We present an imaging scheme that takes advantage of the superior lateral resolution of volume holographic imaging (VHI) and a-priori surface information about the object to build a profilometer that can resolve 50µm longitudinal features at a working distance of ≈ 50 cm with a single VHI camera. We discuss the scheme and present experimental results of surface profiles of MEMS devices.
©2003 Optical Society of America
Volume holographic imaging (VHI) is an imaging technique in which optical information is processed by a volume hologram present in the imaging system. VHI is a recent development  and VHI systems can achieve very rich functionalities for diverse imaging applications. This has been demonstrated in the context of a confocal microscope , a depth selective telescope  and a real-time hyperspectral microscope . In , we have derived the impulse response of a VHI system with a volume hologram that is recorded using planar reference and signal beams and then read out with a point source collimated by an objective lens placed in front of the hologram; we refer to this system as Planar reference VHI (PR–VHI). The Bragg selectivity of the volume hologram can be utilized to optically “slice” the object space and then image 3D objects one slice at a time. The VHI system can only “see” objects located within the Bragg slice; the rest of the object space is invisible to the hologram. For PR–VHI the Bragg slice is a virtual slit located at the front focal point of the objective optics.
The depth resolution of VHI systems degrades quadratically with increasing object distance in a manner similar to most 3D imaging systems. For a PR–VHI system, it is possible to design the objective optical system to optimize the depth resolution for a particular working distance. For instance, a PR–VHI system with telephoto objective optics can resolve longitudinal features of 1 mm at a distance of 50 cm  when nothing is known in advance about the object being imaged. At the optimal value, the depth resolution is diffraction limited and so, it is not possible to improve the resolution without incorporating some additional information.
In this paper, we discuss an imaging scheme for surface profilometry or  imaging. A surface profilometer returns a height map z=z(x,y) of the object as the “image” as opposed to a 3D imaging system which returns the 3D intensity distribution I=I(x,y,z) of the object. Surface profilometers are used extensively in industry for inspection and metrology applications. Commercially available profilometers utilize a wide range of measurement techniques. For instance, the Atomic Force Microscope (AFM)  uses a tiny stylus tip to scan the object’s surface. The height is obtained from the angular deflection of a light beam reflected by the tip cantilever. The Zygo© profilometer uses interference techniques and a depth scan to measure the height map in a non–contact fashion. These systems can achieve very high depth resolution ≈ 0.1 nm, but can do so only at very small working distances (in the order of a few mm). On the other hand, stereo vision systems use multiple cameras and triangulation to estimate the surface profile at longer working distances. However, the depth resolution is poorer  and in passive systems, one has to establish correspondence in between the multiple stereo images to triangulate correctly.
In this paper, we discuss a PR–VHI based metrology system with active monochromatic illumination that can achieve high depth resolution of ≈ 50µm at long working distances of ≈50 cm using a-priori information about the object surface. In Section 2, we present the theory of PR–VHI and discuss how the depth resolution can be optimized for a particular working distance. In Section 3, we discuss how a-priori object information can be used to improve the depth resolution further. In Section 4, we present experimental PR–VHI surface profiles of some MEMS objects at long working distances and conclude in Section 5 by discussing directions for future work.
2. Imaging Properties of a PR–VHI system
A simplified schematic for PR–VHI is shown in Fig. 1. The volume hologram of thickness L is recorded using two planar beams, a normally incident reference beam and an oblique signal beam incident at an angle θ s ≪1 rad as shown in Fig. 1(a). The readout/imaging setup consists of a point source reference that is collimated by an objective optical system of front focal length f and aperture a before it is incident on the hologram as shown in in Fig. 1(b). The volume hologram diffracts only the Bragg matched components of the reference source and a CCD camera captures the Fourier transform of the diffracted beam. This is the PR–VHI “image” of the point source. For an on–axis point source defocused by a small distance δ from the front focus of the objective lens, the diffracted intensity on the CCD is derived under the weak diffraction approximation ) and is given by
where Ib=I(x′=θ s F,y′=0) is the peak intensity produced by the probe and F is the focal length of the Fourier transforming lens in front of the CCD. The diffraction pattern contains two contributions: A disk representing the defocus due to the lens and a sinc squared term that defines the slit imposed by the Bragg selectivity.We neglect diffraction ripples at the disk edges; these ripples have negligible effect in practice. Figure 2 shows an experimentally observed diffraction pattern on the CCD for a PR–VHI system with the Bragg slit and defocused disk as expected.
Based on (1), we can calculate the depth resolution of the PR–VHI system . The depth resolution (defined in terms of the Full width at half maximum (FWHM) of the point spread function (PSF)) is
Further, the lateral resolution can be defined as the width of the Bragg slit and this can is calculated from (1) to be
The PR–VHI system can image a volumetric object by scanning it completely in 3D just like a confocal microscope. A detailed comparison in between the resolution of a PR–VHI system and a long–range confocal microscope is given in Ref. . Obviously, the total image acquisition time depends on the amount of scanning required and it is beneficial to reduce this as much as possible for most applications.
Since the fringes that constitute the PR volume hologram are invariant in the out of plane direction of Fig. 1, it is possible to image in the PR–VHI geometry using a line source and scanning the object space one line at a time. This reduces the amount of scanning required. For a line source illumination as depicted in Fig. 3(a), the diffracted intensity on the CCD can be derived in a manner similar to (1). The result is
where Ib=I(x′=θ s F,y′) is the peak intensity produced by the probe. We note that (1) is the response to a defocused point source whereas (4) is the response of a defocused line source. The depth and lateral resolutions for the line readout follow a trend similar to that observed for a point source readout on account of the similar scaling factors in Eqs. (1) and (4). However, the depth resolution for the line readout method is poorer than the same optical system with the point readout method. This is shown in Fig. 4 which compares the longitudinal PSFs when the same PR–VHI system is probed first with a point and then a line source. A similar degradation is observed in confocal microscopy when the confocal pinhole is replaced by a narrow slit.
From (2) we observe that the Δz FWHM of the PR–VHI system varies quadratically with f but varies only inverse linearly with a, whereas most lens based imaging systems have an inverse quadratic dependence on a . This property is used to optimize the depth resolution up to the diffraction limit to yield an optimal depth resolution 
Equation 5 specifies the best achievable depth resolution when nothing is known in advance about the object (r min is a constant that results on account of the optimization). In the next section, we discuss a strategy to exploit a-priori object information to improve the depth resolution even further.
3. Exploiting a-priori object information to enhance depth resolution
We have seen that the optimal depth resolution for PR–VHI is given by (5). This is a worst case result applicable when we do not have any prior information about the object being imaged. In many cases, such as surface metrology and inspection applications, prior information is available and can be utilized to yield improved resolution. For instance, most microchips and MEMS devices have only flat surface features of various heights since silicon manufacturing technology is predominantly planar.
Consider the case when the object is known to consist only of flat surfaces, as shown in Fig. 5. A PR–VHI system can obtain a surface profile of the object by a complete point/line scan as described in section 2. The direction along which scanning is done has a significant bearing on the system’s longitudinal resolution.
In Fig. 5(a), the object (or equivalently the PR–VHI sensor) is scanned such that both the longitudinal scan direction and the angle of incidence of the active illumination is parallel to the optical axis of the sensor. In this case, the resolution of the system is indicated by the dashed ellipse. Notice that the quadratic dependence of depth resolution results in the depth resolution (Δz) being poorer than the lateral resolution (Δx).
However, we can take advantage of the fact that the object consists only of flat surfaces by altering the scan direction. This is shown in Fig. 5(b). The object with flat surfaces is scanned in a direction inclined (or equivalently the PR–VHI sensor and/or the active illumination direction are inclined) at an angle ϕ with respect to the optical axis. The depth resolution of the PR–VHI system along the optic axis is still Δz, but since the surfaces of the object are inclined at an angle ϕ with the axis, the PR–VHI can resolve lateral features of size ≈ Δx cosϕ and surface height features ≈ Δx sinϕ. In other words, the superior lateral resolution (the width of the Bragg slit) manifests itself as an “apparent” depth resolution of the object’s surface features. It is important to note that this benefit is possible only because we know the nature of the object surface and are using this knowledge. Note that an inclined (non-flat) surface would reflect the incident active illumination in a different direction (see Fig. 9) and no light would enter the PR–VHI system.
Figure 6 compares the resolution between a PR–VHI system imaging a flat surface for the case of scanning the object along the optical axis and scanning the object at an angle ϕ=30° inclined with respect to the optical axis. Notice that the inclined sensor has much better resolution on account of the sinc squared envelope imposed by the Bragg selectivity of the volume hologram.
4. Experimental results
We implemented a PR–VHI based surface profilometer with long working distance two microfabricated objects based on the principle discussed in Section 3. The experimental setup used was the same as shown in Fig. 5(b). The telephoto objective system was designed so that the the front focal length of the system was 460 mm and the effective focal length of the system was 95 mm. The volume hologram was a 2 mm thick crystal of LiNbO3. It was recorded using a doubled Nd:Yag CW laser (λ=532 nm) which was split to create two mutually coherent planar beams, a normally incident reference and an oblique signal with θ s=25°. The hologram had a typical diffraction efficiency η=5% and its aperture was 2 mm and the focal length of the Fourier transforming lens in front of the camera was 50 mm. The PR–VHI system was placed such that the optical axis was inclined at an angle ϕ=30° with respect to normal to the object surface. The object was mounted on a 2D actuation stage controlled by Newport CMA-25CCCL actuators. During imaging, the object was scanned using the line method described in Section 2. This was achieved by placing a cylindrical lens in the path of the laser illumination such that the object surface was located at the focus of the cylindrical lens. We used a Jai CV235 industrial CCD camera to capture the diffraction pattern at each location. The scanning and subsequent reconstruction of the object was controlled using Matlab.
Figure 7 shows the experimentally obtained surface profile of a MEMS microturbine. The turbine was manufactured to have surface height features ≈ 225µm. Notice that there is a complete contrast reversal in the PR–VHI images when the focus moves from the top to the bottom of the object. Figure 8 shows the height map of the nanogate, a device used to measure nanofluidic flow rates developed at MIT . The nanogate was microfabricated so that it had surface height features ≈ 150µm. We note a complete contrast reversal as the PR–VHI system is focused at the top and bottom of the object.
5. Conclusions and Future work
We have shown that a PR–VHI system is capable of exploiting a-priori object information to resolve small features at large working distances. We believe that a PR–VHI based metrology system would be useful for various industrial inspection and manufacturing applications. A schematic of such a system is shown in Fig. 9. It is possible to reduce the scanning required for imaging by using broadband illumination. We are currently working on quantifying the trade-offs involved in reducing the scanning required and the resultant reduction in depth resolution.
The authors would like to thank Tina Shih, Wenyang Sun and Kehan Tian for useful discussions; Alexander Slocum and James White for providing the nanogate and Chee-Wei Wong for providing the microturbine. This projected was funded by the Air force research laboratories, Eglin AFB through grant F0830-00-0012 and the Air Force office of scientific research via MURI grant E-16-V91-G1.
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