In-plane deformations can be easily measured by a simple electronic speckle pattern interferometry (ESPI) measuring system^[1]. In such a system, the temporal phase-shifting technique is often applied for phase retrieval to improve the performance^[2]. However, in a standard two-beam configuration, only one displacement component is measured.

In order to measure the two-dimensional (2D) in-plane displacement field [or the whole three-dimensional (3D) displacement field], several solutions have been proposed. The most direct one is to use digital speckle photography (DSP) (or to combine DSP with out-of-plane ESPI for the 3D measurement)^[3]. Nevertheless, the sensitivity of DSP is not as good as that of ESPI. A natural solution is then to combine ESPI measuring systems^[4,5]. More recently, in-plane ESPI measurement systems were notably combined with out-of-plane ESPI to perform 3D analysis using optical switches, though with a limited time resolution (for example, the acquisition time for the “$\mathrm{Q}$-300 3D-ESPI System” is 3.5 s for 3D analysis), due to the iterative process requirement^[6–8]. To solve the time issue, a spatial phase-shifting technique can be applied^[6]; however, it often makes the system substantially more complex and expensive.

In this Letter, we show the possibility of doing simultaneous 2D measurement using the widely recognized ESPI technique and a single laser without switches. The optical arrangement is shown in Fig. 1. There are three coherent laser beams originating from a single laser: Beam 1, Beam 2, and Beam 3. The phases of Beam 1 and Beam 2 can be modulated by the corresponding piezo-actuated mirrors. When two temporal phase modulation functions, $F_1(t)$ and $F_2(t)$, are applied to them, respectively, the scalar light field of the subjective speckles $E(x,y)$ can be expressed as $$\begin{gathered} E(x,y)=A_1(x,y)e^{i[2\pi f_{c}t+{\theta }_1(x,y)+F_1(t)]} \\ +A_2(x,y)e^{i[2\pi f_{c}t+{\theta }_2(x,y)+F_2(t)]} \\ +A_3(x,y)e^{i[2\pi f_{c}t+{\theta }_3(x,y)]}. \end{gathered}$$(1)

$A_m(x,y)$ and ${\theta }_m(x,y)$ are the amplitude and the initial phase of Beam $m$ ($m=1,2,3$) at point ($x$, $y$), respectively, and $f_c$ is the optical frequency of the laser.

On the sample surface, the light intensity $I(x,y)$ can be expressed as $$\begin{gathered} I(x,y)\propto {|E(x,y)|}^2 \\ =[A_1^2(x,y)+A_2^2(x,y)+A_3^2(x,y)] \\ +2A_1(x,y)A_2(x,y)\cos [{\theta }_1(x,y)+F_1(t)-{\theta }_2(x,y)-F_2(t)] \\ +2A_1(x,y)A_3(x,y)\cos [{\theta }_1(x,y)+F_1(t)-{\theta }_3(x,y)] \\ +2A_2(x,y)A_3(x,y)\cos [{\theta }_2(x,y)+F_2(t)-{\theta }_3(x,y)]. \end{gathered}$$(2)

After a small displacement $\mathit{u}(x,y)$, the light intensity turns into $$\begin{gathered} I^{\prime }(x,y)\propto {|E^{\prime }(x,y)|}^2 \\ =[A_1^2(x,y)+A_2^2(x,y)+A_3^2(x,y)] \\ +2A_1(x,y)A_2(x,y)\cos [{\theta }_1^{\prime }(x,y)+F_1(t)-{\theta }_2^{\prime }(x,y)-F_2(t)] \\ +2A_1(x,y)A_3(x,y)\cos [{\theta }_1^{\prime }(x,y)+F_1(t)-{\theta }_3^{\prime }(x,y)] \\ +2A_2(x,y)A_3(x,y)\cos [{\theta }_2^{\prime }(x,y)+F_2(t)-{\theta }_3^{\prime }(x,y)], \end{gathered}$$(3)with $${\theta }_1^{\prime }(x,y)={\theta }_1(x,y)+\frac{2\pi }{\lambda }({\mathit{n}}_1-{\mathit{n}}_s)·\mathit{u}(x,y),$$(4)$${\theta }_2^{\prime }(x,y)={\theta }_2(x,y)+\frac{2\pi }{\lambda }({\mathit{n}}_2-{\mathit{n}}_s)·\mathit{u}(x,y),$$(5)$${\theta }_3^{\prime }(x,y)={\theta }_3(x,y)+\frac{2\pi }{\lambda }({\mathit{n}}_3-{\mathit{n}}_s)·\mathit{u}(x,y),$$(6)where $\lambda$ is the wavelength of the laser, ${\mathit{n}}_m$ is the unit vector along the illumination direction of Beam $m$ ($m=1,2,3$), and ${\mathit{n}}_s$ is the unit vector along the viewing direction. ${\mathit{n}}_m$ and ${\mathit{n}}_s$ can be roughly considered to be the same for every point ($x$, $y$) on the sample surface.

If we choose the following linear (or sawtooth) modulation functions: $$F_1(t)=2\pi f_{1}t,$$(7)$$F_2(t)=2\pi f_{2}t,$$(8)then we have $$\begin{gathered} I(x,y)\propto {|E(x,y)|}^2 \\ =[A_1^2(x,y)+A_2^2(x,y)+A_3^2(x,y)] \\ +2A_1(x,y)A_2(x,y)\cos [{\theta }_1(x,y)-{\theta }_2(x,y)+2\pi (f_1-f_2)t] \\ +2A_1(x,y)A_3(x,y)\cos [{\theta }_1(x,y)-{\theta }_3(x,y)+2\pi f_{1}t] \\ +2A_2(x,y)A_3(x,y)\cos [{\theta }_2(x,y)-{\theta }_3(x,y)+2\pi f_{2}t]. \end{gathered}$$(9)

Obviously, when $f_1$, $f_2$ and $|f_1-f_2|$ are not equal to each other with a lock-in detection at $f_1$, ${\theta }_1(x,y)-{\theta }_3(x,y)$ can be extracted; with a lock-in detection at $f_2$, ${\theta }_2(x,y)-{\theta }_3(x,y)$ can be extracted^[9]. The same procedure can be carried out to obtain ${\theta }_1^{\prime }(x,y)-{\theta }_3^{\prime }(x,y)$ and ${\theta }_2^{\prime }(x,y)-{\theta }_3^{\prime }(x,y)$. If we set $$C_1(x,y)=[{\theta }_1^{\prime }(x,y)-{\theta }_3^{\prime }(x,y)]-[{\theta }_1(x,y)-{\theta }_3(x,y)],$$(10)$$C_2(x,y)=[{\theta }_2^{\prime }(x,y)-{\theta }_3^{\prime }(x,y)]-[{\theta }_2(x,y)-{\theta }_3(x,y)],$$(11)then, according to Eq. (4)–(6), we have $$C_1(x,y)=\frac{2\pi }{\lambda }({\mathit{n}}_1-{\mathit{n}}_3)·\mathit{u}(x,y),$$(12)$$C_2(x,y)=\frac{2\pi }{\lambda }({\mathit{n}}_2-{\mathit{n}}_3)·\mathit{u}(x,y).$$(13)

The $z$ components of ${\mathit{n}}_1$, ${\mathit{n}}_2$, and ${\mathit{n}}_3$ are almost equal, so they will cancel each other out in Eqs. (12) and (13). Concerning the $x$ and $y$ components of ${\mathit{n}}_1$, ${\mathit{n}}_2$, and ${\mathit{n}}_3$, we can see from Fig. 1 that $({\mathit{n}}_1-{\mathit{n}}_3)$ is parallel to the $Y$ axis, and $({\mathit{n}}_2-{\mathit{n}}_3)$ is parallel to the $X$ axis. So, $C_1(x,y)$ and $C_2(x,y)$ can be expressed as $$C_1(x,y)=g·u_y(x,y),$$(14)$$C_2(x,y)=g·u_x(x,y),$$(15)where $g$ is a measurable constant, $u_x(x,y)$ and $u_y(x,y)$ are the $x$ and $y$ components of $\mathit{u}(x,y)$, respectively. This means the 2D in-plane displacement field can be measured. The whole procedure is briefly illustrated in Fig. 2.

This method can also be extended to carry out 3D displacement field measurements without increasing acquisition time and without an additional laser or camera: if the laser is separated into a fourth coherent beam and its direction of illumination is ${\mathit{n}}_4$, then this system can be used to measure the 3D displacement field, as long as ${\mathit{n}}_4-{\mathit{n}}_3$ has a non-zero $z$ component, and this fourth beam is modulated at an appropriate frequency.

It should be noticed that when piezoelectric actuators are driven to make sawtooth displacements, the precision cannot be guaranteed, especially at a high frequency, where the fly-back time of the mirror cannot be neglected. The nonlinearity and noise generated by the sudden return becomes unacceptable when high-speed measurement is required. This issue can be addressed with sinusoidal phase modulations such as $$F_1(t)=a\sin 2\pi f_{1}t,$$(16)$$F_2(t)=a\sin 2\pi f_{2}t,$$(17)where $a$ is the amplitude of phase modulation. It should be noticed that $f_1$ and $f_2$ are not randomly chosen. It is favorable to choose coprime integers, as will be detailed later. Now we have $$\begin{gathered} I(x,y)\propto {|E(x,y)|}^2 \\ =[A_1^2(x,y)+A_2^2(x,y)+A_3^2(x,y)] \\ +2A_1(x,y)A_2(x,y)\cos [\begin{array}{c}{\theta }_1(x,y)-{\theta }_2(x,y)\end{array} \\ \begin{array}{c}+a\sin 2\pi f_{1}t-a\sin 2\pi f_{2}t]\end{array} \\ +2A_1(x,y)A_3(x,y)\cos [{\theta }_1(x,y)-{\theta }_3(x,y) \\ +a\sin 2\pi f_{1}t] \\ +2A_2(x,y)A_3(x,y)\cos [{\theta }_2(x,y)-{\theta }_3(x,y) \\ +a\sin 2\pi f_{2}t]. \end{gathered}$$(18)

In Eq. (18), according to the Jacobi–Anger expansion, in the frequency domain, the third term will be distributed at the integer multiples of $f_1$, and the fourth term will be distributed at the integer multiples of $f_2$. If we set $a=2.4048\mathrm{rad}$, so that $J_0(a)=0$ ($J_0$ is the zeroth Bessel function of the first kind), then both of them will not contain any signal at 0 Hz. So, in the frequency domain, they will not overlap with each other until the least common multiple of $f_1$ and $f_2$, which is 63 Hz in our case ($f_1=9\mathrm{Hz}$, $f_2=7\mathrm{Hz}$, 9 and 7 are coprime integers). This means these two terms can be efficiently separated.

As for the second term in Eq. (18), we can make use of the trigonometric formulas (sum/difference identities) to get $$\begin{gathered} \cos [{\theta }_1(x,y)-{\theta }_2(x,y)+a\sin 2\pi f_{1}t-a\sin 2\pi f_{2}t] \\ =\cos [{\theta }_1(x,y)-{\theta }_2(x,y)]\cos (a\sin 2\pi f_{1}t)\cos (a\sin 2\pi f_{2}t) \\ -\cos [{\theta }_1(x,y)-{\theta }_2(x,y)]\sin (a\sin 2\pi f_{1}t)\sin (a\sin 2\pi f_{2}t) \\ -\sin [{\theta }_1(x,y)-{\theta }_2(x,y)]\sin (a\sin 2\pi f_{1}t)\cos (a\sin 2\pi f_{2}t) \\ +\sin [{\theta }_1(x,y)-{\theta }_2(x,y)]\cos (a\sin 2\pi f_{1}t)\sin (a\sin 2\pi f_{2}t). \end{gathered}$$(19)

If we analyze the first term in Eq. (19) with the Jacobi–Anger expansion when $J_0(a)=0$, then we have $$\begin{gathered} \cos (a\sin 2\pi f_{1}t)\cos (a\sin 2\pi f_{2}t) \\ =4\sum _{p=1}^{\infty }\sum _{q=1}^{\infty }{J}_{2p}(a)J_{2q}(a)\cos (2p·2\pi f_{1}t)\cos (2q·2\pi f_{2}t), \end{gathered}$$(20)with $$\begin{gathered} \cos (2p·2\pi f_{1}t)\cos (2q·2\pi f_{2}t) \\ =\frac{1}{2}\cos (2p·2\pi f_{1}t+2q·2\pi f_{2}t) \\ +\frac{1}{2}\cos (2p·2\pi f_{1}t-2q·2\pi f_{2}t), \end{gathered}$$(21)where $p$ and $q$ are positive integers. Through Eq. (21), we know that this term contains signals at the frequencies of ($|2p{f}_1\pm 2q{f}_2|$). Likewise, by analyzing the other three terms in Eq. (19), it can be concluded that the second term in Eq. (18) contains ($|p{f}_1\pm q{f}_2|$) signals.

Since $p$ and $q$ are positive integers, the solutions for $|p{f}_1\pm q{f}_2|=n{f}_1$ or $|p{f}_1\pm q{f}_2|={nf}_2$ (n is a positive integer) can only be found when $p$ or $q$ is relatively large, which represents the high-frequency signal where the intensity is usually very small [since $J_p(a)\approx 0$ or $J_q(a)\approx 0$]. So, we can estimate that it will not have too much influence on the interesting frequencies. A simple simulation is done with $f_1=9\mathrm{Hz}$, and $f_2=7\mathrm{Hz}$, and the result is shown in Fig. 3 to illustrate this point.

We can now draw a direct link between the four terms in Eq. (18) and the frequency spectrum. The first term corresponds to the signal at 0 Hz, the third term corresponds to signals at $p{f}_1$, the fourth term corresponds to signals at $q{f}_2$, and the second term corresponds to signals at other frequencies. So, just like using linear modulation, information can be easily sorted out so that the 2D displacement field can be measured. In fact, by simply replacing the lock-in detection algorithm with the traditional sinusoidal phase modulation algorithm (see Ref. [10]) or the recently proposed generalized lock-in detection algorithm (see Refs. [9,11,12]), the needed phase information can be obtained. Likewise, while dealing with the same set of data, if we set the demodulation frequency at $f_1$ in the algorithm, then we can get $C_1(x,y)$; if we set it at $f_2$, then we can get $C_2(x,y)$. With $C_1(x,y)$ and $C_2(x,y)$, the 2D displacement field $u_x(x,y)$ and $u_y(x,y)$ can be obtained.

We used a bending specimen, as shown in Fig. 4. It is a test sample fabricated by the company HOLO 3^[13].

First, we make sure that there is already an initial contact between the micrometer screw and the bending specimen. Then, the two phase modulations are turned on, and a short video (1 s, 63 frames per second) is recorded. Likewise, we record another video after turning the micrometer screw so that the deformation state changes. By analyzing these two videos, we can measure the 2D displacement field.

When applying sinusoidal phase modulations described by Eqs. (16) and (17) with $f_1=9\mathrm{Hz}$, and $f_2=7\mathrm{Hz}$, we successfully obtained phase images ($C_1$ and $C_2$), as shown in Fig. 5. The fringe visibility is very good; besides, very fine fringes can be observed on the left part of Figs. 5(a) and 5(b).

From the obtained phase images (Fig. 5), we can quantitatively measure the 2D deformation (Fig. 6). First, the original phase images [Figs. 6(a) and 6(b)] were filtered with the conventional 2D convolution method [see Figs. 6(c) and 6(d)]. Then, they are 2D unwrapped to get smooth phase images, and the displacements $u_y$ and $u_x$ [Figs. 6(e) and 6(f)] can be calculated by Eqs. (14) and (15). The strains ${\epsilon }_y$, ${\epsilon }_x$, and ${\gamma }_{xy}$ can be quantitatively measured [Figs. 6(g), 6(h), 6(i)] for any choice of origins for $u_y$ and $u_x$.

When applying linear/sawtooth phase modulations described by Eqs. (7) and (8), similar fringes (see Fig. 7) are obtained, since the modulation frequencies are quite low ($f_1=9\mathrm{Hz}$, and $f_2=7\mathrm{Hz}$). However, the sawtooth approach will become much less efficient at higher speeds. There are some small differences in the fringe pattern, which are mainly due to phase noise, initial phase adjustment, and the fact that the loading processes were done manually and are not perfectly reproducible.

Compared to previous reports of 2D in-plane displacement field measurements, the proposed approach is much simpler with only one laser and one camera; yet, high-quality fringes have been obtained. A camera with moderate speed (63 frames per second) is used; still, the data acquisition time (1 s for 2D information) is even a little advantageous over some commercialized systems (e.g. 3.5 s for 3D information^[8]). Obviously, this system has the potential to be operated at a high speed while providing accurate results by using the sinusoidal phase modulation together with a high-speed camera. Although the relatively voluminous data may be a challenge for lower-end computers to carry out real-time analysis, it does not seem to be a problem for the future, since the semiconductor industry is developing rapidly. Last but not least, this approach has the potential to carry out simultaneous ESPI measurements of the 3D displacement field.