Opto-Electronic Advances, 2022, 5 (6): 210174, Published Online: Aug. 19, 2022   

Nonlinear optics with structured light Download: 685次

Author Affiliations
School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
Abstract
The interest in tailoring light in all its degrees of freedom is steadily gaining traction, driven by the tremendous developments in the toolkit for the creation, control and detection of what is now called structured light. Because the complexity of these optical fields is generally understood in terms of interference, the tools have historically been linear optical elements that create the desired superpositions. For this reason, despite the long and impressive history of nonlinear optics, only recently has the spatial structure of light in nonlinear processes come to the fore. In this review we provide a concise theoretical framework for understanding nonlinear optics in the context of structured light, offering an overview and perspective on the progress made, and the challenges that remain.

1 Introduction

Structured light1 refers to the modern-day ability to tailor light in all its degrees of freedom (DoFs), spatial and temporal, to create complex optical fields in both the classical2-5 and quantum6, 7 domains. Combining DoFs have given rise to novel and exotic states of light as 2D, 3D and even 4D fields, including optical knots8, 9, skyrmions10, 11, Mobius strips12, spatio-temporal fields13-16, ray-wave structured fields17, 18, quantum-like classical light19-21 and photonic wheels22. But although the progress has been rapid of late, the topic itself can be dated back to Thomas Young and his double slit experiment, where arguably the first structured light was created. Indeed, the very essence of structured light is the notion of superpositions, where interference (not necessarily in intensity) gives rise to the desired structure. Today one can formulate all of structured light as a linear superposition principle1, giving rise to geometric representations of the superpositions, from the orbital angular momentum (OAM)23, to the total angular momentum24 of light, and more recently a generalised framework for multiple DoFs25. For example, even simple plane waves hold the potential for structure: one plane wave may have a phase gradient, two plane waves will give rise to an intensity structure (as done by Young more than 200 years ago), three plane waves can produce an optical phase singularity, while multiple plane waves can give rise to exotic families of structured light, for instance, planes waves travelling on a cone give rise to Bessel beams26. If the interfering plane waves are allowed to hold information in another DoF, say polarization, then just two can create exotic polarization structures27 and if focussed, will create synthetic chiral light in 3D28. It is clear that there is a strong link between interference, a linear effect, and structured light. For this reason, the vast bulk of studies involving structured light have considered linear optical elements, with only much more recent progress in nonlinear optics with structured light, the topic of this review.

The invention of the laser29 is seen as fundamental to the development of the research field of nonlinear optics, with the first nonlinear optical demonstration of second harmonic generation (SHG)30 following quickly, soon after by third harmonic generation31 and SHG carrying spin angular momentum (SAM)32. The reason for the strong historical linkage is simple: nonlinear optical phenomena are weak and typically need a coherent high power source to be observed. This knowledge can be dated back as far a Fresnel, who already understood that wave superpositions could transcend the linear regime. With the development of stronger laser sources and more efficient nonlinear materials, we have overcome this requirement. It is only natural then that the study of nonlinear optical phenomena shifts from asking “how much light is there?” based on efficiency concerns (intensity being the key to address this), to “what does the light look like?” (the structure of the light). Seminal works began analysing the structure of the generated light three decades ago33 with early work demonstrating the doubling of the number of singularities in the generated field34. Following the link between orbital angular momentum (OAM) and these so-called screw dislocations (see ref.35 and references therein), the use of OAM carrying Laguerre-Gaussian (LG) modes in nonlinear optics was demonstrated36 followed a little later by the first production of quantum structured photons by nonlinear optics, demonstrating OAM entangled states37. Although these important works set the scene, further progress has been slow, until only recently.

In this review we follow the progress in the field, from intensity drive processes that serve to alter the frequency of the pump light, to the present day nonlinear toolkit for the creation, manipulation and detection of structured light. We begin with the familiar wave mixing processes of second order, which have been deeply explored and continue to develop to this day, serving to exemplify how counter-intuitive these interactions can be with the introduction of structured light. We then move on to show the types of media that allow these process and how they can also be structured, playing a crucial role in recent advances. We expand into higher-order parametric processes, including third harmonic generation and the generation of optical vortex solitons. Finally, we cite recent developments in high harmonic generation, an extreme non-parametric process, and the unusual applications of nonlinear processes in the quantum regime.

2 Theoretical background

The field of nonlinear optics is a venerable topic, and the reader is referred to excellent textbooks on the topic38-40. For the benefit of the reader, We begin by briefly outlining the core theory needed for the review, and to this end we begin with Maxwell's equations in the presence of a medium. If condensed and rewritten in terms of a wave equation, one finds,

[××+1c22t2]E=4πc22Pt2,

where E is the electric field and P is the polarization of the medium. This describes the response of the medium to the input electric field, and the counter response of the polarised medium on the field. In a analogy to the harmonic oscillator, we can do a perturbative expansion of the medium polarization in a power series of the electric field strength,

Pi=χij(1)Ej+χijk(2)EjEk+χijkl(3)EjEkEl+,

where Pi is the i component of P and χ(n) is the nth order susceptibility tensor, a tensor of the nth order. The first term is responsible for the well known linear optical effects, such as refraction and birefringence. All other terms are referred to as nonlinear parametric effects, which this review will mostly feature. This expansion describes a plethora of nonlinear optical effects, such as wave-mixing, self and cross phase modulation, among many. For example, the second order term, with its second order susceptibility tensor, allows the medium polarisability to be separated in frequency components. Second order wave mixing can result in effects such as second harmonic generation, sum frequency generation, difference frequency generation and optical rectification, with a well understood rule set.

But what are the governing principles when the light has internal structure? In this review we will often use scalar and vectorial combinations of light that carries OAM, a highly topical example. Such modes of light have an azimuthal phase profile given by exp(ilϕ), where ϕ is the azimuthal angle and l is the topological charge, for photons with l of OAM. For brievity we will refer to OAM modes by their topological charge, l. What are the selection rules when such complex light fields are used in nonlinear processes? As we will show, the development of nonlinear optics with structure light has produced complex behaviour with some as yet unanswered questions. Further, it is not always possible to find exact solutions for the coupled equations that describe these phenomena. For instance, nonlinear processes of even the second order can generate coupled wave equations whose number increases directly with the number of transverse modes involved.

With suitable approximations (such as the slowly varying envelope approximation and lossless media) the generated field in three wave mixing can take the form

E3(ω3)(r)zχ(2)E1(ω1)(r)E2(ω2)(r).

Similar differential equations are also derived for each frequency, initially obtaining three coupled differential equations. While these equations were initally derived as plane waves, structured light reveals more intricate interactions. For example, the phases and intensities are all intertwined: reshaping one field can mean completely new dynamics and new structures in all three involved fields. Due to the low conversion efficiency, a characteristic of nonlinear processes, in single-pass geometry we can make use of the non-depletion approximation, where the input fields can be regarded as static and therefore do not change on propagation. These equations show one remarkable feature of nonlinear optics: the nontriviality of the superposition principle. For example, let us associate the generated field E3(r) with input fields {E1(r), E2(r)} and generated field E3(r) with input fields {E1(r), E2(r)}. If now we use as inputs E1(r)=E1(r)+E2(r) and E2(r)=E2(r)+E2(r) it will not follow that E3(r)=E3(r)+E3(r). In this equation, the vector nature of this interaction was omitted for simplicity, but it suffices to say that input fields have the same polarization for type-I and orthogonal polarizations for type-II. Equation (3) describes sum frequency generation (where ω3=ω2+ω1) and, if considered difference frequency generation ( ω3=ω2ω1), one of the fields ( ω1) would be complex conjugated. For second harmonic generation both fundamental fields would have the same frequency ( ω3=2ω1=2ω2) and the two fundamental fields would be identical for type-I but not necessarily for type-II.

The expansion of these fields into propagating waves in different directions gives us the phase-matching quantity

Δk=kω3r3kω1r1kω2r2.

That ensures that the light generated is through a coherent process and interferes constructively at each wavefront generation. When Δk=0, this is referred to as perfect phase-matching.

Figure 1 highlights a few differences between the linear and nonlinear regimes, the latter illustrated using second harmonic generation as an example. We illustrate that the generated spatial structure is not simply a superposition, but the product of input modes. A consequence is that while the original modes may be eigenmodes of free space, the final mode may not. For instance, as shown in Fig. 1(a), if two OAM modes with topological charges of l=4 and l=4 combine, then the superposition creates a petal-like structured which is stable in propagation, while the nonlinear process creates a ring-like structure with no OAM, which is unstable in propagation.

Fig. 1. Linear and nonlinear processes. Using second harmonic generation (SHG), we illustrate the differences between linear and nonlinear processes. (a) Linear processes produce an output mode that is the addition of two input spatial modes of light, while SHG produces the product of the two modes. The linear superposition of two different modes with orthogonal polarization states generates a vector beam, which has a inhomogeneous polarization state. The polarization profile is represented as yellow lines across the transverse profile. In SHG, and wave mixing in general, the polarization profile will dictate where wave mixing happens and thus alter directly the spatial profile. In (b) we show exemplify how path can also be controlled via polarization and the different phase matching conditions of crystals, including the periodic poling of type-0. The mechanism which allows these interactions is sketched in (c). Phase-matching is the condition necessary for wave mixing to occur and exploits birefringence (types I and II) or periodical polling (type-0) to achieve it.

下载图片 查看所有图片

Polarization also has a non-immediate role. In the linear regime, optical beams with orthogonal polarization states do not interfere to produce fringes (but they do produce fringes in polarization27, 41). In contrast, in nonlinear media, they do interact through the coupled interactions with the medium. If the input beam has a inhomogenous polarization profile, i.e. vector beam, then this interaction is different in every point of the transverse profile. We illustrate this in Fig. 1(b) where a vector beam used as input shows that SHG has different efficiencies across the transverse profile. This can be seen as a projection onto one of the crystals axis, generating a beam with a uniform polarization state and its spatial structure is influenced by both polarization and structures of the fundamental beams. This dependency creates states that binds path, input polarization and spatial mode. To understand the connection, one can consider the schematics in Fig. 1(b). The perfect phase-matching condition can be fulfilled for more than one propagation direction at the same time, each as independent processes. The wavevectors in this equation are considered inside the matter (often a crystal), where differently polarized beams would see different refractive indices, crafted specifically to fulfill this condition in types I and II. For type-0, the material is structured with a periodic polling, which gives a contribution of Gm=m2π/Λ to phase-matching where Λ is the domain length.

The combination of Eqs. (3) and (4) exemplifies the role structured light's degrees of freedom (DoF) in wave mixing, encompassing spatial profiles through the coupled wave equations (Eq. (3)), polarization in the phase-matching (both in χ(2) and kx,y,z) and path in the phase-matching Δk. Only by considering all these DoF and their interaction we can grasp a full understanding of nonlinear processes with structured light.

3 Structured dofs and their nonlinear coupling

By choosing sum-frequency generation and breaking wavelength degeneracy, it is possible to encode different structures in each frequency. If one of the fields is physically expanded and thus approximated to be a plane wave, we see the directly transfer and manipulation of the spatial profile of a beam across wavelengths42-44. In this case, the lack of structure of one field enables the generated beam to completely inherit the structure from the other. By using different spatial modes in each frequency, it was possible to perform OAM algebra45. This creates an interesting interaction where the wavelength is used as a control parameter for the spatial structure.

In initial works with SHG it was observed that the generated field would be proportional to the square of the fundamental frequency36, as it is possible to see in Eq. (3) if the conditions ω3=2ω2=2ω1 and E1=E2 are set. This describes type-I phase matching. If the vectorial nature of this interaction with matter is chosen accordingly, it is possible to use type-II phase matching to have different spatial modes in the same frequency but different polarizations46, 47, creating in the SHG a profile composed of the product of two different modes of the same frequency. Even in the collinear geometry configuration, there is an interplay between the spatial and polarization degrees of freedom.

The path degree of freedom can also be used: in Eq. (4) we can see that the phase-matching depends not only on the material but on the propagation direction of the beams. If two beams are crossed inside the crystal so that the phase-matching is fulfilled, a third beam is generated, as illustrated in Fig. 1(c). Using this it was possible to study the transverse structure transfer in SHG48 and off axis singularity combination49. In these cases (the former is shown in Fig. 2(a)) one input wave can be approximated as a plane wave and the other has nonzero OAM. The non-collinear interaction generates the second harmonic of both input modes with the square of the spatial profile but also creates a third beam with the product of them, having the same OAM as the input, but different polarization and wavelength.

Fig. 2. Wave mixing with different degrees of freedom. In (a), the authors show OAM algebra in noncollinear SHG. When type-II phase-matching is used, the same noncollinear geometry allows for polarization switching, shown in (b). This effectively couples multiple degrees of freedom in a single process: path, polarization, radial and angular transverse structures. The radial selection rules of LG modes in wave mixing are demonstrated in (c). There is a intrinsic relation between the radial and angular degrees of freedom, which is manifested in the propagation dependence of the spatial profiles. In (d), a experimental scheme using a Sagnac interferometer achieves faithful frequency conversion of vector light. Spin and orbital angular momentum are combined in second harmonic generation in (e). Figure repoduced with permission from: (a) ref.48, Springer Nature; (b) ref.50, © Optica Publishing Group; (c) ref.54, © American Physical Society; (d) ref.69, American Physical Society; (e) ref.70, under aCreative Commons Attribution 4.0 International License

下载图片 查看所有图片

The three process depicted above are independent and do not interfere with each other. Interestingly, not all nonlinear process are independent. Using polarization as a control parameter in type-II SHG, the authors realized that nonlinear process can interfere destructively50. As illustrated in Fig. 2(b) two input beams with opposite OAM and orthogonal polarization states pass through a half-wave plate (HWP) at an angle θ and impinge at the crystal with a small angle. When θ=0 the phase matching conditions are only satisfied for one path. For θ=22.5° all three paths have equal phase-matching conditions satisfied and therefore equal intensities. However, when θ=45° the phase matching conditions are satisfied for all paths, but the middle one has zero intensity. This happens because there are two wave mixing process occurring on the same path, but interfering destructively. This interplay between path and polarization enabled an opportunity for all-optical switching.

3.6 Scalar structured light

The fields in Eq. (3) can be expanded in the well known spatial modes, and by using orthogonality relations, the right-hand side of this equation becomes a set of three mode overlap integrals. The modal description of this process has resulted in a important result regarding the interaction of light with matter, for example, the conservation of OAM per photon in classical36, 51 and quantum37 nonlinear processes.

Interestingly, the coupling is not only between light and matter, but between differences in structure of the fields themselves, particularly within a given family. For instance, the “untwisting” of the azimuthal phase of an OAM Laguerre-Gaussian (LG) mode in turn altered the radial index52, 53, with the rules governing this interaction only recently unveiled54, and shown to be true for wave mixing processes of any order55. This intricate relation is illustrated in Fig. 2(c). The first row shows a process where two different radial structures are used as inputs and the state generated ends up as a superposition of different radial orders, up to a mode of order equal to the sum of the input orders. On the second row, the azimuthal phases cancel each other, generating higher radial orders. Similar processes have been observed with Hermite-Gaussian (HG) modes. Here, their separable Cartesian form makes the interpretation of the selection rules far more straightforward, aided further by the fact that they are the natural solutions of the anisotropy of a biaxial crystal56, an important aspect in optical cavities. Since these seminal studies, wave mixing with structured light has included Ince-Gaussian57, 58 and Bessel-Gaussian59-62 modes, confirming OAM conservation and exploring the selection rules of these families. OAM conservation was not only shown for integer but also for fractional topological charges63, 64, in an off axis configuration65 and even in plasmonic media66-68. A summary of the different behaviours structured light and its different modes can have in second order nonlinear wavemixing are summarized in Table 1.

Table 1. Behaviour of various structures of light in second order nonlinear wave mixing. Here, nx/ny are the indices for HG modes, , p are the azimuthal/radial indices for LG modes and p/m are the parameters for Ince-Gaussian modes. Indices with primes, such as ℓ′′ are of fundamental field modes and the ones without are of the frequency generated.

StructureBehaviour observedRelations
Laguerre-GaussianOAM operations36, 47, 51 Radial selection rule53, 54 = ℓ′ + ℓ′′pp′+ p(ℓ′× > 0) p ≤ min(|ℓ′|,||) + p′+ p(ℓ′ × < 0)
Hermite-GaussianIndependent selection rules56nxn′x + n′′ (mod 2)
Optimal base for conversion72nyn′y + n′′y (mod 2)
Ince-GaussianDoF coupling 57pp′+ p′′ (mod 2)
m0mp (mod 2)
Helical Ince-GaussianOAM conservation 57pp′+ p′′ (mod 2)
m0mp (mod 2)
mNet = m′ + m′′
Bessel-GaussianOAM doubling in SHG59 Transverse wavenumber superposition 60 = 2ℓ′k = 2k′
Bessel bottle beamsSelf-healing and divergence increase 62-
Airy beamsFocusing distance related to wavelength Vortex phase preservation(Ring-Airy)120 Direction switching in DFG121- -
Fractional OAMTopological charge transfer 122 Birth of vortex and creation of radial orders 63- -
Anti-chiral vorticesRadial-azimuthal coupled diffraction 55-
Vector beamsPolarization singularity doubling in SHG 82 Faithful frequency conversion 69 Phase conjugation in StimPDC 90- - -

查看所有表

One might ask if there is there a recipe for the input to the nonlinear process in order to obtain a desired output structured field? The answer can be trivial, where one or more of the input profiles are plane waves and one of them contains the desired structure. By this approach, LG and HG structured modes have been created, as well as general structured images71. When this is not possible, the HG basis is suggested to be optimal72, and has been used for high fidelity mode generation73. Because wave mixing allows for light modulation by light, the process can be adapted to be used as a detector of structured light71, 74, 75, and has been used to detect LG and HG modes with very little modal cross-talk, in a manner analogous to modal decomposition76. Even complex images can be handled in this manner, with the benefit of noise reduction (squaring a signal will amplify the strong and the decrease weak). For this reason, this has been an emerging application of SHG, with demonstrations including augmented edge contrast77, 78 and contrast enhancement to improve recognition of human faces and QR codes79.

3.7 Vectorial structured light

So far we have considered the case where the structured light is scalar, so that the polarization is homogenous across the field. A complex vectorial structure is achieved by combining orthogonally polarized states such that each has its own unique spatial mode. If the spatial modes are also orthogonal, then the polarization structure of the field will be maximally inhomogeous3. On the right-hand side of Fig. 1(a) we have an example of a vector beam with spin-orbit coupling. Two beams of orthogonal circular polarization states and opposite OAM are combined to form a complex polarization pattern. The yellow lines represent the linear polarization state at every point in the transverse profile at a given angle. Since nonlinear wave mixing depends on both the spatial mode and polarization DoFs, it might seem that a inhomogenous polarization structure would be bound to change when frequency converted. In fact, frequency conversion of vector structured beams has been characterized as producing non-trivial scalar patterns in type-II SHG80, 81 and having an altered vector structure when generated in SHG with sandwiched crystals82 and using a Sagnac loop83. In this sense, the inhomogeneous state of polarization has been proposed as a control parameter for nonlinear process50, 84. Recently an elegant approach was realized using a Sagnac loop, making it possible to convert a vector beam in frequency69, 85, 86 while retaining the polarization structure, as illustrated in Fig. 2(d). Here, one input is a vector beam and the other a auxiliary beam, the latter having a somewhat flat intensity and phase distribution. A polarizing beamsplitter (PBS) is used to separate the vector beam into two components where the loop shape makes them propagate in opposite directions. By inserting a half-wave plate (HWP) in the loop, each component of the vector beam is combined with an orthogonally polarized co-propagating plane wave, which enables faithful frequency conversion of each component independently. Lastly, the same PBS recombines both components back into a vector beam with the same spatially structured polarization but at a converted wavelength. Some observed effects of vector beams in wave mixing process are summarized in Table 1. The examples provided only deal with second order processes. A theoretical approach was already proposed to characterize the full vectorial nature of wave mixing for every nonlinear process order, based on input and output fields87, but has yet to be realised.

A peculiar effect observed in the nonlinear regime is phase conjugation, where the generated beam has the conjugate (negative) phase of a impinging beam. The allure of the nonlinear approach is that no knowledge of the initial phase is required for the process, unlike linear phase conjugation that always requires some wavefront sensing and adaptive control. In nonlinear optics this effect was first achieved and historically associated with four-wave mixing, but it has been shown that a second order effect, Stimulated Parametric Down Conversion, can partially achieve it, conjugating the transverse phase structure88 but not the propagation direction. It has been demonstrated with scalar89 and as well as vector90-92 beams.

3.8 Spin-orbit coupling

In paraxial optics, the spin angular momentum and the orbital angular momentum of a photon are treated as independent degrees of freedom. But even in this regime, we can find instances of these two quantities coupled. A notable example is a special group of vectorial inhomogeneous beams made of spatial modes carrying different OAM in polarization components carrying SAM. Besides these vector vortex beams, conical diffraction93 has been shown to produce optical vortices in the linear regime depending on the input SAM, effectively coupling them. Conical diffraction is a consequence of birefringence and has been reported to excite second harmonic generation in biaxial crystals94-96. The combination of conical diffraction with nonlinear process such as second harmonic generation can be combined to create cascaded processes that operate both on OAM and SAM97. In this interesting example, the SAM is converted into OAM by conical diffraction, but only partially. The two parts (converted and unconverted) then act as fundamental fields for a SHG process of each state. The resulting beams from this conversion also suffer conical diffraction, having their OAM altered according to their SAM. By starting with a simple Gaussian beam with SAM, the authors show these two DoFs can be strongly coupled even in a simple material. However, these two degrees of freedom, while independent and possibly coupled, can interact in a nonlinear process70, as depicted in Fig. 2(e). A spin-orbit coupled beam of OAM lω is combined inside the crystal with another beam only having SAM Sω, resulting in the generation of a beam having OAM l2ω=lω+Sω.

3.9 Intra-cavity dynamics

Lasers are a well known nonlinear device, and here too structured light laser cavities have a long history (see ref.98 for a review), with internal frequency generation used extensively for OAM generation99 and even with wavelength tuneability100. While a full review is beyond the scope of this article (see refs.98-100 for good reviews), we briefly highlight some interesting advances. These include intra-cavity geometric phase101 for helicity control, spin-orbit effects102 with high purity, vortex OPOs103 to move into the mid-infrared, wavelength and OAM tuneable lasers104 based on fibre geometries. Most of these solutions have been at low power. Nonlinear laser amplifiers have been used to raise the power levels, both in bulk crystals105 and disks106 with vectorial light, including parameteric amplification of ultrafast structured light107, and with scalar structured light in Erbium fibre amplifiers108 as well as by Raman amplification109.

Frequency converting cavities for structured light at the source include the use of exotic intra-cavity elements such as spatial light modulators for radial modes110 and metasurfaces for super-chiral OAM modes111, with recent work extending to vortex lattices112 and Poincaré beams113. Nonlinear optical elements are often placed in cavities to enhance the efficiency, but this too can influence modal structure. Nonlinear cavities such as Optical Parametric Oscillators (OPOs) show rich behaviour not seen in free-space propagation. For example, controlling the spatial properties of a Gaussian pumped triple resonant OPO changes its threshold and allows for simultaneous oscillation of several mode pairs with fixed relative phases46, and can result in multiple complex patterns114, 115. A thorough study on the influence of the geometrical properties of the OPO on the generated spatial modes can be seen in116 and their applications in continuous variable entanglement in117. The structured output can be tailored by structuring the pump56, 118, as can the geometry of the cavity itself119, making the cavity selective to specific modes.

4 Structured matter for structured light

The nonlinearity we are discussing refers to the interaction of light and matter. The structure of the output light (created or detected) is therefore tailored by both the input light and the medium, allowing the latter to be tailored. This is achieved when the medium higher-order susceptibility is no longer a constant but instead has a space dependency, e.g., χ(2)(r) for the second order term. The structuring of the medium can be a very important tool to shape the outcome of a nonlinear process. We will now present two of the more prominent structured media in the field: crystals and metasurfaces.

4.2 Crystals

In the past this structuring of crystals has been done through acousto-optic modulators, giving rise to effects such as Bragg and Raman-Nath scattering, modulating the refractive index hence the phase matching conditions as well. The modern toolkit includes more direct manipulation of materials (e.g., structured photonic crystal). Phase matching in nonlinear photonic crystals has been well explained and explored123-125 with periodic poling playing a important role in the past decade126, branching into many applications, including a nonlinear version of the Talbot effect127. By introducing a carefully crafted spatial modulation in a nonlinear crystal, it was shown to be possible to control the amplitude and phase of the generated fields128-130. One highlight is the work illustrated in Fig. 3(a) where the authors carefully exploit the inversion of dipole domains to “twist” light as it is created130. At any point in the fundamental beam’s spatial profile there is light conversion with the same efficiency, but not the same phase. This phase modulation acts as a medium-enabled nonlinear holography.

Fig. 4. Nonlinear optics enabled metasurfaces. These devices were shown to enable non-trivial interactions while frequency converting beams. In (a) a SHG process coupling SAM and OAM. The combination of frequency conversion with holography creates metasurfaces with metalensing properties in (b)144. An application taking advantage of the high damage threshold of these materials can be seen in (c)111 where the inclusion of a metasurface inside an optical cavity creates a laser with OAM from the source. Figure repoduced with permission from: (a) ref.143, © American Chemical Society; (b) ref.144, under a Creative Commons Non-Commercial No Derivative Works (CC-BY-NC-ND) Attribution License; (c) ref.111, Springer Nature.

下载图片 查看所有图片

The phase-matching conditions involves not only material but also energy constrains. The periodic polling can not only enable frequency control131, 132 but when multiplexed it achieves phase-matching for multiple wavelengths in the same crystal133. Recently, a novel pattern in the periodic polling named quasi-periodic polling achieved simultaneous second and third harmonic generation134. Further, the structuring of the media is not restricted to one dimension: by using oblique incidence on a periodically polled crystal it was possible to couple mode selection with phase matching135, coupling DoFs of light and matter. Photonic crystals can be structured so that phase-matching is crafted in both longitudinal and transverse directions136-138 so that light is structured as it is generated. A thorough review on this emerging area can be seen in ref.139.

An interesting combination of birefringence and periodic polling can be seen in ref.140, where the spatial macroscopic structure complements the unit cell structure to achieve both type-0 and type-II phase-matching simultaneously. Besides changing the structure itself, changing the orientation of the medium can achieve interesting results. The sandwich crystal configuration (a combination of two identical crystals optically joined but oriented at 90°) has been employed for the frequency conversion of vector light82.

As much as the structured of the medium dictates phase-matching, the other way around also happens: we can use this property of the medium from a material analysis perspective141 and use these nonlinear process to characterize crystals according to their symmetry groups142.

4.3 Metasurfaces

The structuring of the medium is not exclusive to crystals, as metasurfaces have been employed in many areas and nonlinear optics is no exception. They have seen a lot of atention recently by achieving high conversion efficiencies. The nanostructures composing these crafted surfaces are capable of confining light in volumes smaller than the diffraction limit145, 146, greatly enhancing nonlinear effects. Excellent reviews can be found in ref.145, 147, 148. They are structured by definition and can combine wavelength conversion with wavefront control149-151, spin-orbit interactions152, OAM operations involving SAM143, image encoding153 and optical activity154. Two illustrative cases can be highlighted: OAM-SAM interactions143 and metalensing144. By creating gold meta-atoms with three-fold symmetry, the authors in ref.143 arranged the metasurface to enable azimuthal geometric phase and frequency conversion at the same time, creating devices depicted in Fig. 4(a) that operates on both SAM and OAM. In the second one, illustrated in Fig. 4(b), the authors combine a novel technique that exploits Mie ressonance in all-dielectric metasurfaces and third harmonic generation. The phase of the generated wave inherits a metalens profile from the medium structure. This results in a process that illuminates an aperture with light of a given wavelength and then, after passing through the metasurface, it is converted to its third harmonic and imaged at a focal point. All in a flat and compact optical component. The development of metasurfaces has allowed tremendous growth in nonlinear optics, not only because of their high efficiency, but their fabrication process being scalable and the high damage threshold needed for laser sources integration. In ref.111 the authors demonstrate how a metasurface placed inside a laser cavity can generate high purity OAM modes from the source, depicted in Fig. 4(c).

Fig. 3. Nonlinear Holography. In (a) the structuring of the medium is illustrated: the fundamental field is always the same, but the medium is not. The selective inversion of the electric domain across the transverse plane creates different spatial structures in the second harmonic field. The periodical transverse structure is responsible for multiple phase matching mechanisms, both longitudinally and transversely. In (b) it is shown how non-collinear SHG can transfer a specific intensity pattern from one wavelength to the other. First row shows the imaging arrangement and the second column shows the phase-matching conditions and an example of output modes. Right below is a experimental demonstration that this can be used for real-time frequency conversion of computer generated holograms. Figure repoduced from: (a) ref.130, © American Physical Society; (b) ref.158, © Optica Publishing Group.

下载图片 查看所有图片

5 Nonlinear holography

Since very early in the study of nonlinear optics, it was understood that wave mixing meant modulation and that this could be used for holography155. In the original version, the counter-propagating fields involved in the four-wave mixing formed a grating that changed the generated field. Nowadays, we have more advanced forms of holography. When looking at Eq. (3), it is clear that all fields involved in wave mixing influence each other in amplitude and phase. But more importantly, it has come to a collective understanding: the optical field involved in wave-mixing can be seen as diffracted by the other involved fields. Going back to Eq. (3) we can set E1(ω1) to be a plane wave and E2(ω2) a diffraction pattern, both in a non-depleting regime happening only at a single plane in propagation. This would generate a field E3(ω3) not different than a simple plane wave passing through the same diffraction obstacle.

In this sense, by shaping the fundamental beam as a hologram it is possible to modulate the generated beam as it is created156. This process allows for holograms that are self adaptive and depend on the generating fields, being able to copy or regenerate optical modes157, even complex patterns in real time158. In this example, illustrated in Fig. 3(b), the authors generate a hologram and the light affected by it is filtered in the Fourier domain. Only the first order, containing the intended pattern, and the zero-th order, containing a Gaussian profile, are selected. Those are used as inputs for non-collinear SHG, and the result in the intermediate path is the frequency conversion of any pattern encoded in the hologram in real time. The authors demonstrate this by encoding the holograms with frames of a movie of a running horse in a infrared laser and detecting the same frames on the visible green light.

If the interaction happens in more than a single plane, i.e. the medium is longer than a diffraction length, these approaches can be extended to three dimensions for volume holography159, 160 in nonlinear crystals, and the reader is referred to refs.161-163 for excellent reviews on this topic.

6 Four-wave mixing

As we consider higher-order nonlinear effects, wave-mixing becomes increasingly complex. For example, OAM conservation in a four-wave mixing (FWM) process with third order nonlinearity was observed in cold cesium gas in ref.164 where only one beam was structured with OAM, resulting in the transfer of OAM to the generated beam and similarly with modal superposition165. This was later expanded to include both probe and pump having OAM166, 167, where the phase matching conditions can be fulfilled in more directions than lower order process and this results in the creation of a higher number of states created in different paths, as depicted in Fig. 5(a). Beyond OAM, a hot atomic vapour of 85Rb was used to generate Bessel beams from Gaussian pumps by careful control of the phase matching168. With the same medium, a multimode four-wave-mixing process was established with two pump beams of the same frequency that crossed at a small angle, producing three photons that are highly correlated and could be applied to multipartite entanglement distribution169. The idea exploited the simultaneous fulfillment of two phase matching conditions that reinforce one another.

Fig. 5. Higher order process. In the generation of high harmonic orders, it is possible to generate beams of many different OAM from just two different inputs, as depicted in (a). The process of writting and reading optical memory is depicted in (b) and the diference in time scales depending on the order of the nonlinear process in (c). In (d) it is demonstrated robust self-trapping of a bright vortex beam by exploiting higher order nonlinearities of odd orders. Figure repoduced from: (a) ref.209, Springer Nature; (b) ref.191, © Optica Publishing Group; (c) ref.192, © Optica Publishing Group; (d) ref.204, © American Physical Society.

下载图片 查看所有图片

Using a long medium approximation, radial and angular mode conversion by FWM in a heated Rb vapour was demonstrated, making evident the role of the Gouy phase-matching in this regime170. Beyond just spatial DoFs, the spatial and temporal DoFs are not independent in this process171, where frequency control enables selection of various spatial modes as outputs.

Recent developments with dielectric materials have been shown to enable four-wave mixing with high efficiency. These materials have been crafted in the nanoscale as plasmonic nanoantennas172, 173, metasurfaces145-148, nanodisks174, enabling not only frequency conversion to a wider range of wavelengths but the intrinsic structure also motivated simultaneous wavefront shaping144.

7 High-harmonic generation

High-harmonic generation (HHG) is an extreme process, not regarded as perturbative process and cannot be represented in Eq. (2). This can be seen phenomenologicaly by the fact that all harmonics generated have comparable efficiencies, unlike parametric frequency conversion. Instead, HHG is defined by the ionization of the medium: light impinging in a medium is strong enough that it perturbs an electron bound to an atomic system to the point where it escapes its bounding potential. When this electron is recaptured by an identical atomic system, it liberates the kinetic energy stored, emitting a photon of an energy many times the absorbed ones. This description is known as the recombination model175. The different underlying physics makes this process still a mystery to be studied in the context of structured light interaction with matter. In recent years, there has been considerable progress tackling this problem. One might wonder if OAM would be conserved or how SAM would affect this process. A few studies observed that the polarization of the impinging light can be controlled176 even at isolated pulses timescale177. Regarding the spatial structure, some previously unseen behaviour was demonstraded. When using optical phase vortices, OAM operations happen periodically (along harmonic order)178. Analogous to phase matching conditions in a non-collinear parametric process, it was possible generate many other beams with just two inputs having differing OAM179, demonstrating OAM algebra. Not only the azimuthal degree of freedom was studied, but the radial structure was also studied in ref.180 to show its dependency on the atomic dipole phase. An exciting application is the control of generated spectral domain via structuring the pump to generate an effective blazed active grating in gases181, 182 and the generation of autofocusing intense beams183. While these studies were done separate, others show that these two DoF are not independent in this process and use polarization as a control parameter of this process184, 185. Since the output was made of many different frequencies with different OAM, this effect was characterized as producing Spatio-Temporal vortex in extreme UV186 and even self-torque, a behaviour previously not seen in light187.

8 Space-time coupling

The medium cannot interact instantly with light: first, structured light interacts with a medium that inherits this structure momentarily. When the first light source is no longer there, a second light source interacts with the medium and inherits the structure of the first one. This effect is known as optical memory and is regarded as a possibility for storing quantum information in a multi-dimensional state space. A demonstration of this principle was observed in ref.188, where light interacting with an atomic system (cold cesium gas) induced by a coherence grating lead to OAM conservation, a first step towards the demonstration of optical storage. This spatially dependent coherence transferred to the medium was shown be maintained in time189, reporting storage times of up to 100 μs. It was shown in ref.190 that it is possible to store OAM in the same system and also retrieve it by employing Bragg diffraction. The same effect was also achieved in ref.191 but exploiting a different effect: coherent population oscilation, which uses the long relaxation time of the ground state of an open two-level system to store information carried by a light field. This process is depicted in Fig. 5(b), where a writting stage is the interference of two beams carrying opposite OAM inside the medium. The reading stage is a Gaussian beam that enters the medium and exits with information from a beam which was no longer there. In Fig. 5(c) it is shown that different nonlinearity orders exhibit different time signatures, which can be used as a control mechanism192. Advancing on the path of long lived optical memory storage, by exploring electromagnetic induced transparency, in ref.193 the authors were able to execute OAM storage and retrieval as a reversible process in single photon level.

While the process described above couples a specific structure to another during a time window, there are light structures that are notorious for having its time and space non-separable: the spatiotemporal optical vortices14, 194. These beams exhibit OAM transverse to propagation direction, instead of usual longitudinal OAM of phase vortex beams. One might wonder if these space-time structures would hold in the nonlinear regime. Recent works showed that in SHG the spatiotemporal OAM is also conserved195, 196, while also reporting effects such as time astigmatism and singularity splitting due to group-velocity dispersion.

9 Spatial solitons

The self-focusing action of a medium can balance precisely the diffraction of a beam, resulting in the creation of optical solitons. The first observed optical solitons were dark vortex solitons, which are phase vortices that propagate in a self-defocusing medium with third order nonlinearity χ(3)<0197. This means that the beam modulate itself, with a defocusing effect shaped across the transverse plane by the intensity profile. The dark central of a vortice would naturally increase due to propagation, but a self-defocusing effect in the bright regions would redistribute the intensity of the ring back to the center. The balance of these two process creates a dark soliton: a dark region that does not diffract in propagation.

On the other hand, bright phase vortices suffer from azimuthal modulation instability in self focusing media, which results in their splitting and thus, were hard to be observed. This type of instability in the transverse modulation is similar to one responsible on the filamentation of beams and generation on trains of optical solitons198.

However, by using non-centrosymetric metal-dielectric nanocomposites, higher-order nonlinear effects such as fifth and seventh order become dominant and cause self-phase modulation199, 200. This ultimately allowed for the observation of stable bright vortex solitons in ref.201-204. In Fig. 5(d) this is illustrated in two columns: lower intensity (left) and higher intensity (right). For lower intensities, the natural diffraction of the beam propagation happens as usual as the beam size increases in propagation. For higher intensities, the beam size stays roughly the same in a short propagation distance inside this medium. This happens because the self-modulation effect is caused by nonlinear polarization of odd orders which alternate in sign. The lower orders can saturate, so by increasing intensity, the higher orders nonlinear effects becomes dominant and balances defocusing with focusing. For more detailed information, excellent reviews are found in refs.205-208.

10 Quantum regime

Nonlinear processes have long been associated with quantum optics as the source of entangled photons. The most common source of entangled photons is Spontaneous Parametric Down Conversion (SPDC)210, a nonlinear process at its core. By harnessing entanglement and the transverse structure of the photons it is possible to increase the dimensions of quantum protocols6. This is often achieved by post-selecting a particular state, the choice of which affects the bi-photon entanglement spectrum in both its shape and dimensionality. This was first realized using OAM37 and subsequently many transverse structures were studied211-215, as well as inhomogenously polarized beams92, 216 and multi-path schemes217, 218. Soon after it followed that it was possible to engineer the pump profile to manipulate the bi-photon spectrum and generate a entanglement spectrum straight out of the source219-223. Beyond nonlinear optics for creation, the detection and control of quantum states by nonlinear processes has been far less studies, and very much in its infancy.

Although quantum technologies have experienced rapid development in recent years, with light playing a key role, this has mostly been restricted to linear optical solutions, e.g., the ubiquitous beam splitter. For optical systems, a photon-photon interaction in vacuum is not possible. While this is partially true in matter as well, we observe in the nonlinear regime a photon-photon interaction mediated by the medium. Unfortunately this interaction is very unlikely to happen, but it does not mean impossible as this mixture have seen important advances recently (see ref.224 for a good review), with the building block of single photon wave mixing225. Nonlinear optics have been suggested in various quantum processes226-229 and even used for Bell filters230 for polarization, entanglement swapping231 and a quantum repeater device232, 233. Only recently has structured light entered the equation, with a nonlinear version of spatial teleportation demonstrated with up to 10 modes, overcoming the significant hurdle of ancilliary photons and settting a new state-of-the-art of 10 dimensional teleportation234.

11 Conclusion

In this review we have touched many topics regarding nonlinear optics with structured light. Unlike linear optics, which generally act on only one degree of freedom, these process have the intriguing feature of coupling many DoFs through the properties of the medium. The possibility is for compact solutions for the creation, control and detection of structured light, yet many open questions remain: what structures can we create? How can we transfer structures within and between DoFs? What is the exact input one would need to generate a specific desired output? These questions are still open even in the lowest order of wave mixing. As new light-matter interactions are discovered in the nonlinear regime, it is exciting to see how their structures couple and what insights can be deduced.

From real time holographic transmission to optical memory effects, from bulk crystalline media to sparse gas jets, there are many physical phenomena that are nonlinear optical processes. The development of new materials, techniques and interactions, alongside ever more powerful laser sources, all signal an exciting future for nonlinear control of structured light, and structured light control of nonlinear processes.

References

[1] Forbes A, de Oliveira M, Dennis MRStructured lightNat Photonics20211525326210.1038/s41566-021-00780-4

[2] Otte E, Alpmann C, Denz CPolarization singularity explosions in tailored light fieldsLaser Photonics Rev201812170020010.1002/lpor.201700200

[3] Rosales-Guzmán C, Ndagano B, Forbes AA review of complex vector light fields and their applicationsJ Opt20182012300110.1088/2040-8986/aaeb7d

[4] Willner AE, Huang H, Yan Y, Ren Y, Ahmed N et alOptical communications using orbital angular momentum beamsAdv Opt Photonics201576610610.1364/AOP.7.000066

[5] Padgett MJOrbital angular momentum 25 years on [Invited]Opt Express201725112651127410.1364/OE.25.011265

[6] Forbes A, Nape IQuantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured lightAVS Quantum Sci2019101170110.1116/1.5112027

[7] Erhard M, Fickler R, Krenn M, Zeilinger ATwisted photons: new quantum perspectives in high dimensionsLight Sci Appl201871714610.1038/lsa.2017.146

[8] Larocque H, Sugic D, Mortimer D, Taylor AJ, Fickler R et alReconstructing the topology of optical polarization knotsNat Phys2018141079108210.1038/s41567-018-0229-2

[9] Galvez EJ, Rojec BL, Kumar V, Viswanathan NKGeneration of isolated asymmetric umbilics in light’s polarizationPhys Rev A20148903180110.1103/PhysRevA.89.031801

[10] Zdagkas A, Shen YJ, McDonnell C, Deng J, Li G et al. Observation of toroidal pulses of light. arXiv: 2102.03636 (2021).

[11] Keren-Zur S, Tal M, Fleischer S, Mittleman DM, Ellenbogen TGeneration of spatiotemporally tailored terahertz wavepackets by nonlinear metasurfacesNat Commun201910177810.1038/s41467-019-09811-9

[12] Bauer T, Banzer P, Karimi E, Orlov S, Rubano A et alOptics. Observation of optical polarization Möbius stripsScience201534796496610.1126/science.1260635

[13] Dallaire M, McCarthy N, Piché MSpatiotemporal Bessel beams: theory and experimentsOpt Express200917181481816410.1364/OE.17.018148

[14] Chong A, Wan CH, Chen J, Zhan QWGeneration of spatiotemporal optical vortices with controllable transverse orbital angular momentumNat Photonics20201435035410.1038/s41566-020-0587-z

[15] Kondakci HE, Abouraddy AFDiffraction-free space-time light sheetsNat Photonics20171173374010.1038/s41566-017-0028-9

[16] Shen YJ, Hou YN, Papasimakis N, Zheludev NISupertoroidal light pulses as electromagnetic skyrmions propagating in free spaceNat Commun202112589110.1038/s41467-021-26037-w

[17] Shen YJ, Nape I, Yang XL, Fu X, Gong ML et alCreation and control of high-dimensional multi-partite classically entangled lightLight Sci Appl2021105010.1038/s41377-021-00493-x

[18] Shen YJ, Yang XL, Naidoo D, Fu X, Forbes AStructured ray-wave vector vortex beams in multiple degrees of freedom from a laserOptica2020782083110.1364/OPTICA.382994

[19] Spreeuw RJCA classical analogy of entanglementFound Phys19982836137410.1023/A:1018703709245

[20] Ndagano B, Perez-Garcia B, Roux FS, McLaren M, Rosales-Guzman C et alCharacterizing quantum channels with non-separable states of classical lightNat Phys20171339740210.1038/nphys4003

[21] Forbes A, Aiello A, Ndagano BClassically entangled lightProg Opt20196499153

[22] Aiello A, Banzer P, Neugebauer M, Leuchs GFrom transverse angular momentum to photonic wheelsNat Photonics2015978979510.1038/nphoton.2015.203

[23] Padgett MJ, Courtial JPoincaré-sphere equivalent for light beams containing orbital angular momentumOpt Lett19992443043210.1364/OL.24.000430

[24] Milione G, Sztul HI, Nolan DA, Alfano RRHigher-order poincaré sphere, stokes parameters, and the angular momentum of lightPhys Rev Lett201110705360110.1103/PhysRevLett.107.053601

[25] Shen YJRays, waves, SU(2) symmetry and geometry: toolkits for structured lightJ Opt20212312400410.1088/2040-8986/ac3676

[26] Mazilu M, Stevenson DJ, Gunn-Moore F, Dholakia KLight beats the spread: “non-diffracting” beamsLaser Photonics Rev2010452954710.1002/lpor.200910019

[27] Gossman D, Perez-Garcia B, Hernandez-Aranda RI, Forbes AOptical interference with digital hologramsAm J Phys20168450851610.1119/1.4948604

[28] Ayuso D, Neufeld O, Ordonez AF, Decleva P, Lerner G et alSynthetic chiral light for efficient control of chiral light-matter interactionNat Photonics20191386687110.1038/s41566-019-0531-2

[29] Maiman THStimulated optical radiation in rubyNature196018749349410.1038/187493a0

[30] Franken PA, Hill AE, Peters CW, Weinreich GGeneration of optical harmonicsPhys Rev Lett1961711811910.1103/PhysRevLett.7.118

[31] New GHC, Ward JFOptical third-harmonic generation in gasesPhys Rev Lett19671955655910.1103/PhysRevLett.19.556

[32] Simon HJ, Bloembergen NSecond-harmonic light generation in crystals with natural optical activityPhys Rev19681711104111410.1103/PhysRev.171.1104

[33] Abraham NB, Firth WJOverview of transverse effects in nonlinear-optical systemsJ Opt Soc Am B19907951962

[34] Basistiy IV, Bazhenov VY, Soskin MS, Vasnetsov MVOptics of light beams with screw dislocationsOpt Commun199310342242810.1016/0030-4018(93)90168-5

[35] Shen YJ, Wang XJ, Xie ZW, Min CJ, Fu X et alOptical vortices 30 years on: OAM manipulation from topological charge to multiple singularitiesLight Sci Appl201989010.1038/s41377-019-0194-2

[36] Dholakia K, Simpson NB, Padgett MJ, Allen LSecond-harmonic generation and the orbital angular momentum of lightPhys Rev A199654R3742R374510.1103/PhysRevA.54.R3742

[37] Mair A, Vaziri A, Weihs G, Zeilinger AEntanglement of the orbital angular momentum states of photonsNature200141231331610.1038/35085529

[38] Boyd RW. NonlinearOptics 3rd ed (Elsevier, Oxford, 2008).

[39] Murti YVGS, Vijayan C. EssentialsofNonlinearOptics (John Wiley & Sons, New York, 2014).

[40] Shen YR. ThePrinciplesofNonlinearOptics (John Wiley & Sons, New York, 1984).

[41] Singh K, Buono WT, Chavez-Cerda S, Forbes ADemonstrating arago-fresnel laws with Bessel beams from vectorial axiconsJ Opt Soc Am A2021381248125410.1364/JOSAA.431186

[42] Zhou ZY, Li Y, Ding DS, Jiang YK, Zhang W et alGeneration of light with controllable spatial patterns via the sum frequency in quasi-phase matching crystalsSci Rep201445650

[43] Shao GH, Wu ZJ, Chen JH, Xu F, Lu YQNonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matchingPhys Rev A20138806382710.1103/PhysRevA.88.063827

[44] Steinlechner F, Hermosa N, Pruneri V, Torres JPFrequency conversion of structured lightSci Rep201662139010.1038/srep21390

[45] Li Y, Zhou ZY, Ding DS, Shi BSSum frequency generation with two orbital angular momentum carrying laser beamsJ Opt Soc Am B20153240741110.1364/JOSAB.32.000407

[46] Schwob C, Cohadon PF, Fabre C, Marte MAM, Ritsch H et alTransverse effects and mode couplings in OPOSAppl Phys B19986668569910.1007/s003400050455

[47] Buono WT, Moraes LFC, Huguenin JAO, Souza CER, Khoury AZArbitrary orbital angular momentum addition in second harmonic generationNew J Phys20141609304110.1088/1367-2630/16/9/093041

[48] Roger T, Heitz JJF, Wright EM, Faccio DNon-collinear interaction of photons with orbital angular momentumSci Rep20133349110.1038/srep03491

[49] Bovino FA, Braccini M, Giardina M, Sibilia COrbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beamsJ Opt Soc Am B2011282806281110.1364/JOSAB.28.002806

[50] Buono WT, Santiago J, Pereira LJ, Tasca DS, Dechoum K et alPolarization-controlled orbital angular momentum switching in nonlinear wave mixingOpt Lett2018431439144210.1364/OL.43.001439

[51] Courtial J, Dholakia K, Allen L, Padgett MJSecond-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modesPhys Rev A1997564193419610.1103/PhysRevA.56.4193

[52] Pereira LJ, Buono WT, Tasca DS, Dechoum K, Khoury AZOrbital-angular-momentum mixing in type-II second-harmonic generationPhys Rev A20179605385610.1103/PhysRevA.96.053856

[53] Pires DG, Rocha JCA, Jesus-Silva AJ, Fonseca EJSHigher radial orders of Laguerre-Gaussian beams in nonlinear wave mixing processesJ Opt Soc Am B2020371328133210.1364/JOSAB.384112

[54] Wu HJ, Mao LW, Yang YJ, Rosales-Guzmán C, Gao W et alRadial modal transitions of Laguerre-Gauss modes during parametric up-conversion: towards the full-field selection rule of spatial modesPhys Rev A202010106380510.1103/PhysRevA.101.063805

[55] Buono WT, Santos A, Maia MR, Pereira LJ, Tasca DS et alChiral relations and radial-angular coupling in nonlinear interactions of optical vorticesPhys Rev A202010104382110.1103/PhysRevA.101.043821

[56] Alves GB, Barros RF, Tasca DS, Souza CER, Khoury AZConditions for optical parametric oscillation with a structured light pumpPhys Rev A20189806382510.1103/PhysRevA.98.063825

[57] Pires DG, Rocha JCA, da Silva MVEC, Jesus-Silva AJ, Fonseca EJSMixing Ince-Gaussian modes through sum-frequency generationJ Opt Soc Am B2020372815282110.1364/JOSAB.401001

[58] Yang HR, Wu HJ, Gao W, Rosales-Guzmán C, Zhu ZHParametric upconversion of Ince-Gaussian modesOpt Lett2020453034303710.1364/OL.393146

[59] Jarutis V, Matijošius A, Smilgevičius V, Stabinis ASecond harmonic generation of higher-order Bessel beamsOpt Commun200018515916910.1016/S0030-4018(00)00974-3

[60] Ding DS, Lu JYSecond-harmonic generation of the nth-order Bessel beamPhys Rev E2000612038204110.1103/PhysRevE.61.2038

[61] Shinozaki K, Xu CQ, Sasaki H, Kamijoh TA comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystalsOpt Commun199713330030410.1016/S0030-4018(96)00413-0

[62] Rao AS, Yadav D, Samanta GKNonlinear frequency conversion of 3D optical bottle beams generated using a single axiconOpt Lett20214665766010.1364/OL.413899

[63] Pires DG, Rocha JCA, Jesus-Silva AJ, Fonseca EJSInteraction of fractional orbital angular momentum in two-wave mixing processesJ Opt20202203550210.1088/2040-8986/ab6ae6

[64] Dai KJ, Miller JK, Li WZ, Watkins RJ, Johnson EGFractional orbital angular momentum conversion in second-harmonic generation with an asymmetric perfect vortex beamOpt Lett2021463332333510.1364/OL.428859

[65] Rao ASCharacterization of off-axis phase singular optical vortex and its nonlinear wave-mixing to generate control broad OAM spectraPhys Scr20209505550810.1088/1402-4896/ab7b09

[66] Zhdanova AA, Shutova M, Bahari A, Zhi MC, Sokolov AVTopological charge algebra of optical vortices in nonlinear interactionsOpt Express201523341093411710.1364/OE.23.034109

[67] Wadhwa J, Singh ASecond harmonic generation of self-focused Hermite-Gaussian laser beam in collisional plasmaOptik202020216232610.1016/j.ijleo.2019.01.116

[68] Xiong XYZ, Al-Jarro A, Jiang LJ, Panoiu NC, Sha WEIMixing of spin and orbital angular momenta via second-harmonic generation in plasmonic and dielectric chiral nanostructuresPhys Rev B20179516543210.1103/PhysRevB.95.165432

[69] Wu HJ, Zhao B, Rosales-Guzmán C, Gao W, Shi BS et alSpatial-polarization-independent parametric up-conversion of vectorially structured lightPhys Rev Appl20201306404110.1103/PhysRevApplied.13.064041

[70] da Silva BP, Buono WT, Pereira LJ, Tasca DS, Dechoum K et alSpin to orbital angular momentum transfer in frequency up-conversionNanophotonics20211177177810.1515/nanoph-2021-0493

[71] Sephton B, Vallés A, Steinlechner F, Konrad T, Torres JP et alSpatial mode detection by frequency upconversionOpt Lett20194458658910.1364/OL.44.000586

[72] Pires DG, Rocha JCA, Jesus-Silva AJ, Fonseca EJSSuitable state bases for nonlinear optical mode conversion protocolsOpt Lett2020454064406710.1364/OL.394640

[73] Fang XY, Kuang ZY, Chen P, Yang HC, Li Q et alExamining second-harmonic generation of high-order Laguerre-Gaussian modes through a single cylindrical lensOpt Lett2017424387439010.1364/OL.42.004387

[74] Kumar S, Zhang H, Maruca S, Huang YPMode-selective image upconversionOpt Lett2019449810110.1364/OL.44.000098

[75] Zhang H, Kumar S, Huang YPMode selective up-conversion detection with turbulenceSci Rep201991748110.1038/s41598-019-53914-8

[76] Pinnell J, Nape I, Sephton B, Cox MA, Rodríguez-Fajardo V et alModal analysis of structured light with spatial light modulators: a practical tutorialJ Opt Soc Am A202037C146C16010.1364/JOSAA.398712

[77] Qiu XD, Li FS, Zhang WH, Zhu ZH, Chen LXSpiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illuminationOptica2018520821210.1364/OPTICA.5.000208

[78] Xu DF, Ma TL, Qiu XD, Zhang WH, Chen LXImplementing selective edge enhancement in nonlinear opticsOpt Express202028323773238510.1364/OE.404594

[79] Hong L, Lin F, Qiu XD, Chen LXSecond harmonic generation based joint transform correlator for human face and QR code recognitionsAppl Phys Lett202011623110110.1063/5.0001301

[80] Zhang L, Qiu XD, Li FS, Liu HG, Chen XF et alSecond harmonic generation with full Poincaré beamsOpt Express20182611678118410.1364/OE.26.011678

[81] Liu HG, Li H, Zheng YL, Chen XFNonlinear frequency conversion and manipulation of vector beamsOpt Lett2018435981598410.1364/OL.43.005981

[82] Saripalli RK, Ghosh A, Chaitanya NA, Samanta GKFrequency-conversion of vector vortex beams with space-variant polarization in single-pass geometryAppl Phys Lett201911505110110.1063/1.5111593

[83] Wu HJ, Zhou ZY, Gao W, Shi BS, Zhu ZHDynamic tomography of the spin-orbit coupling in nonlinear opticsPhys Rev A20199902383010.1103/PhysRevA.99.023830

[84] Bouchard F, Larocque H, Yao AM, Travis C, De Leon I et alPolarization shaping for control of nonlinear prop­agationPhys Rev Lett201611723390310.1103/PhysRevLett.117.233903

[85] Yang C, Zhou ZY, Li Y, Li YH, Liu SL et alNonlinear frequency conversion and manipulation of vector beams in a Sagnac loopOpt Lett20194421922210.1364/OL.44.000219

[86] Ren ZC, Lou YC, Cheng ZM, Fan L, Ding JP et alOptical frequency conversion of light with maintaining polarization and orbital angular momentumOpt Lett2021462300230310.1364/OL.419753

[87] Samim M, Krouglov S, Barzda VNonlinear Stokes-Mueller polarimetryPhys Rev A20169301384710.1103/PhysRevA.93.013847

[88] Ribeiro PHS, Caetano DP, Almeida MP, Huguenin JA, dos Santos BC et alObservation of image transfer and phase conjugation in stimulated down-conversionPhys Rev Lett20018713360210.1103/PhysRevLett.87.133602

[89] de Oliveira AG, Arruda MFZ, Soares WC, Walborn SP, Khoury AZ et alPhase conju­gation and mode conversion in stimulated parametric down-conversion With orbital angular momentum: a ge­ometrical interpretationBraz J Phys201949101610.1007/s13538-018-0614-4

[90] de Oliveira AG, Arruda MFZ, Soares WC, Walborn SP, Gomes RM et alReal-time phase conjugation of vector vortex beamsACS Photonics2020724925510.1021/acsphotonics.9b01524

[91] de Oliveira AG, da Silva NR, de Araújo RM, Ribeiro PHS, Walborn SPQuantum optical description of phase conjugation of vector vortex beams in stimulated parametric down-conversionPhys Rev Appl20201402404810.1103/PhysRevApplied.14.024048

[92] da Silva NR, de Oliveira AG, Arruda MFZ, de Araújo RM, Soares WC et alStimulated parametric down-conversion With vector vortex beamsPhys Rev Appl20211502403910.1103/PhysRevApplied.15.024039

[93] Brenier AInvestigation of the sum of orbital angular momentum generated by conical diffractionJ Opt20202204560310.1088/2040-8986/ab76a7

[94] Yu HH, Zhang HJ, Wang ZP, Wang JY, Pan ZB et alExperimental observation of optical vortex in self-frequency-doubling generationAppl Phys Lett20119924110210.1063/1.3670351

[95] Zolotovskaya SA, Abdolvand A, Kalkandjiev TK, Rafailov EUSecond-harmonic conical refraction: observation of free and forced harmonic WavesAppl Phys B201110391210.1007/s00340-011-4484-5

[96] Peet V, Shchemelyov SFrequency doubling with laser beams transformed by conical refraction in a biaxial crystalJ Opt20111305520510.1088/2040-8978/13/5/055205

[97] Tang YT, Li KF, Zhang XC, Deng JH, Li GX et alHarmonic spin-orbit angular momentum cascade in nonlinear optical crystalsNat Photonics20201465866210.1038/s41566-020-0691-0

[98] Forbes AStructured light from lasersLaser Photonics Rev201913190014010.1002/lpor.201900140

[99] Forbes AControlling light’s helicity at the source: orbital angular momentum states from lasersPhilos Trans A Math Phys Eng Sci201737520150436

[100] Omatsu T, Miyamoto K, Lee AJWavelength-versatile optical vortex lasersJ Opt20171912300210.1088/2040-8986/aa9445

[101] Naidoo D, Roux FS, Dudley A, Litvin I, Piccirillo B et alControlled generation of higher-order Poincaré sphere beams from a laserNat Photonics20161032733210.1038/nphoton.2016.37

[102] Wei DZ, Cheng Y, Ni R, Zhang Y, Hu XP et alGenerating controllable Laguerre-Gaussian laser modes through intracavity spin-orbital angular momentum conversion of lightPhys Rev Appl20191101403810.1103/PhysRevApplied.11.014038

[103] Yusufu T, Niu SJ, Tuersun P, Tulake Y, Miyamoto K et alTunable 3 μm optical vortex parametric oscillatorJpn J Appl Phys20185712270110.7567/JJAP.57.122701

[104] Zhou N, Liu J, Wang JReconfigurable and tunable twisted light laserSci Rep201881139410.1038/s41598-018-29868-8

[105] Sroor H, Lisa N, Naidoo D, Litvin I, Forbes ACylindrical vector beams through amplifiersProc SPIE201810511105111M

[106] Ahmed MA, Beirow F, Loescher A, Dietrich T, Bashir D et alHigh-power thin-disk lasers emitting beams with axially-symmetric polarizationsNanophotonics20221183584610.1515/nanoph-2021-0606

[107] Zhong HZ, Liang CC, Dai SY, Huang JF, Hu SS et alPolarization-insensitive, high-gain parametric amplification of radially polarized femtosecond pulsesOptica20218626910.1364/OPTICA.413328

[108] Jung Y, Kang QY, Sidharthan R, Ho D, Yoo S et alOptical orbital angular momentum amplifier based on an air-hole erbium-doped fiberJ Lightwave Technol20173543043610.1109/JLT.2017.2651145

[109] Zhu S, Pidishety S, Feng YT, Hong S, Demas J et alMultimode-pumped Raman amplification of a higher order mode in a large mode area fiberOpt Express201826232952330410.1364/OE.26.023295

[110] Bell T, Kgomo M, Ngcobo SDigital laser for on-demand intracavity selective excitation of second harmonic higher-order modesOpt Express202028169071692310.1364/OE.385569

[111] Sroor H, Huang YW, Sephton B, Naidoo D, Vallés A et alHigh-purity orbital angular momentum states from a visible metasurface laserNat Photonics20201449850310.1038/s41566-020-0623-z

[112] Rao AS, Miike T, Miyamoto K, Omatsu TOptical vortex lattice mode generation from a diode-pumped Pr3+: LiYF4 laserJ Opt20212307550210.1088/2040-8986/ac067d

[113] Rao AS, Miamoto K, Omatsu TUltraviolet intracavity frequency-doubled Pr3+: LiYF4 orbital Poincaré laserOpt Express202028373973740510.1364/OE.411624

[114] Vaupel M, Maître A, Fabre CObservation of pattern formation in optical parametric oscillatorsPhys Rev Lett1999835278528110.1103/PhysRevLett.83.5278

[115] Marte M, Ritsch H, Petsas KI, Gatti A, Lugiato LA et alSpatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effectsOpt Express19983718010.1364/OE.3.000071

[116] Ducci S, Treps N, Maître A, Fabre CPattern formation in optical parametric oscillatorsPhys Rev A20016402380310.1103/PhysRevA.64.023803

[117] Lassen M, Delaubert V, Janousek J, Wagner K, Bachor HA et alTools for multimode quantum information: modulation, detection, and spatial quantum correlationsPhys Rev Lett20079808360210.1103/PhysRevLett.98.083602

[118] Martinelli M, Huguenin JAO, Nussenzveig P, Khoury AZOrbital angular momentum exchange in an optical parametric oscillatorPhys Rev A20047001381210.1103/PhysRevA.70.013812

[119] Barros RF, Alves GB, Tasca DS, Souza CER, Khoury AZFine-tuning of orbital angular momentum in an optical parametric oscillatorJ Phys B At Mol Opt Phys20195224400210.1088/1361-6455/ab4cc8

[120] Qi T, Wang DM, Gao WSum-frequency generation of ring-airy beamsAppl Phys B202212867

[121] Dolev I, Ellenbogen T, Arie ASwitching the acceleration direction of airy beams by a nonlinear optical processOpt Lett2010351581158310.1364/OL.35.001581

[122] Ni R, Niu YF, Du L, Hu XP, Zhang Y et alTopological charge transfer in frequency doubling of fractional orbital angular momentum stateAppl Phys Lett201610915110310.1063/1.4964712

[123] Dmitriev VG, Gurzadyan GG, Nikogosyan DN. HandbookofNonlinearOpticalCrystals 2nd ed (Springer, Berlin, 1997).

[124] Berger VNonlinear photonic crystalsPhys Rev Lett1998814136413910.1103/PhysRevLett.81.4136

[125] Saltiel S, Kivshar YSPhase matching in nonlinear χ(2) photonic crystalsOpt Lett2000251204120610.1364/OL.25.001204

[126] Arie A, Voloch NPeriodic, quasi-periodic, and random quadratic nonlinear photonic crystalsLaser Photonics Rev2010435537310.1002/lpor.200910006

[127] Zhang Y, Wen JM, Zhu SN, Xiao MNonlinear Talbot effectPhys Rev Lett201010418390110.1103/PhysRevLett.104.183901

[128] Shapira A, Juwiler I, Arie ANonlinear computer-generated hologramsOpt Letters2011363015301710.1364/OL.36.003015

[129] Shapira A, Shiloh R, Juwiler I, Arie ATwo-dimensional nonlinear beam shapingOpt Lett2012372136213810.1364/OL.37.002136

[130] Bloch NV, Shemer K, Shapira A, Shiloh R, Juwiler I et alTwisting light by nonlinear photonic crystalsPhys Rev Lett201210823390210.1103/PhysRevLett.108.233902

[131] Shiloh R, Arie ASpectral and temporal holograms with nonlinear opticsOpt Lett2012373591359310.1364/OL.37.003591

[132] Leshem A, Shiloh R, Arie AExperimental realization of spectral shaping using nonlinear optical hologramsOpt Lett2014395370537310.1364/OL.39.005370

[133] Chen PC, Wang CW, Wei DZ, Hu YL, Xu XY et alQuasi-phase-matching-division multiplexing holography in a three-dimensional nonlinear photonic crystalLight Sci Appl20211014610.1038/s41377-021-00588-5

[134] Lou YC, Cheng ZM, Liu ZH, Yang YX, Ren ZC et alThird-harmonic generation of spatially structured light in a quasi-periodically poled crystalOptica2022918318610.1364/OPTICA.449590

[135] Chen Y, Ni R, Wu YD, Du L, Hu XP et alPhase-matching controlled orbital angular momentum conversion in periodically poled crystalsPhys Rev Lett202012514390110.1103/PhysRevLett.125.143901

[136] Wei DZ, Wang CW, Wang HJ, Hu XP, Wei D et alExperimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystalNat Photonics20181259660010.1038/s41566-018-0240-2

[137] Keren-Zur S, Ellenbogen TA new dimension for nonlinear photonic crystalsNat Photonics20181257557710.1038/s41566-018-0262-9

[138] Wei DZ, Wang CW, Xu XY, Wang HJ, Hu YL et alEfficient nonlinear beam shaping in three-dimensional lithium niobate nonlinear photonic crystalsNat Communications201910419310.1038/s41467-019-12251-0

[139] Zhang Y, Sheng Y, Zhu SN, Xiao M, Krolikowski WNonlinear photonic crystals: from 2D to 3DOptica2021837238110.1364/OPTICA.416619

[140] Lee HJ, Kim H, Cha M, Moon HSSimultaneous type-0 and type-II spontaneous parametric downconversions in a single periodically poled KTiOPO4 crystalAppl Phys B201210858558910.1007/s00340-012-5088-4

[141] Zhang WG, Yu HW, Wu HP, Halasyamani PSPhase-matching in nonlinear optical compounds: a materials perspectiveChem Mater2017292655266810.1021/acs.chemmater.7b00243

[142] Jáuregui R, Torres JPOn the use of structured light in nonlinear optics studies of the symmetry group of a crystalSci Repo201662090610.1038/srep20906

[143] Chen SM, Li KF, Deng JH, Li GX, Zhang SHigh-order nonlinear spin-orbit interaction on plasmonic metasurfacesNano Lett2020208549855510.1021/acs.nanolett.0c03100

[144] Schlickriede C, Kruk SS, Wang L, Sain B, Kivshar Y et alNonlinear imaging with all-dielectric metasurfacesNano Lett2020204370437610.1021/acs.nanolett.0c01105

[145] Rahmani M, Leo G, Brener I, Zayats AV, Maier SA et alNonlinear frequency conversion in optical nanoantennas and metasurfaces: materials evolution and fabricationOpto-Electron Adv20181180021

[146] Zhang YB, Liu H, Cheng H, Tian JG, Chen SQMultidimensional manipulation of wave fields based on artificial microstructuresOpto-Electron Adv2020320000210.29026/oea.2020.200002

[147] Pertsch T, Kivshar YNonlinear optics with resonant metasurfacesMRS Bull20204521022010.1557/mrs.2020.65

[148] Grinblat GNonlinear dielectric nanoantennas and metasurfaces: frequency conversion and wavefront controlACS Photonics202183406343210.1021/acsphotonics.1c01356

[149] Wang L, Kruk S, Koshelev K, Kravchenko I, Luther-Davies B et alNonlinear wavefront control with all-dielectric metasurfacesNano Lett2018183978398410.1021/acs.nanolett.8b01460

[150] Li GX, Chen SM, Pholchai N, Reineke B, Wong PWH et alContinuous control of the nonlinearity phase for harmonic generationsNat Mater20151460761210.1038/nmat4267

[151] Gao YS, Fan YB, Wang YJ, Yang WH, Song QH et alNonlinear holographic all-dielectric metasurfacesNano Lett2018188054806110.1021/acs.nanolett.8b04311

[152] Li GX, Wu L, Li KF, Chen SM, Schlickriede C et alNonlinear metasurface for simultaneous control of spin and orbital angular momentum in second harmonic generationNano Lett2017177974797910.1021/acs.nanolett.7b04451

[153] Walter F, Li GX, Meier C, Zhang S, Zentgraf TUltrathin nonlinear metasurface for optical image encodingNano Lett2017173171317510.1021/acs.nanolett.7b00676

[154] Chen SM, Reineke B, Li GX, Zentgraf T, Zhang SStrong nonlinear optical activity induced by lattice surface modes on Plasmonic metasurfaceNano Lett2019196278628310.1021/acs.nanolett.9b02417

[155] Yariv AFour wave nonlinear optical mixing as real time holographyOpt Commun197825232510.1016/0030-4018(78)90079-2

[156] Liu HG, Li J, Fang XL, Zhao XH, Zheng YL et alDynamic computer-generated nonlinear-optical hologramsPhys Rev A20179602380110.1103/PhysRevA.96.023801

[157] Qiu XD, Li FS, Liu HG, Chen XF, Chen LXOptical vortex copier and regenerator in the Fourier domainPhotonics Res2018664164610.1364/PRJ.6.000641

[158] Liu HG, Zhao XH, Li H, Zheng YL, Chen XFDynamic computer-generated nonlinear optical holograms in a non-collinear second-harmonic generation ProcessOpt Lett2018433236323910.1364/OL.43.003236

[159] Liu S, Mazur LM, Krolikowski W, Sheng YNonlinear volume holography in 3D nonlinear photonic crystalsLaser Photonics Rev202014200022410.1002/lpor.202000224

[160] Hong XH, Yang B, Zhang C, Qin YQ, Zhu YYNonlinear volume holography for wave-front engineeringPhys Rev Lett201411316390210.1103/PhysRevLett.113.163902

[161] Trajtenebrg-Mills S, Arie AShaping light beams in nonlinear processes using structured light and patterned crystalsOpt Mater Express201772928294210.1364/OME.7.002928

[162] Shapira A, Naor L, Arie ANonlinear optical holograms for spatial and spectral shaping of light wavesSci Bull2015601403141510.1007/s11434-015-0855-3

[163] Liu HG, Chen XFThe manipulation of second-order nonlinear harmonic wave by structured material and structured lightJ Nonlinear Opt Phys Mater201827185004710.1142/S0218863518500479

[164] Tabosa JWR, Petrov DVOptical pumping of orbital angular momentum of light in cold cesium atomsPhys Rev Lett1999834967497010.1103/PhysRevLett.83.4967

[165] Barreiro S, Tabosa JWR, Torres JP, Deyanova Y, Torner LFour-wave mixing of light beams with engineered orbital angular momentum in cold cesium atomsOpt Lett2004291515151710.1364/OL.29.001515

[166] Prajapati N, Super N, Lanning NR, Dowling JP, Novikova IOptical angular momentum manipulations in a four-wave mixing processOpt Lett20194473974210.1364/OL.44.000739

[167] Offer RF, Stulga D, Riis E, Franke-Arnold S, Arnold ASSpiral bandwidth of four-wave mixing in Rb vapourCommun Phys201818410.1038/s42005-018-0077-5

[168] Danaci O, Rios C, Glasser RTAll-optical mode conversion via spatially multimode four-wave mixingNew J Phys20161807303210.1088/1367-2630/18/7/073032

[169] Knutson EM, Swaim JD, Wyllie S, Glasser RTOptimal mode configuration for multiple phase-matched four-wave-mixing processesPhys Rev A20189801382810.1103/PhysRevA.98.013828

[170] Offer RF, Daffurn A, Riis E, Griffin PF, Arnold AS et alGouy phase-matched angular and radial mode conversion in four-wave mixingPhys Rev A2021103L02150210.1103/PhysRevA.103.L021502

[171] Swaim JD, Knutson EM, Danaci O, Glasser RTMultimode four-wave mixing with a spatially structured pumpOpt Lett2018432716271910.1364/OL.43.002716

[172] Hasan SB, Lederer F, Rockstuhl CNonlinear plasmonic antennasMater Today20141747848510.1016/j.mattod.2014.05.009

[173] Kauranen M, Zayats AVNonlinear plasmonicsNat Photonics2012673774810.1038/nphoton.2012.244

[174] Grinblat G, Li Y, Nielsen MP, Oulton RF, Maier SADegenerate four-wave mixing in a multiresonant germanium nanodiskACS Photonics201742144214910.1021/acsphotonics.7b00631

[175] Corkum PBPlasma perspective on strong field multiphoton ionizationPhys Rev Lett1993711994199710.1103/PhysRevLett.71.1994

[176] Fleischer A, Kfir O, Diskin T, Sidorenko P, Cohen OSpin angular momentum and tunable polarization in high-harmonic generationNat Photonics2014854354910.1038/nphoton.2014.108

[177] Huang PC, Hernández-García C, Huang JT, Huang PY, Lu CH et alPolarization control of isolated high-harmonic pulsesNat Photonics20181234935410.1038/s41566-018-0145-0

[178] Gariepy G, Leach J, Kim KT, Hammond TJ, Frumker E et alCreating high-harmonic beams with controlled orbital angular momentumPhys Rev Lett201411315390110.1103/PhysRevLett.113.153901

[179] Gauthier D, Ribič PR, Adhikary G, Camper A, Chappuis C et alTunable orbital angular momentum in high-harmonic generationNat Commun201781497110.1038/ncomms14971

[180] Géneaux R, Chappuis C, Auguste T, Beaulieu S, Gorman TT et alRadial index of Laguerre-Gaussian modes in high-order-harmonic generationPhys Rev A20179505180110.1103/PhysRevA.95.051801

[181] Chappuis C, Bresteau D, Auguste T, Gobert O, Ruchon THigh-order harmonic generation in an active gratingPhys Rev A20199903380610.1103/PhysRevA.99.033806

[182] Hareli L, Lobachinsky L, Shoulga G, Eliezer Y, Michaeli L et alOn-the-fly control of high-harmonic generation using a structured pump beamPhys Rev Lett201812018390210.1103/PhysRevLett.120.183902

[183] Panagiotopoulos P, Papazoglou DG, Couairon A, Tzortzakis SSharply autofocused ring-Airy beams transforming into non-linear intense light bulletsNat Commun20134262210.1038/ncomms3622

[184] Dorney KM, Rego L, Brooks NJ, Román JS, Liao CT et alControlling the polarization and vortex charge of attosecond high-harmonic beams via simultaneous spin-orbit momentum conservationNat Photonics20191312313010.1038/s41566-018-0304-3

[185] Kong F, Zhang C, Larocque H, Bouchard F, Li Z et alSpin-constrained orbital-angular-momentum control in high-harmonic generationPhys Rev Res2019103200810.1103/PhysRevResearch.1.032008

[186] Géneaux R, Camper A, Auguste T, Gobert O, Caillat J et alSynthesis and characterization of attosecond light vortices in the extreme ultravioletNat Commun201671258310.1038/ncomms12583

[187] Rego L, Dorney KM, Brooks NJ, Nguyen QL, Liao CT et alGeneration of extreme-ultraviolet beams with time-varying orbital angular momentumScience2019364eaaw948610.1126/science.aaw9486

[188] Barreiro S, Tabosa JWRGeneration of light carrying orbital angular momentum via induced coherence grating in cold atomsPhys Rev Lett20039013300110.1103/PhysRevLett.90.133001

[189] Pugatch R, Shuker M, Firstenberg O, Ron A, Davidson NTopological stability of stored optical vorticesPhys Rev Lett20079820360110.1103/PhysRevLett.98.203601

[190] Moretti D, Felinto D, Tabosa JWRCollapses and revivals of stored orbital angular momentum of light in a cold-atom ensemblePhys Rev A20097902382510.1103/PhysRevA.79.023825

[191] de Almeida AJF, Barreiro S, Martins WS, de OliVeira RA, Felinto D et alStorage of orbital angular momenta of light via coherent population oscillationOpt Lett2015402545254810.1364/OL.40.002545

[192] de Oliveira RA, Borba GC, Martins WS, Barreiro S, Felinto D et alNonlinear optical memory for manipulation of orbital angular momentum of lightOpt Lett2015404939494210.1364/OL.40.004939

[193] Veissier L, Nicolas A, Giner L, Maxein D, Sheremet AS et alReversible optical memory for twisted photonsOpt Lett20133871271410.1364/OL.38.000712

[194] Sukhorukov AP, Yangirova VVSpatio-temporal vortices: properties, generation and recordingProc SPIE20055949594906

[195] Gui G, Brooks NJ, Kapteyn HC, Murnane MM, Liao CTSecond-harmonic generation and the conservation of spatiotemporal orbital angular momentum of lightNat Photonics20211560861310.1038/s41566-021-00841-8

[196] Hancock SW, Zahedpour S, Milchberg HMSecond-harmonic generation of spatiotemporal optical vortices and conservation of orbital angular momentumOptica2021859459710.1364/OPTICA.422743

[197] Desyatnikov AS, Kivshar YS, Torner LOptical vortices and vortex solitonsProg Opt200547291391

[198] Kivshar YS, Pelinovsky DESelf-focusing and transverse instabilities of solitary wavesPhys Rep200033111719510.1016/S0370-1573(99)00106-4

[199] Reyna AS, de Araújo CBSpatial phase modulation due to quintic and septic nonlinearities in metal colloidsOpt Express201422224562246910.1364/OE.22.022456

[200] Reyna AS, de Araújo CBNonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearityPhys Rev A20148906380310.1103/PhysRevA.89.063803

[201] Reyna AS, Jorge KC, de Araújo CBTwo-dimensional solitons in a quintic-septimal mediumPhys Rev A20149006383510.1103/PhysRevA.90.063835

[202] Reyna AS, Malomed BA, de Araújo CBStability conditions for one-dimensional optical solitons in cubic-quintic-septimal mediaPhys Rev A20159203381010.1103/PhysRevA.92.033810

[203] Reyna AS, Bergmann E, Brevet PF, de Araújo CBNonlinear polarization instability in cubic-quintic plasmonic nanocompositesOpt Express201725210492106710.1364/OE.25.021049

[204] Reyna AS, Boudebs G, Malomed BA, de Araújo CBRobust self-trapping of vortex beams in a saturable optical mediumPhys Rev A20169301384010.1103/PhysRevA.93.013840

[205] Kivshar YBending light at willNat Phys2006272973010.1038/nphys452

[206] Kivshar YS, Stegeman GISpatial optical solitonsOpt Photonics News2002135963

[207] Chen ZG, Segev M, Christodoulides DNOptical spatial solitons: historical overview and recent advancesRep Prog Phys20127508640110.1088/0034-4885/75/8/086401

[208] Reyna AS, de Araújo CBHigh-order optical nonlinearities in plasmonic nanocomposites—a reviewAdv Opt Photonics2017972077410.1364/AOP.9.000720

[209] Kong FQ, Zhang CM, Bouchard F, Li ZY, Brown GG et alControlling the orbital angular momentum of high harmonic vorticesNat Commun201781497010.1038/ncomms14970

[210] Couteau CSpontaneous parametric down-conversionContemp Phys20185929130410.1080/00107514.2018.1488463

[211] Romero J, Giovannini D, McLaren MG, Galvez EJ, Forbes A et alOrbital angular momentum correlations with a phase-flipped Gaussian mode pump beamJ Opt20121408540110.1088/2040-8978/14/8/085401

[212] Walborn SP, de Oliveira AN, Pádua S, Monken CHMultimode hong-ou-mandel interferencePhys Rev Lett20039014360110.1103/PhysRevLett.90.143601

[213] Yao AMAngular momentum decomposition of entangled photons with an arbitrary pumpNew J Phys20111305304810.1088/1367-2630/13/5/053048

[214] Vicuña-Hernández V, Santiago JT, Jerónimo-Moreno Y, Ramírez-Alarcón R, Cruz-Ramírez H et alDouble transverse wave-vector correlations in photon pairs generated by spontaneous parametric down-conversion pumped by Bessel-Gauss beamsPhys Rev A20169406386310.1103/PhysRevA.94.063863

[215] Torres JP, Deyanova Y, Torner L, Molina-Terriza GPreparation of engineered two-photon entangled states for multidimensional quantum informationPhys Rev A20036705231310.1103/PhysRevA.67.052313

[216] Khoury AZ, Ribeiro PHS, Dechoum KTransfer of angular spectrum in parametric down-conversion with structured lightPhys Rev A202010203370810.1103/PhysRevA.102.033708

[217] Hu XM, Zhang C, Guo Y, Wang FX, Xing WB et alPathways for entanglement-based quantum communication in the face of high noisePhys Rev Lett202112711050510.1103/PhysRevLett.127.110505

[218] Hu XM, Xing WB, Liu BH, Huang YF, Li CF et alEfficient generation of high-dimensional entanglement through multipath down-conversionPhys Rev Lett202012509050310.1103/PhysRevLett.125.090503

[219] Baghdasaryan B, Fritzsche SEnhanced entanglement from Ince-Gaussian pump beams in spontaneous parametric down-conversionPhys. Rev. A202010205241210.1103/PhysRevA.102.052412

[220] Liu SL, Zhang YW, Yang C, Liu SK, Ge Z et alIncreasing two-photon entangled dimensions by shaping input-beam profilesPhys Rev A202010105232410.1103/PhysRevA.101.052324

[221] Chen YY, Zhang WH, Zhang DK, Qiu XD, Chen LXCoherent genera­tion of the complete high-dimensional bell basis by adaptive pump modulationPhys Rev Appl20201405406910.1103/PhysRevApplied.14.054069

[222] van der Meer R, Renema JJ, Brecht B, Silberhorn C, Pinkse PWHOptimizing spontaneous parametric down-conversion sources for boson samplingPhys Rev A202010106382110.1103/PhysRevA.101.063821

[223] Bornman N, Buono WT, Lovemore M, Forbes AOptimal pump shaping for entanglement control in any countable basisAdv Quantum Technol20214210006610.1002/qute.202100066

[224] Chang DE, Vuletić V, Lukin MDQuantum nonlinear optics — photon by photonNat Photonics2014868569410.1038/nphoton.2014.192

[225] Guerreiro T, Martin A, Sanguinetti B, Pelc JS, Langrock C et alNonlinear interaction between Single PhotonsPhys Rev Lett201411317360110.1103/PhysRevLett.113.173601

[226] Molotkov SNQuantum teleportation of a single-photon wave packetPhys Lett A199824533934410.1016/S0375-9601(98)00423-X

[227] Molotkov SNExperimental scheme for quantum teleportation of a one-photon packetJ Exp Theor Phys Lett19986826327010.1134/1.567857

[228] Walborn SP, Monken CH, Pádua S, Ribeiro PHSSpatial correlations in parametric down-conversionPhys Rep20104958713910.1016/j.physrep.2010.06.003

[229] Humble TSSpectral and spread-spectral teleportationPhys Rev A20108106233910.1103/PhysRevA.81.062339

[230] Kim YH, Kulik SP, Shih YQuantum teleportation of a polarization state with a complete bell state measurementPhys Rev Lett2001861370137310.1103/PhysRevLett.86.1370

[231] Sangouard N, Sanguinetti B, Curtz N, Gisin N, Thew R et alFaithful entanglement swapping based on sum-frequency generationPhys Rev Lett201110612040310.1103/PhysRevLett.106.120403

[232] Gisin N, Pironio S, Sangouard NProposal for implementing device-independent quantum key distribution based on a heralded qubit amplifierPhys Rev Lett201010507050110.1103/PhysRevLett.105.070501

[233] Minář J, de Riedmatten H, Sangouard NQuantum repeaters based on heralded qubit amplifiersPhys Rev A20128503231310.1103/PhysRevA.85.032313

[234] Sephton B, Vallés A, Nape I, Cox MA, Steinlechner F et al. High-dimensional spatial teleportation enabled by nonlinear optics. arXiv: 2111.13624, 2021.

Wagner Tavares Buono, Andrew Forbes. Nonlinear optics with structured light[J]. Opto-Electronic Advances, 2022, 5(6): 210174.

本文已被 2 篇论文引用
被引统计数据来源于中国光学期刊网
引用该论文: TXT   |   EndNote

相关论文

加载中...

关于本站 Cookie 的使用提示

中国光学期刊网使用基于 cookie 的技术来更好地为您提供各项服务,点击此处了解我们的隐私策略。 如您需继续使用本网站,请您授权我们使用本地 cookie 来保存部分信息。
全站搜索
您最值得信赖的光电行业旗舰网络服务平台!