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Approximate Values for the Coefficients a, of the R/T Polynomial for a Nine-Layer Stacka
u1
u2
u3
u4
u5
u6 = u4
u7 = u3
u8 = u2
u9 = u1
a1
−2
−2
−2
−2
−2
−2
−2
−2
−2
a3
−56
−28
−8
4
8
4
−8
−28
−56
a5
−140
0
20
0
−12
0
20
0
−140
a7
−56
28
−8
−4
8
−4
−8
28
−56
a9
−2
2
−2
2
−2
2
−2
2
−2
Each layer has the same optical thickness.
Table II
Designs of Multilayers That Have a Substantial Reflectance at λ0 of 2.2 μm and a Relatively Low Reflectance at λ0/4 and at λ0/5a
Design
OK
Layer
Suppressed
β (deg)
Index
Rmax
m
ABCCCCBA
8
345
67.5
1.861
0.979
1
ABBCCBBA
No
8
345
67.5
—
—
—
AABCCBAA
8
345
67.5
1.705
0.977
1
ABCCCCCBA
9
45
80
1.926
0.970
1
ABBBCBBBA
9
45
80
2.016
0.907
3
AABBCBBAA
No
9
45
80
6.007
—
—
AAABCBAAA
9
45
80
1.811
0.920
2
AABCCCBAA
9
45
80
1.850
0.988
1
ABBCCCBBA
No
9
45
80
3.210
—
—
ABCCCCCCBA
10
456
72
1.892
0.964
2
ABBCCCCBBA
No
10
456
72
5.256
—
—
AABCCCCBAA
10
456
72
1.782
0.989
1
AABBCCBBAA
No
10
456
72
0.604
—
—
ABBBCCBBBA
10
456
72
1.782
0.910
3
AAABCCBAAA
10
456
72
1.677
0.957
2
The number of quarter-wave thickness layers (before any are coalesced) appears in the Layer column. A, B, and C represent layers of optical thickness λ0/16, λ0/18, and λ0/20, respectively, for the stacks containing 8, 9, and 10 layers. The refractive indices of A and C are 1.38 and 2.30, respectively. The refractive index of B is listed in the Index column. The Suppressed column lists order numbers of the stop bands that are suppressed, β is the phase thickness of a quarter-wave optical thickness layer at the lower-order stop band that is suppressed. Rmax is the peak reflectance of the first-order stop band of
Table III
Designs of All-Dielectric Multilayers in Which the Reflectances of Higher-Order Stop Bands Are Suppressed.a
Design
Suppressed
Layer
Count
β (deg)
B
C
D
E
F
G
H
0.5 ABCB 0.5 A
2, 3
8
5
45
1.782
2.30*
0.5 ABCDCB 0.5 A
2–5
12
7
30
1.568
2.024
2.30*
0.5 ABCDEDCB 0.5 A
2–7
16
9
22.5
1.487
1.782
2.134
2.30*
0.5 ABCDEFEDCB 0.5 A
2–9
20
11
18
1.449
1.646
1.928
2.19
2.30*
0.5 ABCDEFGFEDCB 0.5 A
2–11
24
13
15
1.428
1.568
1.782
2.024
2.223
2.30*
0.5 ABCDEFGHGFEDCB 0.5 A
2–13
28
15
12.86
1.415
1.519
1.683
1.886
2.089
2.243
2.30*
ABCBA
2, 3
5
5
72
1.892
2.30*
ABCDCBA
2–5
7
7
51.42
1.656
2.079
2.30*
ABCDEDCBA
2–7
9
9
40
1.549
1.85
2.163
2.30*
ABCDEFEDCBA
2–9
11
11
32.72
1.494
1.707
1.975
2.207
2.30*
ABCDEFGFEDCBA
2–11
13
13
27.69
1.462
1.619
1.831
2.056
2.233
2.30*
ABCDEFGHGFEDCBA
2–13
15
15
24
1.442
1.561
1.729
1.924
2.112
2.249
2.30*
The order numbers of the suppressed stop bands appear in the Suppressed column. Layer is the number of quarter-wave thickness layers before any are coalesced to form a degenerate design. Count is the count of layers in the final design. β is the phase thickness of a quarter-wave thickness layer at the lowest-order stop-band whose reflectance is suppressed. In all instances the index of the A layer is 1 38-and columns B–H give the indices of the other layers in the stack. The 2.30 index, which is marked with an asterisk, is constrained
Implementation of an algorithm that generates the coefficients a0, a1, a2, a3, and a4 of the Kard R/T polynomial for four layers of equal phase thickness. The significance of the boldface and italic type is explained in Appendix A.
Tables (4)
Table I
Approximate Values for the Coefficients a, of the R/T Polynomial for a Nine-Layer Stacka
u1
u2
u3
u4
u5
u6 = u4
u7 = u3
u8 = u2
u9 = u1
a1
−2
−2
−2
−2
−2
−2
−2
−2
−2
a3
−56
−28
−8
4
8
4
−8
−28
−56
a5
−140
0
20
0
−12
0
20
0
−140
a7
−56
28
−8
−4
8
−4
−8
28
−56
a9
−2
2
−2
2
−2
2
−2
2
−2
Each layer has the same optical thickness.
Table II
Designs of Multilayers That Have a Substantial Reflectance at λ0 of 2.2 μm and a Relatively Low Reflectance at λ0/4 and at λ0/5a
Design
OK
Layer
Suppressed
β (deg)
Index
Rmax
m
ABCCCCBA
8
345
67.5
1.861
0.979
1
ABBCCBBA
No
8
345
67.5
—
—
—
AABCCBAA
8
345
67.5
1.705
0.977
1
ABCCCCCBA
9
45
80
1.926
0.970
1
ABBBCBBBA
9
45
80
2.016
0.907
3
AABBCBBAA
No
9
45
80
6.007
—
—
AAABCBAAA
9
45
80
1.811
0.920
2
AABCCCBAA
9
45
80
1.850
0.988
1
ABBCCCBBA
No
9
45
80
3.210
—
—
ABCCCCCCBA
10
456
72
1.892
0.964
2
ABBCCCCBBA
No
10
456
72
5.256
—
—
AABCCCCBAA
10
456
72
1.782
0.989
1
AABBCCBBAA
No
10
456
72
0.604
—
—
ABBBCCBBBA
10
456
72
1.782
0.910
3
AAABCCBAAA
10
456
72
1.677
0.957
2
The number of quarter-wave thickness layers (before any are coalesced) appears in the Layer column. A, B, and C represent layers of optical thickness λ0/16, λ0/18, and λ0/20, respectively, for the stacks containing 8, 9, and 10 layers. The refractive indices of A and C are 1.38 and 2.30, respectively. The refractive index of B is listed in the Index column. The Suppressed column lists order numbers of the stop bands that are suppressed, β is the phase thickness of a quarter-wave optical thickness layer at the lower-order stop band that is suppressed. Rmax is the peak reflectance of the first-order stop band of
Table III
Designs of All-Dielectric Multilayers in Which the Reflectances of Higher-Order Stop Bands Are Suppressed.a
Design
Suppressed
Layer
Count
β (deg)
B
C
D
E
F
G
H
0.5 ABCB 0.5 A
2, 3
8
5
45
1.782
2.30*
0.5 ABCDCB 0.5 A
2–5
12
7
30
1.568
2.024
2.30*
0.5 ABCDEDCB 0.5 A
2–7
16
9
22.5
1.487
1.782
2.134
2.30*
0.5 ABCDEFEDCB 0.5 A
2–9
20
11
18
1.449
1.646
1.928
2.19
2.30*
0.5 ABCDEFGFEDCB 0.5 A
2–11
24
13
15
1.428
1.568
1.782
2.024
2.223
2.30*
0.5 ABCDEFGHGFEDCB 0.5 A
2–13
28
15
12.86
1.415
1.519
1.683
1.886
2.089
2.243
2.30*
ABCBA
2, 3
5
5
72
1.892
2.30*
ABCDCBA
2–5
7
7
51.42
1.656
2.079
2.30*
ABCDEDCBA
2–7
9
9
40
1.549
1.85
2.163
2.30*
ABCDEFEDCBA
2–9
11
11
32.72
1.494
1.707
1.975
2.207
2.30*
ABCDEFGFEDCBA
2–11
13
13
27.69
1.462
1.619
1.831
2.056
2.233
2.30*
ABCDEFGHGFEDCBA
2–13
15
15
24
1.442
1.561
1.729
1.924
2.112
2.249
2.30*
The order numbers of the suppressed stop bands appear in the Suppressed column. Layer is the number of quarter-wave thickness layers before any are coalesced to form a degenerate design. Count is the count of layers in the final design. β is the phase thickness of a quarter-wave thickness layer at the lowest-order stop-band whose reflectance is suppressed. In all instances the index of the A layer is 1 38-and columns B–H give the indices of the other layers in the stack. The 2.30 index, which is marked with an asterisk, is constrained
Implementation of an algorithm that generates the coefficients a0, a1, a2, a3, and a4 of the Kard R/T polynomial for four layers of equal phase thickness. The significance of the boldface and italic type is explained in Appendix A.