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cattering of light by a periodic structure in the presencef randomness. III. Limit of the extended matchediltering method for the detection of weak periodicities

ani C. Vani and Sabyasachi Chatterjee

It is known that, for scattering of light by a rough grating, the periodically occurring intensity maximathat are characteristic of the periodic part of the scattering medium get smeared off and hence becomeundetectable if coherence length r0 of the randomness is much smaller than wavelength � of the periodicpart of the grating. It is also known that by conventional intensity measurements the detectabilityregion is r0�� � 0.33, which can be extended to lower regions by intensity interferometry. It isestablished here that an extended matched filtering method enables one to detect the periodic part of themedium, even in the presence of higher roughness �i.e., lower r0�, and the limit is placed at r0�� � 0.11by our numerical experiments. © 2004 Optical Society of America

OCIS codes: 050.1940, 050.1950, 070.2590.

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. Introduction

cattering of light by a periodic surface is of consid-rable interest in various branches of science. Theain method of characterization of a periodic surface

nvolves the identification of regular distinct peaks inhe scattered intensity; it fails, however, if the scat-ering surface is degraded by randomness. It washeoretically established and later proved experimen-ally that if the optical coherence length of the roughart of a surface is r0 and � is the wavelength of theeriodic part, the intensity peaks in the scatteredntensity will become indistinguishable for r0�� �.33. This limit gives the range in which the usualntensity measurements allow the periodic part of aurface to be extracted from the degradation that isue to a rough random part of the surface. Laterxperiments by Dainty and Newman1 showed thathe limit mentioned above can be extended to loweralues of r0�� if the intensity measurement is re-laced by intensity interferometry. The question,owever, still remains as to whether the Baltes limit

V. C. Vani is with the Department of Instrumentation, Indiannstitute of Science, Bangalore 560 012, India. S. Chatterjeechat@bgl.vsnl.net.in� is with Indian Institute of Astrophysics,angalore 560 034, India.Received 30 September 2003; revised manuscript received 8arch 2004; accepted 25 March 2004.0003-6935�04�183664-08$15.00�0© 2004 Optical Society of America

664 APPLIED OPTICS � Vol. 43, No. 18 � 20 June 2004

0�� � 0.33 is really the ultimate limit insofar as thesual intensity experiments are concerned.Our previous work has shown that if a conven-

ional analysis of the intensity profile is made, thealtes result gives a fair estimate; however, the same

ntensity data can expose hidden periodicity if thentensity data are analyzed by a matched filtering

ethod. For the sake of completeness, the theory onhich the technique is based, is repeated Section 2–4f this paper. As will be seen, the matched filterethod introduced by us puts the detectability limit

t r0�� � 0.11 for identification of the n � �1 peaksfter elimination of the n � 0 peak. In Section 5 weddress the question of what the theoretical basis forhe improvement of the detection is and conclude thathe central theme lies with the Rayleigh criterion inn improved circumstance that obtains when the� 0 peak has been removed by matched filtering.e also consider the possibility of detecting the n �2 peaks. It is argued that detection of these n �2 peaks after elimination of the n � 0 and n � �1eaks by matched filtering will extend the detect-bility limit further, to lower r0�� values. Theain advantage of the method, which merits em-

hasis, is that the basic data are contained in thesual intensity profile such that one does not re-uire sophisticated experimentation such as inten-ity interferometry. The theoretical treatment istraightforward, and the numerical work is alsoimple and thus amenable to rapid computationalchemes. In Section 6 we present the numerical

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. Theory

e consider a reflection grating lying in the xy plane,or which elevations in the z direction are given by

��x, y� � a cos�Qx� � ��x, y�, (1)

here Q � 2��.The random part ��x, y� of the ��x, y� profile is

ssumed to be a zero mean Gaussian stationary ran-om process, which does allow correlation function�r� �note the italics� of the randomness to have dif-erent forms, e.g., a Gaussian function or a Cauchyunction. In our numerical work, only a Cauchy-ype correlation function is considered because wessume that, for the same correlation length in f �r�,he Bessel transform f �vx� is broader for a Cauchyunction �y � 3�2� than for a Gaussian function �y ��, implying that a Cauchy function is less useful

han the Gaussian function for the same physicalarameters involved. We thus define

���x, y� � 0, ���x, y���x�, y�� � �2f�r�, (2)

ith

r � ��x � x��2 � � y � y��2��1�2�, (3)

here

f�r� � exp���r�l ���, (4)

ith � � 1 for the Cauchy function and � � 2 for theaussian function and where l is the correlation

ength of the random part of the profile.The scattering geometry is schematically described

n Fig. 1. The directions of the incident and scat-ered rays, following the convention in Refs. 2 and 3,re expressed in the polar coordinates as �� , 0� and

Fig. 1. Geometry of scattering from a rough grating surface.

1

�2, �3�, as shown in Fig. 1. Further, the wave vec-ors of scattering are defined as

vx � k�sin �1 � sin �2 cos �3�,

vy � �k sin �2 sin �3,

vz � �k�cos �1 � cos �2�, vxy2 � vx

2 � vy2, (5)

here k � 2�� and � is the wavelength of light.In what follows, we calculate the scattered inten-

ity under the Kirchhoff approximation, which isalid for 4rc cos � �� �, where rc is the radius ofurvature of the surface. One can estimate the av-rage radius of curvature to be rc � l2��; hence theethod given below is valid only for 4 �l2���cos �1� �. Another condition is �krc�

1�3 cos �1 �� 1, aondition that is maintained in the numerical casesiven below.The surface is thus, on average, flat; i.e., the radii

f curvature in both principal directions are on aver-ge zero. Because of randomness there may be lo-ally nonzero curvatures in both principal radii ofurvature, but these curvatures are equally likely toe �ve or �ve. Isotropy of the randomness meanshat the standard deviation of the radius of curvatures rc in both principal directions. These criteria al-ow the use of Fresnel coefficients for scattered radi-tion, on which the Kirchhoff approximation is based.he details of these arguments may be found in Sec.3 of Ref. 2.A related question may also be dispensed with

ere: What happens if the surface has an averageonzero curvature in both �or one� of the principalirections? In that case the phase calculations willhange and the effects of focusing and defocusing willave to be considered. In the present case, becausehe surface is on average flat, focusing takes place atnfinity, and that is the point at which we are calcu-ating the intensity profile; i.e., we are working underhe condition of the Fraunhofer limit.

We consider the quantity ���* 0 as the intensity ofight scattered by the rough surface in the direction�2, �3� divided by the intensity of light scattered by amooth surface in the specular direction. Followinghe methods indicated in Ref. 4, we can show, onefining

�g � vz���2, �g1 � avz��2,

hat

���* 0 � �J02���2g1�� f �vx, vy; g� ��

n�1

�

Jn2���2g1��

� � f �vx � nQ, vy; g� � f �vx

� nQ, vy; g���B��1, �2�, (6)

here

f �vx, vy; g� � �2�A� � exp��g�1

� f�r���J �v r�rdr, (7)

0 xy

20 June 2004 � Vol. 43, No. 18 � APPLIED OPTICS 3665

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B��1, �2� � �F3��1, �2, �3��2S��1, �2�, (8)

F3��1; �2, �3� � �1 � cos �1 cos �2

� sin �1 sin �2 sin �3���cos �1�cos �1

� cos �2��, (9)

S��1, �2� � S��1�S��2�, (10)

ith

S��� � exp���1�4�tan � erfc�K cot ���, (11)

K2 � �aQ�2 � 4���1�2. (12)

Note has to be taken of the fact that function f �r� isefined in the xy plane, whereas function f �vx, vy; g�s defined in the Fourier plane, i.e., in the �vx, vy�lane.It can easily be seen that, for � � 2, namely, aaussian correlation, one has

f �vx, vy; g� � �l2�2 Ag�exp���vx2 � vy

2�rg2�,

rg2 � l2�8g,

hereas

f �vx, vy; g� � �l2�Ag2��1 � c�vx2 � vy

2��2x�x

n other cases, where c�2x � r0. It is clear that, forindependent of x, in the limit of x3 �we do get f �vx,y; g� � exp��c�vx

2 � vy2��2�; i.e. a Gaussian profile

esults, showing that the choice of x can accommodatewide range of profiles. The dependence of c on l

nd g also depends on the nature of the correlationunction, i.e., on the exponent �. The extendedatched filter method described below gives aethod by which the best fits for c and x are to beade to extract the terms Jn

2���2g1��. In what fol-ows, we restrict ourselves to n � 1. This constraints necessitated by the fact that the parameters chosenor our numerical work are such that highern

2���2g1�� �for n � 2 and above� terms are ex-remely small.

The intensity expression given in Eq. �6� is aaman–Nath-type expression, in which the factors��1, �2� involve certain geometrical factors such as

hose that contribute to shadowing effects. The fac-or B��1, �2� is independent of �, whereas the term � �n Eq. �6� is strongly � dependent. The expressionsiven in Eqs. �6� and �7� show that, as width �1�r0� of�vx, vy; g� increases, the zeroth-order peak is broadnough to encompass other peaks too, and the detec-ion of the periodic part of the grating becomes im-ossible. It was noted by Baltes and co-workers5–14

hat, for a Gaussian correlation function, for random-ess the periodic part is undetectable if r0�� � 0.33,here Q � 2�� and � is the wavelength of therating structure. With the help of the extendedatched filter method we extract amplitude a andavelength � �or wave vector Q� of the periodic partf the grating, even when r0�� �� 0.33, i.e., when theeriodic part can be identified not by normal intensityeasurements but only by intensity interferometry.1

666 APPLIED OPTICS � Vol. 43, No. 18 � 20 June 2004

e described in earlier papers15,16 that the values ofand � found with measurements at different wave-

engths � consistently turn out to be the same when� 1. These observations show the existence of therst lobe. The second lobe is far more difficult tond, because its intensity is of the order of �a���4 and

s extremely low in the case under study. For accu-ate detection of the grating structure one can per-orm experiments at various values of � and showhat the amplitude of the n � �1 lobe follows theependence of �a���2 on �. For the present we studyhe accuracy to which the n � �1 lobe and waveector Q of the grating can be found. We previouslytudied the accuracy with which the n��1 lobes and

can be found at different values of r0��; the detect-bility becomes worse as r0�� becomes smaller.

. Extended Matched Filtering Method

n this paper we give a new method for enhancementf resolution. It is based on an a priori hypothesishat a hidden periodicity, if it exists, is completelyidden by randomness. The nature of the random-ess is not known, as parameters c and x in the profilere unknown but are determined to match the casehat gives the best fit. The nature of the profile f �vx�s, however, assumed a priori to be f �vx� � �1 � �c�x�vx

2�x, as can be the case for widely different typesf randomness, i.e., for widely different forms for theorrelation function f �r�. The presence of sidelobesf shapes similar to f �vx � Q� is the central theme inhe extended matched filtering method described inhe present paper. It is to be understood that f �r� istrictly not known but that we can determine it byssuming f �vx� to be as given by Eq. �13� below andhen finding the best fit for f �vx, vy; g� in the �b, y�arameter space. Ideally this search will give us c�and x � y. The parameters c and x are found

ndependently but must give coherence length r0hrough the solution �c�x� � r0; i.e., coherence length0 gets known when c and x are found from the fit.

We begin by considering that, for �g1 �� 1, themplitudes of the successive nth peaks fall as�g1�

2n, whereas the widths of the peaks vary asvx � r0

�1. As the separation of the peaks is vx �, the central n� 0 peak can subsume all the higher-

rder peaks for �vx �� vx � Q. The extendedatched filtering method advanced here follows the

ame principle as given earlier by us15,16 but uses aore-accurate method to arrive at the best filter. As

efore, this method too envisages the shape f �vx, 0; g�f the central peak and separates it from the totalntensity profile. In this way it tries to identify the� �1 peaks, whose shapes must match that of the� 0 peak that has been eliminated.For the sake of convenience of notation, we write�vx, 0; g� f �vx�. It is indeed obvious that, becausehe periodicity exists in the x direction, its presence isost clearly seen in the quantity ��*�vx, 0; g���vx, 0;� , denoted now ��*�vx, 0; g���vx, 0; g� ��*�vx���vx� ,ith which we work from now on. We begin by as-

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uming �this is the shape of the central peak, whichust be the same for all other peaks too� that

f �vx� � fa�vx� � �1 � �b�2y�vx2�y. (13)

We note that, on defining

Z�vx� � ���*�vx���vx� ���*�0���0� � � fa�vx�, (14)

!�vx� � Z�vx��Zmax, (15)

e find that

!�vx� � �N0 � N1 � N2���D0 � D1 � D2�, (16)

here we have defined

N0 � J02� f �vx� � fa�vx��,

N1 � J12� f �vx � Q� � f �vx � Q� � 2fa�vx� f �Q��,

N2 � J22� f �vx � 2Q� � f �vx � 2Q� � 2fa�vx� f �2Q��,

D0 � J02� f �Q� � fa�Q*��,

D1 � J12� f �Q* � Q� � f �Q � Q*� � 2fa�Q*� f �Q��,

D2 � J22� f �Q* � 2Q� � f �Q* � 2Q�

� 2fa�Q*� f �2Q��, (17)

* is the wave vector at which Z�vx� is a maximum,nd for the sake of brevity in notation we suppresshe arguments in the ordinary Bessel functions andrite Jn � Jn���2g1��.It is clear from Eqs. �16� and �17� that, close to vx�

, the quantity ���*�vx���vx� ���*�0���0� � must follow�vx�. The quantity Z�vx� thus gives a measure of theeviation of fa�vx� from the correct f �vx� term. It isasy to see that, if we succeed in making f �vx� � fa�vx�dentically, the term N0 also vanishes identically.ecause of the presence of a hidden periodicity theeviation !a�vx� indeed increases at points close to theidelobes, e.g., vx��Q,�2Q; however, the structureust follow fa�vx� as given by the terms N1 and N2 inqs. �16� and �17�.The existence of a hidden periodicity demands that

he shape of the sidelobe exactly match that of theentral peak. This matching is sought through aest-fit condition, in which the estimator �2 esti-ates, in the a sense of standard deviation, how far

he hypothesis !a�vx� really differs from the correctituation as given by !�vx�. It is indeed correct that,t vx � Q*, !aD�Q*� � !aN�Q*�, making !a�Q*� � 1dentically, which is an important condition for the

atched filtering that follows.We find that close to the peak we can expand

f �vx� � 1 � �c�2�vx2 � �1�2���c�2x��vx

2��2x�x � 1�

� 1�6��c�2x��vx2��3x�x � 1��x � 2�. (18)

his means that the radius of curvature of f �vx� closeo the peak equals c�1�2, whereas the exponent xescribes essentially the tail of the function. Thiseans that close to the peak the shape of f �vx� is

uadratic in vx, whereas the dependence on x startsor higher values of x. The fit attempted here is a

wo-parameter fit that involves the primary param-ters c and x such that coherence length r0 beyondhich f �vx� falls rapidly is given by r0 � c�2x.It is clear that, when our assumed fa�vx� is very

lose to the actual f �vx�, the terms N0 and D0 becomeegligible and the N1 terms show prominent peaks atx � �Q � �Q*, whereas these lobes should match,n shape, the chosen fa�vx � Q*�. To select the best

atch quantitatively, we note that if the terms N0nd D0 are negligible �because of the match f �vx� �

a�vx�� we must have !�vx� � !a�vx� extremely small,here !a�vx� is defined as

!a�vx� � !aN�vx��!aD�vx�, (19)

here

!aN�vx� � � fa�vx � Q*� � fa�vx � Q*�

� 2fa�vx� fa�Q*��, (20)

!aD�vx� � � fa�Q* � Q*� � fa�Q* � Q*�

� 2fa�Q*� fa�Q*��, (21)

he matched filtering is applied by defining an esti-ator

�2 � �0

Q*

�!a�vx� � !�vx��2dvx, (22)

nd we select the filter by choosing b and y to be thosehat give �2�b, y� to be the minimum. Note must beaken of the fact that, whereas !�vx� is a quantity thats found from the experimental data, !a�vx� is a quan-ity that is given by our hypothesis, so �2 gives aeast-squares estimate of the deviation of the datarom the hypothesis under consideration. Our mainim in the extended matched filtering method is tonticipate f �vx� first such that the shape of fa�vx� thatescribes f �vx� must also match the shapes of theidelobes at n � �1 to be found by searching for theinimum of �2�b, y�. �2 as defined above is intro-

uced in the same spirit as in Refs. 15 and 16 but isifferent from that given in 17 and 18. On redefin-ng �2 as given here, we note that the identification ofhe matched filter is considerably improved.

. Calculation of the Parameters

dentification of the matched filter enables us to de-ect, through a least-squares fit, the parameters forhe rough part of the surface. To find parameters forhe periodic part of the surface we use Eq. �11� to geton defining f��vx� � "f�"vx�

a��Q* � Q� � fa��Q* � Q� � 2fa�Q� fa��Q�

� ��J1�J2�2� f��Q* � 2Q� � f��Q* � 2Q��, (23)

maxJ02 � � fa�0� � fa�2Q*� � 2fa

2�Q*�

� 2fa�Q*�Zmax�J12 � � fa�3Q*� � fa�Q*�

� 2fa�2Q*� fa�Q*� � 2fa�Q*�Zmax�J22 � 0, (24)

he typical results from which are given in Section 5.

20 June 2004 � Vol. 43, No. 18 � APPLIED OPTICS 3667

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. Detectability Criterion

etectability of the periodic part of a grating fromntensity measurements has been discussed byaltes and co-workers.5–14 It is clear �e.g., see Eq.

4�� that if r0 is too small the function f �vx� will be-omes broad, such that the central n � 0 peak willncompass the n � 1 peaks and make them unde-ectable. This result, it must be noted, has a closeelationship to the Rayleigh criterion of detectability.

The detectability criterion for two peaks of equalntensity, according to the Rayleigh criterion, de-

ands that the intensity at the mid point betweenhese two peaks be less than 0.88 times the intensityf either of the peaks. A further refinement by Spar-ow19,20 pushes the detectability to even lower rangesy demanding, in essence, that detection always beossible when 2I�Q�2� � �I�0� � I�Q��� 0, where I�vx�s the intensity at vx. Both of these arguments maket understandable that detectability becomes impos-ible if r0Q � 2r0�� has less than a certain criticalalue.The extended matched filter method begins byaking an estimate of r0 in a self-consistent manner

y demanding that the n � �1 peaks, if they areresent, have exactly the same profile as the n � 0eak, with the same r0. Elimination of the n � 0eak thus brings out the n��1 peaks in !�vx�, whichre separated by 2Q. Hence their detectability islways possible whenever 2!�0� � �!�Q� � !��Q�� �, as follows from the Sparrow criterion. This en-anced separation between the peaks improves theossibility of detection and is hence an advance overhe existing methods of algorithm.

As is clear from the foregoing discussion, the cen-ral question about the efficacy of our extendedatched filtering is the possibility of finding r0 accu-

ately. We determine the accepted value of r0 byenerating the �2�b, y�-versus-b curve and taking thealue of �b, y� at which �2 is a minimum. To do so wehoose a certain value of r0 or b and y and determinehe corresponding �2 by following the steps given inhe text between Eqs. �10� and �16�. It is clear thathe accuracy of the method is decided largely by thexactness with which r0 can be found, as we illustrateelow.In the present paper we have followed the exercise

escribed above by choosing the parameters to beomparable to those as given in the experiments thatre due to Dainty and Newman.1 We found that the2-versus-b curve gives a minimum only up to the

imit r0�� � 0.11. �The next lower value of r0�� �.08 that was tried in the numerical work showed noinimum.�It is also convenient to express the detectability

imit in terms of an average quantity, �r � �0�

rrf �r�dr �0� 2rf�r�dr. For a Gaussian function

�r� � exp��r2�2L2� �r � ��2��L, so the detectabil-ty range L�� � 0.33 implies that �r �� � 0.33��2� � 0.41359. For a Cauchy-type function, i.e.,�r� � exp��r�r0�, we have �r � 2r0, and we haveound numerically that the detectability region lies in

668 APPLIED OPTICS � Vol. 43, No. 18 � 20 June 2004

he range r0�� � 0.33, i.e., �r �� � 0.66. The dif-erence in the limiting r0�� values that define theetectability range occurs specifically as a result ofifferences in the correlation functions in the twoases. These differences result in the profile f �vx, vy;� � exp���vx

2 � vy2�L2�2� for the Gaussian function

nd the profile f �vx, vy; g� � �1 � �vx2 � vy

2�r02��1.5 in

he Cauchy function, as can be seen from the mo-ents �vxy

n , defined as

�vxyn �

��0

�

vxynf �vx, vy; g�2vxydvxy�

��0

�

f �vx, vy; g�2vxydvxy� . (25)

t can be seen that for the Gaussian function �vxyn is

efined for all values of n. For the Cauchy function,owever, it is not defined, even for n� 1, because therofile falls as vx

�3 for vx � �1�r0�, and a logarithmicivergence occurs even for n� 1 unless one puts 2��which it is physically correct to do� as the cutoff inhe upper limit of integration. In addition, the co-erence condition is also reflected in the width of theeam in the vxy plane, i.e., by the n � 1 moment.oreover, the relations among the moments �vxy

n elp to extract the nature of the correlation function

n the beam itself, provided that the moments arealculable quantities themselves.

. Numerical Results

n the computations performed in this study we cal-ulated the intensity value, which is simple in natures explained above. The values of the parametersorrespond to those in the experiments of Dainty andewman,1 with a� 3.28# 10�6,�� 6.25# 10�4 cm,� 632.8 Å��6.328 # 10�5 cm� etc.Computations were carried out for the helium–

eon wavelength 6328 Å. We vary the detectionimit, or r0��, in each case by varying the correlationength of the rough surface. Table 1 gives the valuesf r0�� in various cases. Ten values of r0�� werehosen for numerical calculations ranging from r0�� 0.48 to r0��� 0.08. Until r0��� 0.11 we foundminimum in the b-versus-�2 curve. For values of

0�� � 0.15 and less we found a lot of scatter in theata. However, the existence of the minimum islear until the value of r0�� reaches close to 0.08. Inigs. 2 and 3 the results are summarized. Figure 2hows six curves of the constant b �x axis� versus �2�yxis� where the minimum can be seen; for r0�� �.0810, the minimum vanishes. In all these curvese have kept y fixed at 1.5, i.e., close to the valuehere �2 is a minimum.As was discussed above, the accuracy of the method

s highly sensitive to the value of r0. The system willork if the minimum of �2 is found at a value of a�b,� that also gives b�2y � r0

2. Table 1 shows this toe so when the actual value of r0�� is the one withhich simulations were conducted, whereas the cal-

ulated values of r �� are the values given by b�2y as

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ound through matched filtering when the best fit ishosen. The agreement between actual and calcu-ated r0�� can be seen to be excellent, as is furtherllustrated by the fact that in Fig. 3 the curves for 2n�r0��� versus ln�b� and 2 ln�r0��� versus ln�b�2y�ave a gradient of �1, as expected from the theory.alidation of the above conclusions by detailed nu-erical experiments supports our central contention

n which the method is based, to a high degree ofccuracy. The gradient was also determined by a

ig. 2. Six curves showing different detection limits, which we obtept y � 1.5 for this figure, as this was the value of y at which thehows a case in which a peak is undetectable. The r0�� values a.2591, 0.1943, 0.1457, 0.1134.

Table 1. Check of the Accura

c� and y at �min2 r0��

c� �#10�8� y Actual

26.0 1.5 0.485817.0 1.5 0.388612.0 1.6 0.32398.0 1.5 0.25914.3 1.5 0.19432.9 1.4 0.16192.3 1.5 0.14571.9 1.6 0.12951.3 1.4 0.1134

0.0810

aThe values of r0�� are also given.

east-squares fit, which gave a slope of 1.006 in bothases.

Table 1 also illustrates various results. We haveerified that Q on being calculated from Eqs. �23�ives the greatest error of 7.32% when r0��� 0.1295.he error in Q is as little as 0.06% when r0�� �.2591. Table 1 also shows the striking accuracy ofhe method. It must be noted that the value of a is2.8 nm, i.e., 4a�� � avz* � 0.65, in the argumentsf the Jn

2 terms, where vz* is the value of vz where the

d by plotting b �x axis� versus �2 �y axis� for several y values. Weave a minimum in all cases. The absence of this clear minimumich the minima occur are as given in Table 1: 0.4858, 0.3886,

the Matched Filter Algorithma

a�Calculated;�#10�6�

Error ina �%�alculated

0.4710 326 0.510.3808 325.8 0.610.3100 326 0.510.2612 333 1.630.1915 320 2.350.1630 305 6.960.1400 285 12.950.1233 299 8.950.1089 216 33.5

aine�2 gt wh

cy of

C

20 June 2004 � Vol. 43, No. 18 � APPLIED OPTICS 3669

n0itnwsalu

7

IdrpeottetfTittuoT�RtmSae

wlo�Btpstatfiviltoathaa

to

R

1

1

1

Flsst

3

� 1 peak occurs. This gives J12�avz*��J0

2�avz*� �.10, implying that the n � �1 peaks, are at best �i.e.,n the absence of smearing by the f �vx� terms�, 10% ofhe n� 0 term. It is indeed extremely encouraging toote that for r0��� 0.12 the value of a can be found toithin an accuracy of 10%. In summary, these re-

ults establish the great sensitivity of the method toccurately determine both periodicity Q �or wave-ength �� and amplitude a of a grating structure, evennder extremely unfavorable circumstances.

. Conclusions

t has been shown by numerical experiments that theetectability limit in a periodic structure hidden inandom noise is much lower than that which hadreviously been accepted, provided that we use anxtended matched filtering method of detection aspposed to the conventional methods of analysis ofhe usual intensity measurements. We have provedhat, even with pure intensity data involving simplexperimentation and simple computations, the de-ectability limit can be extended to �r0�� � 0.11, aactor of 3 lower than what was previously accepted.he reason for this enhancement of the detectability

s that extended matched filtering looks for two iden-ical peaks separated by 2Q, whereas the conven-ional methods look to identify two peaks of grosslynequal height separated by only Q, the central onef which, n � 0, overwhelms the n � �1 sidelobes.his technique, however, cannot take us below r0�� 0.11. The explanation is to be sought in the

ayleigh criterion of detectability, i.e., failure to de-ect two equal peaks at n � �1 when the centralinima may be 88% of the peaks. According to theparrow criterion, this would mean that the detect-bility would break down when the intensity, afterlimination of the n � 0 peak, begins to curve down-

ig. 3. Of 2 ln�r0��� �x axis� versus ln �b� �filled circles� andn�b�2y� �stars� �y axis� where �c, y� is the point at which �2�c, y�hows a minimum. The linear trend with the gradient close to 1hows the extreme accuracy that the algorithm is capable of at-aining, with its enhanced detectability range. P � r0��.

670 APPLIED OPTICS � Vol. 43, No. 18 � 20 June 2004

ard for vx � 0. We are working to overcome thisimitation. We believe that an enhancement can bebtained if, on elimination of both n � 0 and the n �1 maxima, one can still pinpoint the n � �2 lobes.eing able to do so would be a distinct advantage,

hough the J22 peaks would be extremely low com-

ared with the J02 and J1

2 peaks. The methodhould thus entail extreme accuracy in finding r0, i.e.,he curvature of f �vx�, and hence J0

2 and J12. We

re pursuing ways to achieve this goal. The poten-ial application of high-quality gratings in variousabrication processes merits consideration.21,22 Anmportant issue that arises from these recent obser-ations is the possibility of recording the scatteredntensity for fixed �1 and �2 while sweeping wave-ength �. As predicted,4 in these types of observa-ion the scattered intensity is seen to havescillations as the wavelength is swept. We envis-ge that the matched filtering method developed inhe current series of papers will be of considerableelp in making these oscillations more pronouncednd will thus be further checks on the values of J0

2

nd J12 as calculated by the method given here.

The authors thank the referees of the paper forheir comments, which helped to improve the qualityf the presentation.

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