noise in optical systems f. x. kärtner high-frequency and quantum electronics laboratory university...
TRANSCRIPT
NOISE IN OPTICAL SYSTEMS
F. X. Kärtner
High-Frequency and Quantum Electronics Laboratory
University of Karlsruhe
Outline
I. Introduction: High-Speed A/D-Conversion
II. Quantum and Classical Noise in Optical Systems
III. Dynamics of Mode-Locked Lasers
IV. Noise Processes in Mode-Locked Lasers
V. Semiconductor Versus Solid-State-Lasers
VI. Conclusions
High-Speed A/D-Conversion (100 GHz)
Voltage
Time
o o
To
t
Voltage
Modulator
Timing-Jitter t:
= 2 tTo
Vo
V
VVo
VVo
: 10 bit
=100 GHz1To
=> t ~ 1 fs
Time
Output @ 1350 - 1550 nm
OutputCoupler
SaturableSemiconductorAbsorber
Cr :YAG - Crystal8mm long, 10 GHz Repetitionrate
4+
DichroicBeam Splitter
Nd:YAG Laseror Diode Laser
Mode-Locked Cr : YAG Microchip-Laser4+
• Compact: Saturable Absorber, Dispersion Compensating Mirrors•10 GHz, 20 fs - 1 ps, @ 1350 - 1550 nm•Very Small Timing-Jitter < 1 fs•Cheap (< 10.000 $)
Classical and Quantum-Noise in Optical Systems
(Modes of the EM-Field)
Length L
mmmm
tzjm m
Lea
LtzA mm
,2
,1
),(
mode, th-m ofEnergy :*mmm aa
kTe
aa kTmnmnm
1
1ˆˆ
/,*
Thermal Equilibrium
0m
mmmm aaaa ˆ,ˆ,:QM *
States and Quadrature Fluctuations)2()1( jaaa
)1(a
)2(a1
)1(a
)2(a
Area=/4
Coherent States(Minium Uncertainty States)
Squeezed States
Area=/4
Balanced Homodyne-Detection
jeLO
a
a
jj eaea
aaI
ˆˆ
ˆˆˆ *
dttattatI )(ˆ)()(ˆ)(ˆ *
Continuum of modes
m
mmmm aaI ˆˆˆ *
Loss- and Amplifier-Noise
a
)'(2)(ˆ)'(ˆ;ˆˆ
zzzzadzad
dz
Loss:
Amplifier: )'(2)(ˆ)'(ˆ;ˆˆˆ
zzgzzagdzad
Necessary noise for maintaining uncertainty circle
Spontaneous emission noise
1),'(2)(ˆ)'(ˆ zzgzz Non-Ideal Amplifier:
Dynamics of Mode-Locked Lasers
cavity roundtrip timeTR : A(T,t)
tsmall changes per roundtrip
GDD D
SPM
Gaing, g
Sat. Loss
AAlt
gAAjt
jDtTATR
Tg
2
2
2
22
2
2
||1
1||),(
l:loss
Energy Conserving Dissipative
Steady-State Solution
,000
s exp),(
jTT
jtt
atTAR
s
200 2:Energy Soliton AW
2020 A
21
:Roundtrip per Shift Phase
D
0
4: WidthSoliton
W
D
If pulses are solitonlike
2g
g
D
000 A2A:TheoremArea WD
00 sech)(
ttAtas
The System with Noise
0000 exp)()(),( jTT
jttattatTAR
s
)'()'(2
)','(),( * ttTTT
ghtTStTS
Rqq
),(||1
1||),( 22
2
22
2
2
tTSAAlt
gAAjt
jDtTATR
Tg
)()1
1()(
),( tatT
TgtTS s
gRg
)(1)(
),( 0 tatd
dv
nc
jT
TLtTS s
gRL
Gain Fluctuations:
Cavity Length or Index Fluctuations:
Amplifier Noise:
Soliton-Perturbation Theory
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4Energ
y P
ert
urb
atio
n,
f w-6 -4 -2 0 2 4 6
x
(a) 1.0
0.8
0.6
0.4
0.2
0.0Ph
ase
Pe
rtu
rba
tion
, -
i f
-6 -4 -2 0 2 4 6
x
(b)
-0.8
-0.4
0.0
0.4
0.8
Fre
quency
Pert
urb
atio
n,
-i f p
-6 -4 -2 0 2 4 6
x
(c)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Tim
ing
Pe
rtu
rba
tion
, f t
-6 -4 -2 0 2 4 6
x
(d)
),()()()()()()()()(),( tTAtfTttfTptfTtfTwtTa ctpw
Energy Phase Center-frequency Timing and Continuum
xaxxW
xf sw tanh11
)(0
xjaxf s)(
xaxxf st tanh1
)(
/:with tx
xaxxW
jxf sp tanh2
)(0
Linearized and Adjoint System
Linearized system is not hamiltonian,it is pumped by the steady-state pulse
)exp(),(),(),( 00
jTT
jtTStTatTaTR
TR
L
Adjoint System L+: Biorthogonal Basis
Scalar Product: ..)()( *
2
1ccdttgtfgf
Interpretation: Field g is homodyne detected by LO f
xaxxW
jxf stanh1
2)(
0
xaxf sw 2)( xaxW
jxf sp tanh2
)(0
xaxxW
xf st tanh2
)(0
Basic Noise Processes
)(20
0 TSR
TW
wTR
T
tpwjtTSfTS jj ,,,,),()(
)(1
TSR
TwwTR
T ww
)(1
TSR
TppTR
T pp
)(20
TSR
TW
wgpDt
TRT x
g
2
000
21AW
Wg
TRw
22341
gRp T
g
Noise Sources
tpwjtTSfTS gjgj ,,,,),()(,
)()(
2)( ,0 TSWT
TgTS gw
Rw
)()()(
)( ,0 TSnTn
LTL
TS g
)()( , TSTS gpp
)()()()(
)( , TSnTn
LTL
TTg
TS gtRg
t
Amplitude- and Frequency Fluctuations
Amplitude- and frequency fluctuations are damped and remain bounded
22
2
2)(
)(~
w
wSw
22
2
,2)(ˆ
)(~
p
gpSp
22)0(ˆ
2 ww Sw
2
,2
)0(ˆ2 gpp Sp
Correlation Spectra Variances
Phase- and Timing Fluctuations
Phase- and timing fluctuations are unbounded diffusion processes
2
2
222
22
02
)(ˆ)(ˆ2
)(~
SS
T w
w
R
2
2
222
22
2)(ˆ)(ˆ
2)(~
t
p
p
R
SS
TD
t
Gordon-Haus Jitter
Timing Fluctuations
Quantum Noise
ggR
g
g
nnn
LLL
ppR
pg
R
TT
T
g
g
g
TT
n
n
TT
L
L
TT
ThW
T
ThW
gTtTTt
exp1
exp1
exp1
exp1/
8
/32
)()(
2
2
2
2
2
2
22
2
22
2
2
22
0
20
2
0
222
00
Classical Noise
Long-Term Timing Fluctuations T >> p, L, n, g
Quantum Noise
22
2
2
2
2
2
2
2
2
2
0
20
2
0
222
00
/8
/32
)()(
R
g
g
n
L
R
pg
T
Tg
g
g
Tn
n
TL
L
T
T
hW
hWg
TtTTt
Classical Noise
Semicondutor versus Solid-State Lasers
Semicon-ductorLaser
Solid-StateLaser
W0/h g g g p/TR g/gn/nn g
107
1010
0.2
0.01
40
THz fs
200
300
10
10
1
375
75
10-3
0
ns
1
0
10-3
10-3
ns
1
1000
10
2
2t
450 fs
1 fs
Semiconductor -Laser: Gordon-Haus-Jitter + Index-Fluctuations
Solid-State Laser: Gain-Fluctuations
Dominant sources for timing jitter:
Other parameters are: T=TR=100 ps, o =1
Conclusions
• Classical and quantum noise in modes of the EM-Field
• Spontaneous emission noise of amplifiers
• Dynamics of modelocked lasers (solitonlike pulses)
• Amplitude-, phase-, frequency- and timing-fluctuations
•Solid-State Lasers: no index fluctuations; possibly small
Gordon-Haus Jitter; very short pulses; superior timing jitter in
comparison to semiconductor lasers
References:
H. A. Haus and A. Mecozzi: „Noise of modelocked lasers,“ IEEE JQE-29, 983 (1993).
J. P. Gordon and H. A. Haus: „Random walk of coherently amplified solitons in optical fiber transmission,“ Opt. Lett. 11, 665 (1986).
H. A. Haus, M. Margalit, and C. X. Yu: „Quantum noise of a mode-locked laser,“ JOSA B17, 1240 (2000).
D. E. Spence, et. al.: „Nearly quantum-limited timing jitter in a self-mode-locked Ti:sapphire laser,“ Opt. Lett. 19, 481 (1994).