## Generation of fractal speckles by means of a spatial light modulator

Optics Express, Vol. 15, Issue 12, pp. 7415-7422 (2007)

http://dx.doi.org/10.1364/OE.15.007415

Acrobat PDF (2558 KB)

### Abstract

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

## 1. Introduction

5. J. Uozumi, H. Kimura, and T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J.Mod. Opt. **38**, 1335–1347 (1991). [CrossRef]

6. K. Uno, J. Uozumi, and T. Asakura, “Correlation properties of speckles produced by diffractal-illuminated dif-fusers,” Opt. Commun. **124**, 16–22 (1996). [CrossRef]

7. J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. **156**, 350–358 (1998). [CrossRef]

10. T. Okamoto, H. Gotou, Y. Kawabata, and M. Takeuchi, “Statistical properties of the three-dimensional intensity distribution formed with orthogonally crossed speckle beams,” in *Technical Digest of the Sixth Japan-Finland Joint Symposium on Optics in Engineering*, J. Uozumi, T. Yatagai, and K.-E. Peiponen, eds. (2005), pp. 45–46.

## 2. Generation of fractal speckles

7. J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. **156**, 350–358 (1998). [CrossRef]

_{1}. 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 P

_{2}. Since the intensity distribution of this scattered field corresponds to the power spectrum of the field just behind the fractal aperture in P

_{1}, the ensemble-average intensity at a point (

*x,y*) in P

_{2}follows a negative power-law,

*q*= (

*x*

^{2}+

*y*

^{2})

^{1/2}is a radial coordinate in the plane P

_{2}, and

*D*is the fractal dimension of the random fractal object.

_{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 P

_{3}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 P

_{3}obeys the Gaussian statistics [11]. 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 [6

6. K. Uno, J. Uozumi, and T. Asakura, “Correlation properties of speckles produced by diffractal-illuminated dif-fusers,” Opt. Commun. **124**, 16–22 (1996). [CrossRef]

7. J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. **156**, 350–358 (1998). [CrossRef]

*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

_{1}, 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

*θ*= 100° for the polarizer,

_{p}*θ*= 144° for the slow axis of the quarter-wave plate and

_{w}*θ*= 77° for the analyzer, respectively [12–14

_{a}12. N. Konforti, E. Maron, and S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. **13**, 251–253 (1988). [CrossRef] [PubMed]

*μ*m. The LCD converts color signals into corresponding 8-bit gray level signals. Two personal computers PC

_{1}and PC

_{2}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 1.5

*π*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 P

_{2}. The intensity distribution is then incident on a ground glass plate placed in the plane P

_{2}. The field immediately behind P

_{2}is Fourier-transformed by the lens L

_{4}to produce a fractal speckle field in the back focal plane of L

_{4}. The intensity distribution of the field is captured by a CCD camera having 1036 × 1362 pixels with a pixel size of 4.65

*μ*m. The central portion of 1024 × 1024 pixels around the optical axis is used in the subsequent data processing.

15. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta **21**, 709–720 (1974). [CrossRef]

*π*<

*θ*< 0.75

*π*, truncation due to the circular aperture C

_{1}and a constant modulus are imposed as the constraints, while the spectral domain corresponds to the Fourier plane P

_{2}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.

*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

*U*

_{0}and

*U*are the desired and calculated modulus distributions, respectively, while

_{i}*α*is a constant to equalize the energy of

*U*

_{0}with that of

*U*. The cost function calculated in the case of

_{r}*D*= 1.5 is shown in Fig.4 as a function of the number of iterations. As is shown in this figure, Δ

*E*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.

*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

_{2}by the SLM with the phase modulation images described above are captured by the CCD camera by locating it in P

_{2}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 C

_{1}. 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.

*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 [7

**156**, 350–358 (1998). [CrossRef]

*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*.

*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 [6

6. K. Uno, J. Uozumi, and T. Asakura, “Correlation properties of speckles produced by diffractal-illuminated dif-fusers,” Opt. Commun. **124**, 16–22 (1996). [CrossRef]

*D*of the fractal speckle is given by

_{f}*D*= 2-

_{f}*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.

## 5. Conclusion

## References and links

1. | H. Hakayasu, |

2. | J. Feder, |

3. | T. Vicsek, |

4. | J. Uozumi and T. Asakura, “Optical fractals,” in |

5. | J. Uozumi, H. Kimura, and T. Asakura, “Fraunhofer diffraction by Koch fractals: the dimensionality,” J.Mod. Opt. |

6. | K. Uno, J. Uozumi, and T. Asakura, “Correlation properties of speckles produced by diffractal-illuminated dif-fusers,” Opt. Commun. |

7. | J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. |

8. | J. Uozumi, “Generation of random fractal fields by a double scattering process,” in |

9. | J. Uozumi, “Fractality of the optical fields scattered by power-law-illuminated diffusers,” |

10. | T. Okamoto, H. Gotou, Y. Kawabata, and M. Takeuchi, “Statistical properties of the three-dimensional intensity distribution formed with orthogonally crossed speckle beams,” in |

11. | J.W. Goodman, “Statistical properties of laser speckle patterns,” in |

12. | N. Konforti, E. Maron, and S.-T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. |

13. | G. Wernicke, S. Krüger, J. Kamps, H. Gruber, N. Demoli, M. Dürr, and S. Teiwes, “Application of a liquid crystal display spatial light modulator system as dynamic diffractive element and in optical image processing,” J. Opt. Commun. |

14. | H. Kim and Y. H. Lee, “Unique measurement of the parameters of a twisted-nematic liquid-crystal display,” Appl. Opt. |

15. | R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: March 26, 2007

Revised Manuscript: May 2, 2007

Manuscript Accepted: May 2, 2007

Published: June 1, 2007

**Citation**

Hideki Funamizu and Jun Uozumi, "Generation of fractal speckles by means of a spatial light modulator," Opt. Express **15**, 7415-7422 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7415

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### References

- H. Takayasu, Fractals in Physical Science (Manchester University, Manchester, 1990).
- J. Feder, Fractals (Plenum, New York, 1996).
- T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1992).
- J. Uozumi and T. Asakura, "Optical fractals," in Optical Storage and Retrieval—Memory, Neural Networks, and Fractals, F. T. S. Yu and S. Jutamulia, eds. (Marcel Dekker, New York, 1996), pp. 283-320.
- J. Uozumi, H. Kimura, and T. Asakura, "Fraunhofer diffraction by Koch fractals: the dimensionality," J. Mod. Opt. 38, 1335-1347 (1991). [CrossRef]
- K. Uno, J. Uozumi, and T. Asakura, "Correlation properties of speckles produced by diffractal-illuminated diffusers," Opt. Commun. 124, 16-22 (1996). [CrossRef]
- J. Uozumi, M. Ibrahim, and T. Asakura, "Fractal speckles," Opt. Commun. 156, 350-358 (1998). [CrossRef]
- J. Uozumi, "Generation of random fractal fields by a double scattering process," in Fourth International Conference on Correlation Optics, O. V. Angelsky, ed., Proc. SPIE 3904, 13-24 (1999).
- J. Uozumi, "Fractality of the optical fields scattered by power-law-illuminated diffusers," in Selected Papers from Fifth International Conference on Correlation Optics, O. V. Angelsky, ed., Proc. SPIE 4607, 257-267, (2002). [CrossRef]
- T. Okamoto, H. Gotou, Y. Kawabata, and M. Takeuchi, "Statistical properties of the three-dimensional intensity distribution formed with orthogonally crossed speckle beams," in Technical Digest of the Sixth Japan-Finland Joint Symposium on Optics in Engineering, J. Uozumi, T. Yatagai and K.-E. Peiponen, eds., (2005), pp. 45-46.
- J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., (Springer, Berlin, 1984), pp. 9-75.
- N. Konforti, E. Maron, and S.-T. Wu, "Phase-only modulation with twisted nematic liquid-crystal spatial light modulators," Opt. Lett. 13, 251-253 (1988). [CrossRef] [PubMed]
- G. Wernicke, S. Kruger, J. Kamps, H. Gruber, N. Demoli, M. Durr, and S. Teiwes, "Application of a liquid crystal display spatial light modulator system as dynamic diffractive element and in optical image processing," J. Opt. Commun. 25, 141-148 (2004).
- H. Kim and Y. H. Lee, "Unique measurement of the parameters of a twisted-nematic liquid-crystal display," Appl. Opt. 44, 1642-1649 (2005). [CrossRef] [PubMed]
- R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709-720 (1974). [CrossRef]

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