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 tG becomes: 2 tz /Vgp Etz(, ) exp tikz2 2" G
2 2"k exp 1 iz tz2 1 22 t 2 G G pulse duration increases with z chirp parameter 3 2 pulse duration t (z)/t G G 1 0 0 123 2 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:
A chirped pulse: the same period • 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:
3 not the same period Propagation of Gaussian pulses 2 1 t z / V E(t, z) E exp i t z / V g p 0 t z p p t z G G
where: t 2 pulse width increases with propagation G z tG 2ik"z
p V p phase velocity k p 1 d V group velocity - g 0 speed of pulse envelope k' p dk p
dk2 d 1 k Group velocity dispersion (GVD) ddV 2 g (different for each material)
If (and only if) GVD = 0, then V = Vg. For most transparent solids, V > Vg 4 GVD yields group delay dispersion (GDD)
The delay is just the medium length L divided by the velocity.
The phase delay:
0 L kL() 0 k() 0 so: t v( 0 ) v( 00 )
The group delay: 1 L k () 0 so: tkLg () 00 () v(g 0 ) v(g 0 ) The group delay dispersion (GDD): GDD = GVD×L
d 1 d 1 k () so: GDD L k () L d v d vg g
Units: fs2 or fs/Hz 5 Dispersion in a laser cavity 0.5 0.7 0.9 1.1
1.77 At = 800 nm: GDD k" L = 2×10-3 fs2 n( ) 1.76 sapphire, L ~ 1 cm There is a small positive
1.75 GDD added to the pulse
0 on each round trip. 1 -5e-5 Material: Al2O3 m
dn/d 3e-7 Pulse width: tp = 100 fsec
-1e-4 2k" 2
5e-7 2e-7 tG cm 2 3e-7 2 2
m d n/d
1e-7
1e-7
0.5 0.6 0.7 0.8 0.9 1 1.1 0.5 0.7 0.9 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
index n( ) Beam width: cos sin itn sin t t DD' cos i
i D' d1 time to propagate the distance d1: d D tan T 1 i phase cc
d2 must equal the time to
propagate the distance d2: n nD 'tan D Td t phase 2 cc
propagation BUT: the pulse front travels at Vg, so direction the travel distance is less by:
phase fronts: perpendicular to propagation dVVT gphase pulse fronts: not necessarily! 8 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). 9 Negative GDD Taking the projection of k () onto the optic axis, a given frequency sees a phase Optic axis delay of z () kr ()optic axis We’re considering only the GDD due to kz( ) cos[ ( )] dispersion and not that of the prism itself. So n = 1 (that of the air after the prism). ( /cz ) cos[ ( )]
dd /(/)cos()(/)sin()/ zc cz dd
2 dzdzdz2 d zd2 2 sin( ) sin( ) cos( ) sin( ) 2 dcdcdc d cd
But << 1, so the sine terms can be neglected, and cos( ) ~ 1. 10 Angular dispersion yields negative GDD.
2 dd2 z 0 2 dcd 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 0.8 n = 1.78 49 n = 1.5 0.6 48 = 67.4º
0.4 47
46 Reflectance 0.2 || deviation angle deviation 0 45 0 10 20 30 40 50 60 70 80 90 45 50 55 60 65 70 75 incidence angle incidence angle 12 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 d 2 c d 2 c d 0 0 0 visible and near-IR) Always This term accounts for This term accounts for the beam
negative! the angular dispersion passing through a length, Lprism, only. of prism material.
Vary Lsep or Lprism 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 The beam’s output path not misalign the beam. 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.
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. But compressors can still be > 1 m long. 18 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. 19 Pulse-compressors have alignment issues.
Pulse- Pulse compressors are front tilt notorious for their large size, alignment complexity, and spatio-temporal distortions.
Unless the compressor is aligned perfectly, the output Spatial pulse has significant: chirp 1. 1D beam magnification 2. Angular dispersion 3. Spatial chirp 4. Pulse-front tilt 20 Why is it difficult to align a pulse compressor?
The prisms are usually aligned using the minimum deviation condition.
Minimum deviation Deviation angle Angular dispersion Prism angle
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 effect to align a 1st-order effect is tricky. 21 Two-prism pulse Coarse GDD tuning compressor
Wavelength tuning Roof mirror Periscope
Prism Prism Fine GDD tuning Wavelength tuning
This design cuts the size and alignment issues in half. 22 Single-prism pulse compressor
Corner cube Periscope Prism
GDD tuning Roof mirror Wavelength tuning 23 Angular dispersion from a sequence of prisms
800
L =14 cm 400 p Example: 4 SF11 prisms ) 2 L =1 cm 0 m " (fsec Lp=16 cm -400
Lp=18 cm -800
0.75 0.8 0.85 Wavelength ( m) " depends on ! For very short pulses, third order dispersion is important. 4 23 12Lnpm 1 nn 2 n nnL 3 n n 2 23c 24 When prisms are not sufficient 3 " L nLn"4'2 2 c2 mp
SF6 glass at 800 nm L 4n'2 n = 1.78 " = 0 if m 0.0633 n' = 59 m Lp n" n" = 0.22 m
3 But then "' Lp × fsec /cm
Minimum value of Lp or Lm determines size of "'
It is not possible to ideally compensate both 2nd and 3rd order dispersion with prisms alone.
Diffraction gratings! 25 Diffraction-grating pulse compressor
C B grating equation: 2 c sin sin ' d ' x P ABC 1 cos( ' ) cos( ' ) A x = perpendicular grating separation d = grating constant (distance/groove) Double pass: 3/2 33xx 2 3/2 22 1 sin 22r cd d cd 3 r sin cr d d 26 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.
27 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:
input 2 prism 2 grating 2 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. 28 Pulse Compression Simulation
Using prism and grating pulse compressors vs. only a grating compressor
Note the cubic Resulting intensity vs. time spectral phase! with only a grating compressor:
Resulting intensity vs. time with a grating compressor and a prism compressor:
Brito Cruz, et al., Opt. Lett., 13, 123 (1988). 29 Chirped mirrors
A mirror whose reflection coefficient is engineered so that it has the form:
re i so that r 1 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 Experiment: Theory:
32 33