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Digital holography was experimentally established in the 90’s27−29. Recently, many fascinating possibilities have been demonstrated: the focus can be chosen freely30, a single hologram can provide amplitude-contrast and phase-contrast microscopic imaging31, image aberrations can be compensated32, properties of materials can be investigated33, and digital color holography28, 34 and time averaging are also possible35. Digital holography can be realized through various architectures such as Fresnel holography, Fourier holography, lens-less Fourier holography, and image-plane holography36, 37. Although these architectures present differences in their implementation, they exhibit the same spatial frequency bandwidth when arranged according to an off-axis approach. In this study, we analyzed a case in which off-axis digital image-plane holograms were recorded. This choice is the most suitable for LBM diagnostics. Indeed, recording a single hologram per instant constitutes a powerful tool for studying dynamic events and carrying out high-speed acquisition. In case of image-plane holography, an imaging system is associated with a variable aperture to produce an image of the scene of interest as close as possible to the sensor area (object projected onto the sensor plane). Note that a slight defocus could be also of interest. At the sensor plane, the reference wave
$ \mathcal{R} $ is mixed with the wave from the imaging system$ \mathcal{A'} $ to produce the digital hologram, expressed as$$ \begin{array}{*{20}{l}} \mathcal{H} = |\mathcal{R}|^2 + |\mathcal{A'}|^2 + \mathcal{R}^*\mathcal{A'}+ \mathcal{R}\mathcal{A'}^* \end{array} $$ (1) The term
$ \mathcal{R}^*\mathcal{A'} $ usually refers to the$ +1 $ order of the hologram, whereas$ \mathcal{R}\mathcal{A'}^* $ refers to the$ -1 $ order. In the off-axis configuration, the reference wave is inclined and impacts the sensor at a certain incidence angle such that its complex-valued amplitude can be written as follows ($ a_R $ is considered a constant and$ (u_0,v_0) $ denotes spatial frequencies):$$ \begin{array}{*{20}{l}} \mathcal{R}(x,y) = a_R \exp(2 i \pi (u_0x+v_0 y)) \end{array} $$ (2) The recovery of the complex-valued amplitude of the image of the object is obtained by filtering the
$ +1 $ order in the Fourier spectrum of the hologram. Filtering can thus be written as follows ($ FT $ means Fourier transform):$$ \begin{array}{*{20}{l}} {A'}_{r}\simeq \mathcal{R}^*\mathcal{A'} = \text{FT}^{-1} \left[\text{FT}\left[\mathcal{H} \right] \times G \right] \end{array} $$ (3) Eq. 3 is a convolution formula, and the transfer function
$ G $ is given by the bandwidth-limited angular-spectrum transfer function in the Fresnel approximation:$$ \begin{array}{*{20}{l}} G(u,v) = \left \{ \begin{aligned} \exp(-i\pi d_r \lambda ((u-u_0)^2+(v-v_0)^2))\\\; \text{if}\; ((u-u_0)^2+(v-v_0)^2) \leq R_u^2 \\ 0\; \; \text{if not} \end{aligned} \right. \end{array} $$ (4) In Eq. 4,
$ \lambda $ is the wavelength of light,$ R_u $ is the cut-off spatial frequency of the imaging system, and$ d_r $ is the refocus distance when the projected image is not perfectly focused at the sensor plane. Note that if the image is perfectly in-focus, then$ d_r = 0 $ and the transfer function$ G $ is simply a binary filter centered at spatial frequency$ (u_0,v_0) $ with bandwidth related to$ R_u $ . The main parameter of interest in the extracted$ +1 $ order is the optical phase,$ \psi $ , which can be used for metrology purposes, such as surface height measurements. -
Two-wavelength digital holography has many advantages over single-wavelength holography. With a unique wavelength, the measured surface height is ambiguous when it is larger than the wavelength. Given that the surface of the object is also rough, the surface shape cannot be reconstructed because the phase is extracted from the filtering results from a speckle pattern. Thus, the ambiguity and randomness of the phase can be mitigated by using a second wavelength, leading to the so-called synthetic wavelength. Consequently, the ambiguity is bypassed and the measurement range is increased from microns to millimeters (or larger). With two wavelengths, namely
$ \lambda_1 $ and$ \lambda_2 $ , the synthetic wavelength is given by$ \Lambda = \lambda_1 \lambda_2/|\lambda_1 -\lambda_2| $ 38. The surface height of the object,$ h(x,y) $ , is calculated using the following equation (here, the illumination and observation of the surface are at normal incidence):$$ \begin{array}{*{20}{l}} h(x,y) = \dfrac{\Lambda}{4\pi}[\psi_2(x,y)-\psi_1(x,y)] = \dfrac{\Lambda}{4\pi}\Delta \psi \end{array} $$ (5) where (
$ \psi_1 $ ,$ \psi_2 $ ) denotes the optical phases extracted from the two holograms at the two wavelengths.To obtain real-time measurements of the surface height at the melt-pool area, the simultaneous recording of the two phases (
$ \psi_1,\psi_2 $ ) is required ($ f_{ex} $ and$ f_{ey} $ refer to the spatial sampling frequencies of the sensor, i.e.,$ f_{ex} = 1/p_x $ and$ f_{ey} = 1/p_y $ , where$ p_x $ and$ p_y $ are the sampling pitches of the sensor). This can be achieved by spatial multiplexing of the two holograms at the two wavelengths, so that a single hologram is recorded for both wavelengths. At the sensor plane, the hologram at$ \lambda_2 $ is recorded simultaneously with that at$ \lambda_1 $ , and the spatial frequencies of its reference wave are adjusted so that its$ +1 $ order is localized in the Fourier spectrum, separately from the$ +1 $ order at$ \lambda_1 $ . Fig. 1 illustrates the spectral distribution in the Fourier spectrum of spatially multiplexed digital holograms at the two wavelengths. The basic arrangement for producing spatially multiplexed digital holograms is shown in Fig. 2. The two reference beams are inclined with different incidence angles to produce the spatial frequencies ($ u_1,v_1 $ ) for$ \lambda_1 $ and ($ u_2,v_2 $ ) for$ \lambda_2 $ .Fig. 1 Spectral distribution of the multiplexed off-axis digital holograms in the Fourier domain at two wavelengths.
Fig. 2 Imaging of the object area through the image-plane holographic system with spatial multiplexing of the two holograms at two wavelengths.
From a computational point of view, the phase difference expressed in Eq. 5 is obtained by calculating
$ \Delta \psi = \arg[C_{12}] = \arg[{A'}_{r2}{A'}_{r1}^*] $ , where$ {A'}_{r1} $ and$ {A'}_{r2} $ are the two image fields from the two wavelengths. According to Eq. (3), each of the recovered complex-valued$ A'_{r1-2} $ image includes a bias due to each spatial carrier frequency. More specifically, this bias is due to the presence of the term$ \mathcal{R}^* $ in the output of the Fourier filtering, which must be compensated. This point is discussed in the next subsection. -
The compensation of the bias in the phase difference requires knowing the spatial frequencies of the two wavelengths, i.e., (
$ u_1,v_1 $ ) and ($ u_2,v_2 $ ). These couples can be estimated with spectral analysis when evaluating the centroids of the two$ +1 $ orders in the hologram spectrum (refer to Fig. 1). This yields approximate values that can be used to start the compensation procedure. The method proposed in this paper is to partially compensate for the two carrier frequencies and then refine their residue by using a power spectrum density (PSD) analysis. Thus, partial compensation retains a portion of the frequencies in the complex data$ C_{12} $ . The ratio of the spatial frequencies is denoted by$ \alpha $ (typ.$ \alpha = 0.5 $ ). The partially compensated complex-valued data$ C_p $ can be calculated using the following expression:$$ \begin{split} C_p = & {A'}_{r2}\exp(2 i \pi \alpha (u_2x+v_2 y)) \times\\&{A'}_{r1}^*\exp(-2 i \pi \alpha (u_1x+v_1 y)) \\ & \propto \mathcal{A'}_{2}\mathcal{A'}_{1}^*\exp(2 i \pi (\Delta u x+\Delta v y)) \end{split} $$ (6) PSD analysis is carried out with data
$ C_p $ by independently calculating the average PSD along each line and column. Then, the peak in each PSD is detected, and its associated spatial frequency is estimated. For accurate peak estimation, zero-padding of the lines and columns is applied up to$ 2^{16} $ data points. These two estimations over lines and columns yield$ (\Delta u,\Delta v) $ . Finally, the residue is compensated according to the expression$ C = C_p \exp(-2 i \pi (\Delta u x+\Delta v y)) $ , and the bias-free phase difference is obtained from$ \Delta \psi = \arg[C] = \arg[\mathcal{A'}_{2}\mathcal{A'}_{1}^*] $ . This compensation is systematically applied to digital holograms. The next section elaborates on the experimental setup. -
To qualitatively appraise the quality of the topographic measurement in real conditions, two parallel aluminum alloy tracks realized with the LBM machine were considered. These are remarkably steady experimental conditions because the manufactured object was static, and its structure was stable after cooling. The hologram was acquired with an exposure time
$ \tau $ = 145 µs, and the optical magnification was$ G_t = 2 $ . From the spatially multiplexed digital hologram, the complex-valued amplitudes for both wavelengths were extracted, and the phase difference was obtained. With the moduli of the complex amplitudes, the autocorrelation of the speckle field was performed to estimate the spatial resolution in the image plane. Fig. 4a shows the autocorrelation function of the speckle field for wavelength$ \lambda_1 $ , and Fig. 4b shows its profile along the horizontal line. Note that the autocorrelation function has a width of$ 0.017 $ mm, thus confirming that the spatial resolution in the object plane is approximately 15-17µm, which is in good agreement with the theoretical estimation.Fig. 4 a Autocorrelation function of the speckle field at wavelength
$ \lambda_1 $ ; b horizontal profile of a; c raw Doppler phase map extracted from the phases at$ \lambda_1 $ and$ \lambda_2 $ ; d noise map from c; e probability density functions of the phase noise from d (blue line: experiments, red line: theory).Owing to the surface roughness at the object plane, the phases and the Doppler phase (difference between the two phases
$ \psi_1 $ and$ \psi_2 $ ) extracted from the holograms became corrupted by speckle decorrelation noise. This phase noise can be estimated using a noise estimator42. Fig. 4c shows the raw phase map for the aluminum tracks, and Fig. 4d depicts the noise map estimated from the raw Doppler phase. The phase noise is in the range$ [-\pi,+\pi] $ and exhibits non-Gaussian statistics. Fig. 4e shows the probability density functions (PDFs) calculated from the noise map in Fig. 4d. By fitting the experimental PDF with L2-norm minimization, the value of$ |\boldsymbol{\mu}| $ can be estimated from the theoretical equation by using the following relation43−48 (with$ \epsilon $ denoting the phase noise induced by the decorrelation noise and$ \beta = |\boldsymbol{\mu}| \cos (\epsilon) $ ):$$ \begin{array}{*{20}{l}} p(\epsilon) = \dfrac{1 - |\boldsymbol{\mu}|^2}{2 \pi} \left(1-\beta^2\right)^{-3/2} \left( \beta \sin^{-1}\beta + \dfrac{\pi \beta}{2} + \sqrt{1 - \beta^2}\right) \end{array} $$ (7) The coherence factor is a quality metric that indicates the reliability of the phase estimation. Vry et al. 49 considered that the standard deviation
$ \sigma_{\epsilon} \leq \pi/4 $ is the criterion required for high-quality measurement of the Doppler phase, which leads to the condition$ |\boldsymbol{\mu}| \ge 0.85 $ . For other authors50,$ |\boldsymbol{\mu}| $ should be higher than 0.37, which is more tolerant to noise but requires a powerful filtering algorithm. In Fig. 4d, the noise standard deviation was found to be$ \sigma_{\epsilon} \simeq 1.01 $ rad$ = \pi/3.11 $ , whereas the modulus of the coherence factor was estimated to be$ |\boldsymbol{\mu}| \simeq 0.79 $ . Thus, with$ |\boldsymbol{\mu}| \simeq 0.79 $ , the measurement can be considered to be of correct quality for the experimental parameters.Fig. 5 shows the experimental results concerning the topography of the two aluminum tracks obtained with the holographic setup. Fig. 5a displays a photograph of the top-view of the aluminum tracks, and Fig. 5b, c show the topography of the tracks. In Fig. 5d, the region of the tracks represented by the dashed black line in Fig. 5c corresponding to the dashed white line in Fig. 5a, b, is zoomed.
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When carrying out measurements during the additive manufacturing process, the effect of motion blur on the image quality has to be investigated. This effect is observed along the length of the tracks, which is parallel to the displacement of the motorized stage. If
$ V_0 $ is the velocity of the scan at the melt pool and$ B $ is the blur effect expressed as a fraction of the spatial resolution, then we have the following expression:$$ \begin{array}{*{20}{l}} B = \dfrac{V_0 \tau}{\rho_x} \end{array} $$ (8) Motion blur is problematic if
$ B $ is close to or larger than one during exposure time$ \tau $ . Considering an exposure time$ \tau $ = 25 µs and a spatial resolution of 15 µm, then for$ B = 0.2 $ , the maximum allowed scan velocity is 150 mm·s−1. For higher velocities, the exposure time must be reduced according to Eq. 8. For$ V_0 $ = 1 m·s−1,$ \tau $ should not exceed 3 µs. In this case, the monitoring laser power must be sufficient to reach an adequate level of the signal-to-noise ratio of the measurement. To qualitatively check the quality of the measurement for various velocities of the scan, different velocities were set: 100 mm·s−1, 250 mm·s−1, and 380 mm·s−1. For$ V_0 = 380 $ mm·s−1 and$ \tau $ = 25 µs, we obtain$ B = 0.63 $ ; and for$ V_0 = 250 $ mm·s−1 and$ \tau $ = 25 µs, we obtain$ B = 0.41 $ . These values of$ B $ were slightly beyond the limit estimated using Eq. 8. However, for$ V_0 = $ $ 100 $ mm·s−1 and$ \tau $ = 25 µs, we have$ B = 0.16 $ , which is reasonable. When processing the digital holograms for the three velocities, the values of$ |\boldsymbol{\mu}| $ were systematically estimated to check the overall quality of the holographic measurements in terms of phase noise. For each experiment, the values of$ |\boldsymbol{\mu}| $ remained higher than 0.85, which means that for this set of velocities, the noise quality criterion was respected. However, this criterion did not prevent the measurement from presenting other artifacts such as motion blur. Fig. 6 shows the manufactured aluminum tracks at the beginning, middle, and end of the process for the three selected velocities. Figs. 6a−c show, for$ V_0 = 100 $ mm·s−1, the topography of the tracks at 4.25 ms (just after the beginning of the recording), at 27.75 ms (middle of the time sequence), and at 61 ms (end of the sequence). Figs. 6d−f depict the topography of the tracks for$ V_0 = 250 $ mm·s−1 at 1.5 ms (just after the beginning of the recording), at 14.5 ms (middle of the time sequence), and at the end at 24.25 ms (end of the sequence). Figs. 6g−i show the topography of the tracks for$ V_0 = 380 $ mm·s−1 at 1.5 ms (just after the beginning of the recording), at 7 ms (middle of the time sequence), and at 16.25 ms (end of the sequence).Fig. 6 Topography of dynamic aluminum tracks. a Topography for
$ V_0 = \; 100 $ mm·s−1 at 4.25 ms after the beginning of the recording, b at 27.75 ms, c at 61 ms; d topography for$ V_0 = \; 250 $ mm·s−1 at 1.5 ms after the beginning of the recording, e at 14.5 ms, f at 24.25 ms; g topography for$ V_0 = \; 380 $ mm·s−1 at 1.5 ms after the beginning of the recording, h at 7 ms, i at 16.25 ms. Movies: visualization of dynamic aluminum tracks for$ V_0 = \; 100 $ mm·s−1,$ V_0 = \; 250 $ mm·s−1, and$ V_0 = \; 380 $ mm·s−1. Movie 1: visualization of the dynamic tracks for$ V_0 = \; 100 $ mm·s−1; Movie 2: dynamic tracks for$ V_0 = \; 250 $ mm·s−1; Movie 3: dynamic tracks for$ V_0 = \; 380 $ mm·s−1. -
Experiments were carried out in open air, and an argon flow from a nozzle was sprayed above the fusion area to reduce the oxidation of the substrate and powder. This means that the melt pool generated during the experiments tended to be unstable. Given that the goal of this study was to realize a proof of concept, it was interesting to evaluate the ability of the holographic system to detect such instabilities. To ensure uniform velocity during fusion, the laser processing was initialized after the start of the motorized stage to avoid any acquisitions during the acceleration phase.
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A sheet of 316L plate was used. The exposure time was set to
$ \tau = 6.25 $ µs, and the frame rate was 4000 fps. This allowed freezing the dynamic of the melt pool and avoided motion blur in the holograms. The optical magnification was set at$ G_t = 4.29 $ , the spatial resolution at the melt pool area was approximately 7.5 µm, obtaining$ B = 0.08 $ . Therefore, blur motion was irrelevant. The processing laser had a power of 75 W for a spot diameter between 100 and 200 µm and a velocity of$ V_0 = 100\; $ mm·s−1. Under such conditions, a keyhole regime may be observed.Fig. 7 shows two 3D plots of the topography of the melt pool obtained at two different instants during the laser fire. Fig. 7a displays the melt pool at 87.25 ms, whereas Fig. 7b depicts the melt pool at 88 ms. Fig. 8 shows bottom views corresponding to 3D plots in Fig. 7a, b.
Fig. 7 Monitoring of melt pool on 316L substrate. a 3D plot of melt pool with keyhole at 87.25 ms after the beginning of the recording; b 3D plot of melt pool with keyhole at 88 ms after the beginning of the recording. Laser scanning is along the
$ x $ direction.Fig. 8 Keyhole in melt pool on 316L substrate. a Bottom view of melt pool with keyhole at 88 ms after the beginning of the recording; b bottom view of the melt pool with a keyhole at 88 ms after the beginning of the recording; c illustration of the Rosenthal keyhole regime (adapted from Ref. 51). Movie 4: visualization of the keyhole regime.
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In this case, a 316L sheet was used with powder 316L. This powder had a thickness of
$ \simeq100\; \mu $ m and absorbed the laser energy. The melt pool was composed of the fusion of the powder and the fusion of the superficial layer of the substrate. The exposure time was set to$ \tau = 25 $ µs, and the frame rate was 4000 fps. The increase in the exposure time was due to the non-cooperative surface of the 316L powder, which weakly reflects photons. It follows that the hottest area where the keyhole might appear could not be observed owing to the dynamics of the melt pool. The processing laser had a power of 150 W for a spot diameter between 100 and 200 µm and a velocity of$ V_0 = 100\; $ mm·s−1. The optical magnification was set to$ G_t = 4.29 $ . Thus, the observed area was approximately 400 × 400 µm2. These experimental parameters led to$ B = 0.33 $ , which means that the recording was not fully immune to motion blur; this was imposed because of the limited power of the two lasers of the holographic setup.Fig. 9 shows six pictures of the topography of the melt pool obtained at different instants during the laser melting. The tail and body formation of the melt pool was defined as shown in Fig. 9g.
Fig. 9 Topography of melt pool on 316L substrate with 316L powder, a−f several 3D plots at different instants during laser melting, g illustration of melt pool in keyhole mode (adapted from Ref. 41). Movie 5: visualization of the melt pool on 316L substrate with powder
Melt pool monitoring in laser beam melting with two-wavelength holographic imaging
- Light: Advanced Manufacturing 3, Article number: (2022)
- Received: 03 September 2021
- Revised: 19 January 2022
- Accepted: 20 January 2022 Published online: 02 March 2022
doi: https://doi.org/10.37188/lam.2022.011
Abstract: Over the past two decades, laser beam melting has emerged as the leading metal additive manufacturing process for producing small- and medium-size structures. However, a key obstacle for the application of this technique in industry is the lack of reliability and qualification mainly because of melt pool instabilities during the laser-powder interaction, which degrade the quality of the manufactured components. In this paper, we propose multi-wavelength digital holography as a proof of concept for in situ real-time investigation of the melt pool morphology. A two-wavelength digital holographic setup was co-axially implemented in a laser beam melting facility. The solidified aluminum tracks and melt pools during the manufacturing of 316L were obtained with full-field one-shot acquisitions at short exposure times and various scanning velocities. The evaluation of the complex coherence factor of digital holograms allowed the quality assessment of the phase reconstruction. The motion blur was analyzed by scanning the dynamic melt pool.
Research Summary
Laser manufacturing process: monitoring of melt pool with digital holography
Metal additive manufacturing processes with lasers requires new in-situ monitoring methods. An optical system based on two-wavelength digital holography is co-axially implemented in a laser beam melting facility. Pascal Picart from Le Mans University in France and colleagues now report development and implementation of their digital holographic system based on a high-speed camera and the recording of spatially multiplexed two-color holograms. The holographic method provides recordings at both temporal and spatial scales of the laser melting process. The team provides the proof of concept with obtaining quantitative data from solidified aluminum tracks and melt pools during the manufacturing of 316L, whereas acquisitions are obtained with full-field one-shot acquisitions at short exposure times and various laser scanning velocities.
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