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/d2 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) / c , calculate: dk /dw = (n + w dn/dw) / c 0 0 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 d Use the chain rule :  0 dw d 0 d w 2 d022  c 0   c 0   0 Now, 002/ 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 nd2 c 000 or :

c00 dn vg  / 1 n n d0 n(w) Schott SF 10 glass c00 dn vg  / 1 n n d0 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 dcw2 0 dw d w d 02  c 0 d  0 dn and v/g cn00 d0 Okay, the GVD is:

 2 2 d110 d dn 0 d dn n 0 2 n 0 dwv2g  c0 d  0 c 0 d  0 2c0 d  0 d  0

2 2 0 dn d n dn 22 0  2c0 d  0 d  0 d  0 Units: Simplifying: 3 dn2 GVD k()w 0 ps2/km 0 2cd22 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:

w0 L kL()w0 k()w0  so: t  v (w0 ) v (ww00 )

The group delay: 1 L k()w0  so: tg ()()ww00 k L vg (w0 ) vg (w0 )

The group delay dispersion (GDD): GDD = GVD L

d 1 d 1 k()w   so: GDD L k()w L dw vg 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 2c  d   2c 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. 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 Why is it difficult to align a pulse compressor?

The prisms are usually aligned using the minimum deviation

condition.

Minimum

deviation

Deviation angle Deviation Angular dispersion Angular 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 a bad idea. 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. Single-prism pulse compressor

Corner cube Periscope Prism

GDD tuning Roof mirror Wavelength tuning Diffraction-grating pulse compressor

The grating pulse compressor also has negative GDD.

Grating #2 w 2 3 d  0 Lsep 2 2 2 2 dw2  c d cos (  ) w w0

where d = grating spacing L (same for both gratings) sep

Note that, as in the prism pulse compressor, the Grating #1 larger Lsep, the larger the negative GDD. 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 1980’s 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 Note the cubic with only a grating compressor: spectral phase!

Resulting intensity vs. time with a grating compressor and a prism compressor:

Brito Cruz, et al., Opt. Lett., 13, 123 (1988). The grism pulse compressor has tunable third-order dispersion.

A grism is a prism with a diffraction grating etched onto it.

The (transmission) grism equation is:

a [sin(m) – n sin(i)] = m

Note the factor of n, which does not occur for a diffraction grating.

A grism compressor can compensate for both 2nd and 3rd-order dispersion due even to many meters of fiber. Chirped mirror coatings also yield dispersion compensation.

Longest wavelengths penetrate furthest.

Such mirrors avoid spatio- temporal effects, but they have limited GDD. Chirped mirror coatings

Longest wavelengths penetrate furthest.

Doesn’t work for < 600 nm Real-life chirped mirror coating

(from Agata Jasik group at IET Warsaw) 160 d =10.83 mm AlAs theor  = 1030 nm 120 theor

80

40

Layer thickness [nm] GaAs 0 0 10 20 30 40 50 60 70 Pair number

R = 99.9% 2000 3000 (a) theor (b) 1.00 1500 2000 1000

R = 98.9% 0.98 Reflectivity ]

exp ] 500 2 1000 2 0.96 0

0 / = 0.2%

theor -500 GDD [fs -1000 0.94 GDD [fs  = 8 nm -1000 IET exp SESAM theory -2000 -1500 - 2450±100fs2 0.92 SESAMIET experiment -2000 -3000 SESAMBATOP experiment 1020 1030 1040 1050 1060 1020 1030 1040 1050 1060 Wavelength [nm] Wavelength [nm] Tunable chirped mirror

a) b) 2000 1040 Experimental data Quadratic polynomail fit 1020 0

1000

] -2000 2 980 -4000 3mm 960 GDD[fs 6mm -6000 9mm 12mm 940

-8000 920 Wavelength of GDDminimum [nm] 980 990 1000 1010 1020 1030 1040 1050 0 5 10 15 20 25 Wavelength [nm] Distance from the center [mm]