Researching | Simultaneous 2D in-plane deformation measurement using electronic speckle pattern interferometry with double phase modulations

Journals >Chinese Optics Letters >Volume 16 >Issue 7 >Page 071201 > Article

Chinese Optics Letters
Vol. 16, Issue 7, 071201 (2018)

Simultaneous 2D in-plane deformation measurement using electronic speckle pattern interferometry with double phase modulations

Yunlong Zhu^1、*, Julien Vaillant^1、2, Guillaume Montay², Manuel François², Yassine Hadjar¹, and Aurélien Bruyant^1、**

Author Affiliations

¹ICD-L2N, UMR CNRS 6281, Université de Technologie de Troyes, 10004 Troyes, France

²ICD-LASMIS, UMR CNRS 6281, Université de Technologie de Troyes, 10004 Troyes, France

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DOI: 10.3788/COL201816.071201 Cite this Article Set citation alerts

Yunlong Zhu, Julien Vaillant, Guillaume Montay, Manuel François, Yassine Hadjar, Aurélien Bruyant. Simultaneous 2D in-plane deformation measurement using electronic speckle pattern interferometry with double phase modulations[J]. Chinese Optics Letters, 2018, 16(7): 071201 Copy Citation Text

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Abstract

Electronic speckle pattern interferometry (ESPI) and digital speckle pattern interferometry are well-established non-contact measurement methods. They have been widely used to carry out precise deformation mapping. However, the simultaneous two-dimensional (2D) or three-dimensional (3D) deformation measurements using ESPI with phase shifting usually involve complicated and slow equipment. In this Letter, we solve these issues by proposing a modified ESPI system based on double phase modulations with only one laser and one camera. In-plane normal and shear strains are obtained with good quality. This system can also be developed to measure 3D deformation, and it has the potential to carry out faster measurements with a high-speed camera.

E (x, y) = A_{1} (x, y) e^{i [2 π f_{c} t + θ_{1} (x, y) + F_{1} (t)]} + A_{2} (x, y) e^{i [2 π f_{c} t + θ_{2} (x, y) + F_{2} (t)]} + A_{3} (x, y) e^{i [2 π f_{c} t + θ_{3} (x, y)]} .

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I (x, y) \propto {| E (x, y) |}^{2} = [A_{1}^{2} (x, y) + A_{2}^{2} (x, y) + A_{3}^{2} (x, y)] + 2 A_{1} (x, y) A_{2} (x, y) \cos [θ_{1} (x, y) + F_{1} (t) - θ_{2} (x, y) - F_{2} (t)] + 2 A_{1} (x, y) A_{3} (x, y) \cos [θ_{1} (x, y) + F_{1} (t) - θ_{3} (x, y)] + 2 A_{2} (x, y) A_{3} (x, y) \cos [θ_{2} (x, y) + F_{2} (t) - θ_{3} (x, y)] .

(2)

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I^{'} (x, y) \propto {| E^{'} (x, y) |}^{2} = [A_{1}^{2} (x, y) + A_{2}^{2} (x, y) + A_{3}^{2} (x, y)] + 2 A_{1} (x, y) A_{2} (x, y) \cos [θ_{1}^{'} (x, y) + F_{1} (t) - θ_{2}^{'} (x, y) - F_{2} (t)] + 2 A_{1} (x, y) A_{3} (x, y) \cos [θ_{1}^{'} (x, y) + F_{1} (t) - θ_{3}^{'} (x, y)] + 2 A_{2} (x, y) A_{3} (x, y) \cos [θ_{2}^{'} (x, y) + F_{2} (t) - θ_{3}^{'} (x, y)],

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θ_{1}^{'} (x, y) = θ_{1} (x, y) + \frac{2 π}{λ} (n_{1} - n_{s}) \cdot u (x, y),

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θ_{2}^{'} (x, y) = θ_{2} (x, y) + \frac{2 π}{λ} (n_{2} - n_{s}) \cdot u (x, y),

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θ_{3}^{'} (x, y) = θ_{3} (x, y) + \frac{2 π}{λ} (n_{3} - n_{s}) \cdot u (x, y),

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F_{1} (t) = 2 π f_{1} t,

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F_{2} (t) = 2 π f_{2} t,

(8)

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I (x, y) \propto {| E (x, y) |}^{2} = [A_{1}^{2} (x, y) + A_{2}^{2} (x, y) + A_{3}^{2} (x, y)] + 2 A_{1} (x, y) A_{2} (x, y) \cos [θ_{1} (x, y) - θ_{2} (x, y) + 2 π (f_{1} - f_{2}) t] + 2 A_{1} (x, y) A_{3} (x, y) \cos [θ_{1} (x, y) - θ_{3} (x, y) + 2 π f_{1} t] + 2 A_{2} (x, y) A_{3} (x, y) \cos [θ_{2} (x, y) - θ_{3} (x, y) + 2 π f_{2} t] .

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C_{1} (x, y) = [θ_{1}^{'} (x, y) - θ_{3}^{'} (x, y)] - [θ_{1} (x, y) - θ_{3} (x, y)],

(10)

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C_{2} (x, y) = [θ_{2}^{'} (x, y) - θ_{3}^{'} (x, y)] - [θ_{2} (x, y) - θ_{3} (x, y)],

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C_{1} (x, y) = \frac{2 π}{λ} (n_{1} - n_{3}) \cdot u (x, y),

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C_{2} (x, y) = \frac{2 π}{λ} (n_{2} - n_{3}) \cdot u (x, y) .

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C_{1} (x, y) = g \cdot u_{y} (x, y),

(14)

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C_{2} (x, y) = g \cdot u_{x} (x, y),

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F_{1} (t) = a \sin 2 π f_{1} t,

(16)

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F_{2} (t) = a \sin 2 π f_{2} t,

(17)

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I (x, y) \propto {| E (x, y) |}^{2} = [A_{1}^{2} (x, y) + A_{2}^{2} (x, y) + A_{3}^{2} (x, y)] + 2 A_{1} (x, y) A_{2} (x, y) \cos [\begin{matrix} θ_{1} (x, y) - θ_{2} (x, y) \end{matrix} \begin{matrix} + a \sin 2 π f_{1} t - a \sin 2 π f_{2} t] \end{matrix} + 2 A_{1} (x, y) A_{3} (x, y) \cos [θ_{1} (x, y) - θ_{3} (x, y) + a \sin 2 π f_{1} t] + 2 A_{2} (x, y) A_{3} (x, y) \cos [θ_{2} (x, y) - θ_{3} (x, y) + a \sin 2 π f_{2} t] .

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\cos [θ_{1} (x, y) - θ_{2} (x, y) + a \sin 2 π f_{1} t - a \sin 2 π f_{2} t] = \cos [θ_{1} (x, y) - θ_{2} (x, y)] \cos (a \sin 2 π f_{1} t) \cos (a \sin 2 π f_{2} t) - \cos [θ_{1} (x, y) - θ_{2} (x, y)] \sin (a \sin 2 π f_{1} t) \sin (a \sin 2 π f_{2} t) - \sin [θ_{1} (x, y) - θ_{2} (x, y)] \sin (a \sin 2 π f_{1} t) \cos (a \sin 2 π f_{2} t) + \sin [θ_{1} (x, y) - θ_{2} (x, y)] \cos (a \sin 2 π f_{1} t) \sin (a \sin 2 π f_{2} t) .

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\cos (a \sin 2 π f_{1} t) \cos (a \sin 2 π f_{2} t) = 4 \sum_{p = 1}^{\infty} \sum_{q = 1}^{\infty} J_{2 p} (a) J_{2 q} (a) \cos (2 p \cdot 2 π f_{1} t) \cos (2 q \cdot 2 π f_{2} t),

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\cos (2 p \cdot 2 π f_{1} t) \cos (2 q \cdot 2 π f_{2} t) = \frac{1}{2} \cos (2 p \cdot 2 π f_{1} t + 2 q \cdot 2 π f_{2} t) + \frac{1}{2} \cos (2 p \cdot 2 π f_{1} t - 2 q \cdot 2 π f_{2} t),

(21)

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Yunlong Zhu, Julien Vaillant, Guillaume Montay, Manuel François, Yassine Hadjar, Aurélien Bruyant. Simultaneous 2D in-plane deformation measurement using electronic speckle pattern interferometry with double phase modulations[J]. Chinese Optics Letters, 2018, 16(7): 071201

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