ol-37-14-2970 Light sources generating far fields

Light sources generating far fieldswith tunable flat profilesSerkan Sahin and Olga Korotkova*Department of Physics,University of Miami,1320Campo Sano Drive,Coral Gables,Florida33146,USA*Corresponding author:korotkova@Received March26,2012;revised May28,2012;accepted May29,2012;posted May30,2012(Doc.ID165410);published July13,2012Planar,scalar,optical Schell-model,and quasi-homogeneous sources with correlations that are Fourier transformsof multi-Gaussian functions are introduced.It is demonstrated that far fields produced by these families of sourcescarry interesting characteristics,being flatlike with adjustable steepness of the edge.Beam conditions for suchsources are also derived.©2012Optical Society of AmericaOCIS codes:030.1640,030.6600,260.0260.It is well known in statistical optics that the structure ofthe correlation function of a field in the source plane isclosely related to the intensity distribution of its far field([1],p.236).Moreover,for some model sources,it is pos-sible to obtain analytic relations between these two quan-tities.For instance,for quasi-homogeneous sources theso-called reciprocity relations are known to be of Fouriertype([1],p.243).For model sources of the Schell type[2],on the other hand,it is possible to obtain formulas for themajor statistical characteristics of the field propagatingto any distance from the source and to analyze howsource correlations gradually affect the field([1],p.276).Surprisingly few analytical models for the planarsource correlation functions have been developed so far:the Gaussian Schell-model(GSM)sources,the J n-Bessel-correlated sources[3],the I n-Bessel-correlated sources[4],the nonuniform GSM sources[5],and the Lambertiansources([1],p.248)might constitute the full list of mod-els.The difficulty in developing new models stems fromthe fact that a genuine correlation function must satisfy anumber of restrictions([1],Subsection4.7.1).To alleviatethis task,a sufficient condition in a simple integral formfor the genuine correlation function was derived in[6](see also[7])and has already led to a model in[5].In this Letter we introduce another model for the cor-relation function of a planar source that is based on themulti-Gaussian family of functions.Such functions havepreviously been employed for modeling of beams[8]andscattering potentials[9].Generally,the multi-Gaussianfunctions make it possible to control the width of the flatcenter of the beam intensity profile and the slope of itsedge by the choice of two parameters.But being employedfor modeling of the correlation,rather than for the field it-self,the multi-Gaussian function can serve as an importanttool for generating far fields with flat intensity profiles.Let us set the spectral degree of coherenceμ ρ1;ρ2;ωat a pair of points in the source plane with position vec-torsρ1andρ2and at the oscillation angular frequencyωin the form(in the following the dependence of all fieldstatistics on the angular frequencyωis implied butomitted for brevity)μ ρ1;ρ2;ω 1C0X Mm 1Mm−1 m−1m exp−jρ2−ρ1j22;(1)where C0PMm 1−1 m−1mMmis the normalization factor,Mmstand for binomial coefficients,andδis a typical cor-relation width.As illustrated in Fig.1,the profile functiondefined by Eq.(1)visually resembles a Bessel-correlatedsource or a Lambertian source(see[1],Figs.5.9and5.10);however,it is given by a different functional form.As in the case of a Bessel function whose Taylor’s expan-sion is a sum of sign-alternating terms,the modifiedmulti-Gaussian function is also represented by a sumof positive and negative exponentials.As is evident from[1,6],not any degree of coherencedefines a physically meaningful random source.For in-stance,for a Schell-model source,for which the cross-spectral density function has the formW 0 ρ1;ρ2S ρ1pS ρ2pμ ρ2−ρ1 ;(2)S ρ being the spectral density atρ,the sufficient condi-tion for the cross-spectral density to be genuine is that itmust be expressed by the integralW 0 ρ1;ρ2Zp υ H ρ1;υ H ρ2;υ d2υ;(3)- 4- 1140.810.60.40.2µ|ρ −ρ |δ12Fig.1.(Color online)Illustration of the degree of coherencecalculated from Eq.(1)as a function of the nondimensionalparameter jρ2−ρ1j∕δfor several values of M:M 1(solidcurve);M 4(dashed curve);M 10(dotted curve),andM 40(dotted–dashed curve).2970OPTICS LETTERS/Vol.37,No.14/July15,20120146-9592/12/142970-03$15.00/0©2012Optical Society of Americawhere H ρ;υ is an arbitrary kernel and p υ is a non-negative,Fourier-transformable function.Following [6]we assume that function H ρ;υ has the formH ρ;υ τ ρ exp −i υ·ρ ;(4)and hence W 0 becomesW 0 ρ1;ρ2 τ ρ1 τ ρ2 ~pρ1−ρ2 ;(5)where τ ρ is a (possibly complex)profile function and the tilde denotes the Fourier transform.The choice of p υ defines a family of sources with dif-ferent correlation functions.On taking the Fourier trans-form of Eq.(1)we arrive atp υδ20X M m 1 −1 m −1 Mm exp −m δ2j υj 2 ;(6)representing a family of flat-top profiles (see Fig.2).While it is a Fourier-transformable function by con-struction,its nonnegativity can be proven as follows.Equation (6)can be represented asp υ −δ2C 0XM m 1 M m−x mδ21− 1−x M ;(7)where x exp −δ2j υj 2∕2 .Function p υ is manifestly nonnegative since 0≤x ≤1.We point out that this form of p υ is particularly simple,since it is clear that 1−x is a saturation function whose slope is controlled by index M .Let us also set the Gaussian profile for function τ:τ ρ exp −j ρj 2∕ 4σ2 :(8)Then,together with the weighting function p υ given by Eq.(6),we obtain on substituting them into Eq.(5)the cross-spectral density function of the formW 0 ρ1;ρ2 10exp −j ρ1j 2 j ρ2j 2×X M m 1 −1 m −1m M m exp −j ρ2−ρ1j 22 ;(9)which may be called multi-Gaussian Schell-model source.We will now determine the cross-spectral density func-tion of the field radiated by the source Eq.(9)to the far zone at two points specified by position vectors r 1 r 1s 1and r 2 r 2s 2,with s 21 s 22 1.The field in the far zone of the source is given by the expression ([1],p.240)W ∞ r 1s 1;r 2s 2 2πk 2cos θ1cos θ2×~W 0 −k s 1⊥;k s 2⊥exp ik r 2−r 1 r 1r 2;(10)where k 2π∕λis the wavenumber of the field,λbeing the wavelength,s ⊥is the projection of s onto the source plane,s ⊥ sin θ,s z cos θ,and~W 0 f 1;f 2 14Z Z W 0 ρ1;ρ2 ×exp −i f 1·ρ1 f 2·ρ2 d 2ρ1d 2ρ2(11)is the four-dimensional Fourier transform.On substitut-ing Eq.(9)first into Eq.(11)and then into Eq.(10),weobtain for the cross-spectral density W ∞ r 1s 1;r 2s 2 in the far field the formula W ∞ r 1;r 21C 0k 2cos θ1cos θ2exp ik r 2−r 1 r 1r 2×X M m 1−1 m −1 M m 1a m −b m ×exp −k 2 αm s 21⊥ αm s 22⊥−2βm s 1⊥·s 2⊥ ;(12)wherea m 1 1212;b m 12;(13)αma m2m 2m ;βmb m2m 2m :(14)The far-field spectral density can be found by the formulaS ∞ r W ∞ r ;r ,and one obtains S ∞ rk 2cos 2θC 0j r j 2×X M m 1−1 m −1 M m exp −2k 2s 2⊥ αm−βm 2m 2m :(15)In order for the function W 0 ρ1;ρ2 to generate abeam,the spectral density in Eq.(15)must be negligible-4-2240.810.60.40.2δ|υ|pFig.2.(Color online)Illustration of the function p calculated from Eq.(6)as a function of nondimensional parameter δj υj for the same values of M as in Fig.1.July 15,2012/Vol.37,No.14/OPTICS LETTERS 2971except for directions within a narrow solid angle about the z axis.This is so if ([1],Eq.[5.6-72])exp −2k 2s 2⊥θ2 αm −βm ≈0(16)for any m 1;…;M ,unless s 2⊥≪1,implying that 2k 2 αm −βm ≪1;17 or,in terms of the source parameters,12 112≪2π22;m 1;…;M:(18)If the inequality holds for m 1,the rest of the set,for m 2;…;M holds automatically.Thus,the beam condi-tion for the multi-Gaussian Schell-model sources is the same as that for the GSM sources ([1],Eq.5.6-73):1 1≪2π2:19Figure 3shows several typical far fields radiated by thesource in Eq.(9).It is clearly seen that a beam with a Gaussian degree of coherence (M 1)in the source plane monotonically decreases with the increase of angle θ,while beams with multi-Gaussian source correlations have flat profiles with different heights and steepness of the edges.Another possible class of sources with the correlation function of the form Eq.(1)can be introduced with thehelp of a quasi-homogeneous ly,the cross-spectral density of such a source is of the formWρ1;ρ2 ≈Sρ1 ρ2μ ρ1−ρ2 ≈Sρ1 ρ22~p ρ1−ρ2 ;(20)under the assumption that function S is a slow function of its argument compared to μ.For this type of field,the far-field spectral density is given by ([1],p.243)S ∞ r12 2πk 2cos 2θ~F 0 ~μ k s ⊥ :(21)We also note that,since the multi-Gaussian Schell-model and multi-Gaussian quasi-homogeneous sourceshave the same correlation function of type Eq.(1),their far fields are qualitatively similar for M >1,especially if δ≪σ.In conclusion,we have introduced a family of func-tions whose Fourier transforms are multi-Gaussian functions for the source degree of coherence and have employed them in the Schell-model and the quasi-homogeneous model.We have confirmed that such sources are physically genuine,determined the far fields they generate,and showed that far fields have desirable features representing flat profiles that are useful in beam shaping,optical trapping,and tweezers.The novel family of correlation functions,while being visually similar to J n -Bessel correlations,result in qualitatively different far-field patterns,possessing a flat intensity profile in the central part with a steep,adjustable drop at its edge.We are indebted to Professor E.Shchepakina and the reviewer for their help in preparation of this Letter.O.Korotkova ’s research is supported by United States Office of Naval Research grant N00189-12-T-0136.References1.L.Mandel and E.Wolf,Optical Coherence and Quantum Optics (Cambridge University,1995).2.A.C.Schell,“The multiple plate antenna,”doctoral disser-tation (MIT,1961),Subsection 7.5.3.F.Gori,G.Guattari,and C.Padovani,mun.64,311(1987).4.S.A.Ponomarenko,J.Opt.Soc.Am.A 18,150(2001).junen and T.Saastamoinen,Opt.Lett.36,4104(2011).6.F.Gori and M.Santarsiero,Opt.Lett.32,3531(2007).7.R.Martinez-Herrero,P.M.Mejias,and F.Gori,Opt.Lett.34,1399(2009).8.F.Gori,mun.107,335(1994).9.S.Sahin,G.Gbur,and O.Korotkova,Opt.Lett.36,3957(2011).-0.004-0.0020.0020.00410.80.60.40.2θ [rad ]S (θ)οο( )Fig.3.(Color online)Far-field spectral density generated by a typical multi-Gaussian Schell-model source,calculated from Eq.(15),as a function of angle θ(in radians),for the same four values of M as in Fig.1and λ 632nm,σ 1mm,δ 0.1mm.2972OPTICS LETTERS /Vol.37,No.14/July 15,2012。