加速多边形激光束

加速多边形激光束 光学快报 OCIS 代码 050 1970 100 5090 140 3300 350 5500 加速光具有不寻常的特性 在张力最大 沿着弯曲的路径下的线性衍射传播 在沿直 线光轴传播中心总光束质量不变 这些所谓的自我弯曲光束收到了极大的关注 因为他们 有多种用途 比如远程光谱微粒子排序 此外 高光的强度 在一般情况下 往往在非线 性光学传播过程中 这些光束弯曲的路径 允许在某些应用中增加空间分辨率 最近加快傍轴波动方程从理想的衍射解决方案的艾里激光束有限能源的示范带动等光 束利益 横向加速激光 随后季风傍除了艾里激光束的波动方程解的唯一完整的家族 是 衍射和加速过程中的传播 就像艾里激光束一样 有限能源的版本仅准衍射免费 在这篇文章中 我要介绍另一个家族的有限能源的加速光束 虽然不无衍射 但在延 长距离的线性衍射下仍倾向于维持其属性 他们有很多 而不是一个 在光束总强度峰加 速运算抽动轴与最大角间距相对的优势 这些光束将被简称为加快的正多边形光束 RPBs 因为峰同样是沿着一个圆的间距 其中心是光轴和其半径生长与繁殖 正因为如此 强峰 是在扩大的圆圈内刻章模块化的多边形的顶点 它将会显示 从理论上讲 任何一个奇数 峰超重力可以在透镜的焦平面与特定的阶段 只在有限的通过一定数量的光阑 只有通过 光阑的数量这个制约因素限制了峰的数量 这个阶段的面具来推导通过傅里叶描述成像 这相当于傍轴的波动方程 也就是说 对于 给平面波浪输入 强度分布 h 的有限的衍射成像系统焦距的 f 上添加了一个阶段 n 在给定瞳 规范化坐标点 1 2 exp dxdyusyxikFutsh 是瞳孔孔径坐标 u yxx 22 tysxyxuyxusxF n 是焦距的强度 z 是焦平面沿着光轴的距离 是 2zffzf fyfxtss ii 正常化的横向坐标 在的空间里 K 是波束 定义的部分学者认为是一个半径 yx ap r 经常固定相近似方程 1 这样 在临界点逼近 就像 或者 c X0 usXF cx 02 suxxx ccn 02 tuyyx ccn 2 这种近似假设的 F 点集不单单是在 Xc 这是经常并非如此奇异 然而 最近表明 h 的重要 特性往往可以推导出临界点的 UTH 在其中的 F 的点集是奇异的 或者 C DX 其中 DET 表示行列式 这个所谓的奇点集 S 投影到图像平面坐0 det 2 usxF cx 标突变理论 CT 被称为分岔集 B 它提供了在图像空间的强度最高的地区 被称作 us 散焦 CT 是致力于研究接近单数 退化变量或关键点函数行为的数学的分支 CT 所产生的 凸极结论是在退化点 函数可被归类于类似于扰动有限数量的拓扑行为方式 例如 F 1 可以被认为是系列函数 这个函数的关键点 n 是在控制变量的 s 的影响下被扰动 的 这也导致了 3h 的演变 这些退化类变量可由在其他属性中 B 的几何学所定义 事实 上 有限能源通风和零阶抛物线是著名规范灾难双曲脐部溶血性尿毒综合病的示例 自我 弯曲度和无衍射属性可由 B 5 衍生 RPBS 的创始人属于另一个接近于相关圆锥截面超 生内镜系列 超生内镜和溶血性尿毒症是分开的实际变量 但结合了一个复杂的转换 所 以这个 RPBS 有相似性质不应感到惊讶 图 1 图 1 一 分岔集设置 B 在 4 83 厘米 二 在 h 在 z 4 83 和 c 在 h 在 z 6 13 厘米 显示与mm 为了指定 ii yx 的分辨率 i 3 ii 5 and iii 7 与平面波输入 7 RPBs 分别为 1 21 0 6 0 277 10 4 厘米 DN 1 个 分别为调制相 位掩模瓦特 行 D 是 C 为高斯光束输入的重复 注意 ii 及 3 4 在规模上的变化 设齐次多项式的阶数 n 3 其中 n 是奇数的形式 此外 限制对 n 的对称性 这是不变 n 的 XY 交换操作和奇数根据坐标反转 或 和 然后 xyyx nn yxyx nn 2 1 0 n k knkkkn kn yxyxcyx 其中 w 是一个可调的调制 代入 3 1 F 产生的一个临界点以下简条件 0det 2 2 2 2 2 2 2 2 yx uy ux yx n n n n 3 N 它表明 选择 3 4 线性 长期消失 CK 或者 取得了刻0 2 2 2 2 yx nn 意对称 FB 在 3 对 n 3 5 尤其是在非零 U 平面与 n 3 B 是欧盟著名的三角肌在这 种所谓的尖点在三角肌形成一个等边三角形的顶点 如图三角肌 1 A 它提出三个的 联锁半三次集抛物线 同样的 实现了类似的结果为 n 除非在一个普通的顶点的尖点 B 是由 5 个环环相扣半三次集抛物线五角大楼 这 B 也如图 1 A 这一结果推广到任何奇 怪的列印 一个小的代数显示 4 中的约束满足 Ck 与 C0 1 如果遵循下面的方程组 2 1 2 nn c 0 1 2 1 2 1 22 kknk ckkckkcknkn 4 其中和 2 1 1 nk20 k2 1 2 nkn 设是一套系数 n 次为了掩体 一些例子是 和 n C 3 1 3 c 10 5 1 5 c 当 5 和 3 4 以一种模式取代 35 21 7 1 7 c 出现 揭示了 4 通过以下来满足 2 2 2 22 1 2 d n R nn u yx 5 方程 6 指出 孔径对于给定的图像平面的退化临界点的位置处于一个圆上 它的半径 是 u 的函数 对于一个给定的 u 在任何图像平面里都能通过 Rd 把 X 转换成圆柱坐标 而轻易发现 B 孔径的坐标可借助 参数化 可被固定点条件 2 取代而发现 s B 图 1 a 显示了在 u 为非零实数 n 3 5 和 7 条件下得典型 B 我们可以看到 他们是 由 n 联锁半三次集抛物线的尖端组成 位置位于正所变形 n sided 的顶点 然而 在焦平面 u 0 中 由于单个独立的 EU 退化对应于等级波包 B 只是简单的轴心 在焦摄作用影响 下分裂形成几何图形 Fig 1 a 将集中在轴强度h密切反映 B 在 U 0时 作为总光束传播 个别奇点不会最初是解 决由于衍射 然而 他们将会很快增加强度 h 的扩大 最终开始主导 本文的例子是一个 平面波输入如图 1 b 具有系统 f 75 厘米 波长 633 nm RAP 1 厘米 h 的计算公 式 1 相当于 B 中的 a c 中行 h 后的进一步传播 图 1 行 C 是一种重复 除了输入与高斯光束的束腰 一点 58 厘米和 的分辨率 会出现一个小峰值强度恶化 但结果是可比的 过 u 的范围大 强度极大值将大幅见顶在风口浪尖上的点 虽然在价值 图 2 彩色在线 产品的高峰期的风口浪尖强度和 n 绘制与图口罩 厘米 用平面波输 入和归一化总束流强度 1 图 2 最大值 当然会减少总束扩张 事实上 一旦出现的模式 它会继续下去 直到路 6 大 于有限瞳孔孔径 过渡 从而开始在 这一点后 会逐渐被吸 221 max 1 2 n ap Rnnu 收为尖的峰到 PSF 图 2 中的 n 倍图的波包超过 z 的范围比尖点强度 总的束流强度 整合为 11 主要之 间的 RPBs 的差异是风口浪尖点之间的权力分工 虽然结果会有所不同 用 w 一般在 w 的 增加将扩大总的光束尺寸 降低初始峰值 INTEN 密度 减少加速 但增加的范围和降低率 峰值强度与 z 脱落 代入 6 到临界点的条件 也揭示了风口浪尖点加速运行的功能 依赖 在风口浪尖点在图像平面上圆的半径将扩大为 U 的电源 或 7 21 21 21 1 2 2 1 2 2 nn n k k n cusp u nn c n R 因此 对于 n 3 时 超重力将加速与 u 二次像艾里激光束 这是一个与其关系密切的 家族关系的反思 产生多个加速极大值有其不利因素 强度最大值将小于最大的空气波束的相同总功率 一个显而易见的原因是 功率将被分为若干个极大值而非一个 然而 这分裂的光束并不 能完全证明在强度上的降低 空气波束角的强度极大值是由于等级零的 HU 退化 RPBs 的 极大值是一个等级 1 退化的结果 阿诺尔曾经调查了振荡积分强度的渐进行为如 1 并发 现了不同类型的退化导致 h 和 的不同顺序 这种退化可以划分一种所谓的奇指数 规 定 0 i e 为即较大的 导致较大的 h 在未退化的临界点 为 1 Oh 0 然而 在尖端 为四分之一在 HU 为三分之一 可能的话 只需简单摇动的光束功率就 可以缓和这种关系 另一个缺点是分辨率不共享他们的 HU 兄弟的自由衍射机制 然而 形 式 3 是众所周知的散焦迟钝 即 衍射 它满足对称约束 确保关于 u 的奇数条件在 h 的泰勒扩展就将消失 事实上 这个 PRGG 光束的 n 3 5 已被证明在一定条件下可以更有 效地达到成像物系的极限 此外 EU 是众所周知的不太敏感的某些离轴像差 可以使其性 能更强大的偏差 例如 在 y 射线的影响下 HU 退化的空气波束会分裂成一个尖点和一个 褶层 其中有一个六分之一的 这种优势将申请唯一的 n 3 液体掩饰物已导致加速峰强度的传播沿奇数边的正多边形的顶点扩大 强度极大值模 式将稳定在一个大范围的 Z 中 功率依赖的加速度和近似计算范围的表达式已经规定了 在某些应用中并联峰可以允许比单一的加速度峰值更高的采样密度 并且其独特的多边形 几何形状可能有不可预见的用途 Accelerating regular polygon beams OPTICS LETTERS OCIS codes 050 1970 100 5090 140 3300 350 5500 Accelerating beams have the unusual property that an in tensity maximum propagates along a curved path under linear diffraction though the center of mass of the total beam propagates in a straight line along the optic axis These so called self bending beams have received a great deal of attention because they have a multitude of uses from remote spectroscopy to micro particle sorting Moreover high optical intensities in general tend to facilitate nonlinear optical processes and the curved paths of these beams allow increased spatial resolution in some applications Interest in such beams was recently spurred by the demonstration of accelerating finite energy Airy beams that were derived from ideal diffraction free solutions of the paraxial wave equation Parabolic beams were subsequently shown to be the only other complete family of solutions of the paraxial wave equation besides Airy beams that is diffraction free and that accelerates during propagation Like Airy beams the finite energy versions are only quasi diffraction free This Letter to my knowledge introduces another family of finite energy accelerating beams that though not diffraction free tends to maintain its properties under linear diffraction over extended distances They have the advantage that many as opposed to one intensity peaks within the total beam accelerate away from the op tic axis with maximal angular spacing relative to one another These beams will be referred to as accelerating regular polygon beams RPBs because the peaks are equally spaced along a circle whose center is the optic axis and whose radius grows with propagation As such the intensity peaks are at the vertices of expanding regular polygons inscribed within the circle It will be shown that in theory any RPB with an odd number of peaks can be produced in the focal plane of a lens with a specific phase only mask at the finite pupil Only manufacturing constraints limit the number of peaks The phase of the masks will be derived by means of the Fourier optics description of imaging which is equivalent to the paraxial wave equation To wit for a plane wave input the intensity distribution h of a diffraction limited imaging system of focal length f with an added phase n at the exit pupil is given in normalized coordinates by 1 2 exp dxdyusyxikFutsh Whereare pupil 22 tysxyxuyxusxF n yxx coordinates is the defocus strength z is the distance from the focal plane u 2zffzf along the optic axis is the normalization of the transverse coordinates fyfxtss ii in image space k is the wave number 2 and defines the finite pupil that will be ii yx assumed to be a circle of radius Rap Equation 1 is frequently approximated by the stationary phase approximation at critical points such that or c X0 usxF cx 02 suxxx ccn 2 02 tuyyx ccn This approximation assumes that the Hessian of F is not singular at xc which is frequently not the case However it was recently shown that important properties of h can often be deduced by the set of critical points at which the Hessian of F is singular or usxc0 det 2 usxF cx where det denotes the determinant The projection of this so called singularity set S onto the image plane coordinates is known in catastrophe theory CT as the bifurcation set B and us it gives the envelope of the highest intensity regions in image space that are known as caustics CT is a branch of mathematics devoted to the study of the behavior of functions near singular or degenerate critical points The salient conclusion arising from CT is that at degenerate points functions can be classified into a finite number of topological classes that behave in a manner similar to similar perturbations For instance F in 1 can be thought of as a family of functions in which the state of the critical points of the phase n is perturbed under the influence of the control variables which leads to the evolution of h These degeneracy classes are us characterized by the geometry of their B among other properties In fact the finite energy Airy and zero order parabolic beams are examples of a well known canonical catastrophe the hyperbolic umbilics HUs Both their self bending and diffraction free properties can be derived from their B The founding member of the RPBs belongs to another closely related conic family the elliptic umbilics EUs EUs and HUs are se parate classes for real variables but merge into one for complex transformations So it should not be surprising that RPBs have similar properties Fig 1 Color online a Bifurcation set B at z 4 83 cm b h at z 4 83 cm and c h at z 6 13 cm are shown versus mm for the RPBs of order i 3 ii 5 and iii 7 with a plane wave input The ii yx modulations w of the phase masks were 1 21 0 6 0 277 10 4 cm n 1 respectively Row d is a repeat of c for a Gaussian beam input Note the change in scale for ii and iii of d Let take the form of a homogeneous polynomial of order n 3 in which n is odd Also n constrain the symmetry of n so that it is invariant in an xy exchange operation and odd under coordinate inversion or Then xyyx nn yxyx nn 3 2 1 0 n k knkkkn kn yxyxcyx where w is an adjustable modulation Substituting 3 into F of 1 yields the following degeneracy condition at a critical point 4 0det 2 2 2 2 2 2 2 2 yx uy ux yx n n n n In 5 it was shown that choosing ck in 3 so that the lin ear u term in 4 disappears or 2 n x2 2 n y2 0 yielded F with desirably symmetric B for n 3 5 in 3 In particular the B in a nonzero u plane with n 3 is the well known deltoid of the EU in which the so called cusp points at the vertices of the deltoid form an equilateral triangle The deltoid is shown in Fig 1 a and it is com posed of three interlocking semicubic parabolas Like wise a similar result is achieved for n 5 except that the B is comprised of five interlocking semicubic parabo las with the cusp points at the vertices of a regular pentagon This B is also shown in Fig 1 a This result generalizes to any odd n A little algebra reveals that the constraint in 4 is met if ck with c0 1 obeys the fol lowing set of equations 2 1 2 nn c 5 0 1 2 1 2 1 22 kknk ckkckkcknkn Where and 2 1 1 nk20 k2 1 2 nkn Let be the set of coefficients for the thorder mask Then some examples are n Cn 3 1 3 c and 10 5 1 5 c 35 21 7 1 7 c When 5 and 3 are substituted into 4 a pattern emerges that reveals that 4 is satisfied by 6 2 2 2 22 1 2 d n R nn u yx Equation 6 states that the positions of the degenerate critical points in the pupil for a given image plane will be located on a circle whose radius Rd is a function of u B in any image plane can easily be found by converting x into cylindrical coordinates with Rd for a given u The pupil coordinates can then be parameterized by and substituted into the stationary point condition 2 to find s B Figure 1 a shows a typical B at nonzero u for n 3 5 and 7 As can be seen they are composed of n interlocking semicubic parabolas in which the cusp points lie on the vertices of an n sided regular polygon In the focal plane u 0 however B is simply the on axis point due to a single isolated EU degeneracy corresponding to a rank 0 Hessian It splits under defocus perturbation to form the geometries shown in Fig 1 a The intensities h closely reflect B At u 0 h will be concentrated on axis As the total beam propagates the individual cusp points will not initially be resolved owing to diffraction However they will quickly increase in intensity as h expands and eventually start to dominate Examples from this region with a plane wave input are shown in Fig 1 b for a system with f 75 cm 633 nm and Rap 1 cm The h were calculated from Eq 1 and correspond to the B in row a Row c shows h after further propagation Row d in Fig 1 is a duplication of row c except that the input was a Gaussian beam with a beam waist of 1 58 cm and Rap A small deterioration in peak intensity occurs but the results are comparable Over a large range of u the intensity maxima will be sharply peaked at the cusp points though the value at the Fig 2 Color online Product of the peak cusp intensity and n is plotted versus z cm for the masks in Fig 1 with a plane wave input and normalized total beam intensity maxima will of course decrease with the expansion of the total beam In fact once the pattern emerges it will persist until Rd in 6 is larger than the finite pupil aperture The transition thus begins at After this point the peaks as the cusps will gradually be 221 max 1 2 n ap Rnnu absorbed back into the PSF Figure 2 plots n times the cusp point intensity over a larger range of z than Fig 1 for a total beam intensity that integrates to 1 The primary difference between the RPBs is the division of power between the cusp points though results will vary with w Generally an increase in w will enlarge the total beam size reduce the initial peak intensity and reduce the acceleration but increase the range and decrease the rate at which the peak intensity falls off with z Substituting Eq 6 into the critical point condition also reveals the functional dependence of the acceleration of the cusp points The radius of the circle on which the cusp points lie in the image plane will expand as a power of u or 21 21 21 1 2 2 1 2 2 nn n k k n cusp u nn c n R Thus for n 3 the RPB will accelerate quadratically with u like the Airy beam which is a reflection of their close family relationship Generating multiple accelerating maxima has its disadvantages The intensity maxima will be less than the maximum of an Airy beam for the same total beam power One obvious reason is that the power will be split into several maxima instead of one However the split ting of the beam does not completely account for the drop in intensity The intensity maximum at the corner of an Airy beam is due to a rank 0 HU degeneracy The maxima for the RPBs result from a rank 1 cusp degeneracy Arnol d investigated the asymptotic behavior of the intensity for oscillatory integrals like 1 and found that different types of degeneracies lead to differ ent orders of h with The degeneracies could be c