Abstract

The angular reflectance of a graded-index layer, of arbitrary refractive-index profile, on the surface of a uniform substrate is calculated exactly by direct numerical integration of the wave equation for both states of polarization of the incident light. A study of the results for a number of selected profiles shows that (1) oscillations in reflectance versus wave-number graphs have an almost constant period, and (2) the period, when plotted against angle of incidence, gives a curve that is quite sensitive to the shape of the refractive-index profile. This sensitivity is the basis of a simple graphical procedure by means of which the inverse problem (i.e., deducing the index profile from reflectance measurements) can be solved. The procedure is applied to published reflectance data to determine both the thickness of a graded-index surface layer and the refractive-index profile.

© 1982 Optical Society of America

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References

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  1. S. F. Monaco, “Reflectance of an inhomogeneous thin film,” J. Opt. Soc. Am. 51, 280–282 (1961).
    [Crossref]
  2. Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 429–479.
  3. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 51–55.
  4. T. H. Elmer and F. W. Martin, “Antireflection films on alkali-borosilicate glasses produced by chemical treatments,” Am. Ceram. Soc. Bull. 58, 1092–1097 (1979).
  5. M. J. Minot, “The angular reflectance of single-layer gradient refractive index films,” J. Opt. Soc. Am. 67, 1046–1050 (1977).
    [Crossref]
  6. M. J. Minot, “Single-layer, gradient refractive index antireflection films effective from 0.35 μ m to 2.5 μ m,” J. Opt. Soc. Am. 66, 515–519 (1976).
    [Crossref]
  7. A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (Wiley, New York, 1967), Vol. 1, pp. 110–120.

1979 (1)

T. H. Elmer and F. W. Martin, “Antireflection films on alkali-borosilicate glasses produced by chemical treatments,” Am. Ceram. Soc. Bull. 58, 1092–1097 (1979).

1977 (1)

1976 (1)

1961 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 51–55.

Elmer, T. H.

T. H. Elmer and F. W. Martin, “Antireflection films on alkali-borosilicate glasses produced by chemical treatments,” Am. Ceram. Soc. Bull. 58, 1092–1097 (1979).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 429–479.

Martin, F. W.

T. H. Elmer and F. W. Martin, “Antireflection films on alkali-borosilicate glasses produced by chemical treatments,” Am. Ceram. Soc. Bull. 58, 1092–1097 (1979).

Minot, M. J.

Monaco, S. F.

Ralston, A.

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (Wiley, New York, 1967), Vol. 1, pp. 110–120.

Wilf, H. S.

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (Wiley, New York, 1967), Vol. 1, pp. 110–120.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 51–55.

Am. Ceram. Soc. Bull. (1)

T. H. Elmer and F. W. Martin, “Antireflection films on alkali-borosilicate glasses produced by chemical treatments,” Am. Ceram. Soc. Bull. 58, 1092–1097 (1979).

J. Opt. Soc. Am. (3)

Other (3)

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 429–479.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 51–55.

A. Ralston and H. S. Wilf, Mathematical Methods for Digital Computers (Wiley, New York, 1967), Vol. 1, pp. 110–120.

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Figures (17)

Fig. 1
Fig. 1

Refractive-index profile.

Fig. 2
Fig. 2

Coordinate system, in which the xy plane is in the surface of the graded-index layer and the yz plane is the plane of incidence. The diagram shows a ray of a TE wave at an angle of incidence ϕ. The ray is in the yz plane. The E field is perpendicular to the yz plane and to the ray. For the case of a TM wave the E field would be in the yz plane and perpendicular to the ray.

Fig. 3
Fig. 3

Reflectance of TE wave at 45° angle of incidence for (a) semibounded linear profile, (b) semibounded concave-parabolic profile, (c) semibounded convex-parabolic profile, (d) semibounded cubic profile.

Fig. 4
Fig. 4

Reflectance of TM wave at 45° angle of incidence for semibounded linear, semibounded concave-parabolic, semibounded convex-parabolic, and semibounded cubic profiles. Differences between individual lines are negligible.

Fig. 5
Fig. 5

TE-wave-reflectance, semibounded linear profile for varying angles of incidence (ϕ = 0, 25, 50, 75°).

Fig. 6
Fig. 6

TE-wave-reflectance, semibounded concave-parabolic profile for varying angles of incidence (ϕ = 0, 25, 50, 75°).

Fig. 7
Fig. 7

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 20° (from Ref. 5). Δ(1/λ) = (2.86 μm−1−1.625 μm−1)/4 cycles = 0.309 μm−1.

Fig. 8
Fig. 8

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 30° (from Ref. 5). Δ(1/λ) = (3.00 μm−1−1.43 μm−1)/5 cycles = 0.314 μm−1.

Fig. 9
Fig. 9

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 40° (from Ref. 5). Δ(1/λ) = (2.90–1.50 μm−1)/4 cycles = 0.350 μm−1

Fig. 10
Fig. 10

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 50° (from Ref. 5). Δ(1/λ) = (2.81−1.67 μm−1)/3 cycles = 0.380 μm−1.

Fig. 11
Fig. 11

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 60° (from Ref. 5). Δ(1/λ) = (2.70–1.45 μm−1)/3 cycles = 0.417 μm−1.

Fig. 12
Fig. 12

Experimental reflectance-versus-reciprocal-wavelength plot, ϕ = 70° (from Ref. 5). Δ(1/λ) = (2.62−1.65 μm−1)/2 cycles = 0.485 μm−1.

Fig. 13
Fig. 13

Experimental function E(ϕ) (derived from Ref. 5).

Fig. 14
Fig. 14

Comparison of E(ϕ) with theoretical functions T(ϕ) for various parabolic profiles.

Fig. 15
Fig. 15

Best-fit parabolic profile for n(s).

Fig. 16
Fig. 16

Comparison of E(ϕ) with theoretical functions T(ϕ) for various power-law profiles.

Fig. 17
Fig. 17

Best-fit power-law profile (p = 4) for n(s).

Tables (2)

Tables Icon

Table 1 Δ(d/λ) versus ϕ for TE Waves from Computer Results

Tables Icon

Table 2 Δ(d/λ) versus ϕ for TE Waves from WKB Theory

Equations (28)

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E x = U ( z ) exp [ i ( k 0 sin ϕ ) y ] ,
d 2 U d z 2 - d d z ( log μ μ 0 ) d U d z + k 0 2 ( n 2 - sin 2 ϕ ) U = 0.
H x = U ( z ) exp [ i ( k 0 sin ϕ ) y ] ,
d 2 U d z 2 - d d z ( log n 2 ) d U d z + k 0 2 ( n 2 - sin 2 ϕ ) U = 0.
U = A exp ( i k 1 z ) + B exp ( - i k 1 z ) ,
k 1 = k 0 ( n 1 2 - sin 2 ϕ ) 1 / 2
U ( 0 ) = A + B
( d U d z ) 0 = i k 1 ( A - B ) .
U = C exp ( i k 3 z ) ,
k 3 = k 0 ( n 3 2 - sin 2 ϕ ) 1 / 2
U ( d + ) = C exp ( i k 3 d ) ,
( d U d z ) d + = i k 3 C exp ( i k 3 d ) .
R = | B A | 2 .
U = exp [ ± i k 0 ( n 2 - sin 2 ϕ ) 1 / 2 d z ] ,
Ψ = 4 π λ 0 d ( n 2 - sin 2 ϕ ) 1 / 2 d z
Ψ = 2 m π ,
Ψ = 2 ( m + 1 ) π ,
Ψ = 4 π λ 0 d ( n 2 - sin 2 ϕ ) 1 / 2 d z .
Δ ( 1 λ ) = 1 2 0 d ( n 2 - sin 2 ϕ ) 1 / 2 d z ,
Δ ( 1 λ ) = 1 λ - 1 λ .
z = s d ,
Δ ( 1 λ ) = 1 2 d 0 1 [ n 2 ( s ) - sin 2 ϕ ] 1 / 2 d s ,
log ( Δ 1 λ ) = log { 1 2 0 1 [ n 2 ( s ) - sin 2 ϕ ] 1 / 2 d s } - log ( d ) .
E ( ϕ ) = T ( ϕ ) - log ( d ) ,
E ( ϕ ) = log ( Δ 1 λ ) .
T ( ϕ ) = - log { 2 0 1 [ n 2 ( s ) - sin 2 ϕ ] 1 / 2 d s } .
n ( s ) = N 0 - ( 3 N 0 - 4 N 1 / 2 + N 1 ) s + ( 2 N 0 - 4 N 1 / 2 + 2 N 1 ) s 2 ,
n ( s ) = N 0 + ( N 1 - N 0 ) s p ,