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


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


− −

= − + +∑

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


“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


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


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


Outline


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


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


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


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


− −

= − + +∑

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


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


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

Self Frequency Modulation of Quantum Cascade … Frequency Modulation of Quantum Cascade Lasers ......

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