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A new dimension for nonlinear photonic crystals

The first three-dimensional nonlinear photonic crystals have been constructed thanks to the use of femtosecond laser writing in quadratic nonlinear materials.

Nonlinear photonic crystals are transparent materials that have a spatially uniform linear susceptibility, yet a periodically or quasi-periodically modulated quadratic nonlinear susceptibility. These engineered materials are used extensively for studying nonlinear wave dynamics and in many scientific and industrial applications that require the generation of light at new frequencies and its control.

The term nonlinear photonic crystals was coined by Vincent Berger in 19981 in a theoretical paper that investigated the nonlinear dynamics in two-dimensionally modulated nonlinear structures, followed by the first experimental realization of two-dimensional (2D) nonlinear photonic crystals by Neil Broderick and colleagues2. Since then, over the past two decades, there has been a continuous effort to find a technique that will enable the construction of 3D nonlinear photonic crystals. Such capability will enable many new schemes of manipulation and control of nonlinear optical interactions. However, a scalable method for the engineering of the quadratic nonlinear coefficient of materials in three dimensions, at the required micrometre resolution, has not been demonstrated to date.

Writing in Nature Photonics, two independent groups now report3,4 two different approaches, based on scalable and high-resolution femtosecond laser writing techniques, that demonstrate 3D periodic modulation of the quadratic nonlinear coefficient in photonic crystals and its use for second-harmonic generation.

The need for spatial modulation of the quadratic nonlinear coefficient of materials emerges from the requirement of phase matching (or the equivalent momentum matching) in frequency conversion processes. This need was first recognized in the seminal work by Nicolaas Bloembergen and colleagues in 1962 that laid the basic theoretical foundations to the field of nonlinear optics5. Owing to natural material dispersion, the rate of phase accumulation by waves of different frequencies travelling in the material is different. The quadratic nonlinearity of materials can coherently couple waves of different frequencies. However, after a certain propagation length, termed the coherence length, the coupled waves in the nonlinear material accumulate a phase difference of π, and the direction of energy flow between the coupled waves is reversed. In a second-harmonic generation process, for example, this limits the effective build-up length of the second-harmonic wave to the coherence length. This length depends on material dispersion and is commonly on the order of a few to tens of micrometres for typical quadratic nonlinear materials. However, as already recognized by Bloembergen and colleagues5, a perfect inversion of the crystal structure leads to inversion of the quadratic nonlinear coefficient without affecting the linear susceptibility. By performing such an inversion after an odd multiple of coherence lengths, the direction of nonlinear energy transfer between the waves is re-adjusted, without suffering from linear reflection or scattering losses due to photonic bandgaps. As a result, an efficient directional energy transfer between the propagating waves can be achieved, as shown in Fig. 1 for a second-harmonic generation process. This imperfect phase-matching method is commonly called quasi-phase-matching (QPM)6.

Fig. 1: The role of phase matching in quadratic nonlinear processes.

The build-up of a second-harmonic field along the propagation path of the optical waves for different phase-matching conditions. lc is the coherence length, l is the propagation length in the crystal and E2ω is the amplitude of the second-harmonic field, where ω is the fundamental frequency.

The advantage of the concept of nonlinear photonic crystals1 to study multidimensional QPM is that the nonlinear interaction can be conveniently analysed on the multidimensional reciprocal lattice. On the reciprocal lattice, the momentum mismatch (and phase mismatch) in the nonlinear interaction can be compensated by an existing reciprocal lattice vector. This conceptual understanding opened the door to new phase-matching schemes that were not accessible before1,7. For example, the reciprocal lattice vectors in 2D lattices may have non-integer relations to the inverse of the lattice constant, in contrast to the 1D case. This gives an additional degree of freedom for compensation of momentum and phase mismatch. In addition, the concept of nonlinear photonic crystals facilitated simultaneous phase matching of multiple processes, and the analogy to 1D quasi-crystals was made as a projection of the 2D periodic modulation to a reduced dimension1,7. In a similar manner, the current demonstrations of 3D nonlinear photonic crystals by Tianxiang Xu and colleagues3 and Dunzhao Wei and colleagues4 paves the way for studying new phenomena and to control nonlinear interactions in materials in ways that were not accessible before.

Figure 2 illustrates the potential applications of 3D nonlinear photonic crystals. The increased dimensionality in 3D structures opens more channels in the nonlinear material to be used for many important applications including dense data multiplexing and de-multiplexing, generation of multidimensional entangled states, simultaneous QPM of different nonlinear processes, volume nonlinear holography and beam shaping, and even for improved generation and control of terahertz (THz) radiation.

Fig. 2: Illustration of potential applications of 3D nonlinear photonic crystals.

The increased dimensionality in 3D structures opens more channels in the nonlinear material to be used for many important applications. \(\chi _{{\mathrm{a}},{\mathrm{b}}}^{\left( 2 \right)}\) mark areas with different values of the quadratic nonlinear coefficient, ω1,2 mark different frequency components such that 2ω, ω1 + ω2 and ω1ω2 represent second-harmonic, sum-frequency and difference-frequency processes, respectively. The unlabelled blue arrows represent additional uncharted applications.

The successful construction of 3D nonlinear photonic crystals reported by Xu and colleagues3 and Wei and colleagues4 was based on two different approaches that use femtosecond laser writing techniques to manipulate the nonlinear coefficients. Xu and colleagues used ferroelectric barium calcium titanate that was composed of small randomly oriented nano- to micrometre-scale domains as the nonlinear material. Irradiation of the barium calcium titanate with a tightly focused femtosecond laser (850 nm centre wavelength, 180 fs pulse duration, 76 MHz repetition rate, 6 nJ pulse energy) led to realignment of the small domains into two larger domains with opposite orientations. This occurred as a result of a thermo-electric effect originating from multiphoton absorption in the area of the focused spot. The process was repeated at each lattice point to create the 3D structure. The writing depth was 50–350 μm below the surface of the crystal. The fabricated 3D structures were used to show second-harmonic nonlinear diffraction that fits the 3D nonlinear photonic crystal structure.

Wei and colleagues used a different approach to create the 3D nonlinear photonic crystal. Instead of relying on laser-induced poling of the ferroelectric crystal, they used the laser to significantly reduce the value of χ(2) in specific areas while minimizing the modification of the linear index of the material. The researchers used lithium niobate (LiNbO3), one of the most commonly used nonlinear crystals for frequency conversion and electro-optic applications. A focused femtosecond laser (800 nm centre wavelength, 104 fs pulse duration, 1 kHz repetition rate, 100–200 nJ pulse energy) was used to reduce the crystallinity in domains of size of ~1.5 × 1.5 × 6 μm3, as a result ‘erasing’ their quadratic nonlinear susceptibility. Attention was paid to minimally affect the linear susceptibility of the material in the process. With a careful analysis of the beam profile and intensity changes of the focused spot throughout the material, the researchers were able to reduce the nonlinearity in a 3D periodic structure. The deletion of the quadratic nonlinear coefficient in areas that would normally lead to inverse energy flow between the waves resulted in an overall directional energy flow, similar to the outcome of conventional QPM, but at a reduced efficiency. Using the nonlinear photonic crystal, the researchers were able to show QPM patterns that fit the 3D structure.

Although the results of Xu and colleagues and Wei and colleagues present a significant breakthrough in the field of nonlinear photonic crystals, several challenges remain. One major challenge is to optimize the laser writing methods in order to be able to write deeper into the nonlinear crystal to support the fabrication of large-scale 3D nonlinear photonic crystals. This can be done, for example, by using beam shaping and holographic techniques to control the interference patterns of the writing femtosecond pulses deep in the nonlinear crystal. Another challenge is to further reduce the modification of the linear index by the laser writing process. These small modifications lead to reflections and scattering of the waves that limit the efficient interaction length in the nonlinear photonic crystal. In addition, it would probably be worthwhile to study the suitability of the demonstrated methods to sculpt the quadratic nonlinearity in other quadratic materials. Finally, based on advancements in nanofabrication capabilities, it would be beneficial to investigate alternative methods to generate 3D configurable nonlinear photonic crystals.

It is now up to the nonlinear optics community to overcome the abovementioned challenges and to explore the new possibilities offered by 3D nonlinear photonic crystals.


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Correspondence to Shay Keren-Zur or Tal Ellenbogen.

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Keren-Zur, S., Ellenbogen, T. A new dimension for nonlinear photonic crystals. Nature Photon 12, 575–577 (2018).

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