Topological materials have been at the forefront of research across various fields of physics in hopes of harnessing properties such as scatter-free transport due to protection from defects and disorder. Photonic systems are ideal test beds for topological models and seek to profit from the idea of topological robustness for applications. Recent progress in 3D-printing of microscopic structures has allowed for a range of implementations of topological systems. We review recent work on topological models realized particularly in photonic crystals and waveguide arrays fabricated by 3D micro-printing. The opportunities that this technique provides are a result of its facility to tune the refractive index, compatibility with infiltration methods, and its ability to fabricate a wide range of flexible geometries.
Topological insulators are materials that are insulating in their bulk but conduct current along their edges without back-scattering even in the presence of disorder and defects. The first manifestation of topology was shown in the quantum Hall effect1 where an electron gas restricted to two dimensions in a magnetic field introduces Landau levels leading to energy gaps. When the Fermi energy lies in such a gap, the conduction only stems from the contribution of edge channels and is pinned to a quantized value that is independent of the amount of disorder. This novel phenomenon can be understood within a branch of mathematics called topology that deals with properties of geometric objects that are preserved under continuous transformations.
In physical systems, topology manifests as a property of the eigenstates of the governing Hamiltonian. The eigenstates of a periodic system that depend on a few independent parameters can be parallelly transported along a closed contour in that space of parameters. On doing so, the eigenstates may acquire a non-zero measurable phase that is connected to a topological invariant. According to the bulk-boundary correspondence, the non-trivial topological invariant of the bulk is linked to a range of rich physical phenomena that manifest on the boundaries of a finite-sized sample. The robustness of such edge states is the most accessible and, thus, the most important feature for applications of topology since the edge states persist even in deformed or perturbed systems, as long as the topology of the system is not changed. More recently, it has been shown that the same principles can be applied to photonics to achieve novel and unusual transport properties of light, resulting in the research area of topological photonics.2–6 In photonics, topological robustness or unidirectional transport is sought to be used in optical devices, making them less prone to local fabrication imperfections, e.g., for robust data transfer7,8 or single-mode laser light sources.9–15
Photonic platforms are ideal test beds for theoretical models, and due to the ubiquitousness of topology across systems, many different photonic platforms have emerged. Among these are various waveguide systems,16 photonic crystals,17–20 ring resonators,10,21 gyromagnetic rods,22 fiber loops,23,24 polaritons,25 and many more.
Adding to that toolbox of platforms, recently, structures fabricated by 3D micro-printing have come into play. This fabrication method uses multi-photon polymerization to create three-dimensional structures on the micrometer scale.26 Industry solutions for micro-printing include Nanoscribe, Multiphoton Optics, UPnano, and Femtika, to name a few. Furthermore, home-built micro-printing setups can also be found in research groups across various universities. We will henceforth focus on the Nanoscribe Photonic Professional GT2 since it is the most widely used system.
The Nanoscribe is a commercially available “ready-to-play” instrument that can be found in many universities today. The ability of the Nanoscribe to fabricate versatile structures spanning orders of magnitude has resulted in a wide range of applications (for an exhaustive overview, we direct the reader to the website27). Particularly for topological photonics, the Nanoscribe has been used to fabricate two types of structures: photonic crystals and waveguide systems. Therefore, this paper is organized as follows: We will first give a short introduction into topological insulators (Sec. II) and the working principle of the Nanoscribe (Sec. III). Then, we will review the topological features that have been observed in photonic crystals (Sec. IV) and waveguide systems (Sec. V) fabricated using the Nanoscribe. At the end, we will comment on recent advances, upcoming approaches, and possible future directions (Sec. VI).
II. TOPOLOGICAL INSULATORS
An insulating system is described by a Hamiltonian H that has a bandgap in its eigenvalue spectrum of some generalized energy E (Fig. 1). The eigenmodes of this Hamiltonian can be characterized by an integer topological invariant that is a global property over the entire momentum space. Under continuous transformation to a different system described by the Hamiltonian H′, this invariant cannot change unless this transformation closes and re-opens the gap. Thus, two Hamiltonians, related by a transformation that keeps the bandgap open, are said to be in the same topological phase. The prototypical example of a topological invariant, the Chern number, can be found in two-dimensional systems in the absence of time-reversal symmetry. At an interface between a topologically trivial and a non-trivial material, the gap must close in real-space at the boundary between the two materials, resulting in the presence of edge states [Figs. 1(a) and 1(b)]. These edge states are prohibited from scattering into the bulk by the gap on both sides of the interface and are exponentially confined to the boundary. They are also protected from back-scattering when their slope is monotonic with respect to momentum and when coupling between edge states with different slopes is forbidden.
Besides the Chern number, a plethora of topological invariants are known to exist, depending on the dimensionality of the system and the presence of additional symmetries.16,28 This leads to novel symmetry-protected topological phases that are only distinguishable from trivial phases when the presence of certain symmetries is enforced. Furthermore, some band degeneracies can also be associated with invariants that result in topological semi-metals.
Since the concept of topology is quite universal, it can be applied across many physical systems. Raghu and Haldane transferred the principles of topological protection to photonics, proposing a photonic analog of the quantum Hall effect.29 While the quantum Hall effect relies on the action of a uniform magnetic field on electrons, in photonic systems, a similar action on uncharged photons needs to be found. This can be done employing the so-called synthetic30–32 or artificial gauge fields (AGFs).33,34 These AGFs control the dynamics of photons such that they behave as if effective external fields were acting on them. AGFs can be engineered by geometric or periodic time-dependent modulations (Floquet modulations). Floquet modulations impose periodic boundary conditions on the energy such that the energy bands of the static systems are now being replicated at energies nℏω, with n being an integer and ω being the frequency of the modulation. This way, states that were separated in energy in the static system can now hybridize when the driving frequency matches their energetic separation [Fig. 1(c)]. Therefore, Floquet modulations can induce topological phases from systems that are topologically trivial in the static case.
III. THE NANOSCRIBE
The Nanoscribe uses near infrared femtosecond laser pulses to cross-link a liquid photoresist via two-photon polymerization.35–41 Typical parameters of the laser used are a pulse duration of 100 fs at a repetition rate of 80 MHz and 780 nm wavelength. The laser light is focused by an objective into the resist applied onto a substrate. Using a 63×, NA 1.4 objective with an aperture of 7.3 mm, an average laser power of less than 50 mW at the entrance pupil of the objective is conventionally deployed for printing.42 Absorption of two photons within a very short time-span in the laser’s focal volume triggers polymerization of the resist. This causes only the material within a certain volume to be cross-linked and solidified, allowing us to fabricate elaborate three-dimensional structures by moving the laser focus with respect to the resist. Using galvanometric mirrors to move the laser focus allows for high writing speeds of up to 20 mm/s. The cross-linking of the molecules leads to a change in the material’s refractive index and solubility in organic solvents. After washing off the un-cured resist in a development step, solid structures remain. Fabricated structures have a minimum feature size of about 160 nm in the x–y direction and 1 µm in the vertical direction (when using the 63× objective), while mesoscopic options allow for up to 8 mm object height.27
IV. PHOTONIC CRYSTALS
Historically, the Nanoscribe was developed to fabricate 3D photonic crystals.37,39 Such crystals are described by using the full Maxwell equations with a spatially periodic dielectric function. While a great variety of photonic crystals have been fabricated using the Nanoscribe alone39,55–58 and in combination with coating or infiltration techniques,18,59–68 only recently have topological structures been examined. 3D photonic crystals host many of the same topological phenomena as conventional solids and are known to host topological degeneracies in momentum space, such as Dirac points, Weyl points, and nodal lines.17
Dirac points and nodal lines are point and line degeneracies, respectively, that occur in the presence of certain symmetries that are required to prevent a gap from opening at the degenerate momenta. In contrast, Weyl points are robust degeneracies that are protected by an integer topological invariant or charge. In other words, any arbitrary periodicity-preserving perturbation merely displaces the Weyl point in momentum space but cannot cause a gap to open at the Weyl point. Furthermore, higher charged Weyl points can exist in the presence of additional spatial symmetries, which cause Weyl points of the same charge to overlap. Charge-2 Weyl points have recently been observed in a chiral woodpile photonic crystal fabricated using the Nanoscribe at low refractive index contrast19 [Figs. 2(a) and 2(b)]. At such a low refractive index contrast, there is no complete bandgap surrounding the Weyl point since the projected bands of a finite structure overlap. Nevertheless, the dispersion of the bands forming the Weyl point can be directly observed in the reflection spectrum of the photonic crystal. This is possible since s- and p-polarized light couples selectively to the bands, providing a way to map out the feature of interest.
Due to the bulk-boundary correspondence, Weyl points are associated with Fermi-arc states that reside on the surface of a finite sample whose dispersion connects Weyl points of opposite charge. A measurement of such Fermi-arc states in photonics would allow for a direct observation of the topological charge of the Weyl point. In the photonic crystal sample made of the photoresist only, projections of other bulk bands do not allow for a clear measurement of the associated Fermi-arc surface states. However, increasing the refractive index using infiltration techniques will likely allow for a clear observation of the surface states in the spectrum.
Weyl points can also be found in gyroid photonic crystals.7,17 While many gyroid photonic crystals have been fabricated using the Nanoscribe,69,70 Weyl points have not been directly observed in such structures consisting only of the bare photoresist. However, Nanoscribe-made photonic crystals can also be coated with or transformed into high-index material, such as titania65 or silicon,59,60,66 and also metals.62,63 Several processes have been developed for that including the use of chemical vapor deposition (CVD) and/or atomic layer deposition (ALD).67 Peng et al. reported on the fabrication of a single gyroid by direct laser writing, subsequent coating with a thick layer of Al2O3 via ALD, and infiltration with silicon using CVD.71 Goi et al. were able to observe the signature of charge-1 Weyl points in direct laser written gyroid crystals coated with antimony telluride by ALD18 [Figs. 2(c) and 2(d)]. The same group also developed a protocol to increase the effective refractive index of a gyroid photonic crystal by 40% using charged layer-by-layer deposition of PbS thin films.64 Gyroids have also been fabricated in chalcogenide glasses72 and out of a resist–silver composite by electroless deposition.73
Photonic crystals with Dirac and linear Weyl points exhibit vanishing density of states at the degeneracies and as such can potentially be used to realize large-volume single-mode lasing.74,75 Furthermore, such degeneracies can be used to achieve long-range algebraic interactions between embedded quantum emitters.76 In Nanoscribe structures, this can be readily achieved by dissolving dyes and quantum dots in the photoresist before writing.49–54 While the usual dimensions of photonic crystals that can be fabricated with the Nanoscribe exhibit features in the near infrared wavelength range, several mechanisms have been examined to shift them to smaller wavelengths. Among these are stimulated emission depletion lithography77 and controlled post-printing shrinking.78–80
The aforementioned woodpile photonic crystals with Weyl points can also be realized using air-in-metal designs that are reasonably described by using a tight-binding model.81 More complicated 3D metallic structures can be realized using the Nanoscribe by either printing a template structure, which is then coated with a metal using vapor deposition or inverted using electroplating techniques,63 or by direct micro-printing of metallic structures.43–46 This allows for tight-binding models with unusual properties such as higher-order topology to be directly implemented using the photonic crystals platform.
Losses and gain are inherent to many optical systems, and the effects of non-Hermiticity on the topological properties of photonic crystals are yet to be sufficiently explored. Recently, it was experimentally shown that a photonic Dirac point transforms into an exceptional ring in the presence of losses. The experiment showing the existence of this Dirac exceptional ring was performed in a photonic crystal slab made out of silicon nitride;82 however, such phenomena should exist at any refractive index contrast in the presence of sufficiently large non-Hermiticity. Moreover, the photoresists commonly used with the Nanoscribe exhibit vibrational resonances in the mid-IR, which could be used to further explore topological effects in the vicinity of these lossy resonances.
V. WAVEGUIDE SYSTEMS
A wide range of topological phenomena has been examined in waveguide systems.16 Light propagation through an array of waveguides can act as a model for quantum mechanical systems due to the mathematical analogy between the paraxial approximation of the Helmholtz equation that governs light propagation in a waveguide and the Schrödinger equation that describes the evolution of a quantum mechanical state.83,84 In solids where the electrons are strongly bound to atoms, the system can be accurately described by a tight-binding Hamiltonian. In photonics, this analogously extends to waveguide arrays where light is strongly confined to the waveguides due to their higher refractive index than the medium in which they are embedded.
The eigenmodes of a cylindrical optical waveguide are transverse electromagnetic modes, with a corresponding propagation constant β. The fundamental mode, i.e., the one with highest β, has a Gaussian shape, and its propagation constant can be written as β0 = 2πneff/λ, with the effective refractive index neff and the wavelength λ. The propagation constant β0 thus is proportional to the inverse of the length along z over which the phase of the wave changes by 2π. It is equivalent to the eigenenergy of an atomic wave function at its own lattice site. In a waveguide, light can be guided inside the core region as long as 2πn0/λ < β0 < 2πnwg/λ, with nwg and n0 being the refractive index of the core and of the surrounding material, respectively. Outside the core material, the amplitude of the light is decaying exponentially. If a second waveguide is placed in close vicinity to the first one, it can pick up the decaying field such that the light couples to this second waveguide while propagating along z. This is called evanescent coupling.
For an array of N coupled equal single-mode waveguides, we can describe the evolution of the light inside the waveguide array by the field amplitudes aq(z) of the eigenmodes in each waveguide q along the propagation direction z with N coupled differential equations,85,86
where the sum is only over nearest neighbors. The coupling cp,q between two waveguides, at distances dx and dy from each other in the x and y direction, is given by the overlap of the transverse field modes and ,
assuming that at waveguide q, the refractive index contrast Δnq = nwg − n0 is small.
In such systems, a single waveguide corresponds to an atomic site, while the waveguide axis z acts analogously to the time variable in the Schrödinger equation. The energy in electronic systems is replaced by the waveguides’ propagation constants β. For an array of waveguides, we thus obtain a band structure of β over the transverse wavevector k. The intensity distribution along z over waveguide sites is equivalent to the time evolution of the probability density of an electron in the periodic potential of a solid state material. It is possible to map this intensity distribution by exciting a certain state at the waveguide array’s input facet and observing the intensity distribution at the output facet. Reciprocal space is also accessible in both addressing the waveguides by the use of a spatial light modulator to define the phase at the input facet and imaging momentum space by introducing an additional lens after the output facet to perform a Fourier-transformation87 [Fig. 3(d)].
Existing methods to fabricate waveguide arrays are fs-writing in glass,84,88–90 surface plasmon polariton waveguides,91,92 optical induction of waveguides in photorefractives,93,94 etc. The Nanoscribe not only broadens the range of accessible systems to do experiments but also has some unique advantages (see Secs. V A, V B, and VI). So far, two different methods are used to create waveguide arrays with the Nanoscribe, explained in Secs. V A and V B.
A. Infiltrated waveguides
The inverse of a waveguide array is printed with the Nanoscribe [Fig. 3(a)], which leads to an array of empty channels after development of the uncured photoresist [Fig. 3(b)]. These empty channels are then infiltrated with a second photoresist, e.g., SU8, with the help of capillary forces [Fig. 3(c)]. The sample is baked on a hotplate to solidify the infiltrated resist. Using IP-Dip as a resist during printing and SU8 as an infiltration material, a refractive index contrast of about 0.05 between the waveguide core and the surrounding material is achieved. The big advantage of this method is the flexibility to choose different infiltration materials (see also Sec. VI A) as well as mixing dyes into the resist.51,54 Typical single-mode waveguides are ≥1 µm in diameter, of lengths up to 1 mm, and have a center-to-center distance to their neighbors of about 1.4 µm. While this fabrication method leads to waveguides with a round cross section—since the writing laser focus is only elongated along the z axis, the propagation axis of the waveguides—the structures are very sensitive to parameters during printing. For example, waveguide channels with a diameter smaller than 1 µm or printed using too high laser power tend to clog during development. This is due to the chemical polymerization process: During printing, photo-initiator molecules that are excited in the area surrounding the empty channels can dissipate into the cross-sectional area of the channels and start polymerization reactions there, even though that area has not been illuminated.96 These reactions can lead to polymer filaments inside the waveguide channels, hindering the infiltration or causing the complete clogging of a waveguide. To minimize this effect, the waveguide radius needs to be ≥1 µm and the structure is written with a laser intensity close to the polymerization threshold to reduce the concentration of excited photo-initiator molecules. In addition, employing aberration pre-compensation of the laser focus used for two-photon polymerization via a spatial light modulator97,98 leads to some improvement, as it further confines the laser focus and concentrates the polymerization reaction to a smaller volume. To avoid deformations during the development process caused by the shrinking of structures written with a low laser intensity, additional supports can be written around the structure [Fig. 3(b-ii)].
The combination of SU8 as a waveguide material and IP-Dip as a surrounding material leads to a refractive index contrast of 0.05, as opposed to a contrast of about 7 · 10−4 for waveguide fabrication in glass.84 Therefore, the light is bound more strongly to the waveguide core, which allows for a tighter bending of helical waveguides. For waveguides with a center-to-center distance of 1.5 µm, the coupling is relatively strong with a hopping distance of around 60 µm,95 which reduces the required overall length z of a sample that is necessary in order to observe enough of the propagation dynamics. This is to our advantage since the biggest limitation of this fabrication method lies in the realizable propagation length. As the infiltration process relies on capillary forces, the height of a structure is limited to a maximum of 1 mm (for which the parameters during printing need to be exactly fine-tuned to allow infiltration), while a height of 500 µm is, in general, easily infiltratable.
2. Implementations using infiltrated waveguide arrays
Waveguide arrays fabricated by infiltration were first used to recreate95 a photonic Floquet topological insulator (FTI).99 The photonic FTI consists of waveguides arranged on a honeycomb lattice [Fig. 3(b)] whose trajectory along z follows a helical path. The helical spin of the waveguides effectively leads to a phase term in the coupling constant, making it complex, such that the system mimics the topological Haldane model.100 Mathematically, this phase term can be derived by writing the tight-binding equations in the coordinate system co-rotating with the waveguides to obtain an artificial vector potential,99 followed by a Peierls substitution.101 The arrangement resembles a graphene lattice in a circularly polarized electric AC-field. Due to the breaking of time-reversal symmetry by this artificial field, the system supports a chiral topological mode that moves along the edge of the array only in one direction (only clockwise or only counter-clockwise). Since the topological edge state has a monotone group velocity at each edge, it cannot back-scatter when it encounters a defect, such as a missing waveguide or a corner. In addition, no scattering to the bulk waveguides is possible since the edge state’s energy lies within a bandgap. To test this robustness against “time”-dependent defects, waveguides with different trajectories along z than the rest of the sites (e.g., opposite helicity or a straight waveguide) were put on the edge to see whether the edge mode still moves around these [Fig. 4(a)]. It was observed that as long as the defect’s rotation-frequency in z is the same as the other waveguides’, the edge mode is robust. FTIs fabricated this way were also sought to be used as a sensor of changes in the refractive index of the infiltrating liquid in the waveguide cores.102
A one-dimensional implementation used the Su–Schrieffer–Heeger (SSH) model and its associated topological edge states.103 In the middle of the SSH chain, a defect is created by interfacing a topologically trivial with a non-trivial chain such that the resulting defect waveguide hosts a topological edge state.104 The waveguide that constitutes that defect is modulated in z (“time”), generating Floquet copies of the edge mode in kz (i.e., “energy”). When the frequency of the defect modulation is such that the kz of a Floquet copy of this edge mode coincides with a bulk band, light guided in the defect waveguide can couple out into the lattice. Otherwise, it is trapped.105
As stated before, topological photonics relies on the creation of artificial gauge fields.33,34 AGFs in waveguide systems can be created by altering the trajectories of waveguides along z. In a waveguide implementation, it was shown that at a boundary between two regions with different AGFs, refraction and reflection occurs, even though the two regions are otherwise of exactly the same photonic medium. Reflection and refraction at such “gauge interfaces” are governed by a generalized Snell’s law.87 One can measure the states’ population in momentum space of the incoming, refracted and reflected light by Fourier-transforming the electric field at the output facet with a lens [Fig. 4(b)].
Despite the range of implementations using infiltrated waveguide arrays, the infiltration method is not applicable to all systems. Some applications or models require longer propagation lengths. This calls for longer waveguides that are tricky to obtain using this infiltration method due to limited stability for very long structures and an increase in difficulty during infiltration for long structures. Additionally, some implementations require the ability to modulate the refractive index profile of a waveguide along its cross section or the effective refractive index of a waveguide along z. This is readily possible using the fabrication method described in Sec. V B.
B. Direct printing of waveguides by using differing laser power
With the Nanoscribe, the refractive index of the fabricated structures depends on the amount of cross-linking achieved in the resist during the writing process. In turn, the amount of cross-linking depends on the dose and thus the laser power of the beam that is focused into the resist.108,109 This way, one can print the core of a waveguide using a higher laser power than for the surrounding material and achieve a refractive index difference of up to 0.008. Skipping the development step after printing and post-curing the excess resist under UV light leads to waveguides of very long lengths (up to 8 mm length have been fabricated so far), only limited by the Nanoscribe’s z-drive range.
2. Implementations of directly printed waveguides
Using to advantage the large lengths of the waveguides possible with this fabrication method, negative next-nearest neighbor (NNN) coupling was shown to exist in angled 1D arrays of coupled waveguides106 [Fig. 5(a)]. Adjusting the angle of a zig-zag chain of waveguides, the value and sign of the NNN-coupling c2 can be tuned. For positive values of c2, the spreading along the array of a wave packet with a wavevector of kx close to zero is large compared to a wave packet at the edge of the Brillouin zone kx = π/dx (dx being the lattice constant), while for negative c2, it is the opposite. Interestingly, it was observed that a negative NNN-coupling exists naturally even in a straight 1D array of waveguides fabricated with this method. Since in some systems the NNN-coupling is necessary for the creation of topologically non-trivial phases,90,110–112 the correct phase of the coupling is very important. In addition, being able to tune c2 to zero allows for decreasing the distance between the waveguides to achieve higher nearest neighbor coupling without introducing unwanted effects caused by NNN-coupling.
The presented fabrication method also allows us to make waveguides of almost arbitrary cross section and trajectory [see Fig. 6(b)], e.g., multimode waveguides that support modes carrying orbital angular momentum (OAM) with ℓ ≠ 0. Beams carrying OAM have a helically shaped phase front and a phase singularity in the beam’s center. By encircling the singularity, the electric field amplitude collects a phase of 2πℓ, , which is why ℓ is also referred to as the topological charge of the beam. In order to support OAM modes, the OAM mode has to be an eigenmode of the waveguide. The OAM mode of ℓ = ±1 is a superposition of the TE01 and TE10 mode, which implies that the waveguides must have a circular shape, as an elliptical shape would lift the degeneracy of the TE01 and TE10 mode, thereby making the OAM mode not a valid eigenmode of the waveguide.
In a diamond lattice of directly printed waveguides [Figs. 5(b) and 5(c)], a gauge field depending on the OAM of the input light was demonstrated using the Aharonov–Bohm effect.107 Inserting a waveguide mode with a non-zero OAM, |ℓ| = 1, leads to non-vanishing flux through a plaquette and thus to Aharonov–Bohm-caging [lower row in Fig. 5(c)], while excitation with a constant phase profile, ℓ = 0, does not achieve caging but only leads to spreading of light in the lattice [upper row of Fig. 5(c)]. This allows us to switch the topology of the lattice from trivial to non-trivial by only changing the OAM of the input beam using the same fabricated structure.
VI. UPCOMING NEW DIRECTIONS
A. Infiltrated waveguides
A key advantage of the fabrication method of waveguides by infiltration is that we are not limited to only one material; infiltration with a wide range of materials is possible. Work-in-progress includes infiltration with materials that show nonlinearity or absorption, by, e.g., mixing dyes into the photoresist.51,113 To actually observe light as it propagates through a waveguide array (in a similar way as in Ref. 114), a laser dye can be solved in the infiltration material. The local fluorescence allows the observer to track the path of the light along z. To image the emitted light from the structure, it is helpful if the waveguides are aligned parallel to the substrate such that the entire propagation length can be imaged with the second objective in Fig. 3(d). Additional prisms were fabricated at the ends of the structure to facilitate in- and out-coupling of light from the waveguides [Fig. 6(a-i)]. The 3D micro-printed structure was infiltrated with a solution of SU8 and the dye Oxazin 1 with a concentration of 1 ml SU8 to 5 mg of Oxazin 1. In the composite fluorescence image in Fig. 6(a-ii), the quantum-walk like dispersion of the light in the array of waveguides can be seen, when light with a wavelength of 680 nm is coupled via the prism into one waveguide at the input facet. Using objectives with higher focal lengths would allow us to use the previous version of waveguide structures, with z normal to the substrate, removing the need for the in- and out-coupling prisms.
Another upcoming direction is to explore the role of non-linearity in topological photonic systems. This is possible in infiltrated waveguides by mixing materials with a strong nonlinear response into the infiltrating resist. One of the main challenges of this approach is the inability of the photoresists to withstand very high laser power115 (unlike glass waveguides used in Refs. 116–119). Therefore, there is a need to find materials with sufficiently high nonlinear indices such that only low peak powers are needed. In addition, the mixture must exhibit low absorption and suitable refractive index at operating wavelengths and low viscosity for infiltration. Besides Kerr nonlinearity, other types of nonlinear interactions could also be introduced depending on the choice of the infiltrating material. Yet another possibility to be considered is to have site-dependent nonlinearity by infiltrating different waveguide sites with different materials (as described below).
Besides mixing dyes into the infiltration material, they can also be added to the resist before printing.113 This has already been exploited to fabricate a waveguide equivalent of Flamm’s paraboloid.120 Adding fluorescein to IP-Dip, the authors were able to observe the evolution of light in a structure inspired by black holes.
Also very recently, a method has been developed to selectively infiltrate different independent waveguides with different materials121 [see Fig. 6(a-iii)], even further expanding the degrees of freedom in waveguide arrays. This method can, for example, be employed to realize non-Hermitian topological systems with patterned gain and/or loss. With loss and gain, the system is inherently open and interacts with its environment. As a result of this non-Hermiticity, the time evolution is not unitary as the eigenvectors are no longer orthogonal and the eigenvalues can be complex. One application for this is a topological laser10 where the imaginary part of the eigenvalue of the topological edge mode is increased so that only in this mode lasing can occur.
Introducing gain and loss into the system does not necessarily implicate that the eigenvalues will become complex. If an eigenvector of the Hamiltonian is also an eigenvector of the parity-time (PT) operator, its corresponding eigenvalue is real. In this phase, the gain/loss only has a small influence compared to the bandgap. Upon increasing gain/loss, the bandgap shrinks. When the bandgap closes, some eigenvalues become complex conjugate pairs and the PT-symmetry is broken. Further increasing gain/loss such that it becomes dominant, the system reaches the anti-PT-symmetric phase, where the bandgap opens again; however, this time all eigenvalues are purely imaginary.122 The point at which this phase transition happens is referred to as an exceptional point, where the eigenvectors of the system coalesce and the eigenvalues switch from being purely real to being complex conjugate pairs. Such non-Hermitian systems harboring exceptional points and their interplay with topology have gained great interest in recent years. For example, it has been recently demonstrated that a Weyl point—a monopole of Berry curvature—expands from a point-degeneracy into an exceptional ring as the non-Hermiticity is increased. The topological charge is preserved but is distributed over the ring of exceptional points.123
To implement such systems in a waveguide structure, the individual amount of loss in each waveguide needs to be adjustable. This has been done (in fs-laser written waveguide systems) either by exploiting that radiation losses can be tuned by the bending of waveguides124 or by introducing scattering points125 or breaks123 along the waveguide trajectories. In those cases, the losses are caused by periodically coupling to the continuum of radiating modes. As our infiltration method allows us to use multiple materials, losses could be realized via different amounts of absorption. This design freedom also opens the door to achieve site-dependent nonlinearity.
B. Directly printed waveguides
Upcoming new directions, facilitated by direct printing of waveguides using different laser powers as described in Sec. V B, include the potential to create extra (artificial) dimensions by the use of multimode waveguides. The radial refractive index profile of the waveguide can be changed almost at will. For example, donut-shaped waveguides can be printed [Fig. 6(b)] that support OAM modes with higher ℓ while suppressing Laguerre Gaussian modes with knots along their radial coordinate. Furthermore, this allows us to experiment with modes other than the ground mode that does not have an interesting phase profile itself. By printing waveguides with elliptical cross sections, the degeneracy between TE01 and TE10 modes can be explicitly lifted, creating direction-dependent positive or negative coupling.
The design freedom in the fabrication process does not end at elliptical or donut-shaped waveguides but allows almost arbitrary cross sections and trajectories of each individual waveguide. Thereby, this platform could be used to provide the experimental proof of principle for many proposed OAM-converting waveguide structures. The discrete rotation symmetry of the waveguide cross section and the helical twist are essential features of these proposals, necessary to lift the degeneracy between OAM modes of equal |ℓ|.126,127 The OAM can also be inverted from ℓ to −ℓ by coupling two helically twisted elliptical waveguides with opposite helicity.128 These waveguide structures would be harder or impossible to implement in other platforms.
Since not only a complex trajectory and freely chosen cross sections but even the refractive index profile can be designed within this fabrication method, this opens the door to fabricate waveguides with a configurable inner mode structure. For example, some works have transferred the principle of supersymmetry to the optical regime. In optical waveguides, a supersymmetric partner system has eigenmodes with propagation constants matching the ones from the original system, but missing the ground mode. This has so far been demonstrated for systems of coupled waveguides129 and its implications on topological states,130 but it has not yet been implemented by adjusting the refractive index profile, as proposed in Refs. 131 and 132.
The ability to tailor the refractive index profile and implement structures with higher mode profiles opens up many new directions of research, and we are looking forward to interesting discoveries in this field.
C. Topological photonics in curved space
Since the Nanoscribe prints 3D structures, it seems optimally suited to fabricate devices using curved space to control the properties of light120,136–138 and exploit its characteristics for topology. Recent works suggested that the interplay between the curvature of space as in general relativity and the topology of the system could lead to new effects in topological physics.134 The metric of a curved surface alters the potential139 and/or coupling between waveguides printed onto the surface of a three-dimensional body134 [see Fig. 6(c)]. Unique to waveguides on curved surfaces is that the phase front between neighboring waveguides can be kept the same while changing the distance between them (and thus the coupling), which is not possible on flat surfaces. In particular, a “time” (i.e., z)-varying curvature of the surface acts as a metric-dependent gauge field and allows for tuning of topological phase transitions via its (periodic) curvature. It has been theoretically demonstrated that Thouless pumping and curvature-induced delocalization in the Andre–Aubry–Harper model can be implemented in such systems.134
D. Single photons
Upcoming directions include doing topological photonics with actual quantum states, i.e., single photons.140–144 Using the Nanoscribe, structures such as waveguides can be printed containing nanodiamonds with a nitrogen-vacancy (NV) center135,145–147 [Fig. 6(d)] to act as embedded single photon sources. While at room temperature, NV-centers might not be directly suitable for quantum experiments due to the low yield in un-distinguishable single photons,148 other single photon sources, including silicon or germanium vacancy centers, or quantum dots148 could be integrated in the same fashion into 3D printed structures.
The Nanoscribe is a flexible instrument that allows for the fabrication of a multitude of structures with unique topological properties. Due to the design freedom that direct laser writing offers, it is possible to readily fabricate interesting 2D and 3D photonic structures even in curved, periodic, or more complicated geometries. This Perspective provided an overview of the topological effects that have been observed in structures fabricated by the Nanoscribe, ranging from photonic crystals to different types of waveguide structures. The fabrication of waveguide systems benefits from the Nanoscribe’s great flexibility and precision. Waveguide structures fabricated with the infiltration method allow us to selectively introduce and benefit from material properties of the chosen infiltration medium, a feature unavailable for other waveguide platforms. The outlined new fabrication method for waveguides, using the dependence of the refractive index on the laser power, allows us to design the cross section of the waveguides to explore the interplay of topology and higher-order waveguide modes. Besides the flexibility in geometry, we suggested how different material properties, such as fluorescence or single photon emission, could be exploited when introduced in these photonic structures. Since the Nanoscribe is widely available at many universities, we are convinced that more people will be able to contribute to the fascinating field of topological photonics.
J.S. acknowledges funding from the Deutsche Forschungsgemeinschaft through CRC/Transregio 185 OSCAR (Project No. 277625399). S.V. acknowledges the support of the U.S. Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) under Grant No. N00014-20-1-2325 on Robust Photonic Materials with High-Order Topological Protection as well as the Packard Foundation under Fellowship No. 2017-66821. C.J. acknowledges funding from the Alexander von Humboldt Foundation within the Feodor-Lynen Fellowship program. The authors are grateful to Georg von Freymann and Mikael C. Rechtsman for their comments and suggestions.
The data that support this article, and are not published elsewhere as cited, are available from the corresponding author upon reasonable request.