1. Wave front and energy front
Group delay
Wave front and energy front db
tan g = l db
g = tilt of energy front
= angular dispersion dl dl
Group velocity dispersion
There is also a relation between GVD and angular dispersion

2. Whether considering group delays or group velocity dispersion (GVD),
we will consider sufficiently broad beams, and sufficiently
short propagation distances Lp behind the element, such that
diffraction effects can be neglected. Q
S0, SW
S’0
S’W a P0 PW rW r0 L

3. Q
S0, SW
S’0
S’W a P0 PW rW r0 L

4. The most widely used optical devices for angular dispersion are
prisms and gratings. To determine the dispersion introduced by
them we need to specify not only
a(W),
but also the optical surfaces between which
the path is being calculated.
The ``dispersion'' of an element has only meaning
in the context of a particular application, that will associate
reference surfaces to that element.
Indeed, we have assumed in the previous calculation
that the beam started as a plane wave (plane reference surface normal to the
initial beam) and terminates in a plane normal to the ray at a reference
optical frequency $\omega_\ell$. The choice of that terminal plane is as arbitrary
as that of the reference frequency $\omega_\ell$ (cf. Chapter 1, Section~\ref{Fourier-rep}).
After some propagation distance, the various spectral component of the pulse
will have separated, and a finite size detector will only record a portion of
the pulse spectrum. 4

5. a h a L A B L g C
(R)

6. The stability of the cavity requires that R > L implying that the coefficients a and b are positive. Therefore, in a cavity with a single prism as sketched the group velocity dispersion is adjustable through the parameter h, but always positive.

7. First mode-locked laser with negative phase modulation and positive adjustable dispersion:
Intracavity pulse compression with glass: a new method of generating pulses shorter than 60 fsec
OPTICS LETTERS / Vol. 8, No. 1 / January 1983
dispersion
Phase modulation

8. For a collimated beam, a logical reference plane is normal
to the beam
Pairs of prisms
2nd Element
(reversed)
Reference
plane B
Reference
plane A
1st Element

9. Prism arrangement from the Diels group:
How to check that the prisms are correct?
Minimum deviation
Material?
Brewster angle?
Alignment?

10. A d q4 q3 q2 q1
p-A

11. The group velocity dispersion is simply the sum of three contributions:
1. The (positive) GVD due to the propagation of the pulse through a thickness
of glass Lg.
2. The negative GVD contribution due to the angular dispersion over the path between prisms
3. The negative GVD contribution due to the angular dispersion -- deflection
of the beam at the first interface Lg in the glass of index n.
For Brewster angle prisms at minimum deviation angle:

12. O H A q0 q3 q2 B q1 D t2 W t
W+dW q6 q4 q5 B’ q0
= q3 A’
A’’ a u O’ s

13. O q3 A q2 B t q4 B’ A’ W
= q3
A’’ t2
W+dW a O’ s

14. dq3 q6 q5
q7 = q0 B’ Q A’
a-q2 R
B’’ T
A’’ S q5 u q4 H’
B’’’
A’’’ a O’

15. W O a X A’ q0 A
W + dW q1
dq1 q1
A’’
O’’’ g

16. { ceo Phase delay For a pair of prisms we found:
For gratings, the simple rule
applies also

17. C P d b’ b b A

18. Pulse stretching/compression with
prisms and grating pairs
1. Devices for adjustable dispersion: Prisms and gratings
2. GVD of prisms pairs
3. GVD of gratings pairs
4. Gratings pairs for pulse stretching – what are the adjustable
Parameters?
5. Application: Pulse shaping by “spectral filtering”
6. Can we have positive GVD with grating pairs?

19. 4. Gratings pairs for pulse stretching – what are the adjustable
Parameters?
Wavelength? l Fixed
Groove spacing d? Larger than l
Diffraction angle large (Littrow configuration)
Size of grating: pulse length.

20. Short pulse lasers phase modulation – Dispersion solitons dispersion
Mode-locking = putting modes in phase AO modulator (active mode-locking)
Role of saturable absorber.

21. Saturable absorbers:
Liquid dye jets
Multiple quantum wells.

22. Saturable absorbers:
Liquid dye jets
Multiple quantum wells.

24. ... Substrate: both sides polished MQW for 1064 nm 145.5 nm 0.5 mm
LT In0.25Ga0.75 As
GaAs
substrate
GaAs
GaAs
GaAs
GaAs
...
GaAs
GaAs
GaAs
AlAs
AlAs
21 pairs
Low Temp. growth (350o C, not annealed)
Thickness:
AlAs nm
GaAs nm
(last layer nm)
Thickness:
InGaAs 15 nm
GaAs nm (last layer 69.3 nm)
(first layer included in the nm)
Substrate: both sides polished
LT In0.25Ga0.75 As
GaAs
substrate
GaAs
GaAs
GaAs
GaAs
MQW for 1064 nm
GaAs
GaAs
Low Temp. growth (350o C, not annealed)
Thickness:
InGaAs 14 nm
GaAs nm (last layer 66.8 nm)

25. Equally spaced modes in phase,
make pulses periodic in time
time
Electric field
amplitude

26. Direct creation of a frequency comb
A perfectly regular frequency comb is formed by nonlinear optics:
w, 2w, 3w, 4w, 5w, ...
But they are not in phase.
If they can be put in phase, a pulse train with zero CEO is created.
Reference:

27. tRT Direct creation of a frequency comb 5w 4w 3w 2w w LASER CEO? CEP?
Pulse duration
tRT
Mode bandwidth
Number of pulses
CEO?
CEP?
Pulse duration (for square spectrum) : x \lambda/4c = 390 as
CEO = 0
CEP changes with propagaton in air w 2w 3w 4w 5w W

28. Direct creation of a frequency comb

29. Short pulse lasers
Putting cavity modes in phase
Active
Passive
Saturable absorbers
Dispersion – Kerr modulation

30. The real thing: the laser
Tuning the wavelength, the mode and the CEO
L. Arissian and J.-C. Diels, “Carrier to envelope and dispersion control
in a cavity with prism pairs”, Physical Review A, 75: (2007).

31. Dilemma: If there is dispersion, the modes are not equally spaced!

32. The laser as an orthodontist
Mode locked laser D l
Tuned cw laser
FREQUENCY
TUNABLE
LASER
Frequency counter
MODE-LOCKED
LASER
SPECTROMETER
700
800
900
100
200
Rep. Rate - Hz
Wavelength [nm]
FREQUENCY
COUNTER
Round
-trip
frequency
wavelength
ORTHODONTIST