## Abstract

We propose a novel axial nano-displacement measuring approach. Based on asymmetrical illumination, the axial drifts of the sample plane can be measured by detecting the position of the centroid of the focal spot. Both CCD and QD are used as the detector in the system and two data processing models are designed. With a relatively simple and applicable configuration, the proposed system can realize a wide measuring range of $>4\lambda $and a high axial resolution of 2nm. Moreover, the presented approach is immune to the influence caused by the energy fluctuation of the laser source. Possessing these advantages, this measuring method has big potential to be applied in modern engineering and scientific researches.

© 2013 OSA

## 1. Introduction

Nano-displacement measurement techniques are drawing increasingly attentions during the past decade in the fields of modern engineering and scientific researches. Among these techniques, optical methods are of vital importance for their unique characteristics of non-contact measurement, fast response and high accuracy [1].

Optical methods can be generally classified into two categories. The first one is based on interferometry. The basic principles of optical displacement measurement based on interferometer are raised by Michelson [2]. A typical displacement measuring interferometer system consists of a frequency-stabilized laser source, measuring electronics and interferometer optics, which splits the laser light into a measurement path and a reference path. The distance change in the measurement path relative to the reference path will cause a phase change in the detected interference signal [3]. To further improve the dynamic range and accuracy of the measurement, heterodyne interferometer [4]was introduced and multiple techniques were presented and discussed, including frequency and power stabilization, nonlinearities, modeling, implementation and simulation of displacement measurement system using two-mode He-Ne laser [5–8]. However, most high-resolution interferometers suffer from either an expensive system setup [9] or complex signal processing model [10].

The other category of the displacement measurement methods is without the using of coherent source. A typical system for displacement measurement without interferometry is confocal microscope invented in the early 1960s [11]. Based on this design, various configurations have been developed to further enhance its displacement discerning capability, including bipolar super-resolution differential confocal microscopy (SDCM) [12], chromatic confocal microscope [13], total internal reflection (TIR) [14], differential internal multi-reflection (DIM) [15] and axial displacement measurement based on astigmatism effect [16]. However, these methods still have their own limitations. SDCM had a complicated structure and a stringent adjustment requirement in the offset positions of its two point detectors. Chromatic confocal microscope is easily affected by the environment, especially the energy fluctuation of the light source. TIR method is not applicable in low-NA conditions. The measuring range of DIM is still limited and the astigmatism method has not been able to achieve an axial resolution better than 30nm.

In this paper, we designed and analyzed a novel axial displacement measurement system based on asymmetric illumination. Both QD (quadrant detector) and CCD are used as the detection unit in the experiment with two signal processing methods designed respectively. The sensitivity and measuring range of this system are analyzed through a series of simulations. Meanwhile, the axial resolution of the proposed approach is estimated by practical experiments. With a relatively simple and applicable configuration, the proposed system achieves a resolution of 2 nm and a measuring range of $>4\lambda $.

## 2. Main theory

The schematic optical diagram of our approach is shown in Fig. 1 . After passing through the optical fiber, laser emitting from a laser diode is collimated by a collimating lens. The beam is then converted into different polarization states through the combination of the polarizer and the polarization converter. The polarized beam is phase modulated by a phase plate and then filtered after passing through a series of lens and two spatial filters. The first filter has the shape of a small hole while the second one has the shape of a semi-circle. After the filtering, the original beam with the circular shape turns into a semi-circle beam. Passing through the beam splitter (BS1), the semi-circle beam is focused onto the sample plane installed on a scanning stage by an objective lens. The beam reflected by the sample plane is collected through the same objective lens and then reflected by the beam splitter (BS2) to be separated from the illumination beam. Two devices, a CCD and a QD, are used as the detection module in the system. Hence, another beam splitter (BS2) is placed into the detection path to split the beam into two arms, which are detected by the CCD and the QD, respectively.

The phase plate used in the system is a one-dimensional step phase plate which introduces a phase shift between the upper and lower halves of the transmitting beam (Fig. 2(a) ). The phase plate is placed that the phase separating line is parallel to the X axis. The orientation of the XYZ coordinate system is shown in Fig. 1 and its origin locates at the focal point of the objective lens. According to Fresnel diffraction equation [17], after the laser beam passes through the phase plate, the beam vector becomes:

here, $E(x,y)$denotes the amplitude distribution of the laser beam illuminating on the phase plate and $O(x,y)$is the transmission function of the phase plate. In our application, $E(x,y)$can be approximated by a Gaussian function:

and for the phase plate used in the system, $O(x,y)$can be expressed as:

Using the above equations, we can calculate the intensity distribution of the beam wavefront after it passes through the phase plate and the result is shown in Fig. 2(b). The result indicates that an intensity distribution that is symmetric to a dark centerline has been achieved. The orientation of the dark centerline is along the X axis. In this case, when the beam passes through the semi-circle spatial filter, a high quality semi-circle beam pattern can be generated since the diffraction at the edge of the semi-circle spatial filter can be largely reduced by the existence of the dark centerline.

When the semi-circle beam being focused by the objective lens, the beam intensity distribution on the sample plane can be calculated by Debye Integral [18]. Based on vectorial diffraction theory, for an objective lens with a high numerical aperture, the electric field vector near the focal spot can be obtained as:

Here, $E({r}_{2},{\phi}_{2},{z}_{2})$is the electric field vector at the point of $({r}_{2},{\varphi}_{2},{z}_{2})$expressed in a cylindrical coordinate whose origin locates at the ideal focal point of the objective lens, C is the normalized constant, ${A}_{1}(\theta ,\varphi )$is the amplitude function of the incident semi-circle light, ${A}_{2}(\theta ,\varphi )$is a function related to the structure of the objective lens and $\left[{p}_{x\text{\hspace{0.17em}};\text{\hspace{0.17em}}}{p}_{y\text{\hspace{0.17em}};\text{\hspace{0.17em}}}{p}_{z}\right]$is a matrix unit vector about the polarization of the incident beam.

With Eq. (4), we can calculate the different beam intensity patterns on the sample plane when the sample plane moves around the focal plane of the objective lens along the Z axis. We assume that the illumination beam is circularly polarized and the axial moving range of the sample plane is from $z=-2\lambda $ to $z=2\lambda $, where $\lambda $ is the wavelength of the illumination beam. The calculation results are shown in Fig. 3 (Media 1). From Fig. 3, we find that the centroid of the intensity pattern moves along the Y axis as the sample plane travels along the Z axis. Only when the sample plane places at the focal plane of the objective lens will the centroid of the corresponding intensity pattern locates on the Z axis, i.e. Y = 0. Hence, by detecting the distance between the centroid of the intensity pattern on the sample plane and the Z axis, we can determine the axial displacement of the sample plane.

Both CCD and QD are used to detect the centroid displacement. In our setup, the beam pattern on the detector can be considered as a magnified view of the beam pattern on the sample plane and there lies a linear relationship between the centroid displacement on the sample plane and the centroid displacement on the image plane of the detector. Hence, in the following discussion, we simply use the centroid displacement on the imaging plane to represent that on the sample plane. While using CCD, the centroid displacement on the image plane can be calculated as:

where ${y}_{ci}$and ${P}_{i}$denotes the Y coordinate and the intensity value of the i-th pixel, respectively, $N$is the total number of pixels on the CCD. In the case of QD, the centroid displacement on the image plane is described by the following expression:

where ${U}_{i}$denotes the output voltage value of the i-th quadrant.

## 3. Simulation

Based on the above theory, we test the validity and performance of our measuring approach by calculating the response curves, i.e. the relationship between ${D}_{c}$and the axial displacement $\Delta z$in the CCD detecting mode or the relationship between ${D}_{q}$and the axial displacement $\Delta z$in the QD detecting mode. Both ${D}_{c}$and $\Delta z$are expressed in the units of the illuminating wavelength. In the calculation, illumination beams of different polarization states are considered, including X direction linear, Y direction linear, circular, radial and azimuthal. All the simulated results shown in Fig. 4 are presented with an objective lens of NA = 1.4. The magnification factor of CCD detection is assumed to be $150\times $.

The sensitivity of the measurement is an important means to evaluate the performance of a measuring system and it can be represented by the slope of the response curve. As illustrated in Fig. 4, while using CCD as the detector, the response curves changes little under different polarization states. The slight differences between the response curves can only be distinguished in the magnified view which indicates that the azimuthally polarized illumination beam produces the best sensitivity while using CCD as the detector. In the QD detecting mode, the response curves are influenced by the polarization states more than that in the CCD detecting mode and it is the X linearly polarized illumination beam that leads to the best sensitivity. Figure 4 also indicates that both ${D}_{c}$ and ${D}_{q}$have a one to one relationship with the axial displacement$\Delta z$, in other words, given a ${D}_{c}$or ${D}_{q}$, there will be a unique$\Delta z$. These characters ensure the validity of our measuring approach in both the CCD and QD detecting mode.

Besides sensitivity, the performance of a measuring system is depending on its measuring range as well. Here, the measuring range of our approach is estimated by curve fitting. Since the polarization states do not influence much in the response curve, for simplicity, we just take circularly polarized beam as the example in the measuring range estimation. The curve fitting results are shown in Fig. 5 . In the CCD detecting mode, the response curve is highly linear. The linear fitting of the response curve yields a fitted equation of ${D}_{c}=48.97\Delta z+2.142\times {10}^{-15}$and correlation coefficient $r=0.9999$which indicates a linear measuring range of $>4\lambda $can be achieved and it is profoundly applicable for displacement measurement. In the QD detecting mode, however, ${D}_{q}$ doesn’t respond linearly to $\Delta z$in the whole simulation range of $-2\lambda $~$2\lambda $. The linearity only occurs in a sub-range. Therefore, we apply cubic polynomial fitting in the displacement range of $-2\lambda $ ~$2\lambda $ and linear fitting in the range of $-0.8\lambda $ ~$0.8\lambda $. Cubic fitting yields a fitted equation of ${D}_{q}=-0.0447\Delta {z}^{3}-3.884\times {10}^{-17}\Delta {z}^{2}+0.5225\Delta z+1.264\times {10}^{-17}$and $r=0.9998$. Linear fitting yields an equation of ${D}_{q}=0.5105\Delta z+6.206\times {10}^{-17}$ and $r=0.9993$. Hence, QD detecting mode renders a smaller linear measuring range of $1.6\lambda $. One thing must be noticed is that although the linearity of the response curve in the QD detecting mode only occurs in the range of $-0.8\lambda $ ~$0.8\lambda $, the one to one relationship exists in the whole simulation range of $-2\lambda $ ~$2\lambda $as shown above. Therefore, our approach is also feasible in this $4\lambda $range after calibration.

## 4. Experiment and analysis

To evaluate the performance of our approach in practical, we built an experimental system. The laser diode used in our system is CL-2000 Diode Pump Crystal Laser (633nm) by Crystal Laser. The polarization converter and its controller are made by Newport. Here, we covert the polarization states of the illumination beam into circularly polarized. The scanning stage is Thorlabs NanoMax-TS (Max311D/M) controlled by BPC203 Benchtop Piezo Controller. A Leica HCX PL APO 100 × /1.40–0.7 Oil is used as the objective lens. The CCD used in our system is BeamOn HR USB 2.0 camera and the QD is a quadrant cell photoreceiver for Newport (model 2921).

First, we test the performance of our method in the large displacement case. In CCD detecting mode, we use a scanning stage to move the sample plane along the Z axis in a range of −1.6μm~1.6 μm continuously for 6 seconds, as shown in Fig. 6(a) . In QD detecting mode, the movement of the sample plane is set in a range of −0.4μm~0.4 μm, as shown in Fig. 6(b). The measurement results for the two modes are shown in Fig. 6(c) and Fig. 6(d), respectively. We find that both ${D}_{c}$and ${D}_{q}$change linearly with $\Delta z$, which verifies the feasibility of our approach. Moreover, these results also demonstrate the stability of our system since the peak and valley values in both detecting modes change little during the measurement.

Next, we carry out two experiments to analyze the resolution of this system in CCD detecting mode and QD detecting mode, respectively. While using CCD as the detector, we move the sample plane along the Z axis in a range of −40nm~40nm continuously for 20 seconds (Fig. 7(a) ). The resulting waveform of ${D}_{c}$is shown in Fig. 7(c). As expected, the value of ${D}_{c}$ changes with the axial movement of the sample plane. Although the shape of the waveform becomes a little bit zigzag which indicates that the system starts to lose its discerning ability, the peak and valley values of the waveform are still easy to be distinguished. Hence, an axial resolution of 40nm is achieved in the CCD detecting mode.

To determine the axial resolution of our approach in the QD detecting mode, we move the sample plane along the Z axis by the scanning stage in a range of −2nm~2nm continuously for 4 seconds (Fig. 7(b)). Similar as the CCD detecting mode, both the peak and valley values of the resulting ${D}_{q}$waveform (Fig. 7(d)) can be easily discerned despite its zigzag shape and this result implies that an axial resolution of 2nm can be obtained by our approach while using QD as the detector.

From the simulation and experiment above, we find that both CCD detecting mode and QD detecting mode has their own advantages: CCD detecting mode has a longer linear measuring range while QD detecting mode can achieve a better axial resolution. Hence, under different measuring aims, we may choose the appropriate detecting mode to maximize the performance of the measurement.

In our approach, the attainable resolution is mainly limited by the response characteristic of the detecting devices. And the measurement error of our system may come from several sources. Firstly, since the geometric path of the light ray is determined by the geometric parameters and positions of the optical components in the system, the stability of these optical components will influence the measuring results. Secondly, the positioning error of the scanning stage which cannot be eliminated will also cause measurement error because the calibration of the system is based on the mechanical scanning of the sample plane by the scanning stage. Thirdly, due to the limited AD resolution of the CCD and QD used in the system, quantization error may occur during data detection and processing. Fourth, the air turbulence and temperature fluctuation in the experimental environment may also introduce some errors into the system. In the proposed approach, the energy fluctuation of the laser source will not influence the measuring results since our measurement is based on the drift of the centroid of the beam other than the energy changes.

## 5. Improvement of the measurement performance

Besides polarization states, the beam shape, i.e. the amplitude distribution of the beam will also influence the response curves in our measuring approach. Apart from the Gaussian beam used in our application, the Bessel-Gauss beam [19] is also widely used in scientific research [20, 21]. Bessel-Gauss beam can simply be generated by making the incident Gaussian beam propagating through an annular aperture, or by using some specific devices like axicon, adaptive optics or optical fiber [22]. Here, we calculate the response curves of circularly polarized Bessel-Gauss illumination beams in both the CCD and QD detecting mode and compare them with their Gaussian shaped counterparts. For simplicity, we ignore the magnification factor of CCD detection in the calculation. From the results shown in Fig. 8 , we find that in both the CCD and QD detecting mode, using Bessel-Gauss beam can improve the measuring sensitivity. Moreover, Bessel-Gauss beam also enlarges the linear measuring range in the QD detecting mode.

In our application, the asymmetrical illumination is realized by using a semi-circle shaped filter to block the illumination beam. Besides this method, the asymmetrical illumination can also be obtained with a quarter-circle shaped filter, as shown in Fig. 9(a) . In this case, only a quarter part of the illumination beam will be focused onto the sample plane by the objective lens and the resulted response curves calculated in the same way as the semi-circle illumination are shown in Fig. 9(b) and Fig. 9(c). For simplicity, the illumination beams are assumed to be circularly polarized Gaussian beams and the magnification factor of CCD detection is ignored. From the results, we find that replacing the semi-circle filter with a quarter-circle filter can realize a better measuring sensitivity in both the CCD detecting mode and the QD detecting mode. Actually, the measuring sensitivity can be further improved if more parts of the illumination beam, especially those low-NA parts, are blocked by using another spatial filter.

## 6. Conclusion

In this paper, a novel axial displacement measuring approach is proposed. Based on asymmetrical illumination, the axial displacement of the sample plane can be measured by detecting the position of the beam centroid. Both CCD and QD are used as the detector in the system. With a relatively simple and applicable configuration, our system can realize a wide measuring range of $>4\lambda $and a high axial resolution of 2nm. The performance of our approach can be further improved when applying Bessel-Gauss beam and quarter-circle filter in the system. Moreover, unlike some former methods, the proposed approach will not be affected by the energy fluctuation of the laser source. We believe this method can be widely applied in modern engineering and scientific researches.

## Acknowledgments

We thank Ding Luo for his help in performing the experiment. This work was financially supported by grants from National Natural Science Foundation of China (No. 61205160), the Qianjiang Talent Project (No.2011R10010), the Doctoral Fund of Ministry of Education of China (No.20110101120061 and No.20120101130006)

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