J. M. Elson, L. F. DeSandre, and J. L. Stanford, "Analysis of anomalous resonance effects in multilayer-overcoated, low-efficiency gratings," J. Opt. Soc. Am. A 5, 74-88 (1988)
A Rayleigh–Fourier method is used in an analysis of anomalous resonance effects on multilayer-overcoated, low-efficiency gratings. The ratio of grating amplitude to wavelength, H/λ, is 0.016. For wavelength-to-grating-period values such that only the specular beam and the −1 diffracted order are reflected from the grating, such H/λ values yield a nominal −1-order efficiency on the order of 2%. However, for carefully selected parameters such that coupling of incident beam energy into guided waves is achieved, the −1-order efficiency may exceed 50% and is accompanied by a corresponding reduction in the specular beam intensity. The range over which this anomalous behavior occurs is quite narrow. Composite dielectric structures of one to ten layers are considered.
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Guided-Wave Numbers kgw(0) Normalized to ω/c = 2π/λ for Various Thin-Film Thicknessesa
Case
Thickness (μm)
kgw(0)/(ω/c)
θi (degrees)
n
1.a
0.075
1.66
49.1
+1
1.b
0.250
1.566
41.5
+1
1.c
0.275
1.642
47.6
+1
1.d
0.300
1.71
53.7
+1
These examples are all m = 0 modes. For a wavelength λ = 0.6328 μm and a grating spacing D = 0.7 μm, the phase-matching angle of incidence θi is also shown. In these examples, all coupling involves a single-grating wave number 2π/D in the positive-x direction (n = +1). The thin-film and semi-infinite substrates have dielectric constants of (5.11, 0.0) and (2.25, 0.0), respectively.
Table 2
Guided-Wave Numbers kgw(m) Normalized to ω/c = 2π/λ for Various Thin-Film Thicknessesa
Case
Thickness (μm)
Mode Number (m)
kgwm/(ω/c)
θi (degrees)
n
2.a
0.15
0
1.779
61.0
+1
2.b
0.20
0
1.94525
7.9
+2
2.c
0.225
0
1.067
9.4
+1
2.d
0.225
1
1.9979
10.9
+2
2.e
0.30
0
1.53095
38.8
+1
2.f
0.30
1
2.095
38.1
−3
2.g
0.30
1
2.095
16.7
+2
2.h
0.30
0
1.53095
16.1
−2
The phase-matching angles are also given as described in Table 1. The substrate is Ag, with dielectric constant (−16.4, 0.52) at λ = 0.6328 μm. Cases 2.a, 2.c, and 2.e involve a single-grating wave number in the positive direction (n = +1). Cases 2.b, 2.d, and 2.g involve two wave numbers in the positive direction (n = +2). Case 2.h involves two wave numbers in the negative direction (n = −2), and Case 2.f involves three wave numbers in the negative direction (n = −3). All n > 0 guided waves propagate in the positive-x direction, and all guided waves for n < 0 propagate in the negative-x direction. Note that Cases 2.c and 2.d consider two different guided-wave modes for the same thickness; Cases 2.e–2.h, for different coupling orders and two mode numbers for a given thickness.
Table 3
Guided-Wave Numbers kgw(m) Normalized to ω/c = 2π/λ for Various High–Low Dielectric-Layer Pairsc
Case
Number of Layers
Mode Number (m)
kgw(m)/ω/c
θi (degrees)
3.a
2
0
1.607
44.7
3.b
4
0
1.477
35.0
4
1
1.764
59.3
3.c
6
0
1.4096
30.4
6
1
1.653
48.5
6
2
1.8132
65.4
3.d
8
0
1.3692
27.7
8
1
1.5715
41.9
8
2
1.7353
56.2
8
3
1.834
68.4
3.e
10
0
1.3427
26.0
3.f
10
1
1.5124
37.5
10
2
1.6664
49.7
10
3
1.7782
61.0
10
4
1.8446
70.2
Also given are the phase-matching angles for the various mode numbers and numbers of layers. Of these examples, six cases are chosen for discussion in the text. The text gives the optical constants and thin-film thicknesses. All these examples are for coupling through the n = +1 grating order.
Tables (3)
Table 1
Guided-Wave Numbers kgw(0) Normalized to ω/c = 2π/λ for Various Thin-Film Thicknessesa
Case
Thickness (μm)
kgw(0)/(ω/c)
θi (degrees)
n
1.a
0.075
1.66
49.1
+1
1.b
0.250
1.566
41.5
+1
1.c
0.275
1.642
47.6
+1
1.d
0.300
1.71
53.7
+1
These examples are all m = 0 modes. For a wavelength λ = 0.6328 μm and a grating spacing D = 0.7 μm, the phase-matching angle of incidence θi is also shown. In these examples, all coupling involves a single-grating wave number 2π/D in the positive-x direction (n = +1). The thin-film and semi-infinite substrates have dielectric constants of (5.11, 0.0) and (2.25, 0.0), respectively.
Table 2
Guided-Wave Numbers kgw(m) Normalized to ω/c = 2π/λ for Various Thin-Film Thicknessesa
Case
Thickness (μm)
Mode Number (m)
kgwm/(ω/c)
θi (degrees)
n
2.a
0.15
0
1.779
61.0
+1
2.b
0.20
0
1.94525
7.9
+2
2.c
0.225
0
1.067
9.4
+1
2.d
0.225
1
1.9979
10.9
+2
2.e
0.30
0
1.53095
38.8
+1
2.f
0.30
1
2.095
38.1
−3
2.g
0.30
1
2.095
16.7
+2
2.h
0.30
0
1.53095
16.1
−2
The phase-matching angles are also given as described in Table 1. The substrate is Ag, with dielectric constant (−16.4, 0.52) at λ = 0.6328 μm. Cases 2.a, 2.c, and 2.e involve a single-grating wave number in the positive direction (n = +1). Cases 2.b, 2.d, and 2.g involve two wave numbers in the positive direction (n = +2). Case 2.h involves two wave numbers in the negative direction (n = −2), and Case 2.f involves three wave numbers in the negative direction (n = −3). All n > 0 guided waves propagate in the positive-x direction, and all guided waves for n < 0 propagate in the negative-x direction. Note that Cases 2.c and 2.d consider two different guided-wave modes for the same thickness; Cases 2.e–2.h, for different coupling orders and two mode numbers for a given thickness.
Table 3
Guided-Wave Numbers kgw(m) Normalized to ω/c = 2π/λ for Various High–Low Dielectric-Layer Pairsc
Case
Number of Layers
Mode Number (m)
kgw(m)/ω/c
θi (degrees)
3.a
2
0
1.607
44.7
3.b
4
0
1.477
35.0
4
1
1.764
59.3
3.c
6
0
1.4096
30.4
6
1
1.653
48.5
6
2
1.8132
65.4
3.d
8
0
1.3692
27.7
8
1
1.5715
41.9
8
2
1.7353
56.2
8
3
1.834
68.4
3.e
10
0
1.3427
26.0
3.f
10
1
1.5124
37.5
10
2
1.6664
49.7
10
3
1.7782
61.0
10
4
1.8446
70.2
Also given are the phase-matching angles for the various mode numbers and numbers of layers. Of these examples, six cases are chosen for discussion in the text. The text gives the optical constants and thin-film thicknesses. All these examples are for coupling through the n = +1 grating order.