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As an alternative to the conventional optical frequency scanning technique, an angular spectrum scanning technique is proposed for absolute interferometry. Instead of sweeping the optical frequency over a wide range of spectrum, we sweep the angular spectrum by changing the incident angle of a monochromatic plane wave with a spatial light modulator (SLM). The use of monochromatic light combined with the SLM enables dispersion-free absolute interferometry that is free from mechanical moving components.
©2006 Optical Society of America
1. Introduction
White-light interferometry [1–3] or low-coherence interferometry [4, 5] is a well established technique for absolute interferometry that can measure objects with large discontinuous gaps or isolated surfaces without the phase ambiguity of integer multiple of 2π. The technique has been widely used for the measurement of complex three-dimensional (3-D) micro structures such as Micro-Electro-Mechanical Systems (MEMS), and for the 3-D biological imaging such as optical coherence tomography (OCT) [6]. To avoid the mechanical scanning involved in white-light (or low-coherence) interferometry for optical path difference (OPD) compensation, spectral interferometry has been proposed that makes use of spectral fringe signals in optical frequency domain [7–14]. The spectral fringes are detected by scanning optical frequency with a spectrometer [7, 8] or an optical-frequency-tunable light source [9–14]. These methods permit one to measure absolute height with submicron resolution without mechanical scanning. However, because both white-light (or low-coherence) interferometry and spectral interferometry use a wide range of optical spectrum, they suffer from spectral absorption and index dispersion problems, particularly when the object and/or the propagation medium have inhomogeneous spectral response as in the case of biological samples submerged in a liquid medium. One may realize that, if the 2π phase ambiguity problem is solved, a monochromatic source can become an ideal source for the objects and the medium that have highly dispersive absorption and/or refractive indices. Another advantage of using a monochromatic light is that, if a highly absorbing medium has a narrow spectral window of low absorption, one can perform the measurement by tuning the spectrum of the light source into this low-absorbing spectrum window.
As an alternative to the conventional optical frequency scanning technique, we propose yet another type of frequency-scanning absolute interferometry, in which the angular spectrum of quasi-monochromatic light is swept with a spatial-frequency-tunable source made of a spatial light modulator (SLM). The proposed technique enables absolute interferometry that is free from dispersion problems and mechanical moving components. Experimental results are presented that demonstrate the validity of the proposed principle.
2. Principle
For convenience of explanation, let us first consider a simplified optical geometry for two-beam interferometry as illustrated schematically in Fig. 1, which is often realized with a Michelson or Fizeau interferometer. A point source S is placed on the focal plane of a lens L whose optical axis is normal to the surface of a reference mirror MR. One of the collimated rays exiting from the lens reaches an observation point A on the surface of the object MO directly, and interferes with another ray coming to the same point after being reflected at point B on the surface of a reference mirror. In Fig. 1, point A’ is the mirror image of point A with respect to the reference mirror MR. The propagation vector k (which will be referred to as the k-vector for short) of the collimated beam and the height vector h are in the direction of the vectors BA′ and CA, respectively. The phase difference Δϕ between these two rays is given by
where kh = k cos θ is the height component of the vector k, and θ is the angle of incidence to the reference surface defined by the angle between the vector k and the height vector h.
It is common practice in conventional optical frequency scanning interferometry to adjust the k-vector k to be parallel to the height vector h so as to maximize the fringe sensitivity such that Δϕ = -2hk with θ = 0. The wavenumber k = ω/c is swept by changing the optical frequency ω with a frequency-tunable light source or a spectrometer. Using the relation h = -(1/2)∂(Δϕ)/∂k , one can determine the height from the phase slope or the frequency of the spectral fringe, which is detected by an appropriate algorithm for fringe analysis such as the Fourier transform method [13]. This characteristic of optical frequency scanning interferometry is illustrated in the k-vector space shown in Fig. 2. As shown in Fig. 2, the object is illuminated with the beam whose k-vector k(0) is parallel to the height vector h such that θ = 0 . The operation of optical frequency sweeping corresponds to stretching or shrinking of the k(0) vector while keeping its direction unchanged. This change in the radius of the k-sphere causes the dispersion problems as it corresponds to the change in the optical frequency. If we take a closer look at Eq. (1), we note an alternative solution in which we change the angle θ while keeping the optical frequency constant. In the k-space shown in Fig. 2, this operation corresponds to changing the cone angle θ of the k(θ) vector while keeping the radius of the k-sphere unchanged. The projected height component–k h of the k(θ) vector plays the role of the k(0) vector in optical frequency scanning interferometry. For example, if one can change θ over 0 ~ 30 degrees for the wavelength of 633nm, one can in principle realize the dispersion-free measurement with the performance comparable to the optical frequency scanning interferometry with the wavelength scanning range as wide as 98nm. Similarly to the optical frequency scanning technique, the height is determined from the relation h = -(1/2)∂(Δϕ)/∂kh. Since the collimated beam with the k-vector k(θ) is an angular spectrum component of the monochromatic optical field, we will refer our technique as the scanning angular spectrum technique to differentiate it from the conventional optical frequency scanning technique.
Once fringe signals are obtained by the proposed scanning angular spectrum technique, the subsequent fringe analysis is exactly the same as that of the optical frequency scanning technique, which has been fully described elsewhere [9, 12–14]. Therefore, let us focus our attentions on the technical issues for the practical implementation of the angular spectrum scanning technique. For our purpose, tuning angular spectrum is equivalent to controlling the angle of the illumination beam, which can be done by moving a point source S on the focal plane of a collimator lens L using, for example, a spatial light modulator (SLM). Suppose the optical axis OO’ of the collimator lens L is parallel to the height vector and normal to the reference mirror surface MR. Then the height component of the angular spectrum kh can be written as
where r is the distance of the point source from the optical axis, and in arriving at the last expression, the paraxial assumption has been made that f ≫ r . Since point sources on a ring centered on the optical axis provides identical phase change to the fringes [15, 16], we can use a ring source rather than a single point source, and also that the increase of the radius of the ring source at equal rate will produce a chirped fringe signal. The use of the ring source has the advantages in the amount of usable light and also in the robustness to the shading problem that occurs when a high and/or deep object is illuminated at a large incidence angle. The shadow-free illumination from the circular source solves the shadowing problem intrinsic to the angular spectrum scanning technique. Because the chirping of the fringe frequency is not desirable for the fringe analysis by the Fourier transform method or the phase-shift technique, as well as for the subsequent processing for the height determination, we correct this nonlinearity by choosing the increment of the h-component of the k-vector Δkh ∝ Δ(r 2) = constant. This can be realized by sequentially increasing the diameter of the ring source according to the relation
where n is the sequential number of the angular spectrum scanning. To avoid the spurious interference fringe noises arising from the interference between the beams from different points on the ring source, the ring source should be a spatially incoherent (and yet temporally coherent) source. Such a source can be realized by placing a rotating ground glass on the source plane to destroy the spatial coherence (but preserve the temporal coherence). The fringes generated by all the point sources on the ring vary in synchronism and are incoherently superposed on intensity basis to give a strong fringe signal
where a(x,y;n) and b(x,y;n) are the background fringe intensity and the fringe amplitude. By applying appropriate one-dimensional fringe analysis and phase unwrapping algorithms to the fringe signal with respect to the variable n , one can obtain the phase
at position (x,y) independently from other positions, and determine the height from the linear slope of the phase
where < > denotes the operation to determine the slope by the least squares fit.
3. Experiments
The schematic diagram of the angular spectrum scanning interferometry is illustrated in Fig. 3. Light from a He-Ne laser (λ = 633nm, 10mW) is expanded and collimated by a beam expander EX and lens L1. A liquid crystal spatial light modulator (LC-SLM) (CRL Opto, XGA) sandwiched by a polarizer P1 and an analyzer P2 is placed between lens L1 and a confocal lens pair consisting of lens L2, L3 and a pinhole PH, which functions as a spatial filter to remove the effect of discrete pixels of SLM. A ring source pattern displayed on SLM is relayed by the confocal lens pair and imaged onto a rotating ground glass GG placed on the front focal plane of lens L4, to generate a spatially incoherent ring source. The light from the ring source created on this rotating ground glass is introduced into a Michelson interferometer composed of a beam splitter BS, a reference mirror MR, and an object GB made of a pair of gauge brocks. Lens L5 images the interference fringe pattern on the object surface onto CCD. A half wavelength plate HWP is placed directly behind the laser for the adjustment of illumination level. The pinhole PH is placed between lenses L2 and L3 to smooth out the pixel structure of SLM. The object is a pair of gauge blocks with heights h 1 = 1.090mm and h 2 = 1.600mm, which are placed on an optical plate. Lens L5 is focused on the surfaces of the gauge blocks. The focal length of lens L4 is f = 150mm, and the magnification of the imaging lens L5 and the confocal lens pair are both adjusted to ~1×. The source patterns with various ring radii were generated in a personal computer PC and transferred to LC-SLM for display.
Before starting the measurement, we first performed alignment by placing the center of the ring sources on the axis of lens L4 so as to ensure that the angular spectrum from each point source on the ring has the same h-component kh = kcosθ. In the experiment, we performed this alignment by observing the fringe visibility. Remember that the detected fringe signal is the result of the incoherent superposition of the individual fringe intensity generated by each source element on the ring source. If the center of the ring source deviates from the optical axis of the lens, the h-component of the angular spectrum kh = k cos θ of the beam from each source element on the ring becomes different, and the superposition of the individual fringe intensity will no longer be constructive. As the result, the total fringe visibility will drop down. Making use of this phenomenon, we adjusted the position of the ring source displayed on LC-SLM by moving the center of the ring pixel by pixel until we found the location of the maximal fringe visibility that ensured the exact centering of the ring to the axis of lens L4.
We set the bottom surface of the gauge block (with h 1 = 1.090mm) to have an offset distance from the effective position of the reference mirror MR, as shown in Fig. 3. Then we generated the ring sources sequentially in computer with the initial radius starting from r 1 =31 pixels. To cope with the transmission power loss of LC-SLM, the ring was made to have a finite width of 3 pixels so that sufficient illumination level was attained. According to Eq. (3), we generated 32 ring sources with nonlinearly increasing radii, and displayed them on the rotating ground glass GG sequentially. At the same time, we synchronized the recording of the interferogram with CCD with the speed of 10 frames / second. The sequential change of the radii of the ring source is shown in Fig. 4. An example of the fringe patterns generated on the surfaces of the two gauge blocks (referred to as top and bottom surfaces) are shown in Fig. 5(a) and Fig. 5(c), respectively, for the first (n = 1) and last (n = 32) ring sources in the sequence. The horizontally stretched dark boundary in the middle of the picture is a gap between the two gauge blocks, from which no light is reflected. The white arrows in the pictures indicate the direction of fringe motion with the angular spectrum scanning performed by increasing the radius of the ring source. An example of the fringe intensity variations at arbitrarily chosen two points, A on the top surface and D on the bottom surface, are plotted as a function of the sequential number, in Fig. 5(b). Note the difference in the period of the two fringe signals, which reflects the difference in the speed of the fringe motion, and also indicates the height difference between the two surfaces relative to the reference mirror surface. In Fig. 6(a), the solid line with red dots and the dashed line with green dots show, respectively, the Fourier spectra of the fringe signals at point A and D shown in Fig. 5(b). The peak positions of the spectra serve as a rough indicator of the height values at A and D. To make better estimation of the height from the phase information, we applied the Fourier transform method [17] to the fringe signals and obtained the unwrapped phase distributions as shown in Fig. 6(b) with red dots for point A and green dots for point D. In Fig. 6(b), the solid line and the dashed line represent the linear fitting to the unwrapped phases at point A and D, respectively, and the slope of the lines give the height information.
Fig. 6. (a) Fourier spectra of the fringe signals. (b) Determination of the phase slope by least squares fit to the unwrapped phase.
Theoretically, we can convert the obtained phase slope to the height value by using Eq. (6). In practice, however, the scale factor in the slope-to-height conversion formula of Eq. (6) needs be calibrated experimentally because it involves the parameters r 1 and f that need be determined experimentally. Generally, it is difficult to know the exact radii of the ring sources on the rotating ground glass because the magnification of the confocal lens pair in Fig. 3 and the focal length of lens L4 may not be precisely determined, and certain alignment errors are also unavoidable. For this reason, we developed the following method of calibration. We gave several displacements of known amount to the reference mirror by using a high precision translation stage, and then performed the angular spectrum scanning by changing the radius of the ring source displayed on LC-SLM. Carrying out the same fringe analysis as described above, we obtained the phase slope values for these known positions. As predicted, the phase slope value increased linearly with the known height value, as shown in Fig. 7. The linearly fitted line gives the actual calibration curve for the experiment setup used. We noted that the extension of this fitted line down to the height equal to zero deviates slightly from the origin by 0.015 radian, which may be attributed to the alignment error in the experiment. From the calibration curve shown in Fig. 7, and the phase slope determined independently for each pixel on the object surface, the 3-D height distribution was obtained. The result of measurement for a step object made of the two gauge blocks is shown in Fig. 8(a), and Fig. 8(b) shows the height profile along the 122nd column. In Fig. 8(a) and Fig. 8(b), we removed the invalid data in the dark gap region shown in Fig. 5(a) and Fig. 5(c). The average height of the top surface is 2.034±0.023mm and that of the bottom surface is 1.526±0.018mm. Therefore, the height difference between these two flat surfaces is Δh = 0.508mm, and it is in good agreement with the nominal step height of 0.510mm. We note periodic errors in the measurement result shown in Fig. 8(a) and Fig. 8(b), which amount to 0.02mm in the peak-to-valley value for the top surface and 0.06mm for the bottom surface. One may note from Fig. 8(a) that the directions and the spatial periods of these errors correspond closely to those of fringe patterns shown in Fig. 5(a) and Fig. 5(c). This fact suggests that the errors be related to the initial phase of the fringe signals, such as those shown Fig. 5 (b). Computer simulations for a set of synthesized fringe data revealed that, when both the number of fringes and the number of sample points are significantly reduced, the Fourier transform method suffers from the errors arising from the fringe discontinuity at the boundaries, and also that the amount of these errors are strongly dependent on the initial phase of the fringe signal. In this preliminary experiment, the number of sample points and the number of fringe cycles were restricted to 32 points and 6~9 cycles, respectively. This restriction came partly from the interferometer that has small apertures with the limited acceptance angle of ~3 degrees, and also from the large scan step of the ring source that has a finite ring width to avoid the loss of light at SLM. However, it should be stressed that these limitations on this proof-of-the-principle experiment are by no means the fundamental limitations of the principle itself. The performance can be improved by an appropriate design of the system.
Fig. 8. Experimental result. (a) Three-dimensional height distribution. (b) Height profile along the 122nd pixel row.
4. Conclusions
Based on the observation of the principle of two-beam interferometry in the k-vector space, we proposed an angular spectrum scanning technique for absolute interferometry, which can serve an alternative to the conventional optical frequency scanning technique. Instead of sweeping the optical frequency over a wide range of spectrum, we swept the angular spectrum by changing the incident angle of a monochromatic plane wave with a spatial light modulator (SLM). The use of monochromatic light combined with the SLM enables dispersion-free absolute interferometry that is free from mechanical moving components. The use of ring source was proposed to improve the shadowing problem. Experimental results were presented that demonstrate the validity of the proposed principle.
Acknowledgments
We thank Dong Wei and Akinori Yamada for their help with experiment. Part of this work was supported by The 21st Century Center of Excellence (COE) Program on “Innovation of Coherent Optical Science” granted to The University of Electro-Communications, from Japanese Government. Zhihui Duan gratefully acknowledges the scholarship given to him from Ministry of Education, Culture, Sports, Science and Technology of Japan.
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