In order to improve the low displacement sensitivity in sensing the 2D deflection of a fiber probe used for measurement of micro cavities with high aspect ratio, a fiber stem with a ball mounted on its end is used as a probe and a small segment of it is used as a cylindrical lens to collimate a point light source and image it to a camera. The deflection of the fiber stem can be inferred from the change in image acquired by the camera with ultrahigh displacement sensitivities of 10,000 and 20,000 in two dimensions using a fiber stem of 44 μm in diameter, and the corresponding resolutions are better than 1 nm and 3 nm respectively.
©2010 Optical Society of America
Micro cavities with high aspect ratio are common in such products as fuel nozzles, fiber optic ferrules, wire drawing dies, etc. However, it is difficult to measure them with nanometer accuracy because of probing force and limited probing space. A traditional probe used on a coordinate measuring machine (CMM) has a large probing force to cause the deformations of both the probe and work piece, which cause a micrometer error in the measurement. So, several alternative probing methods have been suggested for measurement of micro cavities, such as capacitance method [1–4], optical triangulation method , strain gauge method , vibro-scanning method [7,8], opto-tactile method [9,10], laser trapping method [11,12], etc. However, most of these methods have sub-micrometer resolutions only. One of the most important problems is the low displacement sensitivity in sensing the deflection of a probe.
B. Muralikrishnan and J. A. Stone et al. [13,14] reported a fiber deflection probing (FDP) method for CMM to inspect the profile of micro cavities. A thin fiber of 50 μm in diameter with a ball of 75 μm in diameter mounted on its end is used as a probe, and a small segment of it is illuminated by a laser diode. The shadow of the fiber is magnified 35 times and then imaged with a camera. The fiber probe is deflected when it contacts a surface. The deflection can be inferred from the change in image acquired by the camera. The position of the measured surface can be inferred by adding the deflection of a fiber probe to the machine scale reading. A 4 nm uncertainty can be achieved roughly in detecting the position of the fiber probe under ideal conditions, and the measuring depth could reach 5 mm. This method has such advantages as low probing force, easy miniaturization, high measuring aspect ratio and high probing resolution. In such an optical system, image magnification is the displacement sensitivity in sensing the deflection of fiber probe. However, imaging the fiber make it very difficult to improve magnification largely. So, image arithmetic of edge detection is used to improve the probing resolution, which is a drawback for real-time application.
Therefore, a method based on micro focal-length collimation is proposed for improvement of FDP technique by sensing the fiber deflection with ultrahigh displacement sensitivity, which offers another way of measuring micro cavities with high aspect ratio. In this method, a fiber stem of 44 μm in diameter with a ball of 80 μm in diameter mounted on its end is used as a probe, and a small segment of it is used as a cylindrical lens to collimate a point light source and image it to a camera. The deflection of the fiber stem can be magnified by this collimation optical system, and the magnification is correlative to the ratio of the image distance to the focal length of the collimation lens. The magnification increases as the ratio increases. As the diameter of a fiber stem is in micrometers, its focal length as collimation lens is in micrometers, too. While the image distance is in hundreds of millimeters, it is easy to obtain an ultrahigh image magnification of fiber deflection. A fiber stem of 44 μm in diameter can be used to achieve magnifications of 10,000 and 20,000 in two dimensions. Consequently, simple arithmetic can be used to achieve probing resolutions better than 1 nm and 3 nm in two dimensions at a data-output speed of 440Hz.
2. Sensing deflection of fiber stem
As shown in Fig. 1(a) , a point light source emits rays within a divergence angle, a thin fiber stem, with a ball mounted on its end is used as a probe, and a small segment of it is used as a cylindrical lens to collimate the rays from the point light source, and the collimated rays are imaged with a camera. If the fiber stem is deflected, the centroid position and the bright band width of image acquired by the camera will change accordingly. The principal plane of such an optical system is as shown in Fig. 1(b) where a point light source lies at the focus of the cylindrical lens, and the image of the point light source is a bright band shape.
When a small segment of this fiber stem is used as a cylindrical lens, its focal length close to the optical axis can be given by:Eq. (1). The use of a collimation lens with micrometer focal length to collimate a point light source is called a micro focal-length collimation optical system (MFCOP). A point light source and a fiber stem form a MFCOP where the 2D deflection of fiber stem causes a correlative change in image with ultrahigh displacement sensitivity.
2.2 Sensing property in direction Y
When the fiber stem has a deflection in direction Y, the centroid position of the image will change accordingly. We can see the sensing property by tracing the central ray AB as shown in Fig. 2 .
Before the fiber deflection, ray AB passes through the center of fiber stem, and its propagation direction is not changed. After the fiber deflection, the propagation direction of ray AB is changed twice during two times of refraction, and its last propagation direction is at an angle of 2(i-i') to the optical axis. When L >> f, YAB can be given by:
The transversal magnification of fiber deflection can be given by:
When L=330 mm and n=1.5, the relationship graphs of S — YAB and S — β can be described as shown in Fig. 3 . Four different radiuses (r) are calculated, and β is over 10,000 when r=22 μm, and β increases as r decreases.
2.3 Sensing property in direction X
When the fiber stem has a deflection in direction X, the bright band width of image will change accordingly. We can see the sensing property by tracing the edge ray AC as shown in Fig. 4 .
The propagation direction of ray AC is changed twice during two times of refraction, and its last propagation direction is at an angle of [θ-2(i-i')] to the optical axis. When L >> f, H can be given by:
The longitudinal magnification of fiber deflection can be given by:
When L=330 mm, n=1.5 and θ=30°, the relationship graphs of S — H and S — α can be described as shown in Fig. 5 . Four different radiuses (r) are calculated too, and the sensing property graphs are not always smooth. However, α can be over 20,000 in the smooth range when r=22 μm, and α increases as r decreases.
3.1 Making point light source
A point light source is the most important element in this method. If the point light source was too big in size, its image would be too wide to be received effectively. If the space stability of the point light source was bad, the resolution of this probing method would deteriorate. A gas laser (He-Ne) of 15 mW in power is used as a parallel light source, and a micro objective with a magnification of 150 and a numerical aperture (NA) of 0.95 is used to focus the parallel light to get a point light source. However, the parallel light always has a drift angle in the range of 10−4rad~10−6rad , which causes disturbance to the space stability of the point light source as shown in Fig. 6 , and this disturbance can be given by:
3.2 Experiment setup
As shown in Fig. 7 , a point light source is produced using the method mentioned above, a line-array charge coupled device (CCD) camera is used to acquire the image at a sampling speed of 18.90 kHz, a 2D nanopositioning stage is used to offer calibrated movements with a resolution of 0.1 nm, a fiber stem of 22 μm in radius is fixed onto the 2D nanopositioning stage, and the distance between the fiber stem and the CCD camera is 330 mm. In the following experiments, a homoeothermic room and a vibration isolation base are adopted, and a rigid transparent cover is used to decrease the disturbance of air fluctuation.
3.3 Test in direction Y
Centroid arithmetic can be used to evaluate the measured deflection from the CCD camera in direction Y, and the centroid position jc of an image can be given by:
The 2D nanopositioning stage is ordered to produce fiber deflection steps of 1 nm in direction Y while the CCD camera is sampling, and fiber deflection of 1 nm is discriminated as shown in Fig. 8(a) . It can be seen that a fiber deflection of 1 nm causes a measured deflection of approximate 1 pixel. While a pixel of the CCD camera is 10 μm in size, the fiber deflection is magnified 10,000 times by the MFCOS. We can also see some noise in Fig. 8(a), which may be caused by the drift angle of parallel light and the sampling noise of CCD camera. Fiber deflection steps of 1 μm are then ordered in a long range in direction Y as shown in Fig. 8(b), the unit of measured deflection is transformed to μm.
3.4 Test in Direction X
In order to find a way to evaluate the change in image as fiber deflection happens in direction X, an optical emulation of MFCOS is built using FRED (an optical engineering software). The position of fiber stem is adjusted to get the most concentrated intensity distribution of image as shown in image 1 of Fig. 9(a) , and the intensity distribution in transverse section of image 1 is shown in Fig. 9(b). Then, the fiber stem is moved 5 μm away from the point light source twice in direction X, and image 2 and image 3 are obtained respectively. We can see the images have their own special intensity distributions and it is hard to obtain the width of image directly.
Arithmetic of first-order absolute central moment (M) is used to evaluate the change in intensity distribution of image, and M can be given by:Eq. (16), i is for the number of lines, j is for the number of rows, and f(i,j) is the grey level of pixel (i,j). It can be seen that the intensity distribution of image is more concentrated as M decreases and vice versa.
The sensing property in direction X is tested firstly in a long range using the experiment setup as shown in Fig. 10(a) . It can be seen that, when S > 3 μm, the displacement sensitivity improves as M increases. As an increase in M is an increase in the imaging area, the displacement sensitivity is limited by the imaging area of CCD camera. In order to obtain the highest possible displacement sensitivity for our experiment setup, the fiber stem is positioned initially so that the CCD camera can obtain the widest possible intensity distribution of image. Then, the 2D nanopositioning stage is ordered to produce fiber deflection steps of 3 nm in direction X while the CCD camera is sampling, and fiber deflection of 3 nm is discriminated as shown in Fig. 10(b).
4. Measurement of micro cavity
It can be seen from the tests results above that the deflection of fiber stem can be measured with ultrahigh sensitivity and nano-resolution in two dimensions. The image signal sampled by a line-array CCD camera includes 2D deflection information and has little coupling problem. Further more, the 2D deflection of a fiber stem can be output easily at a speed of 440Hz with nano-resolution. A micro cavity detector based on micro focal-length collimation is formed as shown in Fig. 11 . Holes of approximately 100 μm in diameter can be measured to a depth of 5mm by this detector.
We demonstrated the utility of this detector by measuring a micro hole piece of 300 μm in nominal diameter. This micro cavity detector is mounted onto a CMM. The working diameter of probe ball is calibrated first, and then the probe ball is moved into the micro hole to contact it in direction Y. The measured deflection (given by the detector) and Y coordinate (given by a double frequency laser interferometer) are sampled simultaneously at a speed of 440Hz.
As shown in Fig. 12 , the measured deflection will change as soon as the probe ball contacts the micro hole. We select the segment of data including the turning point of the measured deflection from the CCD and use two beelines to fit these data. The coordinate Y of the intersection point of two beelines is used as the coordinate Y of the contact point. In direction Y, we can find a maximum distance between the two contact points in the micro hole serving as a diameter. The diameter is measured 21 times when it is found. The average diameter is 297.53 μm with a standard deviation of 22 nm as shown in Fig. 13 .
It can be seen from the presentation above that, when a micro focal-length collimation optical system is formed using a fiber stem of 22 μm in radius, 2D ultrahigh sensitivities of 10,000 and 20,000 can be achieved in sensing the fiber deflection, and 2D resolutions of 1 nm and 3 nm can be achieved at a data-output speed of 440 Hz. A micro cavity detector is developed using this method, and a micro hole piece of 300 μm in nominal diameter is measured with a repeatability of 22 nm.
Compared with other methods proposed for measurement of micro cavities with high aspect ratio, this method not only has the advantages of low measuring force, easy miniaturization and high measuring aspect ratio, but also has the advantages of ultrahigh displacement sensitivity, capability of real-time measurement and higher resolution. These properties lead to a powerful capability to measure micro cavities with high aspect ratio and meet the needs of nano-accuracy and real-time measurement.
The authors thank National Natural Science Foundation of China (50705026) and National Hi-tech Research and Development Program of China (2008AA04Z308) for their financial support.
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