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  1. K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1938).
    [CrossRef]
  2. K. B. Blodgett, Phys. Rev. 57, 921 (1940).
    [CrossRef]
  3. A. F. Turner, J. App. Phys. 12, 351 (1941).
    [CrossRef]
  4. K. M. Greenland, Nature 152, 290 (1943).
    [CrossRef]
  5. E. O. Hulbert, Approximate Theory of Low Reflecting Films, Naval Research Laboratory, November18, 1944.
  6. M. Born, Optik (Julius Springer, Berlin), p. 125.
  7. C. E. Leberknight and Benjamin Luftman, J. Opt. Soc. Am. 29, 65 (1939).
    [CrossRef]
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), first edition, 511–515.

1943 (1)

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

1941 (1)

A. F. Turner, J. App. Phys. 12, 351 (1941).
[CrossRef]

1940 (1)

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

1939 (1)

C. E. Leberknight and Benjamin Luftman, J. Opt. Soc. Am. 29, 65 (1939).
[CrossRef]

1938 (1)

K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1938).
[CrossRef]

Blodgett, K. B.

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

K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1938).
[CrossRef]

Born, M.

M. Born, Optik (Julius Springer, Berlin), p. 125.

Greenland, K. M.

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

Hulbert, E. O.

E. O. Hulbert, Approximate Theory of Low Reflecting Films, Naval Research Laboratory, November18, 1944.

Langmuir, I.

K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1938).
[CrossRef]

Leberknight, C. E.

C. E. Leberknight and Benjamin Luftman, J. Opt. Soc. Am. 29, 65 (1939).
[CrossRef]

Luftman, Benjamin

C. E. Leberknight and Benjamin Luftman, J. Opt. Soc. Am. 29, 65 (1939).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), first edition, 511–515.

Turner, A. F.

A. F. Turner, J. App. Phys. 12, 351 (1941).
[CrossRef]

J. App. Phys. (1)

A. F. Turner, J. App. Phys. 12, 351 (1941).
[CrossRef]

J. Opt. Soc. Am. (1)

C. E. Leberknight and Benjamin Luftman, J. Opt. Soc. Am. 29, 65 (1939).
[CrossRef]

Nature (1)

K. M. Greenland, Nature 152, 290 (1943).
[CrossRef]

Phys. Rev. (2)

K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1938).
[CrossRef]

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

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), first edition, 511–515.

E. O. Hulbert, Approximate Theory of Low Reflecting Films, Naval Research Laboratory, November18, 1944.

M. Born, Optik (Julius Springer, Berlin), p. 125.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram to identify amplitudes of radiation in the different media with directions of propagation for case of monolayer coating. Regions 1, 2, and 3 are air, coating material, and glass, respectively.

Fig. 2
Fig. 2

Reflectivity versus wave-length of incident light for monolayer quarter wave-length thick coating.

Fig. 3
Fig. 3

Schematic diagram to identify amplitudes of radiation in the different media with directions of propagation for case of two-layer coating. Subscripts 1, 2, 3, and 4 refer to air, the top layer, the layer adjacent to the glass and glass, respectively.

Fig. 4
Fig. 4

Reflectivity versus thickness of layer adjacent to glass for two-layer coating. Comparison of results obtained by an approximation method with results given by Eq. (49).

Fig. 5
Fig. 5

Refractive index matching chart for two-layer coatings. The ordinate and abscissa of any point on one of the curves for a given kind of glass indicate the combination of indices for the coating materials to give zero reflectance at one wave-length.

Fig. 6
Fig. 6

Reflectivity versus wave-length for two-layer coatings. The three curves represent reflectivities for coating thicknesses of one-quarter wave-length for light with with wave-lengths in air of 475 mμ, 515 mμ, and 550 mμ as indicated by the abscissae for zero ordinates.

Equations (75)

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· D ¯ = 4 π ρ ,
· B ¯ = 0 ,
× Ē = - ( 1 / c ) ( B ¯ / t ) ,
× H ¯ = ( 1 / c ) [ ( D ¯ / t ) + 4 π σ Ē ] ,
ϵ · Ē = 4 π ρ = 0 ,
· H ¯ = 0 ,
× Ē = - ( μ / c ) ( H ¯ / t ) ,
× H ¯ = ϵ c ( Ē t + 4 π σ ϵ Ē ) .
· Ē - 2 Ē = - ( μ / c ) ( / t ) ( × H ¯ ) ,
2 Ē = μ ϵ c 2 2 Ē t 2 + 4 π μ σ c 2 Ē t .
2 Ē = ( μ ϵ / c 2 ) ( 2 Ē / t 2 ) .
Ē = Ē 0 e i ω t ,
E z / x = ( μ / c ) ( H y / t ) ,
E y / x = - ( μ / c ) ( H z / t ) .
E y = E 0 y e i ( ω t - k x ) ,             and             E z - E 0 z e i ( ω t - k x ) .
H z = ( c k / μ ω ) E 0 y e i ( ω t - k x ) ,
H y = - ( c k / μ ω ) E 0 z e i ( ω t - k x ) .
H 0 z = ( c k / μ ω ) E 0 y             and             H 0 y = - ( c k / μ ω ) E 0 z ,
H z = H 0 z e i ( ω t - k x ) ,             and             H y = H 0 y e i ( ω t - k x ) .
H 0 z = ( ϵ / μ ) 1 2 E 0 y ,             and             H 0 y = - ( ϵ / μ ) 1 2 E 0 z .
E i = E 0 exp [ i ( ω t - k 1 x ) ] ; E r = E 1 exp [ i ( ω t - k 1 x ) ] ;
E m = ( E 2 + exp ( - i k 2 x ) + E 2 - exp ( i k 2 x ) ) e i ω t ;
E t = E 3 exp [ i ( ω t - k 3 x ) ] .
E 0 + E 1 = E 2 + + E 2 - ,
E 2 + exp ( - i k 2 h ) + E 2 - exp ( i k 2 h ) = E 3 exp ( - i k 3 h ) ,
( ϵ 1 / μ 1 ) 1 2 ( E 0 - E 1 ) = ( ϵ 2 / μ 2 ) 1 2 ( E 2 + - E 2 - ) ,
( ϵ 2 / μ 2 ) 1 2 [ E 2 + exp ( - i k 2 h ) - E 2 - exp ( i k 2 h ) ] = ( ϵ 3 / μ 3 ) 1 2 E 3 exp ( - i k 3 h ) .
E 0 [ 1 + ( μ 2 ϵ 1 μ 1 ϵ 2 ) 1 2 ] + E 1 [ 1 - ( μ 2 ϵ 1 μ 1 ϵ 2 ) 1 2 ] = E 3 [ 1 + ( μ 2 ϵ 3 μ 3 ϵ 2 ) 1 2 ] exp [ - i ( k 3 - k 2 ) h ] .
E 1 = [ ( ϵ 2 μ 2 ) 1 2 + ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 - ( ϵ 2 μ 2 ) 1 2 ] + [ ( ϵ 2 μ 2 ) 1 2 - ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 + ( ϵ 2 μ 2 ) 1 2 ] exp ( - 2 i k 2 h ) [ ( ϵ 2 μ 2 ) 1 2 + ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 + ( ϵ 2 μ 2 ) 1 2 ] + [ ( ϵ 2 μ 2 ) 1 2 - ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 - ( ϵ 2 μ 2 ) 1 2 ] exp ( - 2 i k 2 h ) E 0 ;
E 3 = 4 ( ϵ 1 ϵ 2 μ 1 μ 2 ) 1 2 exp ( i k 3 h ) [ ( ϵ 2 μ 2 ) 1 2 + ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 + ( ϵ 2 μ 2 ) 1 2 ] exp ( i k 2 h ) + [ ( ϵ 2 μ 2 ) 1 2 - ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 - ( ϵ 2 μ 2 ) 1 2 ] exp ( - i k 2 h ) E 0 .
E 1 = ( ϵ 2 1 2 + ϵ 3 1 2 ) ( 1 - ϵ 2 1 2 ) + ( ϵ 2 1 2 - ϵ 3 1 2 ) ( 1 + ϵ 2 1 2 ) exp ( - 2 i k 2 h ) ( ϵ 2 1 2 + ϵ 3 1 2 ) ( 1 + ϵ 2 1 2 ) + ( ϵ 2 1 2 - ϵ 3 1 2 ) ( 1 - ϵ 2 1 2 ) exp ( - 2 i k 2 h ) E 0 ;
E 3 = 4 ϵ 2 1 2 exp ( i k 3 h ) ( ϵ 2 1 2 + ϵ 3 1 2 ) ( 1 + ϵ 2 1 2 ) exp ( i k 2 h ) + ( ϵ 2 1 2 - ϵ 3 1 2 ) ( 1 - ϵ 2 1 2 ) exp ( - i k 2 h ) E 0 .
R = [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) exp ( 2 i k 2 h ) + ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) exp ( - 2 i k 2 h ) ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) exp ( 2 i k 2 h ) + ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) exp ( - 2 i k 2 h ) . R = [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + 2 ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) cos ( 2 k 2 h ) [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + 2 ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) cos ( 2 k 2 h ) .
R min = [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 - 2 ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 - 2 ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) ,
R min = { [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] [ 1 - ( ϵ 2 ) 1 2 ] - [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] [ 1 + ( ϵ 2 ) 1 2 ] [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] [ 1 + ( ϵ 2 ) 1 2 ] - [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] [ 1 - ( ϵ 2 ) 1 2 ] } 2 = { ( ϵ 3 ) 1 2 - ϵ 2 ( ϵ 3 ) 1 2 + ϵ 2 } 2 .
T = 16 ϵ 2 ( ϵ 3 ) 1 2 [ ( ϵ 2 ) 1 2 + ( ϵ 3 ) 1 2 ] 2 [ 1 + ( ϵ 2 ) 1 2 ] 2 + [ ( ϵ 2 ) 1 2 - ( ϵ 3 ) 1 2 ] 2 [ 1 - ( ϵ 2 ) 1 2 ] 2 + 2 ( ϵ 2 - ϵ 3 ) ( 1 - ϵ 2 ) cos ( 2 k 2 h ) .
E 2 + = 2 ( ϵ 1 / μ 1 ) 1 2 [ ( ϵ 2 / μ 2 ) 1 2 + ( ϵ 3 / μ 3 ) 1 2 ] [ ( ϵ 2 μ 2 ) 1 2 + ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 + ( ϵ 2 μ 2 ) 1 2 ] + [ ( ϵ 2 μ 2 ) 1 2 - ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 - ( ϵ 2 μ 2 ) 1 2 ] × exp ( - 2 i ( k 2 - k 3 ) h ) E 0 ;
E 2 - = 2 ( ϵ 1 / μ 1 ) 1 2 [ ( ϵ 2 / μ 2 ) 1 2 - ( ϵ 3 / μ 3 ) 1 2 ] [ ( ϵ 2 μ 2 ) 1 2 + ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 + ( ϵ 2 μ 2 ) 1 2 ] exp ( 2 i k 2 h ) + [ ( ϵ 2 μ 2 ) 1 2 - ( ϵ 3 μ 3 ) 1 2 ] [ ( ϵ 1 μ 1 ) 1 2 - ( ϵ 2 μ 2 ) 1 2 ] E 0 .
R = ( n 2 + n 3 ) 2 ( 1 - n 2 ) 2 + ( n 2 - n 3 ) 2 ( 1 + n 2 ) 2 + 2 ( n 2 2 - n 3 2 ) ( 1 - n 2 2 ) cos ( 2 k 2 h ) ( n 2 + n 3 ) 2 ( 1 + n 2 ) 2 + ( n 2 - n 3 ) 2 ( 1 - n 2 ) 2 + 2 ( n 2 2 - n 3 2 ) ( 1 - n 2 2 ) cos ( 2 k 2 h ) .
T = 16 n 2 2 n 3 ( n 2 + n 3 ) 2 ( 1 + n 2 ) 2 + ( n 2 - n 3 ) 2 ( 1 - n 2 ) 2 + 2 ( n 2 2 - n 3 2 ) ( 1 - n 2 2 ) cos ( 2 k 2 h ) .
E 1 - E 2 + - E 2 - = - E 0 ,
- E 1 - n 2 E 2 + + n 2 E 2 - = - E 0 ,
E 2 + exp ( i k 2 h 2 ) + E 2 - exp ( - i k 2 h 2 ) - E 3 + exp ( i k 3 h 2 ) - E 3 - exp ( - i k 3 h 2 ) = 0 ,
n 2 E 2 + exp ( i k 2 h 2 ) - n 2 E 2 - exp ( - i k 2 h 2 ) - n 3 E 3 + exp ( i k 3 h 2 ) + n 3 E 3 - exp ( - i k 3 h 2 ) = 0 ,
E 3 + exp [ i k 3 ( h 2 + h 3 ) ] + E 3 - exp [ - i k 3 ( h 2 + h 3 ) ] - E 4 exp [ i k 4 ( h 2 + h 3 ) ] = 0 ,
n 3 E 3 + exp [ i k 3 ( h 2 + h 3 ) ] - n 3 E 3 - exp [ - i k 3 ( h 2 + h 3 ) ] - n 4 E 4 exp [ i k 4 ( h 2 + h 3 ) ] = 0.
E 1 = N 8 exp [ - i ( k 3 h 3 + k 2 h 2 ) ] + N 5 exp [ i ( k 3 h 3 - k 2 h 2 ) ] + N 3 exp [ - i ( k 3 h 3 - k 2 h 2 ) ] + N 2 exp [ i ( k 3 h 3 + k 2 h 2 ) ] N 1 exp [ - i ( k 3 h 3 + k 2 h 2 ) ] + N 4 exp [ i ( k 3 h 3 - k 2 h 2 ) ] + N 6 exp [ - i ( k 3 h 3 - k 2 h 2 ) ] + N 7 exp [ i ( k 3 h 3 + k 2 h 2 ) ] E 0 ;
E 4 = - 8 n 2 n 3 E 0 / { N 1 exp [ - i ( k 3 h 3 + k 2 h 2 ) ] + N 4 exp [ i ( k 3 h 3 - k 2 h 2 ) ] + N 6 exp [ - i ( k 3 h 3 - k 2 h 2 ) ] + N 7 exp [ i ( k 3 h 3 + k 2 h 2 ) ] } .
R = E 1 E 1 * / E 0 2 ,
T = n 4 E 4 E 4 * / E 0 2 .
R = N 8 2 + N 5 2 + N 3 2 + N 2 2 + A cos x + B cos ( x + y ) + C cos ( x - y ) + D cos y N 1 2 + N 4 2 + N 6 2 + N 7 2 + A cos x + B cos ( x + y ) + C cos ( x - y ) + D cos y ;
x = 2 k 3 h 3 ,             A = 4 ( n 2 2 + 1 ) ( n 3 2 - n 2 2 ) ( n 4 2 - n 3 2 ) ,             C = 2 ( n 2 2 - 1 ) ( n 3 - n 2 ) 2 ( n 4 2 - n 3 2 ) , y = 2 k 2 h 2 ,             B = 2 ( n 2 2 - 1 ) ( n 3 + n 2 ) 2 ( n 4 2 - n 3 2 ) ,             D = 4 ( n 2 2 - 1 ) ( n 3 2 - n 2 2 ) ( n 4 2 + n 3 2 ) .
T = 64 n 2 2 n 3 2 n 4 / [ N 1 2 + N 4 2 + N 6 2 + N 7 2 + A cos x + B cos ( x + y ) + C cos ( x - y ) + D cos y ] .
F ( x , y ) = A cos x + B cos ( x + y ) + C cos ( x - y ) + D cos y .
F / x = - A sin x - B sin ( x + y ) - C sin ( x - y ) ,
2 F / x 2 = - A cos x - B cos ( x + y ) - C cos ( x - y ) ,
F / y = - B sin ( x + y ) + C sin ( x - y ) - D sin y ,
2 F / y 2 = - B cos ( x + y ) - C cos ( x - y ) - D cos y ,
2 F / x y = - B cos ( x + y ) + C cos ( x - y ) .
( 2 F x y ) 2 - 2 F x 2 2 F y 2 = - 3 B C cos ( x + y ) cos ( x - y ) + A B cos x cos ( x + y ) + A C cos x cos ( x - y ) + A D cos x cos y + B D cos ( x + y ) cos y + C B cos ( x + y ) cos ( x - y ) + C D cos ( x - y ) cos y .
F / x = 0 = F / y ,             2 F / x 2 = 6.98 , 2 F / y 2 = 32.0 ,                         ( 2 F / x y ) 2 = 163 ,
d E = - α E d x ,
E = A e - α x e i ( k x - ω t ) ,
E = B e α x e - i ( k x + ω t ) .
E 1 - E 2 + - E 2 - = - E 0 ,
- E 1 - n 2 E 2 + + n 2 E 2 - = - E 0 ,
E 2 + exp [ ( - α 2 + i k 2 ) h 2 ] + E 2 - exp [ ( α 2 - i k 2 ) h 2 ] - E 3 + exp [ ( - α 3 + i k 3 ) h 2 ] - E 3 - exp [ ( α 3 - i k 3 ) h 2 ] = 0 ,
n 2 E 2 + exp [ ( - α 2 + i k 2 ) h 2 ] - n 2 E 2 - exp [ ( α 2 - i k 2 ) h 2 ] - n 3 E 3 + exp [ ( - α 3 + i k 3 ) h 2 ] + n 3 E 3 - exp [ ( α 3 - i k 3 ) h 2 ] = 0 ,
E 3 + exp [ ( - α 3 + i k 3 ) ( h 2 + h 3 ) ] + E 3 - exp [ ( α 3 - i k 3 ) ( h 2 + h 3 ) ] - E 4 exp [ ( - α 4 + i k 4 ) ( h 2 + h 3 ) ] = 0 ,
n 3 E 3 + exp ( - α 3 + i k 3 ) ( h 2 + h 3 ) - n 3 E 3 - exp [ ( α 3 - i k 3 ) ( h 2 + h 3 ) ] - n 4 E 4 exp [ ( - α 4 + i k 4 ) ( h 2 + h 3 ) ] = 0.
R = N 8 2 exp [ 2 ( α 3 h 3 + α 2 h 2 ) ] + N 5 2 exp [ 2 ( - α 3 h 3 + α 2 h 2 ) ] + N 3 2 exp [ 2 ( α 3 h 3 - α 2 h 2 ) ] N 1 2 exp [ 2 ( α 3 h 3 + α 2 h 2 ) ] + N 4 2 exp [ 2 ( - α 3 h 3 + α 2 h 2 ) ] + N 6 2 exp [ 2 ( α 3 h 3 - α 2 h 2 ) ] + N 2 2 exp [ - 2 ( α 3 h 3 + α 2 h 2 ) ] + [ A cosh 2 α 2 h 2 - 8 ( n 3 2 - n 2 2 ) ( n 4 2 - n 3 2 ) n 2 sinh 2 α 2 h 2 ] cos x + N 7 2 exp [ - 2 ( α 3 h 3 + α 2 h 2 ) ] + [ A cosh 2 α 2 h 2 - 8 ( n 3 2 - n 2 2 ) ( n 4 2 - n 3 2 ) n 2 sinh 2 α 2 h 2 ] cos x + B cos ( x + y ) + C cos ( x - y ) + [ D cosh 2 α 3 h 3 - 8 ( n 2 2 - 1 ) ( n 3 2 - n 2 2 ) n 4 n 3 sinh 2 α 3 h 3 ] cos y , + B cos ( x + y ) + C cos ( x - y ) + [ D cosh 2 α 3 h 3 - 8 ( n 2 2 - 1 ) ( n 3 2 - n 2 2 ) n 4 n 3 sinh 2 α 3 h 3 ] cos y ,
T = 64 n 2 2 n 3 2 n 4 / { N 1 2 exp [ 2 ( α 3 h 3 + α 2 h 2 ) ] + N 4 2 exp [ 2 ( - α 3 h 3 + α 2 h 2 ) ] + N 6 2 exp [ 2 ( α 3 h 3 - α 2 h 2 ) ] + N 7 2 exp [ - 2 ( α 3 h 3 + α 2 h 2 ) ] + [ A cosh 2 α 2 h 2 - 8 ( n 3 2 - n 2 2 ) ( n 4 2 - n 3 2 ) n 2 sinh 2 α 2 h 2 ] cos x + B cos ( x + y ) + C cos ( x - y ) + [ D cosh 2 α 3 h 3 - 8 ( n 2 2 - 1 ) ( n 3 2 - n 2 2 ) n 4 n 3 sinh 2 α 3 h 3 ] cos y } .
T = 64 n 2 2 n 3 2 n 4 / [ N 1 2 + N 4 2 + N 6 2 + N 7 2 - A + B + C - D ] .
T = 4 n 2 2 n 3 2 n 4 / ( n 2 2 n 4 + n 3 2 ) 2 .
n 3 = n 2 n 4 1 2 .