## Photon correlation spectroscopy with incoherent light |

Optics Express, Vol. 19, Issue 27, pp. 26416-26422 (2011)

http://dx.doi.org/10.1364/OE.19.026416

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### Abstract

Photon correlation spectroscopy (PCS) is based on measuring the temporal correlation of the light intensity scattered by the investigated sample. A typical setup requires a temporally coherent light source. Here, we show that a short-coherence light source can be used as well, provided that its coherence properties are suitably modified. This results in a “skewed-coherence” light beam allowing that restores the coherence requirements. This approach overcomes the usual need for beam filtering, which would reduce the total brightness of the beam.

© 2011 OSA

## 1. Introduction

6. O. E. Martinez, “Pulse distorsion in tilted pulse shcemes for ultrashort pulses,” Opt. Comm. **59**, 229–232 (1986). [CrossRef]

8. D. Salerno, O. Jedrkiewicz, J. Trull, G. Valiulis, A. Picozzi, and P. Di Trapani, “Noise-seeded spatiotemporal modulation instability in normal dispersion,” Phys. Rev. E **70**65603 (2004). [CrossRef]

*χ*

^{2}crystals [9

9. P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. **81**, 570–573 (1998). [CrossRef]

10. A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. **88**, 083901 (2002). [CrossRef] [PubMed]

12. O. Jedrkiewicz, M. Clerici, A. Picozzi, D. Faccio, and P. Di Trapani, “X-shaped space-time coherence in optical parametric generation,” Phys. Rev. A **76**, 033823 (2007). [CrossRef]

19. A. Ciattoni and C. Conti, “Quantum electromagnetic X waves,” J. Opt. Soc. B , **24**, 2195–2198 (2007). [CrossRef]

## 2. Beam generation

13. R. S. Conroy, A. Carleton, A. Carruthers, B. D. Sinclair, C. F. Rae, and K. Dholakia, “A visible extended cavity diode laser for the undergraduate laboratory,” Am. J. Phys. **68**, 925–931 (2000). [CrossRef]

*σ*(skew angle) with respect to the normal of the direction of propagation [14

14. Z. Bor and B. Rȧcz, “Group velocity dispersion in prisms and its
application to pulse compression and travelling-wave exitation,”
Opt. Commun. **54**, 165–170
(1985). [CrossRef]

*σ*can be adjusted about from 20° to 50° by rotating the reflection grating and thereby changing the reciprocal angle between the incoming beam and the grating.

*ϑ*and

*φ*are the angles between the normal to the grating of the diffracted and incoming beams, respectively,

*λ*is the wavelength, and

*d*is the grating spacing. The skew angle

*σ*can be calculated through simple geometric considerations by imposing that the optical path traveled by the incoming and diffracted waves should be the same:

*ϑ*–

*λ*spectrum of the beam exiting from the grating. Figure 1 (left column) presents the Δ

*ϑ*– Δ

*λ*data measured from the recorded images in the case of a short coherence beam (obtained as zeroth order diffraction i.e, simple reflection) and a skewed short coherence beam (obtained as first order diffraction). The second column of Fig. 1 present a statistically realistic 2D section (transverse and propagation directions) of the intensity profile of the beam. The figure is obtained via FFT, by adding a random phase to the measured amplitude spectral fields and is scaled by using the data gathered from the angular dispersion measurements (temporal and spatial spectral width), the real dimension of the beam (continuous wave and diameter) and the temporal coherence measured by a interferometric experiment with a Michelson-Morley set-up. The simulated spectra is obtained by Fourier transforming the zero order only (upper figure) and the first order only (bottom figure) of the diffracted beam at zero propagation space.

*f*s (corresponding to about 75

*μ*m, i.e. to an aspect ratio of the coherent regions of ∼ 12), and orientation perpendicular to the direction of propagation. In the case of the first order of diffraction, the spectrum clearly shows the angular dispersion introduced by the reflection grating. This results in a coherence region oriented along a particular inclined direction in real space. The coherence regions are again slabs with the same aspect ratio of the zeroth order diffracted beam, but they are skewed by an angle

*σ*with respect to the perpendicular of the direction of propagation [7

7. M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E **68**, 016613 (2003). [CrossRef]

## 3. Experiment description and results

*f*= 15 cm are arranged in the

*f*– 2

*f*–

*f*configuration (a telescope with magnification 1) conjugating a plane close to the grating to a plane inside the sample, which in turn is contained in a standard 1 cm square section glass cell. The telescope is necessary in order to filter out the unwanted diffraction orders and to image the unpropagated beam into the sample. The scattered light is collected using an optical system consisting of an optical fiber (OF) coupled with a GRIN lens (L4) placed at 90° with respect to the impinging beam. The fiber output is then sent directly to the avalanche photodiode (APD) of a commercial PCS apparatus (ZetaPlus, Brookhaven Instruments), which is able to measure the time correlation function of the scattered intensity.

*C*(

*τ*) =<

*I*(

*t*)

*I*(

*t*+

*τ*) > of the light scattered by a suspension of 150 nm diameter polystyrene particles (Duke scientific) in water at a volume fraction of 8

*X*10

^{−6}. Figure 3a shows the normalized correlation functions obtained with different coherence conditions of the impinging beams. When illuminating with laser light (Fig. 3a, continuous line), a classical exponential decay [4] is obtained with a contrast close to 30%. Such exponential decay is characteristic of the Brownian motion of mono dispersed nano-particles, and their size can be obtained using standard fitting procedures which provides a value very close to the manufacturer data (147nm). With short-coherence light at

*σ*=0 (Fig. 3a, dot-dashed line), the correlation function becomes completely flat. In reality a small contrast is still present in the order of 1%, but the resulting correlation function is too noisy to be conveniently fitted and thus it is not possible to obtain any information about the sample size. If the short-coherence beam is conveniently skewed, without applying any spectral filtering, a contrast of about 13% is recovered (Fig. 3a, dotted line), enough to allow a precise evaluation of the decay time and thus of the particles sizes. The results of the fitting provide data similar to those obtained with the laser beam (145nm), thus showing that the correlation function is not dependent on the beam skewness other than in its contrast. In Fig. 3b we show the contrast of the measured autocorrelation function of the light intensity collected at a fixed scattering angle (90°) as a function of the skew angle. The plot presented in Fig. 3b shows that the maximum contrast is actually achieved at a skew angle (45°) equal to the half of the detection angle (90°).

*σ*, as obtained after reflection on the grating indicated by G in Fig. 4. The coherent slab hits the various scattering objects at different times, thus the path length differences are compensated only if the scattering objects are located along an angle that is twice as large as the skewed angle. It should be noted that the picture in Fig. 4b is a slightly simplified 2D projection and that the real interference occurs on the surface of a cone; therefore, the exact angle at which the coherence condition is restored spans from 0° to 2

*σ*, depending on the azimuthal angle. Practically speaking, the present technique takes advantage of the geometrical shape of the beam coherence to increase, over a direction specified by the skewed angle, the coherence volume (the volume of the sample that contributes to the interference signal) inside the sample. As shown in Fig. 4b the coherence volume is oriented along an angle 2

*σ*with respect the propagation direction.

*σ*of the autocorrelation contrast (see Fig. 3b). Indeed, the contrast of the correlation function depends on the coherent overlapping between the scattered waves. The correlation function for the short coherence beam (

*σ*=0) is flat (contrast=C(0)=0) because the longitudinal coherence length is shorter than the optical path differences between the beams scattered from different regions of the sample. The reciprocal coherence of scattered waves is restored with a skewed beam at a detection angle equal to the twice the value of the beam skewness. The measured contrast shown in Fig. 3b can be theoretically derived assuming that the coherence volume is given by the geometrical intersection between the observed volume and a volume equal to the coherence slab but rotated to an angle

*σ*. The observed volume is the volume of the sample that is actually detectable, i.e., optically imaged on the photodetector, given the geometrical orientation of the collecting optics. Considering the skewed coherence slab as a disk or an oblate spheroid oriented with a variable angle

*σ*, while the observed volume is a fixed 45° oriented stripe of thickness comparable with the temporal coherence length, it is possible to extract the theoretical curve shown in Fig. 3b, by calculating the volume of the geometric intersection of the two regions. By fitting the data with this model and using the actual aspect ratio of the skewed volume as the only free parameter, a value of the aspect ratio of about 11 is obtained, which is very close to the aspect ratio directly obtained via the spectrum and interferometric measurements shown above.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Omar, |

2. | P. M. Chaikin and T. C. Lubensky, |

3. | T. Zemb and P. Lindner, |

4. | B. J. Berne and R. Pecora, |

5. | B. Chu, |

6. | O. E. Martinez, “Pulse distorsion in tilted pulse shcemes for ultrashort pulses,” Opt. Comm. |

7. | M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E |

8. | D. Salerno, O. Jedrkiewicz, J. Trull, G. Valiulis, A. Picozzi, and P. Di Trapani, “Noise-seeded spatiotemporal modulation instability in normal dispersion,” Phys. Rev. E |

9. | P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. |

10. | A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. |

11. | O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, “Emergence of x-shaped spatiotemporal coherence in optical waves,” Phys. Rev. Lett. |

12. | O. Jedrkiewicz, M. Clerici, A. Picozzi, D. Faccio, and P. Di Trapani, “X-shaped space-time coherence in optical parametric generation,” Phys. Rev. A |

13. | R. S. Conroy, A. Carleton, A. Carruthers, B. D. Sinclair, C. F. Rae, and K. Dholakia, “A visible extended cavity diode laser for the undergraduate laboratory,” Am. J. Phys. |

14. | Z. Bor and B. Rȧcz, “Group velocity dispersion in prisms and its
application to pulse compression and travelling-wave exitation,”
Opt. Commun. |

15. | S. G. J. Mochrie, A. M. Mayes, A. R. Sandy, M. Sutton, S. Brauer, G. B. Stephenson, D. L. Abernathy, and G. Grbel, “Dynamics of block copolymer micelles revealed by x-ray intensity fluctuation spectroscopy,” Phys. Rev. Lett. |

16. | M. Sutton, “A review of X-ray intensity fluctuation spectroscopy,” C. R. Phys. |

17. | K. A. Nugent, “Coherent methods in the X-ray
sciences,” Adv. Phys. |

18. | G. Zanchetta and R. Cerbino, “Exploring soft matter with x-rays: from the
discovery of the dna structure to the challenges of free electron
lasers,” J. Phys. Condens. Matter |

19. | A. Ciattoni and C. Conti, “Quantum electromagnetic X waves,” J. Opt. Soc. B , |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(290.5820) Scattering : Scattering measurements

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: October 18, 2011

Revised Manuscript: November 21, 2011

Manuscript Accepted: November 21, 2011

Published: December 12, 2011

**Virtual Issues**

Vol. 7, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

D. Salerno, D. Brogioli, F. Croccolo, R. Ziano, and F. Mantegazza, "Photon correlation spectroscopy with incoherent light," Opt. Express **19**, 26416-26422 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-27-26416

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### References

- A. Omar, Electromagnetic Scattering and Material Characterization (Artech House, Norwood, 2011).
- P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 2000).
- T. Zemb and P. Lindner, Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter (Elsevier, 2002).
- B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Elsevier, 2002).
- B. Chu, Laser Light Scattering: Basic Principles and Practice (Dover Publications, 2007).
- O. E. Martinez, “Pulse distorsion in tilted pulse shcemes for ultrashort pulses,” Opt. Comm.59, 229–232 (1986). [CrossRef]
- M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E68, 016613 (2003). [CrossRef]
- D. Salerno, O. Jedrkiewicz, J. Trull, G. Valiulis, A. Picozzi, and P. Di Trapani, “Noise-seeded spatiotemporal modulation instability in normal dispersion,” Phys. Rev. E7065603 (2004). [CrossRef]
- P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett.81, 570–573 (1998). [CrossRef]
- A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett.88, 083901 (2002). [CrossRef] [PubMed]
- O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, “Emergence of x-shaped spatiotemporal coherence in optical waves,” Phys. Rev. Lett.97, 243903 (2006). [CrossRef]
- O. Jedrkiewicz, M. Clerici, A. Picozzi, D. Faccio, and P. Di Trapani, “X-shaped space-time coherence in optical parametric generation,” Phys. Rev. A76, 033823 (2007). [CrossRef]
- R. S. Conroy, A. Carleton, A. Carruthers, B. D. Sinclair, C. F. Rae, and K. Dholakia, “A visible extended cavity diode laser for the undergraduate laboratory,” Am. J. Phys.68, 925–931 (2000). [CrossRef]
- Z. Bor and B. Rȧcz, “Group velocity dispersion in prisms and its application to pulse compression and travelling-wave exitation,” Opt. Commun.54, 165–170 (1985). [CrossRef]
- S. G. J. Mochrie, A. M. Mayes, A. R. Sandy, M. Sutton, S. Brauer, G. B. Stephenson, D. L. Abernathy, and G. Grbel, “Dynamics of block copolymer micelles revealed by x-ray intensity fluctuation spectroscopy,” Phys. Rev. Lett.78, 1275–1278 (1997). [CrossRef]
- M. Sutton, “A review of X-ray intensity fluctuation spectroscopy,” C. R. Phys.9, 657–667 (2008). [CrossRef]
- K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys.591–99 (2010). [CrossRef]
- G. Zanchetta and R. Cerbino, “Exploring soft matter with x-rays: from the discovery of the dna structure to the challenges of free electron lasers,” J. Phys. Condens. Matter22, 1–21 (2010).
- A. Ciattoni and C. Conti, “Quantum electromagnetic X waves,” J. Opt. Soc. B, 24, 2195–2198 (2007). [CrossRef]

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