The structure of quadratic Gauss sums in Talbot effect
TThe structure of quadratic Gauss sums in Talbot effect
Carlos R. Fern´andez-Pousa ∗ Department of Communications Engineering, Universidad Miguel Hern´andez, Av. Universidad s/n, E03203 Elche, Spain (Dated: November 29, 2016)The field diffracted from a one-dimensional, coherently illuminated periodic structure at fractionalTalbot distances can be described as a coherent sum of shifted units cells weighted by a set of phasesgiven by quadratic Gauss sums. We report on the computation of these sums by use of the propertiesof a recently introduced integer s , which is constructed here directly from the two coprime numbers p that q that define the fractional Talbot plane. Using integer s , the computation is reduced, up toa global phase, to the trivial completion of the exponential of the square of a sum. In addition, it isshown that the Gauss sums can be reduced to two cases, depending only on the parity of integer q .Explicit and simpler expressions for the two forms of integer s are also provided. The Gauss sumsare presented as a Discrete Fourier Transform pair between periodic sequences of length q showingperfect periodic autocorrelation. The relationship with one-dimensional multilevel phase structuresis exemplified by the study of Talbot array illuminators. These results represent a simple means forthe design and analysis of systems employing the fractional Talbot effect. Keywords:
Talbot effect, quadratic Gauss sums, perfect periodic autocorrelation, Talbot arrayilluminators.
I. INTRODUCTION
When a diffraction grating is illuminated by a coherentplane wavefront, a series of self-images can be observedat certain distances beyond the object, at integer multi-ples of a fundamental Talbot distance [1], [2]. A relatedphenomenon occurs at fractional values p/q of this dis-tance, where the field is a coherent superposition of q shifted and weighted copies of the grating’s unit cell [3].These phenomena are consequence of the interference be-tween diffraction orders, which acquire a quadratic phaseunder Fresnel propagation, and have also been observedin other domains, such as the angular spectrum [4] and,through the space-time duality [5], [6], in time [7] and inthe optical spectrum [8], [9]. As simple diffractive effects,they have found widespread application [10], [11].Much of this work was triggered by the publication,twenty years ago, of a seminal paper by Berry and Klein[12], who showed the relationship of the weighting factorsat fractional Talbot distances with the classical quadraticGauss sums of number theory. The solution to theseGauss sums was presented in [12] by considering threeseparated cases, relying on the result of a previous in-vestigation [13]. One of these three solutions containedan error which was corrected later by Matsutami andˆOnishi [14]. More recently [15], it has been shown thata proper rewording of the original threefold form of theGauss sums leads to a compact description in terms of theproduct of two phases, one that is constant and only de-pends on integers p and q and a second that is quadraticand proportional to a new integer s . The observation in[15] can be stated as the fact that the Discrete FourierTransform (DFT) of a quadratic phase sequence is itselfa quadratic phase sequence. In addition to its intrinsic ∗ [email protected] beauty, this correspondence finds immediate applicationsas a simple rule to implement the phases in physical sys-tems employing fractional Talbot effect. However, thecharacterization of the integer s that arises in the Gausssums was obtained in [15] by direct comparison, relyingon the original computations in [13] and [14].In this paper, we revisit the computation of theseGauss sums from a new perspective. Using standardtechniques from number theory, integer s is constructedin an abstract way from the two coprime integers p and q that define the fractional Talbot effect. The proper-ties shown by this integer allows for an almost trivialderivation of the main property of the Gauss sum, namelyits quadratic dependence. It is also shown that both s and the result of the Gauss sums can be expressed onlyin two cases associated to the parity of integer q and,in addition, that both members of the DFT pair thatdescribe the Gauss sums show perfect periodic autocor-relation [16], [17]. These results not only simplify thedescription of self-imaging phenomena, but also permit acompact analysis of systems employing the fractional Tal-bot effect. As an example, the relationship with multi-level phase structures is exemplified by the detailed studyof one-dimensional Talbot array illuminators (TAI) [18],[19], [20], [21], [22], [23].In our presentation we will employ the description ofTalbot effect in the time domain [7], where temporallymodulated periodic waves substitute for diffraction grat-ings and dispersion for diffraction. Although our resultscan be extended to different domains as pointed out be-fore, the use of the temporal formalism is preferred herebecause it conforms to the standard definition of discrete-time transforms of signal theory, which is the naturallanguage upon which the main results are presented.The plan of the paper is the following. In Section IIwe review the basic result of [12] about the structure ofTalbot self-images of periodic objects. In Section III, weconstruct the integer s and derive several properties, and a r X i v : . [ m a t h - ph ] D ec in Section IV we compute the Gauss sums of Talbot effectand derive their autocorrelation properties. Section V isdevoted to the analysis of TAIs and Section VI presentsour conclusions. The paper is completed with three ap-pendices which present some results and definitions fromnumber theory and two lengthy computations. II. THE TEMPORAL TALBOT EFFECT
Let us consider a modulated optical wave, whose elec-tric field is given by E ( t ) = E ( t ) exp( jω t ), with ω thefrequency of the optical carrier and e ( t ) the optical com-plex envelope or baseband representation of the wave.The envelope is assumed periodic with fundamental pe-riod T , and represented as the repetition of function w ( t ): E ( t ) = + ∞ (cid:88) n = −∞ δ ( t − nT ) ⊗ w ( t ) , (1)where ⊗ denotes convolution. Here we assume that w ( t )is contained in a period, 0 ≤ w ( t ) < T , and so it describesthe unit cell of the periodic train. The spectrum of theoptical envelope E ( t ) is contained in a set of equally-spaced angular frequencies, ω n = 2 πn/T with n integer.This field feeds a linear delay line with lowest-orderdispersion. Without loss of generality, the transfer func-tion that describes the linear propagation of the envelope E ( t ) can be assumed quadratic, H ( ω ) = exp( − jφω / φ is the dispersion coefficient, equal to the deriva-tive of the group delay with angular frequency at thecarrier value, ω . The temporal Talbot effect manifestsitself as the recovery of the initial train, or of a periodicstructure directly related to the original train, at certainvalues of dispersion φ characterized by the period T andtwo positive and mutually coprime integers p and q . Thedefining equation is: 2 π | φ | = pq T . (2)At these values, H ( ω n ) = exp( − jπσ φ pq n ), where σ φ isthe sign of φ . The output envelope E (cid:48) ( t ) can be com-puted in the spectral domain by use of the Poisson for-mula in (1) and the values of H ( ω n ), E (cid:48) ( t ) = 1 T (cid:32) + ∞ (cid:88) n = −∞ e − jπσ φ pq n e j πn tT (cid:33) ⊗ w ( t ) . (3)Note that the value of the sum for σ φ = − σ φ = 1, since the change inthe sign of the exponential can be compensated with aninversion n → − n . This first part of the equation can bedecomposed by changing the sum over n to a sum over thepair of integers (cid:96) = 0 , . . . , q − m = −∞ , . . . , + ∞ ,with n = (cid:96) + mq . The sum over m can be performed using again the Poisson formula, and we get [12]: E (cid:48) ( t ) = δ (cid:18) t − e pq T (cid:19) ⊗ + ∞ (cid:88) k = −∞ δ ( t − kT ) ⊗ √ q q − (cid:88) n =0 e jσ φ ξ n δ (cid:18) t − n Tq (cid:19) ⊗ w ( t ) , (4)where e x represents the parity of integer x , so that e x = 0when x is even and e x = 1 when x is odd. In (4), e pq thus represents the parity of the product of integers pq .The phases in (4) are given by the following Gauss sums: e jσ φ ξ n = 1 √ q q − (cid:88) m =0 e − πjσ φ pq m ( − pqm e πj nmq = 1 √ q q − (cid:88) m =0 e − πjσ φ pq (1+ qe q ) m e πj nmq . (5)In the final part of the equation we have used that( − pqm = ( − pe q m σ φ . This second form of the Gausssums will be the starting point of our computation inSection IV.According to (4), the output is composed of a repeti-tive structure of the same period T as the original train,explicitly recognized in the second term of the multipleconvolution. This train is shifted by half a period if pq is odd, as shown by the first delta term. The repetitivestructure that defines the output train is described bythe two last terms in (4), and is composed of the coher-ent sum of q replicas of the input unit cell, w ( t ), mutuallyshifted by T /q and weighted by factors exp( jσ φ ξ n ) / √ q .We point out that, expect in the case that w ( t ) is con-tained in an interval of length T /q , the last line in (4)does not describe the unit cell of the output, since thisexpression may extend beyond an interval of length T .When q = 1, the output train is similar to the original.The resulting wave is referred to as an integer Talbot im-age of order or index p , or simply a self-image of order p of the input train. If, in addition, p is odd, it is usuallyreferred to as a shifted integer Talbot image. In general,the resulting trains for q > p/q , and shifted when the product pq isodd. To proceed with our analysis we devote the follow-ing sections to the characterization of phases exp( jξ n ) interms of an integer s constructed from p and q . III. A PARITY-DEPENDENT MODULARINVERSE
The main results concerning the properties of integer s is stated in the following theorem: Theorem 1.
Given two coprime and positive integers, p and q , there exists a unique integer s such that it verifiesthe following three properties:(a) s lies in the range 1 ≤ s ≤ q − s is a solution of the modular equation: sp = 1 + qe q (mod 2 q ) , (6)(c) s has opposite parity to q .In addition, such a integer s also verifies:(d) s is coprime with q .The theorem will be proved in four steps. First, weshow the existence of integers verifying (a) and (b) bythe explicit construction of solutions to (6). Second, wewill show that these solutions also verify (c). Third, weshow that any other solution of (6) is either out of therange (a) or does not verify (c). Finally, we deduce that s and q must be coprime. For the definition of the symbolsused in the proof we refer to Appendix A. Existence.
For even q , Eq. (6) reduces to sp = 1(mod 2 q ), and since integers p and q are mutually prime,so are p and 2 q . Then the equation can solved as: s = (cid:20) p (cid:21) q ( q even) , (7)where [1 /a ] n is the inverse of a (mod n ). In particular,the inverse of p (mod 2 q ) lies in the range 1 ≤ s ≤ q − q is odd, integers 2 p and q are coprime, and asolution to (6) is given by: s = 2 (cid:20) p (cid:21) q ( q odd) . (8)Indeed, in this case the product sp is: sp = 2 p (cid:20) p (cid:21) q = 1 (mod q ) = 1 + aq, (9)for a certain integer a . But this a must be odd, since s in(8) is even and q is odd, and therefore (9) can be writtenas: sp = 1 + q (mod 2 q ) , (10)thus solving equation (6). Now, since the inverse of 2 p (mod q ) is contained in the interval 1 ≤ [1 / p ] q ≤ q − s lies in the range stated in the theorem. Parity.
As for property (c), the statement is trivialfor the solution found for q odd, since the explicit con-struction (8) shows that in this case s is even. For q evenwe first note that p must be odd. Now, the solution of(6) is of the form: sp = 1 + 2 bq (11)for certain integer b . The right hand side of this equa-tion is odd, and therefore s must be odd. Note that theexplicit solutions (7) and (8) together with property (c)point out that s is an modular inverse of p whose concreteform depends on the parity of q . TABLE I. Values of integer s in the range 0 ≤ s ≤ q − p and q . p q
10 14 16 2 4 8 10
12 6 4 14 20 2 8 18 16 10
Uniqueness.
Integers (7) and (8) are the unique so-lutions to (6) in the range 1 ≤ s ≤ q − q . In fact, given two solutions of (6) for given p and q , namely s given by (7) or (8), and a differentsolution ˜ s , their difference verifies (˜ s − s ) p = 0 (mod 2 q ).Here we need to analyze separately the cases.For q even, and since p and 2 q are coprime, this impliesthat ˜ s − s = 0 (mod 2 q ) and so the uniqueness of s inthe range 1 ≤ s ≤ q − q odd if we further assume that p is also odd,since again p and 2 q are coprime.The only remaining case is q odd and p even. Theequation above, (˜ s − s ) p = 0 (mod 2 q ), only implies that(˜ s − s ) = 0 (mod q ), and so both (8) and ˜ s = s + q aresolutions to (6), as it canbe easily checked. In particular,˜ s may lie in the range 0 ≤ ˜ s ≤ q −
1. For example, for p = 2 and q = 3, both s = 2 and ˜ s = 5 are solutions of (6)in the range 1 ≤ s, ˜ s ≤
5. However, integer s as given by(8) is selected by the fact that it has the opposite parityto q , contrary to ˜ s = s + q . This completes the proof. Coprime.
Finally, to show that integer s is indeedcoprime with q , let us suppose that s and q share a com-mon factor, say c . We will show that c = ±
1. First, notethat c is also a factor or the product sp . Then, dividing(6) by this factor we get: spc = 1 c + qe q c (cid:18) mod 2 qc (cid:19) . (12)Now, since the left hand side of this equation is aninteger and c is a factor of q , then 1 /c should be integer,and thus c = ±
1. The proof of Theorem 1 is complete.We point out that the expressions for s derived in [14]and used in [15] are different from the ones given by (7)and (8). The equivalence is shown in Appendix B.The computation of the values of s , as a functionof p and q , can proceed from the systematic solutionof (7) and (8) or from an equivalent expression suchas those in Appendix B. A table for the lowest valuesof p and q was presented in [15] and reproduced inTable I (see, however, note [24]). The calculus of s is facilitated by the observation that, regardless theparity of a given q , (6) implies that integer s for givencoprime numbers p and q is the same that for p + 2 q and q . This property reflects the exact periodicity,without half-interval shifts, of the temporal Talboteffect. In the case of q odd, it also follows from (6)that the integers s for the pair p, q and for the pair p + q and q , coincide. Unfortunately, the result doesnot hold for q even. Specific solutions can be found forseveral series of integers, as shown by the following result. Proposition 2. (a) for any q : p = 1 (mod 2 q ) ⇒ s = 1 + qe q .(b) for any q : p = q ± q ) ⇒ s = q ± n, p : q = ± np ⇒ s = ± n (mod 2 q ). Proof.
First we note that in these three series integers p and q are indeed coprime, since in all cases they aredefined by a relation of the form ap + bq = ± a and b , and thus any common factor between p and q must be equal to one following a similar reasoningto that in (12). Second, it is a simple observation thatthe parity of s is the opposite to that of q in all cases.Moreover, all proposed values of s lie in the range 1 ≤ s ≤ q −
1, except in case (c) for which the residue mod2 q has been taken. Then, it suffices to check that thevalues of s verify (6) by direct substitution. The result istrivial for (a), whereas for (b) and (c) it can be deducedfrom the observation that (6) is equivalent to any of thefollowing equations: sp = ( q ± (mod 2 q ) . (13)In particular, (13) defines the self-dual integers s = p = q ± p and s coincide.This completes the proof.The calculus of s is also simplified by the followingresult, that relates its value for complementary Talbotlines of order p/q and ( q − p ) /q . Essentially it is a sumrule that applies to pairs of integers in each row of Table I: Proposition 3. If s is the integer corresponding to p and q , with p < q , then the integer s (cid:48) corresponding to p (cid:48) = q − p and q , verifies;(a) s (cid:48) = 2 q − s for q odd,(b) s (cid:48) = q − s for q even and 1 ≤ s ≤ q −
1, and(c) s (cid:48) = 3 q − s for q even and q + 1 ≤ s ≤ q − Proof.
First notice that, if p is coprime with q , then p (cid:48) = q − p is also coprime with q . Moreover, integer s (cid:48) as stated above has the same parity as s , so condition(c) in Theorem 1 is fulfilled. As for condition (a) in thatTheorem, for q odd s (cid:48) + s = 2 q , and therefore if s is in therange 1 ≤ s ≤ q − s (cid:48) . For q even, both (b) and(c) above gives s (cid:48) in the range 1 ≤ s (cid:48) ≤ q −
1. Note that s cannot coincide with q since they must be coprime.To complete the proof it suffices to check (6). Direct computation gives: s (cid:48) p (cid:48) = sp + [ n ( q − p ) − s ] q, (14)with n = 1 , , q even, s , p and n ( p − q ) are odd,and thus s (cid:48) p (cid:48) = sp (mod 2 q ). For q odd, s and n ( p − q )are even, and again s (cid:48) p (cid:48) = sp (mod 2 q ). Therefore, sinceby hypothesis (6) is verified by sp , it is also verified by s (cid:48) p (cid:48) . The proof is complete. IV. THE GAUSS SUMS
Integer s constructed in the previous section providesa simple route to evaluate the Gauss sums (4). Our com-putation is based on the observation that (5) involves theexpression: exp (cid:18) − πjσ φ pq (1 + qe q ) m (cid:19) (15)which can be considered as a q -sequence since, for q evenand odd, is periodic in the running index m with period q . We recall that, given a sequence of q complex numbers, x n with n = 0 , . . . , q −
1, the Discrete Fourier Transform(DFT), denoted by F , is defined as the q -sequence X m given by: X m = F ( { x n } ) m = q − (cid:88) n =0 x n e − πjnm/q . (16)The IDFT is the inverse transform to (16), defined as: x n = F − ( { X m } ) n = 1 q q − (cid:88) m =0 X m e πjnm/q . (17)We state the Gauss sum (5) as a DFT pair: Proposition 4.
For any p , q coprime and s the integerconstructed in Theorem 1, the following DFT pair holds: x n ≡ e jσ φ ξ n = e jσ φ ξ exp (cid:18) πjσ φ sq n (cid:19) DF T −−−→ X m ≡ √ q exp (cid:20) − jπσ φ pq (1 + qe q ) m (cid:21) , (18)with: e jξ = (cid:18) sq (cid:19) e j π ( q − = (cid:18) pq (cid:19) e j π ( q − ( q odd) ,e jξ = (cid:16) qs (cid:17) e − j π s = (cid:18) qp (cid:19) e − j π p ( q even) , (19)and (cid:0) ab (cid:1) is the Jacobi symbol of an arbitrary integer a and an odd and positive integer b .For the definition and properties of Jacobi symbolswe refer to Appendix A. We also point out that thepresent result reduces the number of cases to two,depending only on the parity of q , instead to the conven-tional three cases studied in Refs. [12], [13], [14] and [15]. Proof.
We set σ φ = 1, the general case is consideredat the end of the proof. We have to show that: e jξ n = e jξ e πj sq n = 1 √ q q − (cid:88) m =0 e − πj pq (1+ qe q ) m e πj nmq . (20)First we justify the change of variables m = sk , with k = 0 , . . . , q −
1. Since s and q are coprime, the values ofthe product sk span the residues of q , i. e. , the products sk , with k = 0 , . . . , q −
1, span the set { , . . . , q − } mod q . The change m = sk can thus be performed in(20) since the contribution to the sum of a certain m isinvariant under shifts of the form m → m + q , as it isimmediate to show. With this change (20) reads:1 √ q q − (cid:88) k =0 e − πj pq (1+ qe q )( sk ) e πj sq nk . (21)We use now (6) in the first exponential of (21): e − πj sq (1+ qe q )( sp ) k = e − πj sq (1+ qe q ) k = e − πj sq k ( − sqk = e − πj sq k , (22)where we have used that the product sq is even. Withthese simplifications, the exponents in (21) can be com-pleted to the square of a sum,1 √ q e πj sq n q − (cid:88) k =0 e − πj sq ( k − n ) , (23)and the proof is complete if we show that: e jξ = 1 √ q q − (cid:88) m =0 e − πj sq m . (24)The fact that the sum in the right hand side of (24) isindeed a phase can be shown by changing variables inthe double sum defining its modulus squared, following asimilar argument to that of an equivalent computation in[12]. The explicit evaluation of the sum leading to (19)can be completed by standard techniques of the sametype as those used in [13], as is shown in Appendix C.We observe that the quadratic phases exp( jξ n ) are pe-riodic and symmetric,exp( jξ n ) = exp( jξ n + q ) = exp( jξ − n ) = exp( jξ q − n ) . (25)Now, using the general relationship for the DFT of thecomplex-conjugated sequence, F ( { x ∗ n } ) m = F ( { x n } ) ∗ q − m , (26) it follows from the symmetry property that the DFTof the complex conjugated sequence, exp( − jξ n ) is thecomplex conjugated DFT, as stated in (18). The proofis complete.We now compute the periodic autocorrelation of the q -sequence x k = exp( jξ k ), R ξ ( n ) = q − (cid:88) k =0 exp( − jξ k ) exp( jξ k + n ) . (27)Using (18), this gives a sum of the form: R ξ ( n ) = e jπ sq n q − (cid:88) k =0 e − j π sq kn , (28)which can be evaluated in terms of Kronecker’s delta us-ing that, for any integer a : q − (cid:88) k =0 e − j πak/q = qδ a, q ) . (29)We observe from (28) that the sum is nonzero for integers n which are solutions of sn = 0 (mod q ). The solutionsare n = 0 (mod q ), since s and q are coprime, so: R ξ ( n ) = qδ n, q ) . (30)The phase sequences x k shows perfect periodic autocorre-lation [16], and therefore its spectrum, as given by (18),has magnitude √ q . In fact, it can be shown that both q -sequences, x n and X m in (18), have perfect periodicautocorrelation, since both are examples of Chu’s con-struction [17]. In that paper it was shown that, given N and M coprime integers, the M -sequencesexp( ± jπN m /M ) ( M even)exp( ± jπN m ( m + 1) /M ) ( M odd) , (31)have perfect periodic autocorrelation. For q even, thefact that x k = exp( jξ k ) belongs to this set is immediate.For q odd it follows after the shift in the running index k → k + ( q + 1) /
2, since the resulting sequence is in theform (31) up to a global factor.As for the Fourier-transformed sequence X m , it is al-ready in form (31) for q even. For q odd, the sequencereads; X m = √ qe − jπp ( q +1) m /q = √ qe − jπpm /q ( − pm . (32)Using again the shift m → m + ( q + 1) / √ q factor, to:exp (cid:20) − jπ pq (cid:0) m + ( q + 1) m (cid:1)(cid:21) ( − pm = exp (cid:20) − jπ pq m ( m + 1) (cid:21) , (33)which is the expression in (31). V. TALBOT ARRAY ILLUMINATORS
The previous results provide a simple description ofthe weighted sum of unit cells that defines the fractionalTalbot effect. Using (20) in (4), the output field in aTalbot plane with order p/q is described as the repetitionof the signal:˜ w (cid:48) ( t ) = e jσ φ ξ √ q q − (cid:88) n =0 exp (cid:18) jπσ φ sq n (cid:19) w (cid:18) t − n Tq (cid:19) , (34)composed of the sum of delayed input unit cells, weightedby a quadratic phase factors and the amplitude factor1 / √ q . In this section we use this result to analyze aconcrete example.The Talbot array illuminators (TAI) are multilevelphase gratings that concentrate coherent light in a pe-riodic array of sharp binary irradiance distributions ofrectangular form, at fractional Talbot distances of order p/q after the array [18], [19]. Referring to Fig. 1, we de-note the period of the phase grating as L and the widthof the focusing spots as ∆. The compression ratio, ∆ /L ,is a measure of its concentration capacity, and equals thevalue of integer q [19]. ∆ L Phase AmplitudeAmplitude Phase
X Z Z14 T FIG. 1. Scheme of a Talbot array illuminator for p/q = 1 / The TAI phases can be determined when the propaga-tion is considered in the reversed way [21],[22], as shownin Fig. 2. Starting from a binary amplitude grating ofopening ratio 1 /q , which is illuminated by a collimatedand spatially coherent wavefront, one searches for distri-butions of uniform light intensity at the fractional Talbotplanes after the amplitude grating. At these planes thewavefront is composed of a set of phase levels with a pe-riodicity equal to the original period L of the amplitudegrating. In this inverse TAI scheme, the design prob-lem is simply to determine these phase levels at the frac-tional Talbot planes. Note that in the spatial formalismreversing the propagation direction z is equivalent to acomplex conjugation, since the one-dimensional paraxialwave equation is: 2 ik ∂A∂z = − ∂ A∂x , (35) where A ( x, z ) is the complex amplitude of the paraxialwave and k the wavenumber. Therefore, the TAI phasesare the complex conjugated to those acquired in propa-gation.Alternatively, one can reproduce in a multilevel phasegrating the phases acquired by fractional p/q Talbotpropagation, without complex conjugation, and operatethe phase grating at the complementary ( q − p ) /q frac-tional Talbot distance [22]. The combined effect is thepropagation over a q/q = 1 Talbot distance, and thusthe original amplitude grating is reproduced with a half-period shift. In our presentation, however, we use theformer point of view as it does not require the change ofthe Talbot order. AmplitudePhase
X Z Z14 T PhaseAmplitude
FIG. 2. Scheme of the inverse Talbot array illuminator for p/q = 1 / Within the time-domain formalism the general solutioncan be found as follows. As in the inverse TAI problem,let us consider the temporal analogue to a binary ampli-tude grating with opening ratio 1 /q , which is a pulse ofrectangular form and width T /q . Using (34), we observethat the output unit cell at a fractional Talbot plane oforder p/q is composed of q replicas of the basic rectangu-lar pulse, each with a quadratic phase factor proportionalto the ratio s/q . According to the previous development,the TAI phases should be the complex conjugates. Ex-plicitly,Φ ( s/q ) n = exp ( − jσ φ ξ n ) = exp (cid:20) − jσ φ (cid:18) ξ + π sq n (cid:19)(cid:21) . (36)To check this result, let us consider the direct TAI ge-ometry. Cw laser light of unit amplitude, analogous tothe plane wave that illuminates the grating in Fig. 1, isphase-modulated according to the levels in (36), each ina time interval of duration T /q . The input unit cell canbe presented as: w ( t ) = q − (cid:88) n =0 exp ( − jσ φ ξ n ) rect (cid:18) tqT − n (cid:19) , (37)where the rectangle function is defined as rect( x ) = 1 for | x | < / E (cid:48) ( t ) = 1 √ q + ∞ (cid:88) m = −∞ q − (cid:88) n =0 e jσ φ ( ξ m − ξ n ) rect (cid:18) tqT − n − m (cid:19) . (38)We have omitted the δ ( t − e pq T ) term for simplicity, sinceit is not relevant in the present computation. With thechange n + m = k this reads: E (cid:48) ( t ) = 1 √ q + ∞ (cid:88) k = −∞ q − (cid:88) n =0 e jσ φ ( ξ k − n − ξ n ) rect (cid:18) tqT − k (cid:19) . (39)The sum in n is in the form of a convolution, but canbe written as the autocorrelation R ξ ( − k ) = qδ k, q ) using the symmetry property (25). The autocorrelationis nonzero only for k = 0 (moq q ), so setting k = nq for n integer we finally get: E (cid:48) ( t ) = √ q + ∞ (cid:88) n = −∞ rect (cid:18) tqT − nq (cid:19) . (40)The output envelope is the expected rectangle pulse trainwith period T and pulse width T /q .This result can be described as follows. The multilevelphase modulation (37) defines a series of q time bins ofduration T /q within the fundamental period T , explicitlyshown by the succession of rect functions. The shiftingand weighting of unit cells produced by Talbot effect, to-gether with the periodicity of the the train, translatesinto a shifting and weighting of bins, which leads to theconvolution expressed in Eq. (38). This process has beenexemplified in Fig. 3. If the multilevel phases are cho-sen as the complex conjugated to the Talbot propaga-tion phases, the bin-wise multiple interference becomesthe perfect periodic autocorrelation (30). The multipleinterference is therefore constructive for only only bin perperiod, and is destructive for the rest. In other words, thearray illuminator relies on the autocorrelation propertiesof the quadratic Gauss sum arising in Talbot effect.The TAI operation principle can also be analyzed inthe spectral domain. The Fourier transform of the unitcell (37) is: W ( ν ) = Tq sinc (cid:18) νTq (cid:19) q − (cid:88) n =0 e − jσ φ ξ n e − j πνnT/q . (41)Since the unit cell is repeated with period T , the spec-trum of the envelope E ( t ) is composed of spectral lines ν = m/T , with m integer: E ( ν ) = 1 q sinc (cid:18) νTq (cid:19) + ∞ (cid:88) m = −∞ δ (cid:16) ν − mT (cid:17) × q − (cid:88) n =0 e − jσ φ ξ n e − j πmn/q , (42) where sinc( x )=sin( πx )/( πx ). The sum over n becomesthe DFT (18), and the result can be presented as: E ( ν ) = 1 √ q exp (cid:20) jπσ φ pq ( νT ) (cid:21) × sinc (cid:18) νTq (cid:19) + ∞ (cid:88) m = −∞ δ (cid:16) ν − mT (cid:17) ( − pqm . (43)This is the spectrum of a train of squared pulses of width T /q , amplitude √ q and period T , shifted by half a periodwhen pq is odd, and dispersed by the quadratic phase fac-tor in the first line. The dispersive line after the phasemodulator simply compensates for this phase factor, andrenders the pulse train chirp-free and thus of square tem-poral profile.The peak amplitude in (40) is √ q , which amounts to again factor of q in peak power with respect to the powerof the cw laser light. Notice also that (37) can be inter-preted as the multilevel phase modulation of a perfectrectangular pulse. In general, had we phase modulatedan arbitrary pulse contained in an interval of duration T /q and repeated also in
T /q shifts, we would have ob-tained at the output the same pulse with periodicity T and a gain factor of q in power. This is the principleof the noiseless pulse amplification by coherent additiondemonstrated in [25].It is illustrative to analyze some particular cases de-rived from (36). For instance, it contains the six binaryTAIs described in [20], associated to the fractional Talbotplanes p/q = 1 / s = 1), p/q = 1 / s = 4), p/q = 2 / s = 2), and the corresponding shifted integers of theform ( p + q ) /q . Moreover, for p = 1, and using the firstresult of Proposition 2, phases (36) take the form:Φ (1 /q ) n = ( − ne q exp( jσ φ πn /q ) , (44)which coincide, up to a global phase, with the series ofsolutions originally presented in [19]. The equivalenceis immediate for q even, and for q odd, and using theirnotation, it follows from the substitution n = I + ( q/ I half-integer ranging from − N/ N/ − i. e. , in the broadband spectrum of atrain of ultrashort pulses of period T , where the set ofspectral lines separated by a frequency difference of 1 /T plays the role of a diffraction grating in the usual con-figuration of the Talbot effect. In order to induce thespectral Talbot effect it is thus necessary to generate aquadratic phase transformation in the reciprocal domain,in this case in time. Such a transformation is therefore aquadratic phase modulation of the form exp( jαt /
2) forcertain values of the chirp parameter α . For ultrashortpulses, which are essentially located at definite temporalinstants t n = nT , the spectral Talbot effect is generated exp( − j ξ ) exp( − j ξ )exp( − j ξ ) m = 0 m = 1 m = 2 exp(j ξ )= exp(j ξ − )Talbot phasesexp(j ξ )= exp(j ξ − )exp(j ξ )= exp(j ξ − )exp( − j ξ n )Multilevel phases time exp( − j ξ ) exp( − j ξ ) exp( − j ξ )exp( − j ξ ) exp( − j ξ ) exp( − j ξ ) exp( − j ξ )exp( − j ξ )exp( − j ξ ) w ( t − T /3) w ( t ) w ( t − T /3) 0 T FIG. 3. Scheme of the multiple interference in time bins in a Talbot array illuminator with p/q = 1 / σ φ = +1. Theunit cell is contained in interval [0 , T ) and divided in three time bins, shown with a gray background in the top row. Each bincontains the corresponding multilevel phase. The shifting (left) and phase weighting (right) of bins induced by Talbot effectis shown in successive rows, using the cyclic structure of bins inherited by the periodicity of the unit cell. The result of thebin-wise coherent sum is represented in the bottom. simply by setting α = ± π pq T − . Therefore, the effectcan be induced by the set of phases [9]:Φ ( p/q ) n = exp (cid:18) ± jπ pq n (cid:19) . (45)This sequence of phases has, in general, a period of 2 q ,contrary to (36) where the period is always q . This is be-cause in (45) ratio p/q , which is the order of the (spectral)Talbot effect, can contain two odd numbers. By contrast,ratio s/q in (36) is an irreducible fraction of integers withopposite parity, as shown by the third property of The-orem 1. Notice, however, that the two sets of phasescoincide at the self-dual integers s = p = q ± T /q [23], which pro-vide an alternative form of Chu’s construction [17]. Thisresult is thus sufficient when the illumination shows thissymmetry, as is the case of the TAI problem, but not inmore complex systems where this symmetry is broken,such as in Talbot lines with structured illumination.The TAI phases are written in [23] in terms of themodular inverse of p (mod q ), denoted as r : r = (cid:20) p (cid:21) q . (46)We first analyze the relationship between this r and our s . We notice that, although in some cases they coincide,these integers may differ. For instance, for p = 5 and q = 8 we have r = [1 / = 5 but s = [1 / = 13. Ingeneral, integers r and s are either equal (mod 2 q ) ordiffer by q , since the condition pr = 1 (mod q ) implieseither s = r (mod 2 q ) or s = r + q (mod 2 q ). For q even this means that, since s must be odd, r is also odd,independently of the concrete form of the relationship, s = r or s = r + q . For q odd integer s must be even,and so s = r + q is allowed only when r is odd, i.e. , s = r + qe r (mod 2 q ), since this assures that s , as afunction of r , is always even. We are now in position toexpress the connection: Proposition 5.
The TAI phases can be written as:exp (cid:18) jπ sq n (cid:19) ∼ exp (cid:18) jπ rq m (cid:19) ( q even)exp (cid:18) jπ sq n (cid:19) ∼ exp (cid:18) jπ rq m ( m − (cid:19) ( q odd) , (47)where ∼ stands for equality up to a global phase and apossible shift of index n . Proof.
We must analyze case by case. For q even, theequality is obvious if s = r , and for s = r + q we have:exp (cid:18) jπ sq n (cid:19) = exp (cid:18) jπ rq n (cid:19) ( − n = exp (cid:18) jπ rq n (cid:19) ( − nr = exp (cid:18) jπ rq n ( n − q ) (cid:19) = exp (cid:18) jπ rq (cid:16) n − q (cid:17) (cid:19) exp (cid:16) − jπ qr (cid:17) , (48)where in the second step we have used that r is odd.The equivalence up to a global phase follows after theshift m = n − ( q/ q odd we use the explicit formula derived above, s = r + qe r and compute for both cases. For r odd or s = r + q , and using the same algebra as in (48), we have:exp (cid:18) jπ sq n (cid:19) = exp (cid:18) jπ rq n ( n − q ) (cid:19) =exp (cid:18) jπ rq m ( m − (cid:19) exp (cid:18) − jπ q − (cid:19) (49)after the shift m = n − [( q − / r even or s = r we have:exp (cid:18) jπ sq n (cid:19) = exp (cid:18) jπ rq n (cid:19) = exp (cid:18) jπ rq n ( n − q ) (cid:19) , (50)because the additional term exp( − jπrn ) = ( − rn isunity since r is even. In this form, (50) is equivalentto (49) and leads again to (47). The proof is complete. VI. CONCLUSIONS
In this paper we have presented a novel and simplecomputation of the quadratic Gauss sums that define theTalbot effect. It is based on the properties of integer s , aparity-dependent modular inverse constructed from thetwo coprime numbers p and q that define a general frac-tional Talbot plane. The computation can be compactlydescribed in two cases depending on the parity of q , andshows that the DFT of a quadratic phase sequence is itselfa quadratic phase sequence. In addition, both membersof the DFT pair have perfect periodic autocorrelation.These results were exemplified by the study of Talbot ar-ray illuminators, where the perfect autocorrelation arisesas a multiple interference process in bins within the fun-damental period of the grating. Also, connection withprevious characterizations of TAIs have been provided.The results derived here represent a simple means forthe design and analysis of systems based on fractionalTalbot effect. ACKNOWLEDGMENTS
This paper was written during a stay at INRS-EMT,Montreal, Canada, funded by Generalitat Valenciana,Spain through a BEST/2016/281 grant, and evolved fromconversations with Reza Maram, Luis Romero Cort´es andJos´e Aza˜na, to whom I am indebted.This paper is dedicated to the memory of Profs. Mar´ıaVictoria P´erez, Carlos G´omez-Reino and Felipe Mateos,with whom I learnt the beauty of Talbot effect.
Appendix A: Some results from number theory
A basic result in number theory [26], [27], [28], some-times referred to as B´ezout’s lemma, states that giventwo integers, a and b , with greatest common divisor d ,the modular equation ax + by = n , is solvable for x and y if and only if integer n is multiple of d . In particular,if a and b are coprime, the result guarantees that thereexist solutions to the equation: ax + by = 1 . (A1)The pairs of solutions ( x , y ) of (A1) are not unique,and in fact determined mod b and a , respectively, since( x + mb, y − ma ) for any integer m is also a pair ofsolutions. It can be shown that this series exhaust thepossible pairs of solutions to (A1). Let us define x as thelowest positive integer that belongs to a pair of solutionsto (A1). x is non-zero and bounded, x < b . The modu-lar multiplicative inverse, or simply the modular inverseof a (mod b ), denoted as [1 /a ] b , is this unique integer x that verifies: a (cid:20) a (cid:21) b = 1 (mod b ) (A2)and lies in the range 1 ≤ [1 /a ] b ≤ b −
1. The sameconstruction can be applied to the modular inverse [1 /b ] a ,which in this case lies in the range 1 ≤ [1 /b ] a ≤ a − a and b co-prime, the equation ax + by = n admits solutions for anyinteger n . In particular, this means that the modularequation ax n = n (mod b ) is solvable for any n [28, p.31]. A complete set of solutions is given by x n = n [1 /a ] b (mod b ), including the case n = 0. If we set n in therange of residues mod b , 0 ≤ n ≤ b − x n to their residues mod b , we have thus de-fined a bijective map n (cid:11) x n within the residues mod b .This is the map that has been used to perform changesof variables of the type x → y = ax (mod b ) for a , b coprime in invariant expressions mod b in several partsof the paper.The Jacobi symbol (cid:0) ab (cid:1) of an arbitrary integer a andan odd and positive integer b , which are mutually prime,is defined as the product of the Legendre symbols of theprime factors of b . To be specific, if the decomposition of b in prime factors is b = p α · · · p α N N , then: (cid:16) ab (cid:17) = (cid:18) ap (cid:19) α · · · (cid:18) ap N (cid:19) α N (A3)where for prime p j , the Legendre symbol is: (cid:18) ap j (cid:19) = +1 if there exists an integer x such that x = a (mod p j ) − a/p j )equals +1 if and only if a is a quadratic residue mod p j ,0otherwise it is −
1. In particular, (1 /p j ) = +1 for any p j ,and ( a/p j ) = ( a (cid:48) /p j ) if a = a (cid:48) (mod p j ). Properties andvalues of the Legendre symbols can be consulted in [26],[27], [28] and in the second appendix of [13]. The Jacobisymbol (A3) verifies the same properties as the Legendresymbols, and also verifies that its value is +1 when a is aquadratic residue mod b . The converse, however, is notnecessarily true, since for a being a quadratic residue of b it is necessary that a is quadratic residue of all of itsprime factors p j , and so all factors in (A3) must be +1. Appendix B: Equivalence with alternativeexpressions of integer s The computation of exp( jξ n ) in [13], corrected in [14]and as presented in [15], is similar to (19) by with adifferent expression for s . We show here the equivalenceof the integer s in (7) and (8) with the residues mod 2 q of the three types of integers defined in [14], [15], heredenoted as s (cid:48) , by showing that they verify the conditionsstated in Theorem 1. In the first case we also show theconnection by explicit manipulations on s (cid:48) . (a) p odd and q odd. The integer is given by: s (cid:48) = 8 p (cid:20) (cid:21) q (cid:20) p (cid:21) q (mod 2 q ) . (B1)Using that [1 / q = ( q + 1) / s (cid:48) = 4 p (cid:20) p (cid:21) q (mod 2 q ) . (B2)Incidentally, we mention that the value s (cid:48) in the originalcomputation [13] was: s (cid:48) HB = 4 p (cid:20) p (cid:21) q (mod 2 q ) . (B3)Returning to (B2), we recall the definition of the modularinverse: 2 p (cid:20) p (cid:21) q = 1 (mod q ) , (B4)we can be further simplify (B2) to: s (cid:48) = 2 (cid:20) p (cid:21) q × [1 (mod q )] (mod 2 q )= 2 (cid:20) p (cid:21) q (mod 2 q ) , (B5)which coincides with (8).Alternatively, we can show the equivalence by a directcheck of the hypotheses of Theorem 1. First we rephrase(B4) as: 2 p (cid:20) p (cid:21) q = 1 + aq (B6) for certain integer a , which should be odd since q is oddand the left hand side of the equation is even. Then theproduct s (cid:48) p is: s (cid:48) p = (cid:32) p (cid:20) p (cid:21) q (cid:33) = (1+ aq ) = 1+ q (mod 2 q ) , (B7)which is the defining equation (6) in Theorem 1. Thissecond form of the equivalence is completed by observingin (B1) that the parity of s (cid:48) is even, and is thus oppositeto that of q . (b) p odd and q even. The integer is defined as: s (cid:48) = p (cid:20) p (cid:21) q (mod 2 q ) . (B8)Using the definition of modular inverse, p (cid:20) p (cid:21) q = 1 (mod q ) = 1 + bq (B9)for certain integer b , we have that the product s (cid:48) p is: s (cid:48) p = (1 + bq ) = 1 + 2 bq + ( bq ) = 1 (mod 2 q ) , (B10)since q is even. This is the defining equation (6) for q even. We finally notice from (B9) that [1 /p ] q must beodd, so s (cid:48) is odd and thus of opposite parity to q . (c) p even and q odd. The integer is: s (cid:48) = p (cid:20) p (cid:21) q (mod 2 q ) . (B11)The definition of modular inverse (B9) now implies thatinteger b is odd, since q is odd and p is even. Then, theproduct s (cid:48) p is: s (cid:48) p = (1 + bq ) = 1 + q (mod 2 q ) , (B12)which is again the defining equation (6) for q odd. Theproof is completed by noticing that s (cid:48) is now even since p is even. Appendix C: Computation of phase exp ( jξ ) We compute the Gauss sum (24), which leads to theresult (19). As in [13], the computation is reduced tothe application of the standard form of the Gauss sumfound in texts of number theory [27, p. 86]. This stan-dard form applies to sums of quadratic phases of the typeexp( j πan /b ) where a is an arbitrary integer and b isodd. In the case of q odd, we are in these conditionssince s is even. Then, e jξ = (cid:18) s/ q (cid:19) e jπ ( q − / = (cid:18) sq (cid:19) (cid:18) q (cid:19) e jπ ( q − / = (cid:18) sq (cid:19) e jπ ( q − / , (C1)1where in the second step we have used the numeratorproduct rule of Jacobi symbols and the value (2 /q ) =exp[ − jπ ( q − / p and q : (cid:18) sq (cid:19) (cid:18) pq (cid:19) = (cid:18) spq (cid:19) = (cid:18) qe q q (cid:19) = (cid:18) q (cid:19) = 1 , (C2)and so ( s/q ) = ( p/q ) and the first line of (19) is proved.For q even, s is odd and we have to invert the fractionin the exponential of (24). Using the reciprocity of theGauss sums [13], [14], [26] we are led to: e jξ = 1 √ q q − (cid:88) m =0 e − jπ sq m = e − jπ/ √ s q − (cid:88) m =0 e jπ qs m . (C3)The use of the standard form gives the first expression of(19) for q even: e jξ = (cid:16) qs (cid:17) e − jπs/ . (C4)Since the original Gauss sum in (C3) is invariant undershifts s → s + 2 q , so it is the right hand side of (C4), and therefore: (cid:18) qs + 2 qn (cid:19) = (cid:16) qs (cid:17) ( − qn/ . (C5)for any n ≥
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