1. Introduction

The ability to accurately determine the position of a point on the surface of an object can be very useful in a number of areas of manufacturing. With this in mind, manufacturing processes are becoming increasingly reliant on the ability to make high-speed, precise shape and deformation measurements. While conventional scanning and contact methods offer the required precision, they are often innately slow. Optical methods are becoming increasingly popular as they offer high-speed, precision measurements, without the need for a mechanical contact point.

One rapidly developing optical technique which offers an excellent approach for non-contact, high resolution imaging is digital holography [1,2]. A number of advantages make this technique very appealing for inspection. These include the availability of the quantitative phase information, which can be used to detect height changes on the order of a few nanometers, numerical focusing, full-field information and single-shot recording [3–6]. On the other hand, a limitation with single-wavelength, phase-imaging digital holography in reflection mode, is that surface features which exceed half of the probe wavelength cannot be determined due to phase wrapping. To measure the absolute optical path length changes requires finding the integer part of the fringe number, more commonly known as phase unwrapping. One approach to this problem that has been widely explored over the years is the use of phase unwrapping algorithms[7,8]. These are mostly based on searching for ambiguities and adding integer multiples of 2π, followed by scanning across the two-dimensional array. However, while these algorithms can be used to unwrap a smooth continuous surface, a sharp step is often problematic. In addition, they are often computationally-intensive, and so the capability for real-time analysis of measurements is lost.

A straightforward way to overcome the difficulties associated with the phase unwrapping algorithms and determine optical path lengths beyond the ambiguity limit is to use a two-wavelength optical approach. The technique of using two wavelengths to measure large objects is well known in digital holography, holographic contouring and holographic interferometry [9–16]. Each hologram is analyzed to obtain their individual phase information. The two sets of phase data are then processed to obtain difference-phase data proportional to a scale length, i.e. the synthetic wavelength defined by the first wavelength and the second wavelength. Thus, the phase is measured at an effective wavelength much longer than either of the two probing wavelengths. The synthesized wavelength Λ₁₂ is found by:

The two phase images at the two separate wavelengths, λ₁ and λ₂, can be obtained simultaneously using the same optics and CCD camera [17], with the image noise reduced over taking two independent images, because common-mode noise such as vibrations are correlated and cancel out when the difference image is obtained. In addition, aberration correction techniques can be applied [18], and real-time imaging is attainable because only one image is required at each inspection location.

By proper choice of the two wavelengths, the axial range Λ₁₂ can be adjusted to any value that would fit the axial size of the object being imaged. However, in order to obtain a longer range, the two wavelength values λ₁ and λ₂ must be chosen to be close together. In this case, when the difference between the two images is taken, Λ₁₂ becomes noisy due to error amplification and makes the phase measurement precision poorer than that of either of the individual single-wavelength phase images, thereby reducing image quality.

One way to recover the accuracy in the measurement is to carry out a systematic reduction in the wavelength, using the information from the larger-wavelength measurements to remove 2π ambiguities in the shorter-wavelength data. This process, known as hierarchical phase unwrapping, is analogous to a phase unwrapping process; in short, using the noisier, larger synthetic wavelength Λ₁₂, to unwrap the higher resolution shorter wavelengths, λ₁ or λ₂, thus maintaining the large axial range and high precision [19–22]. For this procedure to work correctly at each reduction step the following condition must be satisfied:

where Λ_n+1 is the reduced wavelength and ε_n+1 is its associated noise, Λ_n is the larger synthetic wavelength and ε_n is its associated noise. The condition imposed by Eq. (2), means that large-difference reduction steps, Λ_n/Λ_n+1, in the presence of a large amount of noise are often difficult to accomplish without creating errors. This is a often a limitation for two-wavelength digital holography when generating relatively large synthetic wavelengths for long range imaging [23, 24].

In order to overcome this problem, we introduce the phase data of a third, shorter wavelength. This generates intermediate synthetic wavelengths, which allows for smaller-difference reduction steps in the phase unwrapping procedure. The corrected phase data for the intermediate step can then be used to correct for the ambiguities in the single-wavelength phase data. In this way, we achieve long range phase imaging while maintaining the precision of the single-wavelength phase measurement. The system allows for the real time capture of the three-wavelength complex wave-front in one digital image, and in addition offers the flexibility of performing profilometry over a range of both small and large gauge heights.

2. Multiple-wavelength phase imaging digital holography

In digital holography, the holographic interference between the object and reference waves is recorded by a CCD camera. The digitized hologram is then input into the computer. Numerical reconstruction of the complex wave-field is performed by the application of the angular spectrum algorithm for off-axis digital holography:

where U(x₀,y₀,0) is the complex wave-front in the hologram plane, f_x and f_y are frequency coordinates in the frequency domain, and filter represents digital filtering which allows the spatial frequencies of each wavelength to be isolated in inverse space as displayed in Fig. 1. In the calculation of Eq. (3), two Fourier transforms are needed. However once the field is known at any one plane, only one additional Fourier transform is needed to calculate the field at different values of the reconstruction distance, z.

 

Fig.1. Three-wavelength digital hologram (a), and corresponding Fourier spectrum (b).

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The angular spectrum has some particularly useful advantages for multiple-wavelength imaging; for example, it has no minimum reconstruction distance requirement, and the pixel size of the reconstructed image does not vary as a function of the reconstruction distance.

2.1 Hierarchical optical phase unwrapping

The process of Hierarchical optical phase unwrapping as described by Schnars and Jüptner in Ref. [22], is illustrated as a flow chart in Fig. 2. The first step in the process is to generate a synthetic wavelength Λ_n which is greater than twice the maximum height variation of the object. As a result of this new unambiguous phase range, the phase noise also becomes amplified. From the resulting quantitative phase data, the corresponding surface height information z_n is found by:

The next step in the process is to reduce the wavelength at which the measurement is taken, and use the information of the unambiguous, larger-wavelength measurements to eliminate the ambiguities by determination of the interference order.

The method follows as: calculate the difference between the unambiguous surface profile, z_n, and the ambiguous surface profile z′_n+x (the apostrophe indicates ambiguous throughout this paper) with related wavelength Λ_n+x:

Next determine the integer component in Δz:

Finally add integer multiples of the wavelength to the original ambiguous profile z′_n+x:

z_n+x is now unambiguous, with the accuracy improved over z_n due to the shorter wavelength. The process may continue further, applying correction steps for smaller wavelengths in a similar fashion.

 

Fig. 2. Flow-chart representation of the process of hierarchical optical phase unwrapping.

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2.1 Simulation of two and three-wavelength digital holography

Figure 3 presents a simulation of two-wavelength digital holography using hierarchical optical phase unwrapping. The theoretical object shown in Fig. 3(a) is a symmetrically curved surface, with a maximum height of h=9.0µm. In order to simulate the type of noise level associated with experimental conditions, a white noise level of ε_i~2% has been added to each single-wavelength profile so that each profile contains a noise level of λ_i.ε_i. In Fig. 3(b), using λ₁=633nm and λ₂=612nm, the corresponding surface profiles z′₁ and z′₂ each consist of a number of discontinuities at each multiple of the wavelength. Subtraction of their respective phase data, φ′₁₂=φ₁-φ₂ and the addition of 2π wherever φ′₁₂<0 produces a new phase profile, φ₁₂(x)=φ′₁₂+2π·(φ′₁₂<0), with a new synthetic wavelength Λ₁₂=18.45µm. The corresponding two-wavelength surface profile, z₁₂=φ₁₂Λ₁₂/4π is also displayed in Fig. 3(b).

While z₁₂ is unambiguous, the phase noise in each single-wavelength phase map has been amplified by a factor equal to the magnification of the wavelengths and as a result z₁₂ is strongly disturbed by noise. In order to recover the accuracy of the measurement, z₁₂ is used to correct for the ambiguities in the high precision, single wavelength profile z′₁ as displayed in Fig. 3(c). However, this step, as can be seen in Fig. 3(d), generates integer addition errors due to the large amount of noise in z₁₂ larger than λ₁/2. The remaining spikes in the final surface profile are due to places where the two-wavelength profile noise is too large and no longer satisfies the condition set by Eq. (2).

 

Fig. 3. Simulation of two-wavelength digital holography. (a) Actual height profile of object with a maximum height of h=9.0µm, (b) surface profile plot for z′₁ derived from φ₁ where λ₁=633nm, z′₂ derived from φ₂ where λ₂=612nm, and the corresponding two-wavelength unambiguous surface profile z₁₂ with synthetic wavelength Λ₁₂=18.45µm, derived from the single wavelength profiles z′₁ and z′₂. (c) 2π ambiguity correction of z′₁ using the two-wavelength surface profile z₁₂. (d) Final ambiguity corrected surface height prfile, z₁.

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The phase-unwrapping technique can be extended to three wavelengths, as presented in Fig. 4. The addition of a third, shorter wavelength into the simulation procedure at λ₃=532nm, generates an extended range synthetic wavelength of Λ_13–23=18.45µm, as well as intermediate synthetic wavelengths at Λ₁₃=3.33µm and Λ₂₃=4.07µm. Figure 4(a) displays the single wavelength surface profile z′₃, the intermediate profiles z′₂₃, z′₁₃, and also the extended range profile z_13–23, derived from z′₂₃ and z′₁₃. Note that z_13–23 is identical to the two-wavelength profile z₁₂ in Fig. 3(b). Instead of correcting for the ambiguities in z′₁ using z_13–23, where the combination of noise and reduction step is too large to give an accurate result, the intermediate step profile z′₁₃ is corrected, as shown in Fig. 4(b). This step substantially recovers the profile accuracy, as displayed in Fig. 4(c). Finally, the corrected result, z₁₃, is used as an intermediate step to correct for the ambiguities in the single wavelength profile z′₃ in Fig. 4(d). The final surface profile result, z₃, with the single-wavelength precision fully restored, is shown in Fig. 4(e).

 

Fig. 4. Simulation of three-wavelength digital holography using the same theoretical object as in Fig. 3(a). (a) single-wavelength surface profile z′₃ derived from φ₃ where λ₃=532nm, twowavelength surface profile z′₂₃ derived from φ₂₃ with synthetic wavelength of Λ₂₃=4.07µm, two-wavelength surface profile z′₁₃ derived from φ₁₃ with synthetic wavelength of Λ₁₃=3.33µm and the unambiguous two-wavelength surface profile z_13–23 with synthetic wavelength Λ₁₂=18.45µm derived from z′₂₃ and z′₁₃, (b) ambiguity correction of the intermediate synthetic wavelength profile z′₁₃ using z_13–23, (c) ambiguity corrected profile, z₁₃, (d) second step ambiguity correction using the phase data from the intermediate profile z₁₃ to correct for the ambiguities in the single-wavelength phase data z′₃. e) Final ambiguity corrected surface height profile z₃.

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3. Experimental

The three-wavelength experimental setup is depicted in Fig. 5. The laser sources are two He-Ne lasers emitting at wavelengths λ₁=633nm and λ₂=612nm and a third diode pumped solid state (DPSS) laser emitting at λ₃=532nm. These sources generate synthetic wavelengths of Λ₁₂=18.45µm, Λ₂₃=4.07µm and Λ₁₃=3.33µm respectively. Λ₁₂ sets the large phase range of the system, with Λ₂₃ and Λ₁₃ providing the intermediate synthetic wavelengths. The three off-axis interferometers in an achromatic setup bring each wavelength into one imaging system. The reference beam of each interferometer is adjusted so that all three holograms are recorded with a different set of spatial frequencies in a single digital image, thereby allowing for real-time capture of the three-wavelength complex wave-front. Once the holograms have been captured onto the CCD camera, they are sent via an IEEE 1394b interface to a computer for processing. Owing to the numerical focusing capability of digital holography, the reconstructed image at each wavelength can be focused correctly, thereby allowing for exact superposition of the object. To correct for residual optical aberrations in the reconstructed image, a flat-field reference hologram is recorded on a flat, blank portion of the reflective object.

A potential difficulty when recording more than two holograms in one digital image is the carryover of information between the spatially-heterodyned holograms and the zero order image information in Fourier space. This problem can be addressed by simply adjusting the spatial frequencies of the individual holograms in order to maximize the separation from each other. Other approaches include placing an aperture in the optical system to reduce the pread in frequency space, or the use of smaller radius digital filters. However, care must be taken to prevent reducing the numerical aperture of the optical system.

 

Fig. 5. Apparatus for three-wavelength digital holography. λ/2: half-wave plate, PBS: polarizing beamsplitter, λ/4: quarter-wave plate, M.O: microscope objective.

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4. Results

Here we present examples of images obtained using three-wavelength digital holography. Figure 6 displays digital holographic images and quantitative cross-sectional profiles of a gold-on-chrome USAF 1951 resolution test target, with structures possessing a height of 6.6µm. The single-wavelength phase images reconstructed from the three-wavelength hologram displayed in Fig. 6(a), are shown in Figs. 6(b), 6(c) and 6(d), each with a root mean squared (rms) noise level of ε_i~8nm. These single-wavelength images are combined to generate the large synthetic wavelength phase image (Λ_13–23=18.45µm) in Fig. 6(e), and the intermediate synthetic wavelength phase image (Λ₁₃=3.33µm) in Fig. 6(f). The corresponding quantitative cross-sectional surface profiles for these images, as taken along the yellow line in Fig. 6(a) are shown in Fig. 6(h). The extended range profile z_12–23 is unambiguous; however the noise has been magnified with the rms noise now at ε_12–23~520nm. The large noise level in z_12–23 makes the accurate correction of the ambiguities in the single-wavelength profile in one step, ineffective, as verified by the use of Eq. (2):

λ ₃(1-4(ε ₃))≥(Λ_12–23)(ε ₁₂)

0.532µm(1-4(0.015))≥(18.45µm)(0.028)

0.504≥0.520

To improve the measurement accuracy, hierarchical phase unwrapping is applied. First, the intermediate step profile z′₁₃, with rms noise ε_12–23~57nm is corrected for ambiguities using z_13–23, as illustrated in Fig. 6(h). This step from Eq. (2) gives:

Λ₁₃(1-4(ε ₁₃))≥(Λ_12–23)(ε ₁₂)

3.33µm(1-4(0.017))≥(18.45µm)(0.028)

3.102≥0.520

Then, in the second step the ambiguity corrected intermediate result, z₁₃, displayed in Fig. 6(i), is used to correct for the ambiguities in the single-wavelength profile z′₃, Fig. 6(j). Again from Eq. (2):

λ ₃(1-4(ε ₃))≥(Λ₁₃)(ε ₁₃)

0.532µm(1-4(0.015))≥(3.33µm)(0.017)

0.504≥0.056

The final result is shown in Fig. 6(k) where the height range is extended and the precision of the single-wavelength measurement very nearly fully restored, with a few remaining spikes left over in places where the combination of the noise and the wavelength reduction step appear to be too large to correctly determine the order of the fringe number. The final corrected image is shown in Fig. 6(g), along with its 3D-rendering in Fig. 6(l). From the image the measured step height is observed to be ~6.6µm, which compares well with the manufacturers specifications for the test target structures.

The results of the hierarchical phase unwrapping process are consistent with the simulation data and highlight the advantages of the three-wavelength technique; low noise in combination with a long measurement range.

 

Fig. 6. Three-wavelength digital holography of a USAF 1951 gold-on-chrome resolution target. Structures possess heights of 6.6µm. (a) three-wavelength digital hologram, (b) single-wavelength phase image with λ₁=633nm, (c) single-wavelength phase image with λ₂=612nm, (d) single-wavelength phase image with λ₃=532nm, (e) two-wavelength phase image with synthetic wavelength Λ₁₂=18.45µm, (f) two-wavelength phase image with synthetic wavelength Λ₁₃=3.33µm, (g) final ambiguity-corrected phase image, h) 2π ambiguity correction of the intermediate synthetic wavelength profile z′₁₃ using z_13–23, i) ambiguity corrected profile z₁₃, j) second step 2π ambiguity correction of single wavelength profile z′₃ using z₁₃, k) final ambiguity corrected surface height profile z₃, and l) surface height 3-D rendering of g).

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As a second example we looked at a MEMS device; a micro-cantilever array designed for infra-red sensing. The cantilevers in the array exhibit a shape deflection as they are exposed to infra-red radiation.

Figure 7 displays digital holographic images of a single cantilever in the array. The Scanning Electron Microscope (SEM) image of the cantilever is shown in Fig. 7(a), illustrating the characteristic bend of the central IR absorber. Figure 7(b) is the amplitude reconstruction from the hologram and Figs. 7(c), 7(d) and 7(e) are the single-wavelength phase images.

The single-wavelength images are combined together, and then optical phase unwrapping is applied in order to create the extended range two-wavelength phase image in Fig. 7(f), and the final three-wavelength image shown in Fig. 7(g) with unambiguous range and low noise. A visual comparison between the two and three-wavelength images highlights the considerable noise reduction afforded by use of the three-wavelength technique. In Fig. 7(g) the red highlighted area is selected and a 3×3 median filter is applied to suppress the few remaining spikes. The 3D rendering of this filtered area is displayed in Fig. 7(h) with the cantilever exhibiting a deflection of around 2µm.

 

Fig. 7. Three-wavelength digital holography of an infra-red, micro-cantilever device with dimensions of 100µm×88µm. (a) SEM image, (b) amplitude reconstruction, (c) single-wavelength phase image with λ₁=633nm, (d) single-wavelength phase image with λ₂=612nm, (e) single-wavelength phase image with λ₃=532nm, (f) two-wavelength phase image with synthetic wavelength Λ₁₂=18.45µm, (g) three-wavelength final phase image with synthetic wavelength Λ₁₂=18.45µm, and (h) 3D rendering of highlighted area in (g).

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Real-time capture of the three-wavelength wave-front allows for long range, high precision measurements of dynamically moving objects. This capability is illustrated in Fig. 8, where a few frames of a thermally activated cantilever undergoing shape deflection are shown.

 

Fig. 8. Three-wavelength quantitative phase movie of the shape deflection exhibited by an infra-red sensing micro-cantilever. (Fig_8_Can.mov, 81KB)

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5. Summary

In this paper, we have described the basic concept of three-wavelength digital holography and demonstrated its performance on a standard resolution test target of height 6.6µm, and a MEMS device producing a shape deflection of several microns. The ambiguity in the phase is eliminated by generating a relatively large Λ=18.45µm synthetic wavelength, with the usual error amplification suppressed thanks to the generation of intermediate synthetic wavelengths and subsequent hierarchical optical phase unwrapping. The experimental results are shown to be in good agreement with the simulation data, although there are still a small number of spikes left over in places where the order of the fringe number has been incorrectly determined due to the combination of phase noise and size of the reduction step. These errors may be eliminated by optimal selection of the intermediate synthetic wavelengths for the optical phase unwrapping process, the use of lower coherence sources to reduce the effects of coherent noise and improved wavelength stability.

As the three-wavelength wave-front can be captured in real-time, the technique can be used to make direct, long-range shape measurements of dynamically moving samples with high resolution and speed as demonstrated using the micro-cantilever array.