Numerical investigation on the effect of extra-feedback target position on the fringe visibility in a dual optical feedback interferometry system

Yangfan Jiang; Bin Liu; Bin Liu; Yu Yang; Yuxi Ruan; Yanguang Yu

doi:10.1364/OE.501520

1. Introduction

Benefiting from the merits of low implementation cost, easy optical alignment, and compact structure as compared with the conventional two-beam interferometers, optical feedback interferometry (OFI), or laser self-mixing interferometry (SMI) has been widely reported in non-contact measurement and engineering applications during the past decades [1–16] . In typical OFI-based applications, the internal electric field of the laser intra-cavity and the radiation field reflected or scattered by the external target are superimposed to form OFI fringes, which can be correlated with the real-time state characteristics of the external target, such as position, motion state, etc. The basic displacement measurement resolution of the OFI technology is half laser wavelength as same with the conventional two-beam interferometer, i.e., each OFI fringe captured by the built-in photodetector of the laser corresponds to a half-wavelength change of the external target displacement [17–19]. However, one significant challenge of OFI technology is that the visibility of the OFI fringe is often lower than that of two-beam interferometers, which may induce low measurement performance, especially when the feedback intensity fed back into the laser cavity is weak.

In recent years, with the implementation of dual optical feedback interferometry (DOFI) system formed by adding an additional feedback channel to the original OFI structure, some researchers have not only extended the application of OFI in engineering, e.g., realizing multi-dimensional position sensing at sub-wavelength scale [20] and displacement measurement with sub-wavelength accuracy based on nonlinear frequency mixing [21], but also improved the sensing performance of an OFI system, e.g., avoiding the error caused by incident angle measurement in angular velocity sensing [22], reducing the phase error caused by optical frequency fluctuation and external environment [23], improving the vibration measurement performance [24] and the signal-to-noise ratio in low power interferometry [25]. In our previous work [26], we utilized the two-cavity optical feedback structure to improve the sensing sensitivity of an OFI displacement measurement system. An open problem in the works [24–26] is that in order to ensure that the DOFI configuration can exhibit superior sensing performance compared with the single-channel OFI, the length of the extra optical feedback (it is called as extra-feedback in the following) channel within the laser half-wavelength scale has to be carefully selected [25,26], or the initial phase of the extra-feedback light fed back to the laser has to be controlled as each half-wavelength variation of the extra-feedback cavity length induce a $2\pi $ variation of the initial optical phase. However, those reported works do not carry out in-depth research analysis and give a reasonable explanation for this phenomenon.

For this reason, in this article, we aim to theoretically investigate the influence of the extra-feedback target position on the OFI fringe visibility to build a high-sensitivity DOFI sensing system. Firstly, a simplified theoretical analytical model for describing a DOFI system at steady state was derived. Secondly, after the expansion and approximation of the modified phase equation in steady state, a method for solving the analytical model was developed. Finally, the effect of small changes in the position of the extra-feedback stationary external target within laser half-wavelength range on the OFI fringe signal strength was investigated. The results in this work give comprehensive theoretical analyses on the influence of extra-feedback position on the enhancement of OFI fringe amplitude enhancement in a DOFI system, which provide helpful guidance for design of high-sensitivity OFI-based sensing system.

2. Theory and analysis of a DOFI system

2.1 Theoretical model

Aiming to clarify the characteristics of a DOFI structure, a simplified schematic diagram of the DOFI configuration is depicted in Fig. 1. As we can see, the light emitted by a laser diode (LD) is separated by a beam splitter (BS) with 50:50 split ratio into two beams. One beam of light is directed to a stationary target (ST) and another beam of light directed to a moving target (MT) to be measured. Part of the light transmitted to the surface of the MT will be fed back to the LD cavity, causing periodic fluctuations in the LD output optical power, and this real-time power change will be collected by the laser's embedded photodetector (EM-PD). The light reflected by the ST reenters the laser cavity to enhance the magnitude of the OFI fringe. In particular, the distance from MT to LD is denoted by ${L_1}$ and the distance from ST to LD is ${L_2}$.

Fig. 1. A simplified schematic diagram of the DOFI configuration.

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A group of nonlinear mutually coupled differential equations known as the Lang-Kobayashi (L-K) equations is commonly used to characterize the dynamics of a single-channel OFI system [18,27]. After adding some feedback delay terms associated with ${L_2}$ to the L-K equations, the equations describing the electric field amplitude $E(t )$, the electric field phase $\phi (t )$ and the carrier density $N(t )$ in the DOFI configuration can be separately given by Eqs. (1)-(3) [26].

(1)$$\begin{aligned} \frac{{dE(t )}}{{dt}} &= \frac{1}{2}\left\{ {{G_N}[{N(t )- {N_0}} ]- \frac{1}{{{\tau_p}}}} \right\}E(t )+ \frac{{{\kappa _1}}}{{{\tau _{in}}}}E({t - {\tau_1}} )\cos [{{\omega_0}{\tau_1} + \phi (t )- \phi ({t - {\tau_1}} )} ]\\ &\quad + \frac{{{\kappa _2}}}{{{\tau _{in}}}}E({t - {\tau_2}} )\cos [{{\omega_0}{\tau_2} + \phi (t )- \phi ({t - {\tau_2}} )} ], \end{aligned}$$

(2)$$\begin{aligned} \frac{{d\phi (t )}}{{dt}} &= \frac{\alpha }{2}\left\{ {{G_N}[{N(t )- {N_0}} ]- \frac{1}{{{\tau_p}}}} \right\} - \frac{{{\kappa _1}}}{{{\tau _{in}}}}\frac{{E({t - {\tau_1}} )}}{{E(t )}}\sin [{{\omega_0}{\tau_1} + \phi (t )- \phi ({t - {\tau_1}} )} ]\\ &\quad - \frac{{{\kappa _2}}}{{{\tau _{in}}}}\frac{{E({t - {\tau_2}} )}}{{E(t )}}\sin [{{\omega_0}{\tau_2} + \phi (t )- \phi ({t - {\tau_2}} )} ], \end{aligned}$$

(3)$$\frac{{dN(t )}}{{dt}} = \frac{J}{{eV}} - \frac{{N(t )}}{{{\tau _s}}} - {G_N}[{N(t )- {N_0}} ]\times {E^2}(t ).$$

Specially, the physical meaning and typical numerical simulation values of these parameters appearing in the above equations can be found in Table 1 [25]. In general, the output optical power of an LD can be expressed by the square term of the electric field amplitude, namely ${E^2}(t )$.

Table 1. Physical meanings and typical simulation values of parameters appearing in Eqs. (1)–(3)

View Table

With a view to exploring the mathematical model between the input excitation and the output response of the DOFI system, it is necessary to calculate the steady-state solutions of the dynamic variables in Eqs. (1)–(3) by setting as below:

(4)$$\frac{{dE(t )}}{{dt}} = 0,\textrm{ }\frac{{d\phi (t )}}{{dt}} = {\omega _s} - {\omega _0},\textrm{ }\frac{{dN(t )}}{{dt}} = 0.$$

Here, ${\omega _s}$ represents the laser angular frequency under dual optical feedback disturbance. Applying the above assumptions to Eqs. (1)–(3), we can get the following results:

(5)$$0 = \frac{1}{2}\left[ {{G_N}({{N_s} - {N_0}} )- \frac{1}{{{\tau_p}}}} \right]{E_s} + \frac{{{\kappa _1}}}{{{\tau _{in}}}}{E_s}\cos ({{\omega_s}{\tau_1}} )+ \frac{{{\kappa _2}}}{{{\tau _{in}}}}{E_s}\cos ({{\omega_s}{\tau_2}} ),$$

(6)$${\omega _s} - {\omega _0} = \frac{\alpha }{2}\left[ {{G_N}({{N_s} - {N_0}} )- \frac{1}{{{\tau_p}}}} \right] - \frac{{{\kappa _1}}}{{{\tau _{in}}}}\sin ({{\omega_s}{\tau_1}} )- \frac{{{\kappa _2}}}{{{\tau _{in}}}}\sin ({{\omega_s}{\tau_2}} ),$$

(7)$$0 = \frac{J}{{eV}} - \frac{{{N_s}}}{{{\tau _s}}} - {G_N}({{N_s} - {N_0}} )\times {E_s}^2,$$

where ${E_s}$ and ${N_s}$ refer to the steady-state solutions of electric field amplitude and carrier density, respectively. A first-order approximation describing the optical power output of the DOFI system in steady state can be obtained as follows:

(8)$${E_s}^2 = \frac{{{\tau _p}}}{{{\tau _s}}} \times \left( {\frac{{J{\tau_s}}}{{eV}} - {N_0} - \frac{1}{{{G_N}{\tau_p}}}} \right) + \frac{{2{\tau _p}^2}}{{{\tau _s}}} \times \left( {\frac{{J{\tau_s}}}{{eV}} - {N_0}} \right) \times \left[ {\frac{{{\kappa_1}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_1}} )+ \frac{{{\kappa_2}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_2}} )} \right].$$

The details of the derivation of Eq. (8) can be found in Appendix A.

As a result, the analytical model of a DOFI system in steady state is composed of Eqs. (20) and (8), which govern the laser optical frequency and intensity respectively. In Eq. (8), the first part is the stationary term representing the average intensity, and the second part is the variation term determining the magnitude and visibility of the OFI fringes. In this work, we mainly consider the case that the optical feedback strength of the target to be measured is weak and relatively larger extra-feedback strength from a stationary target is applied to enhance the magnitude of the OFI fringe. Note that when ${\kappa _2} = 0$, we get the conventional single feedback OFI model. Hence, from Eq. (8), it can be found that the magnitude enhancement of the OFI fringe induced by the extra-feedback mainly results from the term of $({{{{\kappa_2}} / {{\tau_{in}}}}} )\cos ({{\omega_s}{\tau_2}} )$. In the following, we will present the details of how to solve Eq. (20) to get the $\cos ({{\omega_s}{\tau_2}} )$ and further unveil the influence of the initial position and feedback strength from the extra-feedback target on the OFI fringe magnitude and visibility.

2.2 Method for solving the phase equation

Starting from the phase equation in steady state as shown in Eq. (20), by first multiplying the left and right terms of the equation by the constant coefficient ${\tau _2}$ and then adding $\arctan \alpha $, we have:

(9)$$\begin{array}{l} {\omega _s}{\tau _2} + \arctan \alpha = {\omega _0}{\tau _2} + \arctan \alpha - \frac{{{\kappa _1}{\tau _2}}}{{{\tau _{in}}}}\sqrt {1 + {\alpha ^2}} \sin ({{\omega_s}{\tau_1} + \arctan \alpha } )\\ \textrm{ } - \frac{{{\kappa _2}{\tau _2}}}{{{\tau _{in}}}}\sqrt {1 + {\alpha ^2}} \sin ({{\omega_s}{\tau_2} + \arctan \alpha } ). \end{array}$$

To make Eq. (9) more concise, the following definition is used:

(10)$${Q_i} = {{{\kappa _i}{\tau _2}\sqrt {1 + {\alpha ^2}} } / {{\tau _{in}}}},\textrm{ i = 1, 2}$$

(11)$${\Phi _{02}} = {\omega _0}{\tau _2} + \arctan \alpha \textrm{ }$$

(12)$${\Phi _{FBi}} = {\omega _s}{\tau _i} + \arctan \alpha ,\textrm{ i = 1, 2}$$

So the phase equation can also be rewritten as follows:

(13)$${\Phi _{FB2}} = {\Phi _{02}} - {Q_1}\sin ({{\Phi _{FB1}}} )- {Q_2}\sin ({{\Phi _{FB2}}} ).$$

Since the linewidth enhancement factor $(\alpha )$ is a measurable fixed physical quantity of a laser, and the length $({{L_2}} )$ of the extra-feedback channel is unchanged, ${\Phi _{02}}$, ${Q_1}$ and ${Q_2}$ are all constants that can be calculated. Besides, the feedback phase ${\omega _s}{\tau _1}$ can be rewritten as follows:

(14)$$\begin{array}{l} {\omega _s}{\tau _1} = ({{\omega_0} + \Delta \omega } )\times ({{\tau_{01}} + \Delta {\tau_1}} )\\ \textrm{ } = {\omega _0}{\tau _{01}} + {\omega _0}\Delta {\tau _1} + \Delta \omega {\tau _{01}} + \Delta \omega \Delta {\tau _1}, \end{array}$$

where ${\tau _{01}}$ represents the round trip time of light in the range of the initial length ${L_{01}}$ of the motion channel, and $\Delta {\tau _1}$ refers to the change in the round trip time of light in the motion channel caused by the movement of the target MT. Due to the last term on the right-hand side of Eq. (14) is a second-order term and thus can be neglected, the following approximation can be obtained:

(15)$${\omega _s}{\tau _1} = {\omega _s}{\tau _{01}} + {\omega _0}\Delta {\tau _1}.$$

To reduce the computational steps when dealing with Eq. (13), combined with Eqs. (12) and (15), we rewrite the parameter ${\Phi _{FB1}}$ in the form of ${\Phi _{FB2}}$:

(16)$${\Phi _{FB1}} = {\Phi _{FB2}} \times \frac{{{L_{01}}}}{{{L_2}}} + \left( {1 - \frac{{{L_{01}}}}{{{L_2}}}} \right)\arctan \alpha + {\omega _0}\Delta {\tau _1}.$$

Substituting Eq. (16) into Eq. (13), we obtain the following:

(17)$${\Phi _{FB2}} = {\Phi _{02}} - {Q_1}\sin \left[ {\frac{{{L_{01}}}}{{{L_2}}} \times {\Phi _{FB2}} + \left( {1 - \frac{{{L_{01}}}}{{{L_2}}}} \right)\arctan \alpha + {\omega_0}\Delta {\tau_1}} \right] - {Q_2}\sin ({{\Phi _{FB2}}} ).$$

Combined with the fact that the sine function has maximum and minimum values of 1 and -1, respectively, the following equation holds:

(18)$${\Phi _{02}} - {Q_1} - {Q_2} \le {\Phi _{FB2}} \le {\Phi _{02}} + {Q_1} + {Q_2}.$$

Here ${\Phi _{02}}$, ${Q_1}$ and ${Q_2}$ are all positive numbers. With the help of the numerical simulation tool MATLAB, the data set of ${\Phi _{FB2}}$ in Eq. (17) can be easily solved within the range of Eq. (18). Then, combined with Eq. (16) and the definition of ${\Phi _{FB2}}$ in Eq. (12), the phase of extra-feedback light ${\omega _s}{\tau _2}$ and the change of ${\omega _s}$ can be calculated, and the phase of the feedback light in the motion feedback channel ${\omega _s}{\tau _1}$ can be further obtained.

Next, we verify the validation of the above method by comparing the results with the numerical simulation results from the L-K equations. As the operation parameters may influence the behavior of the DOFI system, e.g., chaos may be induced by strong feedback strength, we only consider the occasion that the laser is at steady state and there is no fringe loss occurred. The details of keeping the system in steady state can be referred to our previous work [28]. Here, we set a displacement as shown in Fig. 2(a) on the target MT. In this case, the length of the MT channel is expressed as ${L_1} = {L_{01}} + \Delta {L_1}$, where ${L_{01}}$ is the initial external cavity length for the MT channel, and $\Delta {L_1}$ is the displacement applied on the MT. The length of the ST is expressed as ${L_2} = {L_{02}} + \Delta {L_2}$, where ${L_{02}}$ is the initial external cavity length of the ST channel, and $\Delta {L_2}$ is the ST position within the laser wavelength scale. In the simulations, ${L_{01}}$ and ${L_2}$ are 10 cm and 13 cm respectively. The initial laser wavelength (${\lambda _0}$) is 780 nm, and the linewidth enhancement factor is 3. Besides, we set $J = 1.1{J_{th}}$, where ${J_{th}}$ is the threshold of the injection current. Here we converted the feedback strength ${\kappa _i}$ into the feedback factor ${C_\textrm{i}}$ with the relation ${C_\textrm{i}}\textrm{ = }{\kappa _i}{\tau _i}\sqrt {1\textrm{ + }{\alpha ^2}} /{\tau _{in}}$ since the feedback factor is used more in the OFI community. The feedback factor ${C_1}$ is set with 0.1. As a displacement in Fig. 2(a) is applied on the target MT, we get the laser intensity or called the DOFI signals as shown in Figs. 2(b)-(d), where ${C_2}$ is 0.5, 1.0 and 2.0 respectively. It should be noted that the signal shown by the solid black lines is obtained by solving the L-K equations using the fourth-order Runge-Kutta algorithm, while the signals shown by the dotted red lines are obtained by solving Eqs. (20) and (8) using the above method. Both of them maintain a high degree of agreement from both the DC part and the AC fluctuation. This fully demonstrates the accuracy and feasibility of the proposed feedback phase solving method and lays a solid foundation for the next part of our research.

Fig. 2. Validation of the proposed feedback optical phase solution method: (a) displacement applied on the target MT; (b)-(d) OFI signals when C₂ equals 0.5, 1.0 and 2.0 respectively.

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3. Effect of the length of the extra-feedback channel on the DOFI signal

In this section, we investigate the influence of the length of the extra-feedback channel, i.e., the position of the extra-feedback target or initial optical phase of the extra feedback, on the magnitude of DOFI signals. Under the simulation environment of $J = 1.1{J_{th}},\textrm{ }\alpha = 6$, ${C_1} = 0.2,\textrm{ }{C_2} = 0.5,\textrm{ }{L_{01}} = 10\textrm{ }cm$ and ${L_{02}} = [11.7cm/{\lambda _0}]\ast {\lambda _0}$, where $[{\cdot} ]$ represents the operation of rounding down to the nearest integer. It means ${L_{02}}$ is the quotient of ${L_2}$ with respect to ${\lambda _0}$, while $\Delta {L_2}$ exists as the remainder in our simulation calculations. The detailed procedure is as below:

1. Set a displacement on the target MT.
2. Remove the extra-feedback stationary target ST, i.e., set ${\kappa _2}\textrm{ = }0$. Simulate Eqs. (20) and (8) to get the single-channel OFI signal.
3. Add the extra-feedback target ST and change the position of ST$({\Delta {L_2}} )$ within one laser wavelength with a step of 0.025${\lambda _0}$. Simulate Eqs. (20) and (8) to get the DOFI signal with each position of ST. Compare the DOFI signal and OFI signal in step (2).
4. Record the initial positions of the ST ($\Delta {L_2}$) and the corresponding amplitudes of the DOFI fringes in step (3).

As shown in the simulation results of Fig. 3, the amplitude of DOFI fringe varies periodically with the position of ST $\Delta {L_2}$ with a period of 0.5${\lambda _0}$. In Fig. 3, the peak and valley amplitude points within a period are marked separately by points A and B. Note that in our simulation setup, both in DOFI system and single-channel OFI system, the beam splitter is always present, that is, the only operation for the transition from dual feedback to single feedback is to remove the external target 2, so the value of ${\kappa _1}$ does not change in both DOFI system and single-channel OFI system.

Fig. 3. The influence of the location of the extra-feedback target on the amplitude of DOFI signals.

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We then investigated the influence of C₂ and α on the results. Another set of basic simulation parameters is set to $J = 1.7{J_{th}},\alpha = 5,{L_{01}} = 12cm,{L_{02}} = 55cm$ and ${C_1} = 0.1$. The effects of the extra-feedback channel length on the amplitude of the DOFI signal for the cases of feedback factor C₂ equals to 0.4, 1.1 and 1.5 are shown in Fig. 4. Similarly, based on the simulation environment in Fig. 5(a), the variation of DOFI signal amplitude when α value is set to 3, 4 and 5 is also shown in Fig. 5. All of these simulation results shows that the DOFI fringe can be enlarged most when $\Delta {L_2}$ is around $\textrm{0}\textrm{.125}{\lambda _0}$ and it has the lowest amplitude when $\Delta {L_2}$ is around $\textrm{0}\textrm{.375}{\lambda _0}$. Moreover, Fig. 6 shows the simulation comparison of single-channel OFI signal and DOFI signal when the extra-feedback point is at points A and B, i.e., $\Delta {L_2}\textrm{ = 0}\textrm{.125}{\lambda _0}$ and $\Delta {L_2}\textrm{ = 0}\textrm{.375}{\lambda _0}$, respectively with other parameters the same in Fig. 3. Obviously, in the evaluation criterion of signal amplitude, the DOFI fringe is enlarged compared to the single-channel OFI fringe with $\Delta {L_2}\textrm{ = 0}\textrm{.125}{\lambda _0}$, while it diminishes with $\Delta {L_2}\textrm{ = 0}\textrm{.375}{\lambda _0}$. The results shows that the amplitude of the DOFI fringe is even lower than that of the signal-channel OFI fringe without extra feedback, which means that an optical extra-feedback cannot guarantee the enhancement of OFI fringe amplitude unless a proper position of the extra-feedback target is set.

Fig. 4. The variation of DOFI signal amplitude with different C₂ values. (a) C₂ = 0.4; (b) C₂ = 1.1; (c) C₂ = 1.5.

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Fig. 5. The variation of DOFI signal amplitude with different α values. (a) α = 3; (b) α = 4; (c) α = 5.

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Fig. 6. Comparison of single-channel OFI signal and DOFI signal with different extra-feedback positions: (a) displacement of the target MT; (b) single-channel OFI signal; (c),(d) DOFI signals at point A and point B, respectively.

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In this case, the amplitude of the OFI fringe may be enlarged or diminished. Taking Points A and B in Fig. 3 as the example, the feedback terms $\cos ({{\omega_s}{\tau_1}} )$ and $\cos ({{\omega_s}{\tau_2}} )$ are calculated by applying the method for solving the phase equation proposed in the previous section. When it is at the point of maximum amplitude of DOFI signal (point A), there is basically an in-phase change between the two feedback terms, as shown in Fig. 7, which is also the reason why the amplitude of DOFI fringe is enhanced. Similarly, when the amplitude of the DOFI signal is at the minimum point (point B), an almost anti-phase change between the two feedback terms is shown in Fig. 8, which will lead to a lower fringe amplitude than the single-channel OFI signal.

Fig. 7. Feedback term of DOFI system under simulation conditions at point A.

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Fig. 8. Feedback term of DOFI system under simulation conditions at point B.

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The underpinning reason why the introduction of extra-feedback and its positions affect the amplitude of the OFI fringe may be explained as follows. Firstly, it is worth emphasizing that from the perspective of Eq. (8), the ability of the two-channel optical feedback structure to enhance the sensing sensitivity of OFI fringes originates from the feedback term $\cos ({{\omega_s}{\tau_2}} )$. Because ${\tau _2}$ is a constant, the variation of ${\omega _s}$ contributes to the periodic fluctuations of $\cos ({{\omega_s}{\tau_2}} )$. Meanwhile, it can be further concluded that every $\pi c/{L_2}$ change of ${\omega _s}$ corresponds to $2\pi $ change of $\cos ({{\omega_s}{\tau_2}} )$ phase, but under the limitation of weak external optical feedback discussed in this paper, the fluctuation range of ${\omega _s}$ is much smaller than that of $\pi c/{L_2}$. Specifically, the fluctuation range of the feedback term $\cos ({{\omega_s}{\tau_2}} )$ is much smaller than that of $\cos ({{\omega_s}{\tau_1}} )$, which can be well reflected in Figs. 7 and 8. Then, based on phase equation Eq. (20) and simulations, it can be easily demonstrated that the change of ${\omega _s}$ and the change of term $\sin ({{\omega_s}{\tau_1} + \arctan \alpha } )$ are always approximately oppositely when ${L_1}$ changes. Given that the $\alpha $ factor in Fig. 3 has a simulated value of 6, we have $\arctan 6 \approx 0.45\pi$. In other words, the change in ${\omega _s}$ is also approximately inversely related to the change in $\cos ({{\omega_s}{\tau_1}} )$.

When $\Delta {L_{02}}$ is $0.125{\lambda _0}$, it will provide the cosine term $\cos ({{\omega_s}{\tau_2}} )$ with an initial phase of $\pi /2$. Due to the properties of the cosine function, the slope is maximized when the independent variable $(x )$ is at $\pi /2 + n\ast 2\pi $(n is an integer), which is $- 1$. When the slight periodic fluctuation of ${\omega _s}{\tau _2}$ is passed to this function in this case, the vibration amplitude of the feedback term $\cos ({{\omega_s}{\tau_2}} )$ reaches the maximum and remains in anti-phase with the change of ${\omega _s}$, which also explains the reason why the maximum enhancement of the visibility of OFI fringes can be achieved under the simulation conditions at point A in Fig. 3. Similarly, taking the vicinity of $0.375{\lambda _0}$ as the value of $\Delta {L_{02}}$ will provide the cosine term $\cos ({{\omega_s}{\tau_2}} )$ with an initial phase of $3\pi /2$. It is not difficult to conclude that when the small periodic fluctuation of ${\omega _s}{\tau _2}$ is passed to this function in this case, the feedback term $\cos ({{\omega_s}{\tau_2}} )$ is positively correlated with the change of ${\omega _s}$, and then negatively correlated with the change of $\cos ({{\omega_s}{\tau_1}} )$. In addition, the maximum vibration amplitude of $\cos ({{\omega_s}{\tau_2}} )$ also leads to the lowest DOFI sensing sensitivity under simulation conditions at point B in Fig. 3.

According to Eq. (8), it can be found that the extra-feedback strength ${\kappa _2}$ also influences the amplitude of the DOFI fringe. Still taking points A and B as the example, the influence of ${\kappa _2}$ on the amplitude of the DOFI fringe is investigated. The results are shown in Fig. 9. Meanwhile, based on the simulation conditions at points A and B, the effect of linewidth enhancement factor $\alpha $ on the amplitude of the DOFI signal is also plotted in Fig. 10. It is not difficult to find that the amplitude of the DOFI fringe increases with ${\kappa _2}$ and $\alpha $ at point A, while it decreases with ${\kappa _2}$ and $\alpha $ at point B. Therefore, it can be concluded that when the control is around ${\lambda _0}/8 + N\ast {\lambda _0}/2$(N is a positive integer) of the extra-feedback target position, we can better play the advantage of high sensing sensitivity of DOFI configuration. However, when the extra-feedback target position is around $3{\lambda _0}/8 + N\ast {\lambda _0}/2$(N is a positive integer), this will further reduce the sensing performance of OFI.

Fig. 9. (a) and (b) show the effect of feedback strength ${\kappa _2}$ on DOFI signal amplitude under simulation conditions at points A and B, respectively.

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Fig. 10. (a) and (b) show the effect of linewidth enhancement factor $\alpha $ on DOFI signal amplitude under simulation conditions at points A and B, respectively.

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4. Conclusion

This work investigated the effect of the position of the extra-feedback target on the sensing performance in a DOFI configuration. The obtained results show that the amplitude of the DOFI signal changes periodically with the position of the extra-feedback target with a period of ${\lambda _0}/2$. Through the analytical discussion of the DOFI model, it can be found that this is the result of the feedback terms $({\cos ({\omega_s}{\tau_1})\textrm{ and }\cos ({\omega_s}{\tau_2})} )$ from the two feedback channels jointly modulating the output optical power of the LD. With the help of the proposed method for solving the phase equation, it can be concluded that when the position of the control extra-feedback target is around ${\lambda _0}/8 + N\ast {\lambda _0}/2$(N is an integer), the two feedback terms show in-phase changes, resulting in the largest DOFI signal amplitude. In the meantime, when the position of the extra-feedback target is set near$3{\lambda _0}/8 + N\ast {\lambda _0}/2$(N is a positive integer), the two feedback terms show an anti-phase change, resulting in the smallest DOFI signal amplitude, which is not desired in the sensing measurement. Furthermore, we also find that when the position of the extra-feedback target is controlled near ${\lambda _0}/8 + N\ast {\lambda _0}/2$ (N is an integer), appropriately increasing the amount of feedback provided by the extra-feedback channel and selecting LD with larger $\alpha $ factor will further amplify the high sensitivity advantage of the DOFI system. The results reported in this paper will serve as an indispensable reference to guide the establishment of DOFI applications and further promote the widespread use of dual-channel optical feedback structures in OFI technology.

Appdenix A: derivation of the optical power output of a DOFI system

With further expansion and simplification of Eqs. (5)–(7), we obtain:

(19)$${N_s} = {N_0} + \frac{1}{{{G_N}{\tau _p}}} - \frac{{2{\kappa _1}}}{{{G_N}{\tau _{in}}}}\cos ({{\omega_s}{\tau_1}} )- \frac{{2{\kappa _2}}}{{{G_N}{\tau _{in}}}}\cos ({{\omega_s}{\tau_2}} ),$$

(20)$${\omega _s} = {\omega _0} - \frac{{{\kappa _1}}}{{{\tau _{in}}}}\sqrt {1 + {\alpha ^2}} \sin ({{\omega_s}{\tau_1} + \arctan \alpha } )- \frac{{{\kappa _2}}}{{{\tau _{in}}}}\sqrt {1 + {\alpha ^2}} \sin ({{\omega_s}{\tau_2} + \arctan \alpha } ),$$

(21)$${E_s}^2 = \frac{{{J / {eV}} - {{{N_s}} / {{\tau _s}}}}}{{{G_N}({{N_s} - {N_0}} )}}.$$

Inserting Eq. (19) into Eq. (21), we can obtain:

(22)$${E_s}^2 = \frac{{\frac{{{\tau _p}}}{{{\tau _s}}} \times \left( {\frac{{J{\tau_s}}}{{eV}} - {N_0} - \frac{1}{{{G_N}{\tau_p}}}} \right) + \frac{{2{\tau _p}}}{{{G_N}{\tau _s}}}\left[ {\frac{{{\kappa_1}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_1}} )+ \frac{{{\kappa_2}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_2}} )} \right]}}{{1 - 2{\tau _p}\left[ {\frac{{{\kappa_1}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_1}} )+ \frac{{{\kappa_2}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_2}} )} \right]}}.$$

Considering the stable operation conditions, and assuming that ${\kappa _1} < < 0.01$ and ${\kappa _2} < < 0.01$, while the position of the target MT is related to the physical quantity to be measured, and the static target ST only provides extra-feedback to the LD, based on our simulation results and theoretical analysis, the following equality relation exists:

(23)$$\frac{{2{\tau _p}{\kappa _1}}}{{{\tau _{in}}}}\cos ({{\omega_s}{\tau_1}} )+ \frac{{2{\tau _p}{\kappa _2}}}{{{\tau _{in}}}}\cos ({{\omega_s}{\tau_2}} )< < 1.$$

So using the first-order Taylor expansion of the denominator of Eq. (22), a first-order approximation describing the optical power output of the DOFI system in steady state can be obtained as follows. Here, Eq. (24) is Eq. (8) in the main text.

(24)$${E_s}^2 = \frac{{{\tau _p}}}{{{\tau _s}}} \times \left( {\frac{{J{\tau_s}}}{{eV}} - {N_0} - \frac{1}{{{G_N}{\tau_p}}}} \right) + \frac{{2{\tau _p}^2}}{{{\tau _s}}} \times \left( {\frac{{J{\tau_s}}}{{eV}} - {N_0}} \right) \times \left[ {\frac{{{\kappa_1}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_1}} )+ \frac{{{\kappa_2}}}{{{\tau_{in}}}}\cos ({{\omega_s}{\tau_2}} )} \right].$$

Funding

National Natural Science Foundation of China (62005234); China Scholarship Council (202107230002).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Symbol	Physical Meaning	Value
LD Internal Parameters	$G_{N}$	Model gain coefficient	$8.1 \times 10^{- 13} m^{3} s^{- 1}$
	$N_{0}$	Carrier density at transparency	$1.1 \times 10^{24} m^{- 3}$
	$τ_{p}$	Photon lifetime	$2.0 \times 10^{- 12} s$
	$τ_{s}$	Carrier lifetime	$2.0 \times 10^{- 9} s$
	$τ_{i n}$	Internal cavity round-rip time	$8.0 \times 10^{- 12} s$
	e	Elementary charge	$1 .6 \times 10^{- 19} C$
	V	Volume of the active region	$1 .0 \times 10^{- 16} m^{3}$
	$ω_{0}$	Angular frequency of the solitary laser $ω_{0} = 2 π c / λ_{0}$ , where $c$ is the speed of light, $λ_{0}$ is the wavelength of the LD	$2 .42 \times 10^{15} rad/s$
	$α$	Linewidth enhancement factor
LD External Parameters	$J$	Injection current
	$κ$	Feedback strength, $κ = η r_{3} (1 - {r_{2}}^{2}) / r_{2}$ ,where $r_{2}$ , $r_{3}$ are the amplitude reflectivity of the laser emitting facet and target surface respectively; $η$ is the coupling efficiency characterizing possible loss on re-injection.
	$L$	External cavity length
	$τ$	Light round-trip time in the external cavity $τ = 2 L / c$

Numerical investigation on the effect of extra-feedback target position on the fringe visibility in a dual optical feedback interferometry system

Abstract

1. Introduction

2. Theory and analysis of a DOFI system

2.1 Theoretical model

2.2 Method for solving the phase equation

3. Effect of the length of the extra-feedback channel on the DOFI signal

4. Conclusion

Appdenix A: derivation of the optical power output of a DOFI system

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (1)

Equations (24)

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