文稿超快光学8章chapter

1、Spatio-temporal distortionsRay matricesThe Gaussian beamComplex q and its propagationRay-pulse “Kosten-bauder” matricesThe prism pulse compressorGaussian beam in space and time and the complex Q matrixSpatio-temporal characteristics of light and how to model themOptical system 4x4 Ray-pulse matrixSp

2、atio-temporal distortionsOrdinarily, we assume that the pulse-field spatial and temporal factors (or their Fourier-domain equivalents) separate:where the tilde and hat mean FTs with respect to t and x, y, zExampleAngular dispersion is an example of a spatio-temporal distortion.In the presence of ang

3、ular dispersion, the off-axis k-vector component kx depends on w:where kx0(w) is the mean kx vs. frequency w.PrismInput pulseAngularlydispersed output pulsexzSpatial chirp is a spatio-temporal distortion in which the color varies spatially across the beam.Propagation through a prism pair produces a

4、beam with no angular dispersion, but with spatial dispersion, often called spatial chirp.Prism pairs are inside nearly every ultrafast laser.Prism pairInput pulseSpatially chirped output pulseSpatially chirped output pulseInput pulseTilted windowSpatial chirp is difficult to avoid.Simply propagating

5、 through a tilted window causes spatial chirp!Because ultrashort pulses are so broadband, this distortion is very noticeableand often problematic!How to think about spatial chirpwhere x0 is the center of the beam component of frequency w.zxSuppose we send the pulse through a set of monochromatic fil

6、ters and find the beam center position, x0, for each frequency, w.x0(w1)x0(w2)x0(w3)x0(w4)x0(w5)x0(w6)x0(w7)x0(w8)x0(w9)Pulse-front tilt is another common spatio-temporal distortion.Phase fronts are perpendicular to the direction of propagation.Because the group velocity is usually less than phase v

7、elocity, pulse fronts tilt when light traverses a prism.Prism Angularly dispersed pulse with pulse-front tiltUndistorted input pulseAngular dispersion causes pulse-front tilt.Angular dispersion causes pulse-front tilt even when group velocity is not involved.Diffraction gratings also yield pulse-fro

8、nt tilt.Gratings have about ten times the dispersion of prisms, and they yield about ten times the tilt.The path is simply shorter for rays that impinge on the near side of the grating. Of course, angular dispersion and spatial chirp occur, too.Diffraction gratingAngularly dispersed pulse with pulse

9、- front tiltUndistorted input pulseModeling pulse-front tiltPulse-front tilt involves coupling between the space and time domains:For a given transverse position in the beam, x, the pulse mean time, t0, varies in the presence of pulse-front tilt.Pulse-front tilt occurs after pulse compressors that a

10、rent aligned properly.Angular dispersion always causes pulse-front tilt!Angular dispersion means that the off-axis k-vector depends on w:where g = dkx0 /dwwhich is just pulse-front tilt!Inverse Fourier-transforming with respect to kx, ky, and kz yields:Inverse Fourier-transforming with respect to w

11、(or w-w0) yields:using the shift theoremusing the shift theorem againThe combination of spatial and temporal chirp also causes pulse-front tilt.Dispersive mediumSpatially chirped input pulsevg(red) vg(blue) Spatially chirped pulse with pulse-front tilt, but no angular dispersionThe theorem we just p

12、roved assumed no spatial chirp, however. So it neglects another contribution to the pulse-front tilt.The total pulse-front tilt is the sum of that due to dispersion and that due to this effect.Xun Gu, Selcuk Akturk, and Erik ZeekA pulse with temporal chirp, spatial chirp, and pulse-front tilt.x mmt

13、fs803 nm777 nmy = 4.5 mrad797 nm775 nmx mmt fsy = 11.3 mradSuppressing the y-dependence, we can plot such a pulse:where the pulse-front tilt angle is:Well need a nice formalism for calculating these distortions!Spatio-temporal distortions can be useful or inconvenient.Good:They allow pulse compressi

14、on.They help to measure pulses (tilted pulse fronts).They allow pulse shaping.They can increase bandwidth in nonlinear-optical processes.Bad:They usually increase the pulse length.They reduce intensity.They can be hard to measure.Ray OpticsWell define light rays as directions in space, corresponding

15、, roughly, to k-vectors of light waves.Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation.axisThe Optic AxisA mirror deflects the optic axis into a new direction. This “ring laser” has an optic axis

16、that scans out a rectangle.Optic axisA ray propagating through this systemWe define all rays relative to the relevant optic axis.The Ray VectorA light ray can be defined by two co-ordinates:xin, qinxout, qoutits position, xits slope, qOptical axisoptical rayxqThese parameters define a ray vector, wh

17、ich will change with distance and as the ray propagates through optics.Ray MatricesFor many optical components, we can define 2 x 2 ray matrices.An elements effect on a ray is found by multiplying its ray vector.Ray matricescan describesimple and com-plex systems.These matrices are often uncreativel

18、y called ABCD Matrices.Optical system 2 x 2 Ray matrixRay matrices as derivativesWe can write these equations in matrix form.angular magnificationspatial magnificationSince the displacements and angles are assumed to be small, we can think in terms of partial derivatives.For cascaded elements, we si

19、mply multiply ray matrices.Notice that the order looks opposite to what it should be, but it makes sense when you think about it.O1O3O2Ray matrix for free space or a mediumIf xin and qin are the position and slope upon entering, let xout and qout be the position and slope after propagating from z =

20、0 to z.xin qinz = 0 xout qout zRewriting these expressions in matrix notation:Ray Matrix for an InterfaceAt the interface, clearly: xout = xin. Now calculate qout. Snells Law says: n1 sin(qin) = n2 sin(qout)which es for small angles: n1 qin = n2 qout qout = n1 / n2 qin qinn1qoutn2xinxoutRay matrix f

21、or a curved interfaceAt the interface, again: xout = xin. n1n2xinq1q2qinqsRzqoutqsqs = xin /Rq1 = qin+ xin / R and q2 = qout+ xin / RSnells Law: n1 q1 = n2 q2If qs is the surface slope at the height xin, then q1 = qin+ qs and q2 = qout+ qsTo calculate qout, we must calculate q1 and q2.A thin lens is

22、 just two curved interfaces.Well neglect the glass in between (its a really thin lens!), and well take n1 = 1.n=1R1R2n1n=1This can be written:The Lens-Makers Formulawhere:Ray matrix for a lensThe quantity, f, is the focal length of the lens. Its the single most important parameter of a lens. It can

23、be positive or negative.Its possible to extend the Lens Makers Formula to lenses of greater thickness.If f 0, the lens deflects rays toward the axis. f 0If f 0, the lens deflects rays away from the axis.f 0R2 0R1 0A lens focuses parallel rays to a point one focal length away.ffAt the focal plane, al

24、l rays converge to the z axis (xout = 0) independent of input position.Parallel rays at a different angle focus at a different xout.A lens followed by propagation by one focal length:Assume all input rays have qin = 0For all rays xout = 0!Types of lensesLens nomenclatureWhich type of lens to use (an

25、d how to orient it) depends on the aberrations and application.Ray matrix for a curved mirrorLike a lens, a curved mirror will focus a beam. Its focal length is R/2.Note that a flat mirror has R = and hence an identity ray matrix.qinqoutxin = xoutRzq1q1qsConsider a mirror with radius of curvature, R

26、, with its optic axis perpendicular to the mirror:Laser CavitiesTwo flat mirrors, the “flat-flat” laser cavity, is difficult to align and maintain aligned.Two concave curved mirrors, the “stable” laser cavity, is easy to align and maintain aligned.Two convex mirrors, the “unstable” laser cavity, is

27、impossible to align!Mirror curvatures matter in lasers.A system images an object when B = 0.When B = 0, all rays from a point xin arrive at a point xout, independent of angle.xout = A xinA is the magnification.The Lens LawFrom the object to the image, we have:1) A distance do2) A lens of focal lengt

28、h f3) A distance diSoAnd this arrangementmaps position to angle:Lenses can also map angle to position.From the object to the image, we have:1) A distance f2) A lens of focal length f3) A distance fIf an optical system lacks cylindrical symmetry, we must analyze its x- and y-directions separately: Cy

29、lindrical lensesA spherical lens focuses in both transverse directions.A cylindrical lens focuses in only one transverse direction.When using cylindrical lenses, we must perform two separate ray-matrix analyses, one for each transverse direction.Large-angle reflection off a curved mirror also destro

30、ys cylindrical symmetry.Optic axis before reflectionOptic axis after reflectionThe optic axis makes a large angle with the mirror normal, and rays make an angle with respect to it.Rays that deviate from the optic axis in the plane of incidence are called tangential.Rays that deviate from the optic a

31、xis to the plane of incidence are called sagittal. (We need a 3D display to show one of these.)tangential rayRay Matrix for Off-Axis Reflection from a Curved MirrorIf the beam is incident at a large angle, q, on a mirror with radius of curvature, R: where Re = R cosq for tangential raysand Re = R /

32、cosq for sagittal raysROptic axistangential rayqReal laser beams are localized in space at the laser and hence must diffract as they propagate away from the laser.But lasers are Gaussian beams, not rays.The beam has a waist at z = 0, where the spot size is w0. It then expands to w = w(z) with distan

33、ce z away from the laser.The beam radius of curvature, R(z), also increases with distance far away.xCollimated regionR(z) = wave-front radius of curvaturew(z)zBeam radius w(z)Gaussian beam mathwhere: w(z) is the spot size vs. distance from the waist,R(z) is the beam radius of curvature, andy(z) is a

34、 phase shift.This equation is the solution to the wave equation when we require that the beam be well localized at some point (i.e., its waist). The expression for a real laser beams electric field is given by:xCollimated regionR(z) = wave-front radius of curvaturew(z)zBeam radius w(z)Gaussian beam

35、spot size, radius, and phaseThe expressions for the spot size,radius of curvature, and phase shift:where zR is the Rayleigh Range (the distance over which the beam remains about the same diameter), and its given by: xCollimated region 2zRR(z) = wave-front radius of curvaturew(z)zBeam radius w(z) Col

36、limation CollimationWaist spot Distance Distance size w0 l = 10.6 m l = 0.633 m_.225 cm 0.003 km 0.045 km 2.25 cm 0.3 km 5 km22.5 cm 30 km 500 km_Longer wavelengths and smaller waists expand faster than shorter ones.w0Tightly focused laser beams expand quickly. Weakly focused beams expand less quick

37、ly, but still expand.As a result, its very difficult to shoot down a missile with a laser.Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains collimated.Gaussian Beam Collimationw(z)Gaussian beam divergenceThe beam 1/e divergence half angle is t

38、hen w(z) / z as z :The smaller the waist and the larger the wavelength, the larger the divergence angle.Far away from the waist, the spot size of a Gaussian beam will be:w0qzA lens will focus a collimated Gaussian beam to a new spot size: wfocus l f / pwinputSo the smaller the desired focus, the BIG

39、GER the input beam should be!Focusing a Gaussian beam2winput2wfocusffPhase relativeto a plane wave:p/2-p/2zR-zRy(z)Recall the i in front of the Fresnel integral, which is a result of the Guoy phase shift.The Guoy phase shiftThe phase factor yields a phase shift relative to the phase of aplane wave w

40、hen a Gaussian beam goes through a focus.The Gaussian-beam complex-q parameterWe can combine these two factors (theyre both Gaussians):where:q completely determinesthe Gaussian beam.xCollimated regionR(z) = wave-front radius of curvaturew(z)zBeam radius w(z)Ray matrices and the propagation of qWed l

41、ike to be able to follow Gaussian beams through optical systems.Remarkably, ray matrices can be used to propagate the q-parameter.This relationholds for allsystems for which ray matrices hold:Optical systemJust multiply all the matrices first and use this result to obtain qout for the relevant qin!I

42、mportant point about propagating qUseBut use matrix multiplication for the various components to compute the total system ray matrix.Dont compute for each component.Youd still get the right answer, but youd work much harder than you need to!to compute qout.Propagating q: an exampleFree-space propaga

43、tion through a distance z:Then:The ray matrix for free-space propagation is:Propagating q: an example (contd)So:RHS:So:Does q(z) = q0 + z? This is equivalent to: 1/q(z) = 1/(q0 + z).sowhich is just this.LHS:Propagating q: another exampleFocusing a collimated beam (i.e., a lens, f, followed by a dist

44、ance, f ):A collimated beam has a big spot size (winput) and Rayleigh range (zR), and an infinite radius of curvature (R), so: qin = i zRSo: 2winput2wfocusffThe well-known result for the focusingof a Gaussian beamBut:Now consider the time and frequencyof a light pulse in additionWed like a matrix fo

45、rmalism to predict such effects as the: group-delay dispersion t/w angular dispersion kx /w or q /wspatial chirp x/wpulse-front tilt t/xtime vs. angle t/q.where weve dropped“0” subscripts forsimplicity.This pulse has all of these distortions!Well need to consider, not only the position (x) and slope

46、 (q ) of the ray, but also the time (t) and frequency (w) of the pulse.Propagation in space and time: Ray-pulse Kostenbauder matricesKostenbauder matrices are 4x4 matrices that multiply 4-vectors comprising the position, slope, time (group delay), and frequency.A Kostenbauder matrix requires five ad

47、ditional parameters, E, F, G, H, I.Optical system 4x4 Ray-pulse matrixwhere each vector component corresponds to the deviation from a mean value for the ray or pulse. Kostenbauder matrix elementsAs with 2x2 ray matrices, consider each element to correspond to a small deviation from its mean value (x

48、in = x x0 ). So we can think in terms of partial derivatives.spatialchirptime vs. angleGDDpulse-front tiltangular dispersionthe usual 2x2 ray matrixSome Kostenbauder matrix elements are always zero or one.Kostenbauder matrix for propagation through free space or materialThe ABCD elements are always

49、the same as the ray matrix.Here, the only other interesting element is the GDD: I = tout /nin where L is the thickness of the medium, n is its refractive index,and k” is the GVD:So:The 2p is due to the definition of K-matrices in terms of n, not w.Example: Using the Kostenbauder matrix for propagati

50、on through free spaceApply the free-space propagation matrix to an input vector:Because the group delay depends on frequency, the pulse broadens.This approach works in much more complex situations, too.The position varies in the usual way, and the beam angle remains the same.The group delay increase

51、s by k”LwinThe frequency remains the same.Kostenbauder matrix for a lensThe ABCD elements are always the same as the ray matrix.Everything else is a zero or one.where f is the lens focal length.The same holds for a curved mirror, as with ray matrices.While chromatic aberrations can be modeled using

52、a wavelength-dependent focal length, other lens imperfections cannot be modeled using Kostenbauder matrices.So:no spatial chirp (yet)Kostenbauder matrix for a diffraction gratingGratings introduce magnification, angular dispersion and pulse-front tilt:where b is the incidence angle, and b is the dif

53、fraction angle (note that Kostenbauder uses different angle definitions in his paper).The zero elements (E, H, I) e nonzero when propagation follows.So:time is independent of angleno GDD (yet)pulse-front tiltangulardispersionspatial magnificationangularmagnificationKostenbauder matrix for a general

54、prismAll new elements are nonzero.is the GVD,spatial chirptime vs. angleGDDpulse-front tiltangulardispersionwhereLandspatial magnificationangularmagnificationJust angular dispersion and pulse-front tilt. No GDD etc.Kostenbauder matrix for a Brewster prismIf the beam passes through the apex of the pr

55、ism:(this simplifies the calculation a lot!)whereUse + if the prism is oriented as above; use if its inverted.Brewster angle incidence and exitUsing the Kostenbauder matrix for a Brewster prismThis matrix takes into account all that we need to know for pulse compression.When the pulse reaches the tw

56、o inverted prisms, this effect es very important, yielding longer group delay for longer wavelengths (D 0; and use the minussign for inverted prisms).Pulse-front tilt yields GDD.Dispersion changes the beam angle.Brewster angle incidence and exitModeling a prism pulse compressor using Kostenbauder ma

57、tricesKprismKprismKprismKprismKairKairKair1234567K = K7 K6 K5 K4 K3 K2 K1Use only Brewster prismsFree space propagation in a pulse compressorThere are three distances in this problem. n = 1 in free spaceL1L2L3K = K7 K6 K5 K4 K3 K2 K1K-matrix for a prism pulse compressorSpatial chirp unless L1 = L3.N

58、egative GDD!The GDD is negative and can be tuned by changing the amount of extra glass in the beam (which we havent included yet, but which is easy).Time vs. angle unless L1 = L3.To follow beams that are Gaussian in both space and time:Propagating spot size, radius of curvature, pulse length, and ch

59、irpWe could propagate Gaussian beams in space because theyre quadratic in space (x and y):A Gaussian pulse is quadratic in time. And the real and imaginaryparts also have important meanings (pulse length and chirp):The complex-Q matrixWe define the complex Q-matrix so that the space and time depende

60、nce of the pulse can be written:These complex matrix elements contain all the parameters of beams/pulses that are Gaussian in space and time.The complex-Q matrix (contd)When the off-diagonal components are not zero, there is pulse-front tilt:spatial complex-q parameter for Gaussian beamsWhen the off