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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6255fbaf47503a0a93f3825e/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">V<span class="_ _0"></span>ortex astigmatic Fourier<span class="_ _1"></span>-invariant Gaussian </div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls1 ws1">beams </div><div class="t m0 x1 h3 y3 ff1 fs1 fc1 sc0 ls2 ws1">V.<span class="fs2 ls3"> </span><span class="ls4 ws2">V. K</span><span class="fs2 ls5">OTLYAR</span><span class="ls3">,</span></div><div class="t m0 x2 h4 y4 ff1 fs3 fc1 sc0 ls6 ws1">1,2</div><div class="t m0 x3 h3 y3 ff1 fs1 fc1 sc0 ls7 ws3"> A.<span class="fs2 ls3 ws1"> </span><span class="ls8 ws4">A. K<span class="fs2 ls9 ws1">OVALEV<span class="fs1 ls3">,</span></span></span></div><div class="t m0 x4 h4 y4 ff1 fs3 fc1 sc0 ls6 ws1">1,2,*</div><div class="t m0 x5 h3 y3 ff1 fs1 fc1 sc0 ls7 ws3"> A.<span class="fs2 ls3 ws1"> </span><span class="ls4 ws2">P. P<span class="fs2 lsa ws1">ORFIREV</span></span></div><div class="t m0 x6 h4 y4 ff1 fs3 fc1 sc0 ls6 ws1">1,2</div><div class="t m0 x7 h3 y3 ff1 fs1 fc1 sc0 ls3 ws1"> </div><div class="t m0 x1 h5 y5 ff2 fs4 fc0 sc0 ls3 ws1">1</div><div class="t m0 x8 h6 y6 ff2 fs5 fc0 sc0 lsb ws5">Image Processing Systems Institute of the RAS – <span class="_ _2"></span><span class="lsc ws6">Branch of FSRC “Crystallography & Photonics” of </span></div><div class="t m0 x1 h6 y7 ff2 fs5 fc0 sc0 lsd ws7">the RAS, 151 Mo<span class="_ _2"></span>lodogvardeyskaya St., Samara 443001, Russia </div><div class="t m0 x1 h5 y8 ff2 fs4 fc0 sc0 ls3 ws1">2</div><div class="t m0 x8 h6 y9 ff2 fs5 fc0 sc0 lse ws8">Samara National Research University, 34<span class="lsf"> Moskovskoe Shosse, Samara 443086, <span class="_ _2"></span>Russia </span></div><div class="t m0 x1 h5 ya ff2 fs4 fc2 sc0 ls3 ws1">*</div><div class="t m0 x8 h6 yb ff2 fs5 fc2 sc0 ls10 ws1">alexeysmr@mail.ru </div><div class="t m0 x1 h7 yc ff3 fs6 fc0 sc0 ls11 ws1">Abstract:<span class="ff4 ls12 ws9"> We find a two-parameter family of astig<span class="ls13 wsa">matic elliptical Gaussian (AEG) optical </span></span></div><div class="t m0 x1 h7 yd ff4 fs6 fc0 sc0 ls14 wsb">vortices, which are free space modes up to <span class="_ _2"></span>scal<span class="ls15 wsc">e and ro<span class="_ _1"></span>tation. We calculate total normalized </span></div><div class="t m0 x1 h7 ye ff4 fs6 fc0 sc0 ls11 wsd">orbital angul<span class="_ _2"></span>ar momentum<span class="_ _2"></span> of AEG vortices, whic<span class="_ _2"></span>h can be an integer, <span class="_ _2"></span>fractional and zero<span class="_ _2"></span>, and </div><div class="t m0 x1 h7 yf ff4 fs6 fc0 sc0 ls16 wse">which is equal to the algebraic sum of two terms reflecting the contributio<span class="_ _1"></span>n of the vortex and </div><div class="t m0 x1 h7 y10 ff4 fs6 fc0 sc0 ls17 wsf">astigmatic componen<span class="_ _1"></span>ts of the light field. In an<span class="_ _1"></span>y <span class="ls18 ws10">transverse plane, such a beam has an isol<span class="_ _2"></span>ated </span></div><div class="t m0 x1 h7 y11 ff2 fs6 fc0 sc0 ls3 ws1">n<span class="ff4 ls19 ws11">-fold dege<span class="_ _2"></span>nerate intensity<span class="_ _2"></span> null on the opti<span class="_ _2"></span>cal axis (an optical vo<span class="_ _2"></span>rtex) embedded into a<span class="_ _2"></span>n </span></div><div class="t m0 x1 h7 y12 ff4 fs6 fc0 sc0 ls1a ws12">elliptical Gaussian beam. In addition to the qu<span class="_ _1"></span>adratic elliptical phase, a beam has the ph<span class="_ _1"></span>ase of </div><div class="t m0 x1 h7 y13 ff4 fs6 fc0 sc0 ls1b ws13">a cylindrical lens r<span class="_ _2"></span>otated by an angle of 45 de<span class="_ _2"></span><span class="ls1c ws14">grees with respect to th<span class="ls1d ws15">e principal axes of the </span></span></div><div class="t m0 x1 h7 y14 ff4 fs6 fc0 sc0 ls1a ws16">ellipse of the Gaussian beam inten<span class="_ _1"></span>sity distribu<span class="_ _1"></span>tion. The degenerated cen<span class="_ _1"></span>tral intensity null in </div><div class="t m0 x1 h7 y15 ff4 fs6 fc0 sc0 ls1e ws17">these beams does not split it in<span class="_ _1"></span>to <span class="ff2 ls3 ws1">n</span><span class="ls1f ws18"> spatially separated intensity nu<span class="_ _1"></span>lls, as is usually assumed </span></div><div class="t m0 x1 h7 y16 ff4 fs6 fc0 sc0 ls20 ws19">for elliptical astigmatic beams. </div><div class="t m0 x1 h8 y17 ff5 fs3 fc0 sc0 ls21 ws1a">© 2019 Optical Society of America under the terms of the <span class="fc2 ls22 ws1b">OSA Open Access P<span class="ls23 ws1c">ublishing Agreement<span class="_ _1"></span></span></span><span class="ls3 ws1"> </span></div><div class="t m0 x1 h9 y18 ff1 fs6 fc0 sc0 ls1d ws1d">1. Introduction </div><div class="t m0 x1 h7 y19 ff4 fs6 fc0 sc0 ls24 ws1e">It is known [1–5] t<span class="_ _2"></span>hat a cylindrical lens can be used t<span class="_ _2"></span>o determine the topological<span class="_ _2"></span> charge of an </div><div class="t m0 x1 h7 y1a ff4 fs6 fc0 sc0 ls19 ws1f">optical vortex<span class="_ _2"></span>. This property of<span class="ls13 ws20"> the cylindrical lens was noti<span class="ls14 ws21">ced long ago. For example, a </span></span></div><div class="t m0 x1 h7 y1b ff4 fs6 fc0 sc0 ls25 ws22">Hermite-Gauss<span class="_ _2"></span>ian laser beam of an orde<span class="_ _2"></span>r (0, <span class="ff2 ls3 ws1">n</span><span class="ls26 ws23">) with the zero orbi<span class="_ _2"></span>tal angular m<span class="_ _2"></span>omentum </span></div><div class="t m0 x1 h7 y1c ff4 fs6 fc0 sc0 ls27 ws24">(OAM) was transformed by using a cylindrical le<span class="_ _1"></span><span class="ls28 ws25">ns into a Laguerre<span class="_ _2"></span>-Gaussian laser beam<span class="_ _2"></span> [6] </span></div><div class="t m0 x1 h7 y1d ff4 fs6 fc0 sc0 ls13 ws26">that has <span class="ff2 ls3 ws1">n</span><span class="ls1e ws27">-fold degenerate intensity nu<span class="_ _1"></span>ll and possesses the OAM. Using a cylindrical lens, it is </span></div><div class="t m0 x1 h7 y1e ff4 fs6 fc0 sc0 ls19 ws28">possible to generate<span class="_ _2"></span> vortex-free laser beams <span class="ls20 ws29">with the OAM [7,8]. Th<span class="ls29 ws2a">ere are no isolated </span></span></div><div class="t m0 x1 h7 y1f ff4 fs6 fc0 sc0 ls16 ws2b">intensity nulls (singular points) in such<span class="_ _1"></span> beams. These astigmatic beams are described by a </div><div class="t m0 x1 h7 y20 ff4 fs6 fc0 sc0 ls16 ws2c">superposition of an<span class="_ _1"></span> infinite number of optical vo<span class="_ _1"></span>rtices with only even positive and<span class="_ _1"></span> negative </div><div class="t m0 x1 h7 y21 ff4 fs6 fc0 sc0 ls2a ws2d">topological charges [8]. </div><div class="t m0 x9 h7 y22 ff4 fs6 fc0 sc0 ls2b ws2e">It is also known that a linear combination <span class="ls2c ws2f">of even a<span class="_ _2"></span>nd odd Mathieu [9], Ince-<span class="_ _2"></span>Gaussian </span></div><div class="t m0 x1 h7 y23 ff4 fs6 fc0 sc0 ls2c ws30">[10] and Herm<span class="_ _2"></span>ite [11,12] beams wi<span class="_ _2"></span>th a phase shift of <span class="ff6 ls3 ws1">π</span><span class="ls2d ws31">/2 generates elliptical optical vortices </span></div><div class="t m0 x1 h7 y24 ff4 fs6 fc0 sc0 ls1e ws32">with their OAM depending on the degree of<span class="_ _1"></span><span class="ls1a ws33"> ellipticity. Both vortex<span class="_ _1"></span> and astigmatic </span></div><div class="t m0 x1 h7 y25 ff4 fs6 fc0 sc0 ls19 ws34">component<span class="_ _2"></span>s contribute to t<span class="_ _2"></span>he OAM of such beam<span class="_ _2"></span>s and these contributio<span class="_ _2"></span>ns of both </div><div class="t m0 x1 h7 y26 ff4 fs6 fc0 sc0 ls25 ws35">component<span class="_ _2"></span>s (vortex and ast<span class="_ _2"></span>igmatic) to the O<span class="_ _2"></span>AM can change with the <span class="_ _2"></span>propagation of suc<span class="_ _2"></span>h </div><div class="t m0 x1 h7 y27 ff4 fs6 fc0 sc0 ls17 ws36">elliptical beams [2,1<span class="_ _1"></span>3]. </div><div class="t m0 x9 h7 y28 ff4 fs6 fc0 sc0 ls2a ws37">The OAM of optical vortices, <span class="ls1b ws38">including the fractional<span class="_ _2"></span> OAM [14,15], can be m<span class="_ _2"></span>easured not </span></div><div class="t m0 x1 h7 y29 ff4 fs6 fc0 sc0 ls19 ws39">only by using t<span class="_ _2"></span>he cylindrical lens [1–5]<span class="_ _2"></span>, but also by many<span class="_ _2"></span> other ways, for exampl<span class="_ _2"></span>e, by using </div><div class="t m0 x1 h7 y2a ff4 fs6 fc0 sc0 ls25 ws3a">interferogram<span class="_ _2"></span>s [8,14] and a tri<span class="_ _2"></span>angular apertur<span class="_ _2"></span>e [16]. </div><div class="t m0 x9 h7 y2b ff4 fs6 fc0 sc0 ls2e ws3b">In this paper, we consider new laser beams with combined properties of vortex elliptic </div><div class="t m0 x1 h7 y2c ff4 fs6 fc0 sc0 ls26 ws3c">Gaussian beam<span class="_ _2"></span>s [9–13] and of ast<span class="_ _2"></span>igmatic vortex-<span class="ls1c ws3d">free laser beam<span class="_ _2"></span>s [7,8]. We call such a family </span></div><div class="t m0 x1 h7 y2d ff4 fs6 fc0 sc0 ls12 ws3e">of laser beams as astigmatic elliptical Gaussian (AEG) optical vortices. In the initial plan<span class="_ _1"></span>e, </div><div class="t m0 x1 h7 y2e ff4 fs6 fc0 sc0 ls2f ws3f">the AEG-vortex is an <span class="_ _2"></span><span class="ff2 ls3 ws1">n<span class="ff4 ls30 ws40">-fold degenerate circ<span class="_ _2"></span>ularly symme<span class="_ _2"></span>tric intensity null<span class="_ _2"></span> embedded into the<span class="_ _2"></span> </span></span></div><div class="t m0 x1 h7 y2f ff4 fs6 fc0 sc0 ls31 ws41">center of the waist of an ellip<span class="ls2b ws42">tic Gaussian beam, whose waist radii along the Cartesi<span class="_ _2"></span>an axes </span></div><div class="t m0 x1 h7 y30 ff4 fs6 fc0 sc0 ls1e ws43">are related by a certain relation, and then p<span class="_ _1"></span>a<span class="ls32 ws44">ssed through a cylindrical lens rotated in<span class="_ _1"></span> the </span></div><div class="t m0 x1 h7 y31 ff4 fs6 fc0 sc0 ls16 ws45">initial plane around th<span class="_ _1"></span>e optical axis by an a<span class="_ _1"></span>ngl<span class="ls33 ws46">e of 45 degrees with respect to<span class="_ _1"></span> the Cartesian </span></div><div class="t m0 x1 h7 y32 ff4 fs6 fc0 sc0 ls14 ws47">axes. Such a beam propagates in<span class="ls2f ws48"> free space<span class="_ _2"></span> preserving its struct<span class="ls1c ws14">ure up to scale and rotation. </span></span></div><div class="c xa y33 w2 ha"><div class="t m0 x0 hb y34 ff7 fs7 fc0 sc0 ls3 ws1"> Vol. 27, No. 2 | 21 Jan 2019 | OPTICS EXPRESS 657<span class="fc3 sc0"> </span></div></div><div class="c xb y35 w3 hc"><div class="t m0 x0 hd y36 ff8 fs8 fc0 sc0 ls3 ws1">#347499<span class="fc3 sc0"> </span></div></div><div class="c xc y35 w4 hc"><div class="t m0 x0 hd y36 ff8 fs8 fc0 sc0 ls3 ws1">https://doi.org/10.1364/OE.27.000657<span class="fc3 sc0"> </span></div></div><div class="c xb y37 w5 hc"><div class="t m0 x0 hd y36 ff8 fs8 fc0 sc0 ls3 ws1">Journal © 2019</div></div><div class="c xd y37 w6 hc"><div class="t m0 x0 hd y36 ff8 fs8 fc0 sc0 ls3 ws1">Received 8 Oct 2018; revised 14 Nov 2018; accepted 15 Nov 2018; published 8 Jan 2019<span class="fc3 sc0"> </span></div></div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6255fbaf47503a0a93f3825e/bg2.jpg"><div class="t m0 x1 h7 y38 ff4 fs6 fc0 sc0 ls29 ws49">Moreover, the far field (Fourier transform) for <span class="ls13 ws26">AEG beams is located from the initial plane by </span></div><div class="t m0 x1 h7 y39 ff4 fs6 fc0 sc0 ls1c ws14">a distance of double focal length<span class="ls13 ws4a"> of the cylindrical lens. The normalized total OAM of such </span></div><div class="t m0 x1 h7 y3a ff4 fs6 fc0 sc0 ls1b ws4b">beams is calculate<span class="_ _2"></span>d. It turned out to be equal t<span class="_ _2"></span>o the algebraic sum of two terms, one of w<span class="_ _2"></span>hich </div><div class="t m0 x1 h7 y3b ff4 fs6 fc0 sc0 ls2a ws4c">is equal to the topological charge of the opti<span class="_ _2"></span>cal<span class="ls1b ws4d"> vortex, and the sec<span class="_ _2"></span>ond is equal to the <span class="_ _2"></span>OAM of </span></div><div class="t m0 x1 h7 y3c ff4 fs6 fc0 sc0 ls2d ws4e">the astigmatic elliptical Gaussian beam. These two terms can both strengthen and compensate </div><div class="t m0 x1 h7 y3d ff4 fs6 fc0 sc0 ls3 ws4f">each other up to zero. In cont<span class="_ _2"></span>rast to [13], the magnitude of the <span class="ls2f ws3f">contributi<span class="_ _2"></span>ons of the vortex and </span></div><div class="t m0 x1 h7 y3e ff4 fs6 fc0 sc0 ls16 ws50">astigmatic componen<span class="_ _1"></span>ts to the OAM does not vary with the distance. </div><div class="t m0 x1 h9 y3f ff1 fs6 fc0 sc0 ls29 ws51">2. Beam amplitude at double focal length from cy<span class="_ _2"></span>lindrical lens </div><div class="t m0 x1 h7 y40 ff4 fs6 fc0 sc0 ls29 ws52">When a circularly symmetric optical vort<span class="ls15 ws53">ex with an integer topological charg<span class="_ _1"></span>e <span class="ff2 ls3 ws1">n</span><span class="ls24 ws54">, embedde<span class="_ _2"></span>d </span></span></div><div class="t m0 x1 h7 y41 ff4 fs6 fc0 sc0 ls1f ws55">into the waist of an elliptical Gaussian beam, propagates through a cylindrical lens, rotated in </div><div class="t m0 x1 h7 y42 ff4 fs6 fc0 sc0 ls31 ws41">the transverse plane by an angle <span class="ff6 ls3 ws1">α</span><span class="ls11 ws56">, the complex am<span class="_ _2"></span>plitude of su<span class="ls20 ws57">ch a light field immediately </span></span></div><div class="t m0 x1 h7 y43 ff4 fs6 fc0 sc0 ls20 ws19">after the cylindrical lens has the form: </div><div class="t m0 x1 h7 y44 ff4 fs6 fc0 sc0 ls3 ws1"> </div><div class="t m2 xe he y45 ff9 fs9 fc0 sc0 ls34 ws1">()</div><div class="t m0 xf hf y46 ff4 fsa fc0 sc0 ls35 ws1">22</div><div class="t m0 x10 hf y47 ff4 fsa fc0 sc0 ls36 ws1">22</div><div class="t m0 x11 hf y48 ff4 fsa fc0 sc0 ls37 ws1">22<span class="_ _3"> </span>2<span class="_ _0"></span>2</div><div class="t m0 x12 h10 y49 ff4 fsb fc0 sc0 ls38 ws1">(,<span class="_ _4"> </span>)<span class="_ _5"> </span>e<span class="_ _6"></span>x<span class="_ _6"></span>p</div><div class="t m0 x13 h10 y4a ff4 fsb fc0 sc0 ls39 ws1">cos<span class="_ _7"> </span>sin<span class="_ _8"> </span>sin<span class="_ _9"> </span>2</div><div class="t m0 x14 h10 y4b ff4 fsb fc0 sc0 ls3a ws1">exp<span class="_ _a"> </span>.</div><div class="t m0 x15 h10 y4c ff4 fsb fc0 sc0 ls3b ws1">22<span class="_ _2"></span>2</div><div class="t m0 x16 h11 y4d ff2 fsa fc0 sc0 ls3 ws1">n</div><div class="t m0 x17 h11 y4e ff2 fsa fc0 sc0 ls3 ws1">n</div><div class="t m0 x18 h11 y4f ff2 fsa fc0 sc0 ls3 ws1">n</div><div class="t m0 xf h11 y50 ff2 fsa fc0 sc0 ls3c ws1">xy</div><div class="t m0 x19 h12 y51 ff2 fsb fc0 sc0 ls3d ws1">xy</div><div class="t m0 x1a h12 y52 ff2 fsb fc0 sc0 ls3e ws1">Ex<span class="_ _b"></span>y<span class="_ _c"> </span>w<span class="_ _d"> </span>x<span class="_ _e"></span>i<span class="_ _f"></span>y</div><div class="t m0 x1b h12 y53 ff2 fsb fc0 sc0 ls3f ws1">ww</div><div class="t m0 x1c h12 y54 ff2 fsb fc0 sc0 ls40 ws1">ikx<span class="_ _10"> </span>iky<span class="_ _7"> </span>ikxy</div><div class="t m0 x1d h12 y55 ff2 fsb fc0 sc0 ls41 ws1">ff<span class="_ _2"></span>f</div><div class="t m3 x1e h13 y56 ff9 fsc fc0 sc0 ls42 ws1">αα<span class="_ _e"></span>α</div><div class="t m0 x15 h14 y57 ff9 fsa fc0 sc0 ls3 ws1">−</div><div class="t m0 x1f h15 y58 ffa fsb fc0 sc0 ls43 ws1"></div><div class="t m0 x20 h16 y59 ff9 fsb fc0 sc0 ls44 ws1">=+<span class="_ _11"> </span>−<span class="_ _12"></span>−<span class="_ _13"></span>×</div><div class="t m0 x1f h15 y5a ffa fsb fc0 sc0 ls43 ws1"></div><div class="t m0 x1f h15 y5b ffa fsb fc0 sc0 ls43 ws1"></div><div class="t m0 x1f h15 y5c ffa fsb fc0 sc0 ls43 ws1"></div><div class="t m0 x21 h15 y5d ffa fsb fc0 sc0 ls45 ws1"></div><div class="t m0 x22 h16 y5e ff9 fsb fc0 sc0 ls46 ws1">×−<span class="_ _14"> </span>−<span class="_ _15"> </span>−</div><div class="t m0 x21 h15 y5f ffa fsb fc0 sc0 ls45 ws1"></div><div class="t m0 x21 h15 y60 ffa fsb fc0 sc0 ls45 ws1"></div><div class="t m0 x23 h7 y61 ff4 fs6 fc0 sc0 ls47 ws58"> (1) </div><div class="t m0 x9 h7 y62 ff4 fs6 fc0 sc0 ls1f ws59">We used the following designations in Eq<span class="_ _1"></span>. (1): (<span class="ff2 ls3 ws1">x<span class="ff4 ls48">, <span class="_ _4"> </span></span>y</span><span class="ls31 ws5a">) are Cartesian coordinates in the </span></div><div class="t m0 x1 h7 y63 ff4 fs6 fc0 sc0 ls18 ws5b">transverse plane <span class="ff2 ls3 ws1">z</span><span class="ls28 ws5c"> = 0 (<span class="ff2 ls3 ws1">z</span><span class="ls49 ws5d"> is the longitudinal Cartesian coordina<span class="_ _1"></span>te), <span class="ff2 ls3 ws1">w</span><span class="ls4a ws5e"> is the scaling factor for </span></span></span></div><div class="t m0 x1 h7 y64 ff4 fs6 fc0 sc0 ls15 ws5f">the optical vortex, <span class="ff2 ls3 ws1">w</span></div><div class="t m0 x24 h17 y65 ff2 fsd fc0 sc0 ls3 ws1">x</div><div class="t m0 x3 h7 y66 ff4 fs6 fc0 sc0 ls2f ws60"> and <span class="ff2 ls3 ws1">w</span></div><div class="t m0 x25 h17 y65 ff2 fsd fc0 sc0 ls3 ws1">y</div><div class="t m0 x26 h7 y66 ff4 fs6 fc0 sc0 ls13 ws61"> are the waist radii of the elliptic Gau<span class="_ _1"></span>ssian beam along the </div><div class="t m0 x1 h7 y67 ff4 fs6 fc0 sc0 ls48 ws62">Cartesian coor<span class="_ _2"></span>dinates, <span class="ff2 ls3 ws1">f</span><span class="ls1e ws63"> is the focal length of th<span class="_ _1"></span>e cylindrical lens, <span class="ffb ls3 ws1">α</span><span class="ls1f ws64"> is the tilt an<span class="_ _1"></span>gle of the lens </span></span></div><div class="t m0 x1 h7 y68 ff4 fs6 fc0 sc0 ls1f ws65">axis with respect to the vertical axis <span class="ff2 ls3 ws1">y<span class="ff4 ls48">, <span class="_ _16"> </span></span>k</span><span class="ls20 ws66"> = 2<span class="ff6 ls3 ws1">π<span class="ff4">/<span class="ffb">λ</span><span class="ls33 ws67"> is the wavenumber of light with the </span></span></span></span></div><div class="t m0 x1 h7 y69 ff4 fs6 fc0 sc0 ls30 ws1">wavelength <span class="ffb ls3">λ</span><span class="ls4b ws68">. At a distance </span><span class="ff2 ls3">z</span><span class="ls17 ws69"> from the initial plane, the complex amplitude of the beam of Eq. </span></div><div class="t m0 x1 h7 y6a ff4 fs6 fc0 sc0 ls49 ws6a">(1) is defined by the Fresn<span class="_ _1"></span>el transform: </div><div class="t m0 x1 h7 y6b ff4 fs6 fc0 sc0 ls3 ws1"> </div><div class="t m4 xf h18 y6c ff9 fse fc0 sc0 ls4c ws1">()</div><div class="t m5 x1d h19 y6d ff4 fsf fc0 sc0 ls4d ws1">22<span class="_ _17"> </span>2<span class="_ _18"></span>2</div><div class="t m5 x27 h19 y6e ff4 fsf fc0 sc0 ls4e ws1">22</div><div class="t m5 x22 h19 y6f ff4 fsf fc0 sc0 ls4f ws1">22<span class="_ _3"> </span>2<span class="_ _19"></span>2</div><div class="t m5 x24 h19 y70 ff4 fsf fc0 sc0 ls50 ws1">22</div><div class="t m5 x28 h1a y71 ff4 fs10 fc0 sc0 ls51 ws1">(,<span class="_ _1"></span>,<span class="_ _e"></span>)<span class="_ _1a"> </span>e<span class="_ _6"></span>x<span class="_ _6"></span>p<span class="_ _1b"> </span>e<span class="_ _6"></span>x<span class="_ _6"></span>p</div><div class="t m5 x29 h1a y72 ff4 fs10 fc0 sc0 ls52 ws1">22<span class="_ _1c"></span>2</div><div class="t m5 x29 h1a y73 ff4 fs10 fc0 sc0 ls53 ws1">cos<span class="_ _7"> </span>sin<span class="_ _8"> </span>sin<span class="_ _9"> </span>2</div><div class="t m5 x2a h1a y74 ff4 fs10 fc0 sc0 ls54 ws1">exp</div><div class="t m5 x2b h1a y75 ff4 fs10 fc0 sc0 ls55 ws1">22<span class="_ _2"></span>2</div><div class="t m5 x2c h1a y76 ff4 fs10 fc0 sc0 ls56 ws1">()</div><div class="t m5 x2a h1a y77 ff4 fs10 fc0 sc0 ls54 ws1">exp<span class="_ _1d"> </span>.</div><div class="t m5 x2d h1a y78 ff4 fs10 fc0 sc0 ls57 ws1">22</div><div class="t m5 x2e h1b y79 ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x19 h1b y7a ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x2f h1b y7b ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x27 h1b y7c ff2 fsf fc0 sc0 ls58 ws1">xy</div><div class="t m5 x30 h1c y7d ff2 fs10 fc0 sc0 ls59 ws1">ik<span class="_ _15"> </span>iku<span class="_ _1e"> </span>ikv<span class="_ _1f"> </span>x<span class="_ _20"> </span>y</div><div class="t m5 x31 h1c y7e ff2 fs10 fc0 sc0 ls5a ws1">Eu<span class="_ _b"></span>v<span class="_ _b"></span>z<span class="_ _21"> </span>w<span class="_ _c"> </span>x<span class="_ _4"> </span>i<span class="_ _22"></span>y</div><div class="t m5 x32 h1c y7f ff2 fs10 fc0 sc0 ls5b ws1">zz<span class="_ _23"></span>z<span class="_ _24"> </span>w<span class="_ _25"></span>w</div><div class="t m5 x33 h1c y80 ff2 fs10 fc0 sc0 ls53 ws1">ikx<span class="_ _26"> </span>iky<span class="_ _8"> </span>ikxy</div><div class="t m5 x12 h1c y81 ff2 fs10 fc0 sc0 ls5c ws1">ff<span class="_ _2"></span>f</div><div class="t m5 x34 h1c y82 ff2 fs10 fc0 sc0 ls53 ws1">ikx<span class="_ _1e"> </span>iky<span class="_ _1e"> </span>ik<span class="_ _27"> </span>xu<span class="_ _28"> </span>yv</div><div class="t m5 x35 h1c y83 ff2 fs10 fc0 sc0 ls5d ws1">dxdy</div><div class="t m5 x36 h1c y84 ff2 fs10 fc0 sc0 ls5e ws1">zz<span class="_ _29"> </span>z</div><div class="t m6 x37 h1d y85 ff9 fs11 fc0 sc0 ls3 ws1">π</div><div class="t m6 x38 h1d y86 ff9 fs11 fc0 sc0 ls5f ws1">αα<span class="_ _e"></span>α</div><div class="t m5 x39 h1e y87 ff9 fsf fc0 sc0 ls60 ws1">∞∞</div><div class="t m5 x3a h1e y88 ff9 fsf fc0 sc0 ls3 ws1">−</div><div class="t m5 x3b h1e y89 ff9 fsf fc0 sc0 ls61 ws1">−∞<span class="_"> </span>−∞</div><div class="t m5 x3c h1f y8a ffa fs10 fc0 sc0 ls62 ws1"></div><div class="t m5 x3d h1f y8b ffa fs10 fc0 sc0 ls63 ws1"></div><div class="t m5 x14 h20 y8c ff9 fs10 fc0 sc0 ls3 ws1">−</div><div class="t m5 x3e h20 y8d ff9 fs10 fc0 sc0 ls64 ws1">=+<span class="_ _2a"> </span>+<span class="_ _2b"></span>−<span class="_ _2c"></span>−<span class="_ _2d"></span>×</div><div class="t m5 x3c h1f y8e ffa fs10 fc0 sc0 ls62 ws1"></div><div class="t m5 x3d h1f y8f ffa fs10 fc0 sc0 ls63 ws1"></div><div class="t m5 x3c h1f y90 ffa fs10 fc0 sc0 ls62 ws1"></div><div class="t m5 x3d h1f y91 ffa fs10 fc0 sc0 ls63 ws1"></div><div class="t m5 x3c h1f y92 ffa fs10 fc0 sc0 ls62 ws1"></div><div class="t m5 x3f h1f y93 ffa fs10 fc0 sc0 ls65 ws1"></div><div class="t m5 x40 h20 y94 ff9 fs10 fc0 sc0 ls66 ws1">×−<span class="_ _14"> </span>−<span class="_ _15"> </span>−<span class="_ _2e"> </span>×</div><div class="t m5 x3f h1f y95 ffa fs10 fc0 sc0 ls65 ws1"></div><div class="t m5 x3f h1f y96 ffa fs10 fc0 sc0 ls65 ws1"></div><div class="t m5 x3f h1f y97 ffa fs10 fc0 sc0 ls67 ws1"></div><div class="t m5 x41 h20 y98 ff9 fs10 fc0 sc0 ls3 ws1">+</div><div class="t m5 x40 h20 y99 ff9 fs10 fc0 sc0 ls68 ws1">×+<span class="_ _2f"></span>−</div><div class="t m5 x3f h1f y9a ffa fs10 fc0 sc0 ls67 ws1"></div><div class="t m5 x3f h1f y9b ffa fs10 fc0 sc0 ls67 ws1"></div><div class="t m5 x39 h21 y9c ffa fs12 fc0 sc0 ls69 ws1"></div><div class="t m5 x42 h7 y9d ff4 fs6 fc0 sc0 ls47 ws6b"> (2) </div><div class="t m5 x9 h7 y9e ff4 fs6 fc0 sc0 ls15 ws6c">If the tilt angle of the cylindrical lens is 45 degrees (<span class="ffb ls3 ws1">α<span class="ff2 ls6a"> =<span class="_ _0"></span><span class="ff4 ls3"> <span class="_ _30"></span><span class="ff6">π</span><span class="ls6b ws6d">/4) and the <span class="_ _2"></span>distance after the </span></span></span></span></div><div class="t m5 x1 h7 y9f ff4 fs6 fc0 sc0 ls6c ws6e">lens equals the d<span class="_ _1"></span>ouble focal length (<span class="ff2 ls3 ws1">z</span><span class="ls20 ws6f"> = 2<span class="ff2 ls3 ws1">f</span><span class="ls31 ws70">), then the Fresnel transf<span class="ls6d ws71">orm in<span class="_ _1"></span> Eq. (2<span class="_ _1"></span>) become<span class="_ _1"></span>s the </span></span></span></div><div class="t m5 x1 h7 ya0 ff4 fs6 fc0 sc0 ls6e ws72">Fourier transf<span class="_ _2"></span>orm (up to a phase factor <span class="_ _2"></span>before the integrals) of t<span class="_ _2"></span>he amplitude describi<span class="_ _2"></span>ng an </div><div class="t m5 x1 h7 ya1 ff4 fs6 fc0 sc0 ls13 ws73">optical vortex, embedded into an elliptical Gaussian b<span class="_ _1"></span>eam and passed a cylindrical lens: </div><div class="t m5 x1 h7 ya2 ff4 fs6 fc0 sc0 ls3 ws1"> </div><div class="t m7 x30 he ya3 ff9 fs9 fc0 sc0 ls6f ws1">()</div><div class="t m5 x43 h19 ya4 ff4 fsf fc0 sc0 ls4d ws1">22</div><div class="t m5 x4 h19 ya5 ff4 fsf fc0 sc0 ls70 ws1">22</div><div class="t m5 x44 h19 ya6 ff4 fsf fc0 sc0 ls4e ws1">22</div><div class="t m5 x2d h1a ya7 ff4 fs10 fc0 sc0 ls51 ws1">(,<span class="_ _1"></span>,<span class="_ _29"> </span>2<span class="_ _4"> </span>)<span class="_ _31"> </span>e<span class="_ _6"></span>x<span class="_ _6"></span>p</div><div class="t m5 x45 h1a ya8 ff4 fs10 fc0 sc0 ls71 ws1">44<span class="_ _32"></span>4</div><div class="t m5 x46 h1a ya9 ff4 fs10 fc0 sc0 ls56 ws1">()</div><div class="t m5 x45 h1a yaa ff4 fs10 fc0 sc0 ls54 ws1">exp<span class="_ _33"> </span>.</div><div class="t m5 x47 h1a yab ff4 fs10 fc0 sc0 ls72 ws1">22</div><div class="t m5 x48 h1b yac ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x2c h1b yad ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x18 h1b yae ff2 fsf fc0 sc0 ls3 ws1">n</div><div class="t m5 x44 h1b yaf ff2 fsf fc0 sc0 ls58 ws1">xy</div><div class="t m5 x49 h1c yb0 ff2 fs10 fc0 sc0 ls59 ws1">ik<span class="_ _31"> </span>iku<span class="_ _1e"> </span>ik<span class="_ _2"></span>v</div><div class="t m5 x4a h1c yb1 ff2 fs10 fc0 sc0 ls5a ws1">Eu<span class="_ _b"></span>v<span class="_ _b"></span>z<span class="_ _28"> </span>f</div><div class="t m5 x4b h1c yb2 ff2 fs10 fc0 sc0 ls73 ws1">ff<span class="_ _34"></span>f</div><div class="t m5 x4c h1c yb3 ff2 fs10 fc0 sc0 ls53 ws1">x<span class="_ _20"> </span>y<span class="_ _3"> </span>ikxy<span class="_ _28"> </span>ik<span class="_ _d"> </span>xu<span class="_ _28"> </span>yv</div><div class="t m5 x3e h1c yb4 ff2 fs10 fc0 sc0 ls5d ws1">w<span class="_ _35"> </span>x<span class="_ _36"> </span>iy<span class="_ _37"> </span>dxdy</div><div class="t m5 x4d h1c yb5 ff2 fs10 fc0 sc0 ls74 ws1">ff</div><div class="t m5 x4e h1c yb6 ff2 fs10 fc0 sc0 ls75 ws1">ww</div><div class="t m6 x13 h1d yb7 ff9 fs11 fc0 sc0 ls3 ws1">π</div><div class="t m5 x48 h1e yb8 ff9 fsf fc0 sc0 ls60 ws1">∞∞</div><div class="t m5 x2b h1e yb9 ff9 fsf fc0 sc0 ls3 ws1">−</div><div class="t m5 x4f h1e yba ff9 fsf fc0 sc0 ls61 ws1">−∞<span class="_"> </span>−∞</div><div class="t m5 x44 h1f ybb ffa fs10 fc0 sc0 ls76 ws1"></div><div class="t m5 x15 h20 ybc ff9 fs10 fc0 sc0 ls3 ws1">−</div><div class="t m5 x30 h20 ybd ff9 fs10 fc0 sc0 ls77 ws1">==<span class="_ _38"> </span>+<span class="_ _27"> </span>×</div><div class="t m5 x44 h1f ybe ffa fs10 fc0 sc0 ls76 ws1"></div><div class="t m5 x44 h1f ybf ffa fs10 fc0 sc0 ls76 ws1"></div><div class="t m5 x50 h1f yc0 ffa fs10 fc0 sc0 ls78 ws1"></div><div class="t m5 x51 h20 yc1 ff9 fs10 fc0 sc0 ls3 ws1">+</div><div class="t m5 x52 h20 yc2 ff9 fs10 fc0 sc0 ls79 ws1">×+<span class="_ _39"></span>−<span class="_ _3a"></span>−<span class="_ _3b"></span>−<span class="_ _3c"></span>−</div><div class="t m5 x50 h1f yc3 ffa fs10 fc0 sc0 ls78 ws1"></div><div class="t m5 x50 h1f yc4 ffa fs10 fc0 sc0 ls78 ws1"></div><div class="t m5 x50 h1f yc5 ffa fs10 fc0 sc0 ls78 ws1"></div><div class="t m5 x48 h22 yc6 ffa fs13 fc0 sc0 ls7a ws1"></div><div class="t m5 x53 h7 yc7 ff4 fs6 fc0 sc0 ls47 ws74"> (3) </div><div class="t m5 x9 h7 yc8 ff4 fs6 fc0 sc0 ls15 ws75">Our next goal is to calculate the integral in Eq. (3). We rewrite it in the dimensionless </div><div class="t m5 x1 h23 yc9 ff4 fs6 fc0 sc0 ls24 ws1">variables: <span class="_ _1"></span><span class="ff2 ls3">x<span class="ff4">/</span>w<span class="ff4"> <span class="_ _e"></span><span class="ff9">→</span></span>x</span><span class="ls48">, <span class="_ _30"></span><span class="ff2 ls3">y<span class="ff4">/</span>w<span class="ff4"> <span class="_ _1"></span><span class="ff9">→</span></span>y</span>, <span class="_ _e"></span><span class="ff2 ls3">u<span class="ff4">/</span>w<span class="ff4"> <span class="_ _e"></span><span class="ff9">→</span></span>u</span>, <span class="_ _e"></span><span class="ff2 ls3">v<span class="ff4">/</span>w<span class="ff4"> <span class="_ _e"></span><span class="ff9">→</span></span>v</span>, <span class="_ _30"></span><span class="ff2 ls3">w<span class="ff4">/<span class="_ _2"></span><span class="ff2">w</span></span></span></span></div><div class="t m5 x54 h17 yca ff2 fsd fc0 sc0 ls3 ws1">x</div><div class="t m5 x55 h7 ycb ff4 fs6 fc0 sc0 ls20 ws76"> = <span class="ffb ls3 ws1">γ<span class="ff4 ls48">, <span class="_ _e"></span></span><span class="ff2">w<span class="ff4">/</span>w</span></span></div><div class="t m5 x56 h17 yca ff2 fsd fc0 sc0 ls3 ws1">y</div><div class="t m5 x57 h7 ycb ff4 fs6 fc0 sc0 ls20 ws76"> = <span class="ffb ls3 ws1">β<span class="ff4 ls48">, <span class="_ _e"></span></span><span class="ff2">z</span></span></div><div class="t m5 x58 h24 yca ff4 fsd fc0 sc0 ls3 ws1">0</div><div class="t m5 x59 h7 ycb ff4 fs6 fc0 sc0 ls20 ws76"> = 2<span class="ff2 ls3 ws1">f</span><span class="ls24 ws77">, where <span class="ff2 ls3 ws1">z</span></span></div><div class="t m5 x5a h24 yca ff4 fsd fc0 sc0 ls3 ws1">0</div><div class="t m5 x5b h7 ycb ff4 fs6 fc0 sc0 ls7b ws78"> = <span class="ff2 ls4a ws1">kw</span></div><div class="t m5 x5c h24 ycc ff4 fsd fc0 sc0 ls3 ws1">2</div><div class="t m5 x5d h7 ycb ff4 fs6 fc0 sc0 ls7c ws1">/2. </div><div class="t m5 x1 h7 ycd ff4 fs6 fc0 sc0 ls7d ws79">Then, Eq. (1<span class="_ _1"></span>) reads as </div><div class="t m5 x1 h7 yce ff4 fs6 fc0 sc0 ls3 ws1"> </div><div class="t m8 x5e h25 ycf ff9 fs14 fc0 sc0 ls7e ws1">()<span class="_ _3d"></span>(<span class="_ _3e"></span>)</div><div class="t m9 x16 h26 yd0 ff9 fs15 fc0 sc0 ls7f ws1">()</div><div class="t m5 x5f h27 yd1 ff4 fs16 fc0 sc0 ls80 ws1">22<span class="_ _29"> </span>2<span class="_ _e"></span>2<span class="_ _29"> </span>2<span class="_ _3f"> </span>2</div><div class="t m5 x60 h28 yd2 ff4 fs17 fc0 sc0 ls81 ws1">,,<span class="_"> </span>0<span class="_ _40"> </span>e<span class="_ _41"></span>x<span class="_ _41"></span>p<span class="_ _42"> </span>2<span class="_"> </span>.</div><div class="t m5 x61 h29 yd3 ff2 fs16 fc0 sc0 ls3 ws1">n</div><div class="t m5 x62 h29 yd4 ff2 fs16 fc0 sc0 ls3 ws1">n</div><div class="t m5 x63 h2a yd2 ff2 fs17 fc0 sc0 ls82 ws1">Ex<span class="_ _43"></span>y<span class="_ _6"></span>z<span class="_ _1a"> </span>x<span class="_ _0"></span>i<span class="_ _23"></span>y<span class="_ _44"> </span>x<span class="_ _3"> </span>y<span class="_ _45"> </span>i<span class="_ _23"></span>x<span class="_ _45"> </span>i<span class="_ _23"></span>y<span class="_ _46"> </span>i<span class="_ _23"></span>x<span class="_ _47"></span>y</div><div class="t ma x5 h2b yd2 ff9 fs18 fc0 sc0 ls83 ws1">γβ</div><div class="t m5 x12 h2c yd2 ff9 fs17 fc0 sc0 ls84 ws1">==<span class="_ _2"></span>+<span class="_ _15"> </span>−<span class="_ _48"> </span>−<span class="_ _49"> </span>−<span class="_ _4"></span>−<span class="_ _4"></span>−<span class="_ _4a"> </span><span class="ff4 fs6 ls47 ws7a"> (4) </span></div><div class="c xa y33 w2 ha"><div class="t m5 x0 hb y34 ff7 fs7 fc0 sc0 ls3 ws1"> Vol. 27, No. 2 | 21 Jan 2019 | OPTICS EXPRESS 658<span class="fc3 sc0"> </span></div></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
像散椭圆高斯光
