| With the development of Integrated optics, widely used of optical network and optical communications,optical fibers and waveguides become more and more useful in practical application. It is essential to investigate their various characteristics (such as bandwidth, dispersion, transport conditions, etc.), and all of these characteristics are related to the refractive index (RI), so it is particularly important to focus on measuring of RI. In recent years, research teams at home and abroad have made a lot of breakthroughs and have proposed many valuable measurements and theories. However, the existing methods always assume optical fibers or waveguides with symmetrical structure, that bring in some drawbacks in reality. In order to solve this problem, we proposed using two-dimensional inverse matrix method in polar coordinates(TDIMMPC) to measuring the asymmetric refractive index profile (RIP) of optical waveguide.It is easy to understand the principle of the method ,and the operation is simple and fast, with low cost, and it is very easy to build the equipments , and moreover, it has high accuracy. Compared with traditional methods, this method does not either require iterative calculations or assume the initial approximation function of unknown quantity. It also can be solved linearly. This method avoided the error caused by the assumption that the material of medium is symmetry. Therefore it can be used to measure the RI of the asymmetric waveguide, such as the step fiber, polarization maintaining fiber, planar waveguide and strip waveguide. In addition, it can provide a intuitive, three-dimensional and multi-angle graphic.TDIMMPC is combined with the basic principles of near-field(NF) measurement, It has high quality requirements in near-field pattern of shooting to minimize the system error, that ensures the accuracy of the method. We use the concept of two-dimensional finite difference to give the treatment of two-dimensional Laplacian operator in the polar coordinate, to discrete wave equations, and avoided the calculation of two-dimensional wave equation. In addition, we solved the problem that the RI difference of waveguide region and the substrate is small. We transform the primary data from Cartesian coordinate into polar coordinates by using Interpolation method, which reduces the computational error in the process of consideration , accurately obtained the RIP of single-mode step-index fiber (SMF) and channel waveguide .This method can show the RI of any point of the cross section, so it can be used to correctly characterize waveguide with irregular RI variation over their cross sections.1. TheoryBecause of the small RI variation between the between guided wave area and substrate medium, we use NF method to obtain the transmitted optical intensity from the propagation mode near-field collection to get the distribution of the waveguide.The transverse electric or magnetic field: Where, I ( x , y )is the measured intensity profile and I max is the maximum of I ( x , y ). Maxwell's equations can be approximated to look for the scalar wave equation: k =2π/λis the wave number in vacuum with the light wavelengthλ, n ( x,y)is the core RI,βis the wave number in medium .We can obtain from(2)that:Δn ( x,y)the RI difference between guided wave area and substrate medium: Perfect square both sides of(4), ignoringΔn ( x,y)in high terms, there is By substituting(5)into(3),we have : (6)can be obtained as: By substituting(8)into(7), the matrix obtained as:Specific calculation process can be achieved by MATLAB programming. Where ns and k are constant, A is a matrix which is available. In order to obtain the RI distribution, what is equal to get the matrix Nj×1 , we use inverse matrix method to solve the problem. The two-dimensional finite-difference governing equation can be expressed as the following recursive form: 2. Experiment and analysesThe experimental setup used is shown in Fig.1.The experimental system can be applied to observe the near-field image of the waveguide, the wavelength of the diode laser is 1.55μm. Fig.1 Schematic diagram of the experimental setup1.He-Ne laser; 2. Lens; 3. Fiber; 4. 1.55μm semiconductor laser; 5. Couplers; 6. Measured waveguide; 7. Microscopes; 8. Infrared camera; 9. Imaging surveillance equipment; 10. camera controller; 11. computer; 12. plotters (A). Measure RI of SMF G.652The measured NF pattern of the single-mode step fiber (SMF) is shown in Fig.2. Fig.2 The measured pattern of a SMF Fig.3 The optical intensity distribution of SMF Fig.4 Two-dimensional refractive-index distribution of SMF Light intensity is proportional to the brightness of the image collection. Light intensity information of the spot Image will be transferred into data by Pic To DaTa software. on each pixel on the picture is converted into a light intensity data, that is, distance between adjacent points is 0 .092μm.As it shown in Fig. 3 ,the original intensity distribution is close to Gaussian distribution.Using the interpolation method to do the data processing, we transform the primary data from Cartesian coordinate into polar coordinates ,then substitute the result into equation (10), using MATLAB to calculate. 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.As it is shown in Fig.4, we can obtain that the minimum value of the core RI is similar to the cladding RI which is1.4682, and the maximum value of the core RI is 1. 4731, the resolution of the RI change is lower than 10?4 . And it also can be seen from Fig.4, that we can get the value of the RI at any point.Either take a RI distribution curve which is through the fiber core, compare with commercial RIP specification of the SMF G.652, the results shown in Fig.5.Dotted line is from commercial RIP specification, solid line represents the experimental RI distribution curve. It is clear that two lines are primely fit. The maximum value and minimum value of two pieces of RIF are 1.4733 and 1.4682.And as R increases, the RI decreases, and the more close to the edge the faster it reduces. Because of the leakage mode phenomena of the optical fiber, we can not obtain an ideal state of Fig.6 shows the change in the RI which means the situation that arbitrary take an R,θturns from ?πtoπ. We can see from Fig.6, optical fiber has a small heterogeneity. Therefore, in different directions of the same radius, RI has a certain bias, but the bias is very small, in magnitude of 10?5 ,therefore it is not significant. This may be caused by the production process, such as being bent, being distorted, residual stress caused by vibration.(B) Measure RI of Burial Channel Waveguide (BCW) Ibid, the measured NF pattern of the BCW is shown in Fig.7. Transfer light intensity information of the spot Image into data.Transform the primary data from Cartesian coordinate into polar coordinates by using Interpolation calculation, then substitute the result into equation (10), using MATLAB to calculate. We can obtain the Two-dimensional refractive-index distribution of BCW ,as it is shown in Fig.9. It is easy to see that the minimum value of the core RI is similar to the cladding RI which is1.5048, and the maximum value of the core RI is 1.502. RI of the BCW is higher in the middle of waveguide and lower in the edge. It depends on the characters of ion exchange. In the process of BCW Fabricating, because of the drive of electric field, Ag+ remove into waveguide inside gradually, that leads to the phenomenon that RI of the BCW is higher in the middle of waveguide and lower in the edge. And RI here is asymmetric, because during the Fabrication, all kinds of parameters affect the results, including the thickness of photoresist, exposure time, ion-exchange time, the proportion of corrosive ions and so on. These factors caused an increase of asymmetric of the waveguide. As it is shown in Fig.10, we arbitrary take an R,θturns from ?πtoπ,there is a change in the RI. Compare with Fig.6,we know that the variation of N significantly increased, to 10?4 ,it shows that BCW is more asymmetric than SMF.So far, we finished the construction of RI profile of optical waveguide by two-dimensional inverse matrix method in polar coordinates.3. ConclusionThe proposed method here to construction RI of optical waveguides is easy to understand, and the operation is simple and fast, with low cost, and it also very easy to build the equipments , moreover, it has high accuracy. This method does not either require iterative calculations or assume the initial approximation function of unknown quantity. It also can be solved linearly., and does not require the sample length, what is more, it is a non-destructive method. It is not required to assume that medium is symmetric, so it can be used to measure the RI of the asymmetric waveguide. In addition, it can provide the pattern what is more convenient to read the RI. Therefore this method has great value in the practical application. |
PDF Full Text Request