Mode Conversion Between Beams Carrying Orbital Angular Momentum With Opposite Topological Charges Using Two-Dimensional Multimode Interference Waveguides

Recently, the implementation of rectangular waveguides for generation and manipulation of orbital angular momentum (OAM) modes, has been developed to exploit the outstanding features of these modes in more complex integrated devices. Multimode interference (MMI) structures have been widely used in both one and two dimensions as the basic elements in many integrated optical devices like optical beam splitters, mode converters, couplers, wavelength-division (de)multiplexers, and switches. According to the various applications of OAM modes in quantum and classical communications, the study of their propagation properties in MMI waveguides is useful in order to facilitate the way into higher security and capacity systems. This paper presents the results of numerical investigation of a proposed integrated optical OAM mode converter which is performed based on the mode decomposition properties of propagating OAM modes in 2D MMI waveguides, as well as the fact that any order of OAM mode can be represented as superposition of an odd mode and a quarter-wave shifted even mode. The proposed device, which consists of two MMI waveguides connected by phase shifters and linear waveguides, provides mode conversion between beams carrying OAM with odd opposite topological charges. The design procedure is performed for OAM modes with topological charge values of l = ±1 and l = ±3 using beam propagation method. The proposed device is passive with reciprocal behavior and has the output mode purity of 94% and 82% for beams carrying l = ±1 and l = ±3, respectively.

1 Abstract-.Recently, the implementation of rectangular waveguides for generation and manipulation of orbital angular momentum (OAM) modes, has been developed to exploit the outstanding features of these modes in more complex integrated devices.Multimode interference (MMI) structures have been widely used in both one and two dimensions as the basic elements in many integrated optical devices like optical beam splitters, mode converters, couplers, wavelength-division (de)multiplexers, and switches.According to the various applications of OAM modes in quantum and classical communications, the study of their propagation properties in MMI waveguides is useful in order to facilitate the way into higher security and capacity systems.In this work, mode conversion between beams carrying OAM with opposite topological charges in the two-dimensional (2D) MMI waveguides is investigated.In this investigation, the OAM modes with odd topological charges are considered.Using the self-imaging properties of MMI structures, an integrated device including two MMI waveguides connected by phase shifters and linear waveguides is designed.The design procedure is performed for OAM modes with topological charge values of l=±1 and l=±3 using beam propagation method.The proposed device is passive with reciprocal behavior and has the output mode purity of 94% and 82% for beams carrying l=±1 and l=±3, respectively.

I. INTRODUCTION
AISED by Allen in 1992 [1], light beams with the phase dependence of e ilφ carry orbital angular momentum (OAM) independent of the polarization state, where φ is the azimuthal angle, and l indicates the topological charge that can take any integer value, positive or negative.Topological charge represents the number of light twists in one wavelength [2].OAM beams with different values of l, are orthogonal to each other.This property introduces a new degree of freedom that makes OAM very powerful tool for applications in classical and quantum optical communication systems for higher capacity and security [3].
The new degree of freedom introduced by OAM can be effectively exploited to enhance the integrated optical devices such as optical beam splitters [4], mode convertors [7], couplers [8], wavelength-division (de)multiplexers [9] and capacity enhancement of the optical communication links through mode division multiplexing (MDM) technique which loads channel's data on different simultaneous carrying modes [10].Hence, in order to implement their commercial communication systems, switching and conversion of OAM modes are essential operations.
Multimode interference (MMI) [11] structures with many interesting features like their compact size, low sensitivity to fabrication parameters, and ease of fabrication, are efficient candidates for using them in the integrated optical structures.MMI structures, based on the interference between the modes of a multimode waveguide, are now widely used in both one and two dimensions as the basic elements in many switches [12].In one dimensional (1D) MMI devices, the waveguide is single-mode in the transverse dimension and multimode in the other dimension, whereas in two dimensional (2D) ones are multimode in both horizontal and vertical directions [13].In order to carry the power by higher order modes with 2D field distributions, 2D MMI devices are required.From this point of view, a 2D MMI structure can also be utilized for OAM mode transmission [14].
Accordingly, in this work, mode conversion between OAM modes with opposite topological charges in 2D MMI Mode Conversion between Beams Carrying Orbital Angular Momentum with Opposite Topological Charges using Two-Dimensional Multimode Interference Waveguides Afsoun Soltani, Zaker Hossein Firouzeh, S. Faezeh Mousavi, Abolghasem Zeidaabadi Nezhad, and Rahman Nouroozi waveguides is investigated.This investigation is performed for OAM modes with odd values of l.For this propose, using the self-imaging properties of MMI structures, an integrated device including two MMI waveguides connected by phase shifters and linear waveguides is designed.The lengths of MMI waveguides are chosen according to the behavior of OAM modes in these waveguides.It should be noted that any order of OAM mode can be decomposed into odd and even mode field components after passing a certain length of the MMI waveguide [15].This issue and the fact that OAM modes with opposite values of l have imaginary parts with π phase difference, is utilized to design a two stage MMI waveguide joint with tapered phase shifters in between, in order to investigate mode conversion between the odd order of OAM modes with opposite topological charges.The design procedure is performed for OAM modes with l=±1 and l=±3 using beam propagation method (BPM).

II. PRINCIPLE
Based on the multimode interference theory [13], selfimaging in a 2D MMI waveguide produces N×M images at the distances L that can be expressed as [13]: where N and M are positive integers without common divisors with the positive integers Sx and Sy, respectively, which are the positional numbers in the X and Y directions, respectively.Lπx and Lπy also represent the coupling lengths between the two lowest order modes; the former in X direction and the latter in Y direction.For simplicity, in the following discussion, Sx= Sy=1 which is also a common practice for the shortest device length.For a 2D square cross sectional MMI waveguide, the coupling lengths in the X and Y directions are equal and can be expressed as [13]: where nf, λ0 and W are the waveguide material's refractive index, the working wavelength, and the waveguide's width, respectively.
The imaging properties of OAM modes in 2D square cross sectional MMI waveguides indicate that an incident OAM mode, exhibit field-splitting at 3Lc/4 and OAM-maintaining image at 3Lc/2 [15].It is worth mentioning that any order of OAM modes can be decomposed into odd-and even-mode field components as [16]: where ± sign is determined by the sign of the OAM order.For OAM modes with odd topological charges, the location of split fields at field splitting position is at the four sides of the 2D MMI waveguide cross section as shown in Fig. 1.In this figure, the split fields at the top and bottom sides are related to the real part of (3), and the right and left side fields are associated with the imaginary part of (3).
In accordance with (3), OAM modes with opposite topological charges have identical odd parts and π phase difference in even parts.This means conversion between OAM modes with opposite topological charges can occur by applying π phase shift between even parts.This fact is used to design an integrated device to do OAM mode conversion by applying π phase shift between the sided decomposed field components at 3Lc/4 length of a 2D MMI waveguide.

III. DESIGN AND SIMULATION
The proposed device (Fig. 2) consists of three main parts including two identical MMI waveguides and four intermediate ports in between.The first MMI waveguide with field splitting length of 3Lc/4 decomposes the input OAM mode.Intermediate ports in the second part, guide up and down field patterns with no change in phase, and apply π phase shift to the sided decomposed fields.The second MMI waveguide composes the manipulated fields in the last part and makes OAM mode with opposite topological charge.
The simulation of the structure performance is done using the commercially available simulation software package OptiBPM 13.1.All the simulations assume a vacuum wavelength of λ0=1550 nm and a silicon waveguide (nf=3.45)surrounded by silica (nc=1.45)which is compatible with silicon photonics technology.
In order to define the device's dimensions, the MMI waveguides with the widths of 20, 25, 30 and 35 µm are studied.It is initially considered that all the intermediate ports of the proposed structure are linear waveguides.In this situation, the efficient width and height of the ports are determined for each MMI waveguide in such a way that they can transfer maximum amount of the power from the first to third part of the structure.The obtained geometrical parameters are summarized in Table I for input OAM modes with l=±1.The last column of this table indicates the calculated power overlap integral (POI) between the field patterns before and after of the intermediate ports.As shown in Fig. 1, the field distributions in the top-bottom sides are different from right-left sides just in their orientations.Therefore, the width and height of the phase shifters are the same as the height and width of the linear waveguide ports, respectively.The proposed phase shifters make desired phase difference equal to π by gradually reducing the width by half until their half-length, and then increasing the width to the end of their length.It should be noted that the phase difference can be achieved by changing either the height or the width, however a simple way that guarantees an adiabatic transition with very low loss structure is to gradually change the width, as used here.The simulation results imply that the shorter the phase shifter's length, the higher the purity of the output mode.On the other hand, as shown in Table I, the calculated POI is approximately 98% for all of the considered MMI waveguides.Therefore, the dimensions of the proposed structure are selected based on the shortest length for the intermediate ports which are summarized in Table Ⅱ.The same procedure is performed for OAM modes with l=±3.In this case, the optimum dimensions for the MMI waveguides are same as the case of OAM modes with l=±1 (Wx= Wy =20 µm, and L=901 µm).However, the sizes of the intermediate ports are different as listed in Table Ⅲ.In order to evaluate the performance of the proposed structure, the purity of output modes, E(ρ,θ), are calculated using [16]    For odd order OAM modes with |l|>3, the quality of the generated OAM-maintaining image at length 3Lc/2 of the MMI waveguide with Wx= Wy =20 µm is studied by calculating POI along the waveguide (Fig. 4).In general, as the order of input OAM modes in 2D MMI waveguides increases, the value of POI between input mode and generated image decreases.Fig. 4 implies that this decrease is significant for the waveguide with width of 20 µm.Hence, the design of the proposed converter is no longer optimal.As shown in Fig. 5, the performance of MMI waveguides can be improved by enlarging the width of them.Therefore, in order to design proposed mode convertor for higher order OAM modes, it is efficient to use wider MMI waveguides.

IV. CONCLUSION
In this paper, mode conversion between odd order of OAM modes with opposite topological charges in the 2D MMI waveguides is investigated.An integrated optical convertor consisting of three main parts including two MMI waveguides and four intermediate ports in between is introduced.Using the imaging properties of OAM modes in 2D square cross sectional MMI waveguides, the first part of the device decomposes the OAM modes into odd-and even-mode field components.As OAM modes with opposite topological charges have identical odd parts and π phase difference in even parts, phase shifters controlled by waveguide width have been used in the second part to apply appropriate phase shift.In the end, the third part of the proposed device produces OAM mode with opposite topological charge by composing the manipulated fields in the last part.The simulations are performed using BPM method for silicon waveguides surrounded by silica which are compatible with silicon on insulator technology.The proposed mode convertor is passive and the input and output ports can be swapped.The ability to produce high purity output mode (94% and 82% for OAM modes with l=±1 and l=±3, respectively) and reciprocal behavior of the proposed device are two remarkable features that make it utilizable in many classical and quantum communication systems.

Fig. 1 .
Fig. 1.Normalized power distributions of the split fields at the field-splitting length of an MMI waveguide for the input OAM modes with (a) l=±1, (b) l=±3, and (c) l=±5.

Fig. 3 (
a) and (b), show simulation results based on the parameters represented in Table Ⅱ and Table Ⅲ for input OAM modes with l=+1 and l=+3, respectively.In this figure, left and right columns display the normalized power distribution and phase pattern of the input (top rows) OAM modes, and output (bottom rows) generated modes.Comparing the phase patterns of the input and output modes, that are respectively clockwise and counterclockwise for each twist, it is clear that the topological charges are reversed.

Fig. 3 .
Fig. 3. Normalized power distributions (left) and phase patterns (right) of the input (top rows) OAM modes of (a) l = +1, and (b) l = +3, and output (bottom rows) generated modes of (a) l = -1, and (b) l = -3 for the proposed structure.A comparison between the phase pattern of the input and output modes, clearly implies that the topological charges are reversed. : POI between output generated mode and the input OAM mode are studied.

Fig. 5 .
Fig. 5.The calculated POI between the input OAM modes with l = ±1 to ±9 and the generated OAM modes in square cross sectional 2D MMI waveguides with width in the range of 15 to 60 µm.

TABLE III GEOMETRICAL
PARAMETERS OF THE INTERMEDIATE PORTS FOR INPUT OAM MODES WITH l=±3 Table Ⅳ summarizes the calculated purity and POI for the proposed structure.