It was shown in previous studies that, when a diffuser is illuminated by coherent light with an average spatial intensity distribution obeying a negative power function, the scattered field in the Fraunhofer diffraction region exhibits random fractal properties. The method employed so far for producing such fields has a disadvantage in that generated speckle intensities are low due to small transmittance of fractal apertures used in the illumination optics. To overcome this disadvantage, a generation of fractal speckles by means of a spatial light modulator is proposed. The principle is explained and experimental results are also shown.
© 2007 Optical Society of America
It is known nowadays that the concept of fractals plays an important role in describing and understanding a large number of complex geometries and phenomena in various areas of science [1–3]. In the field of optics, optical fields produced by illuminating objects having fractal properties have been studied in view of interest in optical phenomena produced by various objects with fractal structures and optical sensing of fractal objects . In optical sensing of a fractal object, fractal parameters, such as fractal dimension, scaling parameter of the self-affinity and multifractal dimension, are extracted from optical fields produced by the object. Fractal properties of such optical fields were also discussed [4, 5].
Meanwhile, it is also intriguing to consider more direct applications of the fractal concept to optical technology. To this end, it is desirable that certain optical fields having fractal properties can be generated with their fractal dimension controllable in a certain manner. A possibility of achieving this aim has been suggested and demonstrated by Uno et al.  and Uozumi et al. [7–9]. They showed that, when coherent light with an average spatial intensity distribution obeying a negative power function is incident on a diffuser such as a ground glass plate, fields scattered in a diffraction region exhibit random fractal properties. Such random fields, being called fractal speckles, have extremely long tails in the spatial correlation function of the intensity distributions in comparison with ordinary speckles, which implies that they may extend measurement ranges in various metrological applications based on the spatial correlation of speckles. Another interesting application of fractal speckles is found in optical formation of fractal random structures. By illuminating transparent photopolymer with fractal speckles, random media with desired fractal dimension may be produced, and such an approach was actually employed to generate fractal media in view of a random laser .
So far, theoretical, experimental and computer simulation studies were performed on this subject, showing that fractal speckles can be produced in the Fraunhofer and Fresnel diffraction regions, as well as in the image plane of the diffuser, and that intensity distributions in those regions share the same fractal properties such as long power-law tails in their spatial correlation functions .
In the generation of fractal speckles proposed in Ref. , random mass fractal apertures were used to produce in a Fraunhofer diffraction plane intensity distributions to be incident on a planar diffuser because such an intensity distribution obeys a negative power-law with an exponent related with fractal dimension of the aperture. However, random fractal apertures have a disadvantage in that the total transmittance of the aperture is strongly reduced with a decrease in it’s fractal dimension. Since the use of the aperture with small fractal dimension results in extremely dark fractal speckles, it becomes difficult to apply the field to the optical metrology and to optical formation of fractal structures. To overcome the disadvantage, we propose in the present paper an alternative method for generating power-law intensity distributions by means of a phase-only spatial light modulator(SLM). This method realizes a power-law illumination of higher intensity, as well as much easier control of intensity profile, giving rise to bright fractal speckles with controllable fractal dimension.
In Section 2, we review briefly the previous experimental setup and the theoretical background of the generation of fractal speckles. In Section 3, we describe the proposed method for producing higher power-law intensity distributions experimentally. Experimental results of generated speckle intensities are shown in Section 4, where their autocorrelation functions are also discussed.
2. Generation of fractal speckles
Consider an optical system shown in Fig.1 . A random fractal object is placed in the object plane P1. When this object is illuminated by a coherent and uniform plane wave such as an expanded and collimated laser beam, a random field is produced in the Fraunhofer diffraction plane P2. Since the intensity distribution of this scattered field corresponds to the power spectrum of the field just behind the fractal aperture in P1, the ensemble-average intensity at a point (x,y) in P2 follows a negative power-law,
where q = (x 2 + y 2)1/2 is a radial coordinate in the plane P2, and D is the fractal dimension of the random fractal object.
Next, consider that an ordinary plane diffuser such as a ground glass plate is placed in P 2 so that the intensity distribution of Eq. (1) is incident on it. Due to a property of speckle phenomenon similar to the van Citter–Zernike theorem in the coherence theory, the amplitude correlation function of the scattered field in the observation plane P3 is obtained by the Fourier transform of the intensity distribution just behind the diffuser. The correlation function of speckle intensity fluctuations is given by the squared modulus of the amplitude correlation function, as far as the speckle field in P3 obeys the Gaussian statistics . Since the diffuser is illuminated by the intensity distribution following the power function of Eq. (1), the autocorrelation function of complex amplitude distributions produced in the Fraunhofer diffraction plane of the diffuser is expressed in terms of the Fourier transform of Eq. (1), as long as the diffuser is a pure phase screen. Therefore, the intensity correlation function can be shown to obey again a negative power law ,
where r is a radial coordinate in the plane P 3, provided 1<D<2. Note that 2(D-2) < 0 since D < 2. A negative power-law correlation is a typical evidence of the fractality for the mass fractal type of objects . Thus, the scattered intensities in the observation plane P3 are regarded to have a fractal property, and can be called fractal speckles. The fractal dimension of mass fractals is related to the exponent of the power-type correlation function, and can be derived from the slopes of the correlation curves in a log-log plot. In the case of fractal speckles, the intensity correlation function in the range of 1<D<2 is given by Eq. (2), from which the fractal dimension of the speckle, Df , is given by Df = d - γ = 2D-d, where γ = 2(d -D) is the absolute exponent of the correlation function and d is the dimension of the Euclidean space where fractal speckles are distributed and is equal to 2 in the present case .
Fractal speckles produced in the above method have been reported in Ref. . It is noted, however, that the intensity of the fields is extremely low due to small and still reducing transmittance of fractal apertures with a decrease in D. As a solution to this problem, an alternative method for producing power-law intensity distributions is proposed in the next section.
3. Generation of power-law intensities by means of a spatial light modulator
Consider an optical configuration shown in Fig. 2. In comparison with Fig. 1, a solid state laser with the wavelength of 532nm is used as an optical source, and a random fractal aperture is replaced by a phase-only SLM, which consists of a polarizer P, a twisted nematic liquid crystal display (TN-LCD), a circular aperture C1, a quarter-wave plate W and an analyzer A. To make the SLM operate in a phase-only mode, angles of the optical elements with the u-axis are set at θp = 100° for the polarizer, θw = 144° for the slow axis of the quarter-wave plate and θa = 77° for the analyzer, respectively [12–14]. The TN-LCD (LC2002, HoloEye Photonics) has 832 × 624 image pixels with a filling factor of 85% and a pixel pitch of 32μm. The LCD converts color signals into corresponding 8-bit gray level signals. Two personal computers PC1 and PC2 are used to control the LCD and to provide static images of spatial phase modulations as videos to be displayed on the LCD, respectively. Under the above conditions, the SLM generated the intensity and phase modulations shown in Fig. 3, where the intensity modulation is normalized to unity: the peak-to-peak change of the intensity modulation is 0.12 and the maximum phase modulation is 1.5π radians. When a suitable phase modulation image is displayed on the LCD, the SLM diffracts the incident light to produce a power-law intensity distribution on P2. The intensity distribution is then incident on a ground glass plate placed in the plane P2. The field immediately behind P2 is Fourier-transformed by the lens L4 to produce a fractal speckle field in the back focal plane of L4. The intensity distribution of the field is captured by a CCD camera having 1036 × 1362 pixels with a pixel size of 4.65μm. The central portion of 1024 × 1024 pixels around the optical axis is used in the subsequent data processing.
The phase modulation image to be displayed on the TN-LCD can be produced by means of an iterative Fourier transform method known as the Gerchberg–Saxton method . The method iterates the determination of two complex amplitude distributions alternately in object and spectral domains, imposing certain constraints in each domain. In the present case, the phase modulation plane of the LCD corresponds to the object domain, where the limited phase modulation of -0.75π< θ < 0.75π, truncation due to the circular aperture C1 and a constant modulus are imposed as the constraints, while the spectral domain corresponds to the Fourier plane P2 of the object plane, where a power-law modulus distribution is forced to be satisfied. The iterative process starts with the complex amplitude distribution in the spectral domain with the initial condition of the power-law modulus combined with a random phase distribution across the spectral domain. By inverse-Fourier-transforming this complex amplitude, another complex amplitude distribution is obtained in the object domain, where the complex amplitude is converted into a phase-only distribution by imposing the constrains described above. By Fourier-transforming the resultant distribution, a new complex amplitude is produced in the spectral domain, where the amplitude modulus distribution is replaced by the power-law modulus with the phase distribution retained. This completes one turn of the iteration and we repeat this process until sufficient convergence is observed. In this procedure, no pixels in the image are omitted or scaled.
A phase modulation image is produced as a square matrix of size W = 480 and the iterative process is repeated 1000 times. The convergence of the calculated modulus distribution in the spectral domain is evaluated after every iteration by the cost function defined as
where U 0 and Ui are the desired and calculated modulus distributions, respectively, while α is a constant to equalize the energy of U 0 with that of Ur. The cost function calculated in the case of D = 1.5 is shown in Fig.4 as a function of the number of iterations. As is shown in this figure, ΔE is sufficiently converging at about 500 iterations. The calculated phase modulation image is quantized in 5-bit levels due to the difficulty in controlling the phase modulation of the SLM in larger quantization levels and is transformed into the image with gray levels in accordance with the phase modulation curve shown in Fig.3. Figures 5(a)–5(c) show resultant phase modulation images for D = (a) 1.2, (b) 1.5 and (c) 1.8, respectively.
This new approach to the generation of fractal speckles using the phase-only SLM solves the problem in the previous method by realizing fractal speckles of higher intensities and by removing the dependence of the brightness of fractal speckles on the fractal dimension D of the aperture. This method has another advantage that fractal dimension of the fractal speckles is easily controllable by simply changing the phase modulation image displayed on the LCD.
4. Experimental results and discussion
Intensity distributions produced in the plane P2 by the SLM with the phase modulation images described above are captured by the CCD camera by locating it in P2 to examine their properties. Figures 6(a)–6(c) show the spatial distributions of logarithmic intensities for D = (a) 1.2, (b) 1.5 and (c) 1.8, respectively. As is seen from Fig. 6, each intensity distribution is anisotropic due to the anisotropy of the envelope function produced by the Fourier transform of a single pixel in an inequilateral square shape of the LCD. However, the effect of the envelope function on the intensity distribution can be neglected except for marginal regions corresponding to higher spatial frequencies in Fig. 6. On the contrary, the deviation of lower spatial frequencies from the power-law intensity distribution is seen in this figure due to the specular component and the finite extent of the circular aperture C1. Figure 7 stands for logarithmic plots of angular averages of the intensity distributions shown in Fig. 6. It is seen from this figure that the generated intensities obey the power law in the range from 20 to 250 pixels with exponents accordant with theoretical ones.
Fractal speckle intensity distributions produced finally are shown in Figs. 8(a)–8(c) for D = (a) 1.2, (b) 1.5 and (c) 1.8, respectively. Appearances of these patterns are in good accordance with those produced by the optical system of Fig. 1 as reported previously . To obtain statistical properties and fractal dimensions of these intensity distributions, their autocorrelation functions are calculated, then ensemble-averaged over four fractal speckle patterns for each D, and finally angular-averaged. The results are shown in Fig. 9 in a logarithmic plot for D = 1.2, 1.5 and 1.8. As is seen from this figure, the correlation functions have good linearity in a certain region of r.
As explained in Section 2, the correlation function of Eq. (2) is derived from the relationship analogous to the van Cittert–Zernike theorem, which allows us to estimate the range of the power-law correlation tail to be from 4 to 50 pixels from the scaling range of the intensity distributions given in Fig. 7. The results in Fig. 9 exhibit good correspondence with this prediction. However, the linear portion of the correlation decay is seen to become shorter in length than that of the above prediction with an increase in D. Such a trend can be explained qualitatively as follows. From the Fourier transform relationship, the shrinkage of the correlation decay is caused by a deviation from the power-law in the central spot of the intensity distribution incident on the diffuser. Such the deviation effect of the central spot in the intensity profile increases with an increased in D as is seen in Fig. 7 .
As mentioned in Section 2, the fractal dimensions Df of the fractal speckle is given by Df = 2-γ. The inclination γ of the autocorrelation function plotted logarithmically is obtained by fitting correlation data in the range from 4 to 50 data points into a linear line. Table 1 shows theoretical and experimental values of the fractal dimension of the fractal speckles, demonstrating a good agreement between corresponding values except for D = 1.8. The appreciable difference of the corresponding fractal dimensions for D = 1.8 is caused by the effect of the central spot on the correlation profile as is discussed above.
A new method for generating fractal speckle fields by means of a spatial light modulator was proposed and demonstrated experimentally. In the method, a power-law intensity distribution was generated using a phase-only SLM on which a phase modulation image was displayed, and was incident on a ground glass plate as a diffuser. By Fourier-transforming the field immediately behind the diffuser by a lens, fractal speckle fields were observed in the back focal plane of the lens. To confirm the fractality of the generated intensity distribution, its autocorrelation function was calculated and was shown to obey a power law in the scaling range as predicted by the scaling range of the power law intensity of illumination. Fractal dimensions were obtained from the inclination of the autocorrelation in the logarithmic plot and were found to agree well with the theoretical prediction. The use of a phase-only SLM realizes fractal speckles of higher intensities as well as a controllable fractal dimension of the produced speckles. The method provides us with fractal speckles of sufficient intensities for their applications.
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