# Phase _amp; Frequency Noise Metrology

Document Sample

```					Phase & Frequency
Noise Metrology
Enrico Rubiola

Outline
•   Introduction
•   Measurement methods
•   Microwave photonics
•   Electronic and optical components
•   AM noise and RIN

Though frequency standards are                                                                    2

moving to optics (and beyond),
eftf-fcs 2003                                                                                       12

RF and microwaves are inevitable
Example of Measured Spectrum

−164
Sα|ϕ ( f )
P0 = 19 dBm
dB[rad 2 ]/Hz                        avg 10 spectra
single channel
ν0 = 9.2 GHz
−174
Narda CNA 8596          Microwave
s.no. 157               circulator
−184

−194
instrument noise

Fourier frequency, Hz
−204
10                      102         103                     104

Lower phase noiselikeis required
no post-processing is used to hide stray signals, vibrations or the mains
Phase noise & friends                                                                                                        3

v(t) = Vp [1 + α(t)] cos [1 + ϕ(t)]                                             random walk freq.
Sϕ(f)                    b−4 f−4                                     signal sources only

random phase ﬂuctuation
Sϕ (f ) = PSD of ϕ(t)                                                                  flicker freq.
b−3 f−3
both signal sources
and two-port devices
power spectral density                                                               white freq.
b−2 f−2
it is measured as                                                                                                 flicker phase.
b−1 f−1
Sϕ (f ) = E {Φ(f )Φ∗ (f )}             (expectation)                                                                              white phase   b0
f
Sϕ (f ) ≈ Φ(f )Φ∗ (f ) m               (average)
f2/ ν2
1                                                                                       x         0

L(f ) = Sϕ (f ) dBc
2                                             Sy (f)
h−2 f−2                                         h2 f2

random                                                     white phase
walk freq.        h−1 f−1
random fractional-frequency ﬂuctuation                                                                   h1 f
h0
˙
ϕ(t)                   f2                                            flicker freq.                       flicker phase
y(t) =              ⇒     Sy = 2 Sϕ (f )                                                    white freq.
f
2πν0                   ν0

2
Allan variance                               σy (τ)
(two-sample wavelet-like variance)
1
freq.
2
σy (τ )
2
=E       y k+1 − y k             .                     flicker phase
white phase
drift
2                                                           white freq.
flicker freq.            random walk freq.
(2π ) 2
approaches a half-octave bandpass ﬁlter (for white noise),                          h0 /2 τ        2ln(2)h −1
6
h−2 τ
τ
hence it converges for processes steeper than 1/f
4
Mechanical stability
Sϕ (f )
Any phase ﬂuctuation can be
h−                                  converted into length ﬂuctuation
1 /f
1 c
f                           L=
2π ν0
1 c                        b–1 = –180 dBrad2/Hz and ν0 = 10 GHz is equivalent to
L=                                       SL = 1.46x10–23 m2/Hz at f = 1 Hz
2π ν0
SL (f )   1.5x10–23 m2/Hz @ 1 Hz
h−                                   Any ﬂicker spectrum h–1/f can be
1 /f
converted into a ﬂat Allan variance
f
σL = 2 ln(2) h−1
2

σ 2 = 2 ln(2) h−1

A residual ﬂicker of –180 dBrad2/Hz at f = 1 Hz
σL (τ )                                         off the 10 GHz carrier is equivalent to

4.5x10–12 m
σ2 = 2x10–23 m2 thus            σ = 4.5x10–12 m
τ        for reference, the Bohr radius of the electron is R = 0.529 Å

•    Don’t think “this is just engineering” !!!
•    Learn from non-optical microscopy (bulk matter, 5x10–14 m)
•    Careful DC section (capacitance and piezoelectricity)
•    The best advice is to be at least paranoiac
1 – Measurement methods
6
Correlation measurements
x = c–a     Two separate instruments
measure the same DUT.
Only the DUT noise is common
DUT             c(t)           FFT
a(t), b(t) –> instrument noise
c(t) –> DUT noise
y = c–b
b(t)
phase noise measurements
basics of correlation                                       DUT noise,        a, b    instrument noise
normal use        c       DUT noise
Syx (f ) = E {Y (f )X ∗ (f )}                               background,       a, b    instrument noise
= E {(C − A)(C − B)∗ }                             ideal case        c=0     no DUT
background,       a, b    instrument noise
= E {CC ∗ − AC ∗ − CB ∗ + AB ∗ }                   with AM noise     c≠0     AM-to-DC noise
= E {CC ∗ }
0  0      0                                       single-channel
Syx (f ) = Scc (f )

S!(f)
in practice, average on m realizations
Syx (f ) = Y (f )X ∗ (f ) m                                                                   1/"m
= CC ∗ − AC ∗ − CB ∗ + AB ∗ m
correlation
0 as
= CC ∗     m   + O(1/m)                  1/√m
frequency
E. Rubiola, The magic of cross-spectrum measurements from DC to optics, http://rubiola.org
7

Cross-spectrum, increasing m
|Re{Syx}| with C≠0,
#!                                               #!                                               #!                                               #!
m=1 g=0.32                                       m=2 g=0.32                                       m=4 g=0.32                                        m=8 g=0.32
|Sxx|                                            |Sxx|                                            |Sxx|                                             |Sxx|
#        |Re{Syx}|                               #                                                #                                                #
|Re{Syx}|                                        |Re{Syx}|                                         |Re{Syx}|
!%#                                              !%#                                              !%#                                              !%#
|Scc|                                            |Scc|                                            |Scc|                                             |Scc|

!%!#                                             !%!#                                             !%!#                                             !%!#

frequency                                        frequency                                        frequency                                                     frequency
!%!!#                                            !%!!#                                            !%!!#                                            !%!!#
!           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!           !             "!          #!!          #"!       \$!!
#!                                               #!                                               #!                                               #!
m=16 g=0.32                                      m=32 g=0.32                                      m=64 g=0.32                                       m=128 g=0.32
|Sxx|                                            |Sxx|                                            |Sxx|                                             |Sxx|
#                                                #                                                #                                                #
|Re{Syx}|                                        |Re{Syx}|                                        |Re{Syx}|
!%#                                              !%#                                              !%#                                              !%#
|Re{Syx}|
|Scc|                                            |Scc|                                            |Scc|                                             |Scc|

!%!#                                             !%!#                                             !%!#                                             !%!#

frequency                                        frequency                                        frequency                                                     frequency
!%!!#                                            !%!!#                                            !%!!#                                            !%!!#
!           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!           !             "!          #!!          #"!       \$!!
#!                                               #!                                               #!                                               #
m=256 g=0.32                                     m=512 g=0.32                                     m=1024 g=0.32                                                 aver
age
|Re{Syx}|
|Sxx|                                            |Sxx|                                            |Sxx|
#                                                #                                                #

|Re{Syx}|                                        |Re{Syx}|                                        |Re{Syx}|                             !%#
!%#                                              !%#                                              !%#                                                                       devi
|Scc|                                            |Scc|                                            |Scc|                                                                ation

!%!#                                             !%!#                                             !%!#                                                       &'()*+,)-./0!+)1!##!#!\$2!!3#4!05+6)789
m
:%6;5'<(0=*0,/*\$!!>
!%!#
frequency                                        frequency                                        frequency                #                 #!              #!!             #!!!
!%!!#                                            !%!!#                                            !%!!#
!           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!           !           "!   #!!    #"!        \$!!

Increasing m:
ﬁrst, Syx decreases => single-channel noise rejection
then, Sxx shrinks => increased conﬁdence level
E. Rubiola, The magic of cross-spectrum measurements from DC to optics, http://rubiola.org
8

The thermal noise is rejected as any signal.
The limit Sφ = P0/kT does not apply

A                      X=A+B
T2                   0º

0º
Syx = k (T2 – T1) / 2
B           180        X=A–B
T1                        º
0º

X and Y are uncorrelated
The cross spectrum is proportional to the temperature difference

C. M. Allred, A precision noise spectral density comparator, J. Res. NBS 66C no.4 p.323-330, Oct-Dec 1962
Application to AM/PM noise: E. Rubiola, V. Giordano, Rev. Sci. Instrum. 71(8) p.3085-3091, Aug 2000
9
Carrier recirculation

gas cell

Invented by J. Hall for gas spectroscopy.
The gain is increased by the number of times the light beam circulates in the cavity

DUT
Also works with RF/microwave carrier, provided the DUT be “transparent”.
For small no. of roundtrips, gives the appearance of “real-time”
10

Bridge (interferometric) method
V0 cos("0t)                                                             ﬂuctuating error δZ => noise sidebands
pump
ℜ{δZ} => AM noise x(t) cos(ω0t)
0º         –90º       ℑ{δZ} => PM noise –y(t) sin(ω0t)
(microwave)
x(t)
bridge    –        error ampliﬁer
!(t)

FFT
!
+    hybrid                                           y(t)
junction
Z            Z DUT

x(t) cos("0t) – y(t) sin("0t)

Basic ideas
• Carrier suppression => the error ampliﬁer cannot ﬂicker: it does know ω0
• High gain, due to the (microwave) error ampliﬁer
• Low noise ﬂoor => the noise ﬁgure of the (microwave) error ampliﬁer
• High immunity to the low-frequency magnetic ﬁelds due to the microwave
ampliﬁcation before detecting
• Rejection of the master oscillator’s noise
• Detection is a scalar product => signal-processing techniques
Derives from H. Sann, MTT 16(9) 1968, and F. Labaar, Microwaves 21(3) 1982
Later, E. Ivanov, MTT 46(10) oct 1998, and Rubiola, RSI 70(1) jan 1999
11

Actual block diagram
E. Rubiola, V. Giordano, Rev. Sci. Instrum. 73(6) pp.2445-2457, June 2002

channel b (optional)
rf virtual gnd                RF I−Q                  G          B
null Re & Im                     detect Q v2        matrix     matrix w2
g ~ 40dB     LO
inner interferometer                                                    pump
x( t )                                                                                     FFT
CP1                      CP2                       R0       CP4
channel a                     analyz.
DUT
Δ’        CP3                            RF
R 0 =50 Ω

γ                                                           I−Q                       G           B
detect Q v2               matrix      matrix w2
10−20dB                             g ~ 40dB     LO
R0
coupl.
arbitrary phase                                                pump                          G: Gram Schmidt ortho
atten                                                                                                        normalization
B: frame rotation
power splitter

manual carr. suppr.
automatic carrier
atten                     ’    γ’                                      suppression control                   I−Q detector/modulator
var. att. & phase                       atten                     diagonaliz.                                  I
RF
RF     I u1               z1                                            Q
arbitrary phase          pump      LO     I−Q               dual         D
atten                                                    modul Q u2        integr z 2   matrix                    LO −90°      0°

Im                                        Concepts
Up                            • Coarse and ﬁne carrier suppression reduces the ﬂicker noise
v(t)/!2
• Scalar product gives v1(t) and v2(t) in Cartesian frame. Linear algebra
v(t)
ﬁxes the arbitrary phase, gain asymmetry and quadrature defect
null ﬂuct                    Re
v(t)/!2                             • Closed-loop control of the carrier suppression works as a RF VGND
Dn
• Correlation is possible, using two ampliﬁers and two detectors
1                                                    • Correlating the signals detected on two orthogonal axes (±45º)
Sud (f ) =   Sα (f ) − Sϕ (f )                                    eliminates the ampliﬁer noise. Works with a single ampliﬁer!
2
12

Example of results
Noise of a by-step attenuator                                                    Background noise of the ﬁxed-value bridge
!!K"C\$                                                                                           !!L"AK
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@A&.'3.%B/,,C                                                                                   ?B'3(43&C2::A
N'3
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53333333333,-6./07                                       2(63!2E3!F
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>!!FKCM?                                                                                         5!!KLA%7                                             89:;*)32(6

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!!L"C\$                                                                                           !!="AK
>!!JKCM?                                                                                         5!!MLA%7

1&2'+('.3'(42(,%56.78                                                                                           G'D(9)(34()HD):&IE3/0
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>!!IKCM?                                                                                         5!!JLA%7
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Noise of a pair of HH-109 hybrid junctions                                               Background noise of the ﬁxed-value bridge (larger m)
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S!, f &                                                    .F@/!G/?H8;B-.                                                                                 0?81&\$@15A*'B)0
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1!!%MEC3                                                                                         :!!II>%;                                                                BC(1)<1'D077>
?7:@A8/.-2                                                                                                                           \$        0)91!0F1!G
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1!!>MEC3                                                                                         :!!JI>%;
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!!GDE\$                                                                                           !\$"">#
1!!CMEC3                                                                                         :!!HI>%;

7:?B-628:B/:57?8
456-78-/0-8968:;<=/*+                                                                         L(E)6*)1<)*ME*7'NF134
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1!!DMEC3                                                                                         :!!KI>%;
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Averaged spectra must be smooth
Average on m spectra: conﬁdence of a point improves by 1/m1/2
interchange ensemble with frequency: smoothness 1/m1/2
13

The complete machine (100 MHz)
14

A 10 GHz experiment
(dc circuits not shown)
15

Comparison of the background noise
−140                                                                                  correl. & avg.
mixer, interferometer
−150                   sat
ura
ted                                                  nested interferometer
int           mix
dou        e rfe         e
−160                   ble          rom r
int         ete
cor        erf         r                                       saturated mixer
ero
rel           me
. sa
−170                                t. m ter
res                         ix.
idu
al f
res           lick                                                  interferometer
idu            er,
−180        res al f                by−
idu lick                 ste
res      al f      er,            p in                                  correl. saturated mixer
idu lick fix                       ter
al f       er, ed i                fer
−190                    lick       fix nte
er, ed i rfer
om
ete
fix      nte      om           r                           double interf.
ed       rfe
int       rom eter
erf
−200                                            ero eter                          measured floor, m=32k
me
ter
, ±4
5°d
−210                                                                 ete
ctio    expected, m=4x106
n
Fourier frequency, Hz
−220
1                         10                   10 2                 10 3             104               105
16
AM-PM calibration

• In most cases the phase detector is a double-balanced
mixer saturated at both inputs
• In a double-balanced mixer, the offset is affected by power,
• AM noise is detected as it was PM noise, which is
conceptually incorrect
• kφ/kα can be as low as 5

• In the bridge, the effect of AM noise is divided by the
microwave gain
17
Primary AM-PM calibration                                  E. Rubiola – unpublished

A                                                                                       RF input                                                  RF output

− y sin( ω 0t)
nal
ted sig

n
odula                                                                     V0 cos(ω 0t)                                 V0 cos(ω 0t) + x cos(ω 0t) − ysin( ω 0t)

atio
m                                                                                                     carrier

dul
carrier

mo
V0 cos(ω 0t)

x cos(ω 0t)
LO                                                                    I−Q modulator
B                   V0                                                                                      I−Q RF               modulation sidebands
α=                                                                                          modul                                                                    LO pump
x                                                                                                               x cos(ω 0t) − y sin( ω 0t)
amplitude−modulated signal                                                                          I       Q
in−phase                                                                                                                                    90°       RF
carrier                modulation
x    y       modul. input                                                            output
V0 cos(ω 0t)              x cos(ω 0t)

C                    V0
ϕ=
y                                                                AM and PM can be deﬁned as                                                               I         Q

αrms =
− ysin( ω 0t)

signal
modulation

P0 = power of the carrier
Px /P0

ulated
−mod
phase                         ϕ                                                                                         Px = power of the in-phase
carrier
V0 cos(ω 0t)                                                                                                       sidebands
ϕrms =             Py /P0            Py = power of the quadrature
sidebands

Basic ideas
•    Generate reference AM and PM by adding sidebands to a carrier
•    Power detectors provide absolute reference of PM as a null of AM
•    Accurate (0.02 dB) power-ratio measurement with commercial power-meters
•    Correct IQ modulators and detectors with linear algebra (2x2 matrices)
•    Transfer the accuracy of a LF (kHz range) lock-in ampliﬁer to RF/microwave
•    Worst-case accuracy 0.3 dB => improvement in progress
18

Bridge (interferometric) instrument
power                                    AM out
meter

inner interferometer                                     RF                                                       channel b
virtual gnd
CP1                                 CP2                                                              ampli & detector                  readout
I   v 1b                w 1b
°           DUT                   −9                                    Δ"         CP4
−90                                     0   °       Δ’           CP3                                          RF    I−Q                  R
−9                                               °
0°          g           detect Q v 2b        matrix     w 2b
0

180 0°
0°                   γ               −9

°
−20 dB
LO
0°

by−pass
dual channel
FFT analyzer
fine carrier control
channel a
−90°         I       0°                                                                         ampli & det.                      readout
LO I     v 1a                w 1a
0° 80°
0°0°

1

Q                                                                                          RF    I−Q                  R
0°                                            g           detect Q v 2a        matrix     w 2a
−90°                 0°                             180 0°
°

from AM out
0°
to AM/PM
automatic carrier control                                                         modul input

RF                 I            u1                       z1
LO           I−Q   Σ                                                       D                                              osc lock−in in
modul Q u2                                z (t)dt        z2   matrix
Σ                                                                                       amplifier

AM
matrix   PM     modulation input

Light blue: work in progress
The dual-bridge contains almost all the blocks needed to calibrate the measurement
E. Rubiola, V. Giordano, Rev. Sci. Instrum 73 6 p.2445–57, jun 2002. Also arXiv:physics/0503015
2 – Microwave photonics
20
Opto-electronic discriminator
Rubiola, Salik, Huang, Yu, Maleki, JOSA-B 22(5) p.987–997 (2005)

Pλ     τd = 1.. 100 µ s                                                           Laplace transforms
phase
(0.2−20 km)                  detector
laser
EOM
1.55 µm                           _
τd∼ 0            R0 20−40                 vo (t)                        Φ(s) = Hϕ (s)Φi (s)
(calib.)              dB                    out

analyz.
100

FFT
microwave          mW                                  10
mW                                               |Hϕ (f )|2 = 4 sin2 (πf τ )
input                            _
τ∼ 0                                   52 dB
power ampli         Qeq = πν0 τ               90° adjust

The short arm can be a microwave cable or a photonic channel                                          Sy (f ) = |Hy (f )|2 Sϕ i (s)
4ν02
Laplace transforms                                                            |Hy (f )|2 = 2 sin2 (πf τ )
f
e−s τ                            mixer                                       990   J. Opt. Soc. Am. B / Vol. 22, No. 5 / May 2005

Φi (s)                              −                          Vo(s) = k ϕ Φo (s)
Φo(s)                                                                                                        tur
Σ                 kϕ                                                                                                 the
+                                                                                                                log
str
Φo (s) = (1−e−sτ ) Φ i(s)                                                                                    rel
lim
mo
•     delay –> frequency-to-phase conversion                                                           10 GHz, 10 μs
10 GHz, 10 μs                             sec
•     works at any frequency
•     long delay (microseconds) is necessary for high sensitivity                                                                                4.
Th
•     the delay line must be an optical ﬁber                                                                                                     sh
ﬁber: attenuation 0.2 dB/km, thermal coeff. 6.8 10-6/K
cable: attenuation 0.8 dB/m, thermal coeff. ~ 10-3/K                                                                                       In
21
Measurement of a sapphire oscillator
Volyanskiy & al., JOSAB (in press). Also arXiv:0807.3494v1 [physics.optics] July 2008.
Modulateur
Diode Laser
Mach-Zehnder                                  Ampli
JDS Uniphase                                                      Photodiode
laser
=1,5 µm                             EOM
JDS Uniphase
DSC40S
AML
SiGe ampli
8-12GHz

Contrôleur                                                                        Analyseur FFT
2 2 Km
Fibrekm                        5 dBm
(HP 3561A)
de polarisation                                                  RF

Att      3dB                                           DC       FFT
sapphire oscillator                                                         LO
Ampli DC
10 dBm

Coupleur               Déphaseur
phase
10 dB

Ampli RF

ISO            ISO

Oscillateur Saphir

• 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
22
Dual-channel (correlation) measurement
Volyanskiy & al., JOSAB (in press) and arXiv:0807.3494v1 [physics.optics] July 2008.
Derives from: E. Salik, N. Yu, L. Maleki, E. Rubiola, Proc. Ultrasonics-FCS Joint Conf., Montreal, Aug 2004 p.303-306

J.Cussey 20/02/07 Mesure200avg.txt
!#"
J.Cussey 20/02/07 Mesure200avg.txt
–20
!#"
!#"
J.Cussey 20/02/07 Mesure200avg.txt
J.Cussey 20/02/07 Mesure200avg.txt
;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5
!#"                                                                          J.Cussey 20/02/07 Mesure200avg.txt
!#"
!%"
residual phase noise (cross-spectrum),
6A.01H.*IE<.J1K!"F3419C.01-/1*.@9*717.1#"DB12E?>*.1%FG51
;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5
;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5
;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5
6A.01H.*IE<.J1K!"F3419C.01-/1*.@9*717.1#"DB12E?>*.1%FG51
–40
!%"
!%"
short delay ("!0), m=200 averaged spectra,
;<9/0=.*17.1>*-?@17.1A=9B.19C.011-/1*.@9*717.1!"DB12E?>*.1#FG5
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6A.01H.*IE<.J1K!"F3419C.01-/1*.@9*717.1#"DB12E?>*.1%FG51
!%"
!%"                                    unapplying the delay eq. with "=10 "s (2 km)
!("
–60
!("                               !
!("  !("
!("                                y   =1
!'"
0 –12
–80                                                      ba
!'"
!'"                                                          sel

6!12781*97#:345
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!'"                                                            ine

6!12781*97#:345
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6!12781*97#:345
6!12781*97#:345
6!12781*97#:345
–100
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!!#"                                                                                                                     FFT e
ag
aver ect

–120
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!!#"   !!#"                                                                                              eff
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!!%"                                                           ag
–140                                     aver ect
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!!%"                                      ef f
!!%"
!!%"
!!("
–160                                                                    FFT e
!!("
ag
aver ect
!!("   !!("
!!("
!!'"                                                                                     eff
–180
!!'"
!!'"
!  !!'"
!" !!'"                                        !"
#
J.Cussey, feb 2007
!"
\$               Fourier frequency, Hz
!"
%
!"
&

)*+,-./0.12345 \$
101                              102                      103                    104                   105
!                                 #                                              %                       &
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!"! !                             !"
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!"# #                     !"
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!"% %                  !"
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Improvements
• Understanding flicker (photodetectors and amplifiers)
• SiGe technology provides lower 1/f phase noise
• CATV laser diodes exhibit lower AM/FM noise
• Low Vπ EOMs show higher stability because of the lower RF power
• Optical fiber sub-mK temperature controlled
23
Delay-line oscillator – operation
H(s)                                                        jω
model output Vo (s)

....                   ....
noise
Vi (s)
+                    free                    V’(s)
o                                            l=+3                    +6π/τ d
initial conditions,                Σ           A
noise, or locking              +                                          true                                            l=+2                    +4π/τ d
signal                                                                  oscillator
output                                           l=+1                    +2π/τ d
delay
β (s) = e−sτd                                                                 l=0                     0
σ
l=−1                    −2π/τ d
in practice, delay + selector
delay
βd(s) = e−sτd
selector
βf (s)
Barkhausen condition                                                     l=−2                    −4π/τ d

for oscillation: Aß = 1                                                  l=−3                    −6π/τ d

....                   ....
General feedback theory                                                                                                              1 ln(A)
V o (s)      1                                                                                                               τd
H (s) =          =
V i (s)   1 A (s)                                                                              20
19
18
delay−line loop, no selection filter
file le−calc−hdly−flt                          A=1
src allplots−leeson
17
16
Delay-line oscillator

|H(jf)|^2
15                                                      A=0.75
14

1
13
12
H (s) =                                                                                                 11                                                      A=0.71

transfer function
1 A esτd
10
9
8
7                                                      A=0.65
6
Location of the roots                                                                                    5
4
A=0.50
1           2                                                                                       3
2
sl =   ln(A) + j    l                       integer l          (        ,      )                         1
A=0.30
0
0                           1                  2              3
d                           d                                                                                                       f * tau

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN 978-0521-88677-2
24
Delay-line oscillator – phase noise
noise                         ω m = (2π/τd ) m             jω
Ψ (s)                                              free     Φ (s)                                                                        ....
+
Σ                 1
Δω = −
2Q µ                    (2π/τd ) µ             µ=+2
phase noise                +                                 phase noise                      τd m
input                                                       output                                                                       µ=+1
delay                                                                                         µ=0
B(s) = e−sτ d                     ωm                                                                      σ
2Q                                                            µ=−1
µ
in practice, delay + selector                                              Δω = − 2Q                        (2π/τd ) µ             µ=−2
τd m
delay                 selector
B(s) = e−sτ d      B(s) = 1 / (1+s τ f )                                                                                              ....

2Q2 µ 2
σ=−
τd m

General feedback theory
Φ(s)        1                                                     1e+08
H(s) =       =                                                                 phase noise |H(jf)|^2                      file le−calc−dly−hphase

Ψ(s)    1 − B(s)
src allplots−leeson
1e+07 delay−line oscillator with selector
1e+06 parameters:
tau = 2E−5 s
Delay-line oscillator                                                    1e+05    m = 2E5
nu_m = 10 GHz
Q = 2000
1 + sτf
10000

H(s) =                                                                    1000
1 + sτf − e−sτd                                                    100
10
Location of the roots                                                        1

2Q2 µ 2        2π 2Q µ                                              0.1
sµ = −             +j µ−                                                  0.01
frequency, Hz
τd m          τd τd m                                                1000                   10000            1e+05                      1e+06

E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge 2008, ISBN 978-0521-88677-2
25
Delay-line oscillator

ν0          Qeq = πν0 τ            Qeq=3×105 ← L=4km
fL =
2Q
1                               h−1 = b−3 /ν0
2
6.3×10–24
fL =                    fL=8kHz
4π 2 τ 2
10–11                  σy = 2 ln(2) h−1
2
8.8×10–24
2
Leeson              fL
formula   Sϕ (f )       S (f ) for f
2 ψ
fL
f
σy      2.9×10−12 Allan deviation
b–3 = 6.3×10–4 (–32 dB)
26
Delay-line oscillator - measured noise
Phase noise of the opto-electronic oscillator (4 km)
0

–20
our OEO
–40            b–3=10–3 (–30dB)

Wenzel 501-04623
–60                                                             OCXO 100 MHz
Agilent E8257c, 10
mult. to 10 GHz
GHz, low-noise opt.
–80

–100

–120

–140
E. Rubiola, apr 2008
OEO: Kirill Volyanskiy, may 2007
–160
101                             102           103                104           105
frequency, Hz

•   1.310 nm DFB CATV laser
•   Photodetector DSC 402 (R = 371 V/W)‫‏‬
•   RF ﬁlter ν0 = 10 GHz, Q = 125                                                               expected phase noise
•   RF ampliﬁer AML812PNB1901 (gain +22dB)‫‏‬                                                     b–3 ≈ 6.3×10–4 (–32 dB)
27
Optical-ﬁber 10 GHz oscillator
!"#
3<,&6%A&=%</.
3425.,&'+6   ></=6<,,+6                                           (&6%&),+.
<5=%>&,.><45,+6

*+,&-.                      *+,&-.
\$%&'          78.96%:%/;.   ,%/+.                     ,%/+.
()%&'         5<6=         B.12                       0.12

78.&25,%@%+6 78.8%,=+6

CC      3?<=<9+=+>=<6

D==                             3?<=<9+=+>=<6

• use positive feedback with a short cable (3-5 ns) in the feedback path to implement
the mode selector
!"#\$#%&\$&'\$#()*+,-*./, ﬁlter
• the positive feedback also increase the ampliﬁer gain
(AML SiGe parallel ampliﬁers exhibits lowest ﬂicker, but low have gain 22 dB)
0
• use the 2-km (10 µs) path to eliminate the 50-kHz noise peak due to the 4-km (20 µs)
• the microwave power is changed by adjusting the laser power
-2 0
• high noise ﬁgure, due to the two power splitters/combiners
-4 0                        Kiryll Volyanskiy, jan 2008
e                                                 a
Le r´sultat de mesure du bruit de phase d’oscillateur ` trois boucles
28
Regenerative optical-ﬁber 10 GHz oscillator
e    e
pr´sent´ sur la ﬁgure 4.

0
Prf is given, thus V0 =(2RP)1/2                                     b–4 = –7 dBrad2/Hz
ﬁle 923-kirill-oeo
E.R & K.Voliansky, jan 2008

Vπ is estimated (4.5 V at 10                               !20

GHz)                                                       !40

Use
!60

πV0
m = 2J1                                                !80
Vπ                                                                                                          the power is measured
at the ampliﬁer output
9dBm
Get                                                       !100
11dBm
8dBm
P, dBm   Vp, V   πV0/Vπ      m                            !120     b–2 = –50.5 dBrad2/Hz (8 dBm)

11     1.122   0.860    0.783                                                b–2 = –53 dBrad2/Hz (9 dBm)
!140
9     0.891   0.683    0.644                                    (S!)min = –142 dBrad2/Hz (11 dBm)         b–2 = –57 dBrad2/Hz (11 dBm)

!160
8     0.794   0.6.09   0.581                              10
1
10
2
10
3
10
4
10
5

Frequency (Hz)

The oscillator phase noise                                                                   a
Fig. 4 – Courbes de mesure de bruit de phase d’oscillateur ` trois bou
Feeding 8dBm – puissance qui est mesur´model
(11dBm, 9dBm, the available data in theee ` la sortie de l’ampliﬁ
a
minima are 6 dB lower than        teur)                                  2
b0=N/P0 (white noise)                              8    F k T hν            1        hν 1    B 0                 l                              l
(Sϕ )min =                                                                              2   +2
Nous observons que 2 pic `R0
m le                 qη     Pl e       η Pl
e
a 50kHz est supprim´ consid´rablement.
m = 0.725 (Prf=11 dBm)                we get                        e e            e
On conclut que nous avons am´lior´ la stabilit´ de phase d’un oscilla
(Sφ)min = –142 dB                     e                                                    e
opto´lectronique par l’utilisation des deux boucles suppl´mentaires.
P0 = 6.4 μW (-22 dBm)
F = 10 dB (incl. couplers)
η = 0.6                               Pl ≈ 0.71 mW                                                           3
νl = 194 THz                          There is room for engineering
3 – Electronic and optical
components
E. Rubiola – FCS 2004 2004
E. Rubiola – FCS                                                                                                4    430

Flicker electronic &microwave components
Flicker noise in and optical amplifiers
Flicker in noise in RFRF and microwave amplifiers
no carrier
no carrier                                                        noise
noise
S(f)   S(f)                                     S(f) S(f)
near-dc flicker
near-dc flicker                                                      up-conversion
up-conversion
near-
no ﬂicker
dc                        f  f                                                          f   f
noise              ω0 = ?                                                         ω0
random modulation near-dc noise
random modulation fromfrom near-dc noise
stopband                                                    output bandwidth
output bandwidth , stopband, !t " , , !t "
n, n
n , !t " n !t "                  noise-free
noise-free
vi !t "#V i cos!\$
vi !t "#V i cos!\$0 t " 0 t "                                                 v
v o !t " o !t "
AM AM             PM PM         a1   a1
carrier       near-dc noise
v v !t
o
jω0 t
i
i      %
% substitute
modulated signal: v i (t) o#VVe cos!\$ (t) 0 t "&m(t) !t " cos!\$is t "'m , !t "sin"sin !\$ 0 a"
cos!\$ + , n , " ,
modulated signal: !t " = " i # V + n t "&mjn, !t ncos!\$ t "'m , , n ,in ,n’ and n”
0
carrier
the parametric nature ,of 1/f
noise 0 hidden   n , !t !\$ t " t
0                              0   ( (a
1     1
carrier                    AM
AM noisenoise                PM
PM noisenoise
(careful, this hides the down-conversion)

the (t) =
vo simplest i # + a vi (t)2+ . . .
the simplest a1 vv (t) v o #2a1 x&a 2 x 2withwithx # V # V i cos!\$0 t n &"n!t " *n !t**+1+1
2
o
a1 x&a 2 x                   x cos!\$ t " & " !t             "n!t "*
nonlinearity
nonlinearitynon-linear (parametric) ampliﬁer              i          0
carrier near-dc
carrier            near-dc
yields:              expand and select the ω0
yields:
% %                                                    ( (
terms
v o# " a# a1 V i cos!\$0 t "&a2 cos2cos2 !\$0 t "&2 V i"n !t " cos!\$0 t "&n"2 !t "
2
v o !t "  !t     V cos!\$0 t "&a 2 V i2     V i !\$ t "&2 V n !t cos!\$ t "&n 2 !t
1 i                              0         i              0
The noise sidebands are
2 (t) +
vo (t) = Vi a1 + 2a2 n a n!t "jn (t) ejω0 t
2 a 2 n!t "
2                                   to 2 2 2
proportional 2 atheainput carrier
AM noise: #
AM noise: )!t " )!t " #                modulation index: m
modulation index: m # # a
a1 a1
get AM and PM noise                            a1 1
PM noise originates in the same way, but for a phase shift in the product
PM noise originates in the same way, but for a 90°90° phase shift in the product
a2                      a2
α(t) = 2          n (t)          ϕ(t) = 2              n (t)              The AM and the PM noise are
a1                                   a1                    independent of Vi , thus of power
There is also a linear parametric model, which yields the same results
Avantek UTC573, 10 MHz

!130       2Amps ! Pin=!5dBm
23 Jan 07
31

Ampliﬁer ﬂicker noise – experiments
2Amps ! Pin=!22dBm
!140
2Amps ! Pin=!19dBm

2      !150

!160
−120
−125       1Amp ! Pin=!5dBm                                           SiGe LPNT32
bias 2V, 20 mA
Phase noise vs. power
!170
−130                                                       3 dB            NMS floor
R. Boudot 2006
• The 1/f phase noise b–1 is about independent of
−135
!180
−140 1                 10            100              Pin=−15dBm 10000
1000                   100000               power
Pin=−10dBm
−145                                            f, Hz
!50
−150
Pin=−5dBm                   • The white noise b0 scales up/down as 1/P0, i.e., the
JS2
inverse of the carrier power
Pin= !9dBm
−155       29 Jan 2007
F IG                                                     Pin= !6dBm
!70 . 18 – Phase noise of 1-stage and 2 stages cascaded ampliﬁers at 10MHz.
−160                                                       Pin= !4dBm
• Describing the 1/f noise in terms of fc is misleading

−165                                                      Pin= !2 dBm
!90      b!1= !106 dB.rad2 /Hz                            Pin= !1dBm
−170
−175
!110 1
Pin=0dBm
because fc depends on the input power
10              100            1000          10000         100000
f, Hz
!120
!130
Avantek UTC573, 10 MHz
23 Jan 07
• The expected ﬂicker of a cascade increases by:
!130
!150
2Amps ! Pin= !5dBm
/Hz

3 dB, with 2 ampliﬁers
3Amps ! Pin= !24dBm
!140
!170

1              10             100                 1000         10000          100000

!150
f, Hz
5 dB, with 3 ampliﬁers
!160         F IG . 10 – Phase noise of 1 ampliﬁer JS2 vs Pin at 10 GHz.
1Amp ! Pin=!5dBm                                                                     Regenerative ampliﬁers
!170                                                        5!6 dB       NMS floor              • Phase noise increase as the squared gain because
!180
1               10            100                 1000         10000            100000
the noise source at each roundtrip is correlated
f, Hz
!50
JS2, 10 GHz
!70     15 Feb 2007
F IG . 19 – Phase noise of 1-stage, 2-stages and 3-stages cascaded ampliﬁers at 10MHz.                   Phase noise of paralleled ampliﬁers
• Connecting two ampliﬁers in parallel, the phase-
/Hz

!90
noise ﬂicker is expected to decrease by 3 dB

Single amplifier
!110                                                         Pin= −6 dBm
12
!130

!150                                    2 parallel amplifiers
Pin= −5 dBm
!170
1              10             100                 1000         10000          100000
f, Hz
32
Photodetector 1/f noise                                                                                                2
Rubiola, Salik, Yu, Maleki, MTT 54(2) p.816-820, Feb 2006
infrared
1.32 µ m
iso    P!             Pµ r(t) hybrid %
YAG (13dBm) EOM                                                                  !90°                             =6dB
laser                                                   (!3dBm)      (!26dBm)                                   RF            v(t)

0° 0°
photodiodes                            g=37dB                                   FFT
50% coupler             under test                                                IF                  analyz.
iso                         s(t)                &             LO
22dBm                                                                                                          g’=52dB
monitor                                                                                !90°
output                                  power                         phase & aten.                        phase \$
meter
(carrier suppression)               (detection of " or #)

100                          power
MHz 9.9GHz                   ampli
PLL
synth.                                                        microwave                                                    near!dc
Table 1: Flicker noise of the photodiodes.
Fig. 1.   Scheme of the measurement system.
photodiode               Sα (1 Hz)                          Sϕ (1 Hz)
estimate uncertainty               estimate    uncertainty
analyzer measures the output spectrum, Sϕ (f ) or Sα (f ). The where q is the electron charge, is the detector responsivity,
or kd =
gain, deﬁned as kd = v/α HSD30 v/ϕ, is −122.7                                                modulation, and P λ the average
−7.1 m the index of intensity −8.6
−127.6
+3.4                            +3.6
optical power. This is proved by dividing the spectrum density
gPµ R0
kd =             − dissipative ,
DSC30-1K   loss −119.8           (3) Si = 2qı −120.8 λ of the the output 2
−3.1
+2.4
= 2q P           −1.8
+1.7 2
current i by the average
square microwave current i2 = P λ 1 m2 . The background
ac          2
noise
where g is the ampliﬁer gain, Pµ the microwave power, R0 −1.5 amplitude and phase white−1.7 take the same value because
QDMH3           −114.3            =            −120.2
+1.4                            +1.6
50 Ω the characteristic resistance, and the mixer ssb loss. they result from additive random processes, and because the
Under the conditions of our setup (see below) the gain is 43 instrument gain/Hz is the same. The residual ﬂicker noise is
unit            dB/Hz             dB         dBrad2 kd            dB
dBV[/rad], including the dc preampliﬁer. The notation [/rad] to be determined experimentally.
means that /rad appears when appropriate.                          The differential delay of the two branches of the bridge is
Calibration the ∑ ampliﬁer is not detected [Rubiola, Salik, Yu, Maleki, Electron. Lett. 39(19) p.1389-1390 (2003) ]
The noise ofinvolves the assessment of kd and the adjustment kept small enough (nanoseconds) so that a discriminator effect
33
Photodetector 1/f noise
Rubiola, Salik, Yu, Maleki, MTT 54(2) p.816-820, Feb 2006

•    the photodetectors we measured are similar in
AM and PM 1/f noise
•    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 ).

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
W:
waving a hand
0.2 m/s,    S:
single spectrum, with optical A:
average spectrum, with optical           F:
after bending a ﬁber, 1/f
Figure is low enough for the detectors toexperimental a linear regime. This
3: Examples of environment eﬀects operate in mistakes around
which Examples of environment eﬀects andand can increase unpredictably
Figure 3:                                    noise experimental mistakes around
3 m far from the system       connectors and no isolators        connectors and no isolators
B:
background noise          B:
background noise                   the corner. high carrier the instrument Background noise which is stable
the possible a the noise show suppression (50–60 dB) in ∆, (spectrum B) B) f
corner. All All the plots show the instrumentbackgroundnoise (spectrum
B:
background plots                        B:
Background noise
P:
photodiode noise                   the the noise spectrum of Photodiode an (spectrum also Plot 1 spectrum
and noise spectrum of the the Photodiode hour), andP). Plot 1 spectrum
andduration of the measurement (half pairpair (spectrum P).provides a high re
P:
photodiode noise
P:
photodiode noise                                             P:
photodiode noise
of experimentalist andWaves a noise of(≈ (≈ m/s), 3 m3far far away from
W: the experimentalist of hand gently the 0.2 m/s), m The coherence
W: the the laser RIN Waves a the hand gently 0.2∆ ampliﬁer. away from thethe len
34
Sϕ (f )
• Let’s be optimistic: a 10 GHz link is
b−
1 /f                                       limited by the 1/f phase noise of a single
component, –120 dBrad2/Hz @ f=1 Hz
f
• Well known rules give σy(τ) = 4x10–17/τ
f2
Sy (f )   Sy (f ) = 2 Sϕ (f )
ν0
• Realistically, –100 dBrad2/Hz @ f=1 Hz
yields σy(τ) = 4x10–16/τ
h 1f
10–32 @ 1 Hz           f
h1 1
σy (τ )
2
= [1.038 + 3 ln(2πfH τ )]
(2π)2 τ 2
σy (τ )
4x
10 –1
7
/!
τ
4 – AM noise and RIN
36

Amplitude noise & laser RIN
• In PM noise measurements, one can validate the instrument by
AM noise of RF/microwave sources
Pa            va                                      feeding the same signal into the phase detector

FFT analyzer
dual channel
• In AM noise this is not possible without a lower-noise reference
Pb           vb
source
under test
• Provided the crosstalk was measured otherwise, correlation
monitor                                                              enables to validate the instrument
power                                                                                                                       Wenzel 501−04623E 100 MHz OCXO
meter
−123.1                                                        P0 = −10.2 dBm
avg 2100 spectra

Laser RIN
optical                                dc                                                      −133.1

dB/Hz
Pa                 va
source

FFT analyzer
−143.1
dual channel

Sα ( f )
under
test                                                 R
coupler       coupler
Pb                vb
−153.1

monitor                             R
power                                                                                                                    Fourier frequency, Hz
meter                                                                                                 −163.1
10                  102              103           104                105

AM noise of photonic RF/microwave sources                                                                                      !("
=>)'\$&1789

optical                   microwave                          dc                                                                                        Kirill Volyanskiy
!'"
source                        Pa                                 va
FFT analyzer
dual channel

under
test                                                                                                                     !!""
R0                       R                                         678912:;<345
coupler    coupler
Pb                                 vb                                                                                                  60mA
!!!"                                              40mA
monitor                 R0                       R
power                                                                                                                  80mA                                                20mA
meter                                                                                                         !!#"
monitor                                                                                                                                        30mA
power
meter                                                                      !!\$"

E. Rubiola, the measurement of AM noise, dec 2005                                                                           !!%" !                #               \$
100mA
%                     &
!"               !"           !"               !"                !"
arXiv:physics/0512082v1 [physics.ins-det]                                                                                                          )*+,-+./012345
37
AM noise of some sources
source                                        h−1 (ﬂicker)       (σα )ﬂoor
Anritsu MG3690A synthesizer (10 GHz)     2.5×10−11   −106.0 dB   5.9×10−6
Marconi synthesizer (5 GHz)              1.1×10−12   −119.6 dB   1.2×10−6
Macom PLX 32-18 0.1 → 9.9 GHz multipl.   1.0×10−12   −120.0 dB   1.2×10−6
Omega DRV9R192-105F 9.2 GHz DRO          8.1×10−11   −100.9 dB   1.1×10−5
Narda DBP-0812N733 ampliﬁer (9.9 GHz)    2.9×10−11   −105.4 dB   6.3×10−6
HP 8662A no. 1 synthesizer (100 MHz)     6.8×10−13   −121.7 dB   9.7×10−7
HP 8662A no. 2 synthesizer (100 MHz)     1.3×10−12   −118.8 dB   1.4×10−6
Fluke 6160B synthesizer                  1.5×10−12   −118.3 dB   1.5×10−6
Racal Dana 9087B synthesizer (100 MHz)   8.4×10−12   −110.8 dB   3.4×10−6
Wenzel 500-02789D 100 MHz OCXO           4.7×10−12   −113.3 dB   2.6×10−6
Wenzel 501-04623E no. 1 100 MHz OCXO     2.0×10−13   −127.1 dB   5.2×10−7
Wenzel 501-04623E no. 2 100 MHz OCXO     1.5×10−13   −128.2 dB   4.6×10−7

worst
best

```
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