Freeform measurement method based on differential confocal and real-time comparison

Yingqi Tang; Yuhan Liu; Wenbin Wang; Chao Liu; Yun Li; Yin Song; Kemi Xu; Lirong Qiu; Weiqian Zhao

doi:10.1364/OE.453932

1. Introduction

Freeform surface refers to a surface that lacks rotational or translational symmetry about axes normal to the mean plane and can often be described by a sum of a quadric term and high-order polynomials [1–4]. Freeform optics can reduce aberration of optical systems, and exhibits better imaging quality and performance than traditional spherical or aspherical optics [5–7]. However, the lack of the rotational symmetry and the high degrees of freedom pose new challenges to the fabrication and the high-precision measurement of freeform surface, particularly during manufacturing when the local inclination and roughness are continuously changing [8,9]. Typical measurement methods for freeform surfaces include null interferometry, adaptive null interferometry, non-interferometric areal measurement method, and single-point probe profile measurement method, etc. These techniques are briefly summarized in Table 1.

Table 1. Brief summary of freeform surface measurement methods

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The null interferometry makes use of interferometric fringes to measure the FSUT and the CGH is the first choice of null optics for interferometric test of FSUT [10,11]. However, during the manufacturing process, the wavefront distortion may exceed the compensation capacity of the CGH [12,13]. To address this issue, the adaptive null interferometry was proposed. In this method, adaptive optical element such as DM and SLM is used for the compensation of the wavefront distortion [14,15]. However, the deflection error must be calibrated before use [8,16]. In addition, limited by the adjustable ranges, there is still relatively large restriction of the inclination. The non-null interferometry does not use null optics to compensate the distortion of the FSUT, and still can generate interferometric fringes through sub aperture testing or shearing interferometry and so on [17–21]. Unfortunately, this type of method has a large measurement error when measuring a freeform surface with a large aspheric departure. The non-interferometric areal measurement method measures freeform surface avoids the calibration of the wavefront distortion through the wavefront reconstruction [22,23]. These methods include SH wavefront test [24,25] and PMD and so on. However, they are limited to the freeform surface with large inclination [26] and high reflectivity, respectively [8,9,27,28].

The profile measurement method based on the single-point probe, in principle, promises both the versatility and the measurement accuracy. This type of method has achieved a repeatable measurement accuracy of ±50 nm for freeform surface profiles through recent years research [29,30]. The key point that the single-point probe can achieve high-precision measurement is an Abbe error-free or an independent metrology frame. TU Ilmenau and SIOS Meßtechnik GmbH has developed NPMM-200 with 20 pm resolution and 25 mm measurement range [31,32]. The NPMM-200 extended six degrees of freedom Abbe-comparator principle with interferometers and a focus sensor or an atomic force probe. Its positioning repeatability and the smallest step can reach 4 nm and sub-nanometer, respectively. This machine is suitable not only for positioning but also for measuring freeform surface. Its focus sensor has 1 nm resolution, 20° inclination measuring capability and about 1 µm working range, and it exhibits excellent performance when measuring flat and smooth reflective surfaces. Luphoscan makes use of three interferometers to determine the coordinates of another interferometer which is used to measure the normal height of the FSUT [33]. However, the inclination measuring range is only about 8° and it does not monitor the rotation error of the rotation axes. To address these issues, Nanomefos used capacitive sensors to monitor the rotation errors. The dynamic resolution and the measuring range of the local inclination angle has been improved to 20 nm and ±20°, respectively [34]. However, its dynamic resolution can only be achieved at the mid of the measuring range, which is difficult to guarantee in actual measurement. And its nonlinear error is as high as ±150 nm, resulting in a measurement accuracy of about ±50 nm. When the local inclination of the FSUT reaches ±20°, its measurement accuracy and resolution will be further reduced, making it difficult to meet the measurement accuracy requirement of 50 nm.

To further improve the measurement accuracy and the measuring range of the inclination angles, we refer to the Abbe error-free frame and the independent metrology frame based on reference plane real-time comparison (such as homodyne interferometer [35]) measurement, we propose a freeform surface profile measurement method based on the laser differential confocal trigger sensor (LDCTS) and the real-time comparison of the reference planes (Abbe error-free). This method takes advantages of the LDCTS [36–39] to achieve high-precision measurement of the freeform surface with large inclination angles during manufacturing. By rigidly connecting the reference planes to the FSUT and the real-time comparisons of their coordinate changes, the straightness and the rotational errors can be separated, thereby significantly improving the measurement accuracy of the freeform surface. Thus, this method provides a new strategy for the high-precision contactless measurement method of freeform surfaces when the roughness and local inclination are continuously changing. In addition, benefiting from the hundreds of nanometer-level lateral resolution of LDCTS, nanometer precision measurement of micro-area topography of freeform surfaces can also be achieved.

2. Freeform surface measurement principle based on real-time comparison

2.1 Real-time comparison measurement principle based on reference planes

To simplify the discussion, we will first take the Z-axis as an example to explain the working principle of the real-time comparison measurement method and then generalize the discussion to the other two axes. The design principle is depicted in Fig. 1. The reference plane and the FSUT are rigidly installed on the same mobile stage while the LDCTS and the Z-axis reference plane monitoring sensor (RS) are installed coaxially but on a different frame. The error sources of Z-axis mainly include positioning error, straightness error of X-axis (and Y-axis), and rotation error. During the measurement when the reference plane and the FSUT move along the X-axis (or Y-axis), the coordinates change of the RS relative to the reference plane reflects the straightness error δ_z and the rotation error ɛ_z of the X-axis (or Y-axis) guide rail along the Z-axis. By real-time comparison of the reference plane and the freeform surface, the measurement errors of the freeform surface caused by the movement of the guide rail can be partially eliminated, thereby effectively improving the measurement accuracy of the freeform surface.

Fig. 1. Real-time comparison measurement principle based on the reference plane. RS is Reference plane monitoring sensor, δ_z is straightness error, and ɛ_z is the rotation error.

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The Reference sensor and LDCTS in Fig. 1 are rigidly connected. When measuring along the X-axis, the reference plane moves relative to the Z-axis reference monitoring sensor in the X-axis, which can effectively monitor the error caused by the movement in the X-axis; When measuring along the Y-axis, the Z-axis reference monitoring sensor and the reference plane move relatively in the Y-axis, which can also effectively monitor the error caused by the Y-axis movement. Likewise, the X-axis and Y-axis reference monitoring sensors can monitor the motion error in this direction, respectively.

Because objects move in space with 6 degrees of freedom, during the movement of the FSUT relative to the LDCTS, there are 6 degrees of freedom errors in each of the X, Y, and Z axes. In addition, there are 3 perpendicularity errors between the X, Y and Z axes, totaling 21 geometric errors are shown in Table 2.

Table 2. The 21 geometric errors sources of the 3-axis coordinate measuring machine

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In the table, δ represents the position error, ɛ represents the rotation error, and α represents the perpendicularity. For example: δ_z(x) and ɛ_z(x) are the geometric errors of the FSUT relative to the Z-axis when moving along the X axis.

Because the FSUT is rigidly installed on the X-axis and the LDCTS is rigidly installed on the Z-axis, we consider the error transfer matrix from the FSUT coordinate system to the X coordinate system and the Z coordinate system to the LDCTS coordinate system as the identity matrix. According to the principle of error transfer, the error transfer matrix from the FSUT coordinate system to the LDCTS coordinate system can be simply expressed as [40]:

(1)$${_{S}^{F}}{{\boldsymbol T}^e} = {_R^X}{{\boldsymbol T}^e}\cdot {_Y^R}{{\boldsymbol T}^e}\cdot {_Z^Y}{{\boldsymbol T}^e}$$

where $_R^X{{\boldsymbol T}^e}$, $_Y^R{{\boldsymbol T}^e}$, $_Z^Y{{\boldsymbol T}^e}$ are the error transfer matrix from the X-axis coordinate system to the base coordinate system, the base coordinate system to the Y-axis coordinate system, and the Y-axis coordinate system to the Z-axis coordinate system. To simplify the derivation process without loss of generality, $_S^F{{\boldsymbol T}^e}$ can be expressed as:

(2)$$_S^F{{\boldsymbol T}^e} = \left[ {\begin{array}{{cc}} {_S^F{{\boldsymbol R}^e}}&{_S^F{{\boldsymbol t}^e}}\\ 0&1 \end{array}} \right] = \left[ {\begin{array}{{cc}} {_R^X{{\boldsymbol R}^e}}&{_R^X{{\boldsymbol t}^e}}\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{cc}} {_Y^R{{\boldsymbol R}^e}}&{_Y^R{{\boldsymbol t}^e}}\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{cc}} {_Z^Y{{\boldsymbol R}^e}}&{_Z^Y{{\boldsymbol t}^e}}\\ 0&1 \end{array}} \right]$$

where R is the rotation error transfer matrix, and t is the position error transfer matrix. After ignoring the higher-order terms in t, the position error model of the XFYZ structure is obtained:

(3)$$_Z^Y{{\boldsymbol t}^e} = {({\Delta x,\textrm{ }\Delta y,\textrm{ }\Delta z} )^T} = \left( {\begin{array}{{c}} { - {\delta_x}(x) + {\delta_x}(y) + {\delta_x}(z) + y{\varepsilon_z}(x) - z[{{\varepsilon_y}(x) - {\varepsilon_y}(y)} ]- y{\alpha_{xy}} + z{\alpha_{xz}}}\\ { - {\delta_y}(x) + {\delta_y}(y) + {\delta_y}(z) + z[{{\varepsilon_x}(x) - {\varepsilon_x}(y)} ]- z{\alpha_{yz}}}\\ { - {\delta_z}(x) + {\delta_z}(y) + {\delta_z}(z) - y{\varepsilon_x}(x)} \end{array}} \right)$$

where Δx, Δy and Δz are position errors.

Therefore, the measurement result of the LDCTS is:

(4)$${{\boldsymbol z}_{{\boldsymbol LDCTS}}} = {{\boldsymbol z}_{{\boldsymbol FSUT}}} + \Delta z = z - {\delta _z}(x) + {\delta _z}(y) + {\delta _z}(z) - y{\varepsilon _x}(x)$$

where ${z_{FSUT}}$ is the true value of a point on the freeform surface. Because the Z-axis reference monitoring sensor does not move in the Z axis, the error monitoring results related to the Z-axis are all 0, and we obtained the measurement error model ${z_{RS}}$ of the Z-axis reference monitoring sensor is:

(5)$${{\boldsymbol z}_{{\boldsymbol RS}}} ={-} {\delta _z}(x) + {\delta _z}(y) - y{\varepsilon _x}(x)$$

After the real-time comparison, the Z-axis measurement results $z{^{\prime}_{FSUT}}$ of the freeform surface can be obtained as:

(6)$${\boldsymbol z}{{^{\boldsymbol \prime}}_{{\boldsymbol {FSUT}}}} = {{\boldsymbol z}_{{\boldsymbol {LDCTS}}}} - {{\boldsymbol z}_{{\boldsymbol RS}}} = z + {\delta _z}(z)$$

From Eq. (6), we can see that both the straightness and the rotation errors can be removed. Similarly, the spatial motion errors of the X and Y axes can also be significantly reduced using the real-time comparison method and the coplanarity principle. Specifically, we first install the sensors monitoring the reference planes of the X and Y axes in the same plane as the upper surface of the motion stage while keeping the normal of these two sensors’ plane orthogonal to the optical axis of the LDCTS. Since the X-axis and Y-axis reference planes monitoring sensors are independent, the measurement error models of the LDCTS and the reference plane monitoring sensors along the X axis and the Y axis are, respectively, shown as follows:

(7)$$\begin{aligned} {{\boldsymbol x}_{{\boldsymbol LDCTS}}} &= {{\boldsymbol x}_{{\boldsymbol FSUT}}} + \Delta x\\ &= x - {\delta _x}(x) + {\delta _x}(y) + {\delta _x}(z) + y{\varepsilon _z}(x) - z[{{\varepsilon_y}(x) - {\varepsilon_y}(y)} ]- y{\alpha _{xy}} + z{\alpha _{xz}} \end{aligned}$$

(8)$$\begin{aligned} {{\boldsymbol y}_{{\boldsymbol LDCTS}}} &= {{\boldsymbol y}_{{\boldsymbol FSUT}}} + \Delta y\\ &= y - {\delta _y}(x) + {\delta _y}(y) + {\delta _y}(z) + z[{{\varepsilon_x}(x) - {\varepsilon_x}(y)} ]- z{\alpha _{yz}} \end{aligned}$$

The measurement error models ${x_{RS}}$ and ${y_{RS}}$ of the X-axis and Y-axis reference monitoring sensors are:

(9)$$\left\{ \begin{array}{l} {{\boldsymbol x}_{{\boldsymbol RS}}} = {\delta_x}(y) + y{\varepsilon_z}(x)\\ {{\boldsymbol y}_{{\boldsymbol RS}}} ={-} {\delta_y}(x) \end{array} \right.$$

According to the principle of the real-time comparison, the measurement results $x{^{\prime}_{FSUT}}$ and $y{^{\prime}_{FSUT}}$ of the X and Y axes can be obtained as:

(10)$$\left\{ \begin{array}{l} {\boldsymbol x}{{^{\boldsymbol \prime}}_{{\boldsymbol FSUT}}} = {{\boldsymbol x}_{{\boldsymbol LDCTS}}} - {{\boldsymbol x}_{{\boldsymbol RS}}} = x - {\delta_x}(x) + {\delta_x}(z) - z[{{\varepsilon_y}(x) - {\varepsilon_y}(y)} ]- y{\alpha_{xy}} + z{\alpha_{xz}}\\ {\boldsymbol y}{{^{\boldsymbol \prime}}_{{\boldsymbol FSUT}}} = {{\boldsymbol y}_{{\boldsymbol LDCTS}}} - {{\boldsymbol y}_{{\boldsymbol RS}}} = y + {\delta_y}(y) + {\delta_y}(z) + z[{{\varepsilon_x}(x) - {\varepsilon_x}(y)} ]- z{\alpha_{yz}} \end{array} \right.$$

According to Eqs. (6) and (10), after the real-time comparison, the freeform surface measuring device based on the single-point detection eliminates the influence of δ_z(y) and ɛ_x(x) along the Z axis, the influence of δ_x(y) and ɛ_z(x) along the X axis, and the influence of δ_y(x) along the Y axis, thereby significantly improving the measurement accuracy of the freeform surface profile.

2.2 LDCTS with high precision and large local inclination measurement capability

LDCTS with the axial resolution better than 1 nm and the measuring range of the inclination angles of 25°, exhibits the unique axial tomographic measurement capability in the field of optical measurement [37–39]. Furthermore, its stroke in profile measurement is mainly determined by the nano drive control system and is larger by several orders of magnitude than the interferometric measurement method and non-interferometric areal measurement. With the help of the laser interferometer, its repeatability measurement accuracy can also be improved to be better than 5 nm.

The freeform profile trigger measurement principle of the LDCTS with high axial resolution and large local inclination range is shown in Fig. 2. The light emitted by the laser passes through the beam expander, the beam splitter BS1 and reflector, and then is focused on the FSUT by the measurement objective. When the FSUT is near the focal point of the measurement objective, the reflected beam carries the height information of the freeform surface and is then divided into two beams after passing through the measurement objective, the reflector, the beam splitter BS1 and the beam splitter BS2. Subsequently, these two beams respectively pass through the front focus condenser lens L_F and the back focus condenser lens L_B to form the focused beams and enter the pinholes Ph_F and Ph_B which are located in front of the focus of the converging lens L_F and behind the focus of the converging lens L_B. The corresponding axial response curves I_F and I_B are detected by the detectors PMT_F and PMT_B. After normalizing the two axial responses I_F and I_B, the differential confocal axial response curve I_D (u, u_M) with high axial resolution and high dynamic range is obtained. The zero position of I_D (u, u_M) is the height value of the FSUT sampling point.

Fig. 2. Differential confocal measurement principle of freeform surface profile based on real-time comparison. RP_X, RP_Y, RP_Z are the reference plane of X-axis, Y-axis and Z-axis. RS_X RS_Y and RS_Z are the reference plane monitoring sensors of X-axis, Y-axis and Z-axis.

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During the measurement, the normal of different sampling points and the optical axis have different inclination angles, so it is necessary to analyze the differential confocal axial response under different inclination angles. When the inclination angle is β, the schematic diagram of the optical path of the LDCTS is shown in Fig. 3.

Fig. 3. Schematic diagram of simplified optical path for differential confocal measurement of freeform surface profile. n₁ is the optical axis of reflected light on the FSUT, n₂ is the optical axis of incident light in front of detector, d₁ is the focal length of object lens, d₂ is the focal length of image lens, u_M is the normalized axial offset.

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According to the Fresnel diffraction integral formula and the confocal incoherent three-dimensional imaging theory, when measuring freeform surface of the LDCTS with finite pinhole sizes, the intensity points spread function (PSF) I_F (u, +u_M) and I_B (u, -u_M) are:

(11)$$\left\{ \begin{array}{l} {I_F}(u, + {u_M}) = \frac{{{{\cos }^4}\beta }}{{{{\cos }^2}\beta + M{{\sin }^2}\beta }}{\left[ {\frac{{\sin ((u + (u + {u_M})\cos \beta )/4)}}{{(u + (u + {u_M})\cos \beta )/4}}} \right]^2}\\ {I_B}(u, - {u_M}) = \frac{{{{\cos }^4}\beta }}{{{{\cos }^2}\beta + M{{\sin }^2}\beta }}{\left[ {\frac{{\sin ((u + (u - {u_M})\cos \beta )/4)}}{{(u + (u - {u_M})\cos \beta )/4}}} \right]^2} \end{array} \right.$$

where M = d₁/d₂. We note that the light intensity value of the axial response curve decreases as the local inclination of the FSUT increases. In order to suppress the influence of the reflectivity changes induced by the surface flatness and the local inclination, the light intensity signals I_F (u, +u_M) and I_B (u, -u_M) are normalized before calculating the normalized differential confocal axial response I_D (u, u_M) as shown below [39]:

(12)$$\begin{aligned} {I_D}(u,{u_M}) &= \frac{{{I_F}(u, + {u_M}) - {I_B}(u, - {u_M})}}{{{I_F}(u, + {u_M}) + {I_B}(u, - {u_M})}}\\ \textrm{ } &= \frac{{{{\left[ {\frac{{\sin ((u + (u + {u_M})\cos \beta )/4)}}{{(u + (u + {u_M})\cos \beta )/4}}} \right]}^2} - {{\left[ {\frac{{\sin ((u + (u - {u_M})\cos \beta )/4)}}{{(u + (u - {u_M})\cos \beta )/4}}} \right]}^2}}}{{{{\left[ {\frac{{\sin ((u + (u + {u_M})\cos \beta )/4)}}{{(u + (u + {u_M})\cos \beta )/4}}} \right]}^2} + {{\left[ {\frac{{\sin ((u + (u - {u_M})\cos \beta )/4)}}{{(u + (u - {u_M})\cos \beta )/4}}} \right]}^2}}} \end{aligned}$$

According to Eqs. (11) and (12), the slope and the goodness of fit (GOF) of the approximate linear segment of the non-normalized I_F (u, +u_M)-I_B (u, -u_M) and normalized differential confocal axial response I_D (u, u_M) with the local inclination β = 0°∼25° are shown in Figs. 4(a) and (b), respectively. As the inclination angle β increases, the full width at half maxima (FWHM) of I_F (u, +u_M) and I_B (u, -u_M) increases whereas the slope of the approximate linear segment of I_D (u, u_M) and I_F (u, +u_M)-I_B (u, -u_M) decreases, resulting in the reduction of the axial measurement sensitivity. As shown in Figs. 5(a) and (b), we also note that I_D (u, u_M) is much less sensitive to the inclination angles, and the measurement sensitivity of I_D (u, u_M) at 25° is 8.5 times that of I_F (u, +u_M)-I_B (u, -u_M), suggesting that the normalization of the response curve can significantly improve the measuring range of the inclination angle. Furthermore, we find that the GOF of the approximate linear segment of I_D (u, u_M) is also better than I_F (u, +u_M)-I_B (u, -u_M) when β=0°∼25°, which can effectively reduce the axial measurement error of freeform surface in large inclination range.

Fig. 4. Simulated axial response curves of LDCTS at different inclination. (a) Non-normalized axial response of different inclinations. (b) Normalized axial response of different inclinations

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Fig. 5. Simulated axial response curves of LDCTS at different inclination. (a) Axial sensitivity of approximate linear segments at different inclination. (b) Coefficient of determination of approximate linear segments at different inclinations.

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2.3 Analysis of fixed-focus characteristics of laser differential confocal

With the increase of the FSUT inclination, the axial response sensitivity of LDCTS decreases. In order to obtain better performance of LDCTS, we need to analyze the axial response characteristics of LDCTS. This provides a theoretical basis for the design of the optimal normalize offset (u_M), signal noise ratio (SNR), numerical aperture (NA) and laser wavelength (λ) of LDCTS. In order to obtain Eqs. (11) and (12), we have introduced the axial optical coordinates u and radial optical coordinates v into the distribution of the light field of LDCTS, the distribution of the light field is derived from the Fresnel formula. When the radial offset v is 0, the axial response function of the LDCTS as shown in Eqs. (11) and (12) can be obtained by using the Hankel transform [38,39]. Therefore, in order to ensure that the axial resolution of LDCTS is better than 1 nm, we need to further analyze the introduced axial optical coordinate u which is proportional to the axial offset z. It is known that the relationship between the introduced axial optical coordinate u and the axial offset z is:

(13)$$u = \frac{\pi }{{2\lambda }}{(\frac{D}{{{f_c}}})^2}Z$$

where λ is the laser wavelength, D is the aperture of the objective lens, and f_c is the focal length of the objective lens.

The slope at zero-crossing can be obtained by taking the partial derivative of u of Eq. (10), and the resolution of axial response can be obtained using:

(14)$$\Delta u\textrm{ = }\frac{\lambda }{{2\pi \cdot N{A^2} \cdot SNR \cdot \left|{\frac{{\partial {I_D}(u,{u_M})}}{{\partial u}}{|_{u = 0}}} \right|}}$$

The selected λ is 532 nm, NA is 0.80, and the SNR is 270:1. According to Eqs. (13) and (14), the axial response curves and the axial resolution curves of I_D (u, u_M) with different axial offset u_M are obtained as shown in Figs. 6(a) and (b), respectively. We find that the zero-crossing slope of the I_D (u, u_M) corresponding to the noise immunity decreases with the increment of u_M while the axial resolution improves with the increment of u_M_. Considering both effects, we find that u_M = 5.2 is the optimal axial offset for the measurement [38]. According to Eq. (14), the axial resolution can reach 9.97×10⁻⁴ µm.

Fig. 6. Property curves for different offset u_M. (a) Axial response curve I_D (u, u_M). (b) Axial resolution curve Δu.

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The axial resolution calculated according to Eq. (14) [39] is:

(15)$$\Delta \textrm{z} = \frac{{2\lambda }}{\pi } (\frac{{{f_c}}}{D}{) ^2}\frac{{\delta I}}{{{S_{\max }}}} \approx \frac{\lambda }{{3.32 \times N{A^2} \times SNR}} = \frac{{0.532}}{{3.32 \times {{0.80}^2} \times 270}}\mathrm{\mu} \textrm{m} \approx 9.97 \times {10^{ - 4}}\mathrm{\mu} \textrm{m}$$

Then, the anti-reflectivity differential confocal curve obtained using Eq. (12) is shown in Fig. 7. We find that the normalized linear range is u ≈ ±1.6, and then z ≈ ±0.16 µm.

Fig. 7. Anti-reflectivity differential confocal curve when u_M = 5.2.

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Taking into account mechanical interference, the recommended inclination measurement range is 20°, preferably no more than 25°. If we improve the SRN to be better than 640:1, the axial resolution can be achieved 1 nm using an objective lens with NA = 0.5. Then, the working distance of LDCTS is 10.6 mm, and the measuring range of inclination will exceed ± 40°.

3. Experiment and analysis

3.1 Measurement system

According to the real-time comparison measurement principle based on the reference plane, a freeform surface profile measurement system is constructed with the structure shown in Fig. 8. The freeform surface profile measurement system mainly includes a Computer & Controller system A, a laser differential confocal trigger measurement sensor B, a reference plane monitoring system C, and a precision guide rail system D. The LDCTS and the Z-axis capacitance sensors are installed according to the co-axis principle, and the reference plane and the FSUT are installed on the same mobile frame. The Z-axis mobile platform is used to drive the LDCTS during the measurement of the freeform surface. To measure the freeform surface profile, the freeform surface and the LDCTS move relatively along the X and Y directions through the mobile platform. According to the co-plane principle, the reference planes are set for the X-axis guide rail and the Y-axis guide rail, and the straightness error of the X-axis guide rail in the Y direction and the straightness error of the Y-axis guide rail in the X direction are separated in real time.

Fig. 8. Freeform surface real-time comparison measurement system structure based on reference plane.

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The LDCTS uses a semiconductor laser with a wavelength of λ=532 nm. The objective lens with numerical aperture NA = 0.80 and magnification N = 100× is selected. The objective lens driver is a high-precision nano driver with a resolution of 0.125 nm, a stroke of 4 mm, a position stability of ±0.4 nm, and a minimum step of 1 nm. The reference plane is a microcrystalline mirror with RMS≤5 nm. The reference plane monitoring sensor is DT6530 of Micro-Epsilon with the minimum resolution of 0.15 nm and the nonlinear error of 0.025% full scale output (FSO). A laser interferometer and a grating ruler are used for full closed loop. The laser interferometer is Renishaw XL80, and the grating ruler is LIP281 of HEIDENHAIN with the grating period of 2.048 µm and the interpolation error of 0.4 nm.

The working distance of LDCTS is 3.4 mm which is same as the working distance of the objective lens. The measurement range of our system is 150 mm ×150 mm × 50 mm, and its axial measurement frequency can reach 30 Hz in the work range of ±50 µm. When the sampling interval is 1 mm, the position error can be adjusted into a 25 nm window within 80 ms, and the measurement speed is about 120 ms per sample point. When the number of sampling points is 100 × 100, the measurement time is about 20 minutes. Furthermore, through iterative prediction, the LDCTS can follow the profile of the isocline of the FSUT during the measurement process. Then, the height of sampling points can be obtained directly by using the corresponding relationship between voltage of approximate linear section of axial response curve and axial position, thereby significantly improving the sampling speed. The maximum measurable inclination angle of the device is better than 35°, but considering the problem of collision with the FSUT, we only analyze the inclination measurement capability within 25°. In addition, on the basis of not affecting the measurement speed, we recommend an inclination measurement range of 20°. If the local inclination of the FUST exceeds 20°, in order to prevent mechanical interference, we will reduce the measurement speed by at least 5 times, or replace the objective lens with a working distance of 10.6 mm. The freeform surface real-time comparison measurement system is shown in Fig. 9.

Fig. 9. The Freeform surface real-time comparison measurement system. (1) Computer and control system. (2) Laser interferometer. (3) LDCTS. (4) Capacitance sensor. (5) Voice coil linear stage with air bearing. (6) FSUT. (7) RP_z (8) RP_x. (9) RP_y. (10) X-axis air bearing guide rail. (11) Y-axis air bearing guide rail. (12) Z-axis air bearing guide rail.

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3.2 Performance of LDCTS

Since the axial resolution of the LDCTS is better than 1 nm, the measuring range and linearity of the LDCTS are mainly determined by the nanometer precision objective lens positioning and scanning system. Figure 10 shows the measured axial response and resolution characteristic curves of the LDCTS. The results show that its axial resolution is better than 1 nm, which can meet the requirement of freeform surface nanometer accuracy measurement.

Fig. 10. Performance experiment of the LDCTS. (a) Normalized axial response curve of the LDCTS. (b) Axial resolution of the LDCTS.

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In order to verify the inclination measurement capability of the LDCTS, the axial response curve at different inclination angles is measured as shown in Fig. 11(a). As the measured inclination increases, the slope of the approximate linear segment of the normalized differential confocal axial response curve gradually decreases, which is consistent with the simulation as shown in Fig. 4(b). This finding proves that the LDCTS has an inclination measurement capability of 25°. The repeated focusing measurement results of 0°∼25° are shown in Table 3.

Fig. 11. Anti-reflectivity experiment of LDCTS. (a) Inclination measurement capability. (b) Anti-roughness measurement capability.

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Table 3. Repeated measurement results of reflectors at different inclination

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In order to verify the anti-roughness measurement capability of the LDCTS, the roughness flat samples with Ra = 0.1 µm, 0.2 µm, 0.4 µm, 0.8 µm, 1.6 µm, 3.2 µm are measured respectively. The obtained normalized axial response curves are shown in Fig. 11(b). We find that the slopes of the approximate linear segment do not change much for surfaces with different Ra values, which proves that the LDCTS can be adapted to the measurement of surfaces with different roughness.

When the local inclination of the sampling point is different, the walk-off of the beam will change the effective numerical aperture. Due to the existence of aberrations such as primary spherical aberration, primary astigmatism and field curvature, the zero-crossing of the axial response curve of LDCTS will be offset. However, aberrations are systematic errors, which can usually be eliminated by calibration and do not affect the axial repeat measurement accuracy of LDCTS. Since the subdivision error of LIP 281 is 0.4 nm, coupled with the influence of environmental disturbance, the axial measurement STD results at different inclination angles in Table 3 are about 2 nm.

3.3 Real-time comparison error separation analysis based on reference planes

The measurement errors of the real-time comparison method using the reference planes mainly include the rotation errors, straightness errors and perpendicularity errors of the guide rails, the measurement errors of the LDCTS and the capacitance sensors, and the measurement errors of the reference monitors induced by the surface roughness changes. The perpendicularity errors of the guide rails are related to the processing and assembly of the mechanism. The X-axis and Y-axis, X-axis and Z-axis, Y-axis and Z-axis guide rails can be regarded as rigid bodies. After the processing and assembly, the perpendicularity errors are fixed and unchanged which can be calibrated by a laser interferometer. Before measurement, the three perpendicularity errors obtained by the interferometer calibration are 1.79″, 1.81″ and 1.76″, respectively. It is known that the measurement range of the X-axis and Y-axis air bearing guide rail is 150 mm×150 mm, the δ_y(x), δ_z(x),δ_x(y) and δ_z(y) are less than 100 nm, the ɛ_x(x), ɛ_y(x), ɛ_y(y) and ɛ_x(y) are less than 0.2 µrad.

According to the measurement results in Table 3, the measurement std error of LDCTS is less than 2 nm. Because the relative position changes between the reference planes and the capacitive sensors within the full scanning range do not exceed 10% of the full scalar output (FSO). It is known that the FSO of the capacitance sensor is 200 µm, and the non-linear error is less than 0.025% of the FSO. Both the measurement errors of the capacitance sensors and the roughness of the reference planes are less than 5 nm. According to Eqs. (6) and (10), the synthetic position errors of the freeform surface profile measurement system after real-time comparison and measurement are Δx′ ≤ 25 nm, Δy′ ≤ 25 nm and Δz′ ≤ 10 nm. Figures 12(a)∼(c) show the origin of local inclination error Δz_s, local curvature radius error Δz_c and position error Δz, respectively. According to the analysis of the measurement error source, the position error Δx (or Δy) is more than 100 nm. When the simulation range of the local inclination is 0°∼20°, the maximum measurement error Δz_s caused by it is 36 nm, as shown in Fig. 12(d). When the simulation range of the local curvature radius is 200∼1000 mm, the corresponding maximum measurement error Δz_c is 25 nm, as shown in Fig. 12(e). Without the real-time comparison method, the position error Δz will directly cause the Z-axis height measurement error, up to 200 nm, as shown in Fig. 12(f), which is the most sensitive direction during the freeform surface measurement. After careful analysis, we find that the real-time comparison measurement based on the reference planes can reduce the measurement errors induced by Δz to be less than 10 nm, and the measurement errors induced by Δz_s and Δz_c to be 20 nm.

Fig. 12. Influence of coordinate errors on freeform surface measurement. (a) The origin distance measurement error Δz_s resulting from local inclination. (b) The origin distance measurement error Δz_c resulting from local radius of curvature. (c) The origin distance measurement error Δz. (d) Error as a function of local inclination. (e) Error as a function of local radius of curvature. (f) Error as a function of straightness error.

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3.4 Real-time comparison measurement experiment of the plane reflector

In order to verify the error separation method of the real-time comparison measurement method along the Z axis, a plane reflector with effective aperture 120 mm×120 mm and PV<100 nm is tested using the null interferometry method and the proposed method with and without the real-time comparison. Because the object to be measured is a plane reflector, the measurement speed we choose is 10 mm/s, the sampling interval is 1 mm, the number of sampling points is 14400, then the measurement time is about 30 minutes. The experimental results are shown in Fig. 13. We find that the trend of the measurement results using the real-time comparison base on the reference plane is consistent with null interferometry method. The deviation of the proposed method from the null interferometry method measurement result is only about 8 nm while the measurement result without the real-time comparison is deviated by 46 nm from the measurement result of null interferometry method because of the straightness and rotational errors of the guide rails. This finding proves that the real-time comparison method can effectively reduce the measurement error of the sensitive Z-axis.

Fig. 13. Plane reflector measurement results. (a) Measurement result of null interferometry method. (b) Residual diagram of the proposed method. (c) The result of without real-time comparison.

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3.5 Freeform surface profile measurement experiment

In order to verify the profile measurement capability of FSUT with the large inclination range, a generatrix of aspheric surface with the diameter of D = 102 mm has been tested, the design model of the aspheric is z = 0.00365 × (x² + y²). After partial derivation of the design model, it is calculated that the maximum inclination of the aspheric is 20.4°. Due to the variation of the aspheric inclination is more than 20°, in order to prevent collision with the FSUT, we reduced the measurement speed to 2 mm/s, set the sampling interval to 0.1 mm, the number of sampling points was about 1000, and the sampling time was about 10 minutes. Figure 14(a) shows the residual of the measurement data and the calibration data of the measured aspheric generatrix profile after data processing. The calibration data is obtained using UA3P. We find that the PV of the measurement result is 305 nm and the RMS of the measurement result is 69 nm, which are similar to the calibration result. Since the material of the aspheric is aluminum, the thermal stability is relatively poor, and it is impossible to guarantee that the generatrix measured by us and UA3P is the same, so the maximum difference of the residual data of the measurement result and the calibration result is about 60 nm, as shown in Fig. 14(b).

Fig. 14. The residual measurement results of the aspheric surface. (a) The residual of the measurement data and the calibration data. (b) The difference of the residual of the measurement data and the calibration data.

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In order to further verify the measurement accuracy, a model system of the freeform surface (i.e., the off-axis aspherical surface with an effective aperture of Φ150 mm and PV<100 nm) is tested. The measurement results of the FSUT by null interferometry method and the proposed method are shown in Figs. 15(a) and (b), respectively. We find that the PV and RMS values obtained from null interferometry method are 92.4 nm and 8.8 nm, respectively, while our method gives the PV value of 101 nm and the RMS value of 10 nm. The deviation of the PV measurement result from null interferometry method is 8 nm, and the RMS measurement result is better than 11 nm. The experimental results show that the high-precision measurement of the freeform surface with an effective aperture of Φ150 mm can be achieved using our method, and the repeated measurement accuracy is better than 10 nm. Figure 15(c) shows the PV and RMS of the residuals of 15 repeated measurements, suggesting that our method also has an excellent reproducibility.

Fig. 15. Measurement results of FSUT. (a) Measurement result of null interferometry method. (b) Residual diagram of the proposed method. (c) The PV and RMS of the residuals of 15 repeated measurements.

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3.6 Discussion

We have compared the measurement results of our method with the null interferometry method and UA3P as shown in Table 4.

Table 4. The comparison measurement results of different tested surfaces

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We find that the repeated measurement accuracy of our method is comparable to that of the contact measurement method, UA3P and only slightly lower than null interferometry. However, our method is superior to null interferometry in versatility and cost. Furthermore, our method can measure almost any form of freeform surface, while the interferometric method has great limitations in measuring freeform monolithic multi-surface, convex freeform surface or refractive freeform surfaces. Compared with the traditional Abbe error-free freeform profiler such as UA3P and Nanopositioning- and Nanomeasuring Machines [31,32], we use LDCTS to measure the FUST, while the scanning sensors used in UA3P and NPMM-200 are the contact-type AFP and the optical probe with 1um working range, respectively. As far as we know, the contact-type AFP and the optical probe cannot be directly used for fast measurement of freeform surfaces. However, our LDCTS can follow the profile of the isocline of the FSUT during measurement, and utilize the working range of ±50 µm to realize fast measurement of freeform surface. In addition, our system adds three independent reference planes, and the capacitive sensors and LDCTS are used for real-time comparison measurement. Compared with the laser interferometer used in the NPMM-200, the capacitor sensor has a lower cost, RMS noise meets the requirements, and the working distance does not increase with the increase of the measurement range. Besides, our LDCTS can also measure the roughness of freeform surfaces from 0.1 µm to 3.2 µm while the NPMM-200 only shows excellent performance in measuring the surface with small roughness and good profile. Furthermore, the focused spot size of the optical probe of NPMM-200 is about 10 µm, which affects its lateral resolution. However, the lateral resolution of LDCTS is only a few hundred nanometers, which has great advantages in measuring the topography of microstructures.

4. Conclusion

To conclude, we have proposed a high precision measurement method of freeform surfaces based on the LDCTS and the real-time comparison principle. Taking advantages of the LDCTS, our method exhibits the excellent axial resolution of 1 nm and a large measuring range of the inclination angle of ±25°. We find that the real-time comparison measurement method can significantly reduce the spatial motion errors along all three axes and improve the coordinate measurement accuracy. Moreover, through iterative prediction, the LDCTS can follow the profile of the isocline of the FSUT during the measurement process, and the measurement of millions of sampling points within tens of minutes.

The traditional spatial motion errors separation is generally performed by calibration or using multiple sensors simultaneously. The former is limited by the calibrating method and cannot achieve online errors separation. The latter will increase the size of the sensor which is not conducive to measuring freeform surfaces with complex structures or large inclination. In our method, we utilize an independent metrology framework, which can better separate spatial motion errors on the basis of real-time comparative measurement. In particular, the measurement accuracy of our method is 5 times better than the counterpart without the real-time comparison along the most error-sensitive direction (i.e., the Z axis). Compared with null interferometry method, we further find that the deviation between the measured results of the plane reflector and the freeform surface is less than 10 nm. We prospect that the measurement accuracy of the current method can be further improved by increasing the number of reference monitoring sensors to further separate the rotation errors. We believe the proposed method provides a promising strategy for the high-precision measurement method of freeform surface during manufacturing in near future.

Funding

National Key Research and Development Program of China (2017YFA0701203); Special Fund on Scientific Instruments of the National Natural Science Foundation of China (61827826).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Technique	Typical method	Dynamic range^a	Accuracy^a	Versatility^a
Null interferometry	Null optics such as CGH^b is used to compensate the wavefront distortion introduced by the FSUT.	≈ 2 µm	< λ/100 RMS	T4
Adaptive null interferometry	DM^c is used for the compensation of the wavefront distortion by controlling the mirror deflection through the actuators.	≈ 15 µm	100 nm PV	T3
Adaptive null interferometry	SLM^d is used for the compensation of the wavefront distortion.	≈ 25 µm	100 nm PV	T2
Non-null interferometry	TWI^e uses a point source array to generate necessary tilt rays to the FSUT and the inclination at sub apertures are compensate, then the interference fringes can be obtained.	≈ 0.9 mm	100 nm PV	T1
Non-null interferometry	Shearing interferometry measures the difference between the original wavefront and the sheared copy wavefront, this difference is approximately proportional to the slope of the FSUT wavefront.	≈ 1 mm	≈ 100 nm PV	T1
Non-interferometric areal measurement	SH^f utilizes the relationship between the aberrated wavefront and the deviated spots on the CCD after the wavefront passes through the micro-lens to measures the FSUT.	≈ 250 µm	λ/40 RMS	T2
Non-interferometric areal measurement	PMD ^g reconstructs freeform by unwrapping the distorted fringe pattern, that is produced by a regularized fringe pattern reflected from the freeform.	≈ 1 mm	Micro -meter	T2
Single-point probe profile measurement	Abbe error-free 3D profiler	100mm	100 nm PV	T0
Single-point probe profile measurement	3D profiler with independent metrology frame	100mm	50 nm	T0

Axis	X			Y			Z
Position error	δ_x(x)	δ_y(x)	δ_z(x)	δ_x(y)	δ_y(y)	δ_z(y)	δ_x(z)	δ_y(z)	δ_z(z)
Rotation error	ɛ_x(x)	ɛ_y(x)	ɛ_z(x)	ɛ_x(y)	ɛ_y(y)	ɛ_z(z)	ɛ_x(z)	ɛ_y(z)	ɛ_z(z)
Perpendicularity	α_xy			α_yz			α_xz

Inclination angle	0°	5°	10°	15°	20°	25°
PV/nm	6.8	5.7	5.8	4.3	6.9	5.2
STD/nm	1.4	1.2	1.6	1.4	1.8	1.5
Linear segment slope	−0.0043	−0.0043	−0.0040	−0.0035	−0.0033	−0.0030
GOF	0.9936	0.9948	0.9962	0.9973	0.9990	0.9992

Technique	Typical method	Dynamic range^a	Accuracy^a	Versatility^a
Null interferometry	Null optics such as CGH^b is used to compensate the wavefront distortion introduced by the FSUT.	≈ 2 µm	< λ/100 RMS	T4
Adaptive null interferometry	DM^c is used for the compensation of the wavefront distortion by controlling the mirror deflection through the actuators.	≈ 15 µm	100 nm PV	T3
Adaptive null interferometry	SLM^d is used for the compensation of the wavefront distortion.	≈ 25 µm	100 nm PV	T2
Non-null interferometry	TWI^e uses a point source array to generate necessary tilt rays to the FSUT and the inclination at sub apertures are compensate, then the interference fringes can be obtained.	≈ 0.9 mm	100 nm PV	T1
Non-null interferometry	Shearing interferometry measures the difference between the original wavefront and the sheared copy wavefront, this difference is approximately proportional to the slope of the FSUT wavefront.	≈ 1 mm	≈ 100 nm PV	T1
Non-interferometric areal measurement	SH^f utilizes the relationship between the aberrated wavefront and the deviated spots on the CCD after the wavefront passes through the micro-lens to measures the FSUT.	≈ 250 µm	λ/40 RMS	T2
Non-interferometric areal measurement	PMD ^g reconstructs freeform by unwrapping the distorted fringe pattern, that is produced by a regularized fringe pattern reflected from the freeform.	≈ 1 mm	Micro -meter	T2
Single-point probe profile measurement	Abbe error-free 3D profiler	100mm	100 nm PV	T0
Single-point probe profile measurement	3D profiler with independent metrology frame	100mm	50 nm	T0

Axis	X			Y			Z
Position error	δ_x(x)	δ_y(x)	δ_z(x)	δ_x(y)	δ_y(y)	δ_z(y)	δ_x(z)	δ_y(z)	δ_z(z)
Rotation error	ɛ_x(x)	ɛ_y(x)	ɛ_z(x)	ɛ_x(y)	ɛ_y(y)	ɛ_z(z)	ɛ_x(z)	ɛ_y(z)	ɛ_z(z)
Perpendicularity	α_xy			α_yz			α_xz

Freeform measurement method based on differential confocal and real-time comparison

Abstract

1. Introduction

2. Freeform surface measurement principle based on real-time comparison

2.1 Real-time comparison measurement principle based on reference planes

2.2 LDCTS with high precision and large local inclination measurement capability

2.3 Analysis of fixed-focus characteristics of laser differential confocal

3. Experiment and analysis

3.1 Measurement system

3.2 Performance of LDCTS

3.3 Real-time comparison error separation analysis based on reference planes

3.4 Real-time comparison measurement experiment of the plane reflector

3.5 Freeform surface profile measurement experiment

3.6 Discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Tables (4)

Equations (15)

Optics Express

Measurement method		The proposed method		The null interferometry		UA3P
		PV/nm	RMS/nm	PV/nm	RMS/nm	PV/nm	RMS/nm
Surface under tested	Plane	99.1	14.2	91.8	12.7	–	–
	Aspheric	305	69	–	–	306	65
	Freeform	101.2	10.5	92.4	8.8	–	–