Lock-in amplifiers

A short tutorial by R. Scholten

Measuring something

Common task: measure light intensity, e.g. absorption spectrum

Need very low intensity to reduce broadening

Noise becomes a problem

Rb cell Photodiode

Laser

Rb spectrum

Frequency0

2

4

6

8

The principle

Fundamental law of communication theory:Wiener-Khinchin theorem

Reduction of noise imposed upon a useful signal with frequency f0, is

proportional to the square root of the bandwidth of a bandpass

filter, centre frequency f0

Noise

Typical photodetector noise spectrumPhotodector intensity

1.0 10 100 1kHz 10kHz 100kHzFrequency (Hz)

-150

-140

-130

-120

-110

-100

-90

On resonance

Off resonance

Dark noisePow

er s

pect

ral d

ensi

ty (d

B)

Better here!

Bad noise here…

90Hz Δfhere...

…and here!

dc measurements: • broad-spectrum (bad) • at low frequency (bad)

Measuring something

Need to measure at high frequency, where noise is low

Modulate signal, look for component oscillating at modulation frequency

Rb cell Photodiode

Laser

Chopper

Lock-inamplifier

Rb spectrum

Frequency0

2

4

6

8

Signal from noise

A lock-in amplifier is used to extract signal from noiseIt detects signal based on modulation at some known frequencyPremise:

much noise at low frequency (e.g. dc), less noise at high frequencymeasure within narrow spectral range, reduce noise bandwidth

Hence shift measurement to high frequency

Figures from Bentham Instruments document 225.02 Lockin amplifiers

Demodulator or PSD (phase-sensitive detector)

Buffer

Reference

Demodulator

InputMixer

Low-pass filter Output

Mathematical description

Signal VS(t) varies relatively slowlye.g. absorption spectrum scan over 10 seconds

Modulate at relatively high frequency ω (e.g. chopper):

Reference (local oscillator) of fixed amplitude:

phase φ is variableoscillator frequency ω same as modulation frequency

Multiply modulated signal by REF :

Second term at high frequency (2ω)

Low-pass filter (cutoff ~ ω/2 or lower)

( ) ttVV S ωcossig =

( )φω += tV cosref

( ) ( )( ) ( ) ( )φωφ

φωω

++=

+=

ttVtV

tttVVV

SS

S

2coscos

coscos

21

21

refsig

( ) φcosfilter 21

refsig tVVV S=××

Note phase-sensitive detection!

Some details

Simple trig

( ) ( ) ( )( ) [ ]( )[ ]( ) ( )[ ]( )[ ]( ) ( )[ ]( ) ( ) ( ) ( )φωφωφ

φωφωφφωφωφφωφω

φωωφω

φωφωω

φωφωω

++++=

+−+=+−+=+−+=

+−=

+−=

+++=

ttnttVtVtttVtV

tttVtttV

ttttV

ttttV

ttntttVVV

SS

SS

S

S

S

S

S

cos)(2coscos...sin2sincos2coscos

...sin2sincos2coscos...sin2sincos2cos

...sinsincoscoscos

...sinsincoscoscos

cos)(coscos

21

21

21

21

21

21

21

21

2

sigref

Noise

Noise reduces with frequency (1/f noise is major problem)

Shift signal to higher frequency

Noise within given bandwidth reduces as we measure at higher frequency

Laser frequency noise

-100

-90

-80

-70

-60

-50

-40

1.0 10.0 100.0 1000.0 10000.0 100000.0

Frequency (Hz)

dbV/

sqrt

(Hz)

www.technion.ac.il/technion/chemistry/courses/w05/127421

With noise

Signal has noise:

Multiply reference by modulated signal:

Third term – noise – at frequency ωLow-pass filter, frequency less than ω/2, leaves signal componentsWe win twice:

less noise at ωreduce bandwidth

( ) )(cossig tnttVV S += ω

( ) ( ) ( )( ) ( ) ( ) ( )φωφωφ

φωφωω++++=

+++=ttnttVtV

ttntttVVV

SS

S

cos)(2coscoscos)(coscos

21

21

sigref

Using PSD oscillator to modulate

Buffer

Reference

Phase-sensitive detector

Signal

MixerLow-pass filter Output

Mod

Expe

rimen

t

External modulator: true “lock-in”

Buffer

PLL = phase-locked loop

Lock-in amplifier

Signal

MixerLow-pass filter Output

vcoExpe

rimen

t

Ref

Integrator

∫

Further details

True lock-in amp can work with external oscillator for Reference:Input reference from external experimentUse phase-locked-loop to generate stable local oscillator

Lock-in amp has variable post-multiplier (low-pass) filterTime constants: what time constant is appropriate?Shapes (6th, 12th, … order): which is best?

If input signal has harmonics (e.g. due to imperfect modulation) then will detect spurious signal

Use input filter to minimise

Dynamic reserve?

Other applications

Often use lockin to measure response function of actuator (or similar)Two-channel lockin – measure signal and phasePhase → resonances

0°Ref

Phase-sensitive detector

Signal

In-phase and quadrature mixers

ModActu

ator

(e.

g. p

iezo

)

90°

90°

0°

Experiments

Photodiode + LEDSRS FFT spectrum analyserOscilloscopeSwitch LED on/off, e.g. with hand to blockHP function generator to modulate LEDAnd/or chopperSRS lock-in amp

The other half of the story

Frequency modulation

Derivatives: Lock-in amps & feedback servos

So far, we have modulated amplitude, and used LIA to demodulatePSD (lockin) = fancy bandpass filter?!

Can also use frequency modulation (like FM radio)

Let’s measure VS(ω) i.e. a spectrum, where we slowly vary ω(t)

Frequency-modulate: where Ω is the modulation (Fourier) frequency

Using Taylor-series expansion:

Note two things immediately:dc component is same as un-modulated spectrumac component is proportional to derivative of spectrum

( ) tt ΩΩ+= cos00ωω

( ) ( ) ( ) ...cos000

+ΩΩ+==

tddVVV S

SSωωω

ωω

Extract derivative with PSD/lock-in amp

We now multiply our signal by our reference, as before:

Note modulation at Ω, and fixed ω0 i.e. slowly varying laser frequency

Again: low-pass filter (cutoff ~ ω/2 or lower)

We have a measurement proportional to the derivativeMeasurement changes sign if slope changes sign: dispersionNote: modulation depth Ω0 must not be larger than peak in spectrum!Higher-order terms in Taylor expansion: can measure 2nd deriv, 3rd deriv, etc.

( ) ( )

( ) ( ) φω

φω

φ

φω

φ

cos2coscos

...

coscoscos

021

021

0refsig

ddVt

ddVtV

ttddVtVVV

SSS

SS

Ω++ΩΩ++Ω=

=

+ΩΩΩ++Ω=

( ) ( ) ...cos00sig0

+ΩΩ+==

tddVVV S

Sωωω

ω ( )φ+Ω= tV cosref

φω

cos021

refsig ddVVV SΩ≈

Lock-in amplifiers and feedback servos

Example: Lorentzian peak in atomic absorption spectrum

ω ωSmaller slope = smaller signal

On centre:• small• double frequency

Note opposite phase: cosφ <0

ω

PSD out

Modulated output from detector

Demodulated output from lock-in

Lock-in amps in servos

ω

t

t

t

t

t

ω

PSD out φω

cos021

refsig ddVVV SΩ≈

Yet other half of the story

Spread-spectrum

PRBS: Lock-in on steroids!

Lock-in uses small part of spectrumCan use broad spectrum and still separate signal from noisePseudo-random bit sequence

Spread-spectrum communicationscomputer 802.11 wireless, etc.CDMA telephonesModems

Security/encryptionAcoustics

Spread-spectrum modulation

• all frequencies present simultaneously in modulation function

SPREAD-SPECTRUM MODULATION

• phases adjusted so that components add in quadrature

• truly random phases cause excursions out of range⇒ use pseudo-random functions

20 frequenciesrandom phases

http://www.chm.bris.ac.uk/pt/mcinet/sum_schl_02_docs/tof.ppt

http://www.chm.bris.ac.uk/pt/mcinet/sum_schl_02_docs/tof.ppt

Spread-spectrum history

http://www.ncafe.com/chris/pat2/index.html

• Hedy Lamarr (1913-2000), composer George Antheil (1900-1959) patented submarine communication device

• Synchronized frequency hopping to evade jamming

• Original mechanical action based upon pianolas

• Used today in GPS, cellphones, digital radio

http://www.chm.bris.ac.uk/pt/mcinet/sum_schl_02_docs/tof.ppt

Also famous as first nude in cinema-release movie!

Binary pseudo-random sequences

1 2 3 4 5 6 7 8

D

Clock

SHIFT REGISTER

time

clock input

output• sequence length bits

with n bit shift register12 −n

× =

AUTOCORRELATION

( )121 −± n

1

http://www.chm.bris.ac.uk/pt/mcinet/sum_schl_02_docs/tof.ppt

PRBS: Lock-in on steroids!

Generate signal in pseudo-random bit sequence, for example:6-bit (64-bits long)

011010101111110000010000110001010011110100011100100101101110110

8-bit (256 bits long):000110110111001000101101100101100111011010111010100110111100111110101100010100001111010011110001000111010001001100100111001101010110100101011111101110111110000011001100001000010101010001100011111111001010010000000101111011000000111000110100000100100101110

Record signalMultiply by PRBS (auto-correlate)Very much like a lock-in! But uses broad spectrum

Reference

Computer

Signal

Multiplier

Output

Mod

Expe

rimen

t

Record/average

13-bit (8192 bits long) MLS single scan