Lockin Rob Web

8/12/2019 Lockin Rob Web

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Lock-in amplifiers

A short tutorial by R. Scholten


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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

Frequency

0

2

4

6

8


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The principle

Fundamental law of communication theory:Wiener-Khinchintheorem

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

proportional to the square root of the bandwidth of a bandpassfilter, centre frequency f0


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Noise

Typical photodetector noise spectrumPhotodector intensity

1.0 10 100 1kHz 10kHz 100kHz

Frequency (Hz)

-150

-140

-130

-120

-110

-100

-90

On resonance

Off resonance

Dark noisePowerspec

traldensity(dB)

Better here!

Bad noisehere

90Hz fhere...

and here!

dc measurements:

broad-spectrum (bad) at low frequency (bad)


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Measuring something

Need to measure at high frequency, where noise is low

Modulate signal, look for component oscillating at modulation frequency

Rb cellPhotodiode

Laser

Chopper

Lock-in

amplifier

Rb spectrum

Frequency

0

2

4

6

8


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Signal from noise

A lock-in amplifieris used to extract signal from noise

It detects signal based on modulation at some known frequency

Premise:

much noise at low frequency (e.g. dc), less noise at high frequency

measure within narrow spectral range, reduce noise bandwidth

Hence shift measurement to high frequency

Figures from Bentham Instruments document 225.02 Lockin amplifiers


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Demodulator or PSD (phase-sensitive detector)

Buffer

Reference

Demodulator

Input

Mixer

Low-pass filter Output


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Mathematical description

Signal VS(t) varies relatively slowly

e.g. absorption spectrum scan over 10 seconds

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

Reference (local oscillator) of fixed amplitude:

phase is variable

oscillator frequency same as modulation frequency

Multiply modulated signal by REF :

Second term at high frequency (2)

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

( ) ttVV S cossig =

( ) += tV cosref

( ) ( )

( ) ( ) ( )

++=

+=

ttVtV

tttVVV

SS

S

2coscos

coscos

21

21

refsig

( ) cosfilter21

refsig tVVV S=

Note phase-sensitivedetection!


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Some details

Simple trig

( ) ( ) ( )

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

( )[ ]

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

++++=++=

++=

++=

+=+=

+++=

ttnttVtV

tttVtV

tttV

tttV

ttttVttttV

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


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Noise

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

Shift signal to higher frequency

Noise within given bandwidthreduces 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. t echni on. ac. i l / t echni on/ chemi st r y/ cour ses/ w05/ 127421


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With noise

Signal has noise:

Multiply reference by modulated signal:

Third term noise at frequency

Low-pass filter, frequency less than /2, leaves signal components

We win twice:

less noise at reduce bandwidth

( ) )(cossig tnttVV S +=

( ) ( ) ( )( ) ( ) ( ) ( )

++++=

+++=

ttnttVtV

ttntttVVV

SS

S

cos)(2coscos

cos)(coscos

21

21

sigref


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Using PSD oscillator to modulate

Buffer

Reference

Phase-sensitive detector

Signal

Mixer


Mod

Experim

ent


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External modulator: true lock-in

Buffer

PLL =phase-locked loop

Lock-in amplifier

Signal

Mixer


vcoExperiment

Ref

Integrator


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Further details

True lock-in amp can work with external oscillator for Reference:

Input reference from external experiment

Use phase-locked-loop to generate stable local oscillator

Lock-in amp has variable post-multiplier (low-pass) filter

Time constants: what time constant is appropriate?

Shapes (6th, 12th, order): which is best?

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

Use inputfilter to minimise

Dynamic reserve?


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Other applications

Often use lockin to measure response function of actuator (or similar)

Two-channel lockin measure signal andphase

Phase resonances

0

Ref

Phase-sensitive detector

Signal

In-phase and quadraturemixers

ModActuator(e.g.p

iezo)

90

90

0


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Experiments

Photodiode + LED

SRS FFT spectrum analyser

Oscilloscope

Switch LED on/off, e.g. with hand to block

HP function generator to modulate LED

And/or chopper

SRS lock-in amp


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The other half of the story

Frequency modulation


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Derivatives: Lock-in amps & feedback servos

So far, we have modulated amplitude, and used LIA to demodulate

PSD (lockin) = fancy bandpass filter?!

Can also use frequency modulation (like FM radio)

Lets measure VS() i.e. a spectrum, where we slowly vary (t)

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

Using Taylor-series expansion:

Note two things immediately:

dc component is same as un-modulated spectrum

ac component isproportional to derivativeof spectrum

( ) tt += cos00

( ) ( ) ( ) ...cos00 0 ++= = tddV

VV

S

SS


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Extract derivativewith PSD/lock-in amp

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

Note modulationat , and fixed 0 i.e. slowly varying laser frequency

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

We have a measurement proportional to the derivative

Measurement changes signif slope changes sign: dispersion Note: modulation depth 0 must not be larger than peak in spectrum! Higher-order terms in Taylor expansion: can measure 2nd deriv, 3rd deriv, etc.

( ) ( )

( ) ( )

cos2coscos

...

coscoscos

021

021

0refsig

d

dVt

d

dVtV

ttd

dVtVVV

SSS

SS

++++=

=

+++=

( ) ( ) ...cos00sig0

++==

td

dVVV SS

( )+= tV cosref

cos021refsig

d

dVVV S


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Lock-in amplifiers and feedback servos

Example: Lorentzian peak in atomic absorption spectrum

Smaller slope =

smaller signal

On centre:

smalldouble frequency

Note oppositephase: cos


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Modulated output from detector

Demodulated output from lock-in

Lock-in amps in servos

t

t

t

t

t

PSD out

cos021

refsig

d

dVVV S


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Yet other half of the story

Spread-spectrum


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PRBS: Lock-in on steroids!

Lock-in uses small part of spectrum

Can use broad spectrum and still separate signal from noise

Pseudo-random bit sequence

Spread-spectrum communications computer 802.11 wireless, etc.

CDMA telephones

Modems

Security/encryption Acoustics


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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 usepseudo-randomfunctions

20 frequenciesrandom phases

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

ht t p: / / www. chm. br i s. ac. uk/ pt / mci net / sum_schl _02_docs/ t of . ppt


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Spread-spectrum history

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

Hedy Lamarr (1913-2000), composer George Antheil(1900-1959) patented submarine communicationdevice

Synchronized frequency hopping to evade jamming

Original mechanical action based upon pianolas

Used today in GPS, cellphones, digital radio


Also famous as firstnude in cinema-release movie!


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Binary pseudo-random sequences

1 2 3 4 5 6 7 8

D

Clock

SHIFT REGISTER

time

clock input

output

sequence length bits

with nbit shift register

12 n

=

AUTOCORRELATION

( )121 n

1



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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): 00011011011100100010110110010110011101101011101010011011110011111010110001010000111101001

1110001000111010001001100100111001101010110100101011111101110111110000011001100001000010101010001100011111111001010010000000101111011000000111000110100000100100101110

Record signal

Multiply by PRBS (auto-correlate)

Very much like a lock-in! But uses broad spectrum

Reference

Computer

Signal

Multiplier

Output

Mod

Experiment

Record/average


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13-bit (8192 bits long) MLS single scan

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