A nonlinear photonic quasicrystal design method based on vogel's spiral structure

Abstract

The application discloses a nonlinear photonic quasicrystal design method based on a Vogel's spiral structure, which comprises the following steps: designing a crystal lattice vector of a crystal in a two-dimensional space, generating a crystal in a two-dimensional space, and constructing a nonlinear photonic quasicrystal based on a Vogel's spiral structure. The nonlinear photonic quasicrystal designed by the application can generate more abundant inverse lattice vectors in the transverse direction, so that multiple wide-band Cherenkov or Raman Nish rings can be generated by transverse modulation, and more abundant nonlinear Cherenkov diffraction and nonlinear Raman Nish diffraction can be realized. Meanwhile, the nonlinear photonic quasicrystal designed by the application also retains the most important feature of the radial symmetric structure, that is, the isotropy of the inverse space. The nonlinear photonic quasicrystal designed by the application can be used for multi-channel ring-shaped frequency-doubled signal output, and the nonlinear photonic quasicrystal with a sunflower spiral structure can be used for many other devices, which provides the possibility for the realization of new optical devices.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear photonic device design, specifically relating to a nonlinear photonic quasicrystal design method based on Vogel's helical structure. Background Technology

[0002] Nonlinear photonic crystals are a type of second-order nonlinear optical crystal with a coefficient χ. (2) A quasicrystal is a structure with periodic modulation but a uniform refractive index distribution. This structure can provide reciprocal lattice vectors to compensate for phase mismatch during second harmonic conversion, thus satisfying the quasi-phase matching condition for efficient optical frequency conversion. Simultaneously, the transverse reciprocal lattice vectors can also modulate the phase of the second harmonic, achieving nonlinear wavefront modulation. Nonlinear photonic crystals can be classified into one-dimensional, two-dimensional, and three-dimensional nonlinear photonic crystals. These three types can provide one, two, and three reciprocal lattice vectors, respectively. The richer the reciprocal lattice vectors, the richer the second harmonic diffraction patterns that can be generated. A quasicrystal is a structure that is disordered in the short range and ordered in the long range. A quasicrystal can be viewed as a projection of a crystal in a higher-dimensional space onto a lower-dimensional space; therefore, a quasicrystal possesses the characteristics of a crystal in a higher-dimensional space. For example, a one-dimensional quasicrystal in one-dimensional space can be viewed as a projection of a crystal in two-dimensional space onto a one-dimensional space, so a one-dimensional nonlinear photonic quasicrystal in one-dimensional space can provide two reciprocal lattice vectors. The advantage of a quasicrystal over a crystal is that it can provide a richer variety of reciprocal lattice vectors.

[0003] Vogel's helix is ​​renowned for its perfectly circular symmetry in Fourier space. For example, the most important structure within Vogel's helix is ​​the gold helix (also called the sunflower structure), which exhibits sharp loops in the Fourier spectrum. Based on this unique property, gold helix photonic crystals and photonic crystal fibers demonstrate excellent performance in terms of full photonic bandgap and birefringence. For instance, photonic crystal fibers (PCFs) with pores arranged in a gold helix pattern exhibit large birefringence and tunable dispersion. Currently, only one nonlinear photonic crystal based on Vogel's helix has been reported. Utilizing the reciprocal isotropic nature of Vogel's helix, broadband Cherenkov second harmonic emission enhanced in the fundamental wavelength range of 1220–1450 nm was achieved. Nonlinear photonic crystals based on Vogel's spiral can only achieve single-channel ring frequency doubling signal output and are powerless in multi-channel applications. While simple nonlinear photonic quasicrystals can provide multiple reciprocal lattice vectors to achieve multiple phase matching processes, the signal of each frequency doubling channel is only a point, not a ring, which limits its application prospects.

[0004] This invention combines Vogel's spiral and quasicrystal design concepts to design a nonlinear photonic quasicrystal that can be used for multi-channel ring frequency doubling signal output. Summary of the Invention

[0005] The technical problem this invention aims to solve is to address the shortcomings of existing technologies by providing a nonlinear photonic quasicrystal design method based on Vogel's helical structure. The designed nonlinear photonic quasicrystal can generate richer reciprocal lattice vectors in the transverse direction, allowing for richer nonlinear Cherenkov diffraction and nonlinear Ramannix diffraction. Simultaneously, the nonlinear photonic quasicrystal designed in this invention retains the most important characteristic of radially symmetric structures: the isotropy of reciprocal space. The nonlinear photonic quasicrystal designed in this invention can be used for multi-channel ring frequency doubling signal output, while the sunflower helical structure nonlinear photonic quasicrystal can be used in many other devices, providing possibilities for the realization of novel optical devices.

[0006] Vogel's helical structure satisfies:

[0007]

[0008] θ v =n·α v

[0009] Here r v ,θ v These are the polar coordinate parameters of a point in Vogel's spiral structure, representing the polar radius and polar angle, respectively; n = 0, 1, 2, 3…, which is a series of integers; b v It is the magnification factor, which is a constant; α v It is a fixed angle, if α v =2π / φ, That is, α v When the radius is approximately 2.4 radians, this structure is called a sunflower spiral (e.g., ...). Figure 1 As shown), its inverted space image is as follows. Figure 2 As shown, if α v At other angles, it is a regular Vogel's spiral.

[0010] Here, we introduce Vogel's helical structure into the design of nonlinear photonic quasi-crystals, that is, two different α-columns of the Vogel's helical structure. v In other words, α1 and α2 are arranged in a quasi-periodic manner, which is equivalent to designing a one-dimensional nonlinear photonic quasicrystal in a one-dimensional space with fundamental periods of α1 and α2. We use a high-dimensional projection method to arrange the angles of the Vogel's spiral structure in a quasi-periodic manner to obtain a nonlinear photonic quasicrystal based on the Vogel's spiral structure. The specific process is as follows:

[0011] The technical solution adopted in this invention is: a nonlinear photonic quasicrystalline design method based on Vogel's helical structure, comprising the following steps:

[0012] 1) Design the lattice vector of a crystal in two-dimensional space.

[0013] Let the lattice vector A of the crystal in two-dimensional space be... Here, the parallel space component of the lattice vector A The angle is known and can be used to generate the coordinates of the point where the quasicrystal is located; the perpendicular spatial component is... It can be used to determine whether a point in a crystal in a high-dimensional space should be projected into a low-dimensional space, since this is unknown; when using the high-dimensional projection method, it is required that α... / / ·α ⊥ =0, therefore, take α ⊥ For α / / Find the unit orthogonal vectors; find the homogeneous general solution of α1·q1+α2·q2=0 to obtain the lattice vectors of the crystal in two-dimensional space;

[0014] 2) Generate a crystal in a two-dimensional space

[0015] S = (n1n2)·A = (n1·α1 + n2·α2n1·q1 + n2·q2), where S is a point of the crystal in two-dimensional space, n1 = 0, 1, 2, 3…, n2 = 0, 1, 2, 3…, and n1 and n2 are integers;

[0016] A quasicrystal can be viewed as a projection of a crystal in a high-dimensional space onto a low-dimensional space. However, not all points can be projected onto the low-dimensional space; only points within the projection window can.

[0017] Using a high-dimensional projection method, α1 and α2 are arranged in a quasi-periodic manner. That is, all points within the projection window in two-dimensional space are projected onto one-dimensional space. Since the perpendicular spatial component of each point to the one-dimensional space is a one-dimensional vector, the projection window is simply a matter of upper and lower bounds. The calculation method for the projection window is as follows: take n1 and n2 as binary 00, 10, 01, and 11 respectively, and find the maximum and minimum values ​​of the four values ​​0, q1, q2, and q1+q2 to obtain the projection window (W). min W max When n1·q1+n2·q2∈(W) min W max When n1·α1+n2·α2 is an angle value when α1 and α2 are arranged in a quasi-periodic manner; the set of all angle values ​​that meet the requirements is denoted as Θ, and all angle values ​​in Θ are arranged in ascending order, thus obtaining a series of quasi-periodic angle values, where the (q+1)th value is θ. q+1 q = 0, 1, 2, 3…;

[0018] 3) Constructing nonlinear photonic quasicrystals based on Vogel's helical structure

[0019]

[0020] θ Q =θ q+1

[0021] Where r Q ,θ Q b is the polar coordinate parameter of a point in Vogel's spiral structure. Q It is a constant.

[0022] When a laser is incident on the nonlinear photonic quasicrystal based on the Vogel's spiral structure designed in this invention, nonlinear Bragg diffraction of the second harmonic can be observed. In this case, when the nonlinear Bragg diffraction condition is k... 2ω -2k ω -G=0, nonlinear Bragg diffraction condition as follows Figure 3 As shown. Furthermore, the second harmonic can be achieved using only the transverse component that satisfies the nonlinear Bragg law, i.e., k 2ω sinθ-G=0, this type of second harmonic generation is called nonlinear Ramanniz diffraction, and the condition for nonlinear Ramanniz diffraction is as follows: Figure 4 As shown. Similarly, the second harmonic can also be realized by satisfying only the longitudinal component of the nonlinear Bragg law, i.e., k. 2ω cosθ-2k ω -G=0, this type of second harmonic generation is called nonlinear Cherenkov diffraction, and the nonlinear Cherenkov diffraction condition is as follows: Figure 5 As shown.

[0023] The nonlinear photonic quasicrystal based on Vogel's spiral structure designed in this invention can provide two circumferential reciprocal lattice vectors in the transverse direction, thus generating multiple broadband Cherenkov rings or Ramannix rings during second harmonic generation experiments.

[0024] Compared with existing technologies, this invention has the following advantages: The nonlinear photonic quasicrystal based on Vogel's spiral structure designed in this invention can generate richer reciprocal lattice vectors in the transverse direction, thus enabling transverse modulation to generate multiple wideband Cherenkov or Ramannix rings, achieving richer nonlinear Cherenkov diffraction and nonlinear Ramannix diffraction. Simultaneously, the nonlinear photonic quasicrystal designed in this invention retains the most important characteristic of radially symmetric structures, namely the isotropy of reciprocal space. The nonlinear photonic quasicrystal designed in this invention can be used for multi-channel ring frequency doubling signal output, while the sunflower spiral structure nonlinear photonic quasicrystal can be used in many other devices, providing possibilities for the realization of novel optical devices. Attached Figure Description

[0025] Figure 1 A dot matrix image of a sunflower's spiral structure;

[0026] Figure 2 An inverted space image of a sunflower spiral structure;

[0027] Figure 3 This is the condition for nonlinear Bragg diffraction;

[0028] Figure 4 For nonlinear Ramannix diffraction conditions;

[0029] Figure 5 This is the nonlinear Cherenkov diffraction condition;

[0030] Figure 6 The image shows a lattice image of a nonlinear photonic quasicrystal based on Vogel's helical structure designed in this embodiment.

[0031] Figure 7 The image shown is a reciprocal space image of a nonlinear photonic quasicrystal based on Vogel's helical structure designed in this embodiment. Detailed Implementation

[0032] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0033] This invention presents a nonlinear photonic quasicrystal based on Vogel's helical structure. The design method includes the following steps:

[0034] 1) Design the lattice vector of a crystal in two-dimensional space.

[0035] Let the lattice vector A of the crystal in two-dimensional space be... Here, the parallel space component of the lattice vector A Given the angle, the perpendicular spatial component is... Take α ⊥ For α / / unit orthogonal vectors;

[0036] In this embodiment, α1 = 2.4 and α2 = 1.5708 are preferred, resulting in Find the homogeneous general solution of 2.4·q1 + 1.5708·q2 = 0, and obtain the following: Here, k can be any real number; we take k = 1, and get... Then α / / With α ⊥ By combining, a two-dimensional matrix is ​​obtained. That is, the lattice vector of the crystal in two-dimensional space;

[0037] 2) Generate a crystal in a two-dimensional space

[0038] S = (n1n2)·A = (n1·α1 + n2·α2n1·q1 + n2·q2), where S is a point of the crystal in two-dimensional space, n1 = 0, 1, 2, 3…, n2 = 0, 1, 2, 3…, and n1 and n2 are integers;

[0039] Using a high-dimensional projection method, α1 and α2 are arranged in a quasi-periodic manner. That is, all points within the projection window in two-dimensional space are projected onto one-dimensional space. Since the perpendicular spatial component of each point to the one-dimensional space is a one-dimensional vector, the projection window is simply a matter of upper and lower bounds. The calculation method for the projection window is as follows: Take n1 and n2 as binary 00, 10, 01, and 11 respectively, and find the maximum and minimum values ​​of the four values ​​0, q1, q2, and q1+q2, that is, the maximum and minimum values ​​of the four values ​​0, -0.6545, 1, and 0.3455. It can be seen that the maximum value is 1 and the minimum value is -0.6545. Therefore, the projection window is (-0.6545, 1]. When n1·q1+n2·q2∈(W min W max When n1·α1+n2·α2 is an angle value when α1 and α2 are arranged in a quasi-periodic manner; the set of all angle values ​​that meet the requirements is denoted as Θ, and all angle values ​​in Θ are arranged from smallest to largest, thus obtaining a series of quasi-periodic angle values: 0, 1.57, 3.97, 6.37, 7.94, 10.34…, where the (q+1)th value is θ. q+1 q = 0, 1, 2, 3…;

[0040] 3) Constructing nonlinear photonic quasicrystals based on Vogel's helical structure

[0041]

[0042] θ Q =θ q+1

[0043] Where r Q ,θ Q b is the polar coordinate parameter of a point in Vogel's spiral structure. Q b is a constant; in this embodiment, b is taken as... Q =2.

[0044] The lattice image of the nonlinear photonic quasicrystal based on the Vogel's helical structure designed using the above method is shown below. Figure 6 As shown, this nonlinear photonic quasicrystal can provide two radii of 1.50 μm each. -1 and 1.93μm -1 The cyclic reciprocal lattice vector, such as Figure 7 As shown.

[0045] Next, the diffraction angles of the nonlinear Cherenkov ring and the nonlinear Ramanniz ring corresponding to the two toroidal reciprocal lattice vectors are calculated respectively. The experiment preferably uses z-strontium barium niobate (Sr... 0.61 Ba 0.39 Nb₂O₆ was used as the processing material. Ferroelectric domain inversion was induced by femtosecond laser direct writing. 1560 μm was used as the fundamental frequency light. Based on the material's correlation coefficient and the incident light wavelength, k could be calculated. ω =8.97μm -1 k 2ω =18.36μm -1 Based on the conditions of nonlinear Ramanniz diffraction, two radii of 1.50 μm can be calculated. -1 and 1.93μm -1 The nonlinear Ramannix diffraction angles corresponding to the ring reciprocal lattice vectors are 10.72° and 13.87°, respectively. According to the nonlinear Cherenkov diffraction condition, the Cherenkov diffraction angle can be calculated to be 29.18°.

Claims

1. A nonlinear photonic quasicrystal design method based on Vogel's helical structure, characterized in that, Includes the following steps: 1) Design the lattice vector of a crystal in two-dimensional space. Let the lattice vector A of the crystal in two-dimensional space be... Here, the parallel space component of the lattice vector A Given the angle, the perpendicular spatial component is... Take α ⊥ For α / / Find the unit orthogonal vectors; find the homogeneous general solution of α1·q1+α2·q2=0 to obtain the lattice vectors of the crystal in two-dimensional space; 2) Generate a crystal in a two-dimensional space S = (n1 n2)·A = (n1·α1 + n2·α2 n1·q1 + n2·q2), where S is a point of the crystal in two-dimensional space, n1 = 0, 1, 2, 3…, n2 = 0, 1, 2, 3…, and n1 and n2 are integers; Using a high-dimensional projection method, α1 and α2 are arranged in a quasi-periodic manner. That is, all points within the projection window in two-dimensional space are projected onto one-dimensional space. Since the perpendicular spatial component of each point to the one-dimensional space is a one-dimensional vector, the projection window is simply a matter of upper and lower bounds. The calculation method for the projection window is as follows: take n1 and n2 as binary 00, 10, 01, and 11 respectively, and find the maximum and minimum values ​​of the four values ​​0, q1, q2, and q1+q2 to obtain the projection window (W). min W max When n1·q1+n2·q2∈(W) min W max When n1·α1+n2·α2 is an angle value when α1 and α2 are arranged in a quasi-periodic manner; the set of all angle values ​​that meet the requirements is denoted as Θ, and all angle values ​​in Θ are arranged in ascending order, thus obtaining a series of quasi-periodic angle values, where the (q+1)th value is θ. q+1 q = 0, 1, 2, 3…; 3) Constructing nonlinear photonic quasicrystals based on Vogel's helical structure i Q =θ q+1 Where r Q ,θ Q b is the polar coordinate parameter of a point in Vogel's spiral structure. Q It is a constant.

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