Pure Appl. Opt. 7 (1998) 1221 1230. Printed in the UK PII: S0963-9659(98)90684-5 Convergence criteria and optimization technique for beam moment G Gbur and P S Carney Department of Phyic and Atronomy and the Rocheter Theory Center for Optical Science and Engineering, Univerity of Rocheter, Rocheter, NY 14627, USA Received 13 January 1998, in final form 24 March 1998 Abtract. We conider the problem of the characterization of beam by moment of the field intenity in the aperture and it moment in the far field. The well known beam propagation factor, MP 2, i conidered. We give convergence criteria for thee factor and alo dicu a new approach to optimization of the even moment of the far-field intenity. 1. Introduction The beam propagation factor M 2 P = 4πσ xσ (1.1) i frequently ued a a imple meaure of laer beam quality 1]. In thi formula the variance of the intenity in the tranvere plane at the beam wait and the variance of the far-field intenity ditribution are denoted by σx 2 and σ 2, repectively. Thi factor ha been the ubject of much dicuion ince it wa firt introduced, mainly becaue of quetion of computational tability 2] and becaue it diverge in ome cae, i.e. the cae of diffraction from a hard-edge aperture. Attempt have been made to overcome thee problem by truncating the intenity ditribution in the far field or otherwie limiting the range of integration when calculating the moment 3, 4]. Some author 3, 5, 6] have propoed additional tandard involving higher-order moment. However, thee higherorder moment diverge in certain cae jut a the MP 2 factor doe. Apart from convergence ome author have conidered the problem of optimization of beam moment, ubject to certain contraint 10, 11]. We addre the quetion of convergence criteria for the beam propagation factor and for higher-order moment and dicu the problem of moment optimization. We will demontrate that the convergence criteria play a crucial role in the olution of the optimization problem. For implicity, we will retrict our dicuion to a beam with one tranvere degree of freedom, which we denote by x; imilar reult hold for two tranvere degree of freedom. The field in the aperture plane will be pecified by a function f(x)which vanihe identically outide the range a<x<b. We aume that f(x) i quare integrable and ha N derivative on thi open interval. 0963-9659/98/051221+10$19.50 c 1998 IOP Publihing Ltd 1221
1222 G Gbur and P S Carney 2. Convergence of the econd-order moment Apart from multiplicative factor, the field F() in the far zone i related to the field f(x) in the aperture by the expreion F() = f(x)e 2πix dx. (2.1) The variance of the intenity in the far field i given by the formula where σ 2 = 2 2 (2.2) 2 = N 2 F() 2 d (2.3) and N i the normalization factor defined by the formula N 1 = F() 2 d = We chooe our coordinate ytem uch that f(x) 2 dx. (2.4) =0. (2.5) The econd-order moment i then given by equation (2.3); it i thi quantity for which we wih to determine convergence criteria. Since f(x) i a bounded function of x which vanihe outide the interval a<x<band the endpoint of the interval form a et of meaure zero, the integral (2.1) i equivalent to b ε F() = lim f(x)e 2πix dx. (2.6) a+ε Integrating thi expreion repeatedly by part we obtain an aymptotic erie valid for large value of 8, 9]: N 1 F() lim (2πi) n 1 e 2πi(b ε) f (n) (b ε) e 2πi(a+ε) f (n) (a + ε) ] + o ( N) n=0 (2.7) where f (n) (x) = d n f(x )/dx n x =x i the nth derivative of f. Since F() i the Fourier tranform of a function of finite upport, it i bounded on any finite interval and thu the convergence of the econd moment of F() 2 i determined olely by the behaviour for large of the integrand in equation (2.3). From equation (2.7), it then follow that N 1 N 1 F() 2 lim (2πi) n m 2 ( 1) m+1 m=0 n=0 e 2πi(b ε) f (n) (b ε) e 2πi(a+ε) f (n) (a + ε) ] e 2πi(b ε) f (m) (b ε) e 2πi(a+ε) f (m) (a + ε) ] + o ( N). (2.8) The lowet-order term in negative power of in the integrand 2 F() 2 i given by the term m = n = 0: 2 F() 2] m=n=0 = lim(2π) 2 e 2πi(b ε) f(b ε) e 2πi(a+ε) f(a+ε) 2. (2.9)
Convergence criteria and optimization technique for beam moment 1223 Thi term i ocillatory and non-negative for large. It contribution to the integral (2.3) will therefore only converge if limf(b ε)] = 0 (2.10) and limf(a+ε)] = 0 (2.11) i.e. in order for the econd-order moment of the far-zone intenity to be finite the field mut approach zero continuouly at the edge of the aperture. If equation (2.10) and (2.11) are atified, the next non-zero term in the expanion (2.8) i the n = 1, m = 1 term. Thi term and all higher-order term in the integrand of equation (2.3) fall off a 2 or fater for large and therefore converge. Equation (2.10) and (2.11) contitute neceary and ufficient condition for the variance of the far-zone intenity, and therefore the MP 2 factor, to exit. 3. Convergence of higher even-order moment The preceding treatment may be extended to all moment of the form 2k = N 2k F() 2 d (3.1) where k i a poitive integer. We may again make ue of equation (2.8) and conider the n = m = k 1 term of the integrand of equation (3.1): 2k F() 2] n=m=k 1 = lim(2π) 2k e 2πi(b ε) f (k 1) (b ε) e 2πi(a+ε) f (k 1) (a + ε) 2. (3.2) Thi term, like that given by equation (2.9), i ocillatory and i non-negative for large value of : it will add a diverging contribution to equation (3.1) unle f (k 1) (b ε) ] = 0 (3.3) lim and lim f (k 1) (a + ε) ] = 0. (3.4) Conequently, for the 2kth-order moment to exit, the (k 1)t derivative of f(x) mut decreae continuouly to zero at the edge of the aperture. Clearly, all lower order term in mut alo vanih for the 2kth-order moment to converge. Thi i only poible if lim f (j) (b ε) ] = 0 (3.5) and lim f (j) (a + ε) ] = 0 (3.6) for all j<k. A in the cae of the econd moment, thee condition are alo ufficient. Hence a neceary and ufficient condition for the 2kth-order moment, 2k, to exit i that the aperture field and it firt k 1 derivative go continuouly to zero at the endpoint x = a and x = b. The zeroth-order moment i defined without normalization o that it i proportional to the energy of the beam, 0 F() 2 d = f(x) 2 dx.
1224 G Gbur and P S Carney 4. Convergence of the odd-order moment The odd-order moment are omewhat more difficult to deal with becaue a ymmetric function F() 2 will alway have vanihing odd-order moment, at leat in the ene of the principal value 7]. Let u define a new function A() by the expreion A() = F() 2 F( ) 2. (4.1) Then 2k+1 = N A() 2k+1 d. (4.2) 0 It can be hown by argument imilar to thoe preented above that A() ha an aymptotic expanion N 1 A() j=0 k j A jk () (2πi) j+k+2 (4.3) valid for large value of. The function A jk () are given by the expreion A kk () = ( 1) k 4I f (k) (a + ε) f (k) (b ε) ] in 2π(b a 2ε) (4.4) A jk () = ( 1) k+1 4I f (k) (a + ε) f (j) (b ε) + f (j) (a + ε) f (k) (b ε) ] in 2π(b a 2ε) for j k, even value of j + k (4.5) = ( 1) k+1{ 4iI f (k) (b ε) f (j) (b ε) + f (j) (a + ε) f (k) (a + ε) ] +4iI f (k) (a + ε) f (j) (b ε) + f (j) (a + ε) f (k) (b ε) ] co 2π(b a 2ε) } for odd value of j + k (4.6) and I denote the imaginary part. In order that a moment of order 2m + 1 exit, the term in the above expanion for which j + k 2m mut vanih order by order. Thi i a much le retrictive requirement than the condition for the even moment. For intance, the condition lim f (j) (a + ε) = lim f (j) (b ε) = 0 for j m (4.7) i ufficient for the moment of order 2m + 1 to exit, but it i not a neceary condition. Furthermore, if f(x) i real or complex ymmetric, all the odd moment will vanih identically. Some remark about convergence criteria eem appropriate here. It i not difficult to how, uing the method of the preceding ection, that dicontinuitie of the field or it derivative anywhere within the aperture lead to divergent even-order moment. Specifically, a dicontinuity in a derivative of order M of the aperture function lead to a divergence of the (2M +2)th-order moment of the far-zone intenity. Evidently, care hould be taken in uing moment to characterize beam where convergence i not guaranteed a priori. For intance, in ituation where only the field and it firt derivative need be continuou, one would expect that only the econd- and fourth-order moment will be ueful in characterization. 5. From convergence to optimization The problem of minimizing a ratio of moment, in our notation 2n / 2m, wa dicued by McCutchen 10]. Uing an approach baed on higher-dimenional generalization of the diffraction problem, McCutchen wa able to obtain optimal olution for cae where
Convergence criteria and optimization technique for beam moment 1225 n = m + 1. Aakura and Ueno 11] olved the problem of minimizing the econd-order moment while holding the total energy (the zeroth-order moment) contant. Uing a calculu of variation approach, we will conider here the general problem of minimizing 2n, while holding N other even-order moment contant. Thee N moment may be of leer or higher order than the quantity to be minimized. We note that the 2kth moment may be expreed in the form 2k ( ) 1 2k f = N (k) (x) 2 dx (5.1) 2π where N i defined in equation (2.4). Uing thi repreentation for the moment, we begin with a functional of the form J = { N f (n) (x) 2 + N λ k N f (m k ) (x) } 2 dx (5.2) with ioperimetric contraint equation 12] ( ) 1 2mk f (m N k ) (x) 2 dx = µ k. (5.3) 2π Here 2m k and µ k are the order and the value repectively of the kth contrained moment. We may rewrite the functional (5.2) a { N+1 ] 1 J = λ k f(x ) 2 dx f (mk) (x) }dx 2 (5.4) with λ N+1 1 m N+1 n. (5.5) We will aume f to be real; the final differential equation which we obtain are alo valid when f i complex. Taking the firt variation of the functional J defined in equation (5.4) we find that { N+1 δj = 2 λ k N 2m k f(x)(δf)+f (mk) (x)(δf ) (m k) ]} dx. (5.6) One might try to integrate expreion (5.6) by part, et δj equal to zero and thu obtain a differential equation for f. Becaue of the explicit appearance of (2m k) in equation (5.6), thi i only poible if the convergence criteria are atified for all even-order moment up to the highet-order even moment involved. More explicitly, if we et M = max{m k } (5.7) then the aperture field and firt M 1 derivative of the field mut go continuouly to zero at the edge of the aperture. Our optimization problem therefore require that an additional 2M contraint be atified, in the form of 2M boundary condition on the aperture field and it derivative. If thee condition are atified, we obtain the Euler Lagrange equation: N+1 ( 1) m k λ k f (2mk) (x) = 0 (5.8) k=0 where, for convenience, we have defined N+1 λ 0 λ k 2m k m 0 0. (5.9) See the firt footnote.
1226 G Gbur and P S Carney Equation (5.8) ha olution which depend upon 2M +N +2 parameter, 2M contant of integration and N +2 undetermined multiplier, λ k. Equation (5.3) and (5.5) provide N +1 contraint, while equation (5.9) provide another contraint. Thi i generally not enough to pecify all the free parameter. It i now neceary to invoke the 2M boundary condition given by equation (3.5) and (3.6), which mut be impoed to atify the convergence criteria. Thee condition form a et of 2M linear, homogeneou equation. It i clear from the preceding dicuion that the convergence criteria are important for olving the optimization problem. It hould be noted, though, that becaue our contraint are ioperimetric, the 2M + N + 2 equation for the 2M + N + 2 unknown do not form a linear ytem and thu the equation may not have a olution in the pace of N + 1 Lagrange multiplier. In the cae that no moment i pecified, N = 0, and a olution alway exit. However, one cannot olve a problem where moment are contrained to value le than ome minimum poible value. For intance, conider the problem of minimizing the fourth-order moment 4. If one pecifie the energy 0, then one cannot alo require that the econd-order moment 2 be le than the minimum value obtained by olving the variational problem to minimize the econd moment with the energy fixed. One poible olution i to weaken the contraint. We could eek to minimize the expreion 4 / 2, or perhap 4 2 without further contraint. Such minimization problem, involving minimization of product of the form N J = 2m k νk (5.10) (of which the McCutchen problem 10] i a pecial cae), can be olved uing method equivalent to thoe pecified in the above dicuion. The Euler Lagrange equation for uch a functional reult, after a long calculation, in a differential equation which i identical to the differential equation (5.8) found for the functional J, but with N = N + 1, and with the Lagrange multiplier given in term of the a yet undetermined moment: and λ k = (2π) 2m k ν k 2m k 1 (5.11) N λ 0 = ν k. (5.12) The ame boundary condition mut be applied and if thi till leave undetermined contant then we can eek the extrema of J in the pace of the undetermined multiplier {λ} by mean of the normal calculu of everal variable, λ J = 0. (5.13) The poible optimization problem of thi type are numerou and the appropriate choice of the functional J depend upon the intended application. 6. Some example We will now illutrate the method developed above by ome example. following cae we will aume that a = δand b = δ. In all of the
Convergence criteria and optimization technique for beam moment 1227 6.1. The econd-order moment We conider minimization of the econd-order moment, while holding the energy 0 = µ contant. We mut then olve the econd-order differential equation which reult from minimizing J = 2 + λ 0 (6.1) with the appropriate boundary condition, i.e. lim f(±δ ε) = 0 (6.2) and uing the energy contraint. The method outlined above reult in an optimum olution given by the expreion f(x)= ( ) πx µ/δ co (6.3) 2δ which agree with the olution found by Aakura and Ueno 11]. The reulting far field i given by the expreion F() = 4 µδ co 2πδ π(1 4δ)(1 + 4δ). (6.4) Uing thi reult we find the minimum value of the econd-order moment for a fixed energy i 2 min = 1 16δ. (6.5) 2 The value of the propagation factor for thi example i MP 2 = 1.136. 6.2. The ratio 4 / 2 The functional to be conidered here i J = 4 2 1. (6.6) With the ame interval a above and requiring that the olution atifie the convergence criteria, one can find that the aperture function which minimize thi quantity i given by the expreion ( ) πx f(x)=ɣco 2 (6.7) 2δ where Ɣ i an undetermined contant. The far field i then given by the expreion Ɣin 2πδ F() = (6.8) 2π(1 2δ)(1 + 2δ) and the functional J defined by equation (6.6) take the value J min = 1 (6.9) 4δ 2 and the propagation factor i MP 2 = 1.026. It hould be noted that thi function atifie our convergence criteria for the fourthorder moment, while the olution obtained in the previou ection only atifie the criteria for the econd-order moment and, in fact, diverge for the fourth-order moment. For comparion, in the idealized cae of a Gauian beam propagating from an infinite aperture, the propagation factor take on it minimum value of unity.
1228 G Gbur and P S Carney 6.3. The fourth-order moment The problem of minimizing the fourth-order moment 4 differ from thoe encountered in the previou two example in that it i not olvable by either the Aakura and Ueno method 11], or by the method outlined by McCutchen 10]. The functional appropriate to thi ituation i given by the expreion f J (x) 2 dx f ] = (2π) 4 f(x) 2 dx. (6.10) We find from equation (5.8) that in thi cae the aperture function mut atify the differential equation with the boundary condition f (4) (x) = λ f (x) (6.11) lim f(±δ ε) = lim f (±δ ε) = 0. (6.12) In olving thi ytem of equation we find olution of two baic form. Let u denote by λ n olution of the trancendental equation The function tan(λ n δ) = tanh(λ n δ). (6.13) φ n (x) = Ɣco(λ n δ) coh(λ n x) coh(λ n δ) co(λ n x)] (6.14) are then olution of the differential equation and atify the boundary condition (6.12). Let u denote by λ an olution of the trancendental equation The function tan(λ an δ) = tanh(λ an δ). (6.15) ψ n = Ɣinh(λ an δ) in(λ an x) in(λ an δ) inh(λ an x)] (6.16) are then alo olution of the differential equation and atify the boundary condition. Figure 1. The aperture function minimizing 2, 4 / 2 and 4 hown a long-broken, hort-broken and full curve, repectively.
Convergence criteria and optimization technique for beam moment 1229 Figure 2. The angular pectra reulting from the aperture function. Figure 3. The tail of the angular pectra reulting from the aperture function. We find that, in thi cae, J φ n ] = and that ( λn 2π ) 4 (6.17) J ψ n ] = ( λan 2π ) 4. (6.18) It can be een that λ 1 i the lowet eigenvalue and thu the true global (nontrivial) minimum correpond to the φ 1 olution which ha a propagation factor of M 2 P = 1.037. The three aperture function found in thi ection are hown in figure 1 and the correponding far field are hown in figure 2 and 3. The function have been normalized o that in all three cae the energy i 0 = δ.
1230 G Gbur and P S Carney Acknowledgment The author would like to thank Dr Emil Wolf for hi ueful uggetion and comment, and alo wih to congratulate him on reaching thi early miletone in hi career. We alo wih to thank Dr Pierre-André Bélanger and Dr Claude Paré for their helpful comment on thi work. Thi reearch wa upported by the Air Force Office of Scientific Reearch under grant F49620-96-10400 and F49620-97-1-0482 and by the Department of Energy under grant DE-FG02-90E12. Reference 1] Siegman A E 1990 New development in laer reonator Proc. SPIE ed D A Holme (Chicago, IL: SPIE) pp 2 14 2] Lawrence G N 1994 Propoed international tandard for laer-beam quality fall hort Laer Focu World July 109 14 3] Bélanger P, Champagne Y and Paré C 1994 Beam propagation factor of diffracted laer beam Opt. Commun. 105 233 42 4] Martinez-Herrero R and Mejia P M 1993 Second-order patial characterization of hard-edge diffracted beam Opt. Lett. 18 1669 71 5] Martinez-Herrero R, Majia P M, Sanchez M and Neira J L H 1992 Third- and fourth-order parametric characterization of partially coherent beam propagating through ABCD optical ytem Opt. Quantum Electron. 24 S1021 6 6] Weber H 1992 Propagation of higher-order intenity moment in quadratic-index media Opt. Quantum Electron. 24 S1027 49 7] Bremermann H 1965 Ditribution, Complex Variable and Fourier Tranform (Reading, MA: Addion- Weley) 8] Bleitein N and Handelman R A 1975 Aymptotic Expanion of Integral (New York: Holt, Rinehart and Winton) 9] Erdelyi A 1956 Aymptotic Expanion (New York: Dover) 10] McCutchen C W 1969 Two familie of apodization problem J. Opt. Soc. Am. 59 1163 71 11] Aakura T and Ueno T 1976 Apodization for minimizing the econd moment of the intenity ditribution in the Fraunhofer diffraction pattern Nouv. Rev. Opt. 7 199 203 12] Lanczo C 1986 The Variational Principle of Mechanic 4th edn (New York: Dover)