CN217034967U - Fano resonance experiment appearance based on coupling resonance circuit - Google Patents
Fano resonance experiment appearance based on coupling resonance circuit Download PDFInfo
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- CN217034967U CN217034967U CN202122916188.2U CN202122916188U CN217034967U CN 217034967 U CN217034967 U CN 217034967U CN 202122916188 U CN202122916188 U CN 202122916188U CN 217034967 U CN217034967 U CN 217034967U
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Abstract
The utility model discloses a Fano resonance experimental instrument based on a coupling resonance circuit. The utility model comprises the following steps: 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; 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 form a second loop; the first oscillator is an oscillator with wide spectrum resonance of a Lorentzian line type, and the second oscillator is an oscillator with narrow spectrum resonance of the Lorentzian line type; the first oscillator and the second oscillator are coupled through a mutual inductor to form a coupled resonant circuit; the utility model is suitable for being developed as high-level contents in college physical experiments.
Description
Technical Field
The utility model relates to a common physical experiment instrument, in particular to 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 under the external driving of different frequencies, an isolated damped oscillation system with a single resonance frequency shows a typical single resonance phenomenon, the amplitude of the system changes along with the external driving frequency and shows a response spectrum of a lorentz line type, and the line type shows 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 discrete-to-discrete state transition) and the other transition path exhibits a broad spectral resonance (corresponding to discrete-to-continuous state transition), interference between the two transition paths causes the overall absorption line of the system to exhibit an asymmetric Fano resonance profile. Later, people observed 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, echo wall micro-cavities and the like, and found 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 nano photonics and the like in recent years.
Disclosure of Invention
In order to show the Fano resonance phenomenon at the common physical experiment level and establish the connection between a basic experiment and a forward research, the utility model provides 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 one special condition, and the experiment result can be explained through the calculation result of a theoretical formula.
The Fano resonance experiment instrument based on the coupling resonance circuit comprises: 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; 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 form a second loop; the resistance value of a first resistor of the first oscillator is larger than that of a second resistor of the second oscillator, so that the first oscillator is a oscillator with Lorentz line type wide spectrum resonance, and 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.
Setting the resistance value of a first resistor of a first oscillator to determine a quality factor of the first oscillator, and further determining the spectrum width of the first oscillator, 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 an oscillator with Lorentz line type wide spectrum resonance; and setting the resistance value of a second resistor of the second oscillator to determine the quality factor of the second oscillator and further determine the spectrum width of the second oscillator, so that the second oscillator is an oscillator with Lorentz line type narrow spectrum resonance. 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 value of the first resistor of the first oscillator is, the smaller the quality factor of the first oscillator is, and the larger the spectrum width of the first oscillator is; similarly, the larger the resistance of the second resistor of the second oscillator is, the smaller the quality factor of the second oscillator is, and the larger the spectrum width of the second oscillator is.
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 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 vibrator and the second vibrator 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 vibrator and the second vibrator is adjusted by adjusting the mutual inductance value of the mutual inductor, so that the coupling strength is between loss factors of the first vibrator and the second vibrator, the coupling resonance circuit generates obvious Fano resonance, and the situation is shown that the first vibrator has an asymmetric Fano resonance linear response spectrum, and the second vibrator is still a narrow-spectrum Lorentz linear response spectrum;
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 half of 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 oscillator is respectively greater than and less than the resonance frequency of the first oscillator, the asymmetric Fano resonance line type asymmetry of the Fano resonance is opposite; b) the resonance frequency of the second oscillator is equal to the resonance frequency 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 condition of Fano resonance, and at the moment, a narrow-band valley occurs in the middle of a symmetrical wide peak of a response spectrum. The resonant frequency is related to the capacitance of the capacitor and the inductance of the inductor.
F in the vicinity of the resonance frequency of the second vibrator2-3Δf2～f2+3Δf2Wherein, f2Is the resonance frequency of the second vibrator, Δ f2The spectral width of the second oscillator.
In the case a), when the resonant frequency of the second oscillator is greater than the resonant frequency 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 coupled resonant circuit in the first loop generates an asymmetric fano resonant line type from a valley to a peak; 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.
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.
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 f2-3Δf2Change to f2+3Δf2I.e. the frequency f of the excitation signal satisfies: f. of2-3Δf2≤f≤f2+3Δf2Wherein, f2Is the resonance frequency of the second vibrator, Δ f2The 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: the frequency of the excitation signal being less thanWhen the deviation amount is far larger 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 is much greater than the spectrum width of the second oscillator, the phase of the second oscillator tends to be-pi/2, and 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 variable 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 reached. 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 f2-3Δf2To f2+3Δf2In the process of (1), the phase of the second oscillator changes from atan (6) 0.45 pi to-atan 6-0.45 pi, and the phase changes to
The utility model has the advantages that:
the utility model 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 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 deeply understand the physical meaning of the phase in the resonance, and is suitable for being developed as high-order content in college physical experiments.
Drawings
FIG. 1(a) is a schematic diagram of a Lorentzian line type according to the present invention, and FIG. 1(b) is a schematic diagram of a Fano resonance line type according to the present invention;
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(a) is a graph of experimental results of the mode (amplitude-frequency characteristic curve) of the reciprocal complex impedance of the first vibrator (solid line) and the second vibrator (dotted line) isolated in one embodiment of the coupled resonant circuit-based fano resonant experiment apparatus of the present invention, wherein the amplitude of the second vibrator is compressed to its original value 1/25 for convenience of display, fig. 3(b) is a graph of experimental results of the phase (phase-frequency characteristic curve) of the first vibrator (solid line) and the second vibrator (dotted line) isolated in one embodiment of the coupled resonant circuit-based fano resonant experiment apparatus of the present invention, fig. 3(c) is a graph of experimental results of the amplitude-frequency characteristic curve of the first vibrator in the coupled resonant circuit composed of the first vibrator and the second vibrator in one embodiment of the coupled resonant circuit-based fano resonant experiment apparatus of the present invention, and fig. 3(d) is a graph of experimental results of the phase-frequency characteristic curve of the first vibrator in the coupled resonant circuit composed of the first vibrator and the second vibrator A result graph;
FIG. 4(a) is a schematic diagram of the complex impedance Z of the first oscillator calculated by a theoretical formula in an embodiment of the Fano resonance experiment instrument based on the coupled resonant circuit of the present invention1(solid line) and complex impedance Z of the second oscillator with respect to the feedback of the first oscillator under coupling21(dotted line)) Fig. 4(b) is a graph of frequency characteristics of the coupled resonant circuit, where the total equivalent complex impedance Z ═ Z of the coupled resonant circuit in the first loop is calculated by using a theoretical formula in an embodiment of the coupled resonant circuit-based fanuo resonance experimental apparatus of the present invention1+Z21Fig. 4(c) is a complex impedance Z of the first oscillator calculated by a theoretical formula in an embodiment of the franco resonance experiment instrument based on the coupled resonant circuit of the present invention1(solid line) and complex impedance Z of the second oscillator with respect to the feedback of the first oscillator under coupling21FIG. 4(d) is a reciprocal Z of the total equivalent complex impedance of the coupled resonant circuit in the first loop, which is calculated by a theoretical formula in one embodiment of the Fano resonance experiment apparatus based on the coupled resonant circuit-1A frequency characteristic diagram of the mode of (a);
FIG. 5(a) is a diagram of an embodiment of a Fano resonance experiment apparatus based on a coupled resonant circuit according to the present invention2Fig. 5(b) is a complex impedance Z of the corresponding first oscillator calculated by using a theoretical formula in an embodiment of the franco resonance experiment instrument based on the coupled resonant circuit according to the present invention1(solid line) and complex impedance Z of the second oscillator for feedback action of the second oscillator on the first oscillator under coupling action21(dotted line) frequency characteristic diagram of phase, and FIG. 5(C) is a diagram of the frequency characteristic of the coupled resonant circuit-based Fano resonance tester of the present invention2Fig. 5(d) is a complex impedance Z of the corresponding first oscillator calculated by using a theoretical formula in an embodiment of the coupled resonant circuit-based fano resonant experimental apparatus of the present invention1(solid line) and complex impedance Z of the second oscillator with respect to the feedback of the first oscillator under coupling21Frequency characteristic diagram of phase (dotted line).
Detailed Description
The utility model 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 wide-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 cancelled by constructive change or is constructive by destructive change, the phases are nearly the same and are coherent constructive, the phase difference is nearly pi and is coherent cancellation, and the reciprocal of the total equivalent impedance of the first oscillator in the coupled resonance circuit in the first 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, and the situation is shown that the first oscillator has an asymmetric Fano resonance linear response spectrum, and the second oscillator is still a narrow-spectrum Lorentz linear response spectrum.
1. Experimental device
In this embodiment, as shown in fig. 2, the inductance of the first inductor is L1The capacitance value of the first capacitor is C1The resistance value of the first resistor is R1The self-inductance value of the primary coil of the mutual inductor is LM1Mutual inductance value of the mutual inductor is M, and inductance value of the second inductor is L2The capacitance value of the second capacitor is C2The resistance value of the second resistor is R2The self-inductance value of the secondary coil of the mutual inductor is LM2And the line end voltage output outwards by the alternating current signal source is US. Reciprocal Z of total equivalent complex impedance Z of coupled resonance circuit formed by first oscillator and second oscillator in first loop-1Characterizing 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 source1) More specifically, the modulus | Z of the reciprocal of the complex impedance-1The 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 circuit end voltage U output by the AC signal source is measuredSAnd a voltage U across the first resistorR1As a function of the frequency f of the excitation signal and using Z-1＝I1/US＝UR1/(R1·US) To obtain Z-1As 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. The AC signal source and the measuring instrument can adopt a common sine wave signal generator and an oscilloscope, an automatic measuring system is designed for improving the measuring speed, the 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 voltage U at the output end of the AC signal source in the first loopSAnd a voltage U across the first resistorR1And LabVIEW programming is used for processing data, so that experimental measurement results of the amplitude-frequency and phase-frequency characteristic curves of the system can be 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 appears in a weak coupling area. The self-inductance value LM of the primary coil of the mutual inductor1And self-inductance value LM of secondary coil2Also both 8 mH. Other parameters of the first oscillator are: l is a radical of an alcohol1＝32mH、C1＝0.08μF、R1The 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 and a small quality factor, and the corresponding resonance spectrum is wide and corresponds to wide-spectrum resonance required in Fano resonance. Other parameters of the second oscillator are: l is2＝32mH、C2＝0.02μF、 R2The mode of the inverse complex impedance measured when the second oscillator is in the isolated state is shown by the dotted line in fig. 3(a), the resistance value of the second resistor of the second oscillator is small, the quality factor is large, the corresponding resonance spectrum is narrow,corresponding 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. The measured resonance frequencies of the isolated first and second vibrators were about 2804Hz and 5551Hz, respectively, while the resonance frequencies theoretically calculated from the element parameter values were 2813Hz and 5627Hz, respectively, and the measured resonance frequencies substantially agreed with the theoretically calculated values with deviations of 0.3% and 1.4%, respectively, which were mainly attributed to the errors in the individual element parameter values used in the experiments.
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 the positive and negative of the phase are opposite to the rule adopted in the common resonance circuit), the phase at the resonance frequency is equal to 0. The phase place change of the first oscillator near resonance frequency in the phase-frequency characteristic curve is comparatively mild, the phase place change of second oscillator is comparatively rapid, and is consistent with wide spectrum and narrow spectrum resonance action that first and second oscillator showed respectively in the amplitude-frequency characteristic curve, embodies the general law of resonance system: i.e. the resonance behavior of amplitude with frequency and the phase variation with frequency are related to each other.
Next, a coupled resonant circuit formed by the first and second transducers, that is, the circuit shown in fig. 2 was measured, and the amplitude-frequency characteristic curve obtained was shown in fig. 3 (c). It can be seen that, in the vicinity of the resonance frequency of the second oscillator, the amplitude-frequency characteristic curve of the coupled resonant circuit rapidly changes from a valley to a peak, and a significantly asymmetric fano resonant line shape is exhibited. 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 fano resonant line type in the coupled resonant circuit is located close to the resonant 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 theoretical formulas and related 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 the theoretical expression is as follows:
wherein the content of the first and second substances,
L′1＝L1+LM1；
L′2＝L2+LM2。
here, Z1And Z2Respectively represents complex impedance L 'of isolated first oscillator and second oscillator'1And L'2Each 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 can be 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 oscillator1I.e. the complex impedance of the isolated first oscillator itself; complex impedance Z of the other part and the second oscillator2In relation to, reflecting the feedback effect of the second oscillator on the first oscillator under the coupling effect, using Z21To represent this portion of the complex impedance, i.e. Z21Defining a complex impedance for the feedback of the second oscillator to 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 effect21Complex impedance Z with isolated second oscillator2In an inverse relationship.
Substituting the element parameters used in the previous experiment into the theoretical formula to calculate to obtain Z1And Z21The results are shown as a solid line and a broken line in fig. 4(a), respectively. As can be seen, | Z1The 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 | Z1I is in inverse proportion; | Z1The change of | is more gentle because the first oscillator is wide spectrum resonance; i Z1At the resonant frequency (f) of the first vibrator12813Hz) because the total impedance of the series resonant circuit is minimal at the resonant frequency. In contrast, | Z21The 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 | Z21L is proportional to the modulus of the reciprocal of the complex impedance of the second oscillator; i Z21At the resonance frequency (f) of the second vibrator25627Hz) shows a very narrow peak because the resonance of the second oscillator is a narrow spectrum resonance. Comparison of | Z in 4(a)1I and I Z21The numerical value of | Z can be found only in the vicinity of the resonance frequency of the second vibrator21Numerical value of and | Z1I can be compared, and the second oscillator has obvious feedback effect on the first oscillator; and at other frequencies, | Z21|＜＜|Z1I.e. Z is approximately equal to Z1In this case, the feedback effect of the second oscillator on the first oscillator is negligible, and the response of the entire coupled resonant circuit is almost the same as that of the isolated first oscillator. Fig. 4(b) shows | Z | ═ Z calculated by a theoretical formula1+Z21The result of | the curve of the variation with the frequency f of the excitation signal is consistent with the analysis and prediction: curve only in the secondThe vicinity of the resonance frequency of the vibrator is significantly different from that of the isolated first vibrator, which explains why the experimentally measured region of the coupled resonance circuit in fig. 3(c) other than the vicinity of the resonance frequency of the second vibrator is almost identical to that of the isolated first vibrator in fig. 3 (a).
For further analysis of the vicinity Z of the resonance frequency of the second vibrator21The contribution to the coupled resonant circuit is calculated by using a theoretical formula to obtain Z1And Z21The 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 (c) 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 is21And
is in direct proportion. Arg (Z) of the first oscillator in wide-spectrum resonance in the vicinity of the resonance frequency of the second oscillator in narrow-spectrum resonance1) 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 time1) Close to pi/2; and arg (Z) of a second vibrator of narrow spectrum resonance21) The total equivalent impedance | Z | of the coupled resonant circuit will rapidly change from Z | to-pi/2 as the frequency of the excitation signal increases1And Z21To a maximum state given by the coherent constructive sum of1And Z21Given the minimum state. This is confirmed by the | Z | curve calculated by the theoretical formula in fig. 4(b), and in the vicinity of the resonance frequency of the second oscillator, | Z | surely changes from a peak to a valley quickly. Corresponding to it, | Z-1I.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-1I followThe change curve of the frequency f of the excitation signal and the theoretical calculation result are basically consistent with the experimental measurement result in fig. 3(c), 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 the second oscillator is coupled with the first oscillator, the vibration of the second oscillator is driven by the first oscillator through coupling action, and the second oscillator is 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 oscillator, the second oscillator has obvious feedback effect on the first oscillator, and the oscillation phase of the second oscillator has a change close to pi in a narrow frequency range, and the feedback effect and the coherent superposition of the oscillation of the first oscillator correspondingly have a change from cancellation to constructive, so that the amplitude of the first oscillator 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 was set20.02 muf, i.e. the resonance frequency of the isolated second vibrator is greater than the resonance frequency of the isolated first vibrator. 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 capacitor2The amplitude-frequency characteristic curve of the coupling resonant circuit measured experimentally with the resonance frequency of the second oscillator made smaller than the resonance frequency of the first oscillator at 0.16 μ F is shown in fig. 5(a) and is located on the left side of the resonance peak of the first oscillatorA pronounced asymmetric fano resonant line pattern appears. 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. In contrast to the foregoing, the farno resonance line type in fig. 5(a) no longer changes from a valley to a peak but from a peak to a valley as the frequency of the excitation signal increases. 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(b) show the capacitance value C of the second capacitor, respectively2Arg (Z) calculated by theoretical formula under the parameter of 0.16 μ F1) And arg (Z)21) As the frequency varied, since the resonance frequency of the second vibrator was smaller than that of the first vibrator at this time, arg (Z) was observed in the vicinity of the resonance frequency of the second vibrator1) 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 signal1And Z21Instead, 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-1I 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 set2The 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 and dashed lines in FIG. 5(d) show the capacitance value C of the second capacitor, respectively2Arg (Z) calculated by theoretical formula under parameter of 0.08 μ F1) And arg (Z)21) Depending on the change in the frequency of the excitation signal, it can be seen that since the resonance frequencies of the second and first transducers are equal at this time, arg (Z) is near the common resonance frequency of the two transducers1) 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 signal1And Z21Will experience a mismatch between (when arg (Z)21) π/2) to coherent constructive (when arg (Z)21) 0) to incoherent (when arg (Z)21) H-pi/2), the corresponding | Z | changes from valley to peak to valley, and | Z |-1The | is changed from peak to valley to peak, thereby showing the EIT-like phenomenon.
The utility model 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 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. It should be noted that although the famo resonance phenomenon and EIT-like phenomenon are shown in the coupled resonant circuit, as mentioned in the background section, the above phenomena are common in many physical systems, and the famo resonance phenomenon and EIT-like phenomenon can be realized by using the classical mechanical coupling system and the optical coupling system in the similar way.
It is finally noted that the disclosed embodiments are intended to aid in the further understanding of the utility model, but that those skilled in the art will appreciate that: various substitutions and modifications are possible without departing from the spirit and scope of this disclosure and the appended claims. Therefore, the utility model should not be limited by the disclosure of the embodiments, but should be defined by the scope of the appended claims.
Claims (7)
1. A coupled resonant circuit-based Fano resonance experimental instrument is characterized by comprising: 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; 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 form a second loop; the resistance value of a first resistor of the first oscillator is larger than that of a second resistor of the second oscillator, so that the first oscillator is a oscillator with Lorentz line type wide spectrum resonance, and 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.
2. The coupled resonant circuit-based fanuo resonance tester as recited in claim 1, wherein the resistance of the first resistor of the first oscillator is set to determine the quality factor of the first oscillator, and further determine the spectral width of the first oscillator, i.e. the full width at half maximum of the resonance peak of the response spectrum, so that the first oscillator is an oscillator having a wide-spectrum resonance of a lorentz line type; and setting the resistance value of a second resistor of the second oscillator to determine the quality factor of the second oscillator and further determine the spectrum width of the second oscillator, so that the second oscillator is an oscillator with Lorentz line type narrow spectrum resonance.
3. The coupled resonant circuit-based fanuo resonance experimental instrument as claimed in claim 2, wherein 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 larger the resistance value of the first resistor of the first oscillator is, the smaller the quality factor of the first oscillator is, and the larger the spectrum width of the first oscillator is; similarly, the larger the resistance of the second resistor of the second oscillator is, the smaller the quality factor of the second oscillator is, and the larger the spectrum width of the second oscillator is.
4. The coupled resonant circuit-based fano resonance experimental instrument as claimed in claim 1, wherein the parameters are set so that the coupled resonant circuit has the following two conditions: 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 half of 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; f in the vicinity of the resonance frequency of the second vibrator2-3Δf2～f2+3Δf2Wherein f is2Is the resonance frequency of the second vibrator, Δ f2Is the spectral width of the second vibrator; 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.
5. The coupled resonant circuit-based fanuo resonance experimental instrument according to claim 4, wherein in the case a), when the resonant frequency of the second vibrator is greater than the resonant frequency of the first vibrator, the inverse number of the total equivalent impedance of the first vibrator in the coupled resonant circuit in the first loop generates an asymmetric fanuo resonant line type from valley to peak during the process of scanning the frequency of the excitation signal 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.
6. The coupled resonant circuit-based fano resonance experimental instrument of claim 5, wherein the frequency of the excitation signal is swept through the second oscillatorThe resonance frequency of a sub-unit is the frequency f of the excitation signal from f2-3Δf2Change to f2+3Δf2I.e. the frequency f of the excitation signal satisfies: f. of2-3Δf2≤f≤f2+3Δf2Wherein, f2Is the resonance frequency of the second vibrator, Δ f2The spectral width of the second oscillator.
7. The coupled resonant circuit-based Fano resonance tester as recited in claim 6, wherein the sweep range of the frequency of the excitation signal is from f2-3Δf2To f2+3Δf2In the process of (3), the phase of the second oscillator changes from atan (6) to-atan 6 of-0.45 pi, and the phase changes to 0.9 pi.
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