## Abstract

A single-shot common-path phase-stepping radial shearing interferometer (RSI) is proposed for wavefront measurements. In the proposed RSI, three quarter-wave plates are used as phase shifters to produce four spatially separated phase-stepping fringe patterns that are recorded simultaneously by a single CCD camera. The proposed RSI can measure the wavefront under test in real-time, and it is also insensitive to environmental vibration due to its common-path structure. Experimentally the proposed RSI is applied to detect the distorted wavefronts generated by a liquid crystal spatial light modulator. The measured aberrations are in good agreement with that obtained with (by) a Hartmann-Shack wavefront sensor, indicating that the proposed RSI is a useful tool for wavefront measurements.

©2011 Optical Society of America

## 1. Introduction

Radial shearing interferometer (RSI) was proposed firstly by Hariharan *et al.* in 1961 [1]. It has been commonly used in optical testing [2–5], corneal topographic inspection [6–8], wavefront sensing for adaptive optics [9–11] and high power laser beam characterization [12–15]. Radial shear can be induced by classical optical components [16,17] and the other methods [5,18–22] including gratings [18,19], zone plate [5,20], speckle techniques [21,22] and so on. Comparing to other self-referencing interferometers, such as the point diffraction interferometer (PDI) [23,24] which uses a pinhole to generate a planar reference wavefront, RSI needs no planar reference wavefront to achieve absolute wavefront under test [25], so RSI has high energy efficiency and high fringe contrast. Moreover, due to its common-path configuration, the RSI is insensitive to vibration of environmental relative to PDI. The RSI need no special alignment because no pinhole is used. In recent years, RSI has been becoming an important tool for diagnosing the wavefronts of laser beams and the other applications. The disadvantages of the conventional RSIs are that the distorted wavefront is hard to extract from one radial shearography due to the phase extract algorithm is complex. However, fortunately, many of these applications can receive benefits from the accuracy of the methods of phase-shifting interferometry, and phase-shifting shearography is more sensitive than simple digital shearography [26]. Some authors [27–29] have proposed RSIs based on phase-shifting techniques for different applications. Kothiyal *et al*. [27] developed a cyclic RSI for phase shifting interferometry with a polarization phase shifter. Chung *et al.* [28] described a RSI based on a Mach-Zehnder configuration using liquid-crystal-device as the phase shifter. Naik *et al.* [29] designed a Sagnac RSI with geometric phase-shifting technique. However, the temporal phase shifting technique was used in all these developed RSIs. In this case, the measured wavefront must be essentially stationary during the duration of acquiring several phase-stepping images. Therefore, the ability to measure dynamic wavefronts is significantly compromised.

In this paper, we propose a single-shot common-path phase-stepping RSI by using four radial shearing interferograms. A spatially multiplexed four-channel polarization phase-stepping setup is used to realize the measurement of dynamic wavefront aberrations. Comparing with the two-step phase-shifting method [30–32], four-step phase-shifting technique can measure the amplitude and wavefront of one light beam in real time, and it needs no a stable and regular intensity distribution. It is very important for the wavefronts measurement of laser beam whose amplitude and phase distribution are very irregular and unstable. In the proposed RSI, four phase-stepping images are obtained simultaneously by a single CCD camera. The shearing wavefront is calculated from these four fringe patterns by simple four-frame phase-shifting algorithm. According to the principle of RSI, one laser beam is split into one magnified interference beam and one demagnified interference beam. The shearing wavefront is the wavefront difference between the magnified and the demagnified wavefronts in their common area. So we must reconstruct the real wavefront from the shearing wavefront by using an appropriate wavefront reconstruction algorithm.

This paper is organized as follows. In section 2 we introduce the system description and the principle of the proposed RSI. In section 3 we will analysis the adjustment of the fringe contrast. In section 4 we describe the experimental system, the experiment results and the corresponding analysis, respectively. The conclusion of this paper is made in section 5.

## 2. Experimental setup and theoretical description

The schematic diagram of the experimental is shown in Fig. 1 . The system can be divided into three parts: wavefront simulator (WS), Hartmann-Shack wavefront sensor (HS WFS) and common-path phase-shifting RSI (The function of WS and HS WFS in our experiment will be introduced in section 4).

The core part of this system is the common-path phase-stepping RSI. It includes a four-channel polarization phase stepper (FCPPS) and a cyclic RSI, which comprise a polarizing beam splitter (PBS1), two lenses (L5 and L6) and two reflective mirrors (M1 and M2). Actually, the cyclic RSI is a Keplerian telescope system comprised L5 and L6, and the focal lengths of them are *f*
_{5} = 250*mm* and *f*
_{6} = 300*mm*, respectively. One light beam with unknown polarization state is polarized when it pass through the polarizer P2 and is induced into the cyclic RSI system. After then, the polarized light beam is split into two beams with orthogonal polarization directions (vertical direction of the reflected light beam (defined as Beam1) and the horizontal direction of the transmitted light beam (defined as Beam2)). The Beam1 traverses the path PBS1-L5-M1-M2-L6-PBS1, and the size of it is magnified because *f*
_{5} is less than *f*
_{6}. Similarly, the size of the Beam2 which traverses along the opposite path is demagnified. Finally, the magnified beam and the demagnified beam are reflected and transmitted into FCPPS. However, no interference fringe would be observed because these two concentric beams have orthogonal polarization states. A complex amplitude *E*
_{0}(*x*, *y*) can be used to describe the tested light beam, and *A*
_{0}(*x*, *y*) and ${\phi}_{0}(x,y)$ are the amplitude distribution and the phase distribution (i.e. wavefront). We can get

The radial shearing ratio of this cyclic RSI is *s* = *f*
_{6} /*f*
_{5} > 1, and the complex amplitudes of the two beams exiting from the PBS1, *E*
_{1}(*x*, *y*) and *E*
_{2}(*x*, *y*), can be written by

*E*

_{1}(

*x*,

*y*) and

*E*

_{2}(

*x*,

*y*) respectively, and their definitions are shown as

If we use *D* to denote the size of the tested laser beam, the sizes of the magnified and demagnified beams which ejected from the cyclic RSI are *D* × *s* and *D/s*, respectively. The aperture sizes of these two beams are limited by the diaphragm A2,and then the magnified and demagnified beams with *D/s* aperture size are projected into FCPPS (shown in Fig. 1).

The main function of FCPPS is to generate the desired phase shift of $\pi /2$ between each generated interferograms [33]. Firstly, the magnified and demagnified beams, which generated from the cyclic RSI, are equally divided into two channels (a and b, shown in Fig. 1) by the BS3. Quarter wave plates QW1 (fast axis orientated parallel to the polarization direction of the Beam1) and QW2 (fast axis orientated at 45°with respect to both fast axis of Beam1 and Beam2) are placed in channel a. QW1 generates a $\lambda /4$ phase delay between the Beam1 and the Beam2, and QW2 converts two beams into circularly polarized lights. QW3 is placed into the channel b, and the fast axis is the same as QW2, and two circularly polarized lights are generated again. The beams of channel a and b is split into pairs beams a1, a2, b1 and b2 (all shown in Fig. 1) by PBS2, and 180°phase shift between the pairs images of the channel 1 and channel 2 (i.e. beam pairs a1 and a2, beam pairs b1 and b2). Four channels (i.e. a1, a2, b1 and b2) are arranged into four pair beams with the same direction, and then they are projected onto the image plane of only one CCD camera. The beam pairs a2 and b1 are projected onto a proper position of the image plane of the CCD camera firstly (i.e. upper or nether two interferograms). Then the altitude of the other beam pairs a1 and b2 can be adjusted by tilting flat mirror M6 and beam splitter BS4, and the corresponding two interferograms is projected onto the leaving position of the image plane of the CCD camera. Four interference fringes are formed simultaneously. In this part, the lenses L7 and L8 are used to relay the wavefront from the A2 to the photo surface of CCD camera

The Jones formulas can be used to describe the principle of the polarization phase-stepping [33]. The Jones matrices for a horizontal linear polarizer, *P*
_{0}, a vertical linear polarizer, *P*
_{90}, and a quarter-wave plate with horizontal fast axis, *Q*
_{0}, and 45°fast axis, *Q*
_{45}, are respectively [33]

The change in the polarization state for each of the four channels [33] can be obtained by

From Eqs. (4) and (5), one can getWavefront difference $\mathrm{\Delta}\phi (x,y)$ between wavefronts of Beam1 and Beam2 in their common area can be written as

*i*= 1,2.

According to the four interferograms detected by the CCD camera, the wavefront difference $\mathrm{\Delta}\phi (x,y)$can be calculated by four-frame phase shifting algorithm [23], which given by

However, from Eq. (7), the relation between the wavefront under test ${\phi}_{0}(x,y)$ and the wavefront difference $\mathrm{\Delta}\phi (x,y)$ is implicit. So one special wavefront reconstruction method must be employed to reconstruct the wavefront under test ${\phi}_{0}(x,y)$ from the measured wavefront difference$\mathrm{\Delta}\phi (x,y)$. In this paper, we use the modal wavefront reconstruction method [36] to obtain the real wavefront in our optics configuration. This method uses matrix formalism to calculate the Zernike coefficients of a wavefront under test, and it can reconstruct successfully an arbitrary wavefront aberration from a wavefront difference taken by a RSI.

## 3. The relation of the fringe visibility with the angle of P2 and the radial shear

In this section, we will analysis the relation between the fringe visibility of these four interferograms and the angle *θ* of polarizer P2 with respect to horizontal direction. For simplicity, we assume that the light intensity has a uniform distribution in all path of our configuration, and the constant number *I*
_{0} is used to denote the light intensity at the output of P2. As our description in section 2, we can take the PBS1 as a special polarizer which comprises of two independent polarizers with orthogonal polarization directions, i.e. the horizontal direction and the vertical direction. We use *I*
_{t} and *I*
_{r} to define the light intensity distributions which transmitted (*i.e.* Beam2 defined as before) and reflected (*i.e.* Beam1) by PBS1 respectively. According the Malus’ Law [34], we can get

After then, the Beam1 and Beam2 are magnified and demagnified respectively by the cyclic RSI, and the light intensity distributions of output light beam pair in their common area are defined as ${I}_{r}^{\text{'}}$and${I}_{t}^{\text{'}}$. In addition, two beams (i.e. Beam1 and Beam2) have the same ratio of energy loss when they pass through the FCPPS and finally are projected onto the surface of CCD camera. This ratio is defined as *η*. According the principle of conservation of energy, we can get

*I*, can be expressed as

According the definition of contrast [34], the fringe visibility *K* can be written as

*K*and the angle

*θ*can be shown as

*K*when the relation between the shear ratio

*s*and the angle

*θ*satisfy

This peak value is equal to 1.0. Particularly, if the polarizer P2 has a horizontal or vertical polarizing direction, i.e. *θ* = 0°or 90°respectively, the fringe visibility will be equal to zero. In the other word, no fringe patterns can be observed when *θ* equals to 0°or 90°. It can be explained easily in this two extremely situations. If *θ* is equal to 0°(or 90°), there is no Beam1 (or Beam2) but only Beam2 (or Beam1), so four identical intensity distribution but not four interferograms will be obtained simultaneously.

In our experimental the radial shear *s* = 1.2, and the change of fringe visibility *K* with the angle *θ* is plotted in Fig. 2
. From Fig. 2, the best angle of P2 with respect to horizontal direction for the best fringe visibility should be 55.22°in our practical system. From Eq. (15), the best angle *θ* of P2 is different for different radial shear *s*, and the change of the best angle *θ* with different radial shear *s* is shown in Fig. 3(*a*)
. From this curve, we can get that a larger angle *θ* of P2 is required when the system has a larger radial shear *s*. The fringe visibility *K* is not very sensitive to the angle *θ* when the system has a small radial shear *s*, and for a large radial shear *s*, the fringe visibility *K* is very sensitive to a little change of the angle *θ*. For clarity, we plot the relations between the fringe visibility *K* and the angle *θ* in Fig. 3(*b*) for some different radial shears (here are 1.1, 1.4, 1.7, 2.0, 3.0, 5.0, respectively). From Fig. 3(*b*), we can get the change of sensitivity of the fringe visibility with the changing of radial shear *s*.

Certainly, in our practical system, if the radial shear is not very large, so accurate angle of P2 is not required because a little adjustment error of the angle of P2 will not decline obviously the fringe visibility.

## 4. Experiment results and analysis

As the description in the first paragraph of the section 2, besides the common-path phase-shifting RSI, the overall system also includes a wavefront simulator (WS) and a Hartmann-Shack wavefront sensor (HS WFS) (shown in Fig. 1). In the WS, an electrically addressed phase-only liquid crystal spatial light modulator (LC SLM) is used to generate aberrated wavefronts. The aberrated wavefront is then measured by the proposed RSI, as well as by the HS WFS. By adjusting the polarizer a phase-only modulation mode in the LC SLM can be achieved. The HS WFS is used to measure separately the aberrated wavefront generated by WS, and to compare with that obtained by the proposed RSI. The lenses L3 and L4 placed in HS WFS system are used to relay the wavefront from the surface of LC SLM to the HS WFS.

The experiment is performed with a He-Ne laser, and its wavelength *λ* = 632.8*nm*. Four fringe patterns are recorded with one 8 bit, 576 × 768 pixels CCD camera. The HS WFS has a 32 × 32 micro-lens array, and the aberrated wavefronts are generated by a 512 × 512 pixel LC SLM.

In our experiment, we change the angle of polarizer P2 and find that the angle for the best fringe visibility is about 50°-60°, and we cannot observe distinctly the change of fringe visibility at this range. So we take the median value as the best angle of P2, and we need to adjust the angle of P2 with respect to horizontal direction to about 55° before experimental measurement. Then the LC SLM is controlled to generate a random aberration, which is a combination of rotationally symmetric and non-rotationally symmetric Zernike polynomials. Four fringe patterns recorded simultaneously by the CCD camera are shown in Fig. 4
. The image shown in Fig. 4 is averaged from ten images during several seconds. The wrapped wavefront difference $\mathrm{\Delta}\phi (x,y)$ can be calculated by Eq. (9) from these four interferograms in Fig. 4. The unwrapped result is shown in Fig. 5(*b*)
.

We use the modal wavefront reconstruction method [36] to reconstruct the wavefront under test from the unwrapped wavefront difference (shown in Fig. 5(*b*)). The maximum order of Zernike polynomials is set to 45. The reconstructed result is shown in Fig. 6(*a*)
. Let *RMS* and *PV* represent the root-mean-square value and peak-to-valley value of the wavefront under test. The *RMS* value and the *PV* value of the reconstructed wavefronts are 0.4753 *λ* and 3.3222 *λ*, respectively. When we normalize the *RMS* value of each order Zernike polynomial to 1.0*λ*, the modal decomposition coefficients of the wavefront under test can be obtained by the least square method, and it is shown in Fig. 6(*b*).

For comparison, we measured separately the aberrated wavefront by using HS WFS. A spot array is detected when a light beam reflected from LC SLM is split into HS WFS, and the corresponding gradient distributions along horizontal and vertical directions is also calculated. A modal reconstruction method [37] is employed to get the wavefront under test again. The maximum order of Zernike polynomials also is 45. The reconstructed result is shown in Fig. 7(*a*)
, and the *RMS* value and the *PV* value of it are 0.4798*λ* and 3.5953*λ*, respectively. The corresponding coefficients for each order Zernike polynomial of the wavefront under test measured by HS WFS is plotted in Fig. 7(*b*).

The residual error, which is the difference between these two wavefronts under test measured by the proposed RSI and the HS WFS, is shown in Fig. 8(*a*)
, and the *RMS* value and the *PV* value are 0.0348*λ* and 0.3149*λ*, respectively. The modal decomposition coefficients for each order Zernike polynomial of the residual error is shown in Fig. 8(*b*).

From the Fig. 8, a small residual error is obtained, and the result measured by the proposed RSI accords with that given by the HS WFS. The difference is mainly due to the influence of the detecting noise of the CCD camera and the grating effect of the LC SLM. If a better CCD camera and a continuous surface wavefront simulator are employed, better results would be obtained. As we said above, the detecting noise of CCD camera and the grating effect of the LC SLM is the main error source of the proposed RSI.

In practical experiment system, however, they may be not the only two error sources if we don’t adjust accurately this system. Some special notes are listed as follows:

Firstly, the magnified beam (*i.e.* Beam1) must be aligned with the demagnified beam (i.e. Beam2). In the other word, if a lateral offset is existing between Beam1 and Beam2, the under test wavefront would be hard to reconstructed accurately.

Secondly, the accuracy of the proposed RSI is also decided by the accuracy of phase step. As the description in section 2, the phase-shifter in the proposed RSI comprise three quarter wave plates QW1-QW3, so the accuracy of the phase step is decided by the accuracy of the fast axis direction of these three quarter wave plates. In experiment we need to calibrate carefully the fast axis of quarter wave plates.

Thirdly, the polarization direction of the polarizer P1 decides the work state of the LC SLM. So, in our practical experiment, we must adjust the polarization direction of P1 along the fast axis LC SLM and ensure that the LC SLM is working at phase-only modulation mode. If not, an unexpected illumination, which is not modulated by LC SLM and has orthogonal polarization direction with the modulated light beam, will enter into the proposed RSI, and the measurement error is produced.

Finally, the aberrated wavefront generated by WS must be relayed to the photo surface of CCD camera. Special focal lengths of L5-L8 are required. In our experimental system, the aberrated wavefront is relayed to the position of A2 firstly by lenses L5 and L6, and then it is relayed again from the position A2 to the photo surface of CCD camera by L7 and L8.

Certainly, we can avoid the error coming from these four error sources by adjusting carefully our optics configuration. Once it is achieved, the second adjustment is not needed.

## 5. Conclusion

We have demonstrated a single-shot common-path phase-shifting RSI for wavefront measurements. The principle of this system is also introduced in detail, and the performance of the proposed RSI is tested experimentally. An aberrated wavefront is generated by the LC SLM, and it is measured by both the proposed RSI and an HS WFS for comparison. Experiment results show that a good agreement has been achieved between them. We also analyze the relation between the fringe visibility and the angle of P2. The results show that the fringe visibility can be adjusted by rotating P2, and the most proper angle is always existent. In addition, the fringe visibility is more sensitive to a little rotation of P2 when the system has a larger radial shear.

In summary, the proposed RSI is insensitive to environmental vibration due to its common-path configuration, so it has lower demand for measurement environment. In addition, due to only one CCD camera is required to achieve simultaneous acquisition of four phase-stepping interferograms, the proposed RSI can be used to measure the dynamic wavefronts, especially the wavefronts of high power laser beam whose amplitude and phase distribution are very irregular and extremely unstable. Based on these advantages we reasonably believe that the proposed RSI is an accurate and useful tool for wavefront measurements, and it would be applied in adaptive optics system as a wavefront sensor.

## Acknowledgments

The authors thank Prof. Wenhan Jiang and the reviewers for their helpful suggestions.

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