Modelling of measurement of thickness of the mirror layer by method Fizo
Sergey Stepanenko, Zolotaryova О. V, Shevchenko L. V.
Donetsk national technical university
Источник: The 2nd young scientists' scientific and technical conference / Збірник матеріалів науково-технічної конференції
для молодих вчених — тези доповідей. — Донецьк, ДонНТУ — 2006. — 127 с. С. 124–127
At mirrors manufacturing on a polished surface a thin layer of the metal which reflects the light very well is put down. The most used of all are the mirrors made of glass with silver layer.
But silver, as it is known, is rather expensive metal. To check the charge of silver that is to measure the thickness of its layer on a smooth surface, the method offered by Fizo in 1861 has been used. This method is based on chemical interaction of iodine with silver, and is still put into practice.
For measurement of the thickness of a mirror silver layer a grain of iodine is put. Almost at once in a vicinity of a grain a color of a covering starts to change. In 5−10 minutes (depending on a layer thickness) around a grain some coloured rings are formed, and in that place where the grain laid, the mirror becomes transparent. If to shine a mirror with red light or to look through red glass around a grain alternating red and dark rings are visible.
It means that iodine pairs (I2), extending from a grain, interact with silver (Ag) and transfer it in iodide silver (AgI) — transparent firm substance. Around a grain of iodine the layer of iodide silver of variable thickness is formed. An interference of light in this layer explains the occurrence of coloured rings.
When on a mirror the light wave partially reflects from an external surface of Agl layer, partially passes inside of a layer, reflects from its second surface that is from border with silver, and leaves outside. The reflected waves are coherently received as they are interfered. The result of this interference depends on a geometrical difference of a course 8 which can be counted up under the formula
If the corner of the light fall is equal to zero or closer to it, AB = BC, therefore
Where d is the thickness of "lens" in a place of a light beam fall. From the edge to the center of a layer its thickness (d) gradually grows, the difference of a course also increases. At illumination by monochromatic light (for example, red) maxima of light exposure (red rings) are observed where the difference of a course of beams is equal to an integer of the light waves length in iodide silver. Dark rings (minima of light exposure) are formed where the difference of a course of beams is multiple to a half of a wave length in this substance:
Here k = 1, 2, 3... is a number of a dark ring. It is considered to be the first an extreme external dark ring.
From equation (1) and (2) fqllows that where the dark ring with number k is observed, the thickness of AgI layer can be counted up under the formula
If the last dark ring with number kmax coincides with the edge of the central transparent stain the maximal thickness of a layer dmax is
Actually the last dark ring usually does not coincide with the edge of the central transparent stain, and is in some distance from it (as, for example, on model with h=216 nm). Therefore actually dmax is more than the value calculated under the formula (4). We shall try to estimate this discrepancy. The difference of "lens" thickness is equal to the places matching the two next and to a half of a wave length. Hence, the error in definition dmax does not exceed λ/4. Thus, it is possible to consider that
The length of the light wave in substance λ is connected with the length of the light wave in vacuum λ0 equation λ = λ0/n, where n is a parameter of refraction of substance. Considering it
It turns out that the thickness (h) of a silver layer on a mirror is connected with a certain image with the maximal thickness of iodide silver dmax:
For iodide silver n = 2,15. Having substituted this value in (7), we receive the formula for definition of the thickness of a silver layer:
The error of measurement of the thickness of AgI layer by method Fizo as it is shown above is λ/4 = λ0/4n. The thickness of Ag layer is 4 times less than the thickness of AgI layer, therefore the error in the definition of a silver layer thickness is 1/4· λ0/4n = λ0/34,4.
The program which can be used as a working model for the control of silver quantity is developed for modeling measurement of the thickness of a mirror layer in multimedia Flash environment. The convenient menu for work with the program is represented on the screen: the user can change the thickness of a silver covering, simultaneously with the work results the program also varies. The real conditions of physical experiment are recreated. Automatic scaling allows the user to observe an interference picture in as convenient kind as possible.
The application of the given program will help to reduce the calculations spent on the mirrors manufacturing and, as a consequence, will definitely lower expenditures of labour. Accordingly - the price of mirrors will fall down but their quality will improve.
Figure 1 — Model.
Interactive animation, 640×480 px, 11 kB.