John Elie Sader, "Analysis of arbitrarily perturbed circular profiles by implementation of integral-equation theory," J. Opt. Soc. Am. A 7, 2094-2099 (1990)
Perturbation techniques have been limited until now to the approximate analysis of profiles that could be described in terms of small perturbations on a given profile with a known solution. A method of analysis is presented, incorporating perturbation, Green’s function, and integral-equation techniques in the analysis of circular profiles defined by arbitrary perturbations and resulting in extremely high accuracy (<10−5%).
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Comparison of the Normalized Propagation Coefficient U Obtained by the Exact Analytical Solution with That Obtained by the New Analysis for the W Profile with Fiber Parametersa
s
V
A
Uint
Uexact
% error in U
1.5
2
0.1
1.55177070
1.55177070
<10−5
1.5
2
1
1.71225041
1.71225041
<10−5
1.5
3
0.1
1.79366360
1.79366360
<10−5
1.5
3
1
1.92769077
1.92769077
<10−5
2
2
0.1
1.55796763
1.55796763
<10−5
2
2
1
1.73658613
1.73658613
<10−5
2
3
0.1
1.79541415
1.79541415
<10−5
2
3
1
1.93126444
1.93126443
<10−5
A = (ncl2 − nd2)/(nco2 − ncl2). The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Errors less than 10−5% have not been shown but are indicated by <10−5.
Table 2
Comparison of Field Results for the W Profile Obtained from the Exact Analytical Expression and the New Analysis for V = 2, A = 0.1, and s = 1.5a
R
Eint(R)
Eexact(R)
% error in E(R)
0
1.00000000
1.00000000
0
0.5
8.55069203 × 10−1
8.55069203 × 10−1
<10−5
1.0
4.82764815 × 10−1
4.82764815 × 10−1
<10−5
1.3
2.86267235 × 10−1
2.86267235 × 10−1
<10−5
1.5
2.07078237 × 10−1
2.07078237 × 10−1
<10−5
2.0
9.65663850 × 10−2
9.65663849 × 10−2
<10−5
3.0
2.26135367 × 10−2
2.26135366 × 10−2
<10−5
4.0
5.58332921 × 10−3
5.58332919 × 10−3
<10−5
Fiber parameters are as described in Section 3, and A = (ncl2 − nd2)/(nco2 − ncl2). Errors less than 10−5% have not been shown but are indicated by <10−5. The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Note that the error at R = 0 is zero since the field results obtained were normalized to unity at R = 0.
Table 3
Comparison of Field Results for the W Profile Obtained from the Exact Analytical Expression and the New Analysis for V = 2, A = 1, and s = 2a
R
Eint(R)
Eexact(R)
% error in E(R)
0
1.00000000
1.00000000
0
0.5
8.20214442 × 10−1
8.20214442 × 10−1
<10−5
1.0
3.76811128 × 10−1
3.76811128 × 10−1
<10−5
1.3
1.73004579 × 10−1
1.73004579 × 10−1
<10−5
1.5
1.06077928 × 10−1
1.06077928 × 10−1
<10−5
2.0
4.04837420 × 10−2
4.04837420 × 10−2
<10−5
3.0
1.24472642 × 10−2
1.24472642 × 10−2
<10−5
4.0
4.03046007 × 10−3
4.03046006 × 10−3
<10−5
Fiber parameters are as described in Section 3, and A = (ncl2 − nd2)/(nco2 − ncl2). Errors less than 10−5% have not been shown but are indicated by <10−5. The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Note that the error at R = 0 is zero since the field results obtained were normalized to unity at R = 0.
Tables (3)
Table 1
Comparison of the Normalized Propagation Coefficient U Obtained by the Exact Analytical Solution with That Obtained by the New Analysis for the W Profile with Fiber Parametersa
s
V
A
Uint
Uexact
% error in U
1.5
2
0.1
1.55177070
1.55177070
<10−5
1.5
2
1
1.71225041
1.71225041
<10−5
1.5
3
0.1
1.79366360
1.79366360
<10−5
1.5
3
1
1.92769077
1.92769077
<10−5
2
2
0.1
1.55796763
1.55796763
<10−5
2
2
1
1.73658613
1.73658613
<10−5
2
3
0.1
1.79541415
1.79541415
<10−5
2
3
1
1.93126444
1.93126443
<10−5
A = (ncl2 − nd2)/(nco2 − ncl2). The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Errors less than 10−5% have not been shown but are indicated by <10−5.
Table 2
Comparison of Field Results for the W Profile Obtained from the Exact Analytical Expression and the New Analysis for V = 2, A = 0.1, and s = 1.5a
R
Eint(R)
Eexact(R)
% error in E(R)
0
1.00000000
1.00000000
0
0.5
8.55069203 × 10−1
8.55069203 × 10−1
<10−5
1.0
4.82764815 × 10−1
4.82764815 × 10−1
<10−5
1.3
2.86267235 × 10−1
2.86267235 × 10−1
<10−5
1.5
2.07078237 × 10−1
2.07078237 × 10−1
<10−5
2.0
9.65663850 × 10−2
9.65663849 × 10−2
<10−5
3.0
2.26135367 × 10−2
2.26135366 × 10−2
<10−5
4.0
5.58332921 × 10−3
5.58332919 × 10−3
<10−5
Fiber parameters are as described in Section 3, and A = (ncl2 − nd2)/(nco2 − ncl2). Errors less than 10−5% have not been shown but are indicated by <10−5. The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Note that the error at R = 0 is zero since the field results obtained were normalized to unity at R = 0.
Table 3
Comparison of Field Results for the W Profile Obtained from the Exact Analytical Expression and the New Analysis for V = 2, A = 1, and s = 2a
R
Eint(R)
Eexact(R)
% error in E(R)
0
1.00000000
1.00000000
0
0.5
8.20214442 × 10−1
8.20214442 × 10−1
<10−5
1.0
3.76811128 × 10−1
3.76811128 × 10−1
<10−5
1.3
1.73004579 × 10−1
1.73004579 × 10−1
<10−5
1.5
1.06077928 × 10−1
1.06077928 × 10−1
<10−5
2.0
4.04837420 × 10−2
4.04837420 × 10−2
<10−5
3.0
1.24472642 × 10−2
1.24472642 × 10−2
<10−5
4.0
4.03046007 × 10−3
4.03046006 × 10−3
<10−5
Fiber parameters are as described in Section 3, and A = (ncl2 − nd2)/(nco2 − ncl2). Errors less than 10−5% have not been shown but are indicated by <10−5. The subscripts int and exact refer to results obtained by implementation of the analysis presented in this paper and that of the implementation of the exact analytical expression, respectively. Note that the error at R = 0 is zero since the field results obtained were normalized to unity at R = 0.
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