Single Shot Dual-Wavelength Digital Holography: Calibration Based on Speckle Displacements

Abstract

We present a calibration method which allows single shot dual wavelength online shape measurement in a disturbed environment. Effects of uncontrolled carrier frequency filtering are discussed as well. We demonstrate that phase maps and speckle displacements can be recovered free of chromatic aberrations. To our knowledge, this is the first time that a single shot dual wavelength calibration is reported by defining a criteria to make the spatial filtering automatic avoiding the problems of manual methods. The procedure is shown to give shape accuracy of 35 µm with negligible systematic errors using a synthetic wavelength of 1.1 mm.

Keywords:

• calibration
• correlation
• digital holography
• dual-wavelength
• on-line shape measurement
• single-shot
• speckle displacement

NOMENCLATURE

CAD	=	Computer-Aided Design
2-D	=	2 Dimension
FFT	=	Fast Fourier Transform
IFFT	=	Inverse Fast Fourier Transform
CCD	=	Charge-Coupled Device
NA	=	Numerical Aperture
ROI	=	Region of Interest

1. INTRODUCTION

In the manufacturing industry there is a high demand for on line quality control to minimize the risk of incorrectly produced objects. Interferometry and digital holography in combination with computers become faster, more reliable and highly accurate as a well-established technique for industrial shape measurement.^[ 1-5 ^] For example^{[ 6 , 7 ]} has recently presented a method where a single shot shape evaluation has been used for real-time evaluation of free-form objects in comparison with their prescribed CAD-model. Digital holography allows defining numerical optics to perform optical processing of the recorded data.^[ 8-11 ^] Conventional single wavelength profiler techniques can only handle smooth profiles with step heights of less than half a wavelength. Therefore, multi wavelength interferometry has been introduced. On the other hand, use of several wavelengths in the same optical apparatus can lead to the occurrence of severe chromatic aberrations.^{[ 12 ]} For the measurement of the shape of optically rough surfaces, dual wavelength digital holography, produces a longer synthetic wavelength^{[ 13 ]} for retrieving the phase without ambiguity over an extended range. However, disturbances in the form of phase aberration will result in erroneous results and need to be corrected. One possible solution is mechanical adjustment of the set-up or using adaptive focus optics to compensate the focal shift of the image plane. However, the intrinsic flexibility of digital holography gives a possibility to compensate these aberrations and remove errors, fully numerically without mechanical movements.

The spatial frequency content in a hologram, depends on the wavelength and angular content of object and reference light, respectively. Then different spatial frequency filters are needed to filter out the corresponding holograms in the Fourier plane.^{[ 14 ]} Furthermore, if the optics are not totally achromatic, the reconstruction distance for each hologram will change and different image shift and magnification could occur in the reconstruction.^{[ 15 ]} Valuable solutions of spatial frequency filtering for multi wavelength measurements have been discussed in.^{[ 8 , 16 ]} By calculating the scale of spatial frequency in different wavelengths with respect to each other, the filter size is controlled to extract the holograms. Furthermore, they compensate the phase aberration by using polynomial fitting on flat areas or by using a reference hologram. However, these methods fail when a single shot measurement is used for online quality control purposes as the frequency components of each wavelength are embedded in different places of the image spectrum.^[ 17-19 ^] Different laser sources in the single shot measurements, induce extra aberration and pseudo phase changes that can be suppressed by applying the modified flat fielding method.^{[ 9 , 11 , 20 ]} In the flat fielding method they use a single dark frame and a flat frame captured from the system containing dual wavelengths to calibrate the measured data. One possible problem with the flat fielding method is an increase of the noise level because of division with intensity values close to zero. In this article, we use only phase subtraction to prevent magnification of noise. However, this compensation itself can lead to some errors if it is done out of focus.

This article is a continuation of the work presented in.^{[ 6 , 21 ]} In this method the correspondence between speckle displacements and phase gradients is utilized to find the shape without problems with phase unwrapping. However, for this method to work properly it is necessary to have a well collimated set-up. In previous single shot calibrations, spatial frequency filtering is performed manually without any definite criteria.^{[ 11 , 16 ]} Here we present the effect of uncontrolled carrier frequency filtering in the reconstruction process. The objective is to determine a calibration process that allows single shot dual wavelength online shape measurement in a disturbed environment. We demonstrate that a single shot dual wavelength phase map and speckle displacements can be recovered free of chromatic aberrations. A flowchart of the proposed method is shown in Figure 1. The method consists of a loop that includes three main steps. The loop starts with initial guesses of masks in the spectral domain to extract the holograms. Secondly, curvature of the fields is corrected and the fields are then numerically propagated to the focus plane. Thirdly, the phase map is calculated by suppressing the aberration terms. The loop iterates until it finds the best estimation of the mask.

Figure 1 Flowchart of the proposed algorithm for calibrated single shot dual wavelength digital holography. IFFT and ΔA represent inverse Fourier transform and average speckle displacements, respectively.

In the following, we describe the experimental set-up and the calculation procedure to do the calibration The procedure is demonstrated by a test object including a step height. In the comparison a CAD-model which describes the ideal shape of the measured object is utilized.

2. EXPERIMENTAL SET-UP

The single shot dual wavelength experimental set-up is shown in Figure 2 left. The light source is a dual core diode pumped fiber laser that produces pulsed light with wavelengths λ₁ = 1030.44 nm and λ₂ = 1031.4 nm, respectively. This gives a synthetic wavelength of 1.1 mm. A flat metal object is used as a reference plane and the test object is a cylinder with a diameter of 48 mm. A trace of depth 3 mm is machined off the cylinder. We are measuring a 18 mm × 18 mm section including the trace and the cylindrical section (see Figure 2 right). The two beams are combined using a mirror and a beam splitter and are adjusted to be collinear. Afterwards, a negative lens is used to diverge the combined collinear beam to illuminate the object. A 12 bit CCD camera (pco. Sensicam) with a resolution of 1280 × 1024 pixels and a pixel size of 6.45 × 6.45 µm is used to record the image. A circular aperture with a diameter of 4 mm is used to regulate the spatial frequency content of the object light passed on to the detector. The reference beams are taken from a beam splitter (10/90). The angle of each of the reference beams to the detector is adjusted by separate mirrors. These angles are adjusted in a way to have whole and separate spatial frequencies of each hologram in the spectrum of the single shot recorded image. Then all necessary information of two holograms are resolved in a single image.

Figure 2 Set-up (left). Test object (Right).

3. CALCULATION PROCEDURE

The principle of dual wavelength single shot off-axis digital hologram acquisition is shown in Figure 3. Define as the object waves and as the reference waves in the detector plane, where i is 1 or 2, representing the two different wavelengths. The recorded image can then be represented by

where it is assumed that A _ri (x, y, λ _i ) = A _ri is a constant, is the phase difference between the object wave U _oi and reference wave U _ri with the wavelength of λ _i . Further, the reference waves are assumed to be point sources in the exit pupil plane and the coherence between the two wavelengths is zero. The first four intensity terms of Equation (1) correspond to the zero order and are slowly varying in space. These terms are independent of the correlation between the object and reference wave. The last four terms are the interference terms, which are sensitive to the phase difference By adjusting the aperture size to control the spatial frequency content of the object wave the different interference terms can be positioned at different parts of the recorded Fourier plane by appropriate angles of the reference waves. Therefore, it is straightforward to isolate and filter out each of the spectrums that corresponds to the different holograms. The dashed box in Figure 1 represents the required steps to recover the different holograms from a recorded single shot image.

Figure 3 The recording of single shot dual wavelength digital holography.

Consider the third and fourth terms in Equation (1). A modified version, of the object waves are retrieved as

is a model of the correction wave that represents the leading parts of the calibration. The term ax + by indicates a shift of the object carrier frequency center in the spectral domain caused by the off-axis reference wave and C(x ² + y ²) is the combined curvature of the reference wave and correction terms to be discussed in Section 3.2. In the following we will further assume the amplitude to be constant and is therefore ignored from now on.

3.1. Carrier Frequency Calibration

The first step of the reconstruction procedure consists of applying a careful spatial frequency filter in the Fourier domain to filter out the corresponding interference terms. To do this, two masks P _i (ν _k , ν _l , r _i ) defining the spatial filtering are introduced where ν _k , ν _l are the spatial frequency coordinates and r _i is the radius of the cut area. These masks should be of sufficient size to cover the carrier frequencies of the lobes given the numerical aperture NA of the imaging system. The radius of the filters is given by

The central position of the filters are determined by the angles of the reference waves and are represented by the term in Equation (3). In general, there is a need to define different masks P _i (ν _k , ν _l , r _i ) to cover and filter all carrier frequencies of the interference term for λ_i. Let us define a mask P _i (ν _k , ν _l , r _i ) for each wavelength,

where k and l are coordinates of the mask center and r _i is the radius of the mask. To obtain a high accuracy in the chromatic aberration compensation and calibration of the speckle displacements, two important requirements should be met. Firstly, the extracted region of interests (ROI's) should be centered in a way that frequency components and their centers match each other. Secondly, the masks should include the whole but only the required frequency components. The first requirement guarantees that systematic errors caused by the off-axis reference wave are cancelled and the second requirement minimize random noise that may lead to higher decorrelation and lower accuracy in the speckle displacement calculation according to the theoretical expression of the random error in the speckle displacement,^{[ 22 ]}

where σ is the average speckle size, N is sub image size in one dimension and γ the speckle correlation value.

To meet those requirements, a technique that sets the speckle displacements between the two reconstructed speckle patterns to zero in the focus plane is utilized.^{[ 21 , 23 , 24 ]} First the focus plane needs to be found. This is done manually by cutting out the frequency components of the holograms and search for the maximum correlation along the optical axis in the reconstructed holograms.^{[ 25 ]} Since the purpose of this step is only to determine the focus plane position there is no need to accurately cut out the lobes. Once the focus plane is found the exact position of the lobe centers should be found. This is the procedure of estimating the term (ax + by) in Equation (3) which in general includes four unknowns, two for each wavelength. However, as only the difference in central positions is of importance one of the centers is fixed and we search for the correct position of the other. For this purpose the correction vector Δ P _c  = (k ₀, l ₀) as a miss-match between the two hologram contents, is introduced (red arrow in Figure 4).

Figure 4 Fourier transform of the recorded image. Arrow in the right image corresponds to a miss-match of Δ P _c when the centered frequency components of λ₁ and λ₂ do not match each other.

According to the shift properties of the Fourier transform, Δ P _c  = (k ₀, l ₀) will introduce phase and spatial shift in the retrieved holograms f(x,y):

where F (ν _k , ν _l ) represents the Fourier transform of the holograms and F ⁻¹ denotes the inverse Fourier transform operator. Consequently, speckle displacement aberrations will be introduced. Figure 5 shows the effect of phase and spatial shifts as the result of unmatched carrier frequencies. Holograms of the flat reference object have been retrieved in the focus plane where it is expected to have zero speckle displacements. In the first row, both carrier frequencies matched each other. Then Δ P _c  = (0, 0) as a difference in the central position of centered frequency content of the two holograms. The standard deviation for the phase along an arbitrary vertical line is 0.09 radians. The second row shows the result when the carrier frequencies are not matched and Δ P _c  = (5, 5) pixels from the correct position. For this case, speckle displacement aberration is introduced and for the same vertical line, the standard deviation of the phase measurement is increased to 0.15 radians.

Figure 5 Effect of carrier frequency matching on the speckle displacement (left) and corresponding measured phase (right) of flat reference plane in the focus plane. Results of matched and unmatched carrier frequencies are shown in top and bottom rows, respectively. It is expected that both the speckle displacements and the phase are zero.

The automatic procedure is as follows: We first start to define a mask P ₁ (ν _k , ν _l , r) manually where subscript 1 refers to the shorter wavelength. The objective is then to estimate P ₂ (ν _k  + k ₀, ν _l  + l ₀, r) where reasonably good starting estimates of (k ₀, l ₀) are obtained manually. We then search for the set of (k ₀, l ₀) that gives the minimum speckle displacements in the focus plane. See Figure 1. Notice, that possible small errors caused by the manual definition of the first mask will be canceled by the loop iteration to find the second mask. Figure 6 shows two different criteria to find the best shift values of k ₀ and l ₀. A typical shift value of k ₀ = 49 pixels and l ₀ = 41 pixels relative to the original guess of the second mask, was calculated for the mask used in our experimental data. Comparison of the correlation value and the speckle displacements, shows that the speckle displacements represent a sharp minimum for proper mask definition while the correlation has a much more flat profile. The speckle displacements hence provide a more liable norm for defining the best mask position.

Figure 6 The correlation values (left) and absolute speckle displacements (center) from a measurement of the flat reference object for different shift values of k ₀ and l ₀. In the case of k ₀ = 49 pixels, a 2-D plot of both correlation and speckle displacements as a function of l ₀ are shown to the right.

3.2. Curvature of the Reference Waves

In many applications, as in this one, there is a need to propagate the wave and refocus the holograms in different planes. Here we follow the procedure described in.^{[ 26 ]} The term C(x ² + y ²) in Equation (3), will act as a positive lens on the object wave, with a focal length f ₁ equal to the distance between the reference wave point source and the detector.^{[ 27 ]} Figure 7 represents the geometry of the reconstruction configuration where z and z _i are distances in object and image space, respectively, while l is the distance between the detector plane and the exit pupil of the imaging system having effective focal length f ₀. Then a numerical lens f ₁ = l − f ₀ is directly determined from the set-up and used to compensate for the curvature of the reference wave as well as to make the refocusing telecentric. In this way a constant magnification is reassured for the refocused holograms and the effect of the term C(x ² + y ²) in Equation (3) is cancelled.

Figure 7 Geometry of the reconstruction configuration.

3.3. Phase Aberration Compensation

Assume that the specimen does not introduce aberration but only a phase delay for each wavelength. For the known flat area, these terms, are constant for any plane, neglecting the diffraction pattern due to defocus. Therefore, assuming that the steps in the previous two sections are properly performed only a small phase aberration will remain. The final phase difference map will simply be calculated by suppressing the aberration term from the reference flat. Figure 1 shows the flowchart of our method. With the calibration properly performed the final phase map will be

where and are the phase maps corresponding to the object phase lag for the wavelengths of λ₁ and λ₂, respectively. In this way, a phase calibration of the set-up is done that allows a compensation of phase chromatic aberration once and for all. Figure 8 represents the use of a flat plane phase difference to compensate the phase chromatic aberration of the measured object. After defining proper masks and doing telecentric propagation to the focus plane, the final object phase map is obtained by subtraction of the flat plane phase from the object phase lag. Arbitrary vertical lines on the same place in the cylindrical part of the test object have been chosen for comparison of the results before and after calibration. The standard deviation for the phase along the vertical line before calibration is 1.81 radians and has improved to 0.38 radians when using the method described.

Figure 8 Phase calibration using Equation (9). The final phase of the measured object is calculated by the subtraction of the reference phase difference from the object phase (top right) in the focus plane. The standard deviation of the measured phase has improved from 1.81 radians to 0.38 radians using the proposed method.

4. SHAPE MEASUREMENT

In order to find the shape deviations between the measured surface and a prescribed CAD-model we are using the method described in.^{[ 6 ]} A rigid body transformation, that is a rotation and a translation, is to be found such that the transformed CAD-model is matched to the shape gradients, obtained from the speckle displacements, and the phase map. This is an iterative method based on a combination of the iterative closest point algorithm and a phase unwrapping algorithm. It results in that the shape gradients and the phase map are aligned to the CAD-model which is in an orientation given from the obtained transformation. The values in the phase map represent wrapped height values of the measured object. Using the phase map and the transformed CAD-model an accurate shape representation of the measured object can be obtained.

Figure 9 shows the results of the shape measurement using the CAD-model, before and after calibration, respectively. In the second and third column the importance of doing the phase calibration (last step of Figure 1) is presented. Before calibration (first column) the phase and speckle displacements include aberration and consequently the standard deviation of the measurement from the CAD-model is too high. In the second column, phase subtraction has been done without propagating the holograms to the focus plane. The phase map is improved but speckle displacements differ from the direction of the fringes, which introduce noise. The highest accuracy of the measurement and highest agreement between speckle displacements and phase map are achieved when the calibration is done after propagating the image to the focus plane. The standard deviation of the measurement without calibration is 200 µm which is improved to 69 µm and 35 µm when the calibration has been done before and after propagating the holograms to the focus plane, respectively. Furthermore, the random errors in speckle correlation calculation for the used data equals to 0.264 pixel which can be the source of errors.

Figure 9 Experimental results using the method described in Bergström et al.^{[ 6 ]} The upper row shows the measured phase difference, the central row the corresponding speckle displacements in the focus plane and the lower row the estimated shape. The left column are results from an uncalibrated system, the central column are results from when the calibration is performed out-of-focus, and the right column are results from using the calibration procedure described in this article.

5. DISCUSSION AND CONCLUSIONS

In this article we have described a method for dual wavelength single shot holography to compensate and calibrate phase aberration and speckle displacements, which can be used in shape measurement applications. Speckle displacements in the focus plane are used to define a mask in the Fourier plane that gives the correct phase compensation between the reconstructed waves. Since the correlation peak search is broad, and in contrast, the speckle displacement is sharp, we rely on localization of the minimum speckle displacements. In general, interference terms recorded with different wavelengths require different mask sizes. For our recorded image where the wavelength difference is less than 1 nm we have kept the mask size constant.

We need to emphasize that, it is of great importance to do the calibration in the focus plane. Even if the image is recorded out of focus, it should be propagated numerically to the focus plane. Firstly, we have accurate expectation of phase and speckle displacements in the focus plane and secondly, in the focus plane, the calibration avoids unknown error sources.

In summary, the shape measurement method that we are using is based on a single-shot dual-wavelength holographic acquisition. In contrast to what is customary the shape estimate is based on speckle displacements as well as phase difference. In that way, problems associated with the phase wrapping is avoided and objects with larger depth can be measured. In addition, if a CAD-model of the object to be measured is used, it is important to have a good correspondence between the speckle displacements and the phases. If we are having a constant drift between the speckle displacements and the phases, then we will measure or estimate the wrong shape as is seen in Figure 5-bottom row and Figure 9-left column. So in our method that first uses speckle displacements and then the phases, it is much important than in the normal cases to have a calibrated system. In this paper we calibrate both phase and speckle displacements for single shot measurements. The standard deviation of the measurement without calibration is 200 µm which is improved to 69 µm and 35 µm when the calibration has been done before and after propagating the holograms to the focus plane, respectively.

ACKNOWLEDGMENTS

The Authors would also like to thank Dr. Henrik Lycksam, Dr. Per Gren and Dr. Erik Olsson for valuable discussions.

Notes

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uopt.