Abstract

Generalized Brewster-angle conditions for either the s or p plane of polarization are derived for periodic quarter-wave multilayers on a substrate, when the two layers of the basic period are matched to have equal effective optical thickness. By use of these relations, graphs for the design of reflecting polarizers for either the s or p plane of polarization are presented; two examples made with ZnS and MgF2 layers are treated in detail. The multilayer systems under consideration may also be used as transmitting polarizers, antireflection coatings for linearly polarized light, and for polarization-sensitive interference filters.

© 1974 Optical Society of America

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References

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  1. C. J. Broomer, Opt. Spectra 6(7), 34 (1972).
  2. R. J. King and S. P. Talim, J. Phys. E 4, 93 (1971).
    [Crossref]
  3. F. Abelès, J. Phys. Radium 11, 403 (1950).
    [Crossref]
  4. L. A. Catalan, J. Opt. Soc. Am. 55, 857 (1965).
    [Crossref]
  5. M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).
    [Crossref]
  6. M. Banning, J. Opt. Soc. Am. 37, 792 (1947).
    [Crossref] [PubMed]
  7. H. Schröder and R. Schläfer, Z. Naturforsch. 4a, 576 (1949).
  8. J. J. Vera, Opt. Acta 11, 315 (1964).
    [Crossref]
  9. P. B. Clapham, M. J. Downs, and R. J. King, Appl. Opt. 8, 1965 (1969).
    [Crossref] [PubMed]
  10. A. F. Turner and P. W. Baumeister, Appl. Opt. 5, 69 (1966).
    [Crossref] [PubMed]
  11. P. Kard, Izv. Akad. Nauk Eston. SSR 9, 26 (1960).
  12. H. Schröder, Optik 3, 499 (1948).
  13. R. Messner, Feinwerktechnik 57, 142 (1953).
  14. W. W. Buchman, S. J. Holmes, and F. J. Woodberry, J. Opt. Soc. Am. 61, 1604 (1971).
    [Crossref]
  15. W. W. Buchman, S. J. Holmes, and F. J. Woodberry, J. Opt. Soc. Am. 62, 1329 (1972).
    [Crossref]
  16. D. Kermisch, J. Opt. Soc. Am. 62, 1010 (1972).
    [Crossref]
  17. P. Baumeister, Opt. Acta 8, 105 (1961).
    [Crossref]
  18. H. F. Mahlein, Opt. Acta 20, 687 (1973).
    [Crossref]
  19. H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).
  20. H. F. Mahlein, Opt. Laser Tech. 5, 60 (1973).
    [Crossref]
  21. E. Ritter, in Laser Handbook, edited by F. T. Arecchi and E. O. Schulz-Dubois (North–Holland, Amsterdam, 1972), Vol. 1, p. 897.

1973 (2)

H. F. Mahlein, Opt. Acta 20, 687 (1973).
[Crossref]

H. F. Mahlein, Opt. Laser Tech. 5, 60 (1973).
[Crossref]

1972 (4)

1971 (2)

1969 (1)

1966 (1)

1965 (1)

1964 (1)

J. J. Vera, Opt. Acta 11, 315 (1964).
[Crossref]

1961 (1)

P. Baumeister, Opt. Acta 8, 105 (1961).
[Crossref]

1960 (1)

P. Kard, Izv. Akad. Nauk Eston. SSR 9, 26 (1960).

1953 (1)

R. Messner, Feinwerktechnik 57, 142 (1953).

1950 (1)

F. Abelès, J. Phys. Radium 11, 403 (1950).
[Crossref]

1949 (1)

H. Schröder and R. Schläfer, Z. Naturforsch. 4a, 576 (1949).

1948 (1)

H. Schröder, Optik 3, 499 (1948).

1947 (1)

Abelès, F.

F. Abelès, J. Phys. Radium 11, 403 (1950).
[Crossref]

Banning, M.

Baumeister, P.

P. Baumeister, Opt. Acta 8, 105 (1961).
[Crossref]

Baumeister, P. W.

Broomer, C. J.

C. J. Broomer, Opt. Spectra 6(7), 34 (1972).

Buchman, W. W.

Catalan, L. A.

Clapham, P. B.

Downs, M. J.

Holmes, S. J.

Kard, P.

P. Kard, Izv. Akad. Nauk Eston. SSR 9, 26 (1960).

Kermisch, D.

King, R. J.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

Mahlein, H. F.

H. F. Mahlein, Opt. Laser Tech. 5, 60 (1973).
[Crossref]

H. F. Mahlein, Opt. Acta 20, 687 (1973).
[Crossref]

Messner, R.

R. Messner, Feinwerktechnik 57, 142 (1953).

Ritter, E.

E. Ritter, in Laser Handbook, edited by F. T. Arecchi and E. O. Schulz-Dubois (North–Holland, Amsterdam, 1972), Vol. 1, p. 897.

Ruiz-Urbieta, M.

Schläfer, R.

H. Schröder and R. Schläfer, Z. Naturforsch. 4a, 576 (1949).

Schröder, H.

H. Schröder and R. Schläfer, Z. Naturforsch. 4a, 576 (1949).

H. Schröder, Optik 3, 499 (1948).

Sparrow, E. M.

Talim, S. P.

R. J. King and S. P. Talim, J. Phys. E 4, 93 (1971).
[Crossref]

Turner, A. F.

Vera, J. J.

J. J. Vera, Opt. Acta 11, 315 (1964).
[Crossref]

Woodberry, F. J.

Appl. Opt. (2)

Feinwerktechnik (1)

R. Messner, Feinwerktechnik 57, 142 (1953).

Izv. Akad. Nauk Eston. SSR (1)

P. Kard, Izv. Akad. Nauk Eston. SSR 9, 26 (1960).

J. Opt. Soc. Am. (6)

J. Phys. E (1)

R. J. King and S. P. Talim, J. Phys. E 4, 93 (1971).
[Crossref]

J. Phys. Radium (1)

F. Abelès, J. Phys. Radium 11, 403 (1950).
[Crossref]

Opt. Acta (3)

J. J. Vera, Opt. Acta 11, 315 (1964).
[Crossref]

P. Baumeister, Opt. Acta 8, 105 (1961).
[Crossref]

H. F. Mahlein, Opt. Acta 20, 687 (1973).
[Crossref]

Opt. Laser Tech. (1)

H. F. Mahlein, Opt. Laser Tech. 5, 60 (1973).
[Crossref]

Opt. Spectra (1)

C. J. Broomer, Opt. Spectra 6(7), 34 (1972).

Optik (1)

H. Schröder, Optik 3, 499 (1948).

Z. Naturforsch. (1)

H. Schröder and R. Schläfer, Z. Naturforsch. 4a, 576 (1949).

Other (2)

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

E. Ritter, in Laser Handbook, edited by F. T. Arecchi and E. O. Schulz-Dubois (North–Holland, Amsterdam, 1972), Vol. 1, p. 897.

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Figures (15)

Fig. 1
Fig. 1

Generalized Brewster-angle conditions for matched multilayers made of ZnS and MgF2. ns = 1.5 (glass), nH =2.35 (ZnS), nL = 1.38 (MgF2), n0 = 1 (air); k ≥ 0, integer. First layer (on substrate) was ZnS.

Fig. 2
Fig. 2

Same as Fig. 1, but first layer was MgF2.

Fig. 3
Fig. 3

Narrow-band reflecting polarizer for the p component, S(LH)2LA, ns = 1.5 (glass), nH = 2.35 (ZnS), nL =1.38 (MgF2), n0 = 1 (air), θ0 = 88.15°.

Fig. 4
Fig. 4

Narrow-band reflecting polarizers for the s component, S(HL)3HA, ns = 1.5 (glass), nH = 2.35 (ZnS), nL = 1.38 (MgF2), n0 = 1 (air), θ0 = 86.45°.

Fig. 5
Fig. 5

Schematic representation of narrow-band reflecting polarizers. Top: Reflecting polarizer for the p component, S(HL)kA or S(LH)kLA. Bottom: Reflecting polarizer for the s component, S(HL)kHA or S(LH)kA. (ns = 1.5, nH = 2.35, nL = 1.38, n0 = 1.)

Fig. 6
Fig. 6

Determination of n1, n2, and θ0 for the design 1.5 (n1n2) 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 7
Fig. 7

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)2 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 8
Fig. 8

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)3 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 9
Fig. 9

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)4 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 10
Fig. 10

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)5 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 11
Fig. 11

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)n1 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 12
Fig. 12

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)2n1 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 13
Fig. 13

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)3n1 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 14
Fig. 14

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)4n1 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Fig. 15
Fig. 15

Determination of n1, n2, and θ0 for the design 1.5 (n1n2)5n1 1 to obtain reflecting polarizers for the p component (solid lines) or s component (dashed lines).

Equations (17)

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n 1 t 1 cos θ 1 = n 2 t 2 cos θ 2 .
R m = ( 1 - P 1 + P ) 2 .
design             n s ( n 1 n 2 ) k n 0 ,             P = η 0 η s · ( η 1 η 2 ) 2 k ,
design             n s ( n 1 n 2 ) k n 1 n 0 ,             P = η 1 2 η 0 η s · ( η 1 η 2 ) 2 k .
s component             η i = n i cos θ i ,             i = 0 , 1 , 2 , s
p component             η i = n i / cos θ i ,             i = 0 , 1 , 2 , s .
cos θ i = [ 1 - ( n 0 2 / n i 2 ) sin 2 θ 0 ] 1 2 ,             i = 1 , 2 , s .
k = 1 2 · ln ( n s 2 - n 0 2 sin 2 θ 0 ) - ln ( n 0 2 - n 0 2 sin 2 θ 0 ) ln ( n 1 2 - n 0 2 sin 2 θ 0 ) - ln ( n 2 2 - n 0 2 sin 2 θ 0 ) ,
k = 1 2 · 4 ln n s + ln ( 1 - sin 2 θ 0 ) - 2 ln n 0 - ln ( n s 2 - n 0 2 sin 2 θ 0 ) 4 ln n 1 + ln ( n 2 2 - n 0 2 sin 2 θ 0 ) - 4 ln n 2 - ln ( n 1 2 - n 0 2 sin 2 θ 0 ) .
k = 1 2 · ln ( n s 2 - n 0 2 sin 2 θ 0 ) + l n ( n 0 2 - n 0 2 sin 2 θ 0 ) - 2 ln ( n 1 2 - n 0 2 sin 2 θ 0 ) ln ( n 1 2 - n 0 2 sin 2 θ 0 ) - ln ( n 2 2 - n 0 2 sin 2 θ 0 ) ,
k = 1 2 · 4 ln n s + 2 ln n 0 + 2 ln ( n 1 2 - n 0 2 sin 2 θ 0 ) - 8 ln n 1 - ln ( n s 2 - n 0 2 sin 2 θ 0 ) - ln ( 1 - sin 2 θ 0 ) 4 ln n 1 + ln ( n 2 2 - n 0 2 sin 2 θ 0 ) - 4 ln n 2 - ln ( n 1 2 - n 0 2 sin 2 θ 0 ) .
tan θ 0 = n s n 0
n 2 2 = A 1 ( n 1 2 - n 0 2 sin 2 θ 0 ) + n 0 2 sin 2 θ 0 ,
A 1 = ( n 0 2 - n 0 2 sin 2 θ 0 n s 2 - n 0 2 sin 2 θ 0 ) 1 / ( 2 k )             for             n s ( n 1 n 2 ) k n 0 , A 1 = ( ( n 1 2 - n 0 2 sin 2 θ 0 ) 2 ( n 0 2 - n 0 2 sin 2 θ 0 ) ( n s 2 - n 0 2 sin 2 θ 0 ) ) 1 / ( 2 k )             for             n s ( n 1 n 2 ) k n 1 n 0 .
n 2 2 = n 1 2 · n 1 2 ± [ n 1 4 - 4 A 2 n 0 2 sin 2 θ 0 ( n 1 2 - n 0 2 sin 2 θ 0 ) ] 1 2 2 A 2 ( n 1 2 - n 0 2 sin 2 θ 0 ) ,
A 2 = ( n s 4 n 0 2 · 1 - sin 2 θ 0 n s 2 - n 0 2 sin 2 θ 0 ) 1 / ( 2 k )             for             n s ( n 1 n 2 ) k n 0 , A 2 = ( n 0 2 n s 4 n 1 8 · ( n 1 2 - n 0 2 sin 2 θ 0 ) 2 ( 1 - sin 2 θ 0 ) ( n s 2 - n 0 2 sin 2 θ 0 ) ) 1 / ( 2 k )             for             n s ( n 1 n 2 ) k n 1 n 0 .
n i v = 2 n 0 sin θ 0 ,             i = 1 , 2.