Reconstruction of Digital Color Holograms and ... - OSA Publishing

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Reconstruction of Digital Color Holograms And Application to Full Field Metrology Pascal Picart1,2, Denis Mounier3, Eudes-Evrard Bobboh-Ebo2, Jean-Michel Desse4 1

Laboratoire d’Acoustique de l’Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France 2 Ecole Nationale Supérieure d’Ingénieurs du Mans, rue Aristote, 72085 Le Mans Cedex 9, France 3 Laboratoire de Physique de l’État Condensé, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France 4 Office National d’Etudes et de Recherches Aérospatiales, 5, Boulevard Paul Painlevé, 59045, Lille, France [email protected], [email protected], [email protected]

Abstract: This paper focuses on new opportunities given by high spatial resolution multiwavelength digital holographic metrology. The method and its application to simultaneous deformation measurement of an object submitted to mechanical loading are described. ©2007 Optical Society of America

OCIS codes: (090.0090) Holography; (090.2880) Holographic interferometry; (120.3180) Interferometry; (120.0120) Instrumentation, measurement, and metrology

1. Introduction Recently, I. Yamaguchi et al demonstrated phase shifting digital color holography using a multi-wavelength HeCd laser and a color CCD [1]. Digital holograms were recorded with a 1636×1238 pixels matrix each of size 3.9×3.9μm2 at three wavelengths (636nm, 537.8nm and 441.6nm). The sensor was equipped with a Bayer mosaic as chromatic filter allowing the spectral selection in the CCD plane. The authors demonstrated the possibility for the reconstruction of color images by choosing an appropriate weighting for each RGB component. Experimental results have a relatively low spatial resolution since the effective pixel number along each wavelength was 818×619, leading to an effective pixel pitch of 7.8μm. In 2003, Demoli et al presented the first study on fluids using digital color Fourier holography [2]. They used a monochrome 8 bits CCD sensor with 1008×1018 pixels with size 9×9μm2 and three wavelength issued from a krypton laser (647nm), a 2ω NdYAG laser (532nm) and an argon laser (476nm). The proposed strategy consisted in a sequential recording at each wavelength using the subtraction method. These results can only be obtained with very slow phenomena because of the sequential recording. Note that these works do not use the optical phase along each wavelength. This paper proposes an alternative strategy by using a stacked color pixel strategy and its demonstration concerns two-wavelength simultaneous multidimensional metrology. 2. Reconstruction of color holograms At distance dR between an object plane and a recording plane, the diffracted field is related to the object field by a convolution relation between the initial field U(x,y,z) and the convolution kernel, according to Eq. 1 [3]: U (X ,Y , z + d R ) =

jd R

λ

exp ⎡⎢2 jπ / λ d R2 + ( X − x )2 + (Y − y )2 ⎤⎥ ⎣ ⎦ dxdy U ( x, y , z ) 2 2 2 d R + ( X − x ) + (Y − y ) −∞

+∞ +∞

∫ ∫

−∞

(1)

in which the convolution kernel is the impulse response of free space propagation which will be noted h(x, y, d R ) in what follow. In the case of digital holographic metrology, one needs to compute phase differences between computed optical phase. With the computation of Eq. 1 as a convolution formula, the pixel pitch in the reconstructed plane remains invariant and equal to that of the detector, whereas it is not the case when computing a discrete Fresnel transform [4]. This strategy is the more appropriate for digital color holography. In terms of spatial frequency bandwidth, it is necessary that the bandwidth of the kernel covers at least the full bandwidth of the object. If the object bandwidth is greater than that of the kernel, the numerical reconstruction must be implemented with a scanning of the spatial spectrum. In this case, one has to compute a filter banc whose trade is to scan the useful object bandwidth in order to allow the reconstruction of the object. The number of scanning is related to the bandwidth to the kernel. If Δuobject is the bandwidth of the object and Δukernel that of the kernel, in the x direction, then the number of scans is given by Eq. 2: nx =

Δuobject Δu kernel

⎛ ΔA = ⎜⎜ ⎝ λ dR

⎞ ⎟⎟ ⎠

⎛ Np x ⎜⎜ ⎝ λ dR

⎞ ΔA ⎟⎟ = ⎠ Np x

(2)


where ΔA is the object size and Npx is the detector size. A similar relation holds for the vertical direction y with number ny. So, if the real object is greater than the detector area, the reconstructed object will be obtained by a juxtaposition of adjacent parcels. To reconstruct a region of the object centered at spatial coordinate {xi,yi} with spatial extension of (Νpx×Μpy), one has to compute the spatial frequency associated to this zone, ie {ui,vi} = {xi/λdR,yi/λdR}. The spectral filter must then be centered at frequency {ui,vi} in the hologram spectrum. The centering of the filter is performed by modulating the convolution kernel of Eq. 1, according to hi (x, y, d R ) = h(x, y, d R ) × exp[− 2 jπ (ui x + vi y )] . Thus, the transfer function associated to this convolution kernel is ~ ~ H i (u , v, d R ) = H (u − ui , v − vi , d R ) . The spatial frequencies of the filter banc are given in x and y directions by ⎧

Mp y ⎫ Np x , v0 + k y ⎬ with k x ∈ {− (n x − 1) / 2,+(n x − 1) / 2} and k y ∈ − n y − 1 / 2,+ n y − 1 / 2 . λ dR λ dR ⎭ ⎩ This method allows a perfect pixel to pixel superposition between reconstructed objects with different wavelengths.

{ui , vi } = ⎨u0 + k x

{(

)

(

) }

3. Experimental set-up and results The setup is described in Fig. 1 in the case of two wavelength digital holography. The set-up is composed of a twin wavelength Mach-Zehnder interferometer and a three channel (RGB) color camera with a single 8-bit per channel digital output containing (M×N) = (1060×1420) pixels of size px = py = 5μm in the form of three stacked layers of photodiodes. Unlike color filter mosaic sensors, the image sensor detects full color at every pixel location. In a typical Bayer mosaic sensor, one half of the pixels detect green and only one-quarter detect red or blue. This wastes light and creates gaps in the color data, producing a loss of resolution in digital holography [1]. The three layers of photodetectors provide every pixel location with three stacked photodetectors so that every pixel location detects full color. So, the resolution of the digital hologram is optimized because it is that of the full sensor. In the setup, two wavelengths are issued from a 2ω NdYAG laser (λG = 532nm) and a HeNe laser (λR = 632.8nm). The object symmetrically illuminated by the two beams is an aluminum washer of 25mm diameter placed at 750mm in front of the CCD area. Off-line holographic recording is realized using the two spatial filters (SF1 and SF2) in which each collimating lens is displaced out of the afocal axis by means of two micrometric transducers (not represented in figure 1) [4]. This allows the production of two reference waves with appropriate spatial frequencies such that there is no overlapping and no mismatch of the Shannon theorem along each wavelength. In the set-up θG = θR = θ.

Fig. 1. Recording of two wavelength digital color holograms

Since the object is 25mm in diameter, this leads to {nx,ny} = {4,5} and a full reconstructed object of size (5300×5680) pixels along each channel. In order to reduce the memory allocated to the reconstructed object was under sampled to (1325×1420) pixels by keeping 1 pixel over 4. Fig. 2 shows the green and red objects reconstructed with the filter banc. Note that the reconstructed size is the same for the two wavelength and the speckle in the image is bigger in the red channel than in the green channel, as can be expected. When applying a loading along the x direction, the object is deformed [4]. With the recording of color holograms before and after the loading, the phase map of the deformation can be computed along the red channel, ie ΔϕR, and along the green channel, ie ΔϕG. Then, the displacement field along the x and z direction can be computed according to Ux = (λRΔϕR−λGΔϕG)/2πsin(θ) and Uz = (λRΔϕR+λGΔϕG)/2π(1+cos(θ)).


Computation with the filter banc allows extracting the red and green phases, whose particularity is that they have the same pixel pitches, allowing the in plane and out of plane displacement fields to be evaluated. Fig. 3 illustrates the obtained results.

Fig. 2. Green (left) and Red (right) objects reconstructed with the filter banc

Fig. 3. Out of plane displacement (left) and in plane displacement (right) issued from the two wavelength phase maps

4. Conclusion Potentialities of digital color holographic metrology using a strategy based on a three stacked layers of photodiodes has been demonstrated. A multi-wavelength algorithm based on a filter banc was described. It allows invariance of the pixel pitches in the reconstructed planes and a pixel to pixel correspondence between each wavelength. Extension of this methodology will give new opportunities for multidimensional metrology in the field of mechanics, vibrations or fluids. 5. References [1] I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. 27, 1108-1110 (2002). [2] N. Demoli, D. Vukicevic and M. Torzynski, ‘‘Dynamic digital holographic interferometry with three wavelengths,’’ Opt. Express 11, 767-774 (2003). [3] J.W Goodman, Introduction to Fourier Optics (Mc Graw Hill, second edition, New York, 1996). [4] P. Picart, B. Diouf, E. Lolive and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169-1176 (2004).