Main

In optics, the effect of an external electric field on the motion of charged particles in periodic potentials can be mimicked by the propagation of a laser beam in an array of curved waveguides13. A schematic diagram of a one-dimensional waveguide array is shown in Fig. 1a. In such a structure, light propagation is governed by coupling between the modes of neighbouring waveguides, similar to wave dynamics in discrete lattices14. Constant waveguide curvature corresponds to a d.c. field, and in this case the beam experiences Bloch oscillations13, which have also been observed in straight optical lattices with transversely modulated parameters 15,16,17,18. Dynamic localization due to a.c. fields can then be observed in arrays of periodically curved optical waveguides with alternating curvature 5,6,8. A zigzag bending leads to similar behaviour4, and arrays with optimized discontinuous waveguide curvature also enable us to compensate for the long-range coupling between the non-nearest waveguide modes7. In straight waveguide arrays, optical beams experience broadening due to discrete diffraction14, whereas in the regime of Bloch oscillations or dynamic localization a periodic reconstruction of the initial light distribution occurs. Whereas the effective d.c. driving field always leads to Bloch oscillations with a period proportional to the inverse driving amplitude, dynamic localization is a resonant effect that occurs only for certain relations between the a.c. field profiles and the wave-particle parameters. By adjusting the detuning from the dynamic localization resonance, it becomes possible to control the rate of wave transport. For light propagating in curved waveguide arrays, the detuning depends on the wavelength, and application of this effect for optical filtering has been suggested19. By exploring the shaping of polychromatic light, we identify and characterize different dynamic localization resonances in lattices of various geometries and dimensionalities.

Figure 1: Polychromatic light in modulated photonic lattices.
figure 1

a, Schematic diagram of a one-dimensional periodically curved waveguide array, in which a white-light supercontinuum is launched. b, Schematic diagram of the femtosecond writing procedure. Insets: The resulting waveguide refractive index modulation Δn (bottom) and mode (top) profiles. c, The broadband supercontinuum spectrum spanning the entire visible region.

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For our experiments, we use the laser direct-writing method in fused-silica glass (see the schematic diagram of the writing set-up in Fig. 1b) to fabricate one- and two-dimensional waveguide arrays with required bending profiles. To characterize comprehensively light propagation in the fabricated arrays, we use two complementary methods: (1) direct visualization of the laser beam propagation at λ=633 nm through fluorescence imaging, and (2) spectrally resolved imaging of the output optical field distribution for the broadband supercontinuum excitation that spans over the entire visible (λ=450–800 nm) (see Fig. 1c). First, we measure beam evolution in straight waveguide arrays (see Fig. 2a). The beam propagation calculated theoretically for the supercontinuum light and registered experimentally for the wavelength λ=633 nm is shown in the second and third columns, respectively. In the fourth and fifth columns, we present the calculated and measured spectrally resolved output beam profiles, respectively. Note that diffraction increases at longer wavelengths, in agreement with previous studies20,21.

Figure 2: Dynamic localization of white light.
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a, Diffraction in a straight waveguide array. b, First-order dynamic localization at the wavelength λ=550 nm. c, Second-order dynamic localization at the wavelength λ=550 nm. d, Broadband dynamic localization in the spectral region 450–730 nm in a two-segment curved array. First column: waveguide axes bending profiles. Second and third columns: numerical simulations of the polychromatic beam propagation and corresponding fluorescent images measured at the wavelength λ=633 nm. Fourth and fifth columns: numerically calculated and experimentally measured spectrally resolved output beam profiles. e, Effective couplings in the curved arrays shown in b–d as a function of the wavelength.

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To provide a link with the classical dynamic localization effect in one-dimensional lattices, we consider planar arrays with harmonic waveguide bending profiles. Then, after each bending period the beam diffraction in the periodically curved waveguide array is characterized by the effective coupling coefficient between the neighbouring waveguides Ceff=C(λ)J0(ξ/λ), where λ is the optical wavelength, C(λ) is the coupling coefficient between straight waveguides, J0 is the Bessel function and ξ is a parameter that depends only on the lattice geometry. Dynamic localization occurs when the effective coupling is reduced to zero, which happens at a series of resonances between the wavelength of light and the lattice parameters: ξ/λ=ρn, where ρn are the roots of the Bessel function6 (see the Methods section for details). Accordingly, for a given structure, dynamic localization is possible only for specific colours of light. The first-order dynamic localization was previously observed experimentally6, yet higher-order resonances were not explored. Below, we demonstrate that by accessing higher-order dynamic localization and introducing mixed resonances, it becomes possible to realize a new regime of dynamic localization with strongly broadened response.

We create two sets of waveguide arrays with the same parameters except for different bending amplitudes, which are chosen to satisfy the first- and second-order dynamic localization resonance condition at the wavelength of λ0=550 nm. The corresponding results are presented in Fig. 2b and c, respectively. We observe that, in both structures, the frequency components in a narrow spectral region around λ0 return to a single waveguide at the output, whereas significant diffractive broadening occurs for other wavelengths. This is in sharp contrast to the light propagation in the same straight waveguide array without bending, where all spectral components experience strong diffraction (see Fig. 2a). To understand the difference between the dynamic localization of different orders, we present in Fig. 2e wavelength dependencies of the corresponding effective coupling coefficients. For the first-order dynamic localization, Ceff changes sign from negative to positive at the resonance when we move from shorter to longer wavelengths, corresponding to the transition from anomalous to normal discrete diffraction. In contrast, the sign changes from positive to negative around the second-order dynamic localization resonance.

The difference between dynamic localization of different orders for polychromatic light suggests that multi-colour beams can be manipulated through a combination of several dynamic localization resonances22. Specifically, we find that it is possible to achieve approximate dynamic localization in an extremely broad spectral region by combining two successive segments of the curved waveguide array that are tuned to the first- and second-order dynamic localization at the same wavelength (see the corresponding waveguide bending profile in Fig. 2d).

Then, although different colours detuned from the exact dynamic localization wavelength exhibit non-zero diffraction after the propagation in the first segment, the input beam profile is restored through the reversed diffraction in the second segment. The overall effective coupling in the two-segment structure almost completely vanishes in a very broad spectral region around the dynamic localization resonance wavelength (Fig. 2e, red line). In Fig. 2d, light propagation and output beam profiles are shown for the two-segment curved waveguide array. We observe that all spectral components, from λ=450 to 730 nm, are localized in a single waveguide at the array output (Fig. 2d, fourth and fifth columns), despite the highly non-trivial evolution of different spectral components inside the structure (Fig. 2d, second and third columns).

Next, we explore new features of the dynamic localization effect that arise in higher dimensions. Although dynamic localization was predicted for three-dimensional electronic systems1, only one-dimensional dynamic localization has been observed experimentally so far 6,7. Using the laser direct-writing technique, we can create arbitrary two-dimensional lattices. As an example, we consider a hexagonal lattice (see the microscope image of the fabricated structure in Fig. 3a). In a straight lattice, the diffraction pattern has the six-fold symmetry according to the underlying lattice, see Fig. 3b. To compare the results with one-dimensional lattices, we consider the simple harmonic bending profile in the x–z plane: x0(z)=Acos[2πz/L], where A and L are the bending amplitude and period, respectively, and x0(z) is the lattice transverse shift along the x axis. Although the waveguides are modulated in one plane, this affects coupling between the neighbouring waveguides along different directions, defined by the coefficients Ceff(1), Ceff(2) and Ceff(3) as shown in Fig. 4a. The effective coupling coefficients can be expressed as follows23: Ceff(1)=C(λ)J0(ξ/λ), Ceff(2)=Ceff(3)=C(λ)J0(ξ/2λ), and their wavelength dependencies for the fabricated array are presented in Fig. 4b. We see that the horizontal and diagonal coupling coefficients may vanish at different wavelengths, corresponding to partial dynamic localization along particular lattice directions. In the fabricated hexagonal lattice, the diagonal couplings are suppressed at the wavelength λ=633 nm (see the red line in Fig. 4b). In this spectral region, light effectively experiences one-dimensional diffraction, see Fig. 4e. On the other hand, for λ=583 nm, all three couplings are reduced simultaneously by the same factor −0.1, and the diffraction symmetry of the original hexagonal lattice is preserved. Output diffraction profiles at the closely tuned wavelength λ=594 nm are shown in Fig. 4d, where a pronounced hexagonal diffraction pattern is visible (corresponding effective couplings are marked with the yellow line in Fig. 4b). Finally, at λ=550 nm the horizontal coupling is cancelled (Ceff(1)=0). At this wavelength, the beam can still spread across the whole lattice, yet the diffraction pattern is similar to those of square or rectangular lattices where each lattice site is coupled to its four immediate neighbours (see Fig. 4c). Effective couplings at the wavelength λ=543 nm are marked with the green line in Fig. 4b.

Figure 3: Hexagonal lattice.
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a, Microscope image of the fabricated hexagonal lattice. b, Output diffraction pattern observed in the straight hexagonal array at λ=543 nm.

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Figure 4: Spatial–spectral beam shaping in a two-dimensional lattice.
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a, Schematic diagram of the effective couplings in the modulated hexagonal lattice. b, Wavelength dependence of the effective couplings. c–e, Output diffraction profiles of light beams measured experimentally (top) and calculated numerically (bottom) in the same modulated hexagonal lattice at three different wavelengths: 543 nm (c), 594 nm (d) and 633 nm (e).

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Our results demonstrate novel fundamental features of the effect of dynamic localization based on higher-order and mixed resonances in one- and two-dimensional lattices. Polychromatic dynamic localization and localization-induced transformation of lattice geometry, which we observe for the first time to our knowledge, open up new avenues for applications of the dynamic localization effect in various physical contexts. In particular, our work suggests new approaches for flexible shaping of polychromatic light with ultrabroadband or supercontinuum spectra11,12, which can be enhanced further through the introduction of structure tunability and optical nonlinearities.

Methods

Theory of dynamic localization in curved waveguides.

To understand the origins of the dynamic localization effect, we note that light propagation in an array of weakly coupled waveguides can be considered as effectively discretized14, and it is primarily characterized by coupling due to the overlap between the fundamental modes of the nearest-neighbouring waveguides. Second-order coupling between non-nearest neighbours is weak under our experimental conditions and is not taken into account. In a one-dimensional array, light propagation can be described by a set of equations for the mode amplitudes ψm(z) (ref. 6), . Here, z is the propagation distance along the waveguides, m is the waveguide number, the coefficient C(λ) defines the coupling strength between the neighbouring waveguides that depends on the wavelength λ, n0 is the refractive index of the medium, d is the waveguide spacing, the function x0(z) describes periodic waveguide bending in a curved array and dots stand for the derivatives. By the substitution z→t, the spatial light evolution in a curved optical waveguide array becomes fully analogous to the temporal evolution of a quantum particle in an externally driven periodic potential, where ψm(t) have the role of the probability amplitudes, C(λ) is replaced by the nearest-neighbour transfer-matrix element and the driving force x0(t) is determined by the external electric field1.

In a straight waveguide array (with x0(z)≡0), the dispersion relation reads kz=2C(λ)cos(kxd), where kz and kx are longitudinal and transverse components of the wave vector, respectively4. Accordingly, the diffraction strength for a broad beam is determined as D=∂2kz/∂ kx2=−2C(λ)d2cos(kxd) (ref. 4). In a periodically curved array, wave diffraction after each bending period (z→z+L) can be described by a similar relation where C(λ) is replaced by the effective coupling, which is a functional of a specific bending profile6, For a harmonic bending profile x0(z)=Acos[2πz/L] with bending amplitude A, the effective coupling reads Ceff=C(λ)J0(ξ/λ), where ξ=4π2n0d A/L. Thus, when ξ/λ=ρn, where ρn≃2.40,5.52,… is the nth root of the Bessel function J0, the effective coupling vanishes, Ceff=0. This regime corresponds to the periodic diffraction cancellation, Deff=−2Ceffd2cos(kxd)=0, which is called dynamic localization1,6. Note that the dynamic localization resonances do not depend on the dispersion of the coupling coefficient C(λ). The effective coupling method can also be generalized to higher dimensions1,23.

In our numerical simulations, we use a continuous (1+1)- and (2+1)-dimensional finite-difference beam propagation method to model light propagation in one- and two-dimensional waveguide arrays, respectively 22,23 (the numbers in brackets denote the number of transverse and longitudinal spatial dimensions, respectively). For polychromatic beams, we use a superposition of 50 frequency components with an equidistant flat spectrum from 450 to 730 nm as input. Slight differences between the theoretical and experimental results appear for the one-dimensional arrays because in numerical modelling we do not account for the two-dimensional mode reshaping. Nevertheless, this has no effect on the position of dynamic localization resonances, as they do not depend on the mode overlaps as detailed above.

Laser writing.

Our samples are fabricated in fused silica using the femtosecond laser direct-writing technique24. When ultrashort laser pulses are tightly focused into a transparent bulk material, nonlinear absorption takes place leading to optical breakdown and the formation of a microplasma, which induces a permanent change in the molecular structure of the material. In the case of silica glass, the density is locally increased. By moving the sample transversely with respect to the beam, a continuous modification of the refractive index is obtained enabling light guiding (see the schematic diagram in Fig. 1b). This technique can be used to create large waveguiding structures with almost arbitrary three-dimensional topology25. Details of the fabrication technique can be found elsewhere26. All of our samples are 105 mm long with the waveguide spacing d=26 μm. Laser-written waveguides possess an elliptical transverse cross-section of approximately 4×13 μm2. The profiles of the waveguide refractive index distribution and the corresponding mode measured at λ=633 nm are shown as insets in Fig. 1b. To study one-dimensional dynamic localization, we created three curved waveguide arrays consisting of 19 waveguides each. Curved arrays for the first- and second-order dynamic localization contain one full bending period equal to the sample length (L=105 mm), and have bending amplitudes A=93 and 214 μm, respectively. The curved array for the broadband dynamic localization consists of two successive segments of length L1=63 mm and L2=42 mm with bending amplitudes A1=56 μm and A2=86 μm, respectively. To study dynamic localization in two-dimensional lattices, we created a curved hexagonal array with a bending period L=105 mm and bending amplitude A=215 μm (see the microscope image of the fabricated array in Fig. 3a). To characterize the diffraction strength in our samples, we also fabricated one- and two-dimensional straight arrays (see the measured diffraction patterns in the straight arrays in Figs 2a and 3b).

Fluorescence imaging.

To directly observe the light propagation within our one-dimensional arrays (see Fig. 2, third column), we use a fluorescence technique27. For the fabrication of the waveguides, we use fused silica with a high content of hydroxide. This leads to a massive formation of non-bridging oxygen hole centres during the writing process, resulting in a homogeneous spatial distribution of these colour centres along the waveguides. When launching light from a HeNe laser at λ=633 nm into the waveguides, the non-bridging oxygen hole centres are excited and the resulting fluorescence (λ=650 nm) can be directly observed8. As the colour centres are formed exclusively inside the waveguides, this technique yields a high signal-to-noise ratio. In contrast to fluorescent polymers (see, for example, ref. 28), the bulk material causes almost no background noise in our case.

Supercontinuum characterization.

A white-light continuum was generated using a 25-cm-long photonic-crystal fibre (PCF NL-1.7-650, Crystal Fibre). Ultrashort laser pulses with a length of 1 ps from a Ti:sapphire oscillator (Spectra Physics) were coupled into the photonic-crystal fibre. Owing to the high peak power of these pulses, supercontinuum radiation was generated with a spectrum shown in Fig. 1c. This white light is coupled into the central waveguide of the arrays using a ×10 microscope objective (numerical aperture: 0.25). The output facet is imaged onto a CCD (charge-coupled device) camera by a ×10 microscope objective; the spectral resolution is achieved by a prism placed between the objective and the camera. Owing to the chromatic dispersion, the output spatial profiles are resolved at the individual wavelengths29,30 (see Fig. 2, fifth column).