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1、Angular dispersion and group-velocity dispersionPhase and group velocitiesGroup-delay dispersionNegative group- delay dispersionPulse compressionChirped mirrors,Dispersion and Ultrashort Pulses,Dispersion in Optics,The dependence of the refractive index on wavelength has two effects on a pulse, one
2、in space and the other in time.,Group delay dispersion or Chirpd2n/dl2,Angular dispersiondn/dl,Both of these effects play major roles in ultrafast optics.,Dispersion also disperses a pulse in time:,Dispersion disperses a pulse in space (angle):,vgr(blue) vgr(red),qout(blue) qout(red),When two functi
3、ons of different frequency interfere, the result is beats.,taking E0 to be real.,Adding waves of two different frequencies yields the product of a rapidly varying cosine (wave) and a slowly varying cosine (Dw).,Let:,When two waves of different frequency interfere, the result is beats.,Indiv-idual wa
4、vesSumEnvel-opeIrrad-iance,When two waves of different frequency interfere, they also produce beats.,taking E0 to be real.,Traveling-Wave Beats,Indiv-idual wavesSumEnvel-opeInten-sity,z,time,Seeing Beats,Its usually very difficult to see optical beats because they occur on a time scale thats too fas
5、t to detect. This is why we say that beams of different colors dont interfere, and we only see the average intensity.,However, a sum of many frequencies will yield a train of well-separated pulses:,Indiv-idual wavesSumIrrad-iance,Pulse separation: 2p/Dwmin,Group Velocity,vg dw /dk,Light-wave beats (
6、continued):E tot(z,t) = 2E0 cos(kavezwavet) Pulse(DkzDwt)This is a rapidly oscillating wave: cos(kavezwavet) with a slowly varying amplitude: 2E0 Pulse(DkzDwt)The phase velocity comes from the rapidly varying part: v = wave / kaveWhat about the other velocitythe velocity of the pulse amplitude?Defin
7、e the group velocity: vg Dw /Dk Taking the continuous limit, we define the group velocity as:,carrier wave,amplitude,Group velocity is not equal to phase velocityif the medium is dispersive (i.e., n varies).,Evaluate the group velocity for the two-frequency case:,where k1 and k2 are the k-vector mag
8、nitudes in vacuum.,Phase and Group Velocities,vg dw /dkNow, w is the same in or out of the medium, but k = k0 n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So its easier to think of w as the independent variable:Using k = w n(w) / c0, calculate: dk /dw = (n + w dn/dw) /
9、 c0 vg = c0 / (n + w dn/dw) = (c0 /n) / (1 + w /n dn/dw )Finally:So the group velocity equals the phase velocity when dn/dw = 0, such as in vacuum. But n usually increases with w, so dn/dw 0, and: vg vf,Calculating the group velocity,The group velocity is less than the phase velocity in non-absorbin
10、g regions.,vg = (c0 /n) / (1+ w dn/dw) = vf / (1+ w dn/dw)Except in regions of anomalous dispersion (near a resonance and which are absorbing), dn/dw is positive. So vg vf for most frequencies!,Calculating group velocity vs. wavelength,We more often think of the refractive index in terms of waveleng
11、th,so lets write the group velocity in terms of the vacuum wavelength l0.,Recall that the effect of a linear passive optical device (i.e., windows, filters, etc.) on a pulse is to multiply the frequency-domain field by a transfer function:,where H(w) is the transfer function of the device/medium:,Si
12、nce we also write E(w) = S(w) exp-ij(w), the spectral phase of the output light will be:,We simply add spectral phases.,Spectral Phase and Optical Devices,Note that we CANNOT add the temporal phases!,for a medium,The Group-Velocity Dispersion (GVD),The phase due to a medium is: jH(w) = n(w) k0 L = k
13、(w) L To account for dispersion, expand the phase (k-vector) in a Taylor series:,is the group velocity dispersion.,The first few terms are all related to important quantities.The third one is new: the variation in group velocity with frequency:,The effect of group velocity dispersion,GVD means that
14、the group velocity will be different for different wavelengths in the pulse.,vgr(blue) vgr(red),Because ultrashort pulses have such large bandwidths, GVD is a bigger issue than for cw light.,Calculation of the GVD (in terms of wavelength),Recall that:,and,Okay, the GVD is:,Simplifying:,Units:ps2/km
15、or(s/m)/Hz or s/Hz/m,GVD in optical fibers,Sophisticated cladding structures, i.e., index profiles, have been designed and optimized to produce a waveguide dispersion that modifies the bulk material dispersion,Note that fiber folks define GVD as the negative of ours.,GVD yields group delay dispersio
16、n (GDD).,The delay is just the medium length L divided by the velocity.The phase delay:,The group delay:,The group delay dispersion (GDD):,so:,so:,so:,Units: fs2 or fs/Hz,GDD = GVD L,Dispersion parameters for various materials,Manipulating the phase of light,Recall that we expand the spectral phase
17、of the pulse in a Taylor Series:,So, to manipulate light, we must add or subtract spectral-phase terms.,and we do the same for the spectral phase of the optical medium, H:,For example, to eliminate the linear chirp (second-order spectral phase), we must design an optical device whose second-order sp
18、ectral phase cancels that of the pulse:,i.e.,group delay,group delay dispersion (GDD),phase,Propagation of the pulse manipulates it.,Dispersive pulse broadening is unavoidable.If j2 is the pulse 2nd-order spectral phase on entering a medium, and k”L is the 2nd-order spectral phase of the medium, the
19、n the resulting pulse 2nd-order phase will be the sum: j2 + k”L.A linearly chirped input pulse has 2nd-order phase:Emerging from a medium, its 2nd-order phase will be:,(This result pulls out the in the Taylor Series.),A positively chirped pulse will broaden further; a negatively chirped pulse will s
20、horten. Too bad material GDD is always positive in the visible and near-IR,This result, with the spectrum, can be inverse Fourier-transformed to yield the pulse.,So how can we generate negative GDD?,This is a big issue because pulses spread further and further as they propagate through materials.We
21、need a way of generating negative GDD to compensate.,Negative GDD Device,Angular dispersion yields negative GDD.,Suppose that an optical element introduces angular dispersion.,Well need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse).,In
22、putbeam,Opticalelement,a,Optic axis,Here, there is negative GDD because the blue precedes the red.,Negative GDD,q(w),Optic axis,Taking the projection of onto the optic axis, a givenfrequency w sees a phase delay of j(w),Were considering only the GDD due to the angular dispersion q(w) and not that of
23、 the prism material. Also n = 1 (that of the air after the prism).,q(w) 1,But q 1, so the sine terms can be neglected, and cos(q) 1.,z,Angular dispersion yields negative GDD.,The GDD due to angular dispersion is always negative!Also, note that it doesnt matter where the angular dispersion came from
24、(whether a prism or a grating or whatever).And the negative GDD due to prism dispersion is usually much greater than that from the material of the prism.,Copying the result from the previous slide after simplification:,A prism pair has negative GDD.,How can we use dispersion to introduce negative ch
25、irp conveniently?Let Lprism be the path through each prism and Lsep be the prism separation.,This term assumesthat the beam grazes the tip of each prism,This term allows the beam to pass through an additionallength, Lprism, of prism material.,Vary Lsep or Lprism to tune the GDD!,Always negative!,Alw
26、ays positive (in visible and near-IR),Its routine to stretch and then compress ultrashort pulses by factors of 1000.,Pulse Compressor,This device, which also puts the pulse back together, has negative group-delay dispersion and hence can compensate for propagation through materials (i.e., for positi
27、ve chirp).,Angular dispersion yields negative GDD.,What does the pulse look like inside a pulse compressor?,If we send an unchirped pulse into a pulse compressor, it emerges with negative chirp.,Note all the spatio-temporal distortions.,What does the pulse look like inside a pulse compressor?,If we
28、send a negatively pulse into a pulse compressor, it emerges unchirped.,Note all the spatio-temporal distortions.,Adjusting the GDD maintains alignment.,Any prism in the compressor can be translated perpendicular to the beam path to add glass and reduce the magnitude of negative GDD.,Remarkably, this
29、 doesnot misalign the beam.,Output beam,Input beam,The output path is independent of prism position.,The required separation between prisms in a pulse compressor can be large.,Its best to use highly dispersive glass, like SF10, or gratings. But compressors can still be 1 m long.,Kafka and Baer, Opt.
30、 Lett., 12, 401 (1987),Compression of a 1-ps, 600-nm pulse with 10 nm of bandwidth (to about 50 fs).,The GDD the prism separation and the square of the dispersion.,Four-prism pulse compressor,Also, alignment is critical, and many knobs must be tuned.,All prisms and their incidence angles must be ide
31、ntical.,Pulse compressors are notorious for their large size, alignment complexity, and spatio-temporal distortions.,Pulse-front tilt,Unless the compressor is aligned perfectly, the output pulse has significant: 1D beam magnification Angular dispersion Spatial chirp Pulse-front tilt,Pulse-compressor
32、s have alignment issues.,Why is it difficult to align a pulse compressor?,The prisms are usually aligned using the minimum deviation condition.,The variation of the deviation angle is 2nd order in the prism angle.But what matters is the prism angular dispersion, which is 1st order!Using a 2nd-order
33、effect to align a 1st-order effect is a bad idea.,Two-prism pulse compressor,This design cuts the size and alignment issues in half.,Single-prism pulse compressor,Beam magnification is always one in a single-prism pulse compressor!,The total dispersion in a single-prism pulse compressor is always ze
34、ro!,The dispersion depends on the direction through the prism.,So the spatial chirp and pulse-front tilt are also identically zero!,Diffraction-grating pulse compressor,The grating pulse compressor also has negative GDD.,Lsep,w,w,where d = grating spacing(same for both gratings),Grating #1,Grating #
35、2,Note that, as in the prismpulse compressor, thelarger Lsep, the largerthe negative GDD.,2nd- and 3rd-order phase terms for prism and grating pulse compressors,Grating compressors yield more compression than prism compressors.,Note that the relative signs of the 2nd and 3rd-order terms are opposite
36、for prism compressors and grating compressors.,Compensating 2nd and 3rd-order spectral phase,Use both a prism and a grating compressor. Since they have 3rd-orderterms with opposite signs, they can be used to achieve almost arbitrary amounts of both second- and third-order phase.,This design was used
37、 by Fork and Shank at Bell Labs in the mid 1980sto achieve a 6-fs pulse, a record that stood for over a decade.,Given the 2nd- and 3rd-order phases of the input pulse, jinput2 and jinput3, solve simultaneous equations:,Pulse Compression Simulation,Resulting intensity vs. time with only a grating com
38、pressor:,Resulting intensity vs. time with a grating compressorand a prism compressor:,Brito Cruz, et al., Opt. Lett., 13, 123 (1988).,Using prism and grating pulse compressors vs. only a grating compressor,The grism pulse compressor has tunable third-order dispersion.,A grism is a prism with a diff
39、raction grating etched onto it.,A grism compressor can compensate for both 2nd and 3rd-order dispersion due even to many meters of fiber.,a sin(qm) n sin(qi) = ml,The (transmission) grism equation is:,Note the factor of n, which does not occur for a diffraction grating.,Chirped mirror coatings also yield dispersion compensation.,Such mirrors avoid spatio-temporal effects, but they have limited GDD.,Longest wavelengths penetrate furthest.,Chirped mirror coatings,Longest wavelengths penetrate furthest.,Doesnt work for 600 nm,
