## Optical correlation diagnostics of rough surfaces with large surface inhomogeneities

Optics Express, Vol. 14, Issue 16, pp. 7299-7311 (2006)

http://dx.doi.org/10.1364/OE.14.007299

Acrobat PDF (5179 KB)

### Abstract

The feasibilities for optical correlation diagnostics of rough surfaces with large surface inhomogeneities by determining the transformations of the longitudinal coherence function of the scattered field are substantiated and implemented.

© 2006 Optical Society of America

## 1. Introduction

2. D. Pantzer, J. Politch, and L. Ek, “Heterodyne profiling instrument for the angstrom region,” Appl. Opt. **25**, 4168–4172 (1986). [CrossRef] [PubMed]

7. O.V. Angelsky, D.N. Burkovets, P.P. Maksimyak, and S.G. Hanson “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. **42**, 4529–4540 (2003). [CrossRef] [PubMed]

8. O.V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Interference diagnostics of white-light vortices,” Opt. Express **13**, 8179–8183 (2005). [CrossRef] [PubMed]

9. H. Fujii, T. Asakura, and Y Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. **66**, 1217–1222 (1976). [CrossRef]

10. R. A. Spraque, “Surface roughness measurement using white light speckle,” Appl. Opt. **11**, 2811–2819 (1972). [CrossRef]

11. J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and J. Brangaccio, “Digital wavefront measuring interfereometer for testing optical surfaces and lenses,” Appl. Opt. **13**, 2693–2703 (1974). [CrossRef] [PubMed]

12. C.T. Farrell and M.A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. & Technol. **3**, 953–958 (1992). [CrossRef]

## 2. Theory

*h*(

*x*,

*y*) by a normally incident monochromatic plane wave results in a spatial phase modulation of the reflected wave at the boundary field, 4

*πh*(

*x*,

*y*)/

*λ*. The longitudinal coherence function of the scattered field is not assumed to be spatially restricted. If the same surface is illuminated by a polychromatic beam with a finite longitudinal coherence length, reflection from a rough surface is accompanied by spreading of the longitudinal coherence function of the resulting field [5, 6] due to various delays of the partial signals, which are determined by the roughness relief. As a result, the modulus of the complex degree of coherence of the field decreases. The longitudinal coherence function of the field reflected from a rough surface is determined from the Fredholm’s integral equation of the 1

^{st}kind:

_{0}(∆

*z*) is the longitudinal coherence function of the probing beam, and

*f*(

*z*) ≡

*f*(

*h*) is the function of the partial signal delays determined by the height distribution function of the surface inhomogeneities. It has earlier been shown [13] that one can determine the distribution of the partial signal delays,

*f*(

*z*), based on the experimentally obtained coherence function of the scattered field and the known coherence function of the probing beam. Here, we use the techniques given in [14, 15] for solving the Fredholm’s equation of the 1

^{st}kind for determining the height distribution for the surface inhomogeneities,

*f*(

*h*).

*z*). In the Michelson interferometer, see Fig. 1, a monochromatic or a polychromatic field formed by a micro-objective at the plane of the image of a rough surface is mixed with a monochromatic or a polychromatic reference field, respectively.

*z*).

## 3. Experiments

*x*,

*y*) is the intensity distribution at the surface image in monochromatic light.

*z*is described by the following relation:

*I*

_{0}is the reference wave intensity,

*I*

_{s}(

*x*,

*y*) is the intensity distribution at the surface image in polychromatic light, and

*z*

_{0}is an arbitrary starting position.

*I*

^{m}(

*x*,

*y*,

*z*) and

*I*(

*x*,

*y*,

*z*) is determined by the following relation:

*λ*/2 (266 nm, in our case), which is convenient for the use as marking for the coherence function with respect to the roughness height parameter. The degree of coherence of the field for the relative longitudinal shift

*z*of the corresponding distributions

*I*

^{m}(

*x*,

*y*,

*z*) and

*I*(

*x*,

*y*,

*z*) is determined by the magnitudes of the extrema for the MLICF, namely:

*μm*- diameter diaphragm, D, positioned at the focus of the objective O3 (focal length 200 mm) by the objectives O1 and O2. In this way, we form a white-light beam with a degree of spatial (transverse) coherence of 95%. If the mirror M is used, the laser beam is spread using the telescope T and focused on the same diaphragm D. The beamsplitter BS divides the incident beam into the reference and the object beam. The images of the surface of interest, O, and the reference mirror, M2, are projected onto the CCD-camera by the micro-objectives MO1 and MO2, and the objective O4. The mirror M2 with the micro-objective MO2 is shifted along the direction of propagation of the beam, facilitating a change in the optical path delay in the interferometer. Optical phase difference (within a wavelength) is provided by a piezo-ceramic translator, PC, mounted with a mirror M2. By moving the surface of interest along the direction of propagation of the beam, one can obtain an image of the surface using a monochrome CCD-camera.

*n*is the frame number,

*N*and

*M*are the number of pixels in the two directions,

*I*

_{ij}(

*n*) are the intensities of the pixels with the coordinates

*i*,

*j*of the corresponding frames of the specified frame of the monochromatic sequence and the

*n*-th frame of the polychromatic frame, respectively.

*λ*/4 , giving a spacing of 133 nm in Fig. 4.

_{0}(∆

*z*), is determined using the same arrangement, cf Fig. 1, by measuring the visibility of the resulting interference field at the infinitely extended interference fringe for varying optical path difference in the interferometer. In this case, the tested rough surface is replaced with a high-quality mirror. The experimentally found longitudinal coherence function of the probing beam fitted with a Gaussian function is shown in Fig. 8 and is adequately represented by

**α**= 0.83

*ρ*/

*π*,

*ρ*being the coherence length of the probing beam, which in our case was equal to 0.8

*μm*. Thus, Eq. (1) can be written in the form:

10. R. A. Spraque, “Surface roughness measurement using white light speckle,” Appl. Opt. **11**, 2811–2819 (1972). [CrossRef]

*λ*, the number of points determining the coherence function is small. To increase the accuracy in determination of

*f*(

*z*) in this case, one can specify the points of the longitudinal coherence function for smaller separation of the frames, or even for subsequent ones. Therefore, one detects the frames recorded in white light and finds the covariance with the flow recorded in a monochromatic field. While the monochromatic flow is recorded for scanning of the reference wave within a wavelength, the mutual covariance function obviously has maxima and minima, based on which one determines the degree of coherence of the object field. The dependence of the degree of coherence on the frame number is scaled on the mutual intensity correlation function of the field, cf. Fig. 4, where the position of the extremum is determined by the frame number, and the spacing between the adjacent maximum and minimum equals 133 nm.

*λ*/4 (or 133 nm). The surface relief structure reconstructed by processing of the videos avi_1 and avi_2 is shown in Fig. 10.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J.M. Bennett and L. Mattson, |

2. | D. Pantzer, J. Politch, and L. Ek, “Heterodyne profiling instrument for the angstrom region,” Appl. Opt. |

3. | P. Beckmann and A. Spizzichino, |

4. | R.S. Sirohi (Ed), |

5. | O. V Angelsky, S. G. Hanson, and P. P. Maksimyak, |

6. | O.V. Angelsky and P.P. Maksimyak. “Optical Correlation Diagnostics of Surface Roughness” in: |

7. | O.V. Angelsky, D.N. Burkovets, P.P. Maksimyak, and S.G. Hanson “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. |

8. | O.V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “Interference diagnostics of white-light vortices,” Opt. Express |

9. | H. Fujii, T. Asakura, and Y Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. |

10. | R. A. Spraque, “Surface roughness measurement using white light speckle,” Appl. Opt. |

11. | J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and J. Brangaccio, “Digital wavefront measuring interfereometer for testing optical surfaces and lenses,” Appl. Opt. |

12. | C.T. Farrell and M.A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” J. Meas. Sci. & Technol. |

13. | O. V. Angelsky and P. P. Maksimyak, “Transformation of the longitudinal correlation function of a field propagating in a scattering medium,” Opt. Spectrosc. |

14. | A. N. Tikhonov and V. Ya. Arsenin, |

15. | A. V. Goncharovsky, A. M. Cherepashchuk, and A. G. Yagola, |

16. | H.V. Bogatyryova, Ch.V. Felde, and P.V. Polyanskii “Referenceless resting of vortex optical beams” Optica Applicata |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

(170.6960) Medical optics and biotechnology : Tomography

(240.5770) Optics at surfaces : Roughness

(290.0290) Scattering : Scattering

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: May 5, 2006

Revised Manuscript: July 10, 2006

Manuscript Accepted: July 22, 2006

Published: August 7, 2006

**Citation**

Oleg V. Angelsky, Alexander P. Maksimyak, Peter P. Maksimyak, and Steen G. Hanson, "Optical correlation diagnostics of rough surfaces
with large surface inhomogeneities," Opt. Express **14**, 7299-7311 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7299

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### References

- J. M. Bennett and L. Mattson, Introduction to Surface Roughness and Scattering, (Optical Society of America, Washington, D.C, 1999).
- D. Pantzer, J. Politch, and L. Ek, "Heterodyne profiling instrument for the angstrom region," Appl. Opt. 25, 4168-4172 (1986). [CrossRef] [PubMed]
- P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon Press, London 1963).
- R. S. Sirohi ed., Speckle Metrology, (Marcel Deker, New York, 1993).
- O. V Angelsky., S. G. Hanson and P. P. Maksimyak, The Use of Optical-Correlation Techniques for Characterizing Scattering Object and Media - Bellingham, (SPIE Press PM71, Bellingham, 1999).
- O. V. Angelsky, and P. P. Maksimyak, "Optical correlation diagnostics of surface roughness" in: Handbook of Coherent Domain Optical Methods. Biomedical Diagnostics, Environmental and Material Science, V. V. Tuchin, ed., (Kluwer Academic Publishers,Boston, 2004), Vol. 1, 43-92 (2004).
- O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, and S. G. Hanson "Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces," Appl. Opt. 42, 4529-4540 (2003). [CrossRef] [PubMed]
- O. V. Angelsky, S. G. Hanson, A. P. Maksimyak and P. P. Maksimyak, "Interference diagnostics of white-light vortices," Opt. Express 13, 8179-8183 (2005). [CrossRef] [PubMed]
- H. Fujii, T. Asakura and Y Shindo, "Measurement of surface roughness properties by using image speckle contrast," J. Opt. Soc. Am. 66, 1217-1222 (1976). [CrossRef]
- R. A. Spraque, "Surface roughness measurement using white light speckle," Appl. Opt. 11, 2811-2819 (1972). [CrossRef]
- J. H. Bruning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and J. Brangaccio, "Digital wavefront measuring interfereometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
- C. T. Farrell and M. A. Player, "Phase step measurement and variable step algorithms in phase-shifting interferometry," Meas. Sci. Technol. 3, 953-958 (1992). [CrossRef]
- O. V. Angelsky and P. P. Maksimyak, "Transformation of the longitudinal correlation function of a field propagating in a scattering medium," Opt. Spectrosc. 60, 331-336 (1986).
- A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving of Incorrect Problems, (Nauka, Moscow, 1986, in Russian).
- A. V. Goncharovsky, A. M. Cherepashchuk, and A. G. Yagola, Incorrect Problems in Astrophysics, (Nauka, Moscow, 1985 in Russian).
- H. V. Bogatyryova, Ch. V. Felde, and P. V. Polyanskii, "Referenceless resting of vortex optical beams," Opt. Appl. 33, 695-708 (2003).

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