Abstract

Optical properties of two-layer antireflection systems are discussed. Theory is applied to study the optical performance of bismuth-oxide/magnesium-fluoride bilayers. The theoretical study includes graphs and numerical tables showing the spectral reflectances for different glass substrates, the effect of film-thickness mismatching and the spectral reflectance for plane-polarized incident light.

© 1962 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), Chap. 4.
  2. J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).
  3. H. Osterberg and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952); G. E. Pride and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952).
  4. W. Weinstein, Vacuum,  IV, 3 (1954).
    [Crossref]
  5. K. Schuster, Ann. Phys. 4, 352 (1949).
    [Crossref]
  6. O. S. Heavens and S. D. Smith, J. Opt. Soc. Am. 47, 469 (1957).
    [Crossref]
  7. J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
    [Crossref]
  8. L. Holland and T. Putner, J. Sci. Instr. 36, 81 (1959)
    [Crossref]

1959 (1)

L. Holland and T. Putner, J. Sci. Instr. 36, 81 (1959)
[Crossref]

1957 (1)

1955 (1)

J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).

1954 (2)

W. Weinstein, Vacuum,  IV, 3 (1954).
[Crossref]

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
[Crossref]

1952 (1)

H. Osterberg and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952); G. E. Pride and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952).

1949 (1)

K. Schuster, Ann. Phys. 4, 352 (1949).
[Crossref]

Berning, J. A.

J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).

Berning, P. H.

J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).

Cox, J. T.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
[Crossref]

Epstein, L. I.

J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).

Hass, G.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
[Crossref]

Heavens, O. S.

O. S. Heavens and S. D. Smith, J. Opt. Soc. Am. 47, 469 (1957).
[Crossref]

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), Chap. 4.

Holland, L.

L. Holland and T. Putner, J. Sci. Instr. 36, 81 (1959)
[Crossref]

Kashdan, W. H.

H. Osterberg and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952); G. E. Pride and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952).

Osterberg, H.

H. Osterberg and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952); G. E. Pride and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952).

Putner, T.

L. Holland and T. Putner, J. Sci. Instr. 36, 81 (1959)
[Crossref]

Rowntree, R. F.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
[Crossref]

Schuster, K.

K. Schuster, Ann. Phys. 4, 352 (1949).
[Crossref]

Smith, S. D.

Weinstein, W.

W. Weinstein, Vacuum,  IV, 3 (1954).
[Crossref]

Ann. Phys. (1)

K. Schuster, Ann. Phys. 4, 352 (1949).
[Crossref]

J. Opt. Soc. Am. (3)

O. S. Heavens and S. D. Smith, J. Opt. Soc. Am. 47, 469 (1957).
[Crossref]

J. A. Berning, P. H. Berning, and L. I. Epstein, J. Opt. Soc. Am. 45, 407 (1955).

H. Osterberg and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952); G. E. Pride and W. H. Kashdan, J. Opt. Soc. Am. 42, 291 (1952).

J. Sci. Instr. (1)

L. Holland and T. Putner, J. Sci. Instr. 36, 81 (1959)
[Crossref]

Vacuum (2)

W. Weinstein, Vacuum,  IV, 3 (1954).
[Crossref]

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum,  IV, 445 (1954).
[Crossref]

Other (1)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

Optical thicknesses of antireflection systems onto glass (n0=1.50) formed of a dielectric (n1 variable) and a MgF2 coating (n2=1.38). For a particular n1, g1′ gives the optical thickness for the dielectric and g2′ for the MgF2 coating. Similarly, g1″ and g2″ refer to the second solution.

Fig. 2
Fig. 2

Spectral response of single and two-layer antireflection coatings onto glass substrates of different refractive indices. (The optical thicknesses for the two-layer system are given in Table I.)

Fig. 3
Fig. 3

Reflectances of single and two-layer antireflection coatings for oblique incidence. (1) s reflectances (electrical vector perpendicular to the plane of incidence). (2) p reflectances (electrical vector parallel to the plane of incidence).

Fig. 4
Fig. 4

Values of p-reflectance of two-layer antireflection systems (n0=1.50; n1=2.45; n2=1.38; n3=1.00).

Fig. 5
Fig. 5

Values of s reflectance of two-layer antireflection systems (n0=1.50; n1=2.45; n2=1.38; n3=1.00).

Tables (2)

Tables Icon

Table I Two-layer antireflection coatings (n1=2.45, n2=1.38). Optical thicknesses, in degrees and fractions of a wavelength, giving zero reflectance for different refractive indices n0 of the glass substrate (R2 is the reflectance of the Bi2O3 coating before the deposition of the MgF2 layer).

Tables Icon

Table II Film-thickness mismatching. Variations in the minimum reflectance of a two-layer antireflection coating onto glass substrate (n0=1.50).

Equations (11)

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R = | u p E p - H p u p E p + H p | 2 ,
u j = n j / cos φ j             or             u j = n j cos φ j             ( j = 0 , 1 , 2 , 3 , , p )
( E j + 1 H j + 1 ) = ( cos g j ( i sin g j ) / u j i u j sin g j cos g j ) ( E j H j ) ,             j = 1 , 2 , 3 , ( p - 1 ) ,
g j = ( 2 π / λ ) n j h j cos φ j
R = ( u 2 - u 0 ) 2 cos 2 g 1 + ( u 0 u 2 / u 1 - u 1 ) 2 sin 2 g 1 ( u 2 + u 0 ) 2 cos 2 g 1 + ( u 0 u 2 / u 1 + u 1 ) 2 sin 2 g 1
R = | u 3 E 3 - H 3 u 3 E 3 + H 3 | 2
u 3 E 3 ± H 3 = { ( u 3 ± u 0 ) cos g 1 cos g 2 ( u 2 u 0 u 1 ± u 1 u 3 u 2 ) sin g 1 sin g 2 } E 0 + i { ( u 0 u 3 u 1 ± u 1 ) sin g 1 cos g 2 + ( u 0 u 3 u 2 ± u 2 ) sin g 2 cos g 1 } E 0
tan 2 g 1 = ( u 0 - u 3 ) ( u 2 2 - u 0 u 3 ) u 1 2 ( u 0 u 2 2 - u 3 u 1 2 ) ( u 0 u 3 - u 1 2 ) , tan 2 g 2 = ( u 0 - u 3 ) ( u 0 u 3 - u 1 2 ) u 2 2 ( u 0 u 2 2 - u 3 u 1 2 ) ( u 2 2 - u 0 u 3 ) .
tan g 1 tan g 2 > 0             for             u 2 ( u 0 u 3 ) 1 2 > u 1 < ( u 0 u 3 ) 1 2
tan g 1 tan g 2 < 0             for             u 2 ( u 0 u 3 ) 1 2 < u 1 > ( u 0 u 3 ) 1 2 .
g λ = ( λ 0 / λ ) g λ 0 ,