Construction Of Refractive-index Profile Of Asymmetric Dieletric Optical Waveguide By Two-dimensional Inverse Matrix Method

With the fast development of optical communication, optical fibers and waveguide devices play a dominant role in practical application. It is essential to investigate their characters, including the bandwidth, single mode transmission condition, dispersion, mode-field diameter and birefringence effect and so on, and their calculations are based on refractive index (RI). In particular, the wave division multiplex system will be popularized gradually, and the single channel communication velocity tends to rise to40Gbit /s , so dispersion, especially polarization modes dispersion, has been emphasized to such an extent that it could not be neglected. Thus it becomes very important to study the asymmetric refractive index profile (RIP) of optical waveguide. In order to measure RI, there have been many theoretical and experimental methods. However, the most methods can not be used to measure refractive index for irregular and asymmetric optical waveguides.In order to give more intuitional graphics and calculate the RIP of asymmetric waveguides, we set up two-dimensional inverse matrix method (TDIMM). The method is combined with near-field (NF) measurement, which is robust, rapid and high in spatial resolution. The method does not assume the functional form of unknown quantities, and does not need iterative calculations. Moreover, the unknown index profile can be constructed in linear domain, and the accuracy can be ensured.In this paper, the accurate, simple and rapid determinations of the RIPs of asymmetric waveguides are demonstrated by application of the TDIMM to the single-mode step-index fiber (SMF) and channel waveguide. This method can be used to correctly characterize waveguide with small or irregular refractive-index variation over their cross sections. Especially, it can resolve the experimental difficulties caused by narrow guiding area and low refractive-index difference. And it can, in principle, be performed at the communication wavelength (1.55μm). We avoid the difficulty of calculating the two-dimensional scalar wave equation, and accurately give the two-dimensional RIP of the waveguide. In the numerical process, a two-dimensional finite-difference method is used to discretize the governing equation, and a linear two-dimensional inverse matrix is constructed to identify the unknown profiles of RI. The approach is able to rearrange the matrix form of the two-dimensional differential governing equation and construct the unknown two-dimensional RIP of the SMF and channel waveguide based on transmitted optical intensity from the propagation mode near-field collection.1. Theory The RI can be calculated by the RI difference ?n ( x, y), which is expressed as: (1), where n ( x,y)is the core RI; k =2π/λis the wave number in vacuum with the light wavelengthλ; n s is the cladding RI; is the transverse Laplacian operator; is the transverse electric or magnetic field, I ( x , y )is the measured intensity profile and I_max is the maximum of I ( x , y ). According to the finite-difference method, the RI of waveguide's end surface can be determined by the NF information along x axis and y axis. We adopt the square-grid partition, namely, ?x and ?y are the vertical distances along the x axis and y axis between data points respectively, and they are equal to h, the two-dimensional finite-difference governing equation can be expressed as the following recursive form: (2), where , the subscripts i and j are the discrete grid points along the x axis and y axis, respectively. We obtain optical intensity from near-field intensity measurement and calculate the normalized electric-filed intensity distribution A(x, y).We assume no leaky mode, with the boundary conditions:By substituting (3) into (2) and rearranging the recursive forms consisting of the governing equation, we have a linear equation that can be expressed as where when n = j corresponds to the boundary condition, and ( ) ( )For the inverse equation,α_j×j is the coefficient matrix of measured vectorΑ_j×1.The components of vectorΝ_j×1 are the functions of the unknown RI at discrete grid points of the end surface.The two-dimensional inverse equation can be obtained asAs we know, the right side of the Eq.(5) is the measurable matrix. The two-dimensional refractive-index difference is determined by the inverse model from Eq.(5). Consequently, the unknown RIP is constructed by the inverse analysis, which resolves the measurement of unknown index profiles at discrete square grid points with no prior knowledge of the functional forms of these unknown quantities.2. Experiment and analysesThe experimental setup used is shown in Fig.1, where the wavelength of the diode laser is 1.55μm.The experimental system can be applied to observe the near-field image of the waveguide.(A). Measure RI of FiberBy using self-act scaling system, we can measure core-diameter ( 8 .2μm) and cladding-diameter ( 125μm). The measured NF pattern of the single-mode step fiber (SMF) is shown in Fig.2. The pattern is recorded by an infrared video camera. In order to avoid the effect of light scattering caused by the rough end surface, the end surface is carefully polished.By Pic To Data software,the optical image is transformed to optical intensity datum. Fig.3 shows optical intensity distribution from the fundamental mode of the azimuthally asymmetric SMF, which is proportional Fig.2.The measured pattern of a SMF to the brightness value of collected phase image.The sampling interval between data points is 0 .092μm. The intensity distribution is approximate Gaussian profile. It can be seen that we have got very smooth optical intensity profile.The RIP can be calculated by the TDIMM, defined by Eq.(5) from the measured intensity distribution. The result of the two-dimensional RIP of measured the azimuthally asymmetric SMF is shown in Fig.4. The total result gives a very good and visual RIP of the SMF. From the color bar of the profile, we can obtain the maximum value of the core refractive-index which is 1 .4733, and the minimum value of the core refractive index which is similar to the cladding refractive index 1.4682. The relative refractive-index is 0 .35% which is equal to specified values of the SMF. And the resolution of the refractive index change is lower than10 ? 4.Furthermore, as seen from the Fig.4, we can obtain the refractive-index of every point immediately. Because matlab possesses powerful functions, it can displays value of any point when we click that point. In this manner, we have gained the azimuthally asymmetric RIP of the SMF by the TDIMM from the NF measurement. From the above result, it can be seen that the RIP along the y-orientation and the one along the x-orientation are not coincident. Due to interior remnant stress asymmetry factor from industrial manufacture process, in which optical fiber is bent, twisted, vibrated and so on, the core takes on random asymmetry structure.In order to check the result, two pieces of refractive index profiles chosen from Fig.4 along the y-orientation and along the x-orientation are shown in Fig.5. The cross one and the circle one is from the experimental measurement along y-orientation and x-orientation respectively, and the dotted one is from commercial RIP specification. Of course, we can choose any piece of refractive index profile from Fig.4, and then compare the measured RIP with the commercial RIP specification. The maximum value and minimum value of two pieces of refractive-index profiles are 1.4733 and 1.4682, respectively. As seen from comparing the measured results with the commercial one, the cross one and the circle one are primely fit to the commercial one, and they decrease monotonically as the distance to the center increases, and approach a constant. Moreover, two pieces of refractive-index profiles display that the index variation of the SMF is very low.(B) Measure RI of Burial Channel Waveguide (BCW)In this manner, the measured NF pattern's optical intensity distribution of the burial channel waveguide's end surface is shown in Fig.6. By substituting the glass base's RI 1.5 and measured optical intensity into Eq.(5), we can obtainΔn ( x,y),that is, n ( x, y) =Δn( x,y) +nswith the maximum value 1.5078 and the minimum value 1.5. The RIP of BCW is shown in Fig.7 by matlab software.According to the characters of ion exchange, the RIP shows symmetric distribution along x-orientation; the ions do not purely act free diffusion movement along y-orientation, because of the drive of electric field, Ag⁺ remove into waveguide inside gradually ,that shows high ion density in the middle of waveguide and low one in both sides of waveguide,that is, high RI in the middle of waveguide and low one in both sides of waveguide。The reason why the RIP shows asymmetric property along x-orientation and y-orientation is that all kinds of parameters affect the results, including exposure time, exchange time, corroding time and so on. In order to show this result, one RIP along y-orientation is chosen in Fig.8.By now, the RIPs of fiber and channel waveguide have been calculated by the TDIMM.3. ConclusionIn this paper, the NF measurement, combined with the TDIMM, is applied to the two-dimensional RIP reconstruction for the asymmetric waveguide. The technique is fast, robust, highly accurate, and of high resolution, and simple in collecting and processing data. In contrast to traditional approaches, this proposed inverse method requires no prior information on the functional forms of the unknown quantities, no initial guesses and iterations in the calculating process in advance. Moreover, the inverse method also can be applied to the RIP measurement of the asymmetric waveguide such as graded index fibers, single-mode planar waveguides, polarization maintaining fibers and channel waveguides.