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Abstract
The transverse Anderson localization (TAL) can always be observed in one-dimensional (1D) disordered systems as long as the transverse dimension is significantly larger than the localization length. This paper presents a detailed modal analysis in one particular realization of the 1D disordered optical waveguides with wavelength-scale feature size based on the imaginary distance beam propagation method (BPM). The localized modes are independent of the physical properties of the external excitation. Additionally, we investigate how the boundaries of disordered waveguides affect the localized modes, which are only related to the design parameters such as feature size, refractive index contrast, and fill-fraction. Finally, we explore the impact of the design parameters on the average localized mode width in the 1D disordered waveguides.
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1. Introduction
When Anderson observed in 1958 that electron diffusion would vanish in disordered crystals, he developed a novel theory of electron localization in extremely disordered systems [1]. Light will be arrested in a disordered optical medium with a localized state, similar to Anderson localization of electrons in disordered solids. The localization of light is a result of coherent multiple scattering and interference. The disordered microstructures exhibit new effects in both classical optics and quantum electrodynamics [2–3]. Strong coherent scattering in random medium has been effectively applied in random laser [4]. A. D. Mirlin et al. study the statistical intensity distribution for waves propagating in a quasi 1D disordered medium, where one transverse dimension is significantly larger than the other [5]. Subsequently, De Raedt et al. proposed the concept of TAL in the optical system [6] with a randomly distributed transverse refractive index and an unchanged longitudinal refractive index. The TAL criteria is significantly relaxed in the 1D and 2D disordered optical systems. After long distance propagation, light beam will not diffuse further in these systems because of random scattering brought on by the random distribution of the transverse refractive index. In other words, just like in ordinary optical fibers, the beam will propagate along the longitudinal direction with a finite beam radius. The disordered system, which has higher resolution and information capacity than multimode fibers, could be used for direct image transmission in biology and medicine [7–14]. Segev et al. used an optical interference model in photorefractive crystals [15] with a refractive index contrast of ${10^{ - 4}}$ and experimentally observed the presence of TAL for the first time in a 2D system. Karbasi et al. introduced a polymer disordered optical fiber [7,10,13,14] with refractive index fluctuations on the order of 0.1 and observed TAL of light both experimentally and numerically with an effective propagating beam radius comparable to that of a typical index-guiding optical fiber [13,14]. Recently, numerous research have been published that investigate the localization phenomenon from various perspectives, including light field propagation and mode analysis in the 1D and 2D disordered waveguides [16–18].
When propagating in disordered waveguides, the beam expands until its effective width reaches the localization length [6]. The physical properties of the input light source will interfere with the validity of the conclusions when studying the effect of different structural and optical design parameters on the localized beam radius, i.e., localization length. Except for the guiding mechanism, the transmission characteristics of light in disordered waveguides are very similar to those of ordinary dielectric waveguides. We use mode methods to analyze the localized characteristics of disordered waveguides in the same way that we do for ordinary waveguides. Particularly useful is modal analysis, which is frequently used to investigate proper physical intuition and provide reliable answers to the propagation and localization behaviour of in disordered waveguides. All guided modes, which are not localized modes in the real sense, are impacted by the boundary when the transverse dimension of the disordered waveguides is small. It is necessary to change the transverse dimension in order to extract and analyze the localized modes under the same conditions. All localized modes, which operate as independent space channels, cover the whole cross-section of the disordered waveguides. When a narrow Gaussian light wave is incident at the center of the disordered waveguide structure, all of the guided modes involved in the coupling process are localized modes, revealing the true TAL.
In order to numerically simulate the propagation of light beam in disordered waveguides, the scalar wave equation is solved by finite difference beam propagation method (FD-BPM) under the paraxial approximation [19–23]. In order to match this process, we adopt the mode-solving technique referred as the imaginary distance BPM [24] in disordered waveguides. This ensures that the optical field propagation and the corresponding guided modes can be effectively compared under the same structure and simulation parameters.
2. Disordered structure and model analysis
It is worth noting that useful BPM-based mode-solving techniques known as imaginary distance BPM [25,26] have been developed. The imaginary distance BPM technique is formally equivalent to many other iterative mode solving techniques.
In BPM-based mode-solving techniques a given incident field is launched into a geometry that is $z$-invariant. Since the structure is uniform along z, the propagation can be equivalently described in terms of the modes and propagation constants of the structure. Considering 2D propagation of a TE polarized field for simplicity, the incident field ${\phi _{in}}(x )$, can be expanded in the modes of the structure as
The summation should of course consist of a true summation over guided modes and integration over radiation modes, but for brevity the latter is not explicitly shown. Propagation of all guided modes through the structure can be expressed as The propagating field obtained through BPM is conceptually equated with the above expression in the BPM-based mode-solving technique to determine how to extract mode information from the BPM results.As the name implies, in the imaginary distance BPM the longitudinal coordinate z is replaced by $\hat{z} = jz$ so that propagation along this imaginary axis should follow:
The propagation implied by the exponential term in Eq. (2) has become exponential growth in Eq. (3), with the growth rate of each mode being equal to its real propagation constant. The essential idea of the method is to launch an arbitrary field, say a Gaussian, and propagate the field through the structure along the imaginary axis. Since the fundamental mode (m = 0) has by definition the highest propagation constant, its contribution to the field will have the highest growth rate and will dominate all other modes after a certain distance, leaving only the field pattern ${\phi _0}(x )$. The propagation constant of the guided modes can be obtained by the following expression:The 2D structural diagram in one particular realization of the 1D disordered waveguides shown in Fig. 1, which is similar to the 2D disordered system introduced by De Raedt et al. In this 1D disordered structure, the transverse (x-axis) refractive index is randomly distributed, but the longitudinal direction (z-axis) remains uniform. In addition, the structure is independent of the transverse y-axis. The transverse width is D, and N is the total number of slab waveguides covering the transverse profile of the 1D disordered waveguide. The refractive index of each slab waveguide can be either ${n_1}$ (low-index) or ${n_2}$ (high-index), p is the fill-fraction of the low-index material in the higher index medium. All distances in this article are given in units of the wavelength $\lambda $ for convenience. The feature size $d = \lambda $, which is the width of each slab waveguide. The thickness of the cladding ${D_c} = 50\lambda $, whose refractive index ${n_c} = {n_1}$. We only show the cladding structure in Fig. 2(a). The transverse grid size is $\mathrm{\Delta }x = \lambda /20$ in all numerical calculations. In this article we adopted perfectly matched layer (PML) boundary condition [27,28] with thickness is 10$\lambda $.
Fig. 2. Refractive index profiles and typical mode profiles of the 1D slab waveguides. (a) Conventional slab waveguide. (b) and (c) 1D disordered waveguide with width $D = 50\lambda $ and $D = 100\lambda $, respectively.
Figure 2(a) depicts the refractive index profile of a conventional slab waveguide and the corresponding guided mode profiles. The refractive indices of the core and cladding are ${n_2} = 1.01$ and ${n_1} = 1$, respectively. The thickness of the core is $50\lambda $. We discover that the profile of each guided modes completely cover the entire cross section of the core structure. Figures 2(b) and (c) show the disordered refractive index profile and the corresponding guided mode profiles in one particular realization of the 1D disordered waveguide, where the transverse width D is 50$\lambda $ and 100$\lambda $, respectively. The fill-fraction $p = 50\%$. All guided modes, as shown in Fig. 2(b), are the result of disordered distribution and waveguide boundaries, and are not truly localized modes. The introduction of disorder into a conventional slab waveguide narrows the width of the guided modes. Increase the width will reduce the influence of the boundary on the localized modes. In Fig. 2(c), the guided modes $m = 0$ and $m = 1$ are localized modes, but the guide mode $m = 7$ is boundary-dependent in one particular realization of the 1D disordered waveguides. Compared with Fig. 2(b), Fig. 2(c) only expands on both sides of the 1D disordered waveguide structure. The guided mode profiles in Fig. 2(c) clearly illustrate that disorder is behind the localization of the guided modes when the center of the mode field is far from the structural boundary.
In order to make the distribution of all guided modes more intuitive in the optical waveguide, the effective width of guided modes is calculated using the second moment method [16]. For each guided mode, the effective mode width ${\omega _e}$ is defined as:
where x is the transverse coordinates, $\bar{x}$ is the effective center position of the guided modes: where the intensity profile of modes is assumed to be normalized according to $\smallint I(x )dx = 1$.A conventional symmetric slab waveguide in Fig. 2(a), all guided modes are centered and the effective widths of the modes vary with waveguide width, as shown in Fig. 3(a). Figures 3(b) and (c) correspond to Figs. 2(b) and (c) respectively, and the structural parameters are the same with feature size $d = \lambda $. The refractive index contrast $\mathrm{\Delta }n = 0.01$. The effective width of the guided mode in the disordered waveguide is significantly reduced compared to the conventional slab waveguide, and the effective center positions of all guided modes are uniformly distributed throughout the cross section. To facilitate the analysis of the variation trend of the mode width, we use the probability density function (PDF) [18] of all the guided mode width. Figure 3(d) shows that the width of the guided mode reduced with the introduction of disorder. The transverse width increases, the effective width of the guided mode did not increase with it. The vertical axis is in units of $1/\lambda $ so that the total area under the PDF integrates to unity. Under the same condition with refractive index contrast $\mathrm{\Delta }n = 0.01$ and feature size $d = \lambda $, the number of guided modes increases with the increase of the transverse width D. The effective refractive index of all guided modes is distributed between high and low refractive index, as illustrated in Fig. 4. The number of repeated guided modes also increases. Without comparing the mode profile, we cannot tell if guided modes with the same effective refractive index under different width D are localized modes.
Fig. 3. The distribution of the mode width for all the guided modes. (a) Conventional slab waveguide. (b) and (c) 1D disordered waveguide with width $D = 50\lambda $ and $D = 100\lambda $, respectively. (d) The probability density function of the effective guided mode width for 1D disordered waveguides corresponding to (a-c).
Fig. 4. The effective index of all guided modes in 1D disordered waveguides with different width D. The cladding refractive index ${n_c}$ is equal to the lower refractive index ${n_1}$.
Figures 3(b) and (c) show that localized modes are absent in all guided modes of a disordered waveguide with width $D = 50\lambda $. When the width reaches $100\lambda $, all of the guided modes change, indicating that the boundary has an effect on these guided modes [17,29]. It is necessary to successfully separate localized modes from all guided modes, i.e., the modes that are unaffected by boundary. We adjust the width of the disordered waveguide to acquire some modes that do not change in all guided modes which are localized modes. The premise is to ensure that the previous disordered waveguide is unchanged while expanding the width of the waveguide. In Fig. 5(a), we can see that some guided modes overlap in the disordered waveguide with different width $100\lambda $ and $200\lambda $. The overlapping guided modes are the localized modes that arise fro. the introduction of disorder and are independent of structural boundaries. Figures 5(b) and (c) illustrate that as the number of guided modes increases, so does the number of localized modes. The localized modes are distributed over the entire cross-section, and the effective mode widths are significantly less than $50\lambda $. As a result, the average width of all localized modes, which are only relevant for disordered structures, can be used to measure the strength of localization. We can see that, except for the guided modes at the boundary, there are always some guided modes that are disturbed by the boundary, even if they are very weak. The majority of the power is coupled into these localized modes, with the remainder coupled into radiating modes absorbed by the absorbing boundary. Because of the increase in the number of localized modes that are only related to disorder, the PDF of the width of the guided mode tends to stabilize with a peak below 50 wavelengths, as shown in Fig. 5(d). The small peaks that protrude at larger positions are some guided modes that are still influenced by the boundary and thus change indeterminately. The mean value of the effective width of all localized modes can be used as an effective measure under different disorderd conditions in the follow-up.
Fig. 5. (a-c) The distribution of the mode width for all the guided modes in the 1D disordered waveguides with different transverse width D. (d) The probability density function of the effective guided mode width corresponding to (a-c).
In order to simulate the propagation of light in the 1D disordered waveguides and observe TAL phenomenon, a monochromatic Gaussian beam of radius ${\mathrm{\omega }_0}$ is launched into the center at $z = 0$ and propagate along the waveguide by numerically solving the scalar wave propagation Eq. (7) under the paraxial approximation with FD-BPM [24].
The incident electric field with a Gaussian waveform $E = A\textrm{exp}({ - {x^2}/\omega_0^2} )$ and ${\omega _0} = 5\lambda $. The excitation amplitudes of all the guided modes are calculated by using the overlap integrals of the in-coupling Gaussian beam with the guided modes of the structure. ${a_m} = \smallint E{\phi _m}dx$ are the excitation amplitudes of the guided modes, where the guided modes field is assumed to be normalized according to $\smallint \phi _m^2dx = 1$. The optical intensity of the propagated guided modes is given by ${I_g}({x,z} )= {|{{E_g}({x,z} )} |^2}$, where the electric field ${E_g}({x,z} )$ is propagated according to ${E_g}({x,z} )= {a_m}{\phi _m}\textrm{exp}({ - j{\beta_m}z} )$. Here, ${\beta _m}$ is propagation constant correspond to the guided mode number m. The Gaussian beam couples efficiently to those guided modes and the remaining power is coupled to the radiation modes that radiate away from disordered waveguide after propagate long distances.
The coupled power of guided modes and localized modes with the input Gaussian wave for one particular realization of the 1D disordered waveguides with different transverse width D are shown in Fig. 6. There are no localized modes when the width D is $50\lambda $, so the power occupied by the localized modes is 0. The power occupied by the localized modes grows in proportion to the transverse width D. When the transverse width exceeds $500\lambda $, all guide modes excited by the initial input Gaussian wave are localized modes. Approximately 70% of the power in the Gaussian beam is coupled to the localized modes. This is sufficient to demonstrate that the TAL, which can be regarded as a disordered system without boundaries, is entirely responsible for the propagation of a narrow beam along the disordered waveguides.
Fig. 6. The coupled power of guided modes and localized modes with the input Gaussian wave as function of the transverse width D.
As shown in Fig. 7, the modest power is coupled into the radiation modes that emerge from the waveguides when a narrow Gaussian beam is injected into the middle of a disordered waveguide. Although some delocalized modes couple energy in a 1D disordered waveguide with width $100\lambda $, the coupled radiation modes quickly radiate away. All radiating modes must radiate away from the disordered waveguide after considerable propagation distances with transverse width D exceeds $400\lambda $. Each excited guided mode propagates with a different phase velocity, determined by their individual propagation constant. The intensity distribution of all guided modes shows that the beam expands initially and eventually localizes around $40\lambda $, as shown in Fig. 8. The distribution of fields along the waveguide is periodic and the period is approximately 700 wavelength. But the true period of all relevant guided modes is much larger than this value. Part of the guided modes occupy most of the power of the Gaussian input, so the periodicity presented at this time is caused by this part of the guided modes, while the other modes cause the perturbation phenomenon.
Fig. 7. The power of the input narrow Gaussian wave in a disordered waveguide with different transverse width D versus propagation distance.
Fig. 8. The intensity distribution of all guided modes eventually localizes around $40\lambda $ within $700\lambda $ distance, which is periodic with propagation distance.
3. Impact of the design parameters
The modal analysis allows us to explore localization behavior of a disordered lattice independent of the properties of the external excitation. Although it is not the same as the localization length, the average width of the localized modes is closely related to the disordered waveguide structures. Here, we explore the impact of the design parameters of the disordered waveguides on the effective width of localized modes. A more trustworthy result is attained by averaging more than 100 different realizations of the 1D disordered waveguides under the same conditions due to the stochastic nature of the localization process. It should be highlighted that throughout the entire averaging procedure, we only choose the localized modes. Firstly, we need to calculate the effective width of all localized modes take over 100 different individual realizations under the same conditions. Secondly, we sum and average them to calculate the average localized mode width of the 1D disordered waveguides with specific parameters.
Figure 9(a) depicts the average localized mode width of the 1D disordered waveguides with different fill-fraction p as a function of refractive index contrast $\mathrm{\Delta }n$. The statistical data for each disorder level is taken over 100 individual realizations. When the refractive index contrast is lower than 0.5, the polarization characteristic can be ignored. As the refractive index contrast becomes larger, the average localized mode width becomes significantly smaller. The average width of the localized modes varies slowly with fill-fraction of 50% and tends to stabilize as the refractive index contrast increases. The average width of the localized mode directly determines the localization length of the light beam in a disordered waveguide. Here we only show the results more than 30%. In Fig. 9(b), we plot the average localized mode width of the 1D disordered waveguides with different refractive index contrast $\Delta n$ as a function of feature size d. The situation that is the most intriguing for our purposes is when feature size $d\sim \lambda $. There is a good chance that the wave field will tunnel through any potential obstacles and interfere with light that has traveled there by different paths. The effective wavelength is too large relative to the feature size, which will reduce the degree of light scattering. The trend indicates that the influence of refractive index contrast $\Delta n$ on the average width of the localized mode is much larger than feature size d.
Fig. 9. (a) The average localized mode width of the 1D disordered waveguides with different fill-fraction p as a function of refractive index contrast $\mathrm{\Delta }n$. (b) The average localized mode width of the 1D disordered waveguides with different refractive index contrast $\Delta n$ as a function of feature size d. The statistical data for each disorder level is taken over 100 individual realizations.
4. Conclusion
Unquestionably, a useful technique for researching TAL of light is the thorough statistical localized model analysis in the 1D disordered waveguides. Although the localized modes are distributed across the cross-section of the disordered structure, the average width of the modes is not equal to the localization length, but it can effectively characterize the strength of TAL. The localized modes are independent of structural boundary and external excitation, and only are related to the design parameters such as feature size, refractive index contrast and fill-fraction. The results show that high refractive index contrast and fill-fraction result in a smaller average localized mode width. When the feature size is larger in comparison to the wavelength, the average mode width stabilizes with a high refractive index contrast. The same model analysis method will be used to analyze 2D disordered waveguides in the future.
Funding
National Key Technology Research & Development Program (2018YFA0209001, 2021YFB2800203).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]
2. S. John, “Localization of Light,” Phys. Today 44(5), 32–40 (1991). [CrossRef]
3. R. Frank and A. Lubatsch, “Scalar wave propagation in random amplifying media: Influence of localization effects on length and time scales and threshold behavior,” Phys. Rev. A 84(1), 013814 (2011). [CrossRef]
4. A. Lubatsch and R. Frank, “A Self-Consistent Quantum Field Theory for Random Lasing,” Appl. Sci. 9(12), 2477 (2019). [CrossRef]
5. A. D. Mirlin, R. Pnini, and B. Shapiro, “Intensity distribution for waves in disordered media: Deviations from Rayleigh statistics,” Phys. Rev. E 57(6), R6285–R6288 (1998). [CrossRef]
6. H. De Raedt, A. Lagendijk, and P. De Vries, “Transverse localization of light,” Phys. Rev. Lett. 62(1), 47–50 (1989). [CrossRef]
7. 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(12), 2304–2306 (2012). [CrossRef]
8. S. Karbasi, T. Hawkins, J. Ballato, K. W. Koch, and A. Mafi, “Transverse Anderson localization in a disordered glass optical fiber,” Opt. Mater. Express 2(11), 1496–1503 (2012). [CrossRef]
9. S. Karbasi, K. W. Koch, and A. Mafi, “Image transport quality can be improved in disordered waveguides,” Opt. Commun. 311, 72–76 (2013). [CrossRef]
10. S. Karbasi, C. R. Mirr, R. J. Frazier, 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(17), 18692–18706 (2012). [CrossRef]
11. S. Karbasi, R. J. Frazier, C. R. Mirr, K. W. Koch, and A. Mafi, “Fabrication and characterization of disordered polymer optical fibers for transverse Anderson localization of light,” J. Vis. Exp. 77, 50679–50685 (2013). [CrossRef]
12. S. Karbasi, K. W. Koch, and A. Mafi, “Multiple-beam propagation in an Anderson localized optical fiber,” Opt. Express 21(1), 305–313 (2013). [CrossRef]
13. S. Karbasi, R. J. Frazier, K. W. Koch, T. Hawkins, J. Ballato, and A. Mafi, “Image transport through a disordered optical fibre mediated by transverse Anderson localization,” Nat. Commun. 5(1), 3362 (2014). [CrossRef]
14. A. Mafi, J. Ballato, K. W. Koch, and A. Schulzgen, “Disordered Anderson Localization Optical Fibers for Image Transport-A Review,” J. Lightwave Technol. 37(22), 5652–5659 (2019). [CrossRef]
15. 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]
16. S. Karbasi, K. W. Koch, and A. Mafi, “Modal perspective on the transverse Anderson localization of light in disordered optical lattices,” J. Opt. Soc. Am. B 30(6), 1452–1461 (2013). [CrossRef]
17. B. Abaie, S. R. Hosseini, S. Karbasi, and A. Mafi, “Modal analysis of the impact of the boundaries on transverse Anderson localization in a one-dimensional disordered optical lattice,” Opt. Commun. 365, 208–214 (2016). [CrossRef]
18. B. Abaie and A. Mafi, “Modal area statistics for transverse Anderson localization in disordered optical fibers,” Opt. Lett. 43(16), 3834–3837 (2018). [CrossRef]
19. W. P. Huang, C. L. Xu, and S. K. Chaudhuri, “A Finite-Difference Vector Beam Propagation Method for Three-Dimensional Waveguide Structures,” IEEE Photonics Technol. Lett. 4(2), 148–151 (1992). [CrossRef]
20. J. Yamauchi, T. Ando, and H. Nakano, “Propagating beam analysis by alternating-direction implicit finite-difference method,” Electron. Comm. Jpn. Pt. II 75, 54–62 (1992). [CrossRef]
21. J. Chrostowski, W. P. Huang, and S. K. Chaudhuri, “An Unconditionally Stable Vectorial Beam Propagation Method for 3-D Structures,” IEEE Photonics Technol. Lett. 6(4), 549–551 (1994). [CrossRef]
22. W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, “The Finite-Difference Vector Beam Propagation Method: Analysis and Assessment,” J. Lightwave Technol. 10(3), 295–305 (1992). [CrossRef]
23. D. Yevick and B. Hermansson, “Efficient Beam Propagation Techniques,” IEEE J. Quantum Electron. 26(1), 109–112 (1990). [CrossRef]
24. W. P. Huang and C. L. Xu, “Simulation of Three-Dimensional Optical Waveguides by a Full-Vector Beam Propagation Method,” IEEE J. Quantum Electron. 29(10), 2639–2649 (1993). [CrossRef]
25. J. Shibayama, T. Yamazaki, J. Yamauchi, and H. Nakano, “Full-vectorial imaginary-distance BPM analysis of eigenmodes of a light-guiding metal line loaded on a dielectric substrate,” NUSOD ‘05 - Proc. 5th Int. Conf. Numer. Simul. Optoelectron. Devices., 51–52 (2005).
26. S. Jüngling and J. C. Chen, “A Study and Optimization of Eigenmode Calculations Using the Imaginary-Distance Beam-Propagation Method,” IEEE J. Quantum Electron. 30(9), 2098–2105 (1994). [CrossRef]
27. J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
28. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8(5), 649–651 (1996). [CrossRef]
29. D. M. Jović, Y. S. Kivshar, C. Denz, and M. R. Belić, “Anderson localization of light near boundaries of disordered photonic lattices,” Phys. Rev. A 83(3), 033813 (2011). [CrossRef]
