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Symmetry in a Perturbed Optical System

a r X i v :p h y s i c s /0503129v 1 [p h y s i c s .e d -p h ] 16 M a r 2005

Symmetry in a Perturbed Optical System

Jason H.Ste?en ?

University of Washington,Department of Physics

(Dated:February 2,2008)

This paper investigates the e?ects that perturbations to an optical system,such as translations or rotations of the optical elements,have on the ?nal location where a light ray strikes a detector.Symmetry arguments are employed to give selection rules for the combinations of those perturbations that can a?ect the light ray.It is shown that a “Transverse Parity”can be assigned to each type of perturbation and that only combinations with an overall odd parity will produce an e?ect.Some examples of how these results can be applied to undergraduate classroom or laboratory courses are given.I.

INTRODUCTION

The study of physics is replete with the use of symmetry to explain physical phenomena.The conservation of energy,momentum,and electric charge stem from symmetries as do the selection rules for allowed quantum-mechanical in-teractions or multipole moments of a force ?eld or mass distribution.Indeed,in the undergraduate curriculum one ?nds symmetries and their application in nuclear and par-ticle physics,electromagnetism,quantum mechanics,and statistical mechanics to name a few.It comes as no sur-prise that symmetries play a role in optics as well.Indeed,once one moves beyond the nominal placement of the ele-ments of an optical system and studies,instead,the e?ects of perturbations to such systems,the e?ects of symmetry in optics become more subtle and more rich.

This work investigates some roles that symmetry plays in an optical system that has been perturbed from its nom-inal con?guration.This investigation reveals straightfor-ward relations between the perturbations (translations or rotations)of the optical elements that can combine to pro-duce a displacement of the beam spot or a rotation of the incident angle of the beam as it impinges upon a screen or detector.A straightforwared symmetry argument gives the selection rules for the allowed couplings among the pertur-bations.

The advantage of this development for the undergraduate is that,unlike most treatments of symmetry (e.g.quantum mechanics),an optical system is easily manipulated and the consequence of changes to the system are immediately ap-parent.The relative transparancy of the e?ects of changes to an optical system allows the student to focus on under-standing the underlying physical principles instead of the proper use of a given experimental apparatus.With a few basic optical elements and a well designed set of exercizes,a physics student can study one of the most important as-pects of modern physical theory and appreciate the power of using symmetries to describe physical scenarios.The materials presented here are the aspects of a more general study that are readily incorporated into a standard under-graduate lecture or laboratory course.

The original motivation for this work was to examine the systematic e?ects that are introduced into an experiment due to imperfections in the placement and alignment of the optical elements in several,nominally-identical optical sub-systems.In particular,an array of eight optical systems

was used to determine the parameters of motion of a tor-sion pendulum undergoing large-amplitude torsional oscil-lations.Since the experiment required nanosecond timing precision,precise placement of the optical elements was im-portant as well as understanding the leading systematic ef-fects.The results of that investigation are the basis for this work.

This paper will proceed by ?rst introducing the approach that was used for developing the optical system model along with the assumptions that are made about the optical el-ements themselves.Some short examples are provided to show the nature of the model and the results it bears.Next,selection rules for the allowed couplings of optical element perturbations are demonstrated and derived via the intro-duction of a “Transverse Parity”that is assigned to the various ways that an element can be perturbed.Finally,attention is given to the application of this work to the undergraduate classroom or laboratory.

II.

MODEL OF THE OPTICAL SYSTEM

For this study,the optical system is modeled using a quasi object-oriented approach where the position and orientation information of a given lens,mirror,ray,etc.is encapsulated in a lens,mirror,or ray object.For example,a ray object consists of a point of origin and a direction vector,a lens object consists of a focal length,a center point,and a nor-mal vector.The e?ect of a ray refracting through a lens is found using a function that takes a given ray and lens,then returns the refracted ray using analytic geometry.

This method is di?erent from the standard geometrical or two-dimensional matrix formulation that is presented in most undergraduate optics texts 1,2and from more gen-eral three dimensional matrix approaches 3.This approach was chosen because it facilitates the three-dimensional ma-nipulation of the optical elements and allows for a more straightforward interpretation of the analytic results that it produces.Most of the computation was done with the symbol-manipulating software Mathematica.

The following assumptions are made throughout this work:all of the optical elements are perfect,thin,are suf-?ciently large that displacements and rotations of the ele-ments still allow the beam to impinge upon them,and that refraction is not wavelength dependent.It is also conve-nient to assume that the deviations of the optical elements

2 from their nominal position are small so that they can be

treated perturbatively4.That is,rotations are small and

displacements are small compared to the appropriate focal

lengths.Finally,only a single,central ray of the beam is

considered in this investigation.Some comments concern-

ing an extended beam are made near the end of the paper.

A.Example:Linear Displacement I

The?rst example to consider is that of a single lens and

detector nominally placed along the longitudinal(or“op-

tic”)axis.If the lens is displaced vertically a distance dy,as

shown in?gure1,then the location where the beam strikes

the detector is also displaced vertically by an amount

h=

s d?s l

f1 ? s2?s1f

2

dy1

+ s d?s2

3

gives a vertical displacement of the beam on the detector

h = s d ?s l

f

2 dy 2dα?

1f dydα2

.(3)

Unlike the previous examples where all the perturbations were displacements,here there are cross terms between the transverse displacements and the rotations of the lens about the transverse axes.Regardless of the number of

elements in the optical system,each term in equations like (3)will contain an odd number of transverse perturbative factors (i.e.factors that correspond to perturbations in the trans-verse directions).Similar to the previous examples,if there is a rotation about the longitudinal axis,then some cou-plings will contain an even number of perturbative factors.However,the terms that involve longitudinal perturbations still have an odd number of transverse perturbations.The selection rules for terms that ultimately contribute to the displacement of the beam spot are addressed in the next section.

h

S d

FIG.3:A light ray is refracted by a lens of focal length f which is displaced vertically a distance dy and rotated by an angle dαabout the x -axis.The resulting displacement of the beam on the detector is given by h .

III.TRANSVERSE PARITY

The examples presented in the last section show that not all possible combinations of displacements and rotations of the optical elements in a system will change the ?nal posi-tion of the beam spot.The derivation of the selection rules for the allowed terms is relatively straightforward.The dis-placement of the beam spot in the detector (x,y )can be expressed to arbitrary order as a polynomial function of the

displacements and rotations of the set of optical elements in the system

(x,y )=f ({dx },{dy },{dz },{dα},{dβ},{dγ})

(4)

where α,β,and γare rotations about the x ,y ,and z axes respectively and the curly braces represent the set of all perturbations of a given kind (i.e.{dx }represents the x displacements of the set of optical elements).If the entire system is re?ected through the nominal axis of symmetry (or rotated about the axis of symmetry by π)then the beam spot must also re?ect in a similar manner,(x,y )?→(?x,?y ).Therefore,each term in the polyno-mial function (4)must have odd symmetry in the plane transverse to the axis of symmetry.

From this discussion,one can de?ne a preliminary,multi-plicative,transverse parity operator P such that transverse perturbations,whether displacements or rotations,change sign while the signs of longitudinal displacements or rota-tions remain unchanged

P (x,y,z,α,β,γ)?→(?x,?y,z,?α,?β,γ).

(5)

Every possible perturbation to an optical element,whether a lens,detector,or mirror,etc.can be designated as having either even or odd transverse parity,though for a general optical element the transverse perturbations need not have odd transverse parity (hence “preliminary”).The expres-sion for the ?nal location of the beam spot can only be expressed using terms that have overall odd parity.

From the example of two lenses given earlier,each of the displacements of the lenses has odd transverse parity and the result (2)has two terms,both linear in the transverse displacements and thus both with odd transverse parity.A longitudinal displacement of a lens has even parity.Such a displacement would allow a coupling of the form dydz to contribute to the location of the beam spot since the overal parity of such a term is still odd

P (dydz )=(?dy )(dz )=?dydz.

(6)

Each term in the rotation example also has odd parity.The third term in equation (3),for example,is proportional to dydα2which has parity

P (dydα2)=(?dy )(?dα)2=?dydα2.

(7)

One application of transverse parity is to shorten the cal-culation (to a given order)of the ?nal location of the beam spot in a large optical system.Instead of allowing the com-puter to calculate all the terms in the expansion,thereby spending a lot of time on terms that are ruled out because of incorrect parity,one can reject such terms immediately and focus on those that are allowed to contribute.

A.

Other Symmetries

Other symmetries that are inherant in a particular op-tical element can further restrict the allowed terms in (4)and may a?ect the parity of a particular perturbation.For

4

example,a mirror has translational symmetry in the trans-verse plane and a cylindrical lens has translational symme-try along its axis of cylindrical symmetry.A translation of a cylindrical lens along that axis or a rotation about that axis would produce no e?ect on the incident beam and would therefore have even transverse parity.

Symmetry arguments such as this also demonstrate why even-parity combinations of otherwise odd-parity perturba-tions are not allowed.Consider the forbidden combination of dydβfor a standard lens,a vertical translation followed by a rotation about the vertical axis.Since the vertical translation maintains the left-right symmetry of a lens,no rotation about that same axis will produce a change in the location of the beam spot.

The parity of a given perturbation to a given optical ele-ment is ultimately de?ned by how that perturbation a?ects either the ray of light that passes through the geometric center of the optical element for translations or how it af-fects a ray of light that strikes the element at a point that is o?of the rotation axis of a rotation-type perturbation.

A perturbation that produces a result with odd symmetry has odd transverse parity,and one that gives an even result would have even symmetry.These de?nitions of parity only slightly change the parity operation given in(5)in that the odd parity perturbations will change sign and the even per-turbations will not,regardless of the axis involved in the perturbation.

B.A Real Beam

Transverse parity can also describe the behavior of a beam with physical extent.A ray that is displaced from the center of the beam would have an overall odd trans-verse parity.The same is true for a ray that is not paral-lel to the longitudinal axis.That transverse parity applies here is not surprising since,using a?nite number of lenses and a central ray,one can reconstruct any other ray in the physical beam.The physical beam is essentially a superpo-sition of odd-parity,perturbed,central rays and each of the terms that are allowed in this construction must already have overall odd transverse parity.

IV.DISCUSSION

One example of the application of this study to an under-graduate course in optics would be the derivation of equa-tions such as(1)or(3)using either a matrix optics approach or analytic geometry.Another possibility,directly applica-ble to laboratory research,is to determine the sources of systematic error in a given optical system and their relative importance.Such a study could include how those errors scale with the nominal placement of the optical elements (e.g.the separation of a light source and a collomating lens)or with perturbations to the elements.The introduc-tion of a cylindrical lens presents an opportunity to study the translational symmetry of the lens and also the e?ects that arise from the addition of the“preferred direction”along the axis of cylindrical symmetry.

Beyond the direct application to optics,there are many aspects of this work that pertain to many branches of physics.Selection rules,parity,symmetry,dimensionless ratios,and perturbations are ubiquitous in physics and can all be studied with this model.A student could be asked to determine if a given coupling is allowed,to determine the coe?cient that corresponds to that perturbation,or even to empirically?nd the analytic expression for that same coef-?cient by using the dimensionless ratios that can be formed by the separation of the lenses and their focal lengths.Sim-ilarly,a student could study how the coe?cients scale with the nominal separation of the optical elements,such as the expression that multiplies the perturbation to the?rst lens in equation(2)which scales linearly with short separations and quadratically with large separations.

Using the perturbed optical system one can learn about many important tools that physicists use in their e?orts to understand natural law.The apparatus,consisting of a beam of light and a few lenses,is relatively inexpensive and easy to manipulate.The e?ects of a perturbation to the system appear instantly on a screen or detector which transparancy allows for a deeper investigation of the phys-ical principles being studied instead of simply a tutorial on the use of a particular piece of laboratory equipment. This system should prove useful as a pedagogical device for undergraduates,both as a theoretical tool and as an exper-imental investigation of a simple theory.

Acknowledgments

I would like to thank Drs.Michael Moore and Paul Boyn-ton for their comments pertaining to this work.

?jste?en@https://www.wendangku.net/doc/bc17202180.html,

1R.A.Serway,Physics for Scientists and Engineers(Saunders College Pub.,1996),4th ed.

2E.Hecht,Optics(Addison-Wesley Publishing Company,Inc., 1987),2nd ed.3J.B.Almeida,J.Opt.Soc.Am.A16,596(1999).

4When only displacements are considered,the presented for-mulas are exact—there are no higher order terms.

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boring 令人厌烦的,乏味的,无聊的 tedious 乏味的,单调的,冗长的 flat 单调的,沉闷的 dull 乏味的,单调的 troublesome 令人烦恼的,讨厌的,麻烦的 tired 疲劳的,累的 bored 无聊的,无趣的,烦人的 exhausted 极其疲倦的 weary 疲劳的 bright 聪敏的,机灵的 apt 聪明的,反应敏捷的 intelligent 聪明的,有才智的 shrewd 机灵的,敏锐的,精明的(表示生意上的精明) ingenious (人,头脑)灵巧的 alert 警觉的,留神的 cute 聪明伶俐的,精明的 acute/cute acute 指的是视力,感觉的敏锐 dull 愚钝的,笨的 awkward 笨拙的,不灵巧的 absurd 荒谬的 ridiculous 可笑的,荒谬的 idiotic 白痴般的 blunt 率直的,直言不讳的 clumsy 笨拙的,粗陋的 happy 快乐的,幸福的 cheerful 欢乐的,高兴的 content 满意的,满足的 merry 欢乐的,愉快的,快乐的 pleasure 高兴,愉快,满足 enjoyment 享乐,快乐,乐趣 cheer 喝彩 applause 鼓掌,掌声 optimism 乐观,乐观主义 delight 快乐,高兴 kick 极大的乐趣 paradise 天堂,乐园 instant 立即的,即刻的 instantaneous 瞬间的,即刻的 immediate 立即的,即刻的 simultaneous 同时发生的,同时存在的,同步的punctual 严守时刻的,准时的,正点的 pick 挑选,选择 select 选择,挑选 single 选出,挑出 elect 选举,推举 vote 投票,选举 appoint 任命,委派 nominate 提名,任命 propose 提名,推荐 recommend 推荐,举荐 designate 指派,委任 delegate 委派(或选举)…为代表 install(l) 使就职,任命 ballot 使投票表决 dub 把…称为 choice 选择(权) option 选择 selection 选择,挑选 alternative 取舍,供选择的东西 favorite 特别喜爱的人(或物) inclination 爱好 preference 喜爱,偏爱,优先 observe 注意到,察觉到 perceive 认识到,意识到,理解 detect 察觉,发现 appreciate (充分)意识到,领会,体会 alert 使认识到,使意识到 awake 意识到,醒,觉醒 scent 察觉 ancient 古代的,古老的 primitive 原始的 preliminary 预备的,初步的 preliminary trial初审 primary 最初的,初级的 initial 开始的,最初的 original 起初的 former 在前的,以前的 previous 先,前 prior 在前的,优先的 beforehand 预先,事先 medieval 中世纪的,中古(时代)的preceding 在先的,在前的,前面的 senior 资格较老的,地位较高的 following 接着的,下述的 attendant 伴随的 subsequent 随后的,后来的 succeeding 以后的,随后的 consequent 作为结果(或后果)的,随之发生的 resultant 作为结果的,因而发生的therefore 因此,所以 consequently 所以,因此 then 那么,因而 thus 因此,从而 hence 因此,所以 accordingly 因此,所以,于是 thereby 因此,从而

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编校一课丨连接号用法大全 《标点符号用法》新标准中,连接号删除长横线“——”,只保留三种形式:一字线“—”、半字线“-”、波纹线“~”。三种连接号的使用范围各不相同。一字线 一字线占一个字位置,比汉字“一”略长标示时间、地域等相 关项目间的起止或相关项之间递进式发展时使用一字线。例:1.沈括(1031—1095),宋朝人。 2.秦皇岛—沈阳将建成铁路客运专线。 3.人类的发展可以分为古猿—猿人—古人—新人这四个阶段半字线半字线也叫短横线,比汉字“一”略短,占半个字位置。用于产品型号、化合物名称、 代码及其他相关项目间的连接。例:1.铜-铁合金(化合物 名称) 2.见下图3-4(表格、插图编号) 3. 中关园3号院3-2-11室(门牌号) 4.010-********(电话号码) 5.1949-10-01(用阿拉伯数字表示年月日) 6.伏尔加河-顿河运河(复合名词)波纹线波纹线俗称波浪线,占一个字位置标示数值范围的起止时用波纹线,包括用阿拉伯数字表示的数值和由汉字数字构成的数值。例:1.10~30cm 2. 第七~九课常见问题1.在数值间使用连接号时,前后两个数值都需要加上计量单位吗?在标示数值范围时,用波纹线连接号。此时,在不引起歧义的情况下,只在后一数值后计量单位,用波纹线连接的两个

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【英语】英语形容词常见题型及答题技巧及练习题(含答案)及解析 一、初中英语形容词 1.My deskmate is really _____.She likes to attend different activities after school. A. active B. quiet C. lazy D. honest 【答案】 A 【解析】【分析】我的同桌同学非常活跃,放学后喜欢参加很多不同的活动. 句中提到"She likes to attend different activities after school"放学后喜欢参加很多不同的活动,由此推测此人非常活跃,A积极的,活跃的;B安静的;C懒惰的,D诚实的,根据句意可知选择A. 2.Wang Wei speaks English as ________ as Yang Lan. They both study English hard. A. good B. well C. better D. best 【答案】 B 【解析】【分析】句意:王伟的英语讲的和杨澜的一样好。他们学习英语都努力。可知as…as中间用形容词或副词原级;此处是副词修饰动词speak。good好的,形容词原形;well好地,副词原形,better比较级;best最高级,故选B。 【点评】此题考查形容词原级。注意as...as中间跟形容词或副词原级。 3.—If there are ________ people driving, there will be ________ air pollution. —Yes, and the air will be fresher. A. less; less B. less; fewer C. fewer; fewer D. fewer; less 【答案】 D 【解析】【分析】句意:——如果开车的人越少,空气污染越少。——是的,空气将会更新鲜。little少的,形容词,其比较级是less,修饰不可数名词,few几乎没有,形容词,其比较级是fewer,更少,修饰可数名词,people,可数名词,用fewer修饰,air pollution,空气污染,不可数名词,用less修饰,故选D。 【点评】考查形容词的辨析。注意less和fewer意思和用法。 4.—Guess what? The university has accepted my application! —Wow! That's ________ new I've heard this year, Boris! Let's celebrate. A. a worse B. the worst C. a better D. the best 【答案】 D 【解析】【分析】句意:——猜猜什么?那所大学已经接受我的申请了。——哇喔,那是今年我听到的最好的消息,Boris,让我们庆祝一下。A.一个更糟的,比较级;B.最糟的,最高级;C.一个更好的,比较级;D.最好的,最高级。因为大学接受申请了,所以是好消息,排除A、B。根据 I've heard this year,今年我听到的,可知是最高级,故选D。 【点评】考查形容词辨析,注意平时识记最高级结构,理解句意。

梅州_恒大御景半岛_活动方案_“趣味钓虾 欢乐时光”

“趣味钓虾欢乐时光”活动方案 一、活动目的 1、通过趣味摊位互动,营造现场人气,延长客户客户逗留时间,吸引客户 的参与热情,维护老业主满意度,进一步挖掘“老带新“成交,促进项目销售; 2、通过周末暖场主题活动,加强客户对项目的忠诚度。进而增强项目市场 影响力及新客户对项目的认知度,促进新客户成交; 二、活动主题 “趣味钓虾欢乐时光” 三、活动时间 2017年4月15日-16日(上午9:00-12:00,下午14:00-18:00) 四、活动场地布置图 五、活动地点 恒大御景半岛销售中心,老业主、意向客户、新上门客户均可参与; 六、活动内容 1.上门抽奖:客户凭盖了恒大活动专用章的抽奖劵,上门可获得抽奖一次, 以乒乓球抽奖的形式,任意抽取一个球,球上标有数字,可获得对应的礼品; 中奖编号: 客户凭盖了恒大活动专用章的抽奖劵,上门可获得抽奖一次,以乒乓球抽奖的形式,任意抽取一个球,球上标有数字,可获得对应的礼品(100%有

礼); 中奖编号: 编号:888 三开门冰箱(噱头); 编号:88 2g金条(合计1条,具体视现场情况为准); 编号:8 美的搅拌机; 编号:28 恒大油1.8L; 编号:2、18 恒大绿色大米5kg; 编号:1、16、17 恒大玻璃杯; 编号:空白球精美小礼品一份(钥匙扣); 2、乐趣钓真虾:在营销中心内准备一个大游泳圈,里面放满小虾,在工作人员的协助下,来访客户可钓虾;(150份/天);数量有限,赠完截止; 3、沙漏DIY:在销售中心设置一个制作沙漏的DIY摊位,来访客户可自行制作沙漏,(150份/天);数量有限,赠完截止;

4.甜心美食:现场设置美食摊位提供爆米花和可乐供客户品尝,美食不间断。 5. 魔术小丑表演:魔术小丑现场与客户互动。表演魔术; 6.准点抽奖:活动期间,每天准点抽奖时间为下午4:30;同时配合项目推介 会,奖品有(家电、金条等);

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