Abstract

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.

© 1985 Optical Society of America

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References

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  1. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Ellipsometric Function of a Film–Substrate System: Applications to the Design of Reflection-Type Optical Devices and to Ellipsometry,” J. Opt. Soc. Am. 65, 252 (1975).
    [CrossRef]
  2. A.-R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Design of Film–Substrate Single-Reflection Retarders,” J. Opt. Soc. Am. 65, 1043 (1975).
    [CrossRef]
  3. R. M. A. Azzam, M. E. R. Khan, “Single-Reflection Film–Substrate Half-Wave Retarders with Nearly Stationary Reflection Properties over a Wide Range of Incidence Angles,” J. Opt. Soc. Am. 73, 160 (1983).
    [CrossRef]
  4. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 40.
  6. 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.
  7. E. Ritter, “Dielectric Film Materials for Optical Applications,” Phys. Thin Films 8, 1 (1975).
  8. H. K. Pulker, “Characterization of Optical Thin Films,” Appl. Opt. 18, 1969 (1979).
    [CrossRef] [PubMed]
  9. G. Hass, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 3, Chap. 8.
  10. J. H. Apfel, “Graphical Method to Design Internal Reflection Phase Retarders,” Appl. Opt. 23, 1178 (1984).
    [CrossRef] [PubMed]
  11. W. R. Hunter, “Measurement of Optical Properties of Materials in the Vacuum Ultraviolet Spectral Region,” Appl. Opt. 21, 2103 (1982).
    [CrossRef] [PubMed]
  12. S. Kawabata, M. Suzuki, “MgF2–Ag Tunable Reflection Retarder,” Appl. Opt. 19, 484 (1980).
    [CrossRef] [PubMed]

1984

1983

1982

1980

1979

1975

Apfel, J. H.

Azzam, R. M. A.

Bashara, N. M.

Born, M.

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

Hass, G.

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

Hunter, W. R.

Kawabata, S.

Khan, M. E. R.

Pulker, H. K.

Ritter, E.

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

Suzuki, M.

Wolf, E.

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

Zaghloul, A.-R. M.

Appl. Opt.

J. Opt. Soc. Am.

Phys. Thin Films

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.

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Figures (18)

Fig. 1
Fig. 1

Contours of constant film refractive index, n1 = constant, in the n2,k2 plane, where n2-jk2 is the substrate complex refractive index. These equi-n1 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.

Fig. 2
Fig. 2

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 n2 at constant n1 (marked by each curve). n1,n2 specify a particular film–substrate QWR completely, because k2 can be deduced from Fig. 1. The contour AB describes the variation of ζ with n2 along the similarly marked n1 = 1.6 contour in Fig. 1.

Fig. 3
Fig. 3

Polarization-independent reflectance ℛ of film–substrate ρ = j QWRs at ϕ = 70° plotted as a function of n2 at constant n1 (marked by each curve) along the equi-n1 contours of Fig. 1.

Fig. 4
Fig. 4

Same as Fig. 1 except that ρ = −j (QWR with s fast axis).

Fig. 5
Fig. 5

Same as Fig. 2 except that ρ = −j (QWR with s fast axis).

Fig. 6
Fig. 6

Same as Fig. 3 except that ρ = −j (QWR with s fast axis).

Fig. 7
Fig. 7

Same as Fig. 1 except that ρ = −1, corresponding to halfwave retarders (HWR), also at ϕ = 70°.

Fig. 8
Fig. 8

Same as Fig. 2 except that ρ = −1 (HWR).

Fig. 9
Fig. 9

Same as Fig. 3 except that ρ = −1 (HWR).

Fig. 10
Fig. 10

Equi-n1 contours in the n2,k2 plane for ρ = j (QWR, p fast axis) at ϕ = 45° angle of incidence.

Fig. 11
Fig. 11

Normalized film thickness ζ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 10 for ρ = j (QWR, p fast axis) at ϕ = 45°.

Fig. 12
Fig. 12

Polarization-independent reflectance ℛ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 10 for ρ = j (QWR, p fast axis) at ϕ = 45°.

Fig. 13
Fig. 13

Equi-n1 contours in the n2,k2 plane for ρ = −j (QWR, s fast axis) at ϕ = 45°.

Fig. 14
Fig. 14

Normalized film thickness ζ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 13 for ρ = −j (QWR, s fast axis) at ϕ = 45°.

Fig. 15
Fig. 15

Polarization-independent reflectance ℛ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 13 for ρ = −j (QWR, s fast axis) at ϕ = 45°.

Fig. 16
Fig. 16

Equi-n1 contours in the n2,k2 plane for ρ = −1 halfwave retarders (HWR) at ϕ = 45° angle of incidence.

Fig. 17
Fig. 17

Normalized film thickness ζ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 16 for ρ = −1 (HWR) at ϕ = 45°.

Fig. 18
Fig. 18

Polarization-independent reflectance ℛ as a function of n2 at constant n1 (marked by each curve) corresponding to the data of Fig. 16 for ρ = −1 (HWR) at ϕ = 45°.

Tables (4)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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

Equations (15)

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R ν = ( r 01 ν + r 12 ν X ) / ( 1 + r 01 ν r 12 ν X ) , ν = p , s ,
ρ = R p / R s
ρ = ( A + B X + C X 2 ) / ( D + E X + F X 2 ) ,
X = exp ( j 2 π ζ ) ,
A = r 01 p , B = r 12 p + r 01 p r 01 s r 12 s , C = r 01 s r 12 p r 12 s , D = r 01 s , E = r 12 s + r 01 p r 01 s r 12 p , F = r 01 p r 12 p r 12 s .
ζ = d / D ϕ ,
D ϕ = λ 2 ( N 1 2 N 0 2 sin 2 ϕ ) 1 / 2
N 1 / N 0 = n 1 , N 2 / N 0 = n 2 j k 2 .
X = ( B ρ E ) ± [ ( B ρ E ) 2 4 ( C ρ F ) ( A ρ D ) ] 1 / 2 2 ( C ρ F ) ,
| X | = 1 .
f ( n 1 , n 2 , k 2 ; ϕ , ρ ) = 0 .
| X | 1 < 10 6 .
ζ = arg ( X ) / 2 π .
0 < ζ < 1 .
R ν = | R ν | 2 ,

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