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 n1 on an absorbing substrate of complex refractive index n2-jk2, we examine the constraint on n1, n2, and k2 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(n1,n2,k2;ϕ,Δ) = 0, is portrayed graphically by equi-n1 contours in the n2,k2 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 n2 → 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.
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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)
n2
k2
n1
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. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 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 Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I
Δϕ = 0.1°
Δn1 = 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)
n2
k2
n1
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. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 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 Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the HWR Designs at ϕ = 45° Listed in Table III
Δϕ = 0.1°
Δn1 =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)
n2
k2
n1
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. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 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 Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the QWR Designs at ϕ = 70° Listed in Table I
Δϕ = 0.1°
Δn1 = 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)
n2
k2
n1
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. n2,k2 are the real and imaginary parts of the Ag substrate complex refractive index (from Ref. 9). n1 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 Δn1 = 0.01, or a Film-Thickness Error Δd = 1 nm to the HWR Designs at ϕ = 45° Listed in Table III