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Hyperbolic Secant Squared Pulse Shape Dispersion and Ultrashort Pulses Angular dispersion and group-velocity dispersion Phase and group velocities Group-delay dispersion Negative group- delay dispersion Pulse compression Chirped mirrors Refractive index dispersion Sellmeiera formula e.g. SF 10 glass from Schott 2 2 2 2 B1 B2 B3 n 1 2 2 2 C1 C2 C3 B1 = 1,61625977 B2 = 0,259229334 B3 = 1,07762317 C1 = 0,0127534559 C2 = 0,0581983954 C3 = 116,607680 n(0,532 mm) = 1,73673 Dispersion in Optics The dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time. Dispersion disperses a pulse in space (angle): angular dispersion dn/d out ()blue otu ()red Dispersion also disperses a pulse in time: temporal dispersion d2n/d2 vgr(blue) < vgr(red) Both of these effects play major roles in ultrafast optics. Movies! Group velocity v0 vv vs. g phase velocity vvg vvg Calculating the group velocity vg dw /dk Now, 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 it's easier to think of w as the independent variable: 1 v/g dk dw Using k = w n(w) / c0, calculate: dk /dw = (n + w dn/dw) / c0 vg c0 / n w dn/dw) = (c0 /n) / (1 + w /n dn/dw ) Finally: w dn vg v phase / 1 ndw 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 < vphase. Calculating group velocity vs. wavelength We more often think of the refractive index in terms of wavelength, so let's write the group velocity in terms of the vacuum wavelength 0. dn dn d0 Use the chain rule : dw d 0 d w 2 d022 c 0 c 0 0 Now, 002/ c w, so : 22 dw w(2 c0 / 0 ) 2 c 0 c0 w dn Recalling that : vg / 1 n n dw 2 c 2 c0 dn 0 0 we have : vg / 1 n nd2 c 000 or : c00 dn vg / 1 n n d0 n(w) Schott SF 10 glass c00 dn vg / 1 n n d0 The group velocity is lower than the phase velocity in non-absorbing regions. vg = c0 / (n + w dn/dw) Except in regions of anomalous dispersion (near a resonance and which are absorbing), dn/dw is positive. So vg < vphase for most frequencies! Spectral Phase and Optical Devices Recall that the effect of a linear passive ˜ ˜ optical device (i.e., windows, filters, etc.) on Ein (w) Eout (w) a pulse is to multiply the frequency-domain H(w) field by a transfer function: Optical device ˜ ˜ or medium E out(w) H(w) E in (w) where H(w) is the transfer function exp[w ( )L / 2] of the device/medium: for a medium H(w ) BHH ( w ) exp[ i ( w )] Since we also write E(w) = √S(w) exp[-i(w)], the spectral phase of the output light will be: We simply add out(w) H (w) in (w) spectral phases. Note that we CANNOT add the temporal phases! out(t) H (t) in (t) The Group-Velocity Dispersion (GVD) The phase due to a medium is: H(w) = n(w) k L = k(w) L To account for dispersion, expand the phase (k-vector) in a Taylor series: 1 2 k()w L k () w0 L k () w 0 w w 0 L 2 k () w 0 w w 0 L ... w0 1 d 1 k()w0 k()w0 k()w v (w ) v (w ) 0 g 0 dw vg The first few terms are all related to important quantities. The third one is new: the variation in group velocity with frequency: d 1 k()w is the “group velocity dispersion.” dw vg The effect of group velocity dispersion GVD means that 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) d 2 dd d 2 d Recall that: 00 00 dcw2 0 dw d w d 02 c 0 d 0 dn and v/g cn00 d0 Okay, the GVD is: 2 2 d110 d dn 0 d dn n 0 2 n 0 dwv2g c0 d 0 c 0 d 0 2c0 d 0 d 0 2 2 0 dn d n dn 22 0 2c0 d 0 d 0 d 0 Units: Simplifying: 3 dn2 GVD k()w 0 ps2/km 0 2cd22 00 ps/THz·km 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 GVD yields group delay dispersion (GDD). Delays are just velocities times the medium length L. The phase delay: L kL()w0 so: t v (ww00 ) w The group delay:0 1 d 1 k()w0 k()w0 k()w v (w ) v (w ) 0 g 0 L dw vg so: tg ()()ww00 k L vg (w0 ) The group delay dispersion (GDD): GDD = GVD L d 1 so: GDD L k()w L dw vg Units: fs2 or fs/Hz Femtosecond Laser Pulses: Linear Properties, Manipulation, Generation and Measurement Manipulating the phase of light Recall that we expand the spectral phase of the pulse in a Taylor Series: 2 () w 0 1 [ w w 0 ] 2 [ w w 0 ]/2!... and we do the same for the spectral phase of the optical medium, H: 2 HHHH() w 0 1 [ w w 0 ] 2 [ w w 0 ]/2!... phase group delay group delay dispersion (GDD) So, to manipulate light, we must add or subtract spectral-phase terms. For example, to eliminate the linear chirp (second-order spectral phase), we must design an optical device whose second-order spectral phase cancels that of the pulse: 2 2 d d H 2 H2 0 i.e., 2 2 0 dw dw w 0 w 0 Propagation of the pulse manipulates it. Dispersive pulse broadening is unavoidable. nd If 2 is the pulse 2 -order spectral phase on entering a medium, and k”L is the 2nd-order spectral phase of the medium, then the resulting nd pulse 2 -order phase will be the sum: 2 + k”L. /2 (This result nd pulls out the A linearly chirped input pulse has 2 -order phase: 2,in 22 ½ in the Taylor Series.) Emerging from a medium, its 2nd-order phase will be: 3 2 This result, with / 2 / 2 0 dn the spectrum, 2,out 2 2 GDD 2 2 2 2 L can be inverse 2 cd00 Fourier- transformed to A positively chirped pulse will broaden further; yield the pulse. a negatively chirped pulse will shorten. Too bad material GDD is always positive in the visible and near-IR… So how can we generate negative GDD? This is a big issue because pulses spread further and further as they propagate through materials. We need a way of generating negative GDD to compensate. Angular dispersion yields negative GDD. Suppose that an optical element introduces angular dispersion. Optical element Optic Input axis beam Here, there is negative GDD because the blue precedes the red. We’ll need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse). A prism pair has Assume Brewster angle incidence negative GDD. and exit angles. How can we use dispersion to introduce negative chirp conveniently? Let Lprism be the path through each prism and Lsep be the prism separation. 2 2 3 3 2 Always d 0 dn 0 d n 2 4Lsep 2 Lprism 2 2 positive (in dw 2c d 2c d w 0 0 0 visible and near-IR) Always This term assumes This term allows the beam negative! that the beam grazes to pass through an additional the tip of each prism length, Lprism, of prism material. Vary Lsep or Lprism to tune the GDD! 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 positive chirp). Angular dispersion yields negative GDD. It’s routine to stretch and then compress ultrashort pulses by factors of >1000. 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 the unintuitive color variation of the pulse after the first prism. To see the effect on a positively chirped pulse, read right to left! 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 does The output path is not misalign the beam. independent of prism position. Input beam Output beam The required separation between prisms in a pulse compressor can be large. The GDD the prism separation and the square of the dispersion. Compression of a 1-ps, 600-nm pulse with 10 Different prism materials nm of bandwidth (to about 50 fs). Kafka and Baer, Opt. Lett., 12, 401 (1987) It’s best to use highly dispersive glass, like SF10, or gratings. But compressors can still be > 1 m long. Four-prism pulse compressor Also, alignment is critical, and many knobs must be tuned. Wavelength Wavelength tuning tuning Prism Prism Prism Prism Fine GDD Wavelength tuning Wavelength tuning tuning Coarse GDD tuning (change distance between prisms) All prisms and their incidence angles must be identical.
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