NIST Technical Note 1337: Characterization of Clocks and Oscillators

LESAGE AND AUDOIN2 - Vo )]53610- 1 2S,,(v) = V o(21T.6.v)2m\v+ (21T(Vquartz xtal 5 x 10" 4 X 10- 2710 12 3 x 10- 132 (56)H maser 1.4 x 10'He-Ne laser 4 X 10- 27 10 7 3 x 10-'10'· 3 X 10- 29 10- 2 30where t\v is the half width at half maximum ofthe power spectra defmed as10­1(f1(57)We obviously have 21Tt\V . T c= l. If oscillatorsare considered, one has h o = kTI PQ 2 in the~PTTJ(n ithr)lO-T [5]I) 10 1 I)l 1)4 10 5Fig. 23. (Solid line) Two-sample variance of an oscillatorshowill8 a linear frequency drift of 10- 10 per day and flickernoise of frequency given by Sylf) = 1.2 X 10-2> fO'. (Dolledline) The third difference of fractional phase fluctuation isindependent of the drift (according to Lindsey and Chie [1918b].Negligible amplitude noise and gaussian phasefluctuations being assumed, it is well known that10.2. White noise o/phasethe autocorrelation function of v(t) is R,,(T) givenbyPresently availaple good quartz oscillators areaffected by white noise of phase. It is easy to showfrom the deftnition (16) of Y~ ­tthat the followingR,,(T) = - cos 21TV TCXP [-t(21TV )2a2(y )]o o (53) equation is satisfIed:t2t (21TV o TO'(Yt)] 2 = R. (0) - R.. (T) (58)As ~(Yt) is only defIned for stationary phasefluctuations and for phase fluctuations with stationaryfIrst increments, the same is true for R,,(T) and of the stationary phase fluctuations r,p(t).where R ... (T) denotes the autocorrelation functiontherefore S,,(f).The expression of the one-sided S,,(v) then follows[Rutman, 1974a; Lindsey and Chie, 1978a]:10.1. White noise offrequencyv 2Jl roSu(v) = -:; e- • ) [8 (v - vo)This is the simplest to deal with. If the frequencyof the source is perturbed by a broadband whitenoise of frequency, one has S/f) = h o and ~(Yt)+ S.(v - vo) + 1S.(v). S.(v) + ...] (59)= (h o /2T). Whencewhere the asterisk denotes convolution and thebracket contains an inftnite set of multiple-convolutionproducts of S ... (v) by itself. Such an equationR.(T) =2 v~cos 21TIIOTexp(--:;:-ITI) (54)is not easily tractable. It is the reason why thewhere T cis the coherence time of the signaL Wehave(55) TABLE 5. Theoretical values of correlation time and powerspectrum linewidth for various oscillatorsThe one-sided power spectral density is then representedby a Lorentzian given byOscillator5-MHzVO' Hz hOt Hz-' T c, S 2Av, Hzradiofrequency and microwave domain and h o =h vol PQ 2 in the optical frequency domain, whereP is the power delivered by the oscillator and Qthe quality factor of the frequency-determiningelement. Table 5 gives theoretical values of coherencetime and linewidth of good oscillators. Itis only intended to illustrate a comparison, oftenmade, of the spectral purity of oscillators. It mustbe pointed out that, in practice, other noise processesexist which modify these results. Even anymeaning of T cand t\v is removed if R,,(T) is notdefmed.Multiplication of the frequency by n multipliest\v and divides T cby the factor n 2 •TN-186