Orbital angular momentum generator based on spiral twisted photonic crystal fiber

Abstract

The invention discloses an orbital angular momentum generator based on a spiral twisted photonic crystal fiber. The invention discloses the orbital angular momentum generator based on the spiral twisted photonic crystal fiber. A method for simplifying analysis by using an optical transformation principle is provided, and a higher-order orbital angular momentum mode which can be generated by the specially designed photonic crystal fiber under the action of spiral distortion is obtained. The specially-designed photonic crystal fiber is composed of three solid fiber cores and a double-cladding layer, the three solid fiber cores are distributed in an equilateral triangle mode, and air holes in the double-cladding layer are distributed in a hexagon mode. The inner cladding air holes and the outer cladding air holes are different in size, the diameter of the inner cladding air holes is smaller than that of the outer cladding air holes, the hole distances of the two claddings are equal, and a certain distance exists between the two claddings to form an annular core. According to the orbital angular momentum generator, the number of orbital angular momentum generated by the orbital angular momentum generator is large, the orbital angular momentum generator is high in order and low in loss, the orbital angular momentum generator can be better compatible with an optical fiber communication system, and unnecessary loss is reduced.

Description

Technical Field

[0001] The invention relates to the field of optical fiber technology, and in particular to an orbital angular momentum generator based on helical twisted photonic crystal optical fiber. Background Art

[0002] Orbital Angular Momentum (OAM) is a new dimension independent of the traditional physical dimension, carrying an orbital angular momentum of The light is also called a vortex beam, which has a wavefront phase factor exp(ilφ), where i is an imaginary unit, is the reduced Planck constant, l is the topological charge, and φ is the azimuthal coordinate in the spatial angular direction. Currently, optical orbital angular momentum has been applied in many fields, including optical tweezers, sensing, image processing, and communications. Since the OAM topological charge value is theoretically infinite and different OAM modes are mutually orthogonal, orbital angular momentum can be used for multiplexed communication. Among them, optical fiber-based orbital angular momentum multiplexing communication can solve the bottleneck problem of communication capacity, greatly improve the communication capacity and rate, and is a current research hotspot. Studying the generation of high-quality vortex beams is a key issue in improving the application of orbital angular momentum. However, general orbital angular momentum generators have low compatibility with optical fiber communication systems. Therefore, designing an orbital angular momentum generator with high optical fiber compatibility, high order, and high quality is an urgent problem to be solved.

[0003] At present, there are two main methods for generating light beams carrying orbital angular momentum. One is the spatial method, such as spiral phase plates, mode converters and silicon integrated devices. The other is the fiber method, such as fiber couplers, special fiber gratings and helically twisted fibers. Among them, the use of the fiber method to generate orbital angular momentum can be coupled with the fiber system to improve compatibility, thereby reducing unnecessary losses. The relatively simple and low-cost production of helically twisted optical fibers has attracted the attention of experts at home and abroad. At present, the orbital angular momentum generated by helically twisted optical fibers is of low order. It is necessary to design new optical fibers to generate high-order orbital angular momentum through helically twisted fibers. Summary of the invention

[0004] The purpose of the present invention is to provide an orbital angular momentum generator based on a helically twisted photonic crystal fiber, so as to achieve better coupling of the generated orbital angular momentum into an optical fiber orbital angular momentum multiplexing communication system, while reducing the generated orbital angular momentum mode loss and improving the generated orbital angular momentum quality.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] Step 1: A photonic crystal fiber that can be used for orbital angular momentum transmission is designed, characterized in that: the photonic crystal fiber is composed of three real fiber cores and a double cladding, the air holes in the double cladding are circular and distributed in a hexagonal shape, and three air holes distributed at an angle of 120 degrees around the middle air hole in the inner cladding can form three real fiber cores distributed in a regular triangle. There is a certain distance between the two claddings to form a ring core. At this time, the ring core can support intrinsic modes, and the same-order intrinsic odd and even modes with a phase difference of π / 2 can be superimposed into an OAM mode, so that the optical fiber supports orbital angular momentum transmission. The specific superposition formula is:

[0007] Where l is the orbital angular momentum order, m is the radial mode order, and j is the π / 2 phase difference.

[0008] Furthermore, the base material of the optical fiber is silicon dioxide.

[0009] Step 2: Helically twist the photonic crystal fiber designed in step 1 to form an orbital angular momentum generator with a twist period of Λ L , the distortion rate is α=2π / Λ L , and then the three-dimensional problem is converted into a two-dimensional problem through optical transformation for research. Under the helical twisting force, the eigenmode supported by the ring core will carry orbital angular momentum. For a specific twist rate and wavelength, the core supermode and the ring core mode will resonate and couple to form a hybrid mode state. At this time, the hybrid mode carrying orbital angular momentum can also be called the orbital angular momentum mode. By observing the loss peak in the loss spectrum of the core supermode at different twist rates, the wavelength where the resonant coupling occurs can be found, and then the conditions for generating orbital angular momentum can be determined. The calculation formula for mode loss is:

[0010] Where L is the mode loss value, λ is the wavelength, Im(n eff ) is the imaginary part of the effective refractive index of the core supermode, n eff is the effective refractive index, in dB·m -1 .

[0011] Step three: By analyzing the mode electromagnetic field of the orbital angular momentum generator based on the helically twisted photonic crystal fiber in step two, the effective refractive index and electromagnetic field distribution of the core supermode, ring core mode and orbital angular momentum mode can be obtained, and then the influence of helical twist on the effective refractive index of the fiber mode can be known; by analyzing the phase of the orbital angular momentum mode, the corresponding generated orbital angular momentum order can be determined; by studying the changes in the effective refractive index of the core supermode and ring core mode, the matching conditions for resonant coupling can be known.

[0012] The present invention designs an orbital angular momentum generator based on a helical twisted photonic crystal fiber. Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0013] When α = 7853.982 rad / m, the generated OAM modes include "OAM -4,1 , OAM +9,1 , OAM +10,1 , OAM +11,1 , OAM +13,1 ", where "OAM+13,1" is the highest-order OAM mode with the lowest loss that can be generated using helically twisted optical fiber. When α≥8377.58rad / m, the number of OAM modes generated is not less than 10.

[0014] The generated OAM mode loss is low. When α = 7853.982 rad / m, the loss of all modes is less than 1.64×10 -3 dB / m, which is at least two orders of magnitude lower than the OAM mode loss generated by existing helically twisted optical fibers, improving the practical application value of orbital angular momentum generators.

[0015] The generated OAM mode has a high purity. When α=7391.983rad / m, α=7853.982rad / m and α=8377.58rad / m, the OAM mode purity is greater than 93%. The generated orbital angular momentum has a high quality and a low cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 The schematic diagram of the cross section of an orbital angular momentum generator based on a helical twisted photonic crystal fiber in a specific embodiment. In the figure: 1-inner cladding air hole, 2-outer cladding air hole, 3-background material silica, 4-three real fiber cores, d1-inner cladding air hole diameter, Λ1-interval between inner cladding air holes, d2-outer cladding air hole diameter, Λ2-interval between outer cladding air holes. H-width of the annular core, R2-radius of the entire fiber cross section.

[0017] Figure 2 5 is a three-dimensional schematic diagram of an orbital angular momentum generator based on a helical twisted photonic crystal fiber in a specific embodiment. 5 is a cross section of the orbital angular momentum generator, i.e., Figure 1 .

[0018] Figure 3 The circular birefringence of the core supermode in an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment is shown in FIG.

[0019] Figure 4 In a specific embodiment, a ring core of an orbital angular momentum generator based on a helical twisted photonic crystal fiber is supported Mode phase distribution diagram.

[0020] Figure 5 In a specific embodiment, a ring core of an orbital angular momentum generator based on a helical twisted photonic crystal fiber is supported Mode phase distribution diagram.

[0021] Figure 6 It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α=7853.982 rad / m, showing a loss peak at a wavelength λ=1360 nm.

[0022] Figure 7 It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α=7853.982 rad / m, showing loss peaks at wavelengths λ=1820 nm and λ=1900 nm.

[0023] Figure 8 It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α=7853.982 rad / m, showing that a loss peak appears at a wavelength λ=1940 nm.

[0024] Fig. 9 It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α=7853.982 rad / m, showing loss peaks at wavelengths λ=2000nm and λ=2530nm.

[0025] Fig.10 In a specific embodiment, an orbital angular momentum generator based on a helical twisted photonic crystal fiber has a twist rate of α = 7853.982 rad / m, and the core supermodes "M1, M2, M4, M5" and the ring core mode "OAM 10,1 , OAM 11,1 , OAM 13,1 ,” changes with wavelength.

[0026] Fig.11 In a specific embodiment, an orbital angular momentum generator based on a helical twisted photonic crystal fiber has a twist rate of α = 7853.982 rad / m, and the core supermodes "M1, M2, M3, M5, M6" and the ring core mode "OAM 4,1 , OAM 9,1 , OAM 10,1 ” varies with wavelength.

[0027] Fig.12It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α is 7391.983 rad / m.

[0028] Fig.13 It is a loss spectrum of a core supermode of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rate α is 8377.58 rad / m.

[0029] Fig.14 Schematic diagram of the orbital angular momentum mode purity of an orbital angular momentum generator based on a helically twisted photonic crystal fiber in a specific embodiment when the twist rates are α=7391.983 rad / m, α=7853.982 rad / m and α=8377.58 rad / m respectively. DETAILED DESCRIPTION

[0030] The present invention will be further described in detail below with reference to the accompanying drawings and by way of examples. The following examples are intended to explain the present invention but the present invention is not limited to the following examples.

[0031] Figure 1 The figure shows a cross-sectional schematic diagram of an orbital angular momentum generator based on a helically twisted photonic crystal fiber of the present invention. The orbital angular momentum generator mainly includes two parts, namely three real fiber cores and a double cladding structure, wherein there is a certain distance between the double claddings to form a ring core. The background material of the optical fiber is silica, and the rest is air holes, and the radius of the entire optical fiber cross section is R2. The diameter of the inner cladding air hole is d1, the diameter of the outer cladding air hole is d2, and d2>d1. The hole spacing of the inner cladding air holes is Λ1, the hole spacing of the outer cladding air holes is Λ2, and Λ1=Λ2. The twist period of the optical fiber is Λ L , the distortion rate is α.

[0032] In this embodiment, the parameters of an orbital angular momentum generator based on a helical twisted photonic crystal fiber are as follows: the diameters of the air holes in the inner and outer claddings d1 and d2 are 1 μm and 1.6 μm, respectively. The spacings between the air holes in the inner and outer claddings Λ1 and Λ2 are 2.2 μm and 2.2 μm, respectively, and the width of the annular core H between the double claddings is 3.1 μm. The radius R2 of the entire fiber cross section is 19.14 μm, and the twist rate α is a variable under study, with a range of values ​​including α=7391.983 rad / m, α=7853.982 rad / m, and α=8377.58 rad / m.

[0033] During the analysis, the present invention simplifies the three-dimensional problem into a two-dimensional problem through the principle of optical transformation, uses the finite element method and combines the perfect matching layer boundary absorption condition to perform theoretical calculations, obtains the modes supported by the fiber core and the annular core of the present invention, calculates the mode effective refractive index, the thin supermode loss spectrum and the generated OAM mode purity, wherein the specific principle of optical transformation is as follows:.

[0034] The spiral coordinate system for three-dimensional problems is (ξ1, ξ2, ξ3), and the Cartesian coordinate system for two-dimensional problems is (x, y, z). The parameter conversion relationship between the two coordinate systems is:

[0035]

[0035] The Jacobian matrix of the conversion relationship between three-dimensional coordinates and two-dimensional coordinates is:

[0037]

[0038] The transformation of the coordinate system is equivalent to replacing the isotropic material with an inhomogeneous anisotropic equivalent material. At this time, the dielectric tensor ε' in the spiral coordinate system is εT -1 and the magnetic permeability tensor μ'=μT -1 are all tensors, and ε = n 2 and μ=1 are the dielectric constant and magnetic permeability in Cartesian coordinates, respectively, and both are real numbers. n is the refractive index of the material. T is the transformation matrix between the two coordinates, and its inverse matrix is ​​T -1 :

[0039]

[0040]

[0041] When the helically twisted perfect matching layer is formed, the rectangular coordinate system is first converted to the cylindrical coordinate system (ρ, θ, z), and the complex constant s is introduced. ρ ,use represents the complex stretched form of ρ. Then the cylindrical coordinate system is transformed into the spiral coordinate system, and the spiral torsion angle is introduced The transformation matrix and inverse matrix of the two coordinate transformations are T PML and At this time, the dielectric tensor of the replaced material is The magnetic permeability tensor is

[0042]

[0043] in

[0044] In this embodiment, an orbital angular momentum generator based on a helically twisted photonic crystal fiber has three real fiber cores supporting six supermodes, namely, modes "M1, M2, M3, M4, M5, M6", which can be divided into left-handed (LP) polarization modes (M1, M2 and M4) and right-handed (RC) polarization modes (M3, M4 and M5) according to the polarization direction of the supermode. Since the effective refractive index is approximately equal when α=0, the six supermodes can be divided into three module groups, namely, module 1 "M2 and M3", module 2 "M5 and M6", and module 3 "M1 and M4". When the α value increases, circular birefringence will occur in the three modules, such as Figure 3 As shown, observe Figure 3 It can be seen that:

[0045] The orbital angular momentum generator based on helically twisted photonic crystal fiber described in the present invention will have circular birefringence in the three modules as the twist rate increases, that is, the left-handed and right-handed polarization modes in the three modules will split their effective refractive indexes, and will change in a "<" symmetrical trend as the twist rate increases, that is, one of the two supermodes contained in each module increases as the twist rate increases, and the other decreases as the twist rate decreases. The specific change relationship of the effective refractive index is:

[0046]

[0047] Among them, n eff ' is the effective refractive index of the mode after the fiber is twisted, n eff is the effective refractive index of the untwisted fiber mode, k is the total order of the mode (k = l + s), s is the spin angular momentum, λ is the wavelength, and Λ L is the twist period, and α is the twist rate.

[0048] The orbital angular momentum generator ring core based on the helical twisted photonic crystal fiber described in the present invention can support the intrinsic modes (EH mode and HE mode). Figure 4 and Figure 5 The ring core supports and Mode phase diagram, observation Figure 4 and Figure 5 It can be seen that the phase diagram has a spiral phase wavefront for the following reasons:

[0049] When the photonic crystal fiber of the present invention does not have a helical twist, the Poynting vector of the fiber mode precisely points to the direction of the fiber axis. When the photonic crystal fiber is helically twisted, there is a helical channel induction, and the light in the annular core part is forced to propagate along the helical path, then part of the light energy points to the transverse plane of the fiber, that is, at this time, the Poynting vector of the fiber mode partially points to the transverse plane of the fiber, so that the mode carries orbital angular momentum, and at this time, the intrinsic modes HE and EH modes can be regarded as weak OAM modes.

[0050]

[0051] Among them, l is the fiber mode order, m is the radial mode order, which is generally 1, even and odd are the even mode and odd mode of HE and EH modes respectively. The superscript "±" represents the left and right circular polarization directions, and the subscript "±" represents the direction of the spiral wavefront. Figure 4 and Figure 5 If the spiral wavefront rotates counterclockwise, its orbital angular momentum order l>0, and if the spiral wavefront rotates clockwise, its orbital angular momentum order l<0. When the polarization direction rotates counterclockwise, it is the RC polarization state, and its spin angular momentum s=-1. When the polarization direction rotates clockwise, it is the LC polarization state, and its spin angular momentum s=+1.

[0052] When the twist rate of the orbital angular momentum generator based on the helical twisted photonic crystal fiber of the present invention is α=7853.982rad / m, the loss spectrum of the core supermode is as follows: Figure 6-Figure 9 As shown, observe Figure 6-Figure 9 It can be seen that:

[0053] The core supermode of the orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention will have a loss peak at a specific wavelength. This is because the core supermode and the ring core mode resonate and couple at this wavelength, which increases the loss of the core supermode. At this time, the optical fiber is in a hybrid mode state, and both the core supermode and the ring core mode carry orbital angular momentum. Therefore, the hybrid mode is also called the orbital angular momentum mode, and the loss value of the orbital angular momentum mode is relatively ideal, all less than 1.64×10 -3 dB / m. Combining formula (9), we can know:

[0054] The orbital angular momentum generator based on the helical twisted photonic crystal fiber of the present invention can generate "OAM" when the twist rate is α = 7853.982rad / m -4,1 , OAM +9,1 , OAM +10,1 , OAM +11,1 , OAM +13,1 "Orbital angular momentum mode, a total of 9 loss peaks.

[0055] When the twist rate of the orbital angular momentum generator based on the helical twisted photonic crystal fiber of the present invention is α=7853.982rad / m, the relationship between the effective refractive index of the core supermode and the ring core mode and the wavelength is as follows: Fig.10 and Fig.11 As shown, observe Fig.10 and Fig.11 It can be seen that:

[0056] The resonant coupling between the core supermode and the ring core mode in the orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention needs to meet the matching conditions:

[0057] n co =n rc (10)

[0058] where n co and n rc are the effective refractive index of the supermode of the fiber core after twisting and the effective refractive index of the ring core mode, respectively.

[0059] When the twist rates of the orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention are α=7391.983 rad / m and α=8377.58 rad / m, the loss spectra of the core supermode are as follows: Fig.12 and Fig.13 As shown, observe Fig.12 and Fig.13 It can be seen that:

[0060] After the structure of the orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention is determined, it can generate "OAM -4,1 , OAM +9,1 , OAM +10,1 , OAM +11,1 , OAM +13,1 " mode, but the number is different. That is, the OAM mode that can be generated in theory is certain, but the OAM mode that can be generated in practice is related to the twist rate. When α=7391.983rad / m, there are 7 loss peaks in total, and when the twist rate α≥8377.58rad / m, at least 10 loss peaks are generated.

[0061] When the twisting rate of the orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention is different, OAM mode purity changes as Fig.14 As shown, observe Fig.14 It can be seen that:

[0062] The orbital angular momentum generator based on the helical twisted photonic crystal fiber described in the present invention has excellent characteristics, and the purity of the generated OAM mode is greater than 93%.

[0063] The above is only one embodiment of the present invention, not all or the only embodiment. Any equivalent changes made to the technical solution of the present invention by ordinary technicians in this field after reading the specification of the present invention are covered by the claims of the present invention.

Claims

1. An orbital angular momentum generator based on a helical twisted photonic crystal fiber, which applies helical twist to a specially designed photonic crystal fiber to achieve the generation of higher-order orbital angular momentum, characterized in that: The following steps are involved: Step 1: A photonic crystal fiber that can be used for orbital angular momentum transmission is designed, characterized in that: the photonic crystal fiber is composed of three real fiber cores and a double cladding, the air holes in the double cladding are circular and distributed in a hexagonal shape as a whole, and three air holes distributed at an angle of 120 degrees around the middle air hole in the inner cladding can form three real fiber cores distributed in a regular triangle, and there is a certain distance between the two claddings to form a ring core. At this time, the ring core can support intrinsic modes, and the same-order intrinsic odd and even modes with a phase difference of π / 2 can be superimposed into an OAM mode, so that the optical fiber supports orbital angular momentum transmission. The specific superposition formula is: Where l is the orbital angular momentum order, m is the radial mode order, and j is the π / 2 phase difference. Step 2: Helically twist the photonic crystal fiber designed in step 1 to form an orbital angular momentum generator with a twist period of Λ L , the distortion rate is α=2π / Λ L , and then the three-dimensional problem is converted into a two-dimensional problem through the principle of optical transformation for research. Under the action of spiral twisting, the eigenmode supported by the ring core will carry orbital angular momentum. For specific twisting rates and wavelengths, the core supermode and the ring core mode will resonate and couple to form a mixed mode state. At this time, the mixed mode carrying orbital angular momentum can also be called the orbital angular momentum mode. By observing the loss peak in the core supermode loss spectrum at different twisting rates, the specific wavelength where the resonant coupling occurs can be found, and then the conditions for generating orbital angular momentum can be determined. The calculation formula for mode loss is: Where L is the mode loss value, λ is the wavelength, Im(n eff ) is the imaginary part of the effective refractive index of the core supermode, n eff is the effective refractive index, in dB·m -1 . Step three: By analyzing the mode electromagnetic field of the orbital angular momentum generator based on the helically twisted photonic crystal fiber in step two, the effective refractive index and electromagnetic field distribution of the core supermode, ring core mode and orbital angular momentum mode can be obtained, and then the influence of helical twist on the effective refractive index of the fiber mode can be known; by analyzing the phase of the orbital angular momentum mode, the corresponding generated orbital angular momentum order can be determined; by studying the changes in the effective refractive index of the core supermode and ring core mode, the matching conditions when resonant coupling occurs can be known.

2. An orbital angular momentum generator based on a helically twisted photonic crystal fiber according to claim 1, characterized in that: In the step 1, the air holes in the inner and outer cladding layers are of different sizes, and the diameter of the air holes in the inner cladding layer is smaller than the diameter of the air holes in the outer cladding layer.

3. An orbital angular momentum generator based on a helically twisted photonic crystal fiber according to claim 1, characterized in that: In the step 1, the hole spacings between the two cladding air holes are equal.

4. An orbital angular momentum generator based on a helically twisted photonic crystal fiber according to claim 1, characterized in that: In the step 1, the base material of the optical fiber is silicon dioxide.

5. An orbital angular momentum generator based on a helically twisted photonic crystal fiber according to claim 1, characterized in that: In the step 2, the three-dimensional problem is converted into a two-dimensional problem through optical transformation, and then the fiber mode is analyzed. The optical change can reduce the complexity of the research. The specific principle formula is: The spiral coordinate system for three-dimensional problems is (ξ1, ξ2, ξ3), and the Cartesian coordinate system for two-dimensional problems is (x, y, z). The parameter conversion relationship between the two coordinate systems is: Where α = 2π / Λ L is the twist rate, in rad·m -1 The Jacobian matrix reflecting the transformation relationship between the two coordinates is: The transformation of the coordinate system is equivalent to replacing the isotropic material with an inhomogeneous anisotropic equivalent material. At this time, the dielectric tensor ε' in the spiral coordinate system is εT -1 and the magnetic permeability tensor μ'=μT -1 are all tensors, and ε = n 2 and μ=1 are the dielectric constant and magnetic permeability in Cartesian coordinates, respectively, and both are real numbers. n is the refractive index of the material. T is the transformation matrix between the two coordinates, which depends only on ξ1 and ξ2. Its inverse matrix is ​​T -1 :

6. An orbital angular momentum generator based on a helically twisted photonic crystal fiber according to claim 1, characterized in that: a. The eigenmode supported by the ring core after the fiber helical twist carries orbital angular momentum. At this time, the eigenmode can be directly regarded as a discrete weak OAM mode b. When α≥8377.58rad / m, the core supermode and the ring core mode resonate and couple to generate at least 10 OAM modes, among which "+13th order" is the highest order and lowest loss orbital angular momentum mode currently generated by the helical twisted fiber method. The resonant coupling needs to meet the matching conditions: n co =n rc (8) Among them, n co and n rc are the effective refractive index of the supermode of the fiber core after twisting and the effective refractive index of the ring core mode, respectively. c. The effective refractive index change relationship of the twisted fiber mode is: Among them, n eff ' is the effective refractive index of the mode when the optical fiber is twisted, n eff is the effective refractive index of the fiber mode when it is not twisted, k is the total order of the mode (k = l + s), s is the order of spin angular momentum, λ is the wavelength, Λ L is the twist period, and α is the twist rate.

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