Optics & Laser Technology 54 (2013) 68–71
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Generation of dark hollow beam by focusing a sine-Gaussian beam using a cylindrical lens and a focusing lens Huiqin Tang, Kaicheng Zhu n School of Physics and Electronics, Central South University, Changsha 410083, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 February 2013 Received in revised form 8 May 2013 Accepted 14 May 2013
Based on the generalized Huygens–Fresnel diffraction integral, a closed-form propagation equation related to sine-Gaussian beams through a cylindrical lens and a focusing lens is derived and illustrated with numerical methods. It is found that a sine-Gaussian beam through such a system may be converted into a dark hollow beam (DHB) with topological charge index one and its bright enclosure is approximately an elongated ellipse with very high ellipticity. Moreover, the parameter values at which the DHBs have perfect intensity patterns are designed. The optimal relative orientation between the dislocation line of the input sine-Gaussian beam and the axial orientation of the cylindrical lens is speciﬁed. And the ellipticity of the elliptical DHBs is mainly deﬁned by the focal length of the cylindrical lens and the Fresnel number of the optical system. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Dark hollow beam (DHB) Sine-Gaussian beam Vortex
1. Introduction Beams with dark-hollow characteristic (also called dark-hollow beams (DHBs)) have always attracted much attention due to their remarkable usefulness in optical tweezers and cold atom optics [1– 9]. Also, in recent years there has been an increasing interest in developing models to describe DHBs with elliptic symmetry because an elliptic DHB can be regarded as a transition beam between circular and rectangular DHBs [10–23]. For example, Cai et al. [18,21] constructed circular and rectangular DHBs using a suitable superposition of a ﬁnite series of fundamental Gaussian beams. Sun  proposed a model to represent elliptic DHBs with help of the sinh-Gaussian function. Moreover, Chakraborty et al. [15,19] realized elliptic DHBs through designing a special mask illuminated by a plane beam. And Zhao et al.  designed a scheme to convert a circular DHB into an elliptic one with a triangular prism. As is well known, the Hermite–sinusoidal-Gaussian beams are ones of the exact solutions of the paraxial wave equation in the rectangular coordinate system [24–26]. And sinh-Gaussian beams or sine-Gaussian beams are the special cases of the Hermite–sinusoidalGaussian beams, the beams propagating through optical systems had been studied widely [27–29]. In this work, we study the propagation of sine-Gaussian beams through a system including a cylindrical lens and a focusing lens and ﬁnd that a DHB on the focus plane can be realized. The bright enclosure of the DHB is approximately an
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elongated ellipse with very high ellipticity. By virtue of the generalized Huygens–Fresnel diffraction integral, a closed-form propagation equation of sine-Gaussian beam through such a system is derived and illustrated with numerical examples. Furthermore, the vortex properties of the propagating sine-Gaussian beams are analyzed. Our study shows that a vortex with topological charge index one can occur in the propagating sine-Gaussian beam through a combination system including a cylindrical lens and a focusing lens, which provides a new scheme to prepare DHBs with elongated elliptic conﬁgurations experimentally. 2. Propagation equation In our discussion, the incident ﬁeld Eðx0 ; y0 ; 0Þ of a sineGaussian beam at z¼ 0 is assumed to be ! x20 þ y20 Eðx0 ; y0 ; 0Þ ¼ exp − ð1Þ sin ðβx x0 þ βy y0 Þ w20 where w0 is the waist width related to the Gaussian beam, βx and βy are the parameters associated with the sine part. Obviously, differing from the sinh-Gaussian beams introduced in Ref.  which does not carry any vortex and is also not the exact solutions of the paraxial wave equation in the rectangular coordinate system, the beam represented by Eq. (1) carries an edge dislocation and is still a solution of the paraxial wave equation in the rectangular coordinate system [24–26]. And the equation βx x0 þ βy y0 ¼ 0 characterizes the dislocation line orientation. In practice, the beam can be also obtained with a Gaussian beam through a sine-amplitude modulated grating. Suppose that the beam (1)
H. Tang, K. Zhu / Optics & Laser Technology 54 (2013) 68–71
successively propagates through a cylindrical lens and a focusing lens, the additional phase factor introduced by the cylindrical lens whose cylindrical axis is along the y-axis reads as ! ik 2 Tðx0 ; y0 Þ ¼ exp − x ð2Þ 2f Cy 0 where k ¼ 2π=λ is the wavenumber related to the wavelength λ and fCy is the focal length of the cylindrical lens. Therefore, the relative orientation between the axial direction of the cylindrical lens and the dislocation line of the sine-Gaussian beam is characterized as θ ¼ arctanðjβy =βx jÞ, which is shown in Fig. 1. Synthesizing Eqs. (1) and (2), the propagation of light beams through an optical system A B parameterized by the transfer matrix is described by the C D generalized Huygens–Fresnel diffraction integral of the form Eðx; y; zÞ ¼
i ∬ dx0 dy0 E0 ðx0 ; y0 ; 0ÞTðx0 ; y0 Þ λB ik 2 Aðx0 þ y20 Þ−2ðxx0 þ yy0 Þ þ Dðx2 þ y2 Þ exp − 2B
For the propagation through a thin focusing lens with focal length f and followed by a free space, the ABCD matrix becomes ! 1−z=f z A B ¼ ð7Þ −1=f 1 C D where z is the longitudinal coordinate originating at the focusing lens. Scaling the parameters and the coordinates, the focused ﬁeld distributions described by Eq. (6) can be rewritten as " # FN 2 F 2N x2w y2w 2 Eðx; y; zÞ ¼ E0 ðzÞexp i ðx þ yw Þ− 2 þ zf w zf 1 þ iαx 1 þ iαy βyw FN βxw sinh ð8Þ xw þ y zf 1 þ iαx 1 þ iαy w where zf ¼ z=f , uw ¼ u=w0 , βuw ¼ βu w0 ðu ¼ x; yÞ, and F N ¼ πw20 =λf is the Fresnel number, while αy ¼ iðð1=zf Þ−1ÞF N , and αx ¼ αy þ iF N f =f Cy . Because sinhðu þ ivÞ ¼ sinh u cos v þ icosh u sin v for real u and v, therefore, according to Eq. (8) it is noted that the intensity 2 distribution Iðx; yÞ ¼ Eðx; y; zÞ on a given transversal plane may be expressed as
Substituting Eqs. (1) and (2) into this equation we have x2 y2 1 ikD 2 exp − ðx þ y2 Þ ∬ dx0 dy0 exp −ð1 þ iαx Þ 02 −ð1 þ iαy Þ 02 Eðx; y; zÞ ¼ 2λB 2B w0 w0
k k k k x þ β x x0 þ i y þ βy y0 −exp i x−βx x0 þ i y−βy y0 exp i B B B B
F2 Iðx; yÞ ¼ jE0 ðzÞj exp − N z2f (
" FN zf "
and performing tedious integral calculations we easily have " # 2 k w20 ikD 2 x2 y2 2 ðx þ y Þ− þ Eðx; y; zÞ ¼ E0 ðzÞexp 2B 4B2 1 þ iαx 1 þ iαy 2 βy kw0 βx sinh xþ y 2B 1 þ iαx 1 þ iαy
ð4Þ where the notations αx ¼ ðð1=f cl Þ þ ðA=BÞÞðkw20 =2Þ and αy ¼ ðiA=BÞ ðkw20 =2Þ have been used. Recalling the Gaussian integral formulae ! rﬃﬃﬃ Z π β2 exp dxexpð−αx2 þ βxÞ ¼ ReðαÞ 4 0 ð5Þ α 4α
x2w y2w þ 1 þ α2x 1 þ α2y
βyw βxw xw þ y 1 þ α2x 1 þ α2y w βyw βxw xw þ y 1 þ α2x 1 þ α2y w
" sin 2
" FN zf
αy βyw αx βxw xw þ y 1 þ α2x 1 þ α2y w αy βyw αx βxw xw þ y 1 þ α2x 1 þ α2y w
ð9Þ Obviously, on the propagation axis (x ¼y¼0) the intensity is always zero because sin ρ ¼ sinh ρ≡0 for ρ¼ 0.
3. Numerical results and discussions
where in our discussion on the intensity distribution and phase singularity, h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E0 ðzÞ ¼ ðπw20 =λB ð1 þ iαx Þð1 þ iαy ÞÞexp −ðw20 =4Þððβ2x =1 þ iαx Þ þ ðβ2y = 1 þ iαy ÞÞ is a global and unimportant factor at a deﬁned transversal plane.
Fig. 1. The relative orientation between the axial direction of the cylindrical lens (y-axis) and the dislocation line of the input sine-Gaussian beam (along the OA-direction).
Based on Eqs. (8) and (9) the intensity distribution and the phase distribution of the focused beam for given transversal planes can be analyzed with numerical simulations. Without any loss of generality, the intensity distribution will be always normalized to maximum by itself. Carefully choosing the beam parameters and optical system parameters, we perform some numerical calculations. It is found that, on the focal plane (z¼ f), the intensity pattern may form a bright enclosure around the focus point and along the bright enclosure the intensity is almost identical, which can be seen from Fig. 2. Noting that the different scales are used for x and y coordinates we can conclude that the bright enclosure of the intensity pattern is approximately an elongated ellipse. And the long-axis of the elongated ellipse is about 50 times of its short-axis. Moreover, we investigate the real (solid) and imaginary (dashed) zero contours of the focused beam and ﬁnd that
Fig. 2. Intensity distributions (left) and the corresponding real (solid) and imaginary (dashed) zero contours of the ﬁeld (8) for βx w0 ¼ f Cy =f ¼1, βy ¼ 0:85βx and F N ¼ 50 on the focal plane (z ¼ f).
H. Tang, K. Zhu / Optics & Laser Technology 54 (2013) 68–71
the focused beam carries a vortex with topological charge index one because both zero contours have an intersecting point at the origin. In Figs. 3 and 4 the variation of the intensity distribution with βx and βy is analyzed. For example, for given other parameters, when βx w0 ¼ 1 and βy approaches to 0:85βx the obtained elliptic DHB is perfect or, in other words, along the bright enclosure the intensities are almost unanimous, which corresponds to the orientation angle θ≈40:361. And the sine-amplitude gratings should have the ultra-low spatial frequency because of βx w0 ¼ 1. Therefore, to realize a perfect DHB with elliptic geometry, the beam parameters βx and βy should be carefully adjusted. In Fig. 5 the dependence of the intensity distribution I(x,y) on the Fresnel number is drawn. Obviously, the larger Fresnel number leads the long-axis of the dark elliptic region to become longer because its short-axis is shortened although its long-axis seems to be unchanged. And λ ¼ 632:8 nm, f¼ 25 cm and F N ¼ 100 correspond to w0 ≈2:2 mm, which is achievable in practice with the current technology. Moreover, we demonstrate the effect of the focal length of the cylindrical lens on the intensity distribution. For other ﬁxed parameters, the longer focal length of the cylindrical lens will mainly cause the elliptic DHB to have lower ellipticity although its short-axis is not seemly inﬂuenced, which can be seen
from Fig. 6. So we can conclude that the Fresnel number and the focal length of the cylindrical lens mainly deﬁne the ellipticity of the elliptic DHBs. Finally, numerical results show that the perfect DHB considerably degrades when the observation plane quite slightly deviates from the focal plane. But the perfection degradation of intensity patterns will be weakened with a decrease of the parameters f Cy =f and F N , which can be seen from Fig. 7. Therefore, to obtain an elliptic DHB with perfect conﬁgurations by virtue of the scheme it is necessary to make the observation plane exactly locate nearby the focal plane of the focusing lens besides the smaller values of f Cy =f and F N should be chosen.
4. Conclusions In summary, a scheme is proposed with which a sine-Gaussian beam can be transformed into a elliptical DHB with topological charge index one and an elongated elliptic bright enclosure by making use of a cylindrical lens and a focusing lens. Based on the generalized Huygens–Fresnel diffraction integral, a closed-form propagation equation of sine-Gaussian beams through an optical
Fig. 3. Intensity distributions of the focal plane (z ¼ f) for βy ¼ 0:85βx , f Cy =f ¼1, F N ¼ 50 and βx w0 ¼ 0.75 (left), 1.0 (middle) and 1.25 (right).
Fig. 4. Intensity distributions of the focal plane (z ¼ f) for βx w0 ¼ f Cy =f ¼ 1, F N ¼ 100 and βy ¼ 0:8βx (left), 0.85βx (middle) and 0:9βx (right).
Fig. 5. Intensity distributions for βx w0 ¼ f Cy =f ¼1, βy ¼ 0:85βx and F N ¼ 30 (left), 100 (middle) and 200 (right) on the focal plane (z ¼ f).
Fig. 6. Intensity distributions of the focal plane (z ¼f) for βx w0 ¼ 1, βy ¼ 0:85βx , F N ¼ 50 and f Cy =f ¼0.20 (left), 0.85 (middle) and 1.50 (right).
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Fig. 7. (A) Intensity distributions for βx w0 ¼ 1, βy ¼ 0:85βx , f Cy =f ¼ 0.25, F N ¼ 20 and z/f ¼0.995 (left), 1.000 (middle) and 1.005 (right), and (B) same as (A) but f Cy =f ¼ 1.0 and F N ¼100.
system consisting of a cylindrical lens and a focusing lens is derived and illustrated with numerical simulations. And the parameter values at which the DHBs have perfect conﬁgurations are designed. It is found that to obtain a perfect DHB with a unanimous intensity along the enclosure that the beam parameters βx and βy or the orientation angle θ of the input sineGaussian beam should be carefully chosen. And the Fresnel number and the focal length of the cylindrical lens mainly deﬁne the ellipticity of the elongated elliptical region. It is also found that such parameter values are experimentally accessible. In fact, it was found that a rectangular DHB seems to be more suitable than a circular DHB to guide atom laser beams with a particular mode . Especially, if the beam width in one direction of a rectangular DHB is much larger than that in the other direction, the rectangular DHB can be visualized as a onedimensional beam, which may be applied to achieve onedimensional Bose–Einstein condensates . Therefore, compared with these reported in literatures [15,19,22], the scheme presented here is very interesting because it provides a convenient and powerful way to generate the elliptic DHBs with quite high ellipticity. References  Yin J, Gao W, Zhu Y. Generation of dark hollow beams and their applications. In: Wolf E, editor. Progress in optics, vol. 44. Amsterdam: North-Holland; 2003. p. 119–204.  Kuga T, Torii Y, Shiokawa N, Hirano T, Shimizu Y, Sasada H. Novel optical trap of atoms with a doughnut beam. Physical Review Letters 1997;78:4713–6.  Arlt J, Dholakia K. Generation of high-order Bessel beams by use of an axicon. Optics Communications 2000;177:297–301.  Zhu KC, Tang HQ, Sun XM, Wang XW, Liu TN. Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams. Optics Communications 2002;207:29–34.  Cai YJ, Lu XH, Lin Q. Hollow Gaussian beam and its propagation. Optics Letters 2003;28:1084–6.  Ganic D, Gan X, Gu M. Focusing of doughnut laser beams by a high numericalaperture objective in free space. Optics Express 2003;11:2747–52.  Mei ZR, Zhao DM. Controllable dark-hollow beams and their propagation characteristics. Journal of the Optical Society of America A 2005;22:1898–902.  Deng D, Fu X, Wei C, Shao J, Fan Z. Far-ﬁeld intensity distribution and M2 factor of hollow Gaussian beams. Applied Optics 2005;44:7187–90.  Sun QG, Zhou KY, Fang GY, Zhang GQ, Liu ZJ, Liu ST. Hollow sinh-Gaussian beams and their paraxial properties. Optics Express 2012;20:9682–91.
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