18 Lock in Amplifiers

Lock-in amplifiers

http://www.lockin.de/

Signals and noiseFrequency dependence of noise• Low frequency ~ 1 / f

– example: temperature (0.1 Hz) , pressure (1 Hz), acoustics (10 -- 100 Hz)

• High frequency ~ constant = white noise– example: shot noise, Johnson noise, spontaneous emission

noise• Total noise depends strongly on signal freq

– worst at DC, best in white noise region• Problem -- most signals at DC

log(Vnoise)

log(f )

Noise amplitude

1/f noise

0

White noise

0.1 1 10 100 1kHz

log(

Vno

ise)

log(f )

Total noise in 10 Hz bandwidth

1/f noise

0

White noise

0.1 1 10 100 1kHz

Signal at DC

log(

Vno

ise)

log(f )

1/f noise

0

White noise

0.1 1 10 100 1kHz

Signal at 1 kHz

10 Hz

10 Hz

Lock-in amplifiers• Shift signal out to higher frequencies• Approach:• Modulate signal, but not noise, at high freq

– no universal technique -- art– example: optical chopper wheel, freq modulation

• Detect only at modulation frequency– Noise at all other frequencies averages to zero– Use demodulator and low-pass filter

Demodulation / Mixing• Multiply input signal by sine wave• Sum and difference freq generated• Compare to signal addition -- interference• Signal frequency close to reference freq

– low freq beat– DC for equal freq sine waves– DC output level depends on relative phase

Two sine waves

Product

Sum

Signal freq approaches ref freq• Beat frequency approaches DC as signal freq approaches ref freq

1

1.05

1.1

1.15

1.2

1.25

Signal freqvs ref freq

Reference

Mix

er o

utpu

ts

Phase sensitive detection• Signal freq matches reference freq• Reference = sin(2ft) • Signal = sin(2ft + )

– is signal phase shift• Product = cos() - cos(2ft)

Signalphaseshift

0 0.2 0.4

0.6 0.8

Reference wave

Prod

uct w

avef

orm

s--

sign

al ti

mes

refe

renc

eDC part

Low pass filterRemoves noise• Example -- modulate above 1/f noise

– noise slow compared to reference freq– noise converted to slowly modulated sine wave– averages out to zero over 1 cycle

• Low pass filter integrates out modulated noise – leaves signal alone

Reference

Input Output Mixer Low pass

filter Buffer

Lock-in amplifier Demodulated signal

After mixer

Vol

tage

time

After mixer & low pass

Typical LIA low pass filters• For weak signal buried in noise• Ideal low pass filter blocks all except signal• Approximate ideal filter with cascaded low pass filters

18 db/oct

12 db/oct

6 db/oct

Ideal

loggain

frequency

Phase control• Reference has phase control• Can vary from 0 to 360°• Arbitrary input signal phase• Tune reference phase to give maximum DC output

Reference

Phaseshift

Input Output Mixer

Reference options• Option 1 -- Internal reference

– best performance– stable reference freq

• Option 2 -- External reference• System generates reference

– ex: chopper wheel• Lock internal ref to system ref

– use phase locked loop (PLL)– source of name “lock-in amplifier”

Reference

Signal Mixer

Lock-in amplifier System

Reference

Signal Mixer

Lock-in amplifier System

VCO

Integrate

PLL

Analog mixer• Direct multiplication

– accurate– not enough dynamic range– weak signal buried in noise

• Switching mixer– big dynamic range– but also demodulates harmonics

Multiplying mixer

Switching mixer

Harmonic content of square wave

1

1/31/5 1/7 1/9

Switching mixer design• Sample switching mixer• Back-to-back FETs

– example: 1 n-channel & 1 p-channel– feed signal to one FET, inverted signal to second FET

• Apply square wave to gates– upper FET conducts on positive part of square wave– lower FET conducts on negative part

Switching mixer circuit

pn

Signal voltage

source draingate

bias

n-channel FET

Signals with harmonic content• Option 1: Use multi-switch mixer

– approximate sine wave– cancel out first few harmonic signals

• Option 2: Filter harmonic content from signal– bandpass filter at input– Q > 100

Lock-in amp with input filter

Digital mixers

• Digitize input with DAC• Multiply in processor• Advantages:

– Accurate sine wave multiplication– No DC drift in low pass filters– Digital signal enhancement

• Problems:– Need 32 bit DAC for signals buried in noise– Cannot digitize 32 bits at 100 kHz rates

• Should be excellent for slow servos– Ex: tele-medicine, temperature controllers– Digital processing can compensate for certain system time delays ?

Lock-in amps in servos• Lock to resonance peak

– Servos only lock to zero– Need to turn peak into zero

• Take derivative of lineshape– modulate x-voltage– F(x)-voltage amplitude like derivative

• Use lock-in amp to extract amplitude of F(x)– “DC” part of mixer output– filter with integrator, not low-pass

x

F(x)

Take derivative with lock-in

No fundamental• only 2 f signal

Lock-in amps for derivative• Lock-in turns sine wave signal into DC voltage• At peak of resonance

– no signal at modulation freq– lock-in output crosses zero

• Discriminant– use to lock

x

F(x)

Input signal

Lock-inoutput

(derivative)

Zero crossingat resonance

Effect of modulation on lineshape• Start with resonance lineshape• Intensity vs PZT voltage: I = I0 exp( -V2)

• Modulate voltage: V= V0 sin (2 f t)

• Modified lineshape

• Analog to numerical derivatives• Derivative is: I’ = I(V+ V) - I(V) / V

– Set V = 1• Modulation replaces V= V0 sin (2 f t)• Derivative is sine wave part

– Assumes is V0 small

V

I

t

t

V

I

Modulation amplitude

0.05 linewidth

0.1

0.2

0.5 linewidth

1

2

Effect of modulation amplitude• For large modulation amps

– Distortion and broadening• Modulation like a noise source

– Always use minimum necessary

Expanded scan

Modulation amplitude

0.1 linewidth

0.2

0.5 linewidth

1

2

Mixer outputs• Maximum mixer output

– modulation ~ 1 linewidth– saturates and broadens

Mixer out0.1 linewidth

0.2

0.5

1

2

Fabry-Perot servo• Lock to peak transmission of high Q Fabry-Perot etalon• Use lock-in amp to give discriminant

– No input bandpass -- or low Q < 2• Bandpass rolloff usually 2-pole or greater

– No low pass filter -- replace with integrator• Low pass filter removes noise• Need noise to produce correction

• Design tips– reference freq must exceed servo bandwidth by factor of ~ 10– but PZT bandwidth is servo limiter– use PZT resonance for modulation

Acoustic noise

Laser

Fabry-Perot

PD LIA

Sum& HV

reference