Download A| ω - Enrico Rubiola

The Pound-Drever-Hall
Frequency Control
Tutorials of the 2015 EFTF / IFCS
Denver, CO, USA, April 12–16, 2015
Enrico Rubiola
CNRS FEMTO-ST Institute, Besancon, France
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Overview
Basic mechanism
Key ideas
Control loop
Resonators stability
Optimization
Applications
Alternate schemes
home page http://rubiola.org
2
Overview
• Frequency stabilization
to a passive resonator
• Relevant cases:
oscillator / laser
critical
path
control
unit
resonator
critical
path
vco
• The Resonator is
unsuitable to an oscillator
• Cryogenics, vacuum, etc.
frequency
reference
circulator
V~∆𝝼
error
detector
Points of interest
•
•
•
•
•
Power (intensity) detector – available from RF to optics
Compensation of the critical path
Null measurement of the frequency error
Use frequency modulation to get out of the flicker region
One-port resonator –> lowest dissipation –> narrowest linewidth
3
Basic mechanism
Featured article
Black E. D., An introduction to Pound–Drever–Hall laser frequency
stabilization, Am J Phys 69(1) January 2001
Also Technical Note LIGO-T980045-00-D 4/16/98
4
The full scheme
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
ωn = natural (resonant) frequency
ω0 = oscillation frequency
νn
Q
output
2
phase-modulated oscillator
detector
vco
ν0
control
servo
ω0 = ωn
phase
modul
fm
1
power
detector
3
resonator
fm
lock-in amplifier
RF
LO
ve oc ν0 − νn
IF
The error signal is proportional to the frequency error
ve = D(!0 !n )
Phase modulation – Physics
V (t) = Vp cos[!t + m cos(⌦t)]
Phasor diagram
phas
d
e
t
a
ul
d
o
e-m
LSB
carrier
USB
⌦
⌦
Frequency domain
J0
Re
J1
ω
–J1
ω
Im
ω0–Ω
ω0
ω0+Ω
Bessel functions Jn(m)
5
6
Phase modulation – Math
(not too)
small m
small m
V = V0 ei!t eim sin ⌦t
1
X
= V0 ei!t
Jn (m)ein⌦t
i!t
⇥
' V0 e
J0 (m) + J 1 (m)e
h
= V0 J0 (m)ei!t J1 (m)ei(!
' V0
h
m
1+
2
2
J0 (m)
e
i⌦t
+e
i⌦t
LSB
carrier
n= 1
Carrier power
Pc =
phas
ated
l
u
d
e-mo
i⌦t
J1 (m)e
⌦)t
i
e
i⌦t
⇤
+ J1 (m)ei(!+⌦)t
i!t
i
USB
⌦
⌦
J0
Re
Jacobi-Angers expansion
ω
Im
ω0–Ω
ω0
2
m
Ps = J12 (m) P0 '
P0
4
Symmetry (real z)
(
Jn (z)
J–n (z) =
Jn (z)
For moderate m
odd n
even n
ω
–J1
Sideband power
P0 ' P0
J1
J0 (m) ' 1
m2 /4
J1 (m) ' m/2
J0
J1
ω0+Ω
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
The reflection-mode resonator
νn
Q
output
2
detector
vco
ν0
control
phase
modul
1
3
resonator
fm
RF
fm
LO
ve oc ν0 − νn
IF
Featured textbook
D. M. Pozar, Microwave engineering 4th ed, Wiley 2011, ISBN 978-0-470-63155-3
Chapter 6 – Microwave Resonators
Notice that the formalism is suitable to optics
7
Reflection coefficient Γ
reflection coefficient
Γ = V– / V+
incident V +
resonator
reflected V –
Rg
Rp
C
L
g 1 iQ0
=
g + 1 + iQ0
Q0 = unloaded ‘Q’
g = Rp /Rg coupling
!
!n
=
detuning
!n
!
8
9
total
reflection
absorption
Approximations for Γ
total
reflection
Off-resonance
'
Off-resonance
'
1
Recall
g 1 iQ0
=
g + 1 + iQ0
Q0 = unloaded ‘Q’
g = Rp /Rg coupling
!
!n
=
detuning
!n
!
!
'2
close to !n
!n
Resonance, close to ωn
g 1
'
g+1
4Q0 g
!
i
(g + 1)2 !n
even function
resistance mismatch
odd function
frequency error
1
Resonator reflected signal – Physics
phase-modulated incident signal
J0
Re(V + )
incident
Im(V + )
J1
ω0
Re(V – )
reflected
Im(V – )
g 1
'
g+1
ωn
J1
≈0
ω0–Ω
≈0
J0 Im(Γ(ω))
ω0
ω0–ωn
⌦
4Q0 g
!
i
(g + 1)2 !n
reflection coefficient
–J1
≈0
⌦
USB
ω0+Ω
Im(Γ)
Re(Γ)
LSB
carrier
–J1
ω0–Ω
phas
d
ulate
d
o
e-m
ω0+Ω
Γ = V– / V+
incident V +
resonator
reflected V –
10
Resonator reflected signal – Math
phase-modulated incident signal
Incident wave
h
V + = V0 J1 (m)ei(!
⌦)t
+ J0 (m)ei!t + J1 (m)ei(!+⌦)t
LSB
use
(! ± ⌦) '
carrier
1 and
Reflected wave
⇢
V = V0 J1 (m)ei(!
LSB
⌦)t
USB
g 1
(!) '
g+1

g 1
+ J0 (m)
g+1
i
4Q0 !
i
g + 1 !n
4Q0 ! i!t
i
e
g + 1 !n
J1 (m)ei(!+⌦)t
carrier
proportional to ω–ωn
USB
11
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
Power (quadratic) detector
12
νn
Q
output
2
detector
vco
ν0
phase
modul
1
3
fm
control
power
detector
V = kd P
RF
fm
LO
ve oc ν0 − νn
1
1
⇤
Power P = <{V I } = P
<{V V ⇤ }
2
2R0
IF
V and I are
peak values
Power (quadratic) detector
beat –> 2Ω
1
P =
<{V V ⇤ }
2R0
J1
Re(V – )
reflected
Im(V – )
–J1
ω–Ω
ω
beat –> Ω
|V0 |2
P =
2R0
(
≈0
(a+b+c)2 =
ω+Ω
a2 + b2 + c2 + 2ab + 2ac + 2bc
|––––––––––|
dc terms
beat –> Ω
dc terms
J12 (m)

1 2
g 1
+ J0 (m)
2
g+1
2

|––––––––––––––|
beat terms
1 2
4Q0 !
+ J0 (m)
2
g + 1 !n
2
)
+
|V0 |2 2
|V0 |2
4Q0 !
J1 (m) cos(2⌦t) +
2J0 (m)J1 (m)
sin(⌦t)
2R0
2R0
g + 1 !n
diagnostic
error signal
13
14
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
The lock-in amplifier
νn
Q
output
2
detector
vco
ν0
phase
modul
1
3
fm
control
fm
lock-in amplifier
RF
LO
ve oc ν0 − νn
IF
Two-channel lock-in amplifier
x(t) cos(!t)
y(t) sin(!t)
input
x(t)
cos(!t)
y(t)
output
sin(!t)
–90º
2 cos(!t)
at the output, x(t) and y(t)
are low-pass filtered
phase reference
error
diagnostic
|V0 |2
4Q0 !
ve =
2J0 (m)J1 (m)
2R0
g + 1 !n
|V0 |2 2
vd =
J1 (m)
2R0
15
16
In synthesis
The frequency discriminant D is
proportional to
• Oscillator power P0
output
• Power-detector gain kd [V/W]
2
phase-modulated oscillator
ν0
control
servo
ω0 = ωn
phase
modul
fm
power
detector
detector
vco
• Modulation index m
• Resonator’s Q0/ωn
νn
Q
1
3
resonator
fm
lock-in amplifier
RF
LO
ve oc ν0 − νn
• RF gain at the detector output
(not shown)
• Gain of the lock-in amplifier (not
accounted for in equations)
…And affected by the coupling
coefficient g
error signal
ve = D(!0
!n )
IF
17
Key ideas
18
Use a power (intensity) detector
• Power detectors are available in the widest
frequency range
• Sub-audio to UV, and more
• Including the THz band
• The power detector has quadratic response
to voltage – or to electric field
Even function vs odd function
• The detector provides a signal
proportional to the power (intensity)
• Even function at ω0
• Unmodulated signal not suitable
to feedback control
• The modulation mechanism
provides a signal proportional to the
imaginary part
• Odd function at ω0
• Great for feedback control
19
Modulation and flicker
Get out of the flicker & drift region
20
21
Null measurement
😟
😐
• Absolute measurements rely on the “brute
force” of instrument accuracy
• Differential measurements rely on the
difference of two nearly equal quantities,
something like q2–q1. However similar, this
is not our case!
😄
• Null measurements rely on the
measurement of a quantity as close
as possible to zero – ideally zero.
😎
• The Pound scheme detects
• Null of Im(Γ(ω))
• AC regime, after
down-converting to Ω
J0
Re(V + )
incident
Im(V + )
J1
–J1
ω0–Ω
ω0
ω0+Ω
Im(Γ)
Re(Γ)
Re(V – )
reflected
Im(V – )
ωn
J1
≈0
–J1
≈0
≈0
ω0–Ω
J0 Im(Γ(ω))
ω0
ω0–ωn
ω0+Ω
Insensitive to the critical path
oscillator / laser
critical
path
resonator
critical
path
vco
control
unit
frequency
reference
circulator
V~∆𝝼
error
detector
A length fluctuation does not affect
• The phase and amplitude relations between
carrier and sidebands
• In turn, the measurement of Δω
(No longer true in the presence of dispersion)
The mechanism is the same of radio emission
22
23
One-port vs two-port resonator
— one port is better than two —
Rg
Rg
Rp
C
L
C
• Electrical
• Smaller dissipation than the two-port resonator
• Hence higher Q
• Physical / System level – Simpler
• Vacuum
• Cryogenic environment
• Resonator far from the oscillator
RL
Rp
L
24
Control loop
Featured book
K.J. Åström, R.M. Murray, Feedback Systems, Princeton 2008
– Caveat: however outstanding, does not focus on TF applications –
The
–6
10
25
golden rule
• It is generally agreed that a microwave frequency control
loop can lock within 10–6 of the bandwidth
• Cs standard: 10–6 × (100Hz/9.2GHz) ≈ 10–14 stability
• Cryogenic sapphire: 10–6 × (10Hz/10GHz) ≈ 10–15 stability
• In optics, the 10–6 rule yields still unachieved stability
• Optical FP: 10–6 × (10kHz/200THz) ≈ 5×10–19 stability
• The resonator fluctuation
is not a part of the control, and accounted for separately
1
10–6 B
2
|H|
1/2
B
(–3 dB)
O
f
Sweep the oscillator frequency
– High modulation frequency –
J0
Re(V + )
ω0
–J1
Re(V + )
J1
–J1
J0
J1
ω0
Im(Γ)
Im(Γ)
Re(Γ)
Re(Γ)
ωn
1
ωn
ve
0.5
ωn
0.95
0.9
1.05
-0.5
-1
Re(V + )
–J1
J0
ω0
Im(Γ)
Re(Γ)
ωn
ω
J1
1.1
26
Sweep the oscillator frequency
– High modulation frequency –
Courtesy of Alexandre Didier
27
In the conceptual model, we dithered the frequency of the
laser adiabatically, slowly enough that the standing wave inside the cavity was always in equilibrium with the incident
beam. We can express this in the quantitative model by making ! very small. In this regime the expression
Near 28
res
since ! F( $
F( $ ), how
Featured article – Black ED, An introduction to Pound–Drever–Hall
laser frequency stabilization, Am J Phys 69(1) January 2001 – Fig.6
Low modulation frequency
P ref&2
ve
ω0–ωn
Fig. 6. The Pound–Drever–Hall error signal, ' /2!P c P s vs $ /* + fsr , when
the modulation frequency is low. The modulation frequency is about half a
Carrier and PM
#3 sidebands are in the resonance bandwidth
linewidth: about 10 of a free spectral range, with a cavity finesse of 500.
Fig. 7. The P
the modulatio
20 linewidths
500.
Control loop
Franco S, Design with operational amplifier and analog
integrated circuits 2ed, McGraw Hill 1998 – Fig.8.1.
|AB|
arg(AB)
• The control loop must be stable
• |AB| < 1 at the critical frequency where arg(AB) = π
• In practice, ≥ π/4 (45º) phase margin is needed
• Higher dc gain provides higher accuracy
29
30
Pound-detector transfer function
|A|
resonator
halfwidth
–20
dB
/de
lock-in
c cutoff
modulation
ar
ωc
ry
arg(A)
≈ωL
ra
bit
0º
Ω
ω
–90º
arbitrary
frequency
detection
phase
detection
no
detection
• Quasi-static operation at ω < ωL (resonator half-width)
• Oscillator frequency-noise detection (as discussed)
• At ω > ωL, the resonator reflects the noise sidebands
• Oscillator phase-noise detection at ωL < ω < Ω (integrator)
• The internal lock-in filter rolls off at ω > ωc
• The lock-in amplifier stops working at ω ≈ Ω and beyond
Simple servo-loop design
er
h
hig
pe
o
l
s
Proceed from right to left
• Start from Ω (or ωc) and go leftwards
• Set phase margin ≈ π/4 (45º)
• Design the transfer function
31
Improved servo loop
higher slope
Proceed from right to left
• Resonator –20 dB/decade –> 90º phase lag
• Half integrator 10 dB/decade –> 45º phase lag
• 45º phase margin (to 180º), independent of gain
32
Acousto-optic modulator delay
– Warning –
νn
Q
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
output
2
Laser
detector
vco
ν0
e
phase
modul
1
3
sid
M
O
A
in
fm
control
RF
fm
LO
ve oc ν0 − νn
IF
• Acousto-optic modulators are often used to control the
laser frequency (together with piezo modulators
• The AOM introduces a delay – a few µs typical
• The delay limits the maximum speed of the control
33
34
Resonator stability
Temperature and flicker
Temperature compensation
• Most solids (room temperature)
• dielectric permittivity ε –> coefficient of 5–100 ppm/K
• length –> coefficient of 5–25 ppm/K
• Temperature stability < 10–100 µK challenging / impossible
• A turning point is mandatory for high stability
35
and Ronni Basu
36
Thermal compensation – examples
Department of Physics
hnical University of Denmark
0 Kgs. Lyngby, Denmark
1] and in
tra-stable
oved opox”. With
CSO will
flywheel
such as
d Timing
plications
icrowave
ains, and
hed lowthermo–
r design
08 at its
compenmperature
including
Thermo-mechanical
Previous “40K CSO”
32 GHz VCSO Resonator
Dick & al, Proc IFCS 2005
pact cryoional frequiring a
tz oscillairing only
e thermopreviously
sapphire
nalysis of
100 MHz
n–induced
cryogenic
16 GHz Resonator
Paramagnetic
pure
cry
⌫n
stal
Cr3+
doped
turning
point
Sapphire
Elements
1.72 cm
(0.67”)
Silver
Spacer
Sapphire
Support rod
JPL Sapphire (J.Dick)
Fig. Derived
1. The VCSO
resonator
is based
on a previous
“40K CSO”
from
the old
Lampkin
oscillator
Sapphire Cr3+ impurities @ 6K (V.Giordano / M.Tobar)
Also rutile/sapphire compound @ 80 K (V.Giordano)
Natural – Refraction index
Natural – Thermal expansion
design,
retaining the same silver spacer and sapphire support rod. While overall height
is relatively unchanged, reliefs at the sapphire corners effectively shorten the
electrical height of the resonator, and so increase the tuning rate (as MHz/mm
of gap variation) to raise the turnover temperature to 50K. Use of a higher
azimuthal mode number nφ = 12 instead of the previous nφ = 10 increases
Savchenkov
& al, JOSAB
24(12), 200750% reduction for a doubled
the diameter by ≈20%
compared
to the expected
RF frequency. This increased size was found to be necessary to reduce RF
magnetic fields at the exterior of the (now relatively larger) spacer.
smaller than the sapphire material Q limit at 50 K, but high
enough to allow short–term frequency stability of 1×10−14 or
better. A significant modification is the use of the W GE12,1,1
mode instead of the previous W GE10,1,1 which increases the
overall diameter slightly; this change allowed re-use of the
40K CSO silver spacer design without increasing RF losses
due to the spacer.
MgF2 whispering gallery (A. Savchenkov)
A major challenge for this new technology is to reduce
or eliminate the bright–line phase noise spectra that result
Lyon & al, J Appl Phys 48(3), 1977
Semiconductor-grade Si @ 124 K (PTB)
@ 17 K (In progress)
Kessler & al, ArXiv 1112.3854, 2012
37
In some fortunate cases,
the origin of 1/ƒ frequency noise is known
PHYSICAL REVIEW LETTE
Frequency Noise [Hz/rtHz]
PRL 93, 250602 (2004)
1000
Calculation result (case No.1)
100
Spacer contribution
Mirror substrate contribution
Coating contribution
Experimental result (NIST)
10
1
0.1
0.01
Experimental result (VIRGO)
0.001
0.0001
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
found that therm
width and could
stabilization resu
mirrors and a lar
stability. The re
cavity componen
small CTE, mus
ultrastable caviti
experimentalists
ize the importanc
Frequency [Hz]
Numata K, Kemery A, Camp J, Thermal-noise limit in the frequency
FIG.
2 (color
(case250602,
No. 1)Dec
and2004
the
stabilization
of online).
lasers withCalculation
rigid cavities,result
PRL 93(25)
*Electronic a
experimental results (converted from Refs. [20,22] into fre[1] H. B. Callen
quency noises at 563 nm). In this case, the contribution from
[2] Gravitationa
mirrorU.substrate
is dominant,
it is made
of ULE.
The 29(1), 2012
T. Kessler, the
T. Legero,
Sterr, Thermal
noise inbecause
optical cavities
revisited,
JOSA-B
M.-K. Fujim
1/ƒ noise – structural damping
38
1/ƒ noise and FD theorem
A. Holden A, The nature of solids, Dover 1965
Flicker (1/ƒ) dimensional fluctuation is powered by thermal energy
Debye-Einstein theory
for heath capacity
A single theory explains
• Heath capacity
• Elasticity
• Thermal expansion
… and its fluctuations
Fluctuation Dissipation
theorem in a nutshell
S(ƒ)
kT
B
B
P = kTB
thermal bath
ƒ
damping
Thermal equilibrium applies
to all portions of spectrum
39
40
Thermal 1/f from the FD theorem
Look at one vibration mode
• Structural dissipation occurs
at chemical-bond scale
• Virtually instantaneous
• Thermal equilibrium
• P = kT in 1 Hz BW
• x2 ~ 1/ƒ –> flicker
Damping force in phase with x (not with ẋ) –> equivalent to imaginary spring constant
41
Optimization Issues
42
Critical coupling (g = 1)
• Maximum gain.
Immediately seen on Im{Γ}
g 1
'
g+1
4Q0 g
!
i
(g + 1)2 !n
close to !n
• Lowest “useless” power in
the quadratic detector.
Immediately seen on Re{Γ}
J0
Re(V + )
incident
Im(V + )
–J1
ω0–Ω
ω0
ω0+Ω
Im(Γ)
Re(Γ)
• The frequency error due to
residual AM vanishes
Some maths – not shown
J1
Re(V – )
reflected
Im(V – )
ωn
J1
≈0
–J1
≈0
≈0
ω0–Ω
J0 Im(Γ(ω))
ω0
ω0–ωn
ω0+Ω
Detector responsivity
1000
be
500
st
output voltage, mV
200
too low P
saturated
100
linear region
(envelope detector)
100 kΩ
50
100 Ω
20
10
1 kΩ
5
linear region
(power detector)
2
1
−30
−20
−10
0
10
input power, dBm
(a+b+c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
• The error signal comes from the 2ac + 2bc terms
• Highest sensitivity just below the corner
43
44
Maximum power in the resonator
• Dissipated P –> Thermal instability (obvious)
• Traveling P
–> Instability
• Dielectric constant
• Radiation-pressure
Chang & al., …radiation pressure effect…, PRL 79(11) 1997
• Difficult to lock (ωn runaway)
• Control instability and failure
• “Maximum P” applies to the carrier, not to sidebands
• The carrier gets in the resonator, the sidebands are reflected
• Look carefully at the resonator physics
• Loss and dissipation are not the same thing
Modulation index
• The sidebands are reflected
• High modulation index –> high sideband power
• Higher gain without increasing P inside the
resonator
• Effect of higher-order sidebands (±2Ω, ±3Ω, etc.)
• Not documented – though conceptually simple
• DSB modulation, instead of true PM
• A pair of sidebands is simpler than true PM
• Modulator 1/ƒ noise?
45
Modulation frequency
Lower bound for Ω
• Total reflection at ωn ± Ω is necessary
• Thus, Ω >> B/2π,
B = resonator bandwidth
Why to choose the largest possible Ω
• Larger control bandwidth
• Higher dc gain –> higher stability
Why not to choose the largest possible Ω
• Avoid dispersion (PM –> AM conversion)
• Technical issues / Design issues
My experience – at Femto-ST
• 95–99 kHz for the sapphire oscillators (10 GHz, B=10 Hz)
• 22 MHz for the optical FP (193 THz, B ≈ 30 kHz)
46
Residual AM
47
• Residual AM yields a detected signal at the
modulation frequency Ω
• Generally poorer operation
• Frequency error –> ω0 ≠ ωn at the null point
• Frequency fluctuation if the AM fluctuates
Basu R, Wang R, Dick GJ, Novel design of an all-cryogenic RF Pound circuit, Proc IEEE IFCS 2005
Marra G & al, Reduction of residual amplitude modulation …, Proc. EFTF 2004
48
Removing the residual AM
50Ω
• Additional detector
enables nulling the
AM in closed loop
40 kHz Modulation
Signal + dc Bias
Tunnel Diode
Modulator
50Ω
• Reversed, is used as
a variable stub
AM Null
Output
6dB coupler
16 GHz
• The power detector
is reversible
Tunnel Diode
Detectors
φ
Phase
Shifter
Pound
Signal
Output
3dB
couplers
Q
Sapphire
Resonator
Loaded Q of
60 million
Fig. 8.
The new cryogenic Pound circuit. The tunnel diode modu
is DC biased and the phase shifter adjusted so that the upper diode det
0.3
0.45
gives minimal output at the same time as having high power in
Basu R,
design ofsidebands.
an all-cryogenic
Poundincircuit,
Proc IEEE IFCS
2005
oss diode
/ Wang
[V] R, Dick GJ, Novel
modulation
TheRF
power
the sidebands
is monitored
throu
weakly coupled port not shown in the diagram inserted immediately t
49
Filter the detector output
|V0 |2
P =
2R0
(
J12 (m)
dc terms

1 2
g 1
+ J0 (m)
2
g+1
2

1 2
4Q0 !
+ J0 (m)
2
g + 1 !n
2
)
+
|V0 |2 2
|V0 |2
4Q0 !
J1 (m) cos(2⌦t) +
2J0 (m)J1 (m)
sin(⌦t)
2R0
2R0
g + 1 !n
diagnostic
error signal
beat –> 2Ω
Re(V – )
Large, 2Ω
Small, Ω
J1
reflected
Im(V – )
–J1
≈0
ω–Ω
beat –> Ω
ω
ω+Ω
beat –> Ω
Separating the Ω and 2Ω signals with passive filter helps
in getting clean, simple and effective electronics
More optimization issues
• Given the laser power –> best modulation
index (Eric Black)
• Detector saturation power –> best modulation
scheme
• Resonator max power –> best modulation
• Quadrature modulation (µwaves) – does it
really make sense?
50
51
Applications
Resonators and oscillators
52
Microwaves
Whispering gallery resonator
Geometrical optics interpretation
53
Full reflexion
Confinement inside the dielectric
Electromag. fields
H
E
WGH mode
Temperature compensation
Compensation exploiting impurities, ≈6 K
two ideas tested above 30K
E
g
10 !10
Sapphire
Copper
"y (#)
Spring
a)
Rutile
Sapphire
10 !11
b)
10 !12
10 !13
1
10
100
1000
Integration time # (s)
54
Temperature compensation
paramagnetic impurities: Fe3+ Cr3+, Mo3+, Ti3+
Pure Sapphire
Sapphire +
paramag.
ions
55
56
Elisa, before going to Argentina
2-inch sapphire
monocrystal
57
Uliss
ULISS Cryogenic Sapphire Oscillator
Crycooler
Electronics Control
Ultra−stable
and autonomous
frequency reference
(CSO)
5 MHz
Frequency
synthesis
100 MHz
He Compressor
Synthesis
10 GHz
Monitoring
&
Control
PCO
CSO freq.
PCO
PC
100 MHz
Input
Relative frequency stability
ADEV measurement ELISA/ULISS
10−12
3 days measurement without post-processing
Perturbed environment:
- Technical university (ENSMM), ≥ 800 students
- Air conditioning still not operational during
measurements
10−13
10−14
σΛ(τ) 2 units
ELISA
10−15
σy (τ) 1 unit
flicker floor: 4x10–16 10 s < τ < 1,000 s
10−16
1
10
100
1000
10000
Integration Time (s)
3 hours extracted from the entire data set
- Quiet environment, nighttime
- Take away 3dB for two equal units
-Λ-counter compensated: for flicker: σΛ(τ)≃ 1.3xσy(τ)
100000
Flory-Taber Bragg resonator
FLORY AND TABER: DBR MICROWAVE RESONATOR
487
FLORY AND TABER: DBR MICROWAVE RESONATOR
Fig. 1. The dielectric cylinders and plates comprising the DBR structure.
dimensional factorization at this point, and derive simple resonator design rules by analyzing electromagnetic
wave scattering in the radial and axial dimensions independently. By determining the scattering amplitudes and
phase shifts from elements of the full DBR structure, the
resonator dimensions can be easily and accurately specified for a given desired frequency.
A. Scattering from a Dielectric Cylinder
To analyze the physical properties of the dielectric cylinders which make up the radial part of the DBR structure,
scattering of cylindrical electromagnetic waves from a single (infinitely) long dielectric cylinder can be calculated.
This ignores any z-dependence of the structure, and thus
interaction between the radial and axial DBR elements,
and will be justified in a following section. The scattering
to be
is
in Fig. 3’ where the
reflection amplitude and phase will be determined for a
cylindrical wave emanating from the origin at p = 0.
For a source-free region and assuming all fields have a
harmonic time dependence of the form e-Zwt, Maxwell’s
Fig. 2. Top and side cut-away views of the dielectric DBR. structure
inside the metal enclosure.
Fig. 1. The dielectric cylinders and plates comprising the DBR structure.
6.5×105
Fig. 1. The dielectric cylinders and plates comprising the DBR structure.
• Measured Q =
at 9 GHz, and 4.5×105 at 13.2 GHz
• Oscillator stability and noise not reported (yet)
A.
A. • Project dropped
dimensional factorization at this point, and derive simple resonator design rules by analyzing electromagnetic
wave scattering in the radial and axial dimensions independently. By determining the scattering amplitudes and
phase shifts from elements of the full DBR structure, the
resonator dimensions can be easily and accurately specified for a given desired frequency.
Scattering from a Dielectric Cylinder
To analyze the physical properties of the dielectric cylinders which make up the radial part of the DBR structure,
scattering of cylindrical electromagnetic waves from a single (infinitely) long dielectric cylinder can be calculated.
dimensional factorization at this point, and derive simple resonator design rules by analyzing electromagnetic
wave scattering in the radial and axial dimensions independently. By determining the scattering amplitudes and
phase shifts from elements of the full DBR structure, the
resonator dimensions can be easily and accurately specified for a given desired frequency.
Scattering from a Dielectric Cylinder
To analyze the physical properties of the dielectric cylinders which make up the radial part of the DBR structure,
scattering of cylindrical electromagnetic waves from a single (infinitely) long dielectric cylinder can be calculated.
Flory CA, Taber RC, IEEE T UFFC 44(2), March 1997
487
58
V.
CURRENT RESULTS
59
The Bale-Everard Aperiodic Bragg resonator
Using the model and simulation results described in the
previous section a cylindrical Bragg resonator has been
onstructed which utilizes an aperiodic arrangement of
Alumina plates.
Figure 5. A waveguide ABCD parameter model for one half of a six plate Bragg resonator. The air and dielectric plate thicknesses of this structure are
optimised to maximise the magnitude of the input reflection coefficient (S11).
IV.
A-PERIODIC RESONATOR SIMULATION
Breeze, Krupka and Alford[3] have demonstrated that a
significant improvement in the quality factor of a Bragg
resonator can be achieved by utilizing an aperiodic plate
arrangement to re-distribute the energy inside the cavity into
the lower loss air sections. In order to ascertain the reflector
thicknesses required for maximum Q in our resonator a
numerical optimization procedure was adopted.
Incross
this new
model
halfaperiodic
of the Bragg
Figure 6. A
section
viewonly
of theone
6 plate
Braggresonator
resonator is
The resonator has been designed to operate using the TE011
low loss mode with a resonant frequency of 10 GHz. A cross
sectional view of the cavity is shown in fig 6. This initial
design consists of six dielectric plates mounted in an
Aluminum shield. Wire loop probes were used to couple
energy into the cavity. Fig. 7 shows a photo of the central air
filled resonant section and the wire loop coupling probes.
Suitable to Pound lock
Figure 8. A plot of the forward transmission coefficient (S21) for the 6
aperiodic Bragg resonator. A frequency span of 1 GHz is shown.
considered. The structure to be simulated is illustrated in fig.
5. The value of port impedance, Z1, was set equal to the wave
impedance inside an air section. This ensures that we are 235
5
representing a wave travelling
from the central air region
towards a dielectric plate. The value of terminating
due to the small impedance to
impedance, Z2, is less critical
5
ground presented by the end wall. However, its value must be
large enough to avoid reducing the end wall impedance.
• 6-plates 10 GHz resonator
• Q > 3×10 (simulated)
• Q ≈ 2×10 (measured)
The air and dielectricstability
section thicknesses
were initially
set
•to the
Oscillator
and
noise
values used in the periodic design as shown in fig. 3.
Thenot
reflector
section lengths yet
were then optimized until the
reported
magnitude of the input reflection coefficient at port one (S )
11
reached a maximum. When an optimal set of plate thicknesses
was
found two half resonators were combined and then the
Bale S, Everard J - High Q X-band distributed Bragg resonator utilising an aperiodic
plateBragg
arrangement
- Procair
IFCS-EFTF
2009
Figure 7.alumina
A-periodic
resonator central
filled section.
central air section was re-introduced. The final
stage
of
the
Everard J, Proc. IFCS-EFTF 2015
optimization required that length of the central resonant region
A plot of the forward transmission coefficient scattering
+Im[S21 ( f c − f m )])},
(3)
60
where ϵ issuperconducting
the error signal, and k is a constant resonator
that depends on
Small
the conversion efficiencies of the diode and the demodulator.
The net effect of the Pound-loop is that whenever the carSuperconducting
resonator
(NPL,
UK)
rier is off resonance, the transmitted signal is phase shifted
Nb on Al2O3, 300x300 µm2. 7.5 GHz, Q = 5E4,
Lindstrom, Oxborrow & al, Rev Sci Instrum 82, 104706 (2011)
FIG. 2. (Color online) Micrograph and equivalent circuit for one of our
lumped element resonators. The electromagnetic radiation is primarily induc-
61
Optics
Stabilization of the FS comb
The FS comb enables frequency
synthesis from RF/µwaves to optics
• Major breakthrough
• 2005 Nobel prize, R.Glauber, J.Hall,
T.Haensch
Stability and noise
• Low noise in the sub-millisecond
region
• Drift and walk
• Need stabilization
Common practice
• CW laser stabilized to a FP etalon
• PDH control – of course
• Compare/stabilize the FS comb to the
CW laser
Featured book
62
Fabry Perot cavity
e
(al
spacer
Bragg
layer
-
“F=z
C
Optical transmission
mirror
mirror
Bragg
layer
63
bonding
plane
bonding
plane
optical
plane
Ls
Lp < 0
Figure 9.1-6 (a) A lossless
resonator(9 = a) in the steadystate can sustain light waves only
optical
planethe precise resonance frequencies vq. (b) A lossy resonator sustains waves at all frequencies, b
the attenuation resulting from destructive interference increases at frequencies away from t
Lc > 0
resonances.
Lo = Ls+Lp+Lc
optical path
• Smart design of the spacer provides
• Low sensitivity to acceleration
• Temperature compensation
• ULE and Zerodur
• Many materials (Si, Ge, …) have natural turning point
• High Q is possible, ≥ 1010 (≈10 kHz optical bandwidth)
spacer and the optically bonded mirror substrates are
made of ultralow expansion glass (ULE). To maintain
small sensitivity of cavity length to acceleration, we
implemented the following design features. First, because the acceleration sensitivity of fractional cavity
March 15, 2007 / Vol. 32, No. 6 / OPTICS LETTERS
641
length scales with the cavity length, we chose a some10
!7 cm".
Second,
the cavity
what
short
cavity
spacer
Compact, thermal-noise-limited
optical
cavity
for
is held
at itslaser
midplane
to achieve
diode
stabilization
at symmetric
1 Ã 10−15 stretching and compressing of the two halves of the cavity
A. D. Ludlow, X. Huang,* M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye
during
acceleration
to
suppress
vibration
JILA, National Institute10–12
of Standards and Technology, and University of Colorado Department of Physics,
Colorado,cavity
Boulder, Colorado
USA
is 80309-0440,
mounted
vertically to
sensitivity.University ofThe
Received October 30, 2006; accepted November 25, 2006;
exploit the
intuitive symmetry in the vertical direcposted December 20, 2006 (Doc. ID 76598); published February 15, 2007
Wetion
demonstrate
phase
and frequency
stabilization
ofmounting
a diode laser at the is
thermal
noise limit of a passive by
[see
Fig.
1(b)].
This
accomplished
optical cavity. The system is compact and exploits a cavity design that reduces vibration sensitivity. The
subhertz
laser is characterized by comparison
with a secondmidplane
independent system with
similar
fractional fre- on
a monolithically
attached
ring
resting
quency stability !1 ! 10 at 1 s". The laser is further characterized by resolving a 2 Hz wide, ultranarrow
three
Teflon
rods.
Third,
theSociety
cavity
© 2007 Optical
of Americais wider in the
clock transition
in ultracold
strontium.
diode laser optical
OCIS codes: 140.2020, 030.1640, 300.6320.
xternal cavg at 698 nm
Highly frequency-stabilized lasers are essential in
5 MHz. Approximately 10 "W of optical power is into a simple
high-resolution
spectroscopy
and
quantum
cident on the ultrastable cavity for PDH locking. Sta1
2–4
and quanmeasurements, optical atomic clocks,
bilization to this cavity is accomplished via feedback
he PDH sta5
tum information science. To achieve superior stabilto the AOM and a PZT controlling the prestabilization cavity length. The servo bandwidth for this final
ity with high bandwidth control, the laser output is
to the laser
traditionally servo locked to a highly isolated passive
locking stage is #100 kHz. The useful output of the
optical cavity by using the Pound–Drever–Hall
ectric translaser is delivered from a fiber port located near the
6
ultrastable cavity on the same platform. The entire
(PDH) technique. Typical limits to the resulting la3 MHz. The
optical setup occupies less than 1 m3.
ser stability include vibration-induced cavity length
The ultrastable cavity has a finesse of 250,000 and
photon detection shot noise, and
racted byfluctuation,
an
is 7 cm long (cavity linewidth of #7 kHz). Both the
thermal-mechanical noise of the passive cavity
7,8
spacer and the optically bonded mirror substrates are
By thoughtful system design, the relacomponents.
coupled to
a
made of ultralow expansion glass (ULE). To maintain
tive impact of these noise contributions can be adsmall sensitivity of cavity length to acceleration, we
esides. This
justed. Here we report a cavity-laser system that
implemented the following design features. First, beachieves high stability with an appropriate comprovailable pascause the acceleration sensitivity of fractional cavity
mise between acceleration sensitivity and thermal
length scales with the cavity length, we chose a somenoise. The compact, cavity-stabilized diode laser has
esonant fre-
gn, the relacan be adsystem that
ate comprond thermal
de laser has
and is therbility of #1
further eviuse it to inransition in
optical tranwest optical
The JILA bicone spacer
−15
64
65
NIST spherical spacer
Field-test of a robust, portable,
frequency-stable laser
van
(a)
(b)
y
x
David R. Leibrandt,* Michael J. Thorpe, James C. Bergquist, and
Till Rosenband
electronics
National Institute of Standards and Technology, 325 Broadway Street, Boulder, Colorado
80305, USA
*[email protected]
FPGA
ADC
60 cm
I/Q
board
optics
Abstract:
We operate a frequency-stable laser in a non-laboratory
environment where the test platform is a passenger vehicle. We measure the
acceleration experienced by the laser and actively correct for it to achieve a
system acceleration sensitivity of ∆ f / f = 11(2) × 10−12 /g, 6(2) × 10−12 /g,
and 4(1) × 10−12 /g for accelerations in three orthogonal directions at 1 Hz.
The acceleration spectrum and laser performance are evaluated with the
vehicle both stationary and moving. The laser linewidth in the stationary
vehicle with engine idling is 1.7(1) Hz.
accel.
PD
fiber w/ noise
cancellation
(c)
AOM
45 cm
PDH
1070 nm
fiber laser
ultra-stable
1070 nm laser
from lab
60 cm
Fig. 1. Experimental setup. (a) Schematic of the test platform. A cavity-stabilized laser and
the associated electronics are housed in a passenger vehicle in order to evaluate the performance of the laser in a field environment. The phase of the test laser is measured with respect to the phase of a laboratory-based reference laser via a fiber-optic link. (b) Photograph
of the test platform showing the optical breadboard inside the thermal enclosure (open) in
the back of the vehicle. (c) Mounting geometry of the accelerometers (drawn as arrows)
used for acceleration feed-forward. The cube is centered on the reference cavity. FPGA:
field-programmable gate array, ADC: analog-to-digital converter, DDS: direct-digital synthesizer, I/Q: in-phase/quadrature, PD: photodiode, AOM: acousto-optic modulator, PDH:
Pound-Drever-Hall detector.
OCIS codes: (140.3425) Laser stabilization; (140.4780) Optical resonators; (120.7280) Vibration analysis.
References and links
1. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev.
Lett. 82, 3799–3802 (1999).
2. C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two
high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
3. B. Willke, K. Danzmann, M. Frede, P. King, D. Kracht, P. Kwee, O. Puncken, R. L. Savage, B. Schulz, F. Seifert,
C. Veltkamp, S. Wagner, P. Weßels, and L. Winkelmann, “Stabilized lasers for advanced gravitational wave
detectors,” Class. Quantum Grav. 25, 114040 (2008).
4. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger,
T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and
J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,”
Science 319, 1808–1812 (2008).
5. S. Blatt, A. D. Ludlow, G. K. Campbell, J. W. Thomsen, T. Zelevinsky, M. M. Boyd, J. Ye, X. Baillard, M. Fouché,
R. L. Targat, A. Brusch, P. Lemonde, M. Takamoto, F.-L. Hong, H. Katori, and V. V. Flambaum, “New limits
on coupling of fundamental constants to gravity using 87 Sr optical lattice clocks,” Phys. Rev. Lett. 100, 140801
(2008).
6. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye,
“Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1 × 10−15 ,” Opt. Lett. 32, 641–643
(2007).
7. S. Vogt, C. Lisdat, T. Legero, U. Sterr, I. Ernsting, A. Nevsky, and S. Schiller, “Demonstration of a transportable
1 Hz-linewidth laser,” arXiv:1010.2685 (2010).
8. S. A. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
9. J. Millo, D. V. Magalhães, C. Mandache, Y. L. Coq, E. M. L. English, P. G. Westergaard, J. Lodewyck, S. Bize,
P. Lemonde, and G. Santarelli, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79,
053829 (2009).
10. D. Kleppner, “Time too good to be true,” Phys. Today 59, 10–11, March (2006).
DDS
ADC
with respect to the optical axis reduces coupling of acceleration-induced squeeze forces to the
cavity length. Acceleration sensitivity is further reduced by measuring the acceleration of the
cavity and applying real-time feed-forward corrections to the laser frequency [17]. We test the
laser both when the vehicle is stationary and when it is moving with peak accelerations greater
than 0.1 g.
This paper proceeds as follows: Section 2 describes the test setup, including the real-time acceleration feed-forward scheme. Section 3 discusses the acceleration spectrum and laser performance when the vehicle is stationary as well as the acceleration sensitivity of the laser. Section
4 discusses the laser performance when the vehicle is moving. Finally, Section 5 summarizes
and concludes.
2.
Test setup
The setup consists of an optical breadboard, electronics for locking the laser and applying the
acceleration feed-forward, and a computer for data acquisition, all of which are housed in a
66
NIST improved spherical spacer
PHYSICAL REVIEW A 87, 023829 (2013)
LEIBRANDT, BERGQUIST, AND ROSENBAND
Flexure
spring
Invar rod
1070 nm
fiber laser
FPGA
Frequency PDH
DET
servo
VOA
(a)
Optical
axis
Mounting
axis
EOM
OI
Cavity
PBS λ/4
TX
THMS DET
PDH AOM
FPGA
FF AOM
ACC GYRO
Frequency
FS mirrors
reference
and
ULE spacer
environBN DET sensitivity below
Cavity-stabilized laser with acceleration
10−12 g−1
mental
OI*
sensors
ADC
PHYSICAL REVIEW A 87, 023829 (2013)
David R. Leibrandt, James C. Bergquist, and Till Rosenband
Torlon
ball
(b)
National Institute of Standards and Technology, 325 Broadway Street, Boulder, Colorado 80305, USA
(Received 31 December 2012; published 21 February 2013)
We characterize the frequency sensitivity of a cavity-stabilized laser to inertial forces and temperature
Input from
fluctuations, and perform real-time feedforward to correct for these sources of noise. We measure the sensitivity
independent
of the cavity to linear accelerations, rotational
accelerations,
1070
nm laserand rotational velocities by rotating it about three
axes with accelerometers and gyroscopes positioned around the cavity. The worst-direction linear acceleration
Phase
noise over
measurement
g−1 measured
0–50 Hz, which is reduced by a factor of 50 to below
sensitivity of the cavity is 2(1) × 10−11
−12 −1
g for low-frequency accelerations by real-time feedforward corrections of all of the aforementioned
10
inertial forces. A similarFIG.
idea is2.
demonstrated
in which Schematic
laser frequencyofdrift
duefiber-optic
to temperaturefrequency
fluctuations is
(Color online)
the
reduced by a factor of
70 via
real-time
feedforward
temperature
sensor
located
on the
wall of the
servo
and
feedforward.
Afrom
lasera is
stabilized
to the
length
of outer
a spherical
cavity vacuum chamber.
Fabry-Pérot cavity by the PDH method. Environmental sensors detect
DOI:
10.1103/PhysRevA.87.023829
PACS number(s):
42.62.Eh,
42.60.Da,
46.40.−f,of
07.07.Tw
inertial forces and temperature
drift that
perturb
the length
the
FIG. 1. (Color online) (a) Computer-aided design (CAD)
drawing
cavity; the resulting frequency perturbations of the laser are corrected
of the mounted Fabry-Pérot cavity. The spherical cavity spacer is
is stabilized to a passive optical cavity and, given a sufficiently high fidelity of control, the frequency is defined by
The geometry is shown in Fig. 1. Square “cutouts” are
the cavity. Thus, the dimensional stability of the cavity bemade to the underside of a cylindrical spacer and the cavity
comes paramount. Accelerations acting upon the cavity
is supported at four points. The cutouts compensate for vercause it to distort and typically the sensitivity is
tical forces. Vibration insensitivity in the horizontal plane is
#100 kHz/ ms−2. With quiet conditions and vibration isolation, subhertz laser linewidths can be achieved $1–4%. Howa)
x
ever, accelerations remain the dominant source of instability
c
on the 0.1– 10 s time scale. If one is ever to envisage such an
apparatus leaving the confines of a specialized laboratory, the
extreme sensitivity to vibrations must be overcome. Also, the
availability of ultranarrow atomic references $5%, whereRAPID
the COMMUNICATIONS
z
observed Q is determined by the laser linewidth, motivates
PHYSICAL REVIEW A 75, 011801!R" !2007"
the pursuit of even greater stability. One could try to isolate
the cavity further from vibrations, however, given the range
Vibration
insensitive
optical!!10
cavity
of contributing
frequencies
Hz", the difficulty and expense will far outweigh the return in performance. A better
S. A. Webster, M. Oxborrow, and P. Gill
by far
is to reduce
the sensitivity
of theKingdom
cavity to
National Physical Laboratory, approach
Hampton Road,
Teddington,
Middlesex,
TW11 0LW, United
vibrations,
this9 route
already
met with some
"
!Received
31 Octoberand
2006;indeed
published
Januaryhas
2007
success.
An optical cavity is designed and implemented
thatcavity
is insensitive
to vibration
in all directions.
Therefleccavity is
An optical
typically
comprises
two highly
−2
mounted with its optical axis in the horizontal plane. A minimum response of 0.1 !3.7" kHz/ ms is achieved
tive concave mirrors bonded to a spacer with a central bore
for low-frequency vertical !horizontal" vibrations.
and has axial symmetry. The optical mode coincides with the
b)
a small
diameter
at the mirror
surface
DOI: 10.1103/PhysRevA.75.011801symmetry axis and has PACS
number!s":
42.60.Da,
07.60.Ly,
06.30.Ft
relative to the cavity dimensions. To achieve vibration insenA
B
sitivity, one has to ensure that the distance between the two
I. INTRODUCTION
achieved
by removing
material
from
the underside of the
points at the centers
of the mirror
surfaces
remains
invariant
cavity and
using
this approach
we report
vertical accelerawhen a force is applied
to the
support.
This may
simplya be
−2
Ultrastable lasers are a key element in optical frequency
sensitivity !0.1
kHz/ ms
.
achieved through tion
symmetrical
mounting:
a force
applied
standards where they are used both to probe an atomic tranthrough
the mount will cause one half of the cavity to conWEBSTER
al. oscillator
PHYSICAL REVIEW A 77, 033847 !2008"
sition and to provide a measurable
output. et
A laser
FIG.
1.
!a"
Cutout
cavity on mount. The cavity is represented by
DESIGN
while
the other half expands with theII.result
that there is
is stabilized to a passive optical cavity and,tract
given
a suffithe light gray shaded area. The support “yokes” are shown in white.
noisnet
change
in dimension.
An
obvious
implementation
of
ciently high fidelity of control, the frequency
defined
byservo
The geometry is shown in Fig. 1. Square “cutouts”
Spacer are
length, 99.8 mm; diameter,
heatsink60 mm; axial bore diameter,
toREVIEW
hold
its plane of
of asymmetry
its and 21.5
the cavity. Thus, the dimensional stabilityPHYSICAL
ofthis
the iscavity
be-the
made
toabout
the
underside
cylindricalwith
spacer
the cavity
A cavity
77,
033847
!2008"
mm; vent-hole diameter 4 mm. The mirrors are 31 mm in di1
axis the
aligned
with gravity
and,cavity
using
approach,
a reduced
comes paramount. Accelerations acting upon
cavity
is EOM
supported
at fourthis
points.
The cutouts
compensate
for
ameterverand 8 mm thick. The parameters relevant to the design, the
−2
Nd:YAG
has
been
demonacceleration
of
10
kHz/
ms
cause it to distort and typically the
sensitivity issensitivity
tical
forces.
Vibration
insensitivity
in
the
horizontal
plane
is c and the support coordinates x and z are indicated. !b"
cut
depth
Thermal-noise-limited
optical cavity
vibration$6%.
isola#100 kHz/ ms−2. With quiet conditions and strated
A cavity mounted with its axis horizontal sags
Details of the support point
in cross
section. The black shaded areas
vacuum
chamber
tion, subhertz laser linewidths can be achieved
$1–4%. How-1 under
a)
PD
its
weight.
However,
through
op1 asymmetrically
2 own
3
1
represent rubber tubes and spheres. Case A: Diamond stylus set into
A. Webster,
M.ofOxborrow,
S. Pugla, J.x Millo, and PC
P. Gill
ever, accelerations
remain theS.dominant
source
instability
Counter
1
Nd:YAG
timization
of
the
support
positions,
the
distance
between
the
c
a cylindrical ceramic mount,
cushioned
on all sides by a rubber
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom
aluminum
cylinder
on the 0.1–2 10 s time scale. If one is ever to envisage
such
anmirrors can be made invariant, even though the
centers
of
the
Blackett Laboratory, Imperial College London, South Kensington Campus, London, SW7 2BZ, United Kingdom tube, and from below by a rubber sphere. The stylus digs into the
Spectrum
apparatus leaving the confines
of a specialized laboratory,
the
T
3
f-V
deforms
on application
of a Paris,
vertical
force $7% and,
SYRTE, Observatoire decavity
Paris, still
61, Avenue
de l’Observatoire,
75014,
France
underside of the cavity,2 and can be considered to be in rigid contact
analyzer
extreme sensitivity to vibrations must be overcome. Also, the
T1 sphere recessed into a cylindri!Receivedby
31 October
2007; published
27 Marchacceleration
2008"
this method,
a vertical
sensitivity of
with it. Case B: 3-mm-diam rubber
availability of ultranarrow atomic references $5%, where the
oscilloscope
$8%. A zsimilar result is
1.5 kHz/
ms−2 has been demonstrated
cal hole in the yoke.
observed Q isAdetermined
bycavities
the laser
pair of optical
are linewidth,
designed andmotivates
set up so as to be insensitive to both temperature fluctuations and
NPL horizontal cavity
the pursuit of
even greater
stability.
could tryoftothese
isolate
mechanical
vibrations.
WithOne
the influence
perturbations removed, a fundamental limit to the frequency
cavity 2
1050-2947/2007/75!1"/011801!4"
011801-1
the cavity further
vibrations,
however,
given
the
stabilityfrom
of the
optical cavity
is revealed.
The range
stability of a laser locked to the cavity reaches a floor !2
EOM
−15
of contributing
!!10times
Hz",inthe
for averaging
the difficulty
range 0.5−and
100 exs. This limit is attributed to Brownian motion of the mirror
" 10frequencies
substrates
and the
coatings.
pense will far
outweigh
return in performance. A better
servo
approach by far is to reduce the sensitivity of the cavity to
DOI:indeed
10.1103/PhysRevA.77.033847
vibrations, and
this route has already met with somePACS number!s": 42.60.Da, 07.60.Ly, 07.10.Fq, 06.30.Ft
FIG. 1. Experimental scheme for measuring the frequency difsuccess.
cavityThe American Physical Society
©2007
Peltier
67
3572
NPL small cubic cavity
OPTICS LETTERS / Vol. 36, No. 18 / September 15, 2011
68
September 15, 2011 /
Force-insensitive optical cavity
Stephen Webster* and Patrick Gill
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK
*Corresponding author: [email protected]
Received June 20, 2011; revised August 11, 2011; accepted August 11, 2011;
posted August 12, 2011 (Doc. ID 149376); published September 9, 2011
We describe a rigidly mounted optical cavity that is insensitive to inertial forces acting in any direction and to the
compressive force used to constrain it. The design is based on a cubic geometry with four supports placed symmetrically about the optical axis in a tetrahedral configuration. To measure the inertial force sensitivity, a laser is locked
to the cavity while it is inverted about three orthogonal axes. The maximum acceleration sensitivity is 2:5 × 10−11 =g
(where g ¼ 9:81 ms−2 ), the lowest passive sensitivity to be reported for an optical cavity. © 2011 Optical Society of
America
OCIS codes: 140.4780, 140.3425, 120.3940, 120.6085.
September 15, 2011 / Vol. 36, No. 18 / OPTICS LETTERS
Optical cavities find wide application in atomic, optical
and laser physics [1], telecommunications, astronomy,
gravitational wave detection [2], and in tests of fundamental physics [3] where they are used as laser resonators, for power buildup, filtering, spectroscopy, and laser
stabilization. In metrology, they are essential to the operation of optical atomic clocks, both enabling highresolution spectroscopy and providing stability in the
short-term during interrogation of the atomic transition
[4]. In this case, a laser is stabilized to a mode of a passive
optical cavity consisting of two highly reflective mirrors
contacted to a glass spacer. The frequency of the laser
is defined by the optical length of the cavity; thus, for
spectral purity, perturbations to this length must be minimized. To this end, the cavity is both highly isolated from
3573
1:1=0:6=0:4 × 10−11 =g [15]. Here, we present an alternative design based on a cubic geometry with a four-point
tetrahedral support. The cavity is also insensitive to the
forces used to support it and has a maximum passive
acceleration sensitivity of 2:5 × 10−11 =g.
For the cavity to be insensitive to inertial forces, there
must be symmetry both in the geometry of the cavity and
its mount and in the forces acting at the supports. This
ideal must be combined with the requirement for the
cavity to be constrained in all degrees of freedom. Four
identical supports in a tetrahedral arrangement constitute a sufficient and symmetric constraint for a rigid body
in three dimensions and their symmetrical arrangement
Fig. 2.
with respect to an axis results in a cubic geometry.
The(Color o
which a compress
geometry is shown in Fig. 1. The optical axisports.
passes
(a) Fractio
ms and single-ion
d new approaches
avities which are
sitivity than previous vertical cavity designs #13$. Moreover,
a significant improvement of the thermal noise level is demonstrated here #13,16$ by using fused silica mirror substrates,
which minimize the contribution to thermal noise due to the
PHYSICAL
REVIEW
053829
!2009"
higher mechanical
Q factor
ofA 79,
this
material
in comparison to
Ultra Low
Expansion glass !Corning ULE".
Ultrastable lasers based on vibration insensitive cavities
SYRTE horizontal cavity
eutral atoms, the
able clock lasers
y stability via the
the best reportedJ. Millo, D. V. Magalhães, C. Mandache, Y. Le Coq, E. M. L. English,* P. G. Westergaard, J. Lodewyck,
S. Bize, P. Lemonde, and G. Santarelli
magnitude larger
LNE-SYRTE,
Observatoire ELEMENT
de Paris, CNRS, UPMC,
61 Avenue de l’Observatoire,
75014
Paris, France
II. FINITE
MODELING
OF THE
CAVITY
!Received 5 February 2009; published 18 May 2009"
ocks #2$. Improvpresent two ultrastable lasers A.
basedGeneral
on two vibration
insensitive cavity designs, one with vertical optical
considerations
a prerequisite for axisWegeometry,
the other horizontal. Ultrastable cavities are constructed with fused silica mirror substrates,
shown to decrease the thermal noise limit, in order to improve the frequency stability over previous designs.
Vibration sensitivity components measured are equal to or better than 1.5! 10−11 / m s−2 for each spatial
direction, which shows significant improvement over previous studies. We have tested the very low dependence on the position of the cavity support points, in order to establish that our designs eliminate the need for
fine tuning to achieve extremely low vibration sensitivity. Relative frequency measurements show that at least
one of the stabilized lasers has a stability better than 5.6! 10−16 at 1 s, which is the best result obtained for this
length of cavity.
This analysis is restricted to the quasistatic response of
equency noise of
cavities as mechanical resonances are in the 10 kHz range,
effects of residual
while only low frequencies of "100 Hz are of interest in the
minimize the noise
present experiment for application to optical atomic clocks.
generally not sufFurthermore, with commercial compact
isolation systems the
PACS number!s": 42.60.Da, 07.60.Ly, 42.62.Fi
One way to im- DOI: 10.1103/PhysRevA.79.053829
ed lasers is to rez the effect of mirror tilt have been anaI. INTRODUCTION z
support points and
igning the cavity
lyzed. The constructed cavities have then been subjected to
Ultrastable laser light is a key element for a variety of
an extensive study of the vibration response. Both cavity
ps have proposed
applications ranging from optical frequency standards #1,2$,
types exhibit extremely low vibration sensitivity. Sensitivitests of relativity #3$, generation
Zc of low-phase-noise micro- ties are equivalent to previous horizontal cavity designs
cavities #11–14$.
wave signals #4$, and transfer of optical stable frequencies by
but with strongly reduced dependence on support
n thermal noise
in #5,6$, to gravitational wave detection #7–9$. #12,14$
fiber networks
points’ position. The vertical cavity shows a much lower senThese
research
topics,
in
particular
cold
atoms
and
single-ion
y #13$. Moreover,
x sitivity than previous vertical cavity designs
es presentedoptical
here
frequency standards, have stimulated new approaches
a significant improvement of the thermal noise level is demnd thermal tonoise
the design of Fabry-Pérot reference cavities which are
onstrated here #13,16$ by using fused silica mirror substrates,
used to stabilize lasers.
which minimize the contribution to thermal noise due to the
y.
For optical frequency standards with neutral atoms, the
higher mechanical Q factor of this material in comparison to
frequency
esigned based
on noise of state-of-the-art ultrastable clock lasers Ultra Low Expansion glass !Corning ULE".
sets a severe limit to the clock frequency
stability via the
X
Y
Y
c
p
p
nite elementDick
softeffect #10$. Due to this limitation, the best reported
69
PTB transportable laser
Demonstration of a Transportable 1 HzLinewidth Laser
Stefan Vogt, Christian Lisdat, Thomas Legero, Uwe Sterr, Ingo
Ernsting, Alexander Nevsky, Stephan Schiller
APPLIED PHYSICS B: LASERS AND OPTICS
Volume 104, Number 4, 741-745, DOI: 10.1007/
s00340-011-4652-7
70
Si has zero expansion at 17 K and 124 K
71
T = 124 K –> T. Kessler & al., PTB / QUEST - Proc. 2011 IFCS
K. G Lyon & al, Linear thermal expansion measurements on silicon from 6 to 340 K - J Appl. Phys 48(3) p.865, 1977
Swenson CA - Recommended values for the thermal expansivity of Silicon from 0 to 1000 K - JPCRD 12(2), 1983
72
PTB 124-K Si cavity
Christian Hagemann
QUEST - Centre for Quantum Engineering and Space-Time Research
Experimental setup
Silicon cavity is thermally isolated by
two gold-plated copper shields.
4
In many applications of lasers in metrology, sensing, and
in recent experiments [21,22] with crystalline CaF2 WGM
73
ectroscopy, the quality of a measurement critically depends
resonators has the laser frequency stability been quantitatively
n the spectral purity of the employed laser source. For
assessed with sufficient resolution. The lowest relative Allan
is reason, researchers’ efforts have been directed toward
deviations have been determined at the 10−12 level for
e reduction of the laser’s phase or frequency fluctuations
submillisecond integration times. We are able to significantly
nce the early days of laser science [1]. Tremendous progress
improve this performance and explore the thermodynamic
Ei
s been made since then, culminating in light sources with
limits of this approach at room temperature.
j T1
newidths
below 1 Hz, and a work
relative Allan
deviation of their
We fabricate WGM resonators following the pioneering
• Pioneering
by
input
fiber
equency below the 10−15 level [2,3]. Such ultrastable lasers
work of Maleki and coworkers [23,24], combining a shaping
R
Braginsky
and several polishing steps on a home-built1 precision lathe
e operated
in a number of(Moscow)
specialized laboratories and are
[25]. Several MgF2 resonators were produced with a typical
alized by electronically locking the laser frequency to a
radius ofsection
2 mm, one of which is shown in Fig. 1(a). The WGMs
sonance
of an external
reference
defined by two resonator
• Made
popular
bycavity
Maleki
resonator
are located in the rim of the structure, which has been fine
gh-reflectivity mirrors attached to a low-thermal-expansion
and
Ilchenko
roundtrip
polished with a diamond slurry of 25 nm average
grit size in
acer [4–8].
Their
performance (JPL/
has been enabled by sophisτd
timetogether
> 1 dm3 -sized spacers’ material and
cated engineering
of the ∼
the last step. The resulting surface smoothness,
with
OEwaves)
ometry, as well as their vibration and thermal isolation. It is
very low absorption losses in the ultrapure crystalline material
(Corning), enables quality factorsj Tin excess of R22× 109 .
ow understood to be limited by thermodynamic fluctuations
Eo
2
• Similar
Fabry
Perot
the mirror
substrates to
and a
coatings
[9,10].
Several proposals
A high-index prism is used to couple
the beam of an external
overcome this limitation are discussed in the literature,
cavity diode laser into the WGM. For laser stabilization, we
output fiber
9…1011
cluding
of reference
cavities [11–14]. In
choose to work in the undercoupled regime, in which the
• cryogenic
Q = 10operation
has been
alternative method, the optical reference cavity is replaced
presence of the coupling prism has only a weak effect on
reported
y a long (∼km)
all-fiber delay line yielding a simplified setup
Alnis & al., PRA 4, 011804(R) (2011)
the expense of slightly reduced stability [15].
In this
work wepower
explore a handling
different approach [16,17] to
• Poor
ovide a reference for the laser’s frequency, by fabricating
hispering-gallery-mode (WGM) resonators from a single
• magnesium
Temperature
ystal of
fluoride (Fig. 1). The WGM resides in
protrusion
of a MgF2 cylinder, whose
compact dimensions
compensation
is
3
) and monolithic nature inherently reduce the res1
cm
∼
challenging
nator’s sensitivity
to vibrations. It also enables the operation
more noisy and/or space-constrained environments, such
a cryostat or a satellite. Furthermore, in contrast to the
ghly wavelength-selective and complex multilayer coatings
Optical whispering gallery
Spherical FP Etalon
74
Target : σy(τ) ≈ 8x10–16
Phase noise : -104 dBc/Hz state of the
art
σy(τ)
Frequency instability limited by the lab
temperature fluctuations
τ(s)
4
Compact FP Etalon
75
Target : σy(τ) ≈ 2 10-15
Design and machining
completed
Design of vacuum chamber
and thermal isolation
completed
Two options for mirrors
- classical
- crystal
Machining will be done end
November
Light injection in the cavity
To be tested…
First experiments end of 2014
5
76
Silicon FP Etalon (Also Région FC, 70 k€)
Target : σy(τ) ≈ 3x10–17
Relative frequency
sensitivity to vibrations
less than 4.10-12 /
m.s-2
Low vibrations cryocooler:
displacement less than 40
nm
Cavity
Temperature instability less
than 100 µK
6
Thermal effect on frequency
8 mm
5.5 mm
CaF2
optical
resonator
laser scan
•wavelength 1.56 μm (ν0=192 THz)
•Q=5x109 –> BW=40 kHz
•a power of 300 μW shifts the resonant frequency by
1.2 MHz (6x10–9), i.e., 37.5 x BW
•time scale about 60 μs
•[Q = 6x1010 demonstrated with CaF2 (I. Grudinin)]
calibration (2 MHz phase modulation)
77
78
Extreme non-linearity
A
Optical
amplifier
Tunable
cw-laser
Photodiode
Microresonator
Microwave beat note
T.J.Kippenberg,
R.Holzwarth, S.A.Diddams,
Microresonator-based
optical frequency combs,
Science 332:555, Apr.2011
(1)
Non-degenerate (2)
Energy
(2)
Power (a.u.)
fr
νpump
f0
(a.u.)
C
νn-2
νn νWGM
n
WGM
νn-1
WGM
νn-2
νn-1
Frequency
WGM
WGM
νn+1
νn+1
νn+2
κ
Microresonator
modes
Equidistant
comb modes
2π
∆ n-2
νn+2
<
∆ n-1
<
∆n
<
∆ n+1
Frequency
rg on April 28, 2011
B FWM:
Degenerate (1)
VIRGO – Gravitational waves
• Large Michelson
interferometers detect the
space-time fluctuations
79
Cascina, Pisa, Italy
• PDH control is used to lock
ultra-stable lasers to the
interferometer
Vinet JY ed., The VIRGO
physics II, Optics and related
topics, available online
Aug 5, 2014
https://www.cascina.virgo.infn.it
iven a
a tiny
ments
iew of
speed
Morley
encies
which
solely
tested
n 1016
Lorentz invariance
1,2
1
1
1
1
1
1
2
!17
standard model of particle physics down to a level of 10!17 .
DOI: 10.1103/PhysRevD.80.105011
The theory of special relativity formulated in 1905 [1]
revealed Lorentz invariance as the universal symmetry of
local space-time, rather than a symmetry of Maxwell’s
equations in electrodynamics alone. This striking insight
was drawn from two postulates: (i) the speed of light in
vacuum is the same for all observers independent of their
state of motion, and (ii) the laws of physics are the same in
any inertial reference frame. Today, local Lorentz invariance constitutes an integral part of the standard model of
particle physics, as well as the standard theory of gravity,
general relativity. Still, there have been claims that a
violation of Lorentz invariance might arise within a yet
to be formulated theory of quantum gravity [2–7]. Given a
lack of quantitative predictions, the hope is to reveal a tiny
signature of such a violation by pushing test experiments
for Lorentz invariance across the board. An overview of
recent such experiments can be found in [8].
PACS numbers: 03.30.+p, 06.30.Ft, 11.30.Cp, 42.60.Da
rotate
resonator. Thus, to detect an anisotropy of the
speed of
light
!c ¼ cx ! cy we continuously rotate the setup and
servo
look for a modulation of the beat frequency !!. Since the
light in the cavities travels in both directions and c refers to
the two-way speed of light, such an isotropy violation
indicating modulation would occur at twice the rotation
rate.
1
I.c THE EXPERIMENT
y
The experiment applies a pair of crossed optical highcx
finesse resonators implemented
in a single block of fused
frequency
silica
(Fig. 1). This spacer block is a 55 mm # 55 mm #
35comparison
mm cuboid with centered perpendicular bore holes of
10 mm diameter along each
axis. Four fused silica mirror
2
c
substrates coated with a high-reflectivity dielectric coating
y
at " ¼ 1064 nm are optically
contacted to either side. The
length of these two crossed optical resonators is matched to
x
servo
s by 1
design
minicted in
rossed
resoers to
t. The
35 mm cuboid with centered perpendicular bore holes of
10 mm diameter along each axis. Four fused silica mirror
substrates coated with a high-reflectivity dielectric coating
REVIEW D 80, 105011 (2009)
at " ¼ 1064 nm arePHYSICAL
optically
contacted to either side.
The
!17
Rotating optical cavity experiment testing Lorentz invariance at the 10
level
length of these two crossed optical resonators is matched to
S. Herrmann, A. Senger, K. Möhle, M. Nagel, E. V. Kovalchuk, and A. Peters
better thanInstitut2 für#m.
The finesse
ofHausvogteiplatz
each resonator
Physik, Humboldt-Universität
zu Berlin,
5-7, 10117 Berlin (TEM00
ZARM, Universität Bremen, Am Fallturm 1, 28359 Bremen
(Received
10 August 2009;in
published
12 November 2009) of 7 kHz. Two
mode) is 380 000,
resulting
a linewidth
We present an improved laboratory test of Lorentz invariance in electrodynamics by testing the isotropy
Nd:YAG
lasers
atmeasurement
" ¼ compares
1064thenm
stabilized
of the speed
of light. Our
resonanceare
frequencies
of two orthogonal to
opticalthese
resonators that are implemented in a single block of fused silica and are rotated continuously on a
resonators
a Anmodified
Pound-Drever-Hall
precision air using
bearing turntable.
analysis of data recorded
over the course of one year sets a limit method
on an
of the speed of light of !c=c " 1 # 10 . This constitutes the most accurate laboratory test of
[16]. anisotropy
Tuning
and
of ofthe
laser
is
the isotropy of c to
date andmodulation
allows to constrain parameters
a Lorentz
violating frequency
extension of the
80
81
Alternate schemes
The original Pound scheme
All the key ideas are here
However technology, electrical symbols, and writing style are quite different
R. V. Pound, Rev. Sci. Instrum. 17(11) p. 490–505, Nov. 1946
82
83
The Pound-Drever-Hall scheme
The Pound scheme ported to optics
iso
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
laser
phase
modul.
optical
resonator
vco
servo
ampli
photodetector
IF
LO
RF
R.P.V. Drever, J.L. Hall & al., Appl. Phys. Lett. 31(2) p.97–105, June 1983
The Pound-Galani oscillator
84
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
oscillator loop
ν0
• Great VCO for cheap
• Easier to control
Q
output
2
detector
ν
phase
modul
1
3
vco
fm
control
v oc ν −ν 0
• Two-port resonator
• More complex
• Lower Q
RF
LO
IF
Z. Galani & al, Analysis and design of a single-resonator GaAs FET oscillator with noise degeneration, IEEE-T-MTT 39(5), May 1991
Pound-Galani transfer function
|A|
resonator
halfwidth
lock-in
cutoff
ar
modulation
tr
bi
ar
y
0º
≈ωL
ωc
Ω
arg(A)
ω
arbitrary
frequency
detection
phase
detection
• FD region –> full performance
• PD region
• Flat frequency response, not for free
• Poor response of the frequency-error detection
• Higher noise
no
detection
85
Figure from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
The Pound-Sulzer oscillator
phase
modulator
oscillator
loop
bridge
tuning
voltage
audio
oscillator
rf amplifier
integr.
lock-in amplifier
P. Sulzer, Proc. IRE 43(6) p.701-707, June 1955
86
References
• Black ED, An introduction to Pound–Drever–Hall laser frequency
stabilization, Am J Phys 69(1) January 2001
• R.P.V. Drever, J.L. Hall & al., Appl. Phys. Lett. 31(2) p.97–105, June 1983
• Hall JL & al, Laser stabilization, Chapter 27 of Bass et al, Handbook of
optics, McGraw Hill 2001
• Mor O, Arie A, JQE 33(4), April 1997
• R.V. Pound, Rev. Sci. Instrum. 17(11) p. 490–505, Nov. 1946
• Z. Galani & al, IEEE-T-MTT 39(5), May 1991
• P. Sulzer, Proc. IRE 43(6) p.701-707, June 1955
Acknowledgments
87