## 3 Results
### Basic Characteristics
#### 3.1.2 Mode Pattern  
We have previously studied the relations among LP modes, depolarizations, and imaging in SMMF [28]. The refractive index profile \(n_{2}(r)\) of the SMMF is represented by:  
\[n_{2}(r)=n_{20}\bigg{\{}1-2\bigg{(}\frac{r}{a_{n}}\bigg{)}^{p}\Delta\bigg{\}} ^{1/2},r\leq a_{n} \tag{1}\]  
where \(a_{n}\) is the core radius, \(p\) is the grade profile parameter, and \(\Delta\) is the core-cladding index contrast. The value of \(p\) is ~2. The refractive index \(n_{20}\) is calculated using the Sellmeier equation [43]. In weakly guiding fibers, the LP(\(l\), \(m\)) mode denoted as the LP\({}_{l\,m}\) mode is usually used [45; 46]. By solving the basic wave propagation equations in a graded-index fiber using Equation (1), the electric field in the polar coordinate (\(r\), \(\theta\)) is:  
\[E_{lm}(r,\theta)=C_{N}\cdot R_{lm}(r)\sin(l\theta+\theta_{0}), \tag{2}\]  
\[R_{lm}(r)=r^{l}\exp\bigg{(}-\frac{n_{1}k_{0}}{2a_{n}}\sqrt{2\Delta}r^{2}\bigg{)} L_{m-1}^{(l)}\bigg{(}\frac{n_{1}k_{0}}{a_{n}}\sqrt{2\Delta}r^{2}\bigg{)} \tag{3}\]  
where \(l\) and \(m\) are integers, \(C_{N}\) is the normalization constant, \(\theta_{0}\) is the initial phase, \(L_{m-1}^{(l)}\) represents Laguerre polynomials, and \(k_{0}\) is the wave number. When the linearly polarized light (horizontal) is incident into the SMMF and the propagated light is reflected at the facet of the SMMF through the exit polarizer pattern, images can be measured depending on the angles of the exit polarizer owing to propagation modes and depolarization characteristics of the SMMF. The exit image through the polarizer (vertical) is given by:  
\[I_{y}=\Big{[}{A_{01y}}^{2}R_{01}(r)^{2}+{A_{11y}}^{2}R_{11}(r)^{2}\{1-\cos(2 \theta)\}+{A_{21y}}^{2}R_{21}(r)^{2}\{1-\cos(4\theta)\}\Big{]}\delta, \tag{4}\]  
where \(A_{01}\), \(A_{11}\), and \(A_{21}\) are the coefficients for LP\({}_{01}\), LP\({}_{11}\), L\({}_{21}\) modes, respectively, \(\theta\) is the angle with respect to the vertical axis, and \(\delta\) is the depolarization ratio in the core. Comparing the measured profiles with the simulated ones, the coefficients \(A_{01}\):\(A_{11}\):\(A_{21}\) are obtained as 1:0.447:0.837. Figure 2: (a) The right half side of an image is an optical coherence microscopy (OCM) image of metal pattern; (b) the intensity profile indicated by the yellow solid line in (a). Using the sample mirror, the OCM image was obtained as shown in Figure 3a. The OCM image shows the \"dark cross\" owing to the combinations of three LP modes and residual concentric fringes. The profiles at \(\theta=0\) and \(3\pi/4\) are shown in Figure 3c. The asymmetry in the intensity profiles is due to subtle misalignments of the sample mirror and there are dips around the center in the intensity profiles. We simulated the OCM image using Equations (2) and (3). As the OCM image is the interference image with the signal wave and the reference wave, assuming that the reference wave is the plane wave, the OCM image is proportional to the absolute value of the signal wave. The absolute electric fields of the signal light were simulated with the coefficients of \(A_{01}\):\(A_{11}\):\(A_{21}\) of 1:0.447:0.837 as shown in Figure 3b. The profiles at \(\theta=0\) and \(3\pi/4\) are also superimposed in Figure 3c. The measured signal intensity around the center is relatively lower than the simulated ones. As it is known that the distribution of refractive index tends to have local fluctuations in the core and small ripples in the central region of the fiber owing to the fabrication process [47], the signal lights were disturbed to decrease the interference signal and finally, the signal intensity of the OCM image decreased around the center. Consequently, the measured intensity profiles are relatively more deviated than the simulated ones owing to the decrease in the signal intensity around the center. We also obtained the OCM images of the polished aluminum sample 30 times by changing the positions randomly. A typical OCM image is shown in Figure 4a in which vertical scratches can be observed.