光電子英文課件：Chapter 6 Polarization and Modulation of Light

1、Chapter 6 Polarization and Modulation of Light1. Polarization A. State of PolarizationA propagating electromagnetic (EM) wave has its electric and magnetic field at right angles to the direction of propagation.The term polarization of an EM wave describes the behavior of the electric field vector in

2、 the EM wave as it propagates through a medium.If oscillation of the electric field at all times are contained within a well-defined line then the EM wave is said to be linearly polarized (Figure 1a).The field vibrations and the direction of propagation (z) define a plane of polarization (plane of v

3、ibrations) so that linear polarization implies a wave that is plane-polarized.If oscillation of the electric field at all times are contained within a well-defined line then the EM wave is said to be linearly polarized (Figure 1a).The linearly polarized wave has the E oscillations at -45 to x-axis (

4、Figure 1 b).The field vibrations and the direction of propagation (z) define a plane of polarization (plane of vibrations) so that linear polarization implies a wave that is plane-polarized.By contrast, if a beam of light has waves with the E-field in each in random direction but perpendicular to z,

5、 then this light beam is unpolarized. Suppose that we write the Ey and Ex components of a light wave generally as: Ex = Exocos(t kz ) and Ey = Eyocos(t kz + ) is the phase difference between Ey and Ex; can arise if one of components is delayed (retarded).Ex and Ey have the same magnitude but they ar

6、e out of phase by 180. and are the unit vectors along x and y, using = 180,the field in the wave isEquation above state that the vector E0 at -45to the x-axis and propagates along the z-direction. If the magnitude of the field vector E remains constant but its tip at a given location on z traces out

7、 a circle by rotating in a clockwise sense with time, as observed by the receiver of the wave ,then the wave is said to be right circularly polarized (Figure 2).If the rotation of the tip of E is counterclockwise ,the wave is said to be left circularly polarized.7.1 Polarization In circularly polari

8、zed =/2, thenEx = Acos(t kz ) and Ey = -Acos(t kz)It is relatively straightforward to show equations above represent a circle that is Over a distance z, the field E rotates through an angle =k z. Linear and circular polarization are summarized (Figure 3)an elliptically polarized light can result for

9、 any not zero or equal to any multiple of , and when Ex0 and Ey0 are not equal in magnitude (Figure 4).An elliptically polarized light or elliptic light has the tip of E vector trace out an ellipse as the wave propagates through a given location in space.B. Maluss Law(1).A linear polarizer will only

10、 allow electric field oscillation along some preferred direction, called the transmission axis, to pass through the device (Figure 5).(2). The linearly polarized light from the polarizer is now incident on a second identical polarizer. Then by rotating the transmission axis of this second polarizer

11、we can analyze the polarization state of the incident beam; (3).The irradiance (intensity) of light passing through the analyzer is proportional to the square of the electric fieldThe detected intensity of light varies as (Ecos)2.When =0 (E parallel to TA2) , all the electric field will pass , this

12、is the maximum irradiance condition.The irradiance I at any other angle is given by Maluss law: A. Optical AnisotropyAn important characteristic of crystals is that many of their properties depend on the crystal direction, that is crystals are generally anisotropic.Electronic polarization is easier

13、to displace electrons along certain crystal directions. This means that the refractive index n of a crystal depends on the direction of the electric field in the propagating light beam.The velocity of light in a crystal depends on the direction of propagation and on the state of its polarization (th

14、e direction of the electric field).2. Light Propagation In An Anisotropic Medium: Birefringence Most noncrystalline materials such as glasses and liquids, and all cubic crystals are optically isotropic, that is the refractive index is the same in all directions.all classes of crystal excluding cubic

15、 structures are optically anisotropic, that means that the refractive index depends on the propagation direction and the state of polarization.Experiments and theories on “most anisotropic crystals” (highest degree of anisotropy), show that we can describe light propagation in terms of three refract

16、ive indices, called principal refractive indices n1, n2 and n3, along three mutually orthogonal directions in the crystal.Crystals that have three distinct principal indices also have two optic axes and are called biaxial crystals. birefringenceAny unpolarized light ray entering such a crystal break

17、s into two different rays with different polarization and phase velocitiesB. Uniaxial Crystals and Frensnels Optical indicatrixUniaxial Crystals have two of their principal indices the same (n1=n2) and only have one optic axis.Uniaxial crystals, such as quartz, that have n3n1 and are called positive

18、, and those such as calcite that have n3n1 are called negative uniaxial crystals.Basic principles for uniaxial crystals:(1)Any EM wave entering an anisotropic crystal splits into two orthogonal linearly polarized waves which travel with different phase velocities, that is they experience different r

19、efractive indices.These two orthogonally polarized waves in uniaxial crystals are called ordinary (o) and extraordinary (e) waves. ordinary (o) wave: the same phase velocity in all directions and behaves like an ordinary wave in which the field is perpendicular to the phase propagation direction.ext

20、raordinary (e) waves: the phase velocity depends on its direction of propagation and its state of polarization, and the electric field is not necessarily perpendicular to the phase propagation direction. These two waves propagate with the same velocity only along a special direction called the optic

21、 axis.The o-wave is always perpendicularly polarized to the optic axis and obeys the usual Snells law. Birefringence are due to o-waves and e-waves being refracted differently so that when they emerge from the crystal they have been separated. Each ray constitutes an image but the field direction ar

22、e orthogonal.The fact that this is so easily demonstrated by using two polaroid analyzer with their transmission axes at right angle (Figure 7). If we were to view an object along the optic axis of the crystal, we could not see two images because the two rays would experience the same refractive ind

23、ex. the polarization vector and the electric field are parallel along the axespositive uniaxial crystalsn1=n2n3We can represent the optical properties of a crystal in terms of three refractive indices along three orthogonal axes (Figure 8 a).The refractive indices along x, y and z axes are the princ

24、ipal indices n1, n2 and n3 respectively for electric field oscillations along these direction ( not to be confused with the wave propagation direction). The refractive index associated with a particular EM wave in a crystal can be determined by using Fresnels refractive index ellipsoid, called the o

25、ptical indicatrix (a refractive index surface placed in the center of the principal axes Figure 8 a).If all three indices were the same, n1=n2=n3= n0 we would have a spherical surface and all electric field polarization direction would experience the same refractive index n0 .Such a spherical surfac

26、e would represent an optically isotropic crystal Find the refractive indices(1) Place a plane perpendicular to OP and passing through the center O of the indicatrix. This plane intersects the ellipsoid surface in a curve ABAB which is an ellipse.An arbitrary wave vector k (represents the direction o

27、f phase propagation), shown as OP in Figure 8 b, and is at an angle to the z-axis.(2) The major (BOB) and minor (AOA) axes of this ellipse determine the field oscillation directions and the refractive indices associated with this wave. Put differently, the original wave is now represented by two ort

28、hogonally polarized EM waves.The line AOA corresponds to the polarization of the ordinary wave and its semiaxis AA is the refractive index n0=n2 of this o-wave. The electric displacements and the electric field are in the same direction and parallel to AOA . o-wave always experiences the same refrac

29、tive index in all directions. The line AOA corresponds to the polarization of the ordinary wave and its semiaxis AA is the refractive index n0=n2 of this o-wave. The electric displacements and the electric field are in the same direction and parallel to AOA . o-wave always experiences the same refra

30、ctive index in all directions.Oscillations in the extraordinary wave and its semiaxis BB is the refractive index ne() of this e-wave. This refractive index is smaller than n3 but greater than n2(=n0). The e-wave therefore travels more slowly than the o-wave in this particular direction and in this c

31、rystal.(3) Change the direction of OP, we find that the length of the major axis changes with the OP direction. Thus ne() depends on the wave direction.ne=n0, OP is along the z-axis, when the wave is traveling along z (Figure 9 a). This direction is the optic axis and all waves traveling along the o

32、ptic axis have the same phase velocity whatever their polarization.ne() =n3=ne , e-wave traveling along the y-axis or along the x-axis, the e-wave has its slowest phase velocity (Figure 9 b). (4) Along any OP direction that is at an angle to the optic axis, the e-wave has a refractive index ne() giv

33、en by. (1)=0 , ne(0)=n0 =90, ne(90)=ne The major axis BOB in Figure 8(b) determines the e-wave polarization by defining the direction of the displacement vector D and not E.The electric of the e-wave is orthogonal to that of the o-wave, and it is in the plane determined by k and the optic axis. E is

34、 orthogonal to k when the e-wave propagates along one of the principal axes. Construct a wavevector surface The distance from the origin O to any arbitrary point P on a wavevector surface represents the value of k along the direction OP.The o-wave has the same refractive index in all directions, its

35、 wavevector surface will be a sphere of radius n0kvacuum, where kvacuum is the wavevector in free space (Figure 10 a). The wavevector of e-wave depends on the propagation direction and given by ne() kvacuum (Figure 10 a). ConclusionThe electric field E0 of the o-wave is always orthogonal to its wave

36、vector direction k0 and also to the optic axis (Figure 10 a). Since the electric and magnetic fields in the o-wave are normal to k0, The o-wave Poynting vector S0 (the direction of energy flow), is along k0.The polarization of the medium is not parallel to the inducing field in the e-wave and conseq

37、uently the overall electric field Ee in the EM wave is not at right angles to the phase propagation direction ke (Figure 10 a). (3)The energy flow (group velocity) and the phase velocity direction are different (a phenomenon called the Poynting vector “walk-off” effect).(4)The energy flow (the Poynt

38、ing vector Se), direction is taken as the ray direction for e-wave so that the wavefronts advance “sideways” (Figure 10 b) .The group velocity is in the same direction as energy flow (Se).C. Birefringence of CalciteA calcite crystal (CaCO3) is a negative uniaxial crystal and also well known for its

39、double refraction.Cleave the surfaces of a calcite crystal (cut along certain crystal planes).Cleaved form: a attained shape in crystalShape of crystal: rhombohedrons (parallelogram with 78.08 and 101.92)Calcite rhomb: a cleaved form of the crystalPrincipal section: a plane of the calcite rhomb that

40、 contains the optical axis and is normal to a pair of opposite crystal surfaces An unpolarized or natural light entering a calcite crystal (Figure 11), the ray breaks into ordinary (o) and extraordinary (e) waves with mutually orthogonal polarization.The o-wave has its field oscillations perpendicul

41、ar to the optic axis. It enters the crystal undeflected. The o-wave is normal to the optic axis and also to the direction of propagation. The e-wave has a polarization orthogonal to the o-wave and in the principal section. The e-wave polarization is in the plane of the paper, as E (Figure 11). It tr

42、avels with a different velocity and diverges from the o-wave. Cut a plate from a calcite crystal so that the optic axis (along z) would be parallel to two opposite faces of the plate (Figure 12 a), then a ray entering at normal incidence to one of these faces would not diverge into two separate wave

43、s (propagation along the y-direction expect now neno) plate that has the optic axis (take along z) parallel to the plate faces as in Figure 13.When a light ray enters a crystal at normal incidence to the optic axis and plate surface, then the o- and e- waves travel along the same direction.The phase

44、 difference between the orthogonal components E and E of the emerging beam is 3. Birefringent Optical DevicesThe phase difference expressed in terms of full wavelengths is called the retardation of the plate.The polarization of the through beam depends on the crystal type, ne-no and the plate thickn

45、ess L. We know that depending on the phase difference between the orthogonal component s of the field, the EM wave can be lineally, circularly or elliptically polarized as in Figures.Half-wave plate retarder has a thickness L such that the phase difference is or 180, corresponding to a half of wavel

46、ength (/2) of ratardation.Quarter-wave plate retarder has a thickness L such that the phase difference is /2 or 90, corresponding to a half of wavelength (/4) of ratardation.B. Solei-Babinet CompensatorAn optical compensator is a device that allow one to control the retardation (i.e. the phase chang

47、e) of a wave passing through it. In a wave plate retarder such as the half-wave plate, the relative phase change between the ordinary and extraordinary waves depends on the plate thickness and cannot be changed.In compensators is adjustable.The Soleil-Babinet compensator describes below is one such

48、optical device that is widely used for controlling and analyzing the polarization state of light.Consider the optical structure depicted in Figure 15 which has two quartz wedges touching over their large faces to form a block of adjustable height d.Suppose that a linearly polarized light is incident

49、 on this compensator at normal incidence.Its phase change isBut the E2 polarization wave first experience n0 through the wedges (d) and then ne through the plate (D) so that its phase change isThe phase difference is (= 2- 1) between the two polarization isC. Birefringent Prisms Prisms made from bir

50、efringent crystal are useful for producing a highly polarized light wave or polarization splitting of light. Two calcites (or quartz) right angle prisms A and B are placed with their diagonal faces touching to form a rectangular block as shown in Figure.Looking at the cross section of this block, th

51、e optic axis in A is in the plane of the paper and that in B coming out of the paper: the two prisms have their optic axes mutually orthogonal. Further, the optic axes are parallel to the prism sides.Two orthogonally polarized waves that have fields E1 and E2.E1: e-wave.decrease from n0 to neThe ref

52、ractive index changes are opposite which means that the two waves are refracted in opposite direction at the interface.4. Optical Activity And Circular BirefringentOptical activity: the rotation of the plane of polarization by a substrate.Optical activity occurs in materials in which the electron mo

53、tions inducted by the external electromagnetic field follows spiraling or helical paths (orbits).The optical field in light therefore induces oscillating magnetic moments which can be either parallel or antiparallel to the induces oscillating electric dipoles.Wavelets emitted from these oscillating

54、induced magnetic and electric dipoles interfere to consitute a forward wave that has its optical field rotated either clockwise or counterclockwise.If is the angle of rotation of E, then is proportional to the distance L propagated in the optically active medium.The rotation of the plane of polariza

55、tion maybe counter clockThe rotation of the plane of polarization maybe clockThe structure of quartz is such that atomic arrangement spiral around the optic axis either in clockwise or counterclockwise sense.The specific rotatory power (/L) is defined as the extent of rotation per unit length of dis

56、tance traveled in the optically active substance. Specific rotatory power depends on the wavelength.Optical activity can be understood in terms of left and right circularly polarized indices.A vertically polarized light with a field E at the input can be thought of as two right and left handed circu

57、larly polarized waves, EL and ER that are symmetrical with respect to the y-axis, i.e. at any instant =.Optical activity5. Electro-Optic EffectsA. DefinationElectro-Optic effects refer to changes in the refractive index of material induced by the application of an external electric field is not the

58、electric field of any light wave, but a separate external field. The frequency of the applied field has to be such that the field appears static over the time scale it takes for the medium to change its properties, as well as for any light to cross the substance.The electro-optic effects are classif

59、ied according to the first and second order effects. If we were to take the refractive index n to be a function of the applied electric field E, the is n=n (E), we can of course expand this as a Taylor series in E. The new refractive index n would be n =n+a1E+a2E2+ Where the coefficients a1 and a2 a

60、re called the linear electro-optic effect and second order electro-optic effect coefficients.Field induced refractive indexFirst effectsPockels effect: The change in n due to the first E term is called the Pockels effect.Second effectsKerr effect: The change in n due to the second E2 term is called