第一类贝塞尔函数：Jn（x） besselj(nu,Z) computes the Bessel function of the first kind, , for each element of the array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive. 第二类贝塞尔函数：Yn（x） bessely(nu,Z) computes Bessel functions of the second kind, , for each element of the array Z. The order nu need not be an integer, but must be real. The argument Z can be complex. The result is real where Z is positive. 第三类贝塞尔函数：Hn（x） Bessel function of the third kind (Hankel function) besselh(nu,K,Z) 球 贝塞尔函数： function F = sphericalbessel?(n,x) ifx==0, if n==0, F = 1; else F = 0; end else F=sqrt(pi/(2*x))*bessel?(n+0.5,x); end 其中bessel?代表第一类贝塞尔函数或第二类贝塞尔函数
from http://www.st-andrews.ac.uk/seeinglife/science/research/bessel/bessel.html Bessel Beams This is a conversation between Michael, one of our physicists, and James, a member of the public, about the Bessel beam, describing what it is and why it is important for our research! Michael : Hi! I’m a physicist from the University of St Andrews. At the moment we’re very excited about a new type of laser beam that we’ve been doing some research on. It’s called a Bessel beam. James : I see, so what exactly is a Bessel beam? What is it that makes it different? Michael : Well, when you shine a normal laser beam against a wall it shows up as a small point. When you shine a Bessel beam against a wall you get a central dot surrounded by rings of light. Here’s a picture of a Bessel beam. James : I see. What difference do the rings make then? Michael : Well it’s not just the rings. The main difference between a normal laser and a Bessel is the central core. It doesn’t diffract. James : Hmm, I’m not sure I remember what diffraction is. Michael : Diffraction is when light “flares out” after passing through a narrow slit or hole. The thinner the slit the light travels through, the more it flares out. Diffraction doesn’t just happen to light. Ocean waves diffract when they travel through the opening of a harbour wall. Sound diffracts as well. James : So the central core of a Bessel beam doesn’t flare out then? Is that any different to a normal laser beam? They don’t seem to diffract either. Michael : No they don’t seem to diffract but that’s just because of the scale you’re observing them over. You might be able to spot it if you point a laser across a classroom and then put your hand in front. If you were to point a typical laser pen at the moon, then by the time the beam had reached the surface it would have a larger radius than the width of the moon. James : I see, so are we using Bessel beams to point at the moon? That doesn’t seem that useful. Michael : No sorry for the misunderstanding. That’s not what we’re using them for at all. One use we have for them is using them optical injections. James : Optical injections? There’s another phrase I don’t understand. Michael : Don’t worry; it’s definitely not something you would have covered in school. We often want to put new material into a cell. One of the ways we can do this is by using laser light to punch a hole in the cell wall. By placing the cell in a bath of the material we want to insert, the materials should just diffuse in. James : Won’t punching a hole in its wall destroy the cell? Michael : As long as the hole is small enough compared to the cell then it shouldn’t be a problem. Cell walls are actually pretty good at repairing damage to themselves. James : I think I understand optical injections. You use a laser beam like it was a needle to prick the cell wall. Michael : Exactly. James : Well, how does this relate to Bessel beams? Michael : Glad to see you’re paying attention. Well, look at this picture on the right. It shows a close up of a normal laser beam that has been focussed to a point (on top) and a Bessel beam (on the bottom). You can see the difference. James : Yeah. The normal laser beam spreads out hugely compared to the Bessel. Michael : When we’re trying to punch a hole in a cell it’s really important we know exactly how much energy will be hitting the cell. It’s also important to know the area this energy is spread over. James : I think I can see where this is going. With the Bessel beam you could place the cell anywhere along the beam and it would be getting the same energy over the same area. Michael : Exactly. With the normal laser beam it is necessary for the cell to be placed at a very specific place on the beam for it to get the right amount of energy over the right area. This can be quite tricky. Remember, cells are incredibly small. We can fit hundred in just 1 mm. So that means we need to be that precise with our beam. James : Right, well I think I can see how a Bessel beam would make it easier to perform an optical injection but I still have a few more questions. How do we make a Bessel? What about the rings? Do they really never diffract? Michael : I’ll try and take those questions one at a time. Let’s start with how we make a Bessel beam. There are several different methods. The images you can see below are (from left to right) an axicon lens, an annular ring (on the right) and a spatial light modulator (SLM). They all work by altering a normal laser that is either passing through (axicon and the ring) or reflecting off (the SLM). Michael : The details of each method are different but we can visualise the physics of a Bessel beam by thinking about interference patterns of light. James : Remind me how interference works again? Michael : Well, we can think of light acting like a wave. If two light waves meet at a certain point then the result is the two waves added on top of each other. This means that if we get two identical waves meeting when both waves are at peaks then we get a wave with a peak twice as high as the two individual waves. This corresponds to light that is twice as bright. James : I see. So does that mean that if we get two waves meeting when one is at a peak and the other at trough then they cancel each other out and we get a flat line which corresponds to darkness? Michael : Exactly. One final concept to deal with is the idea of extending the wave into all three dimensions. You can see an example to the left. To visualise how a Bessel beam works let’s imagine two extended waves meeting each other travelling at an angle to each other. To the right you can see a top down view of this. The image blue and red lines represent peaks of waves. It is very important to note that the colour in this image is there to make the diagram clearer. In reality both sets of waves would be of identical colour. The two black arrows show us the direction the waves are moving in. We can see that where the two sets of peaks meet we will get positive interference. If we follow the waves back or forward we will begin to build up a line of positive interference. This is represented by the green line and is the central core of the beam. James : So this is how a Bessel beam is made? Michael : Well, first you would have to rotate this image 180˚ out of the page to make a cone of waves but yes, this is one way to think about the physics of a Bessel beam, as a core formed by positive interference. The rings come from further interference. James : About the rings? Do they diffract? What are they useful for? Michael : We don’t use the rings for optical injections but they definitely have other uses. These are in many different fields. The rings do spread out as the beam travels forward but this does not affect the central core. James : This is all beginning to seem a bit too perfect. What about diffraction. Do Bessel beams really never diffract? Michael : A perfect Bessel beam would never diffract however for a perfect Bessel you need infinite energy. This is clearly impossible. So with finite energy we can make approximations of Bessel beams. These approximations to perfect Bessel beams work very well over distances of several cm, which is more than long enough for optical injections. James : Anything else that a Bessel can do that I should be aware of? Michael : Well, there is one more thing. The central core of the Bessel beam can reconstruct itself after being blocked. James : What do you mean by that? Michael : Well, if you were to place an object in the way of a normal laser beam then beyond that object all you would get is shadow. However because a Bessel beam is formed by interference patterns from light travelling at an angle then after a certain distance we get the Bessel central core back again. The gray zone represents the shadow region where we lose the central core. James : This is all a lot to think about! Michael : I agree. There are so many opportunities for Bessel beams beyond the ones you’ve seen today. It can be quite a lot to absorb in one go but if there’s anything you’ve not understood then feel free to get in contact , or better yet come ask us in person at one of our events . This was a contribution from Michael Finlay, one of our undergraduate project students, 2009.
贝塞尔 函数 ( Bessel function, 又称柱函数 ) 是一类在各工程领域中有着广泛应用的特殊函数。贝塞尔函数是贝塞尔方程 ( 一个二阶常微分方程 ) 的解。 一般有三类贝塞尔函数 1 、第一类贝塞尔函数 (Bessel function of the first kind) J v (x) 2 、第二类贝塞尔函数 (Bessel function of the second kind), 又称诺依曼函数（ Neumann function ） :Y v (x)( 或 N v (x)) 3 、第三类贝塞尔函数（ Bessel function of the third kind ），又称汉克尔函数（ Hankel function ） : H 1 v (x) 和 H 2 v (x) 。 当贝塞尔方程为变形贝塞尔方程 ( 虚宗量贝塞尔方程 ) 时，方程的解一般有两类 1、 变形第一类贝塞尔函数（ modified Bessel function of the first kind ） I v (x) 2、 变形第二类贝塞尔函数（ modified Bessel function of the second kind ） K v (x) 另外应用比较多的函数还有球贝塞尔函数 (Spherical Bessel functions) 、艾里函数 (Airy functions ）等等。由于贝塞尔函数的广泛应用，很多软件都提供了现成的函数供用户使用。 Mathematics ：提供了比较多的 Bessel 函数。如： Bessel 函数 BesselJ BesselY BesselI BesselK 球体 Bessel 函数 SphericalBesselJ SphericalBesselY Hankel 函数 HankelH1 HankelH2 SphericalHankelH1 SphericalHankelH2 Airy 函数 AiryAi AiryAiPrime AiryBi AiryBiPrime Kelvin 函数 KelvinBer KelvinBei KelvinKer KelvinKei Struve 函数 StruveH StruveL AngerJ WeberE 零函数 BesselJZero BesselYZero AiryAiZero AiryBiZero Matlab ：提供了五个函数 besselj 、 bessely 、 besselh 、 besseli 和 besselk 分别对应第一类 Bessel 函数、第二类 Bessel 函数、第三类 Bessel 函数、第一类变形 Bessel 函数和第二类变形 Bessel 函数。语法 besselj(nu,z) ： nu 为函数的阶次，可以不为整数但必须为实数。 z 为函数的变量，可以为复数。还有 besselj(nu,z,1) 语法，可查询帮助。其他函数除了 besselh 有些不同外，基本类似。至于球函数、 Airy 函数等可通过这个函数来构造实现。 Excel ：提供了四个函数，即： BESSELJ 、 BESSELY 、 BESSELI 、 BESSELK 。 语法： BESSELJ (x,n) ；其中， x 为参数值， n 为函数的阶数。如果 n 非整数，则截尾取整。即 Excel 中只能计算整数阶次的 Bessel 函数。其他函数类似。 关于贝塞尔函数的专业书籍有： 中文 刘颖，《圆柱函数》国防工业出版社， 1983 。 奚定平，《贝塞尔函数》高等教育出版社和施普林格出版社， 1998. 英文 George Neville Watson 《 A Treatise on the Theory of Bessel Functions 》 Cambridge University Press ， 1995(1944 ， 1922) 最经典的一本书。 BG Korenev 《 Bessel Functions and Their Applications 》 CRC 2002. Milton Abramowitz, and Irene A. Stegun 《 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 》 Dover Publications 1964. I.S. Gradshteyn, I.M. Ryzhik, A Jeffrey, D Zwillinger 《 Table of integrals, series and products 》 (Seventh Edition) Academic Press 2007.
标签: Bessel
