1. Introduction

In the past ten years, researchers from various countries have vigorously developed high-performance external electro-optical modulators to meet the growing demand for broadband and high speed optical communication infrastructure in backbone networks [1]. Especially in optical switching systems such as Optical Add/Drop Multiplexer (OADM) and Optical Time Division Multiplexer (OTDM) [1]. The traditional lithium niobate electro-optical modulator is based on the proton exchange process or the titanium diffusion process [2]. The formed waveguide has weak optical confinement, resulting in conventional modulators are bulky, which is contrary to the current trend of miniaturization and integration of photonic devices [3–5]. At the same time, because the mode field sizes of lithium niobate modulators and single-mode fibers (SMF) are very different, their application is limited by the huge coupling loss between submicron waveguides and single-mode fibers. The insertion loss of existing lithium niobate electro-optical modulators is usually greater than 10 dB [6,7]. Such a high loss will cause a great deterioration in modulation performance.

Compared with traditional modulators, graphene-based modulators have a more compact size, wider bandwidth, smaller optical propagation loss and more insensitive to the ambient variations such as less temperature sensitivity [8]. So far, electro-absorption [9–12] and electro-optic [13–15] modulators based on waveguide integrated graphene have been proposed. Despite the advantages over traditional modulators, there are still problems with the alignment of the fiber pigtail at the coupling, and due to its high insertion loss, it may not be an ideal candidate for practical applications [16]. In order to solve these problems, the research unit has studied the all-fiber structure modulator as an industrial solution. The all-fiber structure (that is, both the input and output interfaces are optical fibers, so there is no need for butt coupling) is very effective in solving the problems of insertion loss and equipment packaging. By using graphene in combination with an all-fiber structure modulator, the advantages of modulators based on graphene and fiber structures can be shared. Highly integrated and simple structure modulator with a small footprint and high modulation depth is expected. In addition, the all-fiber configuration is better compatible with the deployed fiber communication system. So far, several all-fiber optical modulators based on functional materials have been proposed, including graphene [17–20], electro-optic polymers [21,22] and bulk lithium niobate [23]. However, for an optical modulator with an all-fiber structure, there exists a large cladding layer between the external device and the fiber core, which makes the interaction between light and matter weaker, resulting in a longer length (even up to m scale) to realize the modulation effect [24–26]. There are two main types of graphene fiber modulators. One is the electro absorption effect. The current common solution is to use a D-type all-fiber modulator [27]. Since the imaginary part of the effective refractive index represents the loss coefficient, the attenuation of the modulator can be obtained. The imaginary part of the effective refractive index is linearly proportional to the imaginary part of the dielectric constant of graphene. Therefore, the loss value can be changed by adjusting the dielectric constant. The other type is electro-optic effect. The real part of effective refractive index represents electro-optic effect. The real part of the effective refractive index changes by voltage to realize the Pockels effect or Kerr effect. The two polarization directions will produce phase difference. The structure only needs 127um in length to realize the π phase inversion [28]. In order to allow the light field to leak from the core, the D-type fiber will grind the cladding and the core at the same time, which can enhance the light interaction with materials. But it will greatly introduce insertion loss. In response to this problem, we propose an improved double D-type all-fiber modulator. While maintaining low modulation voltage and high modulation efficiency, the device loss is reduced to the order of 0.1 dB/mm. Taking full advantage of the good coupling to the all-fiber system.

In this paper, based on the common D-type all-fiber modulator, we propose an improved double D-type all-fiber modulator. Insertion loss is an important factor affecting modulation efficiency. This manuscript focuses on the analysis of this parameter. Without grinding the core structure, most of the energy is still confined in the core, which can greatly reduce device loss. At the same time, the cladding layers on the upper and lower sides of fiber are cut to the same depth, which can ensure that the mode field does not deviate in one direction. Then the sides are polished. The leaked light field exist on the cut surface in the form of evanescent wave. The graphene layer is coated on both sides to enhance the light-mass interaction and also ensure a certain modulation efficiency. The anisotropy of graphene is fully considered in the calculation, which has been neglected in many previous studies. By combining graphene with an all-fiber structure, the advantages of graphene and all-fiber structure can be combined. Simulation results that the mode loss of the double D-type modulator under X polarization is 0.125 dB/mm, and the mode field mismatch loss is 0.25%. The mode loss in Y polarization is 0.033 dB/mm, and the mode field mismatch loss is 0.32%. This is at a very low loss level. When the modulation voltage is 5 V, the modulation depth is 78.4% under the condition of five-layer graphene, and the modulation speed can reach 15.38 GHz. The all-fiber modulator we proposed will be widely used in future optical fiber communications and all-fiber systems.

2. Modulator structure and parameter optimization

The proposed modulator is schematically illustrated in Fig. 1(a). The cladding of fiber is cut by a femtosecond laser, and its flatness is enhanced by a side polishing process to reduce insertion and back-phase scattering losses. Figure 1(b) is a cross-sectional view of the modulator. The isolation layer length (L), the thickness of the isolation layer (T) and grinding distance (D) are shown in this figure. The thickness of the isolation layer (T) is the thickness of the black part. Limited by the size of the graph, it is shown in an enlarged illustration. The inset shows the mode field diagram of the double-D fiber. It can be seen that most of the mode field is confined in the core, and a small part of it exists in the cutting surface as the form of evanescent waves. Among them, the outer gray part is the substrate of the device, which is used to fix fiber. Yellow part is the electrode. The black part is the isolation layer, including the graphene layer and the Al₂O₃ layer. The blue part is a double-D fiber, the upper and lower cladding layers are cut into planes, while the core remains intact. The Al₂O₃ layer is uniformly deposited on the cut surface as a transition layer, because the adhesion and electrical conductivity of Al₂O₃ are better in line with graphene. Graphene sheets grown by chemical vapor deposition are mechanically transferred to the fibers. One electrode is connected to graphene, and the other electrode is connected to the Al₂O₃ layer. In previous related types of microstructure fiber modulators, the cutting depth of the cladding reaches the center of the core to enhance the direct contact between the core and graphene. However, grinding the core structure will produce a great insertion loss, affecting the low loss advantage of the all-fiber system. According to calculations, if the core structure is destroyed, its mode loss will reach the order of 10 dB/um. Related articles have ignored this key issue. In our work, the upper and lower cladding are cut the same distance. This can ensure that the mode field does not deviate in one direction, so that most of the mode field is still tied to the core, which greatly reduces the device loss. Therefore, with the goal of reducing device loss, the influence of modulator structure parameters on loss is calculated.

 

Fig. 1. (a) Microstructure fiber structure diagram; (b) Double-D cross-section diagram

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The loss of the modulator device mainly comes from two factors. One is that due to the cutting of the cladding, the binding force of the fiber core to the optical field is weakened, and the mode exists on the cutting surface in the form of evanescent wave, resulting in mode loss:

where ${\alpha _{Mode}}$ is the mode loss and ${\mathop{\rm Im}\nolimits} ({{N_{eff}}} )$ represents the imaginary part of the effective mode refractive index. The second is due to the difference between the mode field area and the single mode field area, the mode field mismatch loss during coupling:

where ${d_n}$ is the mode field diameter of the section and ${\alpha _{mis}}$ is the mode field mismatch loss.

We calculate and analyze the influence of each parameter of the device structure on the device loss. The first is grinding distance, that is, the distance D from the core to the cut surface. It can be seen from Fig. 2(a) that within the range of grinding distance from 4 microns to 6 microns, the refractive index of the two polarization states has an imaginary part, that is, the graphene interacts with the optical field leaking out of the core. For this purpose, the closer the grinding is to the core, the greater the leakage field and the greater the value of the imaginary part. When the grinding distance is 6.3 micron, the imaginary part of Y polarization disappears, and the imaginary part of X polarization continues to exist. The reason is that graphene is essentially anisotropic. The dielectric constants of X polarization and Y polarization are different. Y is generally a constant, and X changes with external voltage and is greatly affected by External parameters. The step size is reduced to 0.1 micrometers. By calculation, when the distance is 8.2 micrometers, the imaginary part of the X polarization disappears. It can be seen that under the double-D microstructure, 8.2 microns is the maximum grinding distance. Graphene over 8.2 microns will not interact with the light field. Figure 2(b) shows the relationship between the grinding distance and the attenuation constant. It can be seen that the curve is similar to the curve in Fig. 2(a). It shows that the mode loss comes from the imaginary part of the effective refractive index. When the grinding distance is 4 microns, when it is tangent to the core, the loss is greatest. The loss of X polarization decreases rapidly and enters a flat period at about 8 microns. The loss reduction rate of Y polarization is slower. Combining the previous interaction with graphene, the subsequent calculation selected the grinding distance to be 8 microns. The corresponding X polarization mode loss is 0.60 dB/mm, and the Y polarization mode loss is 0.09 dB/mm.

 

Fig. 2. The relationship of (a) Imaginary part of effective mode refractive index; (b) Mode Loss; (c) Mode field diameter; (d) Mode field mismatch loss; with the grinding distance D.

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When the core is set to 4 microns, that is the mode field diameter of the fundamental mode of a common single-mode fiber is 8.7071 microns. According to Fig. 2(c), when the grinding distance is 4 microns, fiber has the weakest restraint on the mode field. Most of the evanescent waves exist on the cutting surface. The confinement increases and the fiber moves towards being a multi-mode and hence the extension of the evanescence wave is reduced. So the mode field diameter is the smallest. X polarization is 7.26 microns, and Y polarization is 7.43 microns. When the cut surface is far away from the core, the binding capacity increases. The mode field diameter gradually increases to approach 8.7071 microns. As the grinding depth increases, the mode field diameter approaches the fundamental mode. Mode field mismatch loss gradually decreases. When the grinding depth is equal to 8 microns, the X polarization mode field diameter is 8.4738, and the Y polarization mode field diameter is 8.5001. It can be seen from Fig. 2(d) that when the grinding depth is equal to 4, the mode field diameter differs greatly from the fundamental mode. The loss caused by the mismatch of the X polarization mode field is 10.8%, and the Y polarization is 14.3%. When D is equal to 8, the X polarization mode field mismatch loss is 0.25%, and the Y polarization mode field mismatch loss is 0.32%. After D is greater than 9 microns, this loss can be basically ignored.

Secondly, the relationship between the effective refractive index imaginary part, the mode field diameter, the mode loss, the mode field mismatch loss with the length L of the isolation layer is calculated. The first is the trend of the imaginary part of the effective refractive index with the length of the isolation layer. According to the inset in Fig. 3(a), it can be seen that the imaginary part of the X polarization rises slowly and then remains stable, while the imaginary part of the Y polarization is always close to zero. This is because the dielectric constant of Y polarization remains unchanged, and changing L has no effect on it. After calculation, L equal to 3.6 microns is the critical size for graphene to interact with the light field. The X polarization effective mode refractive index produces the imaginary part. As shown in Fig. 3(c), the isolation layer length L ranges from 8–40 microns, with a step size of 2 microns. When L is less than 20 microns, the mode field diameter decreases faster as L increases. After 28 microns, it enters the flat period, and the change of L no longer affects the mode field diameter. When L is 28 microns, the X polarization mode field diameter is 8.5 microns, and the Y polarization mode field diameter is 8.4739 microns.

 

Fig. 3. The relationship of (a) Imaginary part of effective mode refractive index; (b) Mode Loss; (c) Mode field diameter; (d) Mode field mismatch loss; with the Length of isolation layer L.

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The relationship between mode loss, mode field mismatch loss and isolation layer length L is calculated and shown in Fig. 3. As shown in Fig. 3(b) and (d), when L increases, the mode loss and mode field mismatch loss of X polarization increase. It can be seen from Fig. 1(b) that the contact area between graphene and fiber rise with the increase of length. At the same voltage, a larger contact area will enhance the interaction between light and graphene, thus improving the modulation efficiency. Accordingly, to achieve the same modulation effect, the larger the contact area, the smaller the modulation voltage. In this part, we focus on the influence of isolation layer length on insertion loss. Therefore, only qualitative analysis of the modulation voltage is given. In practice, the relationship between the two needs to be balanced. The increase in loss is small, and the difference in the maximum amplitude is less than 0.05 dB/mm in the entire calculated L range. Therefore, in actual production, this parameter is given priority to being easy to produce and satisfying the modulation voltage. Y polarization and X polarization have opposite conclusions on mode loss. This is because graphene is an anisotropic material, and the dielectric constant changes differently when the dielectric constant is fixed and controlled, causing the above phenomenon. When L is 28 microns, the X polarization mode field diameter is 8.5 microns, and the Y polarization mode field diameter is 8.4739 microns. The loss caused by the mismatch of the X polarization mode field is 0.323%, and the Y polarization is 0.251%.

Finally, the relationship between the effective mode refractive index imaginary part, mode field diameter, mode loss, mode field mismatch loss with the thickness T of the isolation layer is calculated. The first is the effective mode refractive index, as shown in Fig. 4(a). The imaginary part of the effective mode refractive index has a period of 0.8 micron. In the period imaginary part first keeps to 0 as the thickness increases, and the mode does not interact with graphene. In the second half of the cycle, the value of the imaginary part rises and then falls. The mode field diameter at the peak is close to the fundamental mode, and the mode field mismatch loss is small. The reason for the periodicity is that when the graphene layer and the isolation layer are regarded as a whole, with the change of thickness, which couples with the mode in the fiber. The period of change is the coupling period. The Y polarization is perpendicular to the transmission direction, and the increase in thickness has little effect on it, so the imaginary part of the Y polarization is always approximately zero. According to Fig. 4(b), the mode loss is basically consistent with the change of the imaginary part of the refractive index. There is also a periodic loss of Y polarization at the peak, but it is smaller than X polarization. Change the step length at the peak to 0.01 microns. It can be accurately calculated that when T=0.5 microns, the X polarization loss is 0.126 dB/mm and the Y polarization loss is 0.033 dB/mm. When T=1.3 microns, the X polarization loss is 0.125 dB/mm, and the Y polarization loss is 0.037 dB/mm, which means that the loss within the cycle is minimal; when T=0.84 microns, the X polarization loss is 1.43 dB/mm. When T=1.61 microns, the X polarization loss is 1.36 dB/mm., The maximum loss during the period appears.

 

Fig. 4. The relationship of (a) Imaginary part of effective mode refractive index; (b) Mode Loss; (c) Mode field diameter; (d) Mode field mismatch loss; with the Thickness of isolation layer T.

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After the above calculation, it is obtained when the grinding depth D is equal to 8 microns, the isolation layer length L is equal to 28 microns, and the isolation layer thickness is 1.55 microns. The mode loss under X polarization is 0.125 dB/mm, and the mode field mismatch loss is 0.25%. The Y polarization loss is 0.037 dB/mm, and the mode field mismatch loss is 0.32%. In the field of micro-structure fiber modulators, our proposed double-D structure can obtain an insertion loss far below the average level when the extinction ratio is ideal.

3. Performance analysis of microstructure fiber modulatior

In this part, we first analyze the optical properties of graphene under different chemical potential conditions. The changes of conductivity and refractive index of graphene with chemical potential are shown in Fig. 5. The real part of conductivity has little change from 0 eV to 0.4 eV. Then it decreases with the increase of chemical potential. It tends to be stable after 0.6 eV. The imaginary part reaches the minimum at 0.4ev. The refractive index shows a similar trend. According to some previous results, we can conclude that the relaxation time is generally in the range of 0.8 ∼ 1.8ps. The relationship between the relaxation time and the conductivity has been calculated. We choose three cases of τ of 0.8, 1.2 and 1.6ps. It can be seen from Fig. 6(a) that the conductivity of graphene don’t have any changes. Other factors may affect the relaxation time, including temperature or doping concentration. This is more in favor of material research, and we have not carried out detailed calculation here. The effect of the number of layers on the effective refractive index of graphene is shown in Fig. 6(b) It can be seen that the shapes of the five curves are basically the same. The relationship is approximately linear. This means that to achieve the same extinction ratio, the length of modulator with 5 layers structure is only 1 / 5 of that of single layer. Therefore, the multilayer graphene structure is helpful to increase the absorption efficiency and modulation efficiency. However, electron relaxation time has little effect on the imaginary part of refractive index. Therefore, the influence on extinction ratio can also be ignored.

 

Fig. 5. The relationship between (a) conductivity and (b) refractive index with chemical potential

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Fig. 6. (a) The relationship between conductivity with chemical potential at different relaxation time; (b) The relationship between imaginary part of refractive index with chemical potential at different layers.

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Next, we use the optimized parameters to calculate the modulation performance of the microstructure fiber modulator. Extinction ratio and modulation speed are two important indicators of the performance of the response modulator. The extinction ratio of the modulator is mainly affected by the loss in the switching state. Divide the part where the imaginary part is not 0 in Fig. 4(a) into several small segments. The calculated data is divided into 4 groups, including the minimum point at both ends, the maximum point at the peak, the rising phase and the falling phase. Fix its structural parameters, change the chemical potential energy in the range of 0–1 eV. The maximum and minimum values of the mode loss are calculated, and the extinction ratio is obtained by the ratio.

The first is the minimum points at both ends. As shown in Fig. 7(a) and (b), when T is equal to 0.5 microns, the X polarization loss is 0.126 dB/mm and the Y polarization loss is 0.033 dB/mm. When T is equal to 1.3 microns, the X polarization loss is 0.125 dB/mm, and the Y polarization loss is 0.037 dB/mm. After calculation, the imaginary part of the refractive index of two polarization states is less than the order of 10⁻¹⁰, which can be ignored. Changing the chemical potential energy, the effective mode refractive index does not change, and the loss change is extremely low. It is proved that graphene does not play a role in absorption. The absolute value of loss is the lowest in the later period of T. However, the difference of the loss within the variation range of the chemical potential energy is too small to achieve the switching effect.

 

Fig. 7. The relationship between mode loss and chemical potential energy when (a) T is 0.5 microns; (b) T is 1.3 microns.

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The second set of data is the point of the rising edge in the cycle. As shown in Fig. 8(a) and (b). When T is equal to 0.8 microns, the minimum value of X polarization loss is 0.607 dB/mm, and the maximum value is 0.871 dB/mm. The extinction ratio is 15.68%. When T is equal to 1.55 microns, the minimum X polarization loss is 0.363 dB/mm and the maximum is 0.515 dB/mm. The extinction ratio is 15.19%. The value is the point on the rising edge of the cycle. The Y polarization remains basically unchanged, and the X polarization changes greatly. When µ_c is equal to 0.4, there is a minimum loss value. When µ_c is equal to 1, there is a maximum loss value, which proves that graphene plays a role in absorption.

 

Fig. 8. The relationship between mode loss and chemical potential when (a) T is 0.8 micron; (b) T is 1.55 micron.

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The third set of data is the peak point within the period. As shown in Fig. 9, when T is equal to 0.84 microns, the X polarization loss has a minimum value of 1.16 dB/mm and a maximum value of 1.43 dB/mm. The extinction ratio is 9.08%. When T is equal to 1.61 microns, the minimum X polarization loss is 1.14 dB/mm, and the maximum is 1.36 dB/mm. The extinction ratio is 7.66%. This value is at the peak point in the cycle. When the chemical potential energy is 0–0.3 eV, the loss remains unchanged. The loss decreases rapidly in 0.3–0.5 eV. Then it decreases slightly afterwards, and remains basically stable.

 

Fig. 9. The relationship between mode loss and chemical potential when (a) T is 0.84 microns; (b) T is 1.61 microns.

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The fourth set of data is the point of the falling edge in the cycle. As shown in Fig. 10, when T is equal to 1 micron, the Y polarization loss has a minimum value of 0.09 dB/mm and a maximum value of 0.08 dB/mm. The extinction ratio is 5.01%. When T is equal to 1.75 microns, the Y polarization loss has a minimum value of 0.66 dB/mm and a maximum value of 0.64 dB/mm. The extinction ratio is 1.33%. The value is the point on the falling edge of the cycle. The X polarization remains basically unchanged, and the Y polarization changes greatly. When µ_c is equal to 0.4, there is a maximum loss. When µ_c is equal to 1, there is a minimum loss, which is the opposite of X polarization. As the thickness increases, the mode field in fiber is gradually coupled into the isolation layer, and the peak is when the coupling is strongest. In this stage, X-polarized light is dominant. In the opposite direction, that is, during the process of decreasing the thickness, the optical field is gradually coupled into the optical fiber from the isolation layer. At this time, Y-polarized light plays a leading role. Under single-layer graphene coating, the extinction ratio is 15.68%. The extinction ratio is proportional to the number of graphene layers. As the number of layers increases, the extinction ratio will increase accordingly.

 

Fig. 10. The relationship between mode loss and chemical potential when (a) T is 1 microns; (b) T is 1.75 microns.

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The traveling wave electrode theory is related to the dielectric constant. While the dielectric constant of graphene is affected by chemical potential. It can be applied to any type of electro-optic modulator. The flat tangent of the fiber is approximated to the waveguide structure. The electrode is approximated to an RC circuit to calculate the high frequency performance of the modulator. The modulation rate of the modulator is mainly affected by the characteristic impedance Z0 and the RF loss of the traveling wave electrode. Using the theoretical model given in [29], the designed traveling wave electrode is calculated as follows: L = 28µm, T = 1.55µm and D = 8µm. As shown in Fig. 11(a), when F>80 GHz, the impedance is close to 100Ω. Finally, the frequency response can be calculated. The results are shown in Fig. 11(b). When the given modulation voltage is 5 V, for example, the 3 dB bandwidth is about 15.38 GHz. In theory, ultra-high-speed modulation can be achieved if the drive voltage is increased or the length of the electrode is reduced.

 

Fig. 11. Under traveling wave electrode conditions. (a) Calculated characteristic impedance Z0. (b) Calculated frequency response.

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4. Conclusions

In this paper, an improved double D-type all-fiber modulator is proposed, and the parameters are optimized and performance analyzed by the simulation software COMSOL Multiphysics. Compared with the common D-type fiber, this structure only cuts the cladding without grinding the core structure. The upper and lower cladding are cut same distance. This can ensure that the mode field does not deviate in one direction, so that most of the mode field is still tied to the core, which greatly reduces the device loss. At the same time, graphene layers are coated on both sides of the cut surface to enhance light-mass interaction and ensure a certain modulation efficiency. Calculations show that the mode loss of the dual-D all-fiber modulator under X polarization is 0.125 dB/mm, and the mode field mismatch loss is 0.25%. The mode loss in Y polarization is only 0.033 dB/mm, and the mode field mismatch loss is 0.32%. In contrast, our proposed double-D structure can reduce the loss by more than 10³ times. This value is in a very low range compared to previous similar work. When the modulation voltage is 5 V, the modulation depth is 78.4% under the condition of five-layer graphene, and the modulation speed can reach 15.38 GHz. While maintaining low modulation voltage and higher modulation efficiency, this structure makes full use of the advantages of good coupling of the all-fiber system, and will be widely used in future optical fiber communications and all-fiber systems.