The dispersion properties of silica and silicon subwavelength-diameter wires with intercore-cladding uniaxial dielectric lithium niobate thin film has been studied numerically in detail. The waveguide dispersion shifts centered around 1550-nm wavelength have been investigated. It shows that the dispersion of optical nanowires with intercore-cladding lithium niobate thin film is highly sensitive to fiber geometry. Moreover, with applied electric field, considerable dispersion shifts without changing its geometric structure can be obtained. Our work may provide an inroad for developing miniaturized functional optoelectronic devices.
© 2015 Optical Society of America
Waveguide dispersion engineering is very important for linear and nonlinear pulse transmission processes in different fields, such as optical communication, optical sensing and nonlinear optics [1–3]. By tailoring the waveguide geometry, different waveguide dispersion profiles can be obtained and the performance of the optical devices, such as optical modulators, switches, super-continuum sources, and wavelength converters, can be improved [4–8]. Among the various waveguides, micro-nano wires have been received increasing attentions for their giant waveguide dispersion, small size, low optical loss, and high tensile strength [9–11]. With the decreasing waveguide dimension, the waveguide characteristics will be different from their conventional counterpart. As a typical representative of one dimensional dielectric micro/nano optical waveguide, silica and silicon nanowires provide a simple and effective way for manufacturing nano-photonic devices, which has a natural compatibility with the existing optical waveguides. Moreover, the emerging semiconductor, polymer, hybrid photon-plasmon nanowires provide a versatile platform for manipulating light at the nanoscale [12–15].
The waveguide dispersion is associated with refractive index of material, the structure size, and the wavelength of the travelling light. Previous works focused on size-dependent waveguide dispersion [16–19]. However, it is difficult to change the structure and the waveguide dispersion once the waveguide geometry is determined. Lithium niobate (LiNbO3) crystal, an excellent electro-optical and nonlinear optical material, can be used in optical switches, electro-optical modulators and nonlinear optical devices [20,21]. By combing the silica/silicon and LiNbO3, the hybrid waveguide can present excellent electro-optical modulation performance . If the dimension of the hybrid waveguides decreases, the waveguide will show totally different waveguide characteristics. Meanwhile, the field-induced change of the refractive index of LiNbO3 provides a flexible method to tailor the dispersion of nanowires waveguide device.
In this paper, the waveguide characteristics of the optical nanowires with intercore-cladding lithium niobate thin film were studied numerically. Considerable dispersion shift of the guided light can be obtained with different fiber geometry or different applied electric field for the same fiber geometry. The results cannot only provide design guidelines for dispersion components, but also shed light on the design of modulator at the nanoscale.
2. Mathematical model
We consider intercore-cladding uniaxial dielectric thin film optical nanowires with rotational symmetry about the z-axis, as shown in Fig. 1. A circular dielectric core (e.g. silica nanowire) with radius a and a uniaxial cylinder dielectric coat (e.g. LiNbO3) with thickness d (d = b-a) are embedded in the infinite air cladding. Refractive indices of the core and air-cladding are assumed to be ns and nc, respectively, whereas the film is uniaxial dielectric material with ordinary and extraordinary refractive indices no and ne, respectively. Solving Maxwell’s equations in cylindrical coordinates (r, , z) leads to the following expressions for the components of the electromagnetic field for the m th mode :
The transverse field components are obtained from the following equations:
Applying boundary conditions at r = a and r = b, a system of eight linear homogeneous equations is obtained that is satisfied by the eight coefficients. The system admits a nontrivial solution only in case its determinant is zero. The propagation constant β is determined by the condition that the determinant of the system of linear equations shall vanish:
Usually, subwavelength-diameter optical nanowires are designed and desired for working as single-mode waveguides . Therefore, here we consider the fundamental modes and thus set m = 1 in Eq. (1) and Eq. (2). With propagation constants (β) obtained by numerically solving Eq. (7), the group velocities (Vg) and waveguide dispersions (Dw) can be obtained as :
For numerical simulations, we choose silica and silicon as core material, respectively. The outermost is air with refractive index about 1.0. LiNbO3 thin film will be investigated to illustrate the behavior of uniaxial thin films. We use the following Sellmeier-type dispersion formula (at room temperature) to obtain the refractive indices of the wire materials for fused silica (SiO2) :25]:
The equations for ordinary and extraordinary refractive indices of bulk LiNbO3 as a function of wavelength λ (in μm) are given by :
3. Results and discussions
3.1 Dispersion shifts of optical nanowires with different fiber geometry
In the simulation, the thickness (d = b-a) of the LiNbO3 thin film was assumed as 20 nm . The diameter- and wavelength- dependent waveguide dispersions of air-clad silica and silicon wires with LiNbO3 thin film are shown in Fig. 2. It shows that, as the fiber core diameter changes, waveguide dispersions will experience a great change. Figure 3 shows the diameter-dependent waveguide dispersion of fundamental mode of air-clad silica wire with LiNbO3 thin film at 633-nm and 1550-nm wavelengths and silicon wire with LiNbO3 thin film at 1550-nm wavelength. We can clearly see that Dw of an 200-nm-diameter silica wire at 1550-nm wavelength approaches to zero, while for 600-nm diameter, it can be about –2200 ps·nm−1·km−1. For silicon wire with LiNbO3 thin film at 1550-nm wavelength, the dispersions shifts are more obvious. The waveguide dispersion achieves the minimum about –72000 ps·nm−1·km−1 around 280-nm diameter and maximum about 12400 ps·nm−1·km−1 around 340-nm diameter. The results show that, at a particular wavelength, the waveguide dispersions of a wire-waveguide can be made zero, positive or considerably negative when a proper diameter is chosen, and thus provide opportunities for achieving enhanced dispersions with small size variations.
To ensure the single mode operation of the nanowire, the core diameters used for calculation here are 400-nm for silica nanowire and 300-nm for silicon nanowire. We change the thickness of LiNbO3 thin film from 20 nm to 40 nm, thickness- and wavelength- dependent waveguide dispersions are shown in Fig. 4 and Fig. 5. With the thickness of the LiNbO3 increasing, the dispersion shift is large, and the dispersion curve moves to the right and down for silica wire while it moves to the right and up for silicon wire. Figure 5 gives the dispersion shift with a increasing thickness of LiNbO3 at 633-nm wavelength and 1550-nm wavelength. Both the dispersion at 633-nm wavelength and 1500-nm wavelength decrease with the thickness augment of the LiNbO3 film for silica wire, while at 1550-nm wavelength it increases with the thickness augment of the LiNbO3 film for silicon wire. For example, at 1550-nm wavelength, the dispersion of the fundamental modes is about –970 ps·nm−1·km−1 for the 20-nm-thickness LiNbO3 thin film silica nanowire and –2226 ps·nm−1·km−1 for the 40-nm-thickness LiNbO3 thin film silica nanowire. At 1550-nm wavelength, the dispersions of the fundamental modes are about –37427 ps·nm−1·km−1 and –3833 ps·nm−1·km−1 for the silicon nanowires with 20-nm-thickness LiNbO3 thin film and 40-nm-thickness LiNbO3 thin film, respectively. Therefore, changing the thickness of thin film will be another feasible and efficient way to adjust the waveguide dispersion .
3.2 Dispersion shifts in nanowires with applied electric field
The LiNbO3 crystal is an electro-optical crystal, whose refractive indices vary with applied electric field. The crystal has a 3 m point group of the trigonal system, in the absence of external field, the corresponding refraction index ellipsoid equation can be expressed as:26]:
The change of the refractive index of thin film will cause the nanowires to present different optical properties. We investigated the waveguide dispersion from the short wavelength side to the IR edge of a 400-nm-diameter silica nanowire and a 300-nm-diameter silicon nanowire with 20-nm-thickness LiNbO3 thin film and the outermost air-cladding. For the lithium niobate thin film, the breakdown field can reach 1000 kV/mm [27–29]. Here, we applied a positive z-direction electric field from 1 × 108 V/m to 7 × 108 V/m range in the simulation, as shown in Fig. 6. As it shows, the dispersion curve blue-shifts for both silica wire and silicon wire with the increasing applied electric field. To make the dispersion shift clear, the insets show the zoomed waveguide dispersions curves from 1540-nm to 1560-nm wavelength to observe the waveguide dispersions shifts.
With 14 V bias voltage of positive z-direction electric field, considerable dispersion shift about 54 ps·nm−1·km−1 and 4300 ps·nm−1·km−1 can be observed for silica wire and silicon wire within 1540-1560 nm wavelength range, respectively. Moreover, the greater the voltage value is, the more evident the dispersion shift will be.
We have also studied the dispersion shift with applying a negative z-direction electric field. We choose the same size as above. After applying a negative z-direction electric field, the calculated dispersions are shown in Fig. 7. As it shows, the dispersion curve red-shifts for silica wire and silicon wire with the increased applied electric field, which is contrary to the results in Fig. 6.
Figure 8(a) shows the applied-electric-filed-dependent dispersion of a 400-nm-diameter silica wire with 20 nm LiNbO3 thin film at the wavelength of 633 nm and 1550 nm. Figure 8(b) shows the applied-electric-filed-dependent dispersion of a 300-nm-diameter silica wire with 20 nm LiNbO3 thin film at the wavelength of 1550 nm. It shows that, only by using an applied electric field, we can achieve large dispersion shifts without changing the size of wires, presenting opportunities for achieving enhanced dispersions in extensive fields such as optical communication and nonlinear optics.
We have numerically studied the dispersion shifts and dispersion tailoring ability in LiNbO3-thin-film coated silica nanowires and silicon nanowires. We found that, with diameter of the core or the thickness of the LiNbO3 thin film variations, considerable dispersion shift of the guided light can be obtained. Besides, after adding external electric field, there still exists a significant shift of the waveguide dispersion. Both the value and the direction of the applied electric field can have influences on the dispersion. Without any changes to the geometric dimension of the nanowire waveguide, we can control the waveguide dispersion with applied electric field, making it a hopeful way for further applications in the optical communication and nano-optics fields.
This work is partially supported by the National 973 Program of China (Grant No. 2012CB315701), the National Natural Science Fund Foundation of China (Grant Nos. 61205125 and 61475102).
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