Focus detection criterion for refocusing in multi-wavelength digital holography

Li Xu; Mike Mater; Jun Ni

doi:10.1364/OE.19.014779

1. Introduction

Applications of focus detection in holographic imaging have been reported in quite a few articles [1–6]. Typical focus detection criteria are based on a serial of reconstructed frames along the axial direction, and determine the best focused frame via a certain kind of “sharpness indicator”. The sharper the image is, the closer its position to the best focus. Commonly adopted indicators include: entropy indicator [1], variance indicator [2], spectral based indicator [3] and correlation coefficient indicator [4]. The majority of focus-finding applications is based on amplitude analysis, even though in many cases it is phase contrast that is actually of interest. As reported in [5], best focus with sharpest phase contrast can be determined, when it corresponds to the lowest amplitude contrast. Such objects are referred to as “pure phase” objects. In more general cases, however, when phase contrast is not strongly dependent on amplitude contrast, the task of determine “the sharpest phase contrast plane” is difficult. Examples of using phase information for refocusing purpose are limited to a few simple cases, such as MEMS structures reported in [7,8], where refocusing distances are determined by counting phase wrapping stripes. This method can be performed only when the object slope is continuous and gentle, so that phase stripes are continuous and clear enough to be counted. It is even believed that phase contrast is not usable for more complicated focus-finding purposes, such as step like height structures, since jumps at 2π caused by phase wrapping would be misunderstood as real sharp features [3]. Phase unwrapping algorithms are available, yet they are not satisfactory solutions, since unwrap errors would significantly reduce the accuracy of the detection.

Digital holographic technology applying dual / multiple wavelengths has experienced substantial development in recent years. M. K. Kim reported an evident improvement of axial resolution by conducting multi-wavelength interference in tomography [9]. Frederic Montfort et al. experimentally demonstrated the constructive effect of multiple complex waves at a selected plane, while destructive effect could be clearly observed at other planes within the axial extent determined by wavelength interval [10]. J. Gass et al. has proposed a novel noise reduction algorithm for reconstructing multi-wavelength topography [11]. A variety of methods have been developed in order to reduce recording time, whose ultimate goal is to achieve real-time multi-wavelength holography [12–14]. Demonstrated applications of multi-wavelength digital holography involves, for example, observing bio samples and bio structures [15–17], contouring mechanical objects [18] and developing endoscopic equipment [19]. Among all these reported development, a core advantage of multi-wavelength digital holography is that, by greatly expanding the unambiguous range, phase wrapping problems can be overcome in practice.

Multi-wavelength interferometry (MWI), which has recently been implemented as a powerful surface quality monitoring tool in the auto industry [20,21], presents a realistic need of phase contrast based focus detection: Over one machined part, there can be several separated surfaces of inspection interest. Due to either measurement time restrictions or geometric structure limitations, not every surface can be measured in-focus. Out-of-focus imaging impedes inspection quality by reducing measurement resolution. Holographic refocusing with correct refocusing distance helps evidently on resolution enhancement of out-of-focus objects [22]. Nevertheless, dependence between amplitude contrast and phase contrast can be fairly loose due to flatness and roughness variation. Thus indirect focus detection criteria are not always a viable choice. In the same time, stripe-counting strategy is also made impossible due to discontinuity and shape-complexity of surfaces.

In this paper, a new phase contrast based focus detection criterion has been proposed by taking the advantage of multi-wavelength digital holography. The major challenge of achieving reliable focus detection lies in how to separate useful phase contrast information from its noisy background, including salt-and-pepper noise and measurement uncertainty. Instead of detecting a maximum sharpness point as many previous criteria do, the developed criterion aims to find a zero-passing point, which is experimentally effective. After introduction of MWI refocusing, a step-by-step derivation of the criterion is provided. Measurement examples of step like height structures are given, demonstrating possible applications of this criterion in the following sections.

2. Multi-wavelength interferometric (MWI) refocusing

MWI imaging generates a set of holograms each of a different wavelength for the same object. A typical system arrangement is shown in Fig. 1 . A wavelength tunable laser (New Focus, TLB-6316, 838nm~853nm wavelength range) serves as the coherent light source. Its output is separated into a reference beam and an object beam. The reference beam contains a controllable phase shifter to generate phase shifted interferograms. In the object beam, part to be measured (object), parabolic mirror (for off-axis aberration correction purpose) and lens, and the camera (Kodak, KAI-4021, 2048 by 2048 pixels) form a reflective imaging system. For each wavelength λ, its hologram can be generated by applying phase shifted interferometry (PSI): Interferograms captured by CCD can be expressed as [20]:

I (φ) = I_{o b j} + I_{r e f} + 2 \sqrt{I_{o b j} I_{r e f}} \cos [(γ_{o b j} - γ_{r e f}) - φ]

where I_obj / γ_obj and I_ref / γ_ref stands for the intensity / phase of the object and reference beam, respectively. φ is a phase shift term generated by the phase shifter. The phase difference term (γ_obj - γ_ref) physically results from the optical path difference H between the object and reference beam:

γ_{o b j} - γ_{r e f} = 2 π \frac{H}{λ} = 2 n π + θ

where n is an integer, and θ is the residual phase angle ranging from –π to π. θ can be mathematically derived from interferograms of designed φ values. For example, if φ is set to be 0, π/2, π and 3π/2, then [23]:

θ = atan 2 (\frac{I (3 π / 2) - I (π / 2)}{I (0) - I (π)})

where atan2 is a four-quadrant inverse tangent function in accordance with the definition in MATLAB [24]. The calculated hologram of wavelength λ can then be represented by a complex field:

E (x, y) = 2 \sqrt{I_{o b j} (x, y) I_{r e f}} \cdot e^{i θ (x, y)}

where

2 \sqrt{I_{o b j} (x, y) I_{r e f}}

is the amplitude term and

e^{i θ (x, y)}

is the phase term for each pixel of the camera.

Fig. 1 MWI imaging system.

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In a single wavelength case, due to the periodicity of the cosine function in Eq. (1), H remains undefined to the additive integer n multiplies 2π as shown in Eq. (2). In other words, θ alone is insufficient to fully represent H, unless n is predetermined. This requirement restricts the varying range of γ_obj - γ_ref to be within –π to π. The unambiguous measurement range of object height h, which equals to half of H in the shown system, is limited to λ/2. For a larger measurement range, at least one more wavelength should be added in order to deduce a correct object height. By generating a synthetic wavelength $λ_{s y n} = λ_{1} λ_{2} / (λ_{1} - λ_{2})$ , the unambiguous measurement range will be expanded to $λ_{s y n} / 2$ . Further introduction of extra wavelengths would make the deduction process more robust, as a Fourier transform based height scan envelope [20]

S (h) = | \frac{1}{P} \sum_{p = 1}^{P} \exp [j (4 π \frac{h}{λ_{p}} - θ (λ_{p}))] |

can be depicted for each and every point over the whole field of view, in order to deduce the correct value of h, which is consistent with the highest peak position of S(h). Experimentally, applying 14 selected wavelengths ranging from 838 to 853nm with pm level accuracy would result in over ± 150mm unambiguous measurement range.

Similar to single wavelength holographic imaging, MWI imaging of an out-of-focus object would result in blurred details. To deblur the image, the captured holograms of each wavelength should be digitally refocused to the focus plane respectively, before the height scanning step of Eq. (5). A practical refocusing equation for MWI holograms is [22]:

E_{2} (x', y') = F_{x', y'}^{- 1} {\exp [\frac{j k d λ^{2}}{2} (u^{2} + v^{2})] \times [F_{u, v}^{+ 1} E_{1} (x, y)]}

where

E_{2} (x', y')

/

E_{1} (x, y)

stands for the out-of-focus / in-focus complex hologram respectively (Fig. 2 ). d is the distance between the planes. k is wave number, which equals to 2π/λ. u and v are spatial frequencies along x and y axis, respectively.

F_{η, ξ}^{\pm 1} f (α, β)

denotes 2D Fourier / inverse Fourier transform:

F_{η, ξ}^{\pm 1} f (α, β) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [\mp j 2 π (α η + β ξ)] f (α, β) d α d β

Comparing with commonly applied single wavelength Fresnel diffraction formula [25], Eq. (6) get the common phase term

\exp (- j k d)

eliminated from the right hand side: Due to enormous magnitude difference between refocusing distance d and wavelength λ, this term would introduce evident phase error even for pm level wavelength inaccuracy. Such phase errors among applied wavelengths would result in phase mismatching in the height scanning step of Eq. (5), making refocused observation no longer valid for metrology purpose. After digitalization, Eq. (6) turns into:

E_{2} (\frac{X' Δ}{M}, \frac{Y' Δ}{M}) = \sum_{X, Y, U, V = 0}^{N - 1} \frac{1}{N^{2}} E_{1} (\frac{X Δ}{M}, \frac{Y Δ}{M}) \times

\times \exp [j \frac{π λ d M^{2}}{N^{2} Δ^{2}} (U^{2} + V^{2})] \times \exp {- j \frac{2 π}{N} [(X - X') U + (Y - Y') V]}

where the whole field captured is of N by N pixels. Δ represents pixel interval. M is the magnification of the imaging system. X’, Y’, X and Y all range from 0 to N-1. Discrete spatial frequencies U and V also range from 0 to N-1.

Fig. 2 Coordinate system for digital refocusing.

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Maximum refocusing range is limited by aliasing error in Eq. (8) [23]. For a small magnification factor of M = 1/20, if N = 2048, Δ = 7.5μm, λ≈850nm, a theoretical maximum refocusing distance d_max can be as large as 27.1m.

A typical MWI refocusing process can be summarized as follows:

● Capture multiple interferograms for each wavelength λ_p
● Generate the individual out-of-focus complex fields, based on Eq. (4)
● Calculate refocused complex fields, using Eq. (8)
● Height scanning (Eq. (5) to reconstruct an in-focus height map h(x,y) from the refocused fields.

As an example, height maps of before and after refocusing a one cent penny is shown in Fig. 3 . The penny is placed 200mm out of focus. The height maps are threshold filtered to emphasize features within the range from 1.250mm to 1.650mm above its background.

Fig. 3 Experimental measurement of a 200mm out-of-focus penny.

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3. Focus detection for MWI imaging

Object height h derived in Eq. (5) originates from the phase term $\cos [(γ_{o b j} - γ_{r e f}) - φ]$ of the captured interferograms, rather than the amplitude term $2 \sqrt{I_{o b j} \cdot I_{r e f}}$ in Eq. (1). The focus detection criterion derived below for h(x,y) is thereby based on phase contrast, which is a significant difference of the proposed work from its predecessors.

A major difficulty of setting up a practical focus-finding criterion on height maps is that the useful height contrast information is relatively weak, comparing with its background noise. Take Fig. 3 as an example, the reconstructed features over the penny are typically around 100μm in height, 1/40 of the experimental height scanning range (−1mm to 3mm). Their occupied area is less than 5% of the whole field of view (200 by 200 pixels). Therefore, an effective feature extraction and enhancement process should be the core part of the criterion.

After generating refocused height maps $h (x, y) {d}$ with a serial of refocusing distance d along z axis, the proposed focus detection criterion can be conducted in the following steps:

● Impulse detector based noise filtering
● Finding the difference between two reconstructed height maps of certain interval
● Eliminate measurement uncertainty
● Feature enhancement
● Implement differential indicator $η {d}$

Step 1: Impulse detector based noise filtering

Figure 3 demonstrates obvious salt-and-pepper noise. This is introduced by practical small errors of λ_ps and θ(λ_p)s. As we are taking a real-time measurement of wavelengths experimentally, the main contributor of errors in λ_ps is considered to be rounding errors. On the other hand, the causes of errors of θ(λ_p) are manifold, including vibration, temporal air turbulence along optical path and speckle induced degradation, etc. Laser speckles due to the roughness of the target object surface randomly degrade intensity modulation, on which we depend to derive θ(λ_p) in Eq. (3). Since the height scanning process (Eq. (5) is ill-posed, small differences of λ_p and θ(λ_p) in $(4 π h / λ_{p} - θ (λ_{p}))$ would result in unpredictable peak position shift. When errors of λ_p and θ(λ_p) are random small values, such shift should be point-by-point: one outlier point will not affect its neighbors. Under such conditions, impulse detector based noise filtering should be effective. The filtered height maps are noted as $\bar{h} (x, y) {d}$ .

Step 2: Finding the difference between two reconstructed height maps of certain interval

Other than focusing on the information of a single frame, the developed criterion here uses two reconstructed frames: the subtraction of two height maps with a certain interval, named “height difference map”, is employed. Assuming the frame interval of the reconstruction serial is δd, then the difference interval between two frames is usually selected to be l times of δd for convenience, and the resulting height difference map:

\bar{h}_d i f f (x, y) {d, d - l δ d} = \bar{h} (x, y) {d} - \bar{h} (x, y) {d - l δ d}

This operation is more or less similar to the correlation coefficient (CC) indicator [4], as it can be viewed as a kind of “robustness enhancement” procedure: larger the interval is, more evident the difference between two frames should be. The idea of using difference instead of the frames themselves has a physical meaning in MWI imaging: An out-of-focus height map provides a “blurred version” of a correct height distribution. Subtraction between reconstructed frames is then just a height difference calculation. The majority points in the two height maps should be similar. After subtraction, they should thereby be around zero. Only those points along the edges of features remain obviously different from zero in a height difference map.

Step 3: Eliminate measurement uncertainty

In the last step, although the majority points are of similar heights in the two frames, their difference may not be exactly zero. This is because of measurement uncertainty introduced by either digitalization error or microscopic surface fluctuation. Although they may take up a considerable percentage of the field of view, such non-zero values do not reflect feature sharpness. They must be eliminated in order not to be mistakenly amplified by the following feature enhancement step. Threshold filtering provides a possible solution: any value between ± ε should be set to zero in the resultant height difference map $\bar{\bar{h}_d i f f} (x, y) {d, d - l δ d}$ .

Step 4: Feature enhancement

After step 3, the remaining non-zero values in height difference maps stand for feature sharpness variation. However, from an image processing point of view, their distribution in gray level (the height scanning range) needs to be further adjusted by a histogram regularization step. The purposes of this step are: 1) enhance weak features / small height differences and 2) reshape the histogram, to make it suitable for a selected sharpness calculation.

In this work, feature enhancement of a height difference map after step 3 is accomplished via a revised histogram equalization process: First, pick out all non-zero values; next, do histogram equalization form these values; then calculate the mean of the equalized values; finally, set all zero points in the original height difference map to be this mean value. The feature-enhanced height difference maps are noted as ${\bar{\bar{h}_d i f f}}_{e q} (x, y) {d, d - l δ d}$

The above described process is developed especially for variance calculation: on one hand, the majority zero valued points contain no feature information, thus should be designed to have no contribution to the variance; on the other hand, variance based indicator would be most efficient when the histogram is reshaped closer to a Gaussian distribution. If another sharpness calculation is to be adopted in step 5, the process of feature enhancement may be adjusted accordingly.

Step 5: Implement differential indicator $η {d}$

The differential indicator described below is based on variance calculation. The major difference is, instead of the maximum of variance, the symmetry around the focus is emphasized: at each position d, two height difference maps, namely, ${\bar{\bar{h}_d i f f}}_{e q} (x, y) {d, d - l δ d}$ and ${\bar{\bar{h}_d i f f}}_{e q} (x, y) {d + l δ d, d}$ are considered. If their variances, $V a r_d i f f_{e q} {d, d - l δ d}$ and $V a r_d i f f_{e q} {d + l δ d, d}$ , are equal, d will be recognized as the location of best focus plane. The regularized indicator to estimate how close a selected plane is to the best focus is thereby:

η {d} = \frac{V a r_d i f f_{e q} {d + l δ d, d} - V a r_d i f f_{e q} {d, d - l δ d}}{V a r_d i f f_{e q} {d, d - l δ d} + V a r_d i f f_{e q} {d + l δ d, d}}

Figure 4 illustrates the idea of

η {d}

. Symmetry detection implements a further subtraction upon height difference maps, which can be viewed as a kind of differential operation to a maximum finding curve. By combining information from three frames:

h (x, y) {d - l δ d}

,

h (x, y) {d}

and

h (x, y) {d + l δ d}

, each point on

η {d}

curve contains more information than the existing indicators, and should therefore be more robust: a sharper indication of best focus with less “false alarms” would be expected. As an extra benefit,

η {d}

directly tells the relative location of the best focus by its sign. This would be especially helpful when either the best focus position or the capture position is unknown.

Fig. 4 Sequential height maps and focus detection curves.

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The proposed focus detection criterion is summarized in flow chart in Fig. 5 .

Fig. 5 Flow chart of the differential focus detection criterion.

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4. Experimental results

Parameter selection

Based on experimental environment, the following parameters in the proposed criterion should be decided in advance:

● δd reconstruction sequence interval

Depend on the nominal depth of view of the imaging system. The depth of view for our system is approximately ± 25mm. Thus δd is selected to be 25mm.

● ± Lδd sequence range

Depend on the maximum possible measurement range. Selected to be ± 750mm in our experiment.

● Type of impulse filter in step 1

Depend on the density of salt-and-pepper noise. In our experiment, the simplest 3-by-3 median filtering is proved sufficient.

● lδd the difference interval between two frames in step 2

Depend on how sensitive the features are towards refocusing. Selected to be 6 × 25mm in our experiment.

● ± ε measurement uncertainty threshold in step 3

Should be related to measurement repeatability. Since difference operation in step 2 amplifies uncertainty by a factor of 2, empirical value of 2ε is usually around 3–6 times of repeatability standard deviation. In our experiment, 5-time repeatability of the majority points is around 2μm. Hence the measurement uncertainty threshold is selected to be ± 5μm.

For comparison purpose, performances of a few other indicators are also displayed:Ent: entropy indicator [1]

E n t = - \sum_{x} \sum_{y} {p (x, y) \times \log [p (x, y)]}

Var: variance indicator [2]

V a r = - \sum_{x} \sum_{y} {[p (x, y) - p_{m e a n}]}^{2}

Spec: spectral indicator [3]

S p e c = \sum_{u} \sum_{v} \log {1 + | [F_{x, y}^{+ 1} p (x, y)] (u, v) |}

F_{x, y}^{+ 1}

: two dimensional Fourier transform CC: correlation coefficient indicator [4]

C C = \frac{\sum_{x} \sum_{y} [p_{1} (x, y) - p_{1}_{m e a n}] [p_{2} (x, y) - p_{2}_{m e a n}]}{\sqrt{{\sum_{x} \sum_{y} {[p_{1} (x, y) - p_{1}_{m e a n}]}^{2}} {\sum_{x} \sum_{y} {[p_{2} (x, y) - p_{2}_{m e a n}]}^{2}}}}

p (x, y)

stands for the height map to be processed, and

p_{m e a n}

stands for its mean value. For variance, spectral and correlation coefficient indicators,

p (x, y)

are median filtered height maps, i.e.,

\bar{h} (x, y) {d}

; For entropy, since available features differences are too weak,

\bar{h} (x, y) {d}

is pre-processed by a direct histogram equalization operation.

In correlation coefficient indicator, the special interval between $p_{1} (x, y)$ and $p_{2} (x, y)$ is selected to be 6 × 25mm, in accordance with lδd for the developed indicator.

Results

Experimental result of the 200mm out-of-focused penny in Fig. 3 is shown. Its height scanning range is from −1mm to 3mm, with a 1μm step size. Features over the penny (typically around 100μm in height) are almost invisible in the original height map $h (x, y) {d}$ in Fig. 6(a) ); on the contrary, salt-and-pepper noises are much more evident. After median filtering, noises are removed in Fig. 6(b)). Difference height map $\bar{h}_d i f f (x, y) {0 m m, - 150 m m}$ is shown in Fig. 7(a) ), and its corresponding histogram is demonstrated in Fig. 7(b)). It can be observed that over 91% of the points are within ± 5μm. In spite of their large percentage, these points are irrelevant towards feature sharpness. Histogram equalization is directly implemented on $\bar{h}_d i f f (x, y) {0 m m, - 150 m m}$ , and the resultant gray image and corresponding histogram are shown in Fig. 7(c)) and Fig. 7(d)). The tiny difference with measurement uncertainty is amplified because of their percentage superiority, while the true feature sharpness information are still crowded in two narrow bands. Figure 7(e)) and Fig. 7(f)) display feature enhanced image ${\bar{\bar{h}_d i f f}}_{e q} (x, y) {0 m m, - 150 m m}$ and its corresponding histogram resulting from step 3 and step 4. It is clear that the grey bars other than zeros are widely spread in the whole range, indicating a more effective feature sharpness enhancement operation.

Fig. 6 height map before and after median filtering.

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Fig. 7 Feature enhancement: (a) $\bar{h}_d i f f (x, y) {0 m m, - 150 m m}$ ; (b) histogram of (a); (c) histogram equalization without uncertainty elimination; (d) histogram of (c); (e) ${\bar{\bar{h}_d i f f}}_{e q} (x, y) {0 m m, - 150 m m}$ feature enhancement after uncertainty elimination; (f) histogram of (e).

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The differential curve $η {d}$ is shown in Fig. 8 . Its zero-passing is at 210.3 mm. Comparing with system’s depth of view, this result is of good accuracy.

Fig. 8 Focus detection curve $η {d}$ for the out-of-focus penny in Fig. 3.

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Figure 9(a) –9(d) demonstrate results by entropy (Ent) indicator, variance (Var) indicator, spectral (Spec) indicator and correlation coefficient (CC) indicator. In Fig. 9(a)), entropy curve does not tell the correct best focus position in this case; Although with limited accuracy, variance curve and spectral curve do show some kind of local maxima in Fig. 9(b) and 9(c). Nevertheless, local maxima are insufficient for automatic focus detection purpose; The CC indicator successfully overcomes the problem of local maxima and finds the correct best focus position. However, it displays a wide top range from approximately 125mm to 300mm in Fig. 9(d)). Such flat top phenomenon indicates that the result is likely to be corrupted by residual noise, and the method is in general less accurate.

Fig. 9 Result of other indicators: (a) entropy indicator; (b) variance indicator; (c) spectral indicator; (d) correlation coefficient indicator.

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Two more examples are provided in Fig. 10 and Fig. 11 . Figure 10 demonstrates a result of “concave” features by measuring a key 200mm out of focus. Different from the penny example, main features over the key (the characters) are below their background. Surface quality of the key is also worse, as more salt-and-pepper noise points can be observed. The detected best focus position is 210.1mm from its capturing origin.

Fig. 10 result of imaging a key 200mm out of focus.

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Fig. 11 result of “coherix” image. For clearance, in (d) the refocused height map is displayed within −60μm to + 35μm range after 3-by-3 median filtering.

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Figure 11 shows a possible solution of “weak feature” recognition. An artificial part with characters “coherix” has been captured 300mm out of focus. The characters are approximately 1mm above their background. The detected refocusing distance is 311.5mm. Over the machined background, there are tooling marks typically of 10μm to 30μm in height. These tooling marks are barely above the uncertainty filtering threshold and containing very limited refocusing information. Tooling marks are of observation interest in machining process analysis. Due to measurement time or structure limitations, some of the machined surfaces have to be observed out-of-focus. Details of tooling marks can therefore be covered by blurring. Since they contribute little in focus detection, a direct focus detection based on tooling marks can be severely deviated. The refocusing result shown in Fig. 11(d)) demonstrates a solution, which relates weak features (tooling marks) to strong features (characters), in order to achieve satisfactory observation quality for the former.

5. Conclusion

By implementing multi-wavelength interferometry, holographic focus detection criterion can be based on phase contrast distribution. A specific criterion has been developed in this article with experimental results on step like height structures proving its feasibility. The developed technology can be especially helpful in practical applications such as surface quality inspection in the auto industry, for it provides a possibility of recognizing micron level tooling marks over surfaces far out of focus. Future studies involve questions such as how the criterion can be improved for a “boundary” condition, with surface features between gentle and step like. Another question of interest is how much the selected parameters in section 4 depend on a specific system and the certain object investigated. Furthermore, studies on how to reduce phase error further in a practical system would also be of remarkable value.

Acknowledgments

The authors are greatly thankful to Prof. Jeffrey A. Fessler and Dr. Carl C. Aleksoff for the fruitful discussions and helpful comments on the manuscript. The presented work builds on the foundation established from NSF as well as NIST funded research projects, including NSF grant (#9453491) and a NIST ATP project: “High definition metrology and processes: 2µm manufacturing.”

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Focus detection criterion for refocusing in multi-wavelength digital holography

Abstract

1. Introduction

2. Multi-wavelength interferometric (MWI) refocusing