Table I
Comparison of the gn Coefficients for a Beam Waist Center Location
| A | B | C | D | E | F | G |
---|
g1 | 0.995 | 0.995 | 0.997 | 1 | 1 | 1 | 0.99943 |
g2 | 0.996 | 0.995 | 0.993 | 0.99898 | 0.99949 | 1 | 0.99841 |
g3 | – | – | 0.992 | 0.99746 | 0.99746 | 0.99746 | 0.99690 |
g4 | – | – | 0.989 | 0.99544 | 0.99595 | 0.99645 | 0.99488 |
g5 | 0.993 | 0.992 | 0.985 | 0.99292 | 0.99292 | 0.99292 | 0.99236 |
g6 | 0.989 | 0.989 | 0.982 | 0.98991 | 0.99041 | 0.99091 | 0.98935 |
g10 | 0.788 | 0.787 | – | 0.79030 | 0.79070 | 0.79101 | 0.79006 |
g45 | 0.585 | 0.589 | – | 0.59191 | 0.59191 | 0.59191 | 0.59191 |
g60 | 0.394 | 0.394 | 0.398 | 0.39555 | 0.39575 | 0.39577 | 0.39567 |
g80 | 0.192 | 0.193 | – | 0.19358 | 0.19368 | 0.19361 | 0.19369 |
g100 | 0.077 | 0.077 | – | 0.07739 | 0.07743 | 0.07737 | 0.07742 |
Note: The beam diameter is 2w
0 = 20 λ, s = 0.015915494. The wavelength is 0.5
μm. Columns: A quadratures, order L; B quadratures, order L
−; C quadratures, order L
− (approximation Q = 1/i); D finite series, order L; E finite series, order L
−; F finite series, order L
− (centered location); G localized interpretation.
Table II
Comparison of the gn Coefficients for a Beam Waist Center Location
| A | D | E | F | G |
---|
g1 | – | 1 | 1 | 1 | 0.999994 |
g2 | – | 0.999990 | 0.999995 | 1 | 0.999984 |
g3 | – | 0.999975 | 0.999975 | 0.999975 | 0.999969 |
g4 | – | 0.999954 | 0.999959 | 0.999965 | 0.999949 |
g30 | – | 0.997652 | 0.997657 | 0.997662 | 0.997646 |
g60 | 0.956 | 0.990777 | 0.990782 | 0.990787 | 0.990771 |
g80 | – | 0.983725 | 0.983730 | 0.983735 | 0.983719 |
g100 | – | 0.974745 | 0.974750 | 0.974755 | 0.974740 |
Note: The beam diameter is 2w
0 = 200 λ, s = 0.0015915494. The wavelength is 0.5
μm. Columns A–G are the same as in
Table I.
Table III
Comparison of the gn Coefficients for a Beam Waist Center Location
| A | D | E | F | G | | |
---|
g1 | – | 1 | 1 | 1 | 0.94465 | | p = 0 |
g2 | – | 0.898678 | 0.949339 | 1 | 0.85358 | |
g3 | – | 0.746697 | 0.746697 | 0.746697 | 0.73322 | | p = 1 |
g4 | – | 0.597951 | 0.630646 | 0.645375 | 0.59873 | |
g5 | – | 0.452445 | 0.452445 | 0.452445 | 0.46475 | | p = 2 |
g6 | 0.330 | 0.327974 | 0.345314 | 0.342192 | 0.34293 | |
g7 | – | 0.227077 | 0.227077 | 0.227077 | 0.24054 | | p = 3 |
g8 | – | 0.150625 | 0.158241 | 0.150333 | 0.16039 | |
g21 | – | 0.468744 | 0.468744 | 0.468744 | 0.82201 | 10−6 | p = 10 |
g22 | – | 0.113411 | 0.170189 | 0.120770 | 0.26967 | 10−6 |
g23 | – | −2.086304 | −2.086304 | −2.086304 | 0.84101 | 10−7 | p = 11 |
g24 | – | −0.264719 | −0.513389 | −4.90853 | 0.24932 | 10−7 |
g25 | – | 10.09847 | 10.09847 | 10.09847 | 0.70262 | 10−8 | p = 12 |
g26 | – | 0.144869 | 1.332550 | 23.43804 | 0.18823 | 10−8 |
Note: The beam diameter is 2w
0 = 2λ, s = 0.15915494. The wavelength is 0.5
μm. Columns A–G are the same as in
Table I.
Table IV
Comparison of the Incident Electric Field Computed Either with gn Coefficients (Right-Hand Side) or Without (Left-Hand Side)
Kr | r.h.s. of Eq 18 | 1.h.s. of Eq 18 |
---|
|
---|
1 | 0.974987 | 0.974987 |
2 | 1.8072855 | 1.8072855 |
10 | 0.7941804 | 0.7941805 |
20 | 0.000795622 | 0.000795622 |
Note: The wavelength is 0.5
μm, z
0 = 0,
θ = 90°, and w
0 = λ.
Table V
Study of the n Domain Where the Finite Series and the Localized Interpretation are Identical
| n even | | n odd | |
---|
|
---|
s | L | L− | F.C. | L | L− | F.C |
---|
|
---|
s0 | 1 | 1 | 1 | 1 | 1 | 1 |
s0/2 | 10 | 1 | 1 | 1 | 1 | 1 |
s0/4 | 28 | 52 | 58 | 53 | 53 | 53 |
s0/8 | 80 | 130 | 136 | 129 | 129 | 129 |
s0/16 | 200 | 276 | 342 | 329 | 329 | 329 |
s0/32 | 468 | 536 | 816 | 823 | 823 | 823 |
s0/64 | 1024 | 1068 | 1670 | 1671 | 1671 | 1671 |
Note: The wavelength is 0.5
μm. Columns: L, order L; L
−, order L
−; F.C., Ferguson and Currie.
Table VI
Comparison of the gn Coefficients for a Location Along the Beam Axis
| A | D | E | G |
---|
g1 | (0.277300, 0.345338) | (0.280006, 0.348707) | (0.280006, 0.348707) | (0.279888, 0.348741) |
g2 | (0.274657, 0.342330) | (0.279808, 0.348750) | (0.279907, 0.348728) | (0.279684, 0.348784) |
g3 | – | (0.279511, 0.348814) | (0.279511, 0.348814) | (0.279387, 0.348848) |
g4 | – | (0.279115, 0.348899) | (0.279214, 0.348878) | (0.278990, 0.348933) |
g30 | – | (0.234722, 0.356328) | (0.234818, 0.356317) | (0.234610, 0.356326) |
g43 | – | (0.181297, 0.359650) | (0.181297, 0.359649) | (0.181207, 0.359610) |
g60 | (0.99474 10−1, 0.320927) | (0. 111850, 0.354374) | (0.111933, 0.354387) | (0.111799, 0.354300) |
g80 | – | (0.639494 10−2, 0.322089) | (0.466088 10−2, 0.322121) | (0.641967 10−2,0.321998) |
g100 | – | (−0.912371 10−1, 0.252183) | (−0.911952 10−1, 0.252228) | (−0.911360 10−1,0.252132) |
Note. The beam diameter is 2w
0 = 20λ and the waist ordinate is z
0 = 10
−1π(
μm). The wavelength is 0.5
μm. Columns A–G are the same as in
Table I.
Table VII
Comparison of the gn Coefficients for a Location Along the Beam Axis
| D | E | G | |
---|
g1 | (0.908000, −0.289025) | (0.711599, −0.453018) | (0.866610, −0.260209) | |
g2 | (0.832928, −0.235844) | (0.696343, −0.420356) | (0.797050, −0.214093) | |
g3 | (0.720320, −0.156074) | (0.635318, −0.289705) | (0.701829, −0.155877) | |
g4 | (0.598260, −0.869411 10−1) | (0.586754, −0.213104) | (0.590465, −0.952384 10−1) | |
g5 | (0.466748, −0. 284455 10−1 | (0.485435, −0. 919815 10−1) | (0.473422, −0.411516 10−1) | |
g6 | (0.347808, −0.115944 10−1) | (0.407329, −0.236618 10−1) | (0.360594, 0.413758 10−1) | w0 |
g7 | (0.245946,0.325540 10−1) | (0.294773, 0.401488 10−1) | (0.259920, 0.270357 10−1) | |
g8 | (0.164718, 0.405967 10−1) | (0.216080, 0.673214 10−1) | (0.176476, 0.396080 10−1) | |
g9 | (0.104385, 0.389802 10−1) | (0.133797, 0.756275 10−1) | (0.112199,0.412818 10−1) | |
g10 | (0.623670 10−1,0.324501 10−1) | (0.827317 10−1, 0.702225 10−1) | (0.662695 10−1, 0.361157 10−1) | |
g11 | (0.366894 10−1,0.236073 10−1) | (0.405030 10−1, 0.562417 10−1) | (0.359451 10−1, 0.278998 10−1) | |
g12 | (0.202714 10−1, 0.168525 10−1) | (0.181172 10−1, 0.419980 10−1) | (0.175703 10−1, 0.194352 10−1) | 2w0 |
Note: The beam diameter is 2w
0 = 2λ and the waist ordinate is z
0 = 1
μm. The wavelength is 0.5
μm. Columns D–G are the same as in
Table I.
Table VIII
Comparison of the gn Coefficients for a Location Along the Beam Axis
| D | E | G |
---|
|
---|
g1 | (0.240798 10−1, −0.153297) | (0.240798 10−1, −0.153297) | (0.253834 10−1, −0.152871) |
g2 | (0.264021 10−1, −0.152549) | (0.252410 10−1, −0.152923) | (0.276870 10−1, −0.152086) |
g3 | (0.298856 10−1, −0.151427) | (0.298856 10−1, −0.151427) | (0.311085 10−1, −0.150847) |
g18 | (0.128509, −0.101413 10−1) | (0.128575, −0.111656 10−1) | (0.125475, −0.108190 10−1) |
g19 | (0.125886, +0.908621 10−2) | (0.125886, 0.908621 10−2) | (0.122822, 0.756862 10−1) |
g20 | (0.120158, +0.281921 10−1) | (0.120518, 0.272750 10−1) | (0.117261, 0.258397 10−1) |
g21 | (0.111131, +0.466865 10−1) | (0.111131, 0.466865 10−1) | (0.108703, 0.434121 10−1) |
Note: The beam diameter is 2w
0 = 2λ and the waist ordinate is z
0 = 10
μm. The wavelength is 0.5
μm. Columns D–G are the same as in
Table I.