Fiber guiding at the Dirac frequency beyond photonic bandgaps

Abstract

Light trap** within waveguides is a key practice of modern optics, both scientifically and technologically. Photonic crystal fibers traditionally rely on total internal reflection (index-guiding fibers) or a photonic bandgap (photonic-bandgap fibers) to achieve field confinement. Here, we report the discovery of a new light trap** within fibers by the so-called Dirac point of photonic band structures. Our analysis reveals that the Dirac point can establish suppression of radiation losses and consequently a novel guided mode for propagation in photonic crystal fibers. What is known as the Dirac point is a conical singularity of a photonic band structure where wave motion obeys the famous Dirac equation. We find the unexpected phenomenon of wave localization at this point beyond photonic bandgaps. This guiding relies on the Dirac point rather than total internal reflection or photonic bandgaps, thus providing a sort of advancement in conceptual understanding over the traditional fiber guiding. The result presented here demonstrates the discovery of a new type of photonic crystal fibers, with unique characteristics that could lead to new applications in fiber sensors and lasers. The Dirac equation is a special symbol of relativistic quantum mechanics. Because of the similarity between band structures of a solid and a photonic crystal, the discovery of the Dirac-point-induced wave trap** in photonic crystals could provide novel insights into many relativistic quantum effects of the transport phenomena of photons, phonons, and electrons.

Introduction

Optical resonators and waveguides are widely used in modern optics¹, including optical couplers/filters^2,3,4, microlasers^5,6, and mainstream optical fibers^7,8,9,10. Any dramatic change to the building blocks will generate a completely new range of optical devices. The study of the fundamental properties of resonators and waveguides thus attracts much of the current research¹. Field confinement in optical fibers is traditionally based on total internal reflection. A high-index core surrounded by a cladding with a lower refractive index has been the most basic requirement⁷. Recently, a novel class of optical fibers that permit the guidance of light in a low-index core region has emerged^8,9,10. These so-called photonic-bandgap fibers operate through bandgap effects of photonic crystals^11,12, which occur because of periodic microstructuring of the dielectric in the cladding region. Photonic crystals have been studied extensively for their bandgap¹ and special in-band dispersion effects such as negative refraction¹³. However, another really interesting feature of photonic crystals is the Dirac points that appear at corners of the Brillouin zone^14,15,16,

Figure 1.

Cross-section and band structures of photonic crystal fiber. (a) A typical cross-section of photonic crystal fibers, where the lattice constant is a=2.21 μm, the hole radius is r_a=0.47a, and the central hollow defect radius is R=1.9a. The fiber is made of either silica glass (ε_r=1.45²), germania glass (ε_r=1.59²), or SF6 glass (ε_r=1.8²). The inset in a is the first Brillouin zone (shaded light gray) of the reciprocal lattice for the triangular lattice of air holes, revealing the high-symmetry points Γ, M, K at the corners of the irreducible Brillouin zone (shaded dark gray). (b–d) Band structures of the fiber cladding made of germania glass, at k_za/2π=0.001 (b), k_za/2π=1.581 (c), and k_za/2π=1.910 (d). The four lowest order Dirac points are indicated by red arrows. A well-isolated, wide-open Dirac spectrum is shown in d in the band structure for k_za/2π=1.910. The inset to d shows an enlarged three-dimensional view around the second Dirac point, where the two bands touch as a pair of linear Dirac cones at the six Brillouin zone corners.

Materials and methods

Calculations of band structures by the plane wave expansion method

A typical cross-section of the photonic crystal fiber is shown in Figure 1a, where the first Brillouin zone of the corresponding reciprocal lattice is shown as an inset. The fiber is formed from either silica glass (ε_r=1.45²), germania glass (ε_r=1.59²), or SF6 glass (ε_r=1.8²). The cladding consists of a triangular lattice of air holes (r_a=0.47a), where a=2.21 μm is the center-to-center distance between the two nearest holes. A central hollow defect (R=1.9a) serves as the core of the fiber. The associated band structures of the triangular lattice for three different k_z are calculated by the plane wave expansion (PWE) method¹, and the results are presented in Figure 1b–1d. Variations of the Dirac point versus k_z are plotted in Figure 2a–2c for the three different fibers. Guided modes of the fibers are also studied by the PWE method, where a 14 × 16 supercell is used for the calculation to include the fiber core as a defect of the photonic crystal. The results for the fundamental guided modes are presented in Figure 2a–2c as red diamonds. Some details of the mode profiles are given in Figure 3. As a very basic property of photonic crystal fibers, the fundamental mode consists of an approximately degenerate pair of orthogonal, quasi-linearly polarized states⁹.

Figure 2

Propagation diagrams and transmission spectra of photonic crystal fibers. All the fibers have the same cross-section as that shown in Figure 1a, where a=2.21 μm, r_a=0.47a, and R=1.9a. (a–c) Propagation diagrams for the silica glass fiber (a, ε_r=1.45²), germania glass fiber (b, ε_r=1.59²), and SF6 glass fiber (c, ε_r=1.8²). The solid red line represents the Dirac frequency of the fiber cladding. The dashed red lines mark the upper and lower edges of the Dirac spectrum. The black line is the theoretically calculated dispersion curve of the fundamental fiber mode obtained using the analytical model (1). The red diamonds are the numerically calculated data points of the fundamental fiber mode obtained using the PWE method. Field distributions of the modes at points 3a–3e, indicated by larger diamonds in b, are presented in Figure 3a–3e. The blue curves mark the edges of the photonic bandgaps. The green line represents the light line k_z=ω/c of air. (d–f) Numerically simulated (by FDTD) transmission spectra of narrow pulses passing through a section (1200 μm in length) of the silica glass fiber (d), germania glass fiber (e), and SF6 glass fiber (f). Propagation scenarios of modes at the four points 4a–4d, marked by red crosses in e, are presented in Figure 4a–4d.

Figure 3

Guided modes in a triangular lattice photonic crystal fiber of germania glass. The fiber structure is shown in Figure 1a, where a=2.21 μm, r_a=0.47a, R=1.9a, and ε_r=1.59². The modes were numerically calculated using the PWE method. The light-blue circles in the dark blue background give the positions of the hole−glass interfaces. (a–e) correspond to the red diamonds labeled 3a–3e in Figure 2b: (a) a gap mode (ωa/2πc= 1.432), (b) a mode at a frequency between the photonic band gap and the lower edge of the Dirac spectrum (ωa/2πc=1.601), (c) the Dirac mode (ωa/2πc=1.689), (d) a mode in the Dirac spectrum (ωa/2πc=1.761), and (e) a mode at a frequency higher than the upper edge of the Dirac spectrum (ωa/2πc=1.950). The electric-field pattern |E| is shown on the left, and the product r^3/2|E| or r|E| on the x- or y-axis is shown on the right.

Finite-difference time-domain (FDTD) numerical simulations

In nonmagnetic media, a complete description of electromagnetic waves in time and space is given by the Maxwell equations . The Maxwell equations can be solved directly by the well-known FDTD method¹⁹ without invoking the functional form exp(iωt−ik_zz). A three-dimensional FDTD model was established with a perfectly matched layer boundary condition on all the edges. The structure is uniform in the longitudinal direction, extending from z=0 μm to z=1200 μm. On the transverse plane, the structure has a profile of the refractive index given by Figure 1a, with an area of 66.3 μm × 65.07 μm to include 30 layers and 34 layers in the x and y directions, respectively. The step sizes in space are Δx=Δz=a/20=0.1105 μm and Δy=a×√3/36≈0.1063 μm. The step size in time is Δt=0.1995 fs, which satisfies the Courant–Friedrichs–Ley condition cΔt≤(1/Δx²+1/Δy²+1/Δz²)^−1/2, where c is the velocity of light. For the study of transmission spectrum, a broadband Gaussian pulse is input from the front end of the fiber. The incident pulse has a Gaussian amplitude profile in the transverse plane, with a waist radius equal to the central hollow core radius of the fiber. The incident pulse has a finite duration in time, with a Gaussian spectrum centered at the Dirac frequency ω_D/2π and a frequency span of 0.5ω_D/2π. This wave is simulated to evolve in the 1200-μm-long germania glass fiber for 8000 fs, which is long enough for the waves to pass through the structure in all directions. The full-time response at the rear end of the fiber is recorded. The transmission spectra are obtained from the time-domain response by Fourier transform. The outcomes presented in Figure 2d–2f reveal peaks at frequencies inside the bandgap and at the Dirac frequency. To investigate evolutions of the different frequency components in detail, waves at specific frequencies are picked out for further simulations. For this purpose, the incident field of a continuous Gaussian beam (single harmonic) is used. The waist radius of the beam in the transverse plane is again set equal to the radius of the fiber core. The wave is simulated to evolve for long enough time to reach a steady state. The field at the end of the simulation (t=8000 fs) is recorded by a monitor in the yz cut plane, and the results are presented in Figure 4 for four different frequency components. Leakage-free propagation of a guided mode at the Dirac frequency is visualized, confirming the transmission spectrum of Figure 2e.

Figure 4

The steady-state electric fields in the germania fiber at the end of the FDTD simulations. The fiber has the cross-section structure shown in Figure 1a, where a=2.21 μm, r_a=0.47a, R=1.9a, and ε_r=1.59². The |E| fields are recorded at t=8000 fs and displayed by a monitor in the yz cut plane that runs directly through the center axis of the fiber at x=0, ranging from z=0 to z=1200 μm in the z-direction and from y=–32.535 to y=32.535 μm in the y-direction. (a–d) correspond to the four red crosses labeled 4a–4d in Figure 2e: (a) |E| field for ωa/2πc= 1.432, (b) |E| field for ωa/2πc=1.624, (c) |E| field for ωa/2πc=1.694, and (d) |E| field for ωa/2πc=1.724.

Results and discussions

Dirac cone for fiber-like propagation

The fiber problem is different from the two-dimensional problem addressed in Ref. 21; thus, outgoing waves are forbidden in the surrounding medium, and field confinement to the center hollow core becomes possible. Because the zero density of states at the Dirac frequency is a special feature of the band structure, and the band structure itself is a consequence of Bragg scattering of light by the photonic crystal lattice, at a deeper level of understanding, the localization phenomenon observed at the Dirac frequency is essentially due to Bragg scattering, which determines the overall band structure. Rather than leading to a complete gap, however, we find an isolated conical singularity where the band gap is vanishingly small. The key feature of this band structure is that the two bands touch as a pair of cones at corners of the Brillouin zone in a linear fashion, with linear dispersion and a linearly vanishing density of states in its vicinity. One would expect to see a true localized mode precisely at the point where the density of states is zero and a guided resonance with finite lifetime away from it. The fact that there is not a band of true localized modes around the Dirac frequency gives the mode special properties such as an algebraic profile and sharp frequency response. This light-trap** phenomenon occurs at a single frequency and should be distinguished from a recent observation of an ‘embedded eigenvalue’^22,23 that also occurs at a single frequency. The existence of the Dirac mode is due to disappearance of radiation states at the Dirac frequency, whereas the trapped light in Refs. 22 and 23 occurs in the presence of radiation states; however, somehow, mutual coupling of the trapped light and radiation states is suppressed by destructive interference. In light of this discussion, the two phenomena are different in how they break up coupling of the trapped light and radiation states.

Numerical confirmation of the guided mode at the Dirac frequency

To confirm the existence of a trapped state at the Dirac frequency, numerical tools²⁴ based on the FDTD method¹⁹ were adopted to simulate the evolutions of waves passing through a section (1200 μm long) of the germania glass fiber. In Figure 4a, a typical gap mode propagating over this distance is presented. The oscillation in amplitude is due to a mismatch between the input wave and the eigenmode of the fiber. When the frequency goes beyond the bandgap, the mode still concentrates much of its field energy in the core. Nevertheless, the mode is no longer a true-bound state but a guided resonance that resembles the so-called ARROW mode^25,26. Like the ARROW mode of a cylindrical waveguide, this mode has a |E|∝1/√r asymptotic profile in space. It is a radiative mode itself (∫|E|²rdrdθ→∞) and is therefore inherently leaky. An example of the guided resonance is shown in Figure 4b. Further away from the bandgap, the resonance is less concentrated within the center core until the resonance broadens away and melts into the continuum. As the localized mode penetrates into the region of the Dirac spectrum, the density of radiation states starts to decrease because of the presence of the Dirac point. Guided resonance similar to that observed in the neighborhood of a bandgap makes an appearance in the Dirac spectrum. The mode, such as that shown in Figure 4c, for example, remains leaky because it is situated inside a band of a continuum of states and mutual coupling is inevitable. When the frequency reaches the Dirac frequency, the density of radiation states vanishes; thus, coupling of the Dirac mode and the radiation states disappears completely. The profile of the mode thus approaches |E|∝r^−3/2, which is square integrable, indicating that the mode becomes trapped again. Mode propagation at the Dirac frequency is shown in Figure 4d, an almost leakage-free propagation is achieved. The leakage-free operational window of the fiber can be best visualized by a transmission spectrum of the fiber itself, as shown in Figure 2d–2f, where the Dirac frequency corresponds to a sharp transmission peak in each case. The simulations presented in Figure 4a–4d correspond to the four points 4a–4d marked by red crosses in Figure 2e. From the observation of the low and high transmissivity and the corresponding locations on the transmission spectrum, Figure 4a–4d is consistent with behaviors expected at the points 4a–4d; therefore, the propagation scenarios of these modes are understood. This finding represents an important discovery, based on which a new type of fiber guiding is established at the Dirac frequency beyond a bandgap.

If the lattice extends to infinity and material absorption is ignored, the Dirac mode in Figure 4d and the gap mode in Figure 4a are both lossless. In practice, however, a real lattice has a finite size, which results in a loss that is associated with field penetration across the boundary into the surrounding medium. Because the tail of the gap mode decays exponentially, whereas the tail of the Dirac mode decays algebraically with distance, for the same lattice extent, the residual field of the Dirac mode on the boundary is stronger than that of the gap mode. Consequently, the Dirac mode is more lossy than a gap mode in general. Because of the leakage to the surrounding medium, propagation at a transmission peak is not truly leakage-free in a fiber of finite cross section, but, as shown in Figure 4a and 4d, declines slowly with distance following an exponential law P=P₀e^−αz for its power. For the germania glass fiber, the attenuation constant α is estimated to be 1.5 × 10⁻⁴ μm⁻¹ for the Dirac mode and ∼0 for the gap mode. Fortunately, the loss due to the finite lattice size can be made as small as desired by increasing the number of surrounding crystal periods.

Practical fiber design, experimental validation, and applications

The design concern for the new proposed fiber is quite similar to the design concern of an ordinary photonic crystal fiber. The complexity of the photonic crystal structure denies an analytical solution; thus, a lot of computer software (see Ref. 24 for some examples) has been developed to facilitate the design of practical photonic crystal fibers. This commercially available software can be directly applied to the design activity of the new proposed fiber because it still belongs to the group of photonic crystal fibers. The only difference is a shift of attention from the bandgap to the Dirac point. To be specific, the first concern of the design is whether a Dirac mode can be supported by a fiber with given parameters. The band structure of a photonic crystal fiber can be calculated using one of the available commercial software products or using a simple MATLAB code based on the PWE method¹. Our studies demonstrate that the Dirac mode is not well supported in a photonic crystal fiber if the refractive index contrast is as small as that of a silica/air (1.45−1) system. We confirm that photonic crystal fibers made of SF6 glass (n_r=1.8) or germania glass (n_r=1.59) are suitable for the Dirac mode propagations. From these studies, the constituent materials of the fiber can be selected, and the air hole radius can be optimized. The second concern for the design is the precise position of the Dirac frequency. As shown in Figure 2b–2c, the curves of the Dirac frequency and the dispersion intersect at ωa/2πc≈1.689 for the germania glass fiber and at 1.415 for the SF6 glass fiber. The frequency at the intersection point can be used as a rough estimate of the operational frequency for leakage-free propagation of the Dirac mode. For the lattice constant a=2.21 μm, these frequencies correspond to wavelengths of approximately 1308 and 1562 nm, respectively. The selected fiber materials, germania and SF6, are transparent at these wavelengths. Note that the Dirac frequency can be scaled to any desired value by adjusting the lattice period a.

Experimental validation of the Dirac mode can be established by the identification of the Dirac transmission peak on spectra measured on the specially designed photonic crystal fibers. From numerical simulations the Dirac peak shows up nicely in transmission spectra of photonic crystal fibers made of SF6 glass or germania glass. The above two fibers are therefore potential media for Dirac peak transmission measurements. In addition, polymer fiber is easy to handle and widely available. The refractive index of polystyrene, which is 1.590, is very close to that of germania glass, so polystyrene fiber is a convenient alternative for the Dirac transmission peak measurement. Experimental verification of the theory presented here is handled smoothly, and this will be the subject of a later paper.

Photonic crystal fibers were traditionally divided into two broad classes, namely, index-guiding fibers and photonic-bandgap fibers, depending on whether they used index guiding or photonic bandgaps for optical confinement. Our finding of Dirac point guiding enriches the varieties of photonic crystal fibers. Bandgap confinement is attractive because it allows light to be guided within a hollow core. This capability not only minimizes the effects of material absorptions, undesired nonlinearities, and any other unwanted properties of the bulk materials but also facilitates applications in biological sensors or organic light-emitting devices, where enhancement of laser interactions with dilute or gaseous media is required. Our new proposed fiber also allows light to be guided within a hollow core; thus, any benefit a photonic-bandgap fiber can offer is also available in the Dirac fiber. If photonic-bandgap fibers already exist; what is the purpose of exploring a new type of guiding phenomenon? The first reason is simple curiosity: the idea of using a Dirac point to trap light in a hollow core is intriguing. Second, the Dirac-point-induced guiding does have unique features. The Dirac mode is different from the gap mode in terms of the resulting frequency characteristics and spatial profile features. Our new proposed fiber can therefore be regarded as a supplement to existing photonic-bandgap fibers. For example, the unique power-law profile of the Dirac mode results in loose lateral confinement, thus enabling long-range interaction between multiple cores. This feature could lead to the creation of new coupling and sensing characteristics for use in optical couplers, laser arrays, and fiber sensors. The sharp frequency response of the Dirac mode can take on the role of a resonant cavity with the establishment of frequency selectivity in applications of coherent random fiber lasers, where resonant cavities are not available but where frequency selectivity is desired²⁷.

Conclusions

A new type of light trap** was demonstrated in fibers at the Dirac frequency in the absence of a bandgap and total internal reflection. This discovery will extend the possibilities of fiber optics and will lead to many new applications in fiber lasing/sensing systems. We believe that the effects discovered in the present study for electromagnetic waves in photonic crystals could occur in other wave phenomenon systems, including acoustic waves in phononic crystals and matter waves in semiconductors. A further search for different systems with the extraordinary localization of photons, as well as phonons and electrons, can expand the applicability of our results.