## Route to strong localization of light: the role of disorder |

Optics Express, Vol. 20, Issue 16, pp. 18156-18164 (2012)

http://dx.doi.org/10.1364/OE.20.018156

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### Abstract

By employing Random Matrix Theory (RMT) and first-principle calculations, we investigated the behavior of Anderson localization in 1D, 2D and 3D systems characterized by a varying disorder. In particular, we considered random binary layer sequences in 1D and structurally disordered photonic crystals in two and three dimensions. We demonstrated the existence of a unique optimal degree of disorder that yields the strongest localization possible. In this regime, localized modes are constituted by defect states, which can show subwavelength confinement properties. These results suggest that disorder offers a new avenue for subwavelength light localization in purely dielectric media.

© 2012 OSA

## 1. Introduction

22. A. Crisanti, G. Paladin, and A. Vulpiani, *Products of Random Matrices in Statistical Physics* (Springer, Berlin, 1993). [CrossRef]

*l*, as well as the existence of an optimal degree of disorder for localizing light. We then considered 2D and 3D geometries, and evaluated through parallel FDTD simulations the behavior of

_{c}*l*in the spectral region where the highest density of localized states is expected to be observed. In particular, we considered different realizations of disordered photonic crystals and analyzed the properties of localized states near the band edge of the photonic bandgap, which is expected to support a high density of localized states [2

_{c}2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

*l*is characterized by a bell-shaped behavior showing a unique minimum associated to the strongest localization possible. For average disorder values different from this optimal condition, strong localization of light is completely lost. This contributes, in part, to explain the elusive character of this phenomenon. Quite remarkably, our analysis reveals that in the condition of maximal localization, electromagnetic modes are constituted by defect states, whose localization length is smaller than one cell of the periodic crystal and has a subwavelength spatial extension, which in both 2D and 3D cases is approximatively 60% smaller than a single internal wavelength in the high index refractive material. This demonstrates that a proper engineering of disorder can be effectively employed for subwavelength control of light within purely dielectrics.

_{c}## 3. 2D and 3D media: defects states and subwavelength effects

*l*, we employed the Inverse Participation Ratio (IPR): with

_{c}*l*̂

*the IPR of a single realization of disorder,*

_{c}*V*the system volume and

**x**=

**x**

_{0}+ Δ, with Δ being a two/three element random vector whose values Δ

*are given by a normal distribution with 〈Δ*

_{i}*〉 = 0 and standard deviation*

_{i}*χ*of disorder accordingly. In our FDTD simulations, in order to get accurate results, we adopted a spatial discretization with at least 20 internal points per wavelengths in the high-index (Si) material.

*μ*m × 66

*μ*m wide along

*x*and

*y*, characterized by air holes of radius

*r*= 0.3

*μ*m. Figure 2(a) shows the refractive index distribution of a single realization for

*χ*= 0.15. In our simulation campaign, we considered on average 20 disorder realizations for each value of

*χ*, which we have verified is sufficiently high to yield a good convergence of the averaged observables. In order to maximize the probability of observing localization effects, we employed a wideband gaussian source with FWHM=250nm centered at

*λ*

_{0}= 4.8

*μ*

*m*[Fig. 2(b) dashed line], whose wavelength is close to one of the gap edges of the photonic crystal, as shown by the Local Density Of States (LDOS) of the system for

*χ*= 0 [Fig. 2(b) solid line]. In each simulation, we launched the gaussian source and then monitored the evolution of

*l*̂

*(*

_{c}*t*) over time

*t*[Fig. 2(c)]. After an initial transient of ≈ 4·10

^{5}timesteps, the source propagated over the system and the localization length

*l*̂

*reached a steady state, characterized by the electromagnetic modes excited in the system after the interaction with the input wave [Fig. 2(c)]. The behavior of*

_{c}*l*, after averaging, is qualitatively identical to the one dimensional case, with a single minimum attained in the dynamics [Fig. 2(d)]. In order to investigate the properties of light in the regime of strong localization, we begin by calculating the LDOS for the strongest localization, which is attained for

_{c}*χ*= 0.15 [Fig. 2(e)]. As seen in Fig. 2, electromagnetic states are composed by a few eigenfrequencies located inside the badgap of the photonic crystal, and they effectively behave as single-defect states. As a result, their spatial localization is nor originated by Anderson interference effects, neither by multiple impurity scattering, but conversely by the localization properties of the electromagnetic bandgap. The consequences of this dynamics are quite interesting, as shown in Fig. 2(f), which displays the spatial distribution of the electromagnetic energy

*ℰ*for one realization at

*χ*= 0.15. As seen, the electromagnetic radiation is able to reach a subwavelength confinement. The FWHM of the energy spot, in fact, is calculated to be

*FWHM*= 0.58

*μ*m, which is 60% smaller than one internal wavelength

*λ*=

*λ*

_{0}/

*n*= 1.38

*μ*m, where

*n*is the refractive index of Si. It is worthwhile remarking that such a subwavelength confinement can be only observed in the narrow range of

*χ*that yields a strong localization regime, i.e., for

*χ*∈ [0.13, 0.18]. As seen in Fig. 2(d), in fact, a variation of a few percent of average disorder from its optimal condition yields an increase of the localization length, with the vanishing of subwavelength confinement.

*μm*× 18

*μm*× 18

*μm*fcc lattice of air holes (with radius

*r*= 0.35

*μ*m) in silicon, and by employing a Gaussian input source with

*λ*

_{0}= 1.5

*μ*m and FWHM = 100nm. As in the 2D case, we have centered the source at one edge of the photonic crystal gap in order to maximize the probability of observing localized states [Fig. 3(a) dashed line]. Figure 3(b)–3(d) summarizes the results arising from numerical simulations. The behavior of the localization length

*l*is qualitatively identical to the 2D case [Fig. 2(d)], with a unique minimum observed as

_{c}*χ*is varied. Quantitatively, the strongest localization is attained at

*χ*= 0.05 [Fig. 3(b)], which corresponds to a small variation of disorder of only 5%. In this regime, as for the two-dimensional case, strongly-localized states are characterized by defect-states originated in the electromagnetic bandgap, as observed by the LDOS reported in Fig. 3(c). In order to further investigate the degree of spatial confinement achieved in the strong localization regime, we considered a single realization for

*χ*= 0.05 and evaluated the FWHM of the hot energy spot [Fig. 4(a)–4(b)], obtaining a value

*FWHM*= 0.19

*μ*m, which is ≈ 56% smaller than one internal wavelength

*λ*

_{0}/

*n*= 0.43

*μ*m in silicon. The three dimensional system therefore shows the same qualitatively scenario of the two dimensional case, with comparable subwavelength confinement properties —i.e., 60% in 2D vs 56% in 3D— although it offers a much narrower disorder window for achieving strong localization effects, as observed by comparing Fig. 3(b) with Fig. 2(d).

## 4. Discussion and conclusions

*l*in these systems, as obtained by FDTD simulations, appears qualitatively identical to the one dimensional case, with the manifestation of a single minimum that sustains the strongest localization of light possible. From a more quantitatively perspective, in both 2D and 3D samples, FDTD simulations predicts the possibility of observing strong localization effects for a relatively small average disorder :

_{c}*χ*= 15% in two dimensions and

*χ*= 5% in the 3D case. In addition, the numerical analysis shows that a small variation of disorder from the optimal condition leads to a complete vanishing of the strong character of localized modes, with the localization length sharply increasing to the initial value pertaining to the unperturbed crystal. The average disorder range for the observation of strong localization effects is quite narrow: approximatively 5% in 2D and only 1% in three dimensions. These results contribute, to a certain extent, to also explain the elusive nature of this phenomenon.

*l*versus disorder in 1D, 2D and 3D systems can be drawn by generalizing the one dimensional RMT analysis to higher manifolds. In 2D and 3D cases, in fact, we can fold the photons dynamics over the scattering trajectory of light rays. A single ray, in the most common configuration where only two materials are employed (i.e., a suspension of particles in a host matrix or air holes on a disordered substrate [26

_{c}26. S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. **100**, 101102 (2012). [CrossRef]

*l*when, on average, light rays scatter along trajectories whose disorder is optimal for their one dimensional evolution. Regardless of the presence of a higher manifold for the dynamics of light, we can therefore qualitatively expect the appearance of a single minimum, if the average disorder is able to optimally localize the dynamics of light rays.

_{c}**k**. The ideal case of an infinite sum of plane waves, which coherently interfere at every angle, yields the smallest spot achievable, i.e., a dirac-delta

*δ*function. Conversely, the opposite case of an incoherent summation of plane waves originates the well-known Rayleigh distribution (see, e.g., [28]) that does not show any hot-spot with subwavelength confinement. Our analysis shows that suitable values the disorder support a coherent combination of plane waves that lives in between these two extrema —i.e., a full coherent combination and a totally incoherent interference—, which results into a subwavelength spatial confinement for light. A proper engineering of disorder can therefore be employed as an avenue for subwavelength light localization, where extremely narrow hot energy spot can be generated to highly enhance absorption and/or emission properties of purely dielectric materials. This is expected to open new applications in the field of high power lasers, sensing and energy harvesting devices.

## Acknowledgments

## References and links

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

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

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

4. | M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. |

5. | S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. |

6. | P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. |

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

8. | A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature |

9. | S. Gentilini, A. Fratalocchi, and C. Conti, “Signatures of Anderson localization excited by an optical frequency comb,” Phys. Rev. B |

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

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. | R. Frank, A. Lubatsch, and J. Kroha, “Theory of strong localization effects of light in disordered loss or gain media,” Phys. Rev. B |

13. | F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light,” Nature |

14. | S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. |

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

16. | A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson localization in one-dimensional randomly disordered optical systems that are periodic on average,” Phys. Rev. B |

17. | V. D. Freilikher, B. A. Liansky, I. V. Yurkevich, A. A. Maradudin, and A. R. McGurn, “Enhanced transmission due to disorder,” Phys. Rev. E |

18. | Y. A. Vlasov, M. I. Kaliteevski, and V. V. Nikolaev, Phys. Rev. B, “Different regimes of light localization in a disordered photonic crystal,” Phys. Rev. B |

19. | M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E |

20. | P. Sheng, |

21. | E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. |

22. | A. Crisanti, G. Paladin, and A. Vulpiani, |

23. | P. Markos and C. M. Soukoulis, |

24. | S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon delocalization transition in dimensional crossover in layered media,” Phys. Rev. Lett. |

25. | M. Ghulinyan, “Periodic oscillations in transmission decay of Anderson localized one-dimensional dielectric systems,” Phys. Rev. Lett. |

26. | S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett. |

27. | J. M. Deutsch and G. Paladin, “Product of random matrices in a microcanonical ensemble,” Phys. Rev. Lett. |

28. | F. Haake, |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(290.4210) Scattering : Multiple scattering

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 15, 2012

Revised Manuscript: July 15, 2012

Manuscript Accepted: July 18, 2012

Published: July 23, 2012

**Citation**

Diego Molinari and Andrea Fratalocchi, "Route to strong localization of light: the role of disorder," Opt. Express **20**, 18156-18164 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18156

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### References

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- S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett.34, 130–132 (2009). [CrossRef] [PubMed]
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- S. Ghosh, N. D. Psaila, R. R. Thomson, B. P. Pal, R. K. Varshney, and A. K. Kar, “Ultrafast laser inscribed waveguide lattice in glass for direct observation of transverse localization of light,” Appl. Phys. Lett.100, 101102 (2012). [CrossRef]
- J. M. Deutsch and G. Paladin, “Product of random matrices in a microcanonical ensemble,” Phys. Rev. Lett.62, 695–698 (1989). [CrossRef] [PubMed]
- F. Haake, Quantum Signatures of Chaos (Springer, New York, 2001).

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