The use of a SLM for the three-wave lateral shearing interference is proposed, and an eight-step phase-shifting scheme is developed for extracting phase information from three-wave interferograms. The two-dimensional 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.
©2009 Optical Society of America
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[1-3]. 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 shearing interferometer[4-6].
To date the SLM has been used for two-wave shearing, 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 shearing 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.
2. 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 L1 and L2 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 t(x,y) = a(x,y)exp[iφ(x,y)] , the field in the output plane with reversed coordinates can be expressed by
where λ is the wavelength of light. The right-hand side of Eq. (2) is composed of three sheared object waves with phase shift among them. Denoting the shear amount by s = λf/T and the phase shift by ϕ = 2πξ 0/T, we get the following intensity distribution from Eq. (2)
where b 0=a 2(x,y) + [β 2/4)[a 2 (x + s,y)+a 2(x−s,y)], b 1 = βa(x,y)a(x + s,y), b 2=βa(x,y)a(x−s,y) and b 3=(β 2/2)a(x + s,y)a(x−s,y). Equations (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 Δφx (x, y) = φ(x + s,y) − φ(x − s,y) of Eq. (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 ϕ = 2πξ 0/T. We employ a scheme of eight-step phase-shift so that
where we have replaced Ij(x,y) with Ij 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)[7-9]. By rotating the orientation of gratings about the optical axis by 90 degrees, we can get the phase difference in the y direction, Δφy(x, y), 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.
3. 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 [10-13]. Here we employ an iterative algorithm based on the least-square estimate. Denoting the phase differences in the discrete form by Δφx(i,j) and Δφy (i, j), we have
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 ∂U/∂φ = 0, so that the following equation is derived
Equation (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 φk in right-hand is current value, and φ k+1 in left-hand is a new value to be updated after one time iteration, and the factor gi,j is expressed by the following equation after taking into account the boundary condition,
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 Δφx,y(i,j) has the same dimension as φ(i,j) 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.
4. 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  to carry out phase unwrapping because it works well in the sense of robustness against noise. Figures (b) and (c) present two orthogonal phase differences. The reconstructed phase from the two orthogonal phase differences after 500 times of iteration is shown in Fig. 2(d). Figure 2(e) 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. Figure 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 10% noise level, as shown in Figs. 3(a) and 3(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.
5. 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. Figure 5 shows the extracted phase differences(in Figs. 5(a) and 5(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 -58.2 mm. 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 6(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/(2πΔn), where Δn 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.
The use of a SLM in the three-wave lateral shearing 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. 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 of shearing interference in two orthogonal directions is conducted; however this system is also suitable for other multiple-direction and multiple-wave shearing interference.
This work is supported in part by the NSFC under Grant Nos. 10874078 and 60608006, and the Natural Science Foundation of Jiangsu Province under Grant Nos. BK2007126 and BK2006110.
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