# Characterization of the transmitted near-infrared wavefront error for the GRAVITY/VLTI Coudé Infrared Adaptive Optics System

Pengqian Yang, Stefan Hippler, Casey Deen, Wolfgang Brandner, Yann Clénet, Thomas Henning, Armin Huber, Sarah Kendrew, Rainer Lenzen, Oliver Pfuhl, et al.

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To cite this version:

Pengqian Yang, Stefan Hippler, Casey Deen, Wolfgang Brandner, Yann Clénet, et al. Characterization of the transmitted near-infrared wavefront error for the GRAVITY/VLTI Coudé Infrared Adaptive Optics System. Optics Express, 2013, 21 (7), pp.9069. 10.1364/OE.21.009069. hal-02552346

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Distributed under a Creative Commons Attribution 4.0 International License
Characterization of the transmitted near-
infrared wavefront error for the
GRAVITY/VLTI Coudé Infrared Adaptive
Optics System

Pengqian Yang,<sup>1,2,3</sup> Stefan Hippler,<sup>2,*</sup> Casey P. Deen,<sup>2</sup> Wolfgang Brandner,<sup>2</sup>
Yann Clénet,<sup>4</sup> Thomas Henning,<sup>2</sup> Armin Huber,<sup>2</sup> Sarah Kendrew,<sup>2</sup> Rainer Lenzen,<sup>2</sup>
Oliver Pfuhl,<sup>5</sup> and Jianqiang Zhu<sup>1</sup>

**Abstract:** The adaptive optics system for the second-generation VLT-interferometer (VLTI) instrument GRAVITY consists of a novel cryogenic near-infrared wavefront sensor to be installed at each of the four unit telescopes of the Very Large Telescope (VLT). Feeding the GRAVITY wavefront sensor with light in the 1.4 to 2.4 micrometer band, while suppressing laser light originating from the GRAVITY metrology system, custom-built optical components are required. In this paper, we present the development of a quantitative near-infrared point diffraction interferometric characterization technique, which allows measuring the transmitted wavefront error of the silicon entrance windows of the wavefront sensor cryostat. The technique can be readily applied to quantitative phase measurements in the near-infrared regime. Moreover, by employing a slightly off-axis optical setup, the proposed method can optimize the required spatial resolution and enable real time measurement capabilities. The feasibility of the proposed setup is demonstrated, followed by theoretical analysis and experimental results. Our experimental results show that the phase error repeatability in the nanometer regime can be achieved.

©2013 Optical Society of America

**OCIS codes:** (010.1080) Active or adaptive optics; (010.7350) Wave-front sensing; (120.3180) interferometry; (050.1960) Diffraction theory; (110.2650) Fringe analysis; (120.3930) Metrological instrumentation.

References and links

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<footnote n="1">¹Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China</footnote>

<footnote n="2">²Max-Planck-Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany</footnote>

<footnote n="3">³Graduate School of Chinese Academy of Sciences, Beijing 100039, China</footnote>

<footnote n="4">⁴LESIA - Observatoire de Paris, 77, Avenue Denfert-Rochereau, 75014 Paris, France</footnote>

<footnote n="5">⁵Max Planck Institute for Extraterrestrial Physics, Postfach 1312, 85741 Garching, Germany</footnote>

<footnote n="7">⁷hippler@mpia.de</footnote>
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# 1. Introduction

The GRAVITY wavefront sensor (WFS) is part of the second generation VLTI instrument GRAVITY [1]. The GRAVITY WFS provides a complete adaptive optics system, which can be generally used at the VLTI due to its stand-alone character. It adds near-infrared (NIR) wavefront sensing at the 8.2-m Unit Telescopes for use with VLTI [2-4] and as such complements the available visible wavefront sensing MACAO systems [5]. The scientific goal of GRAVITY is to measure relative positions on sky, i.e. the separation of 2 objects, at a
level of the order of 10 micro-arc seconds on sky. To achieve this, a sophisticated metrology
system has been designed [6].

<figure><img src="image_3.png" /><figcaption>Fig. 1. Principle of the GRAVITY metrology system shown for two telescopes. A laser beam travels the same path backwards through all the beam combination and the VLTI optical train. Detection of the resulting fringe pattern takes place close to the entrance pupil (M2), above the primary mirror M1, by sampling the pattern at four locations with photodiodes, after reflected by the secondary mirror (M2), the beam was relayed to the delay line. Figure and caption taken from reference [6]. The near-infrared wavefront sensors are located in between the delay lines and the telescope's main optics, precisely at the Coudé focal station of each VLT unit telescope.</figcaption></figure>

The basic idea of the metrology system is to sample the optical path of the incoming telescope beams as complete as possible, with a precision of about 1nm (rms differential optical path difference of 2 objects separated by about 1" on sky). In the GRAVITY design this is accomplished through placing a near-infrared laser between the beam combiner and the
science detector, and projecting its light backwards through the instrument towards the secondary mirror of the VLT unit telescope M2. The principle is shown in Fig. 1. The near-infrared wavefront sensors are located between the delay lines and the telescope main optics, near the Coudé focal station of the telescopes. In a previous paper [2,4], we have shown that the optimal spectral band for the wavefront sensor is in H- and K-band (1.4 to 2.4µm). As the metrology laser's wavelength is at 1.908 microns, the near-infrared wavefront sensors will be sensitive to the very bright laser beacon. Furthermore, the design foresees an injection power into the beam combiner optics of 2 Watts. The implication of this bright beacon is that an entrance window with an optical density of >OD8 at the metrology wavelength of 1.908 microns has to be implemented to protect the wavefront sensor from this artificial signal.

In the current paper, we show the full elaborated results of the transmitted wavefront error (WFE) measurements through the entrance windows of the GRAVITY wavefront sensor system [7]. In order to determine the transmitted WFE performance of the entrance windows in the near infrared wavelength range, we used a slightly off-axis point diffraction interferometer (SOPDI) based on the Michelson [8,9] configuration. This configuration allows optimizing the spatial resolution. In combination with a fast near-infrared camera the complete setup enables real-time measurement capabilities. The advantages of the proposed SOPDI are the possibility to make the measurement with a single shot and that it is built with simple off-the-shelf optical components. In section 2 we describe in detail the optical component under test. The system layout and the SOPDI principle are presented in section 3.1. Following this step, the principle of the windowed Fourier filtering algorithm for phase reconstruction is detailed in section 3.2. Section 3.3 gives a further insight for understanding the proposed slightly off-axis method. To verify the feasibility of the proposed method, we performed an experiment and made stability tests, described in section 3.4. We conclude in section 4.

## 2. Design of the metrology laser blocking filter for the wavefront sensor on top of the cryostat entrance window

<figure><img src="image_4.png" /><figcaption>Fig. 2. Spectral transmission of the MENLO laser, the laser band-pass filter, and Entrance Window with the metrology blocking filter.</figcaption></figure>

The multi-layer coated blocking filters on top of the silicon entrance windows are considered to be one of the key elements for the WFS cryostat of the GRAVITY near-infrared WFS. The importance of the blocking filters to the GRAVITY adaptive optics system is such that all four units of the WFS designs require such coated entrance windows. The blocking filter serves as notch filter for the 1.908 µm metrology laser light, and the suppression should be
better than 10⁻⁸ to prevent the background radiation from saturating the detector and decreasing the signal to noise ratio of the WFS.

The spectral transmission measurements were performed using a spectro-photometer. It is well known that silicon windows do not transmit wavelengths shorter than about 1.3 µm. The detailed spectral transmission within a wavelength range between 1500 nm and 2100 nm is shown in Fig. 2.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Sketch of the GRAVITY adaptive optics system. A beam-splitter inside the Star separator marks the point where common optical path and non-common optical paths between wavefront sensor and beam combiner instrument separate.</figcaption></figure>

The total budget of the WFE in the optical design of the GRAVITY WFS is critical for providing the necessary accuracy wavefront measurement to meet the scientific requirement. The challenge is to avoid that the wavefront distortion introduced by the entrance window is coupled into the detected wavefront. These distortions cannot be corrected with the deformable mirror, since the distortions appear only in the non-common path (see Fig. 3). In addition, the small non-uniformity or imperfections in these multilayer coatings may introduce unacceptable wavefront errors. Moreover, these errors depend on the wavelength in a rather complex manner. The non-common path aberrations would degrade the contrast of final image of WFS. As a result of this design, a high quality of the transmitted wavefront should be maintained to access the promising wavefront detection of the GRAVITY’s AO system.

**3. Characterization of the entrance window in the near-infrared spectral range using a point diffraction interferometer**

Point diffraction interferometry plays an important role in the field of optical science and optical engineering, and has been used extensively in diverse applications such as phase profile quality of wavefront deformations [10–12]. It can provide a noninvasive high resolution phase map of the whole field under test. Off-axis common path point diffraction interferometry was proposed by Medecki [13] in 1996, and latter used to characterize the wavefront quality in the EUV lithographic systems [14]. Recently, P. Gao [15] and R. Guo [16] applied point-diffraction interferometry to microscopy for high-speed applications. The basic principle of the point diffraction interferometer is to generate a diffracted wavefront internally, so called self-referencing. Point diffraction interferometry is much less sensitive to
certain types of environmental disturbances such as mechanical vibration, temperature fluctuations, and air turbulence. These unique features are due to the common path geometry and the reliable phase retrieval algorithm. Although these systems can either provide a common path configuration or be used in the off-axis geometry. However, all of the above methods have been limited to that they either require mechanical phase shifting unit, holographic elements, or polarized optics. To characterize the wavefront propagation error in transmission, we developed a Michelson architecture based near-infrared point diffraction interferometer, and implemented the local windowed Fourier transform algorithm (WFT) for amplitude and phase extraction. This instrument was applied to study the transmitted wavefront quality of the cryostat entrance windows, which were designed for the GRAVITY wavefront sensor. With the optical setup shown in Fig. 4—the slightly off-axis point diffraction interferometer to characterize near-infrared transmitted wavefronts—and the help of the windowed Fourier transform filtering algorithm, we analyzed the phase information of the measured single carrier interferograms.

### 3.1 Optical setup of the IR-point diffraction interferometer

<figure><img src="image_6.png" /><figcaption>Fig. 4. Diagram of the experimental arrangement used for SOPDI for compact setup with common path configuration at 1523 nm wavelength; L1~L5, achromatic lenses; focal lengths of the lenses f1 = 200 mm, f2 = f3 = 100 mm, f4 = 150 mm, f5 = 45 mm; PH, pinhole filter with diameter 25 µm. NPBS, non-polarizing cube beam splitter.</figcaption></figure>

The optical configuration of the proposed infrared SOPDI is sketched in Fig. 4. Briefly, the fiber coupled near infrared frequency-stabilized laser with a wavelength of 1523 nm is collimated and used to illuminate the sample in transmission. The neutral density filters are used to control the intensity of the laser beam. After the collimated beam passes through the test specimen, a lens (L2) is located behind the test specimen to convert the beam into converging beam. The converging beam of the laser beam falls onto the infrared non-polarizing cube beam splitter (NPBS) and then splits into a transmission beam and a reflection beam. The two beams from the beam splitter are incident on the gold-coated mirrors and reflected at their Fourier plane, and recombined by the beam splitter. The reflection beam is low-pass filtered by using the pinhole spatial filter positioned in the spatial filter plane, which is the Fourier plane of L2, and transformed as the reference wave. The accuracy of PDI method largely depends on the accuracy of the diffracted spherical wave [10]. To guarantee the quality of the illumination, the pinhole diameter of 25 µm, which is
approximately 1.4 times the half of the Airy disc given by $d = 1.22\lambda f_2 / D$, where $\lambda = 1523$ nm is the wavelength, D (10 mm) is the input diameter of the collimated beam, and $f_2$ is the focal length of L2 (see Fig. 4). Another lens (L3) was placed in the conjugate position of the first lens (L2) to form a 1:1 4f spatial filtering system. The first lens performs the Fourier transform of the input field of the object, the transformed field is then filtered by the pinhole, and then the product is Fourier transformed again by L3. The M1 mirrors normal has a small angle with respect to the optical axial direction, thus the reflection beam travels in direction with a small angular offset to the optical axis, and the slightly off-axis fringe pattern is formed. A diaphragm in the Fourier plane of the telescope system is used as the field stop to eliminate ghost images and other unexpected stray light.

An infrared camera is located at the back of a 4f imaging system to capture the fringe pattern. Since the sample beam only splits at the end of the optical chain, the proposed setup can be considered as a quasi-common-path interferometer, and its stability will be significantly higher compared to conventional interferometers.

For simplicity, we denote the object wave after passing through the test specimen as $O_o(x, y)$. Considering a tilted angle introduced by mirror M1, the reference wave can be expressed as:

$$O_r(x, y) = FT^{-1}\{FT[O_o(x, y)] \times T_{PH}\} \exp[i\kappa(x, y)]. \quad (1)$$

Where $FT[\cdot]$ and $FT^{-1}[\cdot]$ denotes the 2D Fourier transform and the 2D inverse Fourier transform, respectively. The $FT^{-1}\{FT[O(x, y)] \times T_{PH}\}$ denotes the average or non-diffracted component of the object wave obtained by the pinhole filtering. $T_{PH}$ denotes the transmittance of the pinhole filter; $\kappa(x, y)$ is the spatial frequency induced by the tip/tilt of the M1 respect to the optical axis. Here, the carrier frequencies can be determined from the fringes of the off-axis interferogram.

## 3.2 Wavefront reconstruction

In this section, the wavefront reconstruction method of the slightly off-axis interferometry is introduced. In off-axis configuration, the angle between the object and reference waves introduces a carrier-frequency modulation in the interference pattern. The Fourier transform method has been widely used in optical interferometry for demodulation of carrier fringes [11,17,18]. However, this method has the drawback of poor ability to localize the signal properties, which also lead to a degradation of phase frequencies of the interferograms. To overcome this shortcoming, the method for reconstructing phase and amplitude from a single carrier frequency interferogram by using a Fourier transform method and combining it with localization of filtering were used in our experiment.

The intensity distribution of a carrier-frequency interferogram recorded on the image plane can be described as

$$I(x, y) = |O_o|^2 + |O_r|^2 + O_o^* O_r + O_s O_r^*. \quad (2)$$

In order to reconstruct the fringe with the image free of the zero order, the direct current (DC) term can be suppressed by subtracting the average intensity from the interferogram. The average intensity of all pixels of the interferogram matrix [19] is

$$I_b = \frac{1}{MN} \sum_{n=0}^{N-1} \sum_{m=1}^{M-1} I_i (m\Delta x, n\Delta y). \quad (3)$$

where $\Delta x$ and $\Delta y$ are the pixel sizes horizontal and vertical direction, and m and n denote the corresponding pixel numbers, respectively. By subtracting Eq. (2) from Eq. (1), the DC term of the interferogram is canceled and the interferogram can be derived as
$$I'(x, y) = \gamma(x, y) \cos(\omega_x x + \omega_y y + \varphi(x, y)). \quad (4)$$

$\gamma(x, y)$ denotes the modulation factor; $\varphi(x, y)$ is the phase delay due to the specimen; and $\omega_x$ and $\omega_y$ are the spatial carrier frequency of the fringes in the x-axis and y-axis directions, respectively, which are formed by a tilt between the reference and object waves. In fact, instead of subtracting the average intensity, the DC component of the fringe can also be determined either by the direct measurement or by indirect measurement. However, greater experimental effort has to be done due to the additional measurements required, which also makes the experimental setup more complex. After acquiring the DC term suppressed interferogram via the averaging method, the complex distribution of the object wave in the recording plane can be calculated with the Windowed Fourier Filtering (WFF) method as follows [7,20,21]:

$$O(x, y) = \frac{1}{4\pi} \int_{\eta_1}^{\eta_2} \int_{\xi_2}^{\xi_1} \left\{ \left[ \overline{I'(x, y) \otimes h(x, y, \xi, \eta)} \right] \otimes h(x, y, \xi, \eta) \right\} d\xi d\eta. \quad (5)$$

where $O(x, y)$ denotes the complex amplitude distribution of the object wave; $h(x, y, \xi, \eta) = g(x, y) \exp(j\xi x + j\eta y)$ and $\otimes$ denote the convolution with respect to the coordinates x and y. In this paper, the Gaussian window $g(x, y) = \exp[-(x^2 + y^2)/2\sigma^2]$ is chosen as the window function used for local frequency spectrum filtering, $\sigma$ is the standard deviations of the Gaussian function. $\xi$ and $\eta$ are the spatial coordinates in the frequency domain. Phase ambiguities will still exist after the Fourier transformed local frequency spectrum filtering. This is exactly the same phase unwrapping issue that exists in the standard fringe analysis. Next, an unwrapping algorithm is applied to obtain the final wrapped object phase [22,23]. After removal of the $2\pi$ ambiguity by a phase unwrapping process, the total phase information can be obtained. By using the above mentioned reconstruction method, the phase distribution of the tested specimen is obtained.

Single-shot frame instantaneous interferometry offers an alternative optical test method where environmental noise prohibits conventional phase shifting methods. The advantage of the slightly off-axis interferometry scheme compared with the traditional off-axis interferometry is analyzed in the following. The on-axis interferometry can make full use of the resolving power of the detector [24], thus has the advantage of the high spatial details over the off-axis scheme. However, in this case at least three of the time sequential phase-shifting interferograms have to be acquired to reconstruct the original wavefront under the test. While the WFT spectrum analysis method provides pixel sized frequency information of the whole field, it is impossible for the traditional Fourier transform.

### 3.3 Slightly-off-axis scheme versus traditional off-axis scheme

The Fourier analysis is the most popular method in spatial and temporal phase extraction. A fringe pattern with carrier (carrier frequency $k$) consists of a background and two conjugated fringe patterns. These three items are separable in Fourier domain provided that the carrier frequencies are high enough ($\ge 3f_0$) [17]. Normally the Fourier spectrum is filtered by a bandpass filter and the inverse Fourier transform yields an exponential, from which the phase can be calculated. For a given test object with frequency spectrum lies in the range $[-f_0, f_0]$, then the real image, DC terms, and twin image will lie in $[k - f_0, k + f_0]$, $[-2f_0, 2f_0]$, and $[-k - f_0, -k + f_0]$, respectively [25]. In the traditional Fourier transform method, $k \ge 3f_0$ is required to spatially separate the three terms in the frequency domain, that means a detector with $k_d \ge 4f_0$ to capture the off-axis fringe pattern. This will decrease the resolving power of the detector array at some level, and also introduce ray trace errors due to the off-axis configuration. With our proposed slightly off-axis scheme, the DC term of the
interferogram is canceled by subtracting the mean value of the interferogram. The windowed Fourier transform was implemented to obtain the phase of the real image by pixel sized filtering, the WFT is more tolerant to the quality of the wrapped phase map, and this also makes the filtering more effective to eliminate the low amplitude noise. The slightly off-axis method only requires the carrier frequency $k \ge f_0$ to separate the two conjugated items, resulting in the optimal detector sampling frequency $k_d \ge 2f_0$, only half of that needed in the traditional off-axis interferometry. Thus slightly off-axis interferometry can make better use of the resolving power of the detector to provide a higher spatial resolution of the test wavefront. For the on-axis interferometry, it only requires the detector with a spatial frequency $k_d \ge f_0$ to sequentially record several phase shifted interferograms to obtain the desired phase distribution. To measure the dynamic processes, it is better to acquire the data with single shot to minimize reconstruction errors and data acquisition times. In this aspect, slightly-off-axis interferometry is superior to on-axis interferometry.

In addition, with slightly off-axis scheme described here only one single shot is required, rather than two [24,25]. Kemao Qian et al. [21] demonstrated the advantage of the WFT compared with the traditional Fourier transform interferometry. However, to the best of our knowledge, the method proposed has not been suggested as a tool to optimize the relationship between the acquisition rate and the detector bandwidth for the infrared wavefront measurement. The proposed slightly off-axis scheme enables us to achieve higher spatial resolution as well as real-time measurement capability.

### 3.4 Experimental results and analysis

<figure><img src="image_7.png" /><figcaption>Fig. 5. Photograph of the experimental setup.</figcaption></figure>

To demonstrate the feasibility of the proposed system we used an experimental setup as shown in Fig. 5. The infrared camera (FLIR SC 2000, FLIR Systems Inc.) has a maximum acquisition rate of 206 frames/s at the full resolution of 320X256 pixels and with a pixel size of 30µm (H) x 30µm (V). The imaging system has a magnification of 3.3, so that the cosine modulation of the fringe is approximately sampled by 10 pixels per period. In our experiment, an entrance window designed for the adaptive wavefront sensor cryostat of GRAVITY was used as specimen. Figure 6(b) shows the complex amplitude of the object wave as retrieved by using Windowed Fourier Filtering method, where the phase is still wrapped. The reconstructed OPD of the specimen is given in Fig. 6(c), after the phase has been unwrapped.

To remove the background, the phase retrieval procedure requires measuring the background phase of the setup without the test optics. This background phase subtraction allows us to correct the residual errors associated with the experimental setup. The
measurement is taken without the presence of the specimen. Using these data, the wrapped background phase can be calculated.

<figure><img src="image_8.png" /><figcaption>Fig. 6. Experimental results based on off-axis Window Fourier Filtering interferometry: (a) Typical slightly off-axis interferogram of a multi-layer coated optical window, (b) Wrapped phase distribution extracted as the mean of Windowed Fourier transformed spatial filtering method, (c) Reconstructed object wave.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. The center line profiles of the test of a BK7 optical window measured with SOPDI and a phase shifting interferometer.</figcaption></figure>

A comparison of the centerline profiles obtained with the proposed setup and a prototype step phase shifting interferometer (PSI) from TRIOPTICS GmbH is shown in Fig. 8. A BK7 made optical window, which transmits both in optical and infrared region, was employed to perform the measurements. The resulting height errors of ~3.3nm rms of the centerline profile confirm the excellent performance achieved by the proposed system compared to the PSI measurement.
<figure><img src="image_10.png" /><figcaption>Fig. 8. Measured transmitted wavefront error of the test entrance window. (a) Single slightly off-axis interferogram captured by the near infrared SOPDI, (b) Zernike fit of the wavefront deviation of a central circular part of the entrance window with a diameter of 15 mm. The Zernike fit is calculated to a maximum of the first 36th terms, and only piston and tip/tilt values have been removed.</figcaption></figure>

Figure 8(a) shows the interferogram of the entrance window captured using the near infrared SOPDI. The reconstructed wavefront deformation (see Fig. 8(b)) is represented by a set of the orthogonal functions with suitable coefficients [26,27]. Here the Zernike polynomials were implemented as the set of orthogonal functions. After subtraction of focus and tip-tilt terms, the final WFE is 10.2 nm rms. Since the diameter of footprint of the metrology beam on the entrance window is less than 1.5 mm, the WFE performance of the entrance window meets the GRAVTY specification (which is 60 nm rms).

<figure><img src="image_11.png" /><figcaption>Fig. 9. Stability test for the proposed setup in nanometers associated with the background of the full field view (dashed line) and a single point phase profile (solid line).</figcaption></figure>

To quantify the stability of the instrument against environmental disturbances, and thus identify the repeatability of measurements, we continuously measured sets of 100 frames without the sample, with intervals of 18s (covering in total 30 minutes). The integration time of infrared camera is about 10 ms for each frame. The optical path difference associated with the full field of background phase map had a standard deviation of 3.22 nm, as shown in Fig. 9, which verifies that the proposed setup has high repeatability. An arbitrary 4x4 pixel average point, characterized by standard deviation of 10.8 nm, is also shown in Fig. 9, which shows long term stability of the proposed setup. Obviously, the quasi-common-path design significantly minimizes the systematic errors and ensures the high repeatability and stability.

The intensity of the reference wave is much lower than the intensity of the signal wave after the pinhole filtering. To ensure the best possible signal-to-noise ratio, the fringe contrast
provided by the interferometer should be maximized. Either a variable neutral density filter or
the coating method [12] can be applied, with which adjustable contrast of the fringe pattern
can be realized by adjusting the relative intensities of interfering waves. There are the two
primary error sources that limit the accuracy of the PDI are the errors generated by the
pinhole spatial filter and the systematic error from the system geometry [7]. Basically, the
systematic geometric errors could be simply removed in the background calibration process.
Considering the pinhole induced errors varies randomly as a function of reference pinhole
position as well as the test object can be suppressed through averaging.

It should be noted that the proposed setup can be also used for on-axis parallel phase-
shifting point-diffraction interferometry. When the M1 is placed perpendicular to the optical
axis, on-axis interferograms can be obtained. Compared with the off-axis scheme, the on-axis
configuration can make full use of CCD camera spatial resolving power, and provide more
spatial details of the sample. However, it requires the time series phase-shifting
interferograms to do the phase retrieval, which may make the system more complex. This
configuration could also be used for multi-wavelength without change the main optical
elements.

**4. Conclusion**

In summary, we present near infrared point-diffraction interferometry as a novel approach to studying transmitted wavefront quality in the near-infrared band, with a promising accuracy. The proposed SOPDI is compact, less vibration sensitive, and sufficient for the dynamic process measurement. The major advantage of the setup is the simplicity, high temporal stability and its potential low cost. The carrier frequency is introduced by slightly tilting one mirror with adjustable tip/tilt. Thus, one can directly extract the phase information from a single interferogram with a spatial carrier method in real time. The Windowed Fourier spatial analysis technique permits us to acquire high spatial resolution associated with slightly off-axis setup. The temporal stability enables the investigation of dynamical processes. These advantages could be attractive for dynamical process measurements and advanced wavefront sensing in industrial, biological, and astronomical applications.

**Acknowledgment**

This research is supported by the Max Planck Society (MPS)-Chinese Academy of Sciences (CAS) Joint Doctoral Promotion Program, and funding from the Max-Planck Institute for Astronomy.