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Rotation of Multimode Gauss–Laguerre Light Beams in Free Space V

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Rotation of Multimode Gauss–Laguerre Light Beams in Free Space V Rotation of multimode Gauss–Laguerre light beams in free space V. V. Kotlyar, V. A. So fer, and S. N. Khonina Institute of Image Processing Systems, Russian Academy of Sciences, Samara ~Submitted November 18, 1996! Pis’ma Zh. Tekh. Fiz. 23, 1–6 ~September 12, 1997! An iterative algorithm is presented for calculating diffractive phase optical elements that form light beams which are an effective superposition of a small number of nonradially symmetric Gauss–Laguerre modes with a prescribed energy contribution from each mode. © 1997 American Institute of Physics. @S1063-7850~97!00109-2# 1/2 An important current problem is the development of dif- 2~m2n!! A2r n 2r2 fractive optical elements that are matched with light modes V6 ~r,w!5 Ln mn 2 3 m 2 Fps ~m!! G S s0 D S s D of one type or another. Such diffractive optical elements find 0 0 applications in the problems of parallel injection of radiation r2 into a fiber bundle, selection of transverse laser modes, and 3exp 2 6inw . ~4! formation of nondiffracting beams.1 S s2 D 0 Iterative methods exist for calculating diffractive phase optical elements capable of forming light beams with a An iterative algorithm for calculating the parameters of a prescribed composition of Gauss–Hermite modes,2 diffractive phase optical element forming the complex am- Gauss–Hermite3 and Gauss–Laguerre4 modes in different plitude ~3! has the form diffraction orders, and Bessel modes ~nondiffracting 5 ` beams!. Only radially symmetric Gauss–Laguerre modes 6 ~k! Sk11~r,w!5arg ( BmnVmn~r,w!exp@ivmn# , ~5! are studied in Ref. 4. H m,n50 J The conditions under which rotation of a multimode beam around the propagation axis is observed have also been ` 2p ~k! found, and an expression has been obtained for the total vmn5arg A0~r,w!exp@iSk~r,w!# H E0 E0 number of revolutions. It is known6 that light fields which are a superposition of 6* Gauss–Laguerre modes satisfy the Helmholtz equation. The 3Vmn ~r,w!rdrdwJ, ~6! complex amplitude of such fields in free space can be repre- sented in cylindrical coordinates (r,w,z)as where A0(r,w)5uU0(r,w)u is the amplitude of the illuminat- ing beam, S (r,w) is the phase of the diffractive optical el- 2 2 k ikr r ement calculated at the kth iteration, and B >0 are given U~r,w,z!5exp ikz1 2 mn F 2R s2G numbers which specify the energy contribution of the corre- sponding mode. ` A2r n 2r2 Next, we shall obtain the condition for observing rota- 3 C Ln ( mn m 2 tion of the transverse section of a multimode beam. On the m,n50 S s D S s D basis of Eq. ~1!, we write an expression for the intensity 2 3exp@2ibmn~z!6inw#, ~1! I(r,w,z)5uU(r,w,z)u where ` 2n 2 2r2 A2r 2r2 I~r,w,z!5exp 2 uC u2 Ln 21 2 ( mn s m 2 bmn~z!5~2m1n11!tan ~z/z0!, ~2! F s GH m,n50 S D U S s DU 2 2 n1n R5z(11z0/z ) is the radius of curvature of the parabolic 2r 8 2r2 2 2 2 2 A n front of the light field, s 5s0(11z /z0) is the effective 12 ( ( uCmnCm8n8u Lm 2 S s D U s2 mÞm nÞn S D radius of the beam, 2z052ps0/l is the confocal parameter, 8 8 s is the radius of the beam waist, C are constant coeffi- 0 mn 2r2 cients, and k is the wave number of light with wavelength l. n8 mn 3Lm cosFm n ~r,w! , ~7! To generate a light beam with the amplitude ~1! it is 8S s2 DU 8 8 J necessary to form in the plane z50 the complex amplitude where ` 6 mn U0~r,w!5 ( CmnVmn~r,w!, ~3! F ~r,w!5arg C 2arg C 1@2~m2m8! m,n50 m8n8 mn m8n8 z where we have introduced the following notation for the or- 1~n2n8!#tan21 6~n2n8!w. ~8! thonormalized mode basis functions z0 657 Tech. Phys. Lett. 23 (9), September 1997 1063-7850/97/090657-02$10.00 © 1997 American Institute of Physics 657 All terms in the second sum in Eq. ~7! will rotate at the same rate ~and the entire beam will rotate as a whole! if, in the expression for the polar angle w as a function of the distance z, z w5B tan21 , ~9! z0 the coefficient B is constant: 2~m2m8!1~n2n8! B5 5const. ~10! 7~n2n8! The distance z p over which the beam makes p revolutions can be found from Eq. ~9!: 2pp z 5z tan , p51,2,...,N, ~11! p 0 S B D where N5B/4 is the maximum number of revolutions which the beam can make from z50toz5`(z@z0). The rotation rate n of the transverse section of a multimode beam depends on the distance z as dw z 2 21 v5 5B 11 . ~12! dz F Sz D G 0 It follows from what we have said above that by choos- ing the nonzero terms in Eq. ~3! with numbers satisfying the condition ~10! it is possible to use Eqs. ~5! and ~6! to design phase optical elements which when illuminated by laser light form, with a high efficiency, nonradially symmetric multi- mode Gauss–Laguerre beams whose intensity distribution in the transverse section rotates around the propagation axis. The numerical modeling parameters are: number of mesh points 2563256, waist radius s 50.1 mm, wave- 0 FIG. 1. Phases of diffractive optical elements forming multimode Gauss– length l50.63 mm, radius of the diffractive optical element Laguerre beams ~a! and normalized transverse intensity distributions at dif- R050.5 mm, and confocal parameter z0549.86 mm. ferent distances from the diffractive optical element ~b–f!. Numerical examples are displayed in Fig. 1. Column 1 shows the phase of the diffractive optical element ~1a! cal- culated according to Eqs. ~3!–~6! with two nonzero terms in the sum ~3!, with the coefficients C1,21 and C11,2 is shown in column 1. For two terms condition ~10! obviously holds, and are presented in column 3. Condition ~10! does not hold — B57. The maximum number of revolutions N51.75. It fol- B52, 6, and 11, and different parts of the beam rotate with lows from Eq. ~11! that the beam will have made a complete different rates, which results in distortion rather than rotation revolution over a distance z562.53 mm and another half of the pattern. revolution at z5217.76 mm. Figure 1 shows halftone render- ings of the normalized transverse distributions of the inten- sity of a light beam formed by a diffractive phase optical element ~1a! and the distributions computed using a Fresnel transformation in the planes z562.53 mm ~1b!, z5101.34 1 V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer mm ~1c!, z5140.15 mm ~1d!, z5178.95 mm ~1e!, and Generated Holograms, CRC Press, Boca Raton, 1994. 2 z5217.76 mm ~1f!. The pattern rotates counterclockwise. V. V. Kotlyar, I. V. Nikol’ski, and V. A. Sofer, Opt. Spektrosk. 75, 918 ~1993!. Similar results for a rotating beam consisting of three 3 V. V. Kotlyar, I. V. Nikol’ski, and V. A. Sofer, Pis’ma Zh. Tekh. Fiz. modes with the coefficients C1,21 , C5,0 , and C11,2 are shown 19~20!,20~1993!@Tech. Phys. Lett. 19, 645 ~1993!#. in column 2. Condition ~10! holds: B57, 7, and 7. The phase 4 V. V. Kotlyar, I. V. Nikolsky, and V. A. Soifer, Optik 98,26~1994!. function of such an element ~2a! is shown in halftones ~black 5 V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, J. Mod. Opt. 42, 1231 is 2p, white is 0). The transverse intensity distributions for ~1995!. 6 such a beam ~2b–2f! are shown at the same distances as in A. Yariv, Optical Electronics in Modern Communications, Oxford Uni- column 1. versity Press, New York, 1997, 5th edition @Russian translation, Sov. Ra- dio, Moscow, 1986#. The phase ~3a! and an intensity section ~3b!–~3f! for a three-mode beam with the coefficients C2,22 , C5,0 , and C15,2 Translated by M. E. Alferieff 658 Tech. Phys. Lett. 23 (9), September 1997 Kotlyar et al. 658 Anisotropic scattering of polarized light in a ferrofluid layer A. V. Skripal’ and D. A. Usanov Saratov State University ~Submitted January 22, 1997! Pis’ma Zh. Tekh. Fiz. 23, 7–10 ~September 12, 1997! Previously unknown effects wherein the scattering of plane-polarized light in a ferrofluid layer depends on the orientation of the electric field of the polarized radiation relative to the direction of the applied magnetic field are found experimentally, and their physical nature is explained. © 1997 American Institute of Physics. @S1063-7850~97!00209-7# The scattering of polarized light in ferrofluids has been difference in the intensities of the scattered radiation for Ei 1,2 studied primarily in dilute colloidal solutions. The charac- and E' vanishes. As one can see from Fig. 2, the threshold ter of the scattering of radiation incident on a thin layer of a magnetic field Hni for the parallel orientation of the vectors concentrated ferrofluid was analyzed in Refs. 3 and 4. How- ¯E and H¯, which is determined as shown in the same Fig. 2, ever, the analysis of the results did not include an investiga- turns out to be much lower than the threshold magnetic field tion of the effect of the relative orientation of the electric for the perpendicular orientation of the electric and magnetic field vector of the incident light and the magnetic field ap- field vectors (Hn').
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