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在弱导阶跃折射率光纤中,光场既可以用矢量模描述也可以用标量模描述。要理清这两种描述方法的内在联系就需要先阐明矢量模和标量模之间的简并关系[29]。
事实上,TE0n模、TM0n模与LP1n模的纵向传播常数β是相同的。当m=0时,由矢量模的特征方程(4)得到描述TE0n模、TM0n模纵向传播常数的特征方程,如公式(17a)所示:
考虑到弱导近似,即n1~n2,上式化简为:
根据贝塞尔函数的性质:
公式(17b)可进一步化简为:
对比发现,公式(17e)与当m=1的标量模特征方程(12)是完全相同,也就是说,在弱导近似下,TE0n模、TM0n模的纵向传播常数与LP1n模的纵向传播常数是完全相同的。
事实上,当m>1时,EHm−1,n模与LPmn模的纵向传播常数β是相同的。EHm−1, n模的特征方程(5a)式在弱导近似下可化为:
根据贝塞尔函数的性质公式(18),上式可化为:
得到了LPmn模的特征方程。完全类似的方法可以证明,当m≥0时,HEm+1, n模与LPmn模的纵向传播常数β是相同的。图2描述的矢量模与图3描述的标量模的纵向传播常数的对比如表1所示。
Vevctor modes β/m−1 Linearly polarized modes β/m−1 HE11 8 569 293 LP01 8 569 293 TE01 8 568 310 LP11 8568310 HE21 8 568 310 TM01 8 568 310 EH11 8 567 033 LP21 8 567 039 HE31 8 567 033 HE12 8 566 606 LP02 8 566 608 EH21 8 565 493 LP31 8 565 502 HE41 8 565 493 TE02 8 564 673 LP12 8 564 687 HE22 8 564 672 TM02 8 564 672 EH31 856 3724 LP41 8 563 745 HE51 8 563 723 Table 1. Propagation constants of vector modes and linearly polarized modes
从表1中也可以看出,在弱导近似下,TE0n模、TM0n模与LP1n模的纵向传播常数近似相等,EHm−1, n模与LPmn模的纵向传播常数近似相等,HEm+1, n模与LPmn模的纵向传播常数近似相等。矢量模与标量模的纵向传播常数相同,意味着矢量模与标量模沿着光纤轴向的传输行为相同,在纤芯中横向的振荡形式相同,在包层中横向衰减的趋势相同。或者说,在弱导阶跃折射率光纤中,矢量模和标量模存在着深层次的内在联系。根据公式(1)~(8)可以求得矢量模o态和e态完整的电磁场分布,根据公式(9)~(16)可以求得标量模yo态、ye态、xo态和xe态完整的场分布,对比二者的场分布表达式,在能量归一条件下,可以推导得不同m值下标量模和矢量模之间的组合关系,如公式(20)所示:
公式(20)揭示了弱导光纤中矢量模与标量模之间的简并关系,即:HE1n(o)的场分布与LP0n(ye)相同,HE1n(e)的场分布与LP0n(xe)相同,如公式(20a)所示;LP1n(yo)模、LP1n(xe)模是HE2n(e)模和TM0n模的线性叠加,LP1n(ye)模、LP1n(xo)模是TE0n模和HE2n(o)模的线性叠加,如公式(20b)所示;当m>1时,LPmn(yo)模、LPmn(xe)模是EHm−1, n(e)模和HEm+1, n(e)的线性叠加,LPmn(ye)模、LPmn(xo)模是EHm−1, n(o)模和HEm+1, n(o)模的线性叠加,如公式(20c)所示。或者说,矢量模可以表示成为标量模的线性组合,标量模也可以表示成为矢量模的线性组合。图2描述的矢量模场分布和图3描述的标量模场分布也印证了公式(20)描述的简并关系。限于篇幅,在此不对公式(20)作详细的证明。
Study on mode degeneracy in weakly guiding step index fiber
doi: 10.3788/IRLA202049.0405001
- Received Date: 2019-12-15
- Rev Recd Date: 2020-01-20
- Publish Date: 2020-04-24
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Key words:
- vector mode /
- linearly polarized mode /
- mode degeneracy /
- weakly guiding step index fiber
Abstract: Mode is the fundamental concept in fiber optics, which is one of the most concerned issues in the fiber laser research. However, the mode degeneracy issue has not been clarified in common literature. By the classical electromagnetic theory, the degeneracy of vector mode and linearly polarized modes was analyzed and the relation between vector mode and linearly polarized mode were discussed in this paper. In the weakly guiding step index fiber, the vector mode and the linearly polarized mode were both the orthogonal complete basis for describing the optical field. The optical field described by vector mode [HE1n(o)、HE1n(e)] and linearly polarized mode [LP0n(ye)、LP0n(xe)] were 2 dimensional; the optical field described by vector mode [TE0n、HE2n(o)、HE2n(e)、TM0n]] and linearly polarized mode [LP1n(yo)、LP1n(ye)、LP1n(xo)、LP1n(xe)] were 4 dimensional; the optical field described by vector mode [EHm-1n(o), EHm-1n(e), HEm+1n(o), HEm+1n(e)] and linearly polarized mode [LPmn(yo), LPmn(ye), LPmn(xo), LPmn(xe)] were 4 dimensional.