Generation of moiré by nonlinear transfer characteristics

1828 J. Opt. Soc. Am. A/Vol. 5, No. 11/November 1988

Generation of moire by nonlinear transfer characteristics

Reiner Eschbach*

Department of Electrical and Computer Engineering, University of California, San Diego, La folla, California92093

Received December 9, 1987; accepted June 10, 1988

The intermodulation of signal frequencies, or moire, is an effect generally encountered in situations in whichperiodic or quasi-periodic structures are superimposed. Most familiar are the striking low-frequency. patternsachieved by the multiplicative superposition of two structures. In other situations, the intermodulation of signalfrequencies may be caused by nonlinear distortions of an input signal. This can be a desired feature in measuringapplications or a noise source in transmission or display. Effects of nonlinear transfer characteristics on inputsignals consisting of more than one frequency are shown. The dependence of resulting moire frequencies on thenonlinearity is evaluated, and common nonlinearities are compared by simulating a moire topography experiment.

INTRODUCTION

The superposition of two or more periodic and/or quasi-periodic functions frequently results in the occurrence offunctions with additional frequencies. This phenomenon isknown as moir6 and was first described by Lord Rayleigh forthe superposition of two different gratings.' The additionalfrequencies occurring in this superposition are generally thesum and difference frequencies of the original frequencies ofthe input functions and their higher harmonics. The moir6effect has been used in a vast number of applications sinceits first description. Examples of these are the measure-ment of three-dimensional objects, as it is done in the moiretopography2 "3; stress and strain analysis4' 5; and moire deflec-tometry.6 In addition to the desired moire effect in measur-ing applications, an undesired moire effect, one that is notnecessarily introduced by a multiplicative superposition, isoften encountered in display and printing situations. Here,the generally low-frequency components of the moir6 pat-tern are regarded as noise or disturbance of the desiredoutput. Thus it is desirable to reduce the effect in thesesituations in order to eliminate the visual appearance ofthese frequencies or to increase the effect in measuring ap-plications in order to get a better visualization of the desiredinformation.

The moir6 effect caused by the application of a nonlinear-ity on a signal of more than one frequency is described in thispaper. The evaluation is valid under the assumption thatthe amplitudes of both the input signal and the output signalare limited, which is generally true in physical systems. Theapproach presented here is independent of the strength ofthe nonlinearity, in contrast to the standard power-seriesapproach, which is especially useful for weak nonlineari-ties.7 -9 This deviation from the polynomial expansion is ofspecial interest in hybrid moir6 setups, in which the nonlin-earity can be realized by loading look-up tables inside avideo system in a fast and convenient way independent of itsactual form. This permits the generation of strong nonlin-earities, compared with the nonlinearities generally encoun-tered in amplification and transmission, without calcula-tional effort. Furthermore, the approach uses a sorting cor-responding to the occurring frequencies, compared with the

sorting corresponding to the polynomial expansion of thenonlinearity. The polynomial description is, however, pref-erable in cases in which the nonlinearity is described suffi-ciently by a low-order polynomial.

Common nonlinearities are compared, and the effect ofthe symmetry of the nonlinearity on the resulting frequencyspectrum is shown. Computer simulations are used to dem-onstrate the application of nonlinear transfer characteristicsin moir6 topographic measurements.

GENERATION OF MOIRE

Different ways exist to generate the moire effect. Mostcommon in measuring applications are (a) multiplicationand (b) subtraction and successive rectification of the sig-nals.10 Figure 1 shows these two superpositions for thesinusoidal signals:

I, = S1 + S, sin(27rwlx)

and

I2 = S20 + S2 sin(27rW2 x),

with

I = 1I2 (la)

in Fig. l(a) and

I = li1 - 21 (lb)

in Fig. l(b). The occurrence of a low frequency (W1 - W2) is

clearly apparent in Fig. 1.Although the appearances of the superpositions in Figs.

1(a) and 1(b) are similar, the means by which the moir6effect is achieved are different in nature. This can be seenfrom Eq. (lb), where no moire is obtained for

S,0 >S 8 + S20 + S2,

which is equivalent to omitting the rectifying operation afterthe subtraction. In Eq. (lb) the moir6 effect is achieved by anonlinear operation, in contrast to the linear operation (mul-tiplication) of Eq. (la). Thus undesired moire can occur in

0740-3232/88/111828-08$02.00 © 1988 Optical Society of America

Reiner Eschbach

Vol. 5, No. 11/November 1988/J. Opt. Soc. Am. A 1829

A t~tt^Af\AAAAAA/(a)

(b)

Fig. 1. Generation of the moir6 effect by superposition of twoperiodic structures: (a) the multiplicative superposition (IjI2) and(b) the rectified difference (II, - I21) of the two input signals I, andI2. Both plots show a strong low-frequency component.

situations in which a nonlinear transfer characteristic is in-volved, even if the original superposition would not yieldintermodulation frequencies. This is the origin of the moireeffect in a large number of situations, in which it is regardedas disturbance. Important examples for these cases arenonlinearities in photographic materials or photodetectors,common printing processes, and transmission. 11' 1 2 On theother hand, a nonlinearity can be introduced intentionally toachieve the moire effect in measuring applications. It isnecessary to understand the formation of moir6 by nonlin-earities in order to increase or decrease the strength of theeffect dependent on the situation. The treatment that fol-lows is restricted to the description of moir6 phenomenacaused by a nonlinear transfer characteristic.

MATHEMATICAL DESCRIPTION

The definition of the term moir6 is expanded here to includesignals with a temporal periodicity additional to the spatialperiodicity of optical patterns. The x coordinate in thedescription represents, depending on the desired applica-tion, a temporal or spatial (vector) coordinate. The effect ofthe nonlinearity N on the signal S(x) can be written as

M(x) = N[S(x)],

with M(x) being the result of the nonlinearity including themoir6 terms. Assuming a finite signal Smin < 8(x) < Smax,we can write M(x) by means of a Fourier-series expansion,

M(x) = E ck exp[ikS(x)],k=--

(2)

with

Ck =- N(f)exp[-ikfldf

being the Fourier coefficients of the nonlinearity. The re-normalization of S(x) into the interval [-7r, 7r] has beenperformed, so that the coefficients ck can be taken fromstandard mathematical books. N is assumed to be restrict-ed to the interval [-1, 1]. The Fourier expansion is used toget a signal-independent formulation of the nonlinearity N.In what follows, the signal

S(x) = So + S, sin w1x + S2 sin w2 x,

IS1 + IS11 + IS21 < ir

(3a)

(3b)

is used to demonstrate the effect of the nonlinear transfercharacteristic on the input signal. The treatment of more-complicated inputs, i.e., signals with more than two frequen-cies or signals that underwent a nonlinear distortion, can bedone in a similar but mathematically more complex way.

Inserting S(x) into Eq. (2) gives

M(x) = Ck exp(ikS 0)exp(ikSj sin w1 x)exp(ikS 2 sin W2 X)

x=- x

=ECk exp(ikSO) E J.(kSj)exp(imwjx)k=-- m=-

X E Jn(kS 2)exp(inW2x)no=--

= E E exp[i(mwl + n. 2)x]m=-X n=--

X E cCk exp(ikSo)J.(kSj)Jn(kS 2 )

k=--

= Z E Am,n exp[i(mcoj + nC02)X].m=-X n=--

Here Jj(a) refers to the Bessel function of first kind, argu-ment a, and order L.w

1 3,1 4

Am,n is the complex amplitude ofthe frequency component (mco, + nC2). In this way, theresult of the nonlinearity applied to the input is sorted corre-sponding to the occurring frequencies. The informationabout the nonlinearity is wholly contained in Am,n. Theinformation about the interdependence of Am,n and N per-mits a modification of the nonlinearity to increase or de-crease certain frequency components or to predict the be-havior of the system for a known nonlinearity. Note that,independent of the nonlinearity, only integral multiples ofwi and W2 appear in Eq. (4). Rewriting

(5)Am,n = Z Ck exp(ikS0)Jm(kSi)Jn(kS2 )k=--

and incorporating

JL(a) = (-1)1J_,(a),

one finds that

Amn = (-1)m Amn = (-1)nAmn = (-1)m+nA-mn. (6)

From Eq. (6) it follows that sum and difference frequencies(mw1 i nW2) have the same amplitude, in the absolute-valuesense, independent of the nonlinearity. The difference inthe sign of the Am,n results in a phase shift of differentcomponents relative to each other. The direction of thefrequency can be determined from Eq. (4).

A selection of the sum or difference frequency, as is oftendesired in measuring applications to increase the signal-to-noise ratio of the measurement, cannot be made by thenonlinearity alone. Different approaches must be used here

(4)

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to permit the desired operation. One possibility is the tem-poral multiplexing of several moire patterns, an approachthat requires the temporal stability of the signal.3' 15"16 An-other approach is the introduction of additional degrees offreedom into the system, where color and/or polarization canoften be used advantageously.17 -19 Both approaches can inprinciple be used together with the generation of moire bynonlinearities.

CONDITIONS FOR THE FORMATION OFMOIRE

The amplitude of a given moir6 frequency or the strength ofthe interaction of the frequencies wi, can be calculated ac-cording to Eq. (5) for a known nonlinearity. An importantquestion, however, is whether every nonlinearity producesmoir6 or a mixture of frequencies. This question can beanswered by determining the nonlinearities that do not pro-duce any intermodulation of frequencies. This is done bysolving Eq. (5) for Ck, with the special assumption that

Amn = 0 for all Iml + InI > 1. (7)

Am,n = 0 6m,6n,0 + E expi 2(m + n)l2 m,0 2 1k=1

X {ak cos[kSo + 2 (m + n)]

+ bk sin[kS + (m + n)J} Jm(kS1)Jn(kS2),

(9)

where the Kronecker 6 function is used, with aij = 1 for i = iand bij = 0 otherwise. Equation (9) shows the influence ofthe bias So on the appearance of different frequencies.

With no loss of generality, it is possible to center theinterval of the Fourier expansion [Eq. (2)] at So, and in thiscase to set So = 0, to obtain

r C

| 2em,o0no + EakJm(kS)Jn(kS 2 )Amn = k=1

Is i E bkJm(kSl)Jn(kS 2 )k=1

m + n even

m + n odd

Regarding Eq. (6), we assume in what follows than m, n > 0.The amplitude for m + n = 1 is excluded because it repre-sents the original, input. Additionally, we are interestedonly in coefficients Ck that solve Eq. (7) for all possible valuesS and S2. Coefficients ck that solve Eq. (7) for only acertain set of values would cause moire for other values, andthe corresponding nonlinearity would thus in general causemoir6. To exclude these cases, we set S2 = S - S1 with 0 < S< 7r and integrate over all possible S8 values. It follows that(see Appendix A)

Am,n = 0 if E CkJm+n(kS) = 0k=-X

or, equivalently,

N[S sin(wx)] - sin(wx).

The transition from Eq. (9) to Eq. (10) can also be performedby adjusting the bias So to the symmetry point of the nonlin-earity, which is not necessarily the center of the interval.

Equations (9) and (10) show some interesting properties ofthe nonlinear transfer characteristic. A nonlinearity N thatis an even function with respect to So (bk = 0) gives rise only tofrequencies with (m + n) even, and a nonlinearity N that is anodd function with respect to So (ak = 0) gives rise only to

(8a)

(8b)

The only function that fulfills the requirement of Eq. (7) isthe linear function. Every nonlinearity thus introduces amixing of the signal frequencies.

The actual weights Am,n, and thus the strength of themixing, are dependent on the nonlinearity and on the actualsignals. Equation (5) can be evaluated further to show char-acteristic features of nonlinearities that are independent ofthe signal amplitudes SI and S2.

EVALUATION OF Am,n

The relative strength of the moir6 frequencies is stronglydependent on the form of the nonlinearity. Different non-linearities result in different amplitudes of the moir6 fre-quencies, including the possibility of zero amplitude for awhole set of frequencies. This is no contradiction to theabove calculations, for which zero amplitude for all inter-modulation frequencies was required. In what follows, thenonlinearities are determined that give rise to Am,n = 0 for acertain set Im, n) of frequencies.

Replacing the complex Fourier coefficient ck by the realcoefficients ak and bk of the cosine and sine transformationgives

(a)

r N

f

(b)

N

fFig. 2. Two nonlinearities, (a) squaring and (b) rectification, asfunctions of the input f.

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A0.15

A0.04

(a)

A

(b)

A

(d)

0,025

Fig. 3. Comparison of strengths of the moir6 effect generated by squaring and rectification in the range of 0 < S, + 82 < r. (a), (b) Al,, for the

squaring and the rectification, respectively. (c) Difference between (b) and (a), with the rectification resulting in a larger Al,, for S1, S2 < 2.

(d) A3,1 for the rectification; A3,1 = 0 for squaring. Note that the plots in (c) and (d) are magnified compared with those in (a) and (b), as indicat-

ed by the values of the A axes.

frequencies with (m + n) odd. Common nonlinearities thus

do not produce all moir6 frequencies but have a whole set of

(m, n) combinations for which Am,n = 0. Nonlinearities with-

out a symmetry are capable of producing the complete spec-trum of Eq. (4).

Squaring and RectificationThe results of Eq. (10) are demonstrated here for two impor-

tant nonlinearities, rectification [N(f) = Il] and squaring[N(f) = f2]. Both nonlinearities are even functions with re-

spect to the origin, as shown in Fig. 2. Applying them to the

signal S generates only frequencies with (m + n) even. Con-sequently, the fundamental frequencies wl and W2 do not

appear, but the frequencies 2w,, 2W2, c'1 + W2, 3w, + 2, Cw1 +

3W2, 2wX, + 2W2, etc. may appear. Both nonlinearities producethe frequency w1 + W2, which generally is used for measuring

applications, and are thus applicable to such situations. Theamplitudes Al,, for both nonlinearities are shown in Figs. 3(a)and 3(b). Here

-qu = 4 ( 1 )k1ak 7 k 2

+rect. = 4 1a~k 72 (2k + 1)2

are used as Fourier coefficients for squaring and rectifica-

tion, respectively. Figure 3(c) shows the rectification as thestronger nonlinearity until S8 > 2 or S2 > 2. Here, strengthis measured as relative to the amplitude of the examined

frequency and not to the amplitudes of other frequencies.Squaring results in a stronger amplitude for larger values of

S and S2. Figure 3(d) shows the amplitude of 3wX I W2 for

the rectification. Squaring results in zero amplitude for thisfrequency. This is obvious in the power-series descriptionbut is also shown by Eq. (B4a) of Appendix B:

E k- J 2m(k() = 0 for m > 1.

Applying these nonlinearities to an addition of two gratingsresults in the patterns shown in Fig. 4. Here grating 1 has a

period of 45 pixels in the vertical direction and grating 2 hasa period of 90 pixels at an angle of 20 deg with the vertical.The image field consists of 400 X 400 pixels. The higherharmonics generated by the rectification in Fig. 4(b) resultin a sharper appearance of the fringes compared with thosegenerated by squaring with no higher harmonics in Fig. 4(a).The original frequencies are not visible, but sum (E) anddifference (A) frequencies appear.

Sigmoidlike NonlinearityThe sigmoidlike function is another common nonlinearitythat can often be used to describe the behavior of systems,e.g., television systems. We use

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that permits the change to a cosine for the appropriate biasand a > 1/2. With this definition the Fourier coefficientsare

2 (- 1)'kbk = 2 _1k2

7r a~ -k

As an odd function, the sigmoid does not produce the sum ordifference frequency. The situation changes when So is nolonger the symmetry point of the nonlinearity, which isshown in Eq. (9). In this case even- and odd-frequencycomponents occur for the sigmoid as well as for the two

(1,0)

(a)

(0,1)

(b)

Fig. 4. Moir6 generated by (a) squaring and (b) rectification of twosinusoidel gratings. The gratings have periods of 45 pixels in thevertical direction and 90 pixels at an angle of 20 deg with thevertical, respectively. The sum (E_) and difference (A) frequenciesappear sharper for the case of rectification.

(a)

-.,2L_ i(2, )

(b)

-i A 1"I A _kJA A(c)

(0,2)

(d)

f

Fig. 5. Sigmoidlike nonlinearity described in Eq. (11).

NW a _ sin(af)max[sin(af)]

-1 <a< 1

as the formulation for the sigmoidlike function, which isshown in Fig. 5. Here max[sin(af)] = sin(a7r) is used for 0 <a < 1/2 and max[sin(af)] = 1 is used for 1/2 < a < 1. Notethat Eq. (11) is essentially a masking of the sine function

(e)Fig. 6. Change in the frequency spectrum dependent on the biasvalue So. (a) Spectrum of the input signal with frequency wi denot-ed by (1, 0) and frequency W2 denoted by (0, 1). (b) Spectrumobtained by applying the nonlinearity to the signal. Here the addi-tional frequencies corresponding to (-1, 2) and (2, -1) appear, andno component appears at (-1, 1) or (1, 1), which represent thedifference and sum frequencies. (c)-(e) Spectra resulting from theaddition of a bias So. Spectra in (c) and (d) show no symmetry withrespect to the bias and consist of odd and even doublets (m, n). Thebias in (e) was chosen to convert the sine of the sigmoid into a cosine,thus representing an even nonlinearity with respect to the bias.Only doublets with (m + n) even appear, and the spectra in (e) and(b) are mutually exclusive. Note that spectra in (a)-(e) are normal-ized independently to show the occurring frequencies.

-I-- '

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COMPUTER SIMULATIONS

The above-mentioned nonlinearities have been applied tosimulate a moire topographic measurement. Here, one or

(a)

(a)

(b)

Fig. 7. Sigmoidal nonlinearity applied to the same two gratings asin Fig. 4. (a) Zero bias (So = 0), resulting in an odd symmetry andno sum or difference frequency. The low-frequency pattern corre-sponds to 2w1 - W2. (b) Effect of the same nonlinearity for a bias of37r/4, resulting in an even symmetry and strong sum and differencefrequencies.

nonlinearities mentioned in the preceding subsection. Thisis shown in Fig. 6 for the sigmoid, with a = 2/3. The spec-trum of the original input is shown in Fig. 6(a); Figs. 6(b)-

6(d) represent situations with increasing bias. In Fig. 6(e)the bias is chosen to be So = 37r/4, changing the sine of Eq.(11) into a cosine. Thus Fig. 6(e) represents an even nonlin-earity, and the spectra of Figs. 6(b) (So = 0) and 6(e) aremutually exclusive. The symmetry of the amplitudes ofcorresponding sum and difference frequencies as given byEq. (6) is shown clearly in Fig. 6. In a situation in which w, dW2 is desired, one should thus try to avoid a bias So for therectification and the squaring but should try to introduce abias So #d 0 that does not violate Eq. (3b) for the sigmoid.Figure 7 shows the application of the sigmoid to the samegratings as in Fig. 4. Figure 7(a) corresponds to the odd-symmetry case of Fig. 6(b) showing no sum or differencefrequency, and Fig. 7(b) shows the even-symmetry case cor-responding to Fig. 6(e) with strong sum and difference fre-quencies. The low-frequency pattern in Fig. 7(a) representsthe frequency 2wX - W2 (m + n = 1).

kw/

(C)

Fig. 8. Application of (a) squaring, (b) rectification, and (c) sig-moid in a simulated moir6 topographic setup. The object has beenchosen to yield four contour fringes as a difference frequency. As inFig. 4, the fringes appear sharper for the case of the rectification. In(c) the sigmoid has been used to generate a fringe pattern with twicethe number of contour fringes by selecting the second-order differ-ence frequency [(m, n) = (2, -1)].

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1834 J. Opt. Soc. Am. A/Vol. 5, No. l1/November 1988 Reiner Eschbach

more gratings are projected onto the object under test, andthe resultant phase-modulated grating structures are readinto a video system by using a video camera. By using onlyone projected grating, the image can be described as

I(x, y) 1 + costw[x + f(X, y)]},

with f(x, y) being dependent only on the height z(x, y) of theobject and on system geometries. A reference grating of theform

R(x, y) cc 1 + COS(WRX)

is then added to I(x, y), and the result is nonlinearly distort-ed. Note that the addition of the reference grating can bedone by projecting two gratings onto the object; they act as amutual reference.16'20' 21 Figure 8(a) shows the result ofsquaring as a nonlinearity, after the bias term is removed.The object was chosen to give four concentric fringes as adifference frequency for the case WR = co. Figure 8(b) showsthe result of applying the rectification onto the signal. As inFig. 4(b), the result appears sharper because of the addition-al higher harmonics that are not generated by squaring. Asigmoidal nonlinearity was used to generate Fig. 8(c). HereWR = 2w was used. In this way a doubling of the fringes isachieved by selecting the higher harmonic 2w[x + f(x, y)] -WRX, having twice the phase modulation [2wf(x, y)] as thefundamental frequency.

CONCLUSIONS

Superposition of periodic or quasi-periodic functions gener-ally leads to the formation of moire frequencies or the inter-modulation of different input signals. The most commonsuperposition that gives these frequencies is the multiplica-tion of two or more input signals. This is often done inmeasuring applications, for which the low-frequency moir6component is actually the desired information.

Addition and subtraction of input signals lead to no moir6frequencies in the spectrum of the superposition. In realprocesses, such as display and printing, however, moir6 fre-quencies are generally visible. These frequencies are a con-sequence of the nonlinearities inside the transfer channeland are unavoidably connected to any form of nonlinearity.In this context, it is important to notice that the subtractionof patterns normally results in positive and negative quanti-ties. An additional rectification is thus performed after thesubtraction. This rectification is the reason for the inter-modulation in the subtractive superposition of patterns.

The mathematical description given in this paper pro-vides a way to predetermine the moir6 effect for a givennonlinearity as a function of the input signals. Generalpredictions of the resultant frequency spectrum can be madefrom the form of the nonlinearity. An important parameteris the symmetry of the nonlinearity with respect to the biaspoint of the signal. Even and odd symmetries result inmutually exclusive frequency spectra, with even nonlineari-ties producing only even frequencies and vice versa. Aneven nonlinearity is thus useful in measuring applicationsthat rely on the difference between the two frequencies.Applications that incorporate the difference of three fre-quencies (W1 i W2 d W3 or 2wX + W2 ) require odd nonlinearitiesif the frequency mixing is supposed to be done by the nonlin-earity alone. This was used for the selection of the second

harmonics in the simulation of the moire topography experi-ment.

The additive superposition is the best choice in order toavoid the intermodulation of frequency components. Sub-tractive superposition can be used in the same manner, butcare must be taken to avoid any negative numbers thatwould require rectification.

Nonlinear processes offer a wide variety of possibilities incases in which an intermodulation is desired. Nonlineari-ties give rise to strong intermodulations and may be usedespecially advantageously in measuring applications withthe ability to create and change nonlinearities easily by load-ing look-up tables inside a video system. This has beendemonstrated in a simulated moire measurement, in whichthe nonlinearities have been used to generate the requiredmoir6 patterns.

APPENDIX A

A function fulfilling Eq. (7),

Amn = 0 for all Iml + InI > 1,

would not generate a moir6 effect. We can determine thecorresponding functions by using Eq. (5):

Z ck exp(ikS0)Jm(kS,)Jn(kS2 ) = 0 for all Iml + Inl > 1.

k=--

No assumption has been made about the magnitude of S0 ;thus the functions must fulfill Eq. (5) also for S0 = 0. Fur-thermore, we can set

S1 = S', S2 = S - S'.

By using this special dependence of S and S2, we do not missa solution of Eq. (5), but we may find a function correspond-ing to these weaker requirements but not fulfilling Eq. (5)for all possible S1 and S 2 values. Dividing both sides of theequation by S' and integrating S' from 0 to S (or kS' from 0to kS) gives

0 = Ck | Jm(kS')Jn(kS - kS') (kS') (Al)k=-a

and

0 = > CkJm+n(kS).k=--

(A2)

Only functions given by Eq. (A2) are possible solutions forEq. (5). From the above result, only functions that do notintroduce higher harmonics (m + n)W on a single frequencydo not introduce moir6 of the form (mi 1 + no 2). The linearfunction is thus the only possible solution to the problem.

APPENDIX B

Although Eqs. (2)-(7) have been deduced under the assump-tion that N is a nonlinear function, no explicit use of thenonlinear behavior has been made in the derivations. Thelinear function as well as the constant function is thus stillincluded in the equations. Inserting the Fourier coefficients

Vol. 5, No. 11/November 1988/J. Opt. Soc. Am. A 1835Reiner Eschbach

for the linear function into Eq. (7) and regarding only theterms with (m + n) odd leads to

(Bi)Amn = 2i E (l) Jm(kSk)Jn(kS2)kl kk=1

and comparing this with the input S(x) leads to the interest-ing relation

E (.. 4 )k1 Jm(kSi)Jn(kS 2) = 0 (B2)k=1

for all

IS11 + IS21 < 7r

and

ml + Inl > 1,

and Eq. (A2) generally does not hold for

1S11 + 1S21 >7r.

By using an input signal consisting of M frequencies from w,

to wM, Eq. (B2) can be generalized to

E s 1 -1k JmJ(kSj) = 0 (B3)k=1 i=1

forM

E Imjl > 1

j=1

andM

Isj <j=1

The validity of Eqs. (B2) and (B3) is not intuitively clear,but it can be shown by comparing the equations with theSchl6milch null series.8'9 Inserting N = 1 into Eq. (B3)shows this relation. The Schlomilch null series can be writ-ten as [see Ref. 8, p. 337, Eq. (13a)]

E, k2P J2m(hW) 0, (B4a)k=1

E(-k 1)k1 J2 1(k-) = 0, (B4b)

with

-7 r<<7r, mi>p.

Equations (B4b) and (B3) are identical for p = 0 (i.e., the

linear function), M = 1, and S, = t. Equation (B3) is thus ageneralization of the Schlomilch series to the product of MBessel functions of arbitrary order.

Inserting the Fourier coefficients of the quadratic nonlin-earity, except for constant factors, with

(-1)k-1ak= L2

into Eq. (B4a) shows that only the frequency 2w has a non-

zero amplitude, as expected. Here, again, So is assumed tobe the symmetry point of the nonlinearity.

ACKNOWLEDGMENTS

I wish to thank Olof Bryngdahl of the University of Essen,

Essen, Federal Republic of Germany, for many valuable andstimulating discussions. I would like to thank Sing H. Lee

of the University of California, San Diego, for the opportuni-ty to pursue this research.

This research was supported by the Deutsche Forschungs-gemeinschaft.

* Present address, Xerox Webster Research Center, Web-ster, New York 14580.

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Documents

Generation of moiré by nonlinear transfer characteristics