Topological photonic encoder based on the disclination states

Topological disclination states are highly localized and stable by means of introducing disclination, which provide a robust platform for realizing optical information transition. A photonic encoder, as a kind of optical information transition element, can record, transmit, and protect optical information. However, there is no effective methods to realize topological photonic encoders. In this work, we propose a method to realize topological photonic encoder through topological disclination states. After the introduction of a disclination in the honeycomb structure, four types of disclination states can be generated. To demonstrate the device to carry more information, nine disclination structures with different cylindrical radii are combined, and the disclination states can be denoted by digital signals 1–4 to prepare a topological photonic encoder. In addition, to improve the security of information transition, we build an encryption algorithm based on Morse code. This work provides a new idea for the construction of encoding devices and promotes the practical application of the topological disclination states.


Introduction
Disclination is a naturally occurring and common topological defect structure in crystal materials which can lead to topological phase transitions [1][2][3][4].Topological disclination structure is a kind of topological defect, which can break the rotation symmetry of the original structure and induce fractional charge in the disclination region, and the fractional charge can lead to the topological disclination states with high localization and anti-interference, this phenomenon can be predicted based on the bulk-disclination corresponding [5][6][7].The topological disclination states generated by topological disclination exist in the band gap [5,8] and can be used for optical information transition [9][10][11], which provides a good platform for the preparation of optical information transition devices.However, there are few reports about information transition devices based on topological disclination states.
As a commonly used information transition device, encoder can be used for information recording and communication.In addition, people can use additional encryption algorithms to effectively prevent information leakage, so it is also widely used in secure communication [12][13][14].In recent years, people have made a series of explorations on the preparation strategies of photonic encoding devices, such as fluorescence technology, light-emitting elements, quantum dots, waveguides, and so on [15][16][17][18][19][20][21][22][23].Nevertheless, topological photonic encoders based on topological photonic crystals have not been reported.
In this work, we achieve a topological photonic encoder based on topological disclination states.By introducing controllable deformation in a honeycomb lattice, fractional charges appear at the disclination area, forming a topological disclination structure with four kinds of localized photonic states.By establishing a specific encoding relationship between the disclination states and digital information, we prepare a topological photonic encoder based on topological disclination states.Additionally, we design an algorithm on Morse code that can be used in topological photonic encoder, and the security level of the encoding system is further enhanced to ensure information security.The topological photonic encoder designed in this work provides a new scheme for the preparation of encoder.When combined with encryption algorithms, it has great potential for applications in information storage and communication security.

Design of topological disclination structure
Disclination is a kind of topological defect state, which can be realized by inserting or deleting a sector region of Frank angle Ω = 2π/6 in the hexagonal lattice.In this work, we make the original hexagonal structure into the current topological disclination structure by inserting the sector region of Frank Angle Ω into the honeycomb lattice, then we delete the seven semi-circles in the middle of this part, and a central point defect is formed to achieve the desired topological disclination states at the disclination area.This structure diagram is shown in figure 1(a).In this structure, the blue circular region used in the materials is Si with a refractive index of n = 3.45, and the silver-white region is air with a refractive index of n = 1.The lattice constant of the unit cell a = √ 3 um and the radius of the circles r = 0.2 µm.According to the bulk-disclination correspondence, the disclination breaks the rotational symmetry of the crystal, so the charge configuration protected by the rotation symmetry is also broken, which results in the appearance of fractional charges in the disclination area [5][6][7].
The topological invariant associated with the bandgap of the disclination lattice is defined by χ = (χ M ,χ K ) [4][5][6].In the disclination structure we designed, where χ = (2, 0).Therefore, the band topology is nontrivial for this disclination structure.In the system of disclination, the Kekule distortion coefficient can also be used to determine whether the system is in a topological state [6].We chose r K = l intra /l inter >1 in order to ensure the topology, where ℓ intra and ℓ inter is the intra-cell and inter-cell spacing between waveguides after shift.
After adding the Franck region to the hexagonal structure, it becomes a disclination structure.Since the disclination states only exist at the seven innermost silicon columns, the Hamiltonian describing the disclination states can be written as follows [4]: 0 t 0 0 0 0 t t 0 t 0 0 0 0 0 t 0 t 0 0 0 0 0 t 0 t 0 0 0 0 0 t 0 t 0 0 0 0 0 t 0 t t 0 0 0 0 t 0 Here, t is the coupling of the inter-unit-cell couplings.The eigenvalues and the eigenvectors of this Hamiltonian can be described by the characteristic table of C 7 symmetry.
The dispersion distribution of the unit cell in the transverse magnetic mode is shown in figure 1(b), and a large band gap between 0.48 c/a and 0.72 c/a (where c/a is the normalized unit and c is the speed of light in a vacuum) can be found.Figure 1(c) shows the numerically calculated eigenvalues of the structure.There are four types of disclination states in the band gap of the yellow region, including three types of degenerate disclination states and one type of non-degenerate disclination state.These four types of topological disclination states are our main application objects.Figure 1(d) shows the electric field distribution of the four types of topological disclination states.Here we define the four types of topological disclination states as degenerate Ψ α , ψ β , ψ γ and non-degenerate, Ψ δ , in which the degenerate states can be distinguished by the numbers 1 and 2, such as ψ α1 and ψ α2 .It can be seen that a highly localized topological photonic state appears at the disclination area.According to previous reports, the disclination states not only have highly localized characteristics but also have good anti-interference [4,24].
From figure 1, we know that there is a large absolute band gap in the topological disclination structure, and there are only four types of topological dislocation states in the band gap.If more topological disclination states can be constructed, the structure can be used to store more information.We know that if we change the radii of columns in the unit cell, the position of the band gap will change accordingly [25][26][27].From figure 1(c), we know that all the disclination states are distributed in the band gap, and if the position of the band gap changes, the frequencies of the disclination states will inevitably change accordingly.According to this principle, we can regulate the frequency of the disclination states, to make the disclination states appear in the right place in the band gap, it is convenient to make the disclination states in multiple structures appear in the common band gap.Next, we combine nine different topological disclination structures, as shown in figure 2(a).Here, for convenience, we numbered the nine different disclination structures as 1-9, where the radius of each region from 1 to 9 increased from 1.96 µm to 2.08 µm, and the change in the radius of the column in the adjacent numbered disclination structure is 0.02 µm.The calculated eigenstate distribution is shown in figure 2(b).It can be seen that the four types of disclination states in the nine regions are all distributed together in the common band gap, and there is no other state distribution in the band gap.From the state distribution, the photonic device we designed achieves good frequency division.
Take the state of ψ δ for example.Figure 2(c) shows the electric field distribution of the ψ δ state in each region after the combination.From this figure, we can see that even if the nine disclination structures with different radii are combined, the electric field distribution of the disclination states in each region is still concentrated in the middle disclination region, and the number of the disclination states in the nine disclination regions does not change, in which each region contains three types of degenerate topological disclination states and one type of non-degenerate topological disclination state.The most important thing is that the topological disclination states of each region are independent of each other, and there is no crosstalk due to the combination of different disclination structures.

Establishment of encoding relationship
Topological disclination structures can produce highly localized and anti-interference topological disclination states, and the number of the topological disclination states can be increased by combining different topological disclination structures together, which provides a good platform for preparing high-capacity optical information transition devices.Encoders are commonly used as information transition devices, which have important applications in information recording, information communication and information protection.However, there are no effective methods have been proposed to realize photonic encoders based on topological structure.Here, we prepared a topological photonic encoder based on the topological disclination states.
The construction of the topological photonic encoder depends on establishing a certain encoding relationship between the disclination states and the digital signals.Figure 3 shows the typical mechanism of the topological encoding strategy: using the digital system to label the electric field distribution of the disclination states, and the designed device with multiple sets of disclination structures is used for information recording.As shown in figure 3(a), we use different numbers to mark the disclination structures when it is in different disclination states.First, when there is no electric field distribution in the disclination structure, the number 0 is used to mark it, and when the electric field distribution in the disclination structure is the disclination states ψ α , Ψ β , Ψ γ and Ψ δ , the numbers 1, 2, 3 and 4 are used to mark them respectively.Using this relation, the obtained electric field distribution information is successfully converted into digital encoding information.The large structure corresponds to a 3 × 3 matrix, and when it gets ψ 1α1 in the first position, which corresponds to the figure 3(b) shown, it can be encoded as 100000000.When ψ 3γ2 , Ψ 3γ2 and ψ 8δ are obtained separately on this structure, it can be encoded as 003200040, as shown in figure 3(c).The encoding capacity is an important property of the encoder.For the encoder, the larger the encoding capacity, the more information can be stored.The encoding capacity is an important indicator of whether the encoder can achieve the practical application.The size of the encoding capacity mainly depends on the encoding level and the number of encoding positions.Here, based on the relationship between encoding capacity and encoding level and number of encoding positions, we can define the encoder capacity as C = m n , where m is the encoding level and n is the number of encoding positions [28].At present, the most popular and common photonic crystal encoder is the binary encoder.When the number of binary photonic crystal encoders' encoding positions is the same as that of the topological photonic encoder designed by us, the encoding capacity of the binary encoder is 2 9 according to the formula, while the capacity of the topological photonic encoder designed by us is 5 9 .Compared to the binary photonic crystal encoder, the capacity of the topological photonic encoder is increased by about 3800 times.In this study, the enhanced encoding capacity of the topological disclination encoder stems from the multitude of disclination states inherent in its structure, which improves the coding level of the disclination encoder.The number of disclination states is determined by the system symmetry.For instance, a structure exhibiting C5 symmetry harbors two degenerate disclination states alongside one non-degenerate state [5,6], while the structure designed in this work has C 7 symmetry, and the system contains three pairs of degenerate states and one non-degenerate state.
As a component of information processing, the encoder is very important to ensure information security.In order to further improve the information security level of the encoding system in the process of information processing, we can use additional encryption algorithms to encrypt the encoded information.Here, we use the numbers 1 and 2 to represent the Morse encode according to the Morse code encryption rules, and convert them into ternary codes, and then convert them into quinary codes.As an example, we develop an encryption algorithm for a topological photonic encoder as shown in figure 3(d) to encrypt letters and numbers and use different quinary digital information representations.To illustrate the working mechanism of the encryption algorithm strategy in the topological photonic encoder.Figure 3(e) shows the encryption example of the topological photonic encoder.By establishing encoding relationships between topological misalignments and numbers, as well as additional encryption algorithms, we can use '232001014' to represent information 'B 2' , '400430013' to represent information 'I K M' , and '112100442' to represent information '1 4' .It can be seen that the security of information processing is greatly enhanced by the use of additional encryption algorithms.
In order to demonstrate the anti-interference performance of the topological disclination encoder, we first add interferences to the periphery of the disclination region and the disclination region in a single disclination structure, as shown in figures 4(a) and (c).It can be seen from figure 4(b) that there is no obvious difference between the disclination states with or without interference, which proves that the disclination structure has anti-interference.However, when the interference is introduced into the disclination region, as shown in figure 4(d), although the disclination states exist, the electric field distribution has seriously been influenced.Figure 4(e) shows the interference is added to the topological disclination encoder.After interference is added, we randomly make it generate a string of codes, and its performance does not change, which proves the anti-interference of the topological disclination encoder as shown in figure 4(f).

Discussion
In this work, we use silicon columns with refractive index n = 3.45 to construct a topological disclination structure.From calculated eigenvalues spectrum, we can group up to nine topological disclination structures with radii variation of silicon columns of 0.02 µm as nine encoding positions.In the later work, the encoder can also have a larger encoding capacity by using other materials to increase the band gap size or change the radius of silicon columns.In addition, we can also design encryption algorithms with higher confidentiality to improve the security of information in the communication process.

Conclusion
In summary, we have proposed a topological photonic encoding device based on topological disclination states.By inserting controllable deformation into the honeycomb lattice, we can obtain a topological disclination structure with four kinds of topological disclination states, and then several groups of different disclination structures are combined together to achieve the frequency division effect.By establishing the relationship between the disclination states and digital information, the disclination states are transformed into encoding information, and the topological photonic encoder is constructed.The security of the information in the transmission process is greatly improved by establishing additional encryption algorithms.The topological photonic encoder has good anti-interference and a simple fabrication process.This work provides a new idea for the preparation of encoding devices and promotes the practical application of topological disclination states.

Figure 1 .
Figure 1.(a) Inserting a Frank sector with the angle Ω = 2π/6 to the honeycomb lattice, the disclination structure is constructed.The radius of silicon column r = 0.2 µm, and the lattice constant a = √ 3 um.(b) The TM mode dispersion energy bands of the unit cell (c) the numerically calculated eigenvalues of the disclination structure.The yellow region is the band gap region, where the red dots represent the disclination states.(d) Electric field distribution of three degenerate topological disclination states ψ α , Ψ β , Ψ γ and a non-degenerate topological disclination state Ψ δ .

Figure 2 .
Figure 2. (a) Nine disclination structures with different radii are combined and named No.1 to No.9, according to the radii range from 1.92 µm to 2.08 µm.(b) The numerically calculated eigenvalues after the combination of the disclination structures, where the red eigenstates are the disclination states of the structure.It can be seen that the topological disclination states are all distributed together and without other eigenstates interspersed in the common band gap.(c) The electric field distribution of the combined Ψ δ states in the device and achieved a good frequency division effect.

Figure 3 .
Figure 3. (a) Using the multi-stage digital system to label the electric field distribution of the disclination states.When there is no electric field distribution on the structure, the corresponding number is 0; and the electric field distribution of disclination states ψ α , ψ β , ψ γ and ψ δ correspond to the numbers 1, 2, 3 and 4 respectively.(b) This large structure corresponds to a 3 × 3 matrix, when it gets ψ1α1 in the first position, which corresponds to the figure shown, can be encode as 100 000 000. (c) When ψ 3γ2 , ψ 4β1 and ψ 8δ are obtained separately on this structure, it can be encoded as 003200040.(d) An encryption algorithm for the topological encoder based on the Morse cipher encryption rules, using quinary ciphers to encrypt letters and numbers.(e) According to the encryption algorithm, 'B 2' , 'I K M' and '1 4' are represented by the encoding information '232001014' ,'400430013' and '112100442' respectively.

Figure 4 .
Figure 4. (a), (b) Interference is added to the periphery of the disclination region of the disclination structure and the distribution of its electric field.(c), (d) Interference is added to the disclination region of the disclination structure and the distribution of its electric field.(e), (f) Interference is added to the topological disclination encoder and randomly generates a set of codes, '10 020 020' .