Design of Microwave cavity of PTB

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO. 3 , MARCH 2002 ' 383

Design and Realization of the Microwave Cavity in the PTB Caesium Atomic Fountain

Clock CSFl Roland Schroder, Udo Hiibner, and Dieter Griebsch

Abstract-At t h e Physikalisch-Technische Bundesanstalt (PTB) , t h e caesium atomic fountain clock C S F l was developed. One key element of it is its microwave cavity, which was designed t o have a low transversal phase variation across t h e cavity opening. This usually is achieved by using a cylindrical cavity with a T E O l l field mode, having a high intrinsic quality factor, and by feeding t h e cavity symmetrically via two apertures at opposite position in t h e cavity wall. In contrast t o other solutions, t h e C S F l cavity is tightly coupled t o t h e microwave feeds M 2000). Therefore, detuning of t h e cavity resonance frequency has a reduced impact on the microwave field amplitude in t h e cavity compared t o t h e case of weak coupling. Thus, a temperature stabilization of the cavity can be avoided. T h e ex- tent to which t h e tight coupling may have an impact on t h e transversal phase distribution was studied. This ques- t ion was solved analytically for a simplified cavity model. T h e results were applied t o define t h e coupling geometry of t h e C S F l cylindrical cavity in such a manner t h a t t h e transversal phase distribution should become minimized. T h e key parameter is t h e electric length of t h e waveguides t h a t consti tute t h e junction to the coupling apertures . T h e theoretical studies a re presented in some detail, and s teps of the practical realization of the cavity a re described.

I. INTRODUCTION

T Is EXPECTED that in the near future time keeping in I thc metrological institutes will be based on the operation of frequency standards using laser-cooled atoms in a so-called fountain arrangement. A few fountains have been completed and evaluated [l]-[4], others make significant progress. The best conventional atomic clocks with a thermal atomic beam realize the SI second with an unccrtainty of the order of 1 part in 10'" [5], [6]. Fountains have distinctive advantages compared with the latter devices, which is proven by the fact that, even for the first genera- tion, uncertainties of the order of l part in l O I 5 have been reported [ 11, [a], [4]. There are good reasons to expect even bctter results in the near future.

This papcr deals with the design concept and the realization of the microwave cavity built into the fountain CSFl of the Physikalisch-Technische Bundesanstalt

Manuscript received December 1, 2000; accepted September 24, 2001.

R. Schroder and D. Griebsch are with the Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany (e-mail: [email protected]).

U. Hiibner is with Am Walde 17, D-38179 Grof3-Schwulper, Germany.

(PTB). Apart from the preparation of clouds of laser- cooled atoms (see [1], [3]), the cavity represents a key element for the construction and operation of a fountain. At first sight, the requirements for the cavity are considerably relaxcd, compared with those in a conventional atomic clock. In the latter, the atomic beam iiitcrsects the two arms of a U-shaped, so-called Ramsey cavity, madc of standard X-band waveguide pieces [7]. The inevitable phase diffcrence 'p between the two interaction regions eii- tails a shift Sf, = -'p/(2 . 7r . T ) of the realized atomic transition, whcre T is the time-of-flight betwcen the two arnis. In a fountain the atoms interact twice with the field sustained in a cavity of cylindrical shape, and T , now the time which the atoms spend on their ballistic flight above the cavity, is generally larger by a factor of about 50. Thus, Sf, is considerably reduced. Aiming at a total uncertainty, however, makes a careful design of the cavity indispensable.

As the atomic cloud spreads during its ballistic flight, the atoms will, in general, cross the microwavc field in the cavity at different positions upward and downward. A ma- jor design goal is, thus, to achieve a low transversal phase variation across the cavity opening. Therefore, cylindrical microwave cavities with the field oscillating in the TEOll mode have been used in fountain clocks [3], [8], [9]. This TEOll mode, depicted in Fig. 1, has especially low losses (high intrinsic quality factor Q) which means that there is little field phase dependence on the position of the atom passage. Further reduction of the phase dependence can be achieved by feeding thc microwave power symmetrically into the cavity. Then, in case of perfect coupling symmetry, the radial energy flow is zero in thc center of the cavity, which means that along the symmetry axis the transversal phase dependence on the position is an even function with a minimum.

Some other operational constraints of a fountain need to be considered when designing and manufacturing the cavity. At this point it should be noted that the cavity resonance frequency must be tuned and maintained during operation closely to the caesium hyperfine transition frequency of fo = 9192.6 MHz. This ensures that the microwave field amplitude, which dictates the transition probability and thus thc strength of the resonance signal, will be stable in time. In addition, the potential frequency shift due to cavity pulling is minimized (see [lo] for an explanation of this effect). If the loaded Q of the cavity is high, it is necessary to tune the cavity by variation and

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384 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO. 3 , MARCH 2002

,/--\ \ \ /

/

\ I I I I I

I / L\ I I I \ \ \ /

/ \ / \ '.-/

Fig. 1. Sketch of the magnetic microwave field in the TEOll mode in a fountain cavity. The dotted-dashed line represents the axis of ro- tational symmetry of the cylindrical cavity and the vertical fountain axis.

stabilization of its temperature. If the Q is low enough, the field amplitude will remain sufficiently unchanged even if the cavity is slightly detuned.

In Section I1 the design concept of the CSFl cavity is described; in Section 111 detailed calculations for a simplified cavity model are explained, which corroborate the validity and feasibility of the design concept. Practical realization and manufacturing of the cavity, including some further practical constraints and solutions, are described in Section IV and V. The paper concludes with a summary of the experimental investigations performed to verify the functionality of the cavity in CSF1.

11. CAVITY DESIGN CONCEPT

In the fountain FO-1, developed at the French Labora- toire Primaire du Temps et des Frkquences, high coupling symmetry has been achieved by exciting the two equally sized small coupling areas in the cavity by two mutually decoupled cables, as is illustrated in Fig. 2 [8]. Auxiliary measurements serve to equalize the amplitudes and phases of the waves propagating toward the coupling areas. The requirements on the equality increase with the strength of the coupling of the feeds to the cylindrical cavity. Only a weak coupling (e.g., loaded Q = 15000) results in requirements that can, in practice, be accomplished with reason- able efforts. In this case, the narrow resonance requires the stabilization of the temperature of the cavity if the microwave field amplitude is to be kept fixed and cavity pulling is to be avoided.

In CSFl a different arrangement to feed the coupling slits of the cavity was chosen [3], [4], [ll], which is schemat- ically shown in Fig. 3. The excitation at both coupling sites

Vacu u m tu be

Attenuator

Phase Shifter Attenuator

I N +

Isolator

Fig. 2. Cavity excitation by two mutually decoupled cables, as used in the fountain FO-1 of the Laboratoire Primaire du Temps et des FrBquences, LPTF, France.

I I I I I

I I I I I I

Vacuum tube feeds

I I I I I I

I I I I I I

Vacuum Caupled tube feeds

Isolator P Osci I lator

Fig. 3. Cavity excitation by two coupled cables.

SCHRODER et al.: THE CAESIUM ATOMIC FOUNTAIN CLOCK AND DESIGN OF MICROWAVE CAVITY 385

Fig. 4. Possible realization of the cavity coupling principle shown in Fig. 3. Excitation is achieved by two mutually coupled bent waveguides that are fed symmetrically by an E-plane T.

is done not with decoupled (as in FO-1) but with two short coupled feeds. This guarantees that in the fictive lossless case a pure standing wave field builds up in the cavity and in the coupled feeds. In this case the dependence of the phase on the location inside the cavity perfectly vanishes, even if the coupling symmetry is violated. In the realistic lossy case, one can expect that the position dependence still will be small for a given asymmetry of the coupling. This was the key motivation for the design presented here. It allowed a stronger coupling between the cylindrical cavity and the feeding waveguide. A Q M 2000 was chosen as a design goal. Thus, temperature stabilization of the cavity could be omitted.

A schematic realization of this principle is shown in Fig. 4. A coaxial line is tightly coupled with a waveguide that splits into two branches in an open E-plane T. Each branch guides the wave via a circularly bent waveguide to a coupling slit resulting in symmetrical excitation of the field in the cylindrical cavity. The dimensions of the CSFl vacuum chamber, however, prohibited to use the geometry shown in Fig. 4. A more compact solution had to be chosen and is depicted in Fig. 5. The coaxial line is coupled via a probe directly into the waveguide. In order to achieve the desired in-phasc excitation of the two cavity coupling slits, the lengths of both bent waveguide arms have to differ by A,/2, where A, is the wavelength at the caesium frequency in the bent waveguide. Compared to the E-plane T arrangement of Fig. 4, another modification is that the shorts at the ends of the bent waveguide are placed not directly behind the coupling slits. Instead they are shifted by approximately X,/2 to ease manufacture.

Some further practical advantages can be identified at once. Only one feed cable through the vacuum enclosure has to be installed, and separate adjustments of magnitude and phase of the coupling fields across the slits are not necessary. Moreover, the cavity is tuned before installation and no further adjustments are necessary.

&-+A I

hg I4

It waveguide ! 'Bet- Fig. 5. Schematic view of the cavity arrangement realized in CSFl (horizontal section). One coaxial cable feeds the bent waveguide via a probe. The probe is shifted by X,/4 so that both waveguide arms differ in length by X,/2. The shorts terminating the waveguide are shifted by Xg/2 from the coupling slit to ease the mechanical construction.

In Section I11 we will use a simplified model for the arrangement described in Fig. 5 to analyze qualitatively the phase sensitivity on coupling asymmetries. The findings are confronted with the situation prevailing in the FO-1 arrangement. Later they are used to determine the dimensions of the CSFl cavity arrangement.

111. CALCULATIONS FOR A MODEL CAVITY

A. Motivation of the Proceeding

A quantitative determination of the phase of the magnetic field in the cylindrical cavity as a function of certain asymmetries, which additionally depends on the dimensions of the coupling slits and the lengths of the bent waveguides, would be very interesting; but it is beyond the scope of this paper. Instead, the studies presented here were restricted to the questions whether the contrived coupled feeds showed the desired advantage that, even in a realistic lossy case, phase variations to be expected will be small for a given asymmetry of the coupling and which dimensioning rules were to be obeyed to reach this goal. For an answer of these questions, the inner shape of the cavity is irrelevant, and it is justified to replace the cylindrical cavity by a rectangular waveguide cavity. This model cavity, depicted in Fig. 6, is excited in the TE201 mode, whose magnetic field lines are simi1a.r to those of the TEOll mode of the cylindrical cavity. Using a rectangular waveguide cavity reduces the calculation of the phase from a three-dimensional to a one-dimensional problem.

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Fig. 6. Model cavity, representing the arrangement shown in Fig. 5. The cylindrical cavity is replaced by a rectangular waveguide cavity oscillating in the TE201 mode.

The equivalent circuit of the cavity of Fig. 6 is shown in Fig. 7. The magnitude of the relative susceptances bl and b 2 is a measure for the coupling between the waveguides and the cavity. The length L has to be determined such that the cavity is in resonance at fo for the symmetrical case (bl = b2). The S matrices for 61 and b 2 are

with stl = 1/(1 + ib1/2) , s,1 = -i . s t l b 1 / 2 as follows from straightforward calculations. Elements s t 2 and s,2

are defined by changing index 1 to 2. As usual, voltage U and current I characterize the transversal electric and magnetic field strength, respectively. Uhl and U h 2 are the (voltage) waves taken at the coupling slits propagating toward the cavity, whereas Usl and U,2 are those propagating away from the cavity. The voltage in the central area of the cavity can be expressed as a superposition of two counter-propagating waves:

~ ( z ) = u ~ ( o ) . e-Ycz + u,(o) e+Ycz , ( 2 )

where TC is the complex propagation coefficient ~c = ac + ip for the cavity, ac is the damping factor, /? = 27r/X,, and A, is the waveguide wavelength. In the calculations, the damping factor is chosen to a fictitious value of ac = 5 . mm-l so that the unloaded QC of the cavity is 25,000 and is thus equal to that of the (real) uncoupled cylindrical cavity. This particular choice has no impact on the derivation of (22) but should allow to obtain a rather realistic estimate on the phase variations to be expected. The complex amplitudes Uh(0) and Us(0) of the waves, taken at z = 0, are functions of the incident waves Uhl

and U h 2 1

9 L-i

Fig. 7. Equivalent circuit for the TE201 cavity. Uhl and Uh2 (U,i and Ur2) are the waves propagating toward (away from) the cavity, respectively. bl and bz are the relative susceptances for the coupling.

The common factor F

(4)

reflects (via p) the frequency dependence of the field amplitude near resonance.

B. Calculation of the Dimensions of the Model Cavity

At first, the length of the cavity and the relative sus- ceptance b = 6 1 = b 2 for the case that the loaded Q is &load = 2000 are calculated assuming the conditions that the incident waves u h l and u h 2 are constant, in- dependent of the frequency, that they oscillate in antiphase (Uhl = -Uh2), and that the cavity is coupled symmetrically. This means that indices can be omitted for b = bl = bz and S = SI = S 2 . From this and (3) we get:

with

e - Y C L l 2

1 - S $ ~ - ~ Y C L ' F =

Compared to the Factor F , st and the expression in brack- ets of (5) show a very weak frequency dependence. The magnitude of F will be especially large (in resonance) when s: . exp(-2ycl) becomes positive real. For

we get

SCHRODER et al.: THE CAESIUM ATOMIC FOUNTAIN CLOCK A N D DESIGN OF MICROWAVE CAVITY 387

The term at the right side will be positive real if arctan(2/b) - POL = f2n7r, (n = 0,1, . . .), which for the TE201 resonance is fulfilled for n = 1,

POL = 27r + arctan(2/b), (6)

from which L is obtained if b is known. With POL from (B), one gets exactly for resonance, and approximately within the resonance bandwidth.

At the half maximum points of the resonance curve (Le., for f l p = f o f A f and Pip = PO f Ap, respectively), the condition

(8) JUh(0, P 1 / 2 ) I - - l ~ ( P 1 / 2 ) I def - 1 \Uh(O,PO)/ IF(Po)l Jz'

must be fulfilled, with

1 1----

(F(P1/2)1 1 + 4/b2

I ' ,fi2LAp - l 1 - w

IF@O)l l 1 * iyq

IF(P0) 1

which is simplified to

IF(P1/2)1 1 N

for 2LAp << 1 (e.g., b > 10). The condition (8) is fulfilled if A/3b2/2 = 1, whereof b2 = 2/(ApL) follows, which finally leads to

I

Tcp(Z) I

I 1

I ' I

I

I . I , i :

Fig. 8. Vector diagram of the field I ( z ) (10) at z # 0, explaining the phases S = (6, ~ Sh)/2 and 'p. The magnitudes of I h ( z ) and Ir(z) differ because of the losses and coupling asymmetries.

taking POL x 2n into account (6). With 2 A f / f o = l/Qload, and as AP/& M 2Af/.fr, is valid for X- band waveguide dimensions at f o , b comes out as b = (Qload/7r)1/2 x 25 (for Qload = 2000). Inserting the value for b into (6), the length of the model cavity follows as L = (arctan(2/b) + 2n)/p0 = 47.12 mm using the waveguide wavelength A, = 46.53 mm.

C. Calculation of the Cavity Phase

This paragraph deals with the determination of the phase cp(z) of the transversal magnetic field described by I ( x ) near z = 0, where

I ( Z ) = Ih(0)e-ycz + Ir(0)e+YCz (10)

holds, with

h ( 0 ) = Uh(o)/zg and I r ( 0 ) = -ur(o)/zg, (11)

and uh(0) and U,(O) havc been introduced in (3). Z, is the characteristic impedance. With Ih(0) = IItL(0) \etdrL, IT(0) = lIT(0)\ei6r, a n d y c = ac+ip one gets the relation

I(z)e+ = (IIr(0)lefNcz + I Ih(0)Je-NcZ) cos ( ~ z + 6) si. (IIT(0)le+NCz - lIh(0)le-"cz) sin(pz+S), (12 )

where S = (6, - &)/2 is half the phasc difference phase difference between the wave Ir(0) and the wave I h ( 0 ) at z = 0. A pictorial representation of (10) is given in Fig. 8 for clarity. The calculation shows that S remains below 0.01 for cases relevant here. The phase cp(x) can thus bc calculated, neglecting 6 , from:

http://2Af/.fr

388 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO. 3, MARCH 2002

Using the approximations exp(acz) M 1 + acz and exp(-acz) M 1 - acz and neglecting acx compared to 1 in the denominator one gets:

where the 1/V can be expressed with (11) and ( 3 ) as

Equation (14) shows that the influences of the asymmetry expressed by l / V and of the losses in the cavity (w ac) can be linearly superimposed if one neglects 6. V can be interpreted as a standing wave ratio. In case of asymmetry, one has IUr(0)I # IUh(O)I, and therefore, 1/V # 0, Le., the shape of the phase curve of p(z) is determined not only by the losses ac. To obtain the ratio Uhl/Uh2 in (15) the equations for the total system of Fig. 6 is established and resolved for u h l / u h 2 . The continuity condition of the voltage at the center of the coaxial T imposes:

UTle-yL1 + UhleY" = UTze-y12 + Uh2eyL2. (16)

11 and 12 are the lengths as defined in Fig. 6. y = a + ip is the complex propagation coefficient of the lines where a = 3a, = 1.5. lop5 mm-I and ,6' is as above. The waves Url and Ur2 can be expressed via the waves u h l and Uh2

(Fig. 7) and the s-parameters of the cavity:

(17) UT1 = sT1uhl + StUh2 UT2 = stuhl + STZuh2'

in which the s-parameters are calculated with the cavity parameters ( b l , b2, L, yc) to:

1

1

b \ rl v

x m 0 r i

0.2

0.1

0.0

-0.1

-0.2 40 50 60 70 mm8o

z Fig. 9. 1/V as a function of the length 1, defined in (21) and in Fig. 5 . Remarkable features are the zero crossing at 1 = 58.54 mm, corresponding to I ; and the resonant increases at 1 = 46.83 mm and I = 46.83 mm + X,/2. The parameters are bl = bz = 25, and 61 = 2 mm.

2 &+Af

/

X

0 -1 ro

d

s \ d v

40 50 60 70 mm 80

I Fig. 10. 1/V as a function of the length 1, defined in (21) and in Fig. 5 , and for the parameters 61 = 0 and Sb = (b l - b2)/2 = 2.5. Significant deviation from zero can be seen a t I = 46.83 mm and 1 = 46.83 mm + X,/2.

and

11 = 1 + 61, 12 = 1 - A,/2 - SI, (21)

show 1/V plotted as functions of the length 1 for certain

i 2 a 2

respectively, with A, = X,(fo) = 46.53 mm. Figs. 9 and 10

asymmetries. In Fig. 9 the line length asymmetry of SI = 2 mm is introduced, whereas symmetrical coupling (b l = bz = 25) is assumed. In Fig. 10 a coupling asymmetry of Sb/b = 0.1 ( b = 25) and symmetry (apart of the A,/2 difference) of the bent waveguides arc assumed (61 = 0). In all cases 1/V is calculated for the three frequencies f o and f o f S f with the bandwidth 26 f of the loaded cavity.

The shapes of the curves show remarkable features. First, 1 /V shows resonant increases (discontinuous regions) at I = 46.83 mm. Second, 1/V vanishes in Fig. 9 at 1 = 58.54 mm. Both features can be interpreted phys-

srl = (F sinh(ycl) - - (b l e f yCL + b2e-YcL

sT2 = (F sinh(ycL) - - (ble-YCL + b2e+yCL

Combining (16) and (17) finally leads to:

Uhl - eyzs - ste-yll + sTze-y12 Uh2

(19)

The factor 1 /V shall be calculated as a function of specified asymmetries of the arm lengths and of the coupling slits. To describe the asymmetries we use:

bl = b + 6b,

-- eyll - ste-y12 -t sTle-yll '

b2 ("1 b - 6b

SCHRODER et d.: THE CAESIUM ATOMIC FOUNTAIN CLOCK AND DESIGN OF MICROWAVE CAVITY 389

ically. Assuming symmetry ( U h l = -Uh2, bl = b z ) , and to simplify matters CYC = 0, calculation yields two ref- erence planes E' at small distance (d = 0.3 mm) from the couplings slits where the reflection coefficient equals $1 in case of resonance and -1 far away from resonance. Now, the length 1' between the planes E' and the center of the probe, as defined in Fig. 6, has to be chosen to n . Xg/2 + X,/4 (in our case n = 2) to get the length Zb = 58.54 mm at which 1/V vanishes (second feature), i. e. Fig. 9:

0.00 O . O 1 $ -0.oosl

1; = n . Xg/2 + Xg/4 + d. ( 2 2 )

tures occur: 1/V increases resonantly so that the phase At length values I' = l b i Xg/4 the above mentioned fea- 40 50 60 70 mm 80

shows the unfavorable discontinuous shape. the shapes of the curves in ~ i ~ . 10, one recognizes

small dependencies on the frequency in the region of the resonant increase. This is due to the fact that the exact difference of X,/2 between the line lengths 11 and 12 is

1 Fig. 11. 6 as a function of the length 1. Remarkable features are the zero crossing at 1 = 58.54 mm, corresponding to 1; and the resonant increase at 1 = 46.83 mm and 1 = 46.83 mm + X,/2. The parameters are bl = bz = 25, and 61 = mm,

only realized a t f = f o . Such a frequency dependence is a peculiarity of the direct coaxial to waveguide coupling as shown in Fig. 5. If an E-plane arrangement of Fig. 4 was used, it would not exist (see Fig. 8 in [ll]).

Further calculations of 1/V lead to the following results, which are approximately valid for all values of 1 outside the above mentioned crucial regions and which are given here without proof

the two cases considered, 61 # 0 and 6b # 0, can be

in both waveguides the damping (a) practically does

in first order 1/V is a linear function of the asymme-

superimposed linearly;

not play any role;

tries 61 and 6b. For completeness, the dependence of the angle 6 on 1 is

shown in Figs. 11 and 12 for the same asymmetries and frequencies as before. Fig. 11 and 12 illustrate that S increases significantly to over 0.01 only in the resonant zones, which is outside the interesting regions. The decision to neglect 6 in the calculation of cp(z), thus, was justified.

D. Discussion of the Requirements on Symmetry

Supposing that the phase minimum should not be shifted by more than 1 mm relative to z = 0, one can calculate the resulting restrictions for 1/V. For small x we have tan(pz) M pz so that cp(z) x (1/V + q x ) . pz. This equation represents a quadratic parabola shifted by zo = l / (2acV) relative to z = 0. With QC = 0.5 . lop5 mm-l and the requirement zo < 1 mm we get 1/V < Fig. 10 shows that this is almost fulfilled even for large coupling asymmetries such as ( b l - b2) /b = 0.2. Length asymmetries can be neglected if the length 1 of the bent waveguide is chosen corresponding to 1' = lb (22). Now the same requirements (zo < 1 mm, resp. l / V < shall be fulfilled for the arrangement with decoupled feeds (Fig. 2). Consider the case that the coupling into the cavity is symmetric (bl = b2) and that the waves uhl and

u) X 0 0 0 7

lo r :*1 7 5

0 7 -2.5

40 50 60 70 mm 10

I Fig. 12. 6 as a function of the length 1 for 61 = 0 and 6b = ( b l - b2)/2 = 2.5. Significant deviation from zero can be seen at the length 1 = 46.83 mm and 1 = 46.83 mm+Xg/2.

u h 2 (Fig. 5) at the coupling slit are in antiphase (as in the ideal case) but that they may deviate in their magnitudes. One finds from (15) for the resonant case of the cavity [(6) holds] :

and with b2 = QlOad/7r

(23)

For Qload = 2000 and Il/Vl < this means 11 - Iuhl//luh2ll < 0.013, which is difficult to fulfill. This is

390 IEEE TRANSACTIONS ON UCTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO, 3, MARCH 2002

why Qload = 15750 was chosen in the arrangement described in [8]. Then the amplitudes have to agree only to about 10%. As discussed before, the narrow cavity resonance then requires temperature stabilization, which in CSFl could be avoided.

I v . FINAL DESIGN OF THE C S F l CAVITY

Based on the results achieved so far, the dimensions of the cylindrical cavity were determined, influenced in details by constraints that originate from the manufacturing process. Figs. 9 and 10 show that the best results for 1/V are expected with the choice of I' = 16 (22). We assumed that (22) was valid for a cylindrical cavity as well and determined the dimensions of the waveguide bent around the cavity as it was to be used in CSF1. Measurements done with a prototype of the cylindrical cavity showed that the planes E' are located d = 1.5 mm away from the center of the coupling slits. Therefore, the whole length between both slits is 1; + (16 - A,/2) + 2d = 2X, + 3 mm. The value of A, is determined by the width, 22.8 mm, and to a certain small amount by the curvature of the bent wavc- guide. One calculates A, = 46.72 mm [12]. This value refers to the middle of the waveguide height (10 mm). Starting with the dimensions of the bent waveguide arms and al- lowing for a wall thickness of 1.5 mm, one gets 48.4 mm as the inner diameter of the cylindrical cavity. From this one calculates a height of the cavity of 28.62 mm if the cavity is unloaded. In the loaded case, Qloacl = 2000, determined by slit length 10.8 mm and slit width 2 mm, thc height has to be smaller, namely 28.12 mm. The (theoretical) unloaded Q of the cavity is 28 450 at that particular height-to-diameter ratio, and therefore, smaller than the maximum possible Q of 30 924, which could be achieved for a longer cavity with reduced diameter.

The experimental determination of the height of the cavity (the tuning to the caesium frequency fo) has to bc done with the finally used coaxial line, including the isolator installed outside the vacuum enclosure. The asymmetries of the coupling of the cylindrical cavity also ex- cites the T M l l l mode: which disturbs the desired magnetic field configuration of Fig. 1. The resonance frequency of this mode is affected by the holcs for the atomic passage (diameter 10 mm) in the caps of the cavity and, even stronger, by choke grooves (width 0.6 mm, depth 5 mm) that were included in the design for that purpose. In total, the T M l l l resonance frequency was shifted by 390 MHz to lower frequencies.

V. MECHANICAL DESIGN OF THE CAVITY

In Fig. 13, a vertical section through the cavity is shown. The cavity was machined from OFHC copper, and the individual parts are distinguished in Fig. 13 by different shading. Fig. 14 shows the assembled cavity on the bottom flange of CSFl just before the vacuum drift tube is be-

l

Fig. 13. Vertical section through the CSFl microwave cavity, machined from OFHC copper. Individual parts are distinguished by different shading. The inner ring defines the diameter of the central cylindrical cavity and the diameter of thc bent waveguide and contains the two coupling slits. Before soldering it to the outcr ring, which represents the cover of the bent waveguide, the two shorts (not shown) are put in place. Note that thc orientation of the probe with respect to the coupling slits is rotated to become visible; the correct orientation is shown in Fig. 5. Not shown are thc screws for fastening the two cover plates of thc cylindrical cavity to the outer ring and details of the vacuum seal, mentioned in the text.

ing mounted. So far, the cavity design was described from the view point of microwave component design. Being a constituent part of CSFl, the cavity design is directed by some additional, as well important, considerations. The whole piece has to be totally nonmagnetic and has to be compatible with UHV conditions, Pa being the typ- ical design background gas pressure in thc fountain. This dictates the choice of the material as well as the manufacture steps. Because thc atoms are to be subject to the microwave field only insidc the cavity [13], microwave leakage from the cavity joints as well as from the feed-through and the coupling cable has to be suppressed as much as possible. Measurement capabilities finally allowed to state

SCHRODER et d.: THE CAESIUM A'L'OMIC FOUNTAIN CLOCK AND IIESIGN OF MICROWAVE CAVIlY 39 1

Fig. 14. Mounting of the CSFl cavity on the CSFl basement flange The cavity is elcctrically connccted only through the ground shield of the microwavc coaxial cablc that connccts it to the vacuum basement flange at the vacuum feedthrough. Thc basement Aangc is the single point of grounding the whole vacuum chambcr. Thereby, thc flow of thermocurrcnts inside thc magnetic shield is prohibitcd.

that no fields are detectable at a level largcr than -160 dB below the signal coupled to the cavity in routine operation. One measure cornmonly used to achieve this is to attach cut-off tubes to the cavity openings, in the present case 70 mm in length and 10 nim in diameter.

During assembly, the two rings, detailed in Fig. 13, including the two shorts that define the length of the bent coupling waveguide, are joined by solder. The orientation of the position of the probe with respect to the short is manufactured with an uncertainty of 1 minute of arc, corresponding to a length difference discussed in Section 111-C of less than 10 pm. At this time, the cavity resonance frequency is ad.justed by correctly machining of tlic thickness of the two cover plates. A 10 pm change of the inner distance between the upper and lower covcr (what rnight be called the height of the cavity) shifts the cavity resonance by 1 MHz. This imposes a similar reproducibility of the fastening of the cover plates, in particular as several trials were necessary until the desired tuning was achieved. This

is why the current design docs not include a rigid joint of the covcrs to the rings. Potentially, electron welding or laser welding could be used for that purpose in a future dcsign concept.

Next, a high vacuum bake-out is donc for a few hours at about 180°C, just remaining bclow thc softening temper- aturc of the solder. The two covcr plates then are tightly fixed with titanium screws to thc outer ring, with soldcr rings mounted betwecri thc parts as microwave seals, which is another important measure to prevent microwave leakage. The small coupling structure carrying the probe and visible in Fig. 14 is also held by screws, and another solder ring serves as a microwave seal. Experience has shown that standard rigid coaxial copper cable is not vacuum tight, rncaning that a constant gas flow is present along the cable if it is not sealcd at some point. In the present design, commercial ultra-high vacuum sealant glue is applied at the coupling structure for that purpose. As the vacuum seal affects the resonance frequency of the cavity, it has to be applied before the final tuning.

The risk of a further bakc-out after full assembly was avoided. The pricc to be paid is that the background pressure inside the cavity is not as low as desirable. Recording the fraction of returning atoms after a ballistic flight as a function of the launching height indicates that caesium- background-gas scattering is particularly disturbing when atoms are at apogee inside the cavity. This, of course, is not the normal mode of operating the fountain.

VI. CONCLUSION

PTB's cacsium fountain frequency standard CSFl became fully opcrational in late 1999. A first estirnatc of its uncertainty was made [4], which was, in part, based on the properties of thc microwave cavity discussed in this paper. Some studies have been undertaken up to now from which the performance of the cavity can be infcrred.

No readjustment of the microwave power fed to the cavity for optimuni cxcitation of the atoms was necessary during thc last 10 months of operation. This demonstrates that the small (< 0.5 E() temperature variations at the cavity position, which occurred over the year, did not affect the field amplitude by more than 0.2 dB, the attainable measurement unccrtainty. Frcquency rneasuremcnts made while the microwave power is increased or decreased by 3 dB are sensitive to a detuning of tlic cavity resonance frequency from the cacsium transition frequency. If thc modulation width of the microwave probing frequency is increased from the line width W to 9 . W , as an cxample, this scnsitivity is multiplicd by alniost the samc factor. All measurements carried out hitherto were limitcd by the frequency instability of the fountain and the hydrogen masers used as the laboratory frequency references. The combined relative instability for averaging times 7 = 1 d amounts to 2 . lo-''. No shift excecding the rneasurenient uncertainty of the same order has been

392 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 49, NO. 3, MARCH 2002

noticed, in accordance with expectations as the cavity was tuned to fo with an uncertainty of 500 kHz at the operating temperature during manufacture. Frequency measurements at the ninefold microwave power are particularly sensitive to microwave leakage. Again, no significant shift was detected.

The results of these experiments allow l i s to estimate the magnitude of potential frequency shifts at optimum excitation condition to below 5 . This estimate does not include the effect of the transversal phase distribution. A verification of the desired small transversal phase variation is still missing. Experiments could be envisaged during which the fluorescence signal of only a spatially selected fraction of atoms in the detection area is used. Varying this fraction would allow one to get a rough idea of the phase distribution in the cavity. This procedure, however, would require a precise quantitative knowledge of the imaging properties of the fluorescence detection op- tics. Another approach would be to conduct a numerical calculation of the field map inside the cavity, including the full geometry of feed structure, coupling slits, and bores, similar to what was described in [14].

ACKNOWLEDGMENT

[6] C Thomas, “The accuracy of TAI,” in Proc. 29th Annu. Precase Tame and Tame Interval Applacatzons and Plannzng Meetang,

N. F. Ramsey, Molecular Beams. London: Oxford Univ. Press, 1956. G. Santarelli, “Contribution ?L la rkalisation d’une fontaine atom- ique,” Ph.D. dissertation, Universite Pierre et Marie Curie, Paris, France 1996 (in French). S. R. Jefferts, R. E. Drullinger, and A. DeMarchi, “NIST cesium fountain microwave cavities,” in Proc. Int. Freq. Contr. Symp.,

[lo] J. Vanier and C. Audoin, The Quantum Physacs of Atomac &e- quency Standards. Philadelphia: Adam Hilger, 1989.

[ll] R. Schroder and U. Hubner, “The microwave cavity in the PTB caesium fountain clock,” in Proc. 14th Eur. Freq. Tame Forum,

[12] N. Marcuvitz, Waveguzde Handbook. New York: McGraw Hill,

[13] K. Dorenwendt and A. Bauch, “Spurious microwave fields in caesium atomic beam frequency standards: Symmetry considerations and model calculations,’’ in Proc. Joznt Meetang Eur. Freq. Tame Forum and IEEE Int. Freq. Contr. Symp., pp. 57- 61, 1999.

[14] V. Giordano, L. Pichon, P. CBrez, and G. ThBobald, “Investiga- tion of microwave t transitions in cesium beam clocks operated with U-shaped H plane waveguide cavities,” J . Appl. Phys., vol.

pp. 19-25, 1997. [7]

[8]

[9]

pp. 6-8, 1998.

pp. 480-483, 2000.

1951, pp. 333-334.

78, pp. 1-8, JuI. 1995.

Roland Schroder was born in Helmstedt, Germany, in 1943. He received the Dip1.-Ing. degree in electrical engineering in 1970 from the Technische Universitat Braunschweie. Germanv.

From 1970 to 1971, he was engage; with t h i development of navigation systems at Anschiitz, Kiel, in Germany. He joined the Phvsikalisch-Technische Bundesanstalt. Braunschweie. Germanv. in

The authors are grateful to Dr. Stefan Weyers and Dr. Christian Tamm for stimulating and fruitful discussions -, “ I

during the work, Dr. Andreas Baich was Darticularlv helD- 19j2, where he now is responsible for the development of the elec- - tronics and microwave components of primary clocks. V I

during the preparation of the manuscript.

REFERENCES

A. Clairon, S. Ghezali, G. Santarelli, P. Laurent, S. N. Lea, M. Bahoura, E. Simon, S. Weyers, and K. Szymaniec, “Preliminary accuracy evaluation of a cesium fountain frequency standard,” in Proc. 5th Symp. Freq. Standards and Metrology, pp. 49-59, 1997. S. R. Jefferts, D. M. Meekhof, J. H. Shirley, T. E. Parker, and F. Levi, “Preliminary accuracy evaluation of a cesium fountain frequency standard at NIST,” in Proc. Joznt Meetzng Eur. Freq. Tame Forum and the IEEE Int. Fkeq. Contr. Symp., pp. 12-15, 1999. U. Hiibner, S. Weyers, J. Castellanos, D. Griebsch, R. Schroder, C. Tamm, and A. Bauch, “Progress of the PTB caesium fountain frequency standard,” in Proc. 12th Eur. Freq. Tame Forum, pp,

S. Weyers, U. Hiibner, B. Fischer, R. Schroder, C. Tamm, and A. Bauch, “Investigation of contributions to the stability and uncertainty of the PTB’s atomic caesium fountain,” in Proc. 14th Eur. Freq. Tame Forum, pp. 53-57, 2000. A. Bauch, B. Fischer, T. Heindorff, P. Hetzel, G. Petit, R. Schroder, and P. Wolf, ‘Comparison of PTB’s primary clocks to TAI in 1999,” Metrologaa, vol. 37, no. 6, pp. 683-692, 2000.

544-547, 1998.

Udo Hiibner was born in GroB-Wierau, Silesia, Germany, on Octo- ber 26, 1937. He received the Dip1.-Phys. and Dr.rer.nat. degrees from the Technische Universitat of Braunschweig, Braunschweig, Germany, in 1966 and 1972, respectively.

From 1964 to 1972, he worked at the Technische Universitat of Braunschweig on problems in solid-state theory, especially on con- ductivity and crystal fault problems. Since 1972, he has been with the Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Ger- many, where his research is in the fields of statistical aspects of time scales, laser dynamics (chaos), and interaction of laser light and atoms. He last was engaged in research on atomic fountains until his retirement in 2000.

Dieter Griebsch was born in 1949. He holds a degree in precision mechanics. He has almost 25 years of experience in design, construction, and manufacture of components for use in PTB’s atomic frequency standards. For a few years he was the prime researcher on PTB’s primary clocks CS3 and CS4 before he became engaged in the development of PTB’s trapped-ion optical frequency standard and at the fountain clock CSF1.

Design of Microwave cavity of PTB

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