Abstract

We report the first observation of transverse Anderson localization in a glass optical fiber. The strong localization happens near the outer boundary of the fiber and no trace of localization is observed in the central regions. However, these observations complement previous reports that the boundary of a disordered medium has a de-localizing effect. Our observations can be explained by considering the non-uniform distribution of disorder in the fiber, where the substantially larger disorder near the outer boundary of the fiber offsets the de-localizing effect of the boundary.

© 2012 Optical Society of America

1. Introduction

Anderson localization [1], especially for electromagnetic waves [2], has been of great research interest over the past few years. The observation of strong localization in three dimensional (3D) optical media is quite challenging [3], because optical materials generally cannot provide sufficient scattering strength to satisfy the Ioffe-Regel condition [4]. Wiersma, et al. [5], reported the first experimental evidence of Anderson localization in strongly scattering GaAs powder. The required conditions for localization are considerably relaxed in one dimensional (1D) and two dimensional (2D) systems; in fact, 1D and 2D disordered systems are always localized [6]. In practice, a small localization length, ξ, is easily achievable in 1D and 2D systems, even for a moderate amount of disorder. For example, De Raedt, et al. [7], have shown that if the randomness in the refractive index profile is only limited to the transverse x–y plane, a beam propagating in the z-direction can remain trapped in the transverse direction; a reasonably small localized beam diameter is possible even for small disorder refractive index contrast on the order of 10−4, as was experimentally observed by Schwartz, et al. [8], in a photo-refractive crystal. Transverse Anderson localization has since been actively explored in various systems, e.g., in 1D disordered photonic lattices [9], in optical waveguide arrays with off-diagonal coupling disorder [10], and in amorphous photonic lattices [11].

We recently reported the first observation of transverse Anderson localization in an optical fiber [12], lacking the conventional core/clad structure, where the large index contrast between constituent polymer materials resulted in an effective propagating beam diameter comparable to that of a typical index-guiding optical fiber. The polymer optical fiber composed of about 80,000 randomly placed sites of polymethyl methacrylate (PMMA) and polystyrene (PS), both of which are commodity polymers, with a refractive index contrast of 0.1 and fill-fraction of 50%. The fill-fraction is the material fraction of the lower index polymer (PMMA, n = 1.49) to the total which also includes the higher index polymer (PS, n = 1.59).

We recently carried out a detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers [13]. In general, if transverse Anderson localization is to be used as the waveguiding mechanism in optical fibers, it is desirable to have designs with smaller beam diameters, as well as lower sample-to-sample variation in the value of the beam diameter. The sample-to-sample variation is a natural consequence of the statistical nature of Anderson localization; however, we confirmed that such variations can be suppressed because of “self-averaging” behavior, if the refractive index contrast is increased [13]. In fact, we showed that robust designs (small beam diameter and low sample-to-sample variation in the value of the beam diameter) are possible if the index contrast is increased to that of the fused silica (∼1.5) to air (1.0).

In this paper, we present the first results on the fabrication and analysis of a disordered silica glass-air (porous) optical fiber. Although we obtained a fill-fraction that was substantially below the optimal value of 50%, strong localization of light was observed (only) in regions close to the outer boundary of the fiber; i.e., interface near the porous glass and protective polymer coating. This observation is quite interesting, because Szameit, et al. [14] and Jović, et al. [15] have shown that the boundary of a disordered medium has a de-localizing effect and it is easier to observe strong localization in regions away from the boundary. In this paper, we show that our results do not contradict those of Refs. [14, 15], and that the de-localizing effect of the boundary in our fiber is offset by the substantially higher near-boundary air fill-fraction compared with the center of the fiber, which results in a strong transverse scattering and near-boundary localization effect. Our observations also agree with the results recently published in Ref. [16] on the effect of a non-uniform variation in the strength of the disorder in a 1D lattice on the localization of light. They reported that the tail of the localized light decays faster in a more strongly disordered region, and vice versa.

2. Disordered optical fiber with random air-holes

The optical fiber employed in this work was drawn from “satin quartz” (Heraeus Quartz) which is a porous artisan glass. The starting rod had dimensions of 8 mm in diameter and 850 mm in length and was drawn at Clemson University on a custom-designed Heathway draw tower at a temperature of 1890°C. The fiber was coated with a conventional telecommunications single layer UV-cured acrylate coating. 150 meters of fiber was drawn with an average glass and coated diameter of 250 and 417 μm, respectively. While it was known that the fill-factor; i.e., degree of porosity, was sub-optimal for the idealized level of localization, the satin-quartz was selected as a convenient and extraordinarily inexpensive expedient.

The cross-section of the fiber was imaged using a Hitachi SU-6600 analytical variable-pressure field emission scanning electron microscope (SEM). The fiber was polished to a 1 micron finish and then mounted in the sample holder with carbon tape and placed in the chamber. Images were taken in variable pressure mode using back scattered electron imaging at 20 kV.

The cross-sectional SEM image of the disordered porous optical fiber is shown in Fig. 1(a), which also provides a good estimate of the refractive index profile of the fiber; the light gray background matrix is glass and the black random dots represent the air-holes. The diameter of the fiber is confirmed to be about 250 μm and the average air fill-fraction is about 5.5% with the air-hole diameters varying between about 0.2 μm to 5.5 μm. The distribution of porosity seems to be consistent along the length of the fiber, likely due to the equally consistent distribution of pores along the length of the rod as is what gives “satin quartz” its opalescent qualities.

 

Fig. 1 (a) SEM image of the glass optical fiber with random air-holes. For ease of viewing, the polymer coating has been removed. (b) Refractive index profile used in our simulations.

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To observe the transverse Anderson localization effect, we use the light from a 405 nm diode laser delivered using a 630HP fiber from Thorlabs, where the average mode field diameter of this slightly-multimode fiber is around 4 μm. The 630HP fiber is butt-coupled to the disordered fiber and the output beam profile is measured using a 40x objective on a CCD camera, as explained in detail previously [12,13]. In Ref. [13], we showed that a shorter wavelength results in a stronger localization effect and consequently a smaller localization radius. The disorder fill-fraction in the fiber samples studied in this paper is relatively low. Therefore, we decided to use a 405 nm diode laser, rather than the 633 nm He-Ne laser, which was previously used in Ref. [12].

The finite difference beam propagation method (FD-BPM) was used to carry out the simulations [13]. The refractive index profile is extracted from the SEM image of the fiber in Fig. 1(a) and is directly used in the FD-BPM program; the refractive index profile used in our simulations is shown in Fig. 1(b).

Figure 2 shows a typical result from launching of the beam (405 nm wavelength) into the center of the fiber. It is clear that the disorder is not sufficient to clamp the beam radius to a value smaller than the diameter of the fiber; therefore, the beam fills the entire cross section of the fiber. In the sections to follow, we will show that the beam can remain localized, if the light is coupled near the outer boundary of the fiber. We also show that the stronger localization of the beam near the boundary is due to the higher disorder density (air fill-fraction) in the regions near the boundary of the fiber.

 

Fig. 2 The experimental measurement of the near-field intensity when the beam is launched near the center of the fiber, where no localization is observed.

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3. Localization near the outer boundary

In order to investigate the localization profile of the beam, the 630HP fiber was scanned across the input facet of the disordered fiber near its outer boundary region; i.e., porous glass/coating interface. The near-field intensity at the output facet of the disordered fiber is captured and processed to be compared with the theoretical simulations. We selected 10 different fiber samples, each approximately 10 cm long, and measured the near-field intensity at 10 different locations near the outer boundary of each fiber for a total of 100 independent measurements. The “localized” near-field intensity for four different incident spots near the outer boundary is shown in Fig. 3; in each case, the localized spot consists of multiple peaks, which are located near the boundary.

 

Fig. 3 Near-field intensity measurements at the output facet of the disordered fiber samples for 4 different launch positions.

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The 100 independent measurements should be sufficient to capture the statistical nature of localization and to investigate the exponential decay of the intensity tail.

We repeated the same procedure outlined above in our simulations, using the refractive index profile shown in Fig. 1(b), and collected 100 separate near-field intensity profiles, using incident beams launched at different positions near the outer boundary of the fiber. The beam localization for four different incident spots near the outer boundary are shown in Fig. 4; again, in each case, the localized spot consists of multiple peaks, which are located near the boundary. Similar to the experimental observations, if the beam is launched at the center of the disordered fiber in our simulations, the field fills the entire cross section of the fiber after propagating a distance less than 5 mm.

 

Fig. 4 Near-field intensity simulations at the output facet of the disordered fiber for 4 different launch positions.

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We calculate the beam localization radius (ξ) using the same method described in Refs. [12, 13]. In Fig. 5(a), the region highlighted in black corresponds to the theoretical simulation of the effective beam radius ξavg ± σξ as a function of the propagation distance, where ξavg represents the average value of the effective beam radius over the 100 simulated samples (captured at each point along the fiber in the z-direction), and σξ represents that standard deviation. The effective beam radius expands as the beam propagates along the fiber until it reaches its final localized value, after which the effective beam radius does not change appreciably. For the experimental measurements, we only process the field intensity profiles at the output facet of the fiber; therefore, the region highlighted in red in Fig. 5(a) represents the final stabilized mean value and the standard deviation from the measurements, where reasonable agreement is observed between theory and experiment. The large values of standard deviations signify the statistical nature of the strong localization effect, and can be considerably lowered for disordered fibers with larger air fill-fraction, as discussed in Ref. [13].

 

Fig. 5 (a) The region highlighted in red corresponds to one standard deviation in each direction around the average experimental measurement of the localization length parameter represented by ξavg ± σξ. The region highlighted in black corresponds to theoretical simulation of the effective beam radius ξavg ± σξ as a function of propagation distance. (b) Cross section of the intensity profile of the highest peak in the localized beam averaged over 100 samples of raw data from simulations and 100 samples from experiments in dB units.

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In order to provide evidence for strong localization of the beams, it is common to show an exponential decay of the tails of the localized intensity profiles, as shown recently [12, 13]. However, presenting such a figure in this case is considerably more challenging, because the localized spot is composed of multiple peaks. We observed numerically that the smaller the air fill-fraction, the more separated the peaks are within the localized beam spot, which also results in a larger effective beam diameter. In order to show an exponential decay tail, we selected the highest peak from each sample among the 100 separate measurements. We then averaged the intensity of these highest peaks over the 100 samples, and plotted a cross section of the intensity profile in Fig. 5(b) shown as the solid blue line. We repeated the same procedure for the 100 separate numerical simulations and plotted the cross section of the average intensity profile of the highest peak in Fig. 5(b) marked by the red dots. The experimental and theoretical results are in reasonable agreement. We note that the intensity profile presented in Fig. 5(b) is a cross-section in the radial direction; we observed a similar exponential decay behavior in the angular direction. However, this averaging over the highest peak should only be regarded as for illustration purposes, as emphasized earlier. In order to calculate the localization radius, one must include the intensity of all peaks, as is also considered in Fig. 5(a).

4. Non-uniform distribution of disorder

As discussed above, Szameit, et al. [14] and Jović, et al. [15] have shown that the boundary of a disordered medium has a de-localizing effect and that it is easier to observe strong localization in regions away from the boundary. We claim that our observation presented herein of localization “only” near the boundary of the fiber is not contradictory to those of Refs. [14,15] and arise from the non-uniform distribution of disorder across our disordered fiber. In other words, the air fill-fraction is higher near the outer boundary of the fiber, which offsets the de-localizing effect of the boundary and results in a near-boundary strong localization effect. Shown in Fig. 6(a) is a density plot of the air fill-fraction (disorder density) over the tip of the disordered fiber; the presence of the larger near-boundary air fill-fraction supports our claim that the higher disorder density offsets the de-localizing effect of the boundary. Moreover, we observe spots around the boundary that have a relatively lower disorder density compared with other near-boundary spots, and these spots are likely responsible for the larger effective localized beam radius observed in intensity profiles such as Fig. 3(d) compared with Fig. 3(a).

 

Fig. 6 (a) Density plot shows the air fill-fraction over the tip of the fiber, where the disorder level is generally higher near the outer boundary than the central regions. (b) Segmentation of fiber to different regions for averaging over the angular coordinate, where red and blue colors are used in order to make it easier for the reader to distinguish the regions closer to the center versus regions closer to the outer boundary of the fiber. (c) Air fill-fraction averaged over the angular coordinate as a function of the radial coordinate over the tip of the fiber. The error bars signify the change in the value of the air fill-fraction, if the global image threshold varies by 0.07 around an Otsu’s threshold of 0.37.

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In order to show the (average) radial variation of the disorder density, we average the air fill-fraction of Fig. 6(a) over the angular coordinate and plot the result in Fig. 6(c). The actual regions over which the angular averaging is performed are marked with circles in Fig. 6(b), where we have used two different colors (blue and red), in order to make it easier for the reader to distinguish the regions closer to the center versus regions closer to the outer boundary of the fiber. We note that in Fig. 6(a) and Fig. 6(c), we use a global image threshold of 0.37 using Otsu’s method [17] to deduce a binary refractive index profile from the SEM image of the fiber end. The error bars in Fig. 6(c) show the change in the value of the air fill-fraction, if the Otsu’s threshold is raised or lowered by 0.07; i.e., varying from 0.30 to 0.44. However, we have used the average threshold value of 0.37 in all our simulations in this paper. Again, Fig. 6(c) clearly shows the increase in the air fill-fraction near the outer boundary, compared with the center of the fiber.

We note that under visual examination, the density of the pores in the original satin quartz preform rod appeared uniform and was likely not responsible for the non-uniform distribution of the disorder across the fiber. We speculate that the non-uniformity was caused by the temperature distribution experienced by the glass during the draw process, but further study is warranted to better understand how the porosity and pore distribution changes with processing conditions.

5. Conclusions

We report the first observation of transverse Anderson localization in a glass optical fiber, where the strong localization happens near the outer boundary, rather than the central region. Previous work by Szameit, et al. [14] and Jović, et al. [15] has clearly shown the de-localizing effect of the boundary in a disordered medium. We have shown that the disorder distribution in our fiber samples is not uniform and we observe a substantially larger air fill-fraction in the regions closer to the boundary of the fiber. Therefore, the higher disorder in regions closer to the boundary offsets the de-localizing effect of the boundary. The air fill-fraction is as low as 2% in the central regions, but it reaches almost as high as 8% near the boundary of the fiber.

In Fig. 7, we present our numerical simulations for light localization in the central region of the fiber end at 405 nm wavelength, when the disorder in the refractive index profile is uniformly distributed, for different values of air fill-fraction. Figures 7(a) and 7(d) show that no localization is observed for 3% uniform air fill-fraction. For 6% uniform air fill-fraction, traces of strong localization can be observed in the central regions as shown in Figs. 7(b) and 7(d). For 10% uniform air fill-fraction, strong localization can be clearly observed in the central regions as shown in Figs. 7(c) and 7(d), and the tails of the field decay considerably faster than the case of 6% uniform air fill-fraction. Therefore, our results in Fig. 7 are consistent with our arguments on why localization can only be observed near the outer boundary of this fiber. We note that the difference between the localization strength in the center of the fiber samples as presented in Fig. 7 versus the near-boundary region as explored in Figs. 4 and 5(b) is very small. Although our results do not contradict those of Szameit, et al. [14] and Jović, et al. [15], the difference in the localization strength falls within the margin of error in our 100-element ensemble used in our simulations. We note that the large standard deviation in the localization radius is the result of the small fill-fraction in the fiber samples, as was also discussed in Refs. [12, 13], as well as the large variations in the fill-fraction near the boundary as shown in Fig. 6(c). It is possible to conclusively verify the de-localizing effect of the boundary in samples similar to the disordered fibers presented in this work by exploring similar samples with uniform fill-fraction. However, we expect that this analysis would require many more simulations and massive computational resources, which is beyond the scope this paper. Future efforts are focusing on fibers with higher air fill-fraction (50%) and greater uniformity of the porosity. Potential applications of the disordered optical fibers are in the spatially multiplexed short-haul optical fiber communications, as well as optical imaging. These potential applications will also be explored in the future.

 

Fig. 7 Simulation of the near-field intensity profile when the beam is launched near the center of the fiber, for uniform disorder distribution with (a) 3% air fill-fraction, (b) 6% air fill-fraction, and (c) 10% air fill-fraction. (d) Cross section of the intensity profile for uniformly disordered fibers with 3%, 6%, and 10% air fill-fraction, where the beam is launched near the center of the fiber. All figures are plotted for the intensity profile after propagating 5 cm along the fiber.

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Acknowledgments

S. Karbasi, K. W. Koch, and A. Mafi acknowledge support from the National Science Foundation under the grant number 1029547. Authors also acknowledge Ryan J. Frazier for preparing some of the fiber samples used in these experiments and Stephanie Morris for SEM imaging.

References and links

1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958). [CrossRef]  

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

3. A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009). [CrossRef]  

4. A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

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

6. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).

7. H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989). [CrossRef]   [PubMed]  

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

9. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef]   [PubMed]  

10. L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011). [CrossRef]   [PubMed]  

11. M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).

12. S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Observation of transverse Anderson localization in an optical fiber,” Opt. Lett. 37, 2304–2306 (2012). [CrossRef]   [PubMed]  

13. S. Karbasi, C. R. Mirr, Ry. J. Fraizer, P. G. Yarandi, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Express 20, 18692–18706 (2012). [CrossRef]  

14. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Wave localization at the boundary of disordered photonic lattices,” Opt. Lett. 35, 1172–1174 (2010). [CrossRef]   [PubMed]  

15. D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011). [CrossRef]  

16. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Light localization in nonuniformly randomized lattices,” Opt. Lett. 37, 286–288 (2012). [CrossRef]   [PubMed]  

17. N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. SMC-9, 62–66 (1979).

References

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  1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
    [Crossref]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref] [PubMed]
  3. A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
    [Crossref]
  4. A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).
  5. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
    [Crossref]
  6. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).
  7. H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
    [Crossref] [PubMed]
  8. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
    [Crossref] [PubMed]
  9. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
    [Crossref] [PubMed]
  10. L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
    [Crossref] [PubMed]
  11. M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
  12. S. Karbasi, C. R. Mirr, P. G. Yarandi, R. J. Frazier, K. W. Koch, and A. Mafi, “Observation of transverse Anderson localization in an optical fiber,” Opt. Lett. 37, 2304–2306 (2012).
    [Crossref] [PubMed]
  13. S. Karbasi, C. R. Mirr, Ry. J. Fraizer, P. G. Yarandi, K. W. Koch, and A. Mafi, “Detailed investigation of the impact of the fiber design parameters on the transverse Anderson localization of light in disordered optical fibers,” Opt. Express 20, 18692–18706 (2012).
    [Crossref]
  14. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Wave localization at the boundary of disordered photonic lattices,” Opt. Lett. 35, 1172–1174 (2010).
    [Crossref] [PubMed]
  15. D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
    [Crossref]
  16. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Light localization in nonuniformly randomized lattices,” Opt. Lett. 37, 286–288 (2012).
    [Crossref] [PubMed]
  17. N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. SMC-9, 62–66 (1979).

2012 (3)

2011 (2)

2010 (1)

2009 (1)

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

2007 (1)

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

1997 (1)

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

1989 (1)

H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[Crossref] [PubMed]

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

1979 (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. SMC-9, 62–66 (1979).

1960 (1)

A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

1939 (1)

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).

Abouraddy, A. F.

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Bartal, G.

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

Bartolini, P.

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

Belic, M. R.

D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[Crossref]

Christodoulides, D.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Christodoulides, D. N.

De Raedt, H.

H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[Crossref] [PubMed]

de Vries, P.

H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[Crossref] [PubMed]

Denz, C.

D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[Crossref]

Di Giuseppe, G.

Dreisow, F.

Fishman, S.

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

Fraizer, Ry. J.

Frazier, R. J.

Heinrich, M.

Ioffe, A. F.

A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Jovic, D. M.

D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[Crossref]

Karbasi, S.

Kartashov, Y. V.

Keil, R.

Kivshar, Y. S.

D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[Crossref]

Koch, K. W.

Konotop, V. V.

Lagendijk, A.

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

Lagendijk, A. D.

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[Crossref] [PubMed]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Mafi, A.

Martin, L.

Mirr, C. R.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Nolte, S.

Otsu, N.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. SMC-9, 62–66 (1979).

Perez-Leija, A.

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Rechtsman, M.

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).

Regel, A. R.

A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Righini, R.

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

Saleh, B. E. A.

Schwartz, T.

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

Segev, M.

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

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Szameit, A.

Torner, L.

Tünnermann, A.

van Tiggelen, B.

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

Vysloukh, V. A.

Wiersma, D. S.

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

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

Yarandi, P. G.

Zeil, P.

IEEE Trans. Syst. Man. Cybern. (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. SMC-9, 62–66 (1979).

Nature (2)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 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, 52–55 (2007).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Phys. Rev. A (1)

D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belic, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83, 033813 (2011).
[Crossref]

Phys. Rev. Lett. (4)

H. De Raedt, A. D. Lagendijk, and P. de Vries, “Transverse localization of light,” Phys. Rev. Lett. 62, 47–50 (1989).
[Crossref] [PubMed]

M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Phys. Today (1)

A. D. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62, 24–29 (2009).
[Crossref]

Prog. Semicond. (1)

A. F. Ioffe and A. R. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Other (1)

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic Press, 1995).

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Figures (7)

Fig. 1
Fig. 1 (a) SEM image of the glass optical fiber with random air-holes. For ease of viewing, the polymer coating has been removed. (b) Refractive index profile used in our simulations.
Fig. 2
Fig. 2 The experimental measurement of the near-field intensity when the beam is launched near the center of the fiber, where no localization is observed.
Fig. 3
Fig. 3 Near-field intensity measurements at the output facet of the disordered fiber samples for 4 different launch positions.
Fig. 4
Fig. 4 Near-field intensity simulations at the output facet of the disordered fiber for 4 different launch positions.
Fig. 5
Fig. 5 (a) The region highlighted in red corresponds to one standard deviation in each direction around the average experimental measurement of the localization length parameter represented by ξavg ± σξ. The region highlighted in black corresponds to theoretical simulation of the effective beam radius ξavg ± σξ as a function of propagation distance. (b) Cross section of the intensity profile of the highest peak in the localized beam averaged over 100 samples of raw data from simulations and 100 samples from experiments in dB units.
Fig. 6
Fig. 6 (a) Density plot shows the air fill-fraction over the tip of the fiber, where the disorder level is generally higher near the outer boundary than the central regions. (b) Segmentation of fiber to different regions for averaging over the angular coordinate, where red and blue colors are used in order to make it easier for the reader to distinguish the regions closer to the center versus regions closer to the outer boundary of the fiber. (c) Air fill-fraction averaged over the angular coordinate as a function of the radial coordinate over the tip of the fiber. The error bars signify the change in the value of the air fill-fraction, if the global image threshold varies by 0.07 around an Otsu’s threshold of 0.37.
Fig. 7
Fig. 7 Simulation of the near-field intensity profile when the beam is launched near the center of the fiber, for uniform disorder distribution with (a) 3% air fill-fraction, (b) 6% air fill-fraction, and (c) 10% air fill-fraction. (d) Cross section of the intensity profile for uniformly disordered fibers with 3%, 6%, and 10% air fill-fraction, where the beam is launched near the center of the fiber. All figures are plotted for the intensity profile after propagating 5 cm along the fiber.

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