HYDROGEN MASER : ACTIVE OR PASSIVE ?

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HYDROGEN MASER : ACTIVE OR PASSIVE ?

C. Audoin, J. Viennet, P. Lesage

To cite this version:

C. Audoin, J. Viennet, P. Lesage. HYDROGEN MASER : ACTIVE OR PASSIVE ?. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-159-C8-170. �10.1051/jphyscol:1981818�. �jpa-00221713�

JOURNAL DE PHYSIQUE

CoZZoque C8, supplament au n012, Tome 42, de'cembre 1981 page C8-159

HYDROGEN MASER : A C T I V E OR P A S S I V E ?

C. Audoin, J. Viennet and P. Lesage

Laboratoire de ZrHorZoge Atomique, Equipe de Recherche du C.N.R.S. f*), Bet. 221, Universitg Paris-Sud, 91405 Orsay, France

Abstract.- I t is shown t h a t , f o r a s p e c i f i e d i n t e r r o g a t i o n scheme of t h e atomic t r a n s i t i o n , the c a v i t y p u l l i n g f a c t o r o f passively operated masers i s the same as f o r a c t i v e l y operated ones.

The power s p e c t r a l density of frequency f l u c t u a t i o n s of passive masers i s gi- ven f o r a s p e c i f i e d atomic l i n e i n t e r r o g a t i o n scheme.

The e f f e c t of s p i n exchange l i n e broadening on the frequency s t a b i l i t y capabi- l i t y of passive and a c t i v e masers i s s p e c i f i e d . I t i s shown t h a t i t e x i s t s an optimum value of the atomic f l u x i n t e n s i t y . Design c r i t e r i o u s a r e s p e c i f i e d . The frequency s t a b i l i t y c a p a b i l i t y of p r e s e n t l y designed small o r l a r g e s i z e hydrogen masers i s compared when operated e i t h e r a c t i v e l y o r passively. I t i s shown t h a t l a r g e s i z e a c t i v e masers cannot be surpassed, as long a s ultimate frequency s t a b i l i t y i s considered.

The e f f e c t of temperature on frequency s t a b i l i t y i s s p e c i f i e d .

1.Introduction-The aim of t h i s paper i s t o compare t h e frequency s t a b i l i t y capabili- t i e s of a c t i v e l y and p a s s i v e l y operated hydrogen masers. We w i l l consider

i) Hydrogen masers of c l a s s i c a l design with a f u l l s i z e microwave c a v i t y (1)

.

I n the following, they w i l l be denoted as "large s i z e hydrogen masers", and

ii) Hydrogen masers with a microwave c a v i t y of reduced s i z e , loaded with a - -

high p e r m i t t i v i t y d i e l e c t r i c medium (2'3)

,

They w i l l be l a b e l l e d a s "small s i z e hydrogen masers". Due t o t h e l o s s e s of the materials introduced i n the c a v i t y , they are operated e i t h e r a s passive devices ( 2 ) 1 o r a s a c t i v e ones, b u t with e l e c t r o n i c a l l y achieved enhancement of the c a v i t y q u a l i t y f a c t o r (617)

- -

The reported work has been i n i t i a t e d e a r l i e r ( 8 ) , i n the case of p a s s i v e l y operated masers. I t i s completed here by the consideration of the spin exchange l i n e broadening, and of a more r e a l i s t i c frequency modulation scheme f o r t h e i n t e r - rogation of the atomic t r a n s i t i o n .

I n our comparison we w i l l only be i n t e r e s t e d i n the u l t i m a t e l y achievable medium and long term frequency s t a b i l i t y . We w i l l then focus on the e f f e c t of ther- mal noise i n s i d e the microwave c a v i t y on t h e white component of t h e power s p e c t r a l density of frequency f l u c t u a t i o n s . Although introduced when necessary, t h e e f f e c t of noise added by t h e microwave amplifiers coupled t o t h e microwave cavity i s n o t con- sidered as a fundamental source of frequency s t a b i l i t y l i m i t a t i o n . This i s j u s t i f i e d by continual progress i n the reduction of the noise f i g u r e of these amplifiers.

S.I. Units a r e used throughout t h i s paper.

- - - -

(*I ^Associ.de B 1'Universitd Paris-Sud

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981818

(28-1 60 JOURNAL DE PHYSIQUE

2. I n t e r r o g a t i o n o f t h e atomic t r a n s i t i o n i n a p a s s i v e maser.- There a r e s e v e r a l p o s s i b l e ways t o i n t e r r o g a t e t h e atomic t r a n s i t i o n i n a p a s s i v e maser. We w i l l only c o n s i d e r h e r e t h e very convenient f a s t frequency modulation method ( 2 t 9 )

,

g i v e s a c c e s s t o t h e atomic d i s p e r s i o n l i n e .

The frequency modulated microwave s i g n a l i n j e c t e d i n t o t h e maser c a v i t y i s r e p r e s e n t e d by t h e f o l l o w i n g e q u a t i o n :

where p denotes t h e amplitude o f t h e s i g n a l , w i s t h e a n g u l a r frequency o f t h e c a r - r i e r , w i s t h e modulation a n g u l a r frequency, m i s t h e frequency modulation i n d e x

m

and J i s t h e B e s s e l f u n c t i o n o f o r d e r n.

n

The amplitude p i s d e f i n e d ( 8 ) ( 1 0 ) i n such a way t h a t i t i s n o t r e q u i r e d t o s p e c i f y t h e coupling f a c t o r o f t h e i n p u t loop.

A t t h e o u t p u t p o r t of t h e maser, t h e t r a n s m i t t e d s i g n a l _{- -} i s p r o p o r t i o n a l t o :

j u t jnw t

wt = p e 1 .In) J~ (m) e ( 2 )

n=-m

where G(") i s t h e complex g a i n of t h e atomic medium, i n s i d e t h e microwave c a v i t y , a t frequency w ' = w

+

A t a given a n g u l a r frequency w ' , t h i s g a i n G i s given by (10)

where b is t h e amplitude o f t h e atomic response and i t s phase f o r a s i n u s o i d a l e x c i t a t i o n o f amplitude p ' . We have :

G = f ~ ~ ~ ( e ' c o s y b + j p ' s i n v ) b

where cos 'Q and E' s i n y a r e given i n r e f e r e n c e 10, such a s :

b b

is t h e microwave c a v i t y a n g u l a r resonance frequency, T is t h e microwave c a v i t y

&me c o n s t a n t , TI and T a r e l o n g i t u d i n a l and t r a n s v e r s e g e l a x a t i o n t i m e s of hy- 2

drogen atoms, r e s p e c t i v e l y , a n d cc a parameter, which i s u n i t y a t o s c i l l a t i o n t h r e s - h o l d , given, i n S.I. u n i t s , by :

where u i s t h e magnetic p e r m e a b i l i t y of vacuum, ug i s Bohr magne,ton, fi i s Planck constan? d i v i d e d by 2 a , q i s t h e f i l l i n g f a c t o r , a s d e f i n e d i n r e f e r e n c e 1 , Vc i s t h e volume o f t h e microwave c a v i t y , Q i s t h e q u a l i t y f a c t o r o f t h e c a v i t y (Qc = wcTc/2) and I i s t h e f l u x of hy&ogen atoms e n t e r i n g t h e b u l b i n t h e s t a t e F = 1 , mF = 0 .

The s i g n a l w i s a m p l i f i e d , t h e n r e c t i f i e d i n a square law d e t e c t o r and f i - n a l l y it i s demodulsted by m u l t i p l i c a t i o n by s i n w t. The e r r o r s i g n a l V i s t h e d.c. component of t h e o u t p u t s i g n a l of t h e demodulztor. I t i s then given by :

V = C wt wt

*

where C i s a c o n s t a n t . The b a r means time averaging.

I t w i l l be assumed i n t h e following t h a t i) t h e a n g u l a r frequency d i f f e r e n c e

w

-

l a t i o n urn i s much l a r g e r than t h e atomic linewidth.

I t t h e n comes, from e q u a t i o n s 2 , 4 , 5 , 6 and 8 :

where G and S a r e t h e g a i n and t h e s a t u r a t i o n f a c t o r r e s p e c t i v e l y , f o r t h e c a r r i e r

0

component of t h e i n t e r r o g a t i o n s i g n a l of amplitude p ' = p J (m) a t frequency

( , ) ' = ( , ) = , , ,

0.

3 . Cavity p u l l i n g . - I n a s t e a d y s t a t e regime o f t h e frequency c o n t r o l loop o f t h e q u a r t z c r y s t a l o s c i l l a t o r , t h e e r r o r s i g n a l V i s zero. We then have :

W-W T

0 C

p = - = -

P wc-W (10)

T2

where P i s t h e c a v i t y p u l l i n g f a c t o r o f a p a s s i v e maser, with t h e s p e c i f i e d i n t e r - rogatioR scheme.

Equation 10 shows t h a t t h i s c a v i t y p u l l i n g f a c t o r i s t h e same a s f o r an a c t i - v e l y o p e r a t e d maser ( ' I , a r e s u l t which h a s been i n f e r r e d e x p e r i m e n t a l l y by F.L.

Walls and D.A. H w e (''I. The following d i f f e r e n t r e s u l t was given e a r l i e r ^{( 8 )}

I t i s v a l i d i n d i f f e r e n t c o n d i t i o n s , when t h e atomic d i s p e r s i o n l i n e i s observed u s i n g a p u r e l y s i n u s o i d a l i n t e r r o g a t i o n s i g n a l . T h i s confirms t h a t c a v i t y p u l l i n g f a c t o r s (I0) depend on t h e p a r t i c u l a r method used t o i n t e r r o g a t e t h e atomic t r a n s i - t i o n .

Equation 10 h a s been checked e x p e r i m e n t a l l y . For t h a t purpose, a q u a r t z crys- t a l o s c i l l a t o r h a s been frequency locked t o a p a s s i v e l y o p e r a t e d l a r g e s i z e hy- drogen maser, a s d e p i c t e d on f i g u r e 1. The i n t e r r o g a t i o n s i g n a l is s i n u s o i d a l l y frequency modulated.

Experimental set-up f o r t h e measurement o f t h e c a v i t y p u l l i n g f a c t o r o f a p a s s i v e l y o p e r a t e d H-maser.

Frequency Control

I I

Synchronous Low-Pass Square Ware Integrator

Detector Filler Detector

C8- 1 6 2 JOURNAL Dl$ PHYSIQUE

~i re 2 shows t h a t t h e c a v i t y p u l l i n g f a c t o r h a s t h e expected v a l u e and does n o t g p e n d on a, f o r S << 1. I t h a s a l s o been checked t h a t changes o f t h e sa- t u r a t i o n f a c t o r S

,

P'

Comparison of t h e c a v i t y p u l l i n g f a c t o r P of a p a s s i v e l y o p e r a t e d maser ( w i t h S << 1 and m = 0.1) t o t h e c a v i t y p u l l i n g f g c t o r P o f t h e same maser b u t o p e r a t e d a c t i v e l y ? P i s s e t e q u a l t o t h e r a t i o of s e p a r a t e l y a measured v a l u e s o f T and T2 n

4. Frequency s t a b i l i t y o f a p a s s i v e l y o p e r a t e d hydrogen maser.-

4.1. P m e r s p e c t r a l d e n s i t y o f ~ f ~ e _ q u ~ g c y - f l _ u ~ t u a t i o n s . - A s p r e v i o u s l y ( 8 ) , we assume t h a t thermal n o i s e i n t h e microwave c a v i t y u s e f u l mode i s c r e a t e d by an ap- p r o p r i a t e n o i s e g e n e r a t o r coupled t o a dummy microwave c a v i t y . The o u t p u t s i g n a l then becomes :

where p I n ) + j p l ( n ) r e p r e s e n t s t h e complex amplitude o f t h e u n c o r r e l a t e d thermal 2

n o i s e components i n t h e microwave c a v i t y , a t f r e q u e n c i e s c l o s e t o W + w S i m i l a r i l y , m

.

j p ' 2 ( n ) denotes t h e a d d i t i v e n o i s e components a t t a c h e d t o t h e microwave r e c e i v e r .

The one-sided power s p e c t r a l d e n s i t y (P.S.D.) of t h e s e components a r e g i v e n by :

1 1 kT

- S ( n ) = - S ( n ) = 4 -

b2 '1 b2 p2 P ⁽¹³⁾

and

7

^L-

-

P ' I b2 ~ ' 2 Qc

where k is Boltzman c o n s t a n t , T i s t h e a b s o l u t e temperature o f t h e c a v i t y , P is t h e power d i s s i p a t e d i n t h e microwave c a v i t y a t frequency w = w o r Qext ^and Qc a r e t h e e x t e r n a l and loaded c a v i t y q u a l i t y f a c t o r , r e s p e c t i v e l y , and F is t h e n o i s e f a c t o r of t h e microwave r e c e i v e r .

The one-sided P.S.D. o f f r a c t i o n a l frequency f l u c t u a t i o n s S (£1 = h of t h e c o n t r o l l e d q u a r t z c r y s t a l o s c i l l a t o r t h e n p o s s e s s e s two componentsYdetermi%ed by thermal n o i s e i n t h e maser c a v i t y , and by added r e c e i v e r n o i s e , r e s p e c t i v e l y . I t can b e shown t h a t we have* r

-

h (maser) =

-

- -

fiuO3 T~ a2s0 KQc =Ill

- - - - -

*

+

w i t h

1 ^J (m) 2

Cm= (2J12 ( m ) + [ ~ ~ ( m ) - _ I ( ( ~ l (m) -J3(m) + ( J 2 (m)-J4(m) ) +.

. .

2Jo (m) J (m) Go Go

(16)

and T ( 1 + ~ ~ ) '

8kT 1

h ( r e c e i v e r ) =

- ^- -

I n t h e following, we w i l l f o c u s on t h e e f f e c t o f thermal n o i s e i n t h e maser c a v i t y . We w i l l t h u s determine t h e u l t i m a t e frequency s t a b i l i t y c a p a b i l i t y o f a pas- s i v e l y o p e r a t e d maser. Expected p r o g r e s s i n t h e n o i s e f a c t o r a m p l i f i e r s j u s t i f i e s t h e assumption made.

Equation 16 shows t h a t it e x i s t s an optimum v a l u e m of t h e modulation o p t

i n d e x m, which depends on G However, it can b e shown t h a t f o r Go c l o s e t o u n i t y ,

0 -

we have m = 1.1 and Cm " 4.1 and f o r t h e o t h e r extreme c a s e Go >> 1 , we have OP t

m " 1.1 and C m = 4.3. We then assume m = 1.1 and s e t Cm = 4 i n e q u a t i o n 15.

o p t

W e a l s o s e t So = 1, which i s t h e optimum v a l u e of t h e s a t u r a t i o n f a c t o r f o r Go " 1 and Go >> 1.

The lower l i m i t of t h e P.S.D. o f a p a s s i v e l y o p e r a t e d hydrogen maser i s t h e n

4.2. E_ffZect o f spkn exchange l i n e b r o a d e n i n g . Relaxation times T and T2 depend on t h e atomic beam f l u x I v i a H-H c o l l i s i o n s i n t h e s t o r a g e bulb.' T h i s e f f e c t was f i r s t considered i n r e f e r e n c e 1. We then have :

w i t h 2

T = (TlIo (T2Io

where (T ) and (T2)o a r e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n times f o r I = 0, l o

r e s p e c t i v e l y . q i s t h e s p i n exchange maser parameter and I i s t h e t h r e s h o l d ato- mic f l u x f o r very s m a l l v a l u e s o f q. These two q u a n t i t i e s th a r e d e f i n e d a s ( 1 ) :

C8- 1 6 4 JOURNAL DE PHYSIQUE

a i s t h e s p i n exchange c r o s s - s e c t i o n , vr

-

b t o t

t o t a l f l u x e n t e r i n g t h e bulb.

I t comes, from e q u a t i o n s 7, 19, 20 and 21 :

with

F i g u r e 3 shows t h e v a r i a t i o n of H v e r s u s I/I f o r d i f f e r e n t v a l u e s of

P t h

t h e parameter q. One s e e s t h a t f o r e a c h v a l u e o f q , t h e r e i s a value o f t h e atomic f l u x which minimizes t h e P.S.D. o f f r a c t i o n a l frequency f l u c t u a t i o n s . The l o c u s of t h e minimum of H o c c u r s f o r an optimum v a l u e o f 1/1 given by t h e following

e q u a t i o n : P ^{t h}

-

-

I t h 9

and we have :

H (min.) 2 379 q 2 P

V a r i a t i o n of H (q, 1/1 )

P t h

v e r s u s I/Ith f o r d i f f e r e n t va- l u e s o f t h e p a r a m e t e r q. The d o t t e d l i n e r e p r e s e n t s t h e t h r e s h o l d c o n d i t i o n o f o s c i l - l a t i o n and t h e t h i n l i n e r e p r e - s e n t s t h e l o c u s o f t h e minima of H

.

P

A t t h e minimum o f H

,

a = 0.17/q

I t comes from e q u a t i o n s 23, 25 and 28 :

The d e s i n g o a l o f p a s s i v e l y o p e r a t e d hydrogen masers i s then t o make t h e q u a n t i t y

f n~~~~

9

I n a d d i t i o n , it can be d e r i v e d , from e q u a t i o n s 23, 24 and 27 t h a t t h e o p t i - mum value o f t h e atomic f l u x is t h e f o l l o w i n g :

v b I l

I ( o p t . ) = 1.69

= --

UVr I t o t *bTt

Table 1 shows t h e u l t i m a t e frequency s t a b i l i t y c a p a b i l i t y o f p r e s e n t l y de- s i g n e d p a s s i v e hydrogen masers. I t i s denoted 0 ( T ) , and c a l c u l a t e d f o r T = 100 s.

Y

TABLE 1. A Comparison between a c t i v e l y and p a s s i v e l y operated masers.

vc

"b

17 Qo Qc

17Qc/vc

9

Tt I t h 1/1& (opt)

I (opt) Ha(opt) H (opt) ho a (T = 100 ^S) 0' (T = 100 5)

Y

* . _Wrth a sapphire loaded c a v i t y ( 3 ) , with Qc = 8 500, i t comes U (T = 100 s ) = 2.1 x 1 0 - l 4 Large s i z e maser

Active

15.5 x 2.35

2.8 60 000 45 000 8.1 x l o 6

0.058 0 . 4 7.5 x l o l l

15.4 1.1

0.35

1.4 2.6 x 10-l5 2.6 x l 0 l 5

Small s i z e maser

Passive

15.5 2.35 x

2.8 60 000 30 000 5.4 x 10 6

0.087 0.4 1.1 x 1 0 l 2 (a = 1 ) 1.4

1.5 x loi2

-

15 3.8 x 1.4 x 1 0 - l ~ 2.4 x

Cavity loaded with alumrna*

2.3 1.15

0.5 6 000 3 000 6.5 x 10 5

1.44 0.2 3.7 l o i 3

0.59 2.2 l o i 3

786 2.4 x 3.5 1 0 - l ~ 6 x

( p a s s i v e )

Cavity loaded with c a p a c ~ t o r s

2.35 0.93 x (q1=0.5) 1.25

1 3 000 6 500 3.4 x lo6

0.34 0.2 7 x 1 0 l 2

2.5 1.7 x 10 13

-

44 7.1 x loez6 1.9 x 10-l4 3.2

C8- 1 66 JOURNAL 4E PHYSIQUE

5. Frequency s t a b i l i t y o f an a c t i v e l y o p e r a t e d hydrogen maser.

-

.

f 2

S _Y ( f ) =

*

⁺

Qc Vo

where QR = W T /2 and v i s t h e t r a n s i t i o n frequency.

0 2

We w i l l o n l y c o n s i d e r h

,

0 Y

inines t h e medium term frequency s t a b i l i t y R and, h o p e f u l l y ( I 5 ) l o n g term f r e q ~ e n c y s t a b i l i t y i n t h e absence o f d r i f t i n t h e c a v i t y frequency. One then h a s :

The v a r i a t i o n of P v e r s u s I/Ith i s g i v e n i n r e f e r e n c e 1 a s :

w i t h

P _c =

-

I t r e s u l t s from e q u a t i o n s 21, 33 and 34, t h a t t h e e f f e c t o f s p i n exchange l i n e broadening ( I 6 ) upon t h e v a l u e o f h can b e w r i t t e n a s :

0

w i t h

h =- 8kT 2 "Qc

W 0

F i g u r e 4 shows t h e v a r i a t i o n of H v e r s u s I/I f o r d i f f e r e n t v a l u e s o f t h e

a t h

parameter q. Again, f o r each v a l u e of q 0 , t h e r e i s a v a l u e of t h e atomic f l u x which minimizes t h e P.S.D. of f r a c t i o n a l frequency f l u c t u a t i o n s . The l o c u s o f t h e minimum of H i s d e f i n e d such a s :

and we have :

*1t i s w i t h n o t i c i n g t h a t ho does n o t depend on t h e r e c e i v e r n o i s e f a c t o r , f o r a c t i - v e l y o p e r a t e d masers.

V a r i a t i o n of Ha(q, l / I t h ) v e r s u s 1/1 f o r d i f f e r e n t

t h

v a l u e s o f t h e parameter q.

The t h i n l i n e r e p r e s e n t s t h e l o c u s o f t h e minima of H

It comes, with e q u a t i o n s 23, 36 and 39 :

T I

kT

-

h (min.) = 16

3

- - -

g " ~ - t I "b q2-6q+l

This e q u a t i o n shows t h a t t h e white frequency n o i s e o f a c t i v e l y o p e r a t e d masers i s s m a l l i f V b i s large" and q a s s m a l l a s p o s s i b l e . However, t h e change o f h ( d n ) 0 w i t h q i s s m a l l as l o n g a s q < 0.05.

The optimum v a l u e of t h e atomic f l u x i s given by :

where t h e q u a n t i t y 1-q/l+q does n o t change much i n t h e range 0 < q < 0.172 i n which t h e maser i s a b l e t o o s c i l l a t e ( I ) .

Table 1 shows t h e r e l a t e d frequency s t a b i l i t y c a p a b i l i t y o f a l a r g e s i z e maser of c l a s s i c a l design.

*

P e t e r s ( I 7 ) , which e n a b l e s t o i n c r e a s e t h e b u l b volume.

C8- 1 68 JOURNAL DE PHYSIQUE

6. A comparison o f t h e frequency s t a b i l i t y c a p a b i l i t y o f a c t i v e l y and p a s s i v e l y o p e r a t e d masers.- We apply t h e above r e s u l t s t o e s t a b l i s h a comparison between t h e u l t i m a t e medium and l o n g term frequency s t a b i l i t y c a p a b i l i t i e s U ( T ) = (h /2T) 1/2

v 0

of some p r e s e n t l y designed hydrogen masers.

We c o n s i d e r

i ) a l a r g e s i z e hydrogen maser i n which o n l y a hydrogen s t o r a g e b u l b ( w i t h a diameter of a b o u t 16.5 cm) i s i n c l u d e d i n t h e microwave c a v i t y ; we w i l l assume t h i s maser e i t h e r a c t i v e l y , o r p a s s i v e l y o p e r a t e d ( b u t very c l o s e t o o s c i l l a t i o n t h r e s - h o l d i n t h e second c a s e ) ,

ii) s m a l l s i z e hydrogen maser w i t h a microwave c a v i t y loaded e i t h e r by alumi- na ( 2 ) o r by i n t e r n a l c a p a c i t o r s ( 5 )

We assume t h a t we have : U = 23.5 x 10 -20 m2 a t room temperature (18) (19)

1

Tb/Tt = 1 . 3 , I /I = 2 and t h a t t h e o u t p u t loop i s c r i t i c a l l y coupled f o r p a s s i v e t o t

masers.

Table 1 shows t h e p e r t i n e n t parameters and t h e expected u l t i m a t e frequency s t a b i l i t y measure U f o r 'r = 100 s. Receiver n o i s e degrades t h e c o n s i d e r e d f r e - quency s t a b i l i t y o f p a s s i v e masers only. The frequency s t a b i l i t y c a p a b i l i t y f i g u r e 0'

,

Y e x t

The f o l l o w i n g c o n c l u s i o n s can b e made :

i ) e x p e r i m e n t a l l y measured frequency s t a b i l i t y of a c t i v e l y o p e r a t e d hydrogen masers is c l o s e ( w i t h i n a f a c t o r o f 2 ) t o t h e u l t i m a t e frequency s t a b i l i t y capabi- l i t y i n t h e w h i t e frequency n o i s e r e g i o n ( I 3 ) ( I 4 ) ; t h e same conclusion can be d e r i v e d f o r t h e w h i t e phase n o i s e region.

i i ) e x c e p t f o r l a b o r a t o r y t e s t i n g , t h e r e i s no i n t e r e s t t o o p e r a t e p a s s i v e l y , a l a r g e s i z e H-maser, w i t h i n c o n d i t i o n a < 1.

iii) measured frequency s t a b i l i t y of p a s s i v e s m a l l s i z e masers (20) i s a l s o c l o s e ( w i t h i n a f a c t o r o f 2 ) t o t h e u l t i m a t e frequency s t a b i l i t y c a p a b i l i t y ,

i v ) t h e optimum v a l u e o f t h e atomic f l u x i n t e n s i t y i s s l i g h t l y l a r g e r f o r s m a l l s i z e p a s s i v e masers t h a n f o r l a r g e s i z e a c t i v e masers and

V) p r e s e n t l y designed s m a l l s i z e p a s s i v e masers cannot compete w i t h l a r g e s i z e masers, as l o n g as frequency s t a b i l i t y i s concerned.

The v a l u e o f t h e spin-exchange parameter q v a r y i n g a s u - v this v a l u e w i l l r f

be d r a s t i c a l l y reduced (I8) i n low t e m p e r a t u r e masers ( I 6 ) (21) s o t h a t s m a l l s i z e hydrogen masers w i l l very l i k e l y be a b l e t o o s c i l l a t e i n such c o n d i t i o n s . Equation 40 shows t h a t f o r q 6 0.05, t h e frequency s t a b i l i t y c a p a b i l i t y U-*(T) o f an a c t i v e luaser s c a l e s as T3/4(U/V One may t h e n e x p e c t a spectacu1a;frequency s t a b i l i t y improvement of 2 o r 3 o r 8 e r s o f magnitude a t low temperature. P o s s i b l e frequency s t a b i l i t y l i m i t a t i o n by quantum n o i s e s h o u l d t h e n be considered.

7. E f f e c t o f c a v i t u a l i t f a c t o r enhancement on t h e f r e uen s t a b i l i t

.-

s i z e hydrogen mase:sqhave bYeen o p e r a t e d a c t i v e l y by enhanzing%cticifial:y (6) ( 7 ) , by e l e c t r o n i c means, t h e c a v i t y q u a l i t y f a c t o r . The q u e s t i o n then a r i s e s o f t h e i r frequency s t a b i l i t y c a p a b i l i t y compared t o t h a t of t h e same d e v i c e , b u t o p e r a t e d p a s s i v e l y .

Q u a l i t y f a c t o r enhancement i n c r e a s e s t h e n o i s e temperature TIe of t h e c a v i t y when a feedback loop i s used, we have ( 2 2 ) :

where Qe and Qo a r e t h e enhanced and unloaded c a v i t y q u a l i t y f a c t o r s , r e s p e c t i v e l y ,

B1 i s t h e coupling f a c t o r o f t h e loop connected t o t h e i n p u t of t h e microwave am- p l i f i e r , G i s t h e g a i n i n t r o d u c e d t o achieve t h e v a l u e Q o f t h e q u a l i t y f a c t o r and F is t h e n o i s e f i g u r e of t h e a m p l i f i e r . I t h a s been assuged t h a t t h e microwave ca- v i t y and t h e a m p l i f i e r a r e a t t h e same temperature.

Neglecting t h e n o i s e added by e l e c t r o n i c components which p r o v i d e Q-enhance- ment, t h e c a v i t y n o i s e temperature becomes T given by :

n e

Equation 43 merely means t h a t thermal energy o f t h e c a v i t y mode b e i n g d i s t r i b u t e d o v e r a s m a l l e r bandwidth, i t s ^P.S.D. i s i n c r e a s e d accordingly.

I t i s e a s y t o s e e t h a t e q u a t i o n 36 t h e n becomes :

where H i s given by e q u a t i o n 37, b u t w i t h t h e v a l u e s of q and Ith which a r e r e l a t e d a r e t o Qe.

Equation 44 has been a p p l i e d t o p r e s e n t l y designed s m a l l s i z e hydrogen masers, w i t h c h a r a c t e r i s t i c parameters as given i n Table 1 , e x c e p t f o r t h e c a v i t y q u a l i t y f a c t o r : one assumes now t h a t t h e c a v i t y i s coupled t o e x t e r n a l c i r c u i t s w i t h two l o o p s having coupling f a c t o r s B1 = B2 = 0.2. F i g u r e 5 shows t h a t t h e r e i s an o p t i - mum value o f Qe.

Ultimate frequency s t a b i l i t y c a p a b i l i t y q-, f o r T = 100 s

Y

of p r e s e n t l y designed s m a l l

I 1 s i z e masers, v e r s u s t h e

enhanced c a v i t y q u a l i t y fac-

0 t o r Qe.

a. microwave c a v i t y loaded w i t h aluminia

10-1~ w i t h c a p a c i t o r s .

Fig. 5

Table 2 summarizes t h e r e s u l t s . It i n c l u d e s t h e v a l u e o f t h e frequency s t a - b i l i t y measure U ' (100 s) when t h e e f f e c t of n o i s e o f t h e a m p l i f i e r , i n c l u d e d i n t h e feedback loopY ( c f . e q u a t i o n 4 2 ) , i s taken i n t o account w i t h F = 2.

TABLE 2. Frequency s t a b i l i t y c a p a b i l i t y o f s m a l l s i z e hydrogen masers o p e r a t e d a c t i v e l y w i t h enhanced c a v i t y q u a l i t y f a c t o r .

Qc Qe ( o p t 0 (100 S )

Y

u' (IOOS) Y

a l u m i n i a loaded c a v i t y 4 300 50 000 1.6 x lo-14

2.6 x 10-l4

c a p a c i t o r loaded c a v i t y

9 400 25 000 7.8 x 10 -15

1.1 10-l4

C8- 1 7 0 JOURNAL DE PHYSIQUE

Comparison of r e s u l t s given i n T a b l e s 1 and 2 shows t h a t one may e x p e c t an improvement (by a f a c t o r o f 2 t o 3) i n t h e u l t i m a t e frequency s t a b i l i t y of s m a l l s i z e hydrogen masers when o p e r a t e d a c t i v e l y v i a Q-enhancement. However, t h e c a v i t y p u l l i n g f a c t o r w i l l be i n c r e a s e d accordingly.

8. Conclusion.- The major conclusion o f t h i s work i s t h a t l a r g d s i z e a c t i v e masers remain.

,

Y

w i l l be achieved f o r very l a r g e v a l u e s of 'r, l a r g e r t h a n s e v e r a l days.

The frequency s t a b i l i t y improvement which may be expected from low tempera- t u r e masers h a s been p o i n t e d o u t .

References

( 1 ) KLEPPNER D., BERG H.C., CRAMPTON S.B., RAMSEY N.F., VZSSOT R.F.C., PETERS H.E.

VANIER J., Phys. Rev. A

138

( 2 ) HOWE D.A., WALLS F.L., BELL H.E. and HELLWIG H., Proceedings o f t h e 33rd An- n u a l Symposium on Frequency C o n t r o l p. 554. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n Washington DC, 1979

( 3) MATTISON E.M., BL~MBERG E.L., NYSTROM G.U. and VESSW R.F.C., Proceedings o f t h e 33rd Annual Symposium on Frequency C o n t r o l , p . 549. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n , Washington DC ,1979

( 4) PETERS H.E., Proceedings o f t h e 32nd Annual Symposium on Frequency Control p.

469. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n Washington DC, 1978

( 5 ) WANG H.T.M., LEWIS J.B. and CRAMPTON S.B., Proceedings o f t h e 33rd Annual Sym- posium on Frequency Control p. 543. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n , Washing- t o n DC,1979

( 6 ) WANG H.T.M., Proceedings of t h e 34th Annual Symposium on Frequency C o n t r o l , p.

365. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n , Washington DC, 1980

( 7) PETERS H.E., Proceedings o f t h e 35th Annual Symposium on Frequency C o n t r o l , 1980. To be p u b l i s h e d

( 8 ) LESAGE P., AUDOIN C. and TETU M., Proceedings o f t h e 33rd Annual Symposium on Frequency C o n t r o l , p. 515. E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n , Washington DC, 1979

( 9) BUSCA G., Proceedings o f t h e 33rd Annual Symposium on Frequency C o n t r o l , p. 563 E l e c t r o n i c I n d u s t r i e s A s s o c i a t i o n , Washington DC, 1979

(10) VIENNET J.

,

IM-21 (1972) 204

(11) WALLS F.L.

-

Braunschweig (F.R.G.) June 1980

(12) CUTLER L.S. and SEARtE C.L., Proc. o f t h e IEEE 2 (1966) 136

( 13) VESSOT R. F. C.

,

(14) LESAGE P., These de Doctorat d ' E t a t

-

(16) VESSCfC R.F.C., LEVINE M.W. and MATTISON E.M., Proceedings of t h e 9 t h Annual P r e c i s e Time and Time I n t e r v a l p. 549, Goddard Space F l i g h t C e n t e r , Greenbelt USA, 1977

(17) PETERS H.E., Mc GUNIGAL T.E. and JOHNSON E.H., Proceedings of t h e 22nd Annual Symposium on Frequency C o n t r o l p. 464. N a t i o n a l T e c h n i c a l Information S e r v i c e , S p r i n g f i e l d USA, 1968

(18) ALLISON A.C., Phys. Rev.A 2 (1972) 2695

(19) DESAINTFUSCIEN M. and AUDOIN C.

,

(21) CRAMPTON S.B., SOUZA S.A. and KRUPCZAK G. This i s s u e

(22) LESAGE P. and AUDOIN C., IEEE Trans. on I n s t r . and Meas. IM-30 (1981) 182