1. Introduction

Laser feedback interferometry (LFI) is based on the self-mixing effect of laser can be realized by the laser output coupled back into its laser cavity from an external reflector, has been widely used in various fields because of its unique features of simplicity, auto-alignment, compactness and low cost [1,2]. LFI is similar to the conventional interferometer that an interference fringe corresponds to λ/2 change of the external cavity length. For lasers with 632.8nm wavelength, the fringe resolution is 316.4nm. In order to obtain higher resolution, various methods have been proposed such as phase-shifting methods [3,4], phase-locked methods [5,6], wavelength offset [7] and sinusoidal phase modulating methods [8,9]. However, in previous researches, LFI can work well only at the weak feedback level, and the high-order feedback is mostly avoided and considered as the errors source. Consequently, the optical resolution of conventional LFI is usually less than λ/4 and the complicated circuit processing or optical configuration is required to obtain higher resolution.

In this paper, we present a simple and high-resolution laser high-order feedback interferometry (LHFI) based on the high-density cosine-like intensity fringes with phase quasi-quadrature. The high-density intensity fringes are obtained due to the high-order feedback, the optical resolution of λ/62 is achieved,which is about 31 times higher compared to the conventional optical feedback. Phase quasi-quadrature is generated in a certain displacement range due to the phase shift caused by the changes of external optical path length. Moreover, the directional discrimination can be easily obtained according to the phase relationship of the quasi-quadrature signals. The integrated measuring system is realized, which has the advantages of simple optical configuration and electronics processing.

2. Principle of measurement

The system configuration of the LHFI is shown in Fig. 1(a). The external feedback mirror M₃ is fixed on the PI linear stage (P-621.1CD), which has a range of about 110µm. The laser is composed of a plane mirror M₁, a concave mirror M₂ and the laser capillary. A quartz crystal plate Q is inserted in the laser cavity to split one frequency to two orthogonally polarized modes (e-light and o-light). The phase difference of the two lights is adjusted to 90° by changing the angle between the crystalline axis Q and the laser axis. W is a Wollaston prism to separate the two polarized lights which are detected by the photoelectric detectors D₁ and D₂. The birefringence dual frequency laser and the photoelectric detection elements are packaged by the instrument shell S to construct the optical gauge head. The C is the signal processing circuits that is integrated in an electrical cabinet. Figure 1(b) shows the prototype of this laser high-order feedback interferometer.

 

Fig. 1 Laser high-order feedback interferometer: (a) Schematic diagram; PI: nanopositioning stage; PZT: piezoceramics; M₃: external feedback mirror; M₁: laser cavity mirror; Q: quartz crystal plate; T: laser capillary; M₂: laser cavity mirror; W: Wollaston prism; D₁ and D₂: photoelectric detectors; S: instrument shell; C: signals processing circuits; (b) Instrument prototype.

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The output of laser high-order feedback interferometer can be expressed as [10]

When the feedback mirror M₃ is tuned by PZT, the high density fringes with phase quasi-quadrature are obtained as shown in Fig. 2. During the rise interval of PZT voltage in Fig. 2(a), the deforming length of PZT is about 1μm. So, in conventional weak feedback, the number of interference fringes is less than three. However, in the high-order feedback of our system, the fringes density is much higher compared to that in the conventional weak feedback. Especially, from Fig. 2(b) and 2(c), we can see that the e-light phase is lagging behind the o-light phase when the voltage of the PZT reduces and the o-light phase is lagging behind the e-light phase when the voltage of the PZT increases. It means that the two orthogonally polarized lights have a different pattern of phase lags when the movement of feedback mirror M₃ is in the opposite direction. In addition, the electronic subdivision method is used to further improve the resolution as shown in Fig. 3. It can be seen that the intensity signals of the e-light and o-light are expressed as I_s and I_c respectively, the intensity signal I_-s is antiphase with I_s. First, the three signals are amplified to the same amplitude and input into five subdivision chip. Then the quasi-quadrature square signals I_x and I_y are obtained. Later, the I_x and I_y are input into the electronic circuit for logical four subdivisions. After that, the 20-fold electronic subdivision is realized. Also, the displacement direction is easily discriminated by the output signals of I₊ and I_- which have the different pulses (up or down) when the feedback mirror moves in the opposite direction.

 

Fig. 2 Optical feedback signals, (a): high density fringes (b): the phase relationship of the e-light and o-light on the down-cycle of the PZT voltage (c): the phase relationship of the e-light and o-light on the up-cycle of the PZT voltage.

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Fig. 3 Flow chat of the signal processing for displacement sensor.

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3. Test results

First we test the resolution of LHFI by measuring the displacement of the PI nanopositioning stage which has 0.2nm resolution and 1nm repeatability. The measuring object M₃ is fixed on the PI stage, and the external cavity length is about 80mm. When the phase difference of the two orthogonally polarized lights is about 90°, the frequency difference of them is 315MHz which is measured by the frequency meter. The PI stage is drove by the closed-loop controller E-753.1CD from 0 to 100μm. The pulse numbers are counted and the results are displayed on the LCD of electrical cabinet as shown in Table 1.

Table 1. The pulse numbers of LHFI in 100μm

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From Table 1, we recorded the results of 0~100μm and 100~0μm for 10 times respectively. It can be seen that the pulse numbers are all between 195850 and 195861. The average result of them is 195856, which is regarded as the measurement result in the range of 100μm. So, the pulse equivalent is δ = 100μm/195856, the corresponding resolution of LHFI is about 0.51nm.

Next, LHFI and PI nanopositioning stage are used to measure the displacement synchronously with a step of about 10μm. Figure 4 shows the contrasting results between the LHFI and the PI stage. The linearity in a 100μm is 2 × 10⁻⁵, and the standard deviation is 1.5nm.

 

Fig. 4 The contrast results between LHFI and PI nanopositioning system.

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Last, the stability of LHFI is tested when the target (M₃) remains stationary. The experiments are recorded every second, for 2 minutes. The result, shown in Fig. 5, indicates that an accuracy of 5nm can be achieved in 2 minutes, which is enough for most measurements.

 

Fig. 5 The displacement stability of the LHFI.

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4. Discussion

The principle of this novel displacement sensor is based on the high density quasi-quadrature intensity fringes. The fringe density is determined by the feedback order n (positive integer), and it can be obtained from the pulse equivalent δ. In LHFI, δ can be calculated as$\lambda /(2\times 20\times \delta )=31$. It means that the fringes density of the LHFI is 31 times higher compared to the conventional LFI. So, the LHFI has high resolution. Furthermore, it has the potential value to calibrate the other micro-displacement sensors owing to it can be traced to light wavelength.

The laser high-order feedback interferometer described in this paper cannot be considered fully quadrature in two respects. First, the phase difference of two orthogonally polarized lights is adjusted by rotating the quartz crystal plate, and it is hard to get an ideal phase difference of 90°. Second, the two orthogonally polarized signals lack the quadrature owing to the phase shift caused by the change of external optical path length in the measurement process. The phase quasi-quadrature will lead to the measurement error because the non-uniform five subdivisions in the five subdivisions chip. But this error only exists in the last incomplete period, the reason is that this kind of instrument can self-calibrate to the central wavelength as long as the external object moves $\lambda /62$ according to the principle, i.e., every period of optical fringe corresponding to the$\lambda /62$displacement. We supposed that this error can be neglected when the phase shift is less than$\pi /9$. From Eq. (2), the measurement range of LHFI can be obtained by

5. Error analysis

5.1 non-uniform five zones

The five zones generated by the five subdivision chip do not equal exactly because the phase of two orthogonally polarized lights lack quadrature in the measurement process. Although LHFI can self-calibrate to wavelength reference for every period of $\lambda /62$ without cumulative error, the last incomplete period cannot self-calibrate that may result in measurement error. When the phase shift is less than$\pi /9$, this part of error ${\Delta }_u$can be written as

So, the error generated by the non-uniform five zones is about 0.45nm.

5.2 displacement amounts smaller than $\lambda /1240$

At the beginning and the ending of the measurement, the tiny displacement that is smaller than the resolution ($\lambda /1240$) of LHFI cannot be detected. So, the maximum error of this kind will be smaller than two equivalent pulses, which is${\Delta }_d=1.02\text{nm}$.

5.3 value of the laser wavelength

In practice use, the frequency drift induced by the optical feedback effects and the temperature change of environment is unavoidable and it will lead to the wavelength fluctuation. The maximum frequency drift in four hours is less than 100MHz due to the heat frequency-stabilized technology is used in LHFI. Thus, the maximum error of the wavelength induced by the frequency-drift is$\Delta {\lambda }_{\max }={\lambda }_0^2/c\Delta {\nu }_{\max }$, ${\lambda }_0$ is the wavelength in vacuum, $c$ is the light velocity in vacuum. The relative error of wavelength $\Delta {\lambda }_{\max }/\lambda$ will not exceed$2\times {10}^{-7}$, for the range of 850μm, the measurement error is${\Delta }_{\lambda }=2\times {10}^{-7}\times \text{850}\mu \text{m=0}\text{.17nm}$.

5.4 temperature influence

In common room conditions without the constant temperature method, the temperature changes can induce the variation of the air refractive index, which will also change the laser cavity length. The change of laser cavity length has been considered in the value of laser wavelength. The variation of air refractive index can be estimated by Ellen formula,

Finally, the combined estimation error $\Delta$ is

6. Conclusion

A novel laser feedback interferometry based on high-order feedback in a birefringence dual-frequency laser is demonstrated. This LHFI has high resolution and simple optical configuration and electronics processing. Under typical room conditions, the system’s resolution is 0.51nm in 850μm range, and its 2 min displacement accuracy is 5nm. The LHFI has bright application prospect in precise displacement measurement, particularly, it has potential value to calibrate the other micro-displacement sensors due to the high optical resolution that can be traced to light wavelength.