Abstract

We develop a theory of the dynamics of optically induced conversion processes for samples of arbitrary optical thickness. The result is in the form of a simple first-order differential equation for the spatial variation of the state of the sample with time as an independent parameter. For the practically important case of samples that are initially homogeneous, the state of the sample is expressed in terms of a single function with the characteristics of the material, and its initial state, described by just one parameter. We present tables and plots of the function, which we call the Photochromic Function, with which one can evaluate the detailed behavior of a wide variety of systems, without requiring extensive computation. We also show that from optical transmission measurements on such samples, one can determine at most three independent constants of the material.

© 1976 Optical Society of America

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References

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  1. C. T. Slack, Opt. Acta 17, 547 (1970).
    [Crossref]
  2. E. Mohn, Appl. Opt. 12, 1570 (1973).
    [Crossref] [PubMed]
  3. G. H. Brown, Ed., Photochromism (Wiley, New York, 1971).
  4. W. J. Tomlinson, Appl. Opt. 14, 2456 (1975).
    [Crossref] [PubMed]
  5. W. J. Tomlinson, Appl. Opt. 11, 823 (1972).
    [Crossref] [PubMed]
  6. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 887.
  7. W. J. Tomlinson, E. A. Chandross, R. L. Fork, C. A. Pryde, A. A. Lamola, Appl. Opt. 11, 533 (1972).
    [Crossref] [PubMed]

1975 (1)

1973 (1)

1972 (2)

1970 (1)

C. T. Slack, Opt. Acta 17, 547 (1970).
[Crossref]

Appl. Opt. (4)

Opt. Acta (1)

C. T. Slack, Opt. Acta 17, 547 (1970).
[Crossref]

Other (2)

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 887.

G. H. Brown, Ed., Photochromism (Wiley, New York, 1971).

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Figures (4)

Fig. 1
Fig. 1

Coordinate system used in the calculation.

Fig. 2
Fig. 2

The Photochromic Function Pa(τ). See Fig. 3 for a more detailed plot of the region 0 < τ ≤ 2. Numerical values are given in Table I.

Fig. 3
Fig. 3

The Photochromic Function Pa(τ) for the region 0 < τ ≤ 2., Numerical values are given in Table I.

Fig. 4
Fig. 4

Curves of U(Z,T) as a function of Z for T = 3, for various values of a. For other values of T the curves should be shifted horizontally by Pa(T) − Pa(3).

Tables (1)

Tables Icon

Table I Photochromic Function Pa(τ)

Equations (34)

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d N a ( z , t ) / d t = - F ( z , t ) [ s a N a ( z , t ) - s b N b ( z , t ) ] .
s a N a ( z , ) = s b N b ( z , ) ,
N a ( z , ) / N 0 = s b / ( s a + s b ) .
U ( z , t ) N a ( z , t ) - N a ( z , ) N a ( z , 0 ) - N a ( z , ) = N b ( z , t ) - N b ( z , ) N b ( z , 0 ) - N b ( z , ) ,
d U ( z , t ) / d t = - ( s a + s b ) F ( z , t ) U ( z , t ) .
U ( z , t ) = exp [ - ρ ( z , t ) ] ,
ρ ( z , t ) ( s a + s b ) 0 t F ( z , t ) d t .
T ( s a + s b ) F 0 t
d ρ ( z , t ) / d z = ( s a + s b ) 0 t d F ( z , t ) d z d t ,
d ρ ( z , t ) / d t = ( s a + s b ) F ( z , t ) .
d F ( z , t ) / d z = - A ( z , t ) F ( z , t ) ,
A ( z , t ) = σ a N a ( z , t ) + σ b N b ( z , t ) + α .
d ρ ( z , t ) / d z = - 0 ρ A ( z , t ) d ρ .
α i ( z ) A ( z , 0 ) - α = [ ( σ a - σ b ) N a ( z , 0 ) / N 0 + σ b ] N 0 .
α s A ( z , ) - α = σ a s b + σ b s b s a + s b N 0 ,
Z 0 z [ α i ( z ) + α ] d z .
A ( z , t ) = [ α i ( z ) - α s ] U ( z , t ) + α s + α = [ α i ( z ) - α s ] exp [ - ρ ( z , t ) ] + α s + α ,
d ρ ( z , t ) / d z = - [ α i ( z ) - α s ] { 1 - exp [ - ρ ( z , t ) ] } - ( α s + α ) ρ ( z , t ) ,
ρ ( 0 , t ) = T .
d N a ( z , t ) / d t = - F 0 [ s a N a ( z , t ) - s b N b ( z , t ) ] × exp { - 0 z [ σ a N a ( z , t ) + σ b N b ( z , t ) + α ] d z } .
T r = exp [ - 0 d A ( z , t ) d z ] ,
T r = d ρ d T | z = d .
ρ ( Z , T ) = ln [ 1 + ( e T - 1 ) exp ( - Z ) ] ,
ρ ( Z , T ) = T exp ( - Z ) .
T ρ d ρ ( 1 - a ) [ 1 - exp ( - ρ ) ] + a ρ = - 0 z ( α i + α ) d z ,
a α s + α α i + α
P a ( τ ) 1 τ d ρ ( 1 - a ) [ 1 - exp ( - ρ ) ] + a ρ .
Z = P a ( T ) - P a ( ρ ) ,
ρ ( Z , T ) = P a - 1 [ P a ( T ) - Z ] .
P 0 ( τ ) = ln ( e τ - 1 e - 1 )
P 1 ( τ ) = ln ( τ ) .
V d Z d T = 1 ( 1 - a ) [ 1 - exp ( - T ) ] + a T .
T r = ( 1 - a ) [ 1 - exp ( - ρ ) ] + a ρ ( 1 - a ) [ 1 - exp ( - T ) ] + a T | z = d ,
y ( T r - 1 - 1 ) = ( e z - 1 ) exp ( - T ) ,

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