Modelling the broadband mid-infrared dispersion compensator with hybrid silicon and lithium niobate nanowire

Wudi Cai; Hairong He; Lili Miao; Chujun Zhao

doi:10.1364/OSAC.1.000736

1. Introduction

The ultrafast lasers operating in the mid-infrared (mid-IR) spectral range, and in particular, the range between 2 μm and 5 μm characterized by the presence of strong vibrational absorption lines of atmospheric constituents and gases, have many important applications in ultrasensitive molecular spectroscopy, medical and industrial applications [1–3]. When it comes to the generation and propagation behavior of the mid-IR ultrafast lasers, the dispersion management will be very important for the whole process [4]. The dispersion compensation components in the mode-locked fiber lasers will be indispensable to deliver the high energy dissipative soliton, and the dispersion will play a very important role in the following transmission lines for the specific applications [5–7]. Different methods to realize dispersion compensation have been proposed for the near-infrared regime, such as conventional silica-based single mode fiber at optical wavelength around 1 μm and dispersion compensation fiber at optical wavelength around 1.55 μm [8–10]. However, when translating the laser wavelength into the mid-IR spectral range, it will be more difficult to get large normal dispersion because of the negative dispersion from the commercial fiber components [11,12].

The fiber dispersion and the loss of the host material will dominate the transmission process. The dispersion can be tailored by varying its geometry to modify the waveguide dispersion [13,14]. As for the host material, the silica will experience serious wave absorption when the wavelength is over 2.5 μm although the silica fiber can act as the transmission medium in the optical communication system [15–17]. Considering the waveguide geometry and the low loss of mid-IR material, the emergence of the micro/nano waveguide has provided a flexible way to manipulate the dispersion. As for the various waveguides, the hybrid waveguides can solve many problems that cannot be settled by a single material waveguide structure [18,19]. Lithium niobate (LiNbO₃) crystal, as an outstanding electro-optical and nonlinear optical material, has been widely adopted in optical switches, electro-optical switches, optical body holo-graphic memory device, etc. [20–22]. When combining LiNbO₃ and conventional silicon crystal with transparent window in the mid-IR spectral range, the device of such hybrid structure shows properties beyond the reach of silicon crystals [23]. A hybrid silicon and LiNbO₃ micro-ring resonator modulator take the advantage of the property of silicon, sub-micrometer spatial confinement of light, and the second order nonlinear effects of LiNbO₃ [24]. In addition, the hybrid silicon and LiNbO₃ optical micro-ring resonator with integrated electrodes has been validated to greatly increase the tunability of integrated optical devices [25]. However, the waveguide dispersion behavior of the hybrid waveguide with reduced dimension has not been discussed, and their potential optoelectronic application in the mid-IR regime has not been explored [26,27].

In this paper, we present an analytical model for single-mode operation in hybrid silicon and LiNbO₃ nanowire. We have investigated the waveguide dispersion properties of the hybrid nanowire in the mid-IR regime. By tailoring the parameters of the hybrid optical nanowire, large group-velocity dispersion (GVD) and dispersion shift has been obtained. The results may provide some guidelines for designing both optical dispersion compensator and modulator at the nanoscale.

2. Theoretical model

We assume that the waveguide has the hybrid silicon and lithium niobate configuration with rotational symmetry about the z-axis, as shown in Fig. 1

Fig. 1 The schematic diagram of the microfiber waveguide structure.

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. The core is the silicon with a radius a, and the cladding is the dielectric thin coat of lithium niobate with thickness d (d = b-a). The whole structure is encircled with infinite air, of which refractive index is n_c at core portion. The refractive index of LiNbO₃ film is investigated as n_o (ordinary refractive index) and n_e (extraordinary refractive index) of uniaxial dielectric film, and its optical axis is oriented along the axis of the waveguide.

By solving the Maxwell’s equations in cylindrical coordinates, the following expressions can be obtained:

E_{z} = {\begin{cases} \sum_{m}^{} A^{0}_{m} J_{m} (u r) f_{c}, 0 <r< a \\ \sum_{m}^{} [B^{0}_{m} ℜ m (\frac{n_{e}}{n_{o}} v r) + C^{0}_{m} ℵ m (\frac{n_{e}}{n_{o}} v r)] f_{c} a < r < b \\ \sum_{m}^{} D^{0}_{m} K_{m} (w r) f_{c}, r > b \end{cases}

H_{z} = {\begin{cases} \sum_{m}^{} A^{1}_{m} J_{m} (u r) f_{s}, 0 < r < a \\ \sum_{m}^{} [B^{1}_{m} ℜ_{m} (v r) + C^{1}_{m} ℵ m (v r)] f_{s} a < r < b \\ \sum_{m}^{} D^{1}_{m} K_{m} (w r) f_{s}, r > b \end{cases}

where

ℜ_{m} (u r) = {\begin{cases} J_{m} (u r), β < k_{0} n_{0} \\ I_{m} (u r), β > k_{0} n_{0} \end{cases}

,

ℵ_{m} (u r) = {\begin{matrix} Y_{m} (u r), β < k_{0} n_{0} \\ K_{m} (u r), β > k_{0} n_{0} \end{matrix}

,

u^{2} = k_{0}^{2} n_{c}^{2} - β^{2}

,

v^{2} = \mp k_{0}^{2} n_{0}^{2} \pm β^{2}

,

w^{2} = β^{2} - k_{0}^{2}

,

f_{c} = \cos (m ϕ + φ_{m}) \exp [j (ω t - β z)]

,

f_{s} = \sin (m ϕ + φ_{m}) \exp [j (ω t - β z)]

.

In the above equations, k₀ is the free-space wave number, f_c and f_s are the multiplication factor of E_z and H_z. J_m and Y_m refer to Bessel functions of first kind and second kind of order m. I_m and K_m are the corresponding first kind and second kind modified Bessel function of order m. By applying the relation between the longitudinal and transverse EM field components and considering the boundary conditions at radius a and b, we obtained a system of eight homogeneous equations, respectively. Only if the determinant of the eight equations vanish, the propagation constant β will be determined [28].

\det [M (β)] = 0

In the numerical simulation, the hybrid nanowires are designed for working as single-mode waveguides. Therefore, we consider the fundamental modes and thus set m = 1 in Eqs. (1) and (2). In addition, we assume that refractive index of air cladding is 1.0, and the refractive indices of silicon in the core and LiNbO₃ for the intercore-cladding are given by their Sellmeier functions [13,28]. In addition, the group velocity (V_g), the waveguide dispersions (D_w), and the GVD (β₂) can be calculated [26].

3. Results and discussion

3.1 Typical dispersion compensator behavior

The GVD of the hybrid silicon and LiNbO₃ nanowire from 1500 nm to mid-IR wavelength are shown in Fig. 2

Fig. 2 The GVD curves of the hybrid nanowire with different core size.

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, where the thickness of LiNbO₃ film was assumed as 20 nm. From the simulation, it can be observed obviously that the hybrid waveguide can exhibit large normal or anomalous dispersions. Moreover, the obvious dispersion shifts can be obtained by adjusting the core size of the waveguide. For the waveguide with diameter 600 nm, if the optical wavelength increases from 1500 nm to mid-IR, the GVD for the nanowire will decrease from 0 to −27196.6 ps²·km⁻¹ firstly, then increase to 178440.6 ps²·km⁻¹ at 3.2 μm wavelength, and the dispersion decreases as wavelength goes longer. With the same wavelength range, if the core diameter gets to 700 nm, the GVD will reach the maximum at 208215.0 ps²·km⁻¹ around 3.7 μm. When the core diameter was adjusted to 800 nm, the maximum of the GVD will be 237261.1 ps²·km⁻¹ around 4.2 μm. With the decreasing waveguide dimension, the effective refractive index will approach the cladding index, and the group velocity will approach the light speed since most of the light energy will propagate in air [13]. For a specific nanowire waveguide, the group velocity will also experience a large variation with wavelength. The large variation of group velocity with wavelength will determine the large variation of the dispersion value. The hybrid nanowire here can exhibit large normal dispersion compared to the conventional waveguide, and the spectral range for the normal dispersion operation can be tuned by carefully adjusting the waveguide geometry.

3.2 Broadband dispersion compensation

To present the potential dispersion compensation ability of the hybrid silicon and LiNbO₃ nanowire, the nanowire with different diameters have been discussed. For the 3 μm operating wavelength, the relationship between GVD and waveguide geometry with LiNbO₃ film thickness ranging from 10 nm to 90 nm are plotted in Fig. 3

Fig. 3 The GVD curves for the hybrid nanowire with different waveguide geometry at wavelength 3 μm.

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. It can be found that the dispersion value goes up first, then declines, and at last increases slowly with the increasing diameter. Moreover, with the increasing thickness of LiNbO₃ film, the dispersion gets larger from 400-nm to 520-nm diameter. When set the fiber diameter from 580 nm to 660 nm, the GVD decreases with the increasing LiNbO₃ film thickness. By varying the fiber geometry, the waveguide dispersion can be tuned. For instance, the waveguide dispersions can be tuned in the range of −31977.3 ps²·km⁻¹ to 171328.6 ps²·km⁻¹ with proper fiber diameter. If the nanowire has the diameter smaller than 600 nm with film thickness of 50 nm, the dispersion value present positive, otherwise negative in the large diameter case. With the above considerations, the size of waveguide core should be designed appropriately to realize the dispersion compensation.

When the laser wavelength shifts to 4 μm, the large dispersion value will be obtained which is capable to provide dispersion compensation. As shown in Fig. 4

Fig. 4 The GVD curves for the hybrid nanowire with different waveguide geometry at wavelength 4 μm.

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, the dispersion curves start at 0 ps²·km⁻¹ and then increase over 190000 ps²·km⁻¹ in the thickness range of 10 nm to 90 nm with proper core size. For the fiber with thickness 50 nm, the GVD value goes up to the dispersion maximum at 215873.4 ps²·km⁻¹, then it declines rapidly. What is more, if the fiber geometry is selected as Fig. 4 for 400-nm to 800-nm core diameter, the nanowire dispersion presents positive.

When the wavelength extends to 5 μm, the great normal dispersion can also be obtained as shown in Fig. 5

Fig. 5 The GVD curves for the hybrid nanowire with different waveguide geometry at wavelength 5 μm.

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. The dispersion value approaches to 0 ps²·km⁻¹ and changes very slowly with thickness range from 10 nm to 90 nm and the diameter from 400 nm to 700 nm, which differ from dispersion variations for 3-μm and 4-μm wavelength. When the diameter is larger than 800 nm, the GVD increases quickly with the core diameter increasing. For example, while the film thickness is 50 nm, the dispersion value reaches to 273854.6 ps²·km⁻¹ at the fiber diameter of 940 nm. The large GVD of the waveguide can provide large dispersion compensation.

3.3 Electrical field induced dispersion variation

The LiNbO₃ crystal is an electro-optical material, whose refractive index can be tuned by applied electric field [29]. With the applied electric field, the refractive index of LiNbO₃ thin film can be tuned, and thus the waveguide characteristics can be tailored. We set the nanowire diameter to be 700 nm and film thickness to be 20 nm. For the LiNbO₃ thin film, the breakdown voltage can reach at 1000 kV/mm [30], and we have done the simulation by applying a z-direction positive electric field from 1.0 × 10² kV/mm to 7.0 × 10² kV/mm. As Fig. 6

Fig. 6 Wavelength-dependent GVD curves with z-direction applied positive electric field.

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shows, we investigated the GVD variation from 1.5-μm to 5-μm wavelength. From Fig. 6, it can be seen that with the applied bias increases, the GVD value becomes larger at wavelength range from 3.1 μm to 3.6 μm, then gets less at wavelength from 3.7 μm to 5 μm. Moreover, the largest shifts occur at wavelength around 3 μm to 4 μm. Exactly, with 14-V bias voltage of positive z-direction electric field, we can observe about 20000 ps²·km⁻¹ dispersion shift.

The dielectric film that applied a negative z-direction electric field have also been investigated. We make the parameters to be the same as Fig. 6 but invert the electric field. The simulation result was plotted in Fig. 7

Fig. 7 Wavelength-dependent GVD curves with z-direction applied negative electric field.

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. On the contrary, with the wavelength ranges from 3.8 μm to 5 μm, the more the voltage value is, the more shift of the dispersion will be. For instance, when we apply negative z-direction electric field with −7.0 × 10² kV/mm to LiNbO₃ thin film, the GVD will reach its maximal dispersion value at 217411.0 ps²·km⁻¹. Meanwhile, the value of dispersion without electric field at the same 3.8-μm wavelength is 183968.0 ps²·km⁻¹. From the numerical simulation, it is obvious that electric fields exert impact on the hybrid waveguide dispersion.

To clarify the influences of the waveguide structure and the applied electric field, the GVD curves of the hybrid nanowires with or without z-direction applied electric field have been compared in the Fig. 8

Fig. 8 GVD curves with or without the z-direction applied electric field.

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. In the simulation, we first set the nanowire radius to be 350 nm and film thickness to be 20 nm. The maximum value of the dispersion curve is 208215.0 ps²·km⁻¹. When the electric field with z-direction is applied, the maximum of dispersion value will increase to 217411.0 ps²·km⁻¹ at 3.8 μm. By tailoring the core radius to be 400 nm without electric field, the maximum of dispersion value can reach 23726.1 ps²·km⁻¹ at 4.2 μm. It can be seen from the figure that the dispersion behavior will be more sensitive to waveguide geometry than the applied electric field.

4. Conclusions

In conclusion, we have numerically studied the GVD characteristics of hybrid silicon and LiNbO₃ nanowire in the mid-IR regime. We found that the hybrid waveguide can be made as a broadband compensator. For the 2~5 μm spectral range, large normal dispersion behavior of the hybrid waveguide can be obtained. Besides, by adjusting the core size of the nanowire and thickness of LiNbO₃ film, the waveguide dispersion can be varied and the broadband dispersion compensation can be obtained. Moreover, we can tune the waveguide dispersion without changing geometric dimension by applying the applied bias field. The proposed waveguide may provide some guidelines for designing broadband dispersion compensator, and make an inroad for the miniaturized functional optoelectronic devices.

Funding

National Natural Science Fund Foundation of China (NSFC) (61475102, 11574079 and 61775056); Joint Equipment Pre-Research Foundation of the Ministry of Education of China (Grant 6141A02033404); Natural Science Foundation of Hunan Province (2017JJ1013).

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Modelling the broadband mid-infrared dispersion compensator with hybrid silicon and lithium niobate nanowire

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1 Typical dispersion compensator behavior

3.2 Broadband dispersion compensation

3.3 Electrical field induced dispersion variation

4. Conclusions

Funding

References

Cited By

Figures (8)

Equations (3)

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