Optics and Lasers in Engineering

Use of digital color holography for crack investigation in electronic components

P. Tankam ^a , P. Picart ^a,b,

ⁿ

a

LAUM CNRS, Universite´ du Maine, Av. O. Messiaen, 72085 LE MANS, France

b

ENSIM-E ´cole Nationale Supe´rieure d’Inge´nieurs du Mans, rue Aristote, 72085 LE MANS, France

a r t i c l e i n f o

Article history:

Received 4 April 2011 Received in revised form 27 May 2011

Accepted 27 May 2011 Available online 29 June 2011 Keywords:

Digital holography Digital color holography Deformation measurement Image processing Fringe processing

a b s t r a c t

This paper presents the experimental optical analysis of the crack inside an electronic component. The optical setup is used to carry out multidimensional deformation measurements using digital color holography and the spatial multiplexing of holograms. Since the Fresnel transform method depends on wavelength, a wavelength-dependent-zero-padding algorithm is described and results in a rigorous sizing of each reconstructed monochrome image. The criterion to optimize the parameters is presented and is based on minimizing the widening of the impulse response of the full recording/reconstruction process. The application of the proposed method is illustrated through the analysis of the mechanical deformation of the electronic component, and offers keys to understand its failure mode in industrial conditions.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Holographic interferometry is a very convenient tool to mea- sure full-field surface deformations of naturally rough objects [1,2].

With the availability of high resolution CCD sensors, digital holography was demonstrated in 1994 by Schnars and J ¨uptner [3].

After recording the hologram with suitable conditions [3,5], the numerical reconstruction of the object can be obtained, with a direct discrete Fresnel transform [4,7] or a convolution with FFT [6,7].

Phase shifting techniques can also be implemented and their main advantage is to be free from the limitation imposed by the presence of the zero-order and the conjugate image [8]. Digital holographic interferometry is highly suitable to investigate the mechanical properties of some mechanical components or assem- bly. In order to do this, the phases of two reconstructed wave fields, which were recorded at different loadings of the object, are computed from the complex amplitudes. The displacement field is obtained from the difference between the two optical phases [1,2]. Furthermore, digital holography offers interesting opportunities to get a multidimensional measurement of the displacement field. In order to carry out multidimensional defor- mation measurements, several methods were proposed in the past. In particular, Pedrini et al. [9] described schemes based on

speckle interferometry, in which the recording of the speckle field was obtained with a lens that images the object onto the CCD area. The setup used a single-laser line, whose coherence was broken with a delay line. Multidimensional deformation measure- ment is carried out, thanks to a multidirectional spatial carrier [9,10]. Demonstration was performed with simultaneous two-[9]

and three-[10] dimensional measurements. Two- or three-non- coplanar sensitivity vectors are necessary. In this technique, the evaluation of the phase change, due to a displacement of the object, requires computing a fast Fourier transform first; then, the appropriate part of the Fourier spectrum is filtered; finally the inverse Fourier transform leads to the complex object field.

The spatial frequencies of the spatial carriers need not to be precisely known since they are automatically suppressed when calculating the phase change by a subtraction. In his experiment, the recording of the multiplexed speckle fields used the incoherent superimposition of different holograms and this was obtained by a delay line greater than the coherence length of the laser. In the same way, the adaptation of these methods to digital Fresnel holography was discussed through the spatial multiplexing of monochromatic digital holograms [11]. The application of the concept was given for composite material investigations and 2D-vibration analysis [12–13]. Note that in these works, the coherence of the laser was broken through cross-polarized beams.

Since the use of a single-laser line for spatial multiplexing leads to severe constraints on the optical setup (delay line or non-depolarizing object), the idea of using multicolor digital holography was finally proposed [14–18]. The demonstration Contents lists available at ScienceDirect

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Optics and Lasers in Engineering

0143-8166/$ - see front matter

&

2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.optlaseng.2011.05.018

n

Corresponding author at: ENSIM-E´cole Nationale Supe´rieure d’Inge´nieurs du Mans, rue Aristote, 72085 LE MANS, France.

E-mail address: pascal.picart@univ-lemans.fr (P. Picart).

concerned two-wavelength surface shape measurement [14], two-dimensional deformation measurement [15,17] and three- dimensional deformation measurement [18]. Recording colors is a critical point and several possibilities are available: using a monochrome sensor and sequential recordings [19], spatial multiplexing of colors with a monochrome sensor [14,20], a Bayer mosaic [21], or a stacked color image sensor [15–18]. Lately, various applications have been developed, in particular in contact less metrology: flows analysis in fluid mechanics [16,19], 3D-surface profiling [14,22], phase imaging in digital holographic microscopy [23,24]. Many approaches to record and reconstruct color holograms were discussed [25–29]. In this paper, we report the use of the spatial multiplexing of two-color digital holograms in order to get a simultaneous two-dimensional deformation measurement. The paper is organized as follows:

Section 2 details the main possibilities to record color holograms;

Section 3 explains the experimental setup based on a two-color Mach–Zehnder interferometer with spatial multiplexing of digital holograms; Section 4 discusses the reconstruction method;

Section 5 presents experimental results and Section 6 draws the conclusions of the paper.

2. Digital color recording

The simultaneous recording of digital color holograms can be achieved according to different strategies (cf Fig. 1). First possi- bility is based on the use of chromatic spatial filters organized as a Bayer mosaic [21]. However, in a typical Bayer mosaic sensor, half of the pixels detect green and only one-quarter detect red or blue.

These phenomena waste light and create gaps in the color data, producing a loss of resolution in digital holography. For example, Yamaguchi used a 1636 1238 pixel matrix with pixels sized 3.9 3.9 m m

²

, and the paper showed experimental results with a relatively low spatial resolution. Since the effective pixel number along each wavelength was 818 619, he obtained an effective pixel pitch of 7.8 m m. The second method uses sensors organized as ‘‘triCCD’’ with three matrices of pixels and three-color filters.

Such a sensor guarantees a high spatial resolution and a spectral sensitivity compatible with the constraint of three-color digital

holographic recording and imaging. Desse developed a three- color holographic interferometer devoted to fluid mechanics, with three chips (1344 1024 pixels each), sized 6.45 m m 6.45 m m (ORCA-3CCD Hamamatsu camera). He achieved the field mea- surement of the gas density around a cylinder placed in a wind tunnel with wake flow at Mach 0.45 [30]. A third opportunity consists in using a color sensor based on a stack of photodiodes (http://www.foveon.com) [15]. In such a sensor, the spectral selectivity is related to the average penetration depth of photons in silicon: that of blue photons at 425 nm is about 0.2 m m, that of green photons at 532 nm about 2 m m and that of red photons at 630 nm about 3 m m. So, building junctions at depths around 0.2, 0.8 and 3.0 m m provides workable spectral separation for color imaging. Nevertheless, the spectral selectivity is not perfect because red and green photons can be detected in the blue band, blue and red photons in the green band and green photons in the red band, the probability of finding blue photons in the red band being small. However, such an architecture guarantees a max- imum spatial resolution since the number of effective pixels for each wavelength corresponds to that of the full sensor. Recently, authors have first evidenced that digital color holograms could be recorded/reconstructed, thanks to a sensor based on a strategy using a stack of photodiodes with 1060 1414 pixels sized 5 5 m m

²

[15–18]. A final possibility is the monochrome sensor associated to a spatial multiplexing of holograms. Thus, each reference wave along each color has suitable spatial frequencies that are adjusted separately. The complexity of the optical setup increases with the number of colors: for two-color holography, it is acceptable but with three colors, it becomes prohibitive.

A demonstration of this approach with two- and three-color holograms was given in Refs. [14,20,24]. Two-color digital holo- graphy based on the spatial multiplexing scheme is finally a good compromise between the complexity and the potentialities of the setup.

3. Experimental setup

As indicated in the previous section, the two-color holographic method is based on the spatial multiplexing of digital holograms.

The setup is applied to the investigation of the mechanical causes of the crack of the capacitance belonging to a PCB card. The latter belongs to the electronic monitoring of a sensor related to the engine of the car. Fig. 2 shows the PCB card, whose size is compared to that of a two Euro coin.

Some preliminary investigations revealed that the crack of the component occurs during the wedging of the PCB card inside its electronic box. The card is force pushed inside the box through the loading point indicated in Fig. 2. One can also observe the inspected zone, whose diameter is 15 mm.

Fig. 1. Methods for digital color recording: (a) Bayer mosaic, (b) three sensors

with prism, (c) stack of photodiodes and (d) spatial multiplexing. Fig. 2. PCB component under interest.

The study of the causes of this abnormality is possible, since it was proved that digital holography is well adapted to the contact less measurement of micro-deformations [1,2]. The method adopted for this study consists in estimating the normal and tangential displacement fields of the PCB card behavior during the embedding operation. Thus, a device that reproduces the mechanical behavior of the electronic card during this stage was developed. Embedding is mechanically reproduced, thanks to a progressive loading of the card from the anchor point, thus faithfully replicating the real industrial situation. An identical strength is applied to the component by the loading device described in Fig. 3. The PCB is fixed on its left and right edges and is forced by one screw bound to a nut and fixed behind. Its deformation during the clamping operation is quite similar to flexion. Fig. 3 also shows the two red and green illumination beams with two different directions in order to produce the double sensitivity of the setup [11,15].

The experimental setup is described in Fig. 4. It is based on off- axis holography with the spatial multiplexing of two holograms, as in [11,13].

The recording of holograms is performed by a monochrome sensor CCD (Pixel Fly, 12 bits with N M¼1360 1204 pixels with pitches

p

x

¼p

y

¼4.65 m m). We used a HeNe red laser ( l

R

¼632.8 nm) and a DPSS green laser ( l

G

¼532 nm). Each is separated into a reference and an object beam; both beams are co-polarized with l /2 plates, thus making the interferences possible. The reference waves of the red and green lines are smooth and flat; they are the spatial carriers of their respective holograms. Since the sensor is monochrome, their simul- taneous recording is possible only according to the spatial multi- plexing scheme; thus the spatial frequencies of the two reference waves are independently adjusted in order to create the spectral separation of the holograms [11]. Nevertheless, these spatial frequen- cies must fulfill the Shannon theorem [5]. In the setup, fu

^G_r

; v

^G_r

g ¼ f67:7; 61:1gmm

¹

and fu

^R_r

; v

^R_r

g ¼ f61:2; 65:6gmm

¹

, respectively, for green and red holograms. The distance from object to sensor is d

0

¼908 mm and the illuminating directions are sym- metric in the (x,z) plane ( y

R

¼ y

G

¼ y ).

4. Reconstruction method

It is based on the discrete Fresnel transform, which was exhaustively described by Schnars and J ¨uptner [3] and Kreis et al. [4]. At any distance d

r

, for any wavelength l , the recon- structed field is given by

A

^l_r

ðx,y,d

r

Þ ¼ jexpð2j p d

r

= l Þ l d

r

exp j p

l d

r

ðx

²

þy

²

Þ

^k¼

X

^K=21

k¼ K=2 l¼

X

L=21

l¼ L=2

Hðlp

x

,kp

y

Þexp j p l d

r

ðl

²

p

²_x

þk

²

p

²_y

Þ

exp 2j p l d

r

ðlp

x

xþkp

y

yÞ

ð1Þ where H(lp

x

,kp

y

) is the sampled version of the intensity distribution of the hologram. Note that this expression is a 2D-Fourier trans- form in which the sampling step depends on wavelength. Thus, this results in reconstructed fields with wavelength-dependent sizes. The pitches in the image plane are given by D Z

^l

¼ l d

^l_r

=L

l

p

x

and Dx

^l

¼ l d

^l_r

=K

l

p

y

, with (L

l

,K

l

) being the number of data points used in Eq. (1), [4]. Note that these sampling pitches depend on the Fig. 3. Schematic of the loading device.

Fig. 4. Experimental setup.

reconstruction distance d

^l_r

, the horizon size (L

l

p

x

K

l

p

y

) and the light wavelength. This characteristic constitutes the major draw- back when using the Fresnel transform of Eq. (1) to reconstruct color holograms. Indeed, in the case of multidimensional metrol- ogy, superimposing optical phase differences is necessary in order to measure deformations. To do so, the images at each wavelength must have the same size. Fig. 5 illustrates the reconstruction of the two multiplexed holograms at distance d

^R_r

¼ d

^G_r

¼ d

0

and a horizon with (K

l

L

l

)¼(1024 1024) points. The result is com- puted, thanks to the discrete Fresnel transform.

Fig. 5 exhibits the size difference between the red and green reconstructed images. However, the multi-wavelength holograms must respect this fundamental property: the size of the recon- structed object must be unchanged and the sampling step must be independent of the wavelength. The next section discusses a possible way to overcome this size problem.

5. Principle of the reconstruction method 5.1. Wavelength ratio and rounded value

In this study, we will consider the two lasers mentioned above.

The perfect superimposition of the two reconstructed images must fulfill these conditions:

D Z

^R

¼

^l_L^RdRr

Rpx

¼

^l_L^GdGr

Gpx

¼ D Z

^G

Dx

^R

¼

^l_K^RdRr

Rpy

¼

^l_K^GdGr

Gpy

¼ Dx

^G

8 >

<

> : ð2Þ

In order to simplify the problem, let us consider only the x axis (the y direction is easily obtained by replacing L

l

with K

l

and p

x

with p

_y

) the previous condition becomes l

R

d

^R_r

L

R

¼ l

G

d

^G_r

L

G

ð3Þ where L

R

and L

G

are the number of data points of the recon- structed horizon with discrete Fresnel transform. We assume that the recorded holograms may be padded with zeros. In Eq. (3), we envisage that the reconstruction distance may be different for the red (d

rR

) and green objects (d

rG

): it should lead to a virtual image being in focus for d

^l_r

d

0

, since the recording reference waves are flat [17]. Furthermore, we impose that L

R

and L

G

are even integers. This choice is justified by the zero-padding procedure, which consists in adding the zeros on both sides of the recorded hologram (whose initial size is N by M pixels). An extended

horizon is thus obtained (whose size is L by K). This procedure can be seen as an interpolation technique in the Fourier domain. So, it is necessary to add L/2N/2 zeros (respectively, K/2M/2) on each side of the initial matrix in the x direction (respectively in the y direction). Since N and M are always even (for any standard CCD), L and K are consequently also even, and leads to an integer number of zeros on each side. There are several possibilities to satisfy Eq. (3). The first consists in sequentially recording the holograms with different distances for each wavelength. They are chosen so as to fulfill both Eq. (3) and the best focus of the virtual images (d

^l_r

¼ d

^l₀

) [25–27]. However, such an approach requires moving the object between two illuminations without changing the carrier spatial frequencies of the reference waves. Although the recording distance can be optically modified, the use of optics may introduce some aberrations in the object wave front. Besides, this method is adapted for static objects and is not suitable for dynamic ones, which require a simultaneous recording of colors.

The algorithm presented in this paper is meant as a variant of Ferraro’s [25].

Let us suppose that the recording distance d

₀

is equal to the reconstruction distances: 9 d

^l_r

9 ¼ 9 d

0

9 ; then Eq. (3) leads to L

R

L

G

¼ l

R

l

G

¼ 632:8

532 ¼ 1:189473 ð4Þ

Note that this ratio does not lead to integer values for (L

R

, L

G

).

It can be rewritten by considering a third decimal approximation:

1000 L

R

L

G

¼ integer ð5Þ

with integer¼1189 (rounded off by default) or integer¼1190 (rounded off by excess). Table 1 presents the different possible choices for parameters L

R

and L

G

, according to the condition that L

R

and L

G

are even integers. The corresponding values of D Z

^R

and D Z

^G

with 10

^–2

precision are given, for an object placed at d

0

¼908 mm from the sensor. The bold numbers correspond to the even-integer values.

Table 1 shows that, for the chosen wavelengths, the values of the parameters L

G

and L

R

that fulfill the even-integer conditions can be easily deduced by the following relations:

fL

R

,L

G

g ¼ f2000þ2000k, 2378þ2378kg for integer ¼ 1189 f1000þ200k, 1190þ238kg for integer ¼ 1190 (

ð6Þ with k an integer number that can be freely chosen according to the performances of the processor. Note that according to Table 1,

Fig. 5. Fresnel reconstruction of the two-color holograms: (a) red one in focus and (b) green one in focus.

this assessment does not lead to a rigorous equality D Z

^R

¼ D Z

^G

. To do so, we must consider that the reconstruction distance is slightly different from that of the recording and depends on wavelength. According to the 10

²

approximation of the wave- length ratio, the perfect equality D Z

^R

¼ D Z

^G

can be obtained by changing the reconstruction distance with a fraction of milli- meters with no loss of image quality. Thus, equality D Z

^R

¼ D Z

^G

will be satisfied according to Eq. (3). In order to optimally focus both red and green images simultaneously, d

^R_r

and d

^G_r

are chosen so that d

₀

is their mean value, resulting in

d

^R_r

þd

^G_r

¼ 2d

0

ð7Þ

Now distance d

^R_r

is chosen to be proportional to d

^G_r

so that d

^R_r

¼ L

R

L

G

l

G

l

R

d

^G_r

¼ integer 1000

l

G

l

R

d

^G_r

ð8Þ

The combination of Eqs. (7) and (8) leads to determine d

^R_r

and d

^G_r

with the following equations:

d

^R_r

¼

₁₀₀₀²_l^lGinteger

RþlGinteger

d

0

d

^G_r

¼ 2d

0

d

^R_r

8 <

: ð9Þ

Table 2 shows the possible values for the parameters d

^R_r

, d

^G_r

, L

R

and L

G

obtained by Eqs. (5)–(9), as well as the corresponding values of D Z

^R

and D Z

^G

. The bold numbers also correspond to the even-integer conditions for L

R

and L

G

.

Thus, the same values of L

G

and L

R

, with a slight change in the reconstruction distance, lead to a perfect equality D Z

^R

¼ D Z

^G

at a given precision (here 10

^–2

). In a symmetric way, we obtain l

G

l

R

¼ 0:840707 ð10Þ

The same approximation can be adopted, resulting in 1000 L

G

L

R

¼ integer ð11Þ

with integer ¼840 or integer¼841.

As previously discussed, distances d

^R_r

and d

^G_r

can be computed according to the following equations:

d

^G_r

¼

₁₀₀₀²_l^lRinteger

Gþl_Rinteger

d

0

d

^R_r

¼ 2d

0

d

^G_r

8 <

: ð12Þ

The values of L

G

and L

R

can also be deduced from Eq. (13):

fL

R

,L

G

g ¼ f2000þ50k, 1680þ42kg for integer ¼ 840 f2000þ2000k, 1682þ1682kg for integer ¼ 841

ð13Þ with k an integer number that can be freely chosen.

5.2. Effect of the change in the reconstruction distance

The change in the reconstruction distance introduces a defocus in the image. It is caused by the extension of the impulse response. This point was discussed in Ref. [5]. In order to simplify, let us consider the x direction only. It was shown that the impulse response r

^l_x

of the digital holographic process is affected by the change in the reconstruction distance (i.e. d

r

a d

0

) according to d r

^l_x

¼ Np

x

1 d

^l_r

d

0

ð14Þ

So, the optimal choice for the different parameters, L

R

, L

G

, d

^G_r

, d

^R_r

, and integer ¼{840,841,1189,1190} should lead to the minimum widening of the impulse response. Therefore, Table 1

Reconstruction parameters with a fixed reconstruction distance d

0G

¼d0R

¼d0¼

908 mm and p

x¼py¼4.65

m m.

L

G

D Z

^G

( m m) Integer¼1189 Integer¼1190

L

R¼1.189

K

G

D Z

^R

( m m) L

R¼1.190

K

G

D Z

^R

( m m)

1000 103.88 1189 103.92 1190 103.84

1200 86.57 1426.8 86.60 1428 86.53

1400 74.20 1664.6 74.23 1666 74.17

1600 64.93 1902.4 64.95 1904 64.90

1800 57.71 2140.2 57.74 2142 57.69

2000 51.94 2378 51.96 2380 51.92

2200 47.22 2615.8 47.24 2618 47.20

2400 43.28 2853.6 43.30 2856 43.27

2600 39.96 3091.4 39.97 3094 39.94

2800 37.10 3329.2 37.12 3332 37.08

3000 34.63 3567 34.64 3570 34.61

3200 32.46 3804.8 32.48 3808 32.45

3400 30.55 4042.6 30.57 4046 30.54

3600 28.86 4280.4 28.87 4284 28.84

3800 27.34 4518.2 27.35 4522 27.33

4000 25.97 4756 25.98 4760 25.96

Table 2

Reconstruction parameters with modified reconstruction distance, with d

0¼908 mm and

p

x¼py¼4.65

m m.

LG

Integer¼1189 Integer¼1190

d

rR

¼907.82 mm

d

rG

¼908.18 mm

d

rR

¼908.20 mm

d

rG

¼907.79 mm

LR¼1.189KG

D g

^R

( l m) D g

^G

( l m)

LR¼1.190KG

D g

^R

( l m) D g

^G

( l m)

1000 1189 103.90 103.90 1190 103.86 103.86

1200 1426.8 86.59 86.59 1428 86.55 86.55

1400 1664.6 74.22 74.22 1666 74.19 74.19

1600 1902.4 64.94 64.94 1904 64.91 64.91

1800 2140.2 57.72 57.72 2142 57.70 57.70

2000 2378 51.95 51.95 2380 51.93 51.93

2200 2615.8 47.23 47.23 2618 47.21 47.21

2400 2853.6 43.29 43.29 2856 43.27 43.28

2600 3091.4 39.96 39.96 3094 39.95 39.95

2800 3329.2 37.11 37.11 3332 37.09 37.09

3000 3567 34.63 34.63 3570 34.62 34.62

3200 3804.8 32.47 32.47 3808 32.46 32.46

3400 4042.6 30.56 30.56 4046 30.55 30.55

3600 4280.4 28.86 28.86 4284 28.85 28.85

3800 4518.2 27.34 27.34 4522 27.33 27.33

4000 4756 25.98 25.98 4760 25.96 25.97

contributions can be compared to the intrinsic spatial resolutions given by

r

^R_x

¼

^l_Np^Rd0

x

¼ 90:85 m m

r

^G_x

¼

^l_Np^Gd0

x

¼ 76:38 m m 8 <

: ð15Þ

with N¼1360 and p

x

¼4.65 m m. Table 3 synthesizes the set of parameters obtained for the different values of the integer.

Note that the change in the reconstruction distance contri- butes to widening by no more than 2% the impulse response; this is irrelevant compared to the intrinsic resolution of the sensor.

Furthermore, the case integer¼841 leads to the lowest degrada- tion of the image quality. Thus, this comparison is a key to choose the optimal configuration for the hologram reconstructions.

Indeed, the farther the reconstructing distance is from that of the recording, the wider the impulse response is, and the more the image quality is degraded.

6. Experimental results

To illustrate, the two red and green holograms of Fig. 5 were reconstructed with the following parameters: integer¼841, {K

R

, K

G

}¼{2000,1682}, {L

R

,L

G

}¼{K

R

,K

G

}, d

^R_r

¼ 907:84 mm and d

^G_r

¼ 908:16 mm. Fig. 6 shows the reconstructed red and green holo- grams. Since they have the same size, the two-color hologram is obtained, thanks to their superimposition.

One can observe the capacitance of interest at the top of the images and the loading nut in the center of the images, with the shadows produced by the illumination beams. The method is interesting as it generates a faithful color image, but also measures multidimensional deformations, by adding or subtract- ing the phase map differences extracted from the complex amplitudes of the multicolor holograms [15]. In order to analyze the capacitance crack in the PCB card, 90 two-color holograms were recorded at different loadings. The phase map differences between two successive loadings were then calculated and are illustrated in Fig. 7 for the red and green beams.

The 2 p phase jumps need to be unwrapped. A binary mask is applied to the useless parts of the map (mainly holes and nuts) to get a successful unwrapping process. Then, the phase differences are cumulated to obtain the total one given by the full loading.

Therefore, the out-of-plane and in-plane displacements are calculated according to the set of following equations [15,17]:

u

x

¼

^ðlRDj_2p^R_sinð^lG_y^Dj_Þ ^GÞ

u

z

¼

^ðlR₂^Dj_pð1^R_þcosð^þlGDj_yÞÞ^GÞ

8 <

: ð16Þ

with y

R

¼ y

G

¼ y ¼45 1 . Table 3

Effect of the distance change in the impulse response.

Integer 1189 1190 840 841

d

^Rr

(mm) 907.82 908.20 908.38 907.84

d

^G_r

(mm) 908.18 907.80 907.62 908.16

d r

^R_x

( m m) 1.260 1.398 2.663 1.098

d r

^G_x

( m m) 1.259 1.399 2.665 1.098

d r

^R_x=

r

^R_x

(%) 1.39 1.83 2.93 1.21

d r

^G_x=

r

^G_x

(%) 1.65 1.83 3.49 1.44

Fig. 6. Reconstructed holograms with adapted zero-padding in Fresnel transform: (a) R one, (b) G one and (c) two-color image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Phase map differences between two successive loadings: (a) R phase map difference and (b) G phase map difference.

Fig. 8 shows the in-plane and out-of-plane displacement fields of the deformation of the PCB card.

The out-of-plane result shows that the capacitance is located in the high curvature, i.e. where flexion is maximal and thus undergoes a maximum strain. This could be a first reason for the crack. The global curvatures of the maps were eliminated to analyze the deformation more thoroughly. Fig. 9 shows the displacement maps after removing the curvatures, where (c) is the zoom in the capacitance region of (a) and (d) are zooms of (b).

The zoomed regions contain the capacitance and show that it undergoes a strong non-uniform deformation. This shows a complex stress state, including shear and axial stress in particular, and leading to the failure of the capacitance.

7. Conclusion

In conclusion, this paper presents the use of color holography for simultaneous multidimensional analysis. A numerical method to reconstruct color images is presented. The experimental results are obtained with a two-color interferometer generating simulta- neous recording of two-color images, without Bayer or stacks of

photodiode sensors, but using the spatial multiplexing principle.

The correct image correspondence between reconstructed color images can be achieved by minimizing the widening of the impulse response of the digital reconstruction. With the proposed strategy, the pixel pitch is invariant along each wavelength. The optimal parameters (number of data points, reconstruction dis- tance) must be judiciously chosen to obtain the equality of the pixel pitches. As a result, the reconstructed field size does not depend on the wavelength, although the perfect superimposition is obtained through low modification of the reconstruction dis- tance. This change leads to widening of the impulse response, thus degrading the image quality. Fortunately, this contribution was found negligible, less than 2%, compared to the intrinsic resolution of the sensor. The principle can be extended to three- color holography by analyzing the correct choice of parameters according to the criterion proposed in the paper. The approach is based on an adaptation of zero-padding-dependent Fresnel trans- form. The method was applied to the investigation of the crack cause detection of the capacitance on an electronic PCB card. We showed that the default capacitance was located in a strong curvature region and that it was submitted to non-uniform deformation, causing complex stress situation, and thus failure.

Fig. 9. Displacement fields without curvatures: (a) in-plane displacement, (b) out-of-plane displacement, (c) zoom of (a) and (d) zoom of (b).

Fig. 8. 3D displacement fields of the PCB card: (a) in-plane and (b) out-of-plane .

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