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Apodisation in the time domain
G. Series
To cite this version:
G. Series. Apodisation in the time domain. Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.615-620. �10.1051/jp2:1992156�. �jpa-00247660�
Classification
Physics Absiracts
32.70 42.30
Apodisation in the time domain
G. W. Series (*)
2 Monkton House Close, Old Boars Hill, Oxford, OXI SJQ, G-B-
(Received 2 July J99J, accepted J0 December J99J)
Abstract. The work of Pierre Jacquinot and his colleagues on a method for controlling the
spread of light by diffraction in optical images is recalled. It is possible to perform operations in time-resolved spectroscopy which have the same result, as this article demonstrates.
1. Apodisation.
Among the variety of new and exciting spectroscopic techniques which were investigated and
developed at the Aim£ Cotton Laboratory in Bellevue in the nineteen fifties under the direction of Professor Jacquinot was that known as « apodisation » ; a means of modifying in a
pre-determined way the diffraction image of luminous objects by controlling the distribution of light leaving the exit pupil of the optical system. Particularly troublesome in astronomy and
spectroscopy are the diffraction rings or fringes which normally accompany the image of a star
or spectral line, effects attributable to the sharp cut-off of light at the edge of some element in the optical train. By means of an aperture whose transmission varies over its area these
undesirable « feet » of the image may be largely suppressed at the expense, of course, of reduction of the overall intensity and possible broadening of the central image. Suitable
apodisation of the image of a principal object (a star or spectral line) therefore, can give the benefit of improved resolution of a nearby, subsidiary object (a weak star or satellite spectral line).
Early use of this technique was made by Couder and Jacquinot [I] and by Boughon, Dossier and Jacquinot [2]. Further work, mainly theoretical, was carried out in the late nineteen forties and the fifties : a comprehensive review was written by Jacquinot and Roizen- Dossier [3]. The more recent « image processing » has much in common with apodisation.
As originally conceived and carried out, apodisation was applied to spatial images, as in
photographic astronomy, microscopy and spectroscopy, for example. But new methods in these sciences were opened up in the post-war years : I draw your attention in particular to
spectroscopy based on the pulsed excitation of close-lying states followed by time-resolved
study of the fluorescent light, for apodisation can be applied in the time domain also, as I shall show.
(*) Formerly of the Clarendon Laboratory, Oxford, G-B- and Department of Physics, University of
Reading.
616 JOURNAL DE PHYSIQUE II N° 4
2. Light beats : the so-called quantum beats.
Spectroscopists will feel on familiar ground if I ask them to contemplate the excitation of
mercury atoms by a short pulse of resonance radiation, 253.7 nm. From the ground state, 6~So, atoms are excited to the Zeeman triplet, 6~Pi.
r
r(w
Q ~
~ ~
' decoy:
~
m0Cula1ion al w ~~
o
- j
a) b)
Fig. I. a) Pulse excitation of a pair of Zeeman levels separated in Bohr frequency by
w. b) Modulated, exponential decay of the fluorescent intensity.
Let the atoms be in a magnetic field of a few gauss, 10 G for example, sufficient to separate the Zeeman levels by more than their natural width (about I MHz) ; the fluorescent light seen through a linear analyser will be found to be modulated at the Larmor frequency or at twice that frequency as the fluorescence decays exponentially (Fig. lb). All this is very well known and can be understood in classical terms as the observation of fluorescence from an assembly
of precessing, decaying oscillators, set in motion synchronously by the pulse. The Larmor
frequency at the field in question and hence the g-value can be determined by taking the Fourier transform of the experimental curve I (t).
But now let us shift our focus to that other exemplar of spectroscopists, the sodium
D~ line, 589 nm, 3~Si/~-3~P~j~. We now have to grapple with the complication of hyperfine
structure. The hfs-Zeeman term diagram of the upper state is shown in figure 2. The structure
has been much studied by level-crossing spectroscopy that is, by observation of the steady-
state fluorescence as a function of magnetic field applied to the atoms in the vapour phase.
The intensity of the fluorescence shows significant changes with the field (Fig. 3a for
example) the information sought is the set of values of the magnetic field at which the fluorescence intensity exhibits minima. That will come about when Zeeman levels of different
m cross : for a particular geometrical arrangement one expects to find such minima at the
crossings A, B and C in figure 2.
From the measured values of the magnetic field at these crossing points the hyperfine
interaction constants may be inferred.
But how accurately ? For a pair of levels diverging linearly with the field the level-crossing
curves are readily shown to be Lorentzians whose widths are determined by the slopes of the
levels and their natural widths. But in figure 3a, the experimental curve corresponding to the term diagram in figure 2, it is clear that the features corresponding to A, B, C are not well resolved. The discrete minima we were hoping to find are blurred by the intrinsic widths of the individual features.
m
B lG)
Fig. 2. Energy eigenvalues for the ~P~j~ state of Na Ii
= J
= 3/2) plotted against magnetic field, using
the interaction constants a = 18.7 MHz, b
= 3.0 MHz.
So what can we do to reduce the overlap? Recall that Lorentzians have very long feet : can
we cut them off ? Indeed, we can, up to a point, and at some cost. By elaborating our
experimental technique, and with a little mathematical manipulation, we can convert the Lorentzians into some more favourable functions, Gaussians, for example. Every spectroscop- ist knows that, width for width, Gaussians have daintier feet than Lorentzians. With our level-
crossing curves modified in this way we may expect to find improved resolution of the
important features.
Our classical model directs us to a plausible technique for apodisation. Recall the
modulation (Fig. lb) exhibited by the fluorescence from atoms precessing in a magnetic field.
If we wished to determine the Larmor frequency by Fouder transform of I (t), our accuracy would clearly be limited by the number of oscillations under the decaying exponential. But the exponential represents the behaviour of an assembly of atoms having a distribution of lifetimes. Suppose, by some means, we could bias our observations in favour of the longer-
lived atoms. Such atoms undergo more cycles of precession before they decay. Would not
such biasing lead to a more precise determination of the Larmor frequency, to a narrowing of
level-crossing curves ? Yes, with a reservation, as we shall see.
JOURNAL DE PHYS'QUE 'I T 2, N' 4, APR~ 'W2 24
618 JOURNAL DE PHYSIQUE II N° 4
b)
.
""
. .
.
"
#
w .'
~
w
«
a)
<
©
w
, b~
A
o ~
o B
We write
p(r, w, R)
= exp rR ji + cos w RI, (i)
which represents a curve similar to figure16. Experimental values of p(H ), for different values of w, may be obtained by pulse-counting techniques and stored.
Conventional level-crossing curves are obtained by integrating equation (I) with respect to H, with running from 0 to oD :
~ ~
L (w, r)
= p(H ) dH
= + (2)
o
r r~+ w~
To bias the level-crossing curves in favour of longer-lived atoms we multiply p(H) by a
suitable factor f(H) before carrying out the integration ; this is conveniently done in a computer, but we shall examine the implications analytically.
An obvious form for f(H), and one which is not difficult to realise experimentally, is a step
function increasing from 0 to I at some chosen value H
= Ho. This does indeed secure
narrowing of the level-crossing curves, but it introduces undesirable oscillations into the
wings, for the same reason that fringes in the image plane of an optical system are attributable to the sharp edges of the exit pupil. This and other forms for f(H ) were studied by myself and
colleagues in our first publication [4]. A more desirable form was the displaced Gaussian
f(o, A) 2
= exp (- -AR rid) (3)
2
The full width at half-height of this curve, 2(H~~ Hij~) is equal to 4(In 2)~'~/A (this is the T of Fig. 3c). The displacement of the peak, b~~~, from the exciting pulse,
= 0, has, in
equation (3), been chosen to have a specific relation to A, as follows
H~~
=
2 r/A~, (4)
a choice which secures the removal of oscillations. Thus the function (3) changes equation (2) into the Gaussian
F(w, r, A)
= l~ p(w, r, H) f(H, A) dH
=
~exp §
(I + exp fl j (5)
o A A A
The width of the Gaussian as a function of
w is at our disposal by suitable choice of A (I.e. of 7~. But we must take note of the factor exp (rid )~ if A is too large in relation to
r we lose too much intensity.
The choice A
= r has particular advantages. The width of the curve F(w, r, r) is
approximately the same as that of the Lorentzian L (w, r) and the intensity at the peak has been reduced only by the factor gr~/~le
=
0.66. But we have gained the great advantage of
apodisation.
The technique of time-biasing would not appear to offer much advantage if the objective
was to improve the measurement of the position of an isolated peak ; an improvement in
signal-to-noise ratio by other means would probably be more profitable. But for improving
the resolution of satellites lying on the wing of a principal feature, time-biasing has something
to offer, particularly if the analytical form of the principal feature is not known exactly.
My colleagues and I applied the method to the hyperfine structure of sodium with the results shown in figures 3b and 3c (from Refs. [4, 5], respectively). In these papers various
practical points such as the finite duration of the exciting pulse, background counts, etc. are
620 JOURNAL DE PHYSIQUE II N° 4
treated. The first paper demonstrated the practicality of the method. The results obtained in the later paper provided values for the interaction constants of higher accuracy than had
previously been achieved with the use of conventional lamps for excitation of the fluorescence. Values of comparable accuracy were later obtained by techniques based on laser excitation making use of the vastly greater intensity of fluorescence : nevertheless it
seemed appropriate, in this volume honouring Professor Jacquinot, to recall that, long before lasers were invented, optics and spectroscopy were strongly influenced by his ingenuity, his
enthusiasm, and the encouragement he gave to his younger colleagues.
Acknowledgements.
Acknowledgement is made to IOP Publishing Ltd for permission to copy figures 2 and 3 from the articles in Joumal of Physics B referenced below.
References
[I] COUDER A. and JACQUINOT P., C. R. Acad. Sci. Paris 208 (1939) 1639.
[2] BOUGHON P., DOSSIER B. and JACQUINOT P., C. R. Acad. Sci. Paris 223 (1946) 661.
[3] JACQUINOT P. and ROIzEN-DOSSIER B., Progr. in Optics III Noah-Holland (1964) 29-186.
[4] COPLEY G., KIBBLE B. P. and SERiES G. W., J. Phys. B 1 (1968) 724-35.
[5] DEECH J. S., HANNAFORD P. and SERIES G. W., J. Phys. B 7 (1974) 1131-48.
