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Dispersion and Ultrashort Pulses Angular dispersion and group-velocity dispersion Phase and group velocities Group-delay dispersion Negative groupdelay dispersion Pulse compression Chirped mirrors 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) > out (red ) Dispersion also disperses a pulse in time: vgr(blue) < vgr(red) Both of these effects play major roles in ultrafast optics. “Chirp” d2n/d2 Calculating the group velocity vg d /dk Now, 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 as the independent variable: v g [dk / d ] 1 Using k = n() / c0, calculate: dk /d = ( n + dn/d ) / c0 vg = c0 / ( n + dn/d) = (c0 /n) / (1 + /n dn/d ) Finally: dn v g = v phase / 1 + n d So the group velocity equals the phase velocity when dn/d = 0, such as in vacuum. Otherwise, since n usually increases with , dn/d > 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. Use the chain rule : dn dn d 0 = d d 0 d Now, 0 = 2 c0 / , so : Recalling that : we have : or : d 0 2 c0 2 c0 02 = = = 2 2 d (2 c0 / 0 ) 2 c0 c dn v g = 0 / 1 + n n d 2 c0 2 c0 dn 0 vg = / 1+ n0 d 0 2 c0 n c0 0 dn v g = / 1 n n d 0 The group velocity is less than the phase velocity in non-absorbing regions. vg = c0 / (n + dn/d) Except in regions of anomalous dispersion (which are absorbing), dn/d is positive, that is, near a resonance. So vg < vphase for these frequencies! Spectral Phase and Optical Devices Recall that the effect of a linear passive optical device (i.e., lenses, prisms, etc.) on a pulse is to multiply the frequency-domain field by a transfer function: E˜ in( ) E˜ out( ) = H ( ) E˜ in ( ) where H() is the transfer function of the device/medium: H() E˜ out( ) Optical device or medium exp[ ( ) L / 2] for a medium H ( ) = BH ( ) exp[i H ( )] Since we also write E() = S() exp[-i()], the spectral phase of the output light will be: out ( ) = H ( ) + in ( ) Note that we CANNOT add the temporal phases! out (t) H (t) + in (t) We simply add spectral phases. The Group-Velocity Dispersion (GVD) The phase due to a medium is: () = n() k L = k() L To account for dispersion, expand the phase (k-vector) in a Taylor series: k ( ) L = k ( 0 ) L + k ( 0 ) [ 0 ]L + 0 k ( 0 ) = v ( 0 ) 1 k ( 0 ) = v g ( 0 ) 1 2 k ( 0 ) [ 0 ] L + ... 2 d k ( ) = d 1 v g The first few terms are all related to important quantities. The third one is new: the variation in group velocity with frequency: d k ( ) = d 1 v g is the “group velocity dispersion.” 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 0 02 = d 2 c0 Recall that: d 0 d 02 d d = = d d d 0 2 c0 d 0 dn v g = c / n 0 d 0 and Okay, the GVD is: d d 1 02 d 1 dn 02 d dn n = n = 0 0 2 c d c d v 2 c d d 2 0 0 0 0 0 0 g 02 = 2 c02 Simplifying: dn d 2 n dn 0 2 d d d 0 0 0 GVD k ( 0 ) = d n 2 c02 d 02 3 0 2 Units: ps2/km or (s/m)/Hz or s/Hz/m Dispersion parameters for various materials GVD in optical fibers Note that fiber folks define GVD as the negative of ours. 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). We can define delays in terms of the velocities and the medium length L. The phase delay: 0 k ( 0 ) = v ( 0 ) so: t = k ( 0 ) L L = v ( 0 ) 0 The group delay: 1 k ( 0 ) = v g ( 0 ) so: L t g ( 0 ) = = k ( 0 ) L v g ( 0 ) The group delay dispersion (GDD): d k ( ) = d 1 v g so: d GDD = d GDD = GVD L 1 L = k ( ) L v g Units: fs2 or fs/Hz Manipulating the phase of light Recall that we expand the spectral phase of the pulse in a Taylor Series: ( ) = 0 + 1 [ 0 ] + 2 [ 0 ]2 / 2! + ... and we do the same for the spectral phase of the optical medium, H: H ( ) = H 0 + H 1 [ 0 ] + H 2 [ 0 ]2 / 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 + H2 = 0 i.e., d 2 d 2 + 0 d 2 H d 2 = 0 0 Propagation of the pulse manipulates it. Dispersive pulse broadening is unavoidable. If 2 is the pulse 2nd-order spectral phase on entering a medium, and k”L is the 2nd-order spectral phase of the medium, then the resulting pulse 2nd-order phase will be the sum: 2 + k”L. A linearly chirped input pulse has 2nd-order phase: 2,in Emerging from a medium, its 2nd-order phase will be: 2,out 03 d 2 n /2 /2 L = 2 + GDD = 2 + 2 2 2 2 + + 2 c0 d 0 /2 = 2 +2 (This result pulls out the in the Taylor Series.) This result, with the spectrum, can be inverse Fouriertransformed to yield the pulse. A positively chirped pulse will broaden further; 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 Input beam Optic axis Here, there is negative GDD because the blue precedes the red. Well need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse). Negative GDD Taking the projection of k ( ) () () << 1 onto the optic axis, a given frequency sees a phase delay of (): z ( ) = k ( ) roptic axis = k ( ) z cos[ ( )] = ( / c) z cos[ ( )] Optic axis Were considering only the GDD due to dispersion and not that of the prism itself. So n = 1 (that of the air after the prism). d / d = ( z / c) cos( ) ( / c) z sin( ) d / d 2 d z d z d z z d 2 d = sin( ) sin( ) cos( ) sin( ) 2 2 d c d c d c c d d 2 But << 1, so the sine terms can be neglected, and cos() ~ 1. Angular dispersion yields negative GDD. d d 2 2 0 z 0 c d d 0 2 The GDD due to angular dispersion is always negative! Also, note that it doesnt matter where the angular dispersion came from (whether a prism or a grating). And the negative GDD due to prism dispersion is usually much greater than that from the material of the prism. A prism pair has negative GDD. How can we use dispersion to introduce negative chirp conveniently? Lsep d 2 d 0 2 Always negative! dn 4Lsep 2 2c d 3 0 2 Assume Brewster angle incidence and exit angles. 30 d 2 n + Lprism 2 2 2 c d 0 0 This term assumes that the beam grazes the tip of each prism Vary Lsep or Lprism to tune the GDD! Always positive (in visible and near-IR) This term allows the beam to pass through an additional length, Lprism, of prism material. Pulse Compressor This device has negative group-delay dispersion and hence can compensate for propagation through materials (i.e., for positive chirp). The longer wavelengths traverse more glass. 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 not misalign the beam. Input beam The output path is independent of prism position. 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. Different prism materials Compression of a 1-ps, 600-nm pulse with 10 nm of bandwidth (to about 50 fs). Kafka and Baer, Opt. Lett., 12, 401 (1987) Its 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 tuning Wavelength tuning Prism Prism Prism Wavelength tuning Prism Fine GDD tuning Wavelength tuning Coarse GDD tuning (change distance between prisms) All prisms and their incidence angles must be identical. ! $ "# # % ! "!! " #! " " "% " ! " #! ! " #! #! ! #! ! "!$ ! !! ! " $ ! !! ! ! Two-prism pulse compressor Coarse GDD tuning Wavelength tuning Roof mirror Periscope Prism Wavelength tuning Prism Fine GDD tuning This design cuts the size and alignment issues in half. Single-prism pulse compressor Corner cube Periscope Prism GDD tuning Roof mirror Wavelength tuning Beam magnification is always one in a single-prism pulse compressor! M1 din dout M2 M d out / din 1 1 M1 = = M3 = M2 M4 M tot = M 1 M 2 M 3 M 4 = 1 The total dispersion in a single-prism pulse compressor is always zero! The dispersion depends on the direction through the prism. D1 D2 D1 = M 2 D2 = D3 = M 4 D4 Dtot D3 D1 D2 = + + + D4 = 0 M 2 M 3M 4 M 3M 4 M4 So the spatial chirp and pulse-front tilt are also identically zero! Diffraction-grating pulse compressor The grating pulse compressor also has negative GDD. Grating #2 d 2 d 2 0 Lsep 03 2 c 2 d 2 cos 2 ( ) where d = grating spacing (same for both gratings) Note that, as in the prism pulse compressor, the larger Lsep, the larger the negative GDD. Lsep Grating #1 2nd- and 3rd-order phase terms for prism and grating pulse compressors Grating compressors offer more compression than prism compressors. '' ''' Piece of glass Note that the relative signs of the 2nd and 3rd-order terms are opposite 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-order terms with opposite signs, they can be used to achieve almost arbitrary amounts of both second- and third-order phase. Prism compressor Grating compressor Given the 2nd- and 3rd-order phases of the input pulse, input2 and input3, solve simultaneous equations: input2 + prism2 + grating2 = 0 input3 + prism3 + grating3 = 0 This design was used by Fork and Shank at Bell Labs in the mid 1980s to achieve a 6-fs pulse, a record that stood for over a decade. Pulse Compression Simulation Using prism and grating pulse compressors vs. only a grating compressor Resulting intensity vs. time with only a grating compressor: Note the cubic spectral phase! Resulting intensity vs. time with a grating compressor and a prism compressor: Brito Cruz, et al., Opt. Lett., 13, 123 (1988). The (transmission) grism equation is: a [sin(m) – n sin(i)] = m n Chirped mirror coatings also yield dispersion compensation. Longest wavelengths penetrate furthest. Such mirrors avoid spatiotemporal effects, but they have limited GDD. Chirped mirror coatings Longest wavelengths penetrate furthest. Doesnt work for < 600 nm