Abstract

According to Herpin’s theorem, a symmetrical thin-film combination is equivalent, at one wavelength, to a single film, characterized by an equivalent index and equivalent thickness. If a large number of identical nonabsorbing thin-film combinations are combined, the resulting periodically stratified medium acts as a band-stop filter. In a stop band, the equivalent index of a symmetrical period is a pure imaginary; in a pass band, it is real. Equivalent indices are given for some commonly used periods. These may have values not possessed by any existing substance. These periods have proved useful in the design of optical filters and antireflection coatings.

© 1952 Optical Society of America

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References

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  1. A. Herpin, Compt. rend. 225, 182 (1947).
  2. F. Abelès, Ann. phys. [12]  5, 596 and 706 (1950).
  3. L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill Book Company, Inc., New York, 1946), first edition, Chapter IX.
  4. S. A. Schelkunoff, Bell System Tech. J. 17, 17 (1938).
    [CrossRef]
  5. R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948).
    [CrossRef]
  6. H. A. Kramers, Physica 2, 483 (1935).
    [CrossRef]
  7. See E. L. Ince, Ordinary Differential Equations (Longmans, Green and Company, London, 1926), first edition, pp. 242–248.
  8. See, for example, L. Brillouin, Quart. Appl. Math. VII, 363 (1950).
  9. Rayleigh, Proc. Roy. Soc. (London) A93, 565 (1917).
    [CrossRef]

1950 (2)

F. Abelès, Ann. phys. [12]  5, 596 and 706 (1950).

See, for example, L. Brillouin, Quart. Appl. Math. VII, 363 (1950).

1948 (1)

1947 (1)

A. Herpin, Compt. rend. 225, 182 (1947).

1938 (1)

S. A. Schelkunoff, Bell System Tech. J. 17, 17 (1938).
[CrossRef]

1935 (1)

H. A. Kramers, Physica 2, 483 (1935).
[CrossRef]

1917 (1)

Rayleigh, Proc. Roy. Soc. (London) A93, 565 (1917).
[CrossRef]

Abelès, F.

F. Abelès, Ann. phys. [12]  5, 596 and 706 (1950).

Brillouin, L.

See, for example, L. Brillouin, Quart. Appl. Math. VII, 363 (1950).

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill Book Company, Inc., New York, 1946), first edition, Chapter IX.

Herpin, A.

A. Herpin, Compt. rend. 225, 182 (1947).

Ince, E. L.

See E. L. Ince, Ordinary Differential Equations (Longmans, Green and Company, London, 1926), first edition, pp. 242–248.

Kramers, H. A.

H. A. Kramers, Physica 2, 483 (1935).
[CrossRef]

Muchmore, R. B.

Rayleigh,

Rayleigh, Proc. Roy. Soc. (London) A93, 565 (1917).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, Bell System Tech. J. 17, 17 (1938).
[CrossRef]

Ann. phys. (1)

F. Abelès, Ann. phys. [12]  5, 596 and 706 (1950).

Bell System Tech. J. (1)

S. A. Schelkunoff, Bell System Tech. J. 17, 17 (1938).
[CrossRef]

Compt. rend. (1)

A. Herpin, Compt. rend. 225, 182 (1947).

J. Opt. Soc. Am. (1)

Physica (1)

H. A. Kramers, Physica 2, 483 (1935).
[CrossRef]

Proc. Roy. Soc. (London) (1)

Rayleigh, Proc. Roy. Soc. (London) A93, 565 (1917).
[CrossRef]

Quart. Appl. Math. (1)

See, for example, L. Brillouin, Quart. Appl. Math. VII, 363 (1950).

Other (2)

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill Book Company, Inc., New York, 1946), first edition, Chapter IX.

See E. L. Ince, Ordinary Differential Equations (Longmans, Green and Company, London, 1926), first edition, pp. 242–248.

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Figures (7)

F. 1
F. 1

Equivalent thickness vs actual thickness of some symmetrical thin-film combinations. Refer to Table I for labeling of curves.

F. 2
F. 2

Equivalent index vs actual thickness of some symmetrical thin-film combinations. Refer to Table I for labeling of curves.

F. 3
F. 3

Equivalent thickness vs actual thickness of some symmetrical thin-film combinations. Refer to Table I for labeling of curves.

F. 4
F. 4

Equivalent index vs actual thickness of some symmetrical thin-film combinations. Refer to Table I for labeling of curves.

F. 5
F. 5

Equivalent indices of two periodically stratified media used to form a band-pass filter.

F. 6
F. 6

Calculated reflectance curve for a band-pass filter. For the design of this filter, refer to Table III.

F. 7
F. 7

Calculated reflectance curve (solid) for the design shown in Table IV, intended to simulate R. B. Muchmore’s antireflection coating. The dashed curve gives the reflectance of a single film of magnesium fluoride.

Tables (4)

Tables Icon

Table I Constants of symmetrical thin-film combinations and labeling of curves in Figs. 1 through 4.

Tables Icon

Table II Equivalent index N of symmetrical thin-film combinations for use as antireflection coatings.

Tables Icon

Table III Design for a band-pass filter.

Tables Icon

Table IV Design to simulate R. B. Muchmore’s antireflection coating.

Equations (12)

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( E H ) = ( M 11 M 12 M 21 M 22 ) ( E H ) .
( cos ( 2 π n x / λ 0 ) ( i / n ) sin ( 2 π n x / λ 0 ) i n sin ( 2 π n x / λ 0 ) cos ( 2 π n x / λ 0 ) ) ,
φ = 2 π n x / λ 0 = n ω x / c
M 11 = M 22 = cos γ
N = ( i sin γ ) / M 12 = i M 21 / sin γ .
M 11 = M 22 = cos 2 φ p cos φ q 1 2 ( n q / n p + n p / n q ) sin 2 φ p sin φ q ,
M 12 = ( i / n p ) { sin 2 φ p cos φ q + 1 2 ( n p / n q + n q / n p ) cos 2 φ p sin φ q + 1 2 ( n p / n q n q / n p ) sin φ q } ,
M 21 = i n p { sin 2 φ p cos φ q + 1 2 ( n p / n q + n q / n p ) cos 2 φ p sin φ q 1 2 ( n p / n q n q / n p ) sin φ q } .
E + ( ω / c ) 2 [ n ( x ) ] 2 E = 0 ,
E + [ ω n ( x ) ] E = 0 .
1 2
1 2