Impact of Geometric Input Fibers’ Core Positioning on the Adiabaticity of Photonic Lanterns

Abstract

Photonic lantern is a key device in space division multiplexing (SDM) system. The key challenge of a photonic lantern is mode scalability, which requires the taper length to increase nonlinearly as the mode number scales up. The traditional photonic lantern fabrication method requires stacking the input fibers into the hollow, low-index outer cladding before tapering. It implicitly sets geometric constraints on the input fibers’ core positioning. We propose a photonic lantern design with drilling preform and reduced cladding fibers to lift these constraints and make photonic lanterns more adiabatic. By analyzing the effects of loosening the constraints on the adiabatic requirement of a three-mode photonic lantern, we find further progress could be made to alleviate this adiabatic requirement. The optimal structure for our design is proposed and demonstrated through the beam propagation method (BPM). Our findings could help further improve the mode scalability of photonic lanterns.

1. Introduction

A photonic lantern is a broadband photonic device that maps fundamental modes into ordered modes in a multimode waveguide [1,2]. A fiber-based photonic lantern is viewed to be a key device as a mode multiplexer and demultiplexer for SDM transmission systems [3,4,5,6,7]. Furthermore, it has also found potential applications in the fields of fiber amplifiers [8,9], free space communications [3,10] and wavefront sensing [11,12].

A fiber-based photonic lantern has the merits of incurring very low losses, being compact, having immunity to electromagnetic interference and having the ability to handle high optical power. The mechanism of a photonic lantern is well understood. It maps the fundamental modes of input optical fibers into ordered modes in few-mode or multimode fibers if the geometric requirements are fulfilled [13]. Theoretically, if the tapering process is ideal, a photonic lantern of any mode number can be fabricated with very low loss. In experiments, however, mode scaling is a critical issue due to the limited tapering length. It is found that the required tapering length increases approximately proportionally to N² as the mode number, N, increases [14]. The critical factor is the adiabatic requirement, which determines how slowly a photonic lantern should be tapered. Extensive efforts have been made to loosen the adiabatic requirement. One method is to use input fibers with reduced cladding [15], and another method is to use input fibers with graded-index multimode fibers [16,17]. These methods ensure the successful fabrication of low-loss six-mode photonic lanterns [16] and reasonably low-loss ten-mode photonic lanterns [18]. A further loosening of the adiabatic requirement is achieved by using low-index drilling preforms [19], which greatly alleviate the adiabatic requirement. To further alleviate the adiabatic requirement to facilitate mode scaling, there is one potential dimension left for exploration. This is the geometric constraints of the input fibers’ core positioning by a conventional fabrication method, where input fibers are inserted into hollow, lower-index preforms. This method implicitly sets geometric constraints on the input fibers’ core positioning. Therefore, it is important to explore how to lift these constraints and how loosening these constraints could have an impact on the adiabaticity of photonic lanterns. The paper is organized as follows: first, the constraints on the input fibers’ core positioning are described, and the rationale behind our proposal and approach is discussed; then, the effects of the input fibers’ core positioning on photonic lanterns’ adiabaticity are analyzed, and the optimal geometric structure for a photonic lantern is found. Finally, BPM simulations are performed to verify our proposed structure.

2. Rationale

A schematic diagram of a typical three-mode photonic lantern is illustrated in Figure 1. It consists of three input fibers and a hollow, low-index outer cladding. The three input fibers are stacked into the low-index outer cladding, and the lantern is tapered adiabatically so that fundamental modes from the three input fibers map into the three lowest-order spatial modes or a combination of them (LP₀₁, LP_11a and LP_11b) at the end of the photonic lantern. For a non-mode-selective photonic lantern, three identical fibers are used, while a mode-selective photonic lantern consists of one fiber with a larger effective index and two identical fibers with lower effective indexes.

The traditional fabrication method requires manually inserting the input fibers one at a time into the hollow, low-index cladding. Through this method, the input fibers’ geometric positioning is somewhat constrained. Figure 2a,b show the geometric facets of a three-mode and six-mode photonic lantern made using the traditional fabrication method, with the cores of the input fibers show in black, the claddings in orange and the low-index, outer cladding in blue. For the three-mode photonic lantern, we define the distance from the center of the lantern to the center of the core of the input fiber as r and the inner diameter of the low-index outer cladding as R. It is found that the ratio r/R is fixed at 0.5359, no matter what the diameter of the low-index outer cladding is. For the six-mode photonic lantern, the center to the outer fiber core is defined as r, and the ratio of r/R is always 0.6298. In addition, the ratio of the outer input fiber diameter over the inner input fiber diameter is fixed at 1.4259. This type of constraint applies to photonic lanterns with any number of modes as long as a stacking process is used. To lift these constraints, a drilling preform technique is proposed. Figure 3a shows the schematics of a drilling preform technique based on a three-mode selective photonic lantern. Based on the drilling preform technique [19], the structure consists of a low-index outer cladding (green), an inner drilled-hole cladding (blue) and three inserted input fibers (light blue claddings and black cores). The positions of the drilled holes could be designed with good precision. Furthermore, from a fabrication perspective, since the traditional method requires several manually inserted input fibers into a single low-index outer cladding, it is hard to handle input fibers with small diameters. Our proposed method only requires one input fiber in one individual hole at a time, and input fibers with smaller diameters can be used. Therefore, this structure could potentially break the geometric constraints to a large extent. In other words, for three-mode photonic lanterns, the ratio r/R can be modified.

Figure 3b shows the effective indexes of the lantern modes of an ideal three-mode selective photonic lantern at taper ratios from 1 to 0.075. From an application perspective, the modes of interest are LP₀₁, LP_11a and LP_11b, and other higher-order modes are unwanted. From Figure 3b, the black line represents the effective index evolution of the LP₀₁ mode, while the green and red lines represent that of the LP_11a and LP_11b (degenerate) modes. In addition, the pink area represents the effective index evolution of higher-order modes. Figure 3c shows the intensity profiles of the LP₀₁, LP_11a and LP_11b modes at various taper ratios.

To explore the impact of r/R on the performance of photonic lanterns, the governing function of the adiabatic requirement is applied for an analysis, which is given by Equation (1) [14]:

$$\left|\frac{2\pi }{({\beta }_1-{\beta }_2)}\frac{d\rho }{dz}{\displaystyle \int {\psi }_1\frac{\partial {\psi }_2}{\partial \rho }dA}\right|\ll 1,$$

(1)

where ${\psi }_1$ and ${\psi }_2$ are the normalized local field distributions of the local modes, and ${\beta }_1$ and ${\beta }_2$ are their respective propagation constants. $\rho$ and $A$ are the radii and cross-sectional areas of the local waveguides, $z$ is the coordinate along the $z$ axis of the waveguide and $\frac{d\rho }{dz}$ represents the speed of the taper.

This equation provides guidance for the upper bound of how fast the photonic lantern should be tapered for its lossless fabrication. During the tapering process, if Equation (1) at a certain point is not met, mode coupling could take place, which would result in crosstalk and loss. Crosstalk happens when mode coupling occurs between the LP₀₁ and LP₁₁ modes. If the LP₀₁ or LP₁₁ mode couples to higher-order modes, loss is incurred. From Equation (1), two terms play important roles in the adiabatic requirement. One is the effective index difference between the local modes, which is inversely proportional to the required tapering speed, and the other one relates to the overlap integral, which indicates how fast the mode profile changes.

Quantify crosstalk requires a calculation of the adiabatic requirement between the LP₀₁ and LP₁₁ (LP_11a and LP_11b) modes. As for the issue of loss, the mode coupling of LP₀₁ or LP₁₁ to all higher-order modes needs to be considered. Therefore, the adiabatic requirement equation shall be modified to Equation (2):

$${\displaystyle \sum _j\left|\left.\frac{2\pi }{\left({\beta }_i-{\beta }_j\right)}\frac{d\rho }{dz}{\displaystyle \int {\psi }_i\frac{\partial {\psi }_j}{\partial \rho }dA}\right|\ll 1\right.},$$

(2)

where summation of $j$ includes the modes that are relevant.

3. Results

The calculation was performed. The geometric structure of a three-mode selective photonic lantern is shown in Figure 3a. It consists of three cores: one larger core with a diameter of 16 μm and two smaller cores with a diameter of 9 μm. The refractive indexes of the three cores and their claddings are set to 1.46 and 1.44, respectively. The inner drilled-hole cladding is designed with a diameter of 200 μm and a refractive index of 1.44. The outer cladding has a diameter of 500 μm and a refractive index of 1.42. In principle, the refractive index of the drilled-hole cladding could be designed to be lower than that of the cladding of the input fiber to further alleviate the adiabatic requirement [19]. The adiabatic requirements of the LP₀₁–LP₁₁ modes, LP₀₁ higher-order modes and LP₁₁ higher-order modes at various r/R are calculated.

Figure 4a shows the adiabatic requirements of the LP₀₁- LP₁₁ modes at r/R ratios ranging from 0.3 to 0.075, which are calculated from Equation (2) (left side). For an r/R ratio larger than 0.3, the adiabatic requirement is relatively easy to meet since the modes are guided by the cores and the mode distribution changes slowly. When the taper ratio becomes less than 0.3, the modes begin to leak into the inner cladding very quickly, which is shown in Figure 3c. That is the regime of concern for lantern loss and crosstalk. From Figure 4a, the adiabatic requirements of the LP₀₁-LP₁₁ modes become more stringent as the ratio r/R becomes larger. This means that when the input fibers’ core positioning becomes further away from the center, the photonic lantern is more vulnerable to crosstalk. The effective index difference between the LP₀₁ and LP₁₁ modes and the integral part of Equation (1) are calculated at different values of r/R and shown in Figure 4b,c. Both the inverse of the effective index difference and the integral monotonically increase with r/R.

Figure 4d,e show the adiabatic requirement for the LP₀₁ higher-order modes and the LP₁₁ higher-order modes at different values of r/R. It can be seen that the peak of the adiabaticity requirement of the LP₁₁ higher-order modes is earlier than that of the LP₀₁ higher-order modes, which means that the LP₁₁ mode leaves the core first during the tapering process. It is also found that the adiabatic requirement of the LP₁₁ higher-order modes is much larger than that of the LP₀₁ higher-order modes, which means that the LP₁₁ mode is much more vulnerable to loss, which is consistent with experimental results [16]. When r/R increases, the adiabatic requirement of the LP₀₁ higher-order modes becomes harder to meet, which means that the LP₀₁ mode is more vulnerable to loss when the input fibers’ core positioning increases. The adiabatic requirement for the LP₁₁ higher-order modes is complicated. Figure 4f shows the peak values of the adiabatic requirement of the LP₁₁ higher-order modes as a function of r/R. It is found that the adiabatic requirement first decreases and then increases, which is determined through a combination of the index difference and the overlap integral. The optimal r/R value for the LP₁₁ higher-order modes is around 0.4.

In most fiber communication applications, the insertion loss (IL) and mode-dependent loss (MDL) of a photonic lantern are of concern. The r/R value for the optimal geometric structure is around 0.4 since the loss of the LP₁₁ mode is the dominant source of loss for a three-mode photonic lantern. In some applications [6,20,21,22], where mode selectivity is very important, tradeoffs between crosstalk and LP₁₁ loss (high order mode loss) should be made, and the optimal r/R value might be pushed into the region where r/R is less than 0.4.

To demonstrate that the proposed geometric structure with r/R is optimal, BPM simulations were conducted to provide accurate loss metrics. The taper length was chosen to be 6mm so that the adiabatic requirement is not met. In this case, the optimal geometric structure will have the best loss metrics. In our simulation, we kept the lantern profile the same as shown in Figure 3a and Figure 5a, which show the MDL and IL at various r/R ratios. Both the MDL and IL exhibit similar trends. As r/R increases, both the MDL and IL first decrease, then increase. The lowest loss point is around 0.4, which is consistent with the adiabatic requirement analysis. As in the traditional fabrication method, the r/R ratio of the three-mode photonic lantern is fixed at 0.5359, which is far from 0.4. Therefore, further improvements could be made by loosening the constraints on the input fibers’ core positioning.

The crosstalk characteristics of photonic lanterns were also explored. The mode selectivity (MS) of the LP₀₁ and LP₁₁ modes is defined to measure crosstalk. The MS of the LP₀₁ mode is defined as the ratio between the power received by the LP₀₁ mode and the power received by the two spatial LP₁₁ modes when the LP₀₁ mode is excited The MS of the LP₁₁ mode is defined as the average ratio between the power received by the two spatial LP₁₁ modes and the power received by the LP₀₁ mode when the two spatial LP₁₁(LP_11a and LP_11b) modes are launched each time. Figure 5b shows the MS of the LP₀₁ and LP₁₁ modes as a function of r/R. The MS decreases monotonically as r/R increases, which is consistent with the adiabatic requirement analysis.

4. Discussion

We explored the impact of the geometric input fibers’ positioning on the adiabatic requirement of a three-mode photonic lantern and demonstrated the optimal structure to further improve the adiabaticity. Our results could apply to photonic lanterns with a large number of modes. Making high-quality photonic lanterns with a large number of modes is hard at the current stage since ways to alleviate adiabatic requirements are exhaustive. Our method provides a new method for the alleviation of the adiabatic requirements of photonic lanterns.