Abstract

The design steps for film–substrate single-reflection retarders are briefly stated and applied to the SiO2–Si film–substrate system at wavelength 6328 Å. The criterion of minimum-maximum error of the ellipsometric angle ψ is used to choose angle-of-incidence-tunable designs. Use is made of the (ϕ-d) plane (angle of incidence versus thickness) to determine whether a given film–substrate system with known optical properties and film thickness can operate as a reflection retarder and to determine the associated angles of incidence and retardation angles. This leads to the concept of permissible-thickness bands and forbidden gaps for operation of a film–substrate system as a reflection retarder. Experimental measurements on one of the proposed designs proved the validity of the method.

© 1975 Optical Society of America

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References

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  1. J. M. Bennett and H. E. Bennett, in Handbook of Optics, edited by W. G. Driscoll and W. Vaughan (McGraw–Hill, New York, 1975).
  2. The magnitude of the attenuation is independent of the incident polarization.
  3. R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975).
    [Crossref]
  4. Note that, whereas the use of a film–substrate system as a reflection retarder is new, its use as a reflection polarizer is well known. See, for example, M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972) and other papers in this series.
    [Crossref]
  5. Ellipsometric Tables of the Si–SiO2 System for Mercury and He–Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).
  6. In step 1, instead of using ρ= ejΔ, we use the general form ρ= tanψejΔ; steps 2–5 remain unchanged.
  7. The derivation of Eqs. (2) and (3) is given in Ref. 3.
  8. Alternatively, the angle of incidence ϕ′ can be obtained by any numerical method (e.g., successive bisection) to find the root of the equation |X| = 1.
  9. Our investigation of the effect of substrate absorption on the results of the film–substrate single-reflection retarder designs has shown that the mirror symmetry of the two branches B+ and B−occurs exactly only when the substrate is totally transparent. Deviation from exact symmetry increases with substrate absorption.
  10. Angle-of-incidence-tunable retarders are those with the least thickness. By adding multiples of Dϕ, the performance of the retarder gets worse, refer to Fig. 11(left).
  11. An alternative procedure would be to add the appropriate multiple of Dϕ, at each ϕ, to the image of the unit circle and to obtain the intersection points of this vertically translated image with the straight line d= const. In general, the first method is more convenient.
  12. The existence of exact reflection-retardation modes depends, for a particular system at a given wavelength, on the film thickness only.
  13. dmin and dmax are obtained by extrapolation to Δ = 0° and Δ = ± 18° of the branches shown in Fig. 4.

1975 (1)

1972 (1)

J. Opt. Soc. Am. (2)

Other (11)

Ellipsometric Tables of the Si–SiO2 System for Mercury and He–Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).

In step 1, instead of using ρ= ejΔ, we use the general form ρ= tanψejΔ; steps 2–5 remain unchanged.

The derivation of Eqs. (2) and (3) is given in Ref. 3.

Alternatively, the angle of incidence ϕ′ can be obtained by any numerical method (e.g., successive bisection) to find the root of the equation |X| = 1.

Our investigation of the effect of substrate absorption on the results of the film–substrate single-reflection retarder designs has shown that the mirror symmetry of the two branches B+ and B−occurs exactly only when the substrate is totally transparent. Deviation from exact symmetry increases with substrate absorption.

Angle-of-incidence-tunable retarders are those with the least thickness. By adding multiples of Dϕ, the performance of the retarder gets worse, refer to Fig. 11(left).

An alternative procedure would be to add the appropriate multiple of Dϕ, at each ϕ, to the image of the unit circle and to obtain the intersection points of this vertically translated image with the straight line d= const. In general, the first method is more convenient.

The existence of exact reflection-retardation modes depends, for a particular system at a given wavelength, on the film thickness only.

dmin and dmax are obtained by extrapolation to Δ = 0° and Δ = ± 18° of the branches shown in Fig. 4.

J. M. Bennett and H. E. Bennett, in Handbook of Optics, edited by W. G. Driscoll and W. Vaughan (McGraw–Hill, New York, 1975).

The magnitude of the attenuation is independent of the incident polarization.

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Figures (12)

FIG. 1
FIG. 1

The film–substrate system: ambient (0), film (1) of thickness d, and substrate (2).

FIG. 2
FIG. 2

Design curve, 1 − |X| vs ϕ, for a SiO2–Si film–substrate single-reflection retarder with retardation angle Δ = 30° at 6328 Å.

FIG. 3
FIG. 3

Angle of incidence ϕ′ (in degrees) vs retardation angle Δ (in degrees) for exact operation as reflection retarders, for Si–SiO2 system at 6328 Å.

FIG. 4
FIG. 4

Least film thickness d′ (in angstroms) vs retardation angle Δ (in degrees) for exact operation as reflection retarders for Si–SiO2 system at 6328 Å.

FIG. 5
FIG. 5

The p and s reflectances, p (△) and s(○) vs retardation angle Δ (in degrees) for exact SiO2–Si film–substrate single-reflection retarders at 6328 Å.

FIG. 6
FIG. 6

Performance curves for SiO2–Si film–substrate single-reflection retarder of thickness d = 1010 Å at 6328 Å. Left: ellipsometric angle ψ (in degrees) vs angle of incidence ϕ (in degrees). Middle: ellipsometric angle Δ (in degrees) vs angle of incidence ϕ (in degrees). This is the retarder’s tuning curve. Right: p and s reflectances p (△) and s (○) vs angle of incidence ϕ (in degrees), respectively.

FIG. 7
FIG. 7

Ellipsometric angle ψ (in degrees) versus the angle of incidence ϕ (in degrees) for Si–SiO2 exact reflection retarder designs with retardation angles Δ = 10°–170° with a step of 10° at 6328 Å.

FIG. 8
FIG. 8

The retardation angle Δ (in degrees) vs the angle of incidence ϕ (in degrees) for Si–SiO2 exact reflection retarder designs with retardation angles Δ = 10° and 170° at 6328 Å.

FIG. 9
FIG. 9

Performance curves, as in Fig. 6, but for d = 981.01 Å.

FIG. 10
FIG. 10

Performance curves, as in Fig. 6, but for d = 1041.1 Å.

FIG. 11
FIG. 11

(Right) The CTC’s A and B for film thicknesses d = 18 700 Å and d =3000 Å, respectively, in the complex-ρ plane for Si–SiO2 system at 6328 Å. Curve U, the unit circle, is the locus of all possible reflection retarders. (Left) The images A′, B′, and U′ in the ϕ-d plane of the CTC’s A and B, and the unit circle U of the complex-ρ plane, respectively, for Si–SiO2 system at 6328 Å.

FIG. 12
FIG. 12

Permissible-thickness bands and forbidden-thickness gaps for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.

Tables (4)

Tables Icon

TABLE I This table gives, for each value of the retardation angle Δ, the angle of incident ϕ′ and the least film thickness d′ necessary for operation of the SiO2–Si film–substrate system as an exact reflection retarder at 6328 Å and the associated film-thickness period Dϕ and reflectances p and s (note that p =s).a

Tables Icon

TABLE II Angle-of-incidence-tunable Si–SiO2 retarders at 6328 Å that achieve minimum-maximum error of ψ over selected angle-of-incidence ranges 0°–45°, 45°–90°, and 0°–90°.

Tables Icon

TABLE III Permissible-thickness bands and their widths (in angstroms) and the retardation sign for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.

Tables Icon

TABLE IV Forbidden-thickness gaps and their bandwidths (in angstroms) for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.

Equations (4)

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ρ = e j Δ .
X = - ( B - ρ E ) ± [ ( B - ρ E ) 2 - 4 ( C - ρ F ) ( A - ρ D ) ] 1 / 2 2 ( C - ρ F ) , A = r 01 p , B = r 12 p + r 01 p r 01 s r 12 s , C = r 12 p r 01 s r 12 s , D = r 01 s , E = r 12 s + r 01 p r 12 p r 01 s , F = r 01 p r 12 p r 12 s ,
d = ( α / 2 π ) D ϕ + m D ϕ ,
D ϕ = 1 2 λ [ N 1 2 - N 0 2 sin 2 ϕ ] - 1 / 2 .