zotero/storage/7RDQ5H6A/.zotero-ft-cache

PHYSICAL REVIE% A

VOLUME 4S, NUMBER 1

Sagnac experiment arith electrons: Observation of the rotational phase shift of electron maves in vacuum

JULY 1993

Franz Hasselbach' and Mare Nicklaus~ Institut fiir Angewandte Physik, Universitat Tubingen, Auf der Morgenstelle 18, D 7/00-Tubingen, Germany
(Received 17 November 1992)
A Sagnac experiment with electron waves in vacuum is reported. The phase shift caused by rota-
tion of an electron biprism interferometer placed on a turntable has been measured. It was found to
agree with prediction within error margins of about 30Fo. A compact ruggedized electron interferometer was used. It is based on a high-precision optical bench of 36-cm length. This interferometer is less sensitive by orders of magnitude to mechanical vibrations and electromagnetic stray fields than conventional electron interferometers. A beam of low-energy electrons (150-3000 eV) emitted by a field-emission electron source was used. For the most part, electrostatic electron optical components were employed. The magnified interference fringe pattern was intensified by a dual-stage multichannel-plate intensifier, recorded by a charge-coupled-device video camera, transmitted from the turntable to the laboratory system via a slip ring, and evaluated by an image-processing system. Both the rotation rate and the area enclosed between the two partial waves were varied (up to values
of 0.5 s and 3.9 mm, respectively). Fringe shifts on the order of 5% of a fringe period were attained. Some historical aspects of the Sagnac efFect as well as some aspects of its interpretation are
mentioned. A brief informal discussion is included of the interpretation of the Sagnac phase shift as
a geometric phase ("Berry phase") caused by the global anholonomy of the local phase factor that
is produced by the gauge field induced by rotation.
PACS number(s): 03.65.Bz, 03.30.+p, 06.30.Gv, 41.90.+e

I. INTRODUCTION
The effect of rotation on space-time, as it can be measured in two-beam interferometers in which the beams
enclose a finite area, is called the Sagnac effect. The first proposal of an interferometrical detection of a rotation, made by Sir Oliver Lodge in 1893 [1] (of course in the context of ether theory), has remained largely unknown. In fact, he proposes to detect the Earth's rotation with a large interferometer and derives an expression for the phase shift that yields the correct result for the Sagnac phase if the transformation is made from ether theory to relativity. In a subsequent paper, he proposes to rotate an interferometer on a turntable [2]. In 1913, Sagnac [3] carried out his famous experiments demonstrating the ex-
istence of the efFect with a rapidly rotating light-optical interferometer. A few years earlier [4], however, a graduate student in Jena, Franz Harress, very probably had, unknowingly [5, 6], observed the Sagnac efFect for the first time during his experiments on the Fresnel drag of light. Michelson and Gale [7] demonstrated in 1925 the phase shift caused by the Earth's rotation in a very large light optical interferometer.
After the invention of the laser, the field of light-optical
Sagnac interferometry experienced a dramatic increase in precision as well as in width of application. This is demonstrated by devices such as the "ring laser" [8] and the "fiber-optic gyroscope" [9] which are nowadays used in inertial navigation [10].
After a first proposal in 1961 by Heer [11] of a Sagnac experiment with matter waves, Zimmerman and Mercereau performed a Sagnac-type experiment with elec-

1050-2947/93/48(1)/143(9)/$06. 00

48

tron Cooper pairs in 1965 [12]. Both papers do not seem to have become very widely known. After proposals by Page [13] and Anandan [14], the effect of the terrestrial rotation on the neutron phase was demonstrated in 1979 by Werner, Staudenmann, and Colella [15] in a Si perfectcrystal interferometer. The Sagnac-type experiment, using a neutron interferometer of a similar kind rotating
in the laboratory, was successfully performed in 1984 by Atwood et al. [16]. Very recently, Riehle et aL [17] have performed a Sagnac experiment with (neutral) 4oCa atom
beams.
Clearly then, from a fundamental point of view, an essential gap that remained to be closed in the domain of Sagnac interferometry was the realization of a Sagnac
experiment with charged fermions. The neutron interfer-
ometrical experiments had shown coupling of the neutron mass to both gravitational [18] and accelerational fields [15, 16, 19] at exactly the same value, thereby proving the validity of the classical principle of equivalence in the
quantum limit. Although expected from theory, we felt
it to be of fundamental interest and by no means trivial that the presence of charge, with its coupling to the
electromagnetic field being so many orders of magnitude stronger, does not influence the electrons' coupling to
— — an accelerational field. It therefore seemed worthwhile
to test within the error margins this fundamental as-
sumption directly by using charged fermions in vacuum (and thereby avoiding the conceptual difficulties arising
from using Cooper pairs, i.e., bosons, interacting with a solid-state device). The development of a very compact and rugged electron interferometer [20] made it possible to conceive of a Sagnac experiment with electrons [21,

143

1993 The American Physical Society

FRANZ HASSELBACH AND MARC NICKLAUS

48

22j. The present paper reports on the observation of the phase shift of electron waves in vacuum in a rotating interferometer.

II. THROB%

A. VTKB derivation of the Sagnac efFect

In order to derive an expression for the phase shift of
waves (of any type) in a rotating frame of reference [23], we m.ake the general ansatz

@=%,exp —S

h

~

for the wave function, with the action function (phase) S.
We assume the waves to be propagating on maeroscopieal paths so that they can be treated in a semiclassical way using the WKB approximation. A general experimental setup fulfilling this condition would be a source emitting
two coherent partial waves which then propagate around a finite enclosed area on macroscopically different paths and interfere with each other at a detector. This condition is certainly fulfilled in the case of the electron interferometer described here with its linear dimensions of
tens of centimeters and a separation of the paths of tens of micrometers (whereas the coherence length of the elec-
tron waves was on the order of 100 nrn). We want to investigate the efI'eet of the rotationally
caused change of the Minkowski metric on the phase difference between the partial waves. From the WKB ap-
proximation follows

= ~ g pv PkinrLt, @kine

22

(2)

with the metric g& and the (kinetic) four-momentum
pk;„„. In the general case of charged particles and the presence of electromagnetic fields, the kinetic four-
momentum is

(3)
The relationship between the canonical momentum p„ and S is given by
ppoOSs pe Bx~
We apply the general perturbation ansatz
g„=g„+h„,
s = s(') + s(').
We assume h& to be a small perturbation of a general kind of the unperturbed Minkowski metric of fIat space. This does not yet necessarily have to be a perturbation caused by rotation. From Eqs. (5), (3), and (4) follows

Pk p S(o) 6Ap + S(p1)

and with
S(o) —eh~ —Pk(o. )

we further obtain

+ J klnjLt,

J

(p}
kinp,

~&(pi)

Applying this and (5) to (2), we obtain
(&"+h")(p" +s('l)(p(" +s('l) =~'"

using the contravariant components of the metric. In keeping with the assumption of a small perturbation, we
retain only the terms of first order in g"" and S combined
and obtain

+ p, (&) (o) ,p ~kinv

p (o) (o) ~king, ~kinv

h„, or, equivalently, using the covariant components

~(ko)iPn,

(&) p

2

h

P&

(o)P ~kin

(o) ~kin

(12)

Parametrizing the unperturbed wave path,

(p) p

dh

~kin

(13)

(12) transforms to

Ch"
dA

'("i)

d (i)
dA

2"1 dh~ dh" dA dA

(14)

The phase difference between the two partial waves is given by the integral of the action along the closed path surrounding the encircled area. This yields the phase
shift

1 dS~il

K

dA

1 2h

" dx" dx" dA dA

caused by the perturbation. We now, more specifically, assume the perturbation to
be a rotation of the frame of reference. From the line element of a rotating coordinate system

ds = c dt —dr dr —2(Axr) drdt,

we immediately derive

—2(Axr) drdt = h„dx"dx",

(17)

and further

dx~ dx"

dr dt

(18)

Observing

= (p}p c dt

i

pkin

we obtain the general expression

A(p

=

1 hc2

(A x r) E dr

(20)

for the rotationally induced phase shift between two par-
tial waves whose corresponding particles have the total
energy E. In the general case, E may be variable along the path. In all practical Sagnac interferometers, E is
constant along the path to a very good approximation.
This is certainly the ease in the electron interferometer

48

SACxNAC EXPERIMENT %'ITH ELECTRONS: OBSERVATION. . .

145

used in this experiment, where all changes in kinetic en-
ergy along the beam path are small [24] compared to the rest energy of the electrons (511 keV). Since A, too, is constant along the path in all practical realizations of
Sagnac interferometers, we can pull E and A out of the
integral. We thus obtain the Sagnae phase shift

= A&p

A, c2 A

(rX dr)

2E

hc2

(21)

B. Remarks

A number of comprehensive articles [14, 25—28] have described various aspects of the Sagnac effect and have

also undertaken to elucidate the conceptual diKeulties

that seem to be encountered in its interpretation Va. r

ious authors have derived the Sagnac phase shift in a

number of ways: by optical analogy [13],general relativ-

ity considerations [10, 29, 30], special relativity analyses [28, 31—35], the WKB approximation [15], the Doppler effect of moving media in an inertial frame [36], a classical kinematical derivation [22, 37—39], dynamical anal-

ysis in a noninertial frame [40, 41], by analogy with the

Aharonov-Bohm efFect [42], by extension of the hypothesis of locality [43], by adiabatic invariance [44], using ether concepts [45], and in other ways. This great va-

riety (if not disparity) in the derivation of the Sagnac

phase shift constitutes one of the several controversies

(recounted, e.g. , in [31,46, 47)) that have been surrounding the Sagnac effect since the earliest days of studying

interferences in rotating frames of reference.

The classical kinematical derivation, as it has been

used by many authors (see above), has the advantage of yielding the correct first-order result in a very sim-

ple and intuitive way. Its starting point is a considera-

tion that applies to any type of waves, and it seems en-
tirely classical. Consider a circular path with radius B.

Two counterpropagating coherent wave packets are emit-

ted from the same starting point, and circulate around

the enclosed area A with the velocity u. After a full

circulation by each partial wave, their interferenee pat-

tern is observed at the starting point. In the case of

rotation of the whole system with angular velocity 0,

the co-rotating short distance

point of observation
As = B0 t towards

will have
one of the

moved waves,

the and

„, the same distance bs
time for one circulation

aiws aty—from

the other one. The with the experimen-

tally fulfilled assumption RA (& u. A total path-length
difference Al = 26s = 47r R„Aresults, which transElaAantpde~=s~Eiminct=—o2 mhaan,cp,d2hEaiAsnAe=tAhAe.PpiT/cpha,=seesae&roeff„inAtmhaelAatt.sieunSrhbuesbtrwiestantuivtttlieuyost,ninsg,noonnehcAoloawb=ssetaivchiena/rlps,
steps [48] that introduce the decisive element of relativity. It is this covertly relativistic feature of nonrelativis-

tic quantum mechanics [34] that defines the scope of this

derivation, and Anandan has shown [26] that precisely for this reason there is no Sagnac effect at all if electrons

are treated as classical particles in a nonrelativistic way.
This necessity for limiting the application of the term "Sagnae effect" to the intrinsically relativistic effect, and
for differentiating it from competing effects such as the
Fizeau-Fresnel drag, has been the source of another ongo-
ing controversial discussion [5, 6, 49). This point becomes immediately evident [34, 39] if one would "rroneously-
apply the kinematical derivation to sound waves. It has been argued, therefore, that the Sagnac eff'ect is a purely "topological" effect [50], resulting from the efFect of rotation on the geometry of space-time. This becomes even
more evident if one considers the close analogy [30, 34, 42, 51] of the Sagnac efFect with the Aharonov-Bohm effect [52, 53]; and both, in turn, are manifestations of the "geometric" and/or "topological" phase [54] picked up by a quantum-mechanical system transported adiabatically [55] around a closed circuit in parameter space, the so-called Berry phase [56]. This phase, which is picked up in addition to the usual dynamical phase, depends only on the geometric history of the system, and refiects
the global nonintegrability (or anholonomy) of the local phase factor produced by a gauge field ("phase-shift field" [57]). In the Sagnac experiment, the two partial beams enclosing the finite area A represent the cyclic
evolution of the system, and the rotation provides the
curvature of the (parameter) space around which the par-
allel transport of the electron states takes place. Here, in fact, the parameter space is identical to the threedimensional space (or to space-time; see below). The
Sagnac effect is not as "fully topological" [58] as, e.g. , the Aharonov-Bohm effect [since the electrons are not trav-
eling in (Coriolis-) force field-free regions (see, however, e.g. , [59, 60] for proposals of experiments that would ful-
fill this requirement)]. It can nevertheless be attributed
topological characteristics in the following sense.

(1) The Sagnac phase is valid for any path encircling the enclosed area A, as long as this area (or more exactly, the projection of A onto A) remains constant.
(2) The location of A relative to the axis of rotation is
irrelevant.
(3) For massive particles, it is independent of their kinetic energy in the low-energy ("nonrelativistic") limit
[since in Eq. (21) E denotes the tota/, not only the kinetic

energy].

Chiao derlies

[54] has called this the Sagnac effect,

type "age

aonfhoalnohnoolmonyo, my",

which and has

unalso

pointed out its close analogy with the twin paradox of

relativity (see also [34, 61]).

The question whether the Sagnac efFeet can only be

treated adequately in the framework of general relativity

(as opposed to a special relativistic treatment being suffi-

cient), has perhaps generated the most extensive contro-

versial discussion in this field. Many authors have argued

for either point of view (see the references cited above),

and this has even led some to talk of an "intermediate"

character of the Sagnac effect [62]. To a certain degree,

some of the controversy seems to be a problem of de6ni-

tion rather than of physics. It seems to have become in-

creasingly accepted, however, that the Sagnac effect can

be seen as special relativistic insofar as a rotation alone

146

FRANZ HASSELBACH AND MARC NICKLAUS

(i.e., without masses being present), although causing the system to become noninertial, does not alter a previously flat Minkowski space-time, i.e., the four-dimensional geometry remains pseudo-Euclidean [31,35]. The geometry of the three-dimensional space, however, can be thought of becoming non-Euclidean in this case, i.e., acquiring a curvature. The perturbation of the metric caused by rotation can be shown to be replaceable by a series of Lorentz transformations and translations [26], and both are formally equivalent when expressed in cylinder coordinates [38]. Therefore the Sagnac effect, as measured in an interferometrical setup as illustrated in Fig. 2, can be interpreted as the result of the special relativistic length contraction of the wave paths. The equivalent impossibility of globally synchronizing clocks in a rotating frame of reference was already noted by Einstein [63]. It was experimentally proved with the famous "flying clocks" [64], and with electromagnetic signals transmitted around the earth [33, 65], which both demonstrated the existence of the Sagnac effect on a global scale.
III. EXPERIMENT
A. Experimental conditions and difhculties
The task at hand when performing a Sagnac experiment with free-electron waves (i.e., with electron beams in vacuum, as opposed to, e.g. , electron Cooper pairs traveling in a solid-state device) is to rotate an entire electron optical setup still capable of delivering a detectable electron interference pattern while under rotation. The wave path-length differences to be expected in our experiment were on the order of 10 m, and the ensuing phase shifts on the order of 5%%uo of a fringe period. This meant that the phase had to be measured with a sensitivity of l%%uo of a fringe period or better in order to keep the error introduced by the registration process within acceptable limits, and that the interference pattern had to be stable for the duration of the experiment to a degree comparable to the expected phase shift.
Electron interferometers are substantially more sensitive in two crucial aspects when compared to light optical interferometers and, to a lesser extent, neutron- and atom-beam interferometers. The short electron wavelength (of only fractions of a nanometer) makes them extremely sensitive to mechanical vibrations. The electrons' charge (in contrast to photons, neutrons, and neutral atoms) renders electron interferometers very sensitive to ambient electromagnetic fields, which range from the Earth's magnetic field through ac magnetic stray fields to radio-frequency electromagnetic fields. The lowelectron energies used in our interferometer (the reasons for that are given later) make the influence of such fields even more significant. The unavoidable instabilities and noise of the electronic circuitry supplying the voltages and currents to the electron optical components represent a further limitation in electron interferometry. Specific to a Sagnac electron interferometer, the Aharonov-Bohm effect yields an increasing sensitivity to ac magnetic stray

fields proportional to the enclosed area, which, on the other hand, should be as large as possible for observation of the Sagnac effect.
For these reasons, the maximum attainable enclosed area A between the coherent partial beams is much smaller in electron interferometers (usually on the order of a few mm ) than in light optical or neutron interferometers. This makes relatively high rotation rates on the order of 1 s ~ necessary in order to obtain detectable Sagnac phase shifts. Any electron optical setup can only operate in high vacuum, and the use of a fieldemission electron source in our interferometer, which will be detailed later, even necessitated an ultrahighvacuuin (UHU) environment. The difficulties encountered in UHV with any mechanical devices, such as motors, as well as the constraints imposed by the interference image registration process, made it impossible to rotate solely the electron optical setup within the vacuum chamber. Consequently, the entire vacuum system had to be rotated. The relatively high rotation rates mentioned earlier lead to centrifugal effects, such as minute bending of the vacuum chamber, that can cause shift;s of the whole interference pattern in the registration plane simulating phase shifts of the same magnitude as the expected Sagnac phase shift. Finally, in contrast to electron microscopy with its usual exposure times of a few seconds, a long term stability of at least 10 min was needed because of the relatively long time periods needed for rotationally accelerating and decelerating the whole heavy apparatus, and because of the registration procedure that had to be adopted.
B. Experimental setup
A novel type of electron biprism interferometer was
used. It has been described in detail previously [20]. Its design focuses on rigidity and compactness in order to reduce the sensitivity of the interferometer to the aforementioned influences. Its construction principle is basically that of a high-precision miniaturized optical bench.
All electron optical components are of circular cross section and have a diameter of 28 mm. They are very tightly fixed to two high-precision ceramic rods (8-mm diameter) by a special brace construction. The entire optical bench has a length of only about 36 cm. Due to this compact and rigid design, vibrational eigenfrequencies of the in-
terferometer were achieved that were high enough ()100
Hz) to virtually ehminate its sensitivity to all ambient mechanical vibrations such as those transmitted along the building's floor (typically between 1 and 10 Hz).
The optical bench principle using the geometry described above, when combined with narrow fabrication tolerances, affords an excellent coarse alignment of the electron optical components onto a common optical axis. Tolerances in the diameter of those components on the order of 10 pm were achieved. Therefore, large-scale mechanical alignment is not necessary (and would be difficult to reconcile with the requirement of mechanical rigidity). Consequently, no mechanical alignment facil-

48

SAGNAC EXPERIMENT WITH ELECTRONS: OBSERVATION. . .

147

ities are provided, which eliminates the need for any mechanical feedthroughs. Due to this and to the instrument's compactness, a very efficient magnetic shielding could be achieved. A high permeability alloy cylinder without any lateral bores surrounded the entire optical bench inside the vacuum chamber. A residual orientation-dependent beam deflection caused by the Earth's magnetic field was eliminated by several additional magnetic shields placed inside and outside the vacuum chamber. The last of those was a box [71 cm
(width) xll6 cm (depth) x50 cm (height)] made out of
high permeability material encasing the entire vacuum chamber, The total magnetic shielding factor achieved was on the order of 250000. A schematic diagram of the entire apparatus positioned on the turntable is shown in
Fig. 1. The electron optical setup that was used is shown in
Fig. 2(a), the beam path in Fig. 2(b). Low-energy electrons (150—3000 eV) emitted from a diode field-emission gun were used. The use of a field-emission source has a decisive advantage over a thermionic electron source insofar as it ofFers (1) a much smaller virtual source size, which helps avoid a loss of contrast in the interferogram due to insufficient spatial coherence, and which allows us to do without a demagnification stage; (2) a much higher
beam brightness, which guarantees sufficient intensity in the interference pattern even at high magnifications; and (3) a smaller energy spread, which increases temporal coherence of the partial wave packets. Furthermore, field-
120 cm
FIG. 1. Schematic diagram of the entire apparatus positioned on the turntable. The view is along the rotation axis of
the turntable. The cables supplying voltages and currents to the interferometer as well as the camera cable are connected through holes in the upper plate of the turntable to the electronic circuitry (see text) corotating on a lower plate of equal diameter. Height above ground of the lower plate: 34 cm; distance between lower and upper plate: 60 cm.

(a)

Magnifying

.055piBgcRB $$g-— 1st

2nd

3rd

Wien

Biprism Biprism Biprism Filter

Quadrupole Lenses

Fluorescent Screen

Deflection Element

Deflection Element

Deflection Element

Rotational Alignment Coils

Electron Beam

L]

u

Dual Channel Plate Image intensifier
u

Area Enclosed Between The Two Partial Beams

fl (View Along The Biprism Filaments)

36

4O

4O

f

6t 2i 56 186

f

f

f

f

FIG. 2. (a) Electron optical setup of the interferometer. (b) Electron-beam path viewed along the biprism filaments, i.e., the filaments are to be thought perpendicular to the plane of the drawing. The dimensions of the interferometer are
given in mm along the bottom.

emission sources can be built in a constructionally more
simple form (see below) than thermionic sources. On the other hand, the emission process of most field-emitter
types available today sufFers from unavoidable emission current fluctuations, which are typically on the order of 10%, and often substantially higher. Those fluctuations constituted the major contribution to the error margins of the experiment.
The emitter was a (100)-oriented tungsten single crys-
( tal, which was electrolytically etched [66, 67] into a very
fine tip (curvature at the apex typically 50 nm). Its shape could be influenced and, if necessary, restored to
optimum emission conditions by simultaneously apply-
ing an inverse electric field and heat to the emitter in a so-called "remolding" process [68, 69]. The total field-
emission currents varied widely. An optimal compromise
between image brightness and stability of the emission could, however, often be achieved with emission currents in the range of 20—100 nA. No further acceleration of the electrons was provided besides that produced by the field-emission extraction voltage. This ensures maximum compactness of the electron gun (total length
34 mm), and allowed using very compact components for the subsequent electron optics. Furthermore, higher electron energies would lead to undesirably small wavelengths. Finally, the use of only one anode ensures a minimum virtual source in contrast to the convention-
ally used triode systems where the additional anode unavoidably adds aberrations. On the other hand, the use of a diode system does not allow one to chose the electron energy independent from the emission current. The problem that this poses with the adjustment of the electron optics was solved by electronically coupling [39] the voltage and current supplies with the field-emission extraction voltage. The UHV necessary for the operation of the field-emission source was provided by a 30 l/s ion-
getter pump, which maintained a vacuum of ~5x10
hPa. Most of the electron optical components of the interfer-
ometer use electrostatic rather than magnetic fields. This

FRANZ HASSELBACH AND MARC NICKLAUS

48

was chosen mostly because electrostatic electron optics The impossibility of exactly determining the filament

can be built in a simpler and more compact way. Electro- radii as well as some other effects (such as secondary

static deflection systems are used for fine alignment of the electron emission from and to the filaments) causing un-

electron beam. The coherent wave fronts are made to diverge by a negatively charged filament (Mollenstedt electron biprism [70, 71]). A second filament, which is posi-

certainties as to the effective potential of the filaments [39] resulted in an error margin for A of 10—20%.
A third biprism filament is available for, e.g. , reducing

tively charged, causes them to converge again (and thus the angle of superposition of the two partial waves if a

to enclose a finite area), and to form interference fringes negative voltage is applied. It was, however, not used in

in the region of overlap. The electron wave fronts leaving the experiments that yielded the results reported below.

the first biprism can be rotated by alignment coils [72] which produce a weak longitudinal homogeneous magnetic field. This is necessary in order to align the wave fronts with the direction of the second biprism filament,

With the de Broglie wavelengths of the electron beams
used on the order of A = 0.3 A., and beam separations
achieved of 20—60 pm, the fringe spacings in the primary interference field were on the order of 50—150 nm.

since it is usually impossible to mount the two filaments exactly parallel to each other.
The angle of deflection in a Mollenstedt electron biprism, which is constant for all incoming electron tra-
jectories [71], is given by

1 Uy
2 in(R/r) U '

(22)

In massive particle interferometry, coherence lengths are usually very short (typically in the range of 10—1000 nm) due to the small de Broglie wavelengths and the un-
avoidable energy spread of particle beams (0.3—0.4 eV in the case of the field emitter used here). The intrinsically
dispersive propagation of such de Broglie waves even in vacuum, which causes an increase in the spatial extent of the wave packet as it propagates down the wave path,

where R is the distance between the filament and the earth electrode, r is the filament radius, U is the acceleration voltage, and Uy is the voltage applied to the filament. R was 2 mm, and r was on the order of 1 pm
(but see below). The total size of the enclosed area,
A = Ap + Ai + A2, as shown in Fig. 3, can easily be

calculated as

), A=ar+I 2 tan(o'o+Px)

(1+
~

tan(ae + Pi) an 2 — i —np

does, however, not increase the coherence length available for producing an observable interference pattern [73]. For
this reason, a Mien fi.lter was placed in the beam path in order to allow optimization of the fringe contrast by shifting the two wave packets relative to each other lon-
gitudinally [74]. Its crossed electrostatic and magnetic fields, both perpendicular to each other and the beam path, provoke a shift of the wave packets relative to each other along the beam path without causing any deflection nor phase shift if the fields are suitably matched so that

(23)

the electrostatic (Coulomb) and the magnetic (I orentz) forces exactly cancel. The necessity for such an applica-

where the symbols are the same as the ones used in Fig. 3. Using Eq. (22), and making a few approximations [39],
A can be expressed in experimental parameters as

1

l~ Vyg Uyg

2 ln(R/r)

+'
U~ Upi Uf Q

(24)

where Uyi and Uyz are the voltages at the biprism filaments of the first and the second biprism, respectively.

tion of a Mien filter as well as theoretical background and constructional and experimental details are recounted in
the companion paper [75]. The need of accommodating for a large range of elec-
tron energies on the one hand (see above) combined with
the high stability requirements in electron interferometry on the other hand resulted in very high demands on the stability of the voltage and current supplies. The deflection elements and the Mien filter are the most critical

~ -~ a

— l-
I

components in this respect. In order to limit fringe shifts caused by those voltage and current fluctuations to 1% of a fringe, a relative stability of the supply modules of

on the order of 5 x 10 relative to the maximum out-

Electron Source
1st

adrupole

() put would have been necessary. Stability values of up to

5 x 10 7 for short-term

1 Hz) fluctuations could be

achieved using specifically designed circuitry and rnod-

2nd Biprisrn

ified commercial modules. The fringe shift;s caused by

those fluctuations were averaged out by the registration

a— FIG. 3, Geometry of the enclosed area A. The distance
from the cathode tip to the first biprism filament was 36 mm, the distance between the first and the second biprism
filament was l = 40 mm, and the distance from the second
biprism filament to the first magnifying quadrupole lens was
L = 82 mm. The beam separations achieved were on the order of 2d = 20—60 pm. The distance x between the second
biprism filament and the beginning of the region of overlap is determined by the deflection angles Pi and P2, and is typically
somewhat smaller than l + L.

procedure described below. The long-term drifts could be held sufIiciently low to allow the measurements reported in Sec. IV, but constituted one of the contributions to
the error margins of the experiment.
The primary interference fringes were magnified by two electrostatic quadrupole lenses. The magnified interference image was intensified by a dual-stage channelplate image intensifier which was equipped with a fiberoptic UHV image throughput coated with a P20 phosphor. Prom the intensifier's fiber-optic output, the im-

48

SAGNAC EXPERIMENT WITH ELECTRONS: OBSERVATION. . .

149

age was transferred via a tapered fiber optic to a video camera that was equipped with a high-sensitivity charge-
coupled-device (CCD) sensor. The UHV chamber, including the ion-getter pump, was
mounted on a rotating table (see Fig. 1), as were all in-
terferometer controls, the power supplies providing currents and voltages to the electron optical components,
and the CCD camera control. This was done in order to eliminate the need for the line voltage transmission to the turntable during rotation, which would have re-
quired both increased safety measures and increased mea-
sures to block the ac line frequency from interfering with the experiment. To allow this, the entire interferometer could be switched to power supply from batteries only, and those batteries (12-V lead accumulators) were also
positioned on the rotating table. Details of this setup are described in [39]. The diameter of the turntable was 120 cm. The total rotating mass was about 300 kg.
The rotating table was driven by an electronically controlled dc motor via a U-belt. Centrifugal forces acting on the vacuum chamber caused minute bending of the
interferometer, resulting in lateral shifts of the interferogram in the registration plane. In order to avoid any such inHuence of centrifugal forces on the Sagnac exper-
iments, we measured the phase differences between successive alternating clockwise and counterclockwise rotations at exactly the same absolute rotation rate. The
time for successively accelerating the turntable, register-
ing the interferogram, decelerating the turntable, and inverting the sense of rotation was 30—60 s (depending on
the rotation rate). The CCD sensor used had 576(V) x 384(H) picture el-
ements (pixels). In order to increase the signal-to-noise ratio (SNR) in the interferograms and to average over the mentioned residual, orientation-dependent shifts of the interference field caused by the Earth's magnetic field, the image was accumulated on the CCD sensor during multiple integers of the rotation period.
The video signal was transferred to the laboratory system via a slip ring. It was transmitted to an imageprocessing system that had a video frame memory with an image page size of 512x512x8 bits. The phase information was extracted from the interference pattern in
the following way. The CCD camera was mechanically aligned with the
interference image so that the pixel columns of the CCD sensor were parallel to the fringes. This allowed the information in all pixels of one column to be summed up by the image-processing system to yield a spatial integration
of the interferogram. Thereby the SNR is increased by

10

o& 6-
5—
Q) 6) D) C,'
3

I

~E

~K

l As

I

~~

I

0 0.0

0.5

1.0

1.5

20

3.0

AQi2vr (mm s )

FIG. 4. Experimentally observed fringe shifts for variation of both the enclosed area A and the rotation rate A/2n. , as
listed in Table I. The fringe shifts are given in units of percent
of a fringe period. The horizontal error bars result from the uncertainty in determining the enclosed area A (see text). Also shown is the theoretically predicted Sagnac phase shift
(solid line).

a factor equal to the square root of the number of image
memory rows per interferogram [multiple interferograms (up to 16) were stored in multiple horizontal sections of one image frame, thus reducing the number of image memory rows per interferogram to a fraction of 512]. The numerical result of both the temporal accumulation on the CCD sensor and the spatial integration in the image-
processing system is a one-dimensional low-noise densitometer trace across the interference field. This trace was
Fourier analyzed, and the phase information was calculated via the arctan of the Fourier components of the
intensity distribution I(x),

I(x) cos(2z fpx) dx

(25)

I(x) sin(2vr fpx) dx,

(26)

yielding

p = arctan(S/C),

(27)

TABLE I. Experimental parameters and results for each measurement.

No.

(U)

Uf (U)

Uf, (V)

A (mm2)

A/2vr
(s ')

Rotationsense reversals

&4'theor (%)

&4'expt (%)

-1174

-2.17

+3.35

1.8 + 0.3

0.5

5

3.11 + 0.54

4.4 + 3.0

-1721

-4.60

+6.71

2.8 + 0.3

0.5

7

4.89 + 0.60

4.9 + 1.6

-1721

-6.01

+8.47

3.9 + 0.4

0.5

7

6.81 + 0.78

6.5 + 1.8

-1721

-6.01

+8.47

3.9 + 0.4

0.25

8

3.40 + 0.39

3.6 + 3.7

150

FRANZ HASSELBACH AND MARC NICKLAUS

where fo is the spatial frequency of the interference fringes. By using a Hanning window [76] in the sampling
of I(x), a phase error of less than I'%%uo of a fringe period
could be achieved for the registration process, with a typical total number of electrons in one interferogram of ca. 40 000.
IV. RESULTS
The main diKculties introducing the errors in the experiments were the instabilities of the field-emission current and, to a lesser degree, of the currents and voltages fed to the electron optical components. In order to improve on the statistics of the measurements, phase differences were averaged over a series of successive rotation sense reversals. The maximum number of those reversals per series was limited by the long-term instabilities of the field-emission process. A total of four such series of rotation-sense reversals could be obtained in which the standard deviation of the averaged phase difference did

not substantially exceed the mean value. Details of those measurements have been reported elsewhere [77]. Both
0 the enclosed area A and the rotation rate were var-
ied, and the number of rotation-sense reversals ranged
from 5 to 8. Table I gives the experimental parameters
used, and the expected and the measured phase shifts. The substantially larger error in measurement No. 4 may have been produced by minute mechanical destabilization in the interferometer caused by the preceding rotation experiments. Figure 4 shows that the phase shifts measured are in good agreement with the theoretically expected values for the Sagnac phase shift.
ACKNOWLEDGMENTS
We are indebted to Professor H. Ruder and Professor H. Herold of the Institute of Theoretical Astrophysics at the University of Tiibingen for their valuable contributions to the theory section of this paper. This research project was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ha-1062/2-1, 2,3.

' To whom correspondence should be addressed.
t Present address: National Cancer Institute, National In-
stitutes of Health, Bethesda, MD 20892.
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