Table I
Calculated Complex Spatial Frequencies, Assuming a Correlation Function Composed of Harmonic Components with Real Spatial Frequencies; Calculated
is a Measure of the Uncertainty in Calculated Attenuation Coefficients
Single-line fit
| Multiple-line least squares fit
|
---|
n |
|
| Weight |
|
| 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 |
|
| Weight |
|
|
---|
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;
Value Is a Measure of the Uncertainty in the Computed Mode Attenuation Coefficient in the Presence of Absorbers
Mode (n) |
|
|
---|
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
as a Function of Cladding Absorption Coefficient
Mode (n) | β(cm−1) | Cladding absorption coefficient (cm−1)
|
---|
10−3 | 10−2 | 10−1 | 100 | 101 |
---|
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
Computed as 〈n|γ|n〉 with Correct Eigenfunctions and Those Computed with a Global Least Square Line-Shape Fit (Cladding is Assumed to have an Absorption Coefficient of 1 cm−1)
Mode (n) | 〈n|γ|n〉 (cm−1) | Line-shape 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
in cm−1 as a Function of Jacket Absorption Coefficient
Mode (n) | γJ = 30 cm−1 | γJ = 100 cm−1 | γJ = 500 cm−1 |
---|
Line shape | 〈n|γ|n〉 | Line shape | 〈n|γ|n〉 | Line shape | 〈n|γ|n〉 |
---|
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 |
|
|
---|
0 | 2.071 × 102 | 797.19 | 4.749 × 10−11 |
1 | 8.292 × 101 | 716.23 | 4.868 × 10−11 |
2 | 3.320 × 101 | 635.27 | 4.842 × 10−11 |
3 | 1.329 × 101 | 554.32 | 5.186 × 10−11 |
4 | 5.320 × 100 | 473.37 | 5.000 × 10−11 |
5 | 2.130 × 100 | 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
for Circularly Symmetric α = 1.85 Profile and γJ = 30 cm−1
n | μν |
| 〈n|γ|n〉 | 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 |