ETH Zurich Ultrafast Laser Physics

Ursula Keller / Lukas Gallmann

ETH Zurich, Physics Department, Switzerland www.ulp.ethz.ch

Chapter 10: Noise

Ultrafast Laser Physics

microwave!frequency!

analyzer!

Measurement of laser noise electric current

J(t)!

50 Ω input impedance!photo detector!

light intensity I(t)!

The microwave frequency analyzer measures the power spectral density

autocorrelation function:

SL ω( ) ∝ I ω( ) 2 = F I t( )∗ I t( ){ } = F R τ( ){ }

R −τ( ) = R τ( )

R −τ( ) ≡ I t( ) I t −τ( )∫ dt = I t( )∗ I t( )⇒

FI ω( ) I * ω( )

microwave!frequency!

analyzer!

Measurement of laser noise electric current

J(t)!

50 Ω input impedance!photo detector!

light intensity I(t)!

�

I(t)=PT δ(t−nT )n=−∞

+∞∑

For more details see: U. Keller et al., IEEE J. Quantum.Electron., 25, 280,1989 !

Modelocked laser without any noise

Pulses like delta-functions

I ω( ) = F I t( ){ } = 2πP δ ω − nω L( )

n=−∞

+∞

∑

microwave!frequency!

analyzer!

Measurement of laser noise electric current

J(t)!

50 Ω input impedance!photo detector!

light intensity I(t)!

We measure the power spectral density

SL ω( ) ∝ I ω( ) 2

SL ω( ) = 4π 2P2 δ ω − mω L( )m=−∞

+∞

∑

ω L =2πT

S (ω)L

ωLω

2ωL 3ωL 4ωL

modelocked laser without any noise and any bandwidth limitations of microwave frequency analyzer

microwave!frequency!

analyzer!

Measurement of laser noise electric current

J(t)!

50 Ω input impedance!photo detector!

light intensity I(t)!

�

I(t)= P ⋅T(1+ N(t)) δ(t−nT −ΔT t⎛

⎝ ⎜

⎞

⎠ ⎟ )

n=−∞

+∞∑

�

I(t)=PT δ(t−nT )n=−∞

+∞∑

S (ω)L

ωLω

2ωL 3ωL 4ωL

normalized intensity noise timing jitter!For more details see: U. Keller et al., IEEE J. Quantum.Electron., 25, 280,1989 !

Intensity noise of mode-locked lasers

�

SL(ω)= 4π2P 2 δ(ω−nωL)+ ˜ N (ω−nωL)⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪ n=−∞

+∞∑S (ω)L

ωLω

2ωL 3ωL 4ωL

noise sidebands!

-80

-60

-40

-20

0

dBc

248.4248.2248.0247.8247.6

Frequency [MHz]

filter resolution of microwave spectrum analyzer

intensity noise sideband

-140

-120

-100

-80

-60

dBc

in 1

Hz

band

widt

h

107106105104103102101

Offset Frequenz, Hz

Quantronix Nd:YLF

Units: dB, dBm, dBc

-80

-60

-40

-20

0

dB b

elow

the

first

harm

onic

248.4248.2248.0247.8247.6

Frequency [MHz]

Filter functionIntensity noise

dB ≡ 10log P2P1

dBm ≡ 10log P1 mW

mW dBm

0.1 mW -10 dBm

1 mW 0 dBm

10 mW 10 dBm

100 mW 20 dBm

1000 mW 30 dBm

x 2 + 3 dBm

20 mW 13 dBm

Units: dB, dBm, dBc

-80

-60

-40

-20

0

dB b

elow

the

first

harm

onic

248.4248.2248.0247.8247.6

Frequency [MHz]

Filter functionIntensity noise

dB ≡ 10log P2P1

dBm ≡ 10log P1 mW

mW dBm

0.1 mW -10 dBm

1 mW 0 dBm

10 mW 10 dBm

100 mW 20 dBm

1000 mW 30 dBm

x 2 + 3 dBm

20 mW 13 dBm

dBc “how many dB below the carrier”

carrier is the peak of the harmonic signal

Power in noise sidebands

-80

-60

-40

-20

0

dB b

elow

the

first

harm

onic

248.4248.2248.0247.8247.6

Frequency [MHz]

Filter functionIntensity noise

Measurement with a certain bandwidth B

Psb: power in the intensity sidebands (“two-sided” around harmonics) Pc: peak power of carrier f : offset frequency or noise frequency (i.e. deviation from laser harmonics)

PsbPc

= 2Psb f( ) Pc

Bdf

n fL+ f1

n fL+ f2

∫

Psb f( ) PcB

Psb f( ) Pc

factor 2 because “two-sided” spectral density!

rms intensity noise

Psb: power in the intensity sidebands Pc: peak power of carrier

PsbPc

= 2Psb f( ) Pc

Bdf

n fL+ f1

n fL+ f2

∫

Psb f( ) PcB rms intensity noise or variance of

intensity noise:

σ N ω1 ,ω2[ ] = N t( )2

= 1π

SN ω( )dωω1

ω2

∫

σ N f1 , f2[ ] = PsbPc

Differential Transmission Spectroscopy Ultrafast measurements need some kind of nonlinearities in the

measurement system (i.e. intensity dependent transmission)

Laserbeamsplitter chopper

ωcΔt

probe

pump

device under test

photodetector(PD)

ωωc

JPD noise of probe

signalsignal = [T(Δt, Ipump) —T(Ipump = 0)] Iprobe

Transmission of device under test withpump on pump off

Ultrafast Pump-Probe Techniques

Note: Microwave spectrum analyzer measures power of photocurrent!!

!Need a chop frequency of > 100 kHz!

Example: signal-to-noise 1:10-6

�

20log SN =20log10−6 =−120 dBc

-140

-120

-100

-80

-60

dBc

in 1

Hz

band

widt

h

101 102 103 104 105 106 107

Offset Frequenz, Hz

Quantronix Nd:YLF

Timing jitter

I t( ) = PT δ t − nT − ΔT t( )( )n=−∞

+∞

∑

IΔT ω( ) = F IΔT t( ){ } = −P inω LΔ T ω − nω L( )

n=−∞

+∞

∑

SΔT ω( ) = IΔT ω( ) ⋅ IΔT∗ ω( ) == P2 −i( ) ⋅ +i( ) n2ω L

2

n=−∞

+∞

∑ Δ T ω − nω L( )⎡⎣ ⎤⎦2

SL ω( ) = 4π 2P2 δ ω − nω L( ) + 1

4π 2 n2ω L

2 Δ T ω − nω L( )⎡⎣ ⎤⎦2⎧

⎨⎩

⎫⎬⎭n=−∞

+∞

∑

Timing jitter of mode-locked laser

�

SL(ω)= 4π2P 2 δ(ω−nωL)+ 14π2 n2ωL

2 Δ ˜ T (ω−nωL)⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪ n=−∞

+∞∑

noise sidebands!

S (ω)L

ωLω

2ωL 3ωL 4ωL

Psb4

Pc4

(nωL )2~

Ln f( ) = Psbn f( )Pcn

example : n = 4

L1 f( ) = Psb1 f( )Pc1

= 1n2Ln f( )

normalized power density (two-sided around harmonics)

ω L = 2π frep

Timing jitter of mode-locked laser

�

SL(ω)= 4π2P 2 δ(ω−nωL)+ 14π2 n2ωL

2 Δ ˜ T (ω−nωL)⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪ n=−∞

+∞∑

noise sidebands!S (ω)L

ωLω

2ωL 3ωL 4ωL

Psb4

Pc4

(nωL )2~

-140

-120

-100

-80

-60

dBc

in 1

Hz

Band

brei

te

101 102 103 104 105 106 107

Offset Frequenz, Hz

intensity noise

timing jitter L1(f)

L1 f( ) = Psb1 f( )Pc1

= 1n2Ln f( )

Measurement takes place at the n-th harmonic

Timing jitter sidebands stronger than intensity noise sidebands

Double check: small noise approximation

∝ n2

Intensity noise and timing jitter

I t( ) = P ⋅T 1+ N t( )( ) δ t − nT − ΔT t( )( )n=−∞

+∞

∑

SL ω( ) ≅ 4π 2P2 δ ω − nω L( ) + N ω − nω L( )⎡⎣ ⎤⎦

2+ 14π 2 n

2ω L2 Δ T ω − nω L( )⎡⎣ ⎤⎦

2⎧⎨⎩

⎫⎬⎭n=−∞

+∞

∑

rms timing jitter

�

SL(ω)= 4π2P 2 δ(ω−nωL)+ 14π2 n2ωL

2 Δ ˜ T (ω−nωL)⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪ n=−∞

+∞∑

noise sidebands!

S (ω)L

ωLω

2ωL 3ωL 4ωL

Psb4

Pc4

(nωL )2~L1 f( ) = 1

n2Ln f( ) = 1

n2Psbn f( )Pcn

σ ΔT f1, f2[ ] = ΔT t( )2 = 12π fL

2 L1 f( )dff1

f2

∫ = 12πnfL

2 Ln f( )dff1

f2

∫

normalized power density (two-sided around harmonics)

factor of 2 because L1(f) is “two-sided”

rms and FWHM timing jitter in ps?

�

σΔT f1, f2⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= 12πfL

2 L1( f )dff1

f2∫ = 1

2πnfL2 Ln( f )df

f1

f2∫

�

L1( f )=Psb1 f

⎛

⎝ ⎜ ⎞

⎠ ⎟

Pc1

Ln( f )=Psbn f

⎛

⎝ ⎜ ⎞

⎠ ⎟

Pcn

�

σΔT 130 Hz,20 kHz⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ≈σΔT 130 Hz,∞⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= 9 ps ⇒ ≈21 ps FWHM

�

ΔT(t)= 12πσΔT

exp −t22σΔT2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

=exp −4ln2 tτΔT⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

2⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

�

τΔT = 8ln2 σΔT = 2.355⋅σΔT

S (ω)L

ωLω

2ωL 3ωL 4ωL

Psb4

Pc4

(nωL )2~

rms timing jitter:!

FWHM timing jitter: (Gaussian noise statistics)!

Example:!