Phase-resolved heterodyne shearographic vibrometer for observation of transient surface motion: theory and model

James Perea; George Nehmetallah

doi:10.1364/OE.25.006169

1. Introduction

Shearography is an optical metrology technique utilized for vibration analysis and strain measurements. A basic shearographic sensor utilizes a shearing interferometer, typically using Michelson, Mach-Zehnder, or wedge [1] configurations; each mixes the measured signal with a shifted version of itself. The shearing interferometer generates a self-reference, mitigating sensor vibration and some environmental effects [2,3]. However, the output from a basic shearographic sensor, referred to as a fringe pattern, may only be used for qualitative analysis [4]. Phase resolved measurements, necessary to quantify the measured shearogram, are typically performed by temporal or spatial phase shifting [5]. Temporal phase shifting typically requires stepped motors [2] in the receive optical path to generate additional frames and is not suitable for transients. Observation of transient motion is possible by spatial phase shifting, but requires dividing the received focal plane array into four quadrants and adjusting the polarization state of each quadrant [6, 7], or by creating multiple spatial frequency carriers by adjusting the angle of incidence [8–11]. Additionally, without heterodyning, interrogation of large fields of view at long distances requires significant optical power.

This work seeks to develop a novel technique to perform phase resolved shearographic measurements while incorporating heterodyne gain. Recently, a temporal heterodyne shearing interferometer concept was introduced [12], however the single low rate carrier technique was not suitable for instantaneous phase resolved observations and it did not allow variable shear distance and directions in post-processing. The technique proposed in this work provides the ability to perform instantaneous, adjustable shear, phase resolved measurements for a whole field. The design requires generation of orthogonal polarization states within the shearing interferometer and the reference path by placing a half-wave plate in one segment. The reference leg contains three acousto-optic modulators to create two intermediate frequencies in the kHz range. Mixing the reference and measurement beams on a high-speed focal plane array creates an interference pattern with two carrier frequencies. Each carrier contains information pertaining to one segment of the shearing interferometer, both identically Doppler shifted due to time varying path length changes in the axial dimension of the interrogation beam. Demodulation routines extract the phase shift as a function of time for each pixel. The phase difference between each carrier represents the relative phase of the light reflected from the target surface. Prior work utilized two independent laser sources [13], wavelength modulation of a laser source for polarization multiplexing and phase stepping [14], or polarized spatial frequency carriers and multiple shearing interferometers [9] to observe multiple shearing directions. Because each segment of the shearing interferometer does not coherently interfere, the phase extraction from the carrier frequencies allows for adjustment of the shear direction and amount during post-processing. Separation of each segment also allows for simultaneous whole field Doppler velocity (LDV) and electronic speckle pattern interferometry (ESPI) measurements. Additionally, the phase extraction from the carrier frequencies provides quantitative phase measurements, comparable to a temporal phase shift measurement, but without requiring phase stepping or reduced spatial resolution. Due to the short integration times associated with the dual kHz frequency carriers, the technique is well suited for operation in harsh environments [15] as well as numerous industrial [16,17], medical [18–20], and military applications where high rate observation of surface motion is required.

2. Shearography theory

Shearography measures the gradient of deformation by observing the phase change between deformed and undeformed states of an object. In a basic configuration, the measured electric field before deformation may be described by the mixing of two legs of a shearing interferometer [5]

u_{1} = | M_{1} | \exp [j (ω_{o} t + θ_{1})]

u_{2} = | M_{2} | \exp [j (ω_{o} t + θ_{2})],

where u₁ is the electric field due to one leg of the shearing interferometer and u₂ is the sheared version of u₁. θ₁ and θ₂ are the respective phases, which are random due to reflection from a diffuse surface. |M₁| and |M₂| are the amplitudes of the electric fields and ω_o is the optical carrier frequency. The measured irradiance [5], neglecting the optical frequency following camera integration, is

I = {| M_{1} |}^{2} + {| M_{2} |}^{2} + 2 | M_{1} | | M_{2} | \cos (θ_{1} - θ_{2}) .

θ₂ may also be described in terms of a spatially shifted θ₁

θ_{2} (x) = θ_{1} (x + δ x),

where δ_x is the shear distance in the x-direction in the object frame coordinates. Now, the measured irradiance is

I = {| M_{1} |}^{2} + {| M_{2} |}^{2} + 2 | M_{1} | | M_{2} | \cos (ϕ),

where ϕ = θ₁ − θ₂. Once the object is deformed, a phase shift, Δ is applied to the scattered electric field. The resulting irradiance, following propagation through the shearing interferometer, is

I^{'} = {| M_{1} |}^{2} + {| M_{2} |}^{2} + 2 | M_{1} | | M_{2} | \cos (ϕ^{'}),

where

ϕ^{'} = Δ + ϕ .

Generation of a shearogram is possible by subtracting the two measured irradiances

Δ I = | I - I^{'} | = 2 | M_{1} | | M_{2} | [\cos (ϕ) - \cos (ϕ^{'})] .

Using a trigonometric identity, the resulting shearogram may be defined as

Δ I = 4 | M_{1} | | M_{2} | [\sin (\frac{ϕ + ϕ^{'}}{2}) \sin (\frac{ϕ - ϕ^{'}}{2})] .

Recalling Eq. (7),

Δ I = 4 | M_{1} | | M_{2} | [\sin (ϕ + \frac{Δ}{2}) \sin (\frac{Δ}{2})] .

Assuming shearing occurs in the x direction, and for simplicity, the illumination occurs at normal incidence, for small amounts of deformation, the relative phase change [5] is

Δ = \frac{4 π}{λ} δ_{x} \frac{\partial w_{z}}{\partial x},

where w_z is the component of the deformation vector in the z direction. In the case of Eq. (11), ∂w_z/∂x is the first derivative of the out-of-plane deformation, described by Fig. 1.

Fig. 1 Comparison of surface displacement and the gradient of displacement, ∂w/∂x in the x-direction.

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Simple shearographic measurements yield a representation of the gradient of deformation, shown by Eq. (10). This representation is suitable for visualization, but phase resolved measurements are necessary to quantify the phase change due to deformation.

3. Heterodyne shearography theory

A heterodyne shearographic concept has been developed for phase resolved measurements. The concept utilizes a polarization multiplexed shearing interferometer and multiple temporal local oscillators to apply heterodyne gain and add axial velocity measurements. The concept sensor schematic is shown in Fig. 2.

Fig. 2 Schematic of heterodyne shearographic vibrometer with a dynamic diffuse-scatterer for the target. Lenses have been omitted for clarity. Laser, 200mW SLM 532nm; BS1, non-polarizing beam splitter to generate reference and measurement legs; AOM 1, upshift acousto-optic modulator; BS2, BS3, non-polarizing beam splitter to separate segments for polarization multiplexing; AOM 2, downshift acousto-optic modulator, creating first kHz frequency carrier; u_r1, electric field for reference leg 1; AOM 3, downshift acousto-optic modulator, creating second kHz frequency carrier; u_r2, electric field for reference leg 2; λ/2, half-wave plate to rotate the polarization by 90°; PBS1, PBS2, polarizing beam splitters recombine components with orthogonal polarization states; PF, polarizing filter; u_m1, electric field for measurement leg 1; u_m2, electric field for measurement leg 2; BS4, non-polarizing beam splitter combines measurement and reference legs; FPA, high rate focal plane array.

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The use of a polarizing shearing interferometer allows for each leg of the interferometer to be carried by a different local oscillator frequency. Beginning with the measurement leg, Fig. 2 the electric field from each portion of the interferometer [21] may be described by

u_{m 1} = \hat{x} | M_{1} | \exp [j (ω_{o} t - β z + ψ_{1})]

u_{m 2} = \hat{y} | M_{2} | \exp [j (ω_{o} t - β z + ψ_{2} + π)],

where ω_o is the optical carrier and the π phase delay in u_m2 is the result of a half wave plate present in one leg of the shearing interferometer. ψ_1,2 represents the combination of the Doppler phase shift, ϕ_dop due to target motion and the respective phases of each interferometer segment, θ_1,2, defined by

ψ_{1, 2} = ϕ_{dop} + θ_{1, 2} .

The corresponding reference fields may be described by

u_{r 1} = \hat{x} R_{1} \exp {j [(ω_{o} + ω_{1}) t - β z]}

u_{r 2} = \hat{y} R_{2} \exp {j [(ω_{o} + ω_{2}) t - β z + π]},

where ω₁ and ω₂ are the intermediate local oscillator frequencies.

The total field u_t at the image plane is the superposition of the measurement and reference fields

u_{t} = u_{r 1} + u_{r 2} + u_{m 1} + u_{m 2} .

The irradiance on the detector is

\begin{array}{l} I & = u_{t} u_{t}^{*} \\ = (u_{r 1} + u_{r 2} + u_{m 1} + u_{m 2}) (u_{r 1}^{*} + u_{r 2}^{*} + u_{m 1}^{*} + u_{m 2}^{*}) . \end{array}

The fields with orthogonal polarization states do not mix, i.e. x̂ · ŷ = 0, leaving

u_{r 1} u_{m 1}^{*} + u_{r 1}^{*} u_{m 1} = 2 R_{1} | M_{1} | \cos (ω_{1} t + ψ_{1}),

and

u_{r 2} u_{m 2}^{*} + u_{r 2}^{*} u_{m 2} = 2 R_{2} | M_{2} | \cos (ω_{2} t + ψ_{2}) .

The final irradiance is

I = R_{1}^{2} + R_{2}^{2} + {| M_{1} |}^{2} + {| M_{2} |}^{2} + 2 R_{1} | M_{1} | \cos (ω_{1} t + ψ_{1}) + 2 R_{2} | M_{2} | \cos (ω_{2} t + ψ_{2}) .

Following deformation, the irradiance may be defined as

I^{'} = R_{1}^{2} + R_{2}^{2} + {| M_{1} |}^{2} + {| M_{2} |}^{2} + 2 R_{1} | M_{1} | \cos (ω_{1} t + ψ_{1}^{'}) + 2 R_{2} | M_{2} | \cos (ω_{2} t + ψ_{2}^{'}) .

It is of interest to extract the relative phases of the light scattered from the object, θ₁ and θ₂. However, these are carried by intermediate frequencies and are time varying due to the Doppler shift, ϕ_dop. Using standard demodulation techniques [22], the Doppler and object phases for each local oscillator segment may be extracted

ψ_{1} = Demod {[I]}_{ω_{1}} = ϕ_{dop} + θ_{1}

ψ_{2} = Demod {[I]}_{ω_{2}} = ϕ_{dop} + θ_{2} .

Assuming the Doppler shift acts equivalently on both local oscillators

ψ_{2} - ψ_{1} = θ_{2} - θ_{1} = ϕ .

Repeating the procedure following deformation yields

ψ_{1}^{'} = ϕ_{dop} + θ_{1}^{'},

ψ_{2}^{'} = ϕ_{dop} + θ_{2}^{'},

and

ψ_{2}^{'} - ψ_{1}^{'} = θ_{2}^{'} - θ_{1}^{'} = ϕ^{'} .

Finally, the relative phase change due to deformation is calculated directly by

(ψ_{2}^{'} - ψ_{1}^{'}) - (ψ_{2} - ψ_{1}) = ϕ^{'} - ϕ = Δ .

Where Δ directly corresponds to the gradient of displacement, Eq. (11). The theory indicates it is possible to extract the relative phase difference between undeformed and deformed states without the sign ambiguity of basic shearography and does not require temporal or spatial phase stepping. Additionally, because the electric fields from each segment of the interferometer do not interfere coherently, the shear distance and orientation defined by the optical arrangement is inconsequential and may be adjusted in post-processing. This arrangement provides a significant amount of flexibility when examining spatially and temporally variable surface deformations.

4. Numerical model

A numerical model is used to verify the heterodyne shearographic theory and develop preliminary processing routines to extract the phase shift due to surface deformation. This model generates an electric field at the image plane following illumination of a dynamic rough surface. Spatial information is calculated using discrete Fourier transforms, dependent on Fresnel diffraction equations [22–24]. This transform is independent of time, but motion is introduced as a dynamic phase shift due to path length change and applied to the object field, u_m1, before the transform. The optical frequency and local oscillators are applied following generation of the image irradiance terms.

Initially, a dynamic rough surface was generated. The displacement is defined by a Gaussian function, Fig. 3 and oscillates at frequency f_obj, defined by

w_{z} (ξ, η, t) = Asin (2 π f_{obj} t) \exp {- [\frac{{(ξ^{2} - ξ_{o})}^{2} + {(η^{2} - η_{o})}^{2}}{2 σ^{2}}]},

where ξ and η are object frame coordinates, A is the deformation amplitude, and ξ_o, η_o, and σ define the center and width of the Gaussian function. For the model, the peak displacement was set at 500nm at 159Hz, equivalent to a peak velocity of 500μm/s with a deformation width of 1.5cm.

Fig. 3 Gaussian surface displacement.

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The surface displacement, w_z is converted to an optical phase shift [25], assuming the light is incident and reflected off the surface

Δ p (ξ, η, t) = \exp [j \frac{4 π}{λ} w_{z} (ξ, η, t)] .

The phase shift due to target motion is applied to a complex electric field at the object surface

u_{o} (ξ, η, t) = A_{o} (ξ, η) \exp {j [θ (ξ, η) + Δ p (ξ, η, t)]},

where A_o is the area of illumination, which was set to 90% of the object space, or 2.3cm. A random phase operator, θ = [−π, π], accounts for the rough surface, generating fully developed speckles. The electric field is then applied to a discretized Fresnel diffraction integral [24] and propagated to the front surface of a lens

u_{l} (x, y, t) = κ FFT {[u_{o} (ξ, η, t) \exp [j \frac{π}{λ z} (ξ^{2} + η^{2})]]}_{x, y}

and

κ = \frac{\exp (j k z)}{j λ z} \exp [\frac{j k}{2 z} (x^{2} + y^{2})] .

Following propagation to the lens plane, a lens transfer function is applied

TL = \exp [- j \frac{π}{λ f} (x^{2} + y^{2})],

where f = 1000mm is the focal length. The sytem was defined as a 2f-2f configuration, focusing the electric field onto the image plane following propagation using Eq. (33), yielding Eq. (1) without the optical carrier

u_{1} (x, y, t) = | M_{1} | (x, y, t) \exp [j θ_{1} (x, y, t)] .

The sheared version of u₁, u₂ is generated by shifting the indices of u₁ by 0.5cm. In the heterodyne arrangement, the optical shearing is not necessary, but generation of a standard reference shearogram requires coherent interference of u₁ and u₂, defined by Eq. (8), where I is chosen during the peak deformation and I′ represents the minimum, or negative peak deformation, shown by Fig. 4.

Fig. 4 Non-phase resolved shearographic fringe pattern produced by numerical model.

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The reference beam, u_ref is generated by propagating uniform phase through the Fresnel diffraction equation, Eq. (33). Heterodyning is then included in the numerical model by first generating an arbitrary optical frequency, ω_o and intermediate carrier frequencies, ω_1,2 and applying them to u₁, u₂, and u_r by

u_{m 1} (x, y, t) = | u_{1} | \exp {j [ω_{o} t + ∠ (u_{1} (x, y, t))]}

u_{m 2} (x, y, t) = | u_{2} | \exp {j [ω_{o} t + ∠ (u_{2} (x, y, t))]}

u_{r 1} (x, y, t) = | u_{ref} | \exp {j [(ω_{o} + ω_{1}) t + ∠ (u_{ref} (x, y, t))]}

u_{r 2} (x, y, t) = | u_{ref} | \exp {j [(ω_{o} + ω_{2}) t + ∠ (u_{ref} (x, y, t))]} .

The fields are then mixed separately to account for their intended orthogonal polarization states

I_{sig 1} = (u_{m 1} + u_{r 1}) (u_{m 1}^{*} + u_{r 1}^{*})

I_{sig 2} = (u_{m 2} + u_{r 2}) (u_{m 2}^{*} + u_{r 2}^{*}),

and summed [26],

I = I_{sig 1} + I_{sig 2} .

Equation (43) is now equivalent to Eq. (21).

For the numerical model, and future experimental setup, it is important to consider constraints when selecting carrier frequencies, shear distances, and allowable object deformation. Recall from Eqs. (11), (30), that the amount of phase shift is dependent on shear distance δ_x, deformation width, σ, and deformation amplitude, w_z. The relative phase change, Δ corresponds to the number of 2π phase jumps in the shearographic image. For visualization, a few shearographic fringes are desired. The shear distance, which governs sensitivity, should not exceed half the diameter of the deformation [5]. The deformation amplitude is the one factor that has additional constraints with the heterodyne technique. The velocity of the deformation is proportional to the instantaneous frequency shift, Δω due to Doppler shift on the local oscillator

Δ ω = \frac{d}{d t} (\frac{4 π}{λ} \frac{v (t)}{2 π f_{obj}}),

where v(t) is the surface velocity in the axial direction of the beam, or equivalently related to displacement

w_{z} (t) = \frac{v (t)}{2 π f_{obj}} .

The carrier frequencies should be chosen so the instantaneous frequency shift does not exceed the Nyquist frequency, ω_n and does not drop below the difference, ω₂ − ω₁ which may retain residual phase information from the object if orthogonal polarization components are not completely filtered. Based on this condition, a finite bandwidth may be defined, as illustrated in Fig. 5.

Fig. 5 Constraints on carrier frequency selection due to finite bandwidth limitations.

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Given the constraints, the allowable Doppler frequency shift is limited to one-half the difference between both local oscillators, leading to a maximum measurable surface displacement

w_{z, peak} = \frac{(ω_{2} - ω_{1})}{2} \frac{λ}{4 π ω_{obj}} .

For the numerical model, the carrier frequencies were chosen to be 10kHz and 25kHz. The frequencies are observable using commerically available high speed cameras.

5. Processing routine

Calculation of relative phase and generation of shearographic fringe patterns require extracting the object phase from the Doppler shifted carrier frequencies, Eq. (23). An arctangent demodulation routine is used [22] to extract the phase by first calculating the in-phase and quadrature terms

\begin{array}{l} Q_{1, 2} (t) & = & lowpass [highpass [I (t)] \sin (ω_{1, 2} t)] \\ I_{1, 2} (t) & = & lowpass [highpass [I (t)] \cos (ω_{1, 2} t)] . \end{array}

The in-phase and quadrature terms are then used to estimate the Doppler phase independently for each local oscillator

ψ_{1, 2} = unwrap [\tan^{- 1} (\frac{Q_{1, 2} (t)}{I_{1, 2} (t)})] .

The phase shift due to deformation, Δ, is then determined using Eqs. (24), (28), and (29). Figure 6(a) shows the resolved phase plot prior to image processing where the relative phase due to deformation, Δ has been modified by

Δ = {\begin{array}{l} (ψ_{2}^{'} - ψ_{1}^{'}) - (ψ_{2} - ψ_{1}) & for & (ψ_{2}^{'} - ψ_{1}^{'}) \geq (ψ_{2} - ψ_{1}) \\ (ψ_{2}^{'} - ψ_{1}^{'}) - (ψ_{2} - ψ_{1}) + 2 π, & for & (ψ_{2}^{'} - ψ_{1}^{'}) < (ψ_{2} - ψ_{1}) \end{array} .

The extracted image is corrupted due to speckle, however a single iteration sine-cosine filter

Δ = atan 2 [\sin (Δ, \cos (Δ))]

suppresses some speckle noise, and confines the wrapped phase within (−π, π] [27], allowing for visualization of wrapped phase-resolved fringes, Fig. 6(b).

Fig. 6 (a) Numerically modeled phase resolved shearographic fringe pattern, Δ, generated by extracting phase from two Doppler shifted frequency carriers and applying Eq. (49), (b) Sine-cosine filtered, phase resolved shearographic fringe pattern. Δ was generated by applying Eq. (50).

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To remove additional speckle noise, an iterative sine-cosine smoothing filter [28] was applied. A [3x3] mean filter was applied to the sine and cosine terms of the four-quadrant inverse tangent. This process, repeated 20 times, generates a smooth, wrapped fringe pattern, Fig. 7(a). Additionally, each segment of the shearing interferometer is carried by a different temporal frequency carrier, as a result, the segments do not interfere coherently. Following extraction of the phase terms, and prior to calculating the relative phase from Eq. (29), the relative position of ψ₁ and ψ₂ may be shifted. This allows adjustment of the shearing distance, δ, and direction for simultaneous calculation of ∂w_z/∂x and ∂w_z/∂y. Figure 7(b) shows the wrapped shearogram with reduced shearing in the y-direction.

Fig. 7 (a) Iterative sine-cosine smoothing filtered, phase resolved shearographic fringe pattern and (b) Post-processing adjustment to shearing amount and direction

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The final unwrapped phase map from Fig. 7(a), representing the displacement gradient, is shown in Fig. 8. In preparation for an experimental design, the impact of cross-talk between orthogonal polarization states was considered. The primary concern was due to depolarization from the diffuse surface [29] generating components that could interfere with both frequency carriers. However, modern polarizing filters and polarizing beam splitters have extinction ratios of at least 10,000:1 and 1,000:1, respectively. The resulting amplitude of the undesired cross-component would be orders of magnitude less than the primary phase component, providing negligible contribution.

Fig. 8 Unwrapped shearogram generated by extracting phase from two Doppler shifted frequency carriers.

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In addition to the displacement gradient, it is also possible to extract the Doppler velocity from the heterodyne local oscillators. Figure 9 shows the 2D velocity map, which represents the velocity of the dynamic Gaussian surface shown in Fig. 3. The velocity values for each pixel are calculated using,

v (x, y, t) = \frac{d}{d t} {(\frac{λ}{4 π}) unwrap [ψ (x, y, t)]} .

Recalling, Eq. (23), ψ is the combination of the phase shift due to path length changes plus a fixed offset due to the random object phase. The phase from a single frequency carrier is extracted from the demodulation routine in Eq. (48), which is prone to noise influences from speckle [30]. As the in-phase and quadrature terms are calculated using low SNR dark speckles, low amplitude drop outs occur, leading to erroneous high velocity calculations [31]. To remove the erroneous velocity values, low carrier to noise (CNR) values were identified and replaced with an identifier, in this case non a number (NaN). For each temporal frame, the NaN values were removed by performing a two dimensional median filter that excludes the values from the calculation. The velocity may be extracted from either frequency carrier. The two velocities would be equivalent, however some noise reduction may be possible by averaging the velocities extracted from each carrier.

Fig. 9 Doppler velocity extracted from one carrier frequency, spatially filtered to remove speckle noise.

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6. Conclusion

In this work, we have developed a novel technique for a heterodyne shearographic vibration sensor. The sensor utilizes polarization multiplexing with multiple temporal carrier frequencies to separate each segment of a shearing interferometer. The underlying theory indicates that it is possible to obtain the unambiguous phase associated with surface deformation by extracting the scattered object phase from Doppler shifted carrier frequencies. The supporting numerical model provides the initial processing steps required to extract the phase terms and reconstruct the shearographic image, followed by necessary algorithms to reduce noise and generate an unwrapped phase map. Numerical modeling also presented potential limitations of the technique, which combines aspects of Doppler vibrometery and shearography. Namely, the object under interrogation has peak velocity limits due to finite bandwidth, defined by Eq. (46). This limitation, in turn, implies the shearing distance and object width are particularly important to shearographic fringe pattern generation. Shot noise, a limiting factor in many Doppler vibrometer designs, will also be an issue due to the increased DC from the second local oscillator [32]. However, this sensor concept has numerous benefits, including allowing the observation of adjustable shear phase-resolved measurements of transient surface motion without the requirement of reducing resolution by spatial phase stepping. Additionally, heterodyning the shearographic sensor reduces laser power requirements and provides the opportunity to leverage additional sensing mechanisms, including Doppler velocity.

References and links

1. D. Francis, R. Tatam, and R. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21, 102001 (2010). [CrossRef]

2. F. Languy, J.-F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 μm and shearography at 532 nm,” Opt. Eng. 55, 121704 (2016). [CrossRef]

3. W. J. Bisle, D. Scherling, M. K. Kalms, and W. Osten, “Improved shearography for use on optical non cooperating surfaces under daylight conditions,” Proc. AIPB, 1928–1935 (2001). [CrossRef]

4. W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry (SPIE, 2003).

5. L. Yang and X. Xie, Digital shearography: New Developments and Applications (SPIE, 2016).

6. D. I. Serrano-García, N. I. Toto-Arellano, A. Martínez-García, J. A. R. Álvarez, and G. R. Zurita, “Dynamic phase profile of phase objects based in the use of a quasi-common path interferometer,” Optik 123, 1742–1745 (2012). [CrossRef]

7. S. Yoneyama and S. Arikawa, “Instantaneous phase-stepping interferometry based on a pixelated micro-polarizer array,” Theor. Appl. Mech. Lett. 6, 162–166 (2016). [CrossRef]

8. G. Rodríguez-Zurita, A. García-Arellano, N. Toto-Arellano, V. Flores-Muñoz, R. Pastrana-Sánchez, C. Robledo-Sánchez, O. Martínez-Bravo, N. Vásquez-Pasmiño, and C. Costa-Vera, “One–shot phase stepping with a pulsed laser and modulation of polarization: application to speckle interferometry,” Opt. Express 23, 23414–23427 (2015). [CrossRef]

9. X. Xie, C. P. Lee, J. Li, B. Zhang, and L. Yang, “Polarized digital shearography for simultaneous dual shearing directions measurements,” Rev. Sci. Instrum. 87, 083110 (2016). [CrossRef] [PubMed]

10. X. Xie, L. Yang, N. Xu, and X. Chen, “Michelson interferometer based spatial phase shift shearography,” Appl. Optics 52, 4063–4071 (2013). [CrossRef]

11. C. Falldorf, R. Klattenhoff, and R. B. Bergmann, “Single shot lateral shear interferometer with variable shear,” Opt. Eng. 54, 054105 (2015). [CrossRef]

12. X. Wang, Z. Gao, J. Qin, X. Zhang, and S. Yang, “Temporal heterodyne shearing speckle pattern interferometry,” Optics and Lasers in Engineering 93, 76–82 (2017). [CrossRef]

13. G. L. Richoz and G. S. Schajer, “Simultaneous two-axis shearographic interferometer using multiple wavelengths and a color camera,” Opt. Lasers Eng. 77, 143–153 (2016). [CrossRef]

14. R. M. Groves, S. W. James, and R. P. Tatam, “Polarization-multiplexed and phase-stepped fibre optic shearography using laser wavelength modulation,” Meas. Sci. Technol. 11, 1389 (2000). [CrossRef]

15. T. W. Du Bosq and E. Repasi, “Detector integration time dependent atmospheric turbulence imaging simulation,” Proc. SPIE 9452, 94520B (2015). [CrossRef]

16. P. L. Reu, D. P. Rohe, and L. D. Jacobs, “Comparison of dic and ldv for practical vibration and modal measurements,” Mech. Syst. Signal Pr. 86, 2–16 (2017). [CrossRef]

17. J. G. Chen, R. W. Haupt, and O. Buyukozturk, “Operational and defect parameters concerning the acoustic-laser vibrometry method for frp-reinforced concrete,” NDTE Int. 71, 43–53 (2015). [CrossRef]

18. J. Bencteux, P. Pagnoux, T. Kostas, S. Bayat, and M. Atlan, “Holographic laser doppler imaging of pulsatile blood flow,” Biomed. Opt. 20, 066006 (2015). [CrossRef]

19. F. Tenner, D. Elz, Z. Zalevsky, and M. Schmidt, “Optical tremor analysis with the speckle imaging technique,” J. Imaging. Sci. Techn. 59, 104021 (2015). [CrossRef]

20. M. Khaleghi, J. T. Cheng, C. Furlong, and J. J. Rosowski, “In-plane and out-of-plane motions of the human tympanic membrane,” J. Acoust. Soc. Am. 139, 104–117 (2016). [CrossRef] [PubMed]

21. D.H. Staelin, A.W. Morgenthaler, and J.A. Kong, Electromagnetic waves (Prentice Hall, 1994).

22. J. Perea and B. Libbey, “Development of a heterodyne speckle imager to measure 3 degrees of vibrational freedom,” Opt. Express 24, 8253–8265 (2016). [CrossRef] [PubMed]

23. J. W. Goodman, Introduction to Fourier optics (Roberts and Company, 2005).

24. G. Sendra, H. Rabal, M. Trivi, and R. Arizaga, “Numerical model for simulation of dynamic speckle reference patterns,” Opt. Commun. 282, 3693–3700 (2009). [CrossRef]

25. G.T. Nehmetallah, R. Aylo, and L.A. Williams, Analog and Digital Holography with MATLAB (SPIE, 2015). [CrossRef]

26. P. Hariharan, Basics of Interferometry (Academic, 2010).

27. H. Y. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G. Barbastathis, “Path-independent phase unwrapping using phase gradient and total-variation (tv) denoising,” Opt. Express 20, 14075–14089 (2012). [CrossRef] [PubMed]

28. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999). [CrossRef]

29. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2007).

30. S. Rothberg, “Numerical simulation of speckle noise in laser vibrometry,” Appl. Optics 45, 4523–4533 (2006). [CrossRef]

31. B.K. Park, O. Boric-Lubecke, and V. M. Lubecke, “Arctangent demodulation with dc offset compensation in quadrature doppler radar receiver systems,” IEEE T. Microw. Theory. 55(5), 1073–1079 (2007). [CrossRef]

32. A. Rogalski, Infrared Detectors (CRC, 2010).

Phase-resolved heterodyne shearographic vibrometer for observation of transient surface motion: theory and model

Abstract

1. Introduction

2. Shearography theory

3. Heterodyne shearography theory

4. Numerical model

5. Processing routine

6. Conclusion

References and links

Cited By

Figures (9)

Equations (51)

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