## Abstract

In this paper, a universal LP_{01} to any LP_{lm} mode converter structure is proposed based on a circular waveguide. As examples, nine mode converters from LP_{01} to LP_{02}, LP_{03}, LP_{04}, LP_{05}, LP_{06}, LP_{07}, LP_{11}, LP_{21} and LP_{31} are designed. It is shown that an insertion loss of less than 3 dB is achieved over the entire C-band for all the mode converters. Furthermore, two mode (de)multiplexers based on directional couplers are demonstrated, showing an insertion loss less than 2.5 dB and a crosstalk less than −10 dB over the C-band.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Space/mode division multiplexing (SDM/MDM) has been proposed to overcome the capacity limitations of optical transmission networks based on standard single mode fibers (SMF). The MDM uses the orthogonal guided optical modes of the few mode fibers to carry different optical signals, i.e. one mode corresponding to one optical channel. However, for the MDM, mode converters/(de)multiplexers are usually required at optical transmitters and receivers. In other words, mode conversion from fundamental to higher order modes is required at optical transmitters, while mode conversion from higher-order modes to the fundamental modes is usually required at optical receivers [1].

Mode conversion can be achieved using either free-space or waveguide-based optics. Free-space mode conversion is based on matching the spatial mode profile from an input mode to a desired output mode using phase mask or spatial light modulator such as liquid crystal on silicon (LCOS) [2]. This type of conversion includes optical phase plates, optical beam splitters, optical mirrors and lenses, therefore, it results in bulky structures with high insertion loss. Free-space based mode converters could be achieved in almost any wavelength bands since they are wavelength insensitive.

Mode converter structures based on either fibers or waveguides could be realized through a variety of techniques, such as optical grating, optical couplers, optical tapers, optical lanterns, and photonic crystal fibers, etc. [2–9]. The principle of these mode converters is as follows: The propagation constant of the input mode is matched to a desired output mode by altering the physical characteristics of the fiber or the waveguide such as its cross-section area or its refractive index.

In [10], a mode converter was proposed that is used to convert LP_{01} to LP_{11} based on long period fiber grating (LPFG). The mode converter was designed for the C-band, and an insertion loss of 1.5 dB and an extinction ratio of 22 dB at 1550 nm were achieved. However, the mode converter can operate in a narrow bandwidth (13 nm centered at 1551 nm). The work presented in [11] describes an LP_{01} to LP_{02} mode converter based on multimode interference (MMI) in a fiber, which was realized by interconnecting a single mode fiber (SMF) with a few mode fiber (FMF) using a multimode fiber (MMF). The extinction ratio and insertion loss of the mode converter were 55 dB and 1.8 dB at 1550 nm, respectively.

Directional and tapered velocity mode-selective couplers can also be used to convert and multiplex different spatial modes [12–15]. Moreover, optical couplers can be cascaded to convert and multiplex more modes such as the one illustrated in [16] for converting and multiplexing LP_{01}, LP_{11}, LP_{21} and LP_{02} with low insertion loss. Optical couplers can also be designed using Silica based planar lightwave circuits (PLC) [6,17]. In principle, mode conversion using optical couplers is achieved through matching effective index of one mode in one waveguide to the desired output mode in the other waveguide.

Optical Y-junctions and optical lanterns are another waveguide structures to realize mode converters [17,18]. Waveguides-based mode converters and multiplexers have features of high mode conversion efficiency [19] and compactness in footprint. However, they may in general be wavelength dependent, i.e. broadband is not inherent but may be obtained through optimization of the structures and related parameters.

In this work, we propose a universal mode converter structure that can be used for any mode conversion from LP_{01} to LP_{lm}. Furthermore, the modes can be multiplexed and demultiplexed using proposed mode converter structures combined with directional couplers.

## 2. Mode converter structure

Figure 1 shows the block diagram of the proposed mode converter as well as some cross-sectional views, which consists of an outer tapered circular waveguide section of length *L*_{1} followed by a non-tapered circular section of length *L*_{2}, both sections have a refractive index *n*_{co}.

The tapered section starts with a radius *r*_{0} and ends with a radius *r*_{1}. The radius *r*_{0} is chosen to allow better optical coupling with a single mode fiber used to input the signal from the LP_{01} mode. Inside these two sections, a matrix of small elements is inserted. Each element that is denoted by a row *i* and a column *j*, i.e. element (*i*, *j*), consists of three sections: a tapered section followed by a circular non-tapered section, and then followed by another tapered section. The beginning section of the elements starts with zero radius and tapers to *r ^{i}_{j}* with a length

*L*

^{1}

*. The middle section is circular with a radius*

_{ij}*r*and a length

^{i}_{j}*L*

^{2}

*. The last section is circular with a radius*

_{ij}*r*that is tapered to zero over a length

^{i}_{j}*L*

^{3}

*. The refractive index of element (*

_{ij}*i*,

*j*) is

*n*

_{ij}. Elements (

*i*,

*j*) and (

*i*,

*j*+ 1) (in the same column) are spaced by a gap

*g*. Elements (

^{i}_{j}*i*,

*j*) and (

*i*+ 1,

*j*) (in the same row) are spaced by a gap

*d*. Table 1 illustrates the list of symbol definitions used in Fig. 1.

^{i}_{j}The principle of the mode conversion is to introduce some perturbations in the physical characteristics of the structure. For design of LP_{01} to any LP_{lm} mode converter, a large set of structural and material parameters are used to obtain perturbation, and thus the input LP_{01} mode to the desired output LP_{lm} mode is matched in effective index. The structure perturbations are caused mainly by changing the radii of the different elements through tapering and by changing the core-cladding refractive index difference by inserting (*i*, *j*) elements inside the core.

Two classes of modes are grouped here, i.e. LP_{0m} and LP_{lm} (l≠0). The strategy for grouping these two classes of modes is motivated by the fact that LP_{0m} modes are non-degenerate, circularly symmetric and share a common concentric intensity profile similar to LP_{01} mode. Figure 2 shows the 2D intensity profile of the first four LP_{0m} modes (*m* = 1, 2, 3 and 4). It is seen that all LP_{0m} modes present a peak intensity in the center (red spot in the color figures) surrounded by (*m*-1) circular rings. To convert from an LP_{01} to any LP_{0m} mode, the structure shown in Fig. 1 can be simplified by keeping only one single element (*i*, *j*). However, LP_{lm} (l≠0) modes are different from LP_{0m}, and they are degenerate, symmetrical but not circular. Figure 3 shows the 2D intensity profiles of some LP_{lm} (*l* and *m* = 1, 2) modes.

#### 2.1. LP_{01} to LP_{0m} converters

_{01}

_{0m}

The simple circular symmetry of the intensity profiles of LP_{0m} modes results in simple mode converter structures. Indeed, one single element in the inner section is sufficient to convert LP_{01} to any desired LP_{0m} mode. The proposed LP_{01} to LP_{0m} mode converter is shown in Fig. 4. Based on our knowledge, these structures can be easily fabricated by taking *n*_{co} = 1.4907, and *n*_{cl} = *n*_{1_1} = 1.4877 [17], which are used for this work. Table 2 presents the optimized structural parameters for the LP_{01} to LP_{0m} mode converters for the first six modes (i.e., *m* = 2 ~7), as well as the insertion loss (*IL*) at the center of the C-Band (1550 nm). It is found that the lengths *L _{0}* and

*L*do not have any impact on the conversion efficiency, and in fact they are used for input and output optical coupling. All dimensional parameters in Table 2 are in µm.

_{3}To evaluate the performance of the mode converter, two performance parameters are used, which are the insertion loss (IL) of LP_{0m} mode and the mode extinction ratio (ER), defined as [9]:

_{01}mode at point and the optical output power of mode LP

_{0m}, (m = 1, 2, …) at the output of the mode converter, respectively.

Figure 5 illustrates the insertion loss (*IL*) of the finally designed mode converters as a function of wavelength for LP_{01} converting to LP_{0m} (*m* = 2~7). The insertion loss represents the conversion efficiency from the injected LP_{01} mode to the desired LP_{0m} mode. Figure 5(a) shows the IL from 1.25 to 1.75 μm, covering the O, S, C, L and U bands. Intuitively, it is shown that the increase of the high order modes incurs more power penalty, i.e. worse conversion efficiency. This power penalty is induced due to the leakage to the lower order modes. However, the design is optimized for the C-band, and if a dedicated band is only considered the insertion loss can be reduced by re-optimizing the structures. Figure 5(b) depicts the IL over the C- band only, showing that an IL below 3 dB is achieved for the six mode converters over the entire C- band.

Figure 6 presents the extinction ratio (ER) of the desired output mode (LP_{0m}) with respect to the injected LP_{01} mode. It measures the purity of the converted mode, and in other words, the crosstalk from all other non-desired modes to the desired mode is the inverse of the ER (in the log-scale, the crosstalk will be the negative value of the ER). The higher is the ER, the lower is the crosstalk. Figure 6(a) shows the ER corresponding to Fig. 5(a), covering the O, S, C, L and U bands. It is seen that the ER is strongly dependent on wavelength or band. Figure 6(b) depicts the ER over the whole C-band only, showing that an ER above 8 dB is achieved for the six mode converters. Both Fig. 5 and Fig. 6 show that any converter can be optimized when an individual band is selected.

#### 2.2. LP_{01} to LP_{lm} converters

To convert LP_{01} to LP_{lm} mode (*l* ≠ 0), the structure shown in Fig. 1 is used, where a single inner element is not sufficient. The number of the inner elements depends on the desired output LP_{lm} modes. As an example, we consider three modes only here, i.e. LP_{11}, LP_{21} and LP_{31.} Five inner elements are required to obtain LP_{11} and LP_{31}, and six inner elements are required for LP_{21}, shown in Fig. 7.

All three structures start with an initial radius *r*_{0} of 5 µm for better coupling to a SMF. The LP_{01}-LP_{11} mode converter has a final radius *r*_{1} of 30 µm, whereas the LP_{01}-LP_{21} and LP_{01}-LP_{21} mode converters both have *r*_{1} = 26 µm. The inner elements have different radii ranging from 2 µm (element (2,2) in LP_{01}-LP_{11} MC) to 26 µm (element (2,1) in LP_{01}-LP_{31} MC) and different segment lengths ranging from 50 µm (end segment of element (1,2) in LP_{01}-LP_{21} MC) to 1500 µm (start segment of element (2,2) in LP_{01}-LP_{31} MC). All inner elements have the same refractive index as the cladding (1.4877).

Figure 8 shows the insertion loss of the three mode converters. Figure 8(a) shows the IL from 1.3 to 1.7 μm and Fig. 8(b) presents the IL over the C-band only. It is seen that an IL of less than 2 dB is achieved over the entire C-band for the three mode converters.

Figure 9 shows the extinction ratio (ER) of the three mode converters, where Fig. 9(a) shows the ER over a broadband from 1.3 to 1.7 μm and Fig. 9(b) presents the ER over the C-band only. Over the C-band, an extinction ratio above 10 dB is achieved for the three mode converters. Again, the converters can be further optimized for a dedicated band.

#### 2.3. Discussion

In fact, all simulation results presented in this paper are obtained by Rsoft CAD software [20], a vector mode solver based on Eigen Mode Expansion (EME) [19] method to find the modes inside the waveguide. These results can still be enhanced further with more optimizations of the structure parameters.

It is worth mentioning that mode converters for any other similar modes can be achieved using the structure in Fig. 1. Even though not presented here, LP_{lm} modes can be multiplexed/demultiplexed with a coupling structure. In Appendix, a multiplexer and demultiplexer for LP_{0m} modes (*m* = 1, 2, 3, 4, 5) are given as an example.

Usually, these kind of mode converters can be fabricated by using direct 3D femtosecond laser inscription on a bulk of glass. In the above designs, the fabrication limits of the structural parameters of the mode converters were already considered in optimizing the designs. In our previous work [21], an LP_{01} to LP_{02} mode converter structure was fabricated, validating the working concept and fabrication of the proposed structure.

## 3. Conclusion

In this paper, we have proposed a universal LP_{01} to LP_{lm} mode converter structure. The converter structure is based on tapered waveguides. The converter structure can be simplified when converting to LP_{0m} modes. By converting LP_{01} mode to LP_{02}, LP_{03}, LP_{04}, LP_{05}, LP_{06}, LP_{07}, LP_{11}, LP_{21} and LP_{31}, an insertion loss (IL) is ranged from 0.1 dB to 3 dB and the extinction ratio is larger than 8 dB over the entire C-band.

## Appendix

In this Appendix, an example of two mode (de)multiplexers is given. One (de)multiplexer is for the LP_{0m} modes and the other is for the LP_{lm} modes. We have chosen to use directional couplers (DCs) for multiplexing and demultiplexing. The first DC-based mode multiplexer is shown in Fig. 10, in which (de)multiplexing of the first five LP_{0m} modes (*m* = 1, 2, 3, 4, 5) is considered. However, (de)multiplexing of any number of LP_{0m} modes can be applied in the same manner. The structural parameters of the (de)multiplexer are optimized based on minimizing the insertion loss of (de)multiplexed modes at the output.

Figures 11(a) and 11(b) illustrates the insertion loss of the multiplexer and demultiplexer over the C-band, respectively. It is shown that a worse case of 2.7 dB is obtained for demultiplexing LP_{04}. At the central wavelength of 1550 nm, all the five modes can be (de)multiplexed with an insertion loss of less than 2.3 dB.

Figures 12(a) and 12(b) illustrates the crosstalk of the multiplexer and demultiplexer for the C-band, respectively. Both have less than −15 dB crosstalk at the central wavelength (1550 nm) for the five modes. It is seen also that the mode multiplexer experiences more crosstalk in the longer wavelengths, in particular for the mode LP_{04} that experiences the worst crosstalk between the modes. In general, it is shown that the crosstalk depends on the mode order, so higher order modes suffer from more crosstalk than lower order mode.

In a systematic design, one can optimize the two structures (mode converter and mode multiplexer) to balance the insertion losses from both structures for all the modes.

Figure 13 shows the LP_{lm} (de)multiplex for the three obtained modes (LP_{11}, LP_{21} and LP_{31}). To simplify the structure design, all waveguides have the same radius (30 μm). This choice of using the same radius was motivated by the three radii (*r*_{1}) of the converters (LP_{01}~LP_{11}, LP_{01}~LP_{21} and LP_{01}~LP_{31}) which are 30 μm, 26 μm and 26 μm respectively. Furthermore, the same structure can be used at the transmitter side (for multiplexing) as well as at the receiver side (for demultiplexing) with similar performances.

Figure 14 illustrates the insertion loss (IL) of the LP_{11}, LP_{21} and LP_{31} (d)multiplexer over the C-Band. Figure 14(a) shows that an IL of less than 1.9dB is achieved over the entire C-Band and less than 0.4dB at the design wavelength of 1550nm for the multiplexer. Figure 14(b) gives the simulation results of the demultiplexer over the C-Band. These results are similar to those of the multiplexer, therefore, the device is symmetric, hence it can be used as multiplexer and demultiplexer.

In Fig. 15, the crosstalk caused by the interferences between the three modes over the C-Band is presented. One can see from Fig. 15(a) that a crosstalk below −13.8 dB is achieved over the entire C-Band and below −17.7 dB at the wavelength of 1550nm for the multiplexer. The demultiplexer results are given by Fig. 15(b), where less than −18dB crosstalk is achieved for all three modes at the design wavelength (1550nm). Except for mode LP_{21}, the other modes have a crosstalk below −14dB over the entire C-band.

## Funding

Discovery Grants of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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