Abstract

A general analysis of the propagation of electromagnetic waves through a stratified medium, applicable to the entire class of problems of plane multilayer systems, is developed. The medium is assumed to be divided into m plane-parallel layers, the first and mth of which are semi-infinite. The electromagnetic properties of the m layers are unrestricted and dissimilar, and the thicknesses of the (m−2) finite layers are also different in general. A steady-state monochromatic wave, generated in the mth layer, is assumed to be incident at any angle upon the (m−1)th layer. Formulas for the resulting reflection and transmission are derived. The methods which are employed in the paper facilitate application of these formulas, with a minimum of algebraic complications, to specific problems.

© 1950 Optical Society of America

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References

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  1. Rayleigh, Proc. Roy. Soc. A86, 207 (1912). This paper also appears in Rayleigh’s Scientific Papers (Cambridge University Press, Cambridge, 1920), Vol. VI, p. 71.
    [CrossRef]
  2. A. W. Crook, J. Opt. Soc. Am. 38, 954 (1948).
    [CrossRef] [PubMed]
  3. P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).
    [CrossRef]
  4. L. B. Lockhart and P. King, J. Opt. Soc. Am. 37, 689 (1947).
    [CrossRef]
  5. S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., New York, 1943). This text is recommended for any necessary collateral reading.
  6. O. Heaviside, Electromagnetic Theory (Electrician, London, 1894), Vol. I, Chapter II.
  7. H. Cafferata, Marconi Rev. 6 (No. 64) 12 (1937); Marconi Rev. 6 (No. 67) 21 (1937).A limited quantity of charts of the type described in this reference is available. They may be obtained by requesting Charts H. O. Misc. 9999, 9999a, 9999b, and 9999b-1 from the Technical Information Division, Naval Research Laboratory, Washington 25, D. C.
  8. K. B. Blodgett, Phys. Rev. 57, 921 (1940).
    [CrossRef]
  9. B. Salzberg, Am. J. Phys. 16, 444 (1948).
    [CrossRef]
  10. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–513.
  11. L. Pincherle, Phys. Rev. 72, 232 (1947).
    [CrossRef]

1948 (2)

1947 (2)

1946 (1)

1940 (1)

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

1937 (1)

H. Cafferata, Marconi Rev. 6 (No. 64) 12 (1937); Marconi Rev. 6 (No. 67) 21 (1937).A limited quantity of charts of the type described in this reference is available. They may be obtained by requesting Charts H. O. Misc. 9999, 9999a, 9999b, and 9999b-1 from the Technical Information Division, Naval Research Laboratory, Washington 25, D. C.

1912 (1)

Rayleigh, Proc. Roy. Soc. A86, 207 (1912). This paper also appears in Rayleigh’s Scientific Papers (Cambridge University Press, Cambridge, 1920), Vol. VI, p. 71.
[CrossRef]

Blodgett, K. B.

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

Cafferata, H.

H. Cafferata, Marconi Rev. 6 (No. 64) 12 (1937); Marconi Rev. 6 (No. 67) 21 (1937).A limited quantity of charts of the type described in this reference is available. They may be obtained by requesting Charts H. O. Misc. 9999, 9999a, 9999b, and 9999b-1 from the Technical Information Division, Naval Research Laboratory, Washington 25, D. C.

Crook, A. W.

Heaviside, O.

O. Heaviside, Electromagnetic Theory (Electrician, London, 1894), Vol. I, Chapter II.

King, P.

Lockhart, L. B.

Pincherle, L.

L. Pincherle, Phys. Rev. 72, 232 (1947).
[CrossRef]

Rayleigh,

Rayleigh, Proc. Roy. Soc. A86, 207 (1912). This paper also appears in Rayleigh’s Scientific Papers (Cambridge University Press, Cambridge, 1920), Vol. VI, p. 71.
[CrossRef]

Salzberg, B.

B. Salzberg, Am. J. Phys. 16, 444 (1948).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., New York, 1943). This text is recommended for any necessary collateral reading.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–513.

Am. J. Phys. (1)

B. Salzberg, Am. J. Phys. 16, 444 (1948).
[CrossRef]

J. Opt. Soc. Am. (3)

Marconi Rev. (1)

H. Cafferata, Marconi Rev. 6 (No. 64) 12 (1937); Marconi Rev. 6 (No. 67) 21 (1937).A limited quantity of charts of the type described in this reference is available. They may be obtained by requesting Charts H. O. Misc. 9999, 9999a, 9999b, and 9999b-1 from the Technical Information Division, Naval Research Laboratory, Washington 25, D. C.

Phys. Rev. (2)

K. B. Blodgett, Phys. Rev. 57, 921 (1940).
[CrossRef]

L. Pincherle, Phys. Rev. 72, 232 (1947).
[CrossRef]

Proc. Roy. Soc. (1)

Rayleigh, Proc. Roy. Soc. A86, 207 (1912). This paper also appears in Rayleigh’s Scientific Papers (Cambridge University Press, Cambridge, 1920), Vol. VI, p. 71.
[CrossRef]

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 511–513.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand Company, Inc., New York, 1943). This text is recommended for any necessary collateral reading.

O. Heaviside, Electromagnetic Theory (Electrician, London, 1894), Vol. I, Chapter II.

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Figures (1)

Fig. 1
Fig. 1

The boundary planes of the m layers are parallel to the yz plane. The normal to the incident plane wave, making an angle θm with the x-axis, is in the xz plane, which is the plane of incidence. The situation depicted here corresponds to the case for which the electric vector is perpendicular to the plane of incidence and the magnetic vector has both normal and tangential components.

Equations (28)

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P = ( i ω / c 0 ) [ ( - i ) ( μ - i μ ) ] 1 2 = [ ( σ + i ω 0 ) ( ρ + i ω μ μ 0 ) ] 1 2 A = A 0 [ ( - i ) / ( μ - i μ ) ] 1 2 = [ ( σ + i ω 0 ) / ( ρ + i ω μ μ 0 ) ] 1 2 .
sin θ k = ( P m / P k ) sin θ m p k = P k [ 1 - ( P m / P k ) 2 sin 2 θ m ] 1 2 .
v = ω [ 1 / I m 2 ( p ) + 1 / I m 2 ( q ) ] 1 2 ,
E = E t exp ( x 0 - x ) p + E r exp ( x - x 0 ) p H = H t exp ( x 0 - x ) p - H r exp ( x - x 0 ) p .
( H t / E t ) = ( H r / E r ) = A = A cos θ .
( H t / E t ) = ( H r / E r ) = A = A / cos θ .
E 1 = E 2 cosh p d + ( H 2 / A ) sinh p d H 1 = E 2 A sinh p d + H 2 cosh p d .
( E 1 ) k = ( E 1 ) j cosh ( p d ) k + [ ( H 1 ) j / A k ] sinh ( p d ) k ( H 1 ) k = ( E 1 ) j A k sinh ( p d ) k + ( H 1 ) j cosh ( p d ) k .
Y k = A k [ Y j + A k tanh ( p d ) k ] / [ A k + Y j tanh ( p d ) k ] .
r = ( E r / E t ) m = ( H r / H t ) m = ( A m - Y m - 1 ) / ( A m + Y m - 1 ) .
t = ( E 1 ) m - 1 / ( E t ) m = 1 + r = 2 A m / ( A m + Y m - 1 ) t = ( H 1 ) m - 1 / ( H t ) m = 1 - r = 2 Y m - 1 / ( A m + Y m - 1 ) = ( Y m - 1 / A m ) t .
P 1 = i ω n 1 / c 0 P 2 = i ω n 2 / c 0 A 1 = A 0 n 1 A 2 = A 0 n 2 .
A 1 = Y 1 = A 0 n 1 [ 1 - ( n 2 / n 1 ) 2 sin 2 θ 2 ] 1 2 A 2 = A 0 n 2 cos θ 2
A 1 = Y 1 = A 0 n 1 / [ 1 - ( n 2 / n 1 ) 2 sin 2 θ 2 ] 1 2 A 2 = A 0 n 2 / cos θ 2 .
r = [ n cos θ 2 - ( 1 - n 2 sin 2 θ 2 ) 1 2 ] / [ n cos θ 2 + ( 1 - n 2 sin 2 θ 2 ) 1 2 ]
r = [ n ( 1 - n 2 sin 2 θ 2 ) 1 2 - cos θ 2 ] / [ n ( 1 - n 2 sin 2 θ 2 ) 1 2 + cos θ 2 ]
t = 1 + r ,             t = 1 - r .
P k = i ω n k / c 0 = i 2 π n k / λ A k = A 0 n k ,             k = 1 , 2 , 3
p 2 = i ( 2 π n 2 / λ ) [ 1 - ( n 3 / n 2 ) 2 sin 2 θ 3 ] 1 2
A k = A 0 n k [ 1 - ( n 3 / n k ) 2 sin 2 θ 3 ] 1 2 ,             k = 1 , 2 A 3 = A 0 n 3 cos θ 3 .
Y 1 = A 1 Y 2 = A 2 [ A 1 + i A 2 tan ψ ] / [ A 2 + i A 1 tan ψ ] ,
( n 2 2 / n 1 n 3 ) = [ 1 - ( n 3 / n 1 ) 2 sin 2 θ 3 ] 1 2 cos θ 3 + ( n 3 / n 1 ) sin 2 θ 3 .
n 2 4 [ 1 - ( n 3 / n 1 ) 2 sin 2 θ 3 ] 1 2 cos θ 3 - n 1 n 3 n 2 2 + n 1 n 3 3 sin 2 θ 3 = 0.
n 2 2 = n 1 n 3 , and d 2 = ( n λ / 4 n 2 ) , where n = 1 , 3 , 5 , .
( n m / n 1 ) = ( n m - 1 n m - 3 n 3 ) 2 / ( n m - 2 n m - 4 n 2 ) 2 ,             m = 4 , 6 , 8 , ( n m / n 1 ) = ( n m - 1 n m - 3 n 2 ) 2 / ( n m - 2 n m - 4 n 1 ) 2 ,             m = 3 , 5 , 7 , .
t m , 1 = ( E 1 ) 1 / ( E t ) m = t k = m - 1 k = 2 [ cosh ( p d ) k + ( Y j + A k ) sinh ( p d ) k ] - 1 t m , 1 = ( H 1 ) 1 / ( H t ) m = ( A 1 A m ) t m , 1 .
R = r 2 ,             T = 1 - R = t 2 R e ( Y m - 1 ) / A m .
T m , 1 = R e ( E 1 H 1 * ) 1 / R e ( E t H t * ) m = t m , 1 2 R e ( A 1 ) / A m ,