Lock-in amplifiers http://www.lockin.de/

Signals and noise Frequency 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 Total noise in 10 Hz bandwidth Signal at DC 1/f noise log(Vnoise) 10 Hz White noise 0.1 1 10 100 1kHz log( f ) log(Vnoise) log( f ) Noise amplitude 1/f noise White noise 0.1 1 10 100 1kHz Signal at 1 kHz 1/f noise log(Vnoise) White noise 10 Hz 0.1 1 10 100 1kHz log( f )

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 Reference Signal freq vs ref freq 1 1.05 1.1 1.15 1.2 1.25 Mixer outputs

Phase sensitive detection Signal freq matches reference freq Reference = sin(2pft) Signal = sin(2pft + f) f is signal phase shift Product = cos(f) - cos(2pft) DC part Signal phase shift f 0.2 p 0.4 p 0.6 p 0.8 p p Reference wave -- signal times reference Product waveforms

Low pass filter Removes 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 Demodulated signal After mixer Voltage time After mixer & low pass Reference Input Output Mixer Low pass filter Buffer Lock-in amplifier

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 log gain 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 Phase shift f Input Output Mixer

Reference options Option 1 -- Internal reference System Lock-in amplifier 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” Mixer Signal Reference System Lock-in amplifier Mixer Signal Reference VCO PLL Integrate

Analog mixer Direct multiplication Switching mixer Multiplying mixer 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 1/3 1/5 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 n-channel FET p n Signal voltage source drain gate bias

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 F(x) x 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 Take derivative with lock-in x F(x) 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 Input signal F(x) x Lock-in output (derivative) Zero crossing at resonance

Effect of modulation on lineshape Start with resonance lineshape Intensity vs PZT voltage: I = I0 exp( -V2) Modulate voltage: V= V0 sin (2 p f t) Modified lineshape Analog to numerical derivatives Derivative is: I’ = I(V+ DV) - I(V) / DV Set DV = 1 Modulation replaces DV= V0 sin (2 p f t) Derivative is sine wave part Assumes is V0 small V V t I t

Effect of modulation amplitude For large modulation amps Distortion and broadening Modulation like a noise source Always use minimum necessary Modulation amplitude 0.05 linewidth 0.1 0.2 0.5 linewidth 1 2 Expanded scan

Mixer outputs Maximum mixer output modulation ~ 1 linewidth Modulation amplitude 0.1 linewidth 0.2 Maximum mixer output modulation ~ 1 linewidth saturates and broadens 0.5 linewidth 1 2 Mixer out 0.1 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