Rakesh Kumar Singh, P.Senthilkumaran and Kehar Singh * Department of Physics Indian Institute of Technology Hauz Khas, New Delhi ABSTRACT

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1 distriution in the fol plne of lens illuminted y Fresnel diffrtion of singulr em; Effet of spheril errtion nd defousing Rkesh Kumr Singh, P.Senthilkumrn nd Kehr Singh * Deprtment of Physis Indin Institute of Tehnology Huz Khs, New Delhi 6 ABSTRACT Fresnel diffrtion of ortex infeted em (singulr em) is foused using lens suffering from spheril errtion nd defousing. The resulting intensity distriution in the fol plne of the lens hs een plotted nd its fetures disussed for two lues of the topologil hrge of the ortex nd for three lues of the Fresnel numer. Keywords: Singulr em, Diffrtion, Fresnel numer, Spheril errtion. INTRODUCTION A em whih possesses n isolted point where mplitude eomes zero nd phse undefined, is lled singulr em nd suh n isolted point is lled singulr point. The suffiient riterion for phse singulrity is tht sum of phse hnge round the singulr point must e integrl multiple of 2π. Wefront of singulr em is found to possess helil shpe or system of m helioids, nested on the sme xis. Here the integer m is topologil hrge [, 2]. Seprtion etween onsequent oils of helioids equls 2πm nd etween neighoring helioids 2π. Singulr ems with point ortex n e generted experimentlly y spirl phse grting or y omputer generted hologrms [2, 3]. Propgtion dynmis of optil orties in singulr ems hs een sujet of onsiderle interest reently. Optil orties find pplitions in mny fields suh s in optil sptil filtering, optil testing, optil tweezers, quntum ryptogrphy, ommunitions, high resolution spetrosopy, mirosopy, semiondutor ptterning, nd nonliner optis. In trpping nd mnipultion of smll ojets [8], em oers ertin distne etween perture nd fousing system. Therefore we he onsidered illumintion of lens, in presene of spheril errtion nd defousing, y Fresnel pttern of singulr em. We he onsidered singulr em with two lues of the topologil hrge nd geometry with Fresnel numers,, nd OPTICAL VORTEX Optil ortex is point phse defet. A em infeted y point ortex n e mthemtilly represented s U ( θ, z) = U exp( imθ ikz ) () The mplitude rition U in eq. () n e of ny form with the ondition tht the mplitude hs to e zero t the singulr point. The phse of the we front hs singulrity t point round whih the line integrl ψ. dl = ± 2mπ (2) ICOL-25 where Ψ = mθ must neessrily e n integrl multiple of 2π, for the phse to e single lued. This integer m is lled topologil hrge of the ortex. The totl topologil hrge of gien field onfigurtion is dynmilly onsered quntity euse under norml onditions orties pper nd dispper in pirs in n optil field. A ophsl we front, ontining ortex hs the form of one or multistrt helil surfe round the dislotion xis. The hndedness of the helil surfe deides the polrity of the topologil hrge []. * Corresponding uthor s E-mil:kehrs@physis.iitd.ernet.in Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA

2 3. DIFFRACTION OF A SINGULAR BEAM BY A CIRCULAR APERTURE In trpping nd mnipultion deies, singulr em is usully pssed through finite perture efore fousing. In our study we he used Fresnel-Kirhhoff diffrtion formul to lulte field distriution t different plnes [4-7]. Consider unit mplitude singulr em t z = s U ( θ,) = exp( imθ ) (3) Upon propgtion, the mplitude t the ortex ore eomes null due to the destrutie interferene of seondry wes t the ortex ore. One n lulte diffrted field of eq. (3) y irulr perture t z plne y using the integrl, 2π i N 2 2 U ( ρ, φ, z) = exp ( i kz) exp ( i ρ N ) U exp ( imθ ) exp ( i ρ N i Nρ ρ θ φ ρ dρ dθ π ) exp [ 2 os ( )] (4) In eq. (4) ρ, θ, nd ρ, φ re polr oordintes on the perture nd the osertion plne respetiely. Here ρ nd ρ re normlized y the rdius ' of the irulr perture. N, the Fresnel numer, is defined s N = π' 2 / λ z. Eqution (4) gies the omplex mplitude distriution in the Fresnel field of singulr em diffrted y the irulr perture. Nture nd profile of this diffrted field depends on the Fresnel numer N nd topologil hrge m. With n inrese in the lue of N, resultnt field eomes more osilltory nd Fresnel rings re osered in the intensity pttern. 4. FOCUSING BY LENS The em fter diffrtion t the irulr perture is inident on the lens of rdius ' nd the intensity profile is lulted in the fol plne (Fig. 3) y using integrl 5. Sine Fresnel diffrtion field hnges with N, the intensity profile in the fol plne depends on the position nd size of the lens. Field distriution in the fol plne is gien [7] y ' ' / 2π in f 2 ν U( ν, ϕ, f ) = exp( i kf) exp i U( ρ, φ, z)exp[ iν ρ os( φ ϕ)] ρ dρ dφ (5) π 4N f where N f = π' 2 / λ f nd ν = (2π / λ)(' / f) r f. Here r f is the rdil oordinte nd ν nd φ re polr oordintes in the fol plne of the lens. Cirulr perture Lens Fol plne z f Fig. 3: Singulr em diffrted y irulr perture nd foused using lens fflited with spheril errtion In presene of spheril errtion ( A ) nd defousing ( ), nd for ' =, field distriution on the fol plne is s Ad gien y 2π i N 2 f ν 4 2 U( ν, ϕ, f ) = exp( ikf ) exp( i ) U( ρ, φ, z) exp{ i2π ( As ρ + Ad ρ )} exp[ iν ρ os( φ ϕ)] ρ dρ dφ π 4N f (6) profile in the fol plne of n errtion-free lens illuminted y Fresnel field of irulr perture for different omintions of Fresnel numer nd topologil hrge m = nd 2 is shown in Figs.4 & 4 respetiely. In Fig. 4, ures, nd represent results for Fresnel numer N=,, nd 5 respetiely. These two figures re used s referene to ompre results of errted system with errtion free system for lens whih is illuminted y Fresnel field of the singulr em. In Figs. 5-, we he onsidered the se of errrted system (spheril errtion A s ) in presene of defousing for different omintions of Fresnel numer nd topologil hrge. In Figs. 5-7, defousing prmeter is A d =.5 nd the spheril errtion oeffiient A s ries s ().5 (). nd (). for different omintions of Fresnel numer nd topologil hrge m = nd 2. In Figs.8 -, we took A d =. nd ried A s s (). ().5 nd (). for different omintions of N nd m. ICOL-25 Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA

3 Fig. 4 s. rdil distne t the fol plne of lens illuminted y Fresnel field of singulr em: () N= () N= () N=5. Topologil hrge 4 m =, 4 m = Fig.5 s. rdil distne in the presene of spheril errtion () A s =.5 () A s =. & () A s =. in the presene of defousing (A d =.5) for N =. Topologil hrge 5 m =, 5 m = 2 ICOL-25 Fig.6 s. rdil distne in the presene of spheril errtion () A s =.5 () A s =. & () A s =. in the presene of defousing (A d =.5) for N =. Topologil hrge 6 m =, 6 m = 2 Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA

4 Fig.7 s. rdil distne in the presene of spheril errtion () A s =.5 () A s =. & () A s =. in the presene of defousing (A d =.5) for N = 5. Topologil hrge 7 m =, 7 m = Fig.8 s. rdil distne in the presene of spheril errtion () A s =. () A s =.5 & () A s =. in the presene of defousing (A d =.) for N =. Topologil hrge 8 m =, 8 m = 2 ICOL-25 Fig.9 s. rdil distne in presene of spheril errtion () A s =. () A s =.5 & () As =. in presene of defousing (A d =.) for N =. Topologil hrge 9 m =, 9 m = 2 Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA

5 Fig. s. rdil distne in presene of spheril errtion: () A s =. () A s =.5 & () A s =. in the presene of defousing (A d =.) for N=5. Topologil hrge m =, m = d Fig. s. rdil distne t the fol plne of n errtion free system in presene of defousing: () A d =.25 () A d =.5 () A d =.75 (d) A d =. for N =. Topologil hrge m =, m = 2 ICOL-25 Fig.2 s. rdil distne t the fol plne of n errtion free system in presene of defousing :() A d =.25 () A d =.5 () A d =.75 (d) A d =. for N=5. Topologil hrge 2 m =, 2 m = 2 Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA

6 profile for n errtion free system in the presene of defousing is shown in Figs. & 2 for N = nd 5 with m = nd 2. Cures (), (), (), nd (d) represent the lue of defousing oeffiient (A d ) =.25,.5,.75, nd. respetiely. 5. DISCUSSION In Fig. 4, the intensity profiles for different lues of N nd m re shown when the lens is errtion free. This figure seres s referene to study the effet of spheril errtion in the lens. As the spheril errtion oeffiient is inresed, there is spreding of right ring in ddition to the hnge in the size of the drk ore. It is possile to lne the spheril errtion y introduing pproprite defousing so tht the effet of spheril errtion n e minimized. This lning ours when A d = A s for n optil system [5] when the inident em hs well-defined phse distriution. We studied this ehior in the presene of optil orties nd omputed the intensity profile when oth spheril errtion nd defousing re present simultneously. Fig.5 shows tht suh lning is possile when A d =.5 nd A s = +.5 in se of m = (ure ). Results were lso otined for Fresnel numers nd 5, nd re presented in Figs. 6 nd 7. It is found tht for topologil hrge m = 2 nd Fresnel numer N =, sme type of lning is possile. We lulted results for A d =. for different N nd m nd results re gien in Figs.8-. From these figures it is ler tht the profiles tend towrds the errtion free se when A d = A s. From Figs. nd 2, it seems tht there is present fol shift in the se of Fresnel numer 5 for topologil hrge m = 2. Mgnitude of this shift is found to inrese with derese in the Fresnel numer nd n inrese in the topologil hrge. Also t lower N nd lrge m, lning of spheril errtion with defousing does not our for A d = A s. Further study of this ehior is under inestigtion. For numeril elution of the integrl, we used impulse response funtion method with due onsidertion of the windowing effet. 6. CONCLUSION In this pper we he plotted intensity profiles t the fol plne for errted system in the presene of defousing nd spheril errtion for different omintions of topologil hrge nd Fresnel numer. From these results we n see the hnge in intensity grdient t the fol plne s ffeted y the presene of errtion. Hene trpping pity of drk ore em (singulr em) will e ffeted y n errted system. We he shown tht spheril errtion lning is possile y pproprite defousing, for diffrtion of singulr ems, in some ses. ACKNOWLEDGEMENT Rkesh Kumr Singh thnkfully knowledges the finnil support s JRF (Sntion No. 9/86 (656) 23) from the Counil of Sientifi nd Industril Reserh Indi (CSIR). REFERENCES I.V.Bsistiy, M. S. Soskin nd M. V. Vsnetso, Optil wefront dislotions nd their properties Opt. Commun. 9, (995). 2 M.W.Beijersergen, R.P.C.Coerwinkel, M.Kristensen, J.P.Woerdmn, Helil wefront lser ems produed with spirl phse plte Opt. Commun. 2, (994). 3 N.R.Hekenerg, R.MDuff, C.P.Smith nd A.G.White, Genertion of optil phse singulrities y omputergenerted hologrms Opt.Lett. 7, (992). 4 J.W.Goodmn, Introdution to Fourier optis MGrw Hill, New York (968). 5 V.N.Mhjn, Zernike nnulr polynomils for imging systems with nnulr pupils J.Opt.So.Am.7, (98). 6 J.J.Stmnes, Wes in fol regions Adm Hillger (986). 7 M.Gu nd X. S. Gn, Fresnel diffrtion y irulr plne we with optil phse singulrities nd its effet on the intensity distriution in the fol plne of lens Optik, 5, 5-66 (997). 8 K.T. Ghgn nd G. A. Swrtzlnder, Optil ortex trpping of prtiles Opt. Lett. 2, (996). ICOL-25 Proeedings of Interntionl Conferene on Optis & Optoeletronis, 2-5 De. 25, IRDE, Dehrdun, INDIA