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# OE Instrumentation_03_Interferometry_2.pdf

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### OE Instrumentation_03_Interferometry_2.pdf

1. 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. 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. 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. 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. 5. G. Giuliani - INTERFEROMETRY_02 5 Performance Limits Consine error  Output photocurrent: Iph = I0 {1 + Vcos[2k(sm-sr)]}   The effectively measured pathlength is: ksmcos() 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. 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 = cc [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. 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 + Vcos[2k(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. 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. 9. G. Giuliani - INTERFEROMETRY_02 9 Performance Limits Quantum noise - 2  Iph = I0 {1 + Vcos[2k(sm-sr)]}  Signal term: Is = (I0V)2ksm  Signal-to-noise ratio: (S/N) = I2 s/i2 n = (I0V2ksm)2 / 2qI0B  The NED can be found by letting (S/N) = 1 and solving for sm  NED = (/2V)(qB/2I0)1/2 = = (/2V)(hB/2P)1/2  It is possible to calculate the equivalent quantum phase noise: n = 2kNED = V(2hB/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. 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. 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 = EmEr/(|Em||Er|)  All the above effects, combined, define the final effective visibility: V = (sp pol)exp[-(L/Lc)]
12. 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. 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. 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. 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. 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. 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 = BA = (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. 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 = BA = (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. 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