July 1, 1997 / Vol. 22, No. 13 / OPTICS LETTERS 1027

Double-exposure heterodyne imagingfor observing line-of-sight deformation

David G. Voelz and Lenore McMackin

Imaging Branch, U.S. Air Force Phillips Laboratory, Kirtland Air Force Base, New Mexico 87117

James K. Boger*

Rockwell Power Systems, Albuquerque, New Mexico 87117

Matthew P. Fetrow

Applied Technology Associates, Albuquerque, New Mexico 87117

Received February 24, 1997

An optical heterodyne array imaging system is described for double-exposure interferometric measurementof objects that change because of stress or vibration. The signal measurement and processing approach issummarized, and the method is demonstrated in the laboratory. An advantage of this technique over otherelectronic interferometric imaging methods is that complex exposures are created with digital phase andamplitude values at each image pixel. In addition, measurement of object deformation does not require timesynchronization between the camera exposure time, or laser pulse time, and object vibration frequency.

Holographic interferometry,1 including derived tech-niques such as electronic speckle pattern interfero-metry,2,3 is a powerful method for noncontact andnondestructive testing of parts and materials. A com-mon application of this method is the detection andmeasurement of submicrometer motion in an objectowing to vibration or deformation. Holographic in-terferometry is typically based on a double-exposureapproach in which the coherently illuminated objectis recorded both before and after the deformation.In most methods, an intensity image of the objectis produced, upon which is superimposed a fringestructure related to the deformation. A fringe analy-sis is then done to yield information about the ob-ject changes.4,5 We propose and demonstrate in thelaboratory a double-exposure interferometric imagingapproach based on optical heterodyne array measure-ments.6,7 A heterodyne measurement is made at eachpixel of a sensor array so that a complex image, con-taining amplitude and phase values, is obtained. Us-ing these complex digital images, we can perform acomparison of two object states—for example, beforeand after deformation—in the computer without us-ing fringe analysis algorithms. Movement along theline of sight at each object point is computed completelydigitally.

A schematic of the optical heterodyne imaging sys-tem is shown in Fig. 1. A coherent laser beam is splitinto a probe leg and a reference leg. The probe beamis expanded to illuminate a test object, and an image isformed through a lens on a CCD sensor. The opticalfield at the sensor can be written as

E1sr, td ­ A1srdexpf jf1srdgexpsjvtd , (1)

where r is the two-dimensional spatial coordinate,A1srd is the real field amplitude, f1srd is the f ieldphase, and v is the optical frequency. The expanded

reference beam, written as

Erefstd ­ Aref exps jfrefdexpf jsv 2 Dvdtg , (2)is combined with the image field at the sensor face.In Eq. (2) the amplitude Aref and the phase fref haveno dependence on r. The acousto-optic modulators(AOM’s) in Fig. 1 are used to shift the referencebeam’s optical frequency by 2Dv. Two modulators inseries permit a small frequency shift to be realized inwhich the second modulator removes the base opticalfrequency of the first, leaving only the Dv residualshift.

The intensity function produced at the sensor face is

I1sr, td ­ jE1sr, td 1 Eref stdj2 ­ A1srd2 1 Aref2 1 2A1srd

3 Aref cosfDvt 1 f1srd 1 frefg . (3)In practice, the sensor array and the data-acquisitionsystem collect a sequence of frames such that the cosineterm in Eq. (3) is adequately sampled in time. Thesampled intensity function can be written as

I1srm, tnd ­ A1srmd2 1 Aref2 1 2A1srmdAref cosf2pDntn

1 f1srmd 1 frefg , (4)

Fig. 1. Schematic of the heterodyne array imaging sys-tem: BS’s, beam splitters.

1028 OPTICS LETTERS / Vol. 22, No. 13 / July 1, 1997

where rm denotes a discrete pixel location and tn isthe time at which the nth frame was collected. Ifn ­ 0, 1, 2 . . . N 2 1 and s is the frame rate (framesper second) then tn ­ nys. The beat frequency Dn

has units of cycles per second. If a short exposuretime is important, then the number of frames collected,N , can be as few as four (a four-bin measurement)when the measurements are completed during one beatcycle ss ­ 4Dnd. If measurement noise is significantor high accuracy is required, many more frames can becollected over several beat cycles.

The magnitude 2A1srmdAref and the phase f1srmd 1

fref associated with the cosine term in Eq. (4) areextracted from the frame data. As we are concernedonly about demodulating one particular frequency, wesimply calculate the real R and the imaginary IFourier coefficients associated with Dn:

Rsrmd ­2N

N21Xn­0

I1srm, tndcoss2pDntnd , (5)

Isrmd ­2N

N21Xn­0

I1srm, tndsins2pDntnd . (6)

The amplitude and the phase are calculated at eachpixel rm, with sR2 1 I2d1/2 and tan21sIyRd, respec-tively. So the complete complex digital exposure canbe written as

E1srmd ­ 2A1srmdAref exph jff1srmd 1 frefgj . (7)

This recovered spatial field matches the field in Eq. (1)with the inclusion of a constant magnitude multiplier2Aref and a constant piston phase fref .

After the object is slightly deformed such that thefield amplitude is virtually unchanged but small line-of-sight depth changes Dfsrd occur, a second complexexposure, E2, can be measured:

E2srmd ­ 2A1srmdAref exph j ff1srmd 2 fref 1 Dfsrmdgj .

(8)

We can construct an interference intensity image bycombining E1 and E2 computationally:

I12srmd ­ jE1srmd 1 E2srmdj2

­ 8A1srmd2Aref2h1 1 cosfDfsrmdgj . (9)

Alternatively, we find a quantitative measure of thedeformation of the object directly from the two complexexposures by simply computing

argfE2srmdE1srmdpg ­ Dfsrmd . (10)

I12srmd is analogous to an image produced by holo-graphic interferometry in which the usual intensityimage, A1srmd2, is scaled by a constant and modulatedby fringes, cosfDfsrmdg, that indicate the changes inthe object. On the other hand, the phase-differencevalues obtained in Eq. (10) are a direct measure (mod-ulo 2p) of the object surface deformation that occurredbetween exposures. A phase-unwrapping algorithm8

can be used to remove the 2p ambiguities and buildan image showing continuous deformation across theobject.

To demonstrate the utility of this imaging approachwe set up the system shown in Fig. 1 in the laboratory.

Probe and reference beams were split out from anargon-ion laser operating on a single line (514-nmwavelength and a coherence length of ,6 cm) witha polarizing beam splitter. Half-wave plates beforeand after the splitter were rotated to adjust theintensities of both legs to an appropriate level for thecamera. The AOM’s were driven at 110 MHz and110 MHz–8 Hz, respectively, resulting in an opticaloffset frequency of Dn ­ 8 Hz. The frequency syn-thesizers that drive the AOM’s have a frequencystability of 0.1 Hz. A single complex exposure wasproduced from N ­ 16 consecutive frames taken witha standard RS-170 video format CCD camera with640 3 480 pixels and a pentium-based frame grabber.The magnitude and the phase were computed fromthe 8-Hz Fourier component. Slight errors in theamplitude and phase estimates (maximum of 6% and0.06 rad, respectively) arise because the 30-Hz cameraframe rate is not an integer multiple of the optical beatfrequency. These small errors cause no observableeffect on our results.

Figure 2 shows an intensity image of a hand-heldcompressed air can that was obtained in the lab by theheterodyne array technique. The corresponding objectphase that is also measured is not shown here. Theimage in Fig. 2 exhibits the distinctive speckled naturethat is typical of an image produced with coherent light.Initially, the can was discharged, causing its outsidecasing to cool rapidly. Two exposures, collected sev-eral seconds apart as the casing warmed up, werecombined as prescribed by Eq. (9) to produce the inten-sity image in Fig. 3(a). Deformations in the casing asit warmed between exposures produced the fringes inthis image. Figure 3(b) is the phase-difference imagethat was calculated from the same two frames by useof Eq. (10). The changes in the can are quite clear inthis image, and speckle effects have been nearly elimi-nated because only the relatively smooth line-of-sightchange of the can is being measured. An analysis ofthe smoothness along rows of the phase-difference im-age data suggests that the average error in the mea-surement at each pixel is ,ly20.

In conclusion, we have demonstrated a double-exposure heterodyne array imaging technique for usein measuring object deformation. The heterodyneapproach requires no synchronization among laser

Fig. 2. Coherent intensity image of a hand-held com-pressed air can.

July 1, 1997 / Vol. 22, No. 13 / OPTICS LETTERS 1029

Fig. 3. (a) Interference intensity image and (b) phase-difference image of a hand-held compressed air can thatis warming up. For the difference image, black is 2p andwhite is 1p. A mask created by thresholding the intensityimage has been applied to the phase image for displaypurposes.

pulse times, camera frame rates, and object vibrationfrequency, which is necessary in a common form ofelectronic speckle pattern interferometry.9 Althoughthe most accurate phase measurements for our ap-proach can be made if the camera frame rate is aninteger multiple of the heterodyne beat frequencyat the sensor, the only indispensible timing-relatedrequirement is that the camera frame rate be fastenough to properly sample the heterodyne beat signal.Of course, the object being viewed cannot changesubstantially while the complex exposure is beingcollected. Another advantage of this technique overconventional methods is that a fringe analysis of aninterference image is not required for measurementof quantitative changes in object line-of-sight depth.Phase values are directly computed for each pixel in

an image, and multiple complex images can easily becompared in a computer. Although phase-shifting(or stepping) arrangements also yield direct phasevalues,10 phase-shifting systems require a movingmirror subsystem that is synchronized to the cam-era frame rate. The use of mechanical action mayeventually impose limitations in terms of high-speedframe acquisition and real-time phase measurementapplications.

The time required for a double-exposure heterodyneimage to be produced with our laboratory system isbasically limited by the time required for collectionand construction of the two complex exposures. The16 frames for a single exposure are collected in ,0.5 swith the 30-Hz video rate camera. Computing am-plitude and phase values for all the pixels requires,1.5 s, so the total time for producing two consecutiveexposures is ,4 s. The double-exposure image calcu-lations defined by Eq. (9) or (10) add only a small frac-tion of a second to the processing time. An increasein exposure rate of more than a factor of 10 can eas-ily be gained by use of an off-the-shelf camera with ahigher frame rate and array processing hardware forthe amplitude and phase calculations. Further gainsin speed can be realized with on-chip processing capa-bilities that are just beginning to emerge.11

The authors thank Jim Fienup for his helpful com-ments on this study.

*Present address, Intel Corporation, Rio Rancho,New Mexico 87124.

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