Abstract

The one-to-one correspondence between transmission line theory and the theory of the optical behavior of thin films is discussed in detail. Advantage is taken of this analogy to apply the methods developed for the solution of transmission line problems to problems involving transparent dielectric films. In particular, the use of Smith charts as a convenient way of doing these calculations is suggested. The reflection from a triple film is worked out to illustrate the general method, and the conditions for zero reflectivity of a single film are deduced in a simple fashion. A variation of the method is then used to derive a new way of obtaining the optical constants of opaque metal films. This method has been applied to aluminum films, and the result obtained is in good agreement with that obtained from polarization measurements.

© 1951 Optical Society of America

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References

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  1. S. A. Schelkunoff, Bell System Tech. J. 17, 17 (1938).
    [Crossref]
  2. (a) Konig, Blaisse, and v. d. Sande, Appl. Sci. Res.B1, p. 63. (b)R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948). (c)B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).
  3. P. H. Smith, Electronics (January, 1939); ibid. (January, 1944).
  4. F. Abeles, Compt. rend. 228, 553 (1949).
  5. G. Hass, Optik 1, 2 (1946).

1949 (1)

F. Abeles, Compt. rend. 228, 553 (1949).

1946 (1)

G. Hass, Optik 1, 2 (1946).

1939 (1)

P. H. Smith, Electronics (January, 1939); ibid. (January, 1944).

1938 (1)

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

Abeles, F.

F. Abeles, Compt. rend. 228, 553 (1949).

Blaisse,

(a) Konig, Blaisse, and v. d. Sande, Appl. Sci. Res.B1, p. 63. (b)R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948). (c)B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

Hass, G.

G. Hass, Optik 1, 2 (1946).

Konig,

(a) Konig, Blaisse, and v. d. Sande, Appl. Sci. Res.B1, p. 63. (b)R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948). (c)B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

Schelkunoff, S. A.

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

Smith, P. H.

P. H. Smith, Electronics (January, 1939); ibid. (January, 1944).

v. d. Sande,

(a) Konig, Blaisse, and v. d. Sande, Appl. Sci. Res.B1, p. 63. (b)R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948). (c)B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

Bell System Tech. J. (1)

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

Compt. rend. (1)

F. Abeles, Compt. rend. 228, 553 (1949).

Electronics (1)

P. H. Smith, Electronics (January, 1939); ibid. (January, 1944).

Optik (1)

G. Hass, Optik 1, 2 (1946).

Other (1)

(a) Konig, Blaisse, and v. d. Sande, Appl. Sci. Res.B1, p. 63. (b)R. B. Muchmore, J. Opt. Soc. Am. 38, 20 (1948). (c)B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

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Figures (4)

Fig. 1
Fig. 1

Continuous transmission line with load.

Fig. 2
Fig. 2

Smith chart with sample computation., The points A, B, C, D, E, F, G are obtained in that order. The reflectivity is the square of the distance OG, if the square of the radius of the chart is taken as 100 percent.

Fig. 3
Fig. 3

Three-layer film combination on glass (schematic). The three films are of equal optical thickness.

Fig. 4
Fig. 4

Film combination used for determining the refractive index of aluminum.

Tables (1)

Tables Icon

Table I Comparison of the optical constants of aluminum.

Equations (18)

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d V / d x = - Z S I ,             d I / d x = - Y P V ,
I + = A exp ( - Γ x + i ω t ) , V + = Z 0 I + , Y 0 V + = I + , I - = B exp ( + Γ x + i ω t ) , V - = - Z 0 I - , Y 0 V - = - I - ,
Γ = ( Z S Y P ) 1 2 ,             Z 0 = 1 / Y 0 = ( Z S / Y P ) 1 2 .
Y = ( I + + I - ) / ( V + + V - ) = Y 0 ( 1 - V - / V + ) / ( 1 + V - / V + ) .
Y L = ( I + + I - ) / ( V + + V - ) x = l = Y 0 [ 1 - ( V - / V + ) exp ( 2 Γ l ) ] / [ 1 + ( V - / V + ) exp ( 2 Γ l ) ] .
Y = Y 0 ( Y L cosh Γ l + Y 0 sinh Γ l ) / ( Y 0 cosh Γ l + Y L sinh Γ l ) , Y / Y 0 = ( Y L / Y 0 + tanh Γ l ) / ( 1 + Y L / Y 0 tanh Γ l ) .
Y 0 = I i / V i .
I i + I r = I L ,             and             V i + V r = V L ;
I i = Y 0 V i ,             I r = - Y 0 V r ,             I L = Y L V L .
R V = V r / V i = ( 1 - k ) / ( 1 + k ) ; T V = V L / V i = 2 / ( 1 + k ) .
- d H z / d x = ( i ω κ / c ) E y ;             d E y / d x = - ( i μ ω / c ) H z .
Z x = E y / H z .
Γ = [ ( i μ ω / c ) ( i μ κ / c ) ] 1 2 = i ω n / c , Z 0 = [ ( i μ ω / c ) ( c / i κ ω ) ] 1 2 = ( μ / κ ) 1 2             or             Y 0 = n             if             μ = 1 ,
Y / Y 0 = [ Y L / Y 0 + i tan ( 2 π n l / λ ) ] / [ 1 + i ( Y L / Y 0 ) tan ( 2 π n l / λ ) ] ,
Y / Y 0 = Y 0 / Y L = 2.3 / 1.52.
Y / Y 0 = 1.52 × 1.38 / ( 2.3 ) 2 .
k = Y / Y 0 = ( 2.3 ) 4 / [ ( 1.38 ) 2 × 1.52 ] = 9.67.
Y L / Y 0 = [ Y / Y 0 + i tan ( - 2 π n l / λ ) ] / [ 1 + i ( Y / Y 0 ) tan ( - 2 π n l / λ ) ] .