## Anomalous flow of light near a photonic crystal pseudo-gap |

Optics Express, Vol. 19, Issue 25, pp. 25320-25327 (2011)

http://dx.doi.org/10.1364/OE.19.025320

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### Abstract

Two different transport regimes of light are observed in reflection from the same disordered photonic crystal. A model based on the scaling theory of localization explains the behavior of the path length-resolved reflection at two different probing wavelengths. Our results demonstrate the continuous renormalization of the photon diffusion coefficient measured in reflection from random media.

© 2011 OSA

_{2}powder [11

11. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. **96**(6), 063904–063907 (2006). [CrossRef] [PubMed]

12. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature **390**(6661), 671–673 (1997). [CrossRef]

13. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**(7131), 52–55 (2007). [CrossRef] [PubMed]

14. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. **42**(10), 673–676 (1979). [CrossRef]

_{2}powder [11

11. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. **96**(6), 063904–063907 (2006). [CrossRef] [PubMed]

15. C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, “Exceptional reduction of the diffusion constant in partially disordered photonic crystals,” Phys. Rev. Lett. **101**(12), 123901 (2008). [CrossRef] [PubMed]

*L*much smaller than the slab’s transverse dimensions. The pulse will take on average a time to traverse the slab. The diffusion coefficient

*D*describes the root mean square temporal spread of the photon density within the classical random walk model. Under the condition that the transport mean free path approaches the wavelength λ of light but

*L*remains large compared to λ, the diffusion coefficient will decrease with

*L*according to [8

8. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. **53**(22), 2169–2172 (1984). [CrossRef]

16. P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B **52**(3), 505–509 (1985). [CrossRef]

*v*is the energy transport velocity and λ is the transport mean free path. The quantities

*ξ*and

*L*are known as the coherence and root mean square (RMS) absorption length, respectively, and act as cutoff length scales for the scaling behavior of the diffusion coefficient. Equation (1) is the solution to the differential scaling equation obtained by incorporating coherent backscattering corrections to classical diffusive transport of the wave energy [14

_{abs}14. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. **42**(10), 673–676 (1979). [CrossRef]

*ξ*, whereas on longer scales light resumes diffusive behavior with a reduced diffusion coefficient. The classical diffusion coefficient

17. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. **28**(12), 2331–2336 (1989). [CrossRef] [PubMed]

*μ*is the absorption coefficient and

_{abs}*z*is the depth of the point source inside the medium, usually taken to be one transport mean free path. The extrapolation length ratio

_{0}*z*represents the distance (as a fraction of

_{e}*z*) outside of the medium at which the photon density goes to zero. All other quantities retain their previous meanings.

_{0}*D*and, consequently, contains information about the behavior of the RMS displacement of the energy. After taking the logarithm of both sides, one obtains

*D*is constant, the photon flux will decay with

*t*according to a power law with an exponent −5/2. This behavior is valid only in the regime noted above, and is indeed observed in the diffuse reflectance from many relatively weak, multiple scattering materials such as suspensions of polystyrene microspheres in water (c.f. Figure 2 ) [18

18. G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. **24**(7), 442–444 (1999). [CrossRef] [PubMed]

*D*is given by

*ξ*such that the first term in the sum of Eq. (4) is insignificant, the diffusion coefficient exhibits a scaling behavior like

*ξ*, one should expect a return to a −5/2 exponent and an eventual exponential decay due to absorption losses or finite sample sizes.

19. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel; “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature **405**(6785), 437–440 (2000). [CrossRef] [PubMed]

8. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. **53**(22), 2169–2172 (1984). [CrossRef]

20. C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. **4**(10), 794–798 (2008). [CrossRef]

18. G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. **24**(7), 442–444 (1999). [CrossRef] [PubMed]

21. P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68**(1), 016604 (2003). [CrossRef] [PubMed]

15. C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, “Exceptional reduction of the diffusion constant in partially disordered photonic crystals,” Phys. Rev. Lett. **101**(12), 123901 (2008). [CrossRef] [PubMed]

*µm*to 1

*mm*. The fiber was also oriented at a slight angle with respect to the normal of the sample face to reduce the amount of single-scattered light reflected from the sample’s glass substrates. Furthermore, the impulse response of the entire system was subtracted from the origin of each distribution in order to reduce any ambiguity due to specular reflection from the surface. The two light sources that were used in our study had wavelengths of 1300

*nm*and 1550

*nm*and coherence lengths of 34 ± 2

*µm*and 38 ± 1

*µm*, respectively. The thickness of the sample was measured using a profilometer and was 15.0 ± 2.7

*μm*.

19. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel; “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature **405**(6785), 437–440 (2000). [CrossRef] [PubMed]

*nm*survives the averaging and explains why renormalization is observed at this wavelength. The DOS at 1300

*nm*is larger than at 1550

*nm*and is more sensitive to the structural parameters, resulting in more uncertainty in the estimate. The sample is effectively random at this wavelength and no longer displays crystalline properties.

15. C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, “Exceptional reduction of the diffusion constant in partially disordered photonic crystals,” Phys. Rev. Lett. **101**(12), 123901 (2008). [CrossRef] [PubMed]

22. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A **44**(6), 3948–3959 (1991). [CrossRef] [PubMed]

*D*at 1300 nm was

*ξ*and

*λ*, were used in this case. An initial diffusion coefficient

*D*(before scaling effects begin) was then calculated with an assumed transport velocity of

_{0}**101**(12), 123901 (2008). [CrossRef] [PubMed]

*D*is

_{0}22. J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A **44**(6), 3948–3959 (1991). [CrossRef] [PubMed]

*ξ*was found to be 13.6 ± 1.8

*μm*. Inserting these values into Eq. (1) together with the effective sample lengths

*m*. This is the diffusion coefficient that would appear in path length-resolved transmission measurements. The reported uncertainty reflects both the fluctuations in the sample’s thickness and the effect of the finite coherence lengths of the sources.

^{2}/s21. P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68**(1), 016604 (2003). [CrossRef] [PubMed]

*m*and 3.9 ± 0.6

^{2}/s*m*for wavelengths of 1300

^{2}/s*nm*and 1550

*nm*, respectively. The value for

*D*at 1550

*nm*agrees with the renormalized value estimated from reflection measurements, yet the behavior of the transmission data alone does not indicate that renormalization of the diffusion coefficient occurred. This is because a transmission experiment enforces the interaction with the entire medium. Photons arriving at different times (or path lengths) may have undergone different transport processes but this information is lost since every measured photon has necessarily traveled at least the sample thickness. Alternatively, a transmission experiment integrates over different scattering regimes.

## References and links

1. | A. Ishimaru, |

2. | M. C. W. van Rossum and T. M. N. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. |

3. | Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A |

4. | A. Z. Genack and A. A. Chabanov, “Signatures of photon localization,” J. Phys. Math. Gen. |

5. | F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. |

6. | P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. |

7. | S. John and M. J. Stephen, “Wave propagation and localization in a long-range correlated random potential,” Phys. Rev. B |

8. | S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. |

9. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

10. | S. John and R. Rangarajan, “Optimal structures for classical wave localization: An alternative to the Ioffe-Regel criterion,” Phys. Rev. B Condens. Matter |

11. | M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. |

12. | D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature |

13. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

14. | E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. |

15. | C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, “Exceptional reduction of the diffusion constant in partially disordered photonic crystals,” Phys. Rev. Lett. |

16. | P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B |

17. | M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. |

18. | G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. |

19. | A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel; “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature |

20. | C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. |

21. | P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

22. | J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A |

**OCIS Codes**

(290.1990) Scattering : Diffusion

(160.5293) Materials : Photonic bandgap materials

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: October 11, 2011

Revised Manuscript: November 14, 2011

Manuscript Accepted: November 17, 2011

Published: November 28, 2011

**Citation**

Kyle M. Douglass, Sajeev John, Takashi Suezaki, Geoffrey A. Ozin, and Aristide Dogariu, "Anomalous flow of light near a photonic crystal pseudo-gap," Opt. Express **19**, 25320-25327 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25320

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### References

- A. Ishimaru, Wave Propagation in Random Media (John Wiley & Sons, 1999).
- M. C. W. van Rossum and T. M. N. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71(1), 313–371 (1999). [CrossRef]
- Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1(8), 831–835 (1984). [CrossRef]
- A. Z. Genack and A. A. Chabanov, “Signatures of photon localization,” J. Phys. Math. Gen. 38(49), 10465–10488 (2005). [CrossRef]
- F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81(26), 5800–5803 (1998). [CrossRef]
- P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]
- S. John and M. J. Stephen, “Wave propagation and localization in a long-range correlated random potential,” Phys. Rev. B 28(11), 6358–6368 (1983). [CrossRef]
- S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53(22), 2169–2172 (1984). [CrossRef]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- S. John and R. Rangarajan, “Optimal structures for classical wave localization: An alternative to the Ioffe-Regel criterion,” Phys. Rev. B Condens. Matter 38(14), 10101–10104 (1988). [CrossRef] [PubMed]
- M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96(6), 063904–063907 (2006). [CrossRef] [PubMed]
- D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390(6661), 671–673 (1997). [CrossRef]
- T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). [CrossRef] [PubMed]
- E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42(10), 673–676 (1979). [CrossRef]
- C. Toninelli, E. Vekris, G. A. Ozin, S. John, and D. S. Wiersma, “Exceptional reduction of the diffusion constant in partially disordered photonic crystals,” Phys. Rev. Lett. 101(12), 123901 (2008). [CrossRef] [PubMed]
- P. W. Anderson, “The question of classical localization: a theory of white paint?” Philos. Mag. B 52(3), 505–509 (1985). [CrossRef]
- M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989). [CrossRef] [PubMed]
- G. Popescu and A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. 24(7), 442–444 (1999). [CrossRef] [PubMed]
- A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel; “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405(6785), 437–440 (2000). [CrossRef] [PubMed]
- C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008). [CrossRef]
- P. M. Johnson, A. Imhof, B. P. J. Bret, J. G. Rivas, and A. Lagendijk, “Time-resolved pulse propagation in a strongly scattering material,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(1), 016604 (2003). [CrossRef] [PubMed]
- J. X. Zhu, D. J. Pine, and D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44(6), 3948–3959 (1991). [CrossRef] [PubMed]

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