1. Introduction

Electronic speckle photography, whose basic idea is to compare images of the speckle patterns before and after displacement of an object, has been studied extensively and various techniques have been developed during the past decades [1–5]. Although they differ in specific technical details, these existing techniques are based exclusively on the cross correlation function of intensity of the speckle field, while the underlying phase information has received much less attention. Recently, pseudophase information from the complex analytic signal of speckle field has been proposed, and the improved performance based on the pseudophase-only correlation has been demonstrated [6]. Furthermore, because the pseudophase can be obtained without recourse to interferometry, this technique can be applied to speckle-like artificial markings, such as printed random dots and random textures of natural origin (e.g., biological tissues), under incoherent natural illumination. This versatility of the pseudophase information opens up new possibilities in a wide range of applications beyond those known for laser speckle metrology.

On the other hand, noting that there are a lot of randomly distributed phase singularities in the pseudophase of the analytic signal for a speckle field, we have proposed a method referred to as optical vortex metrology, which is making use of phase singularities for displacement measurement [7]. The proposed optical vortex metrology and conventional speckle metrology [8] share the same philosophy of utilizing the physical phenomena that were initially regarded as obstacles in optical metrology. In early times, speckles were regarded as a nuisance degrading image quality, but now they are well appreciated as a very useful vehicle for optical metrology. Learning from history, we consider that phase singularities can become very useful in optical metrology although they are commonly regarded as obstacles in connection with phase unwrapping [9]. The purpose of this paper is to report a substantial improvement in the techniques for both analytic signal generation and phase singularity localization, and to demonstrate an advantage over our previous technique [7]. Rather than using the information about the local density of phase singularities, which gives rise to a loss of spatial resolution in displacement measurement, we detect the coordinates of the phase singularities in the pseudophase of a complex analytic signal, and then determine the micro-displacement from the differences between the coordinates of the phase singularities before and after displacement.

First, we briefly explain the two-dimensional (2-D) isotropic analytic signal representation of a speckle pattern by using vortex filtering and show how the pseudophase can be extracted from the analytic signal. Then, we describe the experiments of micro-displacement measurement making use of the phase singularities in the pseudophase of the analytic signal, and demonstrate the validity of the principle and the performance of nanometric spatial resolution.

2. Principle

2.1 Two-dimensional isotropic analytic signal of a speckle pattern

Before explaining the proposed optical vortex metrology, we first review the pseudophase retrieval from the complex-valued analytic signal of a speckle pattern along the line of our previous investigation [6,7].

The concept of an analytic signal was introduced to communication theory by Gabor for one-dimensional (1-D) signals [10]. Usually, an analytic signal consists of two parts: The real part is the original signal with its mean value being subtracted, and the corresponding imaginary part is the Hilbert transform of the real part. In our previous approaches, we retrieved the pseudophase from the complex analytic signal of the 2-D speckle pattern by applying the partial Hilbert filter, which passes the rectangular spatial frequency spectrum in the right half plane corresponding to f_x > 0 . However, an anisotropy was found in the generated analytic signal, which was introduced by the difference in the spectral bandwidths in the f_x -and f_y -directions. Although the anisotropic analytic signal does not give rise to a bias or a scale change for the detected displacement, it results in a direction-dependent spatial resolution for the displacement measurement. To obtain a 2-D isotropic analytic signal of a 2-D speckle pattern, we replace the partial Hilbert transform with the Riesz transform or vortex transform [11].

Let I(x,y) be the ac component of the original intensity distribution of the speckle pattern obtained after subtraction of the mean value (average dc component), and let its Fourier spectrum be ℑ(f_x,f_y). We can relate I(x,y) to its isotropic analytic signal Ĩ(x,y) through a vortex filter, which is a pure spiral phase function. Thus, our definition of a 2-D isotropic analytic signal for the speckle pattern is

where V(f_x,f_y) is a spiral phase function with a vortex structure in the frequency domain defined as follows:

Here the phase β = arctan(f_y/f_x) is the polar angle in spatial frequency domain. As pointed out by Larkin et. al. [11], the spiral phase function has the unique property that any section through the origin is a signum function with a π phase jump. After straightforward algebra, we find

where ∗ denotes the convolution operation, and v(x,y) is the 2-D Riesz kernels,

Here, α = arctan(y/x) is the azimuth angle in the spatial polar coordinates defined as usual. The phase θ(x,y) of this complex analytic signal representation of the speckle pattern is referred to as the pseudophase to distinguish it from the true phase of the optical field. Although it is not the true phase of the complex optical field, the pseudophase does provide useful information about the displacement of the object, as will be shown in Section 3.

2.2 Optical vortex metrology

Usually, it will be convenient to separate a complex analytic signal Ĩ(x,y) into its real and imaginary parts,

In analogy with an optical vortex [12] representing the dislocation of wavefronts, the phase singularity in an analytic signal is a point in the plane, where the local phase of analytic signal is undefined. The phase singularities are also the loci of vanishing amplitude determined by the intersection of the two zero crossing lines:

Figure 1(a) shows an example of the real part of the complex analytic signal around a phase singularity, and Fig. 1(b) is the corresponding imaginary part. The reconstructed phase is shown in Fig. 1(c), where a typical 2π helix structure around the phase singularity cannot be observed clearly. This means that the exact location of the phase singularity cannot be identified with subpixel resolution. Note that, in contrast to the discontinuous and complicated phase structure, the real and imaginary parts have an extremely simple structure consisting of a smooth monotonic surface [13]. It is this local monotonicity that has made possible the highly precise reconstruction of the detailed local structure of the complex analytic signal around the phase singularity by a two-dimensional interpolation of the real and imaginary parts. In the immediate vicinity of the phase singularity, the real and imaginary parts of the analytic signal can be expressed as

Applying the least-square method, we can obtain the coefficients a_k,b_k,c_k (k = r,i) that make the planes best fit to the values of the complex analytic signal detected at the pixel grids surrounding the phase singularity.

 

Fig. 1. Real and imaginary parts of analytic signal, and the corresponding phase structure around a phase singularity. Left column: before interpolation; right column: after interpolation.

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The interpolated surfaces for the real and imaginary parts are shown in Fig. 1(d) and Fig. 1(e), with the contour lines Re[Ĩ(x,y)] = 0 and Im[Ĩ(x,y)] = 0 inserted, respectively. From these interpolated real and imaginary parts, we can obtain the detailed phase profile around an optical vortex, as shown in Fig. 1(f). The phase singularity, occurring at the intersection of the zero-contour lines, is a point in the plane. From the coordinate (x,y) of the intersection point, we can identify the location of the phase singularity with subpixel accuracy given by

Just as a random speckle intensity patterns imprint marks on a coherently illuminated object surface, randomly distributed phase singularities in the pseudo-phase information associated with the speckle patterns imprint unique marks, i.e. singularities, related to the object surface with positive and negative topological charges. When observed with focus on the object surface, the displacement of each phase singularity can be directly related to the local displacement of the object surface. Thus, the displacement of an object can be estimated from the histogram of coordinate change for each phase singularity within the whole probing area.

3. Experiments

Experiments have been conducted to demonstrate the validity of the proposed principle, as depicted in Fig. 2. Since a white-light speckle pattern is known to be less prone to decorrelation [14], we generated a white-light speckle pattern by directly illuminating the surface of piezoelectric transducer (PZT) stage (Physik Instrumente P-752.11C NanoAutomation Stage) with a halogen lamp of a microscope (Nikon ECLIPSE ME600); this also serves as a demonstration that the proposed technique can be applied to an object illuminated by incoherent light. The white-light speckle pattern generated on the object surface was imaged by the microscope (with a 20+ objective lens and a 0.45+ relay lens) onto an image sensor. Through introducing controllable micro-displacement with the PZT, we recorded the grayscale images for white-light speckle patterns by a CMOS camera (SILICON VIDEO 9M001) with the pixel size5.2μm+5.2μm. From these nominal magnifications and the pixel separation of the CMOS image sensor, the unit pixel displacement corresponds to an object displacement of 578nm.

 

Fig. 2. Experimental set-up for generation and record of white-light speckle pattern

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To reduce the influence of relatively large detector noise coming from the CMOS image sensor, we used the intensity averaged over 20 frames of speckle patterns taken for each displacement. From the averaged intensity distribution of the white-light speckle pattern, we generated an isotropic signal by the vortex filtering and retrieved the pseudophase information. To adjust the speckle size and control the density of phase singularities, a low-pass filter has been applied to the Fourier spectrum of the averaged intensity distribution before the vortex filtering. In the experiment, we chose the average speckle size carefully so that a single speckle includes about 40 pixels along the line traversing it. After determining the coordinates of all the phase singularities in the probe area, we measured the given displacement by the proposed optical vortex metrology.

Figure 3(a) shows an example of the histogram for the x -coordinate changes of the phase singularities for the displacement when a voltage of 0.008 volt was applied to the PZT, and Fig. 3(b) is the histogram for the y -coordinate changes. From the locations of the maximum of the histograms, we can determine the displacements of the object in the two directions. Based on the histogram, we can also calculate the standard deviation σ, which serves as a reliability measure of the proposed technique.

 

Fig. 3. Histograms of coordinate changes of phase singularities for speckle pattern before and after displacement. (a) x-direction; (b) y-direction. Unit pixel corresponds to 578nm.

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Noting that both histograms look like a symmetric Lorentzian form, we can simply identify the peak locations by calculating the mean values of coordinate differences in the x - and y - directions. In this measurement, the displacement along the x -direction has its mean value equal to 0.03973 pixels (23.0nm) with the standard deviation σ = 0.01518 pixels (8.8nm), which gives an estimation of the uncertainty in one-dimensional displacement estimation based on a single singularity. Similarly, we also get the mean value of Δy equal to -0.01366 pixels (7.9nm,) and the corresponding standard deviation equal to 0.01516 pixels (8.8nm). As expected, the generated analytic signal is indeed isotropic, because the two displacement histograms for phase singularities have equal peak widths with the same standard deviations. From these two mean values, the object displacement distance can be obtained as ∆L = (<∆x>² + <∆y>²)^1/2 = 0.04201 pixels (24.3nm).

Within the linear region of the PZT, we increased the micro-displacement by applying higher voltages. Figure 4 shows how the histograms of the displacements of the phase singularities change with the voltages applied to PZT. As expected, the peak position varied with the applied voltages. The location of the histogram peak for an applied voltage of 0.008 volts is observed at the correct position, but the peak height is less than that for 0.004 volts. This phenomenon may be attributed to two different origins. One is that the speckles begin to change their shapes in addition to their pure lateral displacement; in other words, decorrelation occurs when the displacement is increased. This gives rise to creation or annihilation of the phase singularities, and the newly created or annihilated phase singularities cannot find their counterparts for correct calculation of their coordinate difference. The other origin is the flow of the phase singularities across the boundary of the probe area. The object displacement causes some of the phase singularities to move into (or out of) the probe area across the boundary. This also introduces a decrease in counts of phase singularities in the histogram due to the lack of counterparts in the other pseudophase map. We can observe the same phenomenon for an applied voltage equal to 0.012 volts. As is the case for the shape change of the cross-correlation function in the conventional correlation-based technique, the displacement histogram in the proposed optical vortex metrology becomes broader with a decrease of the peak height for increased applied voltage. In this measure, the standard deviations are 0.0183 pixels, 0.0297 pixels and 0.0340 pixels for voltages equal to 0.004 volts, 0.008 volts and 0.012 volts, respectively. This is understandable if one notes that the decrease of the histogram peak height of phase singularities is closely related to the decrease of the correlation peak in conventional correlation-based speckle metrology, because both are caused by speckle decorrelation.

 

Fig. 4. Variation of the peak positions and peak heights of the displacement histograms of phase singularities, with the amount of voltages applied to piezoelectric transducer. (Unit pixel corresponds to 578nm.)

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Figure 5 shows the relation between the applied voltage to the PZT and the detected nano-scale displacement ∆L , where the dashed line is the linear fitting given by ∆L = 3783.6V-6.7263 with a standard deviation of 1.2nm. With an increase of the applied voltage, the amount of measured displacement increases linearly. Figure 5 serves as an experimental demonstration of the validity of the proposed technique for micro-displacement measurement with nano-scale resolution.

 

Fig. 5. Relation between the applied voltage to PZT and the displacement detected from the peak position of the displacement histogram of the phase singularities. (Unit pixel corresponds to 578nm.)

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4. Conclusions

We have proposed a new technique for displacement measurement, referred to as optical vortex metrology, which makes use of the locations -and subsequent displacement- of phase singularities in the pseudophase of the analytic signal for a speckle pattern. Experiments have been performed that demonstrate the validity of the proposed technique giving nanometric accuracy. Contrary to the common belief in phase unwrapping where the phase singularities are considered as obstacles or nuisances in optical metrology, the proposed technique may serve as yet another evidence for the usefulness of phase singularities in optical metrology, which we believe will open up new possibilities that are worth exploring in the future.