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Frequency measurement and amplification of lidar echo signal based on optomechanical effects

Jun Wen, Youquan Dan, Xuecong Ma, LuoPeng Xu, and Peiyu Sun

Open Access Open Access

Abstract

In this work, we have proposed a new scheme to measure the frequency of lidar echo signals by optomechanically induced transparency or amplification. Our calculation is based on the quantum Langevin equations and the mean-field approximation. The phenomenon that the transmission probabilities of echo signal light of lidar are sensitive to its frequency gives us a method to obtain the probe frequency. The expressions of the external force that keeps the system on the blue or red sidebands are offered, which is the necessary condition for our measurements. For the blue sideband, the process leads to the amplification of the echo signal light if the frequency of the pump field satisfies certain relations. For the red sideband, the echo signal light may be fully absorbed and at this time whose frequency can be obtained by a formula that has been proved in this work. In contrast to the conventional method, our method may have higher accuracy, and cavity optomechanics can amplify the signal under the condition of the blue sideband. Our scheme is based on the quantum theory, which may be easier to associate with quantum Fisher information that can give the highest precision.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The optomechanical system, which consists of optical and mechanical modes, is a hot topic recently. Extensive research efforts about this field have been published, such as optical isolator [1], precision measurements [2,3], topological energy transfer [4], conversion between slow and fast light [5,6], and optomechanically induced transparency(OMIT) [7,8] and amplification(OMIA) [9], which have important applications in both engineering and theories.

Among the applications mentioned above, the OMIT, OMIA and precision measurements have aroused our interest. The OMIT arises from the radiation pressure coupling the optical and mechanical modes, which has been predicated by Agarwal [10] and observed experimentally in the whispering galley cavity [7]. As known to all, the OMIT and OMIA will show the sharp peak if the parameters are reasonable [5,9–11], the position of the peak is much sensitive to the frequency of the probe light (i.e., echo signal light of lidar), so, we may obtain the frequency of the probe light precisely. Recently, some investigative results indicate that the optomechanical system can amplify the strength of probe light and gain large transmission probabilities [12,13]. The advantages above provide a method for precision measurements and detection of signals. The measurements of frequencies and amplification of the signal light are the important tasks for lidar [14,15], and we plan to apply the optomechanical system to the fields of lidar.

Radar has been used in many fields since it appeared during World War II, the investigation about this instrument has never been interrupted. Recently, Doppler radar/lidar has to use three Fabry-Pérot interferometers to detect the frequency of the signals [16]; microwave photonic radars rely on electrical-to-optical (optical-to-electrical) conversions, which may introduce loss and noise [17]. The optomechanical system can measure the frequency and amplify the amplitude of echoes, and the merits are significant for lidar. In this paper, we use OMIT and OMIA by Fabry-Pérot cavity to measure out the unknown frequency of the probe light. The OMIT requires that the system is in the red sideband, and the pump field can convert the mechanical vibration to the probe field via ant-Stokes scattering, which leads to the coherent conversion and induces a transparent window [9]. But OMIA needs the system to meet the condition of the blue sideband which can also amplify the signal. The parameters of this system can also affect the accuracy of the results, and the effects resulting from the tunable parameters will be investigated in the later section. Furthermore, the advantage of our system is that we can measure the echo signals with different frequencies by adjusting the frequency of the pump field and external force that are easy to implement.

This paper is structured as follows: we present the model and the Hamiltonian of our system in Sec. 2. In Sec. 3, we offer the method to measure the frequency of the probe field under the red and blue sideband and discuss the influence resulting from the adjustable parameters. In Sec. 4, the error of the measurements has been analyzed. In the last section, a brief conclusion has been given.

2. Model and Hamiltonian

Figure 1 is the sketch of our system. The system has a two-side Fabry-Pérot cavity whose length is L and driven by a pump field with angular frequency ωd and amplitude εd. The angular frequency and amplitude of the probe field (lidar echo signal) are ωp and εp, respectively. The amplitude of the pump and probe field can be adjusted by their respective power [11]. The moving mirror is regarded as a harmonic oscillator that experiences an external force f with mass m and angular frequency ωm. In order to measure the frequency of the probe field, f needs to be regulated to meet the sideband conditions, but how to change the external force? We can assume that the moving mirror has been charged and put into the electric field, and the electric field can be adjusted precisely. Thus, the external force can be controlled via the electric field.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the system. A Fabry-Pérot cavity field is coupled to a moving mirror. The weak probe field and strong pump field can enter the cavity from the left side. The output field from the fixed mirror has been studied. The whole process is shown as the figure above.

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The Hamiltonian of our system can be written as

$$\begin{aligned} H &= \hbar \omega {}_\textrm{c}{c^ + }c + (\frac{{{p^2}}}{{2m}} + \frac{{m\omega _\textrm{m}^2{q^2}}}{2}) - fq - gq{c^ + }c\\ &+ i\hbar {\varepsilon _\textrm{d}}({c^ + }{e^{ - i{\omega _\textrm{d}}t}} - c{e^{i{\omega _\textrm{d}}t}}) + i\hbar {\varepsilon _\textrm{p}}({c^ + }{e^{ - i{\omega _\textrm{p}}t}} - c{e^{i{\omega _\textrm{p}}t}}), \end{aligned}$$
where $\hbar$ is the reduced Planck constant, p and q are the momentum and position operators respectively, g is the optomechanical coupling strength whose form is g=$\hbar$ωc/L, ωc is the angular frequency of the cavity, and c(c+) is the annihilation (creation) operator. In Eq. (1), the terms express the free energy for the cavity, the energy of the mechanical oscillator, the work arising from the external electric force, the radiation pressure coupling the cavity to the harmonic oscillator, the interaction between the cavity and the pump fields, and the interaction between the cavity and the probe fields, respectively.

In the frame rotating with the angular frequency ωd, the Hamiltonian of our system can be rewritten as

$$\begin{array}{c} {H_{rot}} = \hbar \delta {}_\textrm{c}{c^ + }c + (\frac{{{p^2}}}{{2m}} + \frac{{m\omega _\textrm{m}^2{q^2}}}{2}) - fq - gq{c^ + }c\\ + i\hbar {\varepsilon _\textrm{d}}({c^ + } - c) + i\hbar {\varepsilon _\textrm{p}}({c^ + }{e^{ - i\delta t}} - c{e^{i\delta t}}), \end{array}$$
where δc=ωc-ωd and δ=ωp-ωd are the detuning of the cavity and the probe filed of our system, respectively.

3. Frequency measurement method and measurement accuracy analysis

In order to obtain the output, we have to consider the dynamics of the system, which is governed by quantum Langevin equations [18,19]. The mean values of operators are the important issues for us; under the condition of the mean-field approximation [18,20], the mean value equations are given as

$$\frac{{d\langle p\rangle }}{{dt}} ={-} m\omega _\textrm{m}^2\langle q\rangle + g{\left|{\langle c\rangle } \right|^2} + f - {\gamma _\textrm{m}}\langle p\rangle,$$
$$\frac{{d\langle q\rangle }}{{dt}} = \frac{{\langle p\rangle }}{m},$$
$$\frac{{d\langle c\rangle }}{{dt}} ={-} [\kappa + i({\delta _\textrm{c}} - \frac{{g\langle q\rangle }}{\hbar })]\langle c\rangle + {\varepsilon _\textrm{d}} + {\varepsilon _\textrm{p}}{e^{ - i\delta t}},$$
where γm is the decay rate for the mechanical oscillator, and κ is the total decay rate of the cavity. In this paper, the decay rate of the Fabry-Pérot cavity results from two mirrors, and we assume that the decay rates of two mirrors are the same and its value is equal to κ/2. In order to obtain the solution of Eqs. (3)- (5), we assume the mean values of operators can be divided into three items [21–23]: <O>=Os+εp(O+e-iδt+O_eiδt), where, O can represent p, q, c, and the terms Os, O+ and O- are associated with fields with angular frequencies ωd, ωp and 2ωd-ωp respectively [5], this ansatz uses the perturbation method to solve the nonlinear quantum Langevin equations and the intracavity field around the steady-state values of operators, and the higher sidebands have been ignored. The premise of the ansatz of mean values above is that |εpO±| are much smaller than |Os| [7,13,24]. Inserting the formula of mean values into Eqs. (3)- (5), we can obtain
$${p_\textrm{s}} = 0,\quad {q_\textrm{s}} = \frac{{f + g|{c_\textrm{s}}{|^2}}}{{m\omega _\textrm{m}^2}},$$
$${c_\textrm{s}} = \frac{{{\varepsilon _\textrm{d}}}}{{\kappa + i\Delta }},$$
$${c_ + } = \frac{{({\delta ^2} - \omega _\textrm{m}^2 + i\delta {\gamma _\textrm{m}})[\kappa - i(\Delta + \delta )] - 2i\beta {\omega _\textrm{m}}}}{{({\delta ^2} - \omega _\textrm{m}^2 + i\delta {\gamma _\textrm{m}})[{\Delta ^2} + {{(\kappa - i\delta )}^2}] + 4\beta \Delta {\omega _\textrm{m},}}}$$
where β=g2|cs|2/(2$\hbar$mωm) and Δ=δc-gqs/$\hbar$ are the pivotal objects in our work. We have to obtain the values of β and Δ, which requires that qs and cs should be solved. According to Eqs. (6) and (7), qs satisfies the equation as follow
$$\begin{array}{l} \frac{m}{{{\hbar ^2}}}\omega _\textrm{m}^2{g^2}q_\textrm{s}^3 - (\frac{f}{{{\hbar ^2}}}{g^2} + \frac{2}{\hbar }m\omega _\textrm{m}^2g{\delta _\textrm{c}})q_\textrm{s}^2 + [m\omega _\textrm{m}^2({\kappa ^2}\\ + \delta _\textrm{c}^2) + \frac{2}{\hbar }fg{\delta _\textrm{c}}]{q_\textrm{s}} - [f({\kappa ^2} + \delta _\textrm{c}^2) + g\varepsilon _\textrm{d}^2] = 0. \end{array}$$

Now, the dynamics of our system has been known for given parameters and the general results can be obtained. In this work, we pay attention to the red and blue sideband, and how to get the output probe field? With the application of the input-output relation [7,25] εout1=εin-2(κ/2)<c > and the input field εin=εd+εpe-iδt in the rotating frame, the output field 1 has the following formal

$${\varepsilon _{\textrm{out}1}} = ({\varepsilon _\textrm{d}} - \kappa {c_\textrm{s}}) + {\varepsilon _\textrm{p}}(1 - \kappa {c_ + }){e^{ - i\delta t}} - \kappa {c_ - }{\varepsilon _\textrm{p}}{e^{i\delta t}}.$$

Here we pay much attention to the absorption of the probe field, and the quadrature of the output field has been defined as εT1=κc+. In order to prove some conclusion conveniently, we simplify the form of εT1 [5]

$${\varepsilon _{\textrm{T}1}} = \frac{\kappa }{{\kappa - i(\delta - \Delta ) + \frac{{2i\beta {\omega _\textrm{m}}}}{{({\delta ^2} - \omega _\textrm{m}^2 + i\delta {\gamma _\textrm{m}}) - \frac{{2i\beta {\omega _\textrm{m}}}}{{\kappa - i(\delta + \Delta )}}}},}}$$
the real and imaginary parts of quadrature represent absorption and dispersion of probe fields, respectively; and the transmission probabilities of the probe field is tp=|1-εT1|2. The appearance of peaks shown in Fig. 2 requires that the system meets the condition of the red or blue sideband. The first blue band satisfies Δ=-ωm, and the first red sideband satisfies Δ=ωm [5,26].

The position of peaks in Fig. 2 (a) and (b) is not at δ=-ωm for the blue sideband by the numerical simulation, the maximum value of tp occurs at δ=-ωm-2ωmβ/(κ2+4ωm2), this result can improve the accuracy of measurements and be proved in the Appendix; that tp >1 is OMIA which indicates the probe field has been amplified, and the reason for this phenomenon is that the pump field can generate photon-phonon pairs via the Stokes scattering. For the condition of the red sideband, the peaks take place at δ=ωm-2ωmβ/(κ2+4ωm2) [13] though Re[εT1] is not a negative number which does not meet the requirement Re[εT1]<0 in Ref [13]., this conclusion is still correct, which is verified by numerical calculation; according to Fig. 2(d), tp has two minima approaching to 0 which offers a method to obtain the probe frequency, the abscissae of the two minima of tp are δ=ωm±β1/2-2ωmβ/(κ2+4ωm2) which are proved in the Appendix.

 figure: Fig. 2.

Fig. 2. (a) The Re[εT1] as a function of δ with Δ=-ωm, (b) The tp as a function of δ with Δ=-ωm, (c) The Re[εT1] as a function of δ with Δ=ωm, and (d) The tp as a function of δ with Δ=ωm. The parameters of our system have been set as: m = 250 ng, ωc = 2πc0/λc (the wavelength λc = 1550 nm, c0 is the speed of light in the vacuum), L = 0.1 mm, ωm/2π=12 MHz, εd/2π=7×1010 Hz, κ/2π=4 MHz, δc=-10ωm, γm/2π=7.2 kHz.

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A issue has been proposed that the frequency of the probe field is unknown before it is measured, and we have to change the frequency of the pump field to meet the condition of the blue sideband δ=-ωm[1 + 2β/(κ2+4ωm2)] (or for the red sideband δ=ωm±β1/2-2ωmβ/(κ2+4ωm2)) which corresponds to the tp reaches the maximum and tp > 1(or the minima and tp→0 for the red sideband). This change causes the change of δc, and δc determines whether the system reaches the blue (red) sideband or not. In order to ensure our system satisfies the blue (red) sideband when ωd is changed, other parameters should be corresponding revised. For our system, f, δc and εd are the three tunable parameters, others cannot be adjusted. Thus, the condition of the blue sideband is satisfied, if

$$f = \frac{{\hbar m\omega _\textrm{m}^2({\delta _\textrm{c}} + {\omega _\textrm{m}})}}{g} - \frac{{g\varepsilon _\textrm{d}^2}}{{{\kappa ^2} + \omega _\textrm{m}^2.}}$$
Similarly, the condition corresponding to the red sideband
$$f = \frac{{\hbar m\omega _m^2({\delta _c} - {\omega _m})}}{g} - \frac{{g\varepsilon _d^2}}{{{\kappa ^2} + \omega _m^2,}}$$
The two formulas above provide the relationship between the three tunable parameters considering the sideband condition. Here, we assume that the surface of the moving mirror is charged, and the electric field acting on the mirror can be controlled, so, we can adjust the electric force f to ensure the system reaches the blue or red sideband if the pump frequency and strength are changed. The reason that we do not regulate the εd to meet the sideband condition is that εd affects the sideband much weaker than f.

According to Fig. 3, the curves have almost the same shape for the different δc under the condition of the red and blue sideband, implying that δc has little influence on the accuracy of the measurement of the probe frequency and the transmission probabilities for the certain range.

 figure: Fig. 3.

Fig. 3. (a) The tp as a function of δ with different δc under the condition of Δ=-ωm. (b) The tp as a function of δ with different δc under the condition of Δ=ωm. The parameters have been set as Fig. 2 except for δc.

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The effects of εd should be investigated. As the Fig. 4 shown, as the εd increases, the maximum of tp will decrease and the linewidth increases for the blue sideband; a similar phenomenon also exists in the whispering gallery cavity [9]. For the red sideband, the peak height and its linewidth of tp increase with the εd.

 figure: Fig. 4.

Fig. 4. The parameters have been set as Fig. 2 except for εd. (a) The tp as a function of δ with different εd under the condition of Δ=-ωm. (b) The tp as a function of δ with different εd under the condition of Δ=ωm.

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For the blue sideband (OMIA), increasing transmission probabilities and reducing the linewidth are good for measurements and amplification of the probe field, so we need to reduce the power of the pump field appropriately, and the power of the pump field must ensure that OMIA occurs and the maximum of tp is larger than 1.1. Therefore, we definite that the frequency measured under the blue sideband is equal to

$$\begin{array}{l} {\omega _\textrm{p}} = {\omega _\textrm{d}} - {\omega _\textrm{m}}[1 + {{2\beta } / {({\kappa ^2} + 4\omega _\textrm{m}^2)}}]\\ \;\quad = {\omega _\textrm{d}} - {\omega _\textrm{m}}\left[ {1 + \frac{{{g^2}|{\varepsilon_d}{|^2}}}{{\hbar m{\omega_\textrm{m}}({\kappa^2} + \omega_\textrm{m}^2)({\kappa^2} + 4\omega_\textrm{m}^2)}}} \right],\;\;(\textrm{when}\;{t_p} \ge 1.1) \end{array}.$$
For the red sideband (OMIT), the frequency measured in the red sideband when tp(δrmin) ≤ 1×10−5 and tp (δrmin-2β 1/2) ≤ 1×10−5 turns out to be
$$\begin{array}{l} {\omega _\textrm{p}} = {\omega _\textrm{d}} + {\omega _\textrm{m}} + \sqrt \beta - {{2{\omega _\textrm{m}}\beta } / {({\kappa ^2} + 4\omega _\textrm{m}^2)}}\\ \;\;\; = {\omega _\textrm{d}} + {\omega _\textrm{m}} + \frac{{g|{\varepsilon _\textrm{d}}|}}{{\sqrt {2\hbar m{\omega _\textrm{m}}({\kappa ^2} + \omega _\textrm{m}^2)} }} - \frac{{{g^2}|{\varepsilon _\textrm{d}}{|^2}}}{{\hbar m({\kappa ^2} + \omega _\textrm{m}^2)({\kappa ^2} + 4\omega _\textrm{m}^2)}}, \end{array}$$
and the frequency measured in the red sideband when tp(δrmin) ≤ 1×10−5 and tp (δrmin -2β 1/2)> 1×10−5 is given by
$$\begin{array}{l} {\omega _\textrm{p}} = {\omega _\textrm{d}} + {\omega _\textrm{m}} - \sqrt \beta - {{2{\omega _\textrm{m}}\beta } / {({\kappa ^2} + 4\omega _\textrm{m}^2)}}\\ \;\quad = {\omega _\textrm{d}} + {\omega _\textrm{m}} - \frac{{g|{\varepsilon _\textrm{d}}|}}{{\sqrt {2\hbar m{\omega _\textrm{m}}({\kappa ^2} + \omega _\textrm{m}^2)} }} - \frac{{{g^2}|{\varepsilon _\textrm{d}}{|^2}}}{{\hbar m({\kappa ^2} + \omega _\textrm{m}^2)({\kappa ^2} + 4\omega _\textrm{m}^2)}}\end{array}.$$
It should be noted that the results [i.e.Eqs. (15) and (16)] have high accuracy if εd/2π≤12×1010 Hz for the parameters in this work.

Now, we give the specific process of measurements. In order to get the frequency of the probe field, we don’t need to solve Eqs. (3)- (5) every time if we change the frequency of the pump field and f to satisfy the blue or red sideband condition. For the blue sideband, we can regard ωc-100ωm as the minimum angular frequency of ωd and increase ωd with the step size of 12 kHz until tp > 1.1(the maximum angular frequency of ωd is ωc+100ωm), in this process, we must change the external forces at the same time to ensure that the system is in the blue sideband. According to δ=-ωm-2ωmβ/(κ2+4ωm2), we can get the frequency of the probe field if tp > 1.1. For the red sideband, the process is similar to the blue sideband’s, and the difference is that we get the frequency of the probe field according to the concrete form of δrmin.

4. Error in frequency measurement

For the blue sideband, there are two roots to the equation tp(δb) = 1.1 and we assume the roots are δb1 and δb2, and the error of angular frequency of the probe field does not exceed Db=|δb1-δb2| for the measurement. As the εd increases, the width of the main peak of tp increases, which leads to the increase of Db. On the other hand, as the εd increases, the height of the main peak of tp decreases, and tp = 1.1 is closer to the maximum value, meaning that Db reduces. Consequently, Db is not a monotonic function of εd, and its maximum value is 35.28 kHz (corresponding the maximum wind speed error is less than 0.0274 m/s with λc = 1550 nm) as shown in Fig. 5(a).

 figure: Fig. 5.

Fig. 5. (a) The error Db as a function of εd under the condition of Δ=-ωm. (b) The error Dr1 and Dr2 as a function of εd which correspond to the red solid and blue dashed line under the condition of Δ=ωm respectively. The parameters have been set as Fig. 2 except for εd.

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For the red sideband, tp has two minima whose abscissas are δrmin1 and δrmin2(δrmin1 < δrmin2), and we assume that tp gets its minima if tp≤1×10−5 for the measurements. There are two solutions to the equation tp(δ) = 1×10−5 around δrminj (j = 1,2),and we assume the solutions are δrminj1 and δrminj2, respectively. Then, the errors can be defined as Dr1=|δrmin11-δrmin12| and Dr1=|δrmin21-δrmin22|, respectively. According to Fig. 5(b), the error of the probe frequency is less than 12.66 kHz (corresponding the maximum wind speed error is less than 0.00982 m/s with λc = 1550 nm), and the maximum error is approximately 12.66 kHz.

Three issues should be pointed out. One is that the laser can lead to heating the cavity, which can cause the instability of the cavity, and how to solve this problem? We can use the pulsed laser as the pump field. The second one is that f can affect qs which changes the length of the cavity; thus, g is not a constant. The frequency measurement method we proposed previously is based on g being constant. In order to investigate the error caused by g not being a constant, we can define a new quantity G whose form is G=$\hbar$ωc/[g(L + qs)]-1. And we replace g in Eq. (9) by $\hbar$ωc/(L + qs) and can get the relationship between f and qs precisely. 0.9 mN (or -0.9 mN) corresponds to the f that the system is in sideband when the detuning δc is 100ωm(or -100ωm) approximately. According to Fig. 6, the influence on L and g results from f can be ignored. The last one is that the Langevin equations consist of Brownian noise and vacuum radiation noise [27] which have not been taken into consideration, why? On the one hand, their mean values are 0; on the other hand, the effects of the noise are much small at room temperature [5]. The OMIT can be experimented at room temperature [28].

 figure: Fig. 6.

Fig. 6. (a) The qs as a function of f with δc=-10ωm. (b) The G as a function of f with δc=-10ωm. The parameters have been set as Fig. 2.

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5. Conclusion

In this work, we have offered the method to realize the measurement of the frequency of the probe field via OMIT and OMIA in the optomechanical system and studied the effects of the adjustable parameters, which may provide some simulation for the design of lidar, and the probe field can be amplified when the system is under the condition of the blue sideband. The expression of the external force that keeps the system on the blue (red) sideband is given by us, which is the necessary condition for the measurements of the unknown frequency. The expressions for abscissae of the minima and maximum have been obtained in this paper, whose importance is not limited to measuring the frequencies. Compared to the traditional method based on classical physics, this work offers a new means to measure the frequency of the probe field, which is more convenient to use the quantum Fisher information, which can give the highest accuracy, to optimize parameters and measurement schemes.

Appendix

(1) For the blue sideband, the abscissa of the maxima for the tp is δ=-ωm-2βωm/(κ2+4ωm2), and the process of derivation as follows

According to numerical simulation, the abscissa of the maxima for the tp will lead to the Im[εT1]→0. δ≈Δ=-ωm ensures the tp get the maxima under the condition of the blue sideband, so i(δ-Δ) in Eq. (11) can be ignored, and the following result can be obtained:

$${\mathop{\rm Im}\nolimits} [\frac{{2i\beta {\omega _\textrm{m}}}}{{({\delta ^2} - \omega _\textrm{m}^2 + i\delta {\gamma _\textrm{m}}) - \frac{{2i\beta {\omega _\textrm{m}}}}{{\kappa - i(\delta + \Delta )}}}}] = 0.$$
Taking δ2-ωm2≈-2ωm(δ+ωm) into consideration, the equation above can be simplified to the form
$$\frac{{\delta + {\omega _\textrm{m}}}}{\beta } + \frac{{2{\omega _\textrm{m}}}}{{{\kappa ^2} + 4\omega _\textrm{m}^2}} = 0.$$
Consequently, we can obtain
$$\delta ={-} \omega {}_\textrm{m}(1 + \frac{{2\beta }}{{{\kappa ^2} + 4\omega _\textrm{m}^2}}).$$
(2) For the red sideband, tp has two minima that approach 0, so we can get the conclusion Im[εT1] →0, and the proving process as follow
$$\begin{aligned} {t_\textrm{p}} & = |1 - {\varepsilon _{\textrm{T}1}}{|^2}\\ & = {(1 - Re [{\varepsilon _{\textrm{T}1}}])^2} + {({\mathop{\rm Im}\nolimits} [{\varepsilon _{\textrm{T}1}}])^2} \ge {({\mathop{\rm Im}\nolimits} [{\varepsilon _{\textrm{T}1}}])^2} \ge 0\end{aligned}.$$
Thus, tp→0 can lead to Im[εT1] →0. The approximate form of εT1 [5] is given by
$${\varepsilon _{\textrm{T}1}} \approx \frac{\kappa }{{\kappa - i(\delta - \Delta ) + \frac{{2i\beta {\omega _\textrm{m}}}}{{{\delta ^2} - \omega _\textrm{m}^2 + i\delta {\gamma _\textrm{m}}}}}}.$$
Considering the following approximation that can be simulated:
$$\Delta = {\omega _\textrm{m}},{\delta ^2} - \omega _\textrm{m}^2 = 2{\omega _\textrm{m}}(\delta - {\omega _\textrm{m}}),$$
$$|{\delta ^2} - \omega _m^2|\gg |\delta {\gamma _m}|,$$
we can obtain (δrmin-ωm)2=β by taking Im[εT1] = 0, namely, δrmin=ωm±β1/2. By observing the graphs (Figs. 2, 3 and 4) in this paper, we find the abscissa of the maxima may be the mean of the abscissas of the minima, we have to correct δrmin and ensure their mean is ωm-2βωm/(κ2+4ωm2), that’s to say, δrmin=ωm±β1/2-2βωm/(κ2+4ωm2).

Funding

Foundation of Civil Aviation Flight University of China (J2021-060, J2020-060 and JG 2019-19); Sichuan University Failure Mechanics & Engineering Disaster Prevention and Mitigation Key Laboratory of Sichuan Province Open Foundation (2020FMSCU02).

Acknowledgments

We thank Jian Qi Zhang for the helpful discussion.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (6)
Fig. 1.
Fig. 1. Schematic diagram of the system. A Fabry-Pérot cavity field is coupled to a moving mirror. The weak probe field and strong pump field can enter the cavity from the left side. The output field from the fixed mirror has been studied. The whole process is shown as the figure above.

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Fig. 2.
Fig. 2. (a) The Re[εT1] as a function of δ with Δ=-ωm, (b) The tp as a function of δ with Δ=-ωm, (c) The Re[εT1] as a function of δ with Δ=ωm, and (d) The tp as a function of δ with Δ=ωm. The parameters of our system have been set as: m = 250 ng, ωc = 2πc0/λc (the wavelength λc = 1550 nm, c0 is the speed of light in the vacuum), L = 0.1 mm, ωm/2π=12 MHz, εd/2π=7×1010 Hz, κ/2π=4 MHz, δc=-10ωm, γm/2π=7.2 kHz.

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Fig. 3.
Fig. 3. (a) The tp as a function of δ with different δc under the condition of Δ=-ωm. (b) The tp as a function of δ with different δc under the condition of Δ=ωm. The parameters have been set as Fig. 2 except for δc.

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Fig. 4.
Fig. 4. The parameters have been set as Fig. 2 except for εd. (a) The tp as a function of δ with different εd under the condition of Δ=-ωm. (b) The tp as a function of δ with different εd under the condition of Δ=ωm.

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Fig. 5.
Fig. 5. (a) The error Db as a function of εd under the condition of Δ=-ωm. (b) The error Dr1 and Dr2 as a function of εd which correspond to the red solid and blue dashed line under the condition of Δ=ωm respectively. The parameters have been set as Fig. 2 except for εd.

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Fig. 6.
Fig. 6. (a) The qs as a function of f with δc=-10ωm. (b) The G as a function of f with δc=-10ωm. The parameters have been set as Fig. 2.

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Equations (23)

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H = ℏ ω c c + c + ( p 2 2 m + m ω m 2 q 2 2 ) − f q − g q c + c + i ℏ ε d ( c + e − i ω d t − c e i ω d t ) + i ℏ ε p ( c + e − i ω p t − c e i ω p t ) ,
H r o t = ℏ δ c c + c + ( p 2 2 m + m ω m 2 q 2 2 ) − f q − g q c + c + i ℏ ε d ( c + − c ) + i ℏ ε p ( c + e − i δ t − c e i δ t ) ,
d ⟨ p ⟩ d t = − m ω m 2 ⟨ q ⟩ + g | ⟨ c ⟩ | 2 + f − γ m ⟨ p ⟩ ,
d ⟨ q ⟩ d t = ⟨ p ⟩ m ,
d ⟨ c ⟩ d t = − [ κ + i ( δ c − g ⟨ q ⟩ ℏ ) ] ⟨ c ⟩ + ε d + ε p e − i δ t ,
p s = 0 , q s = f + g | c s | 2 m ω m 2 ,
c s = ε d κ + i Δ ,
c + = ( δ 2 − ω m 2 + i δ γ m ) [ κ − i ( Δ + δ ) ] − 2 i β ω m ( δ 2 − ω m 2 + i δ γ m ) [ Δ 2 + ( κ − i δ ) 2 ] + 4 β Δ ω m ,
m ℏ 2 ω m 2 g 2 q s 3 − ( f ℏ 2 g 2 + 2 ℏ m ω m 2 g δ c ) q s 2 + [ m ω m 2 ( κ 2 + δ c 2 ) + 2 ℏ f g δ c ] q s − [ f ( κ 2 + δ c 2 ) + g ε d 2 ] = 0.
ε out 1 = ( ε d − κ c s ) + ε p ( 1 − κ c + ) e − i δ t − κ c − ε p e i δ t .
ε T 1 = κ κ − i ( δ − Δ ) + 2 i β ω m ( δ 2 − ω m 2 + i δ γ m ) − 2 i β ω m κ − i ( δ + Δ ) ,
f = ℏ m ω m 2 ( δ c + ω m ) g − g ε d 2 κ 2 + ω m 2 .
f = ℏ m ω m 2 ( δ c − ω m ) g − g ε d 2 κ 2 + ω m 2 ,
ω p = ω d − ω m [ 1 + 2 β / ( κ 2 + 4 ω m 2 ) ] = ω d − ω m [ 1 + g 2 | ε d | 2 ℏ m ω m ( κ 2 + ω m 2 ) ( κ 2 + 4 ω m 2 ) ] , ( when t p ≥ 1.1 ) .
ω p = ω d + ω m + β − 2 ω m β / ( κ 2 + 4 ω m 2 ) = ω d + ω m + g | ε d | 2 ℏ m ω m ( κ 2 + ω m 2 ) − g 2 | ε d | 2 ℏ m ( κ 2 + ω m 2 ) ( κ 2 + 4 ω m 2 ) ,
ω p = ω d + ω m − β − 2 ω m β / ( κ 2 + 4 ω m 2 ) = ω d + ω m − g | ε d | 2 ℏ m ω m ( κ 2 + ω m 2 ) − g 2 | ε d | 2 ℏ m ( κ 2 + ω m 2 ) ( κ 2 + 4 ω m 2 ) .
Im [ 2 i β ω m ( δ 2 − ω m 2 + i δ γ m ) − 2 i β ω m κ − i ( δ + Δ ) ] = 0.
δ + ω m β + 2 ω m κ 2 + 4 ω m 2 = 0.
δ = − ω m ( 1 + 2 β κ 2 + 4 ω m 2 ) .
t p = | 1 − ε T 1 | 2 = ( 1 − R e [ ε T 1 ] ) 2 + ( Im [ ε T 1 ] ) 2 ≥ ( Im [ ε T 1 ] ) 2 ≥ 0 .
ε T 1 ≈ κ κ − i ( δ − Δ ) + 2 i β ω m δ 2 − ω m 2 + i δ γ m .
Δ = ω m , δ 2 − ω m 2 = 2 ω m ( δ − ω m ) ,
| δ 2 − ω m 2 | ≫ | δ γ m | ,
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