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THE ACCURACY OF DIFFUSION-CONSTANT MEASUREMENTS BY DIGITAL
AUTOCORRELATION OF PHOTON-COUNTING FLUCTUATIONS
E. Pike
To cite this version:
E. Pike. THE ACCURACY OF DIFFUSION-CONSTANT MEASUREMENTS BY DIGITAL AU- TOCORRELATION OF PHOTON-COUNTING FLUCTUATIONS. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-177-C1-180. �10.1051/jphyscol:1972132�. �jpa-00214922�
JOURNAL DE PHYSIQUE Colloque C1, supplt!ment au no 2-3, Tome 33, Fkurier-Mars 1972, page C1-177
THE ACCURACY OF DIFFU SION-CON STANT MEASUREMENTS BY DIGITAL AUTOCORRELATION OF PHOTON-COUNTING FLUCTUATIONS
E. R. PIKE
Royal Radar Establishment, Malvern, Worcs, England
R&um6. - On discute la prkcision d'une mesure de la largeur de raie spectrale par la mkthode de correlation digitale (( a simple seuil )) (*) des fluctuations du taux de comptage de photons, pour un flux lumineux donne et un temps de mesure h k . Cette precision est limit& par les effets statis- tiques dus a la nature aleatoire du champ lumineux et du processus photo8lectrique ; en cons4 quence, toutes choses kgales par ailleurs, 1e resultat sera d'autant plus precis que l'experience aura durk un temps plus long. On presente des rksultats theoriques et expkrimentaux concernant les precisions que l'on peut espkrer, en fonction du temps d'kchantillonnage, de la surface du detecteur et du seuil.
Abstract. - We discuss the accuracy of a measurement of a spectral linewidth using the method of digital correlation of single-clipped photon counting fluctuations with a given light flux and a given duration of experiment. Statistical effects due to the random nature of the light field and the photoelectric process limit this accuracy, so that the longer the duration of the experiment the more accurate the result, if all other factors are equal. Theoretical and experimental resultsare presented which show the accuracies obtainable as functions of the sample time, detector area and clipping level.
We have investigated, over the past two years, light-scattering spectra from a number of biological molecules from various laboratories, by correlation of clipped photon-counting fluctuations [I]. A list of these is given in an appendix. In many cases we have found satisfactory values for diffusion coefficients under various conditions, while in other cases work is still proceeding. We have also studied Rayleigh scattering from organic liquids and, following Yeh and Keeler [2], have examined some metal-sulphate solutions by the same methods. The techniques have proved to be very powerful, enabling low light fluxes to be studied with high precision. A commercial version of our equipment is now being manufactured under licence to the National Research and Develop-
ment Corporation.
In order to support this activity and to give gui- dance to potential users of the method we have under- taken an extensive study, by both analytical and computational methods, of the statistical varia- tions to be expected in such measurements of spectral linewidth.
The results of the analytical work may be found in detail in a paper by Jakeman et al. [3]. We should like here to summarize these and to present the results of some more recent computations performed by Dr.
A. Hughes, together with new experimental confirma- tion of them by Dr. C. J. Oliver, both of this laboratory.
Special cases of the problem have been studied by
(*) Single clipped D.
Benedek [4], Haus [5], Cummins and Swinney [6], Pusey [7], and Degiorgio and Lastovka 181. The relation of these calculations to ours is discussed by Jakeman et al. [9 1.
We first briefly introduce the ideas behind the method. The light scattered from randomly moving particles has the statistical properties of a Gaussian- distributed complex vector which were studied in detail in connection with radar problems in the early 1940's. Statistical theorems of great practical value were developed connecting the zero crossings of Gaussian radiation fields and their spectra [lo]. It was shown, for instance, that the temporal autocor- relation function of a two-level (one-bit) signal (clipped or hard-limited signal) was simply related to that of a Gaussian signal with the same zero crossings and, further, that little statistical information was lost in passing from one to the other.,We have extended [ l ] these results to the case of optical signal processing where the initial signal consists of a stochastic train of impulses resulting from photodetections. The ana- logue of clipping the electric field at its zero value is the assignation of an integer clipping level k (associated with a fixed sampling time T) such that the signal has one or other of two levels (1 or 0) according as the number of photons detected in a sample time exceeds k or not. In practice we prefer to correlate such a clipped signal not with itself, as in the radar case, but with the original signal (single clipping) since multiplications by ones or zeroes are easy to perform electronically on any number.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972132
C1-178 E. R. PIKE It turns out that the relations between single-clipped correlation functions and unclipped or true autocor- relation functions are even simpler than the original radar formulae. In fact, if a signal is clipped at its mean level these functions are theoretically identical.
Work which had the same motivation, that of reducing the electronic complexity of a true fast parallel corre- lator, was pursued independently by Chen at Harvard and MIT and by Pusey at IBM. Our clipping solution is slightly different from their techniques. Lastovka at MIT was also an early worker in this area and Cummins at John Hopkins used a commercial ana- logue correlator for this application in 1967 [ll].
For most molecules, however, as Cummins discovered, the information recovery rate of commercial electronic correlators is little better, if at all, than that of the wave-analysers first used by himself [I21 and by Benedek's group at MIT [13] for macromolecular studies ; this is particularly true for smaller molecules.
It is usually desireable to work with low concentra- tions of biological molecules due to the concentration dependence of the diffusion coefficients, if not some- times the scarcity of the material. For this reason we have used a clipping level of zero for many practical cases.
The linewidth accuracy can be found analytically in this case. This is done by first calculating the statis- tics of the correlation coefficient estimators and then by calculating an optimised, weighted, least-squares two- parameter fit of the assumed exponential form of the correlation function. The parameters are the decay time and the value at the origin, the value at infinity is given by normalisation channels in the correlator.
For higher clipping levels the analytical work becomes exceedingly cumbersome and we have resorted to computer simulation.
The number of correlation coefficients computed is also an important parameter, determining the bulk of the cost of high-speed real-time equipment. We have made calculations for 20, 100, and an infinite number of channels.
The ratio of the sample time to the coherence time or inverse linewidth of the intensity fluctuation spec- trum must be decided ; for a fixed number of channels there will be an optimum value for this ratio. If too small the count rates will be low and the points will sample the function in a highly correlated fashion, if too large the points will give uncorrelated samples but in the region of low correlation coefficient. In practice, depending on the number of channels and count rates, the optimum occurs when a total delay of two or three coherence times is covered.
The area of the detector is another experimental variable. As this increases so the correlation coeffi- cients decrease 1141, [15], but the light flux increases.
In fact these effects compensate asymptotically [I61 so that little is gained, even with the lowest light fluxes, where photon statistics dominate, by using more than one coherence area. With high values of
light flux the area of the detector can be reduced to well below a coherence area to diminish the averaging effects of the signal statistics without losing accuracy from photon statistics. The break-even point will vary with number of channels and clipping level but a value of 10 photodetections per coherence time is, in prac- tice, effectively an infinite light flux as far as photon statistics affect the accuracy of the linewidth estimate.
We refrain from giving lengthy formulae in this publication but show useful practical results in three figures. In figure 1 the effect of changing the sample timelcoherence time ratio is shown for fixed values of the number of channels (twenty), detector area (0.01 coherence area), experiment duration (lo4 cohe- rence times) and for single clipping at zero. In addi- tion to points already mentioned the loss of accuracy as the count rate gets too high for the zero clipping level can be seen. The ordinate scales as the square root of the number of coherence tinies in the experi- ment duration. The points are experimental values obtained by programming the correlator to complete 25 experiments for each set of parameters chosen, The RMS deviations were then extracted by two- parameter computed curve fitting.
C O R R E L A T I O N
t 2 0 C H A N N E L S CLIPPED AT ZERO -THEORY rOR POINT DETECTOR
lo4 C O H E R E N C E T I M E S
100 COUNTS
C O H E R E N C E TIME ,
FIG. 1. - The linewidth accuracy for 20 channels single-clipped at zero, obtained with an experimenta.1 duration of 104 coherence times, as a function of sample timelcoherence time ratio and counts per coherence time. The solid lines are the theory for a point detector. The points are experimental values using a detector area 0.01 times a coherence area. The ordinate scales as the square root of the number of coherence times in the duration
of an experiment.
THE ACCURACY OF DIFFUSION-CONSTANT MEASUREMENTS C1-179
In figure 2 is shown the effect of increasing the detector area, again with single clipping at zero and with sample timelcoherence time ratio set at 0.1, which is near optimum for this number of channels unless count rates are too high. The behaviour of the points follows reasonably closely that which would be predicted from figure 1 taking into account only the change of count rate. The effects of clipping domi- nate the effects of signal statistics at the large area end of the curves.
I
L I N E W I D T H ACCURACY I N S I N G L E - C L I P P E D C O R R E L A T I O N
2 0 C H A N N E L S CLIPPED AT ZERO
- THEORY FOR P O I N T D E T E C T O R
lo4 COHERENCE T I M E S T/T= = o I
FIG. 2. - The linewidth accuracy for 20 channels single clipped at zero, obtained with an experimental duration of 104 coherence times, as a function of detector arealcoherence area ratio. The solid line shows the theory for a point detector with equivalent count rates. The points are experimental values using a sample time 0.1 times a coherence time. The number of photodetections per coherence time ranged from 0.5 to 23. The accuracy increases with area until the count rate becomes too high for clipping
at zero when it reduces again.
In figure 3 is shown the effect of clipping levels.
We should say that we have now verified our original theoretical formulae [16] for the effects of clipping on the correlation functions. The errors have a minimum value which is very flat at clipping levels between one half and two times the mean counting rate. Increasing the light flux will always iinprove the accuracy to some extent if the appropriate clipping level is chosen but the improvement quickly saturates, depending on the number of channels, as the number of counts per coherence time goes beyond about 10.
I LlNEWlDTH ACCURACY IN SINGLE- CLIPPED
1 CORRELATION 2 0 CHANNELS I
T/T, = 0 . 1 A/A, = 0.01
I . . :
, ' I
1 2 3 4 5 6 7 8 9 1 0
C L I P P I N G LEVEL -
FIG. 3. - The linewidth accuracy for 20 channels obtained with an experimental duration of 104 coherence times, as a function of single-clipping level, and counts per coherence time.
The points are experimental values using a sample time 0.1 times a coherence time and a detector area 0.01 times a coherence area.
The maximum accuracy obtainable does not improve signifi- cantly with count rate above about 10 counts per coherence time.
Theoretical values are shown at zero clipping level only, the behaviour at higher values has been confirmed by computer
simulations.
Appendix : list of biological molecules studied at RRE to date Lysozyme
BSA
Tamm-Horsfall glycoprotein Histone 2 A 1
a-crystalline Haemocyanins
Helix pomatia a Helix pomatia P
Archacatina marginata Pila leopoldvillensis Murex trunculus
Sheep colonic mucosa glycoprotein Adenovirus
Myxovirus (influenza) Thyretin
RNA polymerase y-globulin Aldelase Catalase
Dr. A. Peacocke, Dept of Clinical Biochemistry, Radcliffe Infir- mary, Oxford.
Dr. R. Blagrove, Division of Proteinchemistry CSIRO, Mel- bourne, Australia.
Prof. R. Lontie, Biochemistry Laboratory, Catholic University of Louvain, Belgium.
Dr. E. Wood, Dept of Physiology, Royal University of Malta.
Dr. Kent, Dept of Biochemistry, University of Oxford.
Prof. Belyavin, University College Hospital, London.
Dr. B. Nicholson, Dept of Biochemistry, University of Reading.
E. R. PIKE
Aspartate amino transferase (enzyme) T 3 bacteriophage
Turnip yellow mosaic virus Fibrinogen
Keyhole limpet (haemocyanin) Haemoglobin
Algae (alive and dead)
Dr. P. Bayley, National Institute for Medical Research, Mill Hill London.
Dr. P. Sprag, Dept of Chemistry, University of Birmingham.
Dr. D. Pepper, Blood Transfusion Unit, Royal Infirmary, Edinboro.
Dr. G. Beaver (through).
Dr. W. Gratzer, Kings College, London.
Prof. Guttfreund, Dept of Biochemistr,~ University of Bristol.
Prof. Thonemann, from Marine Biology Dept, University of Swansea.
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411.
YEH (Y.) and KEELER (R. N.), J. Chem. Phys., 1969, 51,1120.
JAKEMAN (E.), PIKE (E. R.) and SWAIN (S.), J. Phys.
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B E N E ~ E ~ (G. ' B.). 1968. Polarisation, Matiere et
~ a ~ o i n e m e n t ' (paris': Presses ~niversitaires de France).
HAUS (H. A.), 1969, Proc. Int. School of Physics
tt Enrico Fermi D, Course XLII (New York : Ac. Press), 111.
CUMMINS (H. Z.) and SWINNEY (H. L.), 1970, Progress in Optics, Vol. VIII Ed. E. Wolf (Amsterdam:
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Private Communication.
Private Communication.
[9] JAKEMAN (E.), PIKE (E. R.) a.nd SWAIN (S.), J. Phys. Ai 1971, 3, L 49.
[lo] VAN vLECK (J. H.), 1943, Harvard University Radio Res. Lab. Rep., No 51.
[ll] CUMMINS (H. Z.), 1969, Proc. Int. School of Physics
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[12] CUMMINS (H. Z.), KNABLE (N.) and YEH (Y.), 1964, Phys. Rev. Letters, 12, 1150.
[13] DUBIN (S. B.), LUNACEK (J. H.) and BENEDEK (G. B.), Proc. US Naf. Acad. Sci., 1967,57, 1164.
[14] SCARL (D. B.), 1968, Phys. Rev., 175, 1661.
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