# Lock-in amplifiers. A short tutorial by R. Scholten

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1 Lock-in amplifiers A short tutorial by R. cholten

2 Measuring something Common task: measure light intensity, e.g. absorption spectrum Need very low intensity to reduce broadening Noise becomes a problem Rb spectrum 8 Rb cell Photodiode 6 Laser 4 0 Frequency

3 The principle Fundamental law of communication theory:wiener-khinchin theorem Reduction of noise imposed upon a useful signal with frequency f 0, is proportional to the square root of the bandwidth of a bandpass filter, centre frequency f 0

4 Noise Typical photodetector noise spectrum -90 Photodector intensity Power spectral density (db) Bad noise -00 here Off resonance Dark noise 90Hz Δf here... On resonance Better here! -50 and here! khz 0kHz 00kHz Frequency (Hz) dc measurements: broad-spectrum (bad) at low frequency (bad)

5 Measuring something Need to measure at high frequency, where noise is low Modulate signal, look for component oscillating at modulation frequency Laser Rb cell Photodiode 8 Rb spectrum 6 Chopper Lock-in amplifier 4 0 Frequency

6 ignal from noise A lock-in amplifier is 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 5.0 Lockin amplifiers

7 Demodulator or PD (phase-sensitive detector) Demodulator Input Mixer Low-pass filter Output Buffer Reference

8 Mathematical description ignal (t) varies relatively slowly e.g. absorption spectrum scan over 0 seconds Modulate at relatively high frequency ω (e.g. chopper): Reference (local oscillator) of fixed amplitude: phase φ is variable oscillator frequency ω same as modulation frequency Multiply modulated signal by REF : sig ref = = = sig ( t) cos t = cos( ω t +φ) ω ref ( t) cosωt cos( ωt + φ) () t cosφ + () t cos( ωt + φ) econd term at high frequency (ω) Low-pass filter (cutoff ~ ω/ or lower) sig ref filter = ( t) cosφ Note phase-sensitive detection!

9 ome details imple trig ref sig = = = = = = = () t cosωt cos( ωt + φ) + n( t)cos( ωt + φ) () t cosωt[ cosωt cosφ sinωt sinφ] +... ()[ t cos ωt cosφ cosωt sinωt sinφ] +... ()( t [ + cos ωt ) cosφ sin ωt sinφ] +... ()[ t cosφ + cos ωt cosφ sin ωt sinφ] +... () t cosφ + ()[ t cos ωt cosφ sin ωt sinφ] () t cosφ + () t cos( ωt + φ) + n( t)cos( ωt + φ) +...

10 Noise Noise reduces with frequency (/f noise is major problem) Laser frequency noise hift signal to higher frequency Noise within given bandwidth reduces as we measure at higher frequency db/sqrt(hz) Frequency (Hz)

11 With noise ignal has noise: sig ( t) cos t n( t) = ω + Multiply reference by modulated signal: ref sig = = ( t) cosωt cos( ωt + φ) + n( t)cos( ωt + φ) () t cosφ + () t cos( ωt + φ) + n( t)cos( ωt + φ) Third term noise at frequency ω Low-pass filter, frequency less than ω/, leaves signal components We win twice: less noise at ω reduce bandwidth

12 Using PD oscillator to modulate Phase-sensitive detector Experiment ignal Mixer Buffer Low-pass filter Output Mod Reference

13 External modulator: true lock-in Lock-in amplifier Experiment ignal Mixer vco Buffer Low-pass filter Output Ref PLL = phase-locked loop Integrator

14 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? hapes (6 th, th, 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?

15 Other applications Often use lockin to measure response function of actuator (or similar) Two-channel lockin measure signal and phase Phase resonances Phase-sensitive detector In-phase and quadrature mixers Actuator (e.g. piezo) ignal Mod Ref

16 Experiments Photodiode + LED R FFT spectrum analyser Oscilloscope witch LED on/off, e.g. with hand to block HP function generator to modulate LED And/or chopper R lock-in amp

17 The other half of the story Frequency modulation

18 Derivatives: Lock-in amps & feedback servos o far, we have modulated amplitude, and used LIA to demodulate PD (lockin) = fancy bandpass filter?! Can also use frequency modulation (like FM radio) Let s measure (ω) i.e. a spectrum, where we slowly vary ω(t) ( ) t Frequency-modulate: ω t = ω 0 + Ω0 cosω where Ω is the modulation (Fourier) frequency Using Taylor-series expansion: ( ω) = ( ω ) + ( Ω cosωt) d dω ω= ω 0 0 Note two things immediately: dc component is same as un-modulated spectrum ac component is proportional to derivative of spectrum

19 Extract derivative with PD/lock-in amp We now multiply our signal by our reference, as before: sig = ( ω ) + ( Ω cosωt) +... = cos( Ωt +φ) 0 d dω ω= ω 0 0 Note modulation at Ω, and fixed ω 0 i.e. slowly varying laser frequency ref sig ref = =... = cos cos d dω ( Ωt + φ) + Ω cosωt cos( Ωt + φ) d dω d dω ( Ωt + φ) + Ω cos( Ωt + φ) + Ω cosφ Again: low-pass filter (cutoff ~ ω/ or lower) sig ref Ω 0 d dω cosφ We have a measurement proportional to the derivative Measurement changes sign if slope changes sign: dispersion Note: modulation depth Ω 0 must not be larger than peak in spectrum! Higher-order terms in Taylor expansion: can measure nd deriv, 3 rd deriv, etc.

20 Lock-in amplifiers and feedback servos Example: Lorentzian peak in atomic absorption spectrum On centre: small double frequency maller slope = smaller signal ω PD out Note opposite phase: cosφ <0 ω ω

21 Lock-in amps in servos Modulated output from detector t t t t Demodulated output from lock-in t ω PD out sig ref Ω 0 d φ dω cos ω

22 Yet other half of the story pread-spectrum

23 PRB: 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 pread-spectrum communications computer 80. wireless, etc. CDMA telephones Modems ecurity/encryption Acoustics

24 pread-spectrum modulation PREAD-PECTRUM MODULATION all frequencies present simultaneously in modulation function phases adjusted so that components add in quadrature truly random phases cause excursions out of range use pseudo-random functions 0 frequencies random phases

25 pread-spectrum history Also famous as first nude in cinemarelease movie! Hedy Lamarr (93-000), composer George Antheil ( ) patented submarine communication device ynchronized frequency hopping to evade jamming Original mechanical action based upon pianolas Used today in GP, cellphones, digital radio

26 Binary pseudo-random sequences clock input Clock D HIFT REGITER time output n sequence length bits with n bit shift register AUTOCORRELATION = n ( ) ±

27 PRB: Lock-in on steroids! Generate signal in pseudo-random bit sequence, for example: 6-bit (64-bits long) bit (56 bits long): Record signal Multiply by PRB (auto-correlate) ery much like a lock-in! But uses broad spectrum Computer Experiment ignal Mod Record/average Reference Multiplier Output

28 3-bit (89 bits long) ML single scan

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