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Superspeckles: a new application of optical superresolution.

The effect of optical superresolution on speckle correlations is studied. Simulations reveal that using a lateral superresolution pupil filter more than twice the out of plane correlation length of the clear pupil can be achieved. This means that the measurement range in speckle correlation measurements doubles. To verify the correlation length an experiment is performed using a liquid crystal (LCD) spatial light modulator as a programmable superresolution filter. The results corroborate the simulation.


Introduction
The extent of a laser speckle in three dimensions is of primary importance in the measurement of deformations or displacement of rough objects. It determines the measurement range within which one can ascertain the deformations in the form of fringe pattern in the case of Digital Speckle Pattern Interferometry (DSPI) or speckle correlation and Young's fringes in case of speckle photography. Speckle correlation length limits the measurement range in DSPI or speckle photography. The latest investigations on the three dimensional size of a speckle was done in Ref. 1-3 by considering the autocorrelation function of the speckle intensity. Previously, image formation along the axial direction was studied by Frieden [4]. The average length of the speckle along the longitudinal direction is evaluated to be 8A. (LID) 2 , where L is the observation distance from the pupil and D is the diameter of the illuminated area or the pupil diameter. The decorrelation in longitudinal direction was experimentally demonstrated by considering the reduction of fringe visibility of Young's fringes from double exposure photographs with successive images taken at defocus positions with constant in-plane displacement.
Therefore, an increase in the average longitudinal length of the speckle without changing the lateral resolution of the imaging system is of great significance in the areas such as DSPI, speckle photography, or even blood flow imaging [5]. Here we examine the increase in the average speckle size along the longitudinal direction (optical axis) using transverse superresolution pupil filters. Simulations show that more than 100% increase in speckle correlation length is easily achievable. This is due to the fact that a reduction in the central spot size of the diffraction pattern results in a decrease ofresolution in the axial direction [6]. An increase in the longitudinal speckle size would also be obtained by just reducing the diameter of the pupil which in turn increases the LID ratio. However, this would reduce the lateral resolution of the image. With the incorporation of a superresolution filter the speckle length is increased together with an increase in spatial resolution. The introduction of superresolution filters is thus destined to extend its applications, which are presently limited to microscopy [7], astronomy [8] and dense DVD writing [9], to areas such as DSPI or speckle photography.

Theory and simulation
To evaluate the degree of longitudinal correlation, speckle fields are first calculated for different object positions obtained by moving the object from its focal position along the optical axis. This is followed by comparing the speckle fields so obtained with respect to the speckle field for the object at the focal position. The image plane and defocused speckle fields are generated using Fresnel approximation of the diffraction integral.
The image formed by an optical system can be considered as the two dimensional convolution of the object function with the impulse response function of the system, expressed as U;(x,y) =ff U 0 (~,17)h(x-~,y-17)d~d17 where P is the pupil function, (u, v) are the pupil plane coordinates, K is a scaling factor, z is the image distance which is the sum of focal length f and defocus, k = 2nl 2 is the wave number and 2 is the wavelength of light.
The integral in Eq. (2) is the Fourier transform of the product of pupil function and an exponential factor, E exp[i ~(-;· -7 } 11 2 + v2 )] . This factor is unity when z is equal to the focal length. For defocused locations this factor accounts fQr the broadening effect of the point spread function (PSF). This is the primary reason for the decorrelation of the speckle patterns obtained at the defocus locations of the object. The energy as well as the phase are spilled into the adjacent pixels.
The object function U 0 of the rough object can be expressed as To perform a simulation the amplitude, a, and the phase, ¢, are generated randomly at every To understand the effect of transverse supenesolution, we compare the unobstrncted pupil to continuous amplitude filters [ 11] that generate relative spot sizes G 90, 85 and 81 %. The details of the design and properties of these continuous amplitude filters are given in Refs. 11 and 12. The amplitude function is represented as where p (u 2 v 2 ) 112 is the radial distance in the pupil plane, b 2 ,,'s are the coefficients of the even terms and k 6. The coefficients b 211 's for amplitude filters are given in Table 1. For the unobstructed pupil function, we consider P(p) 1. Initially, Eq. (2) is evaluated using two dimensional FFT of the product of pupil function P and exponential function E at defocus positions. The lateral resolution in the pupil plane is adjusted to obtain the central spot of the PSF to fit into 4 X 4 pixels of the image. In order to reduce computation time, only the relevant central part of the FFT of the product of P and E is taken to calculate U; according to Eq. (1 ). This pa1t also includes the side lobes of the PSF which extends over several pixels.
Before comparing the speckle correlations, it is interesting to study the axial behavior of the PSF at various relative spot size values. The amplitude distribution U around the focal region of the optical system can be written as [8] 1 U(?) 2J P(p)exp(iup 2 12)p dp (5) 0 where ; = 4kzsin 2 (a / 2) is the axial optical coordinate, and sin a is the numerical aperhlfe.  Figure 2 represents the variation of the correlation coefficient, r, of the speckle images at focus and defocus locations for the unobstructed pupil and amplitude pupil filters. The correlation coefficient for the unobstrncted pupil drops to zero at about 0. 11 mm which is in good agreement with the speckle correlation distance 8A (L/D) 2 106 ~Lin. Further and in line with our discussion relative to Fig. 1, we also note that the correlation distance increases with the decrease in the G value. For G = 81 % the correlation distance doubles as compared to the unobstructed pupil. We tem1 this speckle elongated in the axial direction as 'Superspeckle'. To visualize the increase in longitudinal speckle correlation length, we consider DSPI fringes along the axis at various defocus positions. First, an interferogram is obtained by simulating the interference of the speckle field at focus with a plane wave making an angle with respect to the image plane. A second interferogram is simulated with the speckle field at defocus position and a plane wave propagating along the optical axis. The pixel by pixel intensity difference between these two interferograms produces DSPI correlation fringes. Figure 3 shows the DSPI fringes for the unobstructed pupil obtained at image plane positions ofO mm, 0.04 mm, 0.08 mm and 0.10 mm from the focus, respectively, along with their respective line intensity profiles. The line intensity profile is taken along the central rows by averaging 30 adjacent column pixels. The fringe modulation drops as the defocus increases having a minimum around 0.1 mm. mo '"'  Similarly, Fig. 4 shows the DSPI fringes and their respective line intensity profiles obtained using the amplitude filter with G = 81 %. The modulation drops to almost the same level as in Fig. 3d at around 0.18 mm defocus distance.

Experiment
We used an LCD Spatial Light Modulator (Sony Corp., having 832 x 624 pixels with pixel pitch of 32 µm and 1.3 inch diagonal) to implement the amplitude pupil filter. The characterization is done using two linear polarizers sandwiching the LCD. The procedure for characterization is detailed in Refs. 13 amplitude-only behavior, because any addition of phase values to the random phases of the speckle field cancels out during subtraction correlation. The setup for speckle correlation measurement is shown in Fig. 5. The laser beam (A = 532 run) is expanded and collimated to illuminate a rough object. A mirror and a beam splitter are used for normal incidence on the object. A doublet (L2) with 25.4 mm diameter and 400 mm focal length is used to image the speckled wavefront scattered from the rough object at unit magnification. Therefore the object and image plane distances are equal to 800 mm. The LCD is placed close to L2 in order to minimize the distance to the pupil plane. We have chosen a large focal length for two reasons. First, a large (LID) value means low angle light incidence on the LCD which reduces the error in amplitude modulation from the parallel beam behavior. Second, a linear stage with a manually operated micrometer can be used to move the rough object in the axial direction since the average length of the speckle is about 6.8 mm given by SA. (LID)2. A CCD-camera (IMAC™) with 11 µm pitch is used for recording the speckle patterns. The average size of the speckle in transverse direction covers 4x4 pixels of the CCD. The amplitude filter for G = 81 % shown in Figs. I and 2 is progra1mned onto the LCD. The speckle pattern correlations for defocus positions are obtained using Eq. (5). Figure 6 shows the correlation coefficients obtained from 200x200 pixels. The speckle images are grabbed for each 0.5 mm step along the longitudinal axis. Without amplitude filter, ie., the unobstructed pupil case in Fig. 6, the correlation drops around 6 mm which agrees with the theoretical value. With amplitude filter, the correction extended more than twice the distance and starts falling around 14 mm. This behavior confirms our simulation.

Conclusion
We have applied superresolution filters to produce 'superspeckles' to increase the correlation distance along axial direction. An experiment verifies the simulation. Therefore, we expect to increase the measurement range of DSPI to more than twice the range given by the optical system without superresolution pupil filter, with an increase in the spatial resolution of the image.