THE ABCD-HANKEL TRANSFORMATION IN TWO-DIMENSIONAL FREQUENCY-DOMAIN WITH POLAR COORDINATES

Transcription

1 Jue Phys. hem. News ( 9-34 PN THE BD-HNKEL TRNSFORMTION IN TWO-DIMENSIONL FREQUENY-DOMIN WITH POLR OORDINTES M. Ibchaikh,. Belafhal * Laboatoie de Physique Moléculaie, Dépatemet de Physique, B.P, Faculté des Scieces, Uivesité houaïb Doukkali, 4 El Jadida, Moocco * oespodig autho. Belafhal@ucd.ac.ma Received Novembe ; evised vesio accepted 6 pil bstact This wok is devoted to a theoetical study of the BD-Hakel tasfomatio. s the fequecydomai is vey impotat i optics, we ivestigate a ovel geeal ollis fomula i two dimesios with pola coodiates. We also ivestigate the diect elatioship betwee iput ad output spatial fequecy specta of a light field i this fame system i the case of a Fouie, Fesel ad factioal Fouie ( tasfomatios fo two families of beams: Bessel-Gaussia ad Bessel-modulated Gaussia beams. Keywods: BD-Hakel; Tasfomatio; ollis Fomula; Fouie; Fesel; Factioal Fouie.. Itoductio Oe of the most basic poblems i optics is the detemiatio of the popagatio chaacteistics of beam waves i a paaxial optical system. I 97, ollis [] has foud a impotat elatioship betwee iput ad output spatial fequecy specta of a light field though a optical system chaacteized by, B, ad D elemets of the ay tasfe matix of the system, which is called ollis fomula. We kow that the BD law [- 5] goves the popagatio ad tasfomatio of Gaussia, Hemite-Gaussia, Laguee-Gaussia, o-gaussia ad ospheical light beams though paaxial systems. Ealie, Zalevsky et al [6] have povided some aalytical tools o the BD-Bessel tasfomatio by extedig the -D tasfomatio to a -D oe, by studyig some special cases. I the same yea, Liu et al [7] have poposed a ollis fomula i fequecy-domai i two dimesios with atesia coodiates. This wok still available fo a class of beams that ca be essed i tems of atesia coodiates. Howeve, i ou kowledge the BD-Hakel tasfomatio, which is a Bessel tasfomatio, i two-dimesioal fequecydomai with pola coodiates has ot bee teated befoe. This study is vey impotat whe the popagatio of otatioal symmetic beams is i eed, which is the case fo Bessel-Gaussia ad Bessel-modulated Gaussia beams. s a follow up of pevious eseach, we will attempt to deduce the amplitude distibutio at the output plae of a paaxial optical system, chaacteized by BD matix, illumiated by a light field, i fequecy-domai with pola coodiates. O the othe had, we kow that the Hakel tasfomatio embaces seveal othe optical tasfomatios, so we detemie the coespodig ollis fomula fo seveal cases as Fouie, Fesel, ad RSOS tasfomatios.. ollis fomula i a two dimesioal space domai with the pola coodiates We peset i this sectio the basic theoy of the BD-Hakel tasfomatio deived fom the ollis fomula. Fo this, we coside a BD optical system illumiated by a light field epeseted by a tasvese amplitude distibutio u(x, y at the iput plae ad by u (x, y at the output plae. I this wok, we coside that the iput ad output plaes ae embedded ito the same optical medium. So, the detemiat D B of the coespodig BD matix should be uity. I a space domai, the geealized Huyges-Fesel itegal o the ollis fomula fo oe tasvesal diectio of a othogoal system elates the output field to the iput oe, ad the BD elemets ca be witte as [] iπ u (x, y = ' u(x, y λb [ (x + y + D(x + y (x x + yy ]} dxdy, ( i whee k is the wave umbe ad ' =. With λb pola coodiates, oe fids that the elatioship betwee ( ρ, ψ ad u (, ϕ is give by [7] u 9

2 3 M. Ibchaikh et al, Phys. hem. News ( 9-34 πd π π u (, ϕ = ' i u( ρ, ψ i ρ π i ρ cos( ϕ ψ ρ.. dψ. ( Pactically, we foud the followig essio of the iput amplitude distibutio [8] iψ u ( ρ, ψ = i χ( ρ e, (3 whee is a itege. By usig the well-kow idetity π π iϕ π iψ (4 πj ρi e = i ρ cos( ψ ϕ e dψ λb Eq. ( becomes πd u (, ϕ = π' i + ( ϕ + π λb π H χ( ρ i ρ, (5 whee π π H χ( ρ i ρ = χ( ρ π i ρ ρ., (6 π is the Hakel tasfom of χ( ρ i ρ. If we apply the Paseval equality [8], with > / * * uf (ug (udu = tf (tg(tdt, (7 i o Eq. (, oe obtais the ' = λb essio, which isues the eegy cosevatio. We ca apply Eq. (5 o seveal tasfomatios embaced by the BD-Hakel. The defiitios of the special cases of the BD-Hakel tasfomatio as the Fouie, the Fesel ad the tasfomatios ae listed i Table. I the case of the tasfom, which is i fact a extesio of the covetioal Fouie tasfomatio to the factioal ode, f is a scalig facto ad the agle φ is elated to the factioal idex p by φ = p π. Except i the case whee the phase shift is φ /, this tasfom descibes, i paaxial appoximatio of the diffactio theoy, the evolutio of the complex field amplitude duig popagatio though a quadatic efactive idex medium [9]. The esult coespodig to each tasfomatio is give i Table. oditio Fouie = D =, = / f ad B = f Fesel B = z, = ad = D = B = f si φ, = si φ / f ad = D = cosφ Table : Defiitios of the thee special cases of the BD-Hakel tasfomatio. s a example of the pevious esults, we will ow coside i detail the impotat special case of a otatioal symmetic iput object chaacteized by = ad χ ( ρ = F ( ρ. The above equatios yield the esults summaized i Table 3 which ae the same as those give i Ref. [6] fo the thee tasfomatios. Fouie Fesel u (, ϕ π i i[ ( ϕ + π ] π χ( ρρ. π π i i + ϕ + π λz λz χ( π π ρ i ρ λz λz ρ. π π i i + ( ϕ + π si φ tgφ π π χ( ρ i ρ ρ. si φ tgφ Table : The tasvese amplitude distibutio u (x, y at the output plae fo the thee cosideed cases. I the followig, we will deduce a fomula, which descibes the elatioship betwee the iput ad output agle specta of a paaxial optical system. O the othe had, we will focus ou attetio o the above tasfomatios i the cotext of two-dimesioal fuctios, descibed i pola coodiates. PN

3 M. Ibchaikh et al, Phys. hem. News ( Fouie Fesel F u (, ϕ π π i F ( ρρ. π π i i λz λz π π F ( ρ i ρ ρ. λz λz π π i i si φ tgφ π π F ( ρ i ρ ρ. si φ tgφ Table 3: Like Table, but fo a otatioal symmetic iput beam. 3. Deivatio of BD law i the fequecy space i tem of pola coodiates We kow that, i this domai, the iput ad output agle specta ae espectively give by ( x, y = u (x, y [ iπ( x + y ] dxdy x y, (8a ad ( x, y = u (x, y. (8b [ iπ( x x + y y ] dxdy These agle specta ae eliable by ollis fomula i fequecy domai by [7] λ πλ (, = i. (, i. x y x y [ D( x + y + ( x + y ( x x + y y ] d x d (9 y I a fist step, we will deive the essio of the iput agula spectum i pola coodiates. Fo that, we defie these coodiates as = cos θ x = si θ y, (a ad x = ρ cos ψ. y = ρsi ψ (b With this chage of vaiables, Eq. (8a becomes π (, = u( ρ, ψ, ( x y [ iπ ρ cos( ψ θ ] ρ. dψ which ca be witte, with the help of Eq. (3 ad Eq. (4, as (, θ i ' i = χ θ, ( whee χ' ( = πi χ( ( π ρ ρ.. (3 The secod step cosist of the is to evaluatio of (, θ i fequecy domai i pola coodiates by the use of Eq. (9 ad Eq. (a. With the help of the followig vaiable chage x = cos θ = si θ y, (4 Eq. (9 ca be witte as λ π (, θ = i (, θ πλ i [ D + cos( θ θ ].d.dθ (5 The use of Eq. (4 ad Eq. ( yields (, θ λ λ = iπ iπ πλd ( iθ H χ' i, (6 whee πλd H χ' ( i = πλd πλ χ '( i J.d. (7 (, θ Fouie + ( iπ ( iθ χ' ( J ( π.d Fesel ( = [ ( θ z ] πλ ( π ρ ρ. π( i χ( π ( iπ i si φ tgφ π χ'( i tgφ J π si φ i θ.d Table 4: The output agle specta i the fequecy-domai fo some special cases. PN

4 3 M. Ibchaikh et al, Phys. hem. News ( 9-34 I Table 4, we summaize the diffeet elatioships of seveal tasfomatios. Note that Eq. (6 is valid fo ay beam chaacteized by the adial fuctio χ ad ca be used to study i the fequecy-domai ay tasfomatio of a cosideed beam though a BD axissymmetic optical system. 4. pplicatios I this sectio, we will discuss the popagatio i the fequecy-domai of Bessel-Gaussia ad Q light beams ad povide ew fomulas of some special cases as Fouie, Fesel ad tasfomatios. So, Eq. (6 ca be witte as λ λ iθ (, i i e θ = π π whee δ χ' ( e J ( β d, (8 πλ = β, (9a ad πλd = i δ. (9b Eq. (8 is available fo. If = ad by usig the followig simple iput-output elatioship give i Ref. [7], (a ( x, y iπλb = D D x + y, x y D D oe obtais with the same pocedue as above (, θ = π( i θ πλz χ (π [ ] ρ ρ (. (b I the followig, we will apply Eq. (8 i the case of the two families of beams: Bessel-Gauss ad Q. 4. Bessel-Gauss beams This family of beams ca be essed i tems of the Bessel fuctio J of the fist kid ad th ode [-]. The fuctio χ, fo these beams, is defied by χ ( ρ = J ( αρ ( u ρ, (a whee u =, (b ω α = k si θ, (c ad ω = z R / k. (d I these equatios, k = π / λ is the wave umbe of the field ad the paametes θ, z ad ω ae espectively the coe agle of the ideal o-apodized Bessel field (i the paaxial appoximatio, the Rayleigh age ad the spot size of the fudametal Gaussia mode. If ω oe obtais a pue Bessel field ad if α a odiay Gaussia beam is established. With the help of Eq of Ref. [], oe obtais χ π α + ξ αξ ' = ( i I, ( u 4u u whee I is the modified Bessel fuctio of ode give by I (x = ( i J (ix ad ξ = π. So, Eq. (8 becomes (, θ = ( θ ( R J (h R, (3 whee λω ( θ = iπ Γ, (4a π α ω ω π θ i( 4 Γ πλ πλ = + i Γ R, (4b ad h with π αω λ Γ =, (4c D = π ω + i πλ Γ. (4d To get Eq. (3, we have used Eq of Ref. []. Fo =, by the use of Eq. (b oe obtais α ω θ = πω θ (, i 4 [ ( π ω + iπλz ] I ( παω. (4e We give i Table 5, the essios of (, θ i the thee cases: Fouie, Fesel, ad RSOS tasfomatios. PN

5 M. Ibchaikh et al, Phys. hem. News ( Fouie Fesel ( = [ ] (, θ π i iθ + λ f J ( f αλ ω π ( ω α ω iθ 4 ( iπλz + π ω I παω λωf iπ Γ si φ α ω π ω θ x i 4 Γ iπ π λ f tgφ Γ si φ π αω λ f xj Γ si φ Table 5: The output agle specta i the fequecydomai fo the thee tasfomatios i the case of a Bessel-Gauss beam (with Γ = π ω iπ. tgφ Fom Eq. (3, we ca deduce that the output fields i the fequecy-domai have the same stuctue as the iput field. So, i this domai the studied family of beams costitutes a class of fields whose fom is ivaiat afte popagatio i a paaxial optical system defied by a, B, ad D elemets. O the othe had, the esults of Table 5 ae still accuate i the case of a geealized Bessel- Gauss beam, which is chaacteized by [3] a u [( β aiu ] χ( ρ = e ( u J, (5 whee a is a positive paamete, β is also a positive paamete but smalle tha the wave umbe k ad the agumet of the Bessel fuctio is a complex. So, if we eplace i the esults peseted i this sectio α by β aiu, oe obtais the esults coespodig to the geealized Bessel-Gauss beam. 4. Q beams This class is simila to the Bessel-Gauss beams but the Bessel fuctio agumet is quadatic i the tasvese coodiate [4]. So fa, this family of beams will be efeed to as Q beams. I this case, the iput amplitude is give by χ ( ρ = J ( αρ u ρ, (6 ad χ ' is obtaied, by the use of Eq of Ref. [], as π π u χ' ( = ( i 4 4 α + u α + u π α J. (7 4 α + u Eq. (8 becomes i this case Q (, θ = Q ( θ ( R J ( h whee R ad, (8 Q θ Q h Q with = π λ = 4αω s = Q 4iπ λω 4 ( η + τ ( + α ω iθ Q π, (9a η πλ = h Q + i, (9b τ τε =, (9c η + τ ε, (3 τ, (3 λd η = 4 s + iπ, (3 whee π ω s =. ( α ω The essios of the impotat paametes ε ad η which ae eeded to evaluate Q (, θ fo Fouie, ad RSOS tasfomatios ae: ( ε, η = ( π,4s, π π,4(s i, si φ tgφ ad π π,4(s i, espectively. Eq. ' si φ ' tgφ (3 is idepedet of tasfomatio paametes. Eq. (8 is available fo. I the case of =, Eq. (4e ca be witte as Q (, θ i( z = π θ πλ, (34 whee = ω s π Q, (35a PN

6 34 M. Ibchaikh et al, Phys. hem. News ( 9-34 Q αω R Q = s, ad h s =. (35b s fo the Bessel-Gaussia beams, we deduce that the output fields i the fequecy-domai of the Bessel-modulated Gaussia beams have the same stuctue as the iput field. So, this family of beams costitutes also a class of fields, i the fequecy-domai, whose fom is ivaiat afte popagatio i a paaxial optical system chaacteized by a, B, ad D elemets. 5. oclusio I summay, we have studied the BD- Hakel tasfomatio i two-dimesioal fequecy-domai with pola coodiates. Ou fomulas show that the agle spectum popagatio obeys to a ovel ollis fomula essed i the fequecy-domai, which is simila to that of the complex amplitude popagatio. We ca deduce that the output fields i the fequecy-domai have the same stuctue as the iput Bessel-Gaussia ad Besselmodulated Gaussia fields. O the othe had, we have show that the studied families of beams costitute a class of fields whose fom is ivaiat afte popagatio i a paaxial optical system chaacteized by a, B, ad D elemets. Statig fom the ovel ollis fomula, some tasfomatios ae aalyzed: fee space popagatio, Fouie, Fesel ad tasfomatios i cases: Bessel-Gaussia ad Q. Refeeces [] S.. ollis, J. Opt. Soc. m., 6 (9768. [] H. Kogelik, Bell Syst. Tech., J. 44 ( [3] H. Kogelik, Poc. IEEE, 54 ( [4] J. P. Taché, ppl. Opt., 6 ( [5]. Poas, J. lda, E. Beabeu, ppl. Opt., 3 ( [6] Z. Zalevsky, D. Medlovic,. W. Lohma, Optics omm., 47 ( [7] Z. Liu, X. Wu, D. Fa, Optics omm., 55 ( [8] V. Ditkie,. Poudikov, Tasfomatios Itégales et alcul Opéatioel, Ed. Mi, Moscou (978. [9]. W. Lohma, J. Opt. Soc. m., ( [] I. S. Gadshtey, I. M. Ryzhik, Tables of Itegals, Seies, ad Poducts, 5th Ed., cademic Pess, New Yok, 994. [] F. Goi, G. Guattai,. Padovai, Optics. omm., 64 ( [] V. Bagii, F. Fezza, M. Satasieo, G. Schettii, G. S. Spagolo, J. Mod. Optics, 43 ( [3] M. Satasieo, Optics. omm., 3 (996. [4]. F. R. ao, R. M. Potvliege, Optics. omm., 64 ( PN