Dynamics of photochromic conversion in optically thick samples: theory

W. J. Tomlinson

doi:10.1364/AO.15.000821

Applied Optics
Vol. 15,
Issue 3,
pp. 821-826
(1976)
•https://doi.org/10.1364/AO.15.000821

Dynamics of photochromic conversion in optically thick samples: theory

W. J. Tomlinson

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W. J. Tomlinson, "Dynamics of photochromic conversion in optically thick samples: theory," Appl. Opt. 15, 821-826 (1976)

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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.

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Equations (34)

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τ\a	0.	0.05	0.10	0.20	0.40	0.60	0.80	1.00	2.00	5.00
0.01	−5.141	−5.107	−5.074	−5.011	−4.895	−4.789	−4.693	−4.605	−4.248	−3.596
0.02	−4.443	−4.410	−4.377	−4.314	−4.199	−4.094	−3.999	−3.912	−3.560	−2.922
0.03	−4.033	−3.999	−3.967	−3.905	−3.790	−3.687	−3.593	−3.507	−3.159	−2.535
0.04	−3.740	−3.707	−3.675	−3.613	−3.500	−3.397	−3.304	−3.219	−2.877	−2.266
0.05	−3.512	−3.479	−3.447	−3.386	−3.273	−3.172	−3.080	−2.996	−2.658	−2.061
0.06	−3.325	−3.292	−3.260	−3.199	−3.088	−2.988	−2.897	−2.813	−2.481	−1.896
0.07	−3.165	−3.133	−3.101	−3.041	−2.931	−2.832	−2.742	−2.659	−2.331	−1.760
0.08	−3.027	−2.955	−2.963	−2.904	−2.794	−2.696	−2.607	−2.526	−2.202	−1.643
0.09	−2.904	−2.872	−2.841	−2.782	−2.674	−2.576	−2.488	−2.408	−2.089	−1.542
0.10	−2.793	−2.762	−2.731	−2.673	−2.565	−2.469	−2.382	−2.303	−1.989	−1.453
0.20	−2.049	−2.020	−1.992	−1.939	−1.842	−1.756	−1.679	−1.609	−1.340	−0.908
0.30	−1.592	−1.566	−1.540	−1.493	−1.407	−1.331	−1.264	−1.204	−0.976	−0.629
0.40	−1.251	−1.228	−1.206	−1.164	−1.090	−1.025	−0.967	−0.916	−0.727	−0.452
0.50	−0.974	−0.954	−0.936	−0.900	−0.837	−0.783	−0.735	−0.693	−0.540	−0.326
0.60	−0.737	−0.721	−0.706	−0.677	−0.626	−0.582	−0.544	−0.511	−0.392	−0.231
0.70	−0.528	−0.515	−0.503	−0.481	−0.443	−0.410	−0.381	−0.357	−0.270	−0.156
0.80	−0.338	−0.329	−0.321	−0.306	−0.280	−0.258	−0.239	−0.223	−0.167	−0.095
0.90	−0.163	−0.159	−0.155	−0.147	−0.134	−0.123	−0.113	−0.105	−0.078	−0.044
1.00	0.	0.	0.	0.	0.	0.	0.	0.	0.	0.
1.10	0.154	0.149	0.145	0.137	0.124	0.112	0.103	0.095	0.069	0.038
1.20	0.300	0.291	0.282	0.266	0.239	0.216	0.198	0.182	0.131	0.071
1.30	0.440	0.426	0.412	0.388	0.346	0.313	0.285	0.262	0.187	0.100
1.40	0.576	0.556	0.537	0.504	0.448	0.403	0.367	0.336	0.238	0.127
1.50	0.706	0.681	0.657	0.615	0.544	0.488	0.443	0.405	0.285	0.151
1.60	0.833	0.802	0.773	0.721	0.636	0.569	0.515	0.470	0.328	0.173
1.70	0.957	0.920	0.885	0.823	0.723	0.645	0.582	0.531	0.368	0.192
1.80	1.078	1.034	0.994	0.922	0.807	0.717	0.646	0.588	0.406	0.211
1.90	1.197	1.146	1.100	1.018	0.887	0.786	0.707	0.642	0.441	0.228
2.00	1.313	1.256	1.203	1.111	0.964	0.853	0.764	0.693	0.474	0.244
2.10	1.428	1.363	1.304	1.201	1.039	0.916	0.820	0.742	0.505	0.258
2.20	1.541	1.469	1.403	1.289	1.111	0.977	0.872	0.788	0.534	0.272
2.30	1.653	1.573	1.501	1.375	1.180	1.035	0.923	0.833	0.562	0.285
2.40	1.764	1.675	1.596	1.459	1.248	1.092	0.971	0.875	0.588	0.298
2.50	1.873	1.776	1.690	1.541	1.313	1.146	1.018	0.916	0.613	0.309
2.60	1.981	1.876	1.782	1.621	1.377	1.199	1.063	0.956	0.637	0.320
2.70	2.089	1.974	1.873	1.699	1.439	1.250	1.106	0.993	0.660	0.331
2.80	2.196	2.072	1.962	1.776	1.499	1.299	1.148	1.030	0.682	0.341
2.90	2.302	2.168	2.050	1.852	1.557	1.347	1.189	1.065	0.703	0.350
3.00	2.408	2.264	2.137	1.926	1.615	1.394	1.228	1.099	0.723	0.360
3.50	2.928	2.729	2.558	2.279	1.881	1.608	1.407	1.253	0.814	0.400
4.00	3.440	3.177	2.957	2.606	2.121	1.798	1.564	1.386	0.890	0.434
4.50	3.948	3.612	3.338	2.911	2.340	1.968	1.703	1.504	0.957	0.463
5.00	4.452	4.036	3.704	3.198	2.540	2.122	1.828	1.609	1.016	0.488
5.50	4.955	4.451	4.056	3.469	2.726	2.263	1.942	1.705	1.068	0.510
6.00	5.456	4.856	4.395	3.726	2.898	2.393	2.046	1.792	1.116	0.531
6.50	5.957	5.252	4.724	3.970	3.060	2.514	2.143	1.872	1.160	0.549
7.00	6.458	5.641	5.041	4.202	3.211	2.626	2.232	1.946	1.200	0.566
7.50	6.958	6.022	5.349	4.425	3.354	2.731	2.315	2.015	1.237	0.581
8.00	7.458	6.396	5.648	4.638	3.489	2.830	2.393	2.079	1.271	0.596
8.50	7.958	6.763	5.938	4.842	3.618	2.924	2.467	2.140	1.303	0.609
9.00	8.459	7.123	6.219	5.038	3.740	3.012	2.536	2.197	1.334	0.622
9.50	8.959	7.477	6.493	5.226	3.856	3.096	2.602	2.251	1.362	0.634
10.00	9.459	7.825	6.760	5.408	3.967	3.176	2.665	2.303	1.389	0.645

Abstract

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Figures (4)

Tables (1)

Equations (34)

Applied Optics