Novel gradient focal point (GFP) methods by use of the gradient curvature cylindrical lens, the gradient thickness cylindrical lens and a tilted imaging sensor are proposed for the optical profiler. With the employed simple idea that the different divergence angle of an input beam to the lens generates the different focal position, the height information of one point can be obtained just in a single-shot by GFP approaches. The feasibility of the proposed system is demonstrated to be an alternative to optical profilers.
© 2012 OSA
The optical profilers have several advantages to measure non-destructive and non-contact surface topography. The demand for high speed/high resolution optical profiler has been growing in semiconductor/flat panel display industries, surface/material science, etc. Various optical techniques have been developed to meet the requirement, for example, phase-shifting interferometry (PSI), digital holographic microscopy (DHM), white-light scanning interferometry (WSI) and confocal microscopy (CM).
The configurations of PSI and DHM are a variant Michelson interferometer. As both techniques measure the phase difference caused by the optical path difference between the sample arm and the reference arm, they can take an advantage of full-field imaging which requires no mechanically lateral scanning, non-contact and high sensitive measurement property as the nature of interferometer. These provide PSI and DHM with high speed high resolution performances. Note that the 2π ambiguity by the phase difference restricts the height measurement range. To overcome the limitation of height measurement range, some dual-wavelength methods have been studies [1–5]. However, the measurement range still depends on two wavelengths of lasers, and the wavelengths should be close to each other to measure over hundreds of micrometer height. In addition, the phase unwrapping algorithm to eliminate the 2π ambiguity can limit the measurement speed. These drawbacks make it difficult to fulfill fast and long range height measurement, and allow very restricted applications.
The WSI technique offers not only nanometer height resolution by using a low temporal coherent light source, but also it does not require the lateral scanning as PSI and DHM. In addition, WSI can eliminate the 2π ambiguity and measure long range height thanks to the mechanical axial scanning. On the contrary, high speed optical profiling is difficult due to the axial scanning. A technique of the wavelength scanning interferometry was applied to improve axial scanning speed . However, the wavelength scanning interferometer limits the axial scanning range due to the spectrum bandwidth of light source. Some alternative algorithms were reported to shorten the reconstruction time of three-dimensional (3-D) images [7,8]. These studies still have been employing the mechanically axial scanning methods.
The CM method can construct a 3-D topography thanks to its capabilities of eliminating out-of-focus blur and non-destructive optical sectioning by the spatial filtering effect of a pinhole on the detector. CM illuminates a focal spot on a specimen and detects the point signal through the pinhole. Accordingly, lateral scanning and axial scanning mechanisms are necessary for the reconstruction of 3-D image. The chromatic confocal microscopy (CCM) as a branch of CM has been an emerging technology to enhance axial scanning speed [9–11]. It has an advantage that the broad-band spectrum light source such as supercontinuum laser  is able to extend axial measurement range of CCM. Besides increasing axial scanning speed, beam scanning approach of CCM was introduced  to improve lateral scanning speed. Unfortunately, the spectrum bandwidth of supercontinuum laser and the design of optical system still restrict the axial measurement range. Another representative risk to be considered in CCM technique is the cost issue on account of the broad band spectrum light source with high spatial coherence. Moreover, the complexity of the optical system for aggressively generating axial chromatic aberration and analyzing the spectrum of light signals remains to be resolved. This is an obstacle to be the general high speed optical profilers and to extend its applications to various fields.
Most of the techniques investigated so far cannot meet the requirements of the high speed, high resolution and long range height measurement, simultaneously. In this paper, a novel optical profiler by applying the gradient curvature cylindrical lens, the gradient thickness cylindrical lens or a tilted imaging sensor is addressed to overcome the problems. Note that a light passing through the objective lens generates a corresponding divergence angle with respect to the specimen position. The proposed methods can detect the divergence angle from an input beam with a single illumination of light by applying a new type of lens or the technique, so called gradient focal point (GFP) methods. It means that the proposed approaches have a potential to obtain the specimen position immediately, so that the fast height measurement is possible. The proposed GFP method utilizes a tilted line imaging sensor. Each pixel of the line imaging sensor can eliminate out-of-focus signals as a pinhole. Also, the tilted line imaging sensor can be considered that pinholes are aligned along axial direction. This multi-pixels array can distinguish in-focus signals with out-of-focus signals without axial scanning. Therefore, the basic principle is quite equivalent to the CM, and the optical resolution of the proposed optical profiler is comparable to the level of CM resolution. Furthermore, a simple optical extension enables long range height measurement easily without any negative influences on the speed and the resolution.
2. Principle and analysis
2.1 GFP methods
In fundamental geometrical optics, the curvature radius change of a lens alters the focal length of a lens. Even though a lens has no geometrical change, the focal position shifts when a lens moves along the optical axis. Also, a lens focuses a beam on the different position according to the divergence angle of the input beam. It is obvious from the geometrical optics explained so far that the focal position is determined by the several elements such as radius of curvature, the position of a lens, and the divergence angle of a beam. The key point of proposed GFP methods is to find out, without any mechanical manipulations, the exact focal position of input beam transmitting through a lens, according to the arbitrary divergence angle.
The basic scheme of the first GFP method is illustrated in Fig. 1(a) . A new type of a plano-convex cylindrical lens characterized to have linearly gradient curvature radius with a constant thickness along the x-direction is developed. This lens is first introduced in this paper and called a gradient curvature cylindrical (GCC) lens. Note that the GCC lens has a unique property of linearly varying focal positions owing to gradual growth of the curvature radius. When the imaging sensor parallels and keeps a certain distance to the plano-surface of the GCC lens, the GCC lens focuses an input beam on only a certain sensor position. This generates a bow tie-like spot on the imaging sensor plane displayed in Fig. 1(a). Since only focused light on the imaging sensor has a maximum intensity, the intensity profile similar to the Gaussian shape is obtained on the imaging sensor line. Figure 1(b) describes the second GFP method. Another new plano-convex cylindrical lens is introduced. It has linearly gradient thickness with a constant curvature radius along the x-direction, while the GCC lens has linearly gradient curvature radius. Let us name this lens a gradient thickness cylindrical (GTC) lens. The GTC lens has linearly varying focal positions due to the gradient thickness. The varying thickness generates various focal positions but fixed focal length. Therefore, the GTC lens is analogue to typical plano-convex cylindrical lens which is tilted with respect to the x-axis (see Fig. 1(c)). As a similar principle of the GCC lens, the GTC lens also forms a bow tie-like spot on the sensor plane or a peak intensity profile on the sensor line. Figure 1(d) shows the other GFP method which is quite simpler than the previous two approaches. It is configured by simply tilting the imaging sensor with respect to the x-axis, while a typical cylindrical lens is being placed to parallel to the x-axis. Likewise, a bow tie-like spot or a peak intensity profile also can be obtained.
Figure 2 illustrates in detail how bow tie-like spots and intensity profiles are obtained from the GCC lens according to three different divergence angles for the corresponding input light. The converged light depicted in the left hand side in Fig. 2(a) creates a focal spot inclined to the left side on the imaging sensor plane, so that the peak position of the intensity profile is placed on the left side of the first illustration in Fig. 2(b). On the contrary, the diverged light is focused on a right side of the imaging sensor in the right hand side in Fig. 2(a) and 2(b), and the collimated light is focused on a center of the imaging sensor. Three different approaches for the generation of the gradient focal points were examined so far. It is obvious that the method by the GCC, GTC lens or the tilted imaging sensor yields the bow tie-like image and peak intensity profile. By estimating the maximum intensity position, the focal position of the corresponding input beam can be computed at one time
2.2 Doubly-telecentric condition
In this subchapter, a paraxial GFP system is introduced to examine the relationships between an object plane and an image plane, so that the calibration process for a given system can be easily accomplished by understanding the characteristics. For this analysis, the paraxial raytracing technique  is employed.
Figure 3 describes the GFP system as a paraxial system with thin lenses and paraxial rays in air space. A tedious mathematical manipulation provides a relationship between the object plane and the image plane as13], Eq. (1) becomes as
Equation (3) implies that the image defocus has a negative-linear relation with the specimen defocus and a gradient is the same as a longitudinal magnification of the GFP system as shown in Fig. 3. It is obvious that the ratio of the image defocus to the object defocus is the longitudinal magnification of the optical system , but Eq. (3) additionally denotes that, by the positive and the negative specimen defocuses with same absolute value, two image defocuses also have the same absolute values and opposite directions to each other. Consequently, with only a doubly-telecentric condition, the two estimated peak positions by the two opposite and same amount defocuses of the specimen are symmetric with respect to the in-focus peak position.
Based on Eq. (1), a numerical simulation is performed to verify the numerical relationship between the image defocus and specimen defocus in the condition of the objective lens with the focal length of 10 mm and the cylindrical lens with the focal length of 100 mm. Figure 4 shows the image defocus simulation results according to specimen defocuses. Figure 4(a) shows results when specimen defocuses are ± 1 mm and the distance between the objective lens and the cylindrical lens satisfies the doubly-telecentric condition (solid line), 60 mm shorter (dash line) and 60 mm longer (dot-line) than the doubly-telecentric condition, respectively. Figure 4(b) shows results when specimen defocuses are ± 0.1 mm and the distance between the objective lens and the cylindrical lens has same conditions as Fig. 4(a). It is assured from simulation results that the GFP system designed to meet the doubly-telecentric condition can maintain the linearity of image defocus with respect to the specimen defocus. However, if the specimen is defocused at a short distance such as dozens of micrometer (Fig. 4(b)), the linearity of image defocus is almost maintained even though the GFP system does not meet the doubly-telecentric condition exactly.
2.3 Systematic design
The third GFP method (Fig. 1(d)) is the simplest approach to design the optical profiler among the three GFP approaches. Therefore, the tilted imaging sensor technique with a typical cylindrical lens is selected for the representative systematic design, as illustrated in Fig. 5 .
Two primary specifications to be determined are the focal length of cylindrical lens and the tilt angle of imaging sensor, which is mostly important to design the proposed GFP module. As shown in Fig. 5(b), let us assume that the GFP system consists of two thin lenses in the air. Additionally, the several parameters are already specified such as the focal length and the back aperture diameter of the objective lens, the height measurement range, the measurement resolution and the pixel size of the imaging sensor. Furthermore, for the doubly-telecentric condition, it is supposed that a distance between the objective lens and the cylindrical lens is the sum of two focal lengths of the objective lens and the cylindrical lens. The procedure for determining the two primary specifications is as follows. Firstly, the minimum number of measurement data and the minimum length of the imaging sensor are, respectively, expressed asEq. (3), the image defocus range due to the height measurement range is given byFig. 5(b), the functional relation of the image defocus range and the minimum length of the imaging sensor is given byEq. (5), (6) and (7), the tilt angle is expressed as
The possibility of the mathematical existence of the tilt angle in Eq. (8) leads to an inequality expressed asEq. (9) can be rewritten as
Note that based on the diffraction theory, the focal spot of a lens cannot be an ideal point, but it has a finite spot size, so called Airy disk. Even though the cylindrical lens focuses the light as a line not a spot, the finite width of a line is same as the diameter of Airy disk. The diameter of Airy disk is given by 
Using Eq. (13), it is convenient to compute the focal length of the cylindrical lens, , based on the pre-defined parameters and the doubly-telecentric condition. The tilt angle, , is also calculated by using Eq. (8) and the cylindrical focal length. Even though the design process is basically defined by the tilting image sensor method, it can be extended to obtain not only the gradient of curvature radius and curvature radii of the GCC lens but also the gradient of thickness and the curvature radius of GTC lens. Once the material of GFP lens and the wavelength of the input light source are predetermined, the gradient is identical to the tilt angle, and the curvature radius is easily calculated from the focal length of the cylindrical lens.
3.1 System implementation
The experimental setup of the GFP system depicted in Fig. 6 is based on the first and second GFP methods. A tilted plano-convex cylindrical lens is introduced instead of the GTC lens. The diode laser (QL65F6S, QSI Laser) with a wavelength of 655 nm, the objective lens with a focal length of 10 mm (G Plan Apo 20 × , Mitutoyo) and the CMOS camera (CR-GM00-H6400, Dalsa) with an active resolution of 640 × 480 and a pixel size of 7.4 μm × 7.4 μm as the imaging sensor are employed. The flat mirror (MM1-311-25, Semrock) is used as a specimen. The fabricated GCC lens is utilized to examine the first GFP method. The geometrical dimensions of the fabricated GCC lens are listed in Table 1 . The material of the fabricated GCC lens is PMMA (poly methyl methacrylate).
In order to obtain longer working distance than the focal length of the cylindrical lens, the concave lens (LC1315-A, Thorlabs) with a focal length of −75 mm is employed. The cylindrical lens with a focal length of 100 mm (LJ1567L2-A, Thorlabs) is applied. The CMOS camera being used for the system setup has C-mount with the thread of 25.4 mm diameter and the flange focal distance is 17.5 mm. If the CMOS camera is tilted over 30 degree, the thread mount blocks the light propagating to the CMOS sensor. Unfortunately, the third GFP approach is not feasible to perform the experiments in the given environment. The alternative option is to apply the second GFP method because the effect of tilted cylindrical lens is nearly equivalent to the result by the tilted imaging sensor approach.
3.2 Experimental result of GCC lens method
Let us choose the maximum measurement range as 30 μm and the minimum number of measurement data as three-quarter of horizontal pixel number of the imaging sensor. The measurement resolution is 62.5 nm/pixel by Eq. (4). The input beam diameter to the cylindrical lens is 5.6 mm because the numerical aperture (NA) of the objective lens is 0.28. It is readily considered from Eq. (13) that the possible range of the focal length of the cylindrical lens is defined as . The fabricated GCC lens has gradually increasing focal lengths from 36.9 mm to 49.2 mm at the wavelength of 655 nm. Let us define the representative focal length of GCC lens as 43 mm. Since focal lengths and the length of GCC lens range from 36.9 mm to 49.2 mm and 15 mm, respectively, the gradient becomes about 39 degree. From Eq. (8), the focal length of cylindrical lens is determined to be 86 mm. If the distance between the GCC lens and the concave lens is set to zero, the maximum effective focal length of the compound lens consisting of the GCC lens and the concave lens in the air becomes 82 mm which satisfies the inequality condition in Eq. (13). The working distance of the compound lens is approximately 90 mm. Also, in order to meet the doubly-telecentric condition for the system designed with the objective lens and the compound lens, the distance between the back aperture of the objective lens and the concave lens is determined as approximately 69 mm. Figure 7(a) and 7(b) show three bow tie-like spots exaggerated in vertical direction and three intensity profiles, respectively, with respect to + 15 μm defocus (left), in-focus (center) and −15 μm defocus (right) of the specimen. A low pass filtering technique is applied to estimate the position of maximum intensity. The estimated left, center and right peak are positioned at 72, 288 and 509, respectively. It is noticed that the left and right peak positions are symmetric with respect to the in-focus peak position. The range from the left peak to the right peak is approximately 440 pixels and this yields the measurement resolution of 68.1 nm/pixel. Figure 7(c) displays peak positions as a pixel number of the intensity profile according to ± 15 μm specimen defocuses in GCC lens method. The solid line and the dash line represent measured data and linearly curve-fitted data, respectively. It can be considered that measured data is linear. This result means that the linearity of image defocus is maintained in a doubly-telecentric condition.
3.3 Experimental result of tilted cylindrical lens method
It is specified that the maximum measurement range is 100 μm and the minimum number of measurement data is 320. The measurement resolution is 312.5 nm/pixel by Eq. (4). The input beam diameter to the cylindrical lens is also 5.6 mm. From Eq. (13), the range of the focal length of the cylindrical lens is determined as . When the distance between the cylindrical lens and the concave lens is 195 mm, the effective focal length of the compound lens in the air is 43 mm which satisfies the inequality condition in Eq. (13). The working distance of the compound lens is approximately 158 mm. In order to meet the doubly-telecentric condition, the distance between the back aperture of the objective lens and the concave lens should be −45 mm. However, a negative distance physically does not exist, so we kept the distance as 35 mm because the beam-splitter must be placed between the objective lens and the concave lens (see Fig. 6(b)). From the effective focal length of the compound lens and Eq. (8), a tilt angle of the CMOS camera is given as approximately 51 degree. Instead of tilting the CMOS camera, we employ 51 degree tilted cylindrical lens to set the system up. Figure 8(a) and 8(b) display three bow tie-like spots exaggerated in vertical direction and three intensity profiles, respectively, with respect to + 50 μm defocus (left), in-focus (center) and −50 μm defocus (right) of the specimen. The estimated left, center and right peak positions are 128, 290 and 455, respectively. It is noticed that the left and right peak positions are symmetric with respect to the in-focus peak position. The range from the left peak to the right peak is approximately 330 pixels and this yields the measurement resolution of 303 nm/pixel.
Figure 8(c) shows peak positions as a pixel number of the intensity profile according to ± 50 μm specimen defocuses in tilted cylindrical lens method. The solid line and the dash line represent measured data and linearly curve-fitted data, respectively. It is obvious that the measured data is linear, and the linearity of image defocus is persistent in a short range specimen defocus.
A new type of optical profiler having fast height measurement and high measurement resolution was addressed. Three GFP approaches, i.e. the GCC lens, the GTC lens or a tilted cylindrical lens and a tilted imaging sensor were investigated to meet the requirement. The optical profilers based on the GFP methods are able to detect the specimen height without the z-axial scanning. It was proven that the doubly-telecentric condition of the optical system maintains the symmetricity and linearity of image defocus with respect to the specimen defocus. These characteristics can help calibrating the system easily and simply. Furthermore, the algorithm for determining primary design parameters of GFP system, i.e. the focal length of cylindrical lens and the tilt angle of imaging sensor was presented. It was proven from the experiment that GFPs have the potential to improve the measurement speed and high resolution.
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