Effects of spatial modes on ladar vibration signature estimation

Effects of spatial modes on ladar vibration signatureestimation

Douglas Jameson,1,* Matthew Dierking,1 and Bradley Duncan2

1Sensors Directorate, Air Force Research Laboratory, 3109 Hobson Way St., Wright-Patterson Air Force Base,Ohio 45433-7700, USA ([email protected])

2Electro Optics Program, University of Dayton, 300 College Park, Dayton, Ohio 45469-0245, USA([email protected])

*Corresponding author: [email protected]

Received 28 June 2007; revised 27 August 2007; accepted 28 August 2007;posted 29 August 2007 (Doc. ID 84656); published 10 October 2007

Ladar-based vibrometry has been shown to be a powerful technique in enabling the plant identificationof machines. Rather than sensing the geometric shape of a target laser vibrometers sense motions of thetarget induced by moving parts within the system. Since the target need not be spatially resolved,vibration can be sensed reliably and provide positive identification at ranges beyond the imaging limitsof the aperture. However, as the range of observation increases, the diffraction-limited beam size on thetarget increases as well, and may encompass multiple vibrational modes on the target’s surface. As aresult, vibration estimates formed from large laser footprints illuminating multiple modes on a vibratingtarget will experience a degradation. This degradation is manifest as a spatial low-pass filtering effect:high-order mode shapes, associated with high-frequency vibrations, will be averaged out while low-frequency vibrations will be affected less. A model to predict this phenomenology is proposed for bothpulse-pair and cw vibrometry systems. The cw model is compared to results obtained using an off-the-shelf laser vibrometry system. © 2007 Optical Society of America

OCIS codes: 120.7280, 280.3640, 280.1350.

1. Introduction

The ability to quickly identify targets on the battlefieldis of critical importance. One approach currently beingstudied for use in target identification is remote ladarvibrometry. Target identification utilizing vibrometryis based on the principle that every vehicle type has aunique vibration signature. Each vehicle vibrates withdifferent frequency distributions depending on thetype of engine, the engine state, and the construction ofthe vehicle body and support structures. It has beenshown that the major plant noise vibration frequencyconstituents are relatively independent of ladar to tar-get observation angles and beam spot location but thatthe amplitudes and spatial distributions of vibra-tions are strongly influenced by the structures ofthe target [1].

Vibration frequency measurements are made byobserving target surface displacements, or velocities,

over time. These measurements can be made usingeither cw or pulsed system approaches. Based uponbackscattered light from either of these systems sur-face displacement resolutions on the order of thewavelength of light are achievable. Our interest is indeveloping a vibration measurement system, whichworks at long ranges while providing an accuratevibration spectrum of a target of interest.

Previous work has explored the effects of multimodevibrations on one-dimensional targets using a cw vi-brometer [2]. It was shown that when measuringacross a vibration node, the velocity, and therefore thephase return across the node, will have odd symmetry.Spatially integrating the returned signal across thenode will therefore yield a greatly reduced velocityestimate. In fact, complete extinction of the observedpeak associated with this vibration frequency can beachieved with careful alignment. Our current researchhas expanded this work to examine the effects of spa-tial averaging on two-dimensional vibrating targets.

Close-range industrial laser vibrometry systemsrely on the beam having a small area of interaction

0003-6935/07/307365-09$15.00/0© 2007 Optical Society of America

20 October 2007 � Vol. 46, No. 30 � APPLIED OPTICS 7365

with the target so that it can be assumed that thevelocity across the laser footprint is essentially uni-form. By contrast, a remote sensing ladar vibrometerwill produce a large illumination spot. For example,ignoring turbulence and assuming uniform apertureillumination, a 10 cm diameter beam propagating 10km to a target, at a wavelength of 1.5 �m, will yielda spot size of approximately 37 cm. This spot maytherefore encompass multiple spatial modes of sur-face vibration, and a wide range of velocities may beinterrogated across the target’s surface. The result-ing velocity measurement will therefore be an aver-age of the surface velocities located across thefootprint.

Recall that surfaces with high-order vibrationalmode shapes have correspondingly high-frequency vi-brations. As a result, spatial averaging will be moresevere for high-frequency displacements. We willdemonstrate herein that the larger a footprint is,relative to the period of the surface vibration modes,the lower the velocity estimate will become. We willdevelop models for both a pulse-doublet vibrometeras well as a modified Mach–Zehnder interferometer-based cw system in order to investigate the effects ofspatial averaging on multimode vibrations. The re-sulting cw model will be compared with experimentalresults using a commercial laser vibrometer system.

2. Pulse-Pair Vibrometry Systems

A. Single-Frequency Targets

A pulse-pair vibrometer system operates as a high-resolution range detector and makes rapid measure-ments of the change in range to a target over a knowntime interval. A typical pulse-pair laser vibrometersystem is shown in Fig. 1 [3]. This design representsa monostatic coherent ladar configuration and usesthe same telescope to both transmit and receive sig-nals. The master oscillator (MO) is a laser that seedsthe slave oscillator. As shown in Fig. 2 the slave oscil-lator (SO) creates linearly polarized, high-energyamplitude-modulated pulses shifted by intermediatefrequency (IF) �IF, with pulse spacing �Ts and doubletpulse repetition frequency (PRF) fPRF. Pulses leavethe SO of Fig. 1 and pass through a polarizing-beam-splitter (PBS) cube where the pulse is split. One pulseis sent to the bottom large area single element detec-tor of Fig. 1 to be stored for later comparison as the

monitor pulse. The second pulse travels through thequarter-wave plate and telescope and is sent to thetarget as the signal pulse. Notice that the PBS andthe quarter-wave plate serve as a transmit–receive(T�R) switch. Light is then reflected off the target andpasses back through the T�R switch. The PBS thendirects the signal pulse to the upper large area singleelement detector to be stored. Notice that both thesignal and the monitor pulses are detected after firstmixing each with the MO, resulting in electrical sig-nals having frequency content centered at �IF [3].The signal and corresponding monitor pulses arethen compared digitally to retrieve phase informationdue to the time of flight to the target through a pro-cess known as pulse-pair processing [4].

Referring to Fig. 2, after detection the monitorpulses can be modeled as temporally shifted compo-nents of an IF term according to the relationship

M�t, p, n� � Re�exp�j�IFt�M̃�, (1)

where the phasor M̃ is written as

M̃ � C exp��t2

�2�exp��j�IF� pfPRF

� n�Ts��. (2)

In this relationship �IF is the IF, fPRF is the pulse-pairrepetition frequency, p � 1, 2, 3, . . . , is the pulse-pair number, �Ts is the time between pulses in eachdoublet, n is zero for the first pulse in a given pulsepair and one for the second, and � is the width of eachindividual pulse. Any normalization constants arecontained in C.

The detected signal pulses can be similarly mod-eled by including spatial components due to the tar-get’s surface vibration and the illumination beamshape. If the beam footprint on the target is fullyimaged onto the detector, the resulting electrical sig-nal is found by integrating across the illuminatedarea of the target as followsFig. 1. Block diagram of a pulse-pair laser vibrometer.

Fig. 2. Pulse pairs with doublet separation 1�fPRF, pulse separa-tion �Ts, and width �.

7366 APPLIED OPTICS � Vol. 46, No. 30 � 20 October 2007

S�t, p, n, Z0� � C exp��t2

�2�Re�exp�j�IF�t �p

fPRF

� n�Ts � 2Z0

c ��A

exp��x2 � y2

�2 �� exp�jtarget�x, y��dA, (3)

where the beam shape is assumed to be untruncatedGaussian with on-target width defined by �, the areaof the detector is A, and the distance to the target isZ0 (resulting in a round-trip distance of 2Z0). Thephase target represents the spatially varying phaseinduced in the illumination beam due to the target’svibrating surface. Notice that the IF-signal term isdifferent from the monitor pulse because it accumu-lates extra phase through propagation to and fromthe target. Notice also that the illumination beamweights the individual scatterers’ displacement con-tributions to the phase during integration across thedetector. Assuming a Gaussian-shaped beam meansthat scatterers closer to the beam center will receivea greater weighting. The scatterer contributionsaway from the center of the beam cannot, however, beignored.

Looking more closely, the fourth exponential phaseterm in Eq. (3) represents added path length due totarget vibration and can be expressed as

exp�jtarget�x, y�� exp�j22�Ts

�V�x, y�

� cos��T�t �p

fPRF� n�Ts �

Z0

c ��,

(4)

where the wavelength of the light is given by �, thesurface velocity mode shape is given by V�x, y�, andwhere we have assumed that only one vibration modeat frequency �T is present on the target. A pathlength Z0 (i.e., not 2Z0) is used in this term becausethe velocity can only be estimated at the moment ofinteraction with the target. Therefore the accumu-lated phase need not reflect the time required for thereturn trip. Notice that Eq. (4) in effect represents thephase shift due to the magnitude of the velocityV�x, y�, normalized with respect to the ambiguity ve-locity, which we will now discuss.

Phase can only be processed modulo 2�, so that anyphase accumulated beyond a full wavelength, overthe sampling time �Ts, will be lost. Therefore thetotal additional path length 2�z due to surface vibra-tion, where �z represents the surface displacementof the target, must be less than the wavelength �.Because the time between pulses in each pulse pair is�Ts the additional path length can be written usingthe time between samples and the instantaneous tar-get velocity V as 2�z � 2V�Ts. Applying the condition

that 2�z � � yields 2V�Ts �, the ambiguity velocityVamb is then defined when the last equation becomesan equality giving

Vamb ��

2�Ts. (5)

The ambiguity velocity is clearly a function of theminimum sampling period and represents the peakvelocity beyond which the total phase will passthrough 2�, resulting in information loss.

The final form of the detected signal pulse is foundby substitution of Eq. (4) into Eq. (3) yielding

S�t, p, n, Z0� � Re�exp�j�IFt�S̃�, (6)

where the phasor S̃ is given by

S̃ � C exp��t2

�2�exp��j�IF� pfPRF

� n�Ts � 2Z0

c ��

A

exp��x2 � y2

�2 �exp�j22�Ts

�V�x, y�


fPRF� n�Ts �

Z0

c ��dA. (7)

The signal and monitor pulses are then processedusing zero-lag correlation, which allows the phaseadded due to the target movement to be recovered.The required calculation can be written as [3]

� � angle��

�

�M̃0M̃1*��S̃0S̃1*�*dt�, (8)

where � is the difference between the spatially av-eraged phases induced in the individual pulses ofeach pulse pair, M0 and M1 are the monitor pulses forn � 0 and n � 1, respectively, and S0 and S1 are thesignal pulses for n � 0 and n � 1, respectively. Theoperation of taking the phase angle, in radians, isgiven by the angle (�) notation, and complex conju-gates are denoted by *. Performing this calculationallows the phase due to the changes in path lengthbetween pulses in each pair to be found and is oftenreferred to as phase processing. Since the change inpath length for a single velocity estimate occurs overtime �Ts, an estimate of the surface average velocityover that time is made.

Multiplication of the n � 0 monitor pulse phasor bythe complex conjugate of the corresponding n � 1phasor yields

M̃1M̃2* � �C�2 exp��2t2

�2 �exp�j�IF�Ts�. (9)

Similarly, multiplying the n � 0 signal pulse phasorby the corresponding n � 1 phasor yields


S̃1S̃2* � �C�2 exp��2t2

�2 �exp�j�IF�Ts�

�A

exp��x2 � y2

�2 �exp�j22

��TsV�x, y�


fPRF�

Z0

c ��dAA

exp��x2 � y2

�2 �� exp��j2

2

��TsV�x, y�


fPRF� �Ts �

Z0

c ��dA. (10)

Finally, multiplying Eq. (9) by the complex conjugateof Eq. (10) and substituting into Eq. (8) yields

� � angle�A

exp��x2 � y2

�2 �exp��j22

��TsV�x, y�

� cos��T� pfPRF

�Z0

c ��dAA

exp��x2 � y2

�2 �� exp�j2

2

��TsV�x, y�


� �Ts �Z0

c ��dA�, (11)

where we have dropped the time-integration step dueto the fact that the pulse width � is typically veryshort � 10 ns� compared to the period of vibrations� 10�2–10�5 s�. This allows us to assume the inte-grand of Eq. (8) to be essentially constant.

We now observe that the differential phase ex-pressed in Eq. (11) is directly proportional to thetarget’s spatially averaged instantaneous velocity Vi

according to the relationship

�

�Ts� 2fDi �

2

�

2�zeff

�Ts�

4

�Vi, (12)

where fDi is the instantaneous Doppler frequencyshift experienced by the pth pulse pair and �zeff is thetarget’s spatially averaged, effective surface displace-ment [5]. In practice then, a time series of spatiallyaveraged instantaneous velocities will be constructedfrom multiple observations of �� (i.e., by sequentiallyrecording �� for the p � 0, 1, 2, 3, . . . , pulse pairs).Treating this time history of instantaneous velocitiesas a time-varying amplitude, a fast Fourier trans-form (FFT) operation can then be utilized in order toextract the spatially averaged vibrational signatureof the target [6].

A simple simulation to demonstrate this techniquecan be performed by considering a circular membranetarget with clamped edges illuminated by a truncatedGaussian beam. We will assume that a beam withconstant radius � illuminates the target, at a range

Z0 � 0, through a variable aperture that has amaximum radius at the target’s clamped edge and aminimum radius approaching zero. The target’s cir-cularly symmetric surface velocity mode shapes aregoverned by [7]

V�x, y� � V0J0�Zn

rr0�, r � r0, (13)

where r � �x2 � y2, V0 is the peak velocity, r0 � 2� isthe radius of the target, and Zn is the nth zero of thezero-order Bessel function.

In our simulation we will assume a target vibrationfrequency fT � �T�2 of 250 Hz, a maximum targetvelocity of 1 mm�s, a pulse-pair repetition frequencyof 8000 Hz, a target radius of 5 cm, pulse-pair spac-ing �Ts of 31.64 �s, and a laser wavelength of632.8 nm. We note, however, that the results of thesimulation are independent of these values as long asthe ambiguity velocity (10 mm�s in this case, bychoice of �Ts) is greater than the maximum velocity ofthe simulated target. The only value that is varied isthe radius of the aperture, which vignettes the illu-minating beam.

After examining a series of 80 simulated samples(i.e., p � 0–79) for each of several aperture radiusvalues, the normalized frequency response shown inFig. 3 was obtained. Notice that the frequency re-sponse is reduced as the footprint grows in size, asexpected, and that this effect is more pronouncedwith the higher-order modes. Due to the fact thathigher-order modes correspond to higher vibrationfrequencies Fig. 3 also demonstrates that the antici-pated low-pass filtering effect is taking place.

B. Multiple-Frequency Targets

Up to this point we have assumed that targets vibrateat only a single frequency. This is, however, not gen-

Fig. 3. Normalized frequency response of a pulse-pair vibrometeras a function of normalized aperture radius, shown for three cir-cularly symmetric surface vibration modes.


erally the case. The results of Subsection 2.A can beextended to include targets with even a continuousvibration spectrum if we use an equalization filter. Todetermine the appropriate filter it is helpful to exam-ine the effects on Eq. (11) of a single point scattererlocated at the origin. Assuming V�x, y� � V0��x, y� inEq. (11), where V0 is a constant velocity amplitude,after simplification we find the resulting phase mea-surement to be

� � 24

��Ts sin��T�Ts

2 �V0


��Ts

2 �Z0

c ��

2�. (14)

Recognizing that this phase change is measured overtime �Ts yields the following result similar to Eq.(12),

�

�Ts�

4

�Vi�2 sin��T�Ts

2 ��, (15)

where the instantaneous velocity Vi is given by

Vi � V0 cos��T� pfPRF

��Ts

2 �Z0

c ��

2�. (16)

In general then, for a target with arbitrary vibra-tional frequency content we would once again at-tempt to build up a time history of spatially averagedinstantaneous velocities from multiple observationsof ��, after which we would employ an FFT operationin order to extract the vibration signature of the tar-get. However, as demonstrated by Eq. (15), in order toproperly weight the various vibrational frequencycomponents present on the target we must first mul-tiply the resulting spectrum by the following equal-ization filter:

Heq�� 1

sin��Ts

2 �, �0 � � � ��Ts�. (17)

Notice that the factor of 2 in Eq. (15) is largely incon-sequential and does not show up in Eq. (17) since it isconstant for all target frequencies. In addition, noticethat the ability to measure instantaneous velocities,and therefore target vibrational frequency content, isdirectly related to the pulse-doublet spacing in rela-tion to the vibration frequency �. In particular, themaximum frequency that can be measured is gov-erned by the Nyquist criteria �max � ��Ts.

3. Continuous-Wave Vibrometry Systems

A. System Model

A common method for measuring target vibrations isto use a cw heterodyne interferometer. Such systemsare commonly used for commercial and industrial

purposes, and many systems are currently availablefor purchase [8]. Similar cw systems that simply usehigher power lasers than are typically found in com-mercial systems could be used for ladar target iden-tification.

A common interferometer-based laser vibrometersystem is shown in Fig. 4. The reference beam ismodulated by an acousto-optic modulator (AOM) andin the plane of the detector is written as

UR�x, y, t� � E0�x, y�exp�j��0 � �AOM�t�, (18)

where E0�x, y� is the reference beam field amplitude(typically assumed to be Gaussian), �0 is the laserfrequency, and �AOM is the AOM frequency. Similarly,without regard for target reflectance and propagationlosses, the Doppler-shifted object beam field reflectedby the target and imaged onto the detector is given by[5,9]

UO�x, y, t� � E0�x, y�exp�j�0t � j4

��TV�x, y�cos��Tt��,

(19)

where, as in Section 2, the surface velocity modeshape is given by V�x, y� and we have assumed thatonly one vibration mode at frequency �T is present onthe target. After interfering the object and referencebeams at the detector, the time-varying component ofthe spatially integrated intensity to which the detec-tor responds can be written as

Iac�t� � 2A

�E0�x, y��2

� cos��AOMt �4

��TV�x, y�cos��Tt��dA

� 2A

exp��x2 � y2

�2 �� cos��AOMt �

4

��TV�x, y�cos��Tt��dA, (20)

Fig. 4. Heterodyne cw vibrometer system capable of measuringboth positive and negative velocities.


where again � is the on-target untruncated Gaussianbeam width and A is the area of the single elementdetector.

The short time Fourier transform (STFT) methodis used to demodulate this signal. The recovered sig-nal is first divided into short temporal samples,which may overlap. The individual samples are thenFourier transformed using zero padding to increasethe frequency resolution and reduce aliasing. Next,the location of the frequency peak is found and usedto represent the average Doppler frequency shiftover the sample’s duration. In practice this processis simplified by use of the specgram.m function in theMATLAB signal processing toolbox [10]. The Dopplerfrequency shifts are then converted to velocity esti-mates using the relationship,

Vest�t� ��fD�t�

2 , (21)

where fD�t� is the Doppler frequency shift as a func-tion of time [5]. Finally, as discussed in Section 2, theFourier transform of Vest�t� can be taken in order toextract the spatially averaged vibrational signatureof the target.

The selection of the peak frequency in this processis an approximation to the FM capture effect as de-scribed by [11]. This effect is produced when an FMsignal, which contains multiple frequencies, is sent toa limiter and then demodulated. The instantaneousfrequency of the combined return will be dominatedby the stronger signal. Therefore, when modeling thedemodulated returned signal the peak Doppler fre-quency is chosen to represent the “lock-on” frequencyof the capture effect. The strongest signal will bedetermined by both the number of scatterers at agiven Doppler frequency�velocity and the strength ofthe illuminated weighting on these scatterers. Notethat since the cw model measures instantaneousDoppler frequency shift (and therefore velocity) of thedetected interference signal directly and not as thedifference between two phases that there is no needfor an equalization filter.

To demonstrate the spatial averaging effects of thistechnique, a simulation similar to that used to createFig. 3 was performed for the cw model. In this casethe AOM frequency was assumed to be 20 kHz, andthe surface velocity mode shape was again assumedto be given by Eq. (13), where the peak velocity wasassumed to be 1 mm�s and the vibration frequency fT

was assumed to be 250 Hz. The illumination beamand variable aperture properties of the earlier simu-lation also remain the same. Figure 5 then demon-strates the normalized frequency response for atruncated Gaussian beam illuminating the first threemodes defined by Eq. (13). As with the pulse-pairmodel, truncation of the illuminating beam changesfrom nearly a point to the full footprint where 2�� r0. Notice that in the cw case the low-pass filteringeffect is not as strong as that observed in the pulse-pair model. By comparing Figs. 3 and 5 we see that in

general the normalized frequency response is less forthe pulse-pair�phase processing model than for thecw model, especially for larger beam footprints. Thisis likely caused by a combination of the illuminatingbeam, target surface mode shape, and the FM cap-ture effect discussed above. Therefore, ideally (i.e., inthe absence of noise), cw systems will respond tohigh-frequency vibrations with less relative attenua-tion than would comparable pulse-pair systems.

B. Experiment and Results

Several commercial cw laser vibration systems areavailable from Polytech, two of which were used toverify our cw model [8]. An OFV-503 head unit (withOFV-5000 control) was mounted in line with a 10 cmcollimating beam expander to form a broad illumina-tion beam, while an OFV-343 head unit (with OFV-3001 control) was mounted on a pan-and-tilt motorstage to provide surface velocity point scans acrossthe target. The target was a reflective circular mem-brane made from Reflexite AC1000 reflective sheet-ing material [12]. This material is made of smallmicroreflectors, which reflect a high percentage oflight back toward the source while also remainingflexible enough to support strong vibration modes.During data collection the target was held in aclamped edge aluminum assembly, shown schemati-cally in Fig. 6, with a maximum inside radius r0 of5 cm.

The laser from the OFV-503 was expanded to a 4�diameter of 10 cm and propagated through a variableradius circular aperture. The resulting beam thusapproximated the conditions mentioned for the sim-ulations discussed above. As shown in Fig. 7 the ex-panded and vignetted beam then illuminates thetarget, after which the reflected light returns throughthe beam expander back into the head unit where itis mixed and detected. Recall that heterodyne mixing

Fig. 5. Normalized frequency response of a cw vibrometer as afunction of normalized aperture radius, shown for three circularlysymmetric surface modes.


efficiency drops quickly as the relative tilt betweenthe mixing beams increases [13]. The collected signalwas therefore boosted by use of an 4 m separationbetween the variable aperture and target. Note thatwhile this path length was long enough to assist inproper beam alignment, it was short enough thatdiffraction effects due to truncation of our illumina-tion beam could be ignored.

A true surface velocity profile was measured byfirst finding the frequency corresponding to the 01mode given by Eq. (13). This was accomplished bydriving the target with white noise and measuringthe velocity at the center of the target using the OFV-343 sensor head. The power spectral density of thisvelocity data had several resonance peaks, the first ofwhich was associated with the 01 mode. This wasvalidated by then driving the target at only the 01mode resonance frequency and recording multiple ve-locities across the target’s surface. The grid shown inFig. 8 represents the points at which surface velocitymeasurements were taken. (LABVIEW software wasused to control the pan�tilt of the OFV-343 sensorhead and to automate the data acquisition processes.)For our target an 01 mode peak was found at 375 Hz,and the resulting rms surface velocity map is shownin Fig. 9. Figure 9 itself represents an interpolated fitto the data taken at the grid points shown in Fig. 8.This interpolated velocity profile was then used inour cw model to simulate the expected spatially av-eraged frequency response to which data collectedwith the OFV-503 head could be compared.

When attempting to experimentally verify the cwmodel it became necessary to include additionalterms in order to account for several noise sources. A

normalized simulation of Eq. (20) is shown in Fig. 10.For this figure the frequency of the AOM was as-sumed to be 3800 Hz, the 01 mode velocity profile ofEq. (13) was used, the peak velocity V0 was assumedto be 1 mm�s and the vibration frequency fT was as-sumed to be 250 Hz. These numbers were chosen foraesthetic reasons and apply to Fig. 10 only. Never-theless, these values would have no effect on theaccuracy of the results as long as the Doppler-shiftedsignal falls within the bandwidth of the system.

Figure 10 shows how the detected signal ideallyresponds to constructive and destructive interferenceas the surface velocity passes from zero to peak val-ues. When the surface displacement is zero (and thevelocity is largest) the instantaneous phases fromeach scatterer across the target’s surface add con-structively, and the signal return is strong. This canbe seen in Fig. 10 where the frequency of the signal isat an absolute maximum. When the velocities de-crease from peak to zero (corresponding to maximumabsolute displacement), though, these phases begin

Fig. 6. Cw vibrometer target mount assembly.

Fig. 7. Cw vibrometer experimental setup used for measuring thespatially averaged frequency response due to a truncated Gaussiantarget illumination beam.

Fig. 8. This grid demonstrates the actual locations used in sam-pling the rms velocity across the surface of the target. Positioningerrors occured due to the accuracy and precision of the motorizedpan-and-tilt platform.

Fig. 9. Interpolated rms velocity profile of the lowest-order modepresent on the target. This mode was present at a vibrationalfrequency of 375 Hz.


to add destructively, and the detected signal becomesweak.

In practice, however, as yet unaccounted for targetsurface roughness introduces speckle in the reflectedillumination beam. In addition, the sensor head it-self is not perfectly noiseless. Therefore noise termsmust be added to the cw model in order to reduce themodel’s ability to simulate weak returns. Incorporat-ing these noise terms allows Eq. (20) to be rewrit-ten as

Iac�t� � 2A

exp��x2 � y2

�2 �cos��AOMt �4

��TV�x, y�

� cos��Tt� � Ns�x, y� � Nt�t��dA, (22)

where Ns�x, y� is a spatial phase due to random targetsurface roughness and Nt�t� is a temporal phase noiseterm used to simulate the observed velocity noisefloor of our measurement system. Ns�x, y� was chosento be a zero-mean Gaussian random variable with astandard deviation of 100 cycles of phase, while anrms minimum velocity measurement of 1 �m�s hasbeen previously measured by us. To simulate thisdesired noise floor Nt�t� was empirically determinedto be well represented by a zero-mean Gaussian ran-dom variable with a standard deviation of 0.01 cyclesof phase.

A 10 s point scan of the velocity at the center of thetarget served as the ideal measurement to which wecompared the spatially averaged velocities obtainedfor different illumination footprint diameters. Theexperiment was conducted using aperture diametersof 0.5, 1, 1.5, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75, and4 in. The apertures were applied to the experimentalsetup of Fig. 7 in descending order, and 10 s of datawere taken for each aperture value. The velocity datawere then compared to the ideal central point data bycomparing the peak values of the FFT of each of the

signals. In all cases the target was driven at the 01mode resonance frequency of 375 Hz, and each aper-ture size was used to record a single frequency re-sponse before moving on to the next value. Theprocess was then repeated until a total of ten re-sponses was captured for each aperture. Mean re-sponses as well as the standard deviations of the datawere then calculated for each aperture diameter.

The results of our experiment are shown in Fig. 11.The average experimental data are shown by aster-isks along with error bars, which indicate one stan-dard deviation of the frequency response data. Thesolid curve represents the average of ten runs of thenoisy signal model given by Eq. (22). The statisticalmodel shows very good agreement with the experimen-tal results. In general we observe that as the size of thebeam interrogating a nonuniform target increases, thelower the velocity estimate and corresponding normal-ized frequency response will become.

4. Conclusion

Theoretical models have been created to describeboth pulse-pair and cw ladar vibrometry techniques.In each case the effects of large illumination beamsizes have been included in order to examine how thetarget plane footprint of the interrogating beam af-fects the frequency response of the system. In bothcases an increased beam size yields a decreased fre-quency response for our simulated targets. For thepulse-pair technique an equalization filter was alsodeveloped that will allow a properly weighted relativefrequency response to be determined from instanta-neous phase measurements.

We have also demonstrated that in a noiseless en-vironment the cw vibrometry technique appears to bethe most robust, as the spatial averaging effects due

Fig. 10. Plot of Eq. (16). Peaks occur when surface displacementis zero and all scatterers constructively induce a peak Dopplerfrequency shift. When surface displacement is maximum (and ve-locity approaches zero) destructive interference and surface rough-ness cause a sharp reduction in signal strength. Fig. 11. Experimental results for the normalized frequency re-

sponse as a function of normalized aperture radius. The solid curverepresents the noisy cw model after incorporating the experimen-tal 01 mode shape. The asterisks represent the average of tenexperimental runs at each aperture radius value, while the errorbars indicate one standard deviation in the experimental data.


to large beams do not degrade the frequency responseas severely as in pulse-pair systems. However, webelieve that in long-range noisy environments apulse-pair system will perform better since high peakpower pulses will have greater relative noise immu-nity.

An experiment was designed and performed inorder to verify the cw model for various illuminationconditions defined by a truncated Gaussian beam.To benchmark our data, our ideal theoretical cwmodel was modified to account for target surfaceroughness and detector noise. After incorporatingthe appropriate empirically determined spatial andtemporal phase noise terms our resulting simulatedcw ladar vibrometer performance matched our exper-imental data very well. Future work will focus onexperimentally verifying the pulse-pair model in thepresence of noise.

This effort was supported in part by the U.S. AirForce and Anteon, Inc., of Dayton, Ohio through con-tract F33601-02-F-A581 and by the Ladar and OpticalCommunications Institute (LOCI) at the University ofDayton.

The authors wish to thank John Schmoll, LarryBarnes, Tim Meade, Brian Smith, and Dave Mohler fortheir guidance and technical assistance. The views ex-pressed in this paper are those of the authors and donot reflect on the official policy of the Air Force, De-partment of Defense, or the U.S. Government.

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Effects of spatial modes on ladar vibration signature estimation

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