Time & Frequency Metrology An introduction€¦ · 1 tropical year = 365,2422 solar days = 366,2422 sideral days Definitions of the unit of time CAMAM 2015, N. Dimarcq, « T/F metrology

Time & Frequency MetrologyAn introduction

Cold Atoms and Molecules & Applications in Metrolog y

16-21 March 2015, Carthage, Tunisia

CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 1

Noël DimarcqSYRTE – Systèmes de Référence Temps-Espace, Paris

An introduction

Contents

� Measurement of time with a linear process – Earth rotation

� Measurement of time with a periodic process – The oscillators and their defaults


� Time, frequency and phase – Relations and noise characterization

� Conclusion

Contents





� Conclusion

Flows : Water clock Sand clock

Measuring time with a « linear » process


Combustion : Candles

Rotation : Earth rotation angle

Oil lamp

Measured time = K x measured parameter = Real time?

Gnomons, sundials and meridian telescopes

Measuring time with Earth rotation


Measuring time = Knowledge ot he Earth orientation

�� Measured time = K x θθθθEarth

The SI unit of time – the second – is defined as :

�� until 1956 : the fraction 1/86 400 of the mean solar day

Definitions of the unit of time


The length of the solar day fluctuates due to :

- Tilt of the Earth rotation axis and the ellipticit y of the Earth orbit around the Sun

- Precession of the equinoxes

Irregularities of the solar day


in minutes



�� 1956 to 1967 : the fraction 1/31,556,925.9747 of the tropical year 1900 1 tropical year = 365,2422 solar days = 366,2422 sideral days



1 tropical year = 365,2422 solar days = 366,2422 sideral days

The earth rotation rate fluctuates due to :

- tides (Moon, Sun)

- inner effects (core – mantle interface)

- atmosphere and meteorological effects

- hydrological effects

- seisms (earthquakes, tsunamis, …)

Fluctuations of Earth rotation


seisms (earthquakes, tsunamis, …)

Fluctuations of Earth rotation

0


Leap seconds

�� Next leap second on 30 June 2015

23:59:5823:59:5923:59:6000:00:0000:00:01

(in UTC)

1900Leap seconds

Contents





� Conclusion

�� Analogy with the measurement of a length with a rul er : count the graduations between the start and the end

Measuring time with a periodic process


Physical signal

Oscillator = temporal ruler

Period = elementary temporal graduation

�� Measuring time with an oscillator : count the oscil lations between the start and the end

Measuring time with a periodic process

T : period

ν = 1/T : frequencyT


signal

Counter

1234

t

t

�� The thinner the graduations, the better the measure ment precision

Importance of the size of the graduations


t

t

�� The smaller the period ( = the higher the frequency), the better the measurement precision

Importance of the oscillator frequency


t

νννν [Hz]1 103 106 109 1012 1015

mechanicalOscillator �� quartz microwave laser

kHz – MHz : BAW quartz oscillators (Bulk Acoustic Waves), MEMS Si

MHz – GHz : SAW quartz oscillators (Surface Acoustic Waves), FB AR (Film Bulk-Acoustic wawe Resonator), …

10 GHz : DRO (Dielectric Resonator Oscillators), cryogenic o scillators (whispering modes), OEO-Optoelectronic oscillators…

THz – 1000 THz : Laser

Oscillators families


Which confidence in a measurement ?

Measurement at another moment or with another ruler


Quality of the measurement =

Evaluation of the uncertainty (fluctuations, biases ) +

Necessary comparisons between various standards

t

Which confidence in time measurement ?

Variations of the oscillation frequency during the measurement or oscillators with


Quality of time measurement =

Evaluation of the frequency fluctuations and freque ncy biases +

Necessary comparisons between various standards

oscillators with unequal frequencies

Defaults of oscillators

� The frequency depends on the oscillator dimensions :L

dLd ∝νν

1610−=ννd

If L = 10 cm dL ~ 0.01 fm

� The frequency depends on the environment (temperatu re, pressure, hygrometry, gravity, vibrations, electromagnetic fi elds, radiations, …)


Shielding, fine control, stabilization of the environment (T°, P, vibrations, e.m. fields, …)

Use of materials with low thermal expansion coeffcients (Invar, Zerodur, ULE, …)

Search for an inversion point to cancel the first order temperature dependence

Thermal sensitivity of quartz oscillators


+g-sensitivity

Gravimetric sensitivity of a mechanical pendulum

« En 1672, M. Richer étant allé à l'isle de Cayenne, environ à 5d de l'équateur, pour y faire des observations astronomiques, trouva que son horloge à pendule qu'il avoit reglée à Paris, retardoit de 2' 28''par jour »

l

g

πν

2

1≈


lπ2

Gravimetric sensitivity of a mechanical pendulum


−≈ 20161

12

1 θπ

νl

gPendulum frequency

Other defaults of oscillators

� The frequency depends on the oscillation amplitude (isochronism default)


� Ageing effects

Frequency drifts

Frequency jumps

OSCILLATOR

(quartz, µw, laser, …)

frequency νννν :

Unstable

Inaccurate

νννν

ATOM / ION

SERVO LOOP

correctionfrequency νννν :

Stable

Accurate

= νννν0

Basic principle of atomic clocks / atomic frequency standards


νννννννν0000

E2

E1h νννν0 = E2 –E1

ATOM / ION REFERENCE

νννν0 νννν

CLOCK SIGNAL

2

1



�� 1956 to 1967 : the fraction 1/31,556,925.9747 of the tropical year 1900 1 tropical year = 365,2422 solar days = 366,2422 sideral days



1 tropical year = 365,2422 solar days = 366,2422 sideral days

�� since 1967 : the duration of 9 192 631 770 periods of the radi ation corresponding to the transition between the two hyp erfine levels of the ground state of the cesium 133 atom ( Added in 1999 �� This definition refers to a cesium atom at rest at a temperature of 0 K)

Contents





� Conclusion

An oscillator is never perfect…


( )y(t)ευυ(t) ++×= 100

)()(

νδν t

ty =

Real Ideal Frequency Frequency

( )ttA ).(.2cos. υπSignal delivered by a frequency standards :

Frequency uncertainties


frequency frequency bias fluctuations

Instability : « amplitude » of frequency fluctuations (« type A » uncertainty uA)

Inaccuracy : uncertainty δεδεδεδε on the frequency bias due to systematic effects (« type B » uncertainty uB)

Total frequency uncertainty u total : 222

BAtotal uuu +=

Stability and accuracy


0υυ(t) =Frequency :

tdttυ(t)t

..2')'(2 0νππϕ == ∫Phase :

( )ttASignal ).(.2cos. υπ= ( ) ( ))(cos.).(.2cos. tAttASignal ϕυπ == ( ) ( ) ( ))(..2cos.)(cos.).(.2cos. 0 tTAtAttASignal υπϕυπ ===Frequency, phase and time

Linear evolution of the phase


ttt

tT ===00

1.

2

)(

.2

)()(

υπϕ

υπϕ

Time :

0

0∫

PeriodNumber of counted oscillations

( )ευυ(t) +×= 10Frequency :

( )tdttυ(t)t

.1..2')'(2 0 ενππϕ +== ∫Phase :

Frequency, phase and time

( ) ( ) ( ))(..2cos.)(cos.).(.2cos. 0 tTAtAttASignal υπϕυπ ===


( )ttttT .11.2

)(

.2

)()(

00

ευπ

ϕυπ

ϕ +===Time :

0

0∫

( )y(t)ευυ(t) ++×= 10Frequency :

( )

++== ∫∫ ')'(.1.2')'(2 0 dttytdttυ(t)

tt

ενππϕPhase :

Frequency, phase and time

( ) ( ) ( ))(..2cos.)(cos.).(.2cos. 0 tTAtAttASignal υπϕυπ ===


( ) )(.11.2

)(

.2

)()(

00

txttt

tT ++=== ευπ

ϕυπ

ϕTime :

∫=⇔=t

dttytxdt

tdxty

0

')'()()(

)(

∫∫0

0

0

with

Sy(f)

Sνννν(f) [Hz2.Hz-1]

[Hz-1] σσσσy(ττττ)[dimensionless]

Depending on the applications, the measurement will be sensitive to frequency and/or phase and/or time fluctuations

Characterization of frequency, phase and time noises

Frequency

noise / uncertainty


Phase

noise / uncertainty

Time

noise / uncertainty

Sφφφφ(f) [rad2.Hz-1] σσσσx(ττττ) [s]

Variances and deviations

Characterization on long term (« low » Fourier frequency)

Power Spectral Densities

Characterization on short term (« high » Fourier frequency)

Total measurement noise with a bandpass [f0-∆f/2 , f0+∆f/2] :

dffSff

ff).(noise Total

2/

2/

0

0∫

∆+

∆−= υ

Sν(f) : Power Spectral Density (PSD) of the frequency noise [in Hz2 /Hz]

Spectral description of noise (ex.: frequency noise)


Fourier frequency ff0

∆∆∆∆f

Sν(f)

τ1∝∆f

Averaging over a duration ττττ

PSD for different noise types

∑== αανν fh

fSfSy )(1

)(20

Relative frequency noise

(independent of νννν0)Absolute frequency noise

(dependent on νννν0)


f

Time description of noise

νδν=)(ty


( ) ( )212 21 τττσ kky yy −= +

( ) ( )212 τττσ kky yy −= +

Allan deviation:

Classical variance:

Average over all the

τky

Allan variance and filtering


Relation between PSD and Allan deviation

τ τ τ τ : integration time


Allan deviation behaviour

White frequency noise: the frequency stability (Allan deviation) improves as

τ1


σσσσy(ττττ)

τ τ τ τ [s]

deviation) improves asτ

1

White frequency noise: the frequency stability (Allan deviation) improves as 1

Allan deviation behaviour


Frequency flicker noise:

Stability floor

σσσσy(ττττ)

τ τ τ τ [s]

deviation) improves asτ

1

10-13

100

τ τ τ τ -1/2 τ τ τ τ +1/2

∫=⇔=t

dttytxdt

tdxty

0

')'()()(

)(

Allan deviation (in frequency) and time deviation

�� Case of white frequency noise


10-1 100 101 102 103 104 105 106 107

10-16

10-15

10-14

10

σσ σσ y( ττ ττ

)

ττττ [s]

10-1 100 101 102 103 104 105 106 107

0,1

1

10

σσ σσ x( ττ ττ

) [p

s]ττττ [s]

τ τ τ τ -1/2 τ τ τ τ +1/2

σσσσy(ττττ)σσσσX(ττττ)

ττττ ττττ

[ ]( )ttt .)()(.2cos 21 υυπ −

21 υυ −22

21 σσ +

Mean frequency difference :(for Tuning / Syntonisation)

Total noise :

Comparisons – Beatnote technique

)T-T( 2121 ϕϕ −

If equal frequencies, mean phase (or time) difference (for Synchronization) :


FrequencyStandard

1

Frequency Standard

2

)).(.2cos(. 11 ttA υπ )).(.2cos(. 22 ttA υπ

)).(.2cos().).(.2cos( 2121 ttttAKA υπυπ

Low pass filter

At the beatnote output:

o mean value of the frequency difference �� validation of the frequency accuracy budget

o total noise of the frequency difference �� access to the frequency (or phase) stability

( ) ( ) ( )222 σσσ +=

Comparisons – Beatnote technique


- if standard 1 is much better than standard 2:

- if the two standards are identical:

( ) ( ) ( )22 standard21 standard2notebeat σσσ +=

notebeat 2 standard σσ =

2notebeat

2 standard1 standard

σσσ ==

[ ]( )ttt .)()(.2cos 21 υυπ −

Frequency / Phase locking of a slave oscillator

to a master oscillator

Frequency or Phase Lock Loop (PLL)


Master oscillator

Slaveoscillator

)).(.2cos(. 11 ttA υπ )).(.2cos(. 22 ttA υπ

Low Pass Filter

Contents





� Conclusion

1000 yrs

1 million yrs

1 billion yrs

1 second

error after:

Industrial Cs beam clocks

Cold atom fountain

Optical clocks

Precision of time measurement

Age of universe

10-16

10-18

δν/νδν/νδν/νδν/ν


1000 yrs

1 year

1 hour

1600 1700 1800 1900 2000

1 second

error after:

Harrison clock

Shortt clock

Quartz oscillator

First Cs beam clock

Astronomical, mechanical and electrical era Atomic era

Huygens pendulum

1 day

� Local oscillators in any electronic devices, PLL, f ilters, sensors, …

� Fundamental metrology (SI units), time scales(TAI, UTC, UTC(k), )

� Ranging, positioning, navigation, GNSS

� Network synchronisation, telecom, smart grids, DSN, VLBI, …

Wide spectrum of T/F metrology applications


� Fundamental physics (drift of fundamental constants , gravitational shift, high precision spectroscopy, …)

� Detection of gravitation waves, relativistic geodes y

� Astronomy (pulsars time tagging)

� RADAR, LIDAR, atmosphere analysis, …

� Etc, etc, etc …

Documents

Time & Frequency Metrology An introduction€¦ · 1 tropical year = 365,2422 solar days = 366,2422 sideral days Definitions of the unit of time CAMAM 2015, N. Dimarcq, « T/F metrology