Table I
Calculated Complex Spatial Frequencies, Assuming a Correlation Function Composed of Harmonic Components with Real Spatial Frequencies; Calculated
${{\beta}^{\prime}}_{\mathit{\text{in}}}$ is a Measure of the Uncertainty in Calculated Attenuation Coefficients
Singleline fit
 Multipleline least squares fit


n 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$  Weight 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$  Weight 

1  800.0000  6.91 × 10^{−8}  0.9999992  800.0000  2.14 × 10^{−10}  1.000000 
2  760.0000  −1.14 × 10^{−7}  0.9999981  760.0000  2.01 × 10^{−10}  1.000000 
3  720.0000  −6.79 × 10^{−7}  0.9999984  720.0000  1.97 × 10^{−10}  1.000000 
4  680.0000  −1.57 × 10^{−6}  0.9999994  680.0000  2.09 × 10^{−10}  1.000000 
5  640.0000  −1.64 × 10^{−6}  0.9999996  640.0000  2.33 × 10^{−10}  1.000000 
6  600.0000  −7.09 × 10^{−7}  0.9999985  600.0000  2.34 × 10^{−10}  1.000000 
7  560.0000  −1.03 × 10^{−7}  0.9999980  560.0000  2.22 × 10^{−10}  1.000000 
8  520.0000  6.94 × 10^{−8}  0.9999980  520.0000  2.08 × 10^{−10}  1.000000 
9  480.0000  −1.69 × 10^{−7}  0.9999983  480.0000  1.98 × 10^{−10}  1.000000 
10  440.0000  −3.34 × 10^{−7}  1.0000020  440.0000  1.95 × 10^{−10}  1.000000 
Table II
Comparison Between Calculated and Actual Complex Spatial Frequencies for a Spectrum with Ten Lines; Latter were Arbitrarily Assumed and the Former Calculated by Analyzing the Spectrum
Calculated (least squares fit)
 Actual


Line 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$  Weight 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 

1  800.0000  2.04 × 10^{−10}  1.000000  800.0000  1.00 × 10^{−10} 
2  760.0000  6.01 × 10^{−10}  1.000000  760.0000  5.00 × 10^{−10} 
3  720.0000  1.09 × 10^{−9}  1.000000  720.0000  1.00 × 10^{−9} 
4  680.0000  5.10 × 10^{−9}  1.000000  680.0000  5.00 × 10^{−9} 
5  640.0000  1.01 × 10^{−8}  1.000000  640.0000  1.00 × 10^{−8} 
6  600.0000  5.01 × 10^{−8}  1.000000  600.0000  5.00 × 10^{−8} 
7  560.0000  1.00 × 10^{−6}  1.000000  560.0000  1.00 × 10^{−6} 
8  520.0000  5.00 × 10^{−6}  1.000000  520.0000  5.00 × 10^{−6} 
9  480.0000  1.00 × 10^{−5}  1.000000  480.0000  1.00 × 10^{−5} 
10  440.0000  5.00 × 10^{−5}  1.000000  440.0000  5.00 × 10^{−5} 
Table III
Null Test of Computed Mode Amplitude Attenuation Coefficients for Bound Modes;
${{\beta}^{\prime}}_{\mathit{\text{in}}}$ Value Is a Measure of the Uncertainty in the Computed Mode Attenuation Coefficient in the Presence of Absorbers
Mode (n) 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 

0  730.621  2.45 × 10^{−11} 
2  642.894  2.60 × 10^{−11} 
4  558.536  2.67 × 10^{−11} 
6  475.701  2.77 × 10^{−11} 
8  393.874  2.83 × 10^{−11} 
10  312.804  2.86 × 10^{−11} 
12  232.341  3.14 × 10^{−11} 
14  152.423  3.64 × 10^{−11} 
16  73.463  4.58 × 10^{−11} 
18  2.476  2.92 × 10^{−8} 
Table IV
Mode Amplitude Attenuation Coefficients
${{\beta}^{\prime}}_{\mathit{\text{in}}}$ as a Function of Cladding Absorption Coefficient
Mode (n)  β(cm^{−1})  Cladding absorption coefficient (cm^{−1})


10^{−3}  10^{−2}  10^{−1}  10^{0}  10^{1} 

8  393.875  —  3.149 × 10^{−8}  3.479 × 10^{−7}  3.513 × 10^{−6}  3.516 × 10^{−5} 
10  312.804  6.553 × 10^{−8}  7.867 × 10^{−7}  7.998 × 10^{−6}  8.011 × 10^{−5}  8.012 × 10^{−4} 
12  232.341  1.036 × 10^{−6}  1.036 × 10^{−5}  1.036 × 10^{−4}  1.036 × 10^{−3}  1.036 × 10^{−2} 
14  152.423  7.930 × 10^{−6}  8.057 × 10^{−5}  8.070 × 10^{−4}  8.071 × 10^{−3}  8.069 × 10^{−2} 
16  73.463  4.105 × 10^{−5}  4.127 × 10^{−4}  4.127 × 10^{−3}  4.127 × 10^{−2}  4.124 × 10^{−1} 
18  2.476  2.646 × 10^{−4}  2.654 × 10^{−3}  2.654 × 10^{−2}  2.654 × 10^{−1}  2.609 
Table V
Comparison of Mode Amplitude Attenuation Coefficients
${{\beta}^{\prime}}_{\mathit{\text{in}}}$ Computed as 〈nγn〉 with Correct Eigenfunctions and Those Computed with a Global Least Square LineShape Fit (Cladding is Assumed to have an Absorption Coefficient of 1 cm^{−1})
Mode (n)  〈nγn〉 (cm^{−1})  Lineshape fit (global least square)(cm^{−1}) 

8  3.5162 × 10^{−6}  3.5125 × 10^{−6} 
10  8.0127 × 10^{−5}  8.0112 × 10^{−5} 
12  1.0364 × 10^{−3}  1.0364 × 10^{−3} 
14  8.0717 × 10^{−3}  8.0715 × 10^{−3} 
16  4.1271 × 10^{−2}  4.1271 × 10^{−2} 
18  2.6538 × 10^{−1}  2.6537 × 10^{−1} 
Table VI
Mode Amplitude Attenuation Coefficients
${{\beta}^{\prime}}_{\mathit{\text{in}}}$ in cm^{−1} as a Function of Jacket Absorption Coefficient
Mode (n)  γ_{J} = 30 cm^{−1}  γ_{J} = 100 cm^{−1}  γ_{J} = 500 cm^{−1} 

Line shape  〈nγn〉  Line shape  〈nγn〉  Line shape  〈nγn〉 

16  3.37 × 10^{−9}  1.15 × 10^{−10}  2.07 × 10^{−10}  3.49 × 10^{−10}  4.06 × 10^{−9}  1.17 × 10^{−9} 
18  2.39 × 10^{−2}  2.37 × 10^{−2}  1.98 × 10^{−2}  1.94 × 10^{−2}  1.28 × 10^{−2}  1.16 × 10^{−2} 
Table VII
Null Test for Circularly Symmetric Finite Parabolic Profile with ν = 0 Modes Excited by Input Gaussian Beam
Mode (μ)  Weight 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 
${{\beta}^{\prime}}_{in}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$ 

0  2.071 × 10^{2}  797.19  4.749 × 10^{−11} 
1  8.292 × 10^{1}  716.23  4.868 × 10^{−11} 
2  3.320 × 10^{1}  635.27  4.842 × 10^{−11} 
3  1.329 × 10^{1}  554.32  5.186 × 10^{−11} 
4  5.320 × 10^{0}  473.37  5.000 × 10^{−11} 
5  2.130 × 10^{0}  392.41  5.108 × 10^{−11} 
6  8.528 × 10^{−1}  311.46  1.119 × 10^{−8} 
7  3.419 × 10^{−1}  230.51  2.371 × 10^{−9} 
8  1.383 × 10^{−1}  149.64  3.421 × 10^{−6} 
9  5.844 × 10^{−2}  69.52  3.563 × 10^{−4} 
Table VIII
Mode Amplitude Attenuation Coefficients
${{\beta}^{\prime}}_{\mathit{\text{in}}}\phantom{\rule{0.2em}{0ex}}\left({\mathbf{\text{cm}}}^{1}\right)$ for Circularly Symmetric α = 1.85 Profile and γ_{J} = 30 cm^{−1}
n  μν 
${{\beta}^{\prime}}_{rn}\phantom{\rule{0.2em}{0ex}}\left({\text{cm}}^{1}\right)$  〈nγn〉  Line shape 

15  70  121.520  7.4739 × 10^{−9}  — 
16  71  82.058  3.853 × 10^{−7}  — 
17  80  43.649  4.559 × 10^{−5}  — 
18  81  8.514  2.686 × 10^{−2}  2.809 × 10^{−2} 