Dispersion and how to control it Group velocity versus phase velocity Angular dispersion Prism sequences Grating pairs Chirped mirrors Intracavity and extra-cavity examples 1
Pulse propagation and broadening After propagating a distance z, an (initially) unchirped Gaussian of initial duration t G becomes: Etz (, ) exp tz/v g p t G ik" z exp 1 iz t z pulse duration increases with z G 1 tg pulse duration t G (z)/t G chirp parameter 3 1 k " 0 0 1 3 propagation distance z
Chirped vs. transform-limited A transform-limited pulse: satisfies the equal sign in the relation C is as short as it could possibly be, given the spectral bandwidth has an envelope function which is REAL (phase = 0) has an electric field that can be computed directly from S exhibits zero chirp: the same period A chirped pulse: satisfies the greater than sign in the relation C is longer than it needs to be, given the spectral bandwidth has an envelope function which is COMPLEX (phase 0) requires knowledge of more than just S in order to determine E(t) exhibits non-zero chirp: not the same period 3
4 p p 0 ' 1 dk d k V g p p p k V where: " z ik t z t G G 0 V / V / exp 1 ), ( z t z t z t i z t E z t E G p g p p G pulse width increases with propagation phase velocity group velocity - speed of pulse envelope Group velocity dispersion (GVD) (different for each material) Propagation of Gaussian pulses 1 g dk d k d d V If (and only if) GVD = 0, then V = V g. For most transparent solids, V > V g
GVD yields group delay dispersion (GDD) The delay is just the medium length L divided by the velocity. The phase delay: k( ) 0 0 v( ) 0 so: t L v( ) k( 0) L 0 0 The group delay: k( ) 0 1 v ( ) d 1 k( ) d v g g 0 so: L t ( ) k( ) L g 0 0 v g ( 0) The group delay dispersion (GDD): GDD = GVD L so: d 1 GDD L k( ) L d v g Units: fs or fs/hz 5
Dispersion in a laser cavity 0.5 0.7 0.9 1.1 1.77 n() At = 800 nm: GDD k " L = 10-3 fs 1.76 1.75 0 sapphire, L ~ 1 cm There is a small positive GDD added to the pulse on each round trip. m 1-5e-5 dn/d 3e-7 Material: Al O 3 Pulse width: t p = 100 fsec -1e-4 5e-7 cm e-7 k" t G m 3e-7 d n/d 1e-7 1e-7 0.5 0.7 0.9 1.1 0.5 0.6 0.7 0.8 0.9 1 1.1 (m) 6
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 for the positive GDD that comes from propagating through transparent materials. Negative GDD Device 7
Group and phase velocities - refraction sin i nsin i t d 1 index n() D' t Beam width: D' cost D cos time to propagate the distance d 1 : i D d T phase d1 D tan i c c must equal the time to propagate the distance d : n nd 'tan T d t c c phase propagation direction phase fronts: perpendicular to propagation pulse fronts: not necessarily! BUT: the pulse front travels at V g, so the travel distance is less by: d V V T g phase 8
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. We ll need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse). 9
Negative GDD Taking the projection of onto the optic axis, a given frequency sees a phase delay of z ( ) k( ) r optic axis k ( ) k( ) z cos[ ( )] ( / c) z cos[ ( )] Optic axis We re 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) zsin( ) d / d d z d z d z d z d sin( ) sin( ) cos( ) sin( ) d c d c d c d c d But << 1, so the sine terms can be neglected, and cos() ~ 1. 10
Angular dispersion yields negative GDD. d 0 z d d c d 0 0 The GDD due to angular dispersion is always negative! But recall: In most dielectric materials, k" is positive! (and k" L = ") We can use angular dispersion to compensate for material dispersion: k" total = k" material + k" angular These two terms have opposite sign. 11
Prisms In principle, we are free to specify: the apex angle 0 the angle of incidence 0 These are chosen using two conditions: Brewster condition for minimum reflection loss ( polarization) minimum deviation condition (symmetric propagation) 1 50 Reflectance 0.8 0.6 0.4 0. n = 1.78 deviation angle 49 48 47 46 n = 1.5 = 67.4º 0 0 10 0 30 40 50 60 70 80 90 incidence angle 45 45 50 55 60 65 70 75 incidence angle 1
A prism pair has negative GDD. Assume Brewster angle incidence and exit angles. How can we use dispersion to introduce negative chirp conveniently? Let L prism be the path through each prism and L sep be the prism separation. d 3 4L 0 d sep 0 c Always negative! dn d 0 This term accounts for the angular dispersion only. L prism 0 3 c d n d 0 This term accounts for the beam passing through a length, L prism, of prism material. Always positive (in visible and near-ir) Vary L sep or L prism to tune the GDD! 13
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. The beam s output path is independent of prism displacement along the axis of the apex bisector. Input beam Output beam 14
Four-prism 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. 15
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. 16
Appl. Phys. Lett. 38, 671 (1981) 17
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., 1, 401 (1987) It s best to use highly dispersive glass, like SF10. But compressors can still be > 1 m long. 18
Four-prism pulse compressor Also, alignment is critical, and many knobs must be tuned. Prism Wavelength tuning Wavelength tuning Prism Wavelength tuning Prism Prism Fine GDD tuning Coarse GDD tuning (change distance between prisms) Wavelength tuning All prisms and their incidence angles must be identical. 19
Pulse-compressors have alignment issues. Pulse compressors are notorious for their large size, alignment complexity, and spatio-temporal distortions. Pulsefront tilt Spatial chirp Unless the compressor is aligned perfectly, the output pulse has significant: 1. 1D beam magnification. Angular dispersion 3. Spatial chirp 4. Pulse-front tilt 0
Why is it difficult to align a pulse compressor? The prisms are usually aligned using the minimum deviation condition. Deviation angle Prism angle Minimum deviation Angular dispersion The variation of the deviation angle is nd order in the prism angle. But what matters is the prism angular dispersion, which is 1 st order! Using a nd -order effect to align a 1 st -order effect is tricky. 1
Two-prism pulse compressor Coarse GDD tuning Periscope Wavelength tuning Roof mirror 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 Roof mirror Wavelength tuning GDD tuning 3
Angular dispersion from a sequence of prisms 800 " (fsec ) 400 0-400 L p =14 cm L p =16 cm Example: 4 SF11 prisms L m =1 cm -800 L p =18 cm 0.75 0.8 0.85 " depends on! Wavelength (m) For very short pulses, third order dispersion is important. 4 1L 1 3 3 p n n n n nn L n n c 3 m 4
When prisms are not sufficient 3 " L " 4 ' mn Lpn c SF6 glass at 800 nm n = 1.78 n' = 59 m n" = 0. m L L 4n' n" m " = 0 if 0. 0633 p But then "' L p fsec 3 /cm Minimum value of L p or L m determines size of "' It is not possible to ideally compensate both nd and 3rd order dispersion with prisms alone. Diffraction gratings! 5
B Double pass: Diffraction-grating pulse compressor ' A C P ABC grating equation: sin sin x cos( ' ) ' c d 1 cos( ' ) x = perpendicular grating separation d = grating constant (distance/groove) 3/ 3 3 x x 1 sin r cd d cd 3 r sin cr d d 3/ 6
nd- 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 nd and 3rd-order terms are opposite for prism compressors and grating compressors. 7
Compensating nd 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 nd- and 3rd-order phases of the input pulse, input and input3, solve simultaneous equations: input prism grating 0 input 3 prism 3 grating 3 0 This design was used by Fork and Shank at Bell Labs in the mid 1980 s to achieve a 6-fs pulse, a record that stood for over a decade. 8
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, 13 (1988). 9
Chirped mirrors A mirror whose reflection coefficient is engineered so that it has the form: so that r 1 r e i and is chosen to cancel out the phase of the incident pulse. All 5 mirrors are chirped mirrors 8.5 fsec pulse from the laser 30
Chirped mirror coatings Longest wavelengths penetrate furthest. Doesn t work for < 600 nm 31
Theory: Experiment: 3
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