Simultaneous multidimensional deformation measurements using digital holographic moiré Gannavarpu Rajshekhar, Sai Siva Gorthi,† and Pramod Rastogi* Applied Computing and Mechanics Laboratory, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. *Corresponding author: [email protected] Received 19 April 2011; revised 3 June 2011; accepted 5 June 2011; posted 7 June 2011 (Doc. ID 146186); published 19 July 2011

This paper proposes an elegant technique for the simultaneous measurement of in-plane and out-ofplane displacements of a deformed object in digital holographic interferometry. The measurement relies on simultaneously illuminating the object from multiple directions and using a single reference beam to interfere with the scattered object beams on the CCD plane. Numerical reconstruction provides the complex object wave-fields or complex amplitudes corresponding to prior and postdeformation states of the object. These complex amplitudes are used to generate the complex reconstructed interference field whose real part constitutes a moiré interference fringe pattern. Moiré fringes encode information about multiple phases which are extracted by introducing a spatial carrier in one of the object beams and subsequently using a Fourier transform operation. The information about the in-plane and out-of-plane displacements is then ascertained from the estimated multiple phases using sensitivity vectors of the optical configuration. © 2011 Optical Society of America OCIS codes: 090.1995, 120.2880, 120.5050.

1. Introduction

In recent decades, digital holographic interferometry (DHI) [1] has emerged as a popular technique for deformation analysis in the fields of experimental mechanics and nondestructive testing. The major advantages of DHI are noninvasive behavior, wholefield measurement capability, and good measurement resolution. Moreover, the use of a CCD camera as the recording medium removes the need for the complex and time-consuming chemical processing steps associated with classical holography. The digitalization of recording and processing of holograms greatly facilitates the applicability of DHI for deformation measurements in various areas. For many practical applications, simultaneous measurement of multidimensional, i.e., in-plane and out-of-plane, components of displacement of a 0003-6935/11/214189-09$15.00/0 © 2011 Optical Society of America

deformed object is desired for the complete characterization of deformation mechanisms. For such measurements, various techniques have been proposed in DHI in which the optical configuration relies on object illumination from multiple directions. For multidimensional measurements, methods relying on multiple reference-object beam pairs have been proposed [2–7]. However, the use of multiple reference beams might add complexity to the optical configuration. Recently, methods [8–10] based on the phase-shifting technique have been applied to multidimensional measurements. These methods rely on illuminating the object sequentially by multiple beams, i.e., one beam at a time along different directions. Because of the sequential operation and requirement of multiple frames due to the phaseshifting technique, the method is difficult to apply for simultaneous measurement of in-plane and outof-plane components. The concept of holographic moiré [11,12] was developed in classical holographic interferometry for 20 July 2011 / Vol. 50, No. 21 / APPLIED OPTICS

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the simultaneous measurement of in-plane and out-of-plane displacement components. Various techniques have been proposed for the analysis of holographic moiré fringes in classical holographic interferometry [13–23]. In this paper, we explore the concept of holographic moiré in DHI for the simultaneous measurement of in-plane and out-of-plane displacement components. The proposed method uses a single reference beam and multiple object beams in an off-axis DHI configuration to record a hologram. By digital processing of the recorded holograms before and after deformation, moiré fringes are obtained that contain information about multiple interference phases corresponding to the different object beams. Demodulation of the moiré fringe pattern and subsequent use of sensitivity vectors enable multidimensional measurements. The theory of the proposed method is discussed in the next section. The simulation results are shown in Section 3. The schematic and experimental results are presented in Section 4, followed by discussion and conclusions. 2. Theory

To analyze multidirectional illumination of a diffuse object, consider Fig. 1, where the object is illuminated by two light beams. The beams are assumed to be incident symmetrically on the object with respect to the observation direction. The CCD camera is located at a distance d from the object. In the proposed method, a single reference beam is used in conjunction with the diffusely scattered multiple object beams to record the hologram at the CCD plane x0 y0 . Denoting the reference and object waves at the CCD as Rðx0 ; y0 Þ, O1 ðx0 ; y0 Þ, and O2 ðx0 ; y0 Þ, the intensity recorded at the CCD camera is given as I ¼ jR þ O1 þ O2 j2 ¼ I 0 þ RðO
1 þ O
2 Þ þ R
ðO1 þ O2 Þ;

ð1Þ

where I 0 ¼ jRj2 þ jO1 j2 þ jO2 j2 þ O1 O
2 þ O
1 O2 and ‘
’ denotes the complex conjugate. Here the spatial

coordinate ðx0 ; y0 Þ is omitted for the sake of brevity. The complex amplitude of the object wave at the x–y plane is provided by the Fresnel transformation and is given as [24]
Z Z ∞ ∞ j −j2πd Rðx0 ; y0 ÞIðx0 ; y0 Þ Γ0 ðx; yÞ ¼ exp λd λ −∞ −∞ 
−jπ ððx − x0 Þ2 þ ðy − y0 Þ2 Þ dx0 dy0 : ð2Þ × exp λd Using Eq. (1), we have
Z Z ∞ ∞ j −j2πd Γ0 ðx; yÞ ¼ exp ½RI0 þ R2 ðO
1 þ O
2 Þ λd λ −∞ −∞ þ jRj2 ðO1 þ O2 Þ 
−jπ ððx − x0 Þ2 þ ðy − y0 Þ2 Þ dx0 dy0 : × exp λd

ð3Þ

For an off-axis configuration, the contributions arising from the terms RI 0, R2 ðO
1 þ O
2 Þ, and jRj2 ðO1 þ O2 Þ while calculating the intensity jΓ0 ðx; yÞj2 would be spatially separable. This case is similar to the separability of the dc, real, and virtual images in offaxis digital holography due to the inclination of the reference beam with respect to the optical axis of the object [25]. Hence, focusing on the contribution from jRj2 ðO1 þ O2 Þ, the complex amplitude before deformation can be written as Γ1 ðx; yÞ ¼ a1 ðx; yÞ exp½jϕ1 ðx; yÞ þ a2 ðx; yÞ exp½jϕ2 ðx; yÞ:

ð4Þ

Here ϕ1 ðx; yÞ and ϕ2 ðx; yÞ are random phases corresponding to the two scattered object waves. After object deformation, the complex amplitude can be written as Γ2 ðx; yÞ ¼ a1 ðx; yÞ exp½jðϕ1 ðx; yÞ þ Δϕ1 ðx; yÞÞ þ a2 ðx; yÞ exp½jðϕ2 ðx; yÞ þ Δϕ2 ðx; yÞÞ;

ð5Þ

where Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ are the phase changes in the two object waves due to deformation and are usually referred as interference phases in DHI. Subsequently, the reconstructed interference field can be obtained by multiplying the postdeformation complex amplitude of the object wave by the complex conjugate of predeformation complex amplitude. In other words, we have Γðx; yÞ ¼ Γ2 ðx; yÞΓ
1 ðx; yÞ ¼ a21 ðx; yÞ exp½jΔϕ1 ðx; yÞ þ a22 ðx; yÞ exp½jΔϕ2 ðx; yÞ þ a1 ðx; yÞa2 ðx; yÞ exp½jðΔϕ1 ðx; yÞ þ ϕ1 ðx; yÞ − ϕ2 ðx; yÞÞ þ a1 ðx; yÞa2 ðx; yÞ exp½jðΔϕ2 ðx; yÞ

Fig. 1. Multibeam illumination of the object. 4190

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þ ϕ2 ðx; yÞ − ϕ1 ðx; yÞÞ:

ð6Þ

Here, the last two terms exhibit random behavior due to the presence of random phases ϕ1 ðx; yÞ and ϕ2 ðx; yÞ and hence can be considered part of the noise. Finally, the reconstructed interference field in the above equation can be modeled as

i.e., arctan

Γðx; yÞ ¼ aðx; yÞ exp½jΔϕ1 ðx; yÞ þ bðx; yÞ exp½jΔϕ2 ðx; yÞ þ ηðx; yÞ;

ð7Þ

where ηðx; yÞ is the noise term assumed to be additive white Gaussian noise (AWGN). The reconstructed interference field Γðx; yÞ could be interpreted as the sum of two complex signals whose arguments are the interference phases. Hence, the real part of Γðx; yÞ constitutes a moiré fringe pattern that encodes information about the two interference phases Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ. The relationship between the interference phase and object displacement could be obtained through the sensitivity vector, which is defined as the difference between observation and illumination unit vectors [24]. In Fig. 1, ^s1 and ^s2 denote the unit vectors along the illumination direc^ indicates the unit vector along the tions, whereas u observation direction. For the simplicity of analysis, a two-dimensional (2D) case is considered where all the unit vectors are limited to the x–z plane. So we have ^s1 ¼ − sinðθÞ^x − cosðθÞ^z;

ð8Þ

^s2 ¼ sinðθÞ^x − cosðθÞ^z;

ð9Þ

^ ¼ ^z: u

ð10Þ

Here ^ x and ^z denote the unit vectors along x and z. The displacement of the object is denoted by the vector ~ d ¼ dx ^ x þ dz^z with dz and dx as the out-of-plane and in-plane components. Accordingly, we have 2π ~ d · ð^ u − ^s1 Þ λ 2π ¼ ½d ð1 þ cosðθÞÞ þ dx sinðθÞ λ z

Δϕ1 ¼

and

2π ~ d · ð^ u − ^s2 Þ λ 2π ¼ ½d ð1 þ cosðθÞÞ − dx sinðθÞ: λ z

ð11Þ

ð12Þ

From the above equations, we have 4π d ð1 þ cosðθÞÞ; λ z 4π d sinðθÞ: Δϕ1 − Δϕ2 ¼ λ x

ImfΓðx;yÞg RefΓðx;yÞg

, with ‘Im’ and ‘Re’ indicating

the imaginary and real parts, would be unreliable since Γðx; yÞ is the sum of two complex signals. Similarly, analysis in the frequency domain by calculating the Fourier transform (FT) of Γðx; yÞ would be errorprone due to the overlap between the spectra of the two complex signals. However, this problem could be solved by introducing a spatial carrier in one of the object beams by tilting a mirror in the path of the object beam for one deformation state. For instance, if the spatial carrier is introduced in the first object beam while recording the hologram in one object state, say, after deformation, then Eq. (5) is modified as Γ2 ðx; yÞ ¼ a1 ðx; yÞ exp½jðω1 x þ ω2 y þ ϕ1 ðx; yÞ þ Δϕ1 ðx; yÞÞ þ a2 ðx; yÞ exp½jðϕ2 ðx; yÞ þ Δϕ2 ðx; yÞÞ;

ð15Þ

where ½ω1 ; ω2  is the introduced carrier frequency. Consequently, Eq. (7) is modified as Γðx; yÞ ¼ aðx; yÞ exp½jðω1 x þ ω2 y þ Δϕ1 ðx; yÞÞ þ bðx; yÞ exp½jΔϕ2 ðx; yÞ þ ηðx; yÞ:

ð16Þ

By taking the two-dimensional FT, we have Gðωx ; ωy Þ ¼ FTfΓðx; yÞg ¼ G1 ðωx − ω1 ; ωy − ω2 Þ þ G2 ðωx ; ωy Þ þ Nðωx ; ωy Þ;

ð17Þ

G1 ðωx ; ωy Þ ¼ FTfaðx; yÞ exp½jðΔϕ1 ðx; yÞÞg;

ð18Þ

G2 ðωx ; ωy Þ ¼ FTfbðx; yÞ exp½jΔϕ2 ðx; yÞg;

ð19Þ

where

Δϕ2 ¼

Δϕ1 þ Δϕ2 ¼

demodulation of digital holographic moiré (DHM) fringes. Reliable estimation of Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ from Eq. (7) is a challenging task. The usual “arctan” operation in DHI the interference phase, h for finding i

ð13Þ ð14Þ

It is clear from the above equations that the out-ofplane and in-plane displacement components dz and dx can be estimated from the sum and difference of the interference phases encoded in the moiré fringe pattern. Thus, the problem of measurement of multidimensional displacement components boils down to

Nðωx ; ωy Þ ¼ FTfηðx; yÞg:

ð20Þ

Because of the presence of carrier frequency ½ω1 ; ω2 , the Fourier spectra corresponding to G1 ðωx − ω1 ; ωy − ω2 Þ and G2 ðωx ; ωy Þ are separated in the frequency domain and concentrated around ½ω1 ; ω2  and zero frequencies. Hence, by spectral filtering and inverse FT [26], we obtain the individual complex signals g1 ðx; yÞ ¼ aðx; yÞ exp½jðΔϕ1 ðx; yÞÞ and g2 ðx; yÞ ¼ bðx; yÞ exp½jðΔϕ2 ðx; yÞÞ. Subsequently, the interference phases can be obtained using Δϕ1 ðx; yÞ ¼ anglefg1 ðx; yÞg 
Imfg1 ðx; yÞg ¼ arctan Refg1 ðx; yÞg 20 July 2011 / Vol. 50, No. 21 / APPLIED OPTICS

ð21Þ

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and

3. Simulation Results

Δϕ2 ðx; yÞ ¼ anglefg2 ðx; yÞg 
Imfg2 ðx; yÞg : ¼ arctan Refg2 ðx; yÞg

To analyze the proposed method, a complex reconstructed interference field was simulated as ð22Þ

It is to be noted that the estimates obtained using above equations are wrapped, and hence an unwrapping operation is required to obtain continuous phase distributions. Subsequently, the sum and difference phases can be obtained.

Γðx; yÞ ¼ g1 ðx; yÞ þ g2 ðx; yÞ ¼ exp½jΔϕ1 ðx; yÞ þ exp½jΔϕ2 ðx; yÞ;

ð23Þ

which could be interpreted as the sum of two components g1 ðx; yÞ and g2 ðx; yÞ. Here x and y denote the pixels along the horizontal and vertical directions. Similarly, another complex signal was generated

Fig. 2. (Color online) (a) First phase Δϕ1 ðx; yÞ in radians. (b) Second phase Δϕ2 ðx; yÞ in radians. (c) Moiré fringe pattern corresponding to Γðx; yÞ. (d) Moiré fringe pattern corresponding to Γc ðx; yÞ. (e) Fourier spectrum of Γðx; yÞ. (f) Fourier spectrum of Γc ðx; yÞ. 4192

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by associating a spatial carrier with frequency ½ω1 ; 0 to g1 ðx; yÞ as Γc ðx; yÞ ¼ g1 ðx; yÞ exp½jω1 x þ g2 ðx; yÞ ¼ exp½jðω1 x þ Δϕ1 ðx; yÞÞ þ exp½jΔϕ2 ðx; yÞ:

ð24Þ

For the analysis, noise was added with a signal to noise ratio of 20 dB using MATLAB’s “awgn” function. The simulated phases Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ

in radians are shown in Figs. 2(a) and 2(b). The real part of Γðx; yÞ, which constitutes the moiré fringe pattern, is shown in Fig. 2(c). Similarly, the real part of Γc ðx; yÞ, which corresponds to the moiré fringe pattern in the presence of a carrier, is shown in Fig. 2(d). The 2D Fourier spectra corresponding to Γðx; yÞ and Γc ðx; yÞ are shown in Figs. 2(e) and 2(f). It is clear that the spectra of g1 ðx; yÞ and g2 ðx; yÞ overlap in Fig. 2(e), whereas their spectra are quite separable in Fig. 2(f) due to the presence of a carrier. The main advantage of the inclusion of a carrier in one of the components is the spectral separability in the frequency domain,

Fig. 3. (Color online) (a) Wrapped estimate of phase Δϕ1 ðx; yÞ. (b) Wrapped estimate of phase Δϕ2 ðx; yÞ. (c) Unwrapped estimate of phase Δϕ1 ðx; yÞ in radians. (d) Unwrapped estimate of phase Δϕ2 ðx; yÞ in radians. (e) Sum of estimated phases in radians. (f) Difference of estimated phases in radians. 20 July 2011 / Vol. 50, No. 21 / APPLIED OPTICS

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which permits the retrieval of individual components. By Fourier filtering and inverse FT, the components g1 ðx; yÞ and g2 ðx; yÞ are recovered, and their corresponding wrapped phases are obtained using Eqs. (21) and (22). The wrapped estimates of Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ are shown in Figs. 3(a) and 3(b). The continuous phase distributions in radians after unwrapping are shown in Figs. 3(c) and 3(d). The rms errors for the estimation of Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ were 0.0822 and 0:0802 rad:. The pixels near the borders (10 pixels) were ignored for the calculation of errors. Finally, the sum and difference phases in radians are shown in Figs. 3(e) and 3(f). 4. DHM Schematic and Experimental Results

The general schematic (not drawn to scale) of the DHM optical configuration is shown in Fig. 4. The light from the laser is divided by the beam splitter (BS1) into two beams. One beam is incident on mirror M1 to be used as the object illumination beam 1 through the path M1–BE1–OBJ. The second beam passes through another beam splitter (BS2) to be further divided into another two beams to be used as the reference beam (through M3–BE3–M4–CCD) and the object illumination beam 2 (through M5– M2–BE2–OBJ). The illumination beams are diffusely scattered by the object. The CCD records the hologram formed by the superposition of the reference and diffusely scattered object beams. For the experiment, a Coherent Verdi (Coherent, Inc., USA) laser (532 nm) was used as the light source. A Sony (Sony Corporation, Japan) XCLU1000 (1600 × 1200 pixels) CCD camera was used for the recording of holograms. The object was deformed by applying external load as well as in-plane rotation. The carrier was introduced in the first object wave by the tilt of the mirror M1. Two holograms were recorded, corresponding to the object states before and after deformation. The numerical reconstruction was performed using discrete Fresnel

Fig. 4. (Color online) DHM schematic: BS1–BS2, beam splitters; BE1–BE3, beam expanders; M1–M5, mirrors; OBJ, diffuse object. 4194

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transform [24] to obtain the complex amplitudes Γ1 ðx; yÞ and Γ2 ðx; yÞ. The intensity of the complex amplitude before deformation, i.e., jΓ1 ðx; yÞj2 is shown in Fig. 5(a), where the dc part and the real and virtual images of the object are visible. Without loss of generality, a finite region was selected from the virtual image for analysis, as shown in Fig. 5(a). The complex reconstructed interference field, i.e., Γðx; yÞ was obtained by multiplying the postdeformation complex amplitude with the complex conjugate of the predeformation complex amplitude. The real part of the reconstructed interference field constitutes the DHM fringe pattern shown in Fig. 5(b). The DHM fringes encode information about the desired phases Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ. To recover the phase information, the 2D FT of Γðx; yÞ was calculated, and the corresponding spectrum is shown in Fig. 5(c), where two distinct spectral regions due to individual components are visible. To separate the spectra of individual components efficiently, the carrier frequency was also estimated in the experiment. Since the carrier was introduced in only one of the object waves by the tilt of the mirror, it was estimated by recording two holograms before and after tilting the mirror for the same object state, with the second object wave blocked and subsequently using numerical reconstruction to generate the complex signals. The carrier signal was then obtained by multiplying the post-tilt complex signal with the conjugate of the pretilt complex signal. Mathematically speaking, this operation is equivalent to setting bðx; yÞ ¼ 0 (since the second object wave was blocked) and Δϕ1 ðx; yÞ ¼ 0 (since both holograms were recorded for the same object state) in Eq. (16) to obtain a complex carrier signal cðx; yÞ ¼ aðx; yÞ exp½jðω1 x þ ω2 yÞ, from which the carrier frequency can be estimated using FT. The carrier fringes constituted by the real part of the complex carrier signal thus obtained are shown in Fig. 5(d) for the sake of illustration. It needs to be emphasized that the carrier estimation procedure needs to be performed only once and the obtained information about the carrier frequency could be used for many subsequent deformation measurements of the object, as long as the optical configuration remains the same. Subsequently, the individual components g1 ðx; yÞ and g2 ðx; yÞ were obtained by spatial filtering and inverse FT, and the wrapped phase estimates of Δϕ1 ðx; yÞ and Δϕ2 ðx; yÞ obtained using Eqs. (21) and (22) are shown in Figs. 5(e) and 5(f). The corresponding continuous phase distributions in radians obtained after unwrapping are shown in Figs. 6(a) and 6(b). The sum and difference phase distributions in radians are shown in Figs. 6(c) and 6(d). The wrapped forms of the sum and difference phases are shown in Figs. 6(e) and 6(f) for illustration purposes only. From the estimated sum and difference phases, the information about the out-of-plane and in-plane displacement components could be obtained using Eqs. (13) and (14).

5. Discussion

The simulation and experimental results validate the applicability of the proposed method. In comparison with the existing state-of-the-art techniques for multidimensional deformation measurements in DHI, the proposed method enjoys the following advantages: 1. The proposed method is capable of providing information about the in-plane and out-of-plane displacements in DHI without requiring multiple

data frames, in contrast with the phase-shiftingtechnique-based methods. This improves robustness against vibrations and external disturbances. 2. The proposed method does not require multiple reference beams or CCDs for recording holograms. This simplifies the optical configuration for multidimensional deformation measurements. 3. In the proposed method, a single DHM fringe pattern contains information about multiple phases that are related to the in-plane and out-of-plane displacement components through the sensitivity

Fig. 5. (Color online) (a) Intensity of the numerically reconstructed hologram using discrete Fresnel transform. (b) DHM fringe pattern. (c) 2D FT of Γðx; yÞ. (d) Carrier fringes. (e) Wrapped estimate of first phase Δϕ1 ðx; yÞ. (f) Wrapped estimate of second phase Δϕ2 ðx; yÞ. 20 July 2011 / Vol. 50, No. 21 / APPLIED OPTICS

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Fig. 6. (Color online) (a) Unwrapped phase Δϕ1 ðx; yÞ in radians. (b) Unwrapped phase Δϕ2 ðx; yÞ in radians. (c) Sum of estimated phases in radians. (d) Difference of estimated phases in radians. (e) Wrapped form of sum phase. (f) Wrapped form of difference phase.

vector. Hence, multiple displacement components are simultaneously ascertained by the demodulation of a DHM fringe pattern. This makes the proposed method highly suitable for measurements where multiple displacement components have to be measured at the same time. 6. Conclusions

In this paper, an elegant technique based on the demodulation of digital holographic moiré fringes is presented for the simultaneous measurement of multidimensional displacement components. The analysis and application of the proposed method 4196

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were shown through the simulation and experimental results. The advantages of the proposed method as discussed in this paper validate its application potential. The feasibility of multidimensional deformation analysis from DHM fringes provided by the proposed method could have a strong impact on the use of digital holographic moiré configuration in areas such as experimental mechanics and nondestructive testing. † Currently with Rowland Institute at Harvard University, Cambridge, Massachusetts, 02142, USA

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