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- 1. G. Giuliani - INTERFEROMETRY_02 1 ELECTRO-OPTICAL INSTRUMENTATION Guido Giuliani Università di Pavia e-mail: guido.giuliani@unipv.it Recommended textbook: S. Donati, “'Electro-Optical Instrumentation”, Prentice Hall, 2004
- 2. G. Giuliani - INTERFEROMETRY_02 2 INTERFEROMETRY - Summary 1. Principle 2. Applications 3. Basic configurations Michelson – dual beam Michelson – dual frequency 4. Extension to mechanical/geometrical measurements 5. Limit performance 6. Speckle Pattern & operation on diffusive surfaces 7. Vibrometers: LDV (Laser Doppler Vibrometer) 8. Self-Mixing Interferometry 9. ESPI (Electronic Speckle Pattern Interferometry) 10. White-light interferometry (profilometry) 11. Interferometers for gravitational waves
- 3. G. Giuliani - INTERFEROMETRY_02 3 5. Performance Limits The interferometer performance can be limited by several factors and causes: Limits in the plane Displacement (Velocity) vs. Frequency Cosine error Limited coherence length of the laser source (temporal coherence limit) Quantum noise Spatial coherence and polarization effects Dispersion of the propagation medium Thermodinamic phase noise Speckle-Pattern errors
- 4. G. Giuliani - INTERFEROMETRY_02 4 Performance Limits Limits in the plane Displacement (Velocity) vs. Frequency “Type 1” limits define the minimum measureble displacement, which is often coincident with the resolution. Cause: Quantization (e.g.: /8 counting) Photodetection noise (typ.: quantum noise) “Type 2” limits define the maximum measurable velocity Cause: Limited bandwidth (B), or limited real- rime processing capabilities of the electronics it limits the detection of the Doppler frequency shift: fdoppler = 2(v/c) < B “Type 3” limits (low-frequency) is more severe for the dual-beam scheme baseband signal processing prone to EMI disturbance 1 Hz 1 kHz 1 MHz 1 nm 1 m 1 mm DISPLACEMENT (peak amplitude) FREQUENCY SPEED (m /s) 1 m 0.006 0.6 60 6000 NOISE or Quantization HF cutoff Doppler lim it low-frequency displacement interferometers vibrometers 2 1 1 3
- 5. G. Giuliani - INTERFEROMETRY_02 5 Performance Limits Consine error Output photocurrent: Iph = I0 {1 + Vcos[2k(sm-sr)]} The effectively measured pathlength is: ksmcos() Where: = angle due t residual alignment errors systematic error on the scale factor (responsivity) of the interferometer In addition: variations of the amplitude of the fringes and of the fringe visibility (0 < V < 1) can cause measurement errors Missed fringe countings Larger influence of EMI disturbance He-Ne freq stab laser I I 1 pol,0° 2 PD PD pol,90° /8 plate reference corner cube measurement corner cube sm r s BS x y linear pol at 45° k
- 6. G. Giuliani - INTERFEROMETRY_02 6 Performance Limits Coherence length of the laser source - 1 The e.m. field emitted by a real laser shows phase jumps at random time instants The average duration of the interval between two subsequent phase jumps is called coherence time (c) Definitions: Coherent length: Lc = cc [m] Linewidth: = 1/c [Hz] is the FWHM of the power spectrum of the e.m. field Typical values He-Ne laser: Lc = 300 m = 300 kHz (c = 1 s) Semiconductor laser (good) Lc = 30 m = 3 MHz (c = 100 ns) Semiconductor laser (poor) Lc = 1.5 m = 60 MHz (c = 5 ns) E t 0 Optical power spectrum
- 7. G. Giuliani - INTERFEROMETRY_02 7 Performance Limits Coherence length of the laser source - 2 The coherence length of the laser must be compared with the difference (unbalance) between the reference path and measurement path of the interferometer L = |sm-sr| Interferometric signal: Iph = I0 {1 + Vcos[2k(sm-sr)]} fringe visibility V = exp[-(L/Lc)] |sm-sr| >> Lc no interferometric signal is available (only noise) it is not possible to perform interferometric measurements |sm-sr| < Lc interferometric measurements are OK BUT: phase noise can be generated Non-zero linewidth (0) the instantaneous laser frequency shows temporal fluctuations (t) = 0 + (t) also the interferometric phase shows temporal fluctuations: (t) = 2k(sm-sr) = (4/)(sm-sr) = (4(t)/c)(sm-sr) = (4/c)(sm-sr)[0+(t)] = 0+(t) We have then a phase noise, whose variance is: = (4/c)(sm-sr) = 2 2(sm-sr)/0 /0 we have an uncertainty s of the measurand (sm-sr): s = /2k = (sm-sr) (/0) = (0/) (sm-sr)/Lc V L =|sm-sr| Lc 1
- 8. G. Giuliani - INTERFEROMETRY_02 8 Performance Limits Quantum noise - 1 An interferometer can be used a Vibrometer: an instrument capable of measuring small displacements (<< ), having zero-mean Technique: The “slow” (DC) phase difference between the reference and measurement pathlengths is kept constant (by some specific method) The interferometer operates around the point of maximum sensitivity (half-fringe, linear part of the Iph vs. characteristic) the system acts a linear transducer of small displacements (<< ) The minimum measurable displacement is limited by the photocurrent noise (NED = Noise Equivalent Displacement) Photodiode + load resistance R the noise can be expressed as: i2 n = 2q(Iph+Id)B + 4kTB/R 2qIphB = 2qI0B [A2] (the term I0 = P = (q/h)P can be made arbitrarily large, by increasing P) The term 2qI0B is called quantum noise , or shot noise NOTE: the phase noise limit could be worse than the quantum noise limit !!!! = 2ks t t Iph I0
- 9. G. Giuliani - INTERFEROMETRY_02 9 Performance Limits Quantum noise - 2 Iph = I0 {1 + Vcos[2k(sm-sr)]} Signal term: Is = (I0V)2ksm Signal-to-noise ratio: (S/N) = I2 s/i2 n = (I0V2ksm)2 / 2qI0B The NED can be found by letting (S/N) = 1 and solving for sm NED = (/2V)(qB/2I0)1/2 = = (/2V)(hB/2P)1/2 It is possible to calculate the equivalent quantum phase noise: n = 2kNED = V(2hB/P)1/2 Letting: P = 1 mW ; B = 1 Hz the performance limits are: 10-8 rad and 1 fm 10 10 10 10 -5 -3 -7 -9 1 100 10k 1M 100M Measurement Bandwith B (Hz) Phase noise (rad) or 1/(S/N) ratio n 633 nm =0.9, V=1 Equivalent Power P (W) 0.1 W 10 W 1 mW 100 mW 10 W 1 nm 1 pm 1 fm NED - noise-eqiovalent-displacement (m)
- 10. G. Giuliani - INTERFEROMETRY_02 10 Performance Limits Comparison: phase noise / quantum noise Example 1 He-Ne Laser: P=1mW; =300kHz (Lc=300m); Phase noise: s = (0/) (sm-sr)/Lc = 0.066 nm (for |sm-sr|= 0.1 m ) s = 6.6 nm (for |sm-sr|= 10 m ) Quantum noise: NED = 1 fm (for B = 1 Hz) NED = 1 pm (for B = 1 MHz) whenever the pathlength difference is NOT kept 0, then the phase noise is the main limit to the interferometer performance Example 2 Semiconductor laser: P=10mW; =3MHz (Lc=30m); Phase noise: s = (0/2) (sm-sr)/Lc = 0.33 nm (for |sm-sr|= 0.1 m ) s = 3.3 nm (for |sm-sr|= 1 m ) Quantum noise: NED = 0.31 fm (for B = 1 Hz) NED = 0.31 pm (for B = 1 MHz)
- 11. G. Giuliani - INTERFEROMETRY_02 11 Performance Limits Spatial coherence and polarization effects Need for spatial coherence: it is necessary that the transverse spatial distribution of the fields Em(x,y) e Er(x,y) onto the photodetector be the same Spatial coherence factor: sp = ∫AEm(x,y) Er*(x,y)dxdy /[∫A|Em(x,y)|2dxdy ∫A|Er (x,y)|2dxdy]1/2 sp 1 only for the case of single-mode beams with the same diameter (and radius of curvature, in the gaussian beam approximation) With multi-mode beams (thet contain N modes each) only modes with the same spatial distribution contribute to sp sp 1/N The two beams must have the same polarization state (linear, circular, or elliptical) Polarization factor: pol = EmEr/(|Em||Er|) All the above effects, combined, define the final effective visibility: V = (sp pol)exp[-(L/Lc)]
- 12. G. Giuliani - INTERFEROMETRY_02 12 Performance Limits Dispersion of the propagation medium A laser interferometer with beam(s) propagation in air performs measurements with a scale factor related to: /nair For a He-Ne laser it is possible to determine with a precision of 8 digits Variations of nair ? In standard conditions (T = 15°C, P = 760 mbar): (nair –1)st = (272.6 + 4.608/(m) + 0.061/(m) 2)10-6 = 0.000280 (@ = 632.8 nm) Effects of pressure: the quantity (nair –1) is proportional to the number of moles per unit volume n/V = P/RT: nair –1 = (nair –1)st (P/760)(288/T) The coefficients that account for variations of nair upon temperature and pressure changes are: d(nair –1)/dT =- (nair –1)st (288/T2) ≈ -1 ppm/°C d(nair –1)/dP =- (nair –1)st (1/760) ≈ + 0.36 ppm/mbar Variation of T=10°C and P=10mbar influence the 5th and the 6th digit of the displacement measurement to achieve an acuravy better than 10-6 , temperature and pressure sensors must be used to achieve the correct value of the scale factor of the interferometer
- 13. G. Giuliani - INTERFEROMETRY_02 13 6. Speckle-Pattern Operation on diffusive surfaces In many practical cases it is impossible to use the interferometer onto a cooperative target The operation of laser interferometers (and of most laser instrumentation) on targets with rough, diffusive surfaces involve the phenomen of speckle-pattern When light with high temporal and spatial coherence is projected onto a diffusive surface, the back-diffused light has a granular structure, similar to a bi- dimensional white noise This phenomenon is called speckle-pattern Speckle = small point, or colored stain He-Ne laser on paper
- 14. G. Giuliani - INTERFEROMETRY_02 14 Speckle-Pattern Origin of the Speckles The speckle-pattern is the field emitted in a semi-space by a diffuser illuminated by coherent light Diffusing surface: it has random height variations, with amplitude z >> laser beam D z diffuser z s t l s P P+P' P+P'' E(P) E(P+P') E(P+P'') x y z _ _ _ The resulting field in point P results from the sum of many vectors, each being originated by a different point of the diffuser the phase relation between the different contributions is A displacement from P towareds P+P’ or P+P’’, implies that the field in these points gradually loses coherence with respect to the field in P The spatial contour of a single speckle (grain) is defined as the volume where the fields correlation with point P is > 0.5 Individual speckle grains take the shape of an ellipsoid, with the major axis aligned towards the center of the diffuser area illuminated by the laser
- 15. G. Giuliani - INTERFEROMETRY_02 15 Speckle-Pattern Speckle size - 1 Transverse and longitidinal size of the speckle grains are statistical variables we are interested in knowing the average values of the longitudinal dimension sl (along z) and of the transverse dimension st (in the xy plane) For a diffuser with circular laser illuminating spot with diameter D, it is: st = z/D; sl = (2z/D)2 with z = distance from the center fo the diffuser The longitudinal dimension is much larger than the transverse one The projection along the normal axis of speckle grains that lie outside the normal is identical to that of the in-axis speckle grains laser beam D diffuser z st sl x y z
- 16. G. Giuliani - INTERFEROMETRY_02 16 Speckle-Pattern Speckle size - 2 Each speckle can be considered as a volume corresponding to a single spatial mode, with acceptance a=A = 2 Demonstration: Acceptance: a = Area Solid Angle = A Solid angle under which the source (diffuser) is seen from point P: = (D/2z)2 Area: A = (st/2)2 Letting A = 2 (single mode condition) 2 = (st/2)2 (D/2z)2 st = (4/)(z/D) The set of rays that define is (trasversally) smaller than s for longitudinal extent equal to: st/, con = D/2z si = (2/)(2z/D)2 Formulae shown in previous slides are obtained (apart from multiplicative factors 1) D z source sl s t
- 17. G. Giuliani - INTERFEROMETRY_02 17 Speckle-Pattern Speckle size - 3 Example Plaser = 1 mW D = 2.5 mm z = 0.5 m = 632.8 nm st = z/D = 126 m sl = (2z/D)2 = 25 mm A photodetector with diameter Dfot = 10mm receives a total power given by: Pr = BA = (Plaser/A)A (Dfot/2z)2 = Plaser(Dfot/2z)2 = 0.1 W The photodetector receives N speckles: N =Adet/Aspeckle = (Dfot/2)2/ (st/2)2 = (Dfot/st)2 = 6300, The useful power to generate the interferometric signal is the one that belongs to a single spatial mode (that is, a single speckle grain) Puseful = 0.1 W / 6300 = 15 pW laser beam D diffuser z st sl x y z Dfot
- 18. G. Giuliani - INTERFEROMETRY_02 18 Speckle-Pattern Speckle size - 3 Example Plaser = 1 mW D = 0.25 mm z = 0.5 m = 632.8 nm st = z/D = 1.26 mm sl = (2z/D)2 = 25 mm A photodetector with diameter Dfot = 10mm receives a total power given by: Pr = BA = (Plaser/A)A (Dfot/2z)2 = Plaser(Dfot/2z)2 = 0.1 W The photodetector receives N speckles: N =Adet/Aspeckle = (Dfot/2)2/ (st/2)2 = (Dfot/st)2 = 63, The useful power to generate the interferometric signal is the one that belongs to a single spatial mode (that is, a single speckle grain) Puseful = 0.1 W / 63 = 1.5 nW laser beam D diffuser z st sl x y z Dfot
- 19. G. Giuliani - INTERFEROMETRY_02 19 Performance Limits Speckle-Pattern When performing interferometric measurements on diffusive surfaces there are additional error sources Intensity effect The field Em could represent a “dark” speckle fading of the interferometric signal (“signal drop- out”) Possible solutions: Improve the focusing on the target surface (make D as small as possible) speckle size increases (st 1/D) speckle number N decreases the back-diffused power is distributed over a smaller numberof speckles larger signal-to-noise ratio Use of a second sensor in parallel (sensor diversity) probability of signal drop-out decreases Move the laser spot onto the target surface in the transverse direction a different area of the diffuser is illuminated the speckle distribution changes a “bright speckle” may hit the photodetector (“bright speckle-tracking”) Phase effect Within each speckle, a phase error ( 2) can occur General consequence: It is not possible to measure accurately large target displacements only vibration measuments are possible (laser vibrometry) IphR He-Ne Zeeman laser IphM r s PDm PD r F D wl