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 k_{gw}^{(0)} Normalized to ω/c = 2π/λ for Various Thin-Film Thicknesses^{a}

Case

Thickness (μm)

k_{gw}^{(0)}/(ω/c)

θ_{i} (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 θ_{i} 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 k_{gw}^{(}^{m}^{)} Normalized to ω/c = 2π/λ for Various Thin-Film Thicknesses^{a}

Case

Thickness (μm)

Mode Number (m)

k_{gw}^{m}/(ω/c)

θ_{i} (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 k_{gw}^{(}^{m}^{)} Normalized to ω/c = 2π/λ for Various High–Low Dielectric-Layer Pairs^{c}

Case

Number of Layers

Mode Number (m)

k_{gw}^{(}^{m}^{)}/ω/c

θ_{i} (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 k_{gw}^{(0)} Normalized to ω/c = 2π/λ for Various Thin-Film Thicknesses^{a}

Case

Thickness (μm)

k_{gw}^{(0)}/(ω/c)

θ_{i} (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 θ_{i} 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 k_{gw}^{(}^{m}^{)} Normalized to ω/c = 2π/λ for Various Thin-Film Thicknesses^{a}

Case

Thickness (μm)

Mode Number (m)

k_{gw}^{m}/(ω/c)

θ_{i} (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 k_{gw}^{(}^{m}^{)} Normalized to ω/c = 2π/λ for Various High–Low Dielectric-Layer Pairs^{c}

Case

Number of Layers

Mode Number (m)

k_{gw}^{(}^{m}^{)}/ω/c

θ_{i} (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.