DIGITAL COMPUTING OF THE PATTERN OF A PHOTOELECTRIC RECORDING FABRY-PEROT INTERFEROMETER WITH ADAPTED LINE-SHAPES

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DIGITAL COMPUTING OF THE PATTERN OF A

PHOTOELECTRIC RECORDING FABRY-PEROT

INTERFEROMETER WITH ADAPTED

LINE-SHAPES

H. Hühnermann

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 2, suppliment au no 3-4, Tome 28, mars-avril1967, page C 2

-

260

DIGITAL COMPUTING OF THE PATTERN OF

A

PHOTOELECTRIC RECORDING FABRY-PEROT INTERFEROMETER

WITH ADAPTED LINE- SHAPES

H. HUHNERMANN

Physikalisches Institut der Universitat Marburg (Lahn), Allemagne

Abstract. - We have developed a method of evaluating for hfs recordings obtained by a photoelectric Fabry-Perot. This method utilizes the informations contained in the recordings to a higher degree than the usual ones. Thus the precision of the resulting data increases and the range of application of the optical investigations is enlarged.

A short outline of the computer-program is given and some examples show the precision of derived data.

R6sum6. - Nous avons dkveloppk une mkthode pour depouiller les enregistrements de structure hyperfine obtenus au moyen d'un Fabry-Perot photoCIectrique. Cette mkthode utilise mieux que les mkthodes habituelles les informations contenues dans les enregistrements. Ainsi la precision des resultats est augmentke, et le domaine &application des recherches optiques est 6largi.

Un bref aperGu du programme est donn6, et quelques exemples montrent la precision des resultats obtenus.

The experimental part of optical investigations of hyperfine-structures with the help of a Fabry-Perot interferometer is accurate and is constantly improved by many groups. However, the improvement of the evaluation technique has not kept up with the improvement of the measurement technique. There is usually much more information in the curves measured than one can get out of these curves with the evaluation methods used so far. There will be some examples later. Thus great efforts are made to make some experiments in which one gets smooth measured curves. Yet it would in general be simpler and more timesaving to get many measured curves with bad signal-to-noise ratio and perhaps strongly changing light-intensity. In addition there are many problems which were not attacked up to now, because one could not analyse strongly complex line structures with sufficient accuracy.

One such problem is the investigation of the isotope- shift of different cesium isotopes with unseparated isotope mixtures. The light intensity was low because we intended to work only with small amounts of radio- active substances. In addition the hyperfine structures are strongly unseparated. Of course we could have worked with separated nuclides and would thus have avoided the problem of evaluating unresolved structures. However, we had found the following in previous

experiments : especially with hollow cathodes filled with a few micrograms of cesium there are differences in the burning voltages and the current densities which caused different pressure shifts. These may amount to some tenth of a mK between different hollow cathodes. Thus an error is brought into the evaluation which may be rather than the statistical one given by us. I n addition the work with only one light-source is of course simpler than the work with two. And one is sure that the trajectories of the light beams of both isotopes are really equal. For all these reasons we have attacked the problem to improve the evaluation of measured curves. I shall tell you now about what we have obtained in the work.

First the statistics of the measured curves had to be improved. If we had intended to use RC units they had to be as large as ten seconds because of the low photo currents. Then the time of registration would have amounted to a t least one half hour per interference order to avoid distortions of the curves. In such long times the exitation of the cesium lines chan- ges appreciably as a consequence of the burning down of the cesium. Therefore, we could not get good measured curves with this method. So we have worked out an averaging procedure. Our registrations were done with RC units of only one second and the registration-speed was only four minutes per interference

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PHOTOELECTRIC RECORDING FABRY-PEROT INTERFEROMETER C 2

-

261

order. Therefore, the noise level in the measured curves was hight and the curves were not smooth at all. On the other hand we could register in the same time not only two but 15 orders. These 15 orders were averaged with the help of a digital computer. In addition the slow change of the photo-current caused by the cesium burning down was eliminated in the calculation and also the slow change of the dark current. The resulting curve corresponds to that which one would obtain with an RC unit of 15 seconds, a registration time of one hour and constant excitation of the spectral line. It is available in digital form.

FIG. 1. - Superposition of a registration of Cs 133, Cs 135, Cs 137, A = 8 943 A, length of an interference order 135 mK (above) ; 2 orders of the measured curve (down).

On figure 1 you can see the result of such a superposition and a piece of the original measured curve. Please notice how the disturbing pulse is averaged out.

I will not discuss details of the programme for the computer which was written in ALGOL but figure 2

FIG. 2.

-

Main steps of the superposition (explanations in the text).

will illustrate you the main steps of the superposition. First you see schematically a registration curve with variations of the dark and the photo current. The points symbolize each measured point. The distance between the points corresponds to constant time intervals because a digital voltmeter has measured the current each second and punched out. The second curve represents the measured values after the variations of the photo and dark currents had been eliminated numerically. After this the computer locates the position of a significant component, the so called main component. In the second curve this is shown by

x,

to

x,.

From the positions of the main component the length of the interference order is determined. In addition a constant of nonlinearity is computed which shows approximately the amount by which the distance

x,

-

x,

is larger compared to the distance

x,

-

x,.

These differences are of course only very small. One gets the third curve from the second by eliminating the

nonlinearity.

Moreover, a wave number scale was written at each interference order. Now the true averaging procedure is performed. On the fourth curve of the slide you see how all the interference orders are collected. The abscissa is devided in equal sections, the average is taken of all the points which are above such a section (Fig. _- 2 V). The length of an interval over which is _- averaged depends on the accuracy of the measurement and the line width. We have mostly used an interval length of 1 mK.

Now we come to the evaluation of theSuperimposed curves. The single hyperfine structure components are mostly not separated. Even if only the bases of the lines overlap, also the maxima of the curves and the center of gravities are shifted t o each other, and the distance between the lines seems too small. For the estimation of these attraction effects, computers are used since years. If the components overlap strongly the analysis of the hyperfine structure was difficult up to now and one has not come far beyond the estimation of splitting factors.

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C 2

-

262 H. HUHNERMANN

series one needs more than twenty terms. These 20 terms are physically of no interest. The same is true for all other systems of orthogonal functions. Before I will discuss how we treated the problem of a complete analysis of a hyperfine structure with a digital computer, I will show you with an example how satisfactory the solution is. But one should observe here that it is necessary for a good evaluation that the base of at least one line should be visible. On the left side of figure 3 you can'

Mit den optimalen Parametern Gemittelte MeOkuwe s I errechnete Kurw I

_.

L 11 . . .

_{. .}

_..

.

_.

.

. .

. . .

.

. , .

.

. . . , . . . . . .

.

. .

.

. .

.

. , , . . . . . . . . . . : . . . . . . . : . , . . . . . . . .

FIG. 3. - Superposed curve (I), optimal evaluated curve (II), difference I - I1 in the same scale, and in the scale 10 : 1.

see the superposition of the hyperfine structure of Cu 65. From this curve the B-factor shall be determined newly. On the right side you can see the curve which was computed by the machine with the best adapted parameters. These curves are so similar to each other that one cannot see any difference at the first glance. There- fore the difference between the two curves is given in the same scale at lower left. The differences are almost

approximate function for the shape of a single line. I said that we did not want to use a power series. Two possible approximate functions with a single parameter would be the Gauss function

and the dispersion function

But none of these two describes the real form of the curves satisfactorily. The representation of the centre of the curve is good. This one can show theoretically also by a power series. The flank and the base of the real curve cannot be described this way. The real'curve has a flank which reaches further than the Gauss curve but not as far as the dispersion curve. Therefore, we had to fit the flank with an additional parameter. So we used a modified dispersion curve

The last term of the denominator is noticeable only for larger distances from the centre of the curve because it increases with the fourth power of distance. With this form of the curve all the investigations at copper and cesium in the visible and infrared region of the spectrum were analysed.

The figure 4 shows how much better a fit is when using such a line form instead of a dispersion function or a Gauss function. You see the differences between a measured curve and the three fits obtained with the use of the three different line shapes. The dispersion zero. To make this more clearly visible this curve from t _I _;

-

3M)

lower left is represented at lower right in a scale of !.: 5 .

.;

?i

.. ..

.

:.

:;.

- m

1 : 10. To compute the curve on the right which fits i L s . :

...

:.i,

_,

f ::, _.

\ .-

loo

,

. .

500 points you need only 8 parameters. This are two .Li$?i

,.-.

:.- o

i

:.:

:,, :.:s

;-

;

'.u;'

'.I

A-factors, and the B-factor looked for. A fourth gives . ? V - .

-

-1o0

.

.::

...

the dark current. No. 5 fixes the centre of gravity of

.

.-

-200

the computed curve to that of the measured one. The

..

. . >

1:.

- 4 0 0

r r

6th factor is only a scale factor. The ratios of the . . amplitudes of the components are taken from the

theory. But the intensity ratios could be varied with _;

_.-

100 the programme if necessary. . , -+",I?.,-

C .< r

..

. .

.

..

::*a y : : - 0 The meaning of these six parameters is clear. Only %? $'

.-

-100 the last two which determine the line shape of a single

component have to be discussed in more detail. It is ,;

.-,

..

.

..

a . -100 .A$.;

,

:

!*

:

*&, j? :,.; ?,

known that it is not possible to give a mathematical _' A sc*,. ..- f)

> C

-

expression for the form of a single line. Only the 5 . ,?+

-

-100

distortion of the RC unit of the amplifier can be given _{FIG. 4.}

_-

_{Differences between n~easured}_{and optimal evaluated}

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PHOTOELECTRIC RECORDING F ABRY-PEROT INTERFEROMETER C 2

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263

curve is especially poor. The mean difference is 124 arbitrary units (upper curve). The Gauss function is much more better. Here the mean difference is only 51 (lowest curve). The best fit, however, is that with the modified dispersion curve with a mean difference of 45 (middle curve).

For a time we suspected that a somewhat further reaching flank would lead to even better fits. Therefore, we have tested a curve with three parameters

But it turned out that K was so small in our cases viz. 0.05 and the values of the other parameters were changed so little that it was not worth the additional computation effort.

Let me tell you something about the programme now. The fit starts with the input of the values of the measured curve and rough approximate values for the parameters looked for. In addition the input includes the suspected uncertainties of the parameters P. One can of course suppress the variation of some of the parameters by putting AP = 0. The parameters are varied now one after the other. The varied one assumes three values. These are P - AP, P, P

+

AP. The computer constructs three curves with the help of these three values. This you can see on figure 5. The

F(P- AP) F(P) F(P+AP)

F

FIG. 5. - Main steps of the fitting of a measured curve (hatched). (explanation in the text).

hatched curve represents the measured one. Now it is checked which of the three computed curves gives the best fit to the measured one. For this purpose the deviations D of the computed and from the measured curve are determined and a ( weighted ) sum of the squares of D, called B is found. In our example the sum of the squares for the curves with P

-

AP and P are about equal, for P

+

AP, however, appreciably larger. You can see this on the right side of figure 5. A

parabola is drawn through the three points thus

obtained. As a new approximate value for P the value of the minimum of the parabola is taken. This is indicated by P*. Also a new AP is determined. It can be shown that the interpolation with the parabola gives always a good approximation and sometimes it is even mathematically exact. The variation of a para- meter is finished if the variation of P by AP does not give an appreciable change of B. B is the sum of the squares of the deviations. In the same way the whole fit is finished if the variation of all parameters does not lower the value of B significantly. Even an estimation of the errors of each parameter can be obtained with this criterium. The value of P must be in that region for which B does not show any significant change. To illustrate this look at figure 6. Here you see the parabola

FIG. 6 .

-

Estimation of the error of a fitted value (explanation in the text).

of the squares of the differences as a function of the parameter P. B, is the smallest value of the squares. In the region between P* - AP and P*

+

AP the value of B is larger than B, by less than the significant amount AB. That is, the real P will be somewhere in this region.

The whole procedure depends on the definition of the significant deviation of B. Corresponding to the simple averaging of n numbers we have defined AB = Bo/n. If the average of n numbers is P with a probable error of AP then the sum of the error squares of P is equal to B,. For P AP however the sum of the error squares is exactly larger by B,/n than for P itself. The statement of the probable error AP or the significant change B,/n are equivalent in this case.

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C 2

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264 H. HUHNERMANN

the numbers given for the errors are in general too hight.

Now I will start with the last part of my talk which will deal with the accuracy of the evaluation procedure. One-could perhaps get general criteria for the accuracy which can be reached with the help of information theory. For the present I can only indicate you the possibilities of evaluation and the accuracy which can be reached with the help of some examples. Let us look again at figure 3. We know that all the 500 measured values can be described with 8 parameters. It can be assumed that the ratio of 60 measured values per parameter causes a small error. This is confirmed by our results. For example for the A-Factor of the meta- stable 2 ~ , 1 2 term of Cu we obtained the value

(66.10 f 0.02) mK. Notice that this is the result of a

single registration. The adaptation is more unique for simpler structures and the error even smaller. The splitting factor of the 6 p 2 P , I , term of Cs 133 was obtained from the weighted average of 7 measure- ments as (9.74,

+

0.01 ,) mK. One could measure line distances of 300 mK with the same absolute accuracy if one would know the optical path between the interference mirrors to one part in 100 000. We did not know the latter, but we knew the splitting of the ground state of Cs 133 to arbitrary accuracy. So we determined the distance of the mirrors for an arbitrary adjustment. Nevertheless - and now we come back to figure 3 again - we were disappointed by the fit at first. We stated systematical deviations. If you look very carefully you can see that the right flank of the main component is further reaching than the left one. The explanation was very simple. Not only the Cu

65 lines were excited, but also the Cu 63 ones by traces

of impurities. Therefore the evaluation was repeated, and two data for Cu 63 could be obtained. The contents of Cu 63 is 0.8 f 0.2

%.

The producers value was

0.6

%.

The isotope shift is (76 f 3) mK. The value of

the literature is (75.2

+

0.5) mK. Thus we have the

means of measurements at isotope mixtures of 1 part in 100 if the splitting conditions are favourable.

But the following application of the fitting procedure seems to be more important. Let us look at figure 1 again. There are three components within each line- group and in the largest one even six. The two peaks of each group are caused by Cs 133 and Cs 137. Cs 135 is completely covered. As a consequence of the evaluation the following results were obtained. The isotopic shift of cesium 137 to 133 amounts to

(- 4.81 f 0.07) mK. That of 135 to 133 to

The errors computed seemed to us too small, espe- cially if we remembered that the admixture of Cs 135 was only 17

%.

For this reason with the help of the mass separator at Marburg we produced cesium mixtures which contains only two of the isotopes formed in nuclear fission. The results of the measurements confirmed the values quoted above, but were even more accurate.

A further example are our measurements at Cs 134 which were published recently in Physics Letters. The figure 7 shows the superposition of registrations of

FIG. 7. - Superposition of registrations of Cs 133, Cs 134. Lengh of an interference order 208 mK. Cs 133 (-), Cs 134

(- - -).

both resonance lines. It was possible to determine the isotope shift from the lower line to (

+

1.25

+

0.06) mK.

This value is in accordance with the (+ 1.17 f 0.05) mK

obtained from the other line. In addition it was possible to compute the B-factors for both isotopes (Cs 134 and 133). The small wave on the right side of the left line-group is the only visible indication for the B-factor of Cs 134. This flank is smooth for poor Cs 133 where the B-factor is almost 0. The results for the B-factor of Cs 134 and 133 agree with values given in the literature. The same is true for Cs 135 and 137 measured at other mixtures giving an even poorer resolution. The errors computed by the machine are in general 0.15 mK,

The last example I will show you now is the evaluation of an experiment you would call a failed one. On figure 8 you see the whole registration with not more than 4 112 interference orders. Sometimes the discharge interrupted so that some measured values must be interpolated. The changing of the light- intensity was high.

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PHOTOELECTRIC RECORDING FABRY-PEROT INTERFEROMETER C 2

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265

FIG. 8. - CS 133, Cs 137, 3, = 8 943 A, interference order 208 mK. Complete measured curve (down), superposition (middle, left side), best fitted curve (middle, right side), and difference of superposition and fitted curve (above).

top of the two curves their difference. The fit is of course bad compared with that on figure 3. In spite of this it is surprising how exact the results are.

The isotope shift calculated from this curve of Cs 137 against Cs 133 is - 4.64 mK. Our newest value for the shift is (- 4.81

+

0.07) mK. These two numbers differ only by 0.2 mK. Compare this error with the errors you usually can find in optical reports. Seldom an error of 1/10 mK as result of many measurements has been published.

We have also calculated the splitting factor

of Cs 133 with the help of this bad curve. We found the value 9.63 mK. This value is the second best published up to now. Only the 9.775 _f 0.12, mK