Shearing interferometry has a remarkable advantage over interferometry with a reference beam. Simply, the reference is not required. The object beam interferes with a slightly displaced replica, and for a small shear and slow object phase variations, the interferogram fringes carry information about the first derivative of the tested phase object. Diffraction gratings are attractive as beam-splitting and recombining elements, since they can be applied in a wide spectral range.

The two most important grating lateral shearing interferometry solutions are the Ronchi test [1,2] and Talbot interferometry [3,4]. In the recent Letter [5], their utmost implementation simplicity and the limitations caused by multiple-beam interference, resulting in the so-called self-imaging (Talbot effect) [6–8], were emphasized. Self-imaging imposes interferogram contrast deteriorations and limits the aberration measurement range [1,2,3,9]. This limitation is avoided in various two-beam grating monodirectional [10,11] and at least two-directional [12–16] interferometry setups.

In Ref. [5], we presented a simple one-dimensional (1D) diffraction grating three-beam interferometry solution without resorting to complex hardware. It requires recording two fringe patterns in either the Ronchi or Talbot interferometry configurations with a mutual phase shift of $\pi$ implemented by laterally displacing the beam-splitter grating by half its period. Each pattern was formed by three-beam interference provided either by a grating generating only three orders or a common binary amplitude grating with frequency filtering (this implementation idea can be traced back to the diffraction-grating-based solution of Zernike’s three-slit interferometer reported by Marston [17]). The subsequent addition of the two interferograms gives a synthetic two-beam fringe pattern formed by $+1$ and $-1$ diffraction orders. Its contrast is propagation invariant, i.e., independent of axial positions of the grating and observation plane. That solution, although simple in principle, provides only one direction spatial derivative information using two-shot data acquisition.

This Letter presents the principle and implementation of single-shot, crossed grating, multidirectional $3\times 3$ beam lateral shear interferometry with an extended measurement range. The single Fresnel diffraction pattern of a cross-type structure composed of two multiplicatively superimposed sinusoidal gratings (each generating only three diffraction orders) or a common binary, amplitude, two-dimensional (2D) Ronchi grating with spatial filtering (to pass its $3\times 3$ lowest diffraction orders) is recorded. The single-shot approach represents a significant enhancement in comparison with the above mentioned 1D grating method [5]. In our novel technique, the interferogram pairs contain interferograms along the directions 0, 90, 45, and 135 deg, respectively. They can be digitally separated using the appropriate transform techniques, such as continuous wavelet or Hilbert transforms [5]. The latter one is used in this study. Shearing phase distributions are decoded from the second harmonics of separated fringe families. Since second harmonics correspond to two-beam interference patterns, they allow for an extended wave front aberration measurement range as compared with a common approach analyzing the fundamental harmonic of the grating shearing interferogram. In the case of fast wave front changes and/or larger shears, the calculated orthogonal shearing phases do not correspond to wave front gradients, and dedicated algorithms have to be used for wave front evaluation, using, e.g., the modal method and Zernike polynomials [18–21]. The method principle is analytically described and experimentally corroborated using a crossed-type Ronchi grating and spatial filtering.

The simplicity of our 2D diffraction grating interferometer is to be emphasized. A simple crossed binary amplitude grating with spatial filtering is employed. Digital processing eliminates self-imaging contrast deteriorations met in three and multiple order grating interferometry. Very effective two-beam interference grating solutions were developed [14–16], but they are much more technologically sophisticated and costly.

It is worthy to compare our technique with methods reported in [14]. One of them, corresponding to a classical Hartmann–Shack test (no spatial filtering applied, see Figs. 5–7 [14]), comprises self-imaging. Depending on the observation plane, the classical arrays of spots with intensity distribution varying with defocus are detected. The second method reported in [14], named the three-wave lateral shearing (TWLS) interferometer, uses a bidirectional hexagonal grating with a spatial filter (blocking the zero order and containing three elliptical holes). Three angularly separated two-beam lateral shear interferograms are obtained, but because of blocking the zero order, the intensity pattern does not change with the defocus. The basic difference between these two solutions and the one reported in this Letter is that we encode and later separate two mutually orthogonal three-beam lateral shear interferograms (along the $x$ and $y$ directions), along with an additional two along the 45 and 135 deg directions. From our three-beam interferograms, their intensity second harmonics are used.

Practical remarks concerning the proposed method implementation (both when using sinusoidal grating and binary amplitude grating with spatial filtering) are worthy to be added. The grating frequency should be properly selected to avoid the overlapping of the diffraction spots (point spread functions, PSFs) in the 2D periodic grating spectrum. For example, in the case of turbulent wave front sensing, the angular size of the PSF is approximately equal to $2\lambda /r_0$, where $\lambda$ is the light wavelength, and $r_0$ denotes the so-called Fried’s parameter [22]. Since the angular separation between adjacent PSFs is $\lambda /d$, where $d$ is the grating period, we obtain $d<r_0/2$.

First, we present the analytical studies of the Fresnel diffraction and self-imaging phenomenon in three-beam grating lateral shear interferometry. For simplicity, we conduct the analysis for a 1D Ronchi test. This assumption simplifies the calculations, and permits us to derive general conclusions valid for a broader class of diffraction structures. Comments concerning the 2D system are included at relevant stages of the analysis and experimental works.

Figure 1 shows the schematic representation of the three-beam Ronchi test for lens aberration and phase object examination [9,23]. We will study the case of the grating G placed to the left of the focus O, enabling to employ a binary amplitude grating with spatial filtering to pass its three diffraction orders of 0 and $\pm 1$. In the case of the 2D binary Ronchi grating, the square filter mask should pass the $3\times 3$ lowest orders. These nine orders provide a multidirectional lateral shear operation. Certainly, when using crossed grating with only $3\times 3$ diffraction orders (without higher harmonics), no spatial filtering is required. Selecting the distance $z$ between the grating G and the tested beam focus O influences the spatial frequency of fringes in output plane OP. Fringe frequency is important when employing a single-frame algorithm for automatic fringe pattern analysis.

 

Fig. 1. Schematic geometry of the three-beam Ronchi test. G, diffraction grating; OL1, objective under test; OL2, collimating objective. For presentation clarity, only two diffraction orders are shown, the 0 (solid line) and the $+1$ (dashed line) ones. The axial distance between the grating G and the tested beam focus plane (with spatial filter included when using binary gratings) is denoted as $z$. Focal lengths of OL1 and OL2 are $f_1$ and $f_2$, respectively.

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The mathematical expression describing the complex amplitude of the three-beam field in the plane OP, as Fig. 1, has the form [24]

In the case of significant changes of $\psi (x,y)$ and/or large lateral shears, the intensity distribution in the output plane OP can be described by

It can be seen from Eq. (2) that the intensity distribution is composed of the background, and the first and second harmonic terms. In an overwhelming majority of publications, the analyses exploiting the first harmonic term (described in Eq. (2) by the product of two cosines) were conducted. It readily follows that in the case of larger wave front deformations and/or shear amounts, the first harmonic contrast distribution (described by the first cosine in the product) depends on

The last term of Eq. (2) describes the second harmonic of the pattern intensity distribution. It relates to the two-beam interference of the grating diffraction orders $+1$ and $-1$ free of three-beam self-imaging effects; its contrast distribution is constant over the field of view, it does not change with $z$. We propose extracting the second harmonic term from a recorded single pattern by digital processing instead of recording two $\pi$ shifted frames and adding them [5].

By employing 2D diffraction grating, spatial low-pass filtering of its $3\times 3$ lowest diffraction orders and single-frame recording, we can acquire multidirectional pairs with mutually orthogonal lateral shear intereferograms along directions 0 and 90 deg ($x$ and $y$ directions) and 45 and 135 deg. They can be used for further modal analysis [18–21] to retrieve the searched phase distribution. The two pairs differ by the shear amount by the factor of $2^{1/2}$. Both pairs can be simultaneously acquired for grating axial distances z = Md²/λ.

We conducted experiments in the optical system, shown in Fig. 1. A crossed binary amplitude Ronchi grating of a spatial frequency of 10 lines/mm and an opening ratio of 0.5 was used as the beam-splitter grating G. A He–Ne laser served as the light source. For aberration generation, the objective OL1 ($f=200\mathrm{mm}$) was set opposite to its correction direction to introduce spherical aberration. The square mask was set in the spatial frequency (beam focus) plane to pass $3\times 3$ lowest diffraction orders of G.

We present the results for the testing conditions described by Eq. (2) with the second-order derivative of the aberration influencing the contrast of the grating image. Figure 2 shows the image for $z\approx {Md}^2/\lambda \approx 63\mathrm{mm}$ ($M=4$). Zero contrast bands modulate the crossed grating image. They limit the interferometer measurement range. Additionally, faint parasitic quasi-diagonal curved fringes are also present in Fig. 2. They are generated by a glass plate protecting the CCD matrix. Because of their low spatial frequency as compared with the analyzed grating lines, they are filtered out during the fringe pattern adaptive processing using the Hilbert–Huang transform (HHT). As an alternative technique for filtering those parasitic fringes, the implicit smoothing splines method can be used [29,30].

 

Fig. 2. Interference pattern of the lens under test obtained in the optical system of Fig. 1 for $z\approx {Md}^2/\lambda$ $(M=4)\approx 63\mathrm{mm}$. A square mask inserted in the frequency plane passed $3\times 3$ lowest diffraction orders of the crossed grating G to form the fringe pattern.

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In Fig. 3 we show the modulus of the spectrum of Fig. 2 to illustrate the effect of self-imaging induced zero contrast bands. The second harmonics along the directions $x$, $y$, and 45/135 deg allow to reconstruct the shearing phases. This is not the case for the first harmonics with deeply modulated frequency bands.

 

Fig. 3. Modulus of the Fourier spectrum of the grating image shown in Fig. 2. The comatic shape of the diffraction spots stems from the testing of the lens’ spherical aberration by lateral shear interferometry.

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As already mentioned, the processing and analysis stages of the single-shot image, shown in Fig. 2, and other images that might be recorded for other distances $z$, include extracting the second harmonics along the orthogonal directions and calculating the phases of the corresponding shearing interferences. Because of considerable departures of grating image lines from straightness, shown in Fig. 2, and corresponding frequency spreads, shown in Fig. 3, the Fourier transform method is difficult to apply. We have made calculations using the HHT method [31,32]. Figure 4 shows the calculated phase distributions from the extracted second harmonics with shear along the $x$ and $y$ directions. The intensity distributions of fringe patterns vary with the grating axial distance z [33]. However, we can flexibly apply our HHT algorithm to demodulate either the 0 and 90 deg and/or 45 and 135 deg pairs. For example, for $z={Md}^2/\lambda$ the 45 and 135 deg pairs are readily acquired.

 

Fig. 4. Shearing phase distributions (wrapped, cosine, and continuous) calculated from second harmonics of lateral shear patterns along the $x$ and $y$ directions extracted from the image of Fig. 2. Linear terms related to the grating axial location $z$, shown in Fig. 1, were subtracted. Note the obtained continuous phase distributions in spite of the presence of zero contrast bands and parasitic fringes in Fig. 2.

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A quantitative approach to confirm that the demodulated shearing phase is constant regardless of the grating G location consisted of evaluating the root mean square (RMS) error between phases extracted for different $z$. The obtained $\mathrm{RMS}=0.17\mathrm{r}\mathrm{a}\mathrm{d}$ constitutes only 0.28% of the detected shearing phase dynamic range.

In summary, this Letter reports single-shot, $3\times 3$ beam grating interference-based generation of multidirectional pairs of orthogonal lateral shear interferograms. Extracting second harmonics of the intensity distribution allows for the analyzation of extended range wave front deformations. The self-imaging phenomenon generated contrast modulations are no longer present. The method principle is implemented employing a simple crossed binary amplitude Ronchi grating and spatial filtering. Single-frame automatic fringe pattern processing has been conducted using the HHT algorithm developed by the authors. It provides two orthogonal shearing phase distributions required for subsequent retrieving of the wave front under test by, e.g., the modal method and Zernike polynomials [18–21].