1. Basics of Fourier transform Periodic function

2. The ‘vectors’ are good orthonormal base for every finite energy signal f(t) must be limited: physical signals carry limited amounts of energy FOURIER TRANSFORM of a non-periodic function By definition of Dirac’s delta f(t) must have a finite number of discontinuities f(t) can be split in its even E(t) and odd O(t) components: Hp. Fourier transform Inverse Fourier transform

3. Periodic vs non- periodic functions: Fourier spectrum Time domainFrequency domain Periodic Not periodic |F||F| ω |F||F|

4. It is much easier to describe the transfer function of sequential ‘blocks’ in term of frequency response: only at the last step the time behavior time domain is inferred Convolution and Fourier transform ω→t Time domainFrequency domain The integral in time domain turns into simpler algebraic product in the Fourier frequency domain. Frequency domain Time domain

5. So, if this means a 3dB gain, if the attenuation is -20dB in the signal Logarithmic attenuation and gain ratios Attenuation and gains relative to voltage (V), current (I), power (P), but also pressure and other physical quantities are usually measured as adimensional ratios towards a reference value of the measured quantity, let us say a voltage V 0, current I 0, power P 0 By definition: 20 dB every decade of gain -20 dB every decade of attenuation Otherwise, a -6dB attenuation means If we deal with power

6. Bode’s diagrams If we want to study the behavior of a ‘block’ in the frequency domain responding to sinusoidal stimulus we define: Example: RC network R C V in V out Time constant (TC) ω→0 0.707 ω→¥ 20 dB 1 decade -3 dB ω ωtωt ωtωt ω t : roll-off freq BW

7. IDEAL Frequency/istantaneous signal mixer Signal ‘Reference’ Some simplifications without loss of generality: S(t) and R(t) can be choosen periodic, even with different pulsations ω s and ω r we can choose =0, =0 (Fourier expansion coefficients a 0 =0) LOCK IN COMPONENTS: SIGNAL MIXER a k =0 k>0, b 1 =R a k =0 k>0, b1=S b k =0 n>1

8. Mixer output In our particular case Hp. DC componentAC component If ω s ≠ ω r the DC component vanishes, no matter about phase lags between the two signals LOCK IN COMPONENTS: SIGNAL MIXER Frequency Content

9. The IDEAL low-pass filter (infinite roll-off) |F||F| ω ωrωr ωr-ωsωr-ωs ωsωs ωr+ωsωr+ωs ω 2ω2ω 0 0 02ω2ω |F||F||F||F| ω t : rolloff frequency ONLY DC component pass LP ωtωt ω |A(ω)| 1 Provided that φ could be regulated we have singled out the weight of the component in the signal S(t) at the reference frequency ω r LOCK IN COMPONENTS:LOW-PASS FILTER

10. After the LP filter only the RMS amplitude of the signal will be extracted, acting like a demodulator LOCK IN COMPONENTS: Low-pass filter propertiesLOCK IN COMPONENTS: The LP filter behaves as an integrator for spectral components in the signal with pulsations larger than ω t : this is equivalent to an integration performed up to nTC (up to infinite if roll-off or ‘order’ n of the filter is infinite). The istantaneous mixer output is so integrated to yield by definition self-correlation between the reference and the input signal Signal switched on at t=0 C R Rectifier TC=RC

11. LOCK IN COMPONENTS: The phase-sensitive detector (PSD) LP ωtωt ω |A(ω)| 1 Given a reference R(t) with proper pulsation ω r, whatever the form of the input signal S(t), the DC output component of this block will depend only on the weight of the spectral component of the signal at ω r, apart relative phase lag φ. RMS value of the stimulus is known, then φ has to be tuned to find the max rms value of the signal, in this respect this block is phase-sensitive A zero frequency (very narrow) band-pass filter is obtained with: or S/N ratio (typ. ω r ≈ 1kHz, Δω ≈ 0.01Hz) A zero frequency (very narrow) band-pass filter is obtained with: or S/N ratio (typ. ω r ≈ 1kHz, Δω ≈ 0.01Hz)

12. Lock-in building blocks: The phase shifter and PLL LP ωtωt ω |A(ω)| 1 Φ A phase shifting network (Φ) has to be applied to the reference (most common) to maximize output M x Mixer 1 Mixer 2 reference Reference in phase quadrature Phase of the output signal can be fed back to drive the phase shifter: auto-phase-locking through a Phase Locked Loop (PLL) block sin cos M φ

13. From ideal to a real lockin amplifier IDEAL mixer carries only ω s ± ω r frequency content REAL mixer carries ωs± ωr AND ω s, ω r AND image frequencies ω s ± 2ω r, ω r ± 2ω s But this is the least problem, due to the following LP filter, cutting off/integrating high frequency band of the mixer ouput. IDEAL LP filter has a cutoff frequency ω t =0 and infinite roll-off REAL LP cannot have ω t =0, but only ω t →0, AND roll-off must be finite LP filter output can NOT be noise immune if ouput spectral BW is zero (ω t =0 and infinite roll-off) also output power is zero! ω t =0 means infinite integration time or TC: definitely we won’t wait the eternity to read the output A REAL measuring device will be affected by noise from several sources and in any circuit block: in our case the worst one is the noise at the amplifier input stage

14. Noise sources Intrinsic : mainly noise of the input terminals of the amplifier stages Noise amplitude vs frequency log(V noise ) log(  f ) 1/f noise 0 White noise 0.1 1 10 100 1kHz Johnson noise Shot noise 1/f: ensemble of excitation-deexcitation processes in semi-conductors into environmental thermal bath, almost independent from amplifier input bandwith BW White noise Johnson noiseShot noise Thermal fluctuations of electron density in any resistor R Thermal/quantum fluctuations of discrete number of charge carriers: e=1.6·10 -19 C Extrinsic: MUCH more complex RF/EMI interferences Mains supply lines radiating at 50/100 Hz Capacitive/Inductive coupling with surrounding devices Ground loops Are only most common source of external noise Spectral density has to be recognized for every particular set-up of experiment

15. Lock-in I/O Signal and noise power spectral densities 2 log(V noise ) log(  f ) 0 0.1 1 10 100 1kHz Input BW Typical power spectral density at the input in a ideally good case (no ext noise): colored areas proportional to the power of noise and signal Noise power Signal power (at ω r ) Signal BURIED in noise After PSD detector/integrator 2 log(V noise ) log(  f ) 0 0.1 1 10 100 1kHz LP filter bandwidth Signal Noise Lesser LP filter BW Higher TC Higher S/N ratio ωtωt BUT If TC is too large (ω t →0) not only noise, but also signal power will be lost ! S/N ratios at low frequencies will be poorer in any case: choose reference/ stimulus in the 100 Hz-10 kHz freq. range Input BW Shifted to DC

16. System The ‘classic’ lock-in setup Signal ωtωt ω |A(ω)| 1 Band-pass Filter AC amplifier + Noise G ac Mixer LP / integrator Φ Ref. generator G dc DC Output A BB CD Question : how will an unknown system respond to an external harmonic stimulus? The reference can be generated: internally: a built-in oscillator excites the system directly or through transducers externally: further device excites the system and a PLL circuit has to drive the built-in oscillator to desired stimulus frequency and compensate phase lags Phase shifter PLL tracks φ,ω r ωrωr ωrωr

17. A B C D 60 Hz supply noise Spectral transfer functions of lock-in blocks

18. The response function A(λ) could not be trivially linear (a), and typically is NOT linear (b), or even resonant (c) Response of a physical system to a periodic stimulus I ωrωr λ A(λ) System A will be modulated with respect to the harmonic stimulus at ω r BUT Linear/harmonic term non-linear distortion higher order harmonics Anharmonic terms (c)

19. Response of a physical system to a periodic stimulus II The modulated λ itself could be periodic but not harmonic, like square wave at pulsation ω r : typical of laser beam intercepted by a mechanical chopper wheel Taylor series Fourier series problem: this full series expansion is almost unmanageable ! Hp: Δλ<<λ 0 λ 1 (and λ 2 ) >> λ k for k>1(2) λ 1 >λ 2 | A (k) |<<|A (1) | Small λ modulation First harmonic ω r (and first overtone 2ω r ) dominant Only lower order terms of Taylor series will contribute

20. Discussion of Remember that The k-th order spectral weight will be: Ex.: Which of these terms will give an output after the PSD/integrator?

21. Lock-in output vs. mixer harmonics Most generally N: integer sum of harmonics Δφ N : sum of phases At the output of PSD/demodulator we have: Only surviving terms are for N=0 When the base (k=1) harmonic is fed into the reference input mixer we obtain an estimate of the spectral weight S 1 at the lock-in output

22. Spectral weights S k vs signal derivatives k=1 The dominant contribution comes from: initial hypotheses discard higher order contributions The output is proportional to the first derivative of the input signal, again provided that modulation of the parameter Δλ<<λ 0 k=2 2 terms: Experiment is modulated at ω r but reference input of the mixer is driven at 2ω r by means of a frequency doubler/multiplier: 2f detection Which one will prevail? In most general case the answer is not unique! Not necessarily S 2 is univocally proportional to second derivative only

23. Peculiar cases of λ coupling Linear coupling of λ and A: good approx. For Δλ→0 Only first derivative will be detected Linear coupling of λ and excitation U S k term is proportional mainly to the k-th derivative Notice that 2f detection of 2nd derivative requires a -π/2 phase shift with respect to usual 1f detection

24. Non-linear resonant coupling with λ Only 2ω component For Δλ<Γ G: full width half maximum (FWHM) l res The RMS value of the modulation of A(λ), versus amplitude (AM) or frequency (FM) modulation of λ, yields a term proportional to the first derivative of A(λ), if Δλ<Γ Γ

25. Example: STS point spectroscopy Tip sample The tip is held at fixed distance from the sample Tip or sample bias is scanned with a linear ramp and I(V) is acquired Numerical derivative ‘Lock-in’ derivative MUCH better S/N ratio! The bias is modulated with a small amplitude voltage (some mV)