R. M. A. Azzam and Bruce E. Perilloux, "Constraint on the optical constants of a film–substrate system for operation as an external-reflection retarder at a given angle of incidence," Appl. Opt. 24, 1171-1179 (1985)

Given a transparent film of refractive index n_{1} on an absorbing substrate of complex refractive index n_{2}-jk_{2}, we examine the constraint on n_{1}, n_{2}, and k_{2} such that the film–substrate system acts as an external-reflection retarder of specified retardance Δ at a specified angle of incidence ϕ. The constraint, which takes the form f(n_{1},n_{2},k_{2};ϕ,Δ) = 0, is portrayed graphically by equi-n_{1} contours in the n_{2},k_{2} plane at ϕ = 45, 70° and for Δ = ±90 and ±180°, corresponding to quarterwave and halfwave retarders (QWR and HWR), respectively. The required film thickness as a fraction of the film thickness period and the polarization-independent device reflectance ℛ are also studied graphically as functions of the optical constants. It is found that as n_{2} → 0, ℛ → 1, so that a metal substrate such as Ag is best suited for high-reflectance QWR (ϕ > 45°) and HWR (ϕ ≤ 45°). However, films that achieve QWR at ϕ ≤ 45° are excellent antireflection coatings of the underlying dielectric, semiconductor, or metallic substrate.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 40.

If the incident light is linearly polarized at 45° azimuth with respect to the plane of incidence, so that the p and s components of the incident electric vector are equal and inphase, the p component of the electric vector of the reflected light will lead the s component by 90°, when ρ = j, hence the identification of the p direction as the fast axis. The exp(jωt) harmonic time dependence is assumed.

E. Ritter, “Dielectric Film Materials for Optical Applications,” Phys. Thin Films 8, 1 (1975).

E. Ritter, “Dielectric Film Materials for Optical Applications,” Phys. Thin Films 8, 1 (1975).

Other

G. Hass, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 3, Chap. 8.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 40.

If the incident light is linearly polarized at 45° azimuth with respect to the plane of incidence, so that the p and s components of the incident electric vector are equal and inphase, the p component of the electric vector of the reflected light will lead the s component by 90°, when ρ = j, hence the identification of the p direction as the fast axis. The exp(jωt) harmonic time dependence is assumed.

Contours of constant film refractive index, n_{1} = constant, in the n_{2},k_{2} plane, where n_{2}-jk_{2} is the substrate complex refractive index. These equi-n_{1} contours represent the constraint on the optical constants such that the ratio of the complex p- and s-reflection coefficients of the film–substrate system ρ = j at ϕ = 70° angle of incidence, i.e., a quarterwave retarder (QWR) with p fast axis is realized.

Film thickness as a fraction of the film thickness period ζ, required to achieve ρ = j QWR at ϕ = 70°, plotted as a function of n_{2} at constant n_{1} (marked by each curve). n_{1},n_{2} specify a particular film–substrate QWR completely, because k_{2} can be deduced from Fig. 1. The contour AB describes the variation of ζ with n_{2} along the similarly marked n_{1} = 1.6 contour in Fig. 1.

Polarization-independent reflectance ℛ of film–substrate ρ = j QWRs at ϕ = 70° plotted as a function of n_{2} at constant n_{1} (marked by each curve) along the equi-n_{1} contours of Fig. 1.

Normalized film thickness ζ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 10 for ρ = j (QWR, p fast axis) at ϕ = 45°.

Polarization-independent reflectance ℛ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 10 for ρ = j (QWR, p fast axis) at ϕ = 45°.

Normalized film thickness ζ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 13 for ρ = −j (QWR, s fast axis) at ϕ = 45°.

Polarization-independent reflectance ℛ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 13 for ρ = −j (QWR, s fast axis) at ϕ = 45°.

Normalized film thickness ζ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 16 for ρ = −1 (HWR) at ϕ = 45°.

Polarization-independent reflectance ℛ as a function of n_{2} at constant n_{1} (marked by each curve) corresponding to the data of Fig. 16 for ρ = −1 (HWR) at ϕ = 45°.

Table I Characteristics of Quarterwave Retarders (QWR, ρ = +j) at 70° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

Table II Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ −90°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I

Table III Characteristics of Halfwave Retarders (HWR) at 45° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

Table IV Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ − 180°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the HWR Designs at ϕ = 45° Listed in Table III

Characteristics of Quarterwave Retarders (QWR, ρ = +j) at 70° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

λ(nm)

n_{2}

k_{2}

n_{1}

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.219154

0.277011

71.33

94.09

500

0.050

2.87

1.291322

0.278893

78.72

97.72

600

0.060

3.75

1.349785

0.285499

88.39

98.20

700

0.075

4.62

1.396728

0.292587

99.10

98.40

800

0.090

5.45

1.433078

0.298824

110.48

98.55

950

0.110

6.56

1.471883

0.306027

128.32

98.72

2000

0.480

14.40

1.601287

0.332817

256.69

98.69

λ is the wavelength of light. n_{2},k_{2} are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n_{1} is the film refractive index obtained by solving Eq. (11) with ρ = +j and ϕ = 70°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the QWR.

Table II

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ −90°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I

Δϕ = 0.1°

Δn_{1} = 0.01

Δd = l nm

λ(nm)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

400

0.37 × 10^{−4}

0.329

0.198 × 10^{−2}

2.004

0.613 × 10^{−3}

1.246

500

0.61 × 10^{−5}

0.341

0.066 × 10^{−2}

1.044

0.298 × 10^{−3}

0.961

600

0.19 × 10^{−5}

0.344

0.047 × 10^{−2}

0.681

0.245 × l0^{−3}

0.773

700

0.31 × 10^{−6}

0.345

0.038 × l0^{−2}

0.506

0.214 × 10^{−3}

0.645

800

0.49 × 10^{−6}

0.345

0.033 × 10^{−2}

0.414

0.186 × 10^{−3}

0.553

950

0.10 × 10^{−5}

0.345

0.028 × 10^{−2}

0.335

0.151 × 10^{−3}

0.457

2000

0.25 × 10^{−5}

0.345

0.024 × 10^{−2}

0.185

0.093 × 10^{−3}

0.214

Table III

Characteristics of Halfwave Retarders (HWR) at 45° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

λ(nm)

n_{2}

k_{2}

n_{1}

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.566251

0.417418

59.74

90.89

500

0.050

2.87

1.436138

0.464840

92.97

96.51

600

0.060

3.75

1.385496

0.481921

121.35

97.36

700

0.075

4.62

1.359636

0.489611

147.57

97.76

800

0.090

5.45

1.345145

0.493407

172.49

98.03

950

0.110

6.56

1.333442

0.496091

208.45

98.31

2000

0.480

14.40

1.312227

0.499608

451.99

98.44

^{a} λ is the wavelength of light. n_{2},k_{2} are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n_{1} is the film refractive index obtained by solving Eq. (11) with ρ = −1 and ϕ = 45°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the HWR.

Table IV

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ − 180°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the HWR Designs at ϕ = 45° Listed in Table III

Δϕ = 0.1°

Δn_{1} =0.01

Δd = 1 nm

λ(nm)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

400

0.469 × 10^{−4}

0.018

0.512 × 10^{−3}

0.977

0.101 × 10^{−2}

0.959

500

0.186 × 10^{−4}

0.003

0.116 × 10^{−3}

1.130

0.207 × 10^{−3}

0.716

600

0.138 × 10^{−4}

0.016

0.398 × 10^{−4}

1.179

0.101 × 10^{−3}

0.592

700

0.113 × 10^{−3}

0.024

0.291 × 10^{−5}

1.188

0.604 × 10^{−4}

0.509

800

0.956 × 10^{−5}

0.029

0.169 × 10^{−4}

1.181

0.396 × 10^{−4}

0.446

950

0.781 × 10^{−5}

0.035

0.306 × 10^{−4}

1.163

0.239 × 10^{−4}

0.377

2000

0.586 × 10^{−5}

0.048

0.682 × 10^{−4}

1.058

0.506 × 10^{−5}

0.182

Tables (4)

Table I

Characteristics of Quarterwave Retarders (QWR, ρ = +j) at 70° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

λ(nm)

n_{2}

k_{2}

n_{1}

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.219154

0.277011

71.33

94.09

500

0.050

2.87

1.291322

0.278893

78.72

97.72

600

0.060

3.75

1.349785

0.285499

88.39

98.20

700

0.075

4.62

1.396728

0.292587

99.10

98.40

800

0.090

5.45

1.433078

0.298824

110.48

98.55

950

0.110

6.56

1.471883

0.306027

128.32

98.72

2000

0.480

14.40

1.601287

0.332817

256.69

98.69

λ is the wavelength of light. n_{2},k_{2} are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n_{1} is the film refractive index obtained by solving Eq. (11) with ρ = +j and ϕ = 70°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the QWR.

Table II

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ −90°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I

Δϕ = 0.1°

Δn_{1} = 0.01

Δd = l nm

λ(nm)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

Magnitude error

Phase error (deg)

400

0.37 × 10^{−4}

0.329

0.198 × 10^{−2}

2.004

0.613 × 10^{−3}

1.246

500

0.61 × 10^{−5}

0.341

0.066 × 10^{−2}

1.044

0.298 × 10^{−3}

0.961

600

0.19 × 10^{−5}

0.344

0.047 × 10^{−2}

0.681

0.245 × l0^{−3}

0.773

700

0.31 × 10^{−6}

0.345

0.038 × l0^{−2}

0.506

0.214 × 10^{−3}

0.645

800

0.49 × 10^{−6}

0.345

0.033 × 10^{−2}

0.414

0.186 × 10^{−3}

0.553

950

0.10 × 10^{−5}

0.345

0.028 × 10^{−2}

0.335

0.151 × 10^{−3}

0.457

2000

0.25 × 10^{−5}

0.345

0.024 × 10^{−2}

0.185

0.093 × 10^{−3}

0.214

Table III

Characteristics of Halfwave Retarders (HWR) at 45° Angle of Incidence Using a Dielectric Thin Film on a Ag Substrate at Several Wavelengths ^{a}

λ(nm)

n_{2}

k_{2}

n_{1}

ζ

d(nm)

ℛ(%)

400

0.075

1.93

1.566251

0.417418

59.74

90.89

500

0.050

2.87

1.436138

0.464840

92.97

96.51

600

0.060

3.75

1.385496

0.481921

121.35

97.36

700

0.075

4.62

1.359636

0.489611

147.57

97.76

800

0.090

5.45

1.345145

0.493407

172.49

98.03

950

0.110

6.56

1.333442

0.496091

208.45

98.31

2000

0.480

14.40

1.312227

0.499608

451.99

98.44

^{a} λ is the wavelength of light. n_{2},k_{2} are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n_{1} is the film refractive index obtained by solving Eq. (11) with ρ = −1 and ϕ = 45°. ζ and d are the normalized and actual (least) film thicknesses, respectively, and ℛ is the polarization-independent reflectance of the HWR.

Table IV

Absolute Values of the Magnitude Error (|ρ| − 1) and Phase Error (argρ − 180°) Caused by Introducing, One at a Time, an Angle-of-lncidence Error Δϕ = 0.1°, a Film-Refractive-Index Error Δn_{1} = 0.01, or a Film-Thickness Error Δd = 1 nm to the HWR Designs at ϕ = 45° Listed in Table III