## Abstract

An optimized two-mode optical fiber (TMF) with the graded index (GI) profile is designed and fabricated. We clarify an appropriate region of GI-TMF satisfying DMD = 0 ps/km, the large effective area *A*_{eff}, and the low bending loss for LP_{11} at 1550 nm. According to our fiber design, GI-TMF is successfully fabricated to have the large effective area *A*_{eff} of 150 μm^{2} for LP_{01} mode, and low DMD below 36 ps/km including zero in the C-band. We expect that our design GI-TMF is suitable for MDM and can reduce MIMO-DSP complexity.

© 2013 OSA

## 1. Introduction

The traffic of backbone network has been increasing rapidly corresponding to the growth of broadband users in worldwide. It is reported that the current system utilizing the conventional single mode optical fibers (SMFs) will approach the limit of input power, which is directly related to the transmission capacity in the wavelength division multiplexing (WDM) system, because of the optical nonlinear effects and the fiber fuse [1]. For next generation system, mode division multiplexing (MDM) transmission system using a few-mode fiber (FMF) has been studied actively [2–9].

In the MDM system, multiple-input-multiple-output (MIMO) digital signal processing (DSP) is usually applied to recover the transmitted signals. However, it is known that differential modal group delay (DMD) of FMF increases DSP complexity [2–6]. Then, FMF with low DMD would have advantage to be applied to MDM utilizing the MIMO. Low DMD in the wide wavelength range is required for the WDM applications. Moreover, low bending loss of not only fundamental mode but also higher order modes is essential. Furthermore, enlargement of the effective area (*A*eff) is also desirable for increasing the launched power into the fiber, resulting in increase of the multiplicity of WDM.

It is known that FMF with a graded index (GI) profile minimizes DMD [5, 7–9]. Reference [5] shows the FMF with both of low DMD and low mode coupling, Refs. [8] and [9] shows the optimal value of *Δ* and *α* to minimize DMD. However, there has been no report on FMF design optimizing DMD, bending loss and *A*_{eff}. In this paper, we clarify the suitable profile design of TMF optimizing these properties and fabricate successfully the TMF based on its design.

## 2. Fiber design

Figure 1 shows the refractive-index profile of the graded index (GI) fiber. The GI profile is given by

where *n*_{1} and *n*_{2} are the indices of the core and the cladding, respectively, *Δ* is the relative-index difference between the core and the cladding, which is defined as (*n*_{1}^{2}-*n*_{2}^{2})/2*n*_{1}^{2}, *r* is the distance from the center of the core, *a* is the core radius, and *α* is the index profile parameter. We determined the requirements of TMF that DMD = 0 ps/km, *A*_{eff} ≧150 μm^{2} for LP_{01} mode, and bending loss for LP_{11} mode ≦0.01 dB/km at *R* = 40 mm at the wavelength of 1550 nm. Bending loss at *R* = 40 mm is an equivalent condition for microbending loss in the cable [10]. We think that this value is the important factor to evaluate the cabling adaptability of optical fibers. In addition, bending loss was evaluated with simulation using finite element method [11] and other characteristics were calculated by multilayer division method [12].

Figure 2 shows the relationship between the normalized frequency *T* and the calculated DMD at 1550 nm for the different *α* and *Δ*. Here, DMD is defined as 1/*v*g_{11}-1/*v*g_{01}, where *v*g_{11} and *v*g_{01} are the group velocities of LP11 and LP01 modes, respectively. Normalized frequency *T* is defined by

Here, *k* is the wave number and *A* is a constant value depending on the refractive index profile. Because *Δ* and wavelength are kept to be constant in Fig. 2, increase of *T* means an increase in the core radius. We calculated cutoff frequency of LP_{11} and LP_{21} (or LP_{02}) modes. It is known that the normalized cutoff frequency of the LP_{02} mode of the GI fiber with *α* of 2 is smaller than that of the LP_{21} mode and it is also clear that the normalized cutoff frequencies of two modes depend on the refractive index profile. For the step-index fiber, the normalized cutoff frequency of LP_{21} mode is smaller than that of LP_{02} mode. Moreover, as the difference of the normalized cutoff frequency between LP_{02} and LP_{21} modes is very small, we described the two mode condition in the paper where the cutoff frequency is smaller than that of LP21 mode. Since the cutoff frequencies *T* for GI fiber with different *α* of LP_{11} and LP21 modes are obtained to be 2.5 and 4.5, two-mode propagation region is 2.5 ≤ *T* <4.5. It was confirmed from Fig. 2 that DMD is almost independent of *Δ* in the range of 0.3% to 0.4% and that two-mode propagation with DMD of 0 ps/km is satisfied for *α* ≥ 2.2. In addition, the smaller *α* is, the smaller DMD slope at the normalized frequency of zero DMD is. This means that the value of DMD can be reduced in the whole C band as *α* become smaller. Figure 3 shows the maximum value of DMD for *α* over the entire C band. It is obvious that the maximum DMD value become smaller as *α* is smaller. However, because the LP_{21} mode would propagate in *α* ≦ 2.2, the appropriate range of *α* is 2.2 ≤ *α* ≤ 2.4.

Figure 4 shows calculation results of bending loss for LP_{11} mode at R = 40 mm at 1550nm where DMD = 0 ps/km. It is obvious that the bending loss decreases as *Δ* increase. On the other hand, the bending loss decreases as *α* decrease. The reason is that core radius at the point of DMD = 0 ps/km increases as *α* decreases. Figure 5 shows the calculation results of *A*_{eff} for LP_{01} mode at 1550nm and DMD = 0 ps/km. *A*_{eff} increases as *Δ* and *α* decrease. According to the result of Figs. 3-5, low DMD in the C band and good bending characteristics and large *A*_{eff} can be obtained by decreasing *α* keeping DMD = 0 ps/km. With the calculation results from Figs. 2, 4 and 5, the region satisfying our requirements, that is DMD = 0, *A*_{eff} ≧150 μm^{2} for LP_{01} and bending loss for LP_{11} ≦0.01 dB/km at *R* = 40 mm at 1550 nm, is the hatched area in Fig. 6. The center of the hatched area is shown by the circle in Fig. 6. The fiber parameters on the center are *Δ* = 0.36%, core radius *a* = 11.8 μm and *α* = 2.3. The DMD at 1550 nm is 0 ps/km.

## 3. Characteristics of fabricated fiber

#### 3.1 Refractive index profile

Figure 7 shows the refractive-index profile of the fabricated GI-TMF measured by the refractive near field pattern (RNFP) method. The broken line shows the fitted line to Eq. (1) with the least square method of the fabricated GI-TMF. Table 1 summarizes the structural parameters of the fabricated GI-TMF based on the fitting curve. Though a small central dip was formed, almost designed structural parameters were obtained.

#### 3.2 Optical properties of fabricated GI-TMF

Table 2 shows optical properties of the GI-TMF at *λ* = 1550 nm. The properties of LP_{01} mode were measured on bending to attenuate only LP_{11} mode power. The properties of LP_{11} mode except for bending loss were calculated using multilayer division method and the index profile measured by RNFP. Attenuation for LP_{01} was 0.196 dB/km. The effective areas *A*eff of LP_{01} and LP_{11} modes were obtained to be about 150 μm^{2} and over 200 μm^{2}, respectively.

### 3.2.1 Measurement result of cutoff wavelength

Cutoff wavelength was measured with 2m bending method [13]. Measured spectral loss is shown in Fig. 8. Two peaks were observed in the wavelength range between 1400 to 2400 nm. The edge at longer wavelength of these peaks represent cutoff wavelengths of LP_{11} and LP_{21} modes because those cutoff wavelengths calculated with the index profile measured by RNFP were 1520 and 2315 nm, respectively. The cutoff wavelength of LP_{21} and LP_{11} modes were 1495 nm over 2300 nm, respectively. That means the fabricated GI-TMF can transmit only LP_{01} and LP_{11} modes in C, L and U-band.

### 3.2.2 Measurement result of bending loss for LP_{11} mode

Figure 9 shows the schematic diagram of the experimental setup. Bending loss for LP_{11} mode was measured by exciting only LP_{11} mode using offset-connecting to SMF with a light source of 1550nm laser diode (LD). Offset was about 16 μm and LP_{11} mode power was about 97% of the total input power. Bending loss of GI-TMF was measured with and without bend of radius R_{2}, under the bending condition of bend radius *R*_{1} to eliminate leaky mode. The reason why bend radius *R*_{1} added is because sample fiber length is only 3m and leaky mode can transmit such short length fiber. For the measurement of bending loss, minimum loss change of 10^{−3} dB can be evaluated and from a practical viewpoint, the maximum fiber length wound on the bobbin is about 1 m. In this case, we have to wind the fiber with length of 1 km on the bobbin with a radius of 40 mm but it is impossible. Even if we evaluate the bending loss for the length of 1 m, the bending loss at a radius of 40 mm is about 10^{−5} dB/km so that we cannot measure the bending loss by the current measurement system as mentioned above. Therefore, we estimated the bending loss by utilizing the well-known relationship between the logarithm of the bending loss in dB and the bending radius. Measured value of bending loss of LP_{11} mode for *R*_{2} = 15, 17 and 20 mm are shown by solid circles in Fig. 10. Solid line shows the bending property calculated by finite element method and the measured index profile. Calculated and measured results are in good agreement. Therefore, the bending loss of LP_{11} mode with a radius of 40 mm can be estimated to be 0.016 dB/km from the calculated results as shown in Fig. 10.

### 3.2.3 Experimental setup and results of DMD

Figure 11 shows the experimental setup of the interference method for DMD measurement [14]. To excite the LP_{01} and LP_{11} modes, TMF was spliced with a single mode fiber at the offset of 6.0 μm. Light sources were LEDs with center wavelength at 1300nm, 1450nm, 1550nm and 1650nm. The intensity of the interference pattern depends on the wavelength and is a nearly sinusoidal pattern. The wavelength difference *Δλ* between amplitude peaks is related to DMD, and can be expressed by the following equation.

*λ*

_{0}is the central wavelength of two amplitude peaks,

*c*is the speed of light in vacuum and

*L*is the fiber length. Figure 12 shows the interference spectrum of the GI-TMF with the length of 100 m cut out from one end. Wavelength dependence of the interference pattern was observed. Figure 13 shows the absolute DMD as a function of wavelength which was obtained from Eq. (3) and the result of Fig. 12. It is seen from Fig. 13 that the DMD is 0 ps/km at 1554 nm and less than 36 ps/km in the C-band.

## 4. Conclusions

We have investigated the optimum fiber design of GI-TMF with low DMD, large effective area *A*_{eff}, and low bending loss for LP_{11} mode. We clarified the suitable fiber parameters for the GI-TMF of *Δ* = 0.36% and *α* = 2.3 from the area satisfying requirements that are DMD = 0 ps/km, *A*_{eff} of more than 150 μm^{2} and bending loss for LP_{11} mode of less than 0.01 dB/km at the wavelength of 1550 nm. We successfully fabricated the GI-TMF with *Δ* = 0.363% and *α* = 2.29 according to our fiber design. We confirmed experimentally that the fabricated GI-TMF had a large effective area *A*_{eff} of 150 μm^{2} for LP_{01} mode, and approximately 0.016 dB/km bending loss for LP_{11} mode at R = 40 mm and realized the DMD of less than 36 ps/km including zero in the C-band. We expect that our design GI-TMF is suitable for MDM and has a potential to reduce MIMO-DSP complexity.

## Acknowledgments

This work was partially supported by National Institute of Information and Communication Technology (NICT), Japan under “Research on Innovative Optical fiber Technology”.

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