Three-wave shearing interferometer based on spatial light modulator

The use of a SLM for the three-wave lateral interference is proposed, and an eight-step phase-shifting scheme is developed for extracting phase information from three-wave interferograms. The twodimensional phase of object is reconstructed from two phase differences which are calculated from two orthogonal sheared interferograms. The flexibility of SLM can be fully utilized in the sense of dynamical controlling of the direction and amount of shear, as well as phase shift. The numerical simulation and optical experiment are carried out to demonstrate the feasibility and reliability of the proposed scheme. ©2008 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.5050) Phase measurement; (230.6120) Spatial light modulators. References and links 1. H. H Lee, J. H You, and S. H Park, “phase-shifting lateral shearing interferometer with two pairs of wedge plates.” Opt. Lett. 28, 2243-2245(2003) 2. DeVon W. Griffin, "Phase-shifting shearing interferometer," Opt. Lett. 26, 140-141(2001). 3. P. Liang, J. Ding, Z. Jin, C. Guo, and H. Wang, "Two-dimensional wave-front reconstruction from lateral shearing interferograms," Opt. Express 14, 625-634 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-625 4. S.Zhao and P. Chung , Digital speckle shearing interferometer using a liquid-crystal spatial light modulator, Opt. Eng., Vol. 45, 105606 (2006). 5. Y. Bitou, Digital phase-shifting interferometer with an electrically addressed liquid-crystal spatial light modulator, Opt. Lett. 28, 1576-1578(2003). 6. Victor Arrizón and David Sánchez-de-la-Llave,Common-path interferometry with one-dimensional periodic filters,Opt. Lett. 29, 141-143(2004). 7. Thomas J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A 14, 2692-2701(1997). 8. D. Kerr, G. H. Kaufmann, and G. E. Galizzi, "Unwrapping of interferometric phase-fringe maps by the discrete cosine transform," Appl. Opt. 35, 810-816(1996). 9. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, "Robust phase-unwrapping techniques: a comparison," J. Opt. Soc. Am. A 13, 2355-2366(1996). 10. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861(1986). 11. Clemens Elster, "Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears," Appl. Opt. 39, 5353-5359(2000). 12. A. Dubra, C. Paterson, and C. Dainty, "Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform," Appl. Opt. 43, 1108-1113(2004). 13. X. Tian, M. Itoh, T. Yatagai, "Simple algorithm for large-grid phase Reconstruction of lateral-shearing interferometry," Appl. Opt. 34, 7213-7220(1995). 14. H. Takajo, T. Takahashi, "Least-squares phase estimation from the phase difference," J. Opt. Soc. Am. A 5, 416-425(1988). 15. W. H. Southwell, "Wave-front estimation from wave-front slope measurements," J. Opt. Soc. Am. 70, 9981006(1980). 16. H. Takajo, T. Takahashi, "Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference," J. Opt. Soc. Am. A 5, 1818-1827(1988). 17. A Talmi andE. Ribak,Wavefront reconstruction from its gradients, J. Opt. Soc. Am. A 23, 288-297(2006). 18. J. Dias and J. Leitao. The ZπM algorithm for interferometric image reconstruction in SAR/SAS. IEEE Trans. Image Processing 11, 408–422(2002).


Introduction
Lateral shearing interferometry is a useful technique for evaluating wave front.The unique feature of self-reference of shearing interferometer makes it robust against poorly stabilized environments and more suitable in absence of available reference beam.Besides, coherence restrictions on light source are alleviated in shearing interferometers.Several methods exist for the purpose of shearing interference, including the use of refraction, reflection, or diffraction [1][2][3].Typical shearing implementations also adopt phase-shifting algorithm to extract phase slope from interferograms.In most shearing interferometers, mechanically moving devices are required for adjusting amounts of phase shift and wave-front shear.The accuracy and adaptability of shearing interferometer are limited by mechanical elements.It is well known that the light can also be steered by a spatial light modulator(SLM) controlled by a computer.Thus the SLM provides a promising alternative to mechanical facility for realizing phase shift and wave-front shear in sheering interferometer [4][5][6].
To date the SLM has been used for two-wave sheering, where two diffraction orders (normally the +1 and -1 orders) resulting from a grating pattern displayed on the SLM provide shearing waves.Due to modulation limitation, a grating realized by the SLM generally produces multiple diffraction orders.To ensure only two waves to interfere with each other in the output plane, blocking of unwanted orders is required in optical experiment.Alternative way instead of blocking operation is to optimize the modulation of SLM for generating only the two first orders.However, we found in practice that simultaneously eliminating both the zero order and other orders higher than the first is difficult to achieve for common SLMs.Instead, it can be realized more easily that retaining the zero and first orders while suppressing others.In this paper, we propose a three-wave shearing interferometer.A SLM displaying grating patterns diffracts the incoming object wave into the +1, -1 and zero diffraction orders, resulting in three-wave lateral shearing interference in the detector plane.We present a novel phase-shifting algorithm for extracting phase differences from three-wave interferograms.Validity of three-wave lateral sheering interferometry is shown in numerical simulation and optical experiment.The designed experimental system for three-wave lateral sheering interference enjoys the advantages in common-path, simplicity and dynamical controlling of both shear amount and phase shift.

Principle of three-wave shearing interference
The three-wave shearing principle based on a SLM is shown in Fig. 1.This is a conventional 4f system consisting of two lens L 1 and L 2 with focal length f.A SLM locating in the Fourier plane displays a cosine grating, which is used to diffract the incident light onto the +1, -1 and zero diffraction orders.The grating splits an object wave at the input plane into three copied and sheared waves at the output plane.The shearing interference occurs in the overlapped area of three waves.Assume the transmittance of the cosine grating at the (ξ, η) plane to be as follows, where β and T are the modulation depth and period of the grating fringe, respectively, and ξ 0  denotes a lateral displacement of grating.For a object wave in the input plane with field distribution where λ is the wavelength of light.The right-hand side of Eq.( 2) is composed of three , we get the following intensity distribution from Eq.( 2) where .We employ a scheme of eight-step phase-shift so that , 0, 1, , 7, 4 and accordingly obtain eight shearing interferograms, ( ) 3) and ( 4) we have derived the following relation where we have replaced ( ) , j I x y with I j for simplification.Then a phase-unwrapping algorithm should be performed because the phase difference value calculated from Eq.( 5) is wrapped in the range(-π, π) [7][8][9].By rotating the orientation of gratings about the optical axis by 90 degrees, we can get the phase difference in the y direction, ( , ) y x y ϕ Δ , through the similar procedure.To the best of our knowledge, the above described method provides a novel way for extracting phase difference from three-wave lateral shearing interferograms, which has not been reported.

Wave-front reconstruction from phase differences
The above proposed shearing interference only determines a phase difference of original wave-front.The remaining problem is how to retrieve the wave-front from phase differences.A variety of algorithms have been proposed to reconstruct two-dimensional wave front from two orthogonal phase differences [10][11][12][13].Here we employ an iterative algorithm based on the least-square estimate.Denoting the phase differences in the discrete form by ( , ) where N is the number of the sampling points.We define an error function in terms of the following square norm to reflect the discrepancy between original phase and the measured phase, ] In order to minimize the error U we make 0 , ) x x y y i j i s j i s j i j s i j s Eq.( 9) gives the relationship between the original phases and the measured phase differences in the least-square-error sense.The original phase ϕ(i,j) can be obtained by solving Eq. ( 9).
Many facilities exist for solving such an equation, including non-iterative methods such as transfer function [14,15] and iterative methods [16,17].Here we use the iterative algorithm to search for the optimal phase of Eq. ( 9), and the basis form for iterative scheme is expressed as ] where the value k ϕ in right-hand is current value, and 1 k ϕ + in left-hand is a new value to be updated after one time iteration, and the factor , i j g is expressed by the following equation after taking into account the boundary condition, , 1 1 to 2s 2 to ; 1 to 2s 2 1 to 2 1 to 2s or N-2s+1 to N; 2 +1 to 2 1 1 to 2 or N-2s+1 to N; =2 +1 to 2 3 Following Eqs.(11) and (10) the wave-front under test is reconstructed from two orthogonal phase differences in the iterative procedure.It should be pointed out that because the phase difference has a data domain with a smaller dimension than that of original phase, we need to implement zero filling before the iteration starting so as to enable , ( , ) x y i j ϕ Δ has the same dimension as ( , ) i j ϕ in Eq.( 9).This approximate treatment may yield error, which is, however, small in case of small shear and can also be minimized during the iterative procedure.

Numerical simulations
The performance of the proposed technique is first illustrated by numeric experiments.The numerical simulation is implemented with a synthetic wave front, as shown in Fig. 2(a), whose phase function is in the following form ( ) where x and y are within the range of [-1, 1] with 360×360 grids.For each direction of x and y, eight phase-shifted shearing interferograms with shear amount of 10 pixels are simulated according to Eq.( 3).After extracting phase difference from eight interferograms we choose the ZπM algorithm developed by Dias and Leitao [18] to carry out phase unwrapping because it works well in the sense of robustness against noise.The reconstructed phase from two orthogonal phase differences after 500 times of iteration is shown in Fig. 2(b).Fig. 2(c) serves as an example to demonstrate the convergence speed of the iterative algorithm.In this test we can obtain a stable result through less than 500 iterations.The running time of 500 iterations is about 20 seconds (on a computer with Pentium 4 processor at 1.8 GHz).The effect of shear amounts on reconstruction fidelity is investigated by evaluating the deviation of the reconstructed phase from the original one.Two error measures, the root mean square (RMS) and Peak-to-Valley (PV), are computed for different shears and are listed in Table 1.The unit of error is converted to wavelength λ, taking into account the fact that the 2π radian is equivalent to an optical length of λ.From Table 1 we can see that error increases with the increase of the shear amount.This is the case because the bigger shear yields a smaller spatial support of phase difference and thus leads to more information deficiency in the wave-front reconstruction.Noise is hard to avoid in the practical measuring of phase differences, due to, e.g., speckle, circumstance disturbance, detector noise, and other noise sources.Therefore we add some random noise to the phase differences to investigate its influence on the reconstruction accuracy.The noise level is defined as the ratio between the average absolute value of noise and that of the phase difference.Fig. 3 certifies the performance of the reconstruction algorithm in the case of the existence of noise.The two orthogonal phase differences with a shear of 10 pixels are added by a random noise with with 10% noise level, as shown in Figs.3(a) and (b), respectively.The reconstructed phase is indicated in Fig. 3(c), in which a reconstruction RMS of 3.4×10 -2 λ is achieved.The numerical tests confirm that the proposed three-wave shearing and eight-step phase shifting scheme has satisfactory reconstruction fidelity.

Experimental setup and results
We have conducted optical experiments to verify the effectiveness of the three-wave shearing interferometry proposed above.The experimental setup and shearing interference patterns are shown in Fig. 4. A He-Ne laser provides illumination light with the wavelength of 632.8 nm.A designed cosine grating is displayed on a twisted nematic liquid crystal SLM with 1024×768 pixels(each pixel has a 18μm ×18 μm size) controlled by a computer.The displacement, orientation, as well as period, of the grating can be dynamically changed so that the phase shift value, shear direction, and shear amount are adjusted into appropriate values.The interferograms are detected by a CCD camera with a cell size of 8.6μm×8.6μm.This interferometer enjoys the advantages of common-path, self-interference, and on-axis arrangement, and thus we use a rotating ground-glass diffuser so as to lower spatial coherence of the illumination light for the purpose of speckle elimination.First, we test a concave lens with a nominal focal length of -60.0 mm.The lens under test is placed at the object plane.The interferograms with a shear amount of 4 pixels are captured by the CCD and sampled within a domain of 360×360 pixels.The phase difference distributions in x and y directions are extracted in each direction from eight three-wave shearing interferograms following the phase shifting scheme described.Fig. 5 shows the extracted phase differences( in Figs.5(a) and (b)), and the reconstructed two-dimensional phase (in Fig. 5(c)).From the measured phase which reflects a spherical wave-front, we calculate the focal length of lens to be -58.2mm.We recognize that the discrepancy between measured value and the nominal value is due to the fact: there is a deviation in location between the image plane of the concave and the detector plane; the deduced focal distance starts not from the concave plane itself but from the CCD plane at which the wave-front phase is measured.We also measure the surface of a quartz glass with an etched pattern of the letter E. From the phase differences shown in Figs.6(a) and (b), the two-dimensional phase is reconstructed and shown in Fig. 6 (c).The depth d between the top and bottom of the binary relief relates to the measured phase φ through d=λφ /(2π∆n), where ∆n is the difference between the refractive index of the glass and that of the air.The refractive index of the glass is 1.457 at the 633 nm wavelength.Accordingly, the corresponding step height of the letter is 580±20 nm.For comparison, we also measured the step height with Dektek 3 profilometer from Veeco Instruments Inc.The measured value from the profiler is about 580 nm.The discrepancy between two kinds of measurement is less than λ/30.

Conclusion
The use of a SLM in the three-wave lateral interference has been presented, and an eight-step phase-shifting scheme for extracting phase information from three-wave interferograms has been proposed.The flexibility of SLM can be fully taken advantage of in the system proposed here demonstrated.By using SLM we can dynamically controlled the direction and amount of shear, as well as phase shift.We have demonstrated the feasibility and reliability of the proposed scheme through numerical simulation and optical experiment.In this paper the test

Fig. 1 .
Fig.1.Schematic of the arrangement for generation of three-wave shearing interference.A lateral shearing interferogram is formed in the fringed area.S denotes the shear amount.
sheared object waves with phase shift among them.Denoting the shear amount by s .(3) indicates a three-wave shearing interference existing in the output plane with a shear s in the x direction.Now we will concentrate at the term ( .(3), which represents the phase difference in the x direction, and describe how to extract this phase difference from the interference intensity.By adjusting the lateral displacement of grating, ξ 0 , we can realize a desired value of phase shift represented by

Fig.. 2 .
Fig..2.(a) Original phase used for simulation.(b) phase difference in x direction.(c) phase difference in y direction.(d) Reconstructed phase from the two phase differences after 500 iterations.(e) Relationship between reconstruction error (root mean square) and the number of iterations.

Fig. 3
Fig.3The results of simulation use our method with 10% noise level in phase differences: (a) phase difference in x direction.(b) phase difference in y direction (c) reconstructed wave front with shear amount s=10.

Fig. 5
Fig. 5 Test results of a spherical lens.(a) Phase difference in x direction.(b) Phase difference in y direction.(c) Reconstructed wave front.

Fig. 6
Fig. 6 Results of the optic surface testing: (a) map of the phase difference in x direction; (b) map of the phase difference in y direction; (c) the reconstructed phase distribution.

Table 1 .
Reconstruction error vs. shear amount s.