Self Frequency Modulation of Quantum Cascade … Frequency Modulation of Quantum Cascade Lasers ......
Transcript of Self Frequency Modulation of Quantum Cascade … Frequency Modulation of Quantum Cascade Lasers ......
Self Frequency Modulation of Quantum Cascade Lasers - a path toward coherent
Frequency Combs
Supported by NSF MIRTHE ERC
Jacob B Khurgin, Yamac Dikmelic (JHU) Collaboration with ETH Zurich (J. Faist’s
group)
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
Coherent Frequency Combs
Am
ωm
δωax
A set of equally-spaced lines in frequency domain ( invented in 1990’s by Theodor W. Hänsch and John L. Hall -Nobel Prize 2005)
Applications Metrology: Absolute optical frequency standard (optical clock)
Spectroscopy: fast spectral analysis without tunable laser and using a single detector
T. Udem, R. Holzwarth and T. Hansch, Nature 416 (6877), 233-237 (2002). S. A. Diddams, Journal Of The Optical Society Of America B-Optical Physics 27 (11), B51-B62 (2010). F. Keilmann, C. Gohle and R. Holzwarth, Optics Letters 29 (13), 1542-1544 (2004). S. Schiller, Optics Letters 27 (9), 766-768 (2002). I. Coddington, W. C. Swann and N. R. Newbury, in Phys. Rev. Lett. (2008), Vol. 100, pp. 013902
Normally one uses a tunable source (takes time) or broadband source and detector array How to do it all with a single detector?
Comb 1
δω1
Comb 2
δω2
Ωbm
RF beat frequency
A1m
ω1m
δω1
~100THz
Ωbm
A1m ˑA2m
~100MHz
0 Vernier effect-1:1 correspondence between each line in optical and RF spectra
δω2
A2m
ω2m
~100THz
Control
RF Spectrum Analyzer
Sample
Dual Comb Spectroscopy Absorption
Now we have the optical spectrum translated into RF domain – a single detector and no moving parts.
Why comb and not just many lasers?
Linewidth of frequency comb is very narrow –same as of a single mode laser with the same average power –need locking
First mode locking
Christiaan Huygens (1629-1695)
• Christiaan Huygens noticed that two pendulum clocks would synchronized themselves if hanged at the same mantelpiece
• Due to a small mechanical coupling between the pendulum
Combs in visible, or near IR regions Mode locked Ti: Sapphire , Nd:YAG or Er (Yb) fiber lasers
The common feature of all this medium – very long energy relaxation times τ21 (also called gain recovery times) compared to the cavity round trip τ2 =2L/c therefore the active medium is capable of storing energy between the pulse arrivals
Kerr r lens mode locked Ti Sapphire~780nm Pulse length 10fs Rep rate 70MHz (More than 106 lines)
Fiber laser –Pulse lengh100’s of fs
2
1
L
Quantum cascade laser
Efficient source of Mid-IR (3-5 or 8-12mcm ) radiation –very useful region For chemical analysis –vibrational transitions in many molecules)
But unfortunately in QCL the energy relaxation time is only τ21 ~1ps – shorter than ~10-50ps round trip time. The energy cannot be stored between the pulses.
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
Laser modes
Em(x)
x
L m-th mode
2 mm
n Lk L mc
π ν π= = 2m rtcm mnL
ν τ= =0( ) sin( ) mj t
m m mE x E k x e ω=
Round trip time 2rt
cnL
τ = 11ax m m rtν ν ν τ −
−= − =Inter mode spacing
Spectrum of the laser in the multi-mode regime
Am
ωm
ωax
Multi mode lasing in time domain
( )( 1)/2
0( 1)/2
( ) exp ( )N
n ax nN
tE t A j n j tω ω ϕ−
− −
= − + +∑
Electric Field Am
ωm
ω0
ωax
Power Density
( )2( 1) /22
0( 1) /2
( ) ~ ( ) exp ( )N
n ax nN
S t E t A j n t j tω ω ϕ−
− −
= = − + +∑
( ) ( 1) /2 ( 1) /2 ( 1) /2
2
( 1) /2 ( 1) /2exp exp ( ) ( )
N N N
n n m ax n mn N m n n N
A A A j n m t j t tω ϕ ϕ+ + −
=− − ≠ =− −
= + − − −∑ ∑ ∑
If the phases are random 0
2n
nS A S≈= ∑
S
t
S
Short pulses
0
( 1)/2
( 1)/2( ) axj t jn t
N
nN
eE t e Aω ω− −−
− −= ∑Electric Field
What if all the phases are the same?
Envelope
Carrier
E(t)
t
∆t T
Am ωax
ω ω0
∆ω
S(t)
t
∆t T
Speak
1 / ax rtT ν τ= =Period:
~ 1 / ~ 1 / axt Nν ν∆ ∆Pulse width
2/2 /22
/2 /2
~ ~ ~N N
peak n nN N
S A N A NS− −
∑ ∑
Peak power:
Mode locked pulse
S(t)
t
∆t T
Speak
1 / ax rtT ν τ= =Period:
This is the same pulse going back-forth in the cavity
L
Is this all ?
Mode locking of QCL would allow us to perform fast measurements over the wide spectral range limited only by QCL gain ( a few THZ in mid-IR) – good alternative to the tunable laser
Coherent comb-Mode locking in the “wide sense”
( )( 1)/2
0( 1)/2
( ) exp ( )N
n ax nN
tE t A j n j tω ω ϕ−
− −
= − + +∑
Electric Field Am
ωm
ω0
ωax
What if the relationship between the mode phases is deterministic but not as simple as φn =0?
The frequency comb is still coherent although we do not have a shortest (Fourier Limited pulse)
2 2
2 2 2 2 2 2 2 20 0
( 1)/2
( 1)/2~( ) ax ax ax
t tjCj t n t jC t n jn t j tt tN
Ne e e e eE t e eω ω ω ω ω−− − ∆ ∆ − −∆ ∆
−
− −= ∑
Example 1 – chirped Gaussian pulse: 2 2~ exp( )nA t n−∆ 2 2~n C t nϕ ∆
Instant phase 2
0 2( ) tt t Ct
ϕ ω= +∆
Instant frequency 0 2
( )( ) 2t tt Ct t
ϕω ω∂= = +
∂ ∆
This pulse has advantage in many measurements, like Doppler Radar due to broad spectrum and lower peak power
Frequency modulated signal
t
FM signal
FM Signal: 0( ) cos sin( )FMax
ax
E t A t tωω ωω
∆= +
Instant frequency 0( )( ) cos( )FM axtt t
tϕω ω ω ω∂
= = − ∆∂
Spectrum ( )( 1)/2
0( 1)/2
( ) expFMn
ax
N
axN
tE t J j nωω
ω ω−
− −
∆ = − +∑
Property of Bessel function: ( ) ( 1) ( )nn nJ x J x− = −
ω
Am FM spectrum
Very useful in communications
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
“Hydraulic model of laser”
2
1
UPPER LASER LEVEL
LOWER LASER LEVEL
Pump Energy Pump
decay
Energy Leakage
Threshold!
“Energy Storage Reservoir”
lasing “lasing”-water spill
All the additional pump energy “spills” into the laser after threshold – the laser “wants” to get rid of energy as fast as possible hence it will always “choose” the mode of oscillation with the lowest threshold
Single mode lasing Homogeneous broadening –all the atoms (electrons, molecules) are coupled equally to each mode
Gain
ν
Threshold gain
ν∆
νm
Now the population inversion is clamped at the threshold value- Single mode lasing
Inhomogeneous gain (a) Spectrally inhomogeneous
Gain
ν νm
Gain of group of atoms with resonant frequency ν0k
Gains of group of atoms with different resonant frequencies
Total Gain
Each atom interacts differently with different modes – each mode has its own “reservoir” of atoms
Inhomogeneous gain (b) Spatially inhomogeneous
Em(x) x
Sm(x)
x
L
Em+1(x)
Sm+1(x)
Atom interacts well with the mode m Atom interacts well with the mode m+1
Each atom interacts differently with different modes – each mode has its own “reservoir” of atoms
Multi Mode lasing Inhomogeneous broadening
γ
ν
γt
ν∆
νm
Threshold as a function of pulse shape (phase relation between modes)
In most lasers combination of spectral and spatial inhomogeneity (hole burning) causes multi-mode lasing. QCL is not different.
If there are N mode oscillating then the mode beating would cause changes in instant power on the frequency scale between νax and( N-1) νax
Am
ωm ω0
ωax
If the relaxation time of laser transition τ21 is long compared to the cavity round trip time the medium “does not see” the rapid variations of intensity, and reacts only to the “average” intensity – the gain medium “acts as an integrator”
S(t)
t
S
“Free running laser”
S(t)
t
∆t T
Speak
S
“Mode locked laser”
Free running laser
2
1
UPPER LASER LEVEL
LOWER LASER LEVEL
S(t) S(t)
t
S
S(t)
t
∆tT
Speak
The atom pumped to the upper level can safely remain there waiting until the next intensity peak will “knock it down” by stimulated emission – there is no advantage in any particular relation between the modes, or the temporal shape of the output – chances are the phase relation will be random and the output noisy.
Is it true? Let us check?
Free running laser-simulations (We take a QCL and assume unrealistically long intersubband relaxation time of τ21=5ns while τc=50ps – no FWM terms )
Solving in time domain
0 100 200 300 400 5000
0.05
0.1
0.15
0.2
0.25
Time (τc)
Am
plitu
des
of th
e m
odes
Spectrum
0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Mode Number
Pow
er
Instant Power
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
Time (τc)
Inst
ant P
ower
RF Spectral power density
0 10 20 30 40 50
10-10
10-5
100
Spe
ctra
l Pow
er d
ensi
ty
Frequency (units of τc-1)
Noisy multimode output
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
Active mode locking Gain Loss
The loss is time dependent
T(t)
T0
τrt t
This arrangement makes the periodic train of mode locked pulses preferential because they see lower loss – but pulses are relatively long
S(t)
t
∆t T
Speak
For the QCL the active mode-locking would amount to chopping – the peak power would be the same as CW power without mode-locking
Passive mode locking
The loss is intensity dependent In fast saturable absorber
Gain Loss
0( )1 ( ) /A
sat
tI t Iα
α =+
-2 -1 0 1 2 3
Time (a.u.)
2τp1
Signal (a.u.)
0
0.2
0.4
0.6
0.8
1
Tran
smis
sion
0 ( )abs
d S tdt hα α α σ α
τ ν−
= −
If abs rtτ τ<<
0( )1 ( ) /A
sat
tS t S
αα =+
This arrangement makes the periodic train of mode locked pulses preferential because they see lower loss than free running laser
Passively mode-locked laser We take a QCL and assume unrealistically long intersubband relaxation time of 5 ns +
Saturable absorber with response time of 1 ps). The sign of coefficient G is negative (absorption instead of gain)
Solving in time domain Spectrum Instant Power
RF Spectral power density
0 100 200 300 400 5000
0.05
0.1
0.15
0.2
0.25
Time (τc)
Am
plitu
des
of th
e m
odes
0 50 100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Tota
l Pow
er o
f the
mod
es
Time (τ )
0 20 40 600.005
0.01
0.015
0.02
0.025
0.03
Mode Number
Pow
er
0 0.5 1 1.5 20
10
20
30
40
50
Time (τc)
Inst
ant P
ower
0 10 20 30 40 5010-4
10-3
10-2
10-1
100
101
Spe
ctra
l Pow
er d
ensi
ty
Frequency (units of τc-1)
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
QCL –”reversed fast saturable absorber”
2
1
UPPER LASER LEVEL
LOWER LASER LEVEL
S(t) S(t)
t
S
S(t)
t
∆tT
Speak
The electron pumped to the upper level can only emit photon when the pulse is there – all the electrons put at the upper level between the pulses relax nonradiatively and the energy gets lost. Therefore, the QCL does not like short pulses and cannot be passively modelocked. Even if it forced to emit short pulses their instant power is no larger than the average power of free running QCL – very inefficient energetically because one cannot store energy on the upper laser level and then emit it in one short burst
0( )1 ( ) /A
sat
tI t Iα
α =+
-2 -1 0 1 2 3
Time (a.u.)
2τp1
Signal (a.u.)
0
0.2
0.4
0.6
0.8
1
Gai
n
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
The most favorite arrangement for QCL
On one hand QCL is forced to operate in multi-mode regime by spatial and also spectral hole burning mechanisms
On the other hand QCL has lowest threshold (and highest efficiency) when the intensity is constant in time.
This combination means Frequency Modulation with the period equal to the round trip time
t
FM signal –constant intensity but broad spectrum
Is it so?
No standing wave – no hole burning (unlike a single mode laser)
Free running QCL -simulations (We take a QCL and assume realistic intersubband relaxation time of 1 ps)
Solving in time domain Spectrum Instant Power
RF Spectral power density
0 100 200 300 400 5000
0.05
0.1
0.15
0.2
0.25
Time (τc)
Am
plitu
des
of th
e m
odes
0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Mode Number
Pow
er
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Time (τc)
Inst
ant P
ower
Instant Frequency
0 5 10 15 20 2510-6
10-4
10-2
100
102
Spec
tral
Pow
er d
ensi
ty
Frequency (units of τc-1)
0 0.5 1 1.5 2-20
-15
-10
-5
0
5
10
15
Time (τc)
Inst
ant F
requ
ency
( ) 2 *, , , , ,
,
1nn n n n k kk kn n m k l k l k l k l m n
k k l k
dA G A G A A B G A A A B Cdt
κ κ≠
= − − −∑ ∑
How to “see” FM modulation
ω
Am FM spectrum
( )( 1)/2
0( 1)/2
( ) exp ( )N
n ax nN
tE t A j n j tω ω ϕ−
− −
= − + +∑
Electric Field
There is a very delicate arrangement of amplitudes and phases in the spectrum so that sum of beating terms at any frequency is zero.
( ) 0.n m njn n m
nA A e ϕ ϕ− −
− =∑
( )* . . . .( ) ax x n m njm t j m tn n m n n m
m n m nA A e c c A A e c cP t ω ω ϕ ϕ−+ −
− − + = +
= ∑ ∑ ∑ ∑
Instant RF power
Any disturbance to phase or amplitude (dispersion, frequency dependent loss etc) will cause FM to AM conversion
How to “see” FM modulation
ω
Am FM spectrum
( )1 1
1
( ) . . 0.n m n ax
Nj j t
n nn
P t A A e e c cϕ ϕ ω− −−
=
= + =∑
Filter away half of the spectrum
ω
Am FM spectrum
/2( )
1 11
( ) . . 0.n m n ax
Nj j t
n nn
P t A A e e c cϕ ϕ ω− −−
=
= + ≠∑
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Time (τc)
Inst
ant P
ower
0 5 10 15 20 2510-6
10-4
10-2
100
102
Spec
tral
Pow
er d
ensi
ty
Frequency (units of τc-1)
RF Spectral power density Instant Power
0 5 10 15 20 2510-4
10-2
100
Spe
ctra
l Pow
er d
ensi
ty
Frequency (units of τc-1)0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Time (τc)
Inst
ant P
ower
of 1
/2 m
odes RF Spectral power density
Instant Power
Intensity oscillations before and after polyethylene filter
Sheet of polyethylene acts as an optical discriminator
A. Hugi, et al., Nature, vol. 492, 229–233 (2012)
Outline
•Frequency Combs in “normal” lasers and QCL •Frequency Combs and Ultra-short pulses – related, yet not the same! •Why would laser go multi-mode? •Why would laser mode lock – active and passive? •Why would not QCL mode lock? •Self frequency modulation of QCL’s •Comparison with the Experiment
Results I – Short Wavelength QCL
Mode spectrum and net gain profile Mode magnitude evolution
Gain recovery time: 2 ps Cavity length: 2 mm > 3 modes / FWM halfwidth
Short Wavelength QCL
Intensity modulation Self frequency modulation
After polyethylene filter
Interferograms
Intensity Intermode beat
( ) ( )∑−=
=M
MmmmEI 2cos4,0 22 τωτ
( ) ( ) ( )[ ]∑−
−=+
∗+ +=
1
121 2cos2cos2,M
Mmmcmmc EEI τωτωτω
A. Hugi et al, “Mid-infrared frequency comb based on a quantum cascade laser”, Nature 492, 229–233 (2012)
Results II – Long Wavelength QCL
Mode spectrum and net gain profile Mode magnitude evolution
Gain recovery time: 1 ps Cavity length: 3 mm ~ 10 modes / FWM halfwidth
Long Wavelength QCL
Intensity modulation Self frequency modulation
Interferograms
Intensity Intermode beat
Setup: mid-IR dual comb spectroscopy
Characteristics • Compact setup (65cm x 65cm x 25cm) • No cryogenic cooling needed
6 mm long
Free running heterodyne beat measurements
Important numbers: • Lines = 100; • Spectral coverage = 25 cm-1 • frep = 7.5 GHz • Δfrep = 10 MHz • Sampling = 0.25 cm-1
G. Villares et al., Nat. Comm
Conclusions
Combination of spatial hole burning and fast saturable gain in QCL favors self-frequency modulation The mechanism behind it is four wave mixing Stable frequency comb is thus produced in mid-IR without any additional element inside the cavity – but the exact form of FM is unpredictable Frequency comb can be used in a variety of measurements, with the resolution limited only by the width of the gain – there is no overwhelming need for the short pulses