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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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 2
� Time, frequency and phase – Relations and noise characterization
� Conclusion
Contents
� Measurement of time with a linear process – Earth rotation
� Measurement of time with a periodic process – The oscillators and their defaults
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 3
� Time, frequency and phase – Relations and noise characterization
� Conclusion
Flows : Water clock Sand clock
Measuring time with a « linear » process
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 4
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 5
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 6
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 7
in minutes
The SI unit of time – the second – is defined as :
���� until 1956 : the fraction 1/86 400 of the mean solar day
���� 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
Definitions of the unit of time
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 8
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 9
seisms (earthquakes, tsunamis, …)
Fluctuations of Earth rotation
0
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 10
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
� Measurement of time with a linear process – Earth rotation
� Measurement of time with a periodic process – The oscillators and their defaults
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 11
� Time, frequency and phase – Relations and noise characterization
� 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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 12
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 13
signal
Counter
1234
t
t
���� The thinner the graduations, the better the measure ment precision
Importance of the size of the graduations
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 14
t
t
���� The smaller the period ( = the higher the frequency), the better the measurement precision
Importance of the oscillator frequency
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 15
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 16
Which confidence in a measurement ?
Measurement at another moment or with another ruler
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 17
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 18
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, …)
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 19
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 20
+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≈
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 21
lπ2
Gravimetric sensitivity of a mechanical pendulum
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 22
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 23
−≈ 20161
12
1 θπ
νl
gPendulum frequency
Other defaults of oscillators
� The frequency depends on the oscillation amplitude (isochronism default)
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 24
� 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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 25
νννννννν0000
E2
E1h νννν0 = E2 –E1
ATOM / ION REFERENCE
νννν0 νννν
CLOCK SIGNAL
2
1
The SI unit of time – the second – is defined as :
���� until 1956 : the fraction 1/86 400 of the mean solar day
���� 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
Definitions of the unit of time
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 26
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
� Measurement of time with a linear process – Earth rotation
� Measurement of time with a periodic process – The oscillators and their defaults
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 27
� Time, frequency and phase – Relations and noise characterization
� Conclusion
An oscillator is never perfect…
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 28
( )y(t)ευυ(t) ++×= 100
)()(
νδν t
ty =
Real Ideal Frequency Frequency
( )ttA ).(.2cos. υπSignal delivered by a frequency standards :
Frequency uncertainties
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 29
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 30
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 31
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 ===
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 32
( )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 ===
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 33
( ) )(.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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 34
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)
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 35
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)
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 36
f
Time description of noise
νδν=)(ty
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 37
( ) ( )212 21 τττσ kky yy −= +
( ) ( )212 τττσ kky yy −= +
Allan deviation:
Classical variance:
Average over all the
τky
Allan variance and filtering
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 38
Relation between PSD and Allan deviation
τ τ τ τ : integration time
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 39
Allan deviation behaviour
White frequency noise: the frequency stability (Allan deviation) improves as
τ1
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 40
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 41
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 42
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) :
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 43
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 44
- 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)
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 45
Master oscillator
Slaveoscillator
)).(.2cos(. 11 ttA υπ )).(.2cos(. 22 ttA υπ
Low Pass Filter
Contents
� Measurement of time with a linear process – Earth rotation
� Measurement of time with a periodic process – The oscillators and their defaults
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 46
� Time, frequency and phase – Relations and noise characterization
� 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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 47
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
CAMAM 2015, N. Dimarcq, « T/F metrology – An introduction » 48
� 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 …