Heterodyne speckle interferometry for measurement of two-dimensional displacement

Hung-Lin Hsieh; Po-Chun Kuo

doi:10.1364/OE.382494

1. Introduction

The precision measurement of displacement is crucial in nanotechnology, biotechnology, the semiconductor industry, precision manufacturing, and metrology, particularly in the creation of ever-smaller microchips and surfaces with superior energy or reaction efficiency [1–4]. High-precision manufacturing requires high-precision measurements without any delay, error, or interferences. As manufacturing precision approaches the nanometer scale, optical measurement techniques have become increasingly popular, due to the fact that they are non-contact and results can be obtained quickly.

Heterodyne interferometry involves the extraction of information encoded in the phase and/or frequency of electromagnetic radiation at visible or infrared wavelengths. This approach reduces the effects of low-frequency noise (typically seen in conventional homodyne interferometry), while revealing the direction of displacement and eliminating problems related to current offset. The structural and mathematical complexity of heterodyne interferometry systems necessitates the use of an electronic modulation element, such as a Zeeman laser, electro-optic modulator, or acoustic-optic modulator in conjunction with a lock-in amplifier and or program for signal analysis. A variety of modulation techniques have been proposed. In 2005, La et al. [5] introduced a Michelson interferometer based on a Zeeman laser. In 1997, Miyagi et al. [6] introduced a compact acoustic heterodyne interferometer for measuring out-of-plane displacement, and in 2007 Lee et al. [7] presented an electro-optic modulated heterodyne grating interferometer for measuring in-plane displacement. In the current study, we employed an electro-optical modulator (EOM) over other modulation techniques, due to its simple operation and lower modulation frequency.

The measurement of scattered light (referred to as speckle interferometry) is only practical when using a heterodyne light source to counteract low frequency disturbances, due to the fact that the strength of the signal depends heavily on the characteristics of the measurement surface. In grating interferometers, the fact that the signals are independent from the wavelength [7] provides good resistance to environmental factors, such as temperature, humidity, and pressure [8,9]; however, the small size of optical gratings tends to limit the measurement range. They are also susceptible to imperfections in grating surfaces as well as positioning error when mounted on samples. One alternative to reflective grating interferometers is laser speckle interferometers, which project multiple light sources onto a non-mirror surface to form interference patterns with a phase shift imposed by the Doppler Effect under the effects of surface displacement. Speckle interferometers can also be adapted to many surface types, which further expands the range of measurement [10–12]. The first laser speckle interferometers were developed in the early 1960s [13]. In 1981, Dändliker and Willemin [14] combined heterodyne speckle interferometry with two acousto-optic modulators for the measurement of in-plane and out-of-plane vibrations of surfaces, albeit one dimension at a time. In 2008, Goodman [15] analyzed and sorted various features of the speckle phenomenon, elements that can be affected, physical applications, and ways to avoid interference when not needed. In the same year, Jacquot [16] categorized various applications of speckle interferometry according to the type of incident light, methods of observation, physical parameters, and the methods used to analyze results.

In the current paper, we developed a heterodyne speckle interferometry (HSI) system for the measurement of in-plane displacement in two dimensions. The proposed system combines heterodyne signal modulation to eliminate low-frequency disturbance in conjunction with speckle interferometry to expand the range of measurement. Measurement results along different axes are differentiated in terms of laser polarization. Experiment results shows that the system has the advantages of high measurement resolution and high accuracy, as well as a large measurement range and good measurement repeatability. The measurement principle will be discussed in the following section.

2. Measurement principle

In this section, we examine variations in the optical phase of speckle interference signals resulting from the in-plane (IP) and out-of-plane (OP) displacements in a non-mirror surface. The design of the system used to obtain two-dimensional (2D) measurements of IP displacement was based on the mathematical formulation of optical phase variations.

2.1 Single-beam optical phase variations resulting from displacement of non-mirror surface

When a laser beam with incident angle of θ_i is projected onto a non-mirror surface and scattered in all directions (represented by the hemisphere on the surface), the IP and OP displacements of the surface can be represented as vector L in the following form:

(1)$$L = {\textbf u}\vec{x} + {\textbf w}\vec{z},$$

where u represents IP displacement on the x axis, w represents OP displacement on the z axis, and $\vec{x}$, $\vec{z}$ represents the unit vector along the axes. The relationship between the optical phase variation (δ) of the scattered light at the observation angle of θ_o, IP displacement, and OP displacement can be written as follows [17]:

(2)$$\delta = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_i} - \sin {\theta_o}} )+ {\textbf w}({\cos {\theta_i} + \cos {\theta_o}} )} ],$$

where λ is the wavelength of the laser source. This equation clearly shows that the incident and observation angles are associated with optical phase variations.

2.2 Dual-beam single dimensional optical phase variation resulting from displacement of non-mirror surface

As shown in Fig. 1, laser speckle interferometry involves directing two or more beams with the same wavelength λ toward the same location on the surface from different incidence angles to form interference patterns [18]. Assume that δ₁, δ₂ are phase variations in the two beams resulting from laser Doppler Effect, where θ_i₁, θ_o₁, θ_i₂, and θ_o₂ respectively indicate the incident angle and observation angle of the first and second beams. Note that the incident and observation angle should be negative if the light source or detector is on the negative side of the x-plane. If the incident point of the two beams overlaps and a fixed detector receives the overlapped signals, then the observation angle will be the same (i.e., θ_o₁ = θ_o₂) and the difference in phase variation Φ of the combined heterodyne interference formed by two beams can be written as follows [17]:

(3)$$\begin{array}{l} {\delta _{1}} = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_{i1}} - \sin {\theta_{o1}}} )+ {\textbf w}({\cos {\theta_{i1}} + \cos {\theta_{o1}}} )} ],\\ {\delta _2} = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_{i2}} - \sin {\theta_{o2}}} )+ {\textbf w}({\cos {\theta_{i2}} + \cos {\theta_{o2}}} )} ],\\ \Phi = {\delta _1} - {\delta _2} = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_{i1}} - \sin {\theta_{i2}}} )+ {\textbf w}({\cos {\theta_{i1}} - \cos {\theta_{i2}}} )} ]. \end{array}$$

If the incident beams are symmetrical, then the incident angles of the beams will be the same, such that θ_i₁ = -θ_i₂= θ_i and Eq. (3) can be written as follows:

(4)$$\Phi = {\delta _1} - {\delta _2} = {\textbf u}\frac{{4\pi }}{\lambda }\sin {\theta _i}.$$

The measured phase variation can then be used to calculate in-plane displacement u.

In speckle interferometery, the non-mirror surface corresponds to the reflective grating used in the reflective grating interferometer [19] (with multiple beams incident to the rough surface rather than the grating), and the detector measures the intensity of the scattered light. Unlike a grating, the light reflected in a speckle interferometer is scattered in all directions. This eliminates the need to align the detector to the reflected beam; however, the signal strength tends to vary according to the location of the detector, and alignment in the direction of surface displacement is still necessary. As shown in Eq. (4), the wavelength of the light source is used to calculate the displacement distance represented by each phase variation period (rather than the pitch of a grating).

Fig. 1. Laser Doppler effects and speckle interferometery of two beams.

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2.3 Adapting to spherical coordinates

As shown in Fig. 2, to illustrate the theory underlying measurement with two degrees-of-freedom (DOF), the theory of speckle interferometry can be adapted to a three-dimensional (3D) spherical coordinate system, such that Eq. (1) is written as follows:

(5)$$L = {\textbf u}\vec{x} + {\textbf v}\vec{y} + {\textbf w}\vec{z},$$

where v refers to displacement of the surface on the y-axis. The location of the incident point within a spherical coordinate system can be determined using the follow equations:

(6)$$\begin{array}{l} \vec{x} = r\sin \theta \cos \varphi ,\\ \vec{y} = r\sin \theta \sin \varphi ,\\ \vec{z} = r\cos \theta , \end{array}$$

where r is the radial distance, θ is the polar angle (i.e., the angle between vector and $\vec{z}$ axis, 0° ≤ ≤ 180°), and φ is the azimuthal angle (i.e., the angle between the orthogonal projection of the vector on the xy plane and x-axis, 0°≤φ<360°). Thus, Eq. (2) can be adapted as follows:

(7)$$\begin{array}{l} \delta = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_i}\cos {\varphi_i} - \sin {\theta_o}\cos {\varphi_o}} )+ } \\ {{\textbf v}({\sin {\theta_i}\sin {\varphi_i} - \sin {\theta_o}\sin {\varphi_o}} )+ {\textbf w}({\cos {\theta_i} + \cos {\theta_o}} )} ], \end{array}$$

where θ_i and θ_o are the polar angles of the incidence and reflection vectors, as well as the incidence and reflection angles to the surface, whereas φ_i and φ_o are the azimuthal angles of the incidence and reflection vectors. Thus, the difference in phase variation Φ is derived as follows:

(8)$$\begin{array}{l} \Phi = \frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_{i1}}\cos {\varphi_{i1}} - \sin {\theta_{i2}}\cos {\varphi_{i2}}} )+ } \\ {{\textbf v}({\sin {\theta_{i1}}\sin {\varphi_{i1}} - \sin {\theta_{i2}}\sin {\varphi_{i2}}} )+ {\textbf w}({\cos {\theta_{i1}} - \cos {\theta_{i2}}} )} ], \end{array}$$

where θ_i₁, θ_i₂, φ_i₁, φ_i₂ are the polar angles and azimuthal angles of the incidence and reflection vectors associated with the two beams.

Fig. 2. Incidence and reflection angle under 3D coordinate system.

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2.4 Configuration of heterodyne speckle interferometry system

As shown in Fig. 3, a heterodyne light source is obtained when a laser source with the wavelength λ passes through a polarizer with the transmittance axis set at 45° to the xz plane in conjunction with an electro-optical modulator (EOM) using a sawtooth signal at the preferred frequency. According to Su’s principle and Jones calculation [20], the amplitude of the heterodyne light beam can be written as follows:

(9)$${E_{EOM}} = \left( {\begin{array}{{c}} {{{\mathop{\rm e}\nolimits}^{i\Delta \omega t/2}}}\\ {{{\mathop{\rm e}\nolimits}^{ - i\Delta \omega t/2}}} \end{array}} \right),$$

where Δω is the frequency of the modulation signal.

Fig. 3. Configuration of heterodyne speckle interferometry system.

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The heterodyne laser beam entering a 2D grating is diffracted. The x-axis ± 1^st order beams respectively pass through 0° and 90° polarizers, with the second beam passing through an additional half-wave plate (HWP) with fast axis at 45° to shift its polarization. Thus, both beams become p-beams. Similarly, the y-axis ± 1^st order beams respectively pass through the same polarizers and HWP to become s-beams. The Jones matrix of the beams on each individual surface E_x+₁, E_x-₁ and E_y+₁, E_y-₁ can be written as follows:

(10)$$\begin{array}{l} {E_{x + 1}} = P({0^\circ } )\cdot {E_{EOM}} = \left( {\begin{array}{c} {{{\mathop{\rm e}\nolimits}^{i(\Delta \omega t/2)}}}\\ 0 \end{array}} \right),{E_{x - 1}} = J({180^\circ } )\cdot P({90^\circ } )\cdot {E_{EOM}} = \left( {\begin{array}{c} {{{\mathop{\rm e}\nolimits}^{ - i(\Delta \omega t/2)}}}\\ 0 \end{array}} \right),\\ {E_{y - 1}} = P({90^\circ } )\cdot {E_{EOM}} = \left( {\begin{array}{c} 0\\ {{{\mathop{\rm e}\nolimits}^{ - i(\Delta \omega t/2)}}} \end{array}} \right),{E_{y + 1}} = J({180^\circ } )\cdot P({0^\circ } )\cdot {E_{EOM}} = \left( {\begin{array}{c} 0\\ {{{\mathop{\rm e}\nolimits}^{i(\Delta \omega t/2)}}} \end{array}} \right), \end{array}$$

where J (180°) is the Jones matrix of phase retardation, which is performed in the proposed system using a 45° half-wave plate.

All four beams are reflected onto the measured surface via mirrors to form two sets of speckle interference on their respective planes. The beams are incident on the same point symmetrically in their respective planes; therefore, the incident angles of the beams on both planes are symmetrical (θ_ix₊₁=θ_{i x}₋₁=θ_ix, θ_iy+1=θ_iy−1=θ_iy, φ_ix₊₁=0°, φ_{i x}₋₁=180°, φ_iy+1=90°, φ_iy−1=270°). According to Eq. (7), the phase shift of each beam can be derived as follows:

(11)$$\begin{array}{l} {\delta _{x + 1}} = \exp \left\{ {i\frac{{2\pi }}{\lambda }[{{\textbf u}({\sin {\theta_{ix}} - \sin {\theta_{ox}}} )+ {\textbf w}({\cos {\theta_{ix}} + \cos {\theta_{ox}}} )} ]} \right\},\\ {\delta _{x - 1}} = \exp \left\{ {i\frac{{2\pi }}{\lambda }[{{\textbf u}({ - \sin {\theta_{ix}} - \sin {\theta_{ox}}} )+ {\textbf w}({\cos {\theta_{ix}} + \cos {\theta_{ox}}} )} ]} \right\},\\ {\delta _{y + 1}} = \exp \left\{ {i\frac{{2\pi }}{\lambda }[{{\textbf v}({\sin {\theta_{iy}} - \sin {\theta_{oy}}} )+ {\textbf w}({\cos {\theta_{iy}} + \cos {\theta_{oy}}} )} ]} \right\},\\ {\delta _{y - 1}} = \exp \left\{ {i\frac{{2\pi }}{\lambda }[{{\textbf v}({ - \sin {\theta_{iy}} - \sin {\theta_{oy}}} )+ {\textbf w}({\cos {\theta_{iy}} + \cos {\theta_{oy}}} )} ]} \right\}, \end{array}$$

where δ_x+₁ and δ_x₋₁ are the phase variation of light scattered from E_x+₁ and E_x₋₁ when the surface undergoes displacement in the x-axis, and δ_y₊₁ and δ_y-₁ are the phase variation of E_y+₁ and E_y₋₁ when the surface undergoes displacement in the y-axis.

In measuring x-axis displacement, a detector with a 0° analyzer is used to receive the displacement signal along the xz plane, as only p-polarized light E_x+₁δ_x+₁ and E_x₋₁δ_x₋₁ is required. Likewise, in measuring y-axis displacement, a detector with a 90° analyzer is used to receive the displacement signal along the yz plane, as only s-polarized light from E_y+₁δ_y+₁ and E_y₋₁δ_y₋₁ is required. When the plane undergoes displacement on the x or y axis, the corresponding intensities I_x, I_y are derived as follows:

(12)$$\begin{array}{l} {I_x} = {|{{E_{x + 1}}{\delta_{x + 1}} + {E_{x - 1}}{\delta_{x - 1}}} |^2} = 2 + 2\cos \left( {\Delta \omega t + \frac{{4\pi {\textbf u}}}{\lambda }\sin {\theta_{ix}}} \right),\\ {I_y} = {|{{E_{y + 1}}{\delta_{y + 1}} + {E_{y - 1}}{\delta_{y - 1}}} |^2} = 2 + 2\cos \left( {\Delta \omega t + \frac{{4\pi {\textbf v}}}{\lambda }\sin {\theta_{iy}}} \right). \end{array}$$

The phase variation in each signal Φ_x and Φ_y can be extracted using a lock-in amplifier for phase demodulation and calculation:

(13)$${\Phi _x} = \frac{{4\pi }}{\lambda }{\textbf u}\sin {\theta _{ix}},{\Phi _y} = \frac{{4\pi }}{\lambda }{\textbf v}\sin {\theta _{iy}}.$$

The displacement on each in-plane axis of the surface can then be calculated from the phase variation in each signal, as follows:

(14)$${\textbf u} = \frac{{{\Phi _x}\lambda }}{{4\pi \sin {\theta _{ix}}}},{\textbf v} = \frac{{{\Phi _y}\lambda }}{{4\pi \sin {\theta _{iy}}}}.$$

In-plane displacement on the x-axis (u) and y-axis (v) can then be extracted using a lock-in amplifier (LIA) in the form of hardware or software to measure phase variations.

In the experiment, we used a 2D holographic grating with a specification of 3.33 μm pitch and 21% diffraction efficiency to split the identical light source into four beams for subsequent use. Due to the 2D grating is difficult to acquire, another option is to pair a one-dimensional (1D) grating with another 1D grating or beam splitter to split the beams a second time. It would even be possible to use a polarized beam splitter to eliminate the need for two of the polarizers. The use of mirrors to illuminate the beam symmetrically onto the surface, makes it possible to use the beam splitting method; however, this increases the complexity of the structure, which makes it harder to set up the measurement system.

3. Experiments

The proposed HSI measurement system was evaluated using a number of experiments, including single- and dual-axis in-plane displacements over various ranges as well as the repeatability of the results and time required to obtain measurements. The light source in the experiments was a He-Ne laser (λ = 632.8 nm) with a 45° polarizer and an electro-optical modulator (EOM, Newport co., model: 4002) with the sawtooth heterodyne signal frequency set at 16 kHz. The incident angles were set at ± 45°; however, the system still required calibration to compensate for misalignment error during set-up (see Section 4.4). Experiments on long-range displacement were performed using a long-range displacement platform, whereas measurements of displacement at the mid-, short-, and nano-scales was performed using a 6-DOF positioning stage.

For the measurement of captured signals and the analysis and recording of lock-in data, the detector (Thorlabs, model PDA36A) sends the acquired signal to an analytics program (National Instruments, LabView) using a data acquisition (DAQ) card (National Instruments, BNC-2110). To deal with reference signals, the long-range displacement platform (Measure control Inc, XS-50) has a built-in linear encoder, whereas the positioning stage (Physik Instrument, P-562.6CD with Physik Instrument, E-712.6CDA controller) has an internal capacitance displacement sensor. Both are closed-loop controlled with the measured reference signal.

3.1 X-axis (horizontal) and y-axis (vertical) displacement tests over various ranges

Long-range, mid-range as well as short-range displacement tests were performed to demonstrate the measurement capability of the proposed HSI system across various measurement ranges. Long-range displacement measurements were obtained using a long-range displacement platform with 3000 μm horizontal back-and-forth displacement output. Note that experiment results were compared with the displacements recorded using the linear encoder. The platform is capable of making only horizontal displacements; therefore, the long-range displacement measurement test was performed only on the x-axis. Moreover, the mid- and short-range displacement measurements were obtained using a 6-DOF positioning stage (100 μm for mid-range and 1 μm for short-range displacement) with sinusoidal, triangular, and trapezoidal waveform motions. Note that two random signals (one for each axis) were generated using multiple sinusoidal and ramp waves to provide displacement patterns for the platform. The experiment results were compared with the values obtained using the stage-mounted capacitance displacement sensor.

As shown in Figs. 4 to 7, the long-range, mid-range, short-range and random signals displacements measured using the proposed HSI (lines of colored circles) match the corresponding reference signals obtained using commercial sensors (solid black lines). After calculating the measurement results obtained by the linear encoder, capacitive displacement sensor and our proposed interferometer, the measurement errors between the methods in x-axis and y-axis are approximately 0.02%, 0.03% and 0.2% for the ranges of 3000 μm, 100 μm and 1 μm, respectively. These experiments demonstrate the efficacy of the proposed system in obtaining various range surface measurements of high precision.

Fig. 4. Experiment result of x-axis displacement measurement on the scale of 3000 μm using triangular waves.

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Fig. 5. Experiment results of x-axis and y-axis displacement measurements on the scale of 100 μm using sinusoidal, triangular, and trapezoidal waves.

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Fig. 6. Experiment results of x-axis and y-axis displacement measurements on the scale of 1 μm using sinusoidal, triangular, and trapezoidal waves.

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Fig. 7. Experiment results of x-axis and y-axis random signals displacement.

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3.2 Repeatability tests

Repeatability is an important factor, indicating the reliability of measurement results. When obtaining displacement measurements, less deviation (δ) between the measured start and end positions corresponds to a smaller difference between each of the measurements, i.e., superior repeatability. In this study, we assessed the repeatability of the proposed system by moving the positioning stage in 5 steps of 10 nm (a total of 50 nm) each along both axes. We then compared with the measured displacement values obtained from the interferometer and built-in commercial capacitive displacement sensor. The results are displayed in Fig. 8.

Fig. 8. Experiment results of repeatability in measurement of x-axis and y-axis displacement. 5 steps of 10 nm each (a total of 50 nm) along x-axis and y-axis are shown in (a) and (c). Average value of each step are shown in (b) and (d).

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Figure 8 is the experiment results of repeatability test in measurement, where Figs. 8(a) and 8(c) show the measured results for x-axis and y-axis, respectively. The error (green solid triangle) represents the result of deviation between the capacitive displacement sensor and the proposed HIS. Clearly, the trend and behavior of the displacements measured by our method conform to those measured by the capacitive displacement sensor. The experimental results obtained by the two methods are in agreement with each other along each axis. Figures 8(b) and 8(d) show the average values for every step level (position) of the capacitive displacement sensor (solid black square) and the HSI (solid blue and red circle) in each axis. For example, as can be seen in Fig. 8(b), the average values of the measurement results of the HSI at the 0^th step and the 10^th step are denoted by B0 and B10, respectively; while the average values of the measurement results of the capacitive sensor at the 1^st step and the 9^th step are denoted by A1 and A9, respectively. The deviations (repeatability) between each corresponding positions (e.g. B0 and B10, A1 and A9) can then be obtained by subtracting the two average values. After calculating the experiment result, Table 1 shows the deviations (repeatability) between each corresponding positions of the two methods in x-axis and y-axis. As can be seen from Table 1, δ_CP and δ_HSI represent the deviations of the capacitive sensor and the HSI at the corresponding positions. Note that the smaller the deviation, the higher the repeatability. The maximum deviations of our proposed HSI in x-axis and y-axis are respectively approximately 0.5 nm and 0.9 nm while the values of the internal capacitive sensor are approximately 0.3 nm and 0.2 nm. This strongly demonstrates the high repeatability of the proposed HSI configuration on both axes.

Table 1. Deviations between the corresponding steps (positions) of the repeatability test

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4. Discussion

4.1 Sensitivity

Theoretically, the resolution of laser interferometers is limited only by the detection capabilities of the sensor, whereas the measurement range is limited only by the dimensions of the measured surface. In the proposed HSI, the relationship between measured phase variation Φ and surface in-plane displacement u can be expressed as follows:

(15)$$\frac{\Phi }{{\textbf u}} = \frac{{4\pi \sin {\theta _i}}}{\lambda },$$

where λ is the wavelength (632.8 nm) of the He-Ne laser used in the experiment, and θ_i is the symmetrical incidence angle, where a larger incidence angle corresponds to measurements of higher resolution. With all incidence angles set at 45°, the sensitivity should be 0.805°/nm.

4.2 Analysis of periodic nonlinearity error

Based on the theory of heterodyne measurement, the proposed HSI system is prone to non-linear periodic errors associated with frequency mixing, polarization mixing, and polarization-frequency mixing. The periodic nonlinearity error can result from imperfections in the polarizers or from the misalignment of polarizers and/or half wave plates. In the following discussion, we discuss the periodic nonlinearity error.

The optical elements for each measurement plane comprised a polarizer for each beam, an HWP for one of the beams, and an analyzer in front of the detector for signal separation. Polarization is the only fundamental difference between x-axis and y-axis measurements; therefore, we can use the same calculation method and the calculated periodic nonlinearity error will be the same. Thus, for the sake of simplicity, we describe only x-axis measurement in the following. We assume that the extinction ratio and azimuth angle of the analyzer and polarizers are α, β, γ and δ_AN, δ_P1, δ_P2 respectively. We also assume that the phase retardation error and alignment error of the azimuth angle of the half wave plate are ɛ and δ_HWP, respectively. Using the phase variation from both beams (δ₁ and δ₂), the electric field of the x-axis E_x′ including the periodic nonlinearity error can be written as follows:

(16)$$\begin{aligned}{E_x}^\prime = &\textrm{AN(}\alpha ,0^\circ{+} {\delta _{AN}})[{e^{i{\delta _1}}}P({\beta ,0^\circ{+} {\delta_{P1}}} ){E_{EOM}} + \\ &{e^{i{\delta _2}}}HWP({180^\circ{+} \varepsilon ,45^\circ{+} {\delta_{HWP}}} )P({\gamma ,90^\circ{+} {\delta_{P2}}} ){E_{EOM}}]. \end{aligned}$$

The corresponding intensity I_x′ can be written as follows:

(17)$${I_x}^{\prime} = {|{{E_x}} |^2} \propto AC\cos ({\omega t + {\Phi _x}^{\prime}} ),$$

where Φ_x′ indicates the deformed phase. The corresponding variation in phase error ΔΦ can be obtained using the following equation:

(18)$$\Delta {\Phi _x} = {\Phi _x}^{\prime}({\alpha ,{\delta_{AN}},\beta ,{\delta_{P1}},\gamma ,{\delta_{P2}},\varepsilon ,{\delta_{HWP}}} )- {\Phi _x}.$$

This refers to periodic nonlinearity error in signals obtained for the measurement plane in x-axis. Set-up of the current experiment included precision rotation mounts to allow careful adjustment and minimize deviations from the correct azimuth angle of the analyzers, polarizers and half-wave-plate. Note that this made it possible to control the misalignment angles to within approximately 5′. The extinction ratio of each analyzer and polarizer were 1:4000, and the retardation error of HWP was λ/300 (retardation angle error = 360°/300 = 1.2°). For each phase cycle, the periodic nonlinearity error in measured displacement was approximately ± 0.16°. For example, when all incident angles were set at 45°, the periodic nonlinearity error would be ± 1.25 nm. The difference between estimates and measurement results can generally be attributed to the problem of elliptical polarization from the light source or a lack of precision in the alignment of the experiment set-up.

In addition, since the x-axis and y-axis displacement signals of the proposed interferometer were carried at the same modulation frequency and then separated by using the two corresponding analyzers, a crosstalk phenomenon may induce nonlinear errors in the measurement results of each axis. This means the signal of y-axis would contribute an error to the periodic nonlinearity error in the measured displacement of x-axis signal. In order to analyze the periodic nonlinearity error caused by this cross-talk phenomenon, by substituting the same misalignment parameters into Eq. (16) to Eq. (18) under the saturation of considering this phenomenon, we obtain periodic nonlinearity error of approximately ± 0.163° (±1.27 nm) for every signal period. The result shows that the periodic nonlinearity error caused by the cross-talk phenomenon is extremely small (approximately 1.8% of the total periodic nonlinearity error), which can be regardless in our proposed interferometer.

5. Conclusions

This paper presents a novel heterodyne speckle interferometer for the measuring two degrees-of-freedom in-plane displacements. The system measures in-plane displacement of a non-mirror surface by directing multiple incident beams to one specific point on the sample surface to form speckle interference. The phase of the interference pattern is then shifted by any surface displacement. The use of polarizers and half-wave-plates to separate the displacement signals according to axis makes it possible to obtain measurements of in-plane displacement on two axes simultaneously. Experiments demonstrated the efficacy of the proposed system in measuring x-axis and y-axis displacements as well as the stability, repeatability, and speed of the measurement process. Our results are comparable to those obtained using commercial displacement sensors built into the sample stage. Other important factors related to sensitivity, and periodic nonlinearity error are also discussed.

Funding

Ministry of Science and Technology, Taiwan (107-2221-E-011-104-MY2).

Acknowledgments

We thank Dr. Ju-Yi Lee for his help with the experiment.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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	x-axis		y-axis
Position	δ_CP (nm)	δ_HSI (nm)	δ_CP (nm)	δ_HSI (nm)
0 nm	0.06 (A10-A0)	0.02 (B10-B0)	0.23 (A10-A0)	0.92 (B10-B0)
10 nm	0.18 (A9-A1)	−0.31 (B9-B1)	0.03 (A9-A1)	−0.68 (B9-B1)
20 nm	−0.08 (A8-A2)	0.53 (B8-B2)	−0.11 (A8-A2)	−0.50 (B8-B2)
30 nm	−0.33 (A7-A3)	−0.11 (B7-B3)	−0.08 (A7-A3)	−0.33 (B7-B3)
40 nm	−0.05 (A6-A4)	−0.17 (B6-B4)	0.11 (A6-A4)	0.20 (B6-B4)

Heterodyne speckle interferometry for measurement of two-dimensional displacement

Abstract

1. Introduction

2. Measurement principle

2.1 Single-beam optical phase variations resulting from displacement of non-mirror surface

2.2 Dual-beam single dimensional optical phase variation resulting from displacement of non-mirror surface

2.3 Adapting to spherical coordinates

2.4 Configuration of heterodyne speckle interferometry system

3. Experiments

3.1 X-axis (horizontal) and y-axis (vertical) displacement tests over various ranges

3.2 Repeatability tests

4. Discussion

4.1 Sensitivity

4.2 Analysis of periodic nonlinearity error

5. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (8)

Tables (1)

Equations (18)

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