Optical interferometry has been attractive for characterizing optical phenomena in physics, chemistry, and engineering because of its inherent high precision and well-established techniques for extracting the contrast and phase of an interferogram [1–3]. In dimensional metrology, it also plays a very important role in measuring displacements/distances [4], reconstructing surfaces of specimens [5], and characterizing film thicknesses [6]. The fundamentals of optical interferometry are centered on comparing a measured wavefront to a corresponding reference, according to the wavelength of light. Hence the reference wavefront should be well-defined and accurately determined. In the case of optical surface metrology, the reference wavefront needs to be planar or spherical for measuring the surface profile of a specimen; It can be theoretically and experimentally well-calibrated with several methods, such as spatial filtering [7] and nulling techniques [8].

On the other hand, shearing interferometry has different features compared to typical interferometry, as an aspect of generating its interference. The measurement wavefront in shearing interferometry is split into two, and they are interfered with each other without any reference [9]. Because of this self-interference in shearing interferometry, the effort to prepare the reference wavefront is no longer made, and the technique is more flexible for use in optical metrology. Instead, a device to generate two or more shearing wavefronts should be considered, and a wavefront-reconstruction procedure is additionally needed.

In this review paper, we investigate recent publications regarding shearing interferometry, reported over the last two decades. The type of shearing interferometry is categorized according to its fundamental principle for generating the interference, and each research trend is provided, with a summary of experimental results in its application fields. This review paper consists of the theoretical background for shearing interferometers, the optical configurations, analysis, and perspective. Even though not all recently reported shearing interferometers can be introduced in this paper, we believe this review contains the most important progress and theoretical and experimental results in shearing interferometry. It is noted that this work is restricted and focused on typical shearing interferometers with spatially sheared wavefronts, not exceptional ones, such as the spectral-shearing interferometers used to characterize the ultrashort pulses and materials in laser physics [10, 11], or self-mixing interferometers to measure displacements [12, 13].

When the wavefront of interest is incident upon a shearing interferometer, as shown in Fig. 1(a), it is split into more than two wavefronts, and they are interfered with each other.

Figure 1. Optical layout. (a) Shearing interferometry, and (b) typical interferometry.

Because of the absence of a reference, the wavefront is not compared to any typical shape, such as a plane or a sphere. Instead, the differences between two sheared wavefronts are contained in the interference fringe, and the gradient map of the wavefront can be obtained by phase extraction. Even though further analysis for wavefront reconstruction from the gradient map should be implemented in the shearing interferometer, it is free from the preparation and calibration of a reference wavefront, mostly important in two-arm interferometers, as shown in Fig. 1(b).

In general, a shearing interferometer can be categorized as rotational/azimuthal [14], radial [15], or lateral [16] by the way it shears the wavefronts, as shown in Fig. 2. In an azimuthal shearing interferometer (ASI), the azimuthal angles of two wavefronts are slightly different from each other, and the rotational gradient map can be obtained from the interference fringe. In theory, the wavefronts are not laterally shifted, and their sizes are exactly the same. In this case, axially asymmetric features of the wavefront such as coma and astigmatism aberrations are detected with high sensitivity, while ASI does not provide any phase information for the axially symmetric shapes like defocus and spherical aberrations, as shown in Fig. 3. In a radial shearing interferometer (RSI), one of the wavefronts is radially contracted and the other is extended, to obtain the interference in the overlapping region, as shown in Fig. 2. Based on the radial shearing ratio between two wavefronts, the axially symmetric wavefront can be reconstructed from the radial gradient map, as shown in Fig. 4.

Figure 2. Wavefront shearing by three kinds of shearing interferometers.

Figure 3. Rotational gradient maps by azimuthal shearing interferometry.

Figure 4. Radial gradient maps by radial shearing interferometry.

On the other hand, a lateral shearing interferometer (LSI) generates two sheared wavefronts to obtain the surface gradient map along the x- or y-directions. Even though two orthogonal gradient maps, i.e. both the x- and y-directional gradient maps, should be measured to successfully reconstruct the original wavefront in LSI, as shown in Fig. 5, this approach is more suitable for wavefront reconstruction because the wavefront does not have any of the symmetry issues indicated in ASI and RSI. In addition, LSI is more convenient for measuring two surface gradient maps, with a simple modification of the rotation of the lateral shearing device. In fact, the combination of ASI and RSI could realize reconstruction of the wavefront independent of the wavefront-symmetry issue, but their optical configurations are too different from each other to be configured together. Instead, ASI and RSI have been used respectively in measuring the aforementioned asymmetric and symmetric features of the wavefront.

Figure 5. Lateral gradient maps by lateral shearing interferometry.

In shearing interferometry, a temporally and spatially coherent light source is typically used, because two sheared wavefronts should be interfered. When using a broadband light source, the optical-path difference between two wavefronts needs to be within the temporal coherence length of the source, while the shearing amount is restricted by the spatial coherence of the extended source.

3.1. Azimuthal Shearing Interferometry

Most ASIs are typically implemented with a dove prism for image rotation, as shown in Fig. 6. When the wavefront is incident to a Mach-Zehnder interferometer, two wavefronts are rotated each by the angle of its own Dove prism, and the interference fringe can be obtained.

Figure 6. Optical configuration for azimuthal shearing interferometry: BS, beam splitter; M, mirror.

Based on the detection of only axially asymmetric wavefronts in ASI, it has been recently applied to detecting extrasolar planets [17, 18]. An on-axis (star) and an off-axis (planet) point source were located in the simulator of a planetary system, and the interference fringes were collected as one of the Dove prisms in the ASI was rotated, as shown in Fig. 7. Because the wavefront from the star was axially symmetric, the interference fringe was null, and no fringe variations were observed. In the case of the wavefront from the planet, the interference fringe varied according to the rotation of the Dove prism, due to its axially asymmetric feature. Then the existence of the planet could be predicted by observing the fringe variations in ASI.

Figure 7. The optical layout and the interferograms of the simulated solar system with two point sources: DF, neutral density filter; SF, spatial filter; M, mirror; BS, beam splitter; DP, Dove prism; OP, observation plane. The star beam is aligned with the RSI’s optical axis, and the planet beam is inclined with respect to the star beam. Reprinted with permission from [17] Copyright © 2020, The Optical Society.

In addition, ASI can be also applied to flipping interferometry by using a retroreflector or an equivalent right-angle prism with two beam splitters, for quantitative phase microscopy and asymmetric aberration detection as simple configurations [19, 20]. As shown in Fig. 8, the shearing device was composed of two beam splitters and the two wavefronts generated by this prism assembly were flipped with respect to each other. The wavefront rotation was realized by the rotation of the prism assembly [20]. Another simple way to obtain two azimuthal shearing wavefronts is to use a grating pair. Although it includes moiré fringes similar to the lateral shearing pattern [21, 22], this simple technique was very useful for characterizing an X-ray wavefront, to minimize the optical components.

Figure 8. Flipped/reversal and rotational shearing interferometer with two beam splitters. Reprinted from Opt. Commun. 2004; 233; 245-252, Copyright © 2004, with permission from Elsevier [20].

3.2. Radial Shearing Interferometry

For axially symmetric surface-figure or wavefront measurements, RSI is an effective tool compared to other interferometric techniques, because of its stability and compactness. Moreover, RSI only needs a single radial gradient map, as opposed to LSI, which requires two orthogonal gradient maps. One stable RSI is based on a cyclic interferometer with a zoom-lens system [23–25], i.e. a Sagnac interferometer, as shown in Fig. 9(a), in which the contracted and expanded wavefronts pass through the same optical components and most environmental errors can be reduced because of its common-path configuration. For the instrumentation of RSI, polarizing optical components have been introduced in cyclic interferometers, and a polarization camera where an array of four polarizers with 0°, 45°, 90°, and 135°-rotated transmission axes were pixelated, was used to immediately calculate the phase map, as shown in Fig. 9(b), without temporal phase shifting [24, 25].

Figure 9. Optical layout and principle of cyclic radial shearing interferometers. (a) Cyclic radial shearing interferometer with a zoom-lens system, and (b) snapshot cyclic radial shearing interferometer using a polarization camera: PBS, polarizing beam splitter; QWP, 45°-rotated quarter-wave plate; M1, M2, mirrors; L, lens; PCMOS, polarization pixelated complementary metal-oxide-semiconductor camera. The inset is the structure of the PCMOS. Reprinted with permission from [24] Copyright © 2020, The Optical Society.

Recently, a compact snapshot RSI has been presented as a wavefront-measuring sensor [26] and a surface-figure metrological tool [27] with a geometric phase lens (GPL) and a polarization camera, as shown in Fig. 10(a). Similar to a zone plate, a GPL has the ability to generate two distinguishable wavefronts, one convergent and the other divergent, as shown in Fig. 10(b), which can be used for RSI by the scheme of a GPL pair. In this case, the polarization and diffraction characteristics of a GPL maximize the diffraction efficiency of two diffracted wavefronts, and enable us to instantaneously measure the wavefront using a polarization camera. In this research, the dynamic wavefront measurements were demonstrated [26] and surface-figure measurements with various spherical wavefronts to extend the field of view were verified [27].

Figure 10. Principle of dynamic wavefront sensor. (a) Schematic of a radial shearing wavefront sensor using a geometric phase lens (GPL) pair and polarization pixelated complementary metal-oxide-semiconductor camera (PCMOS), and (b) characteristic response of a GPL. Reprinted with permission from [26] Copyright © 2022, The Optical Society.

3.3. Lateral Shearing Interferometry

Among all shearing interferometers, most researchers have focused on LSI because of their unrestricted applications to measure wavefronts and surface figures. Furthermore, their optical configuration and implementation are more straightforward than for other shearing interferometers. The simplest way to realize LSI is to use a shearing plate or a wedge prism to confirm the collimation of the light beam. However, LSI has been applied to more extensive areas, such as adaptive optics [28–30], freeform surface metrology [31–33], and biomedical diagnosis [34–40].

In LSI, three issues to confirm its benefit in the application areas mentioned above have been considered: Convenient adjustment of the lateral shearing amount, compact common-path configuration, and snapshot capability.

The amount of lateral shearing is the most important parameter in LSI, and measurement performance aspects such as sensitivity and precision are strongly dependent on it [41]. When the measurand is slowly varying, the lateral shearing should increase to obtain a lateral gradient with a high signal-to-noise ratio (SNR), and vice versa in the case of drastically varying objects. To adjust the amount of lateral shearing, a Michelson interferometer with retroreflectors was used in the shearing part [42]. In a cyclic interferometer, the tip-and-tilt motion of a beam splitter or a mirror was introduced to produce the off-axis interferogram [43], as shown in Fig. 11.

Figure 11. Schematic of light beams in cyclic lateral shearing interferometry: (a) P-polarized component, (b) s-polarized component. Reprinted with permission from [43] Copyright © 2017, The Optical Society.

Even though interferometric shearing devices were widely used in LSI because of their intuitive convenience, much effort has been put into the compactness of the optical layout, to extend its applicability in various areas. The use of a birefringent prism [44, 45] and a grating [46–51] significantly reduced the system complexity for observing dynamic physical phenomena. The adoption of LSI with a two-dimensional (2D) grating (so-called quadriwave LSI [47–51] as shown in Fig. 12) makes the system, which simultaneously yields x- and y-directional gradient maps, compact.

Figure 12. Schematic of a quadriwave radial shearing interferometer. Reprinted from T. Ling et al. Sci. Rep. 2017; 7; 9 [51], Copyright © 2017, T. Ling et al.

Besides, a polarization grating (one of the geometric phase components) has been used to generate two orthogonally polarized and laterally sheared wavefronts [41].

One of the main research themes in LSI has been snapshot measurement capability, to reduce the measurement time and minimize the noise caused by environmental variations. The traditional way to extract the phase map from the lateral shearing interferogram is based on temporal phase shifting, which makes snapshot measurement difficult. However, the spatial carrier-frequency method, using a 2D Fourier transformation and spatial filtering, has been adopted [47–55] and the phase map could be obtained with a single image, as shown in Fig. 13.

Figure 13. Spatial carrier-frequency method using a 2D Fourier transformation. Reprinted from T. Ling et al. Sci. Rep. 2017; 7; 9 [51], Copyright © 2017, T. Ling et al.

Although this spatial carrier-frequency method sacrifices lateral resolution in the phase map, it is sufficient for reconstructing the wavefronts and surface figures in LSI based on wedge prism and grating. Another technique to immediately obtain the phase map is to use a polarization camera, as introduced in RSI [41, 56].

3.4. Wavefront-reconstruction Algorithm

Once the gradient maps in shearing interferometry are ready, the original wavefront needs to be reconstructed from them, which is also important in wavefront-measuring devices such as Shack-Hartmann sensors. The wavefront can be reconstructed from its gradient using two different approaches, as shown in Fig. 14. One is direct integration of the gradient values, the so-called zonal method, while the other is based on combination of well-defined mathematical basis functions, the modal method.

Figure 14. Categorization of wavefront-reconstruction methods.

In the zonal method, the calculation requires much effort and time, but the wavefront can be reconstructed in greater detail. Most of the recent approaches using the zonal method have been implemented in LSI and based on the Southwell geometry, where the wavefronts to be calculated coincide with their local slope measurements. From the fundamental theory of Southwell’s reconstruction algorithm, modified algorithms have been proposed to improve the accuracy [57, 58], reduce the computing time [59], and confirm the convergence of the result. To improve reconstruction accuracy, more gradient data were included in the integration formula, with diagonal gradients beyond the horizontal and vertical ones [57], as shown in Fig. 15.

Figure 15. Algorithm of zonal methods. (a) Grid-sampling geometry for the zonal wavefront-reconstruction method, and (b) domain-divided. Reprinted from [57] Copyright © 2022, Optical Society of Korea.

To significantly reduce the calculation time, the wavefront to be reconstructed was divided and an optimal zonal block based on the computational complexity was determined [59]. By reasonable wavefront division, the total computation time was much less than that for the typical zonal method.

On the other hand, wavefront reconstruction can be rapidly implemented by the modal method, although the wavefront should be properly assumed with the basis functions and higher-order functions should be used for complicated wavefront shapes, which leads to careful considerations. The Zernike polynomials have been widely used for modal analysis because of their mathematical expressions, which indicate the optical aberrations [60, 61]. In this case the gradient of the wavefront can be assumed as a combination of the modified Zernike polynomials, and the original wavefront is reconstructed by finding their coefficients. The modal method has been further investigated in ASI and RSI, because misalignment of the components in the optical configuration can induce a small amount of lateral wavefront shift as the unexpected. To calibrate these lateral gradients in the wavefront reconstruction [62, 63], the decentering of the wavefront was included in the Zernike polynomials, and was determined by the optimization process along with the Zernike polynomial coefficients.

4.1. Application Fields of Shearing Interferometry

Because of the absence of a reference wavefront, shearing interferometers have been widely used in science and industry. In astronomical physics, they have an important role in estimating the wavefront and detecting its aberrations, for adaptive optics [28–30, 64, 65]. The wavefront distortion is measured by the shearing interferometer and transferred to a deformable mirror, to cancel it out. These adaptive optics also have been applied to industrial fields, especially EUV lithography systems [66].

Another application of shearing interferometers is measuring the surface figures of optical components [27, 31–33, 41, 42, 44]. As mentioned, RSI can be an effective tool to determine the surface shapes of the spherical and aspherical lenses and mirrors used in digital cameras and smartphone cameras. In the case of LSI, the measurement of freeform surfaces has been attempted [42, 44] because LSI allows us to obtain x- and y-directional gradient maps, which reconstruct the original surface figures.

A shearing interferometer is very useful in observing dynamic phenomena, with its simple optical configuration using a wedge or Wollaston prism to detect the phase changes that lead to the physical variation of a material [67–72]. It has been used to measure diffusion [67], tear film [68, 69], plasma [70, 71], exploding wires [72], and radiative heat transfer [73], for example, which are difficult to be observed by other techniques.

In addition to physical and industrial applications, shearing interferometers recently have proved remarkable for quantitative phase imaging in biomedical applications [74–78]. Similar to phase-contrast microscopy and differential interference microscopy, shearing interferometry is capable of obtaining phase maps of a specimen, including live cells.

4.2. ASI and RSI versus LSI

Regardless of the kind of wavefront, LSI is more convenient than ASI or RSI because it includes all of the gradient information to reconstruct the original wavefront. However, LSI requires two directional gradient maps (x- and y-directional gradients), and in fact can be also realized with a combination of ASI and RSI. The difficulty of combining ASI and RSI is centered on the implementation of two different optical configurations simultaneously. Furthermore, there is a lack of zonal methods for ASI and RSI to reconstruct the wavefront. This is why plenty of research related to LSI has been reported. However, there is no purely axially asymmetric surface in reality, which means RSI has the possibility to measure freeform surfaces with a single radial gradient map [26]. In this case, of course, the asymmetry of the shape should be so small as to be negligible in the measured results, such as a wavefront with a small amount of off-axis aberration, as shown in Fig. 16.

Figure 16. Reconstructed off-axis aberrations of wavefront by a radial shearing interferometer. Reprinted with permission from [26], Copyright © 2022, The Optical Society.

No single type of shearing interferometers dominates the others; Each type has its own advantages for applications. In measuring off-axis aberrations such as coma and astigmatism of the wavefront, for instance, ASI is the most appropriate, because of its high sensitivity to avoid effects caused by the axially symmetric aberrations, while vice versa for RSI. Compared to LSI, ASI and RSI only need a single azimuthal and radial gradient respectively to restore the wavefront, which is competitive with LSI.

4.3. Prospects in Shearing Interferometry

In the measurement of wavefronts, an important issue to be considered in shearing interferometers is the variation in shearing amount caused by the shape [26], as shown in Fig. 17, which makes wavefront reconstruction difficult. In traditional shearing interferometers the shearing ratio is fixed, because the wavefront incident to the interferometer is nearly planar after it is converted by preliminary optics. However, this is not always possible, and the variation in shearing amount should be calibrated according to the wavefront shape. Especially if shearing interferometers are used to measure freeform surfaces, more fruitful research should be carried out to find the proper shearing amount, or to predict it in theoretical and experimental ways.

Figure 17. Radial shearing ratio variation. (a) The variation of the radial shearing ratio with changing the radius of wavefront curvature, and (b) the peak-to-valley (PV) of a quadratic phase, related to varying radius of curvature. Reprinted with permission from [26], Copyright © 2022, The Optical Society.

The recent version of a snapshot shearing interferometer is rapid, robust, and insensitive to environmental conditions. However, most of its application fields are still limited to the measurement wavefronts or surface figure of optical components as traditionally they did. Recently, they have been also used in quantitative phase imaging in biomedical fields, but their approaches are the same as those from wavefront measurements, i.e. reconstructed wavefront. If the gradient directly obtained by shearing interferometers can be used to measure some physical quantities, it is expected that the distinguishing features of shearing interferometers will be further discovered and extended to various areas.

In this review we have discussed recent research in the area of shearing interferometry. As opposed to typical interferometers, which need a reference wavefront, shearing interferometry generates two sheared wavefronts from the original wavefront to be measured, as the surface figure of a specimen in the azimuthal, radial, or lateral direction, to obtain the phase map corresponding to the wavefront gradient along the shearing direction due to the self-interference. In this review, the research trend of each technique was introduced, along with the advanced techniques and experimental results.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

No data were generated or analyzed in the current study.

Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2021R1A2C1008661).