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Abstract
An optical standing-wave interferometer based on the detection of scattered light is proposed in this study. By inserting an ultra-thin scattering plate into the optical standing-wave field and detecting the scattered light, the intensity of the optical standing-wave field can be observed. The phase quadrature detection technique using two scattering plates is developed for measuring the displacement. The experimental results demonstrate that the measurement resolution and range can reach nanometer and micrometer levels, respectively.
© 2017 Optical Society of America
1. Introduction
A laser interferometer is a commonly used measurement tool and offers high measurement resolution and a wide dynamic measurement range [1,2]; therefore, it has been widely used for highly precise displacement measurements within the fields of nanotechnology, semiconductor technology, biotechnology, etc. In a typical interferometer, such as the Michelson or Mach–Zehnder configuration, the laser light is split into two beams by a beamsplitter, and these two beams propagate back to the beamsplitter which then superposes their amplitudes interferometrically. By analyzing the interference signal or fringes, the displacement of the target or another quantity, such as the refractive index can be determined. Because several optical components are used, including the beamsplitter, retroreflectors, and polarization components, and because a two optical arm configuration is required in a typical interferometer, they are limited in their ability to be of compact size. Over the past few years, standing-wave sensors have been developed and widely used for displacement measurement [3–10], spectrum analysis [11–14], laser tuners [15], and wavelength-sensitive detection [15]. By means of utilizing transparent thin photodiodes, a highly compact size was achieved. In general, transparent thin photodiodes are designed as a sandwich structure. p-, i- and n-type semiconductors and a transparent conductive oxide (TCO) are stacked upon a glass substrate by the sputter and plasma-enhanced chemical vapor deposition technique. In order to achieve a higher sensing performance, the thickness of the stacked layers in the transparent thin photodiode must be well controlled. For example, the thickness of the i-layer must be of the order of a quarter wavelength, and the total thickness of the photodiode should be around half a wavelength to avoid absorption losses and to achieve a high response of the photocurrent [6]. Moreover, because the n-i-p diode is ultra-thin, in order to protect the n-i-p diode from ion bombardment, the sputtering rate for stacking the TCO layers must be low. In short, the manufacturing process of the transparent photodetector is highly complex.
In this study, we developed an optical standing wave detection method based on light scattering using a single layer of spherical silicon dioxide nanoparticles. The preparation of the single layer of spherical silicon dioxide nanoparticles is relatively easy. The detection principle of the proposed technique is described in detail below. Moreover, the application of the displacement measurement by quadrature detection technique is introduced, and the feasibility and performance are demonstrated.
2. Principle
In this section, at first, the observation of the optical standing waves using the scattering plate is introduced. Next, the quadrature signals and the calibration process for the displacement measurement are described.
2.1. Observation of the optical standing wave
As shown in Fig. 1, after passing through the optical isolator OI, the laser beam with wavelength λ normally is incident on the mirror M and is reflected back. The forward and backward light beams interfere with one another and build the standing-wave field. Assuming the forward and backward light beams (E1 and E2) meet at position z1, then the electric fields of these two beams can be expressed as [3]
andwhere ω is the optical frequency of the laser beam, and z2 is the position of the mirror M. These two beams are superposed and interfere with each other. The interference intensity of the standing wave at position z1 can be written aswhere I0 = E102 + E202 and γ = 2 E10E20/(E102 + E202) are the main intensity and contrast of the optical standing wave, respectively. This is the intensity distribution of a typical optical standing wave. This optical standing-wave field will be scattered when an ultra-thin scattering plate S is inserted in the optical path. The thickness of the scattering plate should be thinner than one half of the wavelength. The scattered light intensity is proportional to the intensity of the optical standing wave. The scattered light collected by the lens L is received by the photodetector PD. The light intensity Is on the photodetector is used for the displacement measurement. Considering the thickness [4] of the scattering plate and speckle effect [16], the contrast γs of the scattered light intensity Is on the photodetector should be smaller than γ. Is can be rewritten aswhere Is0 is the main intensity of the signal, and ϕs is the initial phase that results from the thickness of the scattering plate. A more detailed discussion about the signal of the scattering light will be provided in section 4.1. Equation (4) indicates that the optical standing-wave field can be observed by the ultra-thin scattering plate, and the displacement z2 of the mirror can be determined by analyzing the variation of the scattered light intensity.
Fig. 1 Observation of the optical standing wave by scattering plate S. OI: Optical isolator, L: Lens, PD: photodetector, M: Mirror.
2.2. Quadrature signals
Equation (4) indicates that the signal resulting from the optical standing wave is a periodic function of distance between the scattering plate S and the mirror M, and the displacement of the mirror M can be determined via detection of the phase variation of the signal. In the present study, phase detection based on the quadrature method was developed. As shown in Fig. 2, two scattering plates Sa and Sb are inserted into the standing-wave field and the scattered lights Isa and Isb are received by photodetectors PDa and PDb, respectively. Similarly, the scattered light intensities on PDa and PDb can be written as
andwhere IsX0, γsX, and ϕsX are the main intensity, contrast, and initial phase of the scattered light on the photodetectors, respectively, and the suffix X is a or b. z1a and z1b are the positions of the scattering plates Sa and Sb. We can select a suitable position for scattering plate Sb to create a π/2 phase shift between Isa and Isb, that iswhere m is an integral. The scattered light intensity Isb on PDb can be rewritten asThe aperture A is used to separate the scattered lights from the two scattering plates Sa and Sb. The signals Isa and Isb, (Eqs. (5) and (8)) are quadrature. By adjusting the DC (Isa0 and I sb0) and AC (I sa0⋅γsa and Isb0⋅γsb) terms, the phase variations of the signals can be solved and used to determine the displacement of the mirror M.Fig. 2 Schematic of the quadrature detection system. OI: Optical isolator, PZT: Piezoelectric actuator, DAQ: Data acquisition card, PC: Personal computer.
2.3. Quadrature signals calibration for displacement measurement
The following calibration process was used to adjust the DC and AC terms of these two signals in Eqs. (5) and (8) to ensure the quadrature condition and avoid measurement error. We first drive the mirror with an increasing displacement. As shown in Fig. 3, the intensity of the interference signal Isa oscillates between the minimum Isam and maximum IsaM (see the lower curve in Fig. 3). The DC and AC terms of the signal Isa can be expressed as
andSimilarly, the DC and AC terms of the signal Isb (see the upper curve in Fig. 3) can be given asandThe signals in Eqs. (5) and (8) can be calibrated and expressed asandFrom the calibrated signals I'sa and I'sb in Eqs. (13) and (14), the relationship between the position z2 of the mirror M and the calibrated interference signals can be written asHere C = z1a + λϕ0a/4π can be regard as a constant. Equation (15) indicates that we can measure the intensity variation of the calibrated interference signals I'sa and I'sb to determine the displacement z2 of the mirror M.3. Performance test
3.1. Preparing of the scattering plate
In this work, SiO2 nano-spheres are prepared by the Stöber process as in our previous study [17]. After the synthesis of the SiO2 spheres, a single layer SiO2 sphere array is obtained using the dip-coating method. Figure 4(a) shows a panoramic view of the scattering plate made with a single layer of SiO2 spheres of 300 nm in diameter on the glass substrate. The SiO2 spheres cover the whole glass substrate and are uniform without significantly aggregates at a glance. Figure 4(b) shows the SEM image of the cross-sectional view of the monolayer of 300 nm SiO2 spheres on a glass substrate. The inset of Fig. 4(b) reveals the arrangement morphology the SiO2 spheres.
Fig. 4 (a) Optical image of the glass substrate covered with 300 nm SiO2 spheres. (b) SEM image of the cross-sectional view of the monolayer of 300-nm SiO` spheres on glass substrates. Inset: Top view of the monolayer spheres arrangement.
3.2. Experimental setup
The configuration of the displacement measurement is shown in Fig. 2. A green laser (model: ADR-1805; SFOLT Co.Ltd) with a wavelength of 532 nm was used as the light source. A mirror M was mounted on a piezoelectric actuator (model: P-611; Physik Instrumente (PI) GmbH), and the displacement of the actuator was measured. The standing-wave field was built between the mirror M and the laser. In order to avoid the laser feedback interference or self-mixing effect resulting from the back-reflection or scattering light, the mirror M and the scattering plates Sa and Sb were titled slightly. In addition, a homemade optical isolator OI consisting of a Savart prism and a quarter-wave plate was used in this system. The scattered lights from the two scattering plates Sa and Sb inserted into the standing-wave field were received by photodetectors PDa and PDb, and the received signals were processed by a data acquisition card (NI6143) and a personal computer. A Labview (version: 7.0, National Instruments Corporation) program was used to calculate the phase variation for the displacement measurement.
3.3. Micrometer- and nanometer-scale displacements testing
To demonstrate that the proposed optical standing-wave interferometer was able to accurately measure the displacement, the piezoelectric actuator moved the mirror M in triangular-wave and sinusoidal forms with several micrometer-scale displacements. Figure 5 shows the measurement results. An internal strain gauge sensor integrated within the piezoelectric actuator was used to simultaneously measure the movement of the actuator and verify the displacements. The symbol “o” and solid curves indicate the measurement results obtained with the proposed method and the results obtained using the strain gauge sensor, respectively. The trend and behavior of the displacement measured by our method conformed to those measured by the strain gauge sensor. From the experimental results it can be seen that displacements of different magnitudes could be measured with satisfactory precision. However, the measurement results suffered inherent nonlinear periodic error, which will be analyzed in the following section.
The experiment of the step-wise motion with nanometric scale was also demonstrated in this study. The strain gauge sensor was also used simultaneously to confirm the displacement measurements. The measurement results obtained with the proposed method and the strain gauge sensor are shown in Fig. 6. The measurement results obtained with our method coincided well with those obtained using the strain gauge sensor.
4. Discussion
4.1. The contrast of the signal
M. Sasaki [4], D. Knipp [13] and H. L. Kung [14] indicated that the signal contrast of the standing-wave sensor based on the thin film photodiode depends on the active layer thickness. Here we propose a rough model that is similar to their idea to describe the signal contrast. As shown in Fig. 7(a), the scattering plate is inserted into the optical standing-wave field, and of course the scattering material scatters the field. The scattering plate is divided into small elements of a layer with thickness dz'. From Eq. (3), the scattered light intensity dIs on the photodetector by the small element layer (dz') can be expressed as
where n and Z are the refractive index and geometric thickness of the scattering material, and nZ is the optical thickness of scattering material. z + z' is the position of the small element layer. Because the scattered light from the individual small elements within the layer are incoherent, their intensities can be summed, that isEquation (17) indicates that the contrast of the intensity signal on the photodetector isThe last term of Eq. (17) is the initial phase ϕs = 2πnZ/λ that results from the thickness of the scattering plate. We used lasers of different wavelengths and scattering plates of different thicknesses to test the contrast of the scattered light. The scattering material consisted of SiO2 nano-spheres and the refractive index was ~1.450 [18]. From Eq. (18), the relationship between the contrast γs and the ratio of the optical thickness and wavelength nZ/λ could be obtained and the curve is shown in Fig. 7(b). Here we assumed that γ = 0.9. The symbol “o” represents the measured contrasts of the standing-wave signal. The measured contrast γs decreased with increasing ratio nZ/λ. In spite of the fact that there were differences between the measured data and the simulation, the measured contrast trend followed the simulation curve. We believe that dust adsorbed on the reflecting mirror or the scattering plate surface disturbed the measurements. Nevertheless, the contrast of the signals and the signal-to-noise ratio (SNR) were good for the displacement measurement.Fig. 7 (a) Light scattered by the small element layer of the scattering plate. (b) Relationship between the contrast of the scattered light and the ratio of nZ to λ.
4.2. Measurement resolution
From Eq. (15), the displacement z2 was determined by the measured intensity of the signals. Therefore, the resolvable intensity was used to determine the minimum measurable displacement Δz2. According to the measurement uncertainty analysis [19] and Eq. (15), the minimum measurable displacement could be written as
Here ΔI'sa and ΔI'sb represent the minimum detectable calibrated signals, and can be regarded as noise. We set them to have the same magnitude ΔI' = ΔI'sa = ΔI'sb. The term is the magnitude of the signals, and the last term of Eq. (19) is the reciprocal of the SNR of the signals. Here we define the SNR as the ratio of the mean of the magnitude of the signal to the standard deviation of the noise [20]. In our experiment, the mean of the magnitude of the signal and the standard deviation of the noise are around 970 mV and 32 mV, respectively. Thus, an SNR of 30 is obtained. Equation (19) indicates that the minimum measurable displacement depends on the SNR, with an estimated measurement resolution of the displacement of 1.4 nm. In addition, the piezoelectric actuator was held stationary for experiment estimation of the noise level of the measured displacement and the measurement resolution. The test results are shown in Fig. 8, and the noise level of the measured displacement is coincident with the value obtained by the measurement uncertainty analysis method.4.3. Nonlinear periodic error
In order to examine the nonlinear periodic error, mirror M was subject to micrometer scale displacement. The displacement was measured simultaneously with our method and a strain gauge sensor. The nonlinear periodic error can be observed by analyzing the difference between the displacement measured using our method and that measured by the strain gauge sensor. The results are shown in Fig. 9. The curve in Fig. 9(a) shows the nonlinear periodic error to have a quantity of 5 nm. We believe that this kind of error resulted from multi-beam interference and imperfect calibration. The scattering plate consisted of a nanosphere and glass substrate. As a consequence of the partial reflection on the glass surface, beams reflected from the scattering plate propagated and oscillated between the scattering plate and the mirror M. These multi-reflected beams are known as ghost beams. The ghost reflection and the multi-beam interference resulted in nonlinear periodic error. P. Hu [21] carried out a complete investigation of the nonlinear periodic error resulting from the multi-order ghost beams in the homodyne interferometer. Here we use Hu’s model to analyze the nonlinear periodic error in our system. The nonlinear periodic error εk resulting from the kth ghost beam can be simply expressed as
where Γ is the intensity of the light beam, and Γk/Γ1 denotes the intensity ratio of the kth ghost beam to the main beam; φ is the phase variation resulting from the displacement of the mirror. In our experiment, the intensity ratio Γ2/Γ1 of the second ghost beam to the main beam is about 11%. From Eq. (20), the peak-to-peak value of the nonlinear periodic error ε2 can be estimated, and it is coincident with the measured results in Fig. 9(a). Given that the intensity of the higher order ghost beams is significantly smaller than that of the main beam, the nonlinear periodic error arising from the higher order ghost beams is limited. The calibration process described in section 2.3 is followed to build the quadrature signals for phase detection. The calibrated (I'sa and I'sb) signals are plotted as Lissajous patterns and are shown in Fig. 9(b). The Lissajous patterns of the calibrated signals did not comprise a real circle, because the calibrated signals I'sa and I'sb contained residual DC, unequal AC terms, and a phase shift not equal to π/2. Due to the ghost reflection and the multi-beam interference, even when we fine-tuned the DC and AC terms of the signals, and the phase shift between the two signals, the residual quantities continued to result in a nonlinear periodic error of 5 nm.Fig. 9 (a) Difference between the measured displacement using our method and that measured using the strain gauge sensor, and (b) Lissajous patterns of the calibrated (I'sa and I'sb) signals.
5. Summary
Based on the detection of light scattering, an optical standing wave detection methodology and corresponding application of the displacement measurement was proposed. In the developed system, the forward- and backward-propagating laser beams make up the optical standing-wave field, scattering plates comprised of a single layer of spherical silicon dioxide nanoparticles are inserted into the optical standing-wave field, and the scattered light is received for the observation of the standing wave. We demonstrated the ability of the system to perform phase quadrature detection by two scattering plates to measure the displacement of the mirror. In our experiment, the measurement resolution and range were found to reach nanometer and micrometer levels, respectively. The periodic nonlinearity error caused by residual DC, unequal AC terms of the signals, and quadrature phase shift error were discussed.
Funding
Ministry of Science and Technology (MOST) of Taiwan (MOST 103-2221-E-008-066-MY3, MOST 106-2221-E-008-073, MOST 106-2218-E-008-006).
Acknowledgments
Thanks go to Mr. Yen-Nung Cheng for preparing the manuscript.
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