Accuracy Improvement of a Miniature Laser Diode Interferometer by Compensating Nonlinear Errors and Active Stabilizing Laser Diode Wavelengths

Abstract

A miniature laser diode interferometer (MLDI), which can be embedded in precision machines or measurement equipment as an on-line measurement sensor, is designed. A compact laser diode (LD) is adopted as the laser source of the MLDI. The measurement accuracy of the MLDI is affected by the nonlinear errors induced by the installation errors and manufacturing errors of the optical elements and the stability and accuracy of the LD wavelength. An arithmetic is applied to eliminate the nonlinear errors, and an error sensitivity analysis is conducted to enhance the understanding of which error components are more important or influence the measurement accuracy of the MLDI. An active wavelength stabilizer based on a compact laser wavelength meter is proposed to improve the stability of the LD wavelength. A group of experiments are carried out to verify the effectiveness of the proposed methods and the capabilities of the MLDI.

1. Introduction

Due to the advantages of a high resolution, high accuracy and direct traceability to the primary standard of length, laser interferometers have been widely applied for measuring the displacement or geometric errors of precision machines or measurement equipment [1,2,3].

A He-Ne laser with a stabilized frequency is generally adopted as the laser source of the laser interferometers. However, the He-Ne lasers are too voluminous, and He-Ne laser-based laser interferometers are usually bulky. Therefore, He-Ne laser-based laser interferometers can only be used as off-line measurement instruments. There is an increasing requirement for on-line measurement. Therefore, recently, lots of research efforts have been directed towards the development of miniature laser interferometers, which can be embedded in precision machines or measurement equipment as an on-line measurement sensor [4,5,6,7,8]. A laser diode (LD), which has the advantages of a small size, a long-operating time and a high-output power, is widely adopted as the laser source of miniature laser interferometers [4,5,6,7,8,9,10]. However, the LD suffers from low stability and low accuracy in terms of wavelength. Since the wavelength is the length unit of the laser interferometer, it is essential to improve the stability and the accuracy of the LD wavelength in order to achieve a high-precision and stable displacement measurement.

There are many approaches and techniques for stabilizing the laser wavelength/frequency [11,12,13,14,15,16,17,18]. Efforts aim to stabilize the frequency of LDs by locking them to the reference frequency of atoms or molecules, in which the LD’s frequency is modulated and turned around the frequency transition of the reference frequency of atoms or molecules [11,12]. The laser frequency can also be stabilized to the resonances of a Fabry–Perot cavity or a photonic crystal cavity [13,14,15]. The stabilization of laser frequency can also be carried out based on a high-accuracy wavelength meter [16,17,18]. However, due to the complexity of the construction, the above-mentioned methods cannot be applied for stabilizing the LD wavelength/frequency in miniature laser interferometers.

In addition, the incomplete separation of the differently polarized partial beams in the interferometer as well as multiple back reflections will cause the optical mixing. Inevitable nonlinear errors, which are primary caused by the optical mixing, will thus restrict the performance of the miniature laser interferometers. A group of studies have been carried out to eliminate the nonlinear errors, such as the ellipse-fitting method [19], the recursive weighted least-squares method combined with the Kalman filter algorithm [20], the optimization of the optical path structure [21], the field programmable gate array [22] and the intermodulation [23]. Although the above-mentioned methods can effectively correct nonlinear errors, they involve time-consuming calculation or a complicated hardware system.

Therefore, methods for improving the measurement accuracy of a miniature laser diode interferometer (MLDI), in which a laser diode is adopted as the laser source, are proposed in this paper. The nonlinear errors of the MLDI induced by the installation errors and the manufacturing errors of the optical elements are investigated. The error sensitivity analysis of the nonlinear errors is analyzed. In order to stabilize the LD wavelength with high stability for a long period of time, an active wavelength stabilization method based on a real-time laser wavelength meter is proposed.

2. Measurement Principle of the MLDI

Figure 1 shows the schematic diagram of the MLDI, which is composed of a moved part and a fixed part. The fixed part consists of a displacement meter (DM), a laser wavelength meter (LWM) and a laser wavelength stabilizer (LWS). The moved part is composed of a corner cube retroreflector (CR1).

A laser beam from a laser diode (LD), whose wavelength is measured and stabilized by the LWM and LWS, is divided into two beams by a beam splitter (BS1) after being bent by a mirror (M1). The transmitted beam from BS1 enters into the LWM, which consists of a grating (G), two focus lenses (Fl1 and FL2), two quadrant photodetectors (QPD1 and QPD2), a thermistor (PT100) and two mirrors (M3 and M4), for real-time measuring of the LD wavelength based on the principle of the diffraction grating. The detailed principles of the LWM can be found in our previous research [6] and are not repeated in this paper for the sake of brevity. The LWS is composed of a thermistor, two thermoelectric coolers (TECs) and an LD holder. The principle of the LWS is introduced in the next section. The reflected beam from BS1 is divided again by a polarized beam splitter (PBS1). The reflected beam from PBS1 is projected onto CR2. The transmitted beam from PBS1 is projected onto the CR1 of the DM, which consists of two beam splitters (BSs, BS2 and BS3), three PBSs (PBS1 to PBS3) and four photodetectors (PDs, PD1 to PD4). The reflected beams from CR1 and CR2 are entered into PBS1 again. When CR1 has a motion of z, the count of the interference fringes, which is caused by the reflected beams from CR1 and CR2, will be changed and further cause the output variations of the PDs (PD1 to PD4). According to the outputs of the PDs, the intensity of the sinusoidal and cosine signals in the ideal case can be expressed by [6,20],

$$I_{\sin }=\sin (\Delta \phi )\mathrm{and}{I}_{\cos }=\cos (\Delta \phi ),$$

(1)

where Δφ is the phase difference between the initial and final phases of the incomplete sinusoidal waves. However, due to the installation errors and the manufacturing errors of the optical elements of the MLDI-induced nonlinear errors, the intensity of the sinusoidal and cosine signals will be varied to

$${I^{\prime }}_{\sin }=E_{\sin }\sin (\Delta \phi +\delta )+F_{\sin }\mathrm{and}{I^{\prime }}_{\cos }=E_{\cos }\cos (\Delta \phi )+F_{\cos },$$

(2)

where E_sin and E_cos, δ and F_sin and F_cos represent the AC amplitudes, the phase delay and the DC offsets of the sinusoidal and cosine signals, respectively. The sinusoidal and cosine signals in this case are shown in Figure 2a. The Lissajous figure with respect to Figure 2a will thus be distorted, as shown in Figure 2b.

In addition, the low wavelength accuracy and stability of the LD will influence the accuracy and the stability of the MLDI. Therefore, the moving displacement z of the moved part can be calculated by [6]

$$z=\left(N+\frac{\Delta \phi }{2\pi }\right)\cdot \frac{\lambda }{2n}+z_{en}+z_{ew}$$

(3)

where N, λ, n and Δφ represent the integer count of the interference fringes, the wavelength of the LD, the refractive index of the air and the phase difference between the initial phase and the final phase, respectively. z_en and z_ew represent the measurement errors caused by the nonlinear errors and the unstable LD wavelength, respectively. In order to improve the measurement accuracy of the MLDI, z_en and z_ew should be eliminated and compensated.

3. Nonlinear Errors Caused by Optical Elements

Nonlinear errors, which are induced by installation errors and manufacturing errors, will limit the performance of the MLDI. Here, PBS1 is taken as an example to analyze the nonlinear errors, which are induced by the installation errors and the manufacturing errors.

In an ideal case, the Jones matrix of PBS1 can be expressed by [24]

$$J_{PBS,S}=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\mathrm{and}{J}_{PBS,P}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$$

(4)

When PBS1 has an installation error α with respect to the X-axis, the Jones matrix of PBS1 is revised to be

$$\{\begin{array}{l}{J^{\prime }}_{PBS,S}=\left[\begin{array}{cc}\cos \arctan (\sin \alpha )& \sin \arctan (\sin \alpha )\\ 0& 0\end{array}\right]\\ {J^{\prime }}_{PBS,P}=\left[\begin{array}{cc}0& 0\\ -\sin \arctan (\sin \alpha )& \cos \arctan (\sin \alpha )\end{array}\right]\end{array}$$

(5)

The sinusoidal and cosine signals of the MLDI can thus be expressed by

$$\begin{gathered} \{\begin{array}{l}{I}_{\sin p1\alpha }=2E_{\sin p1\alpha }^2{a}_{p1\alpha }{b}_{p1\alpha }\cos \arctan (\sin \alpha )\sin (\Delta \phi )\\ I_{\cos p1\alpha }=2E_{\cos p1\alpha }^2[a_{p1\alpha }^2\cos \arctan (\sin \alpha )\sin \arctan (\sin \alpha )+a_{p1\alpha }{b}_{p1\alpha }\cos \arctan (\sin \alpha )\cos (\Delta \phi )]\end{array} \\ here,\{\begin{array}{l}{a}_{p1\alpha }=(\cos \arctan (\sin \alpha )-\sin \arctan (\sin \alpha ))\cos \arctan (\sin \alpha )\\ b_{p1\alpha }=(\cos \arctan (\sin \alpha )+\sin \arctan (\sin \alpha ))\cos \arctan (\sin \alpha )\end{array} \end{gathered}$$

(6)

where E_sinp1α and E_cosp1α represent the AC amplitudes of the sinusoidal and cosine signals when PBS1 has an installation error α.

Therefore, a nonlinear error e_p1α caused by α can be expressed by

$$\begin{array}{ll}{e}_{p1\alpha }& =\frac{\lambda }{4\pi }(\arctan \left(\frac{a_{p1\alpha }{b}_{p1\alpha }\cos \arctan (\sin \alpha )\sin (\Delta \phi )}{a_{p1\alpha }^2\cos \arctan (\sin \alpha )\sin \arctan (\sin \alpha )+a_{p1\alpha }{b}_{p1\alpha }\cos \arctan (\sin \alpha )\cos (\Delta \phi )}\right)\\ & -\arctan \left(\frac{\sin (\Delta \phi )}{\cos (\Delta \phi )}\right))\end{array}$$

(7)

Similarly, the nonlinear errors e_p1β and e_p1γ caused by the installation errors β and γ with respect to the Y- and Z-axis, respectively, can be expressed by

$$\begin{gathered} \begin{array}{ll}{e}_{p1\beta }& =\frac{\lambda }{4\pi }(\arctan \left(\frac{a_{p1\beta }{b}_{p1\beta }\cos \arctan (\sin \beta )\sin (\Delta \phi )}{a_{p1\beta }^2\cos \arctan (\sin \beta )\sin \arctan (\sin \beta )+a_{p1\beta }{b}_{p1\beta }\cos \arctan (\sin \beta )\cos (\Delta \phi )}\right)\\ & -\arctan \left(\frac{\sin (\Delta \phi )}{\cos (\Delta \phi )}\right))\end{array} \\ here,\{\begin{array}{l}{a}_{p1\beta }=(\cos \arctan (\sin \beta )-\sin \arctan (\sin \beta ))\cos \arctan (\sin \beta )\\ b_{p1\beta }=(\cos \arctan (\sin \beta )+\sin \arctan (\sin \beta ))\cos \arctan (\sin \beta )\end{array} \end{gathered}$$

(8)

$$\begin{gathered} e_{p1\gamma }=\frac{\lambda }{4\pi }(\arctan \left(\frac{a_{p1\gamma }\sin (\Delta \phi )}{a_{p1\gamma }{c}_{p1\gamma }\cos \arctan (\sin \gamma )+b_{p1\gamma }{c}_{p1\gamma }}\right)-\arctan \left(\frac{\sin (\Delta \phi )}{\cos (\Delta \phi )}\right)) \\ here,\{\begin{array}{l}{a}_{p1\gamma }=(\cos \gamma -\sin \gamma )(\cos \gamma \cos \arctan (\sin \gamma )-\sin \alpha \sin \arctan (\sin \gamma ))\\ b_{p1\gamma }=({\sin }^3\gamma -\sin \alpha {\cos }^3\gamma )(\cos \gamma +\sin \gamma )\\ c_{p1\gamma }=\cos \alpha ({\cos }^2\gamma -{\sin }^2\gamma )(\cos \gamma +\sin \gamma )\end{array} \end{gathered}$$

(9)

A nonlinear error e_p1m caused by the manufacturing error of PBS1 can be expressed by

$$\begin{gathered} \begin{array}{ll}{e}_{p1m}=\frac{\lambda }{4\pi }(& \mathrm{atan}(\frac{4a_{p1m}^2\sin (\Delta \phi )+(t_{p1}+t_{s1}-r_{p1}-r_{s1})(2a_{p1m}^2+4a_{p1m}{b}_{p1m}\cos (\Delta \phi ))}{(t_{p1}^2+r_{s1}^2)2\cos (\Delta \phi )+2t_{p1}^2{(a_{p1m}+b_{p1m})}^2-2r_{s3}^2{(a_{p1m}-b_{p1m})}^2})\\ & -\mathrm{atan}\left(\frac{\sin (\Delta \phi )}{\cos (\Delta \phi )}\right))\end{array} \\ here,\{\begin{array}{l}{a}_{p1m}=t_{p1}{r}_{s1}\approx 1\\ b_{p1m}=t_{s1}{r}_{p1}\approx 0\end{array} \end{gathered}$$

(10)

where t_p1, t_s1 and r_p1, r_s1 represent the transmission coefficient and the reflection coefficient of PBS1 with respect to the P light and S light, respectively.

If the MLDI has n optical elements, there will be 3n installation errors and 4n manufacturing errors. Therefore, it is time-consuming to adjust the installation errors and test the manufacturing errors one by one. An error sensitivity analysis of the installation errors and the manufacturing errors on the nonlinear errors is proposed to enhance the understanding of which error components are more important or influence the measurement accuracy of the MLDI. The sensitivity coefficient (S_i) of each error component (E_i) can be expressed to be

$$S_i=\frac{\left|\frac{\partial z}{\partial E_i}\right|}{\sum \left|\frac{\partial z}{\partial E_i}\right|}$$

(11)

Each installation error of the optical elements is considered as in the same error range, which is from 3° to −3°. The sensitivity analysis results of the installation errors are shown in Figure 3. α_i, β_i and γ_i (i = p1, p2, p3, q1, q2, q3, n) represent the installation errors around X-, Y- and Z-axis of each optical element (PBS1, PBS2, PBS3, Q1, Q2, Q3, NPBS) of the DM. It can be seen from the figure that the sensitive error components of the installation errors on the nonlinear errors are α_p1, β_p1, γ_p1, α_p2, β_p2, α_p3, β_p3, γ_q3, α_n and β_n. The effects of the installation errors of the quarter-waveplate (Q1, Q2, Q3) on the nonlinear errors were less than the effects of the installation errors of the beamsplitter (PBS1, PBS2, PBS3, NPBS) on the nonlinear errors.

According to the above analysis, the effects of the installation errors and the manufacturing errors of the optical elements can be reduced by carefully adjusting and selecting the optical elements. However, it is difficult to totally eliminate those effects. Therefore, an algorithm was proposed to eliminate the residual nonlinear errors.

The flowchart of the algorithm is shown in Figure 4. The DC offsets and the AC amplitudes are first eliminated by dynamically checking the peak values of I′_sin and I′_cos. The phase delay is then eliminated by using the vector summation and subtraction methods [20]. The normalization method is applied to eliminate the new AC amplitude difference between two quadrature signals, which resulted from the vector summation and subtraction methods. Finally, a sinusoidal signal and a cosine signal without the AC amplitude difference, the DC offset and the phase delay were obtained for measuring the displacement.

Some simulations were conducted to verify the effectiveness of the method for the elimination of the nonlinear errors. Figure 5 shows the corrected sinusoidal and cosine signals and the corresponding Lissajous figure with respect to Figure 2. It can be seen that the AC amplitudes, the phase delay, the DC offsets and the high-order nonlinear harmonics of the sinusoidal and cosine signals were eliminated.

4. Measurement Error Caused by the Stability of the LD Wavelength

The unstable LD wavelength will affect the measurement accuracy of the MLDI. A passive wavelength stabilization method based on the principle of automatic temperature control and the principle of the laser beam drift compensation was proposed to stabilize the LD wavelength in our previous research [6]. However, the stability of the laser diode wavelength can only be controlled at the level of 10⁻⁶ for a short period of time. In order to realize the high stability for a long period of time, an active wavelength stabilization method, which includes two consecutive steps, is proposed, as shown in Figure 6.

Step 1: Temperature control of the LD

The temperature control of the LD is based on the principle of automatic temperature control. A closed-loop control system is adopted to improve the control precision of the LD surface temperature, as shown in Figure 6. The surface temperature of the LD is detected by a thermistor, which is installed on the LD. A differential signal is generated by differing the surface temperature with the desired temperature and input to the PID controller 1. Two thermoelectric coolers (TECs) will be employed to heat or cool according to the outputs of the PID controller, which are the drive voltages of the TECs. The LD’s surface temperature can then be effectively controlled within a commanded range, in which the full width at half maximum (FWHM) is the smallest. The FWHM is an important parameter to characterize the stability of the LD wavelength.

Step 2: Wavelength control of the LD

However, the capability of the wavelength stabilization by using the temperature control is limited. Therefore, a wavelength control method according to the real-time measured wavelength by the laser wavelength meter (LWM) is further proposed, as shown in Figure 6.

A closed-loop control system is adopted to improve the precision of the wavelength control. After comparing the actual wavelength variation with the desired wavelength variation, a differential signal is generated and input into the Backpropagation Neural Networks-based PID (BPNN-PID) controller, which is composed of an input layer with three neurons, a hidden layer with five neurons and an output layer with three neurons. The PID controller parameters K_i, K_p and K_d, which are the three neurons of the output layer, are obtained by training the BPNN according to a group of measured wavelength variations. The obtained PID controller parameters were used to control the drive voltage of the TECs according to the instantaneously measured wavelength variations. Therefore, the LD’s surface temperature will be precision controlled, from which the LD wavelength can be effectively controlled within a commanded range.

5. Experiments and Discussion

A group of experiments were carried out to investigate the capacity of the designed MLDI. Figure 7 shows the experimental setup. The size of the MLDI, including the electric cables and the base plate for mounting the optical components, is 110 mm (x) × 100 mm (y) × 20 mm (z), which is much smaller than commercial He-Ne laser interferometers, such as LDDM (Optodyne, Rancho Dominguez, CA, USA), which is shown in Figure 7.

Prior to using the designed MLDI to measure the displacement of a linear stage, the effectiveness of the proposed active wavelength stabilization method was first investigated. As mentioned, before the LD wavelength control, the surface temperature of the LD should be effectively controlled within a commanded range, in which the FWHM is the smallest. The smaller the FWHM, the better the stability of the LD wavelength. Therefore, the FWHM at various temperatures was detected three times by the LWM with the particular temperate stabilized for 10 min, as shown in Figure 8. It can be seen from Figure 8 that when the temperature is set to be 18 °C and 29 °C, the corresponding FWHMs are the smallest.

The X- and Y-directional outputs of QPD1 and QPD2 of the LWM, from which the LD wavelength can be evaluated, were then monitored when the temperature of the LD holder was set to be 18 °C and 29 °C, respectively. The outputs of QPD1 and QPD2 are shown in Figure 9. It can be seen that when the temperature was set to be 18 °C, the stability of QPD1 and QPD2 is better than that when the temperature was set to be 29 °C. Therefore, 18 °C was selected to be the desired temperature of the active wavelength stabilization method.

The accuracy of the LWM will affect the control accuracy of the active wavelength stabilizer and the measurement accuracy of the displacement. Therefore, the accuracy of the LWM was then tested by using a commercial spectrometer (CCS100, ThorLabs, Newton, NJ, USA) which has a wavelength response range from 350 nm to 750 nm and a measurement accuracy of less than 0.5 nm. The measured wavelengths by using the LWM and the commercial spectrometer are shown in Figure 10. The temperature of the LD holder was set around 18 °C. It can be seen from the figure that the residual between the LD wavelength measured by the LWM and that measured by the spectrograph was less than 4 pm, on the basis of which the effectiveness of the LWM was verified.

The stability of the LD wavelength was then controlled based on the active wavelength stabilization method proposed in this research and the passive wavelength stabilization method proposed in our previous research [6]. Figure 11a shows the results. It can be seen from the figure that the stability of the LD wavelength was improved from 7.96 × 10⁻⁶ to 5.95 × 10⁻⁷ by using the active wavelength stabilization method rather than the passive wavelength stabilization method. The Allan deviation of the stability of the LD wavelength is shown in Figure 11b. The short-term stability of the LD wavelength was improved to be 10⁻⁷ from 10⁻⁶, and a long-term stability of the LD wavelength of 10⁻⁸ can be achieved.

Finally, the measurement accuracy of the MLDI was investigated by a group of comparison experiments. The displacement of a linear stage with a stroke of 100 mm was simultaneously measured by the designed MLDI and a commercial interferometer (LMMD, Optodyne, CA, USA) with a measurement accuracy of ±0.5 ppm and a measurement range of 15 m. Figure 12 shows the experimental results with five times. The measurement accuracy of the designed MLDI was evaluated to be ±800 nm, as shown in Figure 12a, without the nonlinear error compensation and the LD wavelength stabilization. It can be seen from Figure 12b that the measurement accuracy of the designed MLDI was improved to within ±200 nm after the nonlinear error compensation and the wavelength stabilization. It can be seen that almost 85% of the measurement errors primarily caused by the nonlinear errors and the unstable LD wavelength have been compensated. The standard deviation of the measured displacement was improved to 30.44 nm from 325.28 nm, on the basis of which the effectiveness of the proposed methods was verified.

6. Conclusions

A miniature laser diode interferometer (MLDI) was designed for the on-line measurement of the displacement of precision machines or measurement equipment. In order to reduce the size of the MLDI, a compact laser diode (LD) was adopted as the laser source of the MLDI. The factors that affect the measurement accuracy of the MLDI were first analyzed. The effects of the nonlinear errors caused by the installation errors and the manufacturing errors—along with the error caused by the unstable LD wavelength—on the measurement accuracy of the MLDI were investigated. An error sensitivity analysis of the installation errors was proposed. An algorithm was proposed to eliminate the nonlinear errors. In addition, an active wavelength stabilization method based on the laser wavelength meter was proposed to stabilize the stability of the LD wavelength. It has seen verified that the stability of the LD wavelength was improved from 10⁻⁶ to 10⁻⁷ by using the active wavelength stabilization method rather than the passive wavelength stabilization method during a long period of time. The measurement accuracy of the designed MLDI was improved from ±800 nm to ±200 nm after the nonlinear error compensation and the wavelength stabilization. The uncertainty analysis will be carried out in future work.