Abstract

A review of the history of the theory of the reflection of light from a pile of plates is given. The presence of incorrect formulae in some current text and reference books is noted. The formulae for a pile of absorbing plates are derived by a method different from that used by Stokes, giving the results in a form different from his. The equivalence of the two forms is proven. An upper bound to the error caused by assuming an infinite instead of a finite number of multiple reflections is derived in the Appendix.

© 1947 Optical Society of America

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References

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  1. P. Drude, Lehrbuch der Optik (Leipzig, 1912), third edition, pp. 271–272.
  2. R. W. Wood, Physical Optics (New York, 1934), third edition, pp. 340–343.
  3. A. J. Fresnel, Oevres Complètes (Paris, 1866), Vol. 10, pp. 640–648.
  4. R. Clausius, “Ueber die Intensität des durch die Atmosphäre reflecterte Sonnellichts,” Crelle’s J. 36, 185–215 (1848).
    [CrossRef]
  5. F. de la Provostaye and P. Desains, “Memoire sur la Polarisation de la Chaleur par Refraction simple,” Ann. de Chim.et Phys. [3]  30, 158–178 (1850).
  6. H. Wild, “Ueber ein neues Photometer und Polarimeter nebst einigen damit angestellten Beobachtungen,” Pogg. Ann. 99, 235–274 (1856).
  7. C. Bohn, “Zur Polarization des Lichtes durch einfache Brechung,” Pogg. Ann. 117, pp. 117–131 (1862).
  8. F. E. Neumann, Vorlesungen über Theoretische Optik (Leipzig, 1885), pp. 141–151.
  9. G. G. Stokes, “On the intensity of the light reflected from and transmitted through a pile of plates,” Phil. Mag. [4]  24, 480–489 (1862);Proc. Roy. Soc. London 11, 545–556 (1862).
  10. Wüllner, Experimental Physik (Leipzig, 1899), fifth edition, Vol. 4, p. 727.
  11. Winkelmann, Handbuch der Physik (Leipzig, 1906), Vol. 6, p. 1252.
  12. R. W. Wood, “On the emission of polarized light by fluorescent gases,” Phil. Mag. 16, 184–189 (1908).
    [CrossRef]
  13. L. Dunoyer, “Sur la fluorescence des vapeurs des metauxalkalins,” le Radium 9, No. 6, 209–218 (1912).
    [CrossRef]
  14. E. Gaviola and Peter Pringsheim, “Über den Einfluss der Konzentration auf die Polarisation von Farbstoff Lösungen,” Zeits. f. Physik. 24, 24–36 (1924).
    [CrossRef]
  15. Schuster, Theory of Optics (New York, 1924), third edition, pp. 51, 52.
  16. Wien-Harms, Handbuch der Experimental Physik (Leipzig, 1928), Vol. 18, p. 141.
  17. O. D. Chwolson, Lehrbuch der Physik (Braunschweig, 1922), Vol. 2, part 2, pp. 717–718.
  18. Geiger and Scheel, Handbuch der Physik (Berlin, 1928), Vol. 19, p. 971.

1924 (1)

E. Gaviola and Peter Pringsheim, “Über den Einfluss der Konzentration auf die Polarisation von Farbstoff Lösungen,” Zeits. f. Physik. 24, 24–36 (1924).
[CrossRef]

1912 (1)

L. Dunoyer, “Sur la fluorescence des vapeurs des metauxalkalins,” le Radium 9, No. 6, 209–218 (1912).
[CrossRef]

1908 (1)

R. W. Wood, “On the emission of polarized light by fluorescent gases,” Phil. Mag. 16, 184–189 (1908).
[CrossRef]

1862 (2)

C. Bohn, “Zur Polarization des Lichtes durch einfache Brechung,” Pogg. Ann. 117, pp. 117–131 (1862).

G. G. Stokes, “On the intensity of the light reflected from and transmitted through a pile of plates,” Phil. Mag. [4]  24, 480–489 (1862);Proc. Roy. Soc. London 11, 545–556 (1862).

1856 (1)

H. Wild, “Ueber ein neues Photometer und Polarimeter nebst einigen damit angestellten Beobachtungen,” Pogg. Ann. 99, 235–274 (1856).

1850 (1)

F. de la Provostaye and P. Desains, “Memoire sur la Polarisation de la Chaleur par Refraction simple,” Ann. de Chim.et Phys. [3]  30, 158–178 (1850).

1848 (1)

R. Clausius, “Ueber die Intensität des durch die Atmosphäre reflecterte Sonnellichts,” Crelle’s J. 36, 185–215 (1848).
[CrossRef]

Bohn, C.

C. Bohn, “Zur Polarization des Lichtes durch einfache Brechung,” Pogg. Ann. 117, pp. 117–131 (1862).

Chwolson, O. D.

O. D. Chwolson, Lehrbuch der Physik (Braunschweig, 1922), Vol. 2, part 2, pp. 717–718.

Clausius, R.

R. Clausius, “Ueber die Intensität des durch die Atmosphäre reflecterte Sonnellichts,” Crelle’s J. 36, 185–215 (1848).
[CrossRef]

de la Provostaye, F.

F. de la Provostaye and P. Desains, “Memoire sur la Polarisation de la Chaleur par Refraction simple,” Ann. de Chim.et Phys. [3]  30, 158–178 (1850).

Desains, P.

F. de la Provostaye and P. Desains, “Memoire sur la Polarisation de la Chaleur par Refraction simple,” Ann. de Chim.et Phys. [3]  30, 158–178 (1850).

Drude, P.

P. Drude, Lehrbuch der Optik (Leipzig, 1912), third edition, pp. 271–272.

Dunoyer, L.

L. Dunoyer, “Sur la fluorescence des vapeurs des metauxalkalins,” le Radium 9, No. 6, 209–218 (1912).
[CrossRef]

Fresnel, A. J.

A. J. Fresnel, Oevres Complètes (Paris, 1866), Vol. 10, pp. 640–648.

Gaviola, E.

E. Gaviola and Peter Pringsheim, “Über den Einfluss der Konzentration auf die Polarisation von Farbstoff Lösungen,” Zeits. f. Physik. 24, 24–36 (1924).
[CrossRef]

Geiger,

Geiger and Scheel, Handbuch der Physik (Berlin, 1928), Vol. 19, p. 971.

Neumann, F. E.

F. E. Neumann, Vorlesungen über Theoretische Optik (Leipzig, 1885), pp. 141–151.

Pringsheim, Peter

E. Gaviola and Peter Pringsheim, “Über den Einfluss der Konzentration auf die Polarisation von Farbstoff Lösungen,” Zeits. f. Physik. 24, 24–36 (1924).
[CrossRef]

Scheel,

Geiger and Scheel, Handbuch der Physik (Berlin, 1928), Vol. 19, p. 971.

Schuster,

Schuster, Theory of Optics (New York, 1924), third edition, pp. 51, 52.

Stokes, G. G.

G. G. Stokes, “On the intensity of the light reflected from and transmitted through a pile of plates,” Phil. Mag. [4]  24, 480–489 (1862);Proc. Roy. Soc. London 11, 545–556 (1862).

Wien-Harms,

Wien-Harms, Handbuch der Experimental Physik (Leipzig, 1928), Vol. 18, p. 141.

Wild, H.

H. Wild, “Ueber ein neues Photometer und Polarimeter nebst einigen damit angestellten Beobachtungen,” Pogg. Ann. 99, 235–274 (1856).

Winkelmann,

Winkelmann, Handbuch der Physik (Leipzig, 1906), Vol. 6, p. 1252.

Wood, R. W.

R. W. Wood, “On the emission of polarized light by fluorescent gases,” Phil. Mag. 16, 184–189 (1908).
[CrossRef]

R. W. Wood, Physical Optics (New York, 1934), third edition, pp. 340–343.

Wüllner,

Wüllner, Experimental Physik (Leipzig, 1899), fifth edition, Vol. 4, p. 727.

Ann. de Chim.et Phys. (1)

F. de la Provostaye and P. Desains, “Memoire sur la Polarisation de la Chaleur par Refraction simple,” Ann. de Chim.et Phys. [3]  30, 158–178 (1850).

Crelle’s J. (1)

R. Clausius, “Ueber die Intensität des durch die Atmosphäre reflecterte Sonnellichts,” Crelle’s J. 36, 185–215 (1848).
[CrossRef]

le Radium (1)

L. Dunoyer, “Sur la fluorescence des vapeurs des metauxalkalins,” le Radium 9, No. 6, 209–218 (1912).
[CrossRef]

Phil. Mag. (2)

R. W. Wood, “On the emission of polarized light by fluorescent gases,” Phil. Mag. 16, 184–189 (1908).
[CrossRef]

G. G. Stokes, “On the intensity of the light reflected from and transmitted through a pile of plates,” Phil. Mag. [4]  24, 480–489 (1862);Proc. Roy. Soc. London 11, 545–556 (1862).

Pogg. Ann. (2)

H. Wild, “Ueber ein neues Photometer und Polarimeter nebst einigen damit angestellten Beobachtungen,” Pogg. Ann. 99, 235–274 (1856).

C. Bohn, “Zur Polarization des Lichtes durch einfache Brechung,” Pogg. Ann. 117, pp. 117–131 (1862).

Zeits. f. Physik. (1)

E. Gaviola and Peter Pringsheim, “Über den Einfluss der Konzentration auf die Polarisation von Farbstoff Lösungen,” Zeits. f. Physik. 24, 24–36 (1924).
[CrossRef]

Other (10)

Schuster, Theory of Optics (New York, 1924), third edition, pp. 51, 52.

Wien-Harms, Handbuch der Experimental Physik (Leipzig, 1928), Vol. 18, p. 141.

O. D. Chwolson, Lehrbuch der Physik (Braunschweig, 1922), Vol. 2, part 2, pp. 717–718.

Geiger and Scheel, Handbuch der Physik (Berlin, 1928), Vol. 19, p. 971.

F. E. Neumann, Vorlesungen über Theoretische Optik (Leipzig, 1885), pp. 141–151.

Wüllner, Experimental Physik (Leipzig, 1899), fifth edition, Vol. 4, p. 727.

Winkelmann, Handbuch der Physik (Leipzig, 1906), Vol. 6, p. 1252.

P. Drude, Lehrbuch der Optik (Leipzig, 1912), third edition, pp. 271–272.

R. W. Wood, Physical Optics (New York, 1934), third edition, pp. 340–343.

A. J. Fresnel, Oevres Complètes (Paris, 1866), Vol. 10, pp. 640–648.

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Equations (74)

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r n = n ρ / [ 1 + ( n 1 ) ρ ] , t n = ( 1 ρ ) / [ 1 + ( n 1 ) ρ ] .
ρ = sin 2 ( i i ) sin 2 ( i + i ) , ρ = tan 2 ( i i ) tan 2 ( i + i ) ,
g = e k T sec i ,
ρ , ( 1 ρ ) 2 ρ g 2 , ( 1 ρ ) 2 ρ g 2 ( ρ 2 g 2 ) , ( 1 ρ ) 2 ρ g 2 ( ρ 2 g 2 ) 2 ; , ( 1 ρ ) 2 ρ g 2 ( ρ 2 g 2 ) ( h 1 ) ,
r = ρ + ( 1 ρ ) 2 ρ g 2 [ 1 ( ρ 2 g 2 ) h ] / ( 1 ρ 2 g 2 ) .
r = ρ + ( 1 ρ ) 2 ρ g 2 / ( 1 ρ 2 g 2 ) .
t = ( 1 ρ ) 2 g / ( 1 ρ 2 g 2 ) .
r ( m + n ) = r m + t m 2 r n + t m 2 r m r n 2 + t m 2 r n ( r m r n ) ( h 1 ) ,
r ( m + n ) = r m + t m 2 r n / ( 1 r m r n ) .
t ( m + n ) = t m t n / ( 1 r m r n ) .
t n sin α = r n sin n β = 1 sin ( α + n β ) .
t n c c 1 = r n d n d n = 1 c d n c 1 d n ,
c = ( 1 p + q ) / r ,
d = ( p + q ) / t ,
p = ( 1 + t 2 r 2 ) / 2 ,
q = ( p 2 t 2 ) 1 2 = [ ( 1 p ) 2 r 2 ] 1 2 .
r ( n + 1 ) = r + t 2 r n / ( 1 r r n ) .
r n = r a n / b n ,
r a ( n + 1 ) / b ( n + 1 ) = r [ 1 + a n b n t 2 / ( 1 a n b n r 2 ) ] .
a ( n + 1 ) = ( 1 + t 2 r 2 ) a n ( a n b n ) ,
b ( n + 1 ) = ( 1 r 2 ) a n ( a n b n ) ,
a ( n + 1 ) b ( n + 1 ) = t 2 a n .
a ( n + 2 ) = ( 1 + t 2 r 2 ) a ( n + 1 ) [ a ( n + 1 ) b ( n + 1 ) ] .
a ( n + 2 ) ( 1 + t 2 r 2 ) a ( n + 1 ) + t 2 a n = 0 ,
a ( n + 2 ) 2 p a ( n + 1 ) + ( p 2 q 2 ) a n = 0 .
A + B x 1 2 p x + ( p 2 q 2 ) x 2 = a 1 + a 2 x + a 3 x 2 + a 4 x 3 + .
a 2 = ( 1 + t 2 r 2 ) = 2 p .
1 1 2 p x + ( p 2 q 2 ) x 2 = p + q 2 q · 1 1 ( p + q ) x p q 2 q · 1 1 ( p q ) x .
a n = [ ( p + q ) n ( p q ) n ] / 2 q .
b n = a ( n + 1 ) ( t 2 r 2 ) a n = a n t 2 a ( n 1 ) .
t = t t n 1 r r n = ( b n t n ) t b n r 2 a n .
b n r 2 a n = a ( n + 1 ) ( t 2 r 2 ) a n r 2 a n = a ( n + 1 ) t 2 a n = b ( n + 1 ) .
t ( n + 1 ) = ( b n t n ) t / b ( n + 1 ) .
b 1 = 1 , t 1 = t = t 1 / b 1 , b 2 = 1 r 2 , t 2 = t 2 1 r 2 = t 2 b 2 .
t n = t n / b n .
t n / t n = r n / r a n = 1 / b n .
c c 1 t n = d n d n r a n = c d n c 1 d n b n .
( c c 1 ) 2 t 2 n = ( d n d n ) 2 r 2 a n 2 = ( c d n c 1 d n ) 2 b n 2 = 4 q 2 r 2 t 2 n .
p = t = 1 r , q = 0
a n = lim q 0 [ ( p + q ) n ( p q ) n ] / 2 q = lim q 0 ( n p ( n 1 ) + ( n 3 ) p ( n 3 ) q 2 + ) = n t ( n 1 )
b n = a n t 2 a ( n 1 ) = n t ( n 1 ) ( n 1 ) t n .
r n = r a n / b n = n r / [ n ( n 1 ) t ] = n r / [ 1 + ( n 1 ) r ] ,
t n = ( 1 r ) / [ 1 + ( n 1 ) r ] ,
r n = n ρ / [ 1 + ( n 1 ) ρ ] , t n = 1 r n = ( 1 ρ ) / [ 1 + ( n 1 ) ρ ] ,
r ( n + 1 ) = r n + ( 1 r n ) 2 ρ [ 1 + ρ r n + ( ρ r n ) 2 + ( ρ r n ) ( h 1 ) ]
= r n + ( 1 r ρ ) 2 ρ / ( 1 ρ r n ) ( 1 r n ) 2 ρ ( ρ r n ) h / ( 1 ρ r n ) = [ ρ + ( 1 2 ρ ) r n ] / ( 1 ρ r n ) ( 1 r n ) 2 ρ ( ρ r n ) h / ( 1 ρ r n )
r n = [ ρ + ( 1 2 ρ ) r n ] / ( 1 ρ r n ) .
( n + 1 ) = r n r ( n + 1 ) r ( n + 1 ) r n δ r n = r n r ( n + 1 ) r n ρ + ( 1 2 ρ ) r n 1 ρ r n δ r n .
( n + 1 ) = n ( n + 1 ) · 1 + ( n 1 ) ρ 1 + n ρ · δ r n < δ r n .
r ( n + 1 ) = ( n + 1 ) ρ 1 + n ρ n h ( 1 ρ ) ρ ( 2 h + 1 ) [ 1 + ( n 1 ) ρ ] h ( 1 + n ρ ) .
x = n h ( n + 1 ) · ( 1 ρ ) ρ 2 h [ 1 + ( n 1 ) ρ ] h ,
ζ ( n + 1 ) = x / ( 1 x ) .
d ζ ( n + 1 ) / d ρ = ζ ˙ = / ( 1 x ) 2 = 0 .
n h ( 1 ρ ) ρ h < ( n + 1 ) n h ( n + 1 ) [ 1 / ρ + ( n 1 ) ] h , 0 < ρ 1 .
1 ( n + 1 ) [ ( 1 / ρ ) + ( n 1 ) ] h = ρ h ( n + 1 ) [ 1 + ( n 1 ) ρ ] h
( n + 1 ) [ 1 + ( n 1 ) ρ ] ( h + 1 ) / n h ρ ( 2 h 1 ) = ( n 1 ) ( h + 1 ) ρ 2 [ ( n 3 ) h 1 ] ρ 2 h = 0 .
ρ = 2 h / ( 2 h + 1 ) < 1 .
ρ = { + [ ( n 3 · h 1 ) 2 + 8 ( n 1 ) h ( h + 1 ) ] 1 2 + ( n 3 ) h 1 } / 2 ( n 1 ) ( h + 1 ) = { + [ ( n + 1 ¯ · h 1 ) 2 + 8 n h ] 1 2 + ( n 3 ) h 1 } / 2 ( n 1 ) ( h + 1 ) = { + [ ( n + 1 ¯ · h + 3 ) 2 8 ( h + 1 ) ] 1 2 + ( n 3 ) h 1 } / 2 ( n 1 ) ( h + 1 ) .
0 < ( n 1 ) h 1 ( n 1 ) ( h + 1 ) = ρ 0 < ρ < ρ 1 = ( n 1 ) h + 1 ( n 1 ) ( h + 1 ) 1
= x / ρ | n , h = 0
x = x n | h = x n | h , ρ + x ρ | n , h ρ n | h = x n | h , ρ
= ( 1 ρ ) ρ 2 h n ( h 1 ) ( n + 1 ) 2 [ 1 + ( n 1 ) ρ ] ( h + 1 ) { ( n + 1 ) h n [ n ( n 1 ) + ( n + 1 ) h ] ρ } .
2 h 1 2 h · 2 h 2 h + 1 = [ ( 2 h 1 ) ( 2 h + 1 ) 4 h 2 ] / ( 2 h + 1 ) = 1 / ( 2 h + 1 ) < 0 .
( n + 1 ) h n [ n ( n 1 ) + ( n + 1 ) h ] ( n 1 ) h 1 ( n 1 ) ( h + 1 ) = 1 + n n 2 h + 1 .
x = x n h < 0 .
x = 1 2 ( 1 ρ ) ρ 2 h ,
δ r 2 = ζ 2 = ( 1 ρ ) ρ 2 h 2 ( 1 ρ ) ρ 2 h ·
δ r 2 = ζ 2 = [ 2 h / ( 2 h + 1 ) ] ( 2 h + 1 ) 4 h [ 2 h / ( 2 h + 1 ) ] ( 2 h + 1 ) ,
1 / 4 h δ r 2 = ( 1 + 1 2 h ) ( 2 h + 1 ) 1 / 4 h .
d d h ( 1 / 4 h δ r 2 ) = ( 2 h + 1 ) / 2 h 2 + 1 / 4 h 2 = ( 4 h + 1 ) / 4 h 2 .
lim h ( 1 / 4 h δ r 2 ) = e = 2.71828 + .
1 / 4 h δ r 2 > e ,
δ r n < ( n 1 ) / 4 e h < 0.09197 ( n 1 ) / h ,
ρ = ( μ 2 1 ) 2 / ( μ 2 + 1 ) 2 = 0.14793 .