Precision enhancement of three-dimensional displacement tracing for nano-fabrication based on low coherence interferometry

Yu Cheng; Xiangchao Zhang; He Yuan; Wei Wang; Min Xu

doi:10.1364/OE.27.028324

1. Introduction

With the development of nano science and technology, the high precision positioning technique has become a key in advanced technologies such as the IC lithography, micro/nano manufacture and precision instrumentation [1–3]. In some cases, the absolute displacement with respect to a particular reference position, rather than a relative displacement, is demanded. For example, the spot size of the focused laser beam in the femtosecond laser fabrication directly determines the fabricating resolution. But the depth of focus turns out to be very tight, as a consequence precise positioning of the laser beam is a key factor to the high precision fabrication.

Optical techniques are usually adopted in the measurement of nano-displacements for their high resolution and high dynamic range, e.g., the grating interferometry and laser interferometry [4,5]. The former has a large measuring range, but unfortunately its positioning repeatability is in the order of submicron, which is not sufficient for the requirement of nano-stages [6,7]. On the contrary, the homodyne interferometry and heterodyne interferometry are superior on the measuring resolution, thus they have extensive applications in many areas [8,9]. But the measurement accuracy is seriously affected by the stability of the wavelength and refractive index, which in turn vary with the environmental temperature and humidity. More importantly, these methods cannot measure the absolute distances.

The swept-frequency interferometry is becoming popular due to the versatility of its sources and its ability to measure length absolutely. The length of the optical path can be determined when the laser is swept through a frequency range [10–12]. This method can achieve a large measuring range, but the frequency sweeping makes the real-time measurement of displacements infeasible. Frequency combs can achieve accurate long-distance measurement with high dynamic range, e.g., achieving 1.1 µm uncertainty over 50 m [13], or achieving 24 pm resolution over 14 µm [14]. Nevertheless, the complexity and expenses of the frequency comb technique make it difficult to find extensive applications.

The low coherence interferometry (LCI) is currently widely employed in the measurement of surface topographies, called the coherent scanning interferometry [15,16]. By scanning the measured sample along the optical axis, the value of the mutual coherence function varies with the optical path difference (OPD). The maximum of the envelope of the mutual coherence function occurs when the OPD is equal to zero, corresponding a region with the clearest and sharpest fringes in the interferogram. Eventually the surface topographies are obtained from the relative height of each region [17,18]. In order to improve the measuring accuracy of the atomic force microscopy, the probe was proposed to be imaged using LCI, and its absolute position can be specified precisely from the captured interferograms [19,20]. But this method mainly intends to improve the measuring accuracy of AFM and cannot be directly applied to measuring the displacement of samples.

An absolute position sensor was introduced to detect the position of a single point. Although the movement of a sample can be obtained by multi-channel detectors, the assembly difficulty and cost are relatively high, and the resulting Abbe errors can reduce the measurement accuracy [21]. The absolute ranging can also be achieved by measuring the phase slope of the Fourier components in the frequency domain with the low coherence interferometry [22]. Nhue et al reported an approach to measure the vertical movements by identifying the best coherence position of a flat [23]. However, the measurement accuracy is affected by the noise since only a part of the measurement data is used. In addition, the tilt of the sample is not considered in these displacement tracing methods, but this issue is significant in nano-fabrication because of the straightness error of the guide rail and other factors. Thus conventional methods cannot meet the requirements of precise positioning and displacement tracing.

In this paper, a high-precision measuring strategy of three-dimensional displacements is developed based on the low coherence interferometry. The rest of the paper is organized as follows. Section 2 presents the methodology of the proposed measuring method. Section 3 provides numerical and experimental validation, and finally, the paper is summarized in Section 4.

2. Methodology of the three-dimensional displacement tracing

2.1 Measuring setup of the Low Coherence Interferometry

In the low coherence interferometry, the interference intensity can be written as

(1)$$I({x,y,h} )= {A_1} + {A_2}\left\{ {1+{e^{ - {{\left[ {2\pi \frac{{z({x,y} )+ h}}{{{L_c}}}} \right]}^2}}}\cos \left[ {4\pi \frac{{z({x,y} )+ h}}{{{\lambda_0}}}} \right]} \right\},$$

where A₁ is the average intensity, A₂ is the intensity modulation, h is the displacement of the measured surface or the reference mirror, λ₀ is the mean wavelength, and L_c is the length of coherence. In practice, z(x, y) is the topography of the sample to be machined/measured. When the object moves with the displacement table, the captured interferogram changes accordingly.

Due to the low coherence of light, the LCI does not suffer from speckle noise or ghost image, thus the measuring reliability can be effectively improved. Here an LED point source is adopted for the LCI measurement, and a collimated beam is obtained through the objective lens. The real time displacement tracing system for the femto-second laser fabrication is illustrated in Fig. 1(a), where the fabricating beam is depicted in red, and the measuring beam is depicted in green. The reference mirror should be carefully placed to guarantee that the location with zero OPD is coincident with the optimal focusing point of the fabricating beam, and the scanning distance is within the coherence length. The variation of the sample topography leads to different interference intensities, as shown in Fig. 1(b). As the sample/objective moves along the z direction, interferograms are continuously captured using a high speed CMOS camera, as shown in Fig. 1(c). The spectral bandwidth of the CMOS is 400∼800 nm, since the infrared light is generally used in the actual femtosecond laser fabrication. Therefore, they can work simultaneously without interaction. Rather than separating the monitoring and fabricating paths, the same objective lens is used for both purposes to ensure that the displacement of the region being fabricated can be detected precisely, and the Abbe error caused by the inclination of the sample surface can be avoided.

Fig. 1. The real time displacement tracing system for the femto-second laser fabrication. (a) The LCI system. (b) The interference intensity of LCI. (c) The recorded interferograms.

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2.2 Calculation of the three dimensional displacements

The relationship between the interference intensity and the displacement is highly nonlinear, thus the calculation of the displacement is implemented in two steps, namely, the system calibration and the iterative refinement.

2.2.1 Calibration of the system parameters

After setting up the displacement tracing system, a stage/objective is moved in the z direction by a high precision piezoelectric ceramic transducer (PZT) and a series of interferograms are obtained. It should be noted that the displacement of the PZT needs be guaranteed accurate and not inclined in practice. A set of intensity-displacement data I-h can be recorded, with each pixel (u, v) in the captured interferogram sets corresponding to a point z(x, y). The system parameters, namely, the mean wavelength of the light source, coherence length and light intensity can be obtained accordingly by numerical fitting. The flow chart of the system calibration is shown in Fig. 2.

Fig. 2. The flow chart of the system calibration.

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The initial guess is set first, which is critical to the convergence results. Here four variables A₁, A₂, L_c and λ₀ in Eq. (1) can be obtained straightforwardly, e.g., A₁ and A₂ are calculated by the maximum and minimum intensities of all the interferograms; L_c and λ₀ can be obtained by measuring the frequency spectral distribution of the light source. However, the difficult part is to specify the topography z(x, y). Set z(x, y) ≈ 0 to those image pixels with the greatest interference intensity. If the sample surface is sufficiently smooth, z(x, y) can be obtained from its neighboring pixels whose heights have been specified already.

Then the interference intensities of the same pixel (u, v) is fitted successively using the Levenberg-Marquardt algorithm to refine the height of each point z(x, y) . Considering that the number S of data for each point, i.e., the number of interferograms is small, random errors will exert significant effect on the fitted results.

After that, all the data points are utilized in the fitting of the global parameters A₁, A₂, L_c and λ₀ with z(x, y) fixed, subsequently the height z(x, y) is optimized individually for each point using the optimized global parameters. Refine the global parameters and topographies z(x, y) alternatively until convergence, so that reliable system parameters can be obtained and the influence of discrete sampling can be effectively reduced.

2.2.2 Iterative refinement

After the system calibration, subsequent interferograms are captured in the practical measurement, and the displacement h is in Eq. (1) can be worked out accordingly. However, vibration and tilt may occur in the movement of the stage, then the three-dimensional displacement can be described as

(2)$$h(t )= {k_x}(t )x + {k_y}(t )y + d(t ),$$

where k_x and k_y are the tilts about the x and y axes, respectively, and d is the vertical displacement.

Then the three-dimensional displacement h(t) associated with each interferogram is solved with the Levenberg-Marquardt algorithm. Define the set of variables x = [k_x, k_y, d] first, then an initial guess of d⁽⁰⁾ is obtain by a low precision sensor like a grating interferometer. As the actual tilt is very small, the initial guess is set as x⁽⁰⁾=[0, 0, d⁽⁰⁾]. Finally, the iterative refinement is conducted as

(3)$$\begin{array}{l} {{\textbf x}^{({s + 1} )}}={{\textbf x}^{(s)}} - {({{J^T}J+k{\textbf E}} )^{ - 1}}{J^T}{{\textbf r}^{(s )}}\\ \textrm{with}\,{{\textbf r}^{(s )}}={I_m} - I({{{\textbf x}^{(s )}}} )\,\textrm{and}\,J = \frac{{\partial I}}{{\partial {\textbf x}}}, \end{array}$$

where E is the identity matrix, J is the Jacobian matrix, k is the damping factor, and I_m is the measured interferogram.

In the actual calculation, due to the debris generated by the femtosecond laser fabrication, the central region is excluded dynamically, and eventually the displacement of the central region is obtained by interpolating Eq. (2). Through the rapid convergence in the iterative optimization process, the tilt about the x and y axes can be decoupled from the interferograms and fed back to the nano-fabrication system. Meanwhile, the influence of random errors can be suppressed by the averaging effect of the least squares fitting, then the precision can be greatly improved.

In addition, only the variable h is of concern in Eq. (1), which means that the displacements are mainly dependent on the fringe change, instead of the interference pattern itself. Therefore, the wave aberration of the measuring beam arising from the spherical aberrations caused by the beam splitter and the objective lens can lead to irregular interference fringes. But little change occurs in this quasi-parallel beam during the vertical scanning within a small range, thus the influence of such wave aberration to the displacement measurement can be ignored. This implies that the requirement on the component qualities and assembly accuracy of the optical setup can be significantly released.

2.2.3 Precise Focusing

In order to achieve precise focusing for the femtosecond laser fabrication, the position of the reference mirror depicted in Fig. 1(a) will be carefully adjusted to ensure the spot to be fabricated is within the coherence length of LCI, i.e., clear interference fringes appear in the measured sample. In order to establish the relationship between interference fringes of LCI and the optimal focusing position in the laser fabrication, the sample is processed at several positions after the system calibration is completed. Then the characteristic sizes of the fabricated spots are measured by the scanning electronic microscopy, where the optimal focusing is achieved when the spot has a minimal size. The relative departure between the position of optimal focusing and the position producing the greatest mutual coherence function at the central region is recorded.

In practice, beam focusing is conducted in two steps. First, the reference light in the interference setup is blocked by inserting a black baffle. As a consequence, the monitoring path turns out to be a simple microscopic imaging system. In this case, the image at the focused region has higher sharpness than at the defocused region. Consequently, a criterion based on the image sharpness is used for coarse focusing, as calculated by Eq. (4). The maximum value of Var means the best focusing position [24].

(4)$$Var = \frac{1}{{XY}}\sum\limits_{x = 1}^X {\sum\limits_{y = 1}^Y {{{[{I({x,y} )- \overline I } ]}^2}} } .$$

Second, the black baffle is removed after completing the coarse focusing. Several interferograms are captured during scanning along the z direction, and the position of the sample center is calculated by iterative refinement. Then, the relative difference between the current position and the optimal focusing position can be worked out during scanning. The difference is fed back to the control system and precise focusing can henceforth be achieved by adjusting the displacement table.

2.3 Uncertainty estimation

In conventional methods, only the average or maximum interference intensity is used to determine the displacement. The measuring uncertainty changes remarkably with the sample position since the degree of coherence decreases when the measuring position is gradually shifted from the zero-OPD position. The total differential expansion is conducted on Eq. (1) and the resulting uncertainty is presented in Eq. (5) and Fig. 3. When the measuring position is gradually shifted from the zero-OPD position, Δh increases simultaneously. Yet the underlying noise of the detector caused by temperature, dark current and other factors will not change, leading to decrease of the measuring precision.

(5)$$\Delta h = \Delta I\frac{{{\lambda _0}{L_c}^2}}{{4\pi {A_2}{e^{ - {{\left( {\frac{{2\pi }}{{{L_c}}}h} \right)}^2}}}\left|{{L_c}^2\sin \left( {\frac{{4\pi }}{{{\lambda_0}}}h} \right)+8\pi {\lambda_0}h\cos \left( {\frac{{4\pi }}{{{\lambda_0}}}h} \right)} \right|}}.$$

In the proposed method, all the measured points on the surface are utilized to reduce the influence of random errors by the averaging effect. The measurement uncertainty is discussed in two cases. Assuming that there are m measuring points in an interferogram, the first case is that the topography of the sample is completely smooth and unanimous, then the uncertainty will be reduced as ${\Delta }{h_m}\textrm{ = }{{{\Delta }h} \mathord{\left/ {\vphantom {{{\Delta }h} {\sqrt m }}} \right. } {\sqrt m }}$.

Fig. 3. Relationship between the measuring uncertainty and OPD.

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In practice, the sample may be tilted or of complex topography, and the OPD at each point will not be identical. For the sake of clarity, only the displacement h in the z direction in Eq. (2) is considered. The solution of the displacement is a numerical optimization problem.

(6)$$\begin{array}{l} {h^{\ast }} = \arg \min \sum\limits_{k = 1}^m {{{||{{I_k} - f({{z_k} + h} )} ||}^2}} \\ \textrm{with}\,f(x )={A_1} + {A_2}\left[ {1+{e^{ - {{\left( {2\pi \frac{x}{{{L_c}}}} \right)}^2}}}\cos \left( {4\pi \frac{x}{{{\lambda_0}}}} \right)} \right]. \end{array}$$

Therefore, the solution h* conforms to

(7)$$\begin{array}{{c}} {Q = \frac{{\partial {{\sum\limits_{k = 1}^m {||{{I_k} - f({{z_k} + h} )} ||} }^2}}}{{\partial h}} = \sum\limits_{k = 1}^m { - 2[{{I_k} - f({{z_k} + h} )} ]} {f^\prime }({{z_k} + h} )}\\ {{{ {dQ} |}_{{h^{\ast }}}} = \sum\limits_{k = 1}^m { - 2} {f^\prime }({{z_k} + {h^{\ast }}} )d{I_k} + \sum\limits_{k = 1}^m {\{{2{f^\prime }^2({{z_k} + {h^{\ast }}} )- 2f^{\prime\prime}({{z_k} + {h^{\ast }}} )[{{I_k} - f({{z_k} + {h^{\ast }}} )} ]} \}} d{h_m}=0} \end{array}.$$

Due to the symmetric distribution of the residuals of the least squares fitting, we have $\sum\limits_{k = 1}^m {f^{\prime\prime}({{z_k} + {h^{\ast }}} )[{{I_k} - f({{z_k} + {h^{\ast }}} )} ]=0}$. Therefore,

(8)$$dh = \frac{{\sum\limits_{k = 1}^m {{f^\prime }({{z_k} + {h^{\ast }}} )d{I_k}} }}{{\sum\limits_{k = 1}^m {{f^\prime }^2({{z_k} + {h^{\ast }}} )} }}.$$

In practical applications, the uncertainty of I is basically the same and proportional to the coherence length ΔI = pL_c [25], and the z coordinates can be considered to be uniformly distributed over a range of ± H with H≥λ₀,

(9)$$\begin{array}{l} \Delta {h_m}=\Delta I\frac{{\sqrt {\int_{ - H}^H {{f^\prime }^2({z + {h^{\ast }}} )dz\frac{m}{{2H}}} } }}{{\int_{ - H}^H {{f^\prime }^2({z + {h^{\ast }}} )dz\frac{m}{{2H}}} }}=\frac{{p{L_c}}}{{\sqrt {\int_{ - H}^H {{f^\prime }^2({z + {h^{\ast }}} )dz\frac{m}{{2H}}} } }}\\ \textrm{with}\,{{f^{\prime}}^2}(x )= 32{\pi ^2}{e^{-2{{\left( {\frac{{2\pi }}{{{L_c}}}x} \right)}^2}}}{A_2}^2\left[ {\left( {\frac{{4{\pi^2}}}{{{L_c}^4}}{x^2} - \frac{1}{{{\lambda^2}}}} \right)\cos \left( {\frac{{8\pi }}{\lambda }x} \right) + \frac{{4\pi }}{{{L_c}^2\lambda }}x\sin \left( {\frac{{8\pi }}{\lambda }x} \right) + \frac{{2{\pi^2}}}{{{L_c}^4}}{x^2} + \frac{1}{{{\lambda^2}}}} \right]. \end{array}$$

This complex integral function can be simplified. Since h is much less than L_c, several small terms can be ignored and only 1/λ² is remained. The resulting relative error is less than 1%. Consequently, the uncertainty can be expressed as

(10)$$\Delta {h_m}=\frac{{{\lambda _0}p\sqrt {\frac{{{L_c}H}}{m}} }}{{{{({2\pi } )}^{\frac{3}{4}}}{A_2}\sqrt {erf\left( {\sqrt 8 \pi \frac{{{h^{\ast }} + H}}{{{L_c}}}} \right) - erf\left( {\sqrt 8 \pi \frac{{{h^{\ast }} - H}}{{{L_c}}}} \right)} }}.$$

It is found that the uncertainty is affected by the central wavelength, the length of coherence, the intensity modulation, OPD and the range of surface heights. The impacts of the actual OPD and the length of coherence are shown in Fig. 4 with H=λ₀. The measuring uncertainty increases monotonously with the increase of OPD. As for the coherence length, larger coherence noise occurs in a light source of high coherence. On the contrary, a shorter coherence length significantly reduces the intensity modulation, especially when OPD is growing larger. In addition, the measurement range of displacements is proportional to the coherence length. As a result, the wavelength width of the light source needs to be selected carefully according to the specific requirements of practical measurements.

Fig. 4. Influence of the coherence length and OPD to the measuring uncertainty. The central wavelength is λ₀=500 nm.

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3. Numerical and experimental demonstration

The performance and feasibility of the proposed method are verified by the numerical demonstration and experiments.

3.1 Numerical demonstration

Set the system parameters as listed in Table 1 and generate a series of interferograms for system calibration according to Eq. (11), with random errors embedded in accordance with [26]. Here the interference intensity is converted into an integer to mimic the quantization effect of the gray levels. The central wavelength is 540 nm considering the actual light source and CMOS spectral bandwidth. To verify the universality of the proposed method, different light sources, sampling intervals of the displacements and noise levels are tested.

(11)$$I(x,y,h) = \textrm{INT[}{I_{theoretical}}(x,y,h) + n(x,y,h)],$$

the function INT [] denotes the integer nearest to the argument gray level, and the noise n (x, y, h) is Gaussian-distributed with a mean of zero and a standard deviation in the range 1%∼10% of the modulation value A₂.

After system calibration, interferograms at different locations are generated. Then calculate the corresponding displacements and compare the obtained displacements with the pre-set values. The calibration and measurement results are presented in Table 2. To be more intuitive, the measurement results are also presented in Figs. 5(a)–5(d).

Fig. 5. Numerical demonstration results of different system parameters. To make the plots clearer, the red dot lines presenting the solved displacements in (a) and (b) are shifted by 5 µm, and those in (c) and (d) by 0.5 µm.

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Table 1. Parameter setting

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Table 2. System calibration results and displacement measurement results

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The results of system calibration indicate that the calibration method has universality and effectiveness. Meanwhile, the measurement error is mainly caused by the random noise and gray level quantization. It is found that the measurement error associated with a greater coherence length is smaller under the same noise condition since the coherence noise is not considered due to the complexity. Meanwhile, the measurement uncertainty increases when the sample position gradually shifts further from the zero-OPD position, as discussed in Subsection 2.3.

Then the anti-noise capability of the method is also validated by the Monte Carlo simulation of 5000 runs, using the same parameters with those in Fig. 5(a). The results associated with difference noise levels are depicted in Figs. 6(a) and 6(b) and Table 3. The measurement error always conforms to the Gaussian distribution, but the standard deviation increases proportionally with the noise level.

Fig. 6. Monte Carlo testing results. The random errors added are 5% and 10%, respectively.

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Table 3. Specifications of Monte-Carlo results

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In order to demonstrate the measuring ability of three-dimensional displacements, the sample does not only move along the z axis, but also tilts, as depicted in Fig. 7. The red lines represent the ideal trajectories of the sample, and the black lines denote the actual trajectories. The proposed method is applied to separate the translation and tilt, and the calculated three-dimensional trajectory h with the tilt about the x and y axes is shown in Figs. 8(a)–8(d). Here two three-dimensional motion modes, namely a triangular mode and a sinusoidal mode, are set. The actual calculating error of tilt angles is less than 0.01 µrad, implying that the proposed method can credibly track the sample displacement.

Fig. 7. Three-dimensional displacement tracing. The ideal trajectory is depicted in red, and the actual trajectory in black.

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Fig. 8. Measurement results of tilts. To make the plots clearer, the black dot line presenting actual tilts are shifted by 0.1 mrad from the actual values. (a) and (b) are the results of triangular mode, and (c) and (d) for sinusoidal mode.

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3.2 Experimental demonstration

The displacement tracing system shown in Fig. 1(a) is built. The adopted light source is a green LED by CCS Inc. and it works as a point light source through lenses and high-power pinholes. A CMOS camera of EoSens Cube 4 by MotionBLITZ and a PZT of P-611.3 NanoCube XYZ by PI are adopted for capturing images and moving the stage, respectively. Meanwhile, the moving direction of the high precision PZT is adjusted to be parallel to the optical axis. Subsequently 30 interferograms are captured at different positions during the scanning, with an interval of 100 nm. Then the system calibration is conducted, and the obtained global parameters A₁, A₂, L_c and λ₀ are 94.07, 134.26, 7.97 µm and 0.54 µm, respectively. For the purpose of comparison, the calculated surface topography z(x, y) is compared with the measuring result of a coherent scanning interferometer CCI MP. The relative deviation is shown in Fig. 9(b). The PV and RMS of the deviation are 14.37 nm and 2.02 nm, respectively, implying that the system calibration result is reliable.

Fig. 9. System calibration results. (a) Calculated surface (b) deviation.

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After the system calibration, the PZT is reset to the original position, then it moves at 10 µm/s while the CMOS capture 100 frames per second. The proposed method is applied to obtain the displacement of the stage. The results are shown in Fig. 10(a), and the RMS of the displacement measurement error is 1.16 nm. Besides, the tilt measurement caused by environmental vibration is also detected, as shown in Fig. 10(b). The measurement accuracy is comparable to the resolution of the displacement table, implying that the proposed method has good performance. The calculation time of each frame is 0.6 ms, suggesting that this method can be utilized in real-time measurement.

Fig. 10. Experimental results. (a) presents the displacement measurement result. The black dot line presenting actual displacement is shifted by 0.1 µm to make the plots clearer; (b) illustrates the tilt measurement results; (c) depicts the positioning repeatability.

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Furthermore, the positioning repeatability of the proposed method is also verified. 20 interferograms are captured at a fixed position during 5 minutes. Then the stage moves 0.53 µm to another position, and another 20 interferograms are captured in same way. Subsequently the proposed method is applied, with the calculation results shown in Fig. 10(c). The RMS of the positioning error turns out to be 1.75 nm. It is proved that the measurement precision can achieve the nanometer level, which is improved by more than one order of magnitude compared with existing methods, which can fit the demand of nano-fabrication.

4. Summary

This paper presents a high precision and real time measuring strategy of the three dimensional displacements in nano-fabrication based on the low coherence interferometry. The proposed method has the advantage of absolute measurement, because LCI has a unique zero-OPD position. This is critical to the precise focusing of the fabricating beam. Simultaneously, a three-dimensional displacement is calculated by taking advantage of the whole interferograms taken by a high speed camera. The measurement precision can achieve a nanometer level due to the averaging effect of the least squares fitting. At the same time, the measuring stability and applicability can be guaranteed. Consequently, this method is of significance to improve the positioning accuracy and efficiency in nano-fabrication.

Funding

National Natural Science Foundation of China (51875107); Science Challenge Project (JCKY2016212A506-0106); Fudan University-CIOMP Joint Fund (FC2018-007); National Key Research and Development Program of China (2017YFB1104700).

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Noise level	Mean value of error	Standard deviation	Skewness	Kurtosis
5%	0.0033 nm	1.2652 nm	−0.0665	−0.0786
10%	−0.0304 nm	2.5212 nm	0.0533	−0.007

Noise level	Mean value of error	Standard deviation	Skewness	Kurtosis
5%	0.0033 nm	1.2652 nm	−0.0665	−0.0786
10%	−0.0304 nm	2.5212 nm	0.0533	−0.007

Precision enhancement of three-dimensional displacement tracing for nano-fabrication based on low coherence interferometry

Abstract

1. Introduction

2. Methodology of the three-dimensional displacement tracing

2.1 Measuring setup of the Low Coherence Interferometry

2.2 Calculation of the three dimensional displacements

2.2.1 Calibration of the system parameters

2.2.2 Iterative refinement

2.2.3 Precise Focusing

2.3 Uncertainty estimation

3. Numerical and experimental demonstration

3.1 Numerical demonstration

3.2 Experimental demonstration

4. Summary

Funding

References

Cited By

Figures (10)

Tables (3)

Equations (11)

Optics Express

case	length of coherence	sampling interval	noise level
(a)	50 µm	100 nm	1%
(b)	50 µm	10 nm	2%
(c)	5 µm	100 nm	1%
(d)	5 µm	10 nm	2%

case	length of coherence	central wavelength	Root Mean Square (RMS) error
(a)	50 µm	540 nm	0.0229 nm
(b)	50 µm	540 nm	0.0438 nm
(c)	5 µm	540 nm	0.0335 nm
(d)	5 µm	540 nm	0.0689 nm