Intrinsic speckle noise in in-line particle holography

2046 J. Opt. Soc. Am. A/Vol. 10, No. 9/September 1993

Intrinsic speckle noise in in-line particle holography

Hui Meng

Department of Mechanical Engineering, University of Houston, Houston, Texas 77204-4792

W L. Anderson

Department of Electrical Engineering, University of Houston, Houston, Texas 77204-4793

Fazle Hussain and David D. Liu

Department of Mechanical Engineering, University of Houston, Houston, Texas 77204-4792

Received July 15, 1992; revised manuscript received April 5, 1993; accepted April 6, 1993

In-line holography of particle fields suffers from image deterioration caused by intrinsic speckle noise. This isnot accounted for by previous theoretical treatments based on the scattering of a single particle, nor has therebeen any quantitative description of this noise except for an empirical criterion by Royer [Nouv. Rev. Opt. 5, 87(1974)] based on geometrical obscuration. We develop a theoretical model for in-line holography of multiple par-ticles, using diffraction theory and statistical analysis, and show that the virtual image of the particle ensembleis the dominant source of speckle in reconstruction. We quantify the effect of speckle with a signal-to-noiseratio (SNR). The SNR is found to depend on a speckle parameter (which embodies particle diameter, concen-tration, and sample depth) and on the film gamma. Experimental results show reasonably good agreementwith our model. The SNR equation provides prediction of image quality and thence application limits of in-lineholography for particle fields. The fundamental understanding obtained here points not only to constraints butalso to possible improvements in experimental procedures.

1. INTRODUCTION

Although in-line (Gabor) holography provided the key con-ceptual basis' for all developments in holography and co-herent optical information processing, its deficiencieshave confined it to a relatively narrow niche of application,namely, in studies involving small particles. Within thoseareas it provides several significant advantages over off-axis holography, experimental simplicity being the fore-most. By utilizing the cone of relatively small-angleforward scattering (diffraction) it achieves maximum sen-sitivity and makes minimum demands on film resolution.Moreover, the coherence requirements on the laser arequite modest; thus even in-line holograms made with useof the earliest pulsed lasers (which had poor coherence)permitted estimates of the sizes of sparsely distributedaerosol particles on a one-by-one basis,23 either indirectlyfrom individual diffraction patterns or directly by recon-struction of the real images. Since distances from par-ticles to the plane of recording of the hologram weretypically large enough to justify the Fraunhofer approxi-mation per particle, the term Fraunhofer holography hascome to be attached generally to in-line holography ofsmall particles (although only Fresnel approximation isjustified for the total illuminated region). Fraunhoferholography, as defined in this sense, has been widelyapplied in particle sizing 2 8 and particle coordinate 9

and velocity',"-" measurements, among other applica-tions. Recent advances in pulsed laser technology andcomputer data processing have led to its emergence inturbulent-flow measurement,13 '14 a three-dimensional (3D)data-intensive endeavor known as holographic particlevelocimetry (HPV).14

However, in-line particle holography is, unfortunately,subject to speckle noise, which arises when particle distri-bution becomes dense and/or the sample volume includes asufficient depth along the beam to encompass a largenumber of particles. The degradation evidenced in thereconstructed images has been a source of frustration tomany experimenters.' 4 '16 An example is given by Joneset al., 5 who cast doubt on the usefulness of in-line holog-raphy for drop-size measurement in dense fuel sprays.Their experience showed that drops with diameters of afew tens of micrometers at a distance of 30 cm from anatomizer and with density greater than 1 mm-3 producednoisy holographic images of little use for analysis unlessthe sample volume depth was less than 1 mm. A depthconstraint of this nature obviously brings into questionany advantage of holography over conventional photogra-phy. In HPV applications both the particle seeding den-sity and the seeding volume must be large enough toprovide 3D flow information. However, any increase inparticle density to improve resolution of small-scale mo-tions (inherent to turbulent flows) is unavoidably accom-panied by an increase in speckle noise.

Despite the more than three decades since the utilityof in-line holography as an experimental tool for particlestudies was demonstrated by Thompson and co-workers,2its signal-noise characteristics as a function of the rele-vant system parameters have never been analyzed suffi-ciently to provide a sound basis on which designs ofexperiments could proceed or on which corrective mea-sures might be devised. In the past, image deteriorationwith dense particle fields has usually been attributed tobeam obscuration. This effect is reflected in the popu-larly used Royer criterion 6 based on a "shadow density"

0740-3232/93/092046-13$06.00 ( 1993 Optical Society of America

Meng et al.

Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. A 2047

(the ratio of total cross-sectional area of particles to thetotal illuminated area). Royer categorizes image qualityas follows: a shadow density less than 1% produces a"good" hologram, between 1% and 10% produces a"marginal"-quality hologram, and greater than 10%,a "bad" hologram. This empirical criterion seems con-venient but does not provide insight into the real cause ofthe image deterioration, namely, speckle noise, which isintimately related to particle scattering and to the holo-graphic process.

Speculations as to the causes of the image deteriorationother than beam obscuration also have been made. Forexample, Ewan7 suggests such possibilities as reference-wave degradation, interference from virtual images orfrom flare light, and interference from out-of-focus par-ticles. These perceptions are insightful, but no analysishas been given in support of them. Ewan also points outthat, in a Fraunhofer plane, speckle arises from interfer-ence among individual particle transforms (a detailed de-scription for this process is given by Anderson' 7 ), but hefails to relate it to image degradation that is due to par-ticle density.

The principal element missing in previous works,whether speculative or empirical, has been an adequatetreatment of the scattering from an ensemble of particles.This scattering cannot be treated as a simple extension ofthe scattering from a single particle other than for thecase of a sparse distribution of particles in a very thinlayer, such that there is minimal far-field interference ofthe scattered waves from individual particles.

In our study, in-line holography is analyzed by considera-tion of scattering from multiple particles, and the associ-ated speckle process is described. The image degradationby speckle noise is quantified in terms of a precisely de-fined signal-to-noise ratio (SNR). The density of particlesto be considered is moderate: not so sparse that the scat-tered waves of particles in the far field are isolated, yet notso dense that their real images overlap to form a specklefield. The ultrahigh-density regime excluded here, al-though utilized in speckle velocimetry,"8 precludes holo-graphic reconstruction of useful information. Thespeckle process that is relevant primarily to practical in-line particle holography is a far-field effect and should bedistinguished from speckle that is due to overlapping ofparticle images. The considered ranges of particle size(5-50 ptm) and density (1-30 mm-3) are of particular in-terest in holographic particle velocimetry and in manyother applications. Our analysis also addresses the role offilm gamma in SNR.

of wavelength A (Fig. 1). The range from maximum tominimum particle depth is L. Holographic film H isplaced at a distance z >> d2 /A (i.e., in far field) from allparticles. Light scattered by the particles is representedby the object beam 0, and undeviated light by the refer-ence beam R. The optical field at the recording plane (xy)is given by

N

UH(x,y) = R O(x,y) = R + O(XY),k-i

(1)

where O(X, y) is the scattered wave of the kth particle. Ifone chooses, for convenience, R = 1, the intensity distri-bution at H is

IH(X, Y) = UHUH* = 1 + 0*(X,Y) + O(X,Y) + 00*(X,Y) .(2)

The interference between R and 0 is represented by(O + 0*), and that between individual scatterers is 00*.When the number N of contributing scatterers is suffi-ciently large, the latter appear as a random pattern, orspeckle noise. Appropriate exposure and development ofthe photographic film in plane H then yields not only thehologram pattern (O + 0*) but also a speckle pattern 00*.

The coordinate system that we use is shown in Fig. 2.Generally speaking (under the far-field condition), an ob-

(xy)

A HR

o 0 01

@00 0 A0 00 or 020 0.

Ok

L-_

1 - - Z

Fig. 1. Scattering of multiple particles in in-line hologramrecording with collimated illumination.

X

.4- Z -

r=(X2 + y 2)1/2

y

(a)

2. THEORY OF IN-LINE HOLOGRAPHY OFMULTIPLE PARTICLES, ASSOCIATEDSPECKLE, AND SIGNAL-TO-NOISE RATIO

In the absence of any adequate prior theoretical treatmentof in-line holography of multiple particles, it is appropriateto begin with the fundamentals. The speckle phenomenonfor in-line holography of multiple particles occurs both inhologram recording and in reconstruction, and we treatthese two processes separately.

A. Hologram RecordingLet us consider an ensemble of randomly located particles,each of diameter d, illuminated by a collimated laser beam

(4TI) (xy)1 .- ()k

* f.rk, X A- Zk -

(b)Fig. 2. Coordinates for recording. (a) Single-particle case: aparticle is centered in the origin of plane (h), and its diffractionpattern is centered in the origin of plane (x-y) at a distance zaway; (b) multiple-particles case: the kth particle is on theplane (6n)*, with location rk, and its diffraction pattern, centeredat rk, is observed on plane (xy) at a distance Zk-

IbYlll

_r � r orz

_ . AT

Meng et al.

p = (2+ 112)1/2

11_t


ject plane (o) with transmittance 1 - A(,r1) gives rise toa Fraunhofer diffraction field UH(X, y) on the recordingplane (xy) located at a distance z from the object plane. Ais the complement of amplitude transmittance function ofthe object. For illumination by a plane wave with ampli-tude 1, the Fresnel approximation, valid over the illumina-tion region, gives

UH(X,Y) -A exp[i2rz/A]f [1 -A(6,q)]Az _0) _Q0

x exp{iT[x _ )2 + (y - q)2]/Az}dgdq. (3)

Actually, this integration can be regarded as a sum of in-tegrals, each over a small area occupied by an individualparticle, and each with its individual z value. For each ofthese integrals the far-field approximation (f + 2)/Az << 1 can be invoked. The other quadratic phase fac-tor, exp[ilT(x2 + y2 )/Az], is independent of the integrationvariables, so the expression for UH(X, y) comes close tobeing a Fourier integral (again, applicable to each particlebut not to the illuminated region as a whole). The linearphase factor exp[i7rz/A] is unimportant and can be omittedhenceforth. Photographic recording of IH(x, y) results inwhat is called a Fraunhofer hologram.

1. Single ParticleWe first consider a single spherical particle of diameter d.As in previous treatments,' 9 we model it as an opaquedisk, represented by a circ function in the () plane (withradial coordinate p): circ(21p - rkl/d), where, in antici-pation of our multiple-particle analysis, we assign the par-ticle an index k, radial location rk = (ak, bk), and distanceto hologram Zk. For convenience of integration in Eq. (3),the origin of (x, y) as well as (i7) is then set at r. Topresent the result in the same form as in Eq. (2), we define

Ck = rd2 /(4AZk), (4a){2J,(irdlr - rkl)/Azk1

Qk = (rdr - r)/Az I j' (4b)oDk = (r - rkl )/AZk + /2. (4c)

J, is the first-order Bessel function, and r - r =[(x - ak )2 + (y - bk )2 ] 2 . Note that AZkCkQk is theFourier transform of the circ function. Then the objectwave on the (xy) plane can be written as

O(x,y) = CkQk exp(i(Dk) (5)

and the intensity as

IH(r) = 1 + 2CkQk cos(ck) + Ck2 Qk 2 . (6)

The second term, or fringe signal, corresponds to 0 + 0*.Since Ck is roughly the inverse of the so-called far-fieldnumber,8 its magnitude is typically of the order of 10-2.The last term, 00*, for the single-particle case is a halowith maximum intensity Ck2 and thus can be neglected incomparison with the fringe signal.

2. Multiple ParticlesNext we consider the hologram recording of an ensembleof N identical particles (i.e., k = 1, 2, ... N) of the samediameter d with plane-wave illumination. Here we areconcerned only with primary scattering from each par-ticle and neglect multiple scattering, i.e., rescattering of a

primary scattered wave by another particle, as justified inAppendix A. Direct integration of Eq. (3) for multipleparticles would be extremely cumbersome. Instead, weadapt Eq. (2) to multiple particles by identifying the objectwave in the (xy) plane,

N NO(X,y) = E Ok(X,Y) = 2 CkQk exp(i4);

k-1 k-1(7)

hence the recorded holographic fringe intensity for all theparticles is

No + 0* = 2E CkQk cos (Dk.

k-1

The 00* term on the hologram is

(8)

N N

IN 00* = ojok = > CjCkQjQk exp[i(Dj - cbk)].j;k1 j;k1=

(9)

IN is evidently the result of a random-walk complex ampli-tude and, under the assumption that the scattering pro-cess does not cause depolarization, is a linearly polarizedspeckle pattern. The summation in Eq. (9) is much moresensitive to the rapidly varying FDk than to the slowly vary-ing Ck. If we assume that the maximum distance betweenparticles is much smaller than the minimum distancebetween the particles and H, we can write Ck C =7rd2/(4Az), where z is the median distance between a par-ticle and H. Under the conditions of interest here, thephase of the scattered wave is uniformly distributedwithin - to ; thus, if N is large, this random process isessentially Gaussian and IN obeys negative exponentialstatistics.20 A mean of terms in Eq. (9) will be taken overan ensemble of macroscopically similar but microscopicallydifferent realizations of particle fields and will be denotedby (). Since Qk and k are independent random variables,

(QjQk exp[i( - )]) = (QjQk) (exp[i((D - )])

{(Qk2)

Oifj = k ;if j # k. (10)

From Eqs. (9) and (10), the ensemble mean intensity of00* is

N(IN) = E C2(Qk2) = NC2Qk2,

k-1(11)

where the overbar represents an average over particleindex k and the second equality is justified as follows.Since the particles are identical and their positions rk(k = 1, 2,... N) are random, the ensemble average is equalto the average over all k within one realization. The stan-dard deviation is equal to the mean20 :

0N = (IN)- (12)

3. Dependence of Speckle Intensity on Particle FieldIn order to calculate N of the 00* speckle intensity byusing Eq. (11) and Eq. (12), we need to estimate N (theeffective number of scatterers contributing to an observa-tion point P) and Q2 (average of Airy-pattern intensity).As noted above, scattering by the particle of interest hereis essentially Fraunhofer diffraction, so the scattered en-ergy is concentrated mostly in the central lobe of its Airypattern, and the energy contribution from the sidelobes to

Meng et al.


HL

From formulas (17), (19), and (11),

N = (IN) = G/3 = (r 3/48)d 2n3 L.

P

Fig. 3. The Airy cone containing scatterers contributing tospeckle at observation point P.

the speckle pattern can be neglected. The effective num-ber of phasor elements contributing to speckle at P canthen be determined from the expected number of particleswith central lobes overlapping at P. As is shown in Fig. 3,a point P on H sees a conical region in the particle fielddefined by a half-angle

00 = 1.22A/d A/d, (13)

which corresponds to the scattering angle at the first zeroof Jl(x)/x, referred to as the Airy cone. The number ofparticles in the Airy cone is

N = (/3)n 80o2[(z + L/2)3 - (z - L/2)3 ]

= -7rns(A2 z2 /d2 )(L + L 3 /12z 2 ) (14)

where n, is the particle concentration and L is the sampledepth. For the typical case of z >> L, the second termin Eq. (14) can be dropped compared with the firstterm. Thus

N = rrnA2z2L/d2. (15)

Defining

G NC 2, (16)

we get

G = ('7r3/16)d 2n8 L. (17)

G is the active dimensionless parameter in ON = (IN) andis termed the speckle parameter. Although (d2nL) isequal to the shadow density, Eq. (17) relates it directlyto the physics of the scattering process. Note that G im-plicitly carries the meaning of total scattering strength,while C2 is the scattering strength of a single particle.

To find QT2 , we consider the contribution to P from allthe particles in a thin shell of differential angle dO at aconical angle 0 (as viewed from P), with a range from zeroto 0 [see Eq. (13)]. Assuming a uniform random distribu-tion of particles, the total contribution I from N particlesin the cone is

I= - n[(z + L/2)3- (z - L/2)3]3

x J [2Jlt k 0)/ A 0) MOd (18)

Division by N as in Eq. (14) with a (d/A)0 then gives

Tk2= 21 I2J (ct) ada = 0.335 1/3. (19)a rt

(20)

Equation (20) shows directly how speckle noise 00*depends on the particle diameter, concentration, andsample depth.

4. DiscussionAdditional factors need to be considered when the overallnoise problem in hologram recording is being addressed.If we view the whole particle field as one object, then theintermodulation term 00* is the only speckle noise, thefringe term 0 + 0* being the signal. However, fromthe standpoint of reconstruction of the image of an indi-vidual particle, say the ith, the overlapping fringe patternsof all the other particles 2 k=1; k1i CQk cos cF, also consti-tute noise. This part of the noise again has a granularintensity distribution, or speckle, and has a Gaussianprobability distribution with zero mean. It can be shownthat its variance is equal to 2crN; hence the variance of thetotal noise background relative to the fringes of a singleparticle becomes

CH 2 = uN2 + 2 N, (21)

where oN2 is the variance of 00*. The standard devia-

tion of the total is

o0H = o-N(l + 2/0N) 2

= (G/3)(1 + 6/G)112 .

Equivalently, this result can be obtained with the model ofa modified Rician density (see p. 29 of Ref. 20), which de-scribes a random-walk field plus a constant phasor (thereference wave 1).

In some applications10 '12 the fringes are analyzed di-rectly from the hologram to reconstruct individual par-ticles, in which case 0

H is of central concern. However,the following treatment is based on the use of the opticalreconstruction of the hologram. Then the 0 and the 0*parts of the 0 + 0* term are decoupled.

B. Hologram ReconstructionIn the optical reconstruction of an in-line hologram, anilluminating plane wave is diffracted into four componentscorresponding to the four terms of Eq. (2). At a distancez behind the hologram, real images of particles are recon-structed from the 0* term (Fig. 4), and it is usually theseimages that are analyzed. From the 0 term, virtualimages are reconstructed at a distance z in front of thehologram. The far-field scatter waves from the virtualimages are added to the real-image field, thus formingspeckle there. This speckle process is analogous to thatin the hologram recording, except that the particle en-semble is replaced by its virtual image and the observationdistance z is replaced by 2z. The diffraction from the00* pattern recorded on the hologram also contributes asnoise in the real-image field. In what follows we evaluatequantitatively the effects of all the components in the re-constructed field.

The object-to-reference-beam ratio for this type ofhologram is small (of the order of C2 - 10-4). Withinthe small dynamic range of exposure, it is reasonable to

(22)

Meng et al.


Virtual _ - - - l oImage

(a)

Holograrxy)

R

Virtual '. *.Image : °

* - -

*4 - = 2z

The integration of the 0* and the 0 terms can beviewed as a sum of integrals associated with individualparticles at various origin shifts rk (k = 1, 2,...). Withthe approximation Zk Z, the integrations are straight-forward. From 0* we obtainReal

Image(27a)

where

Real* Image

(b)Fig. 4. In-line hologram reconstruction. (a) Single-particlecase, in which the virtual image is seen only as a weak, near-spherical wave background in the real-image plane (x'y');(b) multiple-particle case: speckle arises from interferenceof waves from the virtual image of the particle ensemble.

assume linearity of transmittance; i.e.,

t(x,y) = tb - K[E(x,y) - Eb], (23a)

where t is the amplitude transmittance of the hologram,E is the exposure energy, and - K is the slope of thet-E curve of the film at a bias exposure Eb. Since therecording intensity IH(X, y) is normalized with respect tothe plane-wave illumination, we can write E(x, y) =EbIH(xy); therefore

t(X, y) = tb - KEbAIH(x, y), (23b)

where AIH(x, y) represents the intensity modulation. Ob-viously, tb is the transmittance resulting from the expo-sure to the plane-wave illumination alone.

1. Reconstruction FieldIf we illuminate the hologram with a plane wave, thetransmitted field immediately behind the hologram is pro-portional to t(x, y), and thus

Ut(x, y) B - [0*(x, y) + O(x, y) + 00*(x, y)], (24)

where

B tb/KEb. (25)

For simplicity, we replace the proportionality in re-lation (24) with an equality. The field on a plane (xty') ata distance z (equal to the mean of Zk) is

iUR(X', Y') = - - exp(i2irz/A)ff U(x, y)

x exp{i~n{(x - xt)2 + (y - y') 2]/Az}dxdy.

(26)

Ir - rki = [(X' - ak) 2 + (y' - bk)2] 1/2, (27b)representing the reconstructed real images; and from 0we get

Usp1 = qqk ep(ijsk), (28)k

representing the virtual-image wave front. Here C, Qk,and k are defined for r' - r in the same manner as Ck(later approximated as C) and as Qk and Dk in Eq. (4), butZk is replaced with = 2z, and Tk has a constant phasedelay of -r/2, not 7r/2.

Up,, is a random-walk summation and gives rise to aspeckle pattern on the (x'y') plane. For integration of the00* term, following the approach taken in Ref. 19, we em-ploy the stationary-phase approximation,2' obtaining whatwe call U 2:

Usp2 = A foo* (A Az)

x exp{i kz/A[(x - x')2 + (y - y) 2]}dxdy

_ _ ( y) (29)

This shows that the optical field 00* (x'/Az,y'/Az) is ageometric propagation of the field transmitted throughthe specklegram 00* (x/Az, y/Az). This result is reason-able from a statistical point of view, since diffraction froma specklegram is macroscopically similar to the fieldtransmitted through the specklegram. It follows that thetotal reconstruction field can be expressed as

UR(X',y') = B + Usig + Uspl + Up2- (30)

2. Evaluation of Background NoiseWe single out Ug as Eq. (27) and evaluate the backgroundas the sum of three coherent components: a constantphasor / =_ B (the transmitted plane wave) and the twospeckle fields, U,,, and U,,2- The total speckle field Uphas an ensemble mean intensity,

(Isp) = (lUl + U212)

K > ~~~~~2= ( |E Qqk exp(ipk) - ECQk 4 °°

1 ( *2\X cos %Ik + 6(°°*))16 /

Since (cos 1k) = 0, we obtain

(I4) = (Ip,) + (Isp2),where

M

(i) = Z C2(Qk2),k-1

(I,,2) = 1/16 ((00*)2). (31b)

(31)

(31a)

Meng et al.

U.ig = 2 cirek d/2


The summation in (Ispl) is for M particles contributing toany point P' on the reconstruction plane. These particlesare contained in an Airy cone seen from P' at a distance

= 2z, as shown in Fig. 4(b). Calculation of (I,,) followsin exactly the same manner as for (IN) [see Eq. (20)], andit can be shown that MC2 = NC2 . Thus

= M13 = G/3. (32)

Because

N(oo*) 2 = C4 > QiQkQIQr exp[i~t~ + (Dk - (D1 - m)]

jklm(33)

level, measured by a-BN, is comparable with the imageintensity level.

4. Definition of Signal-to-Noise RatioWe define the signal-to-noise ratio (SNR) for the recon-structed image as the quotient of signal (real-image) in-tensity sig and noise standard deviation:

SNR= Isig/OBN- (40)

With sig = 1 (value of the circ function inside a particleimage) and Eq. (38), it follows that

SNR = {(G/3)[1 + 2B2/(G/3)]12 }-, (41)

it follows that

((oo*)2) = NC4 Qk 4 + 2(N2 N)C4(Qk2)2.

Calculation shows that Q4k4 0.093 and Qk2 = 0.335;hence ((OO*)2) = 0.224G2 . Therefore

(Ip)=0.014G . (35)

In the practice of in-line particle holography, G is smallerthan 1 (as will be seen below); thus (Isp2) << (I,,,) and canbe neglected. This shows that the virtual image of theparticle field is the dominant source of speckle in the re-construction stage. Therefore

(Isp) = (4,,) = G/3. (36)

Since the background field can be described as randomwalk (speckle field Up) plus a constant phasor (VI-), theprobability-density function of the total background in-tensity IBN is a modified Rician function.2 0 The mean in-tensity is

(IN) = (Isp) + Ic G/3 + B2.

The standard deviation can be shown to be

oBN = (I,,) (1 + 2p)1 /2 ,

(37)

(38)

where

I, B2

(39)

3. Speckle SizeTo see how the speckle background affects the recognitionof particle images, we estimate the characteristic specklesize s, defined as a statistical average of the distancebetween adjacent regions of maximum and minimumbrightness.2 2 The reconstruction beam diameter (W)evidently must be large enough to cover the Airy zones onthe hologram for the particles to be reconstructed; i.e.,W >> (260 )z, where 0o is the half-angle of the central-lobeparticle scattering defined in relation (13). Hence theangular aperture determining the speckle size is 20o (seeFig. 3, but note that the distance from the virtual imagefield to the real image is 2z). Therefore, in the absence ofthe transmitted plane wave B, s A/(20o) = d/2. Thepresence of B along the axis doubles this value.22 Becausespeckle size is comparable with the particle size, particlerecognition will become difficult if the background noise

where G = (Or3/16)d2n5L as given in Eq. (17) is the speckleparameter and B = tb/KEb as given in Eq. (25) is a con-

(34) trast parameter.

C. Discussions of the Signal-to-Noise-Ratio equation

1. Parameter BEquation (41) shows that SNR increases with decreasingB. The parameter B measures the dc component (tb) ofthe hologram transmittance relative to the gain KEb;hence B also measures the directly transmitted plane-waveamplitude relative to the image amplitude. These quanti-ties are related to film gamma (Cy) in the following way.In photography, y normally is defined as the slope of thelinear portion of the D-log(E) (Hurter-Driffield) curve.More generally, if y(E) is taken as the slope of the D-log(E)curve at any point, then in any region where y is essen-tially constant the t-E characteristic of the film is gov-erned by

t(E) = cE-Y/2. (42)

Now, by definition,

K = -t'(Eb) = (2)cEb-'Y 2 Eb-';

hence

B = 2/,y.

It follows that

+ 8 1/2 -1SNR = (1G3)[1 2 G

(43)

(44)

where G/3 = 0.646d2 nL.Equation (44) indicates that a larger y leads to a higher

SNR. For conventional coherent light processing includ-ing off-axis holography, one usually confines film exposureto the linear region of the t-E curve2 3 in order to take ad-vantage of the large dynamic range available there. Thislinear region, shown as Region I in Fig. 5(a), is a mappingfrom the toe region of the D-log(E) curve in Fig. 5(b).However, the small object-to-reference intensity ratio(_10-4) for an in-line hologram of particles requires only asmall dynamic range. Since Region II gives the largestpossible by, it is advantageous to expose in-line hologramsof particles in that region so that high image contrast andthus high SNR can be obtained.

For a given film and developer, the y values for bothRegion I and Region II increase with developing time, as

Meng et al.


1

'U

H-E

In

r.

0.8

0.6

0.4

0.2

0

2.5

2

.bc.1Ia

1.5

1

0.5

n

0 20 40 60 80 100 120Exposure Energy E (J/a 2 )

(a)

From Eq. (44), the SNR then approaches the upper bound:

Sup(SNR) = (G/3)-1 = (0.646d2nL)-'.

140

0.8 1 1.2 1.4 1.6 1.8 2 2.2log (E)

(b)Fig. 5. (a) t-E curve and (b) D-log(E) curve obtained by use ofAgfa-Gevaert 8E56 plates and D-19 developer for 2-min develop-ing. Region II gives a higer y and is more suitable for recordingin-line holograms of small particles than Region I.

1 4 i i I I I 1 1 1 1 1 I I I I

12

10

8

7Y6

4

20

1 2 3 4 5 6 7 8

Developing time (min)Fig. 6. Experimental data for film y as a function of developingtime and exposure region defined as in Fig. 5, with 8E56 platesand D-19 developer.

is shown in Fig. 6 (data correspond to conditions given inFig. 5). However, this increase is useful only up to apoint. For example, under the condition mentioned above,a 7-mmn developing of holograms exposed in Region IIgives a value of 10, but the hologram becomes almostopaque, with amplitude transmittance t falling to the orderof iO-'. Also, with increasing 'y, nonlinearity becomes ap-preciable, causing additional noise. 2

14' 2 5

Nonetheless, the asymptotic condition y (or B - 0)can be approached effectively by use of a high-pass spatialfilter that removes the plane wave B in the reconstruction.

(47)

2. Parameter GThe asymptotic equation (45) shows that the ultimatelimitation to the SNR is imposed by the particle field itselfthrough the speckle parameter G. If the G value of a par-ticle field reaches 1, Sup(SNR) 5dB, which (as foundalso in our experiments) is a practical limit for particlerecognition in a speckle background. In-line holographybecomes unsuitable under this condition.

For arbitrary finite values of y, Eq. (44) shows that theSNR decreases monotonically with increasing G (thusni, L, d). The increase of speckle noise through increaseof n8 and L is to be expected because of the greater num-ber of contributing scatterers. Increasing d also accentu-ates speckle noise as scattering is intensified at larger d,and speckle is just a result of interference of these scat-tered waves.

Since (d2nL) G is indeed the shadow density used inRoyer's criterion, 6 the bridge between shadow density andspeckle noise is now established.

3. Effective GammaComparison of measured SNR with that predicted theo-retically requires conversion of the reconstructed optical-image field into a quantifiable form (e.g., photographs,video pictures, CCD array images), but the means bywhich such conversion can be effected are generally non-linear, device dependent, and inherently difficult to char-acterize analytically. Although a slope ' can be definedfor this conversion in analogy with the hologram film y,these two are fundamentally unrelated, since y' does notaffect the individual coherent components as y does. Ourtheoretical study is confined to the optical image recon-structed from the hologram. A quantitative evaluation ofthe nonlinear transformation effects introduced by acamera or other imaging device is beyond the scope ofthis work.

To test Eq. (44) experimentally, however, it is necessaryto accommodate in some way the transformation effects ofthe imaging device at hand (CCD camera). To do so, weintroduce an effective gamma, which we call Ye, and use itto characterize the effects of both y and y'. We then usethe effective gamma to obtain the effective SNR:

SNR = {(G/3)[1 + 8 }/-Ye2(G/3)(44a)

The simplification involved in using Ye is admittedly im-perfect but, in the absence of a rigorous alternative, isunavoidable. Despite its limitations, however, Ye hassome qualitative features in common with . For example,thresholding of a CCD camera increases the SNR, whichcould be viewed as though caused by an increase in Ye inEq. (44a). The increase in image contrast caused bythesholding serves as the clue to the connection of Ye tothe hologram film .

4. Limitations of the Theoretical ModelBesides this simplification, our theoretical model also ne-glects the following factors:

(a) Aperture effects: The principal concern withaperture in this context is not with that of the hologram

-lI I I I lIII II I l II I I I I

..

- . I

_ * u~~~

_ * * -i , 1 1 1 1 1 7 1 1

Meng et al.


recording itself but rather with an aperturing that is indi-vidually operative for each particle. The fringe spacingon the hologram for each particle decreases linearly withthe radius measured from the particle center. As spacingdecreases, the signal becomes increasingly susceptible tolimitations of film resolution and hologram noise. Par-ticle or film motion during exposure can contribute greatlyto aperture effect because, if fringes within the centrallobe of the Airy pattern are smeared, reconstructed signalintensity is significantly affected, to a degree dependingon particle diameter. A discussion of this is given inAppendix B.

(b) Additional noise sources: These include coherentnoise from reflections, scattering of dust, film grain, andother imperfections. The capture effect that is wellknown in the context of frequency or phase-modulatedcommunications 2 6 also may be a factor.

(c) Model deficiency: The opaque disk model fortranslucent spherical particles is subject to error, particu-larly for larger particles.27 28

In addition, noise from the real-image field, i.e., the in-fluence of out-of-focus particles, is neglected in our model.This implies the following:

(a) Moderate shadow density: The quantity d2 n.Lmust be small enough that the waves from out-of-focusreal images of particles will not overlap appreciably toform speckle. For a realistic range in applications (G <<1 or d2n8 L << 0.5), it is usually sufficient to consider onlythe virtual-image speckle.

(b) Moderate particle size: The depth of field of aparticle image scales with the far-field distance d2/A.Therefore, for large particles, even when the shadow den-sity is low and thus there is not much overlap of the recon-structed real images (thus no speckle from them), the highintensity of out-of-focus images can be detrimental to thesignal unless the interparticle distance is larger than thedepth of field. Another effect of large particles is thatmultiple scattering can be appreciable.

(c) L << z: If L is comparable to z, the real-imagefield is almost connected to the virtual-image field, andmodification of the model is required. In our experi-ments, however, we found that the model still works rea-sonably well for L 2z/3.

However, even when the real-image field does contributeappreciably to noise, the strongest source of speckle is stillthe virtual image. A quantitative study of the real-imagespeckle has not been conducted. Complications involvedin such a study include the following: (a) Fraunhoferdiffraction can no longer be applicable everywhere,(b) multiple scattering can become appreciable, and (c) thereal-image speckle depends on the location of the interro-gation plane within the reconstructed volume. Treat-ment of these effects is left for future investigation.

3. EXPERIMENTAL STUDY OF THESIGNAL-TO-NOISE RATIO OF IN-LINEHOLOGRAPHIC IMAGES

On the basis of the definition of SNR [see Eq. (40)], wedeveloped a practical algorithm for calculation of SNRfrom the digitized CCD pictures of a reconstructed par-ticle field, as described below.

A. Experimental SetupSince the experiments were aimed at exploring SNRinfluenced by speckle noise of in-line holograms undergeneral conditions, a simplified setup was suitable. Sus-pensions in water of polystyrene spheres (Duke Scientific)with uniform diameter d were used, and their concentra-tions n8 were well controlled. Their specific gravity of1.05 permitted near-neutral buoyancy, so that a randomdistribution in water was easily obtained. SNR was mea-sured for a range of values of d, ns, and volume depth L(the last-named was fixed by water-tank thickness).

The 514.5-nm cw radiation from an argon laser (Coher-ent Innova 90-4) was used for both hologram recordingand reconstruction. In-line holograms with use of colli-mated illumination were made, as shown in Fig. 7. Theillumination area was shaped by an aperture of 32 mm X32 mm to fit the size of the water tank. Agfa-Gevaert8E56 plates were used for recording the holograms, whichwere exposed to high densities (average 30-90 puJ/cm2,Region II) and developed in D-19 developer. In most ex-periments, the developing time was 3 min, yielding a filmy of -4.

To analyze the reconstructed real image, we used aCCD camera (Kodak Megaplus Model 1400), which hasa limited dynamic range and intrinsically performsthresholding even when the nominal offset of the camerais set to zero. As discussed above, a partial accommoda-tion of these effects is incorporated into the use of effec-tive gamma ye. Views of a thin slice in the reconstructedfield were obtained by the CCD camera through a micro-scope assembly with a 5X objective. A picture of0.5 mm X 0.5 mm was digitized as an array of 256 X256 pixels, which was then transferred to a workstation(SUN 4/260) for analysis. A three-axis traverse(DAEDAL, 0.254-Am resolution) moved the hologramthrough the 3D image field so that different portions ofthe image could be examined. The thresholding levelcould be changed by the offset. Illumination laser powerwas varied to provide proper input level to the camera andespecially to compensate for the effect of low diffractionefficiency of the hologram as a result of heavy filmexposure.

B. Method of Signal-to-Noise Ratio EstimationA typical picture recorded by the CCD camera is composedof in-focus particle images (signal), out-of-focus particle

ParticleSuspension Hologram

Argon Laser

(a)

Hologram

Argon Laser . Cmr

(b)Fig. 7. Experimental setup for SNR study of in-line par-ticle holography: (a) hologram recording, (b) hologramreconstruction.

Meng et al.


images, and speckle (noise). To determine experimen-tally the SNR from the actual images, we measure theaverage intensity of the particle images and the standarddeviation of the background noise (BN), as suggested bythe SNR definition in Eq. (40).

A signal-mask frame was formed to pick out all the in-focus particle images. For generation of the signal mask,the original picture was smoothed by correlation with aparticle template (taken to be a circ function). Thesmoothed picture was mapped into a binary image bythresholding (pixel values set to 1 at in-focus particle loca-tions and 0 otherwise). Multiplying the original picturewith the signal-mask frame yielded all the in-focus

particle images, for which the average intensity was thencalculated. A similar procedure was used to form a noise-mask frame based on the opposite binary logic, but thethreshold was adjusted (equal to or lower than that usedfor the signal mask) to eliminate the out-of-focus images.Multiplication of the original image frame by the noise-mask frame extracted the noise, whose standard deviationwas computed digitally.

C. Measurement ResultsFigure 8 illustrates the appearance of reconstructed im-ages corresponding to high (19 dB), medium (14 dB), andlow (<5 dB) SNR values obtained with different particle

(a) (b)Fig. 8. Reconstructed images of particles at SNR of 19, 14, and 5 dB, corresponding to particle concentrations of 1.5, 6, and 40 mm-3 (topto bottom, respectively). Particle diameter, 21 um; suspension depth, 19 mm. (a) Pictures as viewed by the CCD camera, (b) intensityplot for subsections of the three pictures in (a).

Meng et al.


model. This deviation, discussed in detail in Appendix B,arises primarily because of an aperture effect (fringesmearing) imposed by experimental limitations, namely,appreciable particle motion during exposure. In thesmaller range of particle sizes, loss in resolution and en-

20

15

0 2 4 6 8 10 12 14

Film y

Fig. 9. SNR versus film y. Increase of SNR with v occurs onlyup to a point (- 4.5). Circles are measured data. The con-tinuous curve is a cubic spline fitted to the data. Curves A and Bcorrespond to maximum and minimum CCD thresholding, re-spectively. Particle diameter is 21 gLm; concentration is10 mm- 3 ; depth of the suspension is 15 mm; recording distance is32 mm.

1 0

5 '1 0

00

1.5mm-- -

> 37mm73

6 -

I I I I I I I

50 1 00

Sample Depth L (mm)

25

20

ae

SC,,

1 5

1 0

5

01 1 0

Concentration ns (mmn3)

Fig. 11. Dependence of SNR on depth L of the particle suspen-sion for three particle concentrations. Symbols are measureddata. Solid curves are calculated with Eq. (44a). Particle di-ameter is 21 Aim; recording distance is 50 mm.

30

25

a

1 00

Fig. 10. Dependence of SNR on particle concentration n 8 forthree particle diameters. Symbols are measured data. Solidcurves are calculated with Eq. (44a). Displacements of measureddata to calculated curves are related to the deviation in SNR ver-sus d curve (see Fig. 12). Depth of the particle suspension is19 mm; recording distance z is 32 mm.

concentrations. It is evident that a SNR greater than5 dB is needed for good distinction of particle images.

Figure 9 shows measured SNR versus film y, where theoptimal value of y is seen to be -4.5. Values of y wereobtained from measured D-log(E) curves for differentexposure and developing conditions. For every hologram,the SNR was measured for A, maximum and B, minimumlevels of CCD camera thresholding varied by tuning anoffset on the camera. Thresholding clearly changes theeffective SNR's and hence, as explained above, the value ofye. For the measurements described below, performed at-y 4 for film and a medium-level CCD thresholding, avalue of 20 gives the closest match of the calculated SNRcurves to the experimental data; hence ye is estimated tobe -20.

We present measured SNR (symbols) as a function ofparticle-field parameters in Figs. 10-13, compared withresults of calculation (solid curves) with Eq. (44a), inwhich ye = 20 was used. Figures 10 and 11 show goodagreement of measured SNR dependence on particle con-centration n, and sample depth L with the calculation, butFig. 12 shows a deviation in the slope of the measured SNRdependence on particle diameter d with respect to the

20

1 5

1 0

5I 1 0 1 00

Particle Diameter d (m)Fig. 12. Dependence of SNR on particle diameter d. Filledcircles are measured data. The solid curve is calculated withEq. (44a). Deviation of measured data from the calculated curveis due to the aperture effect imposed by fringe smearing, as isexplained in Appendix B. Depth L of the particle suspension is19 mm; particle concentration n 8 is 3/mm3 ; recording distance zis 50 mm.

20

ae

zSP4

15

1 0

51 0 1 00 1 000

Recording Distance z (mm)

Fig. 13. Dependence of SNR on recording distance z. Circlesare measured data. The solid line is calculated with Eq. (44a).The fall of measured SNR at large z is attributed to loss of modu-lation fringes in recording. Particle diameter d is 21 Am. Par-ticle concentration n8 is 3 mm- 3. Depth L of the particlesuspension is 19 mm.

20

1 5

1 0

5

0

_I I I I I I I I I I . I I I I_

A

B

w I I I I I I0

I I I I I " " I I I I I I 1l, -

0

0

I I . \111111 I 1111

I ' I ' ' " ' ,I '' I ' I I I I-

0 0 0 00 0 0000 0

000

0

I 1 11 1 I I I I 1 1

Meng et al.

I I I I I I I I

* _ ,


ergy that is due to this aperture effect makes identifica-tion of particles difficult; so no measurements were madefor diameters of less than 5 ±m. Above -50 jkm, effectsof out-of-focus particle images become severe since thedepth of field is larger than the interparticle distances.

A parameter that does not appear in our SNR equationis the recording distance z, measured from the centerplane of the particle suspension to the recording plane.Although Eq. (14) indicates that z does not affect specklebecause of the condition z >> L, measured data in Fig. 13show that SNR is independent of z only within a certainlimit. In terms of nondimensional distance (ratio of z tothe far-field distance d2 /A), the SNR begins to drop whenthis number exceeds 250, and it falls below 10 dB at 625.To explain this, we recall that the recorded fringes [seeEq. (6)] are in the form of phase modulation, which is in-sensitive to z as long as the modulation amplitude 2C isabove a threshold set by noise level. In fact, as z in-creases, the area involved in recording the ring patternsgrows larger to compensate for the decrease of C (propor-tional to lz). However, as C becomes so small that themodulation is lost in the noise, the particle image can nolonger be recovered. Another factor is the gradual in-crease of astigmatism with increasing z observed in thereconstructed images. This may also contribute to theabrupt cutoff in Fig. 13.

4. CONCLUSIONS

We have developed a theoretical model, supported by ex-periments, of in-line far-field holography of multiple par-ticles. This model shows that the major source of specklein the reconstructed image field is the interference amongscattered waves from the virtual image of the particlefield (combined with the transmitted plane wave). Wehave defined SNR, which is suitable for predicting thequality of in-line holographic particle images, and haveshown that the SNR depends on a speckle parameter Gand film gamma. The SNR is found to decrease with in-creasing particle concentration, volume depth, and par-ticle diameter, all of which are embodied in G.

The SNR can be predicted by use of Eq. (44) or (44a) fora certain particle field for in-line holography. Given thedesired SNR for an image-analysis system, one can esti-mate beforehand whether the particle field can producethe necessary image quality without performing the ex-periment. An upper limit for in-line holography to beusable is, roughly, G = 1.

It can be deduced from our model that the upper boundof the SNR can be reached by removing the undiffractedlight in the reconstruction with a high-pass filter. Moresignificantly, the SNR can be improved dramatically byusing off-axis holography, in which the virtual-image com-ponents, undiffracted illumination, and real-image com-ponents are spatially separated, thereby eliminating themajor source of speckle. Experiments (not discussedhere) prove this point quite conclusively.

APPENDIX A: EVALUATION OF MULTIPLESCATTERING

It has been noted previously that multiple scattering(higher-order scattering), inevitable in-dense 3D particle

fields, also contributes to speckle in hologram recording. 4

Here we evaluate the significance of noise contributionthat is due to secondary scattering, the simplest butstrongest type of multiple scattering. Let the illumina-tion wave be 1 and the incident field on the kth particle PA(Fig. 14) be 1 + Wk, where Wk is the coherent summationof primary scattered waves reaching Pk and evaluated atthat location. Then the total scattered wave from Pkarriving at the H plane is (1 + WA)OA, and the total objectwave from the particle field is

NO(x, y) = E (1 + W)Ok (X, y).

k-1

Then the intensity on H is

N NIH(Xy) 1 + (k + Ok*) + E OjOk

kil j;k-N

+ E (WOk + Wk*Ok*)k-1N

+ E (WjOjOk* + Wj*Oj*Ok + WjOjWk*Ok*)j;= A +i

~IHX, Y) + IM (X,Y) ,

(Al)

(A2)

where IH(x, y) is the intensity for primary scattering only[Eq. (2)]. The speckle intensity that is due to secondaryscattering is well approximated by

N

IM = (Wkk + Wk*Ok*),k-1

(A3)

the third- and fourth-order cross terms being negligible.Consider the total speckle intensity,

(A4)IN' = IN + IM-

Its mean is equal to (IN), and its variance is

(TN = (IN'2) - (IN') 2 = N2 + M2

where

NOW = 4E (WkI 2)(0ko12),

kil

(AS)

(A6)

Wk and Ok being independent. For random spatial distri-bution of particles, the ensemble average of OA 12 is equal tothe average over k for N particles in the cone shown inFig. 3; i.e.,

(1oAI2) = C2T 2 = C2/3

o0 0 a:0 0

o 0 O o e f e0 e

@0 0 0a0 0a

--* Zp,k , I

* L M

(A7)

H) ))jo

Fig. 14. Secondary scattering at the kth particle Pk. The com-plex sum of primary scattered waves incident upon Pk is Wk; thetotal wave scattered by Pk is (1 + WNA.

Meng et al.

l


Wk is a random-walk sum of waves primary scattered by mAparticles in the Airy cone seen by Pk. The assumption ismade that most of or all the contributions to Wk are due tofar-field diffraction. Then

Wk = E °i ,(A8)i=1

where the number mk is a function of location Zp,A of theparticle Pk. It can be found that

(1WkJ2) = E(oil 2) = mkC 2/3. (A9)i-i

Referring to Fig. 14, one can find mk as

mk = (/3)n.0o (Z,k) 3. (A10)

Let Zpk = eAL, where sk is a random variable having homo-geneous probability distribution in the range (0,1). Then

(IWk 2) = (Or/3)n,002L k 3 C2 /3 = N(L/Z) 2ek

3 C2 /3, (All)

and thus

NU'M

2 = (NC2/3)2(L/Z)2 2 Sk /N. (A12)k-1

Noticing that NC 2/3 = oUN and k_1 Sk3/N = 1/4, we find

the variance of the speckle intensity IM that is due to thesecondary scattering to be

CM2 = N 2 (L/z) 2 /4. (A13)

Therefore the standard deviation of the total speckle in-tensity becomes

UN = _N[1 + (1/4)(L/Z) 2] 2. (A14)

Since z 2 L is always true, we can justify oN' U oN (forthe worst case z = L, OUN' = l.lUN). Therefore the specklecontribution from secondary scattering is negligible, espe-cially when L << z. Because multiple scattering rapidlygets weaker with higher order, we can conclude that it isadequate to consider only primary scattering. However,this conclusion would not be true for very dense and largeparticles (since we assumed the interparticle distance tobe larger than d2/A in the above analysis). If this condi-tion is violated, multiple scattering can be appreciable.

APPENDIX B: APERTURE EFFECT

Aperture effect on reconstructed particle images fromin-line holograms has been studied by other authors inconnection with edge smear as a result of insufficientrecording of sidelobes.2 9 In the context of our SNR mea-surements, we are more concerned with the decrease ofaverage intensity in the reconstructed images that is dueto aperture restriction within the central lobe. We findthat this effect of limiting aperture is the primary sourceof deviation of the measured SNR dependence on diame-ter from that predicted theoretically.

From Eq. (6), the portion of the hologram recording ofimportance for a single particle (centered on the opticaxis) can be written as

sin(ar2)[2Ji(adr)/(adr)],

where a = Ir/Az. The zone plate (the sine term) in itselfwould focus incident plane-wave illumination to a point inthe image plane, but it is modulated by the jinc-functionenvelope so that it reconstructs a circ-function patternwith the proper diameter d. The sidelobes {i.e., portionsof the envelope [2J1(adr)/(adr)] extending beyond its firstzero adr = 3.832} contribute little energy to the imagebut improve resolution of the particle edges. In holo-graphic particle velocimetry, in which only particle loca-tions are of major concern, images reconstructed withoutsidelobe information are still considered good enough; butloss of zone-plate fringes within the central lobe severelyaffects the total energy or the average intensity in thereconstructed image and thus decreases SNR.

Among mechanisms for finite apertures, we are particu-larly interested in fringe-frequency cutoff, which can beimposed by noise on the hologram or by fringe smearingthat is due to particle or film motion during exposure.Since the zone-plate fringes [sin(ar2 )] are independent ofd, such a cutoff (say, at frequency fA) introduces an aper-ture with a fixed radius that is applicable regardless ofparticle diameter:

pc = 7rfl/a. (B2)

The jinc function for a larger particle is more compactthan that of a smaller particle, having a greater portioncontained within the spatial-frequency range imposed bythe aperture. Thus the larger particle is favored; Isig in-creases with d. This is the main reason for the deviationof SNR dependence on diameter in our experiments (ap-proximately as d-') from that predicted theoretically(approximately as d-2 ). Other factors, such as coherent-noise background, also favor larger particles and so mayhave made minor contributions to this deviation, as well.

Obviously the aperture effect, as it depends on fringesmearing, could be improved with the use of a pulsed laserfor recording. As noted above, our SNR measurementsused a cw laser and a mechanical shutter. For our rangeof particle velocities, up to 10-/um smearing was present,so fringes with spacings smaller than or equal to thisamount were lost. With f, = 100 line pairs/mm, z =

50 mm, and A = 0.5 Am, we get a = 125 mm 2 , which cor-responds to an aperture with pc = 2.5 mm.

To evaluate the aperture effect numerically, we repre-sent the reconstructed image field of a particle as theFourier-Bessel integral over the hologram plane (radialcoordinate p):

f(r, pc) = 2 Jo(2arp)H(p, p0)pdp, (B3)

where H(p, pc) is a transfer function that is dependent ona cutoff parameter pc. With scaling of radius such thatp! = pad (and Pc' = pcad) we can define

g(r', pc) = 21 J1(p')JO(rp')H(p, p')dp', (B4)

where r' = 2rnd. We note that if H = 1, g(r') = circ(r').The energy in the reconstruction for a given exposure isgiven by 27r fog 2(r')r'dr'. The average intensity Isig of thereconstructed image is obtained by dividing the energy bythe nominal area of the particle, i.e., vr'2 , where r' = 1

Meng et al.

(Bl)


li(d)

00 50

d (m)

Fig. 15. Numerical simulation of the average intensity sig of areconstructed particle image versus particle diameter d, showingmonotonic increase of Iig with d as a result of aperture effectsbased on frequency cutoff within the central lobe.

(equivalent to r = d/2). Therefore

'sig = 2 g2(r')rdr'. (B5)

We simulated fringe smearing that is due to particle mo-tion by a slowly varying low-pass transfer function, i.e.,one that does not have sharp cutoff. Figure 15 shows theresult of plotting Iig by using H(p',p,') = (1 + p/pC')-(other choices of low-pass transfer function would displaysimilar behavior). We observe a monotonic increase of Iigwith d.

ACKNOWLEDGMENTS

We are grateful to Gabriel Roy for many fruitful discus-sions. This research is supported by the U.S. Office ofNaval Research through grant N00014-93-1-0532. Wesincerely appreciate discussions with Valery Zimin,Leonard Weinstein, and T. Scott Simmons.

REFERENCES

1. D. Gabor, 'A new microscope principle," Nature (London) 161,777 (1948).

2. B. A. Silverman, B. J. Thompson, and J. H. Ward, 'A laser fogdisdrometer," J. Appl. Meteorol. 3, 792-801 (1964).

3. B. J. Thompson, G. B. Parrent, J. H. Ward, and B. Justh,'A readout technique for the laser fog disdrometer," J. Appl.Meteorol. 5, 343-348 (1966).

4. J. D. Trolinger, R. A. Belz, and W M. Farmer, "Holographictechniques for the study of dynamic particle fields," Appl.Opt. 8, 957-961 (1969).

5. B. J. Thompson, "Holographic particle sizing techniques,"J. Phys. E 7, 781-788 (1974).

6. R. A. Belz and R. W Menzel, "Particle field holography atArnold Engineering Development Center," Opt. Eng. 8, 256-265 (1979).

7. B. C. R. Ewan, "Fraunhofer plane analysis of particle fieldholograms," Appl. Opt. 19, 1368-1372 (1980).

8. S. L. Cartwright, P. Dunn, and B. J. Thompson, "Particlesizing using far-field holography: new developments," Opt.Eng. 19, 727-733 (1980).

9. P. R. Hobson, "Precision coordinate measurements usingholographic recording," J. Phys. E 21, 139-145 (1988).

10. R. Menzel and F. M. Shofner, 'An investigation of Fraunhoferholography for velocimetry application," Appl. Opt. 9, 2073-2079 (1970).

11. B. C. R. Ewan, "Particle velocity distribution measurementby holography," Appl. Opt. 18, 3156-3160 (1979).

12. P. H. Malyak and B. J. Thompson, "Particle displacement andvelocity measurement using holography," Opt. Eng. 23, 567-576 (1984).

13. L. W Weinstein, G. B. Beeler, and A. M. Linderman, "High-speed holocine-matographic velocimeter for studying turbu-lent flow control physics," publ. 85-0526 (American Instituteof Aeronautics and Astronautics, New York, 1985).

14. H. Meng and F Hussain, "Holographic particle velocimetry:a 3D measurement technique for vortex interactions, coherentstructures and turbulence," Fluid Dyn. Res. 8, 33-52 (1991).

15. A. R. Jones, M. Sarjeant, C. R. Davis, and R. 0. Denham,'Application of in-line holography to drop size measurementin dense fuel sprays," Appl. Opt. 17, 328-330 (1978).

16. H. Royer, 'An application of high-speed microholography:the metrology of fogs," Nouv. Rev. Opt. 5, 87-93 (1974).

17. W L. Anderson, "Particle identification and counting byFourier-optical pattern recognition," in Applications of Opti-cal Fourier Transforms, H. Stark, ed. (Academic, New York,1982), Chap. 3.

18. R. Meynart, "Digital image processing for speckle flow ve-locimetry," Rev. Sci. Instrum. 53, 110 (1982).

19. G. A. Tyler and B. J. Thompson, "Fraunhofer holography ap-plied to particle size analysis: a reassessment," Opt. Acta23, 688-700 (1976).

20. J. D. Goodman, "Statistical properties of laser speckle pat-terns," in Laser Speckle and Related Phenomena, J. C.Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

21. M. Born and E. Wolf, Principles of Optics (Pergamon,Oxford, 1965), p. 752.

22. A. E. Ennos, "Speckle interferometry," in Laser Speckle andRelated Phenomena, J. C. Dainty, ed. (Springer-Verlag,Berlin, 1984), pp. 207, 210.

23. A. A. Friesem and J. S. Zelenka, "Effects of film nonlineari-ties in holography," Appl. Opt. 6, 1755-1759 (1967).

24. A. Kozma, "Photographic recording of spatially modulatedcoherent light," J. Opt. Soc. Am. 56, 428-432 (1966).

25. J. W Goodman and G. R. Knight, "Effects of film nonlineari-ties on wavefront-reconstruction images of diffuse objects,"J. Opt. Soc. Am. 58, 1276-1283 (1968).

26. P. F. Panter, Modulation, Noise and Spectral Analysis(McGraw-Hill, New York, 1965), p. 350.

27. I. Prikryl and C. M. Vest, "Holographic imaging of semitrans-parent droplets or particles," Appl. Opt. 21, 2541-2547(1982).

28. Z.-K. Lu and H. Meng, "Holography of semi-transparentspherical droplets," in Optics in Engineering Measurement,W F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 599,19-25 (1986).

29. D. M. Robinson, 'A calculation of edge smear in far-fieldholography using a short-cut edge trace technique," Appl.Opt. 9, 496-497 (1970).

Meng et al.

Intrinsic speckle noise in in-line particle holography

Documents

Transcript of Intrinsic speckle noise in in-line particle holography