by the coherent constructive sum of (a) changes to a value of Z 1 And Z 21 The coherence of (c) cancels the given minimum state. This is confirmed by the | Z | curve calculated by the theoretical formula in fig. 4(b), and | Z | does change rapidly from a peak to a valley in the vicinity of the resonance frequency of the second vibrator. Corresponding to it, | Z -1 I.e. the amplitude-frequency characteristic of the coupled resonant circuit, is then in turn rapidly changed from a valley to a peak, i.e. giving the steep asymmetric fano resonant line type observed in the experiment. FIG. 4(d) further shows the | Z calculated by the theoretical formula -1 The theoretical calculation result is basically consistent with the experimental measurement result in fig. 3(c) along with the change curve of the frequency f of the excitation signal, so that the asymmetric Fano resonance line type measured in the experiment is well reproduced, and the effectiveness of the theoretical formula is verified from another angle.
By combining the theoretical analysis, the physical mechanism of the Fano resonance phenomenon in the coupled resonance circuit is summarized, and the phenomenon is caused by the interference between the two mechanisms of the self vibration of the first oscillator and the feedback action of the second oscillator on the first oscillator. The first oscillator is driven by external excitation and is wide-spectrum resonance with large loss; and there is coupling between the second vibrator and the first vibrator, the vibration of which is driven by the first vibrator through coupling action, and the second vibrator is a narrow-spectrum resonance with small loss. When the frequency of the external excitation signal is far away from the resonance frequency of the second oscillator, the vibration of the second oscillator can be ignored, and the response of the coupling resonance circuit is determined by the first oscillator; when the frequency of the external excitation signal is close to the resonance frequency of the second vibrator, the second vibrator has obvious feedback effect on the first vibrator, and the vibration phase of the second vibrator has a change close to pi in a narrow frequency range, and the feedback effect and the coherent superposition of the vibration of the first vibrator correspondingly have a change from cancellation to constructive, so that the amplitude of the first vibrator has a steep asymmetric Fano resonance line type. Here, the rapid change of the phase of the second vibrator with frequency is a key to generating an asymmetric fano-resonance line type.
4. Fano resonance phenomenon and EIT-like phenomenon under changed parameters
In the above experiment, the capacitance C of the second capacitor is set 2 0.02 μ F, i.e. the resonance frequency of the isolated second oscillator is greater than the resonance frequency of the isolated first oscillator. By changing the relative magnitudes of the resonance frequencies of the first and second transducers, different frequency response behaviors can be observed.
Setting the capacitance C of the second capacitor 2 When the resonance frequency of the second oscillator is set to 0.16 μ F, the amplitude-frequency characteristic curve of the coupling resonance circuit measured experimentally is as shown in fig. 5(a), and a pronounced asymmetric fano resonance line pattern appears on the left side of the resonance peak of the first oscillator. This is because the fano resonant line type occurs in the vicinity of the resonant frequency of the second vibrator, and the resonant frequency of the second vibrator is smaller than the resonant frequency of the first vibrator at this time. Contrary to the foregoing, the Fano resonance line type in FIG. 5(a) no longer changes from a valley with an increase in the frequency of the excitation signalTo peaks but from peaks to troughs. This phenomenon can also be explained by the law of phase change of the oscillator. The solid line and the broken line in fig. 5(b) show the capacitance value C of the second capacitor, respectively 2 Arg (Z) calculated by theoretical formula under the parameter of 0.16 μ F 1 ) And arg (Z) 21 ) With the change in frequency, since the resonance frequency of the second vibrator is now lower than that of the first vibrator, arg (Z) is observed in the vicinity of the resonance frequency of the second vibrator 1 ) Approximately in a state close to-pi/2, and then, at arg (Z) 21 ) Z is rapidly changed from pi/2 to-pi/2 along with the increase of the frequency of the excitation signal 1 And Z 21 Instead, the phase changes rapidly from coherent cancellation to coherent constructive state, and the corresponding | Z | changes from valley to peak instead, while | Z | changes from valley to peak -1 I instead varies from peak to trough, i.e. the asymmetry of the fano-resonance line type is exactly the opposite of before.
If the capacitance value C of the second capacitor is set 2 The amplitude-frequency characteristic curve of the coupling resonant circuit measured experimentally with the resonance frequencies of the second and first oscillators being the same at 0.08 μ F is shown in fig. 5 (c). Compared with the normal single-peak response spectrum of the isolated first oscillator, the response spectrum has a sharp valley near the common resonance frequency of the two oscillators, and the response curve is an EIT-like phenomenon, namely a narrow-band transparent window appears in the middle of the original absorption peak. This phenomenon can also be explained by the law of the phase change of the oscillator. The solid line and the broken line in fig. 5(d) show the capacitance value C of the second capacitor, respectively 2 Arg (Z) calculated by theoretical formula under the parameter of 0.08 μ F 1 ) And arg (Z) 21 ) As the frequency of the excitation signal changes, it can be seen that since the resonance frequencies of the second transducer and the first transducer are equal at this time, arg (Z) is near the common resonance frequency of the two transducers 1 ) Approximately at a state close to 0, and then, at arg (Z) 21 ) Z is rapidly changed from pi/2 to-pi/2 in the course of increasing the frequency of the excitation signal 1 And Z 21 Will experience a mismatch between (when arg (Z) 21 ) π/2) to coherent constructive (when arg (Z) 21 ) 0) to incoherent (when arg (Z) 21 ) Approximately-pi/2), the corresponding | Z | changes from a valleyChange to a peak and then to a valley, and | Z -1 The | is changed from peak to valley to peak, thereby showing EIT-like phenomenon.
The invention shows the Fano resonance phenomenon and EIT-like phenomenon in the coupling resonance circuit, and the simple classical circuit system can give out complete theoretical description on the aspect of common physics, thereby giving explanation to the experimental phenomenon by means of theoretical formulas and calculation results and understanding the back physical mechanism. The experimental device and the measuring method required by the experiment are both basic, the displayed phenomenon has rich physical connotation, and the Fano resonance phenomenon and the EIT-like phenomenon can be displayed, so that the connection with the forward-edge research is established, and the learning interest of students is stimulated; meanwhile, the method is very helpful for students to understand the general physical law of the resonance phenomenon, particularly to deeply understand the physical significance of the phase in the resonance, and is suitable for being developed as high-order content in college physical experiments. It should be noted that although the franco resonance phenomenon and the EIT-like phenomenon are shown in the coupled resonant circuit, as mentioned in the background section, the phenomena are common in many physical systems, and the franco resonance phenomenon and the EIT-like phenomenon can be realized by using the classical mechanical coupling system and the optical coupling system in a similar way.
It is finally noted that the disclosed embodiments are intended to aid in the further understanding of the invention, but that those skilled in the art will appreciate that: various substitutions and modifications are possible without departing from the spirit and scope of the invention and the appended claims. Therefore, the invention should not be limited to the embodiments disclosed, but the scope of the invention is defined by the appended claims.
Claims (4)
1. A realization method of a Fano resonance experimental instrument based on a coupling resonance circuit is characterized by comprising the following steps:
1) setting up an experimental instrument:
i. the first inductor, the first capacitor, the first resistor and a primary coil of the mutual inductor are sequentially connected to form a first oscillator, and then connected with an alternating current signal source to form a first loop;
a second inductor, a second capacitor, a second resistor and a secondary coil of the mutual inductor are sequentially connected to form a second oscillator and form a second loop;
the quality factor of the first oscillator is controlled by changing the resistance value of the first resistor of the first oscillator, and the spectrum width of the first oscillator is further changed, wherein the spectrum width is the full width at half maximum of a resonance peak of a response spectrum, so that the first oscillator is a oscillator with Lorentz-type wide-spectrum resonance; the quality factor of the second oscillator is controlled by changing the resistance value of a second resistor of the second oscillator, and the spectrum width of the second oscillator is further changed, so that the second oscillator is a oscillator with Lorentz line type narrow spectrum resonance;
the first oscillator and the second oscillator are coupled through a mutual inductor to form a coupled resonant circuit;
2) the alternating current signal source takes a sine wave as an excitation signal to excite the first vibrator to vibrate;
3) the first vibrator is coupled with the second vibrator through the mutual inductor so as to excite the second vibrator to vibrate, the vibration of the second vibrator reacts on the first vibrator through the coupling of the mutual inductor to form feedback on the vibration of the first vibrator, and the total equivalent impedance of the first vibrator in the first loop in the coupled resonant circuit is changed;
4) in the process that the frequency of the excitation signal scans the resonant frequency of the second vibrator, the second vibrator resonates, and the second vibrator plays a feedback role on the vibration of the first vibrator; the second oscillator is narrow-spectrum resonance, and the vibration phase of the second oscillator changes
The relation with the sweep range of the frequency of the excitation signal satisfies:
n is the ratio of half of the scanning range of the frequency of the excitation signal to the spectral width of the second oscillator; the first vibrator is wide-spectrum resonance, the vibration phase of the first vibrator is regarded as being kept unchanged, and the feedback effect of the second vibrator on the first vibrator is overlapped with the coherence of the vibration of the first vibratorThe adding state is correspondingly changed along with the change of the vibration phase of the second oscillator, namely the coherent adding state is changed from constructive cancellation to destructive or is changed from destructive to constructive cancellation, so that the reciprocal of the total equivalent impedance of the first oscillator in the coupled resonant circuit in the first loop generates an asymmetric Fano resonant line type response spectrum; the coupling strength of the first oscillator and the second oscillator is determined by the mutual inductance value of the mutual inductor, the larger the mutual inductance value of the mutual inductor is, the stronger the coupling strength is, the coupling strength of the first oscillator and the second oscillator is adjusted by adjusting the mutual inductance value of the mutual inductor, so that the coupling strength is between loss factors of the first oscillator and the second oscillator, the coupled resonant circuit generates Fano resonance, the first oscillator has an asymmetric response spectrum of a Fano resonance line type, and the second oscillator still has a response spectrum of a narrow-spectrum Lorentz line type;
the frequency of the excitation signal sweeping the resonant frequency of the second transducer means that the frequency f of the excitation signal is from f 2 -3Δf 2 Change to f 2 +3Δf 2 I.e. the frequency f of the excitation signal satisfies: f. of 2 -3Δf 2 ≤f≤f 2 +3Δf 2 Wherein f is 2 Is the resonance frequency of the second vibrator, Δ f 2 Is the spectral width of the second vibrator;
the frequency of the excitation signal is swept over a range from f 2 -3Δf 2 To f 2 +3Δf 2 In the process of (1), the phase of the second oscillator changes from atan6 ═ 0.45 pi to-atan 6 ═ 0.45 pi, and the phase changes to
5) The coupled resonant circuit has the following two conditions by setting parameters:
a) if the resonance frequency of the second vibrator is far away from the resonance frequency of the first vibrator and the deviation amount is larger than the spectrum width of the first vibrator, the first vibrator in the coupling resonance circuit generates Fano resonance near the resonance frequency of the second vibrator; when the resonance frequency of the second vibrator is respectively greater than and less than the resonance frequency of the first vibrator, the asymmetry of the asymmetric Fano resonance line type of the Fano resonance is opposite;
b) the resonance frequency of the second oscillator is equal to that of the first oscillator, the EIT-like phenomenon occurs on the first oscillator in the coupled resonance circuit, the EIT-like phenomenon is a special case of Fano resonance, and at the moment, a narrow-band valley occurs in the middle of a symmetrical broad peak of a response spectrum.
2. The implementation method of claim 1, wherein in step 1) iii), the relationship between the resistance values of the first resistor of the first oscillator and the second resistor of the second oscillator and the quality factors and the spectral widths of the first oscillator and the second oscillator satisfies: the larger the resistance values of the first resistor of the first oscillator and the second resistor of the second oscillator are, the smaller the quality factors of the first oscillator and the second oscillator are, and the larger the spectrum widths of the first oscillator and the second oscillator are.
3. The method of claim 1, wherein in step 5), when the resonant frequency of the second vibrator is greater than that of the first vibrator in case a), the inverse number of the total equivalent impedance of the first vibrator in the coupled resonant circuit in the first loop circuit generates an asymmetric fanout resonant line type from valley to peak during the frequency of the excitation signal sweeps through the resonant frequency of the second vibrator; when the resonant frequency of the second oscillator is smaller than that of the first oscillator, in the process that the frequency of the excitation signal scans the resonant frequency of the second oscillator, the inverse number of the total equivalent impedance of the first oscillator in the coupling resonant circuit in the first loop generates an asymmetric Fano resonant line type from peak to valley.
4. The method of claim 1, wherein in step 5), f is the resonant frequency of the second vibrator 2 -3Δf 2 ～f 2 +3Δf 2 Wherein f is 2 Is the resonance frequency of the second vibrator, Δ f 2 The spectral width of the second oscillator.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202111412414.1A CN114078353B (en) | 2021-11-25 | 2021-11-25 | Fano resonance experimental instrument realization method based on coupling resonance circuit |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202111412414.1A CN114078353B (en) | 2021-11-25 | 2021-11-25 | Fano resonance experimental instrument realization method based on coupling resonance circuit |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN114078353A CN114078353A (en) | 2022-02-22 |
| CN114078353B true CN114078353B (en) | 2022-08-26 |
Family
ID=80284280
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202111412414.1A Active CN114078353B (en) | 2021-11-25 | 2021-11-25 | Fano resonance experimental instrument realization method based on coupling resonance circuit |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN114078353B (en) |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US2948818A (en) * | 1954-05-28 | 1960-08-09 | Parametron Inst | Resonator circuits |
| US5576674A (en) * | 1995-03-17 | 1996-11-19 | Allen Telecom Group, Incorporated | Optimum, multiple signal path, multiple-mode filters and method for making same |
| CN111106449A (en) * | 2019-11-25 | 2020-05-05 | 西北工业大学深圳研究院 | Device and method for realizing electrical regulation Fano resonance based on asymmetric open resonant ring |
| CN112953391A (en) * | 2015-01-27 | 2021-06-11 | 华为技术有限公司 | Radio frequency oscillator |
Family Cites Families (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7587144B2 (en) * | 2004-01-12 | 2009-09-08 | Oewaves, Inc. | Tunable radio frequency and microwave photonic filters |
| WO2007100760A2 (en) * | 2006-02-27 | 2007-09-07 | The Penn State Research Foundation | Detecting quadrupole resonance signals using high temperature superconducting resonators |
| US10186743B2 (en) * | 2017-01-30 | 2019-01-22 | United Arab Emirates University | Microstrip circuits exhibiting electromagnetically induced transparency and fano resonance |
| US10236860B1 (en) * | 2017-07-19 | 2019-03-19 | Rockwell Collins, Inc. | High selective (brick wall) filters based on Fano resonances |
| CN109781089B (en) * | 2017-11-10 | 2021-04-16 | 中国科学院半导体研究所 | Resonant optical gyroscope based on Fano resonance effect |
-
2021
- 2021-11-25 CN CN202111412414.1A patent/CN114078353B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US2948818A (en) * | 1954-05-28 | 1960-08-09 | Parametron Inst | Resonator circuits |
| US5576674A (en) * | 1995-03-17 | 1996-11-19 | Allen Telecom Group, Incorporated | Optimum, multiple signal path, multiple-mode filters and method for making same |
| CN112953391A (en) * | 2015-01-27 | 2021-06-11 | 华为技术有限公司 | Radio frequency oscillator |
| CN111106449A (en) * | 2019-11-25 | 2020-05-05 | 西北工业大学深圳研究院 | Device and method for realizing electrical regulation Fano resonance based on asymmetric open resonant ring |
Non-Patent Citations (1)
| Title |
|---|
| 浅谈谐波源检测设计;李丹等;《科技创新导报》;20100211(第05期);第1-11页 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN114078353A (en) | 2022-02-22 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Ilinskii et al. | Nonlinear standing waves in an acoustical resonator | |
| Vitanov et al. | Landau-Zener model: Effects of finite coupling duration | |
| JP2019176475A (en) | Bulk acoustic wave (baw) resonator | |
| US20090216467A1 (en) | Sensor, system, and method, for measuring fluid properties using Multi-Mode Quasi-Shear-Horizontal Resonator | |
| CN114078353B (en) | Fano resonance experimental instrument realization method based on coupling resonance circuit | |
| CN108566114A (en) | A kind of ultrasonic transducer excited frequency selection method | |
| CN217034967U (en) | Fano resonance experiment appearance based on coupling resonance circuit | |
| JPS5833309A (en) | Slip wave resonator | |
| Jain et al. | Acoustic radiation from a superconducting qubit: From spontaneous emission to Rabi oscillations | |
| KR20150065723A (en) | Method and circuit arrangement for determining a working range of an ultrasonic vibrating unit | |
| US10608587B2 (en) | Crystal oscillator circuit | |
| JPS6211043Y2 (en) | ||
| CA1228415A (en) | Resonant frequency modification of piezoelectric transducers | |
| Pigache et al. | Modeling and identification of Rosen-type transformer in nonlinear behavior | |
| US7101721B2 (en) | Adaptive manufacturing for film bulk acoustic wave resonators | |
| CN209402485U (en) | SAW resonator | |
| Nazarov et al. | Wave processes in media with hysteretic nonlinearity: Part 2 | |
| RU2707394C2 (en) | Generator | |
| Lozzi et al. | Anchor loss dependence on electrode materials in contour mode resonators | |
| Alekseev et al. | Study of layered structures using modified acoustic resonator spectroscopy | |
| JP2011171888A (en) | Lamb wave resonator and oscillator | |
| JP3948206B2 (en) | Oscillator circuit | |
| Plessky et al. | Surface Transverse Wave (STW) resonators on langasite | |
| Nimets et al. | Controlling the characteristics of nanomechanical resonators | |
| JPS596438B2 (en) | displacement transducer |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |