Table 1
U2/V for LPμ,ν Modes of a Truncated Parabolic Profile with V = 12
| | | U2/V |
---|
| | |
|
---|
μ | ν | 2(2μ + ν + 1) | Evanescent-Wave Theory | Crone and Arnold |
---|
0 | 0 | 2 | 1.9999931 | 2.00014 |
0 | 1 | 4 | 3.999901 | 3.99990 |
0 | 2 | 6 | 5.999268 | 5.99909 |
1 | 0 | 6 | 5.998681 | 5.99942 |
0 | 3 | 8 | 7.996234 | 7.99579 |
1 | 1 | 8 | 7.991059 | 7.99148 |
0 | 4 | 10 | 9.984579 | 9.98352 |
1 | 2 | 10 | 9.956599 | 9.95703 |
2 | 0 | 10 | 9.942477 | 9.94051 |
0 | 5 | 12 | 11.94260 | 11.94051 |
1 | 3 | 12 | 11.81026 | 11.80771 |
2 | 1 | 12 | 11.69238 | 11.69315 |
Table 2
Cladding Influence on τμ,ν of Modes of Table 1
| | | τ(Relative) |
---|
| | |
|
---|
μ | ν | Evanescent-Wave Theory Δτ (nsec/km) | Crone and Arnold (nsec/km) | Evanescent-Wave Theory (nsec/km) |
---|
0 | 0 | − 7.129 × 10−4 | 0.026 | 0.029 |
0 | 1 | − 9.555 × 10−3 | 0.096 | 0.105 |
0 | 2 | − 6.661 × 10−2 | 0.174 | 0.194 |
1 | 0 | − 3.187 × 10−1 | 0.125 | 0.142 |
0 | 3 | − 3.255 × 10−1 | 0.104 | 0.140 |
1 | 1 | − 7.595 × 10−1 | − 0.328 | − 0.294 |
0 | 4 | − 1.294 | − 0.630 | − 0.563 |
1 | 2 | − 3.604 | − 2.902 | − 2.876 |
2 | 0 | − 4.759 | − 4.012 | − 4.034 |
0 | 5 | − 5.031 | − 4.126 | − 3.977 |
1 | 3 | − 18.131 | − 16.774 | − 17.409 |
2 | 1 | − 29.993 | − 28.597 | − 28.984 |
Table 3
β0,02, Δμ of Mode LP0,0 of Profile 1
| Δμ × 1010 | β002 | Δβ002 × 1012 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.248263022239 | 2.248263022240 | 1 | 3 |
Truncated | −4.4 | −4.4 | 2.248263022241 | 2.248263022242 |
Table 4
τ0,0 of Mode LP0,0 of Profile 1
| τ00 (μsec/km) | Δτ00 (nsec/km) |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Untruncated | 5.0035436223568 | 5.0035436223622 | −3.36 10−8 | −4.45 10−8 |
Truncated | 5.0035436223177 | 5.0035436223232 |
Table 5
β5,02, Δμ of Mode LP5,0 of Profile 1
| Δμ × 102 | β502 | Δβ502 × 104 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.2283577248493 | 2.2283580215575 | 2.247 | 2.254 |
Truncated | −5.12612055 | −5.12731527 | 2.228582718372 | 2.228583051824 |
Table 6
τ5,0 of Mode LP5,0 of Profile 1
| τ50 (μsec/km) | Δτ50 (nsec/km) |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Untruncated | 5.006388030635 | 5.006388752832 | −3.778 | −3.892 |
Truncated | 5.0024971 | 5.0026097 |
Table 7
β0,02, Δμ of Mode LP0,0 of Profile 3
| Δμ × 1012 | β002 |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.2475714411110 | 2.2475714411126 |
Truncated | −7.9 | −7.9 | 2.2475714411110 | 2.2475714411125 |
Table 8
τ0,0 of Mode LP0,0 of Profile 3
| τ00 (μsec/km) |
---|
|
|
---|
| Min | Max |
---|
Untruncated | 5.0034673072748 | 5.0034673072869 |
Truncated | 5.0034673072738 | 5.0034673072859 |
Table 9
β4,12, Δμ of Mode LP4,1 of Profile 3
| Δμ × 102 | β412 | Δβ412 × 104 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.2281502794 | 2.2281532874 | 1.340 | 1.404 |
Truncated | −3.706103241 | −3.720033539 | 2.2282873461 | 2.2282907046 |
Table 10
τ4,1 of Mode LP4,1 of Profile 3
| τ41 (μsec/km) | Δτ41 (nsec/km) |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Untruncated | 4.9997404646 | 4.9997640982 | −2.580 | −2.750 |
Truncated | 4.997014 | 4.997160 |
Table 11
β0,02, Δμ, τ0,0 of Mode LP0,0 of Profile 2
| Δμ × 108 | β002 | τ00 (μsec/km) |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.249063256039 | 2.249067265926 | 5.00371100127 | 5.00371769208 |
Truncated | −4.77 | −4.73 | 2.249063256137 | 2.249067266024 | 5.00371099935 | 5.00371769045 |
Table 12
β5,02, Δμ, τ5,0 of Mode LP5,0 of Profile 2
| Δμ × 102 | β502 | τ50 (μsec/km) |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 2.2290194768 | 2.2327874711 | 5.0097666 | 5.0130827 |
Truncated | −0.98029688 | −5.30444056 | 2.2290753316 | 2.2329899054 | 4.997281 | 5.012751 |
Table 13
β0,112, Δμ of Mode LP0,11 of Profile 1
| Δμr × 103 | Δμi × 1012 | β0,112,r | β0,112,i × 1014 | Δβ0,112,r × 105 |
---|
|
|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 0 | 0 | 2.2270333691 | 2.2270335716 | 3.13 | 3.14 | 1.27 | 1.31 |
Truncated | −2.9685468 | −2.9690811 | −5.1 | −5.1 | 2.2270462716 | 2.2270464739 |
Table 14
β0,172, Δμ of Mode LP0,17 of Profile 1
| Δμr × 102 | Δμi × 10 | β0,172,r | β0,172,i × 104 | Δβ0,172,r × 104 |
---|
|
|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |
---|
Untruncated | 0 | 0 | 0 | 0 | 2.2140726940 | 2.2140745145 | 5.68894 | 5.69660 | 2.425 | 2.460 |
Truncated | −4.95990346 | −4.9604860 | −1.158166199 | −1.59506547 | 2.2143169827 | 2.2143187385 |
Table 15
Comparison of WKB and Evanescent Wave Theory Results for the Imaginary Part of the Propagation Constants of Leaky Modes of Profile 1
| | | k × βμνi (μm−1) |
---|
| | |
|
---|
μ | ν | WKB | Evanescent-Wave Theory (Max) | Evanescent-Wave Theory (Min) |
---|
0 | 11 | 0.62981 × 10−13 | 0.55918 × 10−13 | 0.55638 × 10−13 |
0 | 12 | 0.28925 × 10−6 | 0.28339 × 10−6 | 0.28305 × 10−6 |
0 | 13 | 0.16560 × 10−4 | 0.16163 × 10−4 | 0.16145 × 10−4 |
0 | 14 | 0.11958 × 10−3 | 0.11675 × 10−3 | 0.11658 × 10−3 |
0 | 15 | 0.36401 × 10−3 | 0.37378 × 10−3 | 0.37333 × 10−3 |
0 | 16 | 0.68966 × 10−3 | 0.80585 × 10−3 | 0.80506 × 10−3 |
0 | 17 | 0.94378 × 10−3 | 0.13985 × 10−2 | 0.13966 × 10−2 |
1 | 9 | 0.19533 × 10−7 | 0.10317 × 10−7 | 0.10279 × 10−7 |
1 | 10 | 0.77207 × 10−4 | 0.53066 × 10−4 | 0.52971 × 10−4 |
1 | 11 | 0.62079 × 10−3 | 0.47953 × 10−3 | 0.47886 × 10−3 |
2 | 7 | 0.28469 × 10−4 | 0.71817 × 10−5 | 0.71604 × 10−5 |
3 | 5 | 0.86694 × 10−3 | ∞ | ∞ |
Table 16
β0,02, Δμ of Mode LP0,0 of Profile 1 with a Step at the Core–Cladding Interface
| Δμ × 1010 | β002 |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 2.248263022239 | 2.248263022240 |
Clad | 2.8 | 2.8 | 2.248263022238 | 2.248263022241 |
Table 17
β5,02, Δμ of Mode LP5,0 of Profile 1 with a Step at the Core–Cladding Interface
| Δμ × 103 | β502 | Δβ502 × 105 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 2.22835772485 | 2.22835802156 | 3.495 | 3.555 |
Clad | −8.0231013 | −8.0246972 | 2.22839296505 | 2.22839326715 |
Table 18
τ5,0 of Mode LP5,0 of Profile 1 with a Step at the Core–Cladding Interface
| τ50 (μsec/km) | Δτ50 (nsec/km) |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Unclad | 5.00638803064 | 5.00638875283 | −0.806 | −0.831 |
Clad | 5.005558 | 5.005582 |
Table 19
β0,112, Δμ of Mode LP0,11 of Profile 1 with a Step at the Core–Cladding Interface
| Δμ × 104 | β0,112 | Δβ0,112 × 106 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 2.227033369051 | 2.2270335716222 | 1.160 | 1.566 |
Clad | −3.032378 | −3.032952 | 2.227034732346 | 2.2270349350890 |
Table 20
τ0,11 of Mode LP0,11 of Profile 1 with a Step at the Core–Cladding Interface
| τ0,11 (μsec/km) | Δτ0,11 (nsec/km) |
---|
|
|
|
---|
| Min | Max | Min | Max |
---|
Unclad | 5.006070949859 | 5.006071481903 | −0.037 | −0.039 |
Clad | 5.006032 | 5.006034 |
Table 21
β0,02, Δμ of Mode LP0,0 of Profile 1 with a Valley at the Core–Cladding Interface
| Δμ × 109 | β002 | Δβ00 × 1012 |
---|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 2.248263022239 | 2.248263022240 | 4 | 6 |
Clad | −1.366 | −1.366 | 2.248263022244 | 2.248263022245 |
Table 22
β5,02, Δμ of Mode LP5,0 of Profile 1 with a Valley at the Core–Cladding Interface
| Δμr × 10 | Δμi × 10 | β5,02,r | β5,02,i × 104 | Δβ5,02,r × 104 |
---|
|
|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 0 | 0 | 2.2283577248 | 2.2283580216 | 9.75440 | 9.763191 | 6.987 | 6.999 |
Clad | −1.588035692 | −1.588734594 | −2.233159503 | −2.234390205 | 2.2290566591 | 2.2290575756 |
Table 23
β0,112, Δμ of Mode LP0,11 of Profile 1 with a Valley at the Core–Cladding Interface
| Δμr × 103 | Δμi × 104 | β0,112,r | β0,112,i × 107 | Δβ0,112,r × 105 |
---|
|
|
|
|
|
|
---|
| Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |
---|
Unclad | 0 | 0 | 0 | 0 | 2.2270333691 | 2.2270335716 | 8.34 | 8.34 | 3.528 | 3.572 |
Clad | −7.9877221 | −7.9897428 | −1.855671 | −1.857351 | 2.227068849 | 2.227069094 |