Phase noise, oscillators etc.
E. Rubiola,
V. Giordano, K. Volyanskiy, H. Tavernier, Y. Kouomou Chembo, R. Bendoula,
P. Salzenstein, J. Cussey, X. Jouvenceau, R. Boudot, L. Larger, ........
FEMTO-ST Institute, Besançon, France
CNRS and Université de Franche Comté
Outline
•
•
•
•
•
•
•
•
•
Phase noise & friends
Amplifier noise
Correlation
AM noise
Bridge (interferometric) noise measurements
Advanced methods
Delay-line instrument
Optical resonators
Non-linear AM oscillations
home page http://rubiola.org
1 – Phase noise & friends
1 – introduction
X
Clock signal affected by noise
Chapter 1. Basics
Time Domain
V0
v(t)
2
Phasor Representation
amplitude fluctuation
V0 α(t) [volts]
normalized ampl. fluct.
α(t) [adimensional]
√
V0 / 2
t
V0
v(t)
ampl. fluct.
√
(V0 / 2)α(t)
phase fluctuation
ϕ(t) [rad]
phase time (fluct.)
x(t) [seconds]
√
t
polar coordinates
phase fluctuation
ϕ(t)
V0 / 2
Figure 1.1: .
v(t) = V0 [1 + α(t)] cos [ω0 t + ϕ(t)]
where V0 is the nominal amplitude, and α the normalized amplitude fluctuation,
Cartesian
coordinates
= V0 cos frequency
ω0 t + nisc (t) cos ω0 t − ns (t) sin ω0 t
which
is adimensional. v(t)
The instantaneous
1 dϕ(t)
ω0
+
2π
2π dt
It holds that (1.3)
nc (t)
ns (t)
of stable
with
α(t)signals
= of the form
and (1.2),
ϕ(t)
=
|nc (t)| ! VThis
and deals
|nswith
(t)| the
! measurement
V0
0 book
0
main focus on phase, thus frequency and time. This V
involves
several topics, V0
under low noise approximation
ν(t) =
1 – introduction
X
Phase noise & friends
random phase fluctuation
random walk freq.
Sϕ(f)
Sϕ (f ) = PSD of ϕ(t)
b−4 f−4
signal sources only
power spectral density
it is measured as
Sϕ (f ) = E {Φ(f )Φ∗ (f )}
Sϕ (f ) ≈ "Φ(f )Φ∗ (f )#m
1
L(f ) = Sϕ (f ) dBc
2
flicker freq.
(expectation)
white freq.
b−2 f−2
(average)
random fractional-frequency fluctuation
⇒
flicker phase.
b−1 f−1
white phase
Sy (f)
f2
Sy = 2 Sϕ (f )
ν0
f2/ ν20
h2 f2
h−2 f−2
random
walk freq.
white phase
h−1 f−1
h0
h1 f
flicker freq.
flicker phase
white freq.
Allan variance
(two-sample wavelet-like variance)
! "
$
#
2
1
2
σy (τ ) = E
y k+1 − y k
.
2
b0
f
x
ϕ̇(t)
y(t) =
2πν0
both signal sources
and two-port devices
b−3 f−3
f
σy2 (τ)
approaches a half-octave bandpass filter (for white
noise), hence it converges for processes steeper than 1/f
flicker phase
white phase
white freq.
h0 /2 τ
freq.
drift
flicker freq.
2ln(2)h −1
random walk freq.
(2π ) 2
h−2 τ
6
τ
1 – introduction
X
Relationships between spectra and variances
noise
type
white
PM
Sϕ (f )
b0
h2 f
flicker
PM
b−1 f −1
white
FM
b−2 f
−2
b−3 f
−3
flicker
FM
random
walk FM
Sy (f )
b−4 f −4
2
b0
h2 = 2
ν0
σy2 (τ )
3fH h2 −2
τ
2
(2π)
2πτ fH "1
mod σy2 (τ )
3fH τ0 h2 −3
τ
2
(2π)
h1 f
b−1
h1 = 2
ν0
h1 −2
[1.038+3 ln(2πfH τ )]
τ
2
(2π)
0.084 h1 τ −2
n"1
h0
b−2
h0 = 2
ν0
1
h0 τ −1
2
1
h0 τ −1
4
h−1
b−3
= 2
ν0
2 ln(2) h−1
27
ln(2) h−1
20
h−2
b−4
= 2
ν0
(2π)2
h−2 τ
6
(2π)2
0.824
h−2 τ
6
1
(ẏ)2 τ 2
2
1
(ẏ)2 τ 2
2
h−1 f
−1
h−2 f −2
linear frequency drift ẏ
Sϕ ↔ Sy
fH is the high cutoff frequency, needed for the noise power to be finite.
1 – introduction
X
Basic problem: how can we measure a low random signal
(noise sidebands) close to a strong dazzling carrier?
solution(s): suppress the carrier and measure the noise
convolution
(low-pass)
time-domain
product
vector
difference
s(t) ∗ hlp (t)
s(t) × r(t − T /4)
s(t) − r(t)
distorsiometer,
audio-frequency instruments
traditional instruments for
phase-noise measurement
(saturated mixer)
bridge (interferometric)
instruments
2 – Amplifier noise
AM/PM noise
additive
white
parametric
local (flicker)
environmental
E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN13 9780521886772
Noise figure F, Input power P0
V0 cos ω0 t
∑
g
nrf (t)
RF spectrum
Amplifier white noise
2 – amplifier noise
P=FkT 0B
S( ν )
P0
B
ν 0 −f
ν0
power law
Sϕ =
bi f
white
phase noise
i=−4
ν 0 +f
ν
USB
Sφ(f)
i
B
Ne=FkT0
LSB
0
!
3
F kT0
b0 =
P0
P0
low P0
high P0
f
Cascaded amplifiers (Friis formula)
The (phase) noise is chiefly that of the 1st stage
F1
g1
F2
g2
F3
g3
(F2 − 1)kT0
N = F1 kT0 +
+ ...
2
g1
The Friis formula applied to phase noise
F1 kT0
(F2 − 1)kT0
b0 =
+
+ ...
2
P0
P0 g1
E. Rubiola
– FCS– 2004
E. Rubiola
FCS 2004
4
Amplifier
flicker
noiseamplifiers
Flicker
noise
in RF
and
microwave
Flicker
noise
in RF
and
microwave
amplifiers
2 – amplifier noise
S(f)
no carrier
no carrier
near-dc
noise
no flicker
noise
noise
up-conversion
up-conversion
S(f) S(f)
S(f)
near-dc
flickerflicker
near-dc
f
f
f
ω0 = ?
random
modulation
near-dc
noise
random
modulation
fromfrom
near-dc
noise
4
f
ω0
stopband
, , output bandwidth
output bandwidth
stopband
,
n , , !t "n !t " noise-free
noise-free
n , !t " n !t "
vi !ti cos!$
"#V i cos!$
v o !t "v o !t "
vi !t "#V
t" 0t"
0
a
a
1
AM AM
PM PM
1
carrier
near-dc noise
, ,!! , ,
, , nature
, , , , ,of 1/f
jω
t
!
the
parametric
modulated
0
v
!t
"
#
V
cos!$
t
"&m
n
!t
"
cos!$
t
"'m
!t "sin
modulated
signal:signal:
vvoi (t)
!t " o=
#VV
t "&m
n !t(t)
" cos!$noise
t "'m
n !tn"sin
!$ !$
t " 0 at "1
i + n0 (t)
0+ jn
i ei cos!$
0
is0 hidden in n’ and n” 0
% %
carriercarrier
substitute
AM noise
AM noise
PM noise
PM noise
( (a
(careful, this hides the down-conversion)
2 x&a
2 x2
the
simplest
the simplest
v
#
a
x #cos!$
V i cos!$
t " & n!t " *n !t*"*n!t
"*+1
vo (t)
= a1 vvio(t)
+
a
v
(t)
. withx # V
# ao1 x&a
t" &
+1
2 1i 2 x +
2 . .with
0 n !t "
i
0
nonlinearity
nonlinearity
near-dc
non-linear (parametric) amplifier
carriercarrier near-dc
yields:
yields:
expand and select the
ω022
terms2
2
2
2
v
!t
"
#
a
V
cos!$
t
"&a
V
cos
!$
t
"&2
V
n
!t
"
cos!$
t
"&n
!t "
v o !t " o# !a1 V i cos!$
t "&a
!t
"
1 i
0 2 V i2cosi !$ 0 t "&2
0 V i n !ti" cos!$ 0 t "&n
0
0
$
" !
#
The noise sidebands
are
2 a2
vo (t) = Vi a1 + 2a
n2 a(t)
+ "jn!! (t) ejω0 t
2 a22 n!t
"2 n!t
2
a
2 input carrier
proportional
AM noise:
modulation
index:m #m to
# the
AM noise:
)!t " )!t
# "#
modulation
index:
a a1
a a1
% %
1 and PM noise
get AM
( (
1
PM noise
originates
in same
the same
butafor
a 90°
phase
in the
product
PM noise
originates
in the
way,way,
but for
90°
phase
shiftshift
in the
product
a2 !
α(t) = 2
n (t)
a1
a2 !!
ϕ(t) = 2
n (t)
a1
The AM and the PM noise are
independent of Vi , thus of power
There is also a linear parametric model, which yields the same results
1
4
Amplifier flicker noise
S!(f) , log-log scale
2 – amplifier noise
b–1 ! const. vs. P0
b
b0 = FkT0 / P0
–1 f –1
b0 , higher P0
b0 , lower P0
f'c
f"c
f
typical amplifier phase noise
RATE GaAs HBT SiGe HBT Si bipolar
microwave microwave HF/UHF
fair
−100
−120
good
−110
−120
−130
best
−120
−130
−150
unit dBrad2 /Hz
fc = ( b–1 / FkT0 ) P0 depends on P0
The phase flicker coefficient b–1 is about independent of power.
Describing the 1/f noise in terms of fc is misleading because fc depends
on the input power
In a cascade, (b–1)tot does not depend of the amplifier order. Each stage
contributes about equally
b–1 is roughly proportional to the gain through the number of stages
Paralleling m amplifier, (b–1)tot is divided by m
5
2 – amplifier noise
Amplifier flicker noise – experiments
−120
Phase noise, dBrad 2 /Hz
The 1/f phase noise b–1 is
about independent of power
SiGe LPNT32
−125
bias 2V, 20 mA
R. Boudot 2006
−130
−135
Pin=−15dBm
Pin=−10dBm
Pin=−5dBm
−140
−145
−150
The white noise b0 scales
up/down as 1/P0, i.e., the
inverse of the carrier power
−155
−160
−165
−170
−175
Pin=0dBm
1
100
1000
10000
100000
f, Hz
−100
Phase noise, dBrad 2 /Hz
10
P=−50dBm
Amplifier X−9.0−20H at 4.2 K
P=−60dBm
P=−70dBm
Data from IEEE UFFC 47(6):1273 (2000)
P=−80dBm
−110
P=−80dBm
−120
P=−70dBm
P=−60dBm
P=−50dBm
−130
1
101
102
103
104
Fourier frequency, Hz
105
6
2 – amplifier noise
X
S!(f) , log-log scale
Flicker noise in cascaded amplifiers
b–1 ! const. vs. P0
b
A
B
B
A
b0 = FkT0 / P0
–1 f –1
b0 , higher P0
b0 , lower P0
f'c
f
f"c
fc = ( b–1 / FkT0 ) P0 depends on P0
AB and BA have the
same 1/f noise
The phase flicker coefficient b–1 is about independent of power. Hence:
in a cascade, (b–1)tot does not depend of the amplifier order
in practice, in a cascade each stage contributes about equally
m
!
(b−1 )tot =
(b−1 )i
i=1
b–1 is roughly proportional to the gain through the number of
stages
2Amps ! Pin=!22dBm
S ! (f), dB.rad 2/H
2 – amplifier noise
!140
X
2Amps ! Pin=!19dBm
Flicker in cascaded amplifiers – experiments
!150
!50
!160
AML812PNB1901 (SiGe HBT)
1Amp !15
Pin=!5dBm
Feb 2007
S! (f), dB.rad 2/Hz
!70
!170
!90
!180
3 dB
1
10
100
1000
NMS floor
10000
100000
f, Hz
!110
2 cascaded Amplifiers, Pin=−27.5dBm
F IG . 18 – Phase noise of 1-stage and 2 stages cascaded amplifiers at 10MHz.
!150
Single Amplifier, Pin=−4dBm
1
10
100
1000
10000
100000
f, Hz
!120
S!(f), dB.rad 2/Hz
Avantek UTC573, 10 MHz
Jan stage
07 cascaded amplifier at
F IG . 13 – Phase noise of 1 amplifier (parallel SiGe HBT) AML812PNB1901 and of a23double
!130
10 GHz.
2Amps ! Pin= !5dBm
3Amps ! Pin= !24dBm
!140
!150
9
!160
1Amp ! Pin=!5dBm
5!6 dB
!170
!180
1
10
100
1000
f, Hz
NMS floor
10000
3 dB, with 2 amplifiers
5 dB, with 3 amplifiers
!130
!170
The expected flicker of
a cascade increases by:
100000
2 – amplifier noise
Flicker noise in parallel amplifiers
vi (t)
uk (t)
um(t)
A1
v1 (t)
ψ1
Ak
vk (t)
ψk
Am
vm(t)
ψm
m−way power combiner
input
m−way power divider
u1 (t)
output
vo(t)
ϕ
The phase flicker coefficient b–1 is about independent of power
The flicker of a branch is not increased by splitting the input power
At the output,
"
1 !
the carrier adds up coherently
b−1 =
b−1 branch
m
the phase noise adds up statistically
Hence, the 1/f phase noise is reduced by a factor m
Only the flicker noise can be reduced in this way
Gedankenexperiment: join the m branches of a parallel amplifier
forming a single large active device: the phase flickering is
proportional to the inverse physical size of the amplifier active region
X
2 – amplifier noise
input
vi (t)
1
uk (t) = √ vi (t)
m
m
1 !
vo (t) = √
vk (t)
m
k=1
"
%
# !
$
1
vk (t) = √ Vi a1 + 2a2 nk (t) + jn!!k (t) ej2πν0 t
m
a2
ψk (t) = 2 n!!k (t)
a1
1
!!
j2πν0 t
V
2a
n
(t)
e
i
2
k
ϕk (t) = m
a1 Vi ej2πν0 t
1 a2 !!
=
2 n (t)
m a1 k
m
!
1 a22
Sϕ (f ) =
4 2 Sn!!k (f )
2
m a1
k=1
1 a22
Sϕ (f ) =
4 Sn!! (f )
m a21
1
Sϕ (f ) =
Sψ (f )
m
$
1 #
b−1 =
b−1 branch
m
branch-amplifier input
main output
branch → output
branch
branch → output
!
branches → output
m equal branches → output
uk (t)
um(t)
A1
X
v1 (t)
ψ1
Ak
vk (t)
ψk
Am
vm(t)
ψm
m−way power combiner
Parallel amplifiers, mathematics
m−way power divider
u1 (t)
output
vo(t)
ϕ
b!1= !104 dB.rad2 /Hz
S! , dB.rad 2/Hz
2 – amplifier noise!93
Flicker noise in parallel amplifiers
!113
!50
!133
JS2, 10 GHz
15 Feb 2007
S!(f), dB.rad 2/Hz
!70
!153
!90
!173
Pin=!25dBm
Pin=!23dBm
12 Jan 2007
1
10
100
!110
f, Hz
1000
10000
Single amplifier
Pin= −6 dBm
100000
!130
F IG . 6 – Phase noise of 1 amplifier AFS6 vs Pin at 12GHz.
2 parallel amplifiers
Pin= −5 dBm
!150
!170
1
10
100
1000
10000
100000
f, Hz
!50
AFS6, 10 GHz
F IG . 11 – Phase27noise
1 single amplifier JS2 and a double stage parallel amplifier at 10 GHz.
Febof2007
S!(f), dB.rad 2/Hz
!70
!90
Single AFS6 Amplifier
Pin= −45 dBm
8
!110
!130
vibrations
!150
!170
2 Parallel AFS6 Amplifiers
Pin= −33 dBm
1
10
100
1000
f, Hz
10000
100000
Connecting two
amplifiers in parallel,
the expected flicker
is reduced by 3 dB
X
2 – amplifier noise
X
Flicker noise in parallel amplifiers
−140
AML812PNA0901 (100mA)
Phase noise, dBrad 2 /Hz
AML812PNB0801 (200mA)
AML812PNC0801 (400mA)
−150
−160
AML812PND0801 (800mA)
−170
101
102
103
104
105
Fourier frequency, Hz
106
Specification of low phase-noise amplifiers (AML web page)
amplifier
AML812PNA0901
AML812PNB0801
AML812PNC0801
AML812PND0801
unit
gain
10
9
8
8
dB
parameters
F
bias power
6.0
100
9
6.5
200
11
6.5
400
13
6.5
800
15
dB
mA
dBm
102
−145.0
−147.5
−150.0
−152.5
phase noise vs. f , Hz
103
104
−150.0
−158.0
−152.5
−160.5
−155.0
−163.0
−157.5
−165.5
dBrad2 /Hz
105
−159.0
−161.5
−164.0
−166.5
2 – amplifier noise
X
Environmental (parametric) noise in amplifiers
temperature
g
phase
A
B
B
A
vibrations
etc.
input carrier
amplitude
A time constant can be present
φ = φA + φB and α = αA + αB
regardless of the amplifier order
Cascaded amplifiers
Cascading m equal amplifiers, Sα(f)
and Sφ(f) increase by a factor m2.
If the amplifier were independent, Sα
(f) and Sφ(f) would increase only by a
factor m.
let
z(t) = x(t) + y(t)
Phase noise
Sz (f ) = ZZ ∗
= (X + Y ) (X + Y )∗
= XX ∗ + Y Y ∗ + XY ∗ + Y X ∗
= Sx + Sy + Sxy + Syx
!"#$ !"#$
>0
>0
2 – amplifier noise
X
Environmental effects in RF amplifiers
Amplifier phase noise
–5
f
courtesy of J. Ackermann N8UR, http://www.febo.com
comments on noise are of E. Rubiola
–5
f
–5
f
Spectracom 8140T
b–1 = –113.5 dB
-1
DD
TA
HP 5087A and
TADD-1 10 MHz
b–1 = –133 dB
508
7A
in dBrad2/Hz (dBc/Hz + 3 dB)
814
0
T
1
D-
D
TA
HP
b–1 is the 1/f noise coefficient
background
b–1 = –142 dB
TADD-1 5 MHz
b–1 = –138.5 dB
It is experimentally observed that the temperature
fluctuations cause a spectrum Sα(f) or Sφ(f) of the 1/f5 type
Yet, at lower frequencies the spectrum folds back to 1/f
3 – Correlation
3 – correlation
8
Correlation measurements
a(t)
DUT
x = c–a
c(t)
b(t)
FFT
y = c–b
DUT noise,
normal use
background,
ideal case
background,
with AM noise
Syx (f ) = E {Y (f )X ∗ (f )}
= E {(C − A)(C − B)∗ }
= E {CC ∗ − AC ∗ − CB ∗ + AB ∗ }
= E {CC ∗ }
0
0
0
Syx (f ) = Scc (f )
= !CC ∗ "m + O(1/m)
a(t), b(t) –> mixer noise
c(t) –> DUT noise
phase noise measurements
basics of correlation
a, b
c
a, b
c=0
a, b
c≠0
instrument noise
DUT noise
instrument noise
no DUT
instrument noise
AM-to-DC noise
single-channel
S!(f)
in practice, average on m realizations
Syx (f ) = !Y (f )X ∗ (f )"m
= !CC ∗ − AC ∗ − CB ∗ + AB ∗ "m
Two separate mixers
measure the same DUT.
Only the DUT noise is common
1/"m
0 as
1/√m
correlation
frequency
3 – correlation
9
Thermal noise compensation
3 – correlation
10
Thermal noise compensation
100 MHz prototype, carrier
power Po = 8 dBm
mesured noise, dBrad2/Hz
thermal floor
injected noise, dBrad2/Hz
3 – correlation
11
Example of correlation measurement
100 MHz carrier
!!K"C$
S!9 f :
S"9 f :
>!!MKCM?
$
;<'/;..=78
;<=78
N'3
>..........;<0=78?
!!J"C$
>!!FKCM?
!!I"C$
plot 459
*+,-)(./'0
P" H.!#CI.;<0
/E-.#F!.*G(%@'/
@A&.'3.%B/,,C
/'0.!/6.!D
/,-)(.2,%/)C
%&''()
>!!KKCM?
$
kBT" =P" H.!!IJCI.;<>'/;..?=78
!!L"C$
>!!JKCM?
1&2'+('.3'(42(,%56.78
!$""C$
>!!IKCM?
!
!"
!"$
Noise of a by-step attenuator, measured at 100 MHz by
correlation. The mixer is replaced with a bridge.
!"#
3 – correlation
X
Useful schemes
REF
(ref)
LO
DUT
2−port
device
RF
dc
x
RF
RF
dc
arm b
y
arm a
x
dc
y
LO
LO
arm b
REF
LO
(ref)
DUT
dc
FFT
analyzer
arm a
FFT
analyzer
phase
RF
phase
phase lock
meter output
(noise only)
LO
dc
x
RF
DUT
RF
arm b
REF
dc
y
FFT
analyzer
arm a
bridge a
DUT
2−port
device
bridge b
Σ
Δ
µw
Σ
Δ
µw
LO
RF
LO
RF
LO
(ref)
phase and ampl.
phase lock
dc
x
dc
y
FFT
analyzer
phase and ampl.
REF
3 – correlation
12
Pollution from AM noise
(ref)
AM
phase
arm a
DUT
2−port
device
LO
x
dc
RF
AM
RF
y
dc
arm b
FFT
analyzer
file am−correl−2port
The mixer converts power into dc-offset,
thus AM noise into dc-noise,
which is mistaken for PM noise
v(t) = kφ φ(t) + kLO αLO + kRF αRF
LO
(ref)
phase
rejected by correlation and avg
AM
not rejected by correlation and avg
file am−correl−oscillator
phase lock
REF
REF
AM
AM
LO
RF
dc
arm a
LO
AM
arm b
REF
LO
dc
RF
AM
y
FFT
analyzer
DUT
arm a
x
dc
x
RF
DUT
AM
RF
arm b
REF
dc
y
FFT
analyzer
file am−correl−discrim
LO
AM
phase lock
E. Rubiola, R. Boudot, The effect of AM noise on correlation phase noise
measurements, IEEE Tr.UFFC 54(5):926–932 May 2007, and arXiv/physics/0609147
4 - AM noise
E. Rubiola, arXiv/physics....... 2006
4 – AM noise
14
Tunnel and Schottky power detectors
parameter
law: v = kd P
video out
~60
Ω
10−200
pF
vsvr max.
max. input power (spec.)
absolute max. input power
output resistance
output capacitance
gain
cryogenic temperature
electrically fragile
external
50 Ω to
100 kΩ
The “tunnel” diode is actually a
backward diode. The negative
resistance region is absent.
0
Measured
1×10
3.2×102
1×103
3.2×103
1×104
2
conditions: power −50 to −20 dBm
up to 4 decades
10 MHz to 20 GHz
1.5:1
−15 dBm
20 dBm or more
1–10 kΩ
20–200 pF
300 V/W
no
no
1–3 octaves
up to 40 GHz
3.5:1
−15 dBm
20 dBm
50–200 Ω
10–50 pF
1000 V/W
yes
yes
Herotek DZR124AA s.no. 227489
-20
Herotek DT8012 s.no. 232028
Schottky
-20
output voltage, dBV
load resistance, Ω
detector gain, A−1
DZR124AA DT8012
(Schottky) (tunnel)
35
292
98
505
217
652
374
724
494
750
tunnel
Tunnel
output voltage, dBV
rf in
input bandwidth
Schottky
-40
10 kΩ
-60
3.2 kΩ
-40
10 kΩ
3.2 kΩ
-60
1 kΩ
320 Ω
-80
-80
100 Ω
1 kΩ
320 Ω
-100
-100
100 Ω
ampli dc offset
-120
-60
-50
-40
-30
-20
-10
input power, dBm
0
10
ampli dc offset
-60
-50
-40
-30
-20
-10
input power, dBm
0
10
4 – AM noise
15
Noise mechanisms
Shot noise SI (f ) = 2qI0
detector
rf in
amplifier
v
video out
~60
Ω
10−200
pF
Thermal noise
SV (f ) = 4kBT0R
in
external
50 Ω to
100 kΩ
noise−free
n
out
in
Flicker (1/f ) noise is also present
Never say that it’s not fundamental,
unless you know how to remove it
In practice
the amplifier white noise turns out to be higher than the detector noise
and the amplifier flicker noise is even higher
4 – AM noise
Pa
va
Pb
vb
source
under test
dual channel
FFT analyzer
Cross-spectrum method
monitor
va (t) = 2ka Pa α(t) + noise
vb (t) = 2ka Pb α(t) + noise
The cross spectrum Sba(f ) rejects
the single-channel noise because
the two channels are independent.
1
Sba (f ) =
Sα (f )
4ka kb Pa Pb
power
meter
Sα (f)
log/log scale
•Averaging on m spectra, the single-
single channel
1
m
channel noise is rejected by √1/2m
•A cross-spectrum higher than the
averaging limit validates the measure
cross spectrum
•The knowledge of the single-channel
meas. limit
noise is not necessary
f
16
Example of AM noise spectrum
Wenzel 501−04623E 100 MHz OCXO
P0 = −10.2 dBm
avg 2100 spectra
−123.1
dB/Hz
−133.1
Sα ( f )
4 – AM noise
−143.1
−153.1
Fourier frequency, Hz
−163.1
10
flicker:
102
103
104
105
h−1 = 1.5×10−13 Hz−1 (−128.2 dB) ⇒ σα = 4.6×10−7
Single-arm 1/f noise is that of the dc amplifier
(the amplifier is still not optimized)
17
4 – AM noise
18
AM noise of some sources
source
h−1 (flicker)
Anritsu MG3690A synthesizer (10 GHz)
2.5×10−11
Marconi synthesizer (5 GHz)
1.1×10−12
Macom PLX 32-18 0.1 → 9.9 GHz multipl.
1.0×10−12
Omega DRV9R192-105F 9.2 GHz DRO
8.1×10−11
Narda DBP-0812N733 amplifier (9.9 GHz)
2.9×10−11
HP 8662A no. 1 synthesizer (100 MHz)
6.8×10−13
HP 8662A no. 2 synthesizer (100 MHz)
1.3×10−12
Fluke 6160B synthesizer
1.5×10−12
Racal Dana 9087B synthesizer (100 MHz)
8.4×10−12
Wenzel 500-02789D 100 MHz OCXO
4.7×10−12
Wenzel 501-04623E no. 1 100 MHz OCXO
2.0×10−13
Wenzel 501-04623E no. 2 100 MHz OCXO
1.5×10−13
(σα )floor
−106.0 dB
5.9×10−6
−120.0 dB
1.2×10−6
−105.4 dB
6.3×10−6
−118.8 dB
1.4×10−6
−110.8 dB
3.4×10−6
−127.1 dB
5.2×10−7
−119.6 dB
1.2×10−6
−100.9 dB
1.1×10−5
−121.7 dB
9.7×10−7
−118.3 dB
1.5×10−6
−113.3 dB
2.6×10−6
−128.2 dB
4.6×10−7
worst
best
5 – Bridge method
5 – bridge (interferometer)
20
Wheatstone bridge
adj. phase
LO
RF
0
equilibrium: Vd = 0 –> carrier suppression
static error δZ1 –> some residual carrier
real δZ1 => in-phase residual carrier Vre cos(ω0t)
imaginary δZ1 => quadrature residual carrier Vim sin(ω0t)
fluctuating error δZ1 => noise sidebands
real δZ1 => AM noise vc(t) cos(ω0t)
imaginary δZ1 => PM noise –vs(t) sin(ω0t)
IF
synchronous
detection: get
vc(t) vs(t)
(AM or PM noise)
nrico Rubiola – Phase Noise – 55
5 – bridge (interferometer)
Interferometer – scheme
Bridge (interferometric) PM and AM
noise measurement
bridge
detector
optional:
I-Q detection
High immunity to low-f magnetic fields
and rejection of the master-oscillator noise
yet, difficult for the measurement of oscillators
21
5 – bridge (interferometer)
nrico Rubiola – Phase Noise – 57
X
Interferometer – synchronous detection
Synchronous detection
Synchronous in-phase and quadrature detection
5 – bridge (interferometer)
A bridge (interferometric) instrument can
be built around a commercial instrument
You will appreciate the computer interface and the software ready for use
22
6 – Advanced methods
6 – advanced methods
Advanced – flicker reduction
Enrico Rubiola – Phase Noise – 64
Advanced – flicker reduction
Origin of flicker in the bridge
In the early time of electronics, flicker was called “contact noise”
Coarse (by-step) and fine (continuous) adjustment of the bridge null are necessary
24
6 – advanced methods
X
Coarse and fine adjustment of the bridge
null are necessary
Enrico Rubiola – Phase Noise – 65
Advanced – flicker reduction
6 – advanced methods
25
Flicker reduction, correlation, and closedloop carrier suppression can be combined
channel b (optional)
w1
readout
I v1
RF I−Q
G
B
v
detect Q 2
matrix
matrix w2
g ~ 40dB
LO
pump
rf virtual gnd
null Re & Im
inner interferometer
CP1
x( t )
CP2
R0
R 0 =50 Ω
DUT
power splitter
atten
CP3
Δ’
γ
arbitrary phase
10−20dB
coupl.
’
γ’
arbitrary phase
pump
LO
v1
I−Q
detect Q v2
g ~ 40dB
LO
atten
RF
I u1
I−Q
modul Q u2
z1
dual
integr z 2
channel a
readout
G
matrix
FFT
analyz.
w1
B
matrix w2
G: Gram Schmidt ortho
normalization
B: frame rotation
pump
automatic carrier
suppression control
diagonaliz.
var. att. & phase
atten
I
RF
R0
manual carr. suppr.
atten
CP4
D
matrix
I−Q detector/modulator
I
RF
Q
LO −90°
E. Rubiola, V. Giordano, Rev. Sci. Instrum. 73(6) pp.2445-2457, June 2002
0°
6 – advanced methods
Example of results
interferometer
!!L"AK
kBT0
isolation
S"1 f +
S!1 f +
5!!%LA%7
g
isolation
DUT
CP2
26
#
,-(2,33./0
,-./0
N(4
53333333333,-6./07
!!M"AK
89:;*)32(6
5!!KLA%7
kBT0
&'(()*
!!J"AK
g
resistive
terminations
P" @3!%A!3,-6
2<;3=#38>)&?(2
?B'3(43&C2::A
2(63!2E3!F
2:;*)3D:&2*A
5!!LLA%7
#
kBT" .P" @3!!JJA!3,-5(2,337./0
!!="AK
5!!MLA%7
Correlation-and-averaging
rejects the thermal noise
G'D(9)(34()HD):&IE3/0
!#""AK
5!!JLA%7
!
!"#
!"
!"$
!"%
!"K
Residual noise of the fixed-value bridge,
in the absence of the DUT
!!>DE$
S", f &
S!, f &
1!!$MEC3
#
'(-.'//)*+
'()*+
P" L/!DED/'(2
.F@/!G/?H8;B-.
BI5/-0/;J.::E
.-2/!.=/!K
.:@A8/6:;.AE
N-0
1//////////'(2)*+3
!!CDE$
1!!%MEC3
!!J">#
S! , f S" , f -
:!!#I>%;
!!H">#
$
./)0.11234
./234
:1111111111111./9234;
N )<
:!!II>%;
?7:@A8/.-2
kBT" 2P"
!!DDE$
!!K">#
1!!>MEC3
:!!JI>%;
!!GDE$
!$"">#
1!!CMEC3
:!!HI>%;
5678+*10)9
$
=1!!HH>!1./:)0.11;234
P" =1!%>!1./9
0?81&$@15A*'B)0
BC(1)<1'D077>
0)91!0F1!G
078+*1E7'0+>
'())*+
7:?B-628:B/:57?8
456-78-/0-8968:;<=/*+
!!MDE$
1!!DMEC3
!
!"
!"#
!"$
!"%
Noise of a pair of HH-109 hybrid couplers
measured at 100 MHz
L(E)6*)1<)*ME*7'NF134
!$!">#
!">
:!!KI>%;
!"$
!"&
!"%
Residual noise of the fixed-value bridge.
Same as above, but larger m
!"#
6 – advanced methods
27
Microwave circulator in reverse mode
of Measured
Spectrum
(refersExample
to the
Pound
scheme)
eftf-fcs 2003
Sα|ϕ ( f )
−164
12
P0 = 19 dBm
avg 10 spectra
single channel
dB[rad 2 ]/Hz
−174
Narda CNA 8596
s.no. 157
−184
−194
instrument noise
Fourier frequency, Hz
−204
10
102
103
104
no post-processing
is used
totohide
stray
signals,
like vibrations
no post-processing
is used
hide stray
signals,
like vibrations
or the mainsor the mains
6 – advanced methods
±45º detection
DUT noise without carrier
DOWN reference
cross spectral density
-.!/01+"'.*-+"
4"6")6-*.
nc (t) cos ω0 t − ns (t) sin ω0 t
u(t) = VP cos(ω0 t − π/4)
UP reference
#$
28
d(t) = VP cos(ω0 t + π/4)
"
1!
Sud (f ) =
Sα (f ) − Sϕ (f )
2
231451635"'.*-+"
4"6")6-*.
#$
! ns &'(
t +-.&, "!%t(
u
S" @ f ?
S! @ f ?
!!K;F;
<=>34
A,,,,,,,,,,,,,<=9>34B
N)-
A!!HKFKB
u
P" :,!%F!,<=9
8J6,K$H,5L+0C)8
#
<=)8<,,>34
5*/67+,8)9
CD',)-,0E8//F
G%HI,<+C+0C*'/,
8/67+,(/087F
!!;;F;
A!!KKFKB
kBT" :,!!;%,<=9>34
nc&'()*+&,
t
"!%t(
!"
!"
!!M;F;
A!!;KFKB
!!N;F;
0'))+7
A!!MKFKB
d
d
&'()*+),-)+.(+/012,34
!#";F;
A!!NKFKB
!
!"
!"#
!"$
!"%
!"H
Residual noise, in the absence of the DUT
Smart and nerdish, yet of scarce practical usefulness
First used at 2 kHz to measure electromigration on metals (H. Stoll, MPI)
6 – advanced methods
The complete machine (100 MHz)
29
6 – advanced methods
A 9 GHz experiment (dc circuits not shown)
30
6 – advanced methods
Advanced – comparison 31
Comparison of the background noise
Sϕ (f) dBrad2/Hz
real−time
correl. & avg.
−140
sat
−150
mixer, interferometer
ura
int
ted
erf
er
e
r
ble
om
ete
int
e
r
fer r
cor
om
rel
e
. sa
t. m ter
res
ix.
idu
al f
lick
res
idu
er,
res al f
by−
idu lick
ste
al f
res
p in
er,
idu lick fix
ter
al f
er, ed i
fer
lick
om
fix nte
ete
er, ed i rfer
r
nte
fix
o
ed
rfe mete
int
r
erf omet r
ero
e
me r
ter
, ±4
5°
dou
−160
−170
−180
−190
−200
−210
Fourier frequency, Hz
−220
1
10
nested interferometer
mix
10 2
saturated mixer
interferometer
correl. saturated mixer
double interf.
measured floor, m=32k
det
ect
ion
10 3
104
105
6 – advanced methods
Mechanical stability
any flicker spectrum S(f ) = h−1 /f can be transformed
into the Allan variance σ 2 = 2 ln(2) h−1
(roughly speaking, the integral over one octave)
a phase fluctuation is equivalent to a length fluctuation
ϕ
ϕ c
L =
λ =
2π
2π ν0
1 c2
Sϕ (f )
SL (f ) =
2
2
4π ν0
−180 dBrad2 /Hz at f = 1 Hz and ν0 = 9.2 GHz (c = 0.8 c0 ) is equivalent to
√
√
−23
2
−12
SL = 1.73×10
m /Hz
( SL = 4.16×10
m/ Hz)
a residual flicker of −180 dBrad2 /Hz at f = 1 Hz off the ν0 = 9.2 GHz carrier
(h−1 = 1.73×10−23 ) is equivalent to a mechanical stability
!
σL =
1.38 × 1.73×10−23 = 4.9×10−12 m
# don’t think “that’s only engineering” !!!
# I learned a lot from non-optical microscopy
# bulk solid matter is that stable
32
7 – Optical delay line
Delay line theory
7 – optical delay line
34
Rubiola-Salik-Huang-Yu-Maleki, JOSA-B 22(5) p.987–997 (2005)
Pλ
laser
τd = 1.. 100 µ s
(0.2−20 km)
Laplace transforms
phase
detector
EOM
microwave
input
100
mW
_0
τd∼
(calib.)
vo (t)
out
R0 20−40
dB
10
mW
_0
τ∼
Φ(s) = Hϕ (s)Φi (s)
FFT
analyz.
1.55 µm
|Hϕ (f )|2 = 4 sin2 (πf τ )
52 dB
90° adjust
power ampli
Note that here one arm is a microwave cable
Sy (f ) = |Hy (f )|2 Sϕ i (s)
2
4ν
0
|Hy (f )|2 = 2 sin2 (πf τ )
f
Laplace transforms
e−s τ
mixer
−
Φi (s)
Σ
Φo(s)
Vo(s) = k ϕ Φo (s)
990
J. Opt. Soc. Am. B / Vol. 22, No. 5 / May 2005
kϕ
tur
the
log
str
rel
lim
mo
sec
+
Φo (s) = (1−e−sτ ) Φ i(s)
•
•
•
•
delay –> frequency-to-phase conversion
works at any frequency
long delay (microseconds) is necessary for high
sensitivity
the delay line must be an optical fiber
fiber: attenuation 0.2 dB/km, thermal coeff. 6.8 10-6/K
cable: attenuation 0.8 dB/m, thermal coeff. ~ 10-3/K
10 GHz, 10 μs
4.
2
2
Th
sh
"!
In
blo
White noise
7 – optical delay line
intensity modulation
P (t) = P (1 + m cos ωµ t)
photocurrent
qη
i(t) =
P (1 + m cos ωµ t)
hν
microwave power
1 2 ! qη "2 2
P µ = m R0
P
2
hν
shot noise
q2 η
Ns = 2
P R0
hν
thermal noise
Nt = F kT0
total white noise
(one detector)
total white noise
(P/2 each detector)
Sϕ0
35
thermal
! shot
"
# 2 " #2 $
2
hνλ 1
F kT0 hνλ
1
= 2 2
+
m
η P
R0
qη
P
Sϕ0
16
= 2
m
!
hνλ 1
F kT0
+
η P
R0
"
hνλ
qη
#2 "
1
P
#2 $
2
%−156
dBlinerad
7 – optical
delay
kind. Equation (19) deri
36
series expansion
/ Hz&. In practice, the mixer noise can
hardly approach the noise of the microwave amplifier be.
cause of the gain of the latter. The microwave gain, hidsin%z cos -& = 2
den in Eq. (16), is not a free parameter. Its permitted
k=0
range derives from
the
mixer
in
the
! the need of operating
$
"
#2 " # 2
16 hνbelow
F kT
hνλ power.
1
saturation region,
the
maximum
λ 1
0
Sϕ0 = 2
+
holds forThe
twoneglected
detectorsterms “…”
m
ηthePnoiseRfloor
Figure 5 shows
Sqη
function of the
0
!0 as a P
of angular frequency n(%
total optical power for some reference cases.
0 / T / 1. Equation (19)
Threshold power
)
hence the modulation ind
threshold power
thermal
shot
m=
The maximum
F kT0 hνisλ m (
Pλ,t
= V *.
= 0.586
2η
R
q
0
Harmonic distortion
small, but there is no ad
tortion has no first-order
mal). On the other hand
saturation in the photode
only means to increase
new
high-power
signal-to-shot-noise
rati
photodetectors
mW
power and5–10
the dc
bias of
to set and maintain at t
This is due to the possibi
mal sensitivity of the lith
7 – optical delay line
X
Photodetector 1/f noise (1)
2
infrared
iso
(!3dBm)
50% coupler
22dBm
iso
(!26dBm)
photodiodes
under test
power
meter
g=37dB
v(t)
RF
&
(carrier suppression)
FFT
analyz.
IF
LO
phase & aten.
g’=52dB
phase $
(detection of " or #)
power
ampli
PLL
synth.
Fig. 1.
s(t)
=6dB
!90°
monitor
output
100
MHz 9.9GHz
Pµ r(t) hybrid %
!90°
P!
0° 0°
1.32 µ m
YAG (13dBm) EOM
laser
microwave
Scheme of the measurement system.
photodiode
near!dc
Table 1: Flicker noise of the photodiodes.
S (1 Hz)
S (1 Hz)
ϕ
analyzer measures the output spectrum, Sϕ (f ) or Sαα(f ). The where q is the electron
charge, % is the detector responsivity,
estimate
uncertainty
estimate
uncertainty
gain, defined as kd = v/α or kd = v/ϕ, is
m the index of intensity modulation, and P λ the average
!
−7.1 optical power. This is proved
−8.6
"
#
by dividing the spectrum density
HSD30
−122.7
−127.6
gPµ R
0
+3.4 Si = 2qı = 2q%P λ of the the
+3.6
output current i by the average
kd =
− dissipative
,
(3)
loss
#
2 = %2 P 2 1 m2 . The background
square
microwave
current
i
−3.1
−1.8
ac
λ 2
DSC30-1K
−119.8
−120.8
noise take the same value because
where g is the amplifier gain, Pµ the microwave power, R0 +2.4
= amplitude and phase white+1.7
50 Ω the characteristic resistance, and # the mixer ssb loss.
−1.5 they result from additive random
−1.7 processes, and because the
QDMH3
−114.3
−120.2
+1.6 The residual flicker noise is
Under the conditions of our setup (see below) the gain is +1.4
43 instrument gain kd is the same.
experimentally.
dBV[/rad], including the dc
preamplifier. The
notation [/rad]
unit
dB/Hz
dB to be determined
dBrad2 /Hz
dB
The differential delay of the two branches of the bridge is
means that /rad appears when appropriate.
Calibration involves the assessment of kd and the adjustment kept small enough (nanoseconds) so that a discriminator effect
not take place.
With39
this19
conditions,
phase noise of the
The
of through
the ∑ amplifier
is notatdetected
Lett.
p. 1389 the
(2003)
of γ. The gain
is noise
measured
the carrier power
the doesElectron.
7 – optical delay line
•
•
•
•
Photodetector 1/f noise (2)
X
the photodetectors we measured are similar
in AM and PM 1/f noise
the 1/f noise is about -120 dB[rad2]/Hz
other effects are easily mistaken for the
photodetector 1/f noise
environment and packaging deserve
attention in order to take the full benefit
from the low noise of the junction
Figure 2: Example of measured spectra Sα (f ) and Sϕ (f ).
W:
waving a hand
0.2 m/s,
3 m far from the system
B:
background noise
P:
photodiode noise
modulator (EOM) is rejected. The amplitude noise of the source is re
to the same degree of the carrier attenuation in ∆, as results from the g
properties of the balanced bridge. This rejection applies to amplitude noi
to the laser relative intensity noise (RIN).
The power of the microwave source is set for the maximum modulation
m, which is the Bessel function J1 (·) that results from the sinusoidal respo
the EOM. This choice also provides increased rejection of the amplitude n
the microwave source. The sinusoidal response of the EOM results in har
distortion, mainly of odd order; however, these harmonics are out of the s
bandwidth. The photodetectors are operated with some 0.5 mW input
which is low enough for the detectors to operate in a linear regime. This
F:
after bending a fiber, 1/f
S:
single spectrum, with optical A:
averageaspectrum,
withsuppression
optical
possible
high
carrier
(50–60
dB) in mistakes
∆, mistakes
which
isaround
stable f
Figure
3:
Examples
of
environment
effects
and
experimental
Figure
3: Examples
ofisolators
environment effects and
experimental
around
noise
can
increase unpredictably
connectors and no isolators
connectors
and
no
duration
thethe
measurement
(half
anB:
Background
hour),
and noise
also
provides
aB)
high
the corner.
All
plots
instrument
Background
noise
(spectrum
B) re
the
Allofthe
plots
showshow
the the
instrument
(spectrum
background
noise
B:
background noise
B:
corner.
background
noise
RIN
and
of the
thePhotodiode
noise of
the
∆(spectrum
amplifier.
The
coherence
and
thelaser
noise
spectrum
of
pair
P).
Plot
1 spectrum len
andof
thethe
noise
spectrum
of the
Photodiode
pair
(spectrum
P).noise
Plot
1 spectrum
P:
photodiode
P:
photodiode noise
P:
photodiode noise
W: experimentalist
the
experimentalist
Waves
aexperiment
hand
gently
(≈about
0.2 m/s),
3far
m away
far
from
the
YAG
laser used
in
our
is
andaway
all
optical
W: the
Waves
a hand
gently
(≈ 0.2
m/s),
31mkm,
from
thethe sig
Flicker (1/f) noise
7 – optical delay line
37
experimentally determined (takes skill, time and patience)
GaAs: b–1 ≈ –100 to –110 dBrad2/Hz,
SiGe: b–1 ≈ –120 dBrad2/Hz
amplifier
photodetector b–1 ≈ –120 dBrad2/Hz
Rubiola & al. IEEE Trans. MTT (& JLT) 54(2) p.816–820 (2006)
mixer b–1 ≈ –120 dBrad2/Hz
contamination from AM noise (delay => de-correlation => no sweet point
(Rubiola-Boudot, IEEE Transact UFFC 54(5) p.926–932 (2007)
optical fiber
The phase flicker coefficient b–1 is about independent of power
in a cascade, (b–1)tot adds up, regardless of the device order
S!(f) , log-log scale
The Friis formula applies to white phase noise
b–1 ! const. vs. P0
b
b0 = FkT0 / P0
–1 f –1
F1 kT0
(F2 − 1)kT0
b0 =
+
+ ...
2
P0
P0 g1
b0 , higher P0
In a cascade, the 1/f noise just adds up
b0 , lower P0
f'c
f"c
fc = ( b–1 / FkT0 ) P0 depends on P0
f
(b−1 )tot =
m
!
i=1
(b−1 )i
Single-channel instrument
7 – optical delay line
Modulateur
Mach-Zehnder
JDS Uniphase
EOM
Diode Laser
JDS Uniphase
=
1,5 µm
laser
X
Ampli
Photodiode
AML
DSC40S SiGe ampli
8-12GHz
Contrôleur
de polarisation
Fibre
2 Km
2 km
Analyseur FFT
(HP 3561A)
5 dBm
RF
Att
3dB
DC
sapphire oscillator
LO
Ampli DC
10 dBm
Coupleur
10 dB
Déphaseur
phase
Ampli RF
ISO
ISO
Oscillateur Saphir
• The laser RIN can limit the instrument sensitivity
• In some cases, the AM noise of the oscillator under test
turns into a serious problem (got in trouble with an
Anritsu synthesizer)
FFT
7 – optical delay line
Measurement of a sapphire oscillator
Mesure de bruit de phase oscillateur Saphir (Ampli Miteq) avec différents retards optiques
"
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• The instrument noise scales as 1/τ, yet the blue and black plots overlap
magenta, red, green => instrument noise
blue, black => noise of the sapphire oscillator under test
• We can measure the 1/f3 phase noise (frequency flicker) of a 10 GHz
sapphire oscillator (the lowest-noise microwave oscillator)
• Low AM noise of the oscillator under test is necessary
38
7 – optical delay line
nt.
re fast
ors as
Dual-channel (correlation) instrument
Salik, Yu, Maleki, Rubiola, Proc. Ultrasonics-FCS Joint Conf., Montreal, Aug 2004 p.303-306
Fig. 6. The dual photonic delay line cross correlation phase noise measureuses cross spectrum to reduce the background noise
ment setup.
requires two fully independent channels
separate lasers for RIN rejection
version is not useful because of the
there is optical-input
no significant
amount of noise added by the 2.1 km
insufficient rejection of AM noise
fibers used
above
the
thermal
noise
of
the
amplifier.
Therefore,
implemented at the FEMTO-ST Institute
the noise floor of our photonic delay line measurement system
39
7 – optical delay line
40
Dual-channel (correlation) measurement
J.Cussey 20/02/07 Mesure200avg.txt
!#"
J.Cussey 20/02/07
20/02/07 Mesure200avg.txt
J.Cussey
J.Cussey 20/02/07Mesure200avg.txt
Mesure200avg.txt
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!#"
–20
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residual phase noise (cross-spectrum),
short delay ("!0), m=200 averaged spectra,
unapplying the delay eq. with "=10 "s (2 km)
!%"
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7 – optical delay line
Measurement of the optical-fiber noise
•
•
matching the delays, the oscillator phase noise cancels
this scheme gives the total noise
2 × (ampli + fiber + photodiode + ampli) + mixer
thus it enables only to assess an upper bound of the fiber noise
41
7 – optical delay line
Phase noise of the optical fiber
•The method enables only to assess an upper bound of the fiber noise
b–1 ≤ 5×10–12 rad2/Hz for L = 2 km (–113 dBrad2/Hz)
•We believe that this residual noise is the signature of the two GaAs
power amplifier that drives the MZ modulator
42
7 – optical delay line
ν0
fL =
2Q
Delay-line oscillator
Qeq = πν0 τ
1
fL =
4π 2 τ 2
Qeq=3×105 ← L=4km
fL=8kHz
10–11
Leeson
formula
fL2
Sϕ (f ) ! 2 Sψ (f ) for f " fL
f
b–3 = 6.3×10–4 (–32 dB)
43
h−1 = b−3 /ν02
6.3×10–24
σy2 = 2 ln(2) h−1
8.8×10–24
σy ! 2.9×10
−12
Allan deviation
E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN13 9780521886772
Delay-line oscillator
7 – optical delay line
44
Phase noise of the opto-electronic oscillator (4 km)
S!(f), dBrad2/Hz
0
–20
our OEO
b–3=10–3 (–30dB)
–40
–60
Agilent E8257c, 10 GHz,
low-noise opt.
–80
Wenzel 501-04623 OCXO
100MHz mult. to 10 GHz
–100
–120
–140
E. Rubiola, jun 2007
OEO: Kirill Volyanskiy, may 2007
–160
101
•
•
•
•
102
103
frequency, Hz
1.310 nm DFB CATV laser
Photodetector DSC 402 (R = 371 V/W)‫‏‬
RF filter ν0 = 10 GHz, Q = 125
RF amplifier AML812PNB1901 (gain +22dB)‫‏‬
104
105
expected phase noise
b–3 ≈ 6.3×10–4 (–32 dB)
In progress
7 – optical delay line
input F
X
out F
shared fiber
out B
a single fiber is used for both directions
environmental effects are the same
some random effects are independent
input B
8 – Optical resonators
8 – optical resonators
Example of quartz small resonator
46
air
air
Technology: dedicated leathe
• an air-spindle motor for lowest vibration
(from a hard-disk test equipment)
• btw, can you figure out what a hard disk is?
• 3.5” & 7200 rpm => ~ 200 km/h
• 1 (μm)2 bit area, 50 nm head–disk distance
air
•
Small resonators
47
air
8 – optical resonators
air
air
•
Surface metrology: ready
•
A few resonator already made
• quartz, 7 °Mohs (technology training, not for serious oscillators)
• CaF2 4 °Mohs, too soft for serious precision machining
• MgF2 (~6 °Mohs) harder than CaF2, more suitable to machining
•
Achieved Q=3x108 with MgF2 resonator
(still low, but it goes with tapered-fiber coupling)
•
Achieved stable coupling with tapered fiber
8 – optical resonators
Dedicated leathe
48
8 – optical resonators
49
Disk resonator – surface characterization
disk
surface
Fig.F3.1-4: Sillons observables à la surface du disque
surface
rugosity
Fig.F3.1-4: Sillons observables à la surface du disque
10
9
10
(nm)
8
9
7
(nm)
8
6
7
0
10
20
30
40
( m)
50
60
70
80
Rt
Ra
Rq
Rsk
Rku
Rp
Rv
Rz
Rtm
4.5
0.7
0.9
-0.3
2.8
2
2.6
3.7
3.7
Rt
Ra
Rq
Rsk
Rku
Rp
Rv
Rz
Rtm
4.5
0.7
0.9
-0.3
2.8
2
2.6
3.7
3.7
8 – optical resonators
coupling: prism-shaped optical fiber
electron-beam
microscope photo
precision
saw
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Let technologists have fun with their
weird
equipment
0
8
16
24
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I don’t think that the fiber machining is that critical (also experience)
40
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48
8 – optical resonators
Raman oscillations
•The Raman amplification is a quantum phenomenon of nonlinear origin
that involves optical phonons.
•An amplifier inserted in a high-Q cavity turns into an oscillator, like
masers and lasers.
•Oscillation threshold ~ 1/Q2
•In CaF2 pumped at 1.56 μm, Raman oscillation occurs at 1.64 μm
•Due to the large linewidth, the Raman oscillation appears as a bunch of
(noisy) spectral lines spaced by the FSR (12 GHz, or 100 pm in our case)
•Raman phonons modulate the optical properties of the crystal, which
induces noise at the pump frequency (1.56 μm)
51
8 – optical resonators
8 mm
5.5 mm
CaF2
optical
resonator
High temperature gradient
•cross section of the field region 1 μm2
•CaF2 thermal conductivity 9.5 W/mK
•dissipated power 300 μW
•wavelength 1.56 μm
•air temperature 300 K
•still air thermal conductivity 10 W/m2K
•simplification: the heat flow from the mode region is
uniform
bottom plane at a reference temperature
inner bore at a reference temperature
52
Thermal effect on frequency
8 – optical resonators
8 mm
5.5 mm
CaF2
optical
resonator
laser scan
•wavelength 1.56 μm (ν0=192 THz)
•Q=5x109 –> BW=40 kHz
•a dissipated 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>1011 is possible with CaF2 and other crystals!!
calibration (2 MHz phase modulation)
53
8 – optical resonators
Low-power oscillator operation
Assume:
λ = 1560 nm
ρ = 0.8 A/W
Shot noise (m=1)
IRM S
1
= √ ρP λ
2
SI = 2qI = 2q ρP λ
1 ρP λ
SN R =
4 q
In practice, –131 dBrad2/Hz
R = 50 Ohm
(Pλ)peak = 2x10–5 W (20 μW)
Thermal noise (m=1)
1
IRM S = √ ρP λ
2
4kT
4F kT
SI =
or
R
R
2 2
1 ρ P λR
SN R =
8 kT
In practice, –110 dBrad2/Hz
with F=0 dB (!!!)
•Thermal noise is dominant: below threshold, SNR ~ 1/Pλ2
•Thermal noise can be reduced (10 dB or more?) using VGND
amplifiers
•What about flicker of photodetectors with integrated VGND amplifier?
•Dramatic impact on the (phase) noise floor
54
8 – optical resonators
Small resonators
•
Let us dream
• diamond: probably chemical purity may be a problem
(insufficient transparence)
• sapphire: think more about it
(we can learn a lot from the microwave technology)
•
Last-minute news
MgF2 seems to have a turning point of the thermorefractive index
• 74 °C, extraordinary wave
• 176 °C, ordinary wave
X
9 – Non-linear AM
oscillations
9 – non-linear AM
Nonlinear model
X
9 – non-linear AM
A complex envelope equation
X
9 – non-linear AM
Stability of the oscillating solution
X
9 – non-linear AM
A Hopf bifurcation
X
9 – non-linear AM
Hopf bifurcation, obser ved
The Hopf bifurcation leads to the emergence
of robust modulation side-peaks in the Fourier spectrum, which
may drastically affect the phase noise performance of OEOs
X