DC-offset-free homodyne interferometer and its nonlinearity compensation

Pengcheng Hu; Jinghao Zhu; Xiaoyu Zhai; JiuBin Tan

doi:10.1364/OE.23.008399

1. Introduction

Cyclic nonlinearity ranges from several nanometers to tens of nanometers, which limits the application of homodyne interferometry in high-precision displacement measurement. Several compensation methods for reduction of the nonlinearity have been suggested.

In 1981, Heydemann [1] proposed that the nonlinearity could be reduced by using the ellipse-fitting method, which was later improved by Wu and Eom [2,3]. On this basis, various methods [4–7] have been presented to dynamically compensate for the variable cyclic nonlinearity. Under these compensation methods, the quadrature signals can be obtained by processing the raw signal through specific algorithms. In particular, Gregorčič [8] presented a phase-shift correction algorithm to improve the accuracy of displacement measurements. All of these approaches have shown superior abilities to reduce the nonlinearity. However, these compensation methods tend to be time-consuming processes [4,9], which should be implemented in an off-line mode or dynamically with a limited update rate.

To achieve nonlinearity compensation during a high-speed measurement, Keem [9,10] removed the cyclic nonlinearity by adjusting the gains of detectors corresponding to the two quadrature signals in real time. Ann [11] realigned the axes of the wave plates to particular angles to compensate for the nonlinearity caused by the polarizing beam splitter (PBS) in the detection component of the homodyne interferometer cooperated with an active compensation method. However, the DC offsets error and unequal AC amplitudes error still need identification before they can be compensated in real time, which was a limitation to enable high-speed and high-resolution measurements [4,12].

In this paper, we analyze the nonlinearity in a homodyne interferometer starting from the interference principle. An enhanced homodyne interferometer without DC offset was designed, and the gain- and phase-correction methods [9,13] were combined to remove the nonlinearity more effectively.

2. Nonlinearity in a homodyne interferometer

As shown in Fig. 1(a), a conventional homodyne interferometer generally can be divided into three components: an interferometer component, a detector component, and electronics. In the interferometer component, PBS₁ splits a 45° linearly polarized beam of a He-Ne laser into two beams polarized vertically (90°) and horizontally (0°), respectively. As these two beams propagate along their separate paths to the measurement and reference arms, they experience a relative phase difference φ_m, which is dependent on the displacement. After the reflected beams pass through the quarter-wave plates (QWPs) along their respective paths for the second time, their polarization states are rotated by 90°, and then they both propagate through PBS₁ again.

Fig. 1 Schematic diagram of a homodyne interferometer. Polarizing beam splitter (PBS), quarter-wave plate (QWP), target mirror (TM), reference mirror (RM), half-wave plate (HWP), beam splitter (BS), and photodetector (PD).

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Figures 1(b) and 1(c) present two typical detection configurations in a homodyne interferometer. In Fig. 1(b), the horizontally and vertically polarized laser beams are each rotated through 45° by a half-wave plate (HWP) before passing through the BS. In Fig. 1(c), the two beams are instead converted to circularly polarized light by a QWP before passing through the BS. Under either of these two detection schemes, the reference beam and measurement beam will be separated into four beams to interfere with each other on the photoelectric detectors. After coherent signal processing, the phase difference φ_m can be measured.

In the ideal case, the Lissajous trajectory of two quadrature signals should be a perfect circle. However, a nonlinear error including DC offsets error, unequal AC amplitudes error, and the quadrature phase delay error exist at the same time due to 1) imperfections in the optics used in an interferometer such as the PBS, BS, and wave plates and 2) misalignment of the axes of the optics with respect to that of the polarized beam. Furthermore, for a conventional homodyne interferometer, DC offsets can be considerably changed by factors such as laser power drift and laser alignment [2, 3], and result in an error of several nanometers [10]. Without any corrective action taken, the accuracy of the displacement measurement will be reduced.

Because the incident laser beam passes through PBS₁ twice in the interferometer component, the polarization crosstalk between E_r and E_m caused by PBS₁ can be ignored. Moreover, the alignment can be adjusted as precisely as required; therefore, the main source of nonlinearity is the imperfection of the optical components in the detection component of a homodyne interferometer.

In our study, we explain the nonlinearity in the detection component from interference principle. As shown in Fig. 2, due to the imperfection of the optical components, the power of the incident beam will be separated non-uniformly, which is the cause of the nonlinear error. The intensity signal from each detector is given by the equation:

\begin{matrix} I_{n} = (E_{r n} + E_{m n}) \cdot {(E_{r n} + E_{m n})}^{*} \\ = (A_{r n} e^{i φ_{r}_{n}} + A_{m n} e^{i φ_{m n}}) \cdot {(A_{r n} e^{i φ_{r n}} + A_{m n} e^{i φ_{m n}})}^{*} \\ = A_{r n}^{2} + A_{m n}^{2} + 2 A_{r n} A_{m n} e^{i (Δ φ_{m} + α_{n})}, \end{matrix}

where I_n is the intensity detected at the nth detector (where n = 1–4), A_rn and A_mn denote the amplitude of the reference beam and the measurement beam, respectively, Δφ_m is the theoretical value of the phase measurement, and α_n denotes the phase error. Because only the intensity can be detected at the photodetector, we express the intensity relations of the phase quadrature signals as:

I_{x} = I_{1} - I_{3} = (A_{r 1}^{2} - A_{r 3}^{2}) + (A_{m 1}^{2} - A_{m 3}^{2}) + 2 [A_{r 1} A_{m 1} \cos Δ φ_{m} + A_{r 3} A_{m 3} \cos (Δ φ_{m} + α_{x})],

I_{y} = I_{2} - I_{4} = \underset{D C}{\underset{︸}{(A_{r 2}^{2} - A_{r 4}^{2}) + (A_{m 2}^{2} - A_{m 4}^{2})}} + \underset{A C}{\underset{︸}{2 [A_{r 2} A_{m 2} \sin Δ φ_{m} + A_{r 4} A_{m 4} \sin (Δ φ_{m} + α_{y})]}},

where α_x and α_y denote the phase delay errors of the two quadrature signals respectively. The cyclic nonlinearity of the phase measurements can then be expressed as:

Fig. 2 Schematic diagram of the separation of a beam into four beams in an imperfect detection component of a homodyne interferometer. (a) and (b) represent the decompositions of a reference beam and a measurement beam, respectively. (c) represents the corresponding decomposition of the beam formed by combining the reference beam and the measurement beam.

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N L = arc \tan \frac{I_{x} (t)}{I_{y} (t)} - arc \tan \frac{I_{x} (t_{0})}{I_{y} (t_{0})} - [Δ φ_{m} (t) - Δ φ_{m} (t_{0})] .

On the right-hand side of each of the Eqs. (2) and (3), the first and second terms describe the DC part, and the third term of each equation represents the coherent AC signal. Figure 3 depicts the nonlinear error for different cases, and the corresponding parameters are given in Table 1. In the ideal case, the amplitudes of the four parts of the measurement beam and the reference beam are all the same without any phase delay error. Accordingly, no nonlinearity will exist in this ideal case, and the Lissajous trajectory of the two phase-quadrature signals will be a perfect circle. In Case 1, the phase delay error Δα = α_x-α_y is −8°, but the amplitudes are all the same as in the ideal case. Comparing this with Case 2 and Case 3, where

A_{r 1} = A_{r 3} \begin{matrix} , & A_{m 1} = A_{m 3} \end{matrix},

A_{r 2} = A_{r 4} \begin{matrix} , & A_{m 2} = A_{m 4} \end{matrix},

as shown in Fig. 3(a), the DC offsets vanish, and the AC amplitude ratio of the two quadrature signals is constant. In contrast, when Eqs. (5) and (6) does not hold, the DC offsets will cause an increased nonlinearity as can be seen from the plot of Case 3 in Figs. 3(a) and 3(b). Furthermore, the intensity of the measurement beam was reduced to 0.6 times and 0.3 times that of Case 3 respectively; the simulation results shown in Figs. 3(c) and 3(d) indicate that the degree of nonlinearity was influenced by the imbalance between the amplitudes of the reference and measurement beams. The smaller the intensity of the signal beam is, the greater the nonlinearity in a homodyne laser interferometer is. The change in intensity would tend to produce difficulties in the compensation of the nonlinearity. Based on the analysis above, our design provides an improved detection configuration for homodyne interferometers.

Fig. 3 Depictions of the nonlinear error for the different cases in Table 1. (a) Lissajous representation of the four different cases. (b) Nonlinear error for the four different cases in Table 1. (c) Lissajous representation for different intensities of the measurement beam. (d) Nonlinear error for different intensities of the measurement beam.

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Table 1. Parameters for different cases

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3. Enhanced homodyne interferometer and nonlinearity compensation

As shown in Fig. 4, by canceling the HWP and aligning the axes of the QWP to 0° in the improved configuration, the number of sources of nonlinearity is reduced; furthermore, the measurement beam and reference beam are always horizontally or vertically polarized before they encounter the polarizing beam splitters. At the end of the detection component, Wollaston prisms (WPs), which have a higher extinction ratio, are used to replace the PBSs to remove the nonlinearity that they cause. The signals are divided equally by rotating the WPs to 45°. In addition, the four detectors in this homodyne interferometer are all identical.

Fig. 4 Improved configuration of the homodyne laser interferometer. Wollaston prism (WP), quarter-wave plate (QWP), non-polarizing beam splitter (NPBS), photodiode (PD), analog-to-digital converter (ADC), and homemade quadrature-signals processing board (HQSPB).

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3.1 The signal model of the enhanced homodyne interferometer

The horizontally and the vertically polarized electric field vectors output from the interferometer components, E_r and E_m, can be expressed as:

E_{r} = A_{r} [\begin{array}{l} 1 \\ 0 \end{array}] e^{- i φ_{r}}, E_{m} = A_{m} [\begin{array}{l} 0 \\ 1 \end{array}] e^{- i φ_{m}},

where A_r and A_m denote their respective amplitudes and φ_r and φ_m represent their respective phases.

The Jones matrices for a BS can be written as:

B_{T} = [\begin{matrix} t_{p} e^{i τ_{p}} & 0 \\ 0 & t_{s} e^{i τ_{s}} \end{matrix}], B_{R} = [\begin{matrix} r_{p} e^{i δ_{p}} & 0 \\ 0 & r_{s} e^{i δ_{s}} \end{matrix}],

where B_T and B_R denote the transmission and the reflection matrices, respectively. Here, t and r respectively denote the transmissivity and reflectivity, the subscripts p and s represent the polarization states, and τ and δ represent the phase shifts of the BS.

The WPs are rotated 45° around the incident beam; under this condition, the Jones matrices for a WP can be obtained by referring to the Jones matrices of a PBS:

T_{p} = \frac{1}{2} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}], T_{s} = - \frac{1}{2} [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] .

The Jones matrix for an imperfect QWP that is aligned with its fast axis at 0° could be expressed as:

Q W = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 0 \\ 0 & i e^{i σ} \end{matrix}],

where σ is the delay angle error.

The quadrature current signals from the photodiodes are expressed as:

\begin{matrix} I_{x 0} = I_{3} - I_{1} \\ = {[(T_{s} - T_{p}) \cdot B_{R} \cdot (E_{r} + E_{m})]}^{2} G_{0} (I_{x 0}) η \\ = 2 A_{r} A_{m} r_{p} r_{s} \cos [(φ_{m} - φ_{r}) + (δ_{p} - δ_{s})] G_{0} (I_{x 0}) η, \end{matrix}

\begin{matrix} I_{y 0} = I_{4} - I_{2} \\ = {[(T_{s} - T_{p}) \cdot Q W \cdot B_{T} \cdot (E_{r} + E_{m})]}^{2} G_{0} (I_{y 0}) η \\ = 2 A_{r} A_{m} t_{p} t_{s} \sin [(φ_{m} - φ_{r}) + (τ_{p} - τ_{s} - σ)] G_{0} (I_{y 0}) η, \end{matrix}

where G denotes the gain of the detector and η represents the photoelectric conversion efficiency of the detector. As shown in Eqs. (11) and (12), the DC offset error will not exist for the modified configuration shown in Fig. 4. Furthermore, the amplitude ratio and phase delay error of the two quadrature signals is constant in such a scenario.

3.2 Signal correction and nonlinearity compensation for the enhanced interferometer

There are only two systematic nonlinearities in enhanced homodyne interferometers: unequal AC amplitudes and quadrature phase-shift errors. Due to the constant amplitude ratio of the two quadrature signals, unequal AC amplitudes can be corrected for by adjusting the gains of the detectors; the relation between the gains of the two arms can be set according to the equation G₀(I_x₀)/G₀(I_y₀) = t_pt_s/r_pr_s. We then have the following equations:

I_{x 0} = 2 B \cos [(φ_{m} - φ_{r}) + (δ_{p} - δ_{s})],

I_{y 0} = 2 B \sin [(φ_{m} - φ_{r}) + (τ_{p} - τ_{s} - σ)],

where B = ηA_rA_mr_pr_st_pt_s. After conversion by an analog-to-digital converter (ADC), the two signals with equal amplitudes are transmitted to the vector phase correction system, where the additional phase-shift error can be canceled by processing the quadrature signals [13] according to:

I_{x 1} = G_{1} (I_{x 1}) (I_{y 0} + I_{x 0}) = 2 B G_{1} (I_{x 1}) \sin (π / 4 + \frac{Δ τ - Δ δ}{2}) \cos (Δ φ + π / 4 + \frac{Δ τ + Δ δ}{2}),

I_{y 1} = G_{1} (I_{y 1}) (I_{y 0} - I_{x 0}) = 2 B G_{1} (I_{y 1}) \cos (π / 4 + \frac{Δ τ - Δ δ}{2}) \sin (Δ φ + π / 4 + \frac{Δ τ + Δ δ}{2}),

where Δφ = φ_m-φ_r, Δτ = τ_p-τ_s-σ, and Δδ = δ_p-δ_s. Then the ideal signals can be obtained after adjusting the gains of the two quadrature signals once again; the relation between the gains of these two arms is G₁(I_x₁)/G₁(I_y₁) = cos[π/4 + (Δτ-Δδ)/2] / sin[π/4 + (Δτ-Δδ)/2]. The phase difference φ_m can be obtained according to:

\begin{matrix} Δ φ = arc \tan (\frac{I_{y 1}}{I_{x 1}}) - (π / 4 + \frac{Δ δ + Δ τ}{2}) \\ = arc \tan [\frac{\sin (Δ φ + π / 4 + \frac{Δ δ + Δ τ}{2})}{\cos (Δ φ + π / 4 + \frac{Δ δ + Δ τ}{2})}] - (π / 4 + \frac{Δ δ + Δ τ}{2}) . \end{matrix}

These setups can be seen as a calibration process that must be carried out before the measurement. Because both Δτ and Δδ are constant, only one calibration is needed. The final signals contain almost no nonlinear error.

4. Experimental and discussion

4.1 Setup

In order to verify the effectiveness of the enhanced interferometer and the compensation method proposed, experiments were also conducted. The performance of the enhanced interferometer was compared against that of a conventional homodyne interferometer. First, the improved homodyne interferometer with four gain-adjustable detectors was set up as shown in Fig. 4. The target mirror was attached to a one-axis piezo flexure stage (P-753.2CD, Physik Instrument, Germany), which was controlled by a high-speed stage position controller (PI E-709.CP, Physik Instrument, Germany). The stage was driven open-loop by a 0.1 Hz triangular wave with a 10 V amplitude, causing a displacement of approximately 2 μm. The signals from the four detectors were sampled and processed by a homemade quadrature-signals processing board (HQSPB). Lissajous graphs and interference signals were displayed by LabVIEW, which simultaneously performed data processing to measure the nonlinearity. The second step is to adjust the gains of the detectors to make I_x and I_y have equal amplitudes. After correcting the phase delay error between I_x and I_y, the amplitudes of the new signals will be corrected again by adjusting the gain of the amplifying circuit. The nonlinearity will be removed at the end of this process. The steps for the nonlinear compensation are shown in Fig. 5.

Fig. 5 Steps for applying the passive compensation method.

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Moreover, in some applications, the measured target is variable and non-cooperative, such that the intensity from the measurement arm of the homodyne interferometer during the measurement may vary due to the change in target reflectivity; when that happens, the parameters of the nonlinearity model in conventional homodyne laser interferometers may vary and cause difficulties in the nonlinearity compensation. To verify the robustness of the improved homodyne interferometer proposed in such a situation, a continuously variable neutral density filter (Vari-NDF) was placed in front of the target mirror to simulate the change in measured beam reflectivity. The experimental results are shown below.

4.2 Results and discussion

The Lissajous representations of the raw signals from the conventional and the enhanced homodyne interferometer for different intensities of the measurement beam are shown in Fig. 6 without any compensation. As shown in Fig. 6(a), in a conventional configuration, the DC offset of the quadrature signals indeed exists, and changes with the intensity of the measurement beam, the eccentricity of the Lissajous trajectories is caused by the imperfections of the optics used for instance the delay angle error (σ) of the QWP. By contrast, the center of the Lissajous representation of the original quadrature signals in the improved configuration shown in Fig. 6(b) is fixed at the zero point regardless of the intensity, the greater eccentricity of the Lissajous trajectories is mainly caused by the phase shift errors (τ_p-τ_s, δ_p-δ_s) of BS. According to the Eqs. (15) and (16) which corresponding the final signals of the proposed interferometer, the phase delay error which is commonly believed to be constant [6,9,10] can be almost totally removed using a simple method, so it has almost no effect on the interferometric precision.

Fig. 6 The Lissajous representation of the original signals from the conventional and the enhanced homodyne interferometer for different intensities of the measurement beam. (a) Conventional homodyne interferometer. (b) Enhanced homodyne interferometer.

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As shown in Figs. 7(a) and 7(b), after system calibration (checking and adjusting the gain of the module for a fixed amplitude adjustment) and vector phase correction based on hardware, the final corrected signals were obtained. Lissajous trajectories are almost perfect circles, and the nonlinear errors are suppressed to less than 0.5 nm in the improved homodyne interferometer regardless of the intensity in measuring arm. The results of the experimental assessment matched the theory, showing the enhanced homodyne interferometer to be quite effective.

Fig. 7 Final result for different intensities of the measurement beam in improved homodyne interferometer. (a) The Lissajous representation of the corrected quadrature signals. (b) Nonlinear error.

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In the improved interferometer, only one calibration process is needed to adjust the gains of the detectors and the amplifying module before the measurement; no further processing is needed during measurement. The nonlinearity model was fixed and free from the change in intensity of the measurement beam, and so the real-time nonlinear error identification was not required. Therefore, the proposed passive method could be applied in high-speed measurements even if the target is non-cooperative.

5. Conclusion

In this work, we have proposed and demonstrated an enhanced homodyne interferometer without a DC offset. Three types of nonlinear error including two variable types have been changed into two fixed types, so that the nonlinear compensation can be achieved more easily without real-time nonlinear identification. By combining the gain adjustment and phase correction methods, the cyclic nonlinearity error in the homodyne interferometer can be compensated for. Our experimental results indicate that the nonlinearity in a homodyne interferometer was suppressed to less than 0.5 nm after calibrating the system, even if the light intensity in the measurement arm changes.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No. 51105114), and Research Fund for Doctoral Program of Higher Education of China (Grant No. 20102302120006).

References and links

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5. Z. Li, K. Herrmann, and F. Pohlenz, “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003). [CrossRef]

6. T. Požar and J. Možina, “Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer,” Meas. Sci. Technol. 22(8), 085301 (2011). [CrossRef]

7. R. Köning, G. Wimmer, and V. Witkovský, “Ellipse fitting by nonlinear constraints to demodulate quadrature homodyne interferometer signals and to determine the statistical uncertainty of the interferometric phase,” Meas. Sci. Technol. 25(11), 115001 (2014). [CrossRef]

8. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17(18), 16322–16331 (2009). [CrossRef] [PubMed]

9. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43(12), 2443–2448 (2004). [CrossRef] [PubMed]

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		A_r₁²	A_r₂²	A_r₃²	A_r₄²	A_m₁²	A_m₂²	A_m₃²	A_m₄²	δ_x - δ_y (°)
Ideal case		1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	0
Case 1		1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0	−8
Case 2		1.0	1.2	1.0	1.2	0.8	0.6	0.8	0.6	−8
Case 3	A_m² = 1.0	1.2	0.9	0.8	1.1	1.2	0.9	0.8	1.1	−8
	A_m² = 0.6	1.2	0.9	0.8	1.1	0.72	0.54	0.48	0.66	−8
	A_m² = 0.3	1.2	0.9	0.8	1.1	0.36	0.27	0.24	0.33	−8

DC-offset-free homodyne interferometer and its nonlinearity compensation

Abstract

1. Introduction

2. Nonlinearity in a homodyne interferometer

3. Enhanced homodyne interferometer and nonlinearity compensation

3.1 The signal model of the enhanced homodyne interferometer

3.2 Signal correction and nonlinearity compensation for the enhanced interferometer

4. Experimental and discussion

4.1 Setup

4.2 Results and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (17)

Optics Express