The author is with the Institute of Microelectronics and Optical Optoelectronics, Warsaw University of Technology, Koszykowa 75, Warsaw 00-662, Poland.
A practical method for tracing rays in a gradient—index medium (electron rays in a potential field) is described. First, the Euler equation is transformed into a convenient form of four first-order differential equations. Next, a standard numerical technique is used for solving the resulting equations that contain at most two first-order spatial derivatives of a refractive index. A comparison of this method with previous methods is made. Some special distributions of the refractive index and the potential of a spherical field capacitor have been used as a verification of the proposed method.
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Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 3
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6 Δt l0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.63
4 × 10−8
3 × 10−7
<0.06
±10
0.62
7 × 10−8
5 × 10−7
<0.06
±30
0.54
2 × 10−7
2 × 10−6
<0.06
±45
0.44
4 × 10−7
8 × 10−6
<0.06
±60
0.31
6 × 10−7
4 × 10−5
0.20
<0.06
±5
0.50
2 × 10−8
9 × 10−8
<0.06
±10
0.50
3 × 10−8
3 × 10−7
<0.06
±30
0.44
8 × 10−8
9 × 10−7
<0.06
±45
0.36
10−7
3 × 10−6
<0.06
±60
0.25
2 × 10−7
2 × 10−5
0.10
<0.06
±5
0.25
9 × 10−10
6 × 10−9
<0.06
±10
0.25
2 × 10−9
10−8
<0.06
±30
0.22
5 × 10−9
5 × 10−8
<0.06
±45
0.18
8 × 10−9
2 × 10−7
<0.11
±60
0.13
10−8
10−6
0.05
<0.11
±5
0.13
5 × 10−11
4 × 10−10
<0.11
±10
0.12
10−10
8 × 10−10
<0.11
±30
0.11
3 × 10−10
3 × 10−9
<0.11
±45
0.09
5 × 10−10
10−8
<0.17
±60
0.06
9 × 10−10
6 × 10−8
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 4
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 5
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 1, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6 Δt l0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.62
4 × 10−8
3 × 10−7
<0.06
±10
0.61
7 × 10−8
6 × 10−7
<0.06
±30
0.54
2 × 10−7
2 × 10−6
<0.06
±45
0.44
4 × 10−7
9 × 10−6
<0.06
±60
0.31
6 × 10−7
5 × 10−5
0.20
<0.06
±5
0.49
2 × 10−8
2 × 10−7
<0.06
±10
0.49
3 × 10−8
3 × 10−7
<0.06
±30
0.43
8 × 10−8
10−6
<0.06
±45
0.35
10−7
3 × 10−6
<0.06
±60
0.25
2 × 10−7
2 × 10−5
0.10
<0.06
±5
0.25
9 × 10−10
9 × 10−9
<0.06
±10
0.24
2 × 10−9
2 × 10−8
<0.06
±30
0.21
5 × 10−9
6 × 10−8
<0.06
±45
0.18
9 × 10−9
2 × 10−7
<0.11
±60
0.12
10−8
10−6
0.05
<0.11
±5
0.12
5 × 10−11
5 × 10−10
<0.11
±10
0.12
10−10
9 × 10−10
<0.11
±30
0.11
3 × 10−10
4 × 10−9
<0.11
±45
0.09
5 × 10−10
10−8
<0.17
±60
0.06
9 × 10−10
7 × 10−8
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 6
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 1a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 7
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.1, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6Δtl0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.58
4 × 10−8
5 × 10−7
<0.06
±10
0.57
7 × 10−8
8 × 10−7
<0.06
±30
0.50
3 × 10−7
3 × 10−6
<0.06
±45
0.41
3 × 10−7
10−5
<0.06
±60
0.29
6 × 10−7
7 × 10−5
0.20
<0.06
±5
0.46
2 × 10−8
3 × 10−7
<0.06
±10
0.46
3 × 10−8
4 × 10−7
<0.06
±30
0.40
8 × 10−8
10−6
<0.06
±45
0.33
10−7
5 × 10−6
<0.06
±60
0.23
2 × 10−7
3 × 10−5
0.10
< 0.06
±5
0.23
10−9
10−8
<0.06
±10
0.23
2 × 10−9
2 × 10−8
<0.06
±30
0.20
5 × 10−9
7 × 10−8
<0.06
±45
0.16
8 × 10−9
3 × 10−7
<0.11
±60
0.12
10−8
2 × 10−6
0.05
<0.11
±5
0.12
6 × 10−11
8 × 10−10
<0.11
±10
0.11
10−10
10−9
<0.11
±30
0.10
3 × 10−10
5 × 10−9
<0.11
±45
0.08
6 × 10−10
2 × 10−8
<0.17
±60
0.06
9 × 10−9
10−7
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 8
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 2a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 9
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-386 computer with coprocessor 387 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 10
Position Errors a z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-386 computer with coprocessor 387 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 11
Fractional Position Errors of Typical Skew Rays in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2)
Fractional Error at z = 1
Δt
Three-Component Method
Four-Component Method
0.05
6 × 10−10
2 × 10−8
0.10
8 × 10−9
3 × 10−7
0.20
1 × 10−7
5 × 10−6
0.25
4 × 10−7
1 × 0−5
Tables (11)
Table 1
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.1, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6 Δt l0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.63
3 × 10−8
2 × 10−7
<0.06
±10
0.62
7 × 10−8
5 × 10−7
<0.06
±30
0.55
2 × 10−7
2 × 10−6
<0.06
±45
0.45
4 × 10−7
8 × 10−6
<0.06
±60
0.32
6 × 10−7
4 × 10−5
0.20
<0.06
±5
0.50
2 × 10−8
8 × 10−8
<0.06
±10
0.50
3 × 10−8
2 × 10−7
<0.06
±30
0.44
8 × 10−8
10−6
<0.06
±45
0.36
10−7
3 × 10−6
<0.06
±60
0.25
2 × 10−7
10−5
0.10
<0.06
±5
0.25
8 × 10−10
5 × 10−9
<0.06
±10
0.25
2 × 10−9
10−8
<0.06
±30
0.22
5 × 10−9
5 × 10−8
<0.06
±45
0.18
8 × 10−9
2 × 10−7
<0.11
±60
0.13
10−8
9 × 10−7
0.05
<0.11
±5
0.13
5 × 10−11
3 × 10−10
<0.11
±10
0.12
9 × 10−11
7 × 10−10
<0.11
±30
0.11
3 × 10−10
3 × 10−9
<0.11
±45
0.09
5 × 10−10
10−8
<0.17
±60
0.06
9 × 10−10
6 × 10−8
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in Turbo Pascal 5.0 language (on the extended variables).
Table 2
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.1a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 3
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6 Δt l0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.63
4 × 10−8
3 × 10−7
<0.06
±10
0.62
7 × 10−8
5 × 10−7
<0.06
±30
0.54
2 × 10−7
2 × 10−6
<0.06
±45
0.44
4 × 10−7
8 × 10−6
<0.06
±60
0.31
6 × 10−7
4 × 10−5
0.20
<0.06
±5
0.50
2 × 10−8
9 × 10−8
<0.06
±10
0.50
3 × 10−8
3 × 10−7
<0.06
±30
0.44
8 × 10−8
9 × 10−7
<0.06
±45
0.36
10−7
3 × 10−6
<0.06
±60
0.25
2 × 10−7
2 × 10−5
0.10
<0.06
±5
0.25
9 × 10−10
6 × 10−9
<0.06
±10
0.25
2 × 10−9
10−8
<0.06
±30
0.22
5 × 10−9
5 × 10−8
<0.06
±45
0.18
8 × 10−9
2 × 10−7
<0.11
±60
0.13
10−8
10−6
0.05
<0.11
±5
0.13
5 × 10−11
4 × 10−10
<0.11
±10
0.12
10−10
8 × 10−10
<0.11
±30
0.11
3 × 10−10
3 × 10−9
<0.11
±45
0.09
5 × 10−10
10−8
<0.17
±60
0.06
9 × 10−10
6 × 10−8
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 4
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 5
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 1, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6 Δt l0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.62
4 × 10−8
3 × 10−7
<0.06
±10
0.61
7 × 10−8
6 × 10−7
<0.06
±30
0.54
2 × 10−7
2 × 10−6
<0.06
±45
0.44
4 × 10−7
9 × 10−6
<0.06
±60
0.31
6 × 10−7
5 × 10−5
0.20
<0.06
±5
0.49
2 × 10−8
2 × 10−7
<0.06
±10
0.49
3 × 10−8
3 × 10−7
<0.06
±30
0.43
8 × 10−8
10−6
<0.06
±45
0.35
10−7
3 × 10−6
<0.06
±60
0.25
2 × 10−7
2 × 10−5
0.10
<0.06
±5
0.25
9 × 10−10
9 × 10−9
<0.06
±10
0.24
2 × 10−9
2 × 10−8
<0.06
±30
0.21
5 × 10−9
6 × 10−8
<0.06
±45
0.18
9 × 10−9
2 × 10−7
<0.11
±60
0.12
10−8
10−6
0.05
<0.11
±5
0.12
5 × 10−11
5 × 10−10
<0.11
±10
0.12
10−10
9 × 10−10
<0.11
±30
0.11
3 × 10−10
4 × 10−9
<0.11
±45
0.09
5 × 10−10
10−8
<0.17
±60
0.06
9 × 10−10
7 × 10−8
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 6
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 1a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 7
Position Errors at z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.1, h ≃ 1.6 Δt n0 cos ϑa
Δt
Calculation Time (s)
ϑ (deg)
h 1.6Δtl0
Three-Component Method
Four-Component Method
0.25
<0.06
±5
0.58
4 × 10−8
5 × 10−7
<0.06
±10
0.57
7 × 10−8
8 × 10−7
<0.06
±30
0.50
3 × 10−7
3 × 10−6
<0.06
±45
0.41
3 × 10−7
10−5
<0.06
±60
0.29
6 × 10−7
7 × 10−5
0.20
<0.06
±5
0.46
2 × 10−8
3 × 10−7
<0.06
±10
0.46
3 × 10−8
4 × 10−7
<0.06
±30
0.40
8 × 10−8
10−6
<0.06
±45
0.33
10−7
5 × 10−6
<0.06
±60
0.23
2 × 10−7
3 × 10−5
0.10
< 0.06
±5
0.23
10−9
10−8
<0.06
±10
0.23
2 × 10−9
2 × 10−8
<0.06
±30
0.20
5 × 10−9
7 × 10−8
<0.06
±45
0.16
8 × 10−9
3 × 10−7
<0.11
±60
0.12
10−8
2 × 10−6
0.05
<0.11
±5
0.12
6 × 10−11
8 × 10−10
<0.11
±10
0.11
10−10
10−9
<0.11
±30
0.10
3 × 10−10
5 × 10−9
<0.11
±45
0.08
6 × 10−10
2 × 10−8
<0.17
±60
0.06
9 × 10−9
10−7
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Table 8
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 2a
Numerical calculations were carried out on the IBM-286 computer with coprocessor 287 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 9
Position Errors at z = 1 in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-386 computer with coprocessor 387 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 10
Position Errors a z = 1 in a Medium with Refractive-index Distribution n2 = 2.5–0.1 (x2 + y2) at Starting Points z = 0, r = 0.5a
Numerical calculations were carried out on the IBM-386 computer with coprocessor 387 in turbo pascal 5.0 language (on the extended variables).
Asterisks in the second column indicate calculation times with the three-component method.
Table 11
Fractional Position Errors of Typical Skew Rays in a Medium with Refractive-Index Distribution n2 = 2.5–0.1 (x2 + y2)