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1、-338-Chapter 8 Polarization and Propagation of Light in Crystal (Chap 8,Hecht;Chap 7,Zhao)The light field in the classical E-M theory is a vector field.The component has direction,this direction of is called polarization of light.We neglect this vector feature before by treating it as scalar(using s

2、ymbol)in cases such vector nature can be neglected.In this chapter we shall give it a detailed look.The polarization is not only in itself an important property of light which has many applications,it is also related in quantum the spin of photon,and the Jones vector treatment on polarization uses s

3、ame mathematical tools as in quantum.The polarization indeed is a two-level system in quantum mechanics,and our treatment here will pay dividents later in study of quantum mechanics.8-1 Polarization Type of Light(1)Natural light(Unpolarized light)The direction of is randomly distributed,say at some

4、fixed space,measuring the polarization at different times,the result is something like the figure shown:E?E?UE?-339-Question:The field as shown appears to be averaged to zero for the natural light,then why the average intensity of natural light?The answer is that each component is incoherent!It is f

5、rom independent emission of atoms/molecules,belongs to independent wave trains.So there is no fixed phase relation and it is intensity that sums up,a scalar summation instead of vector summation.(2)Partial polarized light.The direction is random and the amplitude depends on direction.The description

6、 of the partial polarized light is actually most difficult in math(compare to the polarized light introduced below).The accurate description in classical physics is Stokes vector which will not be treated in our course,but briefly discussed in Hechts;the formal treatment would be density matrix meth

7、od in statistical mechanics and will not be introduced here.A very useful picture from the density matrix method is that we can treat the partial polarized light as a combination of natural light plus a polarized light.That is why it is called partial polarized.Above are two types of unpolarized lig

8、ht.For polarized light,the E?0-340-direction of has fixed relation over space and time.i.e.it is predictable.These polarized light can be:(3)Linear polarized light.does not change direction over space and time,always along same line Apparently all linear polarized light can be superposition of two o

9、rthogonal linear polarized light(with same frequency)are unit vectors along (or )direction.Previously in the discussion of superposition of waves,we stated that for the sum of waves to show interference,they have to have parallel components of polarization,and the phase difference between the waves

10、determine the intensity.Here we shall see that adding waves with perpendicular(“orthogonal”more precisely)polarization generally changes the polarization of light;the phase difference between the E?E?110 x1y1EE cos(kzt)E icos(kzt)E jcos(kzt)=+?220 x2y2EE cos(kzt)E icos(kzt)E jcos(kzt)=+?xii0iyii0iE|

11、E|cos,E|E|sin=j,iV,H?y,x?-341-orthogonal components will determine the polarization state(type).For a linear polarized light,the phase difference between the two orthogonal components is either 0 in the figure)or.(4)Circular Polarized light.What happens if the two orthogonal polarized light,with pha

12、se difference.i.e.Then what is?We will have circular polarized light if,a quite special case in this sense.For(V leads H)at a fixed space point,say,looking into the direction of lights propagation.for x=0 clearly is not changing over time.But its direction is rotating clockwise with 1(E?2()E?iEHjEV,

13、0)tkz(cosiEExH=?)tkz(cosjEEyV+=?VHEEE?+=xy and EE2=2=0z=)t(cosiEx)tcos(-jEy+2=2y2x2|E|E|E+=-342-angular frequency This is called right circular polarized If fixed in time,say t=0,the polarization over place:We define right-circular polarized in the sense of how it changes over time.Similarly for At

14、fixed space over time,left circular polarized light is shown in the figure.Note means lag in phase(lags)kz(cosiEx)2kz(cosjEy2=0VEHE-343-means lead in phase(leads)This is in accordance of the convention we choose by adopting as wave.So it is conventional to say that if the V component leads in phase

15、by,it is right circular(of course the amplitudes for the H,V components are equal);if V lags by,then it is left circular.(5)Elliptical polarized light For the superposition of or,the polarization state will be what is called elliptical.Linear polarized and circular polarized light are special cases

16、for such superposition.(Fig.8.7,Hecht.Fig.910,Zhaos P244,Vol.1,or Fig.35,Pg.191,Vol.2)Looking into the direction of propagation,at fixed space and the polarization changes over time:(6)Superposition of orthogonal oscillation with different frequencies h,v are integers The superposition at fix space

17、point(say z=0)would be what is known as Lessajou Figure.For example h=2,v=1,would change with 0oenn ooeevcnvcn=-360-Positive crystal or?Huygenss method(drawing)to get lights propagation(dtails in Zhaos book Vol.2 Chap.7;1.3)e light is parallel or perpendicular with OA:Above are the three special and

18、 simplest forms of light propagation in uniaxial crystal.They are also simplest to understand using Huygens principle.These 3 special cases will be all we need for the manipulation of polarization.For the general case(not required for this course),if the angle between e light and OA are not or,the d

19、irection of propagation of energy(ray direction)and the direction of equal-phase wave front no longer overlaps.(Direction of wave vector).oevv 090k?-361-Ray direction:direction of energy flow,given by the Poynting vector,perpendicular to E,H.Wave vector:Normal direction to the equal phase wave front

20、.It is perpendicular to the D,H in a media,and in anisotropic media,the D,E are not along same direction as we shall see below.?Phase velocity,ray velocity,phase velocity surface,ray velocity surface.(optional for this course)Fig.8-1 ,S=EH?k?=ekOA=eSOA=eekS2eiNS=-362-The direction of energy propagat

21、ing(Poynting vector)is different from wave vector.This is caused by that in crystal,generallyare not parallel.:electric displacement.(8-9)where (8-10)polarizability Susceptibility tensor Thus (8-11)Dielectric tensor In isotropic media,is a scalar,and(8-11)shows that in isotropic media.In the anisotr

22、opic media,is generally a tensor.(3 3 matrix)By properly choosing x,y,z direction in crystal,we can have in diagonal form,i.e.For isotropic crystal For uniaxial crystal +=22eiiNkeS?ek?D,E?D?PED0?+=EP0?=EE)1(D00?=+=1=+E|D?x111213xy0212223yz313233zDEDDEEDE=?=zyx000000 xyz=xyz=-363-For biaxial crystal

23、Clearly in anisotropic media,are not parallel.From the Maxwell equation of E-M wave in crystal,for plane wave in forms of (the derivation is quite messy and thus omitted here)We have,So the propagation of energy()and propagation of equal-phase wavefront()is different in direction and velocity in ani

24、sotropic media.For e light in uniaxial crystal,we need to define and distinguish two velocities:Ray velocity,and phase velocity (8-12)Note that in Zhaos book,Fig 1-5,1-7,1-8,1-9,1-10,the velocities used to draw the ellipse are ray velocities.For the e light,in definition of,the (or),(or)are phase ve

25、locities according to definition of n.However in cases of propagation along or perpendicular to OA,the ray velocity and phase velocity are equal.(See Fig 1-12,Zhaos).As derived in the Zhaos book,Pg.174,the angle between and xyzD,E?)trk(ieA?kHD?SHE?HD,E?S?k?rvNv cosvvrN=oovcn=eevcn=ov|vevvS?k?-364-wi

26、th OA(Refer to Fig 8-1 in this note)has relation:(8-13)Example:For positive crystal eonn,.For the index of refraction of e light,whose phase velocity forms angle with the optical axis:phase velocity is the index of refraction defined by phase velocity.For the ordinary light,we have Snells Law.For e

27、light,a similar relation exists between incident angle and angle between normal with wave vector,(refer to Fig 8-1 of this note)(8-14)This however is not Snells law,since is not a constant,it varies as varies.The relation 8-14 will take a simple form if for all,as depicted in fig 1-10,Pg.172(Zhaos),

28、is constant for the e light in this case.cotnncot 2e2o=0 means phase lag,also note my sign is different from the Zhaos and is consistent with the sign convention for phases used in the notes.The physics meaning is same as Zhaos,also relation 2.1 in Zhaos p185 is wrong,the ne,no is reversed)(8-15)o c

29、omponent leads in phase.o component lags in phase.Quarter wave plate(plate):,ovonevenoo02n d=ee02n d=eoeo02(nn)d=0o?e?2=2oenn 0d)nn(2oe=oenn 0d)nn(2oe0 000l)nn(l)kk(21LR0LR=0LR/)nn(=,0LRnn,0LRnnRnLn-385-As we mentioned earlier,different crystalline give rise to levorotatory and dextrorotatory,with s

30、ame power.This means the,in the two crystalline are simply flipped,i.e.:Levo-crystal(left handed crystal line)Symbol:Right circular light.And we have relation:|llLRddRLnnnn-387-One of the isomers will be dextro-,and the other will be levo-,with same value of rotatory power but different sign.Such ch

31、iral molecules play important roles in chemistry and biology.There are many unsolved problems in this area,such as the relation between optical activity and molecular structure;the molecular structure and its chemical,biological functions etc.For example.Some medicine has to be d-isomer to be effect

32、ive,while its l-isomer is completely medical useless.8-6-5 Induced Circular BirefringenceFaraday effect The setup is given in Fig 5-9,Zhaos,Pg223.Fig 8-63,Hechts.(8-36)V:Verdet constant,l is the length of material.is 0 for levorotatory(This is the convention used in crystal).Note：V0,means if the lig

33、ht propagate along,it is left rotatory,if light propagates antiparallel with,it is dextrorotatory.Such change VBl=B?B?-388-of handedness with respect to the light propagation direction is characteristic of the Faraday effect.This means for the reflected beam,its polarization would have angle of with

34、 respect to the original incoming polarization!This is quite different from natural optical active material,where the rotation of polarization is unaffected by the propagation direction of light,i.e.if light passing from left or right to a L-SiO2(or equivalently rotate cystal 180 degree)the result w

35、ill be levorotatary.This is because the handedness cannot be changed by rotation;think about your hand,the right hand cannot overlap with left hand just by rotation.The reason for change of handedness of Faraday effect will be discussed below.Faraday effect is useful in constructing the optical isol

36、ator to block the reflected beams:The output of single pass is rotated by.The reflected would be rotated by another 45 and then orthogonal to the linear polarizer and wont pass.245-389-The reason that the change of handedness of the media towards different light propagation can be explained qualitat

37、ively by the simple picture:The effect of field is to align the magnetic dipoles of the media along one direction:The small magnetic dipoles of media can be thought as created by circulating electrons or charges.The circulating electrons may interact differently with right or left circular light:cir

38、cular birefringence.Assume for light travel along,right circular light interact with magnetic dipole (clockwise)and have larger,so it is levo-rotatory,.However for light propagate antiparallel with,the right circular light in this case,interacts with(viewing against direction of light propagation)(c

39、ounterclockwise),this is exactly the interaction between left circular light with the clockwise dipole in the parallel propagation,so for the antiparallel(with respect to),clearly now the media is dextrorotatory,since.The B?BB?LRnnlLlRnnB?B?,nnlLR=lRLnn=LRnn,|V basis(the Cartesian basis)is clearly a

40、 choice;the|+45 and|-45 linear polarization(a rotated Cartesian)is another,as well as|R,|L for circular basis,you can construct other basis by two independent vectors.(To test independence,we shall refer to the linear algebra,construct a matrix A formed by these vectors,i.e.each base vector is one c

41、olumn in the matrix A;the nxn A in space Rn(real number space)or Cn(complex number)has to be invertible for independent column vectors.You can test these by determinant or elimination of A).You may try this on the circular and elliptical basis in the note.(They are obviously independent due to ortho

42、gonality,but what the heck,test them yourself)-394-P.S.The above question may seem trivial but quite fundamental,it is the heart in linear algebra.You may want to review linear algebra for details.The best linear algebra textbook(for physicist and engineer)in my personal opinion is the book by Prof.

43、Gilbert Strang at MIT.(He has two similar books on introductory linear algebra,one is:Introduction to linear algebra which I have a pdf file;the other is Linear algebra and its applications which I have a hard copy and you could also buy from Amazon with RMB200+.If you cannot find sources and indeed

44、 in need,I would be happy to help)In 2-D,two orthogonal vectors form a complete basis,i.e.any other vector can be written as a superposition of them.Let|1 and|2 to represent the unit vector of this base(I will give you a detailed discussion of this in section 3),the orthonormal condition between the

45、m is defined by direct product between these two:,i,j=1 or 2 (1)And (1 here is identity matrix)is meaning the completeness of the basis.is the Diracs notation(bookkeeping)for a vector,is the notation for the vectors conjugate(complex conjugate if in complex space),the is notation for the scalar(dot)

46、product,i.e.in conventional vector case,and (real space),in linear algebra.More details on this notation are below.1(if i=j)=0(if ij)iji j=1,21iii=|i|i|i ji j?iTi ji j-395-(2)Diracs notation and its matrix form Any vectors expanded on this basis will have component along|1 or|2,lets say a and b are

47、the coefficients:Then (the usual way represent a vector in 2-D Cartesian coordinate is an example:).We could use a ordered array of a,b to represent the vector too(ordered means we already arrange the basis,such as 1st entry of the column matrix is the coefficient of expansion along|1,2nd is along|2

48、,etc)Then:in matrix form (2)where and are the matrix representation of base vector|1 and|2,and the complex-conjugate of the vector|1,|2,the 1|,?is?Well if we represent its conjugate in this form,we would have trouble to conduct the dot product(try it in the current form and you will find trouble usi

49、ng matrix product rules to evaluate the dot product),you may argue we can construct rules of calculation to proceed,but why not using the powerful arsenal already existing in linear algebra?To use the existing rules of linear algebra,the calculation of dot product is extremely straight forward IF we

50、 represent the complex conjugate by a row vector(1 x n matrix),which turns out to be the transposed and complex conjugate matrix of original vectors matrix representation.(In language of linear algebra,the physical vectors live in Column space and its complex conjugate in Row space)So,and Paul Dirac