There has been a growing demand for multimode interference (MMI) devices in many optical systems due to various advantages such as low loss, compact size, large optical bandwidth, high fabrication tolerance, high power uniformity, and polarization insensitivity [1]. MMI devices exhibit the well-known phenomenon of self-imaging, where the input field profile is replicated at periodic intervals along the propagation direction. Several devices based on the MMI phenomenon, such as power divider [2], [3], [4], [5], [6], [7], [8], [9], couplers [10], [11], optical switches [12], [13], Mach–Zehnder interferometers [14] and sensors [15], [16] have been reported in the past decades. A large number of devices reported so far utilize 1D confinement of the field, with a limited literature report on structures with two-dimensional (2D) confinement [17], [18], [19], [20]. In 2D square core waveguide structures, multiple imaging effect has been studied and demonstrated in silica core [17], hollow core [18], silicon core [19], and polymer-based waveguides [20]. The analysis in 2D structures is limited to the imaging spot length, which may not provide the optimum design for an application. For example, the length for uniform N power splits is not identical to the N imaging spot length [21]. Therefore, for accurate design and analysis, it is critical to examine the field at all distances along the propagation direction. The guided mode propagation analysis (MPA) is a popular approach primarily used in the design of 1D confined structures where mode solutions are well known [22]. However, in the case of a 2D structure, exact mode solutions have not been derived so far. Therefore, many approximate methods are used. One such popular method is given by Marcatili, which approximates strongly confined modes in the core region with very less field in the cladding and the four corner regions [23]. The accuracy of this method can be further improved by using the effective index method [24]. Although the above-mentioned analytical methods are easier to use computationally, they are generally less accurate as compared to the numerical methods [25]. Several numerical approaches, such as finite element method (FEM), finite difference method (FDM), beam propagation method (BPM), and finite difference time domain (FDTD), have been used for the design and analysis of 1D/2D MMI based optical devices [26], [27], [28], [29], [30], [31], [32], [33], [34]. Generally, the numerical methods are computationally intensive as compared to the analytical methods.

In addition to the waveguides, the MMI phenomenon has been thoroughly explored in weakly guiding circular core optical fibers for realizing multiple applications [35], [36], [37], [38]. Recently, the MMI effect has been explored in square core multimode fibers (SCMMF) using commercial software tools. Various devices such as power divider [39], [40], high power coherent beam combiner [41], and different sensors [42] are proposed using SCMMF-based designs. The full vector or scalar analytical mode solutions of the circular core optical fibers are well known. However, there is still a knowledge gap in the exact mode solutions for square/rectangular core fibers. Additionally, the implications of weakly guiding approximations have not been studied for square/rectangular core fibers. The solutions of wave equations in any 2D confined optical structures still rely either on Marcatili’s approximation method [17], [23], [24] or the numerical methods [20], [39]. For 2D waveguide cases, modes are calculated as the products of separable solutions in x and y directions. Their launch coefficients are obtained using 2D launch, and propagation constant values are approximated for each 2D mode. This approach completely relies on 2D mode propagation. In this paper, we show that the complete problem of 2D mode propagation can be bifurcated into two 1D problems. This is in contrast to earlier reports [6], [17], [18] as follows. In our method, we do not refer to the product of separable modes as a 2D mode field profile. Therefore, we are not required to find any propagation constants for a 2D mode. Furthermore, we bifurcate the problem into two 1D problems such that 1D launch coefficients are obtained independently in two transverse directions. The propagating fields obtained with our method are compared with the fields obtained from a complete 3D simulation using a commercial BPM software package (RSoft BeamProp). Both results are in agreement, validating our approach.

This work is organized into five sections: In Section 2, detailed field propagation analysis of the field in 2D confined structures (multimode optical fibers) using the proposed method is presented. Additionally, we give details of the method for the center and off-center launching conditions. A comparative study of the results obtained from our method and 3D BPM is also presented. In Section 3, the MMI phenomenon in rectangular and square core multimode fibers is analyzed. Design and performance evaluation of the passive optical devices, namely the power divider and combiner using the proposed method, are discussed in Section 4 followed by the conclusion.