co

O

2

Multiplicative electronic speckle-patterninterferometry fringes

Noe Alcala Ochoa, Fernando Mendoza Santoyo, Carlos Perez Lopez, andBernardino Barrientos

A theoretical analysis of fringe patterns and its experimental corroboration obtained by multiplication oftwo speckled images with electronic speckle-pattern interferometry ~ESPI! are reported. A specificallydesigned digital filter is used to enhance the contrast and visibility of the inherently noisy multiplicationfringes. Phase retrieval is achieved by a phase-stepping technique. Experimental results are pre-sented for the in-plane-sensitive optical ESPI setup; however, out-of-plane and shearing setups may beused as well. The method represents an alternative to the subtraction and addition techniques in ESPI.© 2000 Optical Society of America

OCIS codes: 120.6160, 120.2440, 120.2650, 120.3180.

1. Introduction

It is well known that to obtain fringes in electronicspeckle-pattern interferometry ~ESPI! one shouldorrelate two speckle patterns by either subtractionr addition.1 Because it removes optical noise, the

subtraction process is more widely used, and the re-sultant speckled fringes have higher contrast andvisibility, which thus makes their manipulation aneasier task. The main drawbacks of the subtractionprocess rest in its high sensitivity to environmentalinstabilities that may occur between the acquisitionof the two speckled images to be subtracted and in thepossibility that fast dynamic events may not be seen.The addition process overcomes these two problems,but, because it adds optical noise, the addition fringeshave low contrast and visibility and therefore requirea lengthier image-processing manipulation.

Leendertz2 described a photographic method tomeasure surface displacements by interfering twospeckle patterns. There a recording of a speckle pat-tern from an object was made upon a single frame ofphotographic film placed onto a rigid holder. Afterdeveloping, the negative was returned to the holder,i.e., to its original position, and then any object de-

The authors are with the Centro de Investigaciones en Optica,Apartado Postal 1-948, Leon, Guanajuato, Mexico. N. Alcala

choa’s e-mail address is alon@foton.cio.mx.Received 27 March 2000; revised manuscript received 9 June

000.0003-6935y00y285138-04$15.00y0© 2000 Optical Society of America

5138 APPLIED OPTICS y Vol. 39, No. 28 y 1 October 2000

formation could be observed as live correlationfringes. This fringe-formation mechanism consistedof multiplying the transmittances of the film framebefore and after the object deformation, that is, inutilizing two speckle patterns. The techniqueformed the basis of early ESPI by simply replacingthe photographic negative for a vidicon TV cameraand digitally subtracting the two speckled images ona memory board. Multiplication of the speckle pat-terns was not seen as feasible because multiplicationfringes have the disadvantage of lower contrast andvisibility than subtraction fringes obtained undernormal illumination conditions. Also, multiplicationfringes cannot be obtained within a single frame of aCCD camera, as can the addition fringes obtained,for instance, with a twin-pulsed laser in ESPI.3However, the contrast of multiplication fringes canbe enhanced by use of a digital filter, as describedbelow. Experimental results for an in-plane ESPIsetup are used to corroborate the theoretical anal-ysis presented. Setups for the out-of-plane andshearing ESPI sensitivities give the same results.With the advent of increasing speeds in computinghardware the multiplication fringe method in ESPIrepresents an alternative to the subtraction andaddition methods.

2. Theory

Let I1~x, y! and I2~x, y! be the intensities at any point~x, y! of two speckle patterns that are assumed to becorrelated @the variable dependence on ~x, y! isdropped for brevity; however, it is implied in all vari-

w

br

wa

vfi

w

bl

mipatftrtC

ables#. For a two-beam interferometer, e.g., ESPI, I1and I2 can be expressed as

I1 5 Io 1 Ir 1 2ÎIoIr cos~c!, (1a)

I2 5 Io 1 Ir 1 2ÎIoIr cos~c 1 w!, (1b)

here Io and Ir are the intensities of the object andthe reference beams, respectively. c is the randomphase of the speckle pattern recorded, and w is asmooth function that represents a phase change fromI1 to I2, i.e., the object state change from nondeformedto deformed. Now consider the multiplication of Eq.~1!, viz.,

Im 5 a2 1 ~b2y2!cos~2c 1 w! 1 2ab

3 cos~c 1 wy2!cos~wy2! 1 ~b2y2!cos~w!, (2)

where a 5 Io 1 Ir and b 5 2=IoIr.The last two terms on the right-hand side of Eq. ~2!

correspond to two fringe patterns of different ampli-tudes; the other terms represent optical noise.Hence Im, as a function of w, represents a low-contrast and noisy fringe pattern. It is interestingto note that the third term is apparently a fringepattern of half w; however, its modulation term,which is a function of wy2 also, makes the product afringe pattern of w only.

To enhance the contrast of these multiplicationfringes a purpose-built digital filter4 is used4: De-note by ^Im& the local spatial average of matrix Im andy s2 and sn

2 its variance and normalized variance,espectively:

s2 5 ^Im2& 2 ^Im&2, (3a)

sn2 5 s2y^Im&2. (3b)

Apart from the constant term Ny~N 2 1!, where N isthe number of pixels within the sampling window,Eq. ~3a! represents the sample variance of the image.

Using the fact that Io, Ir, and c are all statistically5

independent, and assuming that ^cos~w!& is constantover a square pixel-window average and that Io and Irare speckled, we can expand by the right-hand side ofEq. ~3a! by considering the square of Eq. ~2! and thensubstituting back into Eq. ~3a! to obtain

s2 5 ^a4& 2 ^a2&2 1 ^a2b2& 1 ~7y32!^b4& 1 ~2^a2b2&

2 ^a2&^b2&!cos~w! 1 ~3y32!^b4&cos~2w!. (4)

When the reference beam is smooth, the relationship^Ir

2& 5 Ir2 should be used instead of ^Ir

2& 5 2 ^Ir&2.

This equation is rather complicated, and a simplifi-cation is needed. To achieve this we first demon-strate that the amplitude of cos~w! is much higherthan that of cos~2w!. Using the relation ^Io& 5 k^Ir&,

here k is a real number, we find the ratio of themplitudes of cos~w! to cos~2w!:

Q~k! 5 ~4y3!~5k2 1 7k 1 5!yk, (5)

which has a minimum for k 5 1 5 22.5, increasing forother values of k. As it stands, Eq. ~5! is proof that

the term that contains cos~2w! is negligible comparedto cos~w! because the amplitude of cos~w! is approxi-mately 20 times larger than that of cos~2w!. To fur-ther simplify Eq. ~4! it is easy to verify that thefollowing relations hold:

^a4& 2 ^a2&2 1 ^a2b2&

1 ~7y32!^b4& 5 5^a2&2 2 2k2^Ir&4, (6a)

2^a2b2& 2 ^a2&^b2& < 5^a2&2 2 20~k4 1 1!^Ir&4. (6b)

Taking into account that 5^a2&2 .. 2k2^Ir&4 for all k

alues, we neglect the latter term in Eq. ~6a! andnally rewrite the variance in Eq. ~4! as

s2 < 5^a2&2F1 1a

a 1 bcos~w!G , (7)

here a 5 2k3 1 3k2 1 2k and b 5 k4 1 1.The fringe pattern in relation ~7! is modulated by

ackground illumination a, a fact that may be a prob-em when Io or Ir is noncollimated or noisy. One way

to reduce this effect is to divide Eq. ~7! by ^Im&2. Itresults that, from Eq. ~2!, ^Im&2 ' ^a2&2. Thus thenormalized variance @Eq. 3~b!# is given by

sn2~w! 5 5F1 1

a

a 1 bcos~w!G . (8)

The visibility of the fringe pattern that results fromrelation ~7! or Eq. ~8! is given by aya 1 b, which hasa maximum value of 7y9 for k 5 1, i.e., when ^Io& 5^Ir&. If this intensity ratio is not 1 the visibility willbe reduced accordingly.

Finally, the phase w can be recovered from Eq. ~8!by any of the known phase-recovery digital tech-niques; for example, with a four-step phase shift thephase may be calculated from

w 5 tan21 Fsn2~w 2 3py2! 2 sn

2~w 2 py2!

sn2~w! 2 sn

2~w 2 p! G . (9)

3. Experimental Results

To corroborate the theory described above, we usedan ESPI in-plane configuration; see Fig. 1. A He–Nebeam launched into an optical fiber illuminates si-multaneously a metallic plate ~object!, of approxi-

ately 4 cm 3 4 cm, and a mirror. To have ann-plane-sensitive-only ESPI setup we set angle u toy4 rad. A piezoelectric translator ~PZT! elementnd a micrometer were attached to the mirror to shifthe reference phase. To capture the reflected lightrom the object we used a zoom lens ~ fy5.6! attachedo a CCD ~640 3 480 pixels!. Finally, a PC computerunning at 200 MHz with a frame grabber was usedo control the PZT through a PZT controller and theCD.A reference image was digitized @I1 in Eq. ~1a!#,

the micrometer moved ;100 mm to introduce theterm w ~deformation!, and the deformed image wasdigitized @I2 in Eq. ~1b!#. This digitization wasdone for various ratios between the object and

1 October 2000 y Vol. 39, No. 28 y APPLIED OPTICS 5139

5llsprvftt

i

5

reference-beam intensities, i.e., for k 5 0.97, 1.85,.37. We achieved these values by decentering theaser beam with respect to the fiber-optic beamauncher. The procedure that we followed to mea-ure the beam intensity ratios was to block out thatart of the beam that goes directly to the object withespect to the part that goes to the mirror and viceersa. The images were shifted in phase six timesor each k value: three times for the reference andhree for the deformed images. We accomplishedhe shifting by moving the previously calibrated6

PZT. Next, the corresponding images were multi-plied ~I1 times I2! and the variance filter applied.Only the results obtained for k 5 0.97 are shown inthe following figures. Figure 2 shows the multipli-cation I1 times I2. The divergent illumination isseen, and no fringes are visually obvious. Figure 3shows the result of applying Eqs. ~3! to Fig. 2. A3 3 3 pixel-window average was used. Larger win-dows will produce smoother fringes. We can ob-serve that the background illumination has beenreduced and the contrast enhanced, with only a

Fig. 1. Optical setup for in-plane ESPI with a 10-mW laser:M.O., optical magnification.

Fig. 2. Multiplication of I1 times I2.

140 APPLIED OPTICS y Vol. 39, No. 28 y 1 October 2000

small dc term remaining. Figure 4 shows thewrapped phase obtained with Eq. ~9!; Fig. 5 showsthe unwrapped phase.7

The processed fringes for other k values showedthat, effectively, the contrast is reduced, as seen fromEq. ~8!, namely, from 0.57 to 0.52 to 0.41 for the kvalues given above. In this case the contrast is de-fined as one minus the ratio of the standard deviationof Eq. ~8! to its mean.

At this point, to assess the performance of themultiplicative method shown above it is convenientto compare the phase values obtained by multipli-cation with those obtained by subtraction. Thefour pairs of images that we used above to obtainthe phase by multiplication were also used to obtainthe phase by subtraction; e.g., instead of multiply-ing the corresponding pairs, we subtracted them.The absolute values gave us four fringe patternsshifted in phase py2 rad among them, with eachmage smoothed with a 3 3 3 pixel window, as used

Fig. 3. Multiplication fringes obtained from Fig. 2 after the dig-itally designed filter is applied.

Fig. 4. Phase map from multiplication fringes obtained from Eq.~9!.

ntaltvosu9tLdwmtt

for the multiplication. The wrapped phase wascalculated with an equation similar to Eq. ~9!,

amely, one that comprises four steps. Finally,he unwrapped phase maps, for both multiplicationnd subtraction fringes, were subtracted in abso-ute value ~Id!. The mean of Id was 2.8 rad, andhe standard deviation was 1.19 rad. The formeralue is a dc term, meaning that the phase maps areut of phase by that amount. The latter valuetands for the similarity of the phases. These val-es changed with the size of the filter used; i.e., a3 9 window modified the average value from 2.8

o 3.11 and the standard deviation from 1.19 to 0.5.arger windows make the average tend to p rad andecrease the standard deviation to 0.33 ~15 3 15!,hich represents approximately 5% of the maxi-um phase value. This comparison shows that

he filtering process must be optimized to reducehe noise and improve the results.

Fig. 5. Unwrapped phase from Fig. 4.

4. Conclusions

The multiplication method that obtains fringes bycorrelation of two speckle patterns in ESPI has beenexamined. It was demonstrated theoretically andexperimentally that enhancement of multiplicationfringes does not require rectification, as is the case forsubtraction. The phase was obtained with a com-mon phase-shifting routine, although any othermethod could be used. The disadvantage that themultiplication fringe method presented here does notwork if only a single frame is recorded will perhaps beovercome with the use of liquid-crystal camera de-vices. Future research toward improving the filter-ing process will mean that multiplication of specklepatterns in ESPI may prove to be an alternativemethod to obtaining fringes through subtraction andaddition.

References1. R. Jones and C. Wykes, Holographic and Speckle Interferometry

~Cambridge U. Press, Cambridge, 1989!, Chap. 4.2. J. A. Leendertz, “Interferometric displacement measurement on

scattering surfaces utilising speckle effect,” J. Phys. E 3, 214–218 ~1970!.

3. A. J. Moore, J. R. Tyrer, and F. Mendoza Santoyo, “Phaseextraction from ESPI addition fringes,” Appl. Opt. 33, 7312–7320 ~1994!.

4. N. Alcala Ochoa and J. M. Huntley, “A convenient method tocalibrate nonlinear phase modulators for use in phase shiftinginterferometry,” Opt. Eng. 37, 2501–2505 ~1998!.

5. J. W. Goodman, “Statistical properties of laser speckle pat-terns,” in Laser Speckle and Related Phenomena, J. C. Dainty,ed. ~Springer-Verlag, New York, 1975!, Chap. 2, p. 17.

6. N. Alcala Ochoa, J. L. Marroquın, and A. Davila, “Phase recov-ery using a twin-pulsed addition fringe pattern in ESPI,” Opt.Commun. 163, 15–19 ~1999!.

7. D. Kerr, G. H. Kauffmann, and G. E. Galizzi, “Unwrapping ofinterferometric phase-fringe maps by the discrete cosine trans-form,” Appl. Opt. 35, 810–816 ~1996!.

1 October 2000 y Vol. 39, No. 28 y APPLIED OPTICS 5141