Frequency, phase, and amplitude changes of the hydrogen maser oscillation

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 4, AUGUST 1992 46 I

Frequency, Phase, and Amplitude Changes of the Hydrogen Maser Oscillation

Claude Audoin and William A. Diener

Abstract-The frequency, the phase, and the amplitude changes of the hydrogen maser oscillation, which are induced by the modulation of the cavity resonant frequency, are consid- ered. The results obtained apply specifically to one of the H- maser cavity autotuning methods which is actually imple- mented, namely the cavity frequency-switching method. The frequency, the phase, and the amplitude changes are analyzed theoretically. The phase and the amplitude variations are measured experimentally. It is shown, in particular, that the phase of oscillation is subjected to abrupt jumps at the times of the cavity frequency switching, whose magnitude is specified. The results given can be used for the design of a phase-locked loop aimed at minimizing the transfer of the phase modulation to the slaved VCXO.

I. INTRODUCTION N the hydrogen maser [I], [2], the cavity-pulling effect I translates any microwave cavity frequency offset,

(U, - w,)/2n, into a corresponding frequency bias, (U - w,)/27r, of the maser oscillation. This effect is described by the equation:

(1) where U,, wc, and w are the angular frequencies of the atomic resonance, of the cavity resonance and of the maser oscillation, respectively. Q, and Q, are the quality factors of the microwave cavity and of the atomic line. In the following, we shall neglect any frequency offset due to the spin-exchange interaction between hydrogen atoms, and we shall assume that the value of Q, does not depend on the cavity resonant frequency.

Different cavity autotuning systems have been imple- mented. Their aim is to control the long term frequency stability of the hydrogen maser [3]. Among them, that based on the cavity frequency-switching method has proven its efficiency [4], [ 5 ] . It ensures an excellent long- term frequency stability, without perturbing significantly the short- and medium-term-frequency stability. This method is founded on the inertia property of the atom assembly, which is related to the large value of the transverse relaxation time T2, of the order of 0.3 s. When the cavity resonant frequency is modulated, with a period of

- U, = Qc(wc - u,>/Qi

Manuscript received January 6, 1992. C. Audoin is with the Laboratoire de I’Horloge Atomique, Unite Propre

de Recherche du CNRS associee a L’Universite Paris-Sud, 91405 Orsay Cedex, France.

W. A. Diener is with the California Institute of Technology, Jet Pro- pulsion Laboratory, Pasadena, CA 91 109.

IEEE Log Number 9201307.

about 0.1 s or shorter, the amplitude of the atomic magnetic moment does not change significantly. If the mean value of the cavity resonant frequency is equal to the atomic transition frequency, the amplitude of the magnetic field sustained by the alternating magnetic moment remains practically a constant. If, in constrast, the mean value of the cavity resonant frequency is not equal to the atomic transition frequency, then the amplitude of the microwave field is modulated, which provides a signal de- pending on the cavity mistuning.

However, the cavity resonant frequency changes pro- duce a periodic modulation of the frequency and of the phase of oscillation, which has been noticed [4], [6]. The amplitude is also very slightly modulated. The purpose of this paper is to give quantitative information on these ef- fects. They are analyzed theoretically in Section I , and experimental results are given in Section 11.

11. THEORETICAL DERIVATION OF THE MASER BEHAVIOR A. General Equations

The properties of the amplitude and of the phase of oscillation can be described using a radioelectric model [3], which takes into account the resonance property of the microwave cavity, as well as the characteristic properties of the atomic medium, i.e., its frequency selectivity and its saturation by the microwave field.

We introduce the same notations as in [3]. The hydrogen maser output voltage is proportional to b cos (ut + cp), where b is the normalized oscillation amplitude, and cp is the phase of oscillation. The quantity b is actually the Rabi angular frequency of the stimulated atomic medium. The oscillating magnetic moment of the atom assembly is represented by m sin (wr +$), where m is the amplitude of the magnetic moment referred to the Bohr magneton, and I) is its phase. The population difference of the two atomic levels involved is M 3 . The cavity response time is T, and the longitudinal and the transverse relaxation times of the atoms are TI and T2, respectively. We have:

T, = 2Qc/w, and T2 = 2Ql/w,. (2)

In practice, we have T, = 10 ,us, T I , T2 = 0.3 s and, therefore, T, << T I , T2. Furthermore, (U, - w,)/w, is always a very small quantity. In general, the parameters b, m, M 3 , cp, and $ can be time dependent. The oscillation angular frequency w will be assumed constant during each half-period of modulation. The instantaneous oscillation

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462 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 4. AUGUST 1992

half-period of modulation. The instantaneous oscillation angular frequency will thus be given by w + $, where the dot means a time derivative.

In the present case, where we neglect the cavity thermal noise and where no external signal feeds the maser cavity, the equations describing the maser behavior are:

(3) T2m + m = M3bT2 cos 8

(4) 0 T M n 'M TIM3 + M3 = -T ,bm cos 8 + TIT

T,b + b = KQ,mcos8 (5 ) (a)

A li/ = U, - w + M 3 ( b / m ) sin 8

e = p - * (8)

(6)

(7) $ = 0, - w - (KQ,m/T,b) sin 8

where I" is the flux of state-selected atoms entering the maser storage bulb, and K is a constant defined in [3].

ting of the cavity resonant frequency, we shall consider I I I

Neglecting, for simplicity, any time constant in the set- Time w

the following cavity frequency modulation: 0 TME TM (b)

~ c ( t ) - wco = wmS(t) (9) where U, is the mean value of the cavity resonant angular frequency, wm is the amplitude of modulation, and g(t) represents a square wave of period TM, whose value is + 1 for 0 < t < T M / 2 and - 1 for T M / 2 < t < TM. We shall assume TM small compared to TI and T2. The variation of the cavity resonant frequency is shown in Fig. l(a).

A

B. Stationary Regime We shall show, in Section I.C, that the duration of the

transient response following an abrupt change of the cavity resonant frequency is of the order of T,. Therefore, during most of each half-period of modulation, a stationary regime is attained, and the parameters of (3)-(8) can then be assumed constant. The subscripts will character- ize the two steady-state values of the oscillation frequency and of the oscillation amplitude occuring during the long- est part of each half-period.

Assuming, at first, that the microwave cavity is tuned in the mean sense, i.e., that we have w, = coo, it can be shown from the preceding equations that the oscillation angular frequency is given by:

(10)

in agreement with the cavity-pulling effect. The amplitude of oscillation, b,, is the same during the two half- periods of modulation. We have:

(1 1)

where a = KQ, TI T21is the oscillation parameter. We see that the amplitude of oscillation is decreased in the pres- ence of modulation, to a degree which depends on the value of a and of U , T,. One may also note that (1 1) pro-

Tc T2

ws(t) - w o = - Umg(t)

TlT2b: = Q - [l + T f w i ]

A

Fig. 1. Periodic variation of parameters of interest. The value of the ratios T,/T,and Tc/T2 has been greatly increased, for the purpose of illustration. It has been assumed U<,, = U,,: (a) Cavity resonant angular frequency offset. (b) Quantity tan 0 . (c) Instantaneous oscillation angular frequency. (d) Phase of the maser oscillation.

vides a simple means of obtaining the value of CY by mea- suring the ratio of the maser output powers for two different values of U,. Equation (11) shows that the amplitude of oscillation is the same during the two half- periods of modulation. This follows from a general property of this amplitude, which is a quadratic function of the cavity resonant frequency.

AUDOIN AND DIENER: CHANGES OF THE HYDROGEN MASER OSCILLATION 463

If, now, the cavity frequency is not tuned in the mean sense, the oscillation angular frequency becomes:

Tr T2

w,(t) - 0, = - [w,, - W O + w,g(t)]. (12)

The amplitude of oscillation is no longer a constant. For a small mistuning, such that Tc(wc, - w,) << 1, the change Ab,(t) is given by:

It is periodic, with period TM, and it is proportional to the mean cavity mistuning w,, - w,. Consequently, a syn- chronous detection of the modulation of the oscillation amplitude provides information which makes possible a feedback control of the cavity resonant frequency [4].

C. Phase Transient It can be shown easily, from (6)-(8) that the angle 8

varies during a transient regime, even in the case where the cavity is tuned to the atomic transition. This greatly complicates the exact solution of the whole set of differential equations. However, a sufficiently accurate expres- sion of the time variation of the phase angles 8, cp, and I) can be obtained by assuming that the quantity T,b/b remains smaller than unity during the transient. Then, the values of 8 and cp obtained will enable us to derive simply the time variation of the measure b of the oscillation amplitude and to verify that T,b/b remains actually small during the transient.

It is appropriate to consider, at first, the differential equation for the angle 8. Neglecting Trb /b , we have from ( 3 ) and (5)-(8):

1 8(l + tan2 0) = w,(t) - U , - - (1 + R) tan 8

where w,(r) is given by (9) and where we have set:

(14) TC

R = Tc/T, = Qc/Qi . (15) The parameter R is typically equal to 3 x lop5. The periodic solution of (14) is:

T

- 2Tcw, exp ( - t / T c ) ( W

+ 2TCw, exp [ - ( t - T,/2)/TC]. (16b)

In these equations and in the subsequent ones, the subscripts 1 and 2 refer to 0 < t < T M / 2 and T M / 2 < t < TM, respectively. It is worth noting that the factor (1 + R) has been omitted in the second term of the right-hand side of (16a) and (16b). This does not change the main con- clusions of this paper. Instead, it cannot be omitted in the

first term, to account for the cavity-pulling effect. The variation of tan 8 versus time is shown in Fig. l(b).

The differential equation describing the behavior of the phase of oscillation is obtained from (5 ) and (7). We have

1 cp = - w,(r) - - tan e(r) (17)

TC where w,(r), w, ( t ) , and tan 8(t) are given by (9), (12), and (16), respectively. It follows that the instantaneous angular frequency, wi(t) = w,(t) + G ( t ) , is given by:

wil(t) - w, = - (w, - w, + w,) + 2w, exp ( - t /T , ) TC T2

( 1 8 4

Tc 772

Wi2(t) - W O = - (w,, - W O - w,)

- 2" eXp [ - ( t - T,/2)/TC]. (18b)

Therefore, the instantaneous oscillation angular frequency shows brief, but very large, deviations, practically equal to 2w,, at the times of the cavity frequency switching. They are depicted in Fig. l(c). Typically, 2w, is of the order of 27r x lo4 rad s- ' .

The mean value of the instantaneous oscillation angular frequency is wi(t) given by:

Equation (19) shows that the mean oscillation frequency obeys the cavity-pulling equation (1) and that it is not biased by the cavity frequency switching.

The time variation of the phase of oscillation is obtained by integrating equation (17). We assume that the phase is measured with respect to that of an oscillator hav- ing a constant angular frequency equal to wi(t) given by (19), and that the mean value of the phase difference ob- served is set equal to zero. We thus have, (o, being a constant, and for T, << T M / 2 :

C P I ( ~ ) - PO = WmTr[I - 2 exp (-t /T-)I

- w , s ( r - 3 T M ) . T2

The variation of the phase of oscillation versus time is shown in Fig. l(d). We see that phase jumps of magnitude 2w,Tc arise at the times of the cavity frequency switching. These phase variations occur with a time equal to the cavity response time, assuming abrupt cavity resonant frequency changes. Afterwards, the phase variation follows a ramp of slope +w, Tc/T2 during each half-period of modulation. It is due to the cavity-pulling effect which causes an offset of the oscillation angular frequency, equal

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 41, NO. 4, AUGUST 1992

to fw, T,/T2, with respect to oi(t). Equations (20a) and (20b) are valid without an approximation on the magnitude of om T,, provided of course that the maser oscillation subsists at the given offset fw, of the cavity resonant angular frequency.

In a simple, but efficient, model of the H-maser, it has been assumed that the oscillating magnetic moment, which excites the cavity electromagnetic field, behaves like an independent source driving a tuned circuit [4]. If q is the total change of the phase shift across this circuit, occuring when its angular resonant frequency is symmet- rically offset from one side of the fixed value w, to the other, with a total range of 2w,, then we have:

q = 2 arctan (w,T,). (21)

For practical values of U, T, of the order of 0.3, we have r] = 2w,T,. r] is thus an excellent estimate of the oscillation phase jump.

The phase $ of the magnetic moment can be obtained from (8), ( 16), and (20). For practical values of w, T, and assuming U, = wo, we have:

$2(t) - po = -U,

(22b) Actually, the variation of $ is noticeably smaller than that of 8 and p.

D. Amplitude Transient

oscillation, by: We define E , the relative change of the amplitude of

Let us consider, at first, the beginning of each half- period of modulation, during a time interval equal to sev- eral times T,. Since the time constant associated with the amplitude m of the magnetic moment is T2 >> T,, the value of this amplitude does not change significantly, and it remains equal to that attained at the end of the preceding half-period i.e., m,. We thus have, from (5 ) : ’

cos 8 cos 8,

T , i + ~ = - - l .

where cos 8, is the steady-state value of cos 8. Assuming U,, = wo, Fig. 2(a) shows the small variation of E versus time, for three different values of parameter om T, around the experimental value U, T, = 0.3. The amplitude transient repeats itself identically after each cavity frequency abrupt change, since cos 8 is an even function. Therefore, its period is T M / 2 . Fig. 2(b) shows that the value of T,b/b remains much smaller than unity. Consequently, the as- sumption made to derive the equations for the phase angles 8, p, and $ was actually justified.

E e b - 1 bs

,Tc = 0.45 0.07 7

0 2 4

Tc i =Tc b / b

Fig. 2. (a) Amplitude transient for three different values of the parameter w,T,. The jump of the cavity resonant frequency occurs at t = 0. It is assumed w,, = U,. (b) Variation of the quantity T,i = T,b/b versus time after a cavity frequency jurirp, for three different values of the parameter U, T,. It is assumed U, = w,.

After the transient is finished, the angle 8 is a constant, and the variation of E is given by:

However, the initial values of E and i are equal to zero, so that the oscillation amplitude is a constant during the remainder of each half-period.

111. MEASUREMENT OF THE PHASE AND OF THE AMPLITUDE TRANSIENTS

A. Experimental Setup The maser used in this experiment, owned by JPL, was

manufactured by the Sigma Tau Standards Corporation. Its cavity quality factor is Q, = 47 000, and thus the cavity response time is T, = 10.5 ps. The atomic flux is set “high,” and the atomic line quality factor is Ql = 1.48 X lo9. The frequency of the modulation of the cavity resonant frequency is 8.5 Hz. The peak-to-peak cavity frequency change is 9.5 kHz, giving 20, = 27r x 9.5 x lo3 rad s-’ and w,T, = 0.31.

The experimental setup is represented in Fig. 3. In its principle, it makes it possible to compare the modulated phase of the maser oscillation at 1.4 GHz to the mean value of that phase. For that purpose, an auxiliary high- quality 5 MHz VCXO is phase locked to the 5 MHz out-

AUDOIN AND DIENER: CHANGES OF THE HYDROGEN MASER OSCILLATION

-

465

HYDROGEN MASER

, 4 2 0 4 M r t ,- Ip :;;;;, , & ISOLATOR USE-LOCKED

AMPLIFIER AUXILIARY

& FILTER

14204 M H r 1400 MHr

MIXER

*EIEF SYNTHESIZER 7 Fig. 3 . Schematic of the experimental setup.

put of the maser, but with a bandwidth of about 0.5 Hz, which is sufficiently narrow to attenuate greatly any resid- ual phase modulation of the maser 5 MHz output. The frequency of the VCXO is multiplied up to 1 400 MHz and mixed with the amplified maser output at 1420. 405 - - - MHz. The gain is of the order of 120 dB. The signal obtained at 20. 405 MHz feeds a balanced mixer used as a phase detector, which also receives a signal from a frequency synthesizer driven by the auxiliary VCXO. The frequency setting of this synthesizer is the same as that of the built-in maser synthesizer. Moreover, the phase of the maser signal can be shifted, so that a voltage with a zero- mean value can be obtained at the output of the phase detector. This voltage varies in the linear part of the phase detector characteristics. It represents the oscillation phase change related to the cavity resonant frequency switching. The phase measurement is calibrated as usual by offset- ting slightly the frequency of the signal delivered by one of the synthesizers.

The changes of the maser oscillation amplitude are ob- served simply by connecting a rectifier to the amplifier maser output, in place of the mixer at 20.4 MHz. Cali- brated attenuators enable measurement of the relative amplitude changes.

B. Experiment Results Fig. 4(a) shows a recorded phase transient. Assuming

an exponential decay, the time constant is of the order of 12 ps, in satisfactory agreement with the theoretical value of 10.5 ps. The slight lengthening of the transient is likely related to the time constant of the order of 3 ~s of the control voltage applied to the varactor coupled to the microwave cavity.

The amplitude change is shown in Fig. 4(b) (the voltage detected was negative). The amplitude increases

\ i !

i-, L

Time (b)

l " " l " " 1

Time (c)

Fig. 4. Examples of experimental records. The observation bandwidth is 1 MHz. The lower trace shows the control voltage applied to the varactor. (a) Oscillation phase transient. (b) Oscillation amplitude transient. (c) Pe- riodic change of the oscillation phase.

slightly after the cavity frequency switching and returns to its initial value after a few times T,, in agreement with theory. However, the relative size of the bump is of the order of 1 % instead of about 3 % as predicted from Fig. 2(a). This disagreement might be due to the additional time constant of the varactor control voltage and to a pos-

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 4. AUGUST 1992

sible mismatch of the calibrated attenuator used to measure the level change. It has been verified that, besides this bump, the modulation of the cavity resonant frequency does not induce oscillation amplitude changes.

Of most interest is Fig. 4(c), which shows an example of recorded periodic phase changes. A phase jump occurs at the times of the cavity frequency switching, and the phase varies linearly with time in the intervals between the cavity frequency changes. The measured magnitude of the phase jumps is 0.66 radian, and the measured slope of the linear phase variations is 0.92 radian per second. The estimated relative accuracy of these values is 5 % for the jumps and 10% for the slopes. The corresponding theoretical values, known with an accuracy of a few %, are 0.63 rad and 0.94 rad s-I, respectively, in good agreement with the measured values.

IV. CONCLUSION The phase and amplitude changes occuring in the cavity

frequency -switching method of the hydrogen maser cavity control have been analyzed and measured. Experimental results support the.theoretica1 ones. The latter can thus be used for the design of a phase-locked loop minimizing the

transfer of the phase modulation of the maser to the slaved vcxo.

ACKNOWLEDGMENT One of the authors (C.A.) is pleased to acknowledge

Dr. Lute Maleki and JPL for their hospitality and for a partial support of his stay at JPL.

REFERENCES [ l ] D. Kleppner, H. M. Goldenberg, and N. F. Ramsey, “Theory of the

hydrogen maser,” Phys. Rev., vol. 126, pp. 603-615, Apr. 1962. [2] D. Kleppner, H. C. Berg, S. B. Crampton, N. F. Ramsey, R. F. C.

Vessot, H. E. Peters and J. Vanier, “Hydrogen-maser principles and techniques,” Phys. Rev., vol. 138, pp. A972-A983, May 1965.

131 J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards. Bristol and Philadelphia: Adam Hilger, 1989, ch. 6, pp.

[4] H. E. Peters, “Design and performance of new hydrogen masers using cavity frequency switching servos,” in Proc. of the 38th Annual Sym- posium on Frequency Control, 1984, pp. 420-427.

[5] H. E. Peters, H. B. Owings, and P. A. Koppang, “Hydrogen masers with cavity frequency switching servos,” in Proc.of the 22nd Precise Time and Time Interval (PTTI). Applications and Planning Meetings,

[6] T. K. Tucker, “Operating and environmental characteristics of Sigma Tau hydrogen masers used in the very long baseline array (VLBA),” TDA Progress Rep. 42-97, pp. 72-80, Jan.-Mar. 1989.

949- 1256.

1990, pp. 283-292.

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Frequency, phase, and amplitude changes of the hydrogen maser oscillation