## Singular-optical coloring of regularly scattered white light

Optics Express, Vol. 14, Issue 17, pp. 7579-7586 (2006)

http://dx.doi.org/10.1364/OE.14.007579

Acrobat PDF (143 KB)

### Abstract

When the surface roughness is comparable with the wavelength of the probing radiation, the scattered field contains both the regular (forward-scattered) component of coherent nature and the diffusely scattered part. Coloring of the regular component of white light scattered by a colorless dielectric slab with a rough surface is considered as a peculiar effect of singular optics with zero (infinitely extended) interference fringes. To explain the observed alternation of colors with respect to the increasing depth of the surface roughness, we apply a model of transition layers associated with the surface roughness. By applying the chromascopic technique, it is shown that the modifications of the normalized spectrum of the forward-scattered white light can be interpreted as the effect of a quarter-wavelength (anti-reflecting) layer for some spectral component of a polychromatic probing beam.

© 2006 Optical Society of America

## 1. Introduction

## 2. Basic observations and interpretation

*h*, are comparable with the wavelength of the probing beam, here the wavelengths of all spectral components of the polychromatic beam. Thus, the assumption of mutual incoherence of partial waves scattered by various surface inhomogeneities is violated, and the phase relations between such waves must be taken into account both for the forward-scattered and for the specularly reflected radiation even in the case of a white-light probing beam from an extended source, when both spatial and temporal coherence is extremely low.

10. M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. **6**, 196.1–196-8 (2004). [CrossRef]

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

5. M. Berry, “Coloured phase singularities,” New J. Phys. **4**, 66.1–66.14 (2002). [CrossRef]

11. O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “On the feasibility for determining the amplitude zeroes in polychromatic fields,” Opt. Express **13**, 4396–4405 (2005). [CrossRef] [PubMed]

*η*, for the

_{l}*l*- th diffraction order of a phase hologram with a harmonic relief (including the zero order,

*l*= 0) characterized by the amplitude transmittance,

*l*-th order Bessel function of the first kind:

*η*= [

_{l}*J*(

_{l}*q*)]

^{2}. Here

*n*and

*n*

_{0}=1 are the indices of refraction of the emulsion and air, respectively, and

*p*being the grating’s period along the

*x*- axis. It is not surprising that the governing parameter for the spectral changes for the forward-diffracted component of a polychromatic radiation contains the ratio

*H/λ*. for the phase-only holographic gratings. Consequently, any limited area of a rough surface can be represented as a set of harmonic gratings with an amplitude transmittance

*λ*leads to vanishing of the product as a whole for the specified wavelength. It means that for an arbitrary height distribution function, a phase singularity associated with the crossing of zero magnitude of the zero-order Bessel function of the first kind certainly takes place for some wavelength of the polychromatic probing radiation and for the corresponding term of the expansion (2). When

_{z}*η*

_{0}approaches zero (viz., undergoes a phase singularity and changes its sign crossing the corresponding amplitude zero) for any

*λ*, its magnitude also becomes small over a finite spectral domain adjacent to this wavelength. In this way, the entire normalized spectrum of the forward-scattered radiation is considerably modified, and the complementary color,

_{z}*λ*(with respect to

_{p}*λ*), including the adjacent spectral domain will prevail in the observed image. Thus, one can observe more or less intense (though always mixed - not pure) colors of the forward-scattered radiation.

_{z}## 3. Surface roughness as a transition layer

*H*, depends on the real height distribution function. As a result, the analysis of the spectral modifications of the forward-scattered component of a polychromatic radiation passing a rough surface is reduced to the problem of matching of impedances of three media [28], viz., in the context of the optical problem, matching of the refraction indices of a glass (

*n*

_{1}), a transition layer (

*n*

_{3}). If the optical thickness of the transition layer,

*n*

_{2}

*H*, equals

*λ*/4,

*λ*being a wavelength in the media with a refraction index

*n*

_{2}, for some spectral component of the probing beam, this layer acts similar to an anti-reflection coating for this component, while under the assumed relation between the indices of refraction of the three media, the waves reflected from two boundaries of the transition layer are in opposite phases and interfere destructively. This certainly happens for some wavelength due to the condition

*λ*<

*H*for all spectral components. As a result, this spectral component and its spectral vicinity will prevail in the forward-scattered light.

*I*and

_{r}*I*are the intensities of the reflected and the incident beams, respectively,

_{i}*λ*is the specified wavelength of the incident beam within the spectral range of the probing radiation, and

_{i}*λ*

_{0}is the wavelength, the amplitude of which vanishes for the reflected radiation. Then the relative intensity of the forward-scattered component at the same wavelength

*λ*, is determined by the difference:

_{i}5. M. Berry, “Coloured phase singularities,” New J. Phys. **4**, 66.1–66.14 (2002). [CrossRef]

6. M. Berry, “Exploring the colours of dark light,” New J. Phys. **4**, 74.1–74.14 (2002). [CrossRef]

7. J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. **5**, 154.1–154.7 (2003). [CrossRef]

5. M. Berry, “Coloured phase singularities,” New J. Phys. **4**, 66.1–66.14 (2002). [CrossRef]

6. M. Berry, “Exploring the colours of dark light,” New J. Phys. **4**, 74.1–74.14 (2002). [CrossRef]

*λ*=435,8 nm (Fig. 1, fragments a and d), or green

_{b}*λ*= 546,1 nm (Fig. 1, fragments b and e), or red

_{g}*λ*= 700 nm (Fig. 1, fragments c and f). The pairs of fragments a and d, b and e, and c and f correspond to the effective depths of the transition layer of 88.36 nm, 110.73 nm, and 141.93 nm, respectively, which are close to 0.1 of the mean diameter of the corundum assumed to be used for obtaining the color effects (∼10

_{r}*μ*m). The results of simulation are in agreement with the alternation of the colors observed experimentally. That is, blue shift takes place for smaller depths of the transition layer, and reddening of the forward-scattered light is observed for larger depths of this layer. It is obvious that the inverse sequence of colors is observed in the specularly reflected light.

- Simulation is performed for a discrete set of spectral components, while in practice one operates with a continuous spectrum. That is why the represented data are only of instructive nature: real colors are strongly dependent on the actual spectral density function of the source, so that one observes different colors induced by a given sample of a rough surface illuminated by sources with different color temperatures. Nevertheless, the general tendency (blue shift to the reddening of the forward-scattered component) is truly predicted by the model of transition layers.
- Comparing the upper and the lower rows of Fig. 1, an inexperienced observer can conclude that the intensity of colors [29] in the specularly reflected radiation is much higher than in the forward-scattered component. However, the apparent higher intensity of colors of the specularly reflected component is the result of the normalization procedure, cf. Eq. (6). Therefore, one must take into account that the colored specularly reflected component is much lower in intensity than the forward-scattered one. This follows from the fact that for a fixed ratio
*H/λ*, the forward-scattered radiation is governed by the multiplier*π*(*n*-*n*_{0}) (see Section 2), while the specularly reflected component of the normally incident beam is governed by the multiplier 4*π*, so that the effective depth of the transition layer in reflection exceeds its effective depth in transmission (for glass) by almost one order of magnitude. As a consequence, the relative intensity of the specularly reflected radiation is much lower (by two orders of magnitude, approximately), than the intensity of the forward-scattered component. It is evident that a surface, which can be regarded as slightly rough for transmitted radiation, cannot be slightly rough for the reflected one. This is the reason, why the possibility for observation of the colored beam specularly reflected from a rough surface was questioned earlier [30].

## 4. Experiments

6. M. Berry, “Exploring the colours of dark light,” New J. Phys. **4**, 74.1–74.14 (2002). [CrossRef]

6. M. Berry, “Exploring the colours of dark light,” New J. Phys. **4**, 74.1–74.14 (2002). [CrossRef]

*n*such that

_{i}*n*

_{1}<

*n*<

_{i}*n*

_{3}. If the immersion rapidly dries out, one observes the alternation of colors in the forward-scattered component of a white-light beam passing the sample as predicted in Section 3, here observed in real time.

## 5. Conclusions

## References and links

1. | G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. |

2. | G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. |

3. | V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata |

4. | S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. |

5. | M. Berry, “Coloured phase singularities,” New J. Phys. |

6. | M. Berry, “Exploring the colours of dark light,” New J. Phys. |

7. | J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. |

8. | I. Freund, “Polychromatic polarization singularities,” Opt. Lett. |

9. | A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. |

10. | M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. |

11. | O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “On the feasibility for determining the amplitude zeroes in polychromatic fields,” Opt. Express |

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

13. | V. Shvedov, W. Krolokowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Yu. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express |

14. | P. V. Polyanskii, “Some current views on singular optics,” in |

15. | S. A. Ponomarenko, “A class of partially coherent vortex beams carrying optical vortices,” J. Opt. Soc. Am. A |

16. | G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. |

17. | H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. |

18. | Ch. V. Felde, “Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams,” Ukr. J. Phys. |

19. | G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A |

20. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in |

21. | M. Born and E. Wolf, |

22. | M. S. Soskin and M. V. Vasnetsov, “Singular Optics” in |

23. | R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, |

24. | F. G. Bass and I. M. Fuks, |

25. | H. C. van de Hulst, |

26. | C. F. Bohren and D. R. Huffman, |

27. | A. Sommerfeld, |

28. | F. S. Grawford Jr., |

29. | R. M. Evans, |

30. | M. Minnaert, |

31. | S. R. Wilk, “Once in a blue moon,” Opt. Photonics. News |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.1670) Coherence and statistical optics : Coherent optical effects

(350.5030) Other areas of optics : Phase

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 26, 2006

Manuscript Accepted: August 3, 2006

Published: August 21, 2006

**Citation**

Oleg V. Angelsky, Peter V. Polyanskii, and Steen G. Hanson, "Singular-optical coloring of regularly scattered white light," Opt. Express **14**, 7579-7586 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7579

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### References

- G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behaviour of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002). [CrossRef] [PubMed]
- G. Popescu and A. Dogariu, "Spectral anomalies at wave-front dislocations," Phys. Rev. Lett. 88,183902 (2002). [CrossRef] [PubMed]
- V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, "Scattering-induced spectral changes as a singular-optical effect," Optica Applicata 32, 843-848 (2002).
- S. A. Ponomarenko and E. Wolf, "Spectral anomalies in a Fraunhofer diffraction pattern," Opt. Lett. 27, 1211-1213 (2002). [CrossRef]
- M. Berry, "Coloured phase singularities," New J. Phys. 4, 66.1-66.14 (2002). [CrossRef]
- M. Berry, "Exploring the colours of dark light," New J. Phys. 4, 74.1-74.14 (2002). [CrossRef]
- J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154.1-154.7 (2003). [CrossRef]
- I. Freund, "Polychromatic polarization singularities," Opt. Lett. 28, 2150-2152 (2003). [CrossRef] [PubMed]
- A. V. Volyar and T. A. Fadeeva, "Generation of singular beams in uniaxial crystals," Opt. Spectrosc. 94, 235-244 (2003). [CrossRef]
- M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, "Computer-synthesized hologram-based rainbow optical vortices," New J. Phys. 6, 196.1-196-8 (2004). [CrossRef]
- O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, "On the feasibility for determining the amplitude zeroes in polychromatic fields," Opt. Express 13, 4396-4405 (2005). [CrossRef] [PubMed]
- O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, "Interference diagnostics of white-light vortices," Opt. Express 13, 8179-8183 (2005). [CrossRef] [PubMed]
- V. Shvedov, W. Krolokowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Yu. S. Kivshar, "Focusing and correlation properties of white-light optical vortices," Opt. Express 13, 7393-7398 (2005). [CrossRef] [PubMed]
- P. V. Polyanskii, "Some current views on singular optics," in Sixth International Conference on Correlation Optics, O. V. Angelsky, ed., Proc. SPIE 5477, 31-40 (2004). [CrossRef]
- S. A. Ponomarenko, "A class of partially coherent vortex beams carrying optical vortices," J. Opt. Soc. Am. A 18, 150-156 (2001). [CrossRef]
- G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, "Partially coherent vortex beams with a separable phase," Opt. Lett. 28, 878-880 (2003). [CrossRef] [PubMed]
- H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young’s interference pattern," Opt. Lett. 28, 968-970 (2003). [CrossRef] [PubMed]
- Ch. V. Felde, "Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams," Ukr. J. Phys. 49, 473-480 (2004).
- G. Gbur, T. D. Visser, and E. Wolf, "Hidden singularities in partially coherent and polychromatic wavefields," J. Opt. A 6, 239-242 (2004). [CrossRef]
- A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.
- M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999.)
- M. S. Soskin and M. V. Vasnetsov, "Singular Optics" in Progress in Optics E, Wolf, ed., (North-Holland, Amsterdam, 2001) Vol. 42, 219-276.
- R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971).
- F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).
- H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- A. Sommerfeld, Optics (Academic, New York, 1954).
- F. S. Grawford, Jr., Waves: Berkley Physics Course (McGraw-Hill, New York, 1968) Vol 3.
- R. M. Evans, An Introduction to Color (Wiley, New York, 1959).
- M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
- S. R. Wilk, "Once in a blue moon," Opt. Photonics News 17, 20-21 (2006).

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