Abstract

The original purpose of this investigation was to explore the possibility of providing a spectrophotometer with automatic means for correcting the error that results when a continuous source is used with a monochromator having slits of finite width. In connection therewith, it was necessary to make what appears to be the first rigorous analysis of slit-width errors. The results of this analysis can be applied to the experimental determination of any functional relationship when the size of the “probe” is significant.

© 1949 Optical Society of America

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References

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  1. Rayleigh, Phil. Mag. 42, 441 (1871).
  2. C. Runge, Zeits. Math. 42, 205 (1897).
  3. F. Paschen, Wied. Ann. 60, 712 (1897).
  4. E. P. Hyde, Astrophys. J. 35, 237 (1912).
    [Crossref]
  5. It goes without saying that the theory developed here can be applied to measurements of spectral reflectance by replacing t(λ) by r(λ) and T(y) by R(y).
  6. R. H. Cameron, Nat. Math. Mag. 15, 331 (1941).
    [Crossref]
  7. According to the convention used here, the Fourier Transform, { }, of a function f(y) is{f(y)}=1/(2π)12∫−∞∞eiωyf(y)dy.
  8. J. A. Van den Akker, J. Opt. Soc. Am. 33, 257 (1943).
    [Crossref]
  9. This does not apply if any of the functions has a structure that is fine by comparison with the width of the slit. In that case, the above solution does not converge.
  10. Arthur C. Hardy, J. Opt. Soc. Am. 25, 305 (1935).
    [Crossref]
  11. For details, see Bachelor’s Thesis of F. Mansfield Young, M.I.T., May, 1948.

1943 (1)

1941 (1)

R. H. Cameron, Nat. Math. Mag. 15, 331 (1941).
[Crossref]

1935 (1)

1912 (1)

E. P. Hyde, Astrophys. J. 35, 237 (1912).
[Crossref]

1897 (2)

C. Runge, Zeits. Math. 42, 205 (1897).

F. Paschen, Wied. Ann. 60, 712 (1897).

1871 (1)

Rayleigh, Phil. Mag. 42, 441 (1871).

Cameron, R. H.

R. H. Cameron, Nat. Math. Mag. 15, 331 (1941).
[Crossref]

Hardy, Arthur C.

Hyde, E. P.

E. P. Hyde, Astrophys. J. 35, 237 (1912).
[Crossref]

Mansfield Young, F.

For details, see Bachelor’s Thesis of F. Mansfield Young, M.I.T., May, 1948.

Paschen, F.

F. Paschen, Wied. Ann. 60, 712 (1897).

Rayleigh,

Rayleigh, Phil. Mag. 42, 441 (1871).

Runge, C.

C. Runge, Zeits. Math. 42, 205 (1897).

Van den Akker, J. A.

Astrophys. J. (1)

E. P. Hyde, Astrophys. J. 35, 237 (1912).
[Crossref]

J. Opt. Soc. Am. (2)

Nat. Math. Mag. (1)

R. H. Cameron, Nat. Math. Mag. 15, 331 (1941).
[Crossref]

Phil. Mag. (1)

Rayleigh, Phil. Mag. 42, 441 (1871).

Wied. Ann. (1)

F. Paschen, Wied. Ann. 60, 712 (1897).

Zeits. Math. (1)

C. Runge, Zeits. Math. 42, 205 (1897).

Other (4)

It goes without saying that the theory developed here can be applied to measurements of spectral reflectance by replacing t(λ) by r(λ) and T(y) by R(y).

According to the convention used here, the Fourier Transform, { }, of a function f(y) is{f(y)}=1/(2π)12∫−∞∞eiωyf(y)dy.

This does not apply if any of the functions has a structure that is fine by comparison with the width of the slit. In that case, the above solution does not converge.

For details, see Bachelor’s Thesis of F. Mansfield Young, M.I.T., May, 1948.

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Equations (38)

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0 E ( λ ) S ( λ ) d λ .
0 E ( λ ) S ( λ ) t ( λ ) d λ .
T = 0 E ( λ ) S ( λ ) t ( λ ) d λ 0 E ( λ ) S ( λ ) d λ .
T ( y ) = 0 E ( λ ) S ( λ ) M ( y , λ ) t ( λ ) d λ 0 E ( λ ) S ( λ ) M ( y , λ ) d λ .
T ( y ) W ( y , λ ) d x = W ( y , λ ) t ( y + x ) d x .
A T ( y ) = W ( y 0 , x ) t ( y + x ) d x ,
A = W ( y 0 , x ) d x
W ( y 0 , x ) = W e ( y 0 , x ) + W 0 ( y 0 , x ) = W e ( y 0 , x ) W 0 ( y 0 , x ) .
A T ( y ) = W e ( y 0 , x ) t ( y x ) d x W 0 ( y 0 , x ) t ( y x ) d x .
A { T ( y ) } = { t ( y ) } { W e W 0 } ( 2 π ) 1 2
{ t } = A { T } ( 2 π ) 1 2 [ { W e } { W 0 } ] .
{ W e } = 1 / ( 2 π ) 1 2 e i ω x W ( y 0 , x ) d x = 1 / ( 2 π ) 1 2 [ 1 + i ω x + + ( i ω x ) n / n ! + ] W ( y 0 , x ) d x = A / ( 2 π ) 1 2 [ 1 + n = 1 e n ( i ω ) n ]
e n = ( 1 / A n ! ) x n W e ( y 0 , x ) d x .
{ W 0 } = A / ( 2 π ) 1 2 n = 1 o n ( i ω ) n
o n = 1 / A n ! x n W 0 ( y 0 , x ) d x ,
α n 1 A n ! x n ω ( y 0 , x ) d x ,
{ W e } { W 0 } = A / ( 2 π ) 1 2 [ 1 + n = 1 ( 1 ) n α n ( i ω ) n ] .
{ t } = { T } 1 + n = 1 ( 1 ) n α n ( i ω ) n = [ 1 n = 1 ( 1 ) n α n ( i ω ) n + [ n = 1 ( 1 ) n α n ( i ω ) n ] 2 ] { T } .
{ d T ( y ) / d y } = i ω { T ( y ) } ,
t ( y ) = T ( y ) n = 1 α n d n T ( y ) d y n + [ n = 1 α n d n d y n ] 2 T ( y ) .
t ( y ) = T ( y ) n = 1 α n [ d n T ( y ) / d y n ] ,
a 1 = α 1 a 2 = α 2 α 1 2 a 3 = α 3 2 α 1 α 2 α 1 3 a 4 = α 4 α 2 2 2 α 1 α 3 + 3 α 1 2 α 2 α 1 4 .
α n = 1 / A n ! x n W ( y , x ) d x .
t ( λ ) = T ( λ ) a 1 [ d T ( λ ) / d λ ] .
t ( λ ) = p + q λ ,
t ( y + x ) = t 0 + t 1 x ,
T ( y ) = t 0 + t 1 W ( y , x ) x d x W ( y , x ) d x = t 0 + t 1 a 1 = t ( y + a 1 ) .
a 1 = W ( y , x ) x d x W ( y , x ) d x
t ( y + a 1 ) = T ( y ) [ a 2 + ( a 1 2 / 2 ) ] [ d 2 T ( y ) / d y 2 ]
t 2 = t 1 n ( d t 2 / d λ ) = n t 1 n 1 ( d t 1 / d λ ) ( d 2 t 2 / d λ 2 ) = n t 1 n 1 ( d 2 t 1 / d λ 2 ) + n ( n 1 ) t 1 n 2 ( d t 1 / d λ ) 2
T 1 = t 1 + α 1 ( d t 1 / d λ ) + α 2 ( d 2 t 1 / d λ 2 ) T 2 = t 2 + α 1 ( d t 2 / d λ ) + α 2 ( d 2 t 2 / d λ 2 ) .
T 2 T 1 n = [ α 2 ( α 1 2 / 2 ) ] n ( n 1 ) t 1 n 2 ( d t 1 / d λ ) 2 .
[ a 2 + ( a 1 2 / 2 ) ] = [ α 2 ( α 1 2 / 2 ) ] = T 2 T 1 n n ( n 1 ) T 1 n 2 ( d T 1 / d λ ) 2 .
2 ( a 2 + a 1 2 2 ) = W ( y , x ) x 2 d x W ( y , x ) d x a 1 2 .
w = ( 24 [ a 2 + ( a 1 2 / 2 ) ] ) 1 2 .
d ( y + a 1 ) = D ( y ) [ a 2 + a 1 2 2 ] × [ d 2 D ( y ) d y 2 2.3 ( d D ( y ) d y ) 2 ]
n = m + 2.3 m ( m 1 ) ( 3 α 4 α 2 2 2 ) 1 D 1 ( d 2 D 1 ( u ) d y 2 ) 2 ,
{f(y)}=1/(2π)12eiωyf(y)dy.