A long-period waveguide grating (LPWG) with a tunable index contrast is proposed. The design features a simple configuration that consists of a two-mode waveguide formed on periodically poled lithium niobate with an angle with respect to its domain wall and a traveling-wave electrode. In the design, the electrical traveling wave introduces a periodic change in the refractive index of waveguide, which functions as a long-period waveguide grating that couples between symmetric and anti-symmetric core modes. The index contrast of grating can be controlled by the traveling-wave intensity. For application to ultrafast device, structural parameters satisfying velocity and impedance matching conditions are numerically calculated.
© 2011 OSA
Optical fiber gratings are well-known components used for wavelength filtering in optical fiber systems. The fiber gratings can be categorized into two; one is a fiber Bragg grating (FBG), the other is a long-period fiber grating (LPFG). The FBG reflects narrow band of light at a particular wavelength while the LPFG converts a particular wavelength of light to higher-order modes that are not guided in the fiber core. Most of the gratings written into the fiber core are difficult to change the index contrast due to the geometrical and material constraints of optical fibers. There were attempts to overcome the limitations. One of them was to use a ring-core fiber and liquid crystal. Liquid crystal that is inserted into the hollow of ring-core fiber introduces a periodic change in the refractive index of core by an external long-period-combed electrode . Another approach to relax the restrictions is to fabricate the gratings in different waveguide structures with different materials [2–4]. For example, the gratings formed on planer waveguides with suitable electrodes can achieve a tunable index contrast based on a thermo-optic effect. Although those gratings show satisfactory controllability, they have not only complex configurations but also need sophisticatedly structured electrodes. Furthermore, the gratings cannot be expected to be used as fast tunable device due to slow response time of related materials.
In this study, we propose a design of long-period waveguide grating (LPWG) with an ultrafast tunable index contrast. The design features a simple configuration that consists of a two-mode waveguide, a periodically poled lithium niobate (PPLN) [5,6], and a traveling-wave electrode [7,8]. The optical waveguide runs at an angle with respect to domain wall of PPLN. In the design, a periodic change in refractive index of the waveguide is induced by an electric field of traveling wave, which functions as the LPWG that couples between symmetric and anti-symmetric core modes. Here, the index contrast of grating can be controlled by the intensity of electric field. For fast operation, structural parameters such as thickness of buffer layer and electrode, and distance between the electrodes that satisfies velocity and impedance matching are also investigated. The proposed LPWG is expected to offer many benefits for handling optical signals including low drive voltage with no bias and no frequency chirping as an ultrafast optical intensity modulator.
2. A waveguide with a periodic index change in a tilted form
Let us consider a two-mode waveguide with a periodic-index change in a tilted form as shown in Fig. 1 .
Here, Λ/2 is period of index change. The periodicity can reach up to a few millimeters from tens of nanometer. n1 and n2 stand for refractive indices of each section (n1 > n2). Let us also assume that the difference between n1 and n2 is about ~10−5. Then, a difference in optical path length of each section can be neglected. For example, assuming that Λ/2 is 500 μm, the optical path difference is 5 nm, which corresponds to a phase difference of about 0.007π at the wavelength of 1.5 μm.
The two-mode waveguide shown in Fig. 1 can be simply reconfigured as follows.
In Fig. 2, n is average refractive index of the waveguide. –δ, and + δ stand for relative phase shifts. βS and βA represent propagation constants for spatially symmetric (fundamental mode) and anti-symmetric modes (second-order mode), respectively. LB denotes optical beatlength. The simplified model describes that the index change in a tilted form leads the refractive indices of the upper and the lower halves of the core to be alternately modulated with a constant phase shift of δ. The axially periodic and spatially anti-symmetric index change induces mode coupling between the symmetric and the anti-symmetric modes at a certain wavelength . Thus, the waveguide can function as an LPWG whose working principle is the same as that of the gratings formed by mechanical pressure or a flexural acoustic wave [10,11].
The LPWG shown in Fig. 2 can be realized based on an optical waveguide running at an angle with respect to the domain wall of PPLN. To make an easy description of the optical and electrical designs of the waveguide grating, the fabrication processes are briefly illustrated in stages (Fig. 3 ). It is worth noting that the LPWG will only affect to the light whose polarization is parallel to the domain wall, which will be described in details in the following section.
3. Optical design (Electrically tunable index contrast of grating)
Lithium niobate (LiNbO3) has been the material of choice for manufacturing fast optical intensity modulator owing to THz-range response speed with a large electro-optic coefficient [12,13]. The linear electro-optic effect of LiNbO3 is used in the grating design. Figure 4 shows the top view of the two-mode waveguide fabricated on a PPLN (Z-cut) where the domain wall is parallel to the extraordinary axis (Refer the second stage in Fig. 3).
The thickness of domain and the width of waveguide are designated by d and 2a. The tilted angle of the waveguide against the domain wall and the grating period are denoted by α and Λ. Then, the grating period Λ is determined by the thickness of domain d and the angle α (Eq. (1).
In this structure, the refractive index variation of each domain when an electric field is applied in the z-direction (extraordinary axis) can be expressed as
Note that only z-polarized light experiences the index variation. In Eq. (2), ne denotes the refractive index of LiNbO3 for extraordinary axis (about 2.15 at the wavelength of 1.3 μm) and r33 represents the electro-optic coefficient (about 30.9 pm/V). In this case, the waveguide come to have alternative refractive index changes, + Δnz and -Δnz along the axial direction when the electric field is applied. For example, the refractive index change of Δnz would be about 3.8 × 10−5 when 5-V driving voltage is applied to the center electrode that is 10 μm away from the near ground electrodes. In this case, a 50% overlap integral factor between the electric field and optical field is assumed . The refractive index change can be tuned by the applied voltage, which enables the LPWG to have an electro-optically tunable index contrast. A few centimeters grating length will be sufficient for efficiently coupling to the second-order mode by considering the number of gratings and the magnitude of phase shift.
To specify the geometrical parameters, the wanted grating period, Λ should be defined in advance. The grating period is related to modal beatlength at the wavelength of our interest. The modal beatlength can be calculated from the effective indices of guided modes and are shown in Fig. 5 (See the fourth stage in Fig. 3).
Here, a rigid structure is employed . In the calculation, the refractive index of buffer layer (SiO2) is 1.457 at 1.3 μm. The refractive indices of LiNbO3 substrate are 2.229 for ordinary axis and 2.15 for extraordinary axis. The refractive indices of optical waveguide are set to be 2.2424 for ordinary axis and 2.163 for extraordinary axis. The thickness of buffer layer is fixed as 3 μm. The top width (2a) and the depth of waveguide are 9 and 4 μm, respectively. In this case, the effective mode indices are calculated to be 2.157689 for fundamental mode and 2.154059 for second-order mode (for z-polarized light) and the optical beatlength (LB) between the modes is about 358 μm. We should note that the other polarization component orthogonal to the z-direction does not experience the periodic index variation.
The waveguides that are shown in Fig. 6 (a) and (b) have the same grating period but different phase shifts. One can easily infer the angle of α and the domain thickness of d that maximizes the net index contrast (phase shift of δ) comparing Fig. 6 (a) and (b).
By taking the optical beatlength of 358 μm in this case and the waveguide width of 9 μm into account, the angle of α can be calculated from Fig. 6 (a), which is about 2.9° (tan−1(9/179)). In this case, the domain thickness d is about 9 μm from Eq. (1) (See also Ref . for interpretation of the blaze angle of LPFG). The beatlength dispersion relation in the waveguide and the corresponding angle of α are plotted in Fig. 7 (We assume no materials’ dispersion here.).
As shown in Fig. 7, the increase of Δn results in the decreases of beatlength and the increase of angle α. The 3-dB optical bandwidth of the proposed LPWG can be calculated by considering the grating length, the beatlength, and its dispersion relation [17,18], which is plotted in Fig. 8 .
In the calculation, the grating length is set to be 30 mm. Here, it is assumed that an electric field is applied so that 100% coupling efficiency can be achieved when the grating length is equal to the coupling length. The 3-dB bandwidth decreases as the refractive index difference of Δn increases.
4. Electrical design (Ultrafast tunability)
The tuning speed is dominated by the electrode characteristics. The basic design of electrodes here is almost the same as that of typical LiNbO3-based modulators with a co-planar waveguide (CPW) electrode ; a concept of traveling-wave electrode. The main factor that limits the tuning speed is the mismatch of group velocity between the optical wave propagating in optical waveguide and the electrical wave propagating along the traveling-wave electrode. Material properties used in numerical calculations are shown in Table 1 . Silica (SiO2) has been used as buffer layer in the typical LiNbO3-based modulators to avoid optical loss by the electrode and lower the microwave index of LiNbO3. The permeability of silica is set to 3.9 in a microwave range. Lithium niobate is a uniaxial crystal whose permeability is 43 for ordinary axis and 28 for extraordinary axis. Typically, the electrodes are made of gold (Au) due to the low conductor loss.
Figure 9 shows the cross-sectional configuration of proposed grating with structural parameters.
Here, Tb, Te, and G indicate thickness of buffer layer and electrodes, and gap between electrodes, respectively. The effective refractive index of electrical wave (Neff) is calculated as a function of the thickness of buffer layer for different G and Te. The velocity-matching condition is that the effective refractive index of electrical wave (Neff) is equals to that of optical wave (n~2.15). Neff can be written as
In Eq. (3), Cm is the capacitance in the given structure and C0 is the capacitance when replacing the materials including LiNbO3 and SiO2 with air. Figure 10 plots Neff as a function of Tb for different Te and G.The result reveals that Neff is reduced as G decreases, and Te and Tb increase.
Impedance matching is also important for the ultrafast operation with effective electrical power consumption. Tb, Te, and G are also the parameters that can determine the impedance. Typically, 50-ohm output and input impedance are standardized in most electrical systems. Here, the characteristic impedance Zc can be calculated on the basis of Eq. (4).
The impedance Zc as a function of Tb for different Te and G is plotted in Fig. 11 .The result indicates that Zc increases when G and Tb increase, and Te decreases.
Based on the results of Fig. 10 and 11, the structural parameters for the velocity and the impedance matching can be obtained. The top figure in Fig. 12 is the plot of impedance as a function of thickness of the buffer layer under condition such that the velocity matching is satisfied. The bottom one is the plot of electrode thickness as a function of the buffer layer under the velocity matching condition.
The results indicate that the impedance and the velocity matching conditions are satisfied at Tb = 1.26 μm, Te = 5.1 μm for G = 10 μm or Tb = 1.15 μm, Te = 14.5 μm for G = 20 μm, or Tb = 1.08 μm, Te = 24 μm for G = 30 μm. Since a thick electrode leads a low-conductor loss and G is little dependent on drive voltage, the thick electrode is desirable for the ultrafast operation.
A long-period waveguide grating (LPWG) with a tunable index contrast is proposed. The LPWG consists of a periodically poled lithium niobate (PPLN), a two-mode waveguide running at an angle with respect to the domain wall of PPLN, and a traveling-wave electrode. The detailed structural parameters of the LPWG are also calculated for efficient coupling and ultrafast operation. The proposed LPWG is expected to offer many benefits for handling optical signals including low drive voltage with no bias and no frequency chirping as an ultrafast optical intensity modulator.
I would like to acknowledge support from T. J. Park Postdoctoral Fellowship.
References and links
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