The short term frequency stability of masers equipped with an external feedback loop to increase the cavity quality factor was studied. The frequency stability of a hydrogen and a rubidium maser were measured and compared with theoretical evaluation. It is shown that the frequency stability passes through an optimum when the cavity Q is varied. Long term fluctuations are discussed and the optimum mid term frequency stability achievably by small size active and passive H-masers is considered

Audoin, C.

Lesage, P.

Petit, P.

Tetu, M.

Tremblay, P.

English

NASA Technical Reports Server

FREQUENCY STABILITY OF MASER OSCILLATORS 
OPERATED WITH E N H A N C E D  C A V I T Y  Q 
Michel TGtu and P i e r r e  Tremblay 
Laboratoire  de Recherches s u r  les O s c i l l a t e u r s  e t  SystSmes 
TJniversit6 Laval,  QuGbec, Canada 
and 
P a u l  Lesage, P i e r r e  P e t i t  and Claude Audoin 
Laboratoire  de 1'Horloge Atomique 
Universiti5 de Paris-Sud, Orsay, France 
ABSTRACT 
This paper presents  an experimental  study of the  s h o r t  term 
frequency s t a b i l i t y  of masers equipped wi th  an e x t e r n a l  feed- 
back loop t o  inc rease  the  cavi ty  q u a l i t y  f a c t o r .  3Xe frequency 
s t a b i l i t y  of a hydrogen and a rubidium maser are measured 
and comparedwith t h e o r e t i c a l  evaluat ion.  It i s  observed t h a t  
the  frequency s t a b i l i t y  passes  through an optimum when the  
cav i ty  Q is  var ied.  Long term f luc tua t ions  are discussed and 
t h e  optimum mid term frequency s t a b i l i t y  achievable  by smal l  
s i z e  a c t i v e  and pass ive  H-masers is considered. 
INTRODUCTION 
Much e f f o r t  has  been devoted r e c e n t l y  t o  hydrogen masers in  order  t o  
reduce t h e i r  s i ze ,  t h e i r  weight, and t o  increase t h e i r  long term f re -  
quency s t a b i l i t y .  New technology and design improvements appl ied t o  COP 
vent iona l  masers [l] l e d  t o  the r e l i z a t i o n  of the  m o s t  s t a b l e  atomic fre- 
quency s tandards  i n  t h e  mid term region of averaging t i m e  121, and to a 
s u f f i c i e n t l y  l i g h t  and rugged device t o  be space-borne L3J. Small s i z e  
masers w i t h  var ious  types of microwave cavities w e r e  proposed and t e s t e d  
[4,5]. 
quency s tandards  w e r e  also inves t iga t ed  [6,7]. The latest development 
i s  a s m a l l  c a v i t y  o s c i l l a t i n g  maser, equipped w i t h  an external loop t o  
enhance the q u a l i t y  f a c t o r  18991. 
Masers w i t h  d i e l e c t r i c  loaded cavities operated as pass ive  f re -  
A t h e o r e t i c a l  model, e s t ab l i shed  t o  eva lua te  the  amplitude noise  and the  
phase noise  i n  a c t i v e l y  and pass ive ly  operated masers El0,113 made poss- 
i b l e  t he  p red ic t ion  of the  s h o r t  term frequency s t a b i l i t y  of  a maser 
equipped wi th  an e x t e r n a l  feedback loop C12j.  Subsequently, the u l t ima te  
performance of such a device  w e r e  evaluated and compared t o  t h e  corres- 
ponding pass ive  s tandard  r131. The following gives  an experimental  check 
of t h a t  theory appl ied t o  a hydrogen maser and a rubidium maser of con- 
vent iona l  design C14,151 ,so equipped. 
667 
https://ntrs.nasa.gov/search.jsp?R=19820012653 2020-03-21T16:31:52+00:00Z
THEORY 
The t i m e  domain frequency s t a b i l i t y  of an  o s c i l l a t o r  is expressed by the  
two sample variance of t h e  relative frequency f luc tua t ions  over an 
averaging t i m e  T, with no dead t i m e  (Allan variance) C161. I n t h e  caseo f  
a maser, the  dominant frequency f luc tua t ions  considered arise from both 
the thermal noise  wi th in  the  electromagnetic cavi ty  and the  thermal noise  
added i n  t h e  rece iver  necessary t o  de t ec t  t he  s i g n a l  C171. When the  
spectrum of the f luc tua t ions  is l imi ted  by a low-pass f i l t e r  of cut-off 
frequency f c ,  i t  can be shown C15,181 t h a t  t he  s h o r t  t e r m  frequency 
s t a b i l i t y ,  i n  the  t i m e  domain, is  expressed by the  r e l a t ion :  
where k is  the Boltzmann constant,  OC, Qc and Qext are respec t ive ly  the  
temperature, the loaded and the  e x t e r n a l  q u a l i t y  f a c t o r s  of the cavi ty ,  
€Ir and Fr are the  temperature and the noise  f a c t o r  of the rece iver ,  wo 
is the  maser angular frequency, QR i s  the  atomic l i n e  Q and Po i s  the  
power delivered t o  the  cavi ty  by the atoms. 
The f i r s t  two terms of equation (1) come from the  thermal noise added t o  
the  maser s i g n a l  by the  cavi ty  and the rece iver  respec t ive ly ;  they cor- 
respond t o  white phase noise.  
r e s u l t i n g  from t h e  st imulated emission of r a d i a t i o n  by t h e  cav i ty ' s  
thermal no i se  wi th in  t h e  atomic linewidth. The atomic power can be 
expressed simply, i n  terms of t h e  cav i ty  Q, by the  f o l l i n g  [19]: 
The t h i r d  term i s  white frequency noise  
Po - -  1 S%w0 Qc - Qt 
QC 
where S is  a f l u x  term, 45 is the  Planck constant divided by 2.rr and Qt is  
a threshold Q value determined by various maser parameters. 
A feedback loop, e x t e r n a l  t o  the  cavity,  can be u t i l i z e d  t o  increase  
the  q u a l i t y  f ac to r ;  a schematic diagram i s  given i n  f i g u r e  1. 
the  maser s i g n a l  is  taken out ,  amplified and re-injected i n t o  the  
cavity.  I f  t h e  phase of t h e  in j ec t ed  s igna l  i s  properly adjusted,  it 
w i l l  add t o  the s i g n a l  already i n  the cavity,  thus simulating a lower 
lo s s  cavity.  In  t h i s  set-up, the  cavity i s  used i n  transmUsion, with 
coupling loop c o e f f i c i e n t s  B, and B2 a t  the i n j e c t i o n  and output po r t s  
respec t ive ly .  The loaded cavity Q becomes C201: 
P a r t  of 
668 
where Qo is the  unloaded cav i ty  Q assoc ia ted  with ohmic losses .  
e x t e r n a l  cav i ty  Q ’ s  are def ined as follows: 
Two 
QO 
62 and Qext , 2 
= -  QO 
Qext , l  6, 
= -  ( 4 )  
The enhanced cav i ty  Q has a maximum value given by: 
(5 1 - QO 
1+ B l +  6, - 2 G q 2  
where G is the  t o t a l  vo l tage  ga in  of the  feedback loop. The atomic power 
of such a maser w i l l  now be: 
Qe - Qt Po = T S * u o  
Qe 
The presence of the  feedback loop alters the  thermal noise  wi th in  the  
cavi ty .  P a r t  of the  cavi ty  no i se  undergoes the  same p r o c e s s a s  the  maser 
s i g n a l  and the loop ampl i f i e r  adds a c e r t a i n  amount of thermal noise .  
These supplementary cont r ibu t ions  give t o  the  cavi ty  an e f f e c t i v e  noise  
temperature which can be w r i t t e n  C211: 
where 6, and Fa are the  temperature and the  noise  f i g u r e  of t h e  loop 
amp li f i e  r . 
I f  we s u b s t i t u t e  equat ions ( 3 ) ,  ( 4 3 ,  ( 5 ) ,  ( 6 )  and (7) i n t o  equat ion (l), 
w e  ob ta in  f o r  the t i m e  domain s h o r t  term frequency s t a b i l i t y  of a maser 
equipped with an  e x t e r n a l  feedback loop, an expression of the  form: 
o ~ ( T )  = K - 2 ~ - 2  + K - 1 ~ - 1  (8) 
where the  white  phase noise  cont r ibu t ion  is: 
Qe 121~kf3 f- r c  - 
K-2 
669 
and the white frequency noise  cont r ibu t ion  is: 
These equations are e x p l i c i t l y  expressed i n  terms of the f ixed  cav i ty  
f3, and Q and its enhanced Q. parameters B 
Computation of equation (8) with the parameters given i n  tab€e 1 f o r  a 
conventional H-maser and a conventional Rb-maser y i e l d s  the  results 
shown i n  f igu re  2. 
1' 0 
0 w 
Q, 
Qt 
QO 
Bl 
$2 
S (normalization 
parameter) 
C 
8 
Fr = Fa 
'r 
'a 
C 
f 
7.15 X lo1 a t .  /sec 
40 000 
6 4  400 
0.157 
0 .171  
300 K 
1.78 
293 K 
300 K 
5 Bz 
5.5 x IO7 
3.65  X 1015 ph. /sec 
22 508 
28 oao 
0.60 
0.55 
337 K 
2.24 
290 K 
290 K 
50 Hz 
Table I: Maser parameters used t o  eva lua te  equation (8). 
The behavior of the  two types of noise is  drawn from the  two asymptotical 
l i n e s  obtained f o r  the  very s h o r t  averaging times and f o r  the  long aver- 
aging times respec t ive ly .  
EWERIHEXTAL STUDIES 
I n  order  to  v e r i f y  the  model j u s t  described, both a n  hydrogen and a ru- 
bidium maser w e r e  equipped with a feedback loop as i l l u s t r a t e d  i n  f igu re  
1. The cavi ty  Q w a s  var ied  by changing the  value of the  a t t enua to r  and 
measured with t h e  usual. technique of the  r . f .  pu lse  decay €191. For t h e  
6 70 
H-maser phase of the loop w a s  adjusted, a t  each at tenuator  s e t t i n g  
t o  cause no frequency s h i f t  from non-enhanced operation. In the case 
of the Rb-maser, the  phase was  adjusted to  make the output power max- 
imum. 
The short  term frequency s t a b i l i t y  for various values of Q with the 
parameters shown in  t ab le  1 are given i n  f igure  3. 
that the s t a b i l i t y  decreases when the  cavi ty  Q reaches high values. 
In the case of the  Rb-maser, a roll-over is  observed i n  the very short  
term region (T <.02 sec). I n  both systems, unpredicted sources of 
i n s t a b i l i t i e s  dominate f o r  long averaging t i m e s .  
bes t  fit  polynomials. 
e 
It is observed 
The so l id  lines are 
From the coeff ic ient  of thepolynomials we can extrapolate the  white phase 
noise contribution, K,2, and the white frequency noise contribution,K-r. 
The square root of each contribution is  shown i n  f igure 4. The so l id  
l ines  are the r e su l t  of the evaluation of equations (9) and (10) as a 
function of Qe. They are normalized with the f lux  term, S, so t h a t  the c2 curve passes over the experimental point indicated by an arrow. 
These r e s u l t s  are experimental evidence tha t  each noise contribution 
can be minimized by choosing the proper value f o r  Q . Consequently, 
i f  the overa l l  frequency s t a b i l i t y  is t o  be optimized, the Q, value w i l l  
be selected according t o  the averaging t i m e  considered, 
e 
A comparison between the experimental observation and the theore t ica l  
evaluation of the frequency s t a b i l i t y  a t  d i f fe ren t  averaging t i m e s ,  
as functions of t he  enhanced Q, is given i n  f igur  5 .  W e  see tha t  
the  s t a b i l i t y  i s  optimum f o r  a ce r t a in  value, Qop'. This value 
depends s l i g h t l y  on the  averaging t i m e  chosen sfnce the  dominant 
type of noise evolves from the white phase noise t o  the w h i t e  frequency 
noise when the averaging t i m e  is  varied from very short  termtomid term. 
In  t h i s  las t  comparison, the averaging t i m e  w a s  l imited t o  values small- 
er  than 300 see f o r  the H-maser and .07 sec f o r  the Rb-maser i n  o r d e r t o  
reduce the influence of the  long term i n s t a b i l i t i e s .  
I n  f igure  3 frequency i n s t a b i l i t i e s  other than the  ones predicted by 
equation ( 8 ) ,  are evident. Since t h e i r  contribution increases with 
the cavi ty  Q, w e  are tempted t o  relate them t o  a cavi ty  pulling 
effect .  
var ia t ion  of the  feedback I.oop components fo r  d i f f e ren t  cavi ty  Q ' s .  
The r e s u l t s  are given i n  f igure  6 f o r  each maser. The coef f ic ien ts  
are approximately 1 x 10-13/oC a t  40 000 f o r  the H-maser and 
3 X lO-"/"C a t  25 000 €or the Rb-maser. These s h i f t s  are due t o  a phase 
pul l ing e f f e c t  re la ted  t o  a change Fn the length of the feedback loop 
and explain some of the long term i n s t a b i l i t i e s .  It is  seen from these 
measurements t ha t  a very precise  control of the loop temperature w i l l b e  
necessary i f  high performance is requlred. Automatic cavity tuningsuch 
as the f a s t  auto-tuning system C7,22f would then improve the long term 
W e  measured the  relative frequency s h i f t  due t o  temperature 
671 
frequency s t a b i l i t y  of masers a c t i v e l y  operated wi th  an e x t e r n a l  feed- 
back loop. 
SMALL SIZE HYDROGEN MASER 
The t h e o r e t i c a l  model seems t o  f i t  w e l l  t h e  r e a l i t y  observed. W e  w i l l  
now use i t  t o  p red ic t  t h e  u l t ima te  frequency s t a b i l i t y  achievable 
by s m a l l  s i z e  active H-maser wi th  enhanced cavity Q. To do so ,  the maser 
parameters found i n  equation (8) are expressed i n  terms of more fundamen- 
ta l  parameters which are assoc ia ted  with t h e  gain and l inewidth of t h e  
atomic system. They are the  s p i n  exchange parameter [ll: 
and the  threshold flux: 
I n  these  equations 
is t h a t  por t ion  of the  f l u x  i n  the  upper ac t ive  quantum state,  V0 is the  
magnetic permeabili ty of vacuum, VB is  the  Bohr magneton, 'l i s  the  f i l l -  
ing  f a c t o r ,  Vc is the  cavity volume, 5 is thespinexchangecross-section, 
vr the  r e l a t i v e  average hydrogen ve loc i ty ,  Tb is  t h e  bulb s torage  t i m e  
constant,  vb i s  the  bulb volume and Tt = 
t ud ina l  and t r ansve r sa l  r e l axa t ion  t i m e  constants.  
Considering only the  e f f e c t  of the  white frequency noise cont r ibu t ion ,  
i t  has been shown [13,eq.441 t h a t  the  frequency s t a b i l i t y  of an active 
maser, with enhanced cavity Q, is expressed by t h e  r e l a t ion :  
Itot is the t o t a l  atomic f l u x  en te r ing  the  bulb, I 
- 
with T! and T: the  longi- 
The func t ion  Ha,,(q,I/Ith) C13,eq. 1 7 1  has t o  be minimized i n  order  t o  
reach the  u l t imate  frequency s t a b i l i t y .  I f  t he  parameters of cur ren t ly  
e x i s t i n g  s m a l l  s i z e  masers are s u b s t i t u t e d  i n  equation (13), w e  obtain: 
The same approach can be appl ied  t o  the  case of a passive H-mase r .  One 
can show C13,eq. 251 t h a t  the optimum frequency s t a b i l i t y  is  expressed 
by : 
672 
Here again t h e  function H (qcl,I/fth) must be minimized by a proper.choice 
of q and I/Itl?. When eva lua t ing  equation (14) with t h e  parameters of 
e x i s t i n g  passive masers [13, t a b l e  11 we have the  value: 
P 
Both types of s m a l l  s i z e  H-masers have then approximately the  same u l t i -  
m a t e  frequency s t a b i l i t y  i n  the  mid t e r m  region. 
CONCLUSION 
This experimental study as w e l l  as the  previous one on amplitude noise 
[lll give g rea t  confidence i n  the  t h e o r e t i c a l  model developedtoconsider 
the e f f e c t  of thermal noise  on t h e  amplitude and the phase of ac t ive ly  
o r  passively operated masers. When a maser is  operated with a feedback 
loop to enhance the cavi ty  Q,an optimum is found where the  frequency 
s t a b i l i t y  is b e s t .  Small s i z e  H - m a s e r s  of the same design can then be 
operated e i t h e r  ac t ive ly  o r  passively; they w i l l  o f f e r  about t he  same 
mid t e r m  frequency s t a b i l i t y  when a l l  the  parameters are optimized. 
Automatic cavi ty  tuning i s  needed on both devices t o  improve the  long 
t e r m  s t a b i l i t y :  t he  electronic complexity w i l l  then be of the  same leve l .  
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674 
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6 75 
1- 
6 76 
- 30,000 
.Of .z i 
AYERAGING TIME 
Fig. 2 Short term frequency. stabi l i ty:  theoretical evaluation. 
677 
Fig. 3 Short t e r m  frequency s t a b i l i t y :  measurement. 
678 
ENHANCED CAVITY Q 
Fig. 4 White phase no i se  and whi te  frequency no i se  con t r ibu t ions .  
679 
680 
+ 
+ 
h 
1 
4, 
+ 


Frequency stability of maser oscillators operated with cavity Q

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