### Arrangement of universal optical setup

The universal optical setup for PS-DSPI and SC-DSPI adopts a three-channel optical arrangement. Each channel consists of an object and reference beam pair derived from an individual laser. Components in each channel are almost the same, but the laser wavelength can be different. The incident angles, or illumination angles, of the three object beams striking the measuring target are artificially arranged to achieve optimal 3D deformation measurement results. The illumination angles will be discussed later.

The optical arrangement of the universal optical setup is depicted in Fig. 1. Considering the similarity of the three channels, the optical arrangement of only one channel is described to show the optical interference process. The laser beam is divided into object and reference beams by a beam splitter. The object beam then strikes the measuring target after being expanded by a negative lens or other optical components or parts with similar function, such as a microscope objective. The scattered light from the target is collected by an imaging lens, such as an aspheric lens, then reaches the image sensor of the camera via an aperture. The aperture works as a regulator of light intensity in PS-DSPI mode and a filter of spatial frequency in SC-DSPI mode. The reference beam is coupled into an optical fiber via a piezoelectric-transducer-driven mirror. The elongation of the piezoelectric transducer (PZT) is automatically controlled by a computer to modulate the optical path of the reference beam, resulting in the phase shift in the PS-DSPI measurement. The emergent light from the fiber strikes the camera sensor at a small angle between it and the optical axis. This angle determines the carrier frequency, a key parameter in the SC-DSPI measurement. The object and reference beams encounter each other on the camera sensor, resulting in optical interference. The generated speckle interferograms are captured by the camera and recorded by the computer for further processing.

The other two channels follow the same principle, but have different illumination angles and reference beam incident angles. The differences in the incident angles of the reference beams guarantee the separation of the interferometric signals from the three channels in the frequency domain, when the setup works in the SC-DSPI mode. The illumination angle differences among the three channels result in different displacement sensitivity coefficients. The combination of these displacement sensitivity coefficients forms a displacement sensitivity matrix with which the relationship between the 3D deformations and the interferometric phases obtained by PS-DSPI and SC-DSPI is built. The phase determination and deformation calculation procedures are discussed in the next section. Various illumination angle combinations among the three channels yield different displacement sensitivity matrices. Among these combinations, right-angle distribution and homogeneous distribution, described in Fig. 2, are the two simplest and optimal arrangements. In both types, the magnitudes of the illumination angles are equal, but the directions differ.

When PS-DSPI is used to measure 3D deformations, the three channels are enabled in turn by opening the shutters in front of each laser. Only one interferogram, generated by a pair of object and reference beams from a channel, is captured by the camera at a time. The implementation of a round of measurements using the three channels in turn yields three equations which express the mathematical relationship between the interferometric phases and image intensities. When SC-DSPI is used for measurement, the three shutters are opened together, resulting in three pairs of object and reference beams emerging on the camera sensor simultaneously. Each object beam- reference beam pair generates an interferogram, resulting in the simultaneous recording of three independent interferograms. The three interferograms are later separated in the frequency domain after a Fourier transform is performed on them. The interferometric phases are extracted from the separated interferograms after an inverse Fourier transform is performed.

### Phase determination using PS-DSPI

The interferogram generated by the PS-DSPI can be expressed as

$$ I\left(x,y\right)={I}_0\left(x,y\right)+B\left(x,y\right) \cos \left[\phi \left(x,y\right)+2\pi {f}_xx+2\pi {f}_yy\right], $$

(1)

where *I*(*x*, *y*) is the intensity distribution of the interferogram, *I*
_{0}(*x*, *y*) is the background light, *B*(*x*, *y*) is a coefficient correlating with the contrast, *ϕ*(*x*, *y*) is the interferometric phase, *f*
_{
x
} and *f*
_{
y
} indicate the carrier frequencies which are introduced by the slightly deflected reference beams, and (*x*, *y*) indicates the two-dimensional distribution.

The interferogram intensity *I*(*x*, *y*) is recorded by the camera, and the carrier frequencies *f*
_{
x
} and *f*
_{
y
} are determined by the incidence angle of the reference beam, but the three remaining variables in Eq. (1) are unknown, making the equation unsolvable. Additional conditions need to be added to resolve this problem. Typically, the additional condition is a series of artificial phase changes. The method to solve the equation by artificially changing the interferometric phase is known as phase shifting. This method can be further divided into temporal and spatial phase shifting. The temporal phase shifting, which changes the phase over time, is the dominant phase determination method in PS-DSPI due to its ease of use and ability to formulate high-quality phase maps. The number of steps and phase change intervals are multifarious [18]. For example, the popular four-step temporal phase shift changes the phase four times with an interval of *π*/2. As a result, four equations are obtained as

$$ {I}_i\left(x,y\right)={I}_0\left(x,y\right)+B\left(x,y\right) \cos \left[\phi \left(x,y\right)+2\pi {f}_xx+2\pi {f}_yy+\left(i-1\right)\frac{\pi }{2}\right],\left(i=1,2,3,4\right). $$

(2)

Solving Eq. (2) for *ϕ*(*x*, *y*) results in the following expression:

$$ \phi \left(x,y\right)={ \tan}^{\hbox{-} 1}\frac{I_4\left(x,y\right)-{I}_2\left(x,y\right)}{I_1\left(x,y\right)-{I}_3\left(x,y\right)}. $$

(3)

After the measuring target has been deformed, the phase shift is carried out again to determine the interferometric phase according to the deformed state. The phase difference is then determined by simply subtracting the phase before deformation from the phase after deformation. This is expressed as

$$ \varDelta {\phi}_1\left(x,y\right)={\phi}_a\left(x,y\right)-{\phi}_b\left(x,y\right), $$

(4)

where *ϕ*
_{
a
}(*x*, *y*) and *ϕ*
_{
b
}(*x*, *y*) are the phase distributions after and before the deformation, respectively. The other two phase differences Δ*ϕ*
_{2}(*x*, *y*) and Δ*ϕ*
_{3}(*x*, *y*) are determined by performing the same procedure on the other channels.

In the proposed universal optical setup, the phase shift is carried out by the PZT. A PZT elongation of *λ*/8, where *λ* is the laser wavelength, causes a phase shift of *π*/2, which is the amount required by the four-step temporal phase shift. Fine control of a well-calibrated PZT aids in the precise determination of the interferometric phase using PS-DSPI.

### Phase determination using SC-DSPI

Due to the simultaneous recording of the three interferograms in the SC-DSPI mode, the image intensity is the sum of all interferograms, which is expressed by

$$ {I}_s\left(x,y\right)={I}_{s0}\left(x,y\right)+{\displaystyle \sum_{i=1}^3{B}_i\left(x,y\right) \cos \left[{\phi}_i\left(x,y\right)+2\pi {f}_{ix}x+2\pi {f}_{iy}y\right]}, $$

(5)

where *I*
_{
s0}(*x*, *y*) is the sum of the background lights.

Aided by Euler’s formula, Eq. (5) can be transformed to

$$ {I}_s\left(x,y\right)={I}_{s0}\left(x,y\right)+{\displaystyle \sum_{i=1}^3\left[{C}_i\left(x,y\right){e}^{j2\pi \left({f}_{ix}x+{f}_{iy}y\right)}+{C}_i^{*}\left(x,y\right){e}^{-j2\pi \left({f}_{ix}x+{f}_{iy}y\right)}\right]}, $$

(6)

where *C*
_{
i
}(*x*, *y*) = *B*
_{
i
}(*x*, *y*)exp[*jϕ*(*x*, *y*)]/2, * denotes the complex conjugate.

After a Fourier transform is performed, Eq. (6) is transformed to

$$ F\left({f}_{\xi },{f}_{\eta}\right)=FT\left[{I}_s\left(x,y\right)\right]=\mathrm{A}\left({f}_{\xi },{f}_{\eta}\right)+{\displaystyle \sum_{i=1}^3\left[{P}_i\left({f}_{\xi }-{f}_{ix},{f}_{\eta }-{f}_{iy}\right)+{Q}_i\left({f}_{\xi }+{f}_{ix},{f}_{\eta }+{f}_{iy}\right)\right]}, $$

(7)

where *FT* denotes the operation of Fourier transform, (*f*
_{
ξ
}, *f*
_{
η
}) are the coordinates in the frequency domain, and

$$ \left\{\begin{array}{c}\hfill \mathrm{A}\left({f}_{\xi },{f}_{\eta}\right)=FT\left[{I}_{s0}\left(x,y\right)\right]\hfill \\ {}\hfill {P}_i\left({f}_{\xi }-{f}_{ix},{f}_{\eta }-{f}_{iy}\right)=FT\left[{C}_i\left(x,y\right){e}^{j2\pi \left({f}_{ix}x+{f}_{iy}y\right)}\right]\hfill \\ {}\hfill {Q}_i\left({f}_{\xi }+{f}_{ix},{f}_{\eta }+{f}_{iy}\right)=FT\left[{C}_i^{*}\left(x,y\right){e}^{-j2\pi \left({f}_{ix}x+{f}_{iy}y\right)}\right]\hfill \end{array}\right., $$

(8)

Eq. (7) shows there are a total of seven components in the frequency domain, where *P*
_{
i
}(*f*
_{
ξ
} − *f*
_{
ix
}, *f*
_{
η
} − *f*
_{
iy
}) and *Q*
_{
i
}(*f*
_{
ξ
} + *f*
_{
ix
}, *f*
_{
η
} + *f*
_{
iy
}) are three pairs of conjugate components and A(*f*
_{
x
}, *f*
_{
y
}) represents the low-frequency background signal. The locations of *P*
_{
i
}(*f*
_{
ξ
} − *f*
_{
ix
}, *f*
_{
η
} − *f*
_{
iy
}) and *Q*
_{
i
}(*f*
_{
ξ
} + *f*
_{
ix
}, *f*
_{
η
} + *f*
_{
iy
}) are determined by the carrier frequencies *f*
_{
ix
} and *f*
_{
iy
}. All seven components can be well separated by fine adjustment of the incidence angles of the reference beams and the aperture in the universal optical setup. To intuitively describe the frequency spectrum obtained by SC-DSPI, Fig. 3 illustrates a distribution of the seven components that was generated by the proposed optical setup in an experiment. More information about the synchronous recording and separation of the multiple interferograms can be found in Refs. [19] and [20].

Since both *P*
_{
i
}(*f*
_{
ξ
} − *f*
_{
ix
}, *f*
_{
η
} − *f*
_{
iy
}) and *Q*
_{
i
}(*f*
_{
ξ
} + *f*
_{
ix
}, *f*
_{
η
} + *f*
_{
iy
}) contain the same interferometric phase, either of them can be used for phase extraction. This is realized by applying an inverse Fourier transform on the selected component and performing further calculations. For example, if *P*
_{
i
}(*f*
_{
ξ
} − *f*
_{
ix
}, *f*
_{
η
} − *f*
_{
iy
}) is chosen, the phase distribution according to the first channel is

$$ {\phi}_1\left(x,y\right)={ \tan}^{\hbox{-} 1}\frac{IM\left[{p}_1\left(x,y\right)\right]}{RE\left[{p}_1\left(x,y\right)\right]}, $$

(9)

where *IM* and *RE* denote imaginary and real parts of the complex number and

$$ {p}_1\left(x,y\right)={\mathrm{FT}}^{\hbox{-} 1}\left[{P}_1\left({f}_{\xi }-{f}_{1x},{f}_{\eta }-{f}_{1y}\right)\right], $$

(10)

where FT^{‐ 1} is the inverse Fourier transform operation.

The phases according to the other two channels, as well as the phases after deformation, are obtained by the same means. Finally, three individual phase difference distributions Δ*ϕ*
_{1}(*x*, *y*), Δ*ϕ*
_{2}(*x*, *y*) and Δ*ϕ*
_{3}(*x*, *y*) are determined by subtracting the phases before deformation from the corresponding phases after deformation.

### Calculation of 3D deformations

The relationship between the deformation and interferometric phase difference in PS-DSPI and SC-DSPI can be expressed by

$$ \varDelta \phi \left(x,y\right)=\frac{2\pi }{\lambda}\overrightarrow{d}\left(x,y\right)\overrightarrow{s}\left(x,y\right), $$

(11)

where Δ*ϕ*(*x*, *y*) is the phase difference, \( \overrightarrow{d}\left(x,y\right) \) is the deformation vector, and \( \overrightarrow{s}\left(x,y\right) \) is the displacement sensitivity vector, which is dependent on the illumination angles.

If the right-angle-distribution optical arrangement is used, Eq. (11) can be transformed to

$$ \left\{\begin{array}{c}\hfill \varDelta {\phi}_1\left(x,y\right)=\frac{2\pi }{\lambda_1}\left[u\left(x,y\right) \sin \alpha +w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \\ {}\hfill \varDelta {\phi}_2\left(x,y\right)=\frac{2\pi }{\lambda_2}\left[v\left(x,y\right) \sin \alpha +w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \\ {}\hfill \varDelta {\phi}_3\left(x,y\right)=\frac{2\pi }{\lambda_3}\left[u\left(x,y\right) \sin \left(-\alpha \right)+w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \end{array}\right., $$

(12)

where *λ*
_{1}, *λ*
_{2}, and *λ*
_{3} are the wavelengths of the three lasers; *u*(*x*, *y*), *v*(*x*, *y*), and *w*(*x*, *y*) are the three components of \( \overrightarrow{d}\left(x,y\right) \) in three dimensions, and *α* is the illumination angle.

To simplify the calculation, all laser wavelengths are assumed to be the same. This assumption, used with Eq. (12), results in the following expressions for the three deformation vector components:

$$ \left\{\begin{array}{c}\hfill u\left(x,y\right)=\frac{\lambda }{4\pi \sin \alpha}\left[\varDelta {\phi}_1\left(x,y\right)-\varDelta {\phi}_3\left(x,y\right)\right]\hfill \\ {}\hfill v\left(x,y\right)=\frac{\lambda }{4\pi \sin \alpha}\left[2\varDelta {\phi}_2\left(x,y\right)-\varDelta {\phi}_1\left(x,y\right)-\varDelta {\phi}_3\left(x,y\right)\right]\hfill \\ {}\hfill w\left(x,y\right)=\frac{\lambda }{4\pi \left(1+ \cos \alpha \right)}\left[\varDelta {\phi}_1\left(x,y\right)+\varDelta {\phi}_3\left(x,y\right)\right]\hfill \end{array}\right.. $$

(13)

For the homogeneous-distribution optical arrangement, Eq. (11) becomes

$$ \left\{\begin{array}{c}\hfill \varDelta {\phi}_1\left(x,y\right)=\frac{2\pi }{\lambda_1}\left[u\left(x,y\right) \sin \alpha \cos {30}^{\bigcirc }+v\left(x,y\right) \sin \alpha \cos {60}^{\bigcirc }+w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \\ {}\hfill \varDelta {\phi}_2\left(x,y\right)=\frac{2\pi }{\lambda_2}\left[v\left(x,y\right) \sin \alpha +w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \\ {}\hfill \varDelta {\phi}_3\left(x,y\right)=\frac{2\pi }{\lambda_3}\left[u\left(x,y\right) \sin \left(-\alpha \right) \cos {30}^{\bigcirc }+v\left(x,y\right) \sin \left(-\alpha \right) \cos {60}^{\bigcirc }+w\left(x,y\right)\left(1+ \cos \alpha \right)\right]\hfill \end{array}\right.. $$

(14)

If the laser wavelengths are assumed to be the same, the deformation vector components have the following expression:

$$ \left\{\begin{array}{c}\hfill u\left(x,y\right)=\frac{\sqrt{3}\lambda }{12\pi \sin \alpha}\left[3\varDelta {\phi}_1\left(x,y\right)-2\varDelta {\phi}_2\left(x,y\right)-\varDelta {\phi}_3\left(x,y\right)\right]\hfill \\ {}\hfill v\left(x,y\right)=\frac{\lambda }{4\pi \sin \alpha}\left[2\varDelta {\phi}_2\left(x,y\right)-\varDelta {\phi}_1\left(x,y\right)-\varDelta {\phi}_3\left(x,y\right)\right]\hfill \\ {}\hfill w\left(x,y\right)=\frac{\lambda }{4\pi \left(1+ \cos \alpha \right)}\left[\varDelta {\phi}_1\left(x,y\right)+\varDelta {\phi}_3\left(x,y\right)\right]\hfill \end{array}\right.. $$

(15)

The solutions of *v*(*x*, *y*) and *w*(*x*, *y*) are the same for both the right-angle-distribution and homogeneous-distribution types, but the solutions of *u*(*x*, *y*) are different.