Formation of higher-order Bessel light beams in biaxial crystals

Transcription

1 15 January 2001 Otics Communications 187 (2001) 407±414 Formation of higher-order Bessel light beams in biaxial crystals T.A. King a,1, W. Hogervorst b,2, N.S. Kazak c,3, N.A. Khilo d,4, A.A. Ryzhevich c, * a Laser Photonics, Deartment of Physics and Astronomy, University of Manchester, M13 9PL Manchester, UK b Laser Centre, Deartment of Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, Netherlands c Institute of Physics, NAS of Belarus, 68 F. Skaryna ave., Minsk, Belarus d Division for otical roblems in information technologies, NAS of Belarus, 1-2 Kurevich str., Minsk, Belarus Received 5 Setember 2000; acceted 15 November 2000 Abstract A transformation of the order of a Bessel light beam (BLB) from zeroth to rst and from rst to second when roagating light through a biaxial crystal has been studied theoretically and exerimentally. The ossibility of the formation of higher-order BLBs by means of this e ect is con rmed. The transformation of beam order is observed when roagating circularly olarized light along the otical axis of the crystal under conditions of internal conical refraction. It is shown that a roer choice of the crystal length or conicity angle of the incident beam ermits comlete transformation of a zero-order Bessel inut beam into a rst-order Bessel beam. Ó 2001 Published by Elsevier Science B.V. PACS: Bs; Ja; Lc Keywords: Bessel light beam; Internal conical refraction; Biaxial crystal 1. Introduction The linear roerties of Bessel light beams (BLBs) have been studied in fair detail by now * Corresonding author. Tel.: ; fax: addresses: terry.king@man.ac.uk (T.A. King), wh@ nat.vu.nl (W. Hogervorst), tol@dragon.bas-net.by (N.S. Kazak, A.A. Ryzhevich), nkhilo@otoinform.bas-net.by (N.A. Khilo). 1 Tel.: ; fax: Tel.: ; fax: Tel.: ; fax: Tel.: (see, for examle, Refs. [1±5]). The main roerties of BLBs stem from the conical structure of their satial sectrum. The result of mutual interference of lane-wave comonents of BLB is its multi-ring satial structure. The total number of BLB rings is usually large. The consequence of this is the fact that the divergence of an individual ring inside a beam is much smaller than the divergence of the whole beam. This roerty is referred to as diffractionless roagation of the BLB and is ronounced in the central zone of the beam at fairly large lengths and at BLB excitation by a collimated Gaussian-tye beam /01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S (00)01124-X

2 408 T.A. King et al. / Otics Communications 187 (2001) 407±414 The ``nondiverging'' otical beams are of interest for alignment and guiding of atoms. For this urose, both zero-order and higher-order BLBs can be used [6±9]. Besides, higher-order BLBs Laguerre±Gaussian beams are of interest for studying the rocesses of creation and annihilation of wavefront dislocations [10±13]. The study and alication of BLBs call for effective methods of their obtaining. At resent, the main method of obtaining zero-order BLBs is based on the use of an axicon [2,14±16]. This method is simle and ermits to convert Gaussiantye beams with an e ciency close to 100%. To obtain higher-order BLBs, a holograhic technique is used now [7,17,18]. Bessel beams of arbitrary order can also be roduced by illuminating an axicon with an aroriate Laguerre±Gaussian mode [9]. In this aer the holograhic technique is used also to transform a Gaussian beam into a higher order Laguerre±Gaussian beam. The holograhic technique is simle and a 40% e ciency can be tyically obtained. From the hysical oint of view, in the holograhic method a transfer of dislocation of the transmission function from a satially inhomogeneous hologram to the light eld is realized. An alternative hysical rincile is used in the method of obtaining elds with wavefront dislocation based on the use of media with dislocation of otical roerties deending on the direction of light roagation. An examle of such media is biaxial crystal. They feature dislocation of the olarization of normal modes at light roagation in the vicinity of binormals. Below we show theoretically and exerimentally that rst-order and higher-order BLBs can be obtained with use of biaxial crystals. formed at internal conical refraction and BLB having a conical structure of the sectrum of satial frequencies. The resence of such a relationshi would ermit obtaining higher-order BLBs by assing laser radiation through a homogeneous crystal. It is known that in the vicinity of binormal, the section of wave vectors surface reresents two coaxial coni (Fig. 1). The geometrical arameters of the coni are such that the wave vectors k of two normal modes roagating at a small angle with binormal are equal k q ˆk x e 1 k y e 2 k bk x bq q 2 =2k e 3 ; 1 where q ˆ kx 2 k2 y 1=2, b ˆ arctg c 2 e 1 3 e 1 2 e 1 2 e 1 1 1=2 =2v 2 Š is the arameter of crystal anisotroy, k ˆ x=v, v is the hase velocity of waves in the direction of binormals, e 1;2;3 are the rincial values of the ermittivity tensor. Indices ˆ1 enumerate the slow and the fast modes, and e 1, e 2, e 3 are the unit vectors of the Cartesian coordinate system with the z-axis arallel to the binormal (Fig. 1). The olarization of normal waves in the vicinity of the binormal is linear, and deending on an azimuth angle u ˆ arctg k y =k x [20]. As the azimuth angle is changed by the value of u, the olarization vectors of the normal waves c ˆ1 rotate by angle u=2 (Fig. 2). These olarization vectors can be exressed in the form 2. Theoretical model It is known that when the light wave roagates in the vicinity of binormals, the e ect of internal conical refraction shows u (see, for examle, Ref. [19]) that is the incident eld forms a cone of directions of the energy ow inside a crystal. On the basis of this, a theoretical model may be roosed, in which there is a relationshi between the elds Fig. 1. Cross-section of the wave vector surface in the vicinity of the binormal by the crystallograhic lane XZ. a is anisotroy arameter and indices ˆ1 enumerate a slow and a fast mode.

3 T.A. King et al. / Otics Communications 187 (2001) 407± Fig. 2. Azimuth deendence of the olarization of the fast and slow eigenmodes in the vicinity of the biaxial crystal binormal. c 1 u ˆ sin u=2 e 1 cos u=2 e 2 ; 2 c 1 u ˆ cos u=2 e 1 sin u=2 e 2 or in an alternative form in terms of vectors of the right and left circular olarization e ˆ e 1 ie 2 = 2 : c 1 u ˆ i e ex iu=2 e ex iu=2 Š; 2 c 1 u ˆ 1 e ex iu=2 e ex iu=2 Š: 2 3 Normally, the e ect of internal conical refraction is considered for the case of linear olarization of the incident light (see, for examle, Refs. [21,22]). It should be noted that due to the azimuth deendence of the olarization direction of normal waves inside the crystal the transmitted radiation has, accordingly, the azimuthally inhomogeneous intensity distribution. Therefore, in order to solve the roblem of formation of azimuthally homogeneous beams, it is necessary to use circularly olarized inut radiation. Seci cally, we shall assume that the incident eld has, for examle, right-hand circular olarization with an amlitude at the inut face of the crystal equal to a 0 q; u ˆa 0 e f in q; u : 4 To nd the refracted waves, we reresent the incident eld olarization vector in the form of a linear suerosition of eigenvectors (2) as follows: e ˆ X a u c u 5 with eigenvalues a 1 u ˆi ex iu=2 = 2 and a 1 u ˆex iu=2 = 2. Then the Fourier sectrum of the eld (4) just before the crystal can be given in the form X A 0 q; u ˆa 0 F in q; u a u c u ; 6 where Z Z F in q; u ˆ f in q; u ex iqqcos u u 1 Šqdq du 1 7 is the Fourier sectrum of the function f in q; u. In a articular case of the azimuth-indeendent inut eld, Z F in q ˆ2 f in q J 0 qq q dq; 8 where J 0 qq is the zero-order Bessel function. When assing over the crystal boundary, each Fourier comonent (6) exeriences re ection deending on the quantity q. This deendence can be neglected for araxial beams. For such beams the Fourier comonents of the eld inside the crystal, near its boundary, are described by exression (6). Consequently, the eld in the crystal is a q; u; z ˆ a0 X Z Z F q; u ; z a u 1 c u 1 ex iqqcos u u 1 ibqzšqdqdu 1 ; where 9

4 410 T.A. King et al. / Otics Communications 187 (2001) 407±414 F q; u 1 ; z ˆF in q; u 1 ex iq 2 z=2k ; q q ˆ x bz 2 y 2 : Let the inut eld be azimuthally symmetric. Then, substituting the olarization vectors from Eq. (3) into Eq. (9), integrating with resect to u 1 and summing over, we obtain a q; u; z ˆ Z a0 F q; z 2 J 0 qq e cos bqz J 1 qq ex iu e sin bqz Šqdq: 10 Now let us assume that the inut eld is lefthand circularly olarized. In this case, in exansion (5) the eigenvalues are equal to a 1 u ˆ i ex iu=2 = 2 and a 1 u ˆ ex iu=2 = 2.A calculation similar to the revious one gives a q; u; z ˆ Z a0 F q; z 2 J 0 qq e cos bqz J 1 qq ex iu e sin bqz Šqdq: 11 As follows from Eqs. (10) and (11), the circularly olarized wave excites in the biaxial crystal the suerosition of two waves with orthogonal olarizations one of which contains wavefront dislocation. Let us consider the articular case of incidence of a zero-order BLB on the crystal. Then f in q ˆJ 0 q in q. Neglecting the in uence of the nite transverse extension of the Bessel beam on its Fourier sectrum we obtain F in q ˆ2d q q in = q in, where d x is the delta function. Integrating Eqs. (10) and (11) we nd the eld amlitudes at the outut face of a crystal of thickness L: a q; u; L ˆa 0 J 0 q in q cos bq in L e J 1 q in q ex iu sin bq in L e 12 for the right-hand olarized incident beam, and a q; u; L ˆa 0 J 0 q in q cos bq in L e J 1 q in q ex iu sin bq in L e 13 for the left-hand olarized one. In Eqs. (12) and (13) the hase factor ex q 2 inz=2k inferred from the de nition of F q; u; z after Eq. (9), which is immaterial for calculating the intensity attern, is omitted. Thus, the outut eld is the suerosition of the zero-order and the rst-order BLBs with orthogonal olarizations. The amlitude ratio of these beams deends on the distance covered in the crystal by the simle harmonic law. When the condition bq in L ˆ 2n 1 =2; 14 where n ˆ 0; 1;... is ful lled, the term in Eqs. (12) and (13) roortional to the Bessel function J 0 q in q becomes zero. Consequently, comlete conversion of the circularly olarized zero-order Bessel beam into the rst-order Bessel beam with orthogonal olarization will take lace. Estimation of the oscillation eriod L 0 ˆ =2bq in at b 0:016 (KTP crystal), q in ˆ 2c in =k, k ˆ 0:63 lm, c in ˆ 0:02 gives L 0 0:5 mm. It is imortant to note that the rocess of increasing the Bessel function order can be continued. To do this, it is necessary to extract from outut eld Eqs. (12) and (13) the rst-order Bessel beam and realize its reeated assage through the crystal. In the general case of incidence on the crystal of a Bessel beam of order m with the hase factor ex imu, integrating Eq. (7), we nd F in q; u ˆ2 i m ex imu d q q in =q in : 15 Substitution of Eq. (15) into Eq. (9) leads to formulas generalizing Eqs. (12) and (13): a q; u; L ˆa0 2 J m q in q cos bq in L e J m 1 q in q ex i m 1 ušsin bq in L e ; 16 a q; u; L ˆa0 2 J m q in q cos bq in L e J m 1 q in q ex i m 1 ušsin bq in L e : 17 Formulas (12), (13), (16), (17) describing the e ect of BLB order transformation hold for araxial BLBs within the alicability limits of the original relation (1). In the case of a KTP crystal, BLBs

5 T.A. King et al. / Otics Communications 187 (2001) 407± with a cone angle not exceeding 10 can be adequately described by formulas (16) and (17). Thus, theoretical analysis indicates the ossibility of formation in biaxial crystals of light beams with screw dislocations and, in articular, of higher-order BLBs. 3. Exerimental results and discussion The e ect of transformation of the order of a BLB was tested exerimentally. The exerimental setu shown in Fig. 3. A collimated, circularly olarized Gaussian beam of a He±Ne laser had a waist size of 2.5 mm and was transformed into a zero-order BLB using an axicon with an internal angle of 2:2 and refractive index of 1.5. Consequently, the cone angle c in ˆ q in =k 0 of the zeroorder BLB was about c in ˆ 1:1. This Bessel beam had illuminated a 12 mm thick KTP crystal oriented erendicularly to the binormal and located at a distance of 16 cm from the axicon. The radius of the BLB formed by the axicon, within the limits of the crystal osition, was about 1.5 mm. In the outut eld, the right and left circularly olarized comonents were searated and investigated indeendently. The intensity distribution in their cross-section was measured and comared to squared zero- and rst-order Bessel functions in accordance with Eq. (12). Fig. 4(a) and (b) show the images of the central art of the beams crosssections beyond the olarizer±analyzer. Fig. 5(a) and (b) give the corresonding intensity distributions in comarison with the squared Bessel functions J 2 0 ck 0q and J 2 1 ck 0q. It is clear that good quantitative agreement between theoretical and exerimental data is obtained. In addition to these radial distributions of intensity, an imortant attribute of rst-order BLBs is their wavefront dislocation. To dislay this, an interference of the BLB with lane and sherical reference waves has been studied. The numerically calculated interference ictures are shown in Fig. 6. It is visible that in the case of interference with a lane reference wave, a characteristic indication of dislocation is the bifurcation of the maximum at the center of the BLB and in case of interference with a sherical wave ± a siral structure (see, also Ref. [23]). The above roerties have been observed in exerimental interference atterns. Images of these atterns are given in Fig. 7(a) and (b) for lane and sherical reference waves resectively. Note that the direction of siralling as well as the orientation of the bifurcated maximum reverse when we change the sign of inut beam circular olarization. After crystal 6 the outut rst- and zero-order Bessel beams by virtue of the small cone angle and relatively small longitudinal dimensions of the crystal, exist ractically in the same region in which the zero-order initial beam would exist if the crystal was removed from the system. This region aroximately corresonds to the gure formed by rotating the shaded rhomb (if the refraction in the crystal is ignored) and has nite dimensions (see Fig. 3). The maximum distance from axicon 5, on which the rst-order outut BLB still exists, z max ˆ R d =c in where R d is the radius of the diahragm oening limiting the Gaussian beam illuminating the axicon. The intensity distributions Fig. 3. Otical system for transformation of the order of the Bessel beam: 1 ± 20 telescoe; 2 ± olarizer; 3 and 7 ± quarter-wave lates, 4 ± diahragm, 5 ± axicon; 6 ± KTP crystal; 8 ± olarizer±analyzer; 9 ± microscoe; 10 ± recording system; 11 ± additional axicon.

6 412 T.A. King et al. / Otics Communications 187 (2001) 407±414 Fig. 5. Exerimental intensity distribution (curve 1) of the outut eld with olarization (a) equal to the inut one and (b) orthogonal to it as comared to design-theoretical zero- and rst-order Bessel beam intensity distributions (curve 2). Fig. 4. Images of the eld beyond the olarizer±analyzer for radiation with circular olarization (a) coinciding with the inut one (zero-order BLB) and (b) orthogonal to it ( rst-order BLB). and interference atterns given by us corresond to one of the cross-section of this region. However, in all the cross-sections of the above region, after the olarizer±analyzer 8 the intensity distribution is analogous to that given in the gure. At distance from axicon 5 larger then z max rst-order BLB is transformed into a ring eld which reserves, nevertheless, the azimuthal hase modulation inherent in the rst-order BLB. Using an additional axicon 11, this ring eld can be transformed into a rst-order BLB again. We also checked the theoretical result of Section 2, indicating the ossibility of stewise increase of the order of Bessel function under reeated assage of light through a crystal. For this urose we used an exerimental set-u with two crystals in series. In the rst stage a zero-order BLB, incident on the KTP crystal, is transformed into a rst-order BLB. Next, the rst-order beam olarization is converted from left-circular to right-circular olarization and the beam is directed

7 T.A. King et al. / Otics Communications 187 (2001) 407± Fig. 7. Images generated by interference of a rst-order Bessel beam (a) with a reference lane wave and (b) with a reference sherical wave. Fig. 6. Distribution of interference maxima, calculated for the case of interference of the eld ex iu with (a) a lane wave and (b) a sherical wave. towards the second biaxial crystal a-hio 3 (iodic acid) along the direction of its otical axis. According to Eq. (16) the outut eld then involves a suerosition of rst- and second-order BLBs. Fig. 8 shows the radial intensity distribution of the outut eld comonent with olarization orthogonal to the inut olarization. It is seen that this distribution is reasonably well aroximated by the squared second-order Bessel function. When incident on the crystal is Gaussian beam instead of Bessel one, it can be transformed into a Laguerre±Gaussian mode LG 01 where radial index is 0 and azimuthal index is 1. Then we generated a rst-order BLB by illuminating an axicon with this mode with total e ciency of about 60%. It can be concluded that the e ect of transformation of Bessel beams when roagating along the biaxial crystal binormal as exressed by Eqs. (12) and (16) is con rmed exerimentally.

8 414 T.A. King et al. / Otics Communications 187 (2001) 407±414 biaxial crystals ermits to transform several light beams simultaneously. Acknowledgements Financial suort from INTAS (INTAS-BEL- ARUS ) is gratefully acknowledged. References Fig. 8. Exerimental intensity distribution (curve 1) of the outut eld behind the second crystal with olarization orthogonal to the inut eld olarization as comared to designtheoretical second-order Bessel beam intensity distribution (curve 2). 4. Conclusions In this work an otical e ect of transformation of transversal structure of light eld roagating along a binormal of biaxial crystal is investigated theoretically and exerimentally. It is shown that circularly olarized inut beam excites two circularly olarized beams in the crystal. The beam olarized orthogonally to the inut beam has a screw wavefront dislocation if the inut beam had no dislocation. When the inut beam contains a dislocation of order m, the above mentioned beam has a dislocation of m 1 or m 1 order. The hysical essence of such wavefront transformation is a transfer of singularity of otical roerties of biaxial crystal in the vicinity of its binormal to the light eld wavefront. Using this method it is ossible to obtain a Bessel beam of rst and higher orders from an inut zero-order BLB. The advantage of the method of rst-order Bessel beams roduction based on biaxial crystals is that transformation e ciency is close to 100%. Besides, because of their high radiation damage threshold, biaxial crystals can be used as intracavity elements. Let us note that transversal invariance of the scheme based on [1] J. Durnin, J. Ot. Soc. Am. A 4 (1987) 651. [2] R.M. Herman, T.A. Wiggins, J. Ot. Soc. Am. A 8 (1991) 932. [3] Y. Lin, W. Seka, J.H. Eberly, H. Huang, D.L. Brown, Al. Ot. 31 (1992) [4] Z. Jiang, Q. Lu, Z. Liu, Al. Ot. 34 (1995) [5] S. Rushin, A. Leizer, J. Ot. Soc. Am. A 15 (1998) [6] M. Florjanczyk, R. Tremblay, Ot. Commun. 73 (1989) 448. [7] C. Paterson, R. Smith, Ot. Commun. 124 (1996) 121. [8] I. Manek, Yu.B. Ovchinnikov, R. Grimm, Ot. Commun. 147 (1998) 67. [9] J. Arlt, K. Dholakia, Ot. Commun. 177 (2000) 297. [10] I.V. Basistiy, V.Yu. Bazhenov, M.S. Soskin, M.Vu. Vasnetsov, Ot. Commun. 103 (1993) 422. [11] K. Dholakia, N.B. Simson, M.J. Padgett, L. Allen, Phys. Rev. A 54 (1996) R3742. [12] A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, A. Stabinis, Ot. Commun. 140 (1997) 273. [13] A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, A. Stabinis, Ot. Commun. 150 (1998) 372. [14] G. Indebetouw, J. Ot. Soc. Am. 6A (1989) 150. [15] G. Scott, N. McArdle, Ot. Eng. 31 (1992) [16] M.V. Perez, C. Gomez-Reino, J.M. Cuadrado, Ot. Acta 33 (1986) [17] A. Vasara, J. Turunen, A.T. Friberg, J. Ot. Soc. Am. A 6 (1989) [18] H.S. Lee, B.W. Steward, K. Choi, H. Fenichel, Phys. Rev. A 49 (1994) [19] M. Born, E. Wolf, Princile of Otics, Pergamon Press, New York, [20] A.G. Khatkevich, Ot. i sektrosko. 46 (1979) 505 (Russian). [21] A.J. Schell, N. Bloembergen, J. Ot. Soc. Am. 68 (1978) [22] J.P. Feve, B. Boulanger, G. Marnier, Ot. Commun. 105 (1994) 243. [23] S. Chavez-Cerda, G.S. McDonald, G.H.C. New, Ot. Commun. 123 (1996) 225.