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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

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72176
10.1038/s41598-024-72176-7
Article
Investigating orbital angular momentum modes in multimode interference (MMI) waveguides and revealing their mode conversion property
Soltani Afsoun 1
Mousavi S. Faezeh 23
Firouzeh Zaker Hossein zhfirouzeh@iut.ac.ir

1
Zeidaabadi Nezhad Abolghasem 1
http://orcid.org/0000-0003-4430-157X
Nouroozi Rahman 4
1 https://ror.org/00af3sa43 grid.411751.7 0000 0000 9908 3264 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 8415683111 Iran
2 https://ror.org/02n742c10 grid.5133.4 0000 0001 1941 4308 Department of Physics, University of Trieste, 34127 Trieste, Italy
3 https://ror.org/02dp3a879 grid.425378.f 0000 0001 2097 1574 CNR-INO, National Institute of Optics, 34149 Trieste, Italy
4 https://ror.org/00bzsst90 grid.418601.a 0000 0004 0405 6626 Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, 45195-1159 Iran
11 9 2024
11 9 2024
2024
14 212725 3 2024
3 9 2024
© The Author(s) 2024
2024
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In this work, the propagation of OAM modes in multimode interference (MMI) waveguides, as the basic elements in many integrated optical devices, is studied to utilize their benefits in integrated OAM applications. OAM modes shape the OAM-maintaining image at the specific length of an MMI waveguide. As the most effective parameters on the properties of the generated image, waveguide’s width (W), topological charge (ℓ) and waist radius (WR) of the input OAM modes are investigated. Power overlap integral (POI) is used to evaluate the quality of images. The investigations show that the calculated POI is enhanced by increasing in WR of the input mode (from 86.91% for WR = 1.5 µm to 98.92% for WR = 3.5 µm; where W = 15 µm and ℓ=±1). Furthermore, the increase in the waveguide’s width leads to decrease the quality of the self-imaged mode (from 98.90% for W = 15 µm to 85.15% for W = 50 µm; where WR = 3 µm and ℓ=±1). It is also demonstrated that mode conversion between even order of OAM modes with opposite topological charges can occur at OAM-maintaining length of the MMI waveguides, which is the most outstanding achievement of this survey for optical communication systems.

Subject terms

Integrated optics
Electrical and electronic engineering
Quantum optics
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Pioneered by Allen in 19921, light beams with the phase dependence of eiℓϕ carry OAM, independent of the polarization state, where ϕ is the azimuthal angle, and ℓ indicates the topological charge (ℓ=±1,±2,…). Topological charge represents the number of twists the light does in one wavelength. In order to process OAM modes, exploiting functionalities such as generation, transmission, and conversion are necessary. To date, generation and manipulation of OAM modes have been developed using several approaches including the spatial light modulators2–4, spiral phase plates5, q-plates6,7, and diffractive phase holograms8. For the special case of OAM mode conversion, which is among the interested functionalities in this paper, using cylindrical lenses1,9,10 and fiber gratings11–15 have been widely reported to realize OAM convertors. For instance, as a seminal reported work, a suspended combination of two cylindrical lenses has been used to transform a Laguerre-Gaussian mode of OAM -ℓħ per photon into one with +ℓħ per photon1, which has been shown the mechanical torque generation using the maximum transformation of OAM. Furthermore, the study of angular momentum interconversion with the aid of optical elements such as the hollow metallic cone, the solid dielectric cone and the metallic wedge has been represented16.

Compared to the mentioned approaches, which suffer from drawbacks such as the complexity of optical alignment12 and the necessity of precise control of parameters17, the integrated implementations have been attracted due to the significant advantages in reliability, miniaturization and scalability18. There are two approaches for generating and manipulating OAM beams using photonic integrated circuits (PICs): out-of-plane and in-plane19. The out-of-plane method involves employing PIC elements or integrated light sources to induce scattered field formation, resulting in the desired OAM modes. Microring resonators20,21, circular phase array emitters22–24, and subwavelength gratings25–28 are the different structural designs for out-of-plane generation of OAM modes. In order to achieve sophisticated OAM functionalities on photonic chips, in-plane approach, which entails on-chip OAM beam generation or its controlled injection into the chip plane, is crucial19. The most common design structures used for in-plain OAM applications are rectangular waveguides. The computational and theoretical investigation into the prospect of OAM modes transmission over dielectric rectangular waveguides has been reported by Lyubopytov et al.29. Furthermore, an on-chip integrated structure including silicon waveguides and couplers has been presented, which can produce OAM modes with ℓ=±130. A three-layer waveguide structure with the functions of the chirality conversion and the torque generation has also been proposed31. The presented convertor converts incident angular momentum into the opposite one. Additionally, an electro-optically active lithium niobate on insulator photonic wire configuration has been presented for manipulation of modes encoded in OAM-SAM states32. Moreover, a rectangular waveguide with a single trench has been designed to generate OAM modes with ℓ=±133. In addition, the integrated rectangular platforms have been represented to generate OAM beam only on the longitudinal component of the electric field which made the application of proposed platforms complicated34–36. Additionally, the investigation of spin and orbital angular momentum of optical fields in a silicon channel waveguide has been proposed based on the superposition of two quasi-TE modes which are limited to the first order of OAM modes37. In another recent work, the design approach to a grating coupler for in-plane generation and propagation of quasi-TE vortex modes with azimuthal order of ℓ=±1 within PICs has been suggested38.

In addition to the mentioned approaches, MMI structures, as a novel and actually neglected choice for integrated OAM applications, can be introduced which have many interesting features, such as their compact size, low sensitivity to fabrication parameters, and ease of fabrication39. In the last few years, MMI structures, based on the interference between the modes of a multimode waveguide, have widely been used in both one and two dimensions as the basic element in many integrated optical devices such as optical beam splitters40–42, mode convertors43, couplers44, wavelength-division (de)multiplexers45, and switches46. 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) devices, MMI waveguides are multimode in both horizontal and vertical directions47. In order to carry the power by higher order modes with 2D field distributions, 2D MMI devices are required. From this point of view, 2D MMI structures can be utilized for OAM modes transmission. The use of 2D MMI structures for OAM modes, was first introduced by describing the self-imaging property of OAM modes in MMI waveguides48, which means the input field profile can be reproduced in single or multiple images at periodic intervals along the propagation direction. 2D MMI waveguides for manipulation of beams carrying OAM has also been utilized to design an OAM mode convertor49. However, these structures still have many unknown potentials in using OAM modes which can make them an attractive and practical part of many circuits.

Accordingly, in this work, the propagation of OAM modes in 2D square cross-sectional MMI waveguides is investigated. OAM modes form the OAM-maintaining image at the specific length of an MMI waveguide. The properties of the OAM-maintaining image are studied by considering three main parameters including the width of the waveguide, the waist radius (WR) of the OAM mode and its topological charge. It is also represented theoretically, and confirmed by simulation results that OAM modes with odd and even values of charge have different behavior in MMI waveguides. For odd order of OAM modes, the generated images at the OAM-maintaining length of an MMI waveguide have the same topological charge as the input, whereas the even order ones are reversed (ℓ→-ℓ). In addition to the topological charge, the propagation of OAM modes in MMI waveguides are affected by the width of MMI waveguide and WR of the input mode. These parameters are chosen in the ranges of 15–50 µm, and 1.5–3.5 µm, respectively for the investigation purposes of this paper. The properties and quality of the images along waveguides influenced by the referred parameters are discussed for OAM modes with odd and even values of ℓ, separately.

Theory

As mentioned in Introduction, 2D MMI waveguides support multiple modes in both horizontal and vertical directions. The analysis of these structures can be performed by extending the guided mode propagation analysis of 1D MMI structures to the 2D case. The guided modes of a 2D MMI waveguide (Fig. 1) with Wx lateral width and Wy vertical width in X and Y directions, respectively, have the form of the following equation50:1 ψuvx,y=ψuxψvy=sinπu+1Wxxsinπv+1Wyy,

where u,v=1,2,3,… are the mode orders in X and Y directions, respectively. The corresponding longitudinal propagation constants are satisfied as:2 βuv2=k2ng2-πu+1Wx2-πv+1Wy2,

where k=2π/λ0 indicates the wave number, λ0 is the working wavelength in vacuum, and ng represents the refractive index of the multimode waveguide. Neglecting the reflected field as well as the power coupled to the radiative modes, the incident field of Ψx,y,0 can be expressed as a superposition of the infinite numbers of guided modes as:3 Ψx,y,0=∑u=0∑v=0Cuvψuvx,y,

where,4 Cuv=4WxWy∫0x∫0yΨx,y,0ψuvx,ydxdy.

After propagating a distance L in the waveguide, the field profile Ψx,y,L can be expressed as:5 Ψx,y,L=∑u=0∑v=0Ψx,y,0ejωt-βuvL,

Fig. 1 Schematic diagram of a 2D MMI waveguide with Wx lateral width and Wy vertical width in X and Y directions, respectively.

Taking the fundamental mode out of the summation and using the paraxial approximation for the propagation constant in Eq. (2), the field profile can be written as:6 Ψx,y,L=ejωt-β00L∑u=0∑v=0Ψx,y,0×expjuu+23LπxπL+jvv+23LπyπL,

where Lπx and Lπy are the coupling lengths between the two lowest order modes in X and Y directions, respectively, as:7 Lπx=πβ00-β10=4ngWxeff23λ0,

8 Lπy=πβ00-β01=4ngWyeff23λ0.

In these equations Wxeff and Wyeff denote the effective waveguide thicknesses, the former in X and the later in Y directions, as:9 Wxeff=Wx+λ0πng2-nc2,

10 Wyeff=Wy+λ0πng2-nc2.

with nc the cladding refractive index51.

In multimode waveguides, an input field profile can be reproduced in single or multiple images at the periodic intervals along the propagation distance of the waveguide, according to the self-imaging property. In Eq. (6), the distance L to produce self-imaging can be expressed as39:11 L=SxN3Lπx=SyM3Lπy.

where N and M are the positive integers without common divisors with the positive integers Sx and Sy, which are the positional numbers in X and Y directions, respectively. For simplicity, in the following discussion, Sx=Sy=1, which is also a common practice for the shortest device length. In a square cross-sectional MMI waveguide (Wx=Wy=W, Lπx=Lπy=Lc, and N=M), if N is an even number (N=2K, where K is an integer), the number of images at the length L=3Lc/N can be decreased. It is worth noting that odd values of K result K self-images for both symmetric and anti-symmetric inputs48. Therefore, as the OAM modes always have the anti-symmetric field components, K=1 leads to produce only one image at the shortest length 3Lc/2 of a 2D square cross-sectional MMI waveguide.

Results

In order to consider the self-imaging phenomenon of OAM modes in MMI waveguides, the mode propagation inside these waveguides is simulated using beam propagation method (BPM) by the commercially available simulation software package OptiBPM 13.1. All the simulations assume a silicon waveguide (ng=3.45) surrounded by silica (nc=1.45) at the working wavelength λ0=1550nm. The detailed consideration of this phenomenon is performed by studying three main parameters. The first one, which is related to the physical structure of the MMI waveguide, is the waveguide’s width. Two other parameters associated with the properties of OAM modes, are WR and the order of the mode’s topological charge.

Width

Width of the MMI waveguide directly specifies the required waveguide’s length to generate the OAM-maintaining image. The considered widths in this study are in a range of 15–50 μm. Table 1 summarizes the calculated OAM-maintaining lengths for the waveguides with the mentioned widths. The results confirms that the wider waveguides need the longer length to produce OAM images.Table 1 The calculated OAM-maintaining lengths for considered waveguides with widths in a range of 15–50 μm.

Waveguide’s width (μm)	15	20	25	30	35	40	50	
OAM-maintaining length (μm) = 3Lc/2	1023	1809	2817	4049	5502	7179	11,199	

WR

In order to investigate the effect of WR, the OAM modes with WR in a range of 1.5–3.5 μm are propagated along the waveguides with the considered widths in Table 1.

As a performance criterion, the power overlap integral (POI) between the input mode (E1) and the output generated image (E2) is calculated as:12 POI=∫SE1x,yE2∗x,ydxdy2∫SE1x,y2dxdy·∫SE2x,y2dxdy.

The results are shown in Table 2 for the calculated POI (%) between the input first order of OAM modes with ℓ=±1 and the output produced image at OAM-maintaining length of the considered waveguides. They imply that for each waveguide’s width, the calculated POI increases with increasing WR of the input mode. In addition, for wider waveguides, a larger input mode’s WR leads to a higher value of POI. These deductions can be clearly inferred from Figs. 2 and 3, which show the normalized power distributions and phase patterns of the input (top rows) first order OAM modes with WR = 2 μm (a), WR = 2.5 μm (b), and WR = 3 μm (c) and the output (bottom rows) generated self-images for MMI waveguides with 15 μm and 20 μm width, respectively.Table 2 The calculated POI (%) between the input OAM modes with ℓ=±1 and the output produced image at OAM-maintaining length of the considered waveguides in Table 1.

Waveguide’s width (μm)	WR (μm)	
1.5	2	2.5	3	3.5	
15	86.91	94.75	97.88	98.90	98.92	
20	71.75	89.65	95.58	97.84	98.82	
25	56.36	81.13	93.79	95.83	97.72	
30	51.44	73.50	87.76	95.40	96.70	
35	41.54	68.32	87.17	93.20	96.64	
40	34.78	64.40	84.12	91.12	94.91	
50	25.03	49.98	72.32	85.18	91.66	

Fig. 2 Normalized power distributions (left) and phase patterns (right) of the input (top rows) first order OAM modes with WR = 2 μm (a), WR = 2.5 μm (b), and WR = 3 μm (c), and output (bottom rows) generated self-images for an MMI waveguide with 15 μm width.

Fig. 3 Normalized power distributions (left) and phase patterns (right) of the input (top rows) first order OAM modes with WR = 2 μm (a), WR = 2.5 μm (b), and WR = 3 μm (c), and output (bottom rows) generated self-images for an MMI waveguide with 20 μm width.

Topological charge

The third considered parameter is the topological charge of OAM modes. In General, any order of OAM mode field can be indicated as the superposition of odd and even mode fields as48:13 fOAMx,y=foddx,y±ifevenx,y,

where the ± sign is determined by the sign of OAM order. The odd and even parts of this equation can be further represented as symmetric or anti-symmetric field functions (fS or fA) in X or Y directions, as24:14 foddx,y=fSxfAy±ifAxfSy,

15 fevenx,y=fAxfAy±ifSxfSy.

Substituting these functions in Eq. (6) and consideration of obtaining one image at the center of a square cross sectional 2D MMI waveguide, the output field profiles at length 3Lc/2K can be written as:16 Ψoddx,y,L=fSxfAyejΘSxΘAy±ifAxfSyejΘAxΘSy,

17 Ψevenx,y,L=fAxfAyejΘAxΘAy±ifSxfSyejΘSxΘSy,

where ΘS and ΘA are the symmetric and anti-symmetric phase terms, respectively, as48:18 ΘSx=ΘSy=K22K+14π,

19 ΘAx=ΘAy=K22K-14π.

Substituting K=1, which corresponds to 3Lc/2, the phase terms become:20 ΘSx=ΘSy=3π4,

21 ΘAx=ΘAy=π4.

Hence, the output fields Ψodd and Ψeven are given as:22 Ψoddx,y,L=ejπfSxfAy±ifAxfSy,

23 Ψevenx,y,L=ejπ2fAxfAy∓ifSxfSy.

The comparison between Eqs. (22) and (16) as well as 23 and 17 clearly imply that the topological charge of odd order OAM modes remains unchanged after passing through the length 3Lc/2 of the MMI waveguide, whereas the charge of even order modes is reversed. Therefore, it can be inferred that an MMI waveguide with the length 3Lc/2 acts as inherent charge converter for OAM modes with even values of ℓ. This fact is schematically shown in Fig. 4.Fig. 4 Schematic diagram of a square cross-sectional 2D MMI waveguide with the length 3Lc/2. The waveguide acts as a charge converter for OAM modes with even values of ℓ.

The performance analysis of produced OAM-maintaining image has already been reported in Table 2 for ℓ=±1. For higher odd OAM modes, the POI graphs are illustrated in Fig. 5 for input modes with ℓ=±1 to ± 9 with WR = 3 μm. The graphs show that the higher the OAM mode order, the lower the calculated POI, especially for wider waveguides. However, the wider waveguides can counteract this decrease in POI by increasing the input mode's WR, as explained in the WR section.Fig. 5 The calculated POI between the input OAM modes of ℓ=±1 to ± 9 with WR = 3 μm and their corresponding output generated images.

For even order OAM modes, in order to declare the mentioned mode conversion property, the simulation results for the second, fourth and sixth order of OAM modes propagating along a 20 μm width MMI waveguide are shown in Fig. 6. In this figure, left and right columns display the normalized power distributions and phase patterns of the input (top rows) OAM modes of ℓ =  + 2 (a), ℓ =  + 4 (b), and ℓ =  + 6 (c), and the output (bottom rows) generated images of ℓ =  − 2 (a), ℓ =  − 4 (b), and ℓ =  − 6 (c). Comparing the phase patterns of the input modes and the generated images, that are respectively clockwise and counterclockwise for each twist, it is clear that the topological charges are reversed.Fig. 6 Normalized power distributions (left) and phase patterns (right) of the input (top rows) OAM modes of ℓ =  + 2 (a), ℓ =  + 4 (b), and ℓ =  + 6 (c), and output (bottom rows) generated images of ℓ =  − 2 (a), ℓ =  − 4 (b), and ℓ =  − 6 (c) for an MMI waveguide with 20 μm width. A comparison between the phase pattern of the input modes and the generated images that are respectively clockwise and counterclockwise for each twist, clearly implies that the topological charges are reversed.

In order to find out how well a 2D MMI waveguide can work as a charge converter for even order of OAM modes, the purity of the generated images, E (ρ, θ), are calculated using49:24 Purity=∫0∞∫02πEρ,θe-ilθ2πdθ2ρdρ∫0∞∫02πEρ,θ2dθρdρ.

The calculated purity for the mentioned waveguide in Fig. 6 with W = 20 μm is reported in Table 3.Table 3 The calculated purity for the mentioned waveguide in Fig. 6.

Topological charge	Purity (%)	
± 2	96.13	
± 4	84.85	
± 6	82.03	

Additionally, in order to illustrate the effect of dimensional variations, the fabrication errors of ± 0.5 μm are introduced into the geometrical parameters of this waveguide. Figure 7 shows the simulation results. For all the three input OAM modes, fabrication errors in the width of MMI waveguide (WMMI) degrade the calculated purity. However, MMI waveguide’s length (LMMI) remains essentially flat over the same error range. Consequently, the fabrication tolerance analysis will be concerned only with variations in WMMI, with errors in LMMI assumed to be negligible.Fig. 7 The effects of the fabrication errors on the calculated Purity; when a deviation of ± 0.5 μm is applied into the width and length of the mentioned waveguide in Fig. 6. The fabrication errors in WMMI degrades the calculated purity. However, LMMI remains essentially flat over the same error range.

In addition to the dimensional variations, surface roughness is another source of error in waveguides. Sidewall roughness effect in multimode rectangular optical waveguides has been already investigated in some references52–56. The theory behind this effect for 2D MMI waveguides can be summarized as follows53,55.

In a real waveguide, there is irregular distribution at the core-cladding interface, which can be characterized by the roughness σ. It is usually assumed that the distortion function of the core boundary is a stationary random process described by the function f(z).

Assuming x-polarized Emn modes in a rectangular waveguide, the coupling coefficients between the guided modes and the radiation modes caused by the aforementioned waveguide irregularities can be expressed as:25 Kmn,ρ=ωε04iP∫∫-∞∞n2-n02E→mn∗xE→ρxdxdy

where ω is the angular frequency of light, ε0 is the dielectric constant of vacuum, P is the power factor, E→mnx and E→ρx are the electric fields of Emn modes and the radiation modes along the x direction, respectively. The function n0(x, y) describes the refractive index distribution of the unperturbed waveguide whereas n(x, y, z) specifies the refractive index dependence of the real, distorted waveguide. The scattering loss coefficient αmn of Emn mode induced by σ is a characteristic parameter to describe the influence on the transmission loss of rectangular optical waveguides. According to the standard perturbation theory, αmn can be expressed as:26 αmn=∬∑s=14Kmn,ρsκ,ν2⟨Fsβmn-βρ2⟩dκdν

The different coupling coefficients Kmn,ρsκ,ν are assigned to the different sidewalls s of the waveguide. The distortion is introduced by the Fourier transform of the autocorrelation function of the stochastic distortion process:27 ⟨Fsβmn-βρ2⟩=∫-∞∞⟨fszfsz-u⟩exp-iβmn-βρudu

In order to determine the scattering loss coefficient αmn induced by the roughness σ, calculation of coupling coefficients Kmn,ρsκ,ν using efficient analytical methods is necessary. Finite-difference time domain (FDTD)57 and ray tracing58 have been used for this propose. Furthermore, a precise analytical method, the radiation-mode Fourier decomposition method (RFDM) has been introduced56. As seen in Eq. 26, regardless of what is the incident field into the waveguide, the analysis of roughness effect is performed by calculating coupling coefficients between guided and radiated modes of the waveguide. On the other hand, as mentioned, these calculations need applying analytical methods, which are elaborative and time-consuming process. Therefore, to maintain the continuity of this paper, which focuses on the behavior of OAM modes in 2D rectangular MMI waveguides (not just the 2D MMI waveguides features per se), and also to avoid content overload, performing the analytical solution is omitted.

Conclusion

This paper presents a study on OAM modes propagation in 2D square cross-sectional MMI waveguides, which leads to be known the mode conversion property of these waveguides. Based on OAM-MMI theory, OAM modes form an OAM-maintaining image at distance 3Lc/2 of an MMI waveguide, where, Lc is the coupling length between the two lowest order modes. In order to investigate the potential of MMI waveguides at this length for OAM integrated applications, the properties of generated images are studied by considering the effects of three parameters including the width of the waveguide, WR of the OAM mode and its topological charge. It is mathematically shown that the topological charge of odd order OAM modes remains unchanged at OAM-maintaining length, whereas the charge of even order modes is reversed. In other words, an MMI waveguide with the length 3Lc/2 acts as a charge converter for OAM modes with even values of ℓ. To numerically confirm this fact, the simulations are performed using BPM method for silicon waveguides surrounded by silica which are compatible with silicon on insulator (SOI) technology. The waveguides are assumed to have width of 15–50 µm, at the working wavelength of 1550 nm. The quality of the generated modes is evaluated by calculation of POI and the purity for input OAM modes with WR in a range of 1.5–3.5 µm. The results demonstrate that the calculated POI is enhanced by increasing in WR of the input mode. However, for a specific WR value, the increase in the waveguide’s width leads to decrease the quality of the self-imaged mode. Therefore, choosing the right size for MMI waveguides to achieve a desired mode quality requires an appropriate input mode’s WR selection. According to the results of this study, 2D cross-sectional MMI waveguides at their OAM-maintaining length can be used in a wide range of integrated OAM applications as the integrated OAM waveguides, mode convertors, switches, and couplers.

Author contributions

A.S., R.N., and Z.H. conceived the idea. A.S. carried out the theoretical analysis, simulation and writing the manuscript. All authors reviewed the manuscript and discussed the results.

Data availability

Data supporting this study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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